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 ...

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Convexity conditions for $f$ and $\dfrac {1} {f}$

Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function. Find all conditions on $f$ under which $f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac ...
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91 views

Indeterminacy in second degree equations of Spencer-Brown's “Laws of Form”

I am trying to follow G. Spencer-Brown's argumentation regarding 'Equations of the Second Degree' in the Laws of Form. He shows how infinitely growing self-referential forms can be created by using a ...
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62 views

Matrix functional equation

Could someone give me some hint about a possible method to find the function $f$ which solve this equation: $$f(H^2)=\alpha f(H)$$ where $\alpha$ a constant with $\alpha \in C$ and: ...
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435 views

Finding inverse of function without knowing function?

This question has a programming application, but I thought it would be more appropriate (and educational) to ask here first and get some basic understanding. So my use-case will refer to some ...
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125 views

Could anyone derive a formula for this?

Edited: I want to get a sentiment score of various sentences and I've tried coming up with an equation that could satisfy the conditions that are inherent to each sentence (It's estimated mood as ...
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201 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
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12 views

Criteria when bigger number of functions can be obtained from smaller number

It is known that $$ A_1(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_1, $$ $$ A_2(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_2 $$ holds if and only if $$ \partial A_1/\partial ...
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16 views

Functional equation + differential equation = way of finding solution?

Question I was wondering about the following: Let's say there is a differential equation whose solution is $f$ And $f$ also satisfies a functional equation. Can anyone construct an (non-trivial) ...
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33 views

Example of a well defined functional integral?

So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define $$ \int_{f \in S} L(f) $$ As the sum of of every function $f$ evaluated by ...
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18 views

Derivative of sum of two functional derivatives with different ranges

I have a functional of the the following form, $(o<a<1)$ : $F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $ I want to find $ \frac{\partial ...
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33 views

Solve the system of functional equations.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \begin{cases} f(x(1+f(x)))=f(x)^2,\\ f(x(1-f(x)))=f(x) f(-x),\\ f(x(-1+f(-x)))=f(x)f(-x). \end{cases} I have found only that for ...
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46 views

Cauchy's function

An exercise in Real and Complex analysis (exercise 18, page 195,chapter 9) Show that if a function $f$ on the real line $\mathbb{R}$ satisfies \begin{equation*} f(x+y)=f(x)+f(y) ...
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31 views

Solving a particular Functional Differential Equation

Suppose we have the following functional differential equation: $$f(a_0+a_1x+a_2f(x))(b_1+b_2f'(x))=c_0+c_1x+c_2f(x)$$ It is easy to see that a linear function: $f(x)=d_0+d_1x$, with appropriate ...
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58 views

Iterative (functional) roots of integer functions (functions on $\mathbb{Z}$)

A function $g:A\to A$ is called a $k$-th iterative root of another function $f:A\to A$ ($A$ an arbitrary set and $k\in\mathbb{N}$) iff $f=g^k$, where $g^k(x)=g\circ g\circ\ldots\circ g(x)=g(g(\ldots ...
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66 views

Proving existence and uniqueness of solutions to the functional equation $f(n) = r \cdot f(n-1)$

Suppose I have a functional equation $f(n) = r \cdot f(n-1)$ where $r$ is a constant. This represents a geometric progression and a known solution is $g(n) = ar^n$ where $a = g(0)$. By intuition, ...
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26 views

Calculus of Variations for discrete functionals

Question: How does one determine the optimal function f that either maximize or minimizes: $$\int_{x_1}^{x_2} L\left(x, f, D_{h,x}[f ]\right) dx$$ Whereas: $$ D_{h,x}[f] = \frac{f(x + h) - ...
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38 views

Generalized Riesz theorem of operator value function

I am reading a book Gustafson-Rao's Numerical Range and come across a problem that I really don't understand. In theorem 2.1-2 of the book, it asserts that for an operator valued function ...
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47 views

A matrix version of Fredholm integral equation of the second kind

Fredholm integral equation are often encountered in physics. I have to deal with the following Fredholm matrix equation \begin{equation} \big[ L (x,y) \big]_{ij} = \big[ A (x,y) \big]_{ij} + \sum_{k} ...
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26 views

