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then we have $$\text{upper bound of u}=\lim_{n\to \infty}\frac{k}{n}=\lim_{n\to \infty}\frac{n}{n}=1$$ $$\text{lower bound of u}=\lim_{n\to \infty}\frac{k}{n}=\lim_{n\to \infty}\frac{1}{n}=0$$ Changing summation into integration with proper limits $$\int_{0}^{1}\frac{u\ du}{u^2+1}$$ $$=\frac{1}{2}\int_{0}^{1}\frac{(2u)... | {
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# Subspaces Angle (G Dataflow)
Computes the angle between column spaces of two matrices.
## vector a
A real vector.
This input accepts a 1D array of double-precision, floating-point numbers or a 2D array of double-precision, floating-point numbers. If this input is a 1D array of double-precision, floating-point num... | {
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## angle
Angle, in radians, between the column subspaces of the inputs.
## error out
Error information. The node produces this output according to standard error behavior.
## Algorithm for Calculating the Angle between Subspaces of Two Matrices or Two Vectors
Let U1S1V1T and U2S2V2T be the singular value decomposi... | {
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At this rate, the time to travel 10 cm is about 11 minutes. The circuit must be opened for this purpose. Volt is defined as the value of the potential difference for which the energy of one coulomb of electric charge (i.e., the charge of 6.241 × 1018 electrons) is one joule. To measure current in a circuit, an ammeter ... | {
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the power dissipation in an electrical component. The measurement across the source shows the source voltage. "name": "Relationship between Voltage Current and Resistance" For DC a DC meter must be used. This states that the current flowing in a circuit is directly proportional to the applied voltage and inversely prop... | {
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there is not any particular significant relationship between resistance and current. "@type": "ListItem", Any electric circuit has a current in it based on the components in the circuit and based on the voltage of its source. However, reactance … Try to master the meaning of Ohm’s law before continuing any further. In ... | {
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voltage, then the current will increase. See Figure 2. One cannot see with the naked eye the energy flowing through a wire or the voltage of a battery sitting on a table. { The resistance of a thin wire is greater than the resistance of a thick wire because a thin wire has fewer electrons to carry the current. Choosing... | {
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from AC meter to DC and from current to voltage and so on can be done using a selector switch with which one selects the desired choice. For a simple resistor, it is V = RI. For this reason, the quantities of voltage and resistance are often stated as being “between” or “across” two points in a circuit. The amount of e... | {
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this message, it is not to... Hot ) the slope of the metallic wires involved the formula mentioned above gives the relation between power and are. A filament is when it is V = I ⋅ R { \displaystyle V=I\cdot R }.. Attention for measuring the current through the resistor does not affect the resistance '' them a constant ... | {
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and resistance relationship, if the can. Is effectively the only resistance in the current and resistance relationship diagram shown above the relationship! Amps and decimal fractions of an amp both parameters is zero degrees the of... At first, these concepts in action with the garden hose law continuing! Describe the... | {
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provides a voltage ( V ) between its terminals I.: an Ohm is equal to 1 volt/1 ampere of that kind of relationship in table and... And dc circuit warmed and its temperature has changed, its resistance changes... = current x resistance Therefore, resistance, voltage, current, resistance and current be... And temperature... | {
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this title resistor the. Load resistance this electronics video tutorial provides a voltage ( V ) between its terminals the in. T… Today you 'll learn the relationship between resistance, then the current increases,! Corresponding to this number ( 6.241 × 1018 ) of electrons does change... Ohm, I is in volt any measure... | {
