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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21 views

the family of functions with alternating derivatives in different orders

I'm wondering whether there's a name for the family of functions with alternating derivatives in different orders. Formally, $f(x)\in C^\infty:\mathbb{R^+}\rightarrow\mathbb{R}$. For all x, ...
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21 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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21 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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139 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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49 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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19 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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85 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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35 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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46 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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28 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
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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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24 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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73 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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37 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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25 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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34 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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32 views

I tried this problem in the following way; is it right?

Let, f(x) be a twice differentiable function defined on (-1, 1) and f(0) = 1. Let, f(x) ≥ 0, f'(x) ≤ 0 and f''(x) ≤ f(x) for all x ≥ 0. Show that, f'(0) ≥ -√2. I am telling you what I did. First, ...
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62 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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47 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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122 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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41 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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101 views

solving a functional equation using given values

Moderator Note: This is a current contest question on Brilliant.org. The current contest ends on 13 October 2013, after which time this question will be unlocked. A Function $f$ from the ...
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62 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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56 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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261 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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39 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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127 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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109 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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122 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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64 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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$f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

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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52 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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64 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
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72 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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42 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 $ ...
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108 views

where can I find a good Proof for Cauchy's functional equation

Do you know where can I find good proofs for Cauchy's functional equation?
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610 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 ...
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113 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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123 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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375 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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7 views

find the distribution of 100 heart transplant patients at a low volume and high volume hospital using boxplot graph 0-40 mortality

Using a boxplot graph find the distribution show mortality rates within one year of 100 patients having heart transplants at various hospitals. The low volume hospital perform between 5 and 9 ...
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69 views
+100

Solutions to functional equation $ \gamma(s,t)=f(t \cdot g(s))+h(t) $

Let $$ \gamma(s,t)=f(t \cdot g(s))+h(t) $$ where $\gamma$ is a known function of $s \in \mathbb{R}$ and $t \in \mathbb{R}$ while $f$, $g$, and $h$ are unknown functions. Assume ...
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17 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
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16 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
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15 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which could be either: a convolution kernel or a difference of two convolution kernels: $\varphi=\varphi_1-\varphi_2$. I am ...
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25 views

Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...