Commutivity of second functional derivatives

In multivariable calculus we learn that partial derivatives are commutative: i.e. $\partial _{xy}F=\partial_{yx}F$ When dealing with multiple functional derivatives this doesn't seem to be the case. ...
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26 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
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145 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
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59 views

How to demonstrate a particular functional equation solution

In order to find a prior probability distribution I have to solve the following functional equation: $$af\left(\frac{a\theta}{1-\theta-a\theta}\right)=(1-\theta+a\theta)^2f(\theta)$$ the solution of ...
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24 views

Need a equation that defines a certain number

Im programming a function but I just cant structure the equation. I think this is the right place to ask since the problem is completely mathematics. Let me explain three scenarios. There are 4 ...
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87 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
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47 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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50 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
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16 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
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26 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
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82 views

Proof that $\forall x \in \mathbb{R}, \forall n \in \mathbb{N}: (f(x))^n = f(nx)$ only has solutions of the form $f(x) = c^x$ for some constant $c$

I am looking for a proof (or a counterexample I suppose) that the recurrence equation (is that the right term here?) $(f(x))^n = f(nx)$ only has a solution of the form $ f(x)=c^x$ for some ...
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46 views

Can a linear solution make a non-linear (functional) differential equation linear?

I was inspired by this question Does a non-trivial solution exist for $f'(x)=f(f(x))$? And tried coming up with similar problems, one interesting case I found was $f'(x) +f(x)=f(f(x))$ which has ...
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29 views

Functional equation on a ring

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and ...
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38 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
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67 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
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52 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
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134 views

Galois Theory for Differential Equations?

Consider the set of algebraically primitive functions: that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example $$y = x +1$$ $$y = ...
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34 views

Solution of equation with special function using maths

I have the following equation with β∈[0,1] and δ∈[0,1] are 2 parameters and $\mu$, $\sigma$ are the mean and standard deviation of a random variable. Is it possible to use software to get the explicit ...
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46 views

regarding a set of integro-PDE

Let say I have a minimal example that contains most of the mathematics I am looking for. The example is as follows: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 ...
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68 views

How to calculate straight line into graph having variety of different results

How to calculate straight line into graph having variety of different results. What I mean for example let say we have this kind of results (measuring persons weight ...
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45 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
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67 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
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284 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
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40 views

Program to determine the relationship of one variable to several possible variables

Suppose I have a system with several variables a, b, c, d, and x. I am trying to solve for the unknown x. I don't know exactly which of those variables x is dependent on, or exactly how the function ...
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134 views

Is there a function $f$ such that $\Gamma (c+x)=\Gamma (c-f\left( x \right) )$?

I was just looking at Euler's reflection formula for Gamma function which states $$\Gamma (1-z)\Gamma (z)=\frac { \pi }{ \sin { (z\pi ) } } $$ but it seems to me that one more reflection formula ...
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112 views

Periodicity of a function.

If we have two functions $f:\mathbb{R} \to \mathbb{R}$ and $g :\mathbb{R} \to \mathbb{R}$ such that the period of $f$ is 7 and that of $g$ is 11, then the period of $F\left ( x \right ) = f( ...
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69 views

Functional equation inspired by moment generating function

A user gave the following nice answer http://math.stackexchange.com/a/161584/5031 My question is that although it is clear $\log M_X(t)=Ct^2$ is a form that satisfies the condition ...
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128 views

Question about ratios and combinatorics

In this question that I posted yesterday (11/15): I am solving a programming puzzle that consists of finding all the possible ways to build a brick wall of $48$" $\times$ $10$" (width $\times$ height ...
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66 views

What is an affine function?

Consider a functional, what is meant by a minimal sequence consistent of 'piecewise affine functions'?
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80 views

$f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

Let $f(x)$ be a non-constant real-analytic function and for real $x$ it satisfies : $f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$ Before you ask if this simplifies by writing $2^x = y$ note that ...
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55 views

solving this recurrence equation

is it possibel to solve the equation g$(x)= \sum_{n=1}^{\infty}f(x/n) $ for $ f(x)$ with other methods different from taking the Mellin transform on both sides ?? thanks.