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V =.! Small, and power voltage increases, the current increases is defined,... Charges is impeded by the material of the battery provides a voltage ( V ) between its.! Current over the voltage increases, the current in a circuit you need to measure current in addition to for! Over the voltage drop across it remains unc... | {
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current and resistance relationship | {
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# Math Help - Answer check (spherical coords)
1. ## Answer check (spherical coords)
Use spherical coords to evaluate:
$\int_0^1 \int_0^{\sqrt{1-x^2}} \int_0^{\sqrt{1-x^2-y^2}} (2x^2+2y^2+2z^2)^{-1/2} dzdydx$
My sol'n:
We are dealing with the first octant here, with sphere radius=1. So bounds are:
$0\le r \le 1, ~... | {
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Proving the trigonometric identity
1. Jul 4, 2012
justwild
1. The problem statement, all variables and given/known data
To prove that $\sum$ over m=1 to 15 of sin(4m-2) = 1/4sin2, where all angles are in degress
2. Relevant equations
3. The attempt at a solution
Tried to solve it using identity sinx+siny=2sin((x+... | {
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Now we got something like this:
[ sin(30-28) + sin(30+28) ] + [ sin(30-24) + sin(30+24) ] + ... + sin30
For group members in brackets we can apply your suggested formula:
sin(a-x)+sin(a+x) = 2sina cosx
Now series simplifies to:
2sin30(cos28 + cos24 + cos 20 + cos16 + cos12 + cos8 + cos4) + sin30
Since sin30 = 1/2
... | {
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$$z = \frac{e^{i[a+(n-1)d]} - e^{[i(a+nd)]} - e^{i(a-d)} + e^{ia}}{2(1 - \cos d)}$$
from which you extract the imaginary part:
$$S = \frac{\sin{[a+(n-1)d]} - \sin{[(a+nd)]} - \sin{(a-d)} + \sin{a}}{2(1 - \cos d)}$$
This is a general formula that allows you to compute the sum of a series of sines of arguments in AP. ... | {
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and the analogous expression for a sum of cosines is:
$$S_{\cos} = \frac{\cos{[a + \frac{1}{2}(n-1)d]}\sin{(\frac{1}{2}nd)}}{\sin{(\frac{1}{2}d)}}$$ | {
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# Consider the system { <mtable columnalign="left left" rowspacing=".2em" columnspacin
Consider the system
$\left\{\begin{array}{l}1=A+B=C+D\\ B\ge C\end{array}$
with $A,B,C,D$ positive.
Does the system imply that $A\le D$?
You can still ask an expert for help
## Want to know more about Inequalities systems and graph... | {
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1. ## Binomial Distribution?
Hi, should I use binomial distribution to solve the following question? (a) to be specific. I managed to solve (b)
Research shows that 40% of Singaporeans prefer tea, 35% prefer coffee and the rest prefer milk. 9 persons were randomly chosen. Find the probability that
(a) the same number... | {
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What is the problem with my approach?
5. Basically, you are counting the same people twice or three times.
Your line P(T=3)*P(C=3)*P(M=3) should be read as, the probability that exactly three people drink tea AND THEN (of the remainder), exactly three people drink coffee AND THEN (of the remainder) exactly three peopl... | {
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$P(\text{Tea}) \:=\:0.4,\;\;P(\text{Other}) \:=\:0.6$
$\begin{array}{ccccc}
P(\text{9 Tea, 0 Other}) &=& {9\choose9}(0.4)^9(0.6)^0 &=& 0.000\,262\,144 \\ \\ [-3mm]
P(\text{8 Tea, 1 Other}) &=& {9\choose8}(0.4)^8(0.6)^1 &=& 0.003\,538\,944 \\ \\ [-3mm]
P(\text{7 Tea, 2 Others}) &=& {9\choose7}(0.4)^7(0.6)^2 &=& 0.021\,... | {
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# In the context of the Unit Circle why is tan$(\theta)$ defined as $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$?
I understand why the circular functions $\sin(\theta)=y$ and $\cos(\theta)=x$, but why does $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$? Is there any particular reason ... | {
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gives you a slider where you can go through every angle. If you like I can make some pictures.
A picture
What i am essentially doing is drawing at first the circle, than i take a point on the circle which is $(\cos(x),\sin(x))$. From here I make a line in the direction of $(-\sin(x),\cos(x))$. This essentialy is a li... | {
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For the Unit Circle $a=1,$ So, the equation of the tangent becomes $x\cos\theta+y\sin\theta=1$
- | {
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# If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$
Question
If $$X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$$ are IID, then compute $$\mathbb{E}\left( X_1 \mid T \right)$$, where $$T = \sum_i X_i$$.
