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 ...
9
votes
1answer
468 views
Adjoint functors as “conceptual inverses”
The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other.
For example, the forgetful functor "ought ...
5
votes
3answers
274 views
Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$
I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given:
$$y=W(e^{ax+b})-W(e^{cx+d})+zx$$
where $W$ is the Lambert $W$ function and ...
4
votes
2answers
278 views
Inverse of symmetric matrix $M = A A^\top$
I have a matrix, generated by the product of a non-square matrix with its own transpose:
$$M = A A^\top.$$
I need the inverse of $M$, assuming $\det(M) \neq 0$.
Given the nature of the matrix $M$, ...
3
votes
1answer
320 views
Is the trace of inverse matrix convex?
Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex.
Actually I know that the trace of a symmetric positive definite ...
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 ...
5
votes
6answers
271 views
What is the inverse function of $\ x^2+x$?
I think the title says it all; I'm looking for the inverse function of $\ x^2+x$, and I have no idea how to do it. I thought maybe you could use the quadratic equation or something. I would be ...
5
votes
1answer
203 views
Inverse function of $\operatorname{li}(x)$ over $x>\mu$?
How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$?
Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
2
votes
2answers
557 views
Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations
Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations
-- i.e. without any reference to higher-order matters like rank, vector spaces or whatever ( ...
2
votes
1answer
602 views
Homework Help - AP Calculus - Inverse of Polynomial
I know it is a simple problem but I am having trouble. Here is what I have so far:
Let $f(x) = x^5 + 2x^3 + x - 1$
a) Find $f(1)$ and $f'(1)$
I have a) done. $f(1)$ is $3$ and $f'(1)$ is ...
2
votes
2answers
234 views
Why is $x^{1/n}$ continuous?
Why is $x^{1/n}$ continuous for positive $x,n$ where $n$ is an integer? I can't see how it follows from the definition of limit. And I don't see any suitable inequalities so is this an application of ...
1
vote
3answers
180 views
Is there a general way to solve transcendental equations?
To make things definite, let's narrow them and call transcendental equation of the form
$$f(x) = 0$$
where $f$ is a real elementary function in the usual sense. For example
$$\cos(\pi x) + x^2 = ...
0
votes
1answer
128 views
What's the limit of coefficient ratio for a reciprocating power series?
I have a question about the coefficient in the inverse of the power series.
Assume
$$
f=1-\sum_{i=1}^{\infty}(ck_i)x^i,
$$
where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
4
votes
3answers
504 views
How to invert this exponential function to solve for x: $y = a \exp(bx) + c \exp(dx)$?
Cheers.
So if I don't make sense, I have a value for $y$, I need to know what $x$ is.
$$y = a \exp(bx) + c \exp(dx)$$
$a = 12.85$,
$b = 0.001857$,
$c = -54.24$,
$d = -0.05316$
1
vote
1answer
65 views
Question related to diagonally dominant matrix
A matrix is said to be positive if each entry in the matrix is positive.
If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
1
vote
2answers
272 views
RYB and RGB color space conversion
I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue).
I managed to solve converting a color from RYB to RGB space based on ...
1
vote
3answers
233 views
Finding The Equivalence Class
Okay, so the question I am working on is, "Suppose that A is a nonempty set, and $f$ is a function that has A as its domain. Let R be the relation on A consisting of all ordered pairs $(x, y)$ such ...
0
votes
2answers
43 views
Reverse rows in a matrix
To rotate a matrix 180 degrees around the center point, what I am planning to do is first transverse the matrix, then reverse the rows and then do it again to produce the final result.
This works and ...
0
votes
1answer
308 views
Derivative of a complicated inverse function
$\Phi(\cdot,0,1)$ and $\phi(\cdot,0,1)$ are cdf and pdf of standard normal distribution.
$$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).$$
...
0
votes
2answers
136 views
Domain of convergence of $f^{-1}: \mathbb R ^N \mapsto \mathbb R^N$ taylor series
In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse ...
0
votes
1answer
321 views
How to find the inverse of F(x), where F is a cumulative distribution function
I have a cumulative distribution function $F(x)= \mathbb P(X < x)$.
Now, for some further purposes, I need to find its inverse. The case is also that I am dealing with a KS test, where I have the ...
-4
votes
1answer
136 views
Please check this inverse this Laplace transform [closed]
I just want to check if my exercise are right:
Inverse of these Laplace transform
$$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)}$$
...
