Elementary questions about functions, notation, properties, and operations such as function composition.
13
votes
0answers
194 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
7
votes
0answers
155 views
Addition formula for $f_n(x+y)$ in closed form.
$n$ is a positive integer.
$$f_n(x)^n+\left(\frac{df_n(x)}{dx}\right)^n=1$$
$f_n(0)=0$,
$f_n'(0)=1$ then
I am looking for the addition formula for $f_n(x+y)$ in closed form.
if $n=1$ then
...
6
votes
0answers
203 views
Does this calculation have a name, or a generic formulation?
Background
I would appreciate help in identifying / explaining this operation:
To calculate each of the $n$ values of $f(\Phi)$:
sample from the distribution of each of $i$ parameters, $\phi_i$
...
6
votes
0answers
184 views
When does the “Zetor function” converge?
Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. ...
5
votes
0answers
229 views
Extension of the Jacobi triple product identity
The Jacobi triple product identity is:
$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$
I would like to extend the idea for ...
5
votes
0answers
189 views
To determine if$f^{-1}(x)$ is periodic function or not? $f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$
$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$
$P(x)$ is polynomial with degree $n$.
$m$ is an positive integer and $m>1$
What is the algoritm to determine $f^{-1}(x)$ is periodic function ...
5
votes
0answers
223 views
expansion for $1-|t|$
Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions
$$
f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)
$$
I am wondering if one can expand ...
5
votes
0answers
256 views
Compositional “square roots”
Let $A$ be a set and $f\colon A\to A$ a function.
The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$?
This is a ...
5
votes
0answers
181 views
Multiplicative Identity analog for absolute value
Is there a standard name for the function:
$$
f(x) =
\begin{cases}
x & \text{if |x|≤1;}\\
1/x & \text{if |x|>1;}\\
\end{cases}
$$
And is there a potential application of this ...
5
votes
0answers
181 views
Is there some official name for this function?
$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$
Like half-circles connected to each other to look like waves:
Its plot looks smooth, but the function is actually not ...
4
votes
0answers
59 views
How can I Create an integral that can only be evaluated via complex contour integration?
In Richard Feynman's book, Surely You're Joking Mr. Feynman!, he says:
One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do."
So Paul ...
4
votes
0answers
79 views
Bijective map from two injective functions
A bijection $f \colon X \to Y$ is constructed from the injective functions $g\colon X \to Y$ and $h \colon Y \to X$. Suppose $X = Y = \mathbb N$. We let $g\colon X \to Y$ and $h \colon Y \to X$ be ...
4
votes
0answers
77 views
Function that is discontinuous only for integer fractions
I have this question:
Find a function $f :\mathbb R \to\mathbb R$ which is discontinuous at the points of the
set $\{\frac1n : n \text{ a positive integer}\} \cup \{0\}$ but is continuous ...
4
votes
0answers
75 views
Domain with $\cosh(x)$
Take the function
$$y=\frac{\sqrt{\cosh\left(\frac{1+x}{x^2}\right) - 1}}{e^{\frac{2}{x-1}\log\left|x-1\right|}+1}$$
I have to find the domain of this function. These are the condition that I set up:
...
4
votes
0answers
129 views
What is this function called (looks like a variant of the exponential function)
We set (as usual) $\displaystyle{x \choose k} := \frac{x \cdot (x-1) \cdots (x-k+1)}{k!}$ for $x\in \mathbb{C}$.
Now we can define a function
$\displaystyle f(x) := \sum\limits_{k=0}^\infty {x ...
4
votes
0answers
100 views
What is the limit of the quotient of Stieltjes Constants?
What is the limit :
$\displaystyle\lim_{n\to \infty }\frac{\gamma_{n-1} }{\gamma_{n}} $
Here $\gamma_{n} $ is the $n$-th Stieltjes Constant.
4
votes
0answers
511 views
Finding general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$.
I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$.
I calculate
$$
\frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad ...
4
votes
0answers
109 views
Why is this change of variables true?
I have an equation $F_{xx}+yF_{yy}+{1\over 2}F_y=0$ defined on $y<0$.
I found that the characteristics are $\alpha={2\over 3}(-y)^{3\over 2}-x,\,\,\,\,\,\beta={2\over 3}(-y)^{3\over 2}+x$ and that ...
4
votes
0answers
43 views
Name of principal root function modified to return real values if possible
Is there a concise name for the function $f_n(x)\colon\mathbb{C}\to\mathbb{C}$ which returns the principal $n$th root of $x$, except in the case when $n$ is odd and $x$ is a negative real number, in ...
4
votes
0answers
235 views
Organizing types of functions by their calculus-related properties, in diagram form?
Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and ...
3
votes
0answers
41 views
What's the mathematical field called where functions create and delete functions?
Motivation
In the field of modular, reconfigurable robotics there are some groups which use term rewriting, or specifically graph rewriting to describe the reconfiguration process of the modular ...
3
votes
0answers
60 views
How do zeros on the complex plane affect the real number line?
Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
3
votes
0answers
29 views
Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.
To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following:
(somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
3
votes
0answers
48 views
Terminology Question: Precompose vs Compose?
I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
3
votes
0answers
86 views
How to compare the similarity between functions?
I'm designing a web service that finds the regression function of a pattern within an image.
I analyzed three images and found the following three regressions:
1) $f(x) = 74.7602 + 0.2005x - ...
3
votes
0answers
154 views
absolute value integrated (2)
Part 1 of the question
Hey again,
I hope that I was able to solve my problem. This is my solution with the example of Part 1 of the question. Is that correct or did I forget anything?
$$ \int ...
