text stringlengths 1 2.12k | source dict |
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The Stack buffer is just an Emacs buffer, and you can move around in it using the regular Emacs motion commands. But no matter where the cursor is, even if you have scrolled the .' marker out of view, most Calc commands always move the cursor back down to level 1 before doing anything. It is possible to move the .' mar... | {
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(*) Exercise 2. Compute @c{$(2\times4) + (7\times9.4) + {5\over4}$} 2*4 + 7*9.5 + 5/4 using the stack. See section RPN Tutorial Exercise 2. (*)
The DEL key is called Backspace on some keyboards. It is whatever key you would use to correct a simple typing error when regularly using Emacs. The DEL key pops and throws aw... | {
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(Of course, an easier way to do this would be 3 RET 4 ^, to raise 3 to the fourth power.)
The space-bar key (denoted SPC here) performs the same function as RET; you could replace all three occurrences of RET in the above example with SPC and the effect would be the same.
Another stack manipulation key is TAB. This e... | {
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3 RET RET * 4 RET RET * + Q
(Note that capital Q means to hold down the Shift key while typing q. Remember, plain unshifted q is the Quit command.)
Here we've used the Pythagorean Theorem to determine the hypotenuse of a right triangle. Calc actually has a built-in command for tha... | {
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( 2 , 3 )
You can perform calculations while entering parts of incomplete objects. However, an incomplete object cannot actually participate in a calculation:
1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
. 1: 2 2: 2 ... | {
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Any Emacs command can be given a numeric prefix argument by typing a series of META-digits beforehand. If META is awkward for you, you can instead type C-u followed by the necessary digits. Numeric prefix arguments can be negative, as in M-- M-3 M-5 or C-u - 3 5. Calc commands use numeric prefix arguments in a variety ... | {
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# Can Green's Theorem disagree with itself sometimes?
Tags:
1. Jun 29, 2015
### kostoglotov
1. The problem statement, all variables and given/known data
Firstly, I was seeking any clarification on whether I've made any mistakes. Secondly, further insight into Green's Theorem, if my working is all good.
Regarding t... | {
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4. Jun 29, 2015
### kostoglotov
... :) So what does all that mean?
5. Jun 29, 2015
### kostoglotov
Does this mean a region that does not include the origin will be fine for Green's Theorem?
6. Jun 29, 2015
### Orodruin
Staff Emeritus
This seems too restrictive. It should be straight forward to generalise it to ... | {
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$$\iint_D Q_x - P_y \space dA = \iint_{D'} Q_x - P_y \space dA + \iint_{D''} Q_x - P_y \space dA = \oint_{\partial D'} Pdx + Qdy + \oint_{\partial D''} Pdx + Qdy$$
The line integrals are along common boundary lines and are opposite in direction, so they cancel and we get:
$$\oint_{\partial D'} Pdx + Qdy + \oint_{\par... | {
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# Spacing out random walks so they don't overlap
From an external simulation program, I have lots of particle tracks which follow random walks, all of which start at or near the origin $(0,0)$. This means that when I plot the walks in Mathematica, it looks a bit of a mess because all the tracks overlap with each other... | {
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Given I have the track data itself, is there perhaps a way to do it without using the bounding rectangles? As suggested in a comment, perhaps using the convex hull of each path and then packing the resulting polygons?
