text stringlengths 1 2.12k | source dict |
|---|---|
syms V(t) a
u = symunit;
T = t*u.s; % time in seconds
A = a*u.m/u.s^2; % acceleration in meters per second
eqn1 = A == diff(V,T)
eqn1(t) =
a*([m]/[s]^2) == diff(V(t), t)*(1/[s])
Because the velocity V is unknown and does not have units, eqn1 has incompatible and inconsistent units.
checkUnits(eqn1)
ans =
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Find the distance traveled in 8 seconds where v0 = 20 and a = 1.3. Convert the result to double.
S = subs(S,[v0 a],[20 1.3]);
dist = S(8);
dist = double(dist)
dist =
201.6000 | {
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# Proving sum of product forms a pattern in n * nnnnnn…
I am consider a problem regarding numbers which are, in decimal, one digit repeated - for instance, $88888888$ is such a number. In particular, I am looking at the following problem:
The sum of the digits of the number $$8\cdot \underbrace{88\ldots 88}_{n\text{ ... | {
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One can note a more general pattern - let us, for convenience, define $$R_n=\underbrace{11\ldots 11}_{n\text{ times}}=\frac{10^n-1}9.$$ Then, we can note that, for fixed $k$, after a point, the sum of the digits of $k\cdot R_n$ increases linearly. Note that, if the digits were $8$'s, then we can express the desired pro... | {
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There are a few productive ways to prove this. The one I think is most elegant would be simply to write this as long multiplication: $$\begin{array} & & & 1 & 1 & 1 & \ldots & 1 & 1 & 1 \\ \times & & & & & & & 6 & 4 \\ \hline && 4 & 4 & 4 &\ldots & 4 & 4 & 4 \\ +&6 & 6 & 6 & 6 &\ldots & 6 & 6 & 0\\\hline \end{array}$$ ... | {
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$\lceil\log_{10}(k)\rceil$ steps and this follows the "irregular" portion of the sum (when zeros are still being added from some terms), which has length at most $\lceil\log_{10}(k)\rceil$ as well. | {
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More directly, however, would be an algebraic approach to prove some identity of the form: $$k\cdot R_n = \alpha \cdot 10^n + \beta \cdot R_{n-m} \cdot 10^m + \kappa$$ for positive integers $\alpha$, $\beta$, $m$, and $\kappa$ with $\kappa < 10^m$ and $0\leq \beta < 10$. Knowing the form of $R_n=\frac{10^n-1}{9}$ makes... | {
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$a^2$ has two digits and the sum of the digits is less than 10. (i.e. $a = 4,5,6, 9$)
$a^2$ has two digits and the sum of the digits is 10 or more. (i.e. $a = 7,8$)
====
If $a^2 = b$ has one digit,$b$, then $a*aaa... = a^2*111... = bbbbb$ and the sum of the digits is $m*b$. That was easy.
===
If $a^2 = 10b + c$ ha... | {
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$m*e = \{1000, 991\}$. $991$ is primes so if $m*e = 991$ then $e = 1; a=8; m=991$. If $m*e = 1000$ then $e$ is 1 or a multiple of 2 or 5 only, so possibilities:
$e = 1;a = 1; m = 1000;$
$e = 4; a = 2; m = 250;$
$e = 4; a =7$ is not possible as that would imply $m = 991$. | {
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### Factoring 2b^2+4b-15 Solution
The variable we want to find is b
We will solve for b using quadratic formula -b +/- sqrt(b^2-4ac)/(2a), graphical method and completion of squares.
${x}_{}=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$
Where a= 2, b=4, and c=-15
Applying values to the variables of quadratic equation -b, a an... | {
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b2 + b + = 0
Solutions
how to factor polynomials?
Polynomials can be factored using this factoring calculator
how to factor trinomials
Trinomials can be solved using our quadratic solver
Can this be used for factoring receivables, business, accounting, invoice, Finance etc
No this cannot be used for that
If you... | {
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# Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$
Calculate $$\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$$ where $$C$$ is straight line segment connecting $$(1,1)$$ to $$(2,2)$$
my question is , after calculating the integra... | {
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$$(1,1),(2,2)$$ are joined by the line-segment $$C:y=x\in[1,2]$$. The integral becomes $$\int_C\frac{xdx+ydy}{x^2+y^2}=\int_C\frac{2xdx}{2x^2}=\int_1^2\frac{dx}x=\ln(2)$$
Alternatively, $$\int_C\frac{xdx+ydy}{x^2+y^2}=\int_C\frac12\cdot\frac{d(x^2+y^2)}{x^2+y^2}=\frac12\int_2^8\frac{dm}m=\frac12\ln(m)\Big|_2^8=\ln(2)$$... | {
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## MDAS Quiz with Solution
The order of operations namely multiplication, division, addition or subtraction or more popularly known as MDAS or PEMDAS is one of the most basic concepts in mathematics and yet many people are totally confused about it. Here is a 15-item quiz with solution to further your understanding ab... | {
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8.) $3 + (9 \div 3) \times 6 - 4$
a.) 8
b.) 17
c.) 20
d.) 42
1.) Perform the operation inside the parenthesis first. That is,
$3 + (9 \div 3) \times 6 - 4 = 3 + 3 \times 6 - 4$
2.) Perform multiplication before addition and subtraction.