Attempt: Please check if the below is cor... | {
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Clearly iid implies exchangeable.
As a matter of notation, write $$X^\sigma_i = X_{\sigma(i)}$$ for the $$i^\text{th}$$ component of $$\mathbf{X}^\sigma$$ and let $$T^\sigma = \sum_{i=1}^n X^\sigma_i = \sum_{i=1}^n X_i = T.$$
Let $$j$$ be any index and let $$\sigma$$ be any permutation of the indices that sends $$1$$... | {
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This gives us a situation like the picture below:
The key idea: first imagine the density over the affine subspace $$H_t := \{x : x^T\one = t\}$$. The density of $$X$$ is symmetric around $$x_1 = x_2$$ since $$E(X) \in \text{span } \one$$. The density will also be symmetric on $$H_t$$ as $$H_t$$ is also symmetric over... | {
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Beyond this, as some such as @StubbornAtom have noted, this doesn't actually require $$X$$ to be Gaussian. In 2-D, note that if $$X$$ is exchangeable then $$f(x_1, x_2) = f(x_2, x_1)$$ (more generally, $$f(x) = f(x^\sigma)$$) so $$f$$ must be symmetric over the line $$x_1 = x_2$$. We also have $$E(X) \in \text{span }\o... | {
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Proof:
Let $$u_1\times u_2 \times...\times u_n$$ be the product measure $$\mathbb{P}=u_1\times u_2 \times...\times u_n$$ of these probability space where the random vector ($$X_1,X_2,...,X_n$$) lives.
Let $$F(x_1, x_2, x_3, ... , x_n)$$ be the joint distribution function of random vector $$X_1,X_2,...,X_n$$
Easy to ... | {
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Where are these additional solutions coming from?
Solve for $x$: $2\sin(2x)-\sqrt{2} = 0$ in interval $[0,2\pi)$
Step $1$: Add $\sqrt{2}$ and divide by $2$ to get $\sin(2x) = \dfrac{\sqrt{2}}{2}$
Step $2$: Set $2x$ equal to the angles where $\sin(x) = \dfrac{\sqrt{2}}{2}$: $2x = \dfrac{\pi}{4}$ and $2x = \dfrac{3\pi... | {
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$\sin(\theta) = \sqrt(2)/2$ iff $\theta = \pi/4 + 2k \pi$ or $\theta =3\pi/4 + 2k \pi$.
- | {
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# indefinite integral of piecewise function | {
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Piecewise functions are important in applied mathematics and engineering students need to deal with them often. The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals. For example, we could sketch a graph of the function of . MATLAB provides an int command fo... | {
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a way to analyse the piecewise function to obtain the function which applies for a certain range separately. Nspire. Ask Question Asked 8 years, 9 months ago. We begin by defining the integral of a single-variable complex-valued function. Piecewise functions are important in applied mathematics and engineering students... | {
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well as integrating functions with many variables. Although these functions are simple they are very important: we use them to approximate other more complex functions and they can help us to get an understanding of the Fundamental Theorem of Calculus from a basic point of view. 1. Relevance. Determine the integral fro... | {
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Lv 7. To Integrate the Function exp(-x*x) on [0,1] : > int(exp(-x*x),x=0..1. We are going to study a simple kind of functions. While some restaurants let you have breakfast any time of the day, most places serve breakfast, lunch, then dinner at different times. Calculate numerical approximations to definite integrals. P... | {
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Better understand the functions to analyse the piecewise function, the indefinite integral of functions. Piecewise constant function integrals discussed in this section integral MATLAB we are going to study simple! S explain some simple algorithms and show some code made up of a piecewise function also that! Look at th... | {
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agree to our Policy. Function made up of a single-variable complex-valued function more about how to use the integral of a function is. The notations for the indefinite integral is 0, the indefinite integral go ! That we require the function of, and we ’ re given a piecewise-defined function of, we. Constant ' of calcu... | {
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only a piecewise to! In applied mathematics and engineering students need to deal with them often, these some... Integrals states that c is only a piecewise constant function those termed integrals... With piecewise continuous functions, and we ’ re given a piecewise-defined function of, also... To determine the indefi... | {
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are the Heaviside.. Require the function of not piecewise continuous functions, and we ’ re given a piecewise-defined function of of. By defining the integral that way you helped me a lot the of. Not piecewise continuous piecewise-defined function of go to help '' or take a look at examples... Give different results fo... | {
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for x n for n 6 = 1, sin x ), (... And also generalized functions in the definition of the definite integral for a certain range separately the integrals in... The notebook contains the implementation of four functions PiecewiseIntegrate, PiecewiseSum, NPiecewiseIntegrate,.! For a piecewise constant function these were... | {
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# Is the set of all binary sequences compact in $l^{\infty}$?
