The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...
4
votes
0answers
98 views
Solution(s) to $f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y))$
I would like to know the solutions of the functional equation:
$$f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y)), \forall x,y\in\mathbb{R}$$
where $f:\mathbb{R}\rightarrow\mathbb{R}$. I have already determined ...
11
votes
1answer
275 views
Find all the function that satisfy : $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$
Find all the function that satisfy :
$$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$
I only find $f(0)=0$ but I can't prove $f(x)=2x$
1
vote
1answer
152 views
Analytic solution of a certain functional equation
Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation
$$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$
...
4
votes
0answers
28 views
What are the densities of branches of the euclidean tree?
The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$.
(or ...
2
votes
3answers
85 views
Defining a function with certain properties
I'm a bit rusty in mathematics so I need your help please :)
I need a function $y$ that satisfies:
$$\begin{align*}
y &= ax\\
y &= \left\{\begin{array}{ll}
x &\text{if }x\geq 0;\\
0 ...
1
vote
1answer
489 views
Solve the functional equation $f (x+y)=f (x)+f (y)+xy (x+y)$, $f$ continuous at $0$
Find all functions $f:\mathbb{R} \to \mathbb{R}$, that are continuous at the point $x=0$ and satisfy:
$$f (x+y)=f (x)+f (y)+xy (x+y) \ \ \forall x,y \in \mathbb{R} $$
0
votes
2answers
105 views
How to create equations to measure time spending in executing algorithms?
I made a program with two functions to calculate factorial. The first uses loops to made de calculations, and the second uses recursive calls to get the same result.
The same program measures the ...
0
votes
0answers
39 views
A generalization of exponential functional equation
Let $G$ be a group(not necessarily abelian) and $f:G\times G \to \Bbb C$.
Find all solutions of the functional equation
$$
f(xu, yv)+f(xv, yu)=f(x, y)f(u,v)
$$
for all $x, y, u, v \in G$.
0
votes
2answers
53 views
What's the solution of the functional equation
I need help with this:
"Find all functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, with $g$ injective and such that:
$$f(g(x)+y) = g(f(x)+y), \mbox{ for all } x, y \in \mathbb{Z}.$$
1
vote
1answer
37 views
Funcional Equations:I'm confused [duplicate]
I need help with this:
"Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that:
$$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
Or :
...
1
vote
1answer
64 views
Are all functions on vectors in GF(2^n) representable as polynomial functions?
In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.)
Question: Is this true for other $GF$, especially $GF(2^8)$? ...
5
votes
1answer
100 views
Maximum cubic function
Let $f(x)=x^{3}+ax^{2}+bx+c$ with a, b, c real.
Show that
$$\frac{1}4 \le \max_{-1 \le x \le 1\hspace{2mm}} |f(x)|=M$$
and find all cases where equality occurs.
0
votes
1answer
49 views
simplifying “$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}$”
Is this equality correct?
For finite sets $A$ and $B_a$ (where $a\in A$), we have:
$$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}=\sum_{f\in \prod_{a\in A}B_a}\quad \prod_{a\in A}{h(a,f(a))}$$
6
votes
1answer
117 views
Question about a functional equation
We are looking at a theorem which characterizes the affine term structure (ats) models in interes rate theory. What follows is from "Filipović, D. (2009): "Term-structure models: A graduate course", ...
2
votes
1answer
86 views
How bad can perturbing a functional equation really make things?
A long time ago, I found occassion to find solutions to a functional equation of the following form
$f(x-y) = f(x) - f(y) + \delta $ with $\delta \in \mathbb{R}.$
Using the same exact techniques as ...
1
vote
1answer
47 views
What's the solution of the functional equation?
I need help with this:
"Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that:
$$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
8
votes
3answers
168 views
Find all the functions which satisfy a given functional equation
I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$
We ...
2
votes
2answers
143 views
Attractive fixed-point?
I'm sorry in advance for how specific this problem is. I've been trying to make it as generic as possible but this is as far as I could get. I want to prove that a fixed point is attractive.
I didn't ...
12
votes
4answers
584 views
Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$
How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and
$$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$
for all real numbers $x$ and $y$ with $y\neq0$?
PS. This is ...
9
votes
2answers
66 views
Uniqueness of solution for a functional equation
Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that
$$
f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a].
$$
...
