Comment by WHLin on Iteration of resolvent operator
Thank you! Just figured out how to do the last step.
View ArticleComment by WHLin on A question on Turing's original reaction-diffusion paper
Thank you! The discriminant < 0 can derived the inequality on (-2), but not this one. I reazlized how to solve it (see edition).
View ArticleComment by WHLin on Is the "Translated average" $(T_rf)(x)$ of an $L^1$...
Consider the function f(x)=1 if x is rational, and f(x)=0 if x is irrational. The local average of Tf(x) = <|f(x)-f(y)|> is 0 if x is irrational, and is 1 if x is rational. Hence, I guess the...
View ArticleComment by WHLin on Is it true that if $(f\circ p)'(0) = \nabla f(0)p'(0)$...
I think it is correct. My intuition is: maybe you could consider radial coordinate, and use the path argument to show that F(p(t)) -> 0 as t ->0 "uniformly" within the neighborhood of 0. Then,...
View ArticleComment by WHLin on How to restrict the area inside a 3D-rotated Box.
math.stackexchange.com/questions/180418/… This may help. Good luck!
View ArticleComment by WHLin on Is this a rule of matrix algebra? (Eigenvectors)
This is called "Laplace expansion" in calculating determinant. en.wikipedia.org/wiki/Laplace_expansion
View ArticleComment by WHLin on Measuring symmetry / Is there a way to estimate the...
You could use centroid as the origin and convert the measure under polar coordinate. A perfect circle/disc will have $\theta$ symmetry, while an irregular shape will have large deviation of $\theta$.
View ArticleComment by WHLin on $L^p$ convergence of certain "average" function
This is related with von Neumann ergodic theorem, for example, see joelmoreira.wordpress.com/2013/01/17/the-ergodic-theorem
View ArticleComment by WHLin on Solving a heat equation
Yes, the eigenfunction (with boundary condition $u(0,t)=0$) are in the form of $sin(m_1\pi x/L) sin (m_2\pi y/L)$, with $m_1=0,1,2,...$ and $m_2 = 0,1,2,...$. In this case $\lambda = m_1^2 + m_2^2$.
View ArticleComment by WHLin on Moment of Ito diffusion computationally
I think it will depend on how $f$ and $\sigma$ behave on $[0, \tau]$ (e.g. Lipchitz condition etc.) There are some discussion here: math.stackexchange.com/questions/4058518/…
View ArticleComment by WHLin on Cross-covariance term of linear stochastic differential...
Your equation is a linear SDE, the general solution has been discussed: math.stackexchange.com/questions/1788853/…
View ArticleComment by WHLin on Where is simple exponential relaxation in relation to...
Let $X_k$ be the expected number of spin that have not change sign at time $k$. You could write down the recursive formula between $X_{k+1}$ and $X_k$. This lead to a equation which should give a...
View ArticleIteration of resolvent operator
I am studying a paper about Markov semigroup and found an equation about resolvent operator. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, and $P(t):\mathcal{M}(\mathbb{R}...
View ArticleAnswer by WHLin for Convergence in $L_p(\mathbb{R})$
Yes, $g_n$ point-wisely converge to zero, but this is not sufficient to say that its integral converges to zero. You would like to first use Dominated Convergence Theorem to say the limiting integral...
View ArticleAnswer by WHLin for whether the function is uniformly continous
I think you could calculate $h'(x)$ and see if $|h'(x)|<M$ for some constant $M>0$ and for all $x \in \mathbb{R}$.
View ArticleAnswer by WHLin for Partial and directional derivatives
Consider 2D case, define$f(0,y) = 1 $ for all $y$.$f(x,0) = 1 $ for all $x$.$f(x,y) = 0$ otherwise. Now, consider partial and directional derivatives at the origin.Consider 2D case, define$f(0,0) = 0...
View ArticleAnswer by WHLin for Why does $f(x,y)= \frac{xy^2}{x^2+y^4}$ have different...