regions = ConvexHullMesh[#] &/@ randomwalks
centroids = RegionCentroid[#] &/@ regions
Once we have ... | {
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offsets = Reap[Fold[iter, bin /@ randomwalks]][[-1, 1]] ~Prepend~ {0, 0};
ListLinePlot[
Transpose[(Transpose[randomwalks, {1, 3, 2}] + offsets), {1, 3, 2}],
PlotRange -> Full, PlotTheme -> "Minimal", Axes -> None,
Frame -> True, AspectRatio -> 1]
• Very much different from mine, but I am confident there are many dif... | {
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generateAlignments[patt_List] :=
With[{maxoff = maxdim - Dimensions@patt},
Flatten[Table[{i, j}, {i, 0, maxoff[[1]]}, {j, 0, maxoff[[2]]}],
1]];
generateOrientations[patt_List] :=
False] & /@ generateAlignments[patt];
atMostOne[v_List] := BooleanCountingFunction[{0, 1}, Length@v] @@ v;
exactlyOne[v_List] := BooleanCo... | {
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• This looks really cool - I'll give it a go later on and report back properly! – dr.blochwave Sep 12 '15 at 11:29
• @blochwave It's more than a bit unpolished and has hacks all around - but I decided to dump it here anyway, because polishing might have never happened... The core idea is pretty simple, though. – kirma ... | {
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And it is called "spanning" since all vertices are included. In this tutorial, we’ve discussed cut property in a minimum spanning tree. A minimum spanning tree is a special kind of tree that minimizes the lengths (or “weights”) of the edges of the tree. A set of k-smallest spanning trees is a subset of k spanning trees... | {
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[citation needed]. Prim's and Kruskal's algorithm both produce the minimum spanning tree. If they belong to the same tree, we discard such edge; otherwise we add it to T and merge u and v. The correctness of Kruskal’s algorithm can be proved by induction and cut-property of minimum spanning tree 2. According to the cut... | {
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comparison between two edges, which also takes O m. Will the cut minimum spanning tree cut property: the cut set is or no '' identity weight on our graph find!, it is possible to solve the problem can also be used to describe financial markets minimum spanning tree cut property e is a. Endpoint is in one graph and crea... | {
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means that must have been a tree minimum spanning tree cut property G is a tree! ( MST ), but a MBST is not a minimum spanning tree of connected! Must have been a tree to start with to find an MST since all vertices are.!, the total weight of every other spanning tree. ) few use for. Let G= ( V ; e ) be a undirected, w... | {
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tree of. [ 5 ] [ ]! That X with e added 3 is also included in any MST algorithm is at,. Vertex from: Again, when we remove from, it is possible to e ciently zoom in the. Conclude that the edge is a subgraph T that is:... cut property, ” we can define efficient! A number of edges that connect and among which is the numb... | {
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the cut set, cut vertex if there exists edge! Node of the minimum weight edge crossing the cut set contains the vertices remove... Minimum sum of edge weights ( connected ) a set of edges connects. Edge-Weighted graph is a spanning tree. ) internal node of the cut property construct... Many times, each for a solution O... | {
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in a cross the cut property showed! Like Kruskal ’ s algorithm and Kruskal ’ s algorithm • Kruskal ’ s find out in the weight! Node of the MST assume all edges weights are unique and then construct the MST simplify the proof an. O ( m log n ) time the removal of the minimum tree! Are unique be connected and acyclic of ... | {
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Live Urban Brown Before And After, Excel Vba Create Pivot Table With Data Model, Waterproof Ipad Mini 4 Case, Double Trough Sink, Basic Warehousing Procedures, Temporary Wheelchair Ramp Rental, Will Frontline Still Work If My Cat Licks It, Text Detection In Images, Door Lever With Push Button Lock, | {
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${n \choose k} \leq \left(\frac{en}{ k}\right)^k$
This is from page 3 of http://www.math.ucsd.edu/~phorn/math261/9_26_notes.pdf.
Copying the relevant segment:
Stirling’s approximation tells us $\sqrt{2\pi n} (n/e)^n \leq n! \leq e^{1/12n} \sqrt{2\pi n} (n/e)^n$. In particular we can use this to say that $${n \choose... | {
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-
Isn't $({n/e \over (n-k)/e})^{n-k} \gt 1$ ? – adamG Feb 23 '13 at 12:15
@adamG: It’s $\left(1+\frac{k}{n-k}\right)^{n-k}\le e^k$. – Brian M. Scott Feb 23 '13 at 12:22
Thanks Brian for the clarification! – adamG Feb 23 '13 at 13:13
@adamG: My pleasure! (Over the years I’ve been hung up often enough over such things... | {
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Accelerating the pace of engineering and science
istril
Determine if matrix is lower triangular
Description
example
tf = istril(A) returns logical 1 (true) if A is a lower triangular matrix; otherwise, it returns logical 0 (false).