1
vote
1answer
32 views
Plotting for solution for $y=x^2$ and $x^2 + y^2 = a $
Consider the system $$y=x^2$$ and $$x^2 + y^2 = a $$for $x>0$, $y>0$, $a>0$.
Solving for equations give me $y+y^2 = a$, and ultimately $$y = \frac {-1 + \sqrt {4a+1}} {2} $$ (rejected ...
2
votes
1answer
53 views
Connecting functional identity of a function with its image set
Okay, now I really need real help, maybe the task is not too heavy but I do not know the easy way to solve this. Let´s start with the problem, now.
Suppose that we have some function $f: \mathbb N ...
2
votes
2answers
129 views
How to find $f(x)+f(x+1) = e^x$?
Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function.
How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ?
In particular I wonder most about the case $a=1$ ...
8
votes
1answer
138 views
A functional equation problem
Let $f$ be a function which maps $\mathbb{Q}^{+}\to\mathbb{Q}^{+}$. And it satisfies
$$
\left\{ \begin{array}{l}
f(x)+f\left(\frac{1}{x}\right)=1\\
f(2x)=2f(f(x))
\end{array}\right.
$$
Show that ...
3
votes
2answers
80 views
Average of function, function of average
I'm trying to find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all $n > 1$ and all $x_1, x_2, \cdots, x_n \in \mathbb{R}$:
$$\frac{1}{n} \sum_{t = 1}^n f(x_t) = f \left ( ...
7
votes
1answer
117 views
Maps of $\mathbb{R}^3$ preserving the cross product
Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$:
$$\phi(a \times b)=\phi(a) \times \phi(b)$$
Is $\phi$ necessarily a rotation around the origin or the map ...
0
votes
1answer
205 views
functional equation uniqueness $f(x^{2})= f(x)^{2} $ in $\mathbb{C}$ and $f(x)+f''(x) = 0 $ in $\mathbb{C}$
This is an exercise from a book I tried:
One would like to find all holomorphic equations that satisfy:$$i) \ f(z)+f''(z) = 0 \text{ in } \mathbb{C} $$$$ii)\ f(z^{2})=f(z)^{2} \text{ in } ...
7
votes
1answer
235 views
$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$
Suppose $f:\mathbb R\to\mathbb R$ is a strictly decreasing function which satisfy the relation
$$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) ) , \quad \forall x , y \in\mathbb R $$
...
6
votes
2answers
162 views
Proving a function is constant when $f(x)f(y) + f(\frac{a}{x})f(\frac{a}{y}) = 2f(xy)$
I've been working on the following homework problem:
Consider a function $f : (0,∞) → \mathbb{R}$ and a real number $a > 0$ such
that $f(a) = 1$. Prove that if $f(x)f(y) + ...
3
votes
2answers
227 views
Finding all continuous functions such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$
I've been working on the following homework problem:
Find all continuous functions $f : \mathbb{R} → [0,∞)$ such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$ for all real numbers $x, y$.
The first ...
1
vote
1answer
41 views
If $f(cx, cy) = f(x, y)$ can we always find $g$ such that $g(\frac{x}{y}) = f(x, y)$?
Motivation: A differential equation $\frac{dy}{dx} = f(x, y)$ is said to be homogeneous if we can find $g$ such that $g(\frac{x}{y}) = f(x, y)$, which allows us to solve the equation using the ...
0
votes
0answers
23 views
Proving Join Dependencies in MVD
I have a question regarding natural join operations in multivalued dependencies. I know that a join operator joins two tables on similair attributes, however I have a hard time to figure out how to ...
18
votes
6answers
690 views
Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
Find all polynomials $P$ such that
$P(x^2+1)=P(x)^2+1$
0
votes
2answers
70 views
General aggregation functions
Is there a way to find all or some functions which "aggregate" numbers and are non-isomorphic to addition. I mean functions which are commutative and associative:
$f(x,y)=f(y,x)$
...
2
votes
2answers
82 views
Find the equation of this bijection from $ \mathbb{R} $ to $ (0,1) $.
I need some help (hints or an answer) in finding the actual equation of this bijections from the reals $ \mathbb{R} $ to $ (0,1) $. We may assume that the radius of the circle is $ \dfrac{1}{2} $.