$3 + 3 \times 6 - 4 = 3 + 18 - 4$
3.) Only addition and subtraction is left, ... | {
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# How do I read and solve equations in the form of $a*b = c \mod d$?
I am trying to find the decryption key of a given RSA problem. I have never solved equations using modulus, and I cannot seem to wrap my head around the equation to find the decryption key.
I am trying to solve this equation:
$43 * d = 1 mod 60$
I... | {
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• Hint: Do you know about modular multiplicative inverse of an integer $a$ modulo $n$ ? – Prasun Biswas Sep 19 '17 at 16:52
• $43d \equiv 1 \mod{60}$ means that when $43d$ is divided by $60$ then the remainder is $1$. It does not mean that $43d = 1$ as $1 \equiv 1 \mod{60}$. – Math Lover Sep 19 '17 at 16:53
• If you co... | {
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The extended Euclidean algorithm gives a way of solving such two-variable equations. In this example we find a solution $d=7$, $e=5$ which gives $d\equiv7\pmod{60}$ as the solution to the congruence.
• Now I understand! My textbook actually wrote it exactly as "Now, solve this equation to find d: 43 * d =1 mod 60" And... | {
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Since $\gcd(43,60) = 1$ there exists $c,d$ that satisfy:
$43 d + 60 c = 1$
Now we apply the Euclidean algorithm. Here is how I think about it.
$\begin{bmatrix} 1\\&1\end{bmatrix}\begin{bmatrix}60\\43\end{bmatrix} =\begin{bmatrix}60\\43\end{bmatrix}$
Now what row opperations do I need to do to reduce the numbers on ... | {
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Thus $\ \color{#c00}{1/43}\equiv 7\pmod{\!60}.\$See this answer for a few other useful methods.
Beware $\$ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus (or, more generally, in certain special contexts such as in the above algorithm). See here for further discus... | {
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# Cubic Spline Interpolation Matrix | {
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Find the cubic spline interpolation at x = 1. Cubic spline with natural boundary conditions. The most common case considered is k= 3, i. axis origin). Predict works as expected. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to ... | {
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at the knots. m calculates divided differences 12) expint. Example of the use of Spline(), Interp(), and Interpolate() functions. , in applications in graphics, numerical methods (e. Let fbe a function from. A little side-note: Bezier-Curves. In interpolating problems, spline interpolation is often preferred to polynom... | {
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are available. 'linear' - linear interpolation 'spline' - cubic spline interpolation 'cubic' - cubic interpolation All the interpolation methods require that X be monotonic. Variable spacing is handled by mapping the given values in X,Y, and XI to an equally spaced domain before interpolating. Part II: Cubic Spline Int... | {
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on the derivatives. We create an “Interpolation_Points” array, and initialize some points we can to draw. CSCE 441 Computer Graphics: Keyframe Animation/Smooth Curves Jinxiang Chai Outline Keyframe interpolation Curve representation and interpolation - natural cubic curves - Hermite curves - Bezier curves Required read... | {
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x =2,4,5 with the cubic spline that. For these reasons quaternion interpolation of the rotational parameters is performed. Then the spline inverse of the matrix Bof the equations for the spline. We consider the problem of shape-preserving interpolation by cubic splines. • This means we have 4n −2 equations in total. Th... | {
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Spline Interpolation. the interpolated result , interpolation kernel , and the scene respectively. I did some simple tests and examples confirming that. These interpolation splines can also be used for extrapolation, that is prediction at points outside the range of x. Interfaces to the BLAS and LAPACK. Then it covers ... | {
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of freedom. Spline interpolation: The existing techniques being not so consistent either with the efficiency or the speed or both, we try to get to the apotheosis of the reconstruction to be Saccomplished by using Cubic-spline interpolation technique. The way of implementing this filter does not involve downsampling,. ... | {
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N 1 t N S 0 (x) S 1 (x) S N 2 (x) S N 1 S(x) (x) x Cubic Spline we want to construct a cubic spline S(x) to interpolate the table presumable of a function f(x). The control point setup can be implemented on MFC interface, can choose cubic spline interpolation or Bezier smoothing. See the example “Spline Interpolation” ... | {
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# Finding the equation of the normal to the parabola $y^2=4x$ that passes through $(9,6)$
Let $$L$$ be a normal to the parabola $$y^2 = 4x$$. If $$L$$ passes through the point $$(9, 6)$$, then $$L$$ is given by
(A) $$\;y − x + 3 = 0$$
(B) $$\;y + 3x − 33 = 0$$
(C) $$\;y + x − 15 = 0$$
(D) $$\;y − 2x + 12 = 0$$
My... | {
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So, the equation of the normal $$\dfrac{y-2t}{x-t^2}=-t \implies xt+y=2t+t^3$$
Here $$2t=6$$
The general equation for a normal to the parabola $$y^2=4ax$$ is
$$y=mx-2am-am^3$$
Putting $$a=1$$ and passing it through $$(9,6)$$, we have $$6=9m-2m-m^3$$ Solving the above cubic for $$m$$ yields three values
$$m=1,2 \te... | {
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# Arrangement of Dominos in a Grid
This the problem from the test and I'm stuck at one part.
Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid to the right so that a path is formed from the upper left-hand corner A to the lower right hand corner B. In a path, consecutive... | {
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"lm_q2_score": 0.8558511396138366,
"openwebmath_perplexity": 352.66315701522564,
"openwebmath_score": 0.7938739061355591,
"ta... |
You need to traverse $7$ cells (you start from $A$, so that cell is a given). You need to make exactly $3$ movements rightwards in total (otherwise we won't reach point $B$'s x-coordinate).
