1
vote
0answers
15 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 ...
3
votes
1answer
116 views

$f,g$ such that $\int fg = \int f \int g$

Suppose $f,g$ are real valued on $\mathbb{R}$ (and no further restrictions apart from the obvious requirement that the integrals exist), then when does $\displaystyle\int f(x)g(x)\,dx = \int f(x) \, ...
0
votes
0answers
52 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
0
votes
0answers
42 views

What is the analytical solution to a Volterra integral equation?

I need to solve a following equation: \begin{equation} r_{k+1} = -\sum\limits_{l=0}^{k-1} r_l \cdot (k-l) \cdot \left(\frac{\omega}{t_c - l}\right)^{2 \beta} + \delta_{k,0} \end{equation} subject to ...
1
vote
0answers
40 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 ...
3
votes
1answer
34 views

A question on functional equations.

Question: If it is given that $$ e^xf(x) = 2 + \int_0^x\sqrt{1+x^4}\,dx $$ then what is the value of $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $ Where I am stuck: Now, since we are to ...
2
votes
1answer
66 views

Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$

Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$. How to prove that: The equation above has a unique solution $U_n$ ...
2
votes
0answers
24 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 $f=0=a$ ...
3
votes
1answer
177 views

Integral Inequality $|f''(x)/f(x)|$

Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$
2
votes
1answer
348 views

Equation with a definite integral - can I differentiate it?

I have an equation like this: $$te^{t} = \int_0^t e^\tau u(\tau)d\tau$$ I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it ...