This has something to do with the uniform continuity around 0. Given a fixed $\epsilon > 0$, if there exist an universal constant $\delta_u$ such that $$|(x,y)|<\delta_u \implies...
View ArticleAutomorphism on 2-sphere
Suppose we have a function define on unit sphere which is parametrized by $u(\theta, \phi)$. Now I apply an automorphism $A: S^2 \rightarrow S^2$ on the sphere that rotates the image of the function...
View ArticleAnswer by WHLin for Discontinuous function $f:\mathbb{R}\to \mathbb{R}$ that...
Define $f(0) = 0$ and $f(x) = log(|x|)$ otherwise. Now, for every $y \neq 0, y \in \mathbb{R}$ there are exactly two values of $x$ such $f(x)=y$. The problem remains is when $y=0$ there are three...
View ArticleAnswer by WHLin for Show that ||A||_2 = √(ρ(AA*)) where ρ(AA*) is the...
Let $T=A^*A$, and we write $|.|$ for Euclidean norm. The matrix $T$ is a "normal matrix" (i.e. $TT^* = T^*T$). Normal matrix has specially nice properties; one of the important property is that any...
View ArticleAnswer by WHLin for How can i show that $\frac{k^3}{3^k}$ is monotonic...
Let $a_k \equiv \frac{k^3}{3^k}$.$a_{k+1}/a_{k} < 1$$\iff$$ \frac{1}{3}(1+\frac{1}{n})^3 <1$$\iff n>2.2614$
View ArticleAnswer by WHLin for How to find a solution of this equation system.
This is just some of my thought: Suppose $\gamma = v_1/v_2$ is finite, we have $ x^T(A-\gamma B)y = 0$. Since $x,y$ are nonnegative and nonzero, at least the matrix $A-\gamma B$ cannot be positive for...
View ArticleAnswer by WHLin for how to prove that Fibonacci sequence is divergent
We prove that $F_n \geq n-2$ by induction.(1) The inequality hold for $n=1,2,3,4$.(2) Suppose it is true for $F_n$ with $n=1,2,...,k+1$ for $k\geq3$, then we have $F_{k+2}=F_{k+1}+F_k \geq (k-1)+(k-2)...
View ArticleAnswer by WHLin for Is this weird function with argument in the integrand...
I think in general the statement is "false". Let $I = [0,2]$, $f: I^2 \rightarrow \mathbb{R}$, and $t_0 = 0$. Define$$ f(x,t) \equiv 0, \;\; \text{if } x \in [0,1) \cup (1,2] $$$$ f(x,t) \equiv 1, \;\;...
View ArticleAnswer by WHLin for A question about determinants an multilinear algebra
This is just some of my thought.Since the dimension is finite, we could represent $\phi$ and $\varphi$ as matrices $A$ and $B$ and consider $F$ as a matrix function mapping from $\Gamma^{n^2}$ to...
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Algebraic equation of a specific six-degree polynomial
During my research, I was analyzing a linear system with characteristic polynomial of six order, $p(x)=x^6+3ax^5+3a^2x^4+a^3x^3 - a^3b_1b_3b_3$. I used Mathematica and it found all roots as the...
View ArticleAnswer by WHLin for Why is $ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy}$...
By definition, we have $sin(t)=t-\frac{t^3}{3!}+\frac{t^5}{5!}-...$ and hence $sin(xyz)=xyz-\frac{(xyz)^3}{3!}+\frac{(xyz)^5}{5!}-...$Therefore, when $(x,y,z)$ approaches to zero, the behavior of...
View ArticleAnswer by WHLin for How do we show that $\lim_{x\to0}(1+x)^{\frac{1}{x}}$ is...
Let $f(x)\equiv (1+x)^{1/x}$, we have $f(x)=exp\{log[(1+x)^{1/x}]\}$. Since $exp\{.\}$ is continuous, we have$$\lim_{x\rightarrow 0} f(x)= exp\{\lim_{x\rightarrow 0} log[(1+x)^{1/x}]\}=...
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