So we are looking for the number of ways we can order $$rrrdddd$$ where $r$ denotes going right, and $d$ denotes going down.
Tha... | {
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"lm_q2_score": 0.8558511396138366,
"openwebmath_perplexity": 352.66315701522564,
"openwebmath_score": 0.7938739061355591,
"ta... |
MathJax reference. Let’s see what it looks like when applying dynamic programming. However I’ve found that simply knowing about dynamic programming and how it fits into a more general problem-solving framework has made me a better engineer, and that in of itself makes it worth the time investment to understand. Thanks ... | {
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"lm_q1_score": 0.9615338123908151,
"lm_q1q2_score": 0.8404548241372259,
"lm_q2_score": 0.8740772433654401,
"openwebmath_perplexity": 450.4392309722117,
"openwebmath_score": 0.37377849221229553,
"... |
am not able to understand this constraint and why we are adding/ subtracting 1 while it is even/odd ? We’ll build both naive and “intelligent” solutions to several well-known problems and see how the problems are decomposed to use dynamic programming solutions. It is of interest therefore to know when such local verifi... | {
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"openwebmath_score": 0.37377849221229553,
"... |
programming is related to a number of other fundamental concepts in computer science in interesting ways. The intuition behind this algorithm is that once you’ve solved for the optimal combination of items at some weight x Why Does De Guiche Want Valvert To Marry Roxane?, Creative Process Pdf, International Hotel San F... | {
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"openwebmath_perplexity": 450.4392309722117,
"openwebmath_score": 0.37377849221229553,
"... |
# ImplicitRegion with RegionPlot Example
Using Mathematica 11.0.1.0 on a MacBook Pro (OSX 10.11.6) I tried this:
Clear[x, y, reg1]
reg1 = ImplicitRegion[
Log[10, 1 + x^2 + y^2] <= 1 + Log[10, x + y], {x, y}];
RegionPlot[reg1]
But got only this output:
RegionPlot[reg1]
But I did get the area.
Area[reg1]
Which g... | {
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"lm_q1q2_score": 0.8404548152886161,
"lm_q2_score": 0.8740772351648677,
"openwebmath_perplexity": 5140.929151670867,
"openwebmath_score": 0.40890368819236755,
"ta... |
# Different results using spline interpolation in Wolfram and MATLAB
I use the same data for interpolation in Mathematica and MATLAB, but the result is different.
x={-1.00,-0.96,-0.65,0.10,0.40,1.00};
y={-1.0000,-0.1512,0.3860,0.4802,0.8838,1.0000};
Interpolation[{x,y}//Transpose,Method->"Spline"][-0.3]
result: -0.... | {
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"lm_q1_score": 0.9615338057771058,
"lm_q1q2_score": 0.84045480889418,
"lm_q2_score": 0.8740772335247532,
"openwebmath_perplexity": 1658.5221578650005,
"openwebmath_score": 0.2078763246536255,
"tags... |
This is a comparison of the results given by Mathematica and MATLAB:
• What methods does splinetx use? The short answer to your first question is no. And the short answer to your second question is yes. The methods used in MATLAB or gnu octave and other similar programs can be realized with WL functions. – CA Trevilli... | {
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"openwebmath_score": 0.2078763246536255,
"tags... |
notAKnotSpline[pts_?MatrixQ] := Module[{dy, h, p1, p2, sl, s1, s2, tr},
h = Differences[pts[[All, 1]]]; dy = Differences[pts[[All, 2]]]/h;
s1 = Total[Take[h, 2]]; s2 = Total[Take[h, -2]];
p1 = ({3, 2}.Take[h, 2] h[[2]] dy[[1]] + h[[1]]^2 dy[[2]])/s1;
p2 = (h[[-1]]^2 dy[[-2]] + {2, 3}.Take[h, -2] h[[-2]] dy[[-1]])/s2;
t... | {
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"lm_q2_score": 0.8740772335247532,
"openwebmath_perplexity": 1658.5221578650005,
"openwebmath_score": 0.2078763246536255,
"tags... |
• (I actually have a more general spline interpolation function than this, but the code is in a hard disk many kilometers away, so releasing that will have to wait.) – J. M.'s technical difficulties Mar 18 at 1:59
• Thanks！It's very helpful. I've read your answers and write a natural cubic spline function. It seems tha... | {
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"openwebmath_score": 0.2078763246536255,
"tags... |
# Help understand
1. Sep 16, 2005
Hi,
A boy wants to knock down a coconut with a rock and his slingshot. He observes that the coconut is about 3.0m above his slingshot and the tree is 4.0m away along the ground.
He knows from experience that the release speed of his rock is 20m/s.
How far above the coconut should he... | {
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"lm_q1_score": 0.9693241991754918,
"lm_q1q2_score": 0.8404387451919326,
"lm_q2_score": 0.8670357615200474,
"openwebmath_perplexity": 1720.3434624095084,
"openwebmath_score": 0.5040285587310791,
"ta... |
7. Sep 16, 2005
### Staff: Mentor
Two things:
(1) I think you have mixed up the 3 and the 4. According to your first post:
(2) Here's a trick that may help you combine those two equations. For each equation, isolate the term with the sin or cos. Then square both sides of each equation. Then add them. (I assume you kn... | {
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"lm_q1_score": 0.9693241991754918,
"lm_q1q2_score": 0.8404387451919326,
"lm_q2_score": 0.8670357615200474,
"openwebmath_perplexity": 1720.3434624095084,
"openwebmath_score": 0.5040285587310791,
"ta... |
Please tell me what is wrong here.