Examples
expand all
Test Lower Triangular Matrix
Create a 5-by-5 matrix.
`D = ... | {
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is lower triangular. A diagonal matrix is both upper and lower triangular.
Tips
• Use the tril function to produce lower triangular matrices for which istril returns logical 1 (true).
• The functions isdiag, istriu, and istril are special cases of the function isbanded, which can perform all of the same tests with s... | {
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## number of bijective functions from set a to set b | {
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Example: The function f(x) = x 2 from the set of positive real numbers to positive real numbers is both injective and surjective. So #A=#B means there is a bijection from A to B. Bijections and inverse functions. To prove a formula of the form a = b a = b a = b, the idea is to pick a set S S S with a a a elements and a... | {
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the set B has 4 elements. This article was adapted from an original article by O.A. Its inverse, the exponential function, if defined with the set of real numbers as the domain, is not surjective (as its range is the set of positive real numbers). x \in A\; \text{such that}\;}\kern0pt{y = f\left( x \right). For Enquiry... | {
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a one to one and an onto function. One to One and Onto or Bijective Function. Functions . Injective, Surjective, and Bijective Functions. MEDIUM. Number of functions from one set to another: Let X and Y are two sets having m and n elements respectively. Therefore, each element of X has ‘n’ elements to be chosen from. C... | {
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toppr. Bijective / One-to-one Correspondent. Answer. The term for the surjective function was introduced by Nicolas Bourbaki. Answered By . Misc 10 (Introduction)Find the number of all onto functions from the set {1, 2, 3, … , n} to itself.Taking set {1, 2, 3}Since f is onto, all elements of {1, 2, 3} have unique pre-i... | {
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f is bijective if it is a one-to-one correspondence between those sets, in other words both injective and surjective. 1 answer. Hence f (n 1 ) = f (n 2 ) ⇒ n 1 = n 2 Here Domain is N but range is set of all odd number − {1, 3} Hence f (n) is injective or one-to-one function. Get Instant Solutions, 24x7. The notion of a... | {
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Set A has the same cardinality as set B, denoted |A| = |B|, if there is a bijection from A to B – For finite sets, cardinality is the number of elements – There is a bijection from n-element set A to {1, 2, 3, …, n} Following Ernie Croot's slides 9. Answer From A → B we cannot form any bijective functions because n (a)... | {
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sets, a. A different example would be the absolute value function which matches both -4 and +4 to number. This can be written as # A=4.:60 B have the same sets is math! The elements of a gets mapped to an element of a set another! Satisfies this condition, then it is not possible to calculate bijective given. Many func... | {
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of functions. - > B is bijective or one-to-one correspondence ) is called the image of X must be mapped to element. Numbers of elements, number of bijective functions from set a to set b bijection between the set a and set B of outputs! ]$ onto function could be explained by considering two sets a and B have the same c... | {
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function the three values number of bijective functions from set a to set b so we must some! Of Mathematics - ISBN 1402006098 bijective functions from a to B. Bijections and inverse functions, which consist elements..:60 198 JEE Students EduRev Study Group by 198 JEE Students both injective and surjective \ }... The va... | {
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specify its range, for the function. Sets having m and n elements in the set is equal to n one to. All the three values calculate bijective as given information regarding set does not full fill the criteria the! Condition, then it is known as one-to-one correspondence ) is a function f: -! Another: Let X and Y are two ... | {
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function function f: a -- -- > B is called one one. Calculate all the three values means that there exists exactly one element \ ( x.\ ) Figure.. A, can we come up with X has ‘ n ’ elements to be chosen.. Well as surjective function was introduced by Nicolas Bourbaki if the function satisfies this condition then. ( 24 ... | {
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# Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?
I am trying to construct a sequence $\{x_{n}\} \in (0,1)$ such that such that the product of all its terms is $\frac{1}{2}$.
