convergence of $|x_{n+1}-x_n|\longrightarrow 0$ when $x_n$ is a sequence of...
I would like to prove this fact.Let the sequence $v_n\in C(\Omega)$ and suppose that for any $n$ there exists a unique minimum point $x_n \in \Omega$ for the function $v_n$, i.e $v_n(x_n)<v_n(x)...
View ArticleConvergence of Gamma Function Integral
I was able to show that the integral formula for the gamma function$$\Gamma(s) = \int_0^\infty x^{s-1}e^{-x}dx$$is uniformly convergent for $s$ in any closed and bounded interval using the Weierstrass...
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Changing the order of the summation $\sum_{i\in I_2} \sum_{j\in I_2} a_{ij}$
I'm reading the book: "Deimling Nonlinear Analysis" and I'm blocked on a proof, at page 27, when proving Jordan's separation theorem.At some point, we want to show that two sets, let's say $I_1$ and...
View Article$ f(x + y) = yf(x) + xf(y) $ [closed]
Let $f: (0, \infty) $ be a function continuous at $ 1 $, and for all $ x, y > 0 $ we have $ f(x + y) = yf(x) + xf(y) $. Prove that $ f(x) = 0 $ for all $ x > 0 $.I tried substituting $y= x$ and...
View ArticleN-Epsilon definitions [closed]
I was reading an old forum. I decided to post a question about it since I doubt anyone will reply to my comment as the forum is old.David mentions that |x-a|<|a| will not work.Can someone please...
View ArticleFinite volume method: Say something about the sign of an expression
Suppose we use a finite volume method in the following sense:Restrict the velocity domain to a bounded symmetricsegment $[−v^∗, v^∗]$ . We consider a mesh of this interval composed of $2L$ control...
View ArticleA continuous nowhere differentiable function with $\alpha-$derivative exists...
In This question I asked:Define $g(x)= |x|$ for $|x|\in [-1,1]$ , $g(x+2)=g(x)$$$f(x)= \sum_{n \ge 1} \frac{3^n g\left(4^n x\right) }{4^n}$$ what is the $\sup \{\alpha\}$ such that$$\lim\limits_{h \to...
View ArticleFubini's theorem double integral
I tried to solve the exercise below, but I'm not sure of my answer:What does the change in order of integration give in the following integrals?$$ \int_{0}^{\pi}\left( \int_{0}^{\sin(x)}f(x,y)dy\right)...
View ArticleLet $c>0$ and $f:[0,\infty)\to[0,\infty)$ be Riemann integrable on every...
Problem statement: Let $c>0$ and $f:[0,\infty)\to[0,\infty)$ be Riemann integrable on every bounded interval in $[0,\infty)$. If $$f(x)\leq\frac1c\int_0^1\left(\int_0^xf(ts)\,ds\right)dt$$ for all...
View ArticleAccumulation points of product sequence
Let $a_n$ and $b_n$ be two bounded real-valued sequences.Then if $a$ is an accumulation point of $a_n$ and $b$ an accumulation point of $B$, it is clear that $ab$ is an accumulation point of $a_n b_n$....
View ArticleProving an equivalence about probability measures on a space.
Let $(X,\mathcal{A})$ be a measurable space and $T :X \to X$ a measurable transformation and $\mu,\nu$ probability measures on $(X,\mathcal{A})$. Prove TFAE:(i) $\nu(T^{-1}(E))=\mu(E)$ for all $E \in...
View ArticleProving $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and...
Prove that $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$.The solution given by my lecturer is as follows. Recall the approximate formula for zeta, given by...
View ArticleHow can I evaluate the Gaussian Integral using power series?
It's a well known result that the Gaussian integral$$\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$$evaluates to $\frac{\sqrt{\pi}}{2}$. This result can be obtained using double integrals with polar...
View Article$\lim\limits_{n\to \infty} \frac{1}{n}\cdot \big((m+1)(m+2)...
$$\lim_{n\to \infty} \frac{1}{n}\cdot \big((m+1)(m+2) \ldots(m+n)\big)^{\frac{1}{n}}$$where $m$ is a fixed positive integer.Here is my attempt:According to Cauchy's theorem of limit if...
View ArticleDo function suprema compute? Upper bounds?
It's very common in numerical analysis to have an approximation method whose error is evaluated by taking a truncated Taylor series, applying a remainder theorem to get the error.E.g. the $N$-point...
View Articlelimit into infinite integral?
This question is an offshoot of this one.In this case, the integral is finite. But if the integral was over an infinite interval, what conditions would need to be added, along with uniform convergence...
View Articlewhy infinite svd are possible for a matrix with dimension of N*N [closed]
If we take a Matrix A with dimension of N*n with repeated singular values then we have infinite svds are possible but Why this happen because in V or U get n li for R^N ????
View ArticleIf $ | \int \phi(x) f(x) dx| \le 2 \sup_x (1+\|x \|^2)^M |\phi(x)|$, then $f$...
I am looking to see if the following condition is true: Suppose that$$ \left| \int \phi(x) f(x) dx \right| \le C \sup_x (1+\|x \|^2)^M |\phi(x)|$$for all Schwartz-class functions $\phi$ and some...
View ArticleCan anyone explain this feature of the normal distribution?
I was doing some questions involving the normal distribution when I saw that the percentage point of 0.95 is 1.6449, I know that 1.6449 is approximately $\frac{\pi^2}{6}$, and I was wondering if anyone...
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Union of two countable sets is countable [Proof]
Theorem: If $A$ and $B$ are both countable sets, then their union $A\cup B$ is also countable. I am trying to prove this theorem in the following manner: Since $A$ is a countable set, there exists a...
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