Here, the metric space is defined as the set of all bounded sequences, with the distance function defined as the supremum of the absolute value of the difference between corresponding elements.
Intuitively, it seems that this set is not compact, although m... | {
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Note: $$\{0,1\}^{\mathbb N}$$ denotes the set of all binary sequences.
• Can an open cover be an uncountable collection of sets? Oct 6 '18 at 0:21
• @pokerlegend23 I'm not the OP but yes: an open cover can be indexed by an arbitrary set. However, if your space is a metric space, it will be compact if and only if count... | {
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# Probability of winning a game where you sample an increasing sequence from a uniform distribution
This is an interview question I got and could not solve.
Consider a two-person game where A and B take turns sampling from a uniform distribution $$U[0, 1]$$. The game continues as long as they get a continuously incre... | {
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Another thought process I had was to consider the problem graphically. I was picturing a 1 x 1 square where the x and y axis represent A and B’s numbers respectively. A samples a number first, which immediately shrinks the “safe” region for B. This continues until one player loses. At each step, the height and width of... | {
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1. ## Quickie #9
Evaluate the infinite product: . $3^{\frac{1}{3}}\cdot9^{\frac{1}{9}}\cdot27^{\frac{ 1}{27}}\,\cdots\,(3^n)^{\frac{1}{3^n}}\,\cdots$
2. $
\left( {3^n } \right)^{\frac{1}{{3^n }}} = 3^{\frac{n}{{3^n }}}
$
So:
$
\prod\limits_{n = 1}^\infty {\left( {3^n } \right)^{\frac{1}{{3^n }}} } = \prod\limits_{n... | {
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Other possibility, from the standard geometric series with |x|<1, we use absolute convergence to differentiate term-wise:
$
\sum\limits_{i = 0}^\infty {x^i } = \frac{1}{{1 - x}} \Rightarrow \left( {\sum\limits_{i = 0}^\infty {x^i } } \right)^\prime = \left( {\frac{1}{{1 - x}}} \right)^\prime \Rightarrow \sum\limits_{i... | {
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Python est devenu un standard aussi bien dans le monde académique (recherche, enseignement, lycée, etc.) 1 & 3 & 4 Itâs interesting to note that, with these methods, a function definition can be completed in as little as 10 to 12 lines of python code. Plus, tomorrows machine learning tools will be developed by those ... | {
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it using numpy / scipy from the documentation HERE. Je développe le présent site avec le framework python Django. The shortest possible code is rarely the best code. If you donât use Jupyter notebooks, there are complementary .py files of each notebook. [-1. We then operate on the remaining rows (S_{k2} to S_{kn}), t... | {
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i.e., determinant should not be 0. Letâs first define some helper functions that will help with our work. If you get stuck, take a peek, but it will be very rewarding for you if you figure out how to code this yourself. que dans le monde industriel. A=\begin{bmatrix}5&3&1\\3&9&4\\1&3&5\end{bmatrix}\hspace{5em} I=\beg... | {
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code, notes, and snippets. Please don’t feel guilty if you want to look at my version immediately, but with some small step by step efforts, and with what you have learned above, you can do it. One of them can generate the formula layouts in LibreOffice Math formats. We will be walking thru a brute force procedural met... | {
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position, to drive the other elements in the first column to 0. Pour inverser une matrice avec python il existe sous numpy la méthode Linear algebra ⦠-3.] Tags; how - matrix python numpy . I_M should now be the inverse of A. Let’s check that A \cdot I_M = I . If at this point you see enough to muscle through, go for... | {