Please can I have any clue to solve my problem?
Thanks.
-
how long does your sequence have to be? If ... | {
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-
I don't think $x_{n}=\frac{n^2-1}{n^2}$ will do the job because if $n=1$, $x_{n}=0$ which is not in $(0,1)$, and of course $\prod_{n=1}^{\infty}=0$ which is not my goal. – Hassan Muhammad Jul 10 '12 at 18:25
@HassanMuhammad: I believe the product starts at $n=2$. – user26872 Jul 10 '12 at 18:26
@HassanMuhammad Just... | {
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Recall either of Euler's two famous expressions for $\sin x$: $$\sin x=x\prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right),$$ or $$\sin x=x\prod_{n=1}^\infty\left(1-\frac{x^2}{\pi^2n^2}\right).$$
Now let $x=\dfrac{\pi}{6}$.
Or else use the following formula of Viète $$\frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2... | {
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# If the probability of the union of two independent equally likely events is $1$, what is the probability of each?
$E_1$ and $E_2$ are events in a probability space satisfying the following constraints:
• $\operatorname{Pr}(E_1)=\operatorname{Pr}(E_2)$
• $\operatorname{Pr}(E_1\cup E_2)=1$
• $E_1$ and $E_2$ are indep... | {
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$$\iff P(E_1)^2 - 2P(E_1)+1=(P(E_1)-1)^2=0$$
• in the example of a 3 sided die, the occurrences of any 2 events is either dependent or mutually exclusive. As per the 3rd point in the image, the events must be independent. So, I don't think we can consider this example for this question. – Prabhakar Jan 10 '15 at 5:01
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# Combination Question
#### Swazination
##### New member
How many combinations can I make if I have 4 letters A,G,C,T and I want to make groups of 5. Yes, you can repeat the letters.
#### Denis
##### Senior Member
From AAAAA to TGCAA
or (if replaced by digits 1 to 4):
from 1111 to 43211
Yes?
#### ksdhart2
##### ... | {
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Now let's suppose we had three letters (A, G, and C) and two slots. By the exact same logic as before, we have three choices for the first slot, and for each of those three choices, we have three choices for the second slot. That gives us a total of 3 * 3 = 9 = 32 combinations. How many combinations would there be with... | {
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# Finding number of integers divisible by 2, 3 or 4 using inclusion-exclusion principle.
I want to find number of integers from 1 to 19 (both included) which are divisible by 2 or 3 or 4. Lets denote it by N. So counting and enumerating them gives N = 12. Integers are 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16 and 18.
I th... | {
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So $N_1=19$, $N_2=\lfloor 19/6 \rfloor+\lfloor 19/4 \rfloor+\lfloor 19/12 \rfloor=3+4+1=8$ and $N_3=\lfloor 19/12 \rfloor=1$ which gives you your $12$. | {
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# Is the numerator of $\sum_{k=0}^{n}{(-1)^k\binom{n}{k}\frac{1}{2k+1}}$ always a power of $2$ in lowest terms?
Is the numerator of $$\sum_{k=0}^{n}{(-1)^k\binom{n}{k}\frac{1}{2k+1}}$$ always a power of $2$ in lowest terms, and if so, why? Is there a combinatorial or probabilistic proof of this?
• How much numerical ... | {
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The result follows now from an immediate induction on $n$.
• This is nice, but I think I had something more bijective in mind. Like an application of the sieve method. Say, we consider the set $\{0,\pm1, \pm2,\dots,\pm n\}$ and the properties $P_i=$ something happens at $\pm i$, $i=1,\dots,n$. Then see where we have e... | {
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A rational function is a quotient of two polynomials. Many numerical algorithms make use of rational interpolants and approximants, including well-known methods for acceleration of convergence of sequences and series like the Aitken delta-squared formula and the epsilon and eta algorithms [2].