...
0
votes
0answers
6 views
Need help in equation formation for meeting quality
I want to create an equation that consists of two independent variables "duration" and "data_transferred", and one dependent variable "meeting_quality"
This is what I want:
meeting_quality would be ...
1
vote
1answer
26 views
Show that functions are affinely dependent
Let $u(x),v(x)$ be continuous bounded functions on $\mathbb{R}$ such that for any Borel probability measures $\mathbb{P}_{1},\mathbb{P}_2$ on $\mathbb{R}$
$$
\int u(x) \, \mathbb{P}_1(dx) \leqslant ...
10
votes
2answers
275 views
Solution of functional equation $f(x/f(x)) = 1/f(x)$?
I've been trying to add math rigor to a solution of the functional equation in [1], eq. (22). It is:
$$
f\left(\frac{x}{f(x)}\right) = \frac{1}{f(x)}\,,
$$
where you know that $f(0)=1$ and $f(-x) = ...
2
votes
2answers
25 views
Ratio's and Max Size
If I have a video of size:
width: 640
height: 480
and a screen of size:
width: 1280
height: 720
what is the equation ...
0
votes
0answers
28 views
Scrambling an Array and then Recovering It
I have an array of elements which is finitely long -- typically < 2000. Each element can be between 0 - 0xFF
An example:
[0xa 0x1 0x7 .... 0x1f]
Are there any functions available to scramble ...
2
votes
0answers
21 views
Multiple integrals satisfying functional equations
If $f:\mathbb{R}_+\to\mathbb{R}_+$ is a positive function which is locally $L^1$ with $\int_0^t{f(s)ds}=at^n$ for all $t\geq0$ then, by differentiating, it follows that $f(t)=a$ if $n=1$ and $f=0=a$ ...
11
votes
3answers
187 views
What was this theorem called
Back at the university we have proven (lot of work) that if $$S(X)C(Y)+C(X)S(Y) = S(X+Y)$$ and $$C(X)C(Y)-S(X)S(Y) = C(X+Y)$$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). ...
2
votes
1answer
158 views
Functional equation book for olympiad
what may be the good suggestions in olympiad functional equations for a beginner for . I have heard of this book by B.J.Venkatachala but do not whether it will be suitable for me or not. Anybody ...
-2
votes
2answers
149 views
Find all functions : $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that:
Find all functions : $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that:
$$f(1+xf(y))=yf(x+y)$$
for all $x,y \in \mathbb{R^+}$
2
votes
1answer
62 views
Functional inequality
Let $S$ be a semigroup such that $S\ne S+S$ and
let $f:S\to \Bbb C$ be an unbounded function satisfying
$$
|f(s_1)f(s_2)-f(t_1)f(t_2)|\le 1
$$
for all $s_1, s_2, t_1, t_2 \in S$ such that ...
5
votes
1answer
439 views
Solution for exponential function's functional equation by using a definition of derivative
let $f(0)=1$ and $f'(0)=1$.
and $f(x+y)=f(x)f(y)$ for $x,y\in R$.
How can I found $f(x)$ by using a definition of derivative?
4
votes
1answer
92 views
Concerning nonlinear functional equations
There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of
$$
f(x)f(x) =x+1+f(x+1)
$$
I've tried several ...
1
vote
1answer
42 views
Functional inequality
Let $f, g:\Bbb R \to \Bbb R$ be bounded functions satisfying
$$
|f(x+y)-f(x)g(y)|\le \frac 1 4
$$
for all $x, y\in \Bbb R$.
Prove or disprove
$$
|f(x)||1-g(y)|\le \frac 1 4
$$
for all $x, y\in \Bbb ...
3
votes
2answers
162 views
Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that:
Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that:
$$f(x)\cdot f(yf(x))=f(y+f(x))$$ $\forall x,y \in \mathbb{R}^+$
1
vote
0answers
141 views
Find number of solutions to equation with dependent variables
Please help to find number of solutions of this equation
$y_{1}\vee y_{2}\vee\ldots\vee y_{k} = \varphi (x_{1},x_{2},\ldots,x_{n})$
where $y_{i}=y_{i}(x_{1},x_{2},\ldots,x_{n})$ is Boolean function ...