13. Sep 17, 2005
### Staff: Mentor
Your work looks correct. Treat the final equation, as I'm sure you did, as a quadratic in t^2 (say X = t^2). The quadratic has two solutions: you just picked the wrong one! (When you take the square root of those solutions, you can ignore the nega... | {
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"lm_q1q2_score": 0.8404387451919326,
"lm_q2_score": 0.8670357615200474,
"openwebmath_perplexity": 1720.3434624095084,
"openwebmath_score": 0.5040285587310791,
"ta... |
Scan through (partial) tuples
I have a list of list of positive integers $$s = \{s_1, s_2, ..., s_k\}$$, each list $$s_i$$ is possibly of different lengths, and I want to find out if there exists a $$k$$-tuple of the $$s_i$$ that sums exactly to $$n$$. In other words, I want to partition $$n$$ into $$k$$ integers, eac... | {
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"lm_q1_score": 0.9693242000616578,
"lm_q1q2_score": 0.8404387426299114,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1631.2728188724054,
"openwebmath_score": 0.3517647385597229,
"ta... |
You could use LinearProgramming.
To use LinearProgramming, convert the list of lists into a single list. For your example we create the list {1, 2, 3, 4, 5, 6, 7, 1, 3, 2, 4, 6}. Since there is no criteria for which tuple to return, I use a cost vector of all 1s. Then, LinearProgramming will try to find a vector v who... | {
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"openwebmath_score": 0.3517647385597229,
"ta... |
findTuple[s_, n_] := Module[{len, lens, left, v, indicators},
lens = Length /@ s;
len = Total[lens];
left = FoldList[Plus, 0, Most[lens]];
v = Quiet[
LinearProgramming[
ConstantArray[1, Total @ lens],
Join[
indicators,
-Differences[indicators],
{Flatten @ s}
],
Join[
Table[{1,-1},Length[s]],
Table[{0,1}, Length[s]-1],
... | {
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"openwebmath_score": 0.3517647385597229,
"ta... |
I will show how similar tasks can be done using (undocumented) Streaming framework. The usual caveat applies: since this is undocumented functionality, there is no guarantee that it will exist in the future versions in the same exact form, or at all. But I thought it may be a nice application to illustrate some of the ... | {
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"lm_q1q2_score": 0.8404387426299114,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1631.2728188724054,
"openwebmath_score": 0.3517647385597229,
"ta... |
This defines tuples with the length 20^5 == 3200000 and total ByteCount about 128Mb:
Times @@ Length /@ lists
ByteCount[Tuples[lists]]
(* 3200000 *)
(* 128000208 *)
Main example
LazyList brief intro
Evaluating LazyTuples[lists] will create a LazyList object, which is a lazy representation of a list of tuples, wit... | {
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"openwebmath_score": 0.3517647385597229,
"ta... |
We now compare that to in-memory computations using Tuples:
Select[Tuples[lists], Total[#]==100&]//Short//AbsoluteTiming
Select[Tuples[lists], Total[#]==100&]//Short//MaxMemoryUsed
Select[Tuples[lists], Total[#] == 100 &, 1]//Short//AbsoluteTiming
(*
{4.55724,{{4,21,17,30,28},{4,21,21,28,26},{4,21,21,30,24},{4,21,21,... | {
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"openwebmath_score": 0.3517647385597229,
"ta... |
ClearAll[createTupleSelect]
createTupleSelect[criteria_] := createTupleSelect[criteria, All]
createTupleSelect[criteria_, numberOfResults_] :=
Replace[
If[
numberOfResults === All,
Hold[Select[tuples, criteria]],
Hold[Select[tuples, criteria, numberOfResults]]
],
Hold[select_] :> Compile @@ Hold[
{{tuples, _Integer, 2}... | {
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"openwebmath_score": 0.3517647385597229,
"ta... |
We will import it now:
Import[StringJoin[
"https://gist.githubusercontent.com/lshifr/",
]]
which will make a new LazyListSelect[list, pred, entireSelect] function available (again, the only reason for StringJoin is the M SE editor bug).
We can now use it:
Normal @ LazyListSelect[ltLrg, None, sel]//Short//Absolute... | {
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"ta... |
Summary
I have outlined how problems such as this can be treated within Streaming framework.
The main advantages of Streaming are built-in memory efficiency and laziness, which can automatically lead to efficient computations when only a subset of data is needed at the end, while allowing one to stay within the decla... | {
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"lm_q2_score": 0.8670357580842941,
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"openwebmath_score": 0.3517647385597229,
"ta... |
• This, of course would be magnificent to use, but I have to rely on vanilla Mathematica at a client's end, thus I have to work with what's in the box. – István Zachar Mar 2 at 8:45
• @IstvánZachar Well, most of Streaming is in the vanilla Mathematica :). But of course, you are right. Being undocumented, it can't be re... | {
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"lm_q2_score": 0.8670357580842941,
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"openwebmath_score": 0.3517647385597229,
"ta... |
It stores some found values, but not many—only one integer per row of s. In particular, it stores a temporary integer p[i] for each i in the range 1, Length[s] representing the partial sum down the current path to that row. One could rewrite this by re-computing the sum down the path each time, though.
