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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44 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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28 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 f}{\...
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149 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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71 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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37 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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91 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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72 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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56 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) + a_1(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, \ \ \ \ f\left(\frac{x}{2}+\frac{x}{2}\cos\left(\frac{v\pi}{x}\right)\right)=\frac{x}{2}\sin\left(\frac{v\...
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94 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 non-...
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51 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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35 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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44 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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58 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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143 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 = x^...
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35 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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47 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 u^0}{\...
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73 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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48 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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73 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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312 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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47 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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63 views

How prove this $\sum_{k=0}^{n-1}\left|\sum_{j=-n}^{n-1}p_{j}e^{ikj\pi/n}\right|^2=2n\sum_{j=-n}^{n-1}|p_{j}|^2$

show that $$\displaystyle\sum_{k=0}^{n-1}\left|\sum_{j=-n}^{n-1}p_{j}e^{ikj\pi/n}\right|^2=2n\displaystyle\sum_{j=-n}^{n-1}|p_{j}|^2=\dfrac{n}{\pi}||\displaystyle\sum_{j=-n}^{n-1}p_{j}e^{ijt}||_{L^2}^{...
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142 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( x)g(\frac{...
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71 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 $M_X^n(t/\sqrt{n})...
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141 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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73 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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89 views

Find a non-constant real-analytic function $f(x)$ such that for $x\in\Bbb R,\;f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

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 $2^...
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56 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.
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129 views

Defining Oblique Lines

Is it correct to classify a line which is neither vertical nor horizontal as oblique. I am trying to classify lines in a plane based on the quadrants through which they pass.
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44 views

if I get the asymptotic solution of a certain equation involving $ f(x)$ does it mean that the solution exists

Let's take a complicated functional equation $ f(g(x))=f(1-x)g(x) $. Let us suppose that by using a) Analytic method b) Numerical method I can prove that for example $ f(x) \sim x $ as $ x\...
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933 views

Solving a functional discrete equation.

I was to solve the following functional discrete equation (I arguing that $a_k$ is a discrete function): \begin{equation}f\left[a_{k+1}\right]-f\left[a_{k}\right]=0\end{equation} where \begin{...
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130 views

solution of d’Alembert’s equation.

i know that equation for d’Alembert’s equation. is looking so $g(x+y)+g(x-y)=2*g(x)*g(y)$ so am trying to find actual solution for this equation,first i took $x=y=0$ and i got $2*g(0)$=$2*g(0)^2$ ...
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521 views

functional derivative of an integral of the function itself

I have the following $$ \frac{d}{dn(x)} \int_{x \in \cal{R}^3} {n(x) dx} $$ I know that this additional relationship holds $$ \int_{x \in \cal{R}^3}{n(x) dx} = N $$ where N is a constant. My ...
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20 views

Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
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25 views

Functional differential equation separatrix

I've been spinning my wheels with the following differential equation, and would greatly appreciate any guidance on ways to attack it. I have $u(x) \geq 0$ for all $x$. Further, $x \geq 0$. The ...
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24 views

General solution for this PDE?

let $f$ be a function maps $\mathbb{R}^2$ to $\mathbb{R}$. let: $u=f^{(1,0)}(x,y)$ $v=f^{(0,1)}(x,y)$ which are partial derivatives w.r.t the first & second argument of $f$. solve $f(h, \...
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A functional equation with an inequality

I have an increasing function on $[0,1]$, $p \mapsto \Pi(p)$, that has the following properties. $$\Pi(0) = 1 - \Pi(1) = 0$$ $$\Pi(p) + \Pi(1-p) < 1 \quad \forall{p} \in (0,1)$$ $$\Pi(p) > p \...
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35 views

What can be said about $f$ in $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$ if $g$ is known?

Suppose you have the equailty $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$, $x,y\in\mathbb{R}$, and $g$ is known. What non-trivial facts about $f$ can be deduced from this? (Assume $f(x,y)\neq f(y,x)$...
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17 views

Uniqueness of a solution to a functional equation

I have two complex-valued functions, $f$ and $g$, that satisfy the following properties. $\overline{x}$ denotes the complex conjugate of $x$ below. $$g(t)\overline{g(t+h)} = f(h) \quad \forall{t,h}\...
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35 views

Simple almot linear functional equation

I'd like to solve functional equation: $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ I've managed to get: $f(0)=0,f(n)=0$ for all $n\in N$; $f(\frac{1}{2})=0$; $f(-x)=-f(x)$. I'll be grateful for any help.
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14 views

How to Compute Discrete Intrinsic Curvature

Given a function $f$, a region $S$ where this function is twice differentiable, and the property that all of its discrete difference series centered in $S$ converge within $S$ We have that $$ f''(x)...
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22 views

Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? \begin{equation} b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
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28 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') u(t'',t';[x])+u(t,t';[\...
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21 views

Solving equation over expectation

I have an expression $(1 + n EX^k)p^{-k}$ which I would like to minimize over $k$. Here $n$ and $k$ are positive numbers, while $X$ and $p$ are in $[0,1]$. Since the expectation converges absolutely, ...
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20 views

Interval between two solutions of an equation

We are in $\mathbb{R}$ and $x\geq 0$. I have an equation: $(1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y)$ Where $\alpha \in (0,1)$, y is a parameter $0<y<1$, W and g are constants in $x$ and ...
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30 views

Solving the functional equation F(z+1)=Q(z)F(z)

I want to solve the functional equation $F(z+1)=Q(z)F(z)$. The $F(z)$ is a matrix function. $Q(z)$ is also a matrix function. But its compoents are all rational functions. i.e. $ F(z)=\begin{matrix} ...
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11 views

Pacejka formula - extracting the curve peak

I don't know if many of you knows the Pacejka Magical Formula, but it looks like this: $D\sin(C\arctan(Bx - E(Bx-\arctan(Bx))))$ As you can see, the formula reaches a peak point(in this case at ~0.6)...