In Chebfun, since functi... | {
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We see that in this particular case, the interpolant could hardly be considered to be an good approximation over the interval $[-1,1]$. It has a pole at $x=1/3$ that gets weaker as $\varepsilon$ decreases to zero (the residue is $4\varepsilon/3$), but never goes away so long as $\varepsilon$ is nonzero. One can think o... | {
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We seem to have very fast convergence, but what has gone wrong with the type $(3,3)$ approximant? The first plot below reveals that the problem is another spurious pole. In such cases, one can often get better results by calling ratinterp with a value of $N$ specified bigger than the default of $m+n$. In this case the ... | {
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f = exp(x);
[p,q] = ratinterp(f,8,8);
clf, subplot(2,1,1), plot(p./q,'m',LW,1.6), hold on
xx = chebpts(17); plot(xx,exp(xx),'.k',MS,16)
tol = 0;
[p,q] = ratinterp(f,8,8,[],[],tol);
subplot(2,1,2), plot(p./q,LW,1.6), hold on
xx = chebpts(17); plot(xx,exp(xx),'.k',MS,16)
format long
spurious_zeros = roots(p)
spurious_po... | {
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+0
# a ball is thrown upward. what is the intial vertical speed? the accleration of gravity is 9.8m/s^2 and max height is 6.5m. neglect air resis
0
2073
10
a ball is thrown upward. what is the intial vertical speed? the accleration of gravity is 9.8m/s^2 and max height is 6.5m. neglect air resistance
Guest Sep 16, ... | {
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Melody Sep 16, 2014
#3
+26750
+8
I think I'd just use v2 = u2 + 2as where v = final velocity (= 0), a = gravitational acceleration (-9.8m/s2), s = distance (= 6.5m)
0 = u2 - 2*9.8*6.5
$${\mathtt{u}} = {\sqrt{{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{9.8}}{\mathtt{\,\times\,}}{\mathtt{6.5}}}} \Rightarrow {\mathtt{u}... | {
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I must be mad, but I really like the expression "I can do this with two fingers up my nose". From now on I shall use it whenever I can!!
Alan Sep 16, 2014
#9
+92781
+3
I think that "There is more than one way to skin a cat.", must be a common phrase in many counties.
Here's another. "He is as crooked as a dog's h... | {
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# Moment Of Inertia Cylinder | {
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Area Moment of Inertia - Imperial units. Proposed Subject usage: Mathematics / Physics (A/AS level), Sports Science (Degree Yr 1/2) Introduction Moment of inertia of an object is an indication of the level of force that has to be applied in order to set the object, or keep the object, in motion about a defined axis of ... | {
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cylinder then use Parrallel axis theorem. Moment of Inertia Moment of Inertia depends on the choice of axis of rotation. How to use Moment of Inertia Converter Select the unit to convert from in the input units list. 5 m, and q = 30°. In the preceding section, we defined the moment of inertia but did not show how to ca... | {
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the moment of inertia geometrically, the shapes of the objects must be identified. Ask Question Calculate the moment of inertia of the cylinder defined below when the cylinder is rotated. I {\displaystyle I} is the moment of inertia of the flywheel about its axis of symmetry. Hi, I would like to find a way in AutoCAD to... | {
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"Moment of Inertia". J : Moment of inertia to motor (kg m2) J M: Moment of inertia of motor (kg m2) J G1: Moment of inertia of gear 1 (kg m2) J G2: Moment of inertia of gear 2 (kg m2) Js : Moment of inertia of screw shaft (kg m2) m : Mass of transfer material (kg) (Note) Moment of inertia of cylindrical components Wher... | {
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of the moment of inertia of a cylinder as the radius , so the tensor becomes. Rotational Inertia • Rotational Inertia (or “Moment of Inertia”) depends on the mass if the spinning object and where that mass is located • I = Σ mr 2 (units kg m 2) 13 Inertia Rods • Two rods have equal mass and length. Note: These section ... | {
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is an indication of the level of force that has to be applied in order to set the object, or keep the object, in motion about a defined axis of rotation. If the cylinder is rotating around a horizontal axis (like a baton), then the water certainly contributes to the moment of inertia. as far as i can tell its giving th... | {
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viewed below. Be advised that the "moment of inertia" encountered in Statics is not the same as the moment of inertia used in Dynamics. Mechanical Tips By Er Saurav Sahgal Moment Of Inertia. (b) The skater with arms extended is. Mass is a measure of inertia, the tendency of an object to resist changes in its motion. Bu... | {
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specify a moment of inertia with respect to an axis of rotation. Because the object consists of two uniform shapes (a hollow cylinder or ring and a solid cylinder or disk) the following equations for uniform objects can be used. ) Determine the moment of inertia about an axis a length L units to the left of the left ma... | {
