Intereting Posts

Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals $(k-1)^{n} + (k-1)(-1)^{n}$
Second pair of matching birthdays
A limit related to super-root (tetration inverse).
How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?
prove a function is continuously differentiable
Generalized Case: Three Consecutive Binomial Coefficients in AP
“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…
dropping a particle into a vector field, part 2
Proving that $\pi(2x) < 2 \pi(x) $
Physical or geometric meaning of complex derivative
How to write zero in the unary numeral system
Non-orientable 1-dimensional (non-hausdorff) manifold
Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$.
Time-optimal control to the origin for two first order ODES – Trying to take control as we speak!
How to think about ordinal exponentiation?

**Yes,this is very similar to a previous question I asked. That was about normal solutions and not weak solutions.**

We define the operator known as the implied derivative denoted as $I(f)(x)(g)$ to be:

- Weak derivative zero implies constant function
- How does integration over $\delta^{(n)}(x)$ work?
- Why is it useful to show the existence and uniqueness of solution for a PDE?
- Questions about weak derivatives
- Prove Corollary of the Fundamental lemma of calculus of variations
- Are weak derivatives and distributional derivatives different?

$$I(f)(x)(g) := g(x) \left(\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} \right) + (1-g(x)) \left(\lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h} \right)$$

Where $g(x)$ is an arbitrary characteristic/indicator function.

I wish to prove whether or not the following conjecture is true. I have no idea how to go about doing that.

Is a function a weak solution to an ordinary differential equation if and only if it is a continuous solution to the corresponding implied differential equation?

By corresponding equations, I just mean that they are corresponding if they are the same except with all of the derivative operators replaced with the implied derivative operator.

- Does convergence in H1 imply pointwise convergence?
- Questions about weak derivatives
- $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$
- function a.e. differentiable and it's weak derivative
- Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$
- Prove Corollary of the Fundamental lemma of calculus of variations
- Distributional derivative of absolute value function
- Are weak derivatives and distributional derivatives different?

- If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
- Divisibility using binomial coefficients
- Example of a domain in R^3, with trivial first homology but nontrivial fundamental group
- Hopf's theorem on CMC surfaces
- modulus calculations & order of operations
- If the union of two sets is contained in the intersection, then one is contained in the other ($\implies A \subseteq B$)
- Number of singular $2\times2$ matrices with distinct integer entries
- finding the combinatorial sum
- Proof that the Lebesgue measure is complete
- Showing that $ \displaystyle \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n} = e^{r} $.
- Calculating the expected values of the min/max of 2 random variables
- Explicit well-ordering of $\mathbb{N}^{\mathbb{N}}$
- A System of Matrix Equations (2 Riccati, 1 Lyapunov)
- Geometric interpretation for sum of fourth powers
- Probability union and intersections