It's a bit slow... | {
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"openwebmath_score": 0.3517647385597229,
"ta... |
(* Note: consider inserting the MaybeSumQ check mentioned below. *)
Module[{a, i = 1, k = Length[s], kmin = kmin0, amax = Length /@ s,
sum, backincrement, p},
If[kmin0 == 0, kmin = k];
Do[a[i] = 1; p[i] = 0, {i, k}];
p[1] = First@First@s;
backincrement[] := If[(a[i] = Mod[1 + a[i], amax[[i]], 1]) == 1,
If[--i == 0, Th... | {
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"lm_q1_score": 0.9693242000616578,
"lm_q1q2_score": 0.8404387426299114,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1631.2728188724054,
"openwebmath_score": 0.3517647385597229,
"ta... |
amax = Table[
Replace[FirstPosition[si, -1, Length[si]],
Dispatch[{{x_} :> x - 1}]]], {i0, k}]
as well as trying handwritten While loops to check each list element one-by-one, and variations of the above with and without Withs and Replaces, and sometimes with /@s. This didn't seem to make much difference.
It compile... | {
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"openwebmath_perplexity": 1631.2728188724054,
"openwebmath_score": 0.3517647385597229,
"ta... |
For completeness, here's the compiled version:
CFindSum0 =
Compile[{{spadded, _Integer, 2}, {n, _Integer}, {kmin0, _Integer}},
Module[{a, i = 1, k = Length[spadded], kmin = kmin0, amax, j, l,
sum, backincrement, p},
If[kmin0 == 0, kmin = k];
amax =
Replace[FirstPosition[si, -1, Length[si]],
Dispatch[{{x_} :> x - 1}]]]... | {
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"lm_q1q2_score": 0.8404387426299114,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1631.2728188724054,
"openwebmath_score": 0.3517647385597229,
"ta... |
Clear[step];
step[ind_, sum_] := Module[{len = Length[ind]},
Which[
sum > n, Nothing,
len >= kmin && sum == n, Throw[ Table[s[[i, ind[[i]]]], {i, len}]],
len < k,
Table[step[Append[ind, i], sum + s[[len + 1, i]]], {i,
Length[s[[len + 1]]]}]
, True, Nothing
]
]
Catch[step[{}, 0]]
(*{3, 7, 3, 6}*) | {
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"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1631.2728188724054,
"openwebmath_score": 0.3517647385597229,
"ta... |
# Why is my regression insignificant when I merge data that produced two significant regressions?
Sorry for the confusing title, I think this is a general statistics question, but I'm working in R. I have a combined dataset of two samples from different countries (n=240 and n=1,010), and when I run a linear regression... | {
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"lm_q1_score": 0.9489172659321807,
"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
• Wow, that's really interesting, I had never heard of Simpson's Paradox! I wonder if you could give me some advice about how to proceed in trying to answer my research question, which is to see whether variable c moderates variable b 's effect on variable a. I'm puzzled as to how I should address something like this, ... | {
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"lm_q1_score": 0.9489172659321807,
"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
• @BenjiKaveladze Yes (see my answer below), but you'll want to verify this is the case by, for example, removing the country with the quadratic relationship from the dataset and see if the observed regression coefficients still change. At any rate, this illustrates how linear regressions can fail to detect nonlinear r... | {
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"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
If your data looks something like this then the reason may be more obvious. Your two original regression lines would be almost parallel and look reasonably plausible but combined they produce a different result which is probably not very helpful.
The data for this chart came from using the R code
exdf <- data.frame(
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9489172659321807,
"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
> summary(fitcombo)
Call:
lm(formula = y ~ x, data = exdf)
Residuals:
Min 1Q Median 3Q Max
-0.8399 -0.4548 -0.0750 0.4774 0.9999
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -9.269561 1.594455 -5.814 0.00017 ***
x -0.007109 0.028549 -0.249 0.80839
---
Signif. code... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9489172659321807,
"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
• In addition to examine the scatterplots, I would try to repeat regression using the country as an additional variable (a~bccountry). This way you will see if some coefficients change significantly between countries.
– Pere
Aug 19, 2017 at 8:47
• @Pere When I include country in the model (a~bccountry), the result prod... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9489172659321807,
"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
In his book, Pearl gives an example very similar to yours. The problem is that there is a confounding variable that is affecting both the independent variable(s) and the dependent variable. In Pearl's example, the question is, Why is an anti-heart attack drug bad for women, bad for men, but good for people? (when the t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9489172659321807,
"lm_q1q2_score": 0.8403909981912566,
"lm_q2_score": 0.8856314753275019,
"openwebmath_perplexity": 755.1924112467557,
"openwebmath_score": 0.4733723998069763,
"tag... |
# L-Norms
$L^p$-norms are some functions which takes a vector as input and output a value. It is written as $\left\|\mathbf x\right\|_ p$.