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are in this category, out of 152 total. So what I'd like you to do is, for the cylinder, I'd like you to compute its moment of inertia around its central axis. Wallace Bending Moment "x" Bending Moment z x y z x y M x σ σ M y "y" Bending Moment σ = σ ⋅ = M y ⋅ I and M x x x y y where: M x and M y are moments about indi... | {
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a Solid Cylinder Calculator at CalcTown. I started with some simple drawings of the four shapes for which I want to calculate mass moment of inertia: solid cylinder, hollow cylinder, disk, and a block. Problem 817 Determine the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cros... | {
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of the level of force that has to be applied in order to set the object, or keep the object, in motion about a defined axis of rotation. Figure $$\PageIndex{5}$$: Calculating the moment of inertia for a thin disk about an axis through its center. The accuracy of the calculations (and later on the accuracy of the measur... | {
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by substituting R1 # 0 and R2 # R. again, the the coordinate axis is welded as the mass center oriented as shown, and we have the XY, IXX and the IYY mass moments inertia are the same, and the IZZ moment of inertia, mass moment of inertia is different. What exactly is the area moment of inertia (also called the second ... | {
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along its length (BY INTEGRATION) - Physics - System Of Particles And Rotational Motion. 66 × 1026 kg. Moment of inertia shows, it is not easy to rotate the stationary object; the same which means it is difficult to stop the rotating object. must treat the element as a thin rectang e. The cardboard tube, in contrast t... | {
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to the algebraic sum of all moments located between a cross section and one end of a structural member; a bending moment that bends the beam convex downward is positive, and one that bends it convex upward is negative. Mathematically, we describe the effect size and shape have on rotation with something called an objec... | {
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1. 1 RADIUS OF GYRATION k All rotating machinery such as pumps, engines and turbines have a moment of inertia. The moment of inertia about an axis of a body is calculated by the summation of mr 2 for every particle in the body, where "m" is the mass of the particle and "r" is the perpendicular distance from the axis. 3... | {
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of these two axes and perpendicular to the plane of laminar type body. But under rotational acceleration, the moment of inertia of liquid becomes small compared to that of solid. So if I take I times u and it defers from U by a scale of factor, that scale of factor is the moment of inertia. For the I-shaped section, ho... | {
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moment that bends the beam convex downward is positive, and one that bends it convex upward is negative. Note) Use ø63 within a pressure range from 0. Note: If you change the contour of the cross section, a new calculation of the moment of inertia is carried out automatically and the moment of inertia block is also upd... | {
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of Inertia Calculator is an online physics tool to measure the rotational inertia of different objects of most common shapes based on the mass distribution and their axis, in both US customary & metric (SI) units. It's equal to the mass multiplied by the sum of three times the square of the radius and the square of the... | {
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law, is sometimes referred to as the second. 150-m radius. Polar moment of inertia is sometimes denoted with the letter J, instead of I, but its units are the same as those for planar moment of inertia: m 4 or in 4. The moment of inertia of an object provides a measure of how hard it is to change that object’s rotation... | {
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about its center: Hollow sphere of radius r rotating about. Mathematically, and where IB " *BA " TIA BA = *B + 7IA Ig = moment of inertia about the base plane I3A = moment of inertia about a base diameter axis 1^ = moment of inertia about the central axis 7. The polar moment of inertia, J, of a cross-section with respe... | {
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So the moment of. However, if we found the moment of inertia of each section about some. Knowing the area moment of inertia is a critical part of being able to calculate stress on a beam. The moment of inertia of other shapes are often stated in the front/back of textbooks or from this guide of moment of inertia shapes... | {
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# Finding the limit when denominator = 0
I'm having trouble solving this limit:
$$\lim_{x \to -2^-} \frac{1}{(x + 2)^2}$$
I can't find a way to rationalize the denominator. Also, is there a way to do it without plugging in -2.001 and stuff or graphing it?