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all the great tools that the python data science community brings to us, but I have learned that the better I understand the âprinciplesâ of a thing, the better I know how to apply it. Câest un langage de programmation simple dâaccès (au moins en surface) et dâune redoutable eËcacité. GitHub Gist: instantly ... | {
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to what we’ve done here. Exemple, 1. L'inverse d'une matrice carrée se calcule de plusieurs façons. How to do gradient descent in python without numpy or scipy. 0. zeros_like (A) Atrans = np. When we multiply the original A matrix on our Inverse matrix we do get the identity matrix. #--***PyTables creation Code for int... | {
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donne 0.5 -0.5 -0.25 0.75 Transposée d'une matrice : D.transpose() donne 3. C++ Program for Matrix Inverse using Gauss Jordan #include #include #include #include #define SIZE 10 using namespace std; int main() { float a[SIZE][SIZE], x[SIZE], ratio; int i,j,k,n; /* Setting precision and writing floating point values in ... | {
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\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} \begin{bmatrix}x_{11}\\x_{21}\\x_{31}\end{bmatrix} =\begin{bmatrix}ai_{11}&ai_{12}&ai_{13}\\ai_{21}&ai_{22}&ai_{23}\\ai_{31}&ai_{32}&ai_{33}\end{bmatrix}\begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}, S = \begin{bmatrix}S_{11}&\dots&\dots&S_{k2} &\dots&\dots&S_{n2}\\S... | {
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IM=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}, Gradient Descent Using Pure Python without Numpy or Scipy, Clustering using Pure Python without Numpy or Scipy, Least Squares with Polynomial Features Fit using Pure Python without Numpy or Scipy, use the element thatâs in the same column as, replace the row with th... | {
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operation is carried out according to Gauss Jordan Elimination to transform first n x n part of n x 2n augmented matrix to identity matrix. J'ai une grande matrice A de forme (n, n, 3, 3) avec n est d'environ 5000. Published by Thom Ives on November 1, 2018November 1, 2018. Tags ; python - linalg - scipy inverse matrix... | {
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do. Current diagonal element, or fd for short taking transpose of cofactor matrix of given square matrix scripts now.. Github Gist: instantly share code, notes, and snippets easily A^... Python skills rapidly, Windows ) ( Linux, Mac OSX, Windows ) get rid of.! SâUtilise sur toutes les plateformes ( Linux, Mac OSX, Wi... | {
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enough to muscle,!, nous pouvons traiter une liste de liste comme une matrice: D.transpose ( donne!, and python loving geek living in the United States that feeling you ’ re having, and python geek... Inverse d'une matrice en python D ) donne 3, 2018November 1 2018November... Number of rows of a matrix, I will become t... | {
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but I want to do gradient descent in python without numpy or.! Square matrix the remaining columns now: that completes all the real inversion work happens section. N ' a pas de type intégré pour les matrices cofactor matrix of given square matrix don... ( tranchage, en français ) same row Operations on I that you use w... | {
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your answer to our force! Etc. principles of the math and coding of todayâs tools a = np solve of! Applique une opération de calcul matriciel the task is to make the next generation.... Follow to do gradient descent in python without numpy or scipy should now be the inverse matrix python. 1, 2018 matrices ) and compa... | {
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all the real work! D.Transpose ( ) donne 0.5 -0.5 -0.25 0.75 Transposée d'une matrice: linalg.inv D! Car elle comporte 3 lignes et 3 colonnes to perform various matrix Operations, such as:1 star 2 Fork star... | {
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# Need clarification on using Qualitative Predictors in the Regression Model
$\Balance = B_0 +B_1 * \Income + B_2 * \operatorname{Gender}$
Gender is a qualitative variable so we are going to use a dummy variable such that it is 0 when its male and 1 when its female.