3
votes
0answers
54 views
Monotonicity of a discrete function
Let $k_1+k_2=k$, where $k, k_1$ are all positive integers with $k_1 \ge 1$. Also let $K=\min\{k_1, \lfloor k_2/9 \rfloor +1\}$.
Define $g(x)=\max\{1 \le i \le k: \lfloor i/9 \rfloor +1=x\}, x=1, ...
3
votes
0answers
164 views
Does a continuous function preserve measurability?
If $f$ is a continuous function on $X$ and E a Lebesgue measurable set, can we conclude that $f^{-1}(E)$ is measurable?
3
votes
0answers
64 views
On which of the following spaces is every continuous (real-valued) function bounded
On which of the following spaces is every continuous (real-valued) function bounded?
i) $X_1 = (0, 1)$;
ii) $X_2 = [0,1]$;
iii) $X_3 = [0, 1)$;
iv) $X_4 =\{t \in [0, 1] : t \mbox{ ...
3
votes
0answers
132 views
Prove an image of function $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$ doesn't contain open ball
$\mathbb R^n\supset[a,b]$ is domain of definition $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$
I need to prove that image of $f$, that means $f[a,b]$, doesn't ...
3
votes
0answers
147 views
Topology of bijective functions between Banach spaces
Suppose $X,Y$ are Banach spaces and consider the space $C(X,Y)$ of bounded continuous functions $X \to Y$ with the supremum norm. Are there any results about the the topological properties of the set ...
3
votes
0answers
214 views
System of equations with Lambert W function
I have a system of equations with two equations containing the Lambert W function as follows,
$$\begin{cases} x = 1 - W_0(\frac{C_1 e^{y + 1}}{y + 1}) \\ y = 1 - W_0(\frac{C_2 e^{x + 1}}{x + 1}) ...
3
votes
0answers
73 views
Function Shape Reference
I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague.
I would like to have a ...
3
votes
0answers
153 views
Definition of functions based on “fuzzy” truth table
I'm stuck on this problem:
I have a "truth-table" (well, I don't know if it can be called truth table, if there aren't true/false values only):
...
2
votes
0answers
27 views
Inferring simplest method to convert bit array 1 to bit array 2.
Consider the set of all bit arrays of length $n$. Now consider the set of all 1-to-1 functions that map from this set to this set.
Now select a single function out of the latter set. Is there any ...
2
votes
0answers
13 views
Repertoire method for solving recursions
I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics:
$g(1) = \alpha;$
$g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1
I have assumed the closed form to ...
2
votes
0answers
35 views
Upper bound for linear function
What may be more surprising is that when $a>0$, any linear function $an +b$ is $\mathcal{O}(n^2)$ which is easily verified by taking $c = a + |b|$ and $n_o = \max (\frac{-b}{a}, 1)$.
$$an + b ...
2
votes
0answers
104 views
Log-concave functions whose sums are still log-concave: possible to find a subset?
Rationale:
I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest.
It is well-known that sums of ...
2
votes
0answers
44 views
Ideas for denoting parameters of a function, as opposed to variables, in the list of arguments?
In general, the list of arguments of a function includes only variables, not parameters.
In some specific cases, a parameter could be incorporated into the function name, like $y$ in $$\log_y (x)$$
...
2
votes
0answers
17 views
Semilattice of functions with meet as “common restriction”
Is there an established name for the operator $\bigwedge$ which takes a nonempty family $F$ of functions and returns their "common restriction":
$$
\bigwedge F = f|_{\bigcap_{f_0, f_1 \in ...
2
votes
0answers
53 views
Extreme Value Theorem and Semicontinuity
Restricting us to function of a single real variable, I was used to prove Extreme value theorem via the short way: show that continuous functions preserve compactness, and the job is done.
Now, I ...
2
votes
0answers
86 views
My solution is right and the book is wrong (parabolas) or did I misunderstand it?
Find the equation of the parabola with the vertex at the origin; directrix 2x = 3
So what I did is, find the equation of the directrix
$$x = \frac{3}{2}$$
and then because its the directrix, the ...
2
votes
0answers
64 views
Scaling function and dilation equation in MRA
In the book I am reading, the author explains how we can use the dilation equation to obtain the scaling function through the process of iteration. In particular, we use the dilation equation, which ...
2
votes
0answers
87 views
how to obtain state space diagram and state space model for transfer function
How do we obtain the state space diagram and state space model for transfer function
for example the question is given
How to draw state variable diagram for the given transfer function
...
2
votes
0answers
210 views
Determining the probability density function from an equation
I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help.
Let me introduce my problem. I have two probability ...
2
votes
0answers
189 views
Implication of Lipschitz continuity
I am a little bit unsure about the following claim. Let $H:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$ be a mapping of the form $H(x,f(x))$, where $f:[0,1]\rightarrow \mathbb{R}$ is some continuous ...
2
votes
0answers
187 views
Additive Functions Puzzle
Puzzle:
Given a function 'solve()' that accepts a single integer parameter, and returns an integer, write a program that determines if this function is an additive function [ solve($x+y$) = ...
2
votes
0answers
63 views
Family of Functions with “Common Restriction”
Is there a discussion in literature of "families of functions with common restrictions"? What I mean is: given a family $F$ of functions we define
$$
D_F = \left\{x \in \bigcup\nolimits_{f \in ...
2
votes
0answers
118 views
Getting an upper bound for a function of two variables
I have a complicated looking expression involving two naturals numbers $n,k$ where $n\ge k\ge 3$:
...
2
votes
0answers
104 views
How does one plot $2^{x^{2}}$?
I need to plot $2^{x^{2}}$ without using calculus. I would like to know how to explain why that function is smooth at $x=0.$