## L0-Norm
It is a measure of how many non-zero values are there in the vector. If have to put it into notations, we need to first define $0^0=0$. The the $L^0$-norm is as follow... | {
"domain": "haifengjin.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8403484037218509,
"lm_q2_score": 0.8479677564567913,
"openwebmath_perplexity": 3305.9059963316336,
"openwebmath_score": 0.8706493377685547,
"tags"... |
# norm
Norm of matrix or vector
## Description
example
norm(A) returns the 2-norm of matrix A. Because symbolic variables are assumed to be complex by default, the norm can contain unresolved calls to conj and abs.
example
norm(A,p) returns the p-norm of matrix A.
norm(V) returns the 2-norm of vector V.
example... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145704512984,
"lm_q1q2_score": 0.8403484000178051,
"lm_q2_score": 0.8479677545357568,
"openwebmath_perplexity": 7978.637078826074,
"openwebmath_score": 0.9318121075630188,
"tags": ... |
norm2 =
(abs(Vx)^2 + abs(Vy)^2 + abs(Vz)^2)^(1/2)
norm3 =
(abs(Vx)^3 + abs(Vy)^3 + abs(Vz)^3)^(1/3)
Compute the infinity norm, negative infinity norm, and Frobenius norm of V:
normi = norm(V, inf)
normni = norm(V, -inf)
normf = norm(V, 'fro')
normi =
max(abs(Vx), abs(Vy), abs(Vz))
normni =
min(abs(Vx), abs(Vy), abs... | {
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"id": null,
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"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145704512984,
"lm_q1q2_score": 0.8403484000178051,
"lm_q2_score": 0.8479677545357568,
"openwebmath_perplexity": 7978.637078826074,
"openwebmath_score": 0.9318121075630188,
"tags": ... |
The Frobenius norm of a vector coincides with its 2-norm.
### Infinity and Negative Infinity Norm of a Vector
The infinity norm of a 1-by-n or n-by-1 vector V is defined as follows:
The negative infinity norm of a 1-by-n or n-by-1 vector V is defined as follows:
## Tips
• Calling norm for a numeric matrix that is ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145704512984,
"lm_q1q2_score": 0.8403484000178051,
"lm_q2_score": 0.8479677545357568,
"openwebmath_perplexity": 7978.637078826074,
"openwebmath_score": 0.9318121075630188,
"tags": ... |
What is an algorithm for describing the partition of this equivalence relation?
Let $\mathbb{R}$ be the set of real numbers,$f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ℝ$ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid f(x) = f(y) \}.$
Describe the partition of $\Bbb{R}$ ... | {
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"openwebmath_score": 0.9788649678230286,
"tag... |
# 矩阵的逆
## 逆
### $A^{-1}$
$$I=AA^{-1}$$
$$I=A^{-1}A$$
### Notes
Note1:
The Inverse exist if and only if elimination produces n pivots(row exchanges are allowed)
Note2:
The matrix A cannot have two different inverse.
Suppose $BA=I$ , $AC=I$ Then $B=C$ :
$$B(AC)=(BA)C$$
Gives
$$BI=IC$$
or
$$B=C$$
Note3:
if A is i... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145695224667,
"lm_q1q2_score": 0.8403483954226394,
"lm_q2_score": 0.8479677506936878,
"openwebmath_perplexity": 1873.4816647210703,
"openwebmath_score": 0.8567298054695129,
"tags": n... |
$$A=LUD$$
$$A^{-1}=D^{-1}U^{-1}L^{-1}$$
### 逆矩阵的性质(Properties)
1：一个矩阵如果是对称的，并且有逆，那么逆也是对称的。
2：三角矩阵的逆如果存在可能是一个稠密矩阵
Subscribe | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145695224667,
"lm_q1q2_score": 0.8403483954226394,
"lm_q2_score": 0.8479677506936878,
"openwebmath_perplexity": 1873.4816647210703,
"openwebmath_score": 0.8567298054695129,
"tags": n... |
# Value of finite product based on empty set
How does one evaluate the following product if the set S happens to be empty?
\begin{aligned} f(n)= n \prod_{x \in S} \left(1-\frac{1}{x}\right) \end{aligned}
Is the value simply n or is it undefined (or zero)??
Thanks.
Edit: It seems rather odd that this question has b... | {
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"lm_q1q2_score": 0.8403395198685125,
"lm_q2_score": 0.8499711832583695,
"openwebmath_perplexity": 496.91625868246905,
"openwebmath_score": 0.852059543132782,
"tags"... |
# Is $\phi(ab)\ge\phi(a)\cdot \phi(b)$ true for every positive integer pair $(a/b)$?
If $\phi(n)$ is the totient-function, does $$\phi(ab)\ge \phi(a)\cdot \phi(b)$$ hold for every pair $(a,b)$ of positive integers ? And does equality hold if and only if $\gcd(a,b)=1$ ?
I defined $$g:=\gcd(a,b)$$ $$a':=\frac{a}{g}$$ $... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682468025243,
"lm_q1q2_score": 0.8403395177060532,
"lm_q2_score": 0.8499711813581708,
"openwebmath_perplexity": 164.02750731950925,
"openwebmath_score": 0.9669567346572876,
"ta... |
Yes.