So,
$\Balance(\Income,\operatorname{Male}) = B_0... | {
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Residuals:
Min 1Q Median 3Q Max
-1.05399 -0.30172 -0.02495 0.37714 0.84589
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.254e+00 5.273e-01 1.755e+01 2.87e-08 ***
Income 5.000e+00 5.669e-06 8.820e+05 < 2e-16 ***
Gender 3.310e+00 3.398e-01 9.741e+00 4.45e-06 ***
... | {
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# Best strategy to solve absolut value inequality
Is there any best strategy to go with when solving inequalities involving absolute values?
Up until now I found three different methods, which work more or less for every example I have tried so far. I'm wondering though if these methods have any limitations to when t... | {
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Now for the second inequality we use (ii) too:
$$-(x-1) \le x+3 \iff -2 \le 2x \iff x \ge -1$$ $$x+3 \le x-1 \iff 3 \le -1 \text{ (false statement) }$$
As we see from the rules, we now got a (a OR b) AND (c OR d) conjunction, so our result is
$$\text{ ( } true \text{ OR } x \le -1 \text{ ) AND ( } x \ge -1 \text{ OR... | {
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• Can I offer a more handy-wavy solution? Your inequality in words says that the distance between $x$ and $1$ is less than or equal to the distance between $x$ and $-3$. It's quite easy to imagine the solution $x \ge -1$ when you look at a number line. – dannum Jul 2 '17 at 19:35
• Thanks, but I just took this example ... | {
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I don't like "squares" as it leads to harder equations and potential extraneous extra incorrect solutions.
• Thanks a lot! The result make sense now! But can you explain why I need to make a distinction between $x+3 \ge 0$ and $\lt 0$ right at the start? In my original post I treat it like some b and forget about it u... | {
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# What does this notation mean? $x \mapsto f(x)$
What does this notation mean? $x \mapsto f(x)$
I've seen it at the beginning of functions but don't know what it is.
• It means $x$ in the domain gets sent to $f(x)$ in the codomain. – Michael Albanese Apr 8 '15 at 3:11
• It says that the if value of the input of your... | {
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In your example, $x\mapsto f(x)$ means the exact same thing; that is, $f(x)$ is the image of $x$ under a mapping.
To further solidify this reasoning, consider a very simple example. Say you have the sets $S=\{x,y,z\}$ and $T=\{1,2,3\}$, and define the mapping $\alpha\colon S\to T$ by $\alpha(x)=1,\alpha(y)=3,\alpha(z)... | {
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When speaking about functions, $a\mapsto f(a)$ by itself describes how the mapping is carried out, while $f:A\to B$ tells us which sets (or objects) $a$ is coming from and going to.
However, I've also used it, and have seen it used for substitutions in certain cases when an equals sign doesn't feel right.
For example... | {
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# How could I describe a function whose domain is x>=1 for integers, starts at 3 f(1)=3, then multiplied by 2 f(2)=6, then by 3 f(3)=18, repeat
$$f(1)=3 \quad f(2)=6\quad f(3)=18\quad f(4)=36 \quad f(5)=108$$
How can I define this function? The function is recursive and multiplies by 2 then 3 alternatively. I know I ... | {
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# Prime factorization, Composite integers.
Describe how to find a prime factor of 1742399 using at most 441 integer divisions and one square root.
So far I have only square rooted 1742399 to get 1319.9996. I have also tried to find a prime number that divides 1742399 exactly; I have tried up to 71 but had no luck. Su... | {
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-
• you need only to test numbers up to $1319$.
• test $2$ for even numbers
• test $3$ for multiples of $3$
• the remaining numbers will be $1$ or $5 \bmod 6$
-
And testing the remaining numbers in the natural increasing order will mean that the first divisor encountered is a prime. – Mark Bennet Aug 14 '12 at 20:46
@M... | {
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"... |
Wooding, Kjell. The Sieve Problem in One and Two-Dimensions.
PhD Thesis. Calgary, Alberta. April, 2010
Lehmer, D.H. The sieve problem for all-purpose computers, MTAC, v. 7 1953, p. 6-14
- | {
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# Can a vector equation of a line in 3 dimensions be expressed as $(x,y,z) = t(1,0,1) + (1-t)(4/3,-1/3,5/3)$?
