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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148 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...
6
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68 views

Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...
5
votes
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118 views

Eigenvectors of difference of inverse matrices

I have two matrices $A$ and $B$, symmetric and positive semi-definite (in fact, they are covariance matrices), and I am interested in computing the eigenvectors of the matrix $A^{-1}-B^{-1}$. From ...
5
votes
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88 views

Explicit quasi-inverse of Künneth-isomorphism?

With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that \begin{align*} A_X \otimes A_Y &\to A_{X \times Y}, \\ (\omega,\eta) &\mapsto {\rm ...
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40 views

Proving whether the following are groups or not.

In each case, I am asked to decide whether the indicated pair is a group or not. If so, prove it; if not, show which group axiom fails. (a) $(\dfrac{1}{2}\mathbb{Z}, +)$ where $\dfrac{1}{2} ...
4
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50 views

Is there always a smooth variant of a homoeomorphism between smooth manifolds?

Let $M$ and $N$ be smooth homeomorphic manifolds. Let $h:M\rightarrow N$ a homeomorphism. Does there exist $r:M\rightarrow N$ that is still a homeomorphism and additionaly smooth? Can it be chosen ...
4
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82 views

Is the inverse of any elementary function asymptotic to some elementary function?

Is the functional inverse of any elementary function asymptotic to some elementary function ? For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See ...
3
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26 views

On generalised inverse

Let $A$ be a positive matrix, may not be invertible. I define its generalised inverse as \begin{equation} A^- = \lim_{n\rightarrow \infty} \left( \frac{1}{n} I + A\right)^{-1}. \end{equation} Lets ...
3
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55 views

The inverse of x!

what is the inverse of a factorial function? Its is not continuous but is modeled by the gamma function which is continuous so must have a inverse any research leads to the inverse gamma function that ...
3
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66 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
3
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78 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
3
votes
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230 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
3
votes
0answers
83 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
3
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396 views

Inverse of identity plus scalar multiple of matrix

Given the matrix $M = ( I + \alpha D P )$, where $I$ is the nxn identity, $D$ is nxn symmetric and invertible, $P$ is nxn symmetric but not always invertible, and $\alpha$ is a scalar, is there a ...
3
votes
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47 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
3
votes
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114 views

Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
3
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2k views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
3
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49 views

Finding the number of the real roots of $a^x=g(x)$ where $g(x)$ is the inverse function of $f(x)=a^x$

Question : Let $a$ be a constant which satisfies $0\lt a\lt 1$. Letting $g(x)$ be the inverse function of $f(x)=a^x$, then find the number $N$ of the real roots of $f(x)=g(x)$. Motivation : This is ...
3
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139 views

Sparse matrix inverse multiplied by sparse matrices

I have the equation $\bf E = Y D^{-1} Y^\top$. $\bf D$ is a potentially large sparse $m \times m$ matrix, and $\bf Y$ is a sparse $n \times m$ matrix, where $n \ll m$. Is there a particularly ...
2
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19 views

Computing one-sided inverse of a matrix over some finite field

Let $M$ be a $k\times n$ matrix with $k < n$, and assume that $\text{rank}(M)=k$. Over $\mathbb{R}$, one can compute a right inverse of $M$ as follows: $$M_\text{right}^{-1} = M^T(MM^T)^{-1}$$ ...
2
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29 views

Matrix inverse series expansion

I want to prove that when $I+K$ is invertible, $$(I+K)^{-1}=I-K+o(K)$$ to establish that the matrix inverse function has derivative $-I$ at $I$. My hope is that this identity carries over from ...
2
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74 views

Inverse of generalized arrow matrix $A = M^T * M + I$

If we have the following linear system: Ax=b And matrix A is created by multiplying a rectangular matrix with it's transpose: $A = M^T * M + I$ What is the best method to solve for x for different b ...
2
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46 views

Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
2
votes
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30 views

Continuity at an inner point of an interval implies the continuity of the inverse

If a function $f$ with domain interval is $1-1$ and continuous at an inner point a of the interval its inverse is continuous at $f(a)$?
2
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20 views

Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
2
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26 views

Differential Equation for Inverse of a Function given the ODE for the function

Assume we have and ODE for a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ in the form of \begin{equation} \mathbf{J}f \times F(x)=D(\lambda_i)\times f(x) \end{equation} where $\mathbf{J}$ denotes ...
2
votes
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50 views

another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
2
votes
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28 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
2
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120 views

Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
2
votes
0answers
56 views

Inversion of a pairing function

I was reading this question on this site and I saw that the following pairing function was mentioned (a modified version of Cantor function): $$\langle x, y\rangle = x * y + ...
2
votes
0answers
99 views

Closed form for elements of inverse matrix of lower triangular matrix of any size

If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{2,1}&a_{2,2}&0&\cdots&0\\a_{3,1} &1_{3,2}&a_{3,3}&\cdots&0\\ ...
2
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295 views

Fastest way to find modular multiplicative inverse

I am looking for a fast way to find the modular multiplicate inverse of an integer $a$ in mod $p$. I am mainly interested in ...
2
votes
0answers
161 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} ...
2
votes
0answers
43 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
2
votes
0answers
624 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
2
votes
0answers
282 views

Self-inverse matrices with integers with pairwise different absolut values.

Let A be a self-inverse matrix ($A=A^{-1}$) with integer values such that no two integers have the same absolut value. Let M be the maximum of the absolut values (maximum-norm) of A. Which M is the ...
2
votes
0answers
191 views

matrix inverse and limit

I would like to get a better understanding of limits and matrix inverses, specifically the relationship between: $\lim_{k\rightarrow \infty}(\mathbf{A}^{-1})$ and $(\lim_{k\rightarrow ...
2
votes
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128 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix ...
2
votes
0answers
666 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
2
votes
0answers
127 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
2
votes
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50 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
2
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27 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
2
votes
0answers
65 views

Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$ f(g(m)) = m $$ I want to understand what all of the ...
2
votes
0answers
138 views

Continuous function that is invertible in one argument---is its inverse continuous in both arguments?

Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with ...
2
votes
0answers
156 views

Question about the Miller Theorem on inverse of sum of two matrices.

The following is a well known theorem on the inverse of $(A+B)$. (Link to the paper: http://www.jstor.org/stable/2690437) Theorem. Let $A$ and $A+B$ be nonsingular matrices, and let $B$ have rank ...
2
votes
0answers
54 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
2
votes
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166 views

Modular Inverse over some given finite field. Which method is more efficient?

I'm trying to do division in some given finite field (let's say mod p). I have 2 Python methods here that are currently doing that, but I'm not sure which is better or if 1 or both is simply wrong. ...
2
votes
0answers
69 views

About the functional inverse of integrals and infinite products.

It seems $\cos(x)$ and $\sin(x)$ are the only entire functions, that are the functional inverse of an integral of some elementary function $f(x)$ , such that they have a simple infinite product ...
2
votes
0answers
571 views

Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
2
votes
0answers
161 views

Optimization problem about large matrices

I'd like to solve the following optimization problem: Find non-negative scalar $a$, $b$, $c$ to minimize $\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$ where ...