Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...
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10 views
Inverse Variation Function as Real Life Example
What is an example of the Inverse Variation in real life? Given the function y = a/x'. I've tried to Google it, and look in all our math books, but I can find no examples.
1
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2answers
71 views
inverse of laplace transform
How to compute this inverse Laplace transform ?
$$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$
Thanks.
7
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1answer
218 views
$\operatorname{arsinh}$ vs $\operatorname{arcsinh}$
I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for ...
4
votes
2answers
74 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
1
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0answers
60 views
General solution for $M^{\circ -1 }(y)=x $ when $g(x)e^{f(x) }=y$
Reading this question $e^{C/x }-1=D/(x + a) $, i found my self completely unable to do anything. This is much more hard for me than my easy exercises about Lambert $W$-function.
So I probably need ...
1
vote
2answers
47 views
What is the inverse z transform of 1/(z-1)^2?
I'd like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$
4
votes
3answers
133 views
How to find inverse of the function $f(x)=\sin(x)\ln(x)$
My friend asked me to solve it, but I can't.
If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$?
I have no idea how to find the solution. I try to find
...
0
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0answers
25 views
question about invers of $csc(x)$
I know that invers of $f(x)=csc(x)$
is
$f(x)=csc^{-1}(x)$
but why wolfram alpha give me $f^{-1}(x)=\pm 2\pi c -0.5(4\pi c +\pi)+csc^{-1}(x)$
for $c \in Z$
please help
thanks for all
3
votes
2answers
29 views
Linear Algebra determinant and rank relation
True or False?
If the determinant of a $4 \times 4$ matrix $A$ is $4$
then its rank must be $4$.
Is it false or true?
My guess is true, because the matrix $A$ is invertible.
But there is ...
1
vote
1answer
169 views
Inverse of upper triangular matrix
I have an upper triangular matrix that I want to solve the inverse for.
I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
2
votes
1answer
23 views
Simplify difference of two arc tangents?
I have a problem, that I am trying to simplify, but there does not seem to be something obvious regarding it.
Very simply, I am trying to figure out if there is a way to 'open' the following:
$$
...
10
votes
5answers
358 views
What's the difference between arccos(x) and sec(x)
My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
0
votes
2answers
52 views
Rings | Homomorphisms | Units
Question
Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is
a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$.
Attempt
...
0
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1answer
44 views
Is there an explicit formula for the inverse of $\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}$?
I apologize if this is trivial but I am stuck.
Given the bijective function $f:(0,2\pi) \to (-2,2)$ with
$$
f(x)=\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}
$$
where $\cot$ is the cotangent, how can ...
4
votes
3answers
82 views
Inverses where $f(g(x))=x\neq g(f(x))$
If $f: \mathbb{Z\to Z}$ and $G: \mathbb{Z\to Z}$, find $f$, and $g$ such that $f(g(x))=x\neq g(f(x))$.
I can find lots of $f$ and $g$ that aren't equal when composed with each other, but I have no ...
2
votes
1answer
35 views
Find the inverse for arbitrary k
I need to find a, b, c, d, e, f, g, h (all of which are not zero)
such that for all k is in Real number, show A is invertible or this can't happen
$$A = \left(\begin{array}{ccc}
...
5
votes
2answers
57 views
Inverse and derivative of a function [duplicate]
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
1
vote
0answers
19 views
Inverse fourier transform involving exponentials
I would like to calculate the inverse fourier transform of:
$$\hat{f}(k)=\exp(-a k^2+ikv)\cdot \frac{\sinh(m\sqrt{(b+ck^2+ikf)^2-d})}{\sqrt{(b+ck^2+ikf)^2-d}}$$
Any clue?
Thanks!
0
votes
0answers
24 views
Convolution of Inverse Gaussian distribution
Im having problems with showing that the sum of two inverse gaussian distributed random variables are stable under convolution, i.e.
let $f_{T_a}(t) = \frac{a}{\sqrt{2 \pi t^3}} e^{-\frac{(a- \nu ...
5
votes
2answers
91 views
How to formally show that $f(z)$ is analytic at $z=0$?
Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$?
I know that for small $z$ we have ...
4
votes
0answers
51 views
Show that there exist a real number $a≠0$, such that the fiber $f^{-1}(a)$ is a finite set
Let $f:ℝ→ℝ$ be a real analytic function. The function $f$ is given by:
$$f(s)=N^{s/2}(2π)^{-s}Γ(s)∑_{n=1}^{∞}a_{n}/n^{s}$$
where $a_{n}$ are the coefficients of a Dirichlet series, $N$ is a natural ...
1
vote
2answers
32 views
Finding the Inverse Laplace Transform
I am having trouble finding the inverse Laplace transform of: $$\frac{1}{s^2-9s+20}$$
I tried writing it in a different way:
...
0
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0answers
22 views
Inversely Proportional
I have been racking my brains on how to do the following question. Do I have to use the formula Pressure = constant x 1/volume ? If so, how to do it?
...
2
votes
3answers
144 views
Inverse function requirements
Let f be an injective function, that is:
$f : X \rightarrow Y$
$f(a) = f(b) \implies a = b$
Now, my question is, does the following need to hold in order for function to be injective:
$(\forall x ...
1
vote
1answer
63 views
Inverse of matrix sum, special case: $(A + x I)$
Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar?
I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on ...
1
vote
1answer
35 views
Linear Algebra — Block Matrix Inversion
Please excuse my formatting...
$X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that
...
2
votes
3answers
402 views
How do we find the inverse of a function with $2$ variables?
$$f(m,n) = (2m+n, m+2n)$$
What do we have to do to find the inverse of this function?