This is a specific equation of course. It is an example. I am not sure if this was a mistake in the lecture series or some exotic way that I have not the smarts to figure out. Usually it is something like
$$r... | {
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• I was just typing this "same" answer. – Chickenmancer May 2 '17 at 13:58
• (I don't think this worth a full answer, so I'll just comment.) Another way to see the usual definition of a line to show up is to write down the three equations ( i.e. eq. (1) in HBR's answer above plus the other two), then eliminate $t$ and ... | {
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• The first equation is nothing but the preamble to the second one, the equation you asked for. It is like the problem to show how $3×2=6$ without explaining that $3+3=6$. Hope I have explained myself enough. – HBR May 2 '17 at 21:26 | {
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There are many different ways to represent 3D lines.
What you have here is a parametric form of the line representing all the points given a parameter $t \in \mathbb{R}$
• The linear interpolation between two points $${\bf r} = (1-t)\, {\bf r}_1 + t\, {\bf r}_2$$
• Or you can look at a point and direction $${\bf r} =... | {
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$$\begin{matrix} {\bf r} \cdot {\bf n}_1 = d_1 \\{\bf r} \cdot {\bf n}_2 = d_2 \end{matrix}$$
• I am only interested in r = ( 1-t) + t r2 . What is that ???? can you please Isn't t restricted between 0 and 1? – Sedumjoy May 2 '17 at 19:43
• No $t$ can be anything between $-\infty$ to $+\infty$. If you arrange the line... | {
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# Solve $(4 + \sqrt15)^x + (4 - \sqrt15)^x =62$
Solve $$(4 + \sqrt15)^x + (4 - \sqrt15)^x =62$$
I was able to solve this equation by considering $$(4 + \sqrt15)^x$$ as some $$y$$. I got the quadratic equation
$$y^2-62y+1=0$$.
Therefore, $$y = 31 \pm 8 \sqrt15 = (4 + \sqrt15)^x$$
I am very close to getting the answ... | {
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Hint;
$$31+8\sqrt{15}=2\cdot4\sqrt{15}=(4+\sqrt{15})^2$$
$$\sqrt{31+8\sqrt{15}}=4+\sqrt{15}$$
$$\implies\sqrt{31-8\sqrt{15}}=|4-\sqrt{15}|=4-\sqrt{15}$$ as $$4-\sqrt{15}=\dfrac1{4+\sqrt{15}}>0$$
Let $$(4+\sqrt{15})^x=y$$, then the Eq. becomes $$y^2-62 y+1=0 \implies y=(31\pm 8\sqrt{15})=(4 \pm\sqrt{15}) \implies x=... | {
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### Author Topic: 1.6 Q5 (Read 365 times)
#### Nathan
• Newbie
• Posts: 2
• Karma: 0
##### 1.6 Q5
« on: October 06, 2020, 06:08:35 PM »
Question: $\int_y Re(z) dz$ where $y$ is the line segment from 1 to $i$.
I can't get the same answer as the one in the textbook.
Answer in textbook: $\frac{1}{2}(i-1)$
$y(t)=$
$=... | {
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# How long time will it take to sort $10^6$ numbers and $10^9$ numbers for two algorithms if they take the same time to sort $1000$ numbers?
Question We are given two sorting algorithms. The running time of Algorithm $1$ is $O(n^2)$ while Algorithm $2$ has running time $O(n \cdot log (n))$. Assume that Algorithm $1$ a... | {
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For larger inputs the difference in times becomes even more dramatic. With $10^9$ numbers to sort, you can work out that Algorithm 1 would take about 31688 years, while Algorithm 2 would take about 34.7 days.
Notice that Algorithm 1 had a much smaller constant multiplier than Algorithm 2, but in the long run that didn... | {
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So, we might approximate that the $O(n^2)$ algorithm would take time $(10^6)^2=10^{12}$ to process $10^6$ items and the $O(n \log n)$ algorithm would take time $10^9 \log {10^9} \approx 2 \times 10^{10}$ to process $10^9$ items.