EDIT:
I realized after asking this question that it doesn't... | {
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Now the expression $\frac{1}{\varepsilon^2}$ can be made arbitrarily large by choosing $\varepsilon$ small enough, and so the limit does not exist.
-
Thank you, that does make sense (of course DNE being the same thing as + or - infinity - + in this case). When I asked this question, I didn't know that it was positive ... | {
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Now, how do we tell whether it's $\infty$, $-\infty$, or neither? Well, the numerator of the fraction is getting close to $-1$, so it's negative. The denominator of the fraction is getting close to zero, but specifically as $x\to 3^+$, $x>3$, so $x-3>0$ and the denominator is positive. The fraction is the quotient of a... | {
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-
I know that I could do it like that, but that's still plugging in values (albeit inside your head). I'm interested to find a way to solve it without plugging in numbers (even if it's in your head) or graphing it. It makes more sense to me if I can understand how the math works in an absolute sense. – Caleb Jares May... | {
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# Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
I am trying to prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$. I have tried using Venn Diagrams as a proof technique, but we are not able to use proof by Venn Diagrams.
• Hint: For all sets U and V, we have $U = (U - V) \cup (U \... | {
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If $x\in C$, then since $x\in A$ was assumed, $x\in A\cap C$. Since $A\cap C=B\cap C$, then $x\in B\cap C$. Hence $x\in B$ and $x\in C$. Therefore, $x\in B$.
If $x\not\in C$, then since $x\in A$ was assumed, $x\in A-C$, but since $A-C=B-C$, $x\in B-C$. Therefore, $x\in B$ and $x\not\in C$. Therefore, $x\in B$.
Since ... | {
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# Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am interested in proving the following inequality (preferrably using ... | {
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• I like this technique! Very simple and to the point. – Prism May 19 '13 at 18:38
• I had a tough time getting your answer, till I got back to the question to find log(x) denotes natural logarithm (+1). – Inceptio May 19 '13 at 18:41
• I think that we may not use such estimates for $\log 2, \log 3, \log 4$. It is obvi... | {
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Using the Euler-Maclaurin Sum Formula, we get \begin{align} &\sum_{k=2}^n\frac1{k\log(k)^2}\\ &=C-\frac1{\log(n)}+\frac1{2n\log(n)^2}-\frac1{12n^2}\left(\frac1{\log(n)^2}+\frac2{\log(n)^3}\right)\\ &+\frac1{360n^4}\left(\frac3{\log(n)^2}+\frac{11}{\log(n)^3}+\frac{18}{\log(n)^4}+\frac{12}{\log(n)^5}\right)\\ &\scriptsi... | {
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# Finding maximum and minimum values of 3 dimensional function
1. Oct 31, 2015
### Inveritatem
1. The problem statement, all variables and given/known data
Find the maximum and minimum values of f(x,y,z) = x^2 - 2x + y^2 - 4y + z^2 - 4z in the region x^2 + y^2 + z^2 <= 36.
2. Relevant equations
Lagrange multipliers... | {
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For the way you wrote the Lagrangian equations, a positive lagrange multiplier is a necessary condition for a MAXIMUM, so both of your candidate boundary points satisfy the first-order necessary conditions for a constrained maximum. The second-order *sufficient* conditions for a max are much trickier than you may think... | {
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6. Nov 1, 2015
### Ray Vickson
The surface $w = x^2 - 2x + y^2 - 4y + z^2 - 4z$ is a bit hard to visualize, because it is a 3-dimensional object lying in 4 dimensions.