I don't even know where to begin.
0
votes
1answer
308 views
Derivative of a complicated inverse function
$\Phi(\cdot,0,1)$ and $\phi(\cdot,0,1)$ are cdf and pdf of standard normal distribution.
$$y=F_\text{mix}(x,\mu,\sigma)=\sum\limits_{i=1}^{K}\lambda_i\Phi\left(\frac{x-\mu_i}{\sigma_i},0,1\right).$$
...
2
votes
1answer
78 views
iteration method for inverse matrix
a)
Consider the iterative scheme
\begin{equation}
x_{n+1} = x_n + c (Ax_n - I) \tag{1}
\end{equation}
When the process converges, show that this scheme (where $c$ is an appropriately chosen real
...
2
votes
1answer
118 views
Power series of matrix which is multiplied by a constant factor $c<1$?
(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole ...
0
votes
0answers
26 views
What is the error in Newton's Method for Matrix Inversion?
I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds.
Suppose we have, as ...
3
votes
3answers
173 views
Evaluate the derivative of an inverse function by using a table of values?
Given the function and derivative values in the table below, evaluate $\frac{d}{dx}f^{-1}(3)$
...
1
vote
1answer
56 views
Linear Algebra: Least-Squares Approximation & “Normal Equation”
I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
-4
votes
1answer
135 views
Please check this inverse this Laplace transform [closed]
I just want to check if my exercise are right:
Inverse of these Laplace transform
$$F^{-1}\left(\frac{1}{p-2}\right)= e^{2s}$$
$$F^{-1}\left(\frac{e^{-2p}}{p^2}\right)=\frac{2}{s^3(s+2)}$$
...
3
votes
4answers
71 views
Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B
I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
0
votes
2answers
67 views
Need help with inverse laplace transform problem
I'm really stuck this problem.
This actually resulted because of equations for a circuit analysis problem, so in case it would help I'll list the equations here too. Although, feel free to ignore ...
1
vote
0answers
57 views
Easy but hard question about Matrix power series! [duplicate]
Assume $W$ is $n\times n$ matrix and $r<1$ is a real number. Let $$Q = \sum_{i=0}^{\infty} (rW)^i=[I_n-rW]^{-1}$$
Now assume that the matrix $W$ is multiplied by a constant real number $c<1$. ...
1
vote
2answers
84 views
What does this syntax mean: “$f^{-1} : N_{10} \Rightarrow N_b $ is the inverse of $f: N N_{b} \Rightarrow N_{10}$?”
I'm trying to solve this but I haven't seen syntax like this before. Can someone please explain the syntax?
http://i.imgur.com/GO1Ki.png
The image is
Show that the one-to-one function $f^{-1} : ...
0
votes
1answer
38 views
Algorithm for root function $[2^{n-1}]$
I am attempting to convert this function $[2^{n-1}]$ into a root function to return original value. Thus far all my attempts have ended in abject failure.
Base : 1 2 3 4 5 6 7 8 9
Result : ...
9
votes
2answers
180 views
On the convexity of element-wise norm 1 of the inverse
Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as
$$
\|A\|_1= \sum_{i,j} |A_{i,j}|.
$$
Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
7
votes
6answers
601 views
If $A^2$ is invertible, then $A$ is also invertible?
True or False: If $A^2$ is invertible, then $A$ is also invertible.
($A$ is a matrix here.)
The answer is true. I was trying to come up with an example that makes this false.
But I couldn't. ...
0
votes
2answers
46 views
Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$
Given the matrix equation:
$$ x^TA^TA = b^TA $$
I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric.
When I solve it ...
0
votes
1answer
66 views
0
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0answers
22 views
Does there exist a basis function where its inverse function is in the same basis?
I'm interested in finding a basis function
$\phi(x)$, which I can use to approximate some function
$y(x) \approx \hat{y}(x) = \sum\limits_i c_i \phi_i((x - d_i)/s_i)$,
where its inverse function, ...
1
vote
1answer
46 views
Let $F(x,y,z) = -c(r/||r||^3)$ be the force resulting from the inverse square law…
$c$ is a constant and $r = (x,y,z)$. Show that $\displaystyle f(x,y,z) = \frac{c}{\sqrt{x^2+y^2+z^2}}$ is a potential function for $F$. What can be concluded from any path from point $A$ to point $B$ ...
1
vote
1answer
65 views
Question related to diagonally dominant matrix
A matrix is said to be positive if each entry in the matrix is positive.
If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
0
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1answer
39 views
Finding upper triangular matrix
I have this question, and im not sure I know how to solve it.
"Find an upper triangular $U$ (not diagonal) with $U^2 = I$ which gives $U=U^{-1}$".
Anybody who can help me getting the first steps of ...
0
votes
1answer
35 views
Did I solve all of the steps of this Trig question properly?
Thanks to some help from the community, I think I did this problem correctly, but I would like someone to confirm that I indeed do it right. Thanks.
Question:
Let $0 \le x \le 1$.
(i.) Find the ...
0
votes
1answer
17 views
How can I solve an equation based off of a quadrant and equation form given an angle?
Given that 3pi/2 < z < 2pi
x = arccos(sin(z))
Given different values for z (which are angles on the unit circle) how would I write the results in these two forms, where C is a constant?:
a.) ...
1
vote
2answers
37 views
How do I write a trig function that includes inverses in terms of another variable?
It's been awhile since I've used trig and I feel stupid asking this question lol but here goes:
Given:
$z = \tan(\arcsin(x))$
Question:
How do I write something like that in terms of $x$?
Thanks! ...