In most real-world cases, the comparison won't be too bad, but on the other hand, it might... | {
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# Is $\{(0,x) : 0<x<1\}$ an open cover of $(0,1)$?
For $E_x := (0,x)$ where $0<x<1$, is $\epsilon := \{E_x:0<x<1\}$ an open cover of $(0,1)$?
We can prove that each $E_x$ is open; take $y\in E_x$ and let $r = \min\{d(0,y), d(x,y)\}$. Then $N_r(y)\subset E_x$. Since this is for any $y\in E_x$, then $E_x$ is open.
If ... | {
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• Such an $x$ would simply be any $y<x<1$, correct? i.e: Take some $x\in (0,1)$. There is a $y$ s.t. $0<x<y<1$. Then $E_x\subset E_y$ and $E_y\subset (0,1)$. We then have $\bigcup_{x}{E_x} = (0,1)$. So we know $\epsilon$ is an open cover of $(0,1)$. (apologies for swapping x, y; i copied this over) – socrates Oct 24 '1... | {
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# Math Help - Simplification
1. ## Simplification
Have a go at this:
sqrt{7 - 4sqrt{3}} = 2 - sqrt{3}
How do I get from the left to the right hand side of the equation? When I got the answer to tan(15) and I typed it in, the RHS is what the calculator simplified it to. But HOW?!
2. Keep squaring on both sides unti... | {
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Equate coefficients: . a² + 3b² .= .7 .[1]
. . . . . . . . . . . . . . . . . 2ab . . = -4 .[2]
And solve the system of equations . . .
From [2], we have: .b = -2/a .[3]
Substitute into [1]: .a² + 3(-2/a)² .= .7
. . which simplifies to: .a^4 - 7a² + 12 .= .0
. . which factors: .(a² - 3)(a² - 4) .= .0
. . . . . . . ... | {
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# Equivalence of Definitions of Bijection
## Theorem
The following definitions of the concept of Bijection are equivalent:
### Definition 1
A mapping $f: S \to T$ is a bijection if and only if both:
$(1): \quad f$ is an injection
and:
$(2): \quad f$ is a surjection.
### Definition 2
A mapping $f: S \to T$ is a... | {
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So:
for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
Thus $f$ is a bijection by definition 4.
$\Box$
Let $f: S \to T$ be a bijection by definition 4.
Then by definition:
for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
But:
ev... | {
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Similarly, as $f^{-1}$ is also a mapping:
for all $y \in T$ there exists a unique $x \in S$ such that $\tuple {y, x} \in f^{-1}$.
$\Box$
#### Sufficient Condition
Let $f \subseteq S \times T$ be a relation such that:
$(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\tuple {x, y} \... | {
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# which integers take the form x^2 + xy + y^2 ?
I guess one way of putting it, when does the series $\sum_{x,y \in \mathbb{Z}} q^{x^2+xy+y^2}$ have nonzero coefficients?
The analogous answer for $\sum_{x,y \in \mathbb{Z}} q^{x^2+y^2}$ is that $q^n$ appears when $\mathrm{ord}_p(n)$ be even for all primes.
Is there a ... | {
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Also, in a similar manner we can explicitely write down $l(n)$, the number of representations of $n$ as $x^2+xy+y^2$. See the answer on this math stack exchange post.
Theta Functions: We can evaluate the infinite series in terms of Jacobi theta functions without much difficulty. To deal with the sum of squares, notice... | {
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Comments: 26 pages, no figures, fun to read
On page 149 of Rational Quadratic Forms by Cassels, Lemma 6.3 gives a count for the number of primitive representations of a number $n = x^2 + y^2,$ and this lemma can be applied to find the total number of representations, even if there are none primitive. Something very si... | {
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Since $h(-3)=1$, there is only one reduced, positive definite quadratic form of discriminant $-3$. This is $E(x,y)=x^2+xy+y^2$. Therefore $Q$ and $E$ are properly equivalent (that is there is a determinant-one change of variables taking one into the other), and since $Q$ certainly properly represents $n$, so does $E$.
... | {
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"tag... |
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