Using the formula $\phi(ab)=\phi(a)\phi(b)\frac{\gcd(a,b)}{\phi(\gcd(a,b))}$, we can see that $$\phi(ab) \geq \phi(a)\phi(b) \iff \frac{\gcd(a,b)}{\phi(\gcd(a,b))}\geq1$$ Denoting $c=\gcd(a,b)$, we just need to prove $c\geq \phi(c)$. However, this is always true, since $\phi(n)$ counts the number of positive inte... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682468025243,
"lm_q1q2_score": 0.8403395177060532,
"lm_q2_score": 0.8499711813581708,
"openwebmath_perplexity": 164.02750731950925,
"openwebmath_score": 0.9669567346572876,
"ta... |
Difference between revisions of "2020 AMC 8 Problems/Problem 20"
A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on ... | {
"domain": "artofproblemsolving.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682464686386,
"lm_q1q2_score": 0.8403395117862619,
"lm_q2_score": 0.849971175657575,
"openwebmath_perplexity": 208.4178524230034,
"openwebmath_score": 0.595799446105957,
... |
Case 2: The length of tree 4 is $44$- So, we have the first $4$ tree lengths as $22, 11, 22, 44$. Now, using quick modular arithmetic, we see that when the length of Tree 5 is $88$, the average of the heights of the 5 trees is $\boxed{24.2}\rightarrow\boxed{\textbf{(B)}}$. This is where our condition is satisfied.
~AT... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682464686386,
"lm_q1q2_score": 0.8403395117862619,
"lm_q2_score": 0.849971175657575,
"openwebmath_perplexity": 208.4178524230034,
"openwebmath_score": 0.595799446105957,
... |
# Difference between revisions of "1999 AHSME Problems/Problem 11"
## Problem
The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four center... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9886682444653243,
"lm_q1q2_score": 0.84033950632617,
"lm_q2_score": 0.8499711718571774,
"openwebmath_perplexity": 1912.0478083010807,
"openwebmath_score": 0.48237520456314087,
... |
# What can the range of a measure be?
Given a measure space $$(X,\mathcal{A},\mu)$$, what can the range of the measure, $$\mu[\mathcal{A}]$$, look like? Clearly it can't be an arbitrary subset of $$[0,\infty]$$ as we know $$0\in \mu[\mathcal{A}]$$. We also know $$\mu[\mathcal{A}]$$ has a maximal element ($$\mu(X)$$).
... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631677250739,
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"openwebmath_score": 0.959776759147644,
"tags"... |
• According to Wikipedia, that theorem is due to Sierpinski. Oct 15, 2020 at 13:20
The range of a measure is either $$[0,\infty]$$, or there exists $$a\in[0,\infty)$$ and $$S\subset[0,\infty]$$ such that $$\mu[\mathcal{A}] = [0,a] + \left\{\sum_{x\in X} x \mid X\subset S\right\}$$ where we interpret the sum of two set... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631677250739,
"lm_q1q2_score": 0.8403348719170827,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 84.75198774038644,
"openwebmath_score": 0.959776759147644,
"tags"... |
General $$\sigma$$-finite case: Any $$\sigma$$-finite measure $$\mu$$ can be decomposed uniquely as $$\mu = \nu+\alpha$$ where $$\nu$$ is non-atomic and $$\alpha$$ is purely atomic. Letting $$C$$ be the set of measures of atoms of $$\alpha$$, we have that for any $$A\in\mathcal{A}$$, either $$\mu(A)=\infty$$ or $$\mu(A... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631677250739,
"lm_q1q2_score": 0.8403348719170827,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 84.75198774038644,
"openwebmath_score": 0.959776759147644,
"tags"... |
General case: Non-$$\sigma$$-finite measures are often weird and messy, but one can show that their ranges are the same shape of sets. Suppose $$(X,\mathcal{A},\mu)$$ is not $$\sigma$$-finite and the range of $$\mu$$ is not $$[0,\infty]$$. Then we can let $$a = \sup\{x\in\mathbb{R}\mid \exists A\in\mathcal{A}\text{ non... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631677250739,
"lm_q1q2_score": 0.8403348719170827,
"lm_q2_score": 0.851952809486198,
"openwebmath_perplexity": 84.75198774038644,
"openwebmath_score": 0.959776759147644,
"tags"... |
Envelope Question: Five letters addressed to individuals 1-5 are randomly placed in five addressed envelopes, one letter in each envelope.
I'm trying to find the probability of:
1. Exactly three letters go in the correct envelopes.
2. Exactly two letters go in the correct envelopes
3. No letters go in the correct env... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631651194373,
"lm_q1q2_score": 0.8403348659895197,
"lm_q2_score": 0.8519528057272543,
"openwebmath_perplexity": 151.78200787978946,
"openwebmath_score": 0.8836026787757874,
"ta... |
Probability that exactly two letters are placed in the correct envelopes
There are $$\binom{5}{2}$$ ways to select which two letters are placed in the correct envelopes. None of the remaining letters go in the correct envelopes. There are just two ways to do this. \begin{align*} &\color{red}{1}, \color{red}{2}, \color... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631651194373,
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"lm_q2_score": 0.8519528057272543,
"openwebmath_perplexity": 151.78200787978946,
"openwebmath_score": 0.8836026787757874,
"ta... |
Dividing that number by $$5!$$ gives the desired probability.