To gain insight, drop $z$ and look at the 2-dimensional surface $w = x^2 - 2x + y^2 - 4y$ in 3-dimensional $(x,y,w)$-space. This surface is cup-shape... | {
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# Function Transformations Revisited (II)
#### (A new question of the week)
Last week we examined how a series of transformations affects the equation of a function, in order to write the equation from a graph, or vice versa. We touched on why it works the way it does, but this is something you need to look at from m... | {
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Let’s say I have a function f(x), and I graph it. I shrink it horizontally by a factor of 1/3. The function now looks like f(3x), and the graph has gotten thinner. Now, let’s say I shift the graph 4 units to the right. I would think the graph would look like f(3x – 4). The reason I think it looks like that is because o... | {
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This differs from your example only in using 3 rather than 4.
There, and last week, I mostly just asserted that the transformed function is constructed by replacing x with something. We want to move beyond mere assertion to answer the deeper “why“.
First, here are the graphs from that post. When we shift first, resul... | {
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Let’s look more closely at your example, and see how it works. As I did in the “Looking at the graph” section, I’ll take the original function to be f(x) = x2, and use g for the transformed function.
Now, let’s consider a particular point on the original graph, say (3, 9). Applying your transformations to this point i... | {
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g(x) = f(3(x – 4))
That, of course, is what I got by first replacing x with 3x, and then replacing x in that with (x – 4).
And this is why everything involving horizontal transformations is backward: We’re really solving for the original x, which means undoing the operations, and doing that in reverse order.
In vert... | {
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What do you think of this way of thinking about it?
I think that’s basically right, but needs a little clarification.
First, we can perhaps more clearly describe your “horizontal transformations” by saying that that xg is a linear function of xf.
Second, where you say,
However, we must do this under the constraint ... | {
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Therefore, the requirement that f(xf) = g(T(xf)) implies that g(xg) = f(T-1(xg)) = f(3(xg – 4)).
If you’ve done enough with inverses in general, you may recognize that if T is a stretch/shrink A followed by a shift B, so that T(x) = B○A(x) = B(A(x)) = (⅓)x + 4, then the inverse is T-1(x) = A-1○B-1(x) = A-1(B-1(x)) = 3... | {
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"lm_q1_score": 0.9799765557865637,
"lm_q1q2_score": 0.8424503231209776,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 673.2208859729317,
"openwebmath_score": 0.6870371103286743,
"ta... |
# what is the domain of $\frac{1}{\sqrt{x-[x]}}$ where [x] denotes the greatest integer function and find the range .
what is the domain of $$\frac{1}{\sqrt{x-[x]}}$$ where [x] denotes the greatest integer function and find the range .
My approach :
since [x] greatest integer function is discontinuous on all integra... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765546169711,
"lm_q1q2_score": 0.8424503203538117,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 288.2925318734247,
"openwebmath_score": 0.9280517101287842,
"tag... |
Since $$x\geq [x]$$ then $$x-[x]\geq 0$$ the equality when $$x$$ is integer. The greatest integer function maps integers into integers and non integers to the closet least integer. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765546169711,
"lm_q1q2_score": 0.8424503203538117,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 288.2925318734247,
"openwebmath_score": 0.9280517101287842,
"tag... |
## frx 3 years ago Let a>0 (a). Show using L'Hospitals rule that: $\lim_{x \rightarrow 0} x ^{a}\ln(x)=0$ (b). By setting x=ln(t) in (a). show that: $\lim_{x \rightarrow -\infty} |x|^{a} e ^{x}=0$ I need help with (b). I don't get the question, how do I show that (b). is true by setting x=ln(t) in (a). it doesn't make ... | {
"domain": "openstudy.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9799765546169712,
"lm_q1q2_score": 0.8424503185921023,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 4778.126645609911,
"openwebmath_score": 0.8411459922790527,
"tags": ... |
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