There are not $$2!$$ ways to organize the lat two letters. There is only $$1$$ way. Because the second way of organizing them would be to put them in their correct envelopes, which wouldn't match up with the constraint of having exactly $$3$$ letters gettin... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631651194373,
"lm_q1q2_score": 0.8403348659895197,
"lm_q2_score": 0.8519528057272543,
"openwebmath_perplexity": 151.78200787978946,
"openwebmath_score": 0.8836026787757874,
"ta... |
So, suppose we want to put letters A and B in the correct envelopes, while the remaining letters C, D, and E go in the wrong envelopes. Well, we start with our count $$(5-2)!=6$$ ways to put A and B in the right places. Now, we need to subtract off the cases in which more than just those two went in the right envelopes... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631651194373,
"lm_q1q2_score": 0.8403348659895197,
"lm_q2_score": 0.8519528057272543,
"openwebmath_perplexity": 151.78200787978946,
"openwebmath_score": 0.8836026787757874,
"ta... |
Ladder and box (and wall)
• November 18th 2006, 05:37 PM
TriKri
Ladder and box (and wall)
A ladder with the length 10 m is leaned against a box (all sides has the length1 m) placed against a wall. But the ground is so slippery that the ladder falls ower the bow and leans against the wall as well. How high up on the wa... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986363162714234,
"lm_q1q2_score": 0.840334862086558,
"lm_q2_score": 0.8519528038477824,
"openwebmath_perplexity": 1601.9710076424208,
"openwebmath_score": 0.8356586694717407,
"tags... |
where
$
f(x) = x^4 + 2x^3 - 98x^2 + 2x + 1
$
$
f'(x) = 4x^3 + 6x^2 - 196x + 2
$
The goal is to make f(x) = 0. Iterating with an initial value of x = 1, I obtained x = 0.111881932. Iterating with an initial value of x = 10, I obtained x = 8.937993689.
So before sliding down, the ladder reaches about 9.94 m up the wa... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986363162714234,
"lm_q1q2_score": 0.840334862086558,
"lm_q2_score": 0.8519528038477824,
"openwebmath_perplexity": 1601.9710076424208,
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Use the Quadratic Formula to solve for $u$,
. . then back-substitute and solve for $y.$
• July 31st 2008, 08:00 PM
Serena's Girl
Wow, that is so cool! *bows down to Soroban*(Bow)
• August 1st 2008, 11:43 AM
TriKri
Congratulations to both of you! You showed two different ways to solve the problem in and both where rig... | {
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"lm_q1_score": 0.986363162714234,
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"lm_q2_score": 0.8519528038477824,
"openwebmath_perplexity": 1601.9710076424208,
"openwebmath_score": 0.8356586694717407,
"tags... |
# XOR of Binary Numbers to Reach a Given Number
Given a set
S = { s1, s2, s3, ... sn}
of Binary Numbers , I need to find if a given Binary Number X with only 1 bit position set as 1 (..00001000...), can be reached by doing bitwise XOR operation.That is ,I need to find out if there is a subset of S such that
X = si... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631659211717,
"lm_q1q2_score": 0.840334861111034,
"lm_q2_score": 0.8519528000888387,
"openwebmath_perplexity": 169.20582574863252,
"openwebmath_score": 0.6406171917915344,
"tag... |
An example run with $S=\{0010, 1001, 1010, 0101, 1110, 1100\}$ and $X=0001$. The augmented matrix is $$\left( \begin{array}{cccccc|c} 0&1&1&0&1&1&0\\ 0&0&0&1&1&1&0\\ 1&0&1&0&1&0&0\\ 0&1&0&1&0&0&1\\ \end{array}\right)$$ We first do some row swaps. Move the third row to the top (need to get that $1$ to top left corner), ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9863631659211717,
"lm_q1q2_score": 0.840334861111034,
"lm_q2_score": 0.8519528000888387,
"openwebmath_perplexity": 169.20582574863252,
"openwebmath_score": 0.6406171917915344,
"tag... |
• IOW, the subset sum problem is much, much harder. – Jyrki Lahtonen Apr 14 '13 at 13:04
• Thank You! Can I get a reference link on how to find whether the augmented matrix is solvable ? The size of the set S can be upto 1000. I just need to find whether a solution exists or not. – Kyuubi Apr 14 '13 at 13:13
• It is th... | {
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# Lecture 15 - Fitting distributions to data
(previous, next)
• Ideal asymptotic distributions
• Examples of fitting data distributions
## More asymptotic distributions
Using methods similar to what we did for the Poisson distribution, we can derive other asymptotic approximations to distributions.
### Normal aka ... | {
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Well, one answer is to just try many values of $$\lambda$$ and look for the one that does the best job of fitting the data. Let's look at some classical examples.
### Deaths from horse-kicks
Here is a classic example of something that seems totally unrelated by can be explained by the Poisson distribution. Notice tha... | {
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Annual deaths from horse kicks in the Prussian army (1875-1894)
Deaths Record
0 144
1 91
2 32
3 11
4 2
5 0
6 0
[Show code] | {
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 #!/usr/bin/env python """ This script creates an animation sho... | {
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observed') xlabel('Number of successes') xlim(-0.5,max(x)+.5) ylim(0,max(y)*1.2) show(block=False) i, di = 0, 1 i_max = len(u) j = 0 while True: marker_fit_error.set_data([ u[i] ], [v[i]]) z = poisson.pmf(x, u[i])*N for rect, h in zip(bar_ob, z): rect.set_height(h) fig.canvas.draw() savefig('frame%04d.tif'%j) j += 1 i ... | {
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### Accidents in a factory
Another example, which seems like it should also be a Poisson distribution, was collected for the rates of accidents in an ammunition factory.(Not a place you want to be having accidents!) In this case, though, the best explanation leaves more than 30 percent of that probability mass unexpla... | {
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Number of munitions accidents in WWI factor per person
Count Observed
0 447
1 132
2 42
3 21
4 3
5 2
Over 5 0
Sum 647
[Show code] | {
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