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Rigrous proof of the rate and order of convergence of bisection method

No, you can not apply this Q-convergence rate formula, in many cases the sequence of quotients has no limit. It is only the interval length, which is an upper error bound, that converges that way to ...
Lutz Lehmann's user avatar
1 vote

Can we convert the following integral equation to a differential equation:$h(r)= \int_0^\infty\frac{f(x)}{e^{r x} + 1} dx$?

Alternatively, you can consider a little bit more general problem by introducing an auxiliary variable and defining the following function : $$ g(r,s) := \int_0^\infty \frac{f(sx)}{e^{rx}+1} \,\mathrm{...
Abezhiko's user avatar
  • 10.3k
0 votes

Estimation of a gamma function-like integral

This is probably not the simplest answer. We need to show that $$ \frac{1}{{k!}}\int_{2k + 2}^{ + \infty } {x^k {\rm e}^{ - x} {\rm d}x} < \frac{1}{{k + 1}} $$ for $k>-1$. (I simply define $k!$ ...
Gary's user avatar
  • 33.2k
1 vote

Example of a series where the ratio test $\limsup |a_{n+1}/a_n|$ can be applied, but $\lim |a_{n+1}/a_n|$ cannot

Consider the series $a_n$ obtained as follows $a_1=1$, $a_2=1$, $a_3=2$ $a_4=1$, $a_5=2$, $a_6=3$ $a_7=1$, $a_8=2$, $a_9=3$, $a_{10}=4$, etc. That is, split $\mathbb{N}$ in blocks $A_k$, each of ...
Mittens's user avatar
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2 votes

Stochastic differential equations with only time integrals

OK - time for an update. Thanks to everyone who commented. As pointed out by some commentators, these types of differential equations are called Random (Ordinary) Differential Equations. They have ...
rufus_lawrence's user avatar
0 votes

How do I get the Jacobian of numerical intengration-methods with half steps?

Usually the symplectic gradient is not computed during integration, it is provided as function in code. In the separable case that is assumed here, one only needs to implement the force or ...
Lutz Lehmann's user avatar
2 votes
Accepted

Bounded linear map on a subspace of a normed vector space that cannot be continuously extended to the subspace's closure.

It suffices to take $U$ to be any non-closed subspace of a normed vector space $V$, $W = U$, and $S$ to be the identity map $\text{id}_U : U \to U$. As an explicit example we can take $V = \ell^2(S)$ ...
Qiaochu Yuan's user avatar
0 votes

How to give this sum a bound?

Expanding on the idea from my comment and the attempt by the OP, I will establish $8\times 15 \times \left(\frac{\pi^4}{90}\right)^2$ is an upper bound for the sum, assuming $x,y\neq0$ in the sum. ...
Steen82's user avatar
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0 votes

Understanding Rudin's PMA Theorem 9.17

The key is to understand the notations here. Each quotient in this sum without $u_i$(i.e. $\frac{f_i(x + te_j)-f_i(x)}{t}$) is the ith coordinate in $R^m$ with respect to $u_i$. So basically, (29) is ...
Kai's user avatar
  • 1
0 votes

A corollary of Rolle's theorem

Since you want to base this on Rolle's theorem: From $f'(a)>0$ then for sufficently small positive $h$, we have ($a<a+h<b$ and) from the limit definition of $f'(a)$, $\frac{f(a+h)-f(a)}{h}>...
Hagen von Eitzen's user avatar
2 votes
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A corollary of Rolle's theorem

This is a specific case of the Darboux theorem. In your case, this is easier, since you are looking for $c$ such that $f'(c)=0$. Since $f$ is differentiable on $[a,b]$, then $f$ is continuous on $[a,b]...
Martigan's user avatar
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4 votes
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Understanding implication of the convexity of a class of distributions

Let's, for the sake of simplicity, assume that both ${X}$, and $Y$ are discrete. Then we can write, by the law of total probability, $$ \mathbb{P}(Y=y) = \sum_x\mathbb{P}(Y=y\vert X = x)\mathbb{P}(X=x)...
MrTheOwl's user avatar
  • 574
2 votes

Nonstandard Analysis research project ideas

There are two approaches to non-standard analysis: (1) the "extension" approach and (2) the axiomatic approach. In the "extension" approach, the real numbers are extended to the ...
Mikhail Katz's user avatar
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4 votes
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Translates of a set of positive Lebesgue measure cover $\mathbb{R}$?

Let $E$ be fat Cantor set (i.e. a Cantor like set of positive measure). Then $E$ has no interior. If $\mathbb R=\bigcup_n (E+x_n)$ then Baire Category Theorem implies that $E+x_n$ has an intetior ...
geetha290krm's user avatar
  • 39.3k
0 votes
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How to show that the limit of a sequence is not equal to some value?

I will clarify my comment. By definition, $l$ is the limit of the sequence $a_n$ if for every $\varepsilon>0$ you can find $N$ such that $\left|a_n-l\right|<\varepsilon$ for all $n>N$. Thus, ...
Davide Masi's user avatar
1 vote
Accepted

Can we convert the following integral equation to a differential equation:$h(r)= \int_0^\infty\frac{f(x)}{e^{r x} + 1} dx$?

You can go part of the way if you substitute $x$ with $u=rx$. You then have $x=u/r$ and $dx=\frac{1}{r}du$. If $r>0$ $h(r)=\frac{1}{r}\int_{0}^{\infty}\frac{f(\frac{u}{r})}{e^u+1}du \\ \Rightarrow ...
Baenazril's user avatar
0 votes

Can the gamma function be generalized to quaternions and how?

The quaternions (1,i,k,l) form an abstract representation of the Lie algebra of the orthogonal group. Its lowest dimensional linear representation are the imaginary multiples of the Pauli matrices. ...
Roland F's user avatar
  • 3,234
3 votes
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Continuity and lebesgue integrability of integral function, proof verification

This seems to be a correct proof. For the convergence $\chi_{E_n}\rightarrow\chi_E$, consider points $t$ in $\mathbb{R}$ different from from $x+h$ and $x-h$; then there is a positive $\delta$ for ...
Susana Santoyo's user avatar
2 votes

Continuity and lebesgue integrability of integral function, proof verification

I will asume that $h>0$. The case $h<0$ is left to you. $$2h\|\phi_1\|=\int_{-\infty}^{\infty} |\int_{x-h}^{x+h} f(t) dt| dx$$ $$ \le \int_{-\infty}^{\infty} \int_{t-h}^{t+h} |f(t)| dx dt$$ $$=\...
geetha290krm's user avatar
  • 39.3k
0 votes

Possibility that all lights $\mathbf{X}=(X_1,X_2,\cdots)$ turn off again with every time turn a light with its number $n\sim\text{geom}(\frac{1}{2})$.

As noted in many other answers, it suffices to show a $\Omega(1/t)$ bound on the chance the process has returned to the starting state at time $t$. This answer gives another way of showing such a ...
Ziv's user avatar
  • 346
2 votes
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Is it true that $\widehat{(\delta_{x_{0}}\otimes T)} = \hat{\delta}_{x_{0}}\otimes \hat{T}$?

This will actually be true even if you replace $\delta_0$ by any distribution $S$ whose Fourier transform is a measurable function with at most polynomial growth, at the very least. It's also probably ...
Bruno B's user avatar
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2 votes
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Courant and John state that theorems proven for functions of two variables can be easily extended to functions of $n$ variables. Why is this so?

Lots of theorems in analysis involving multiple variable functions end up depending on just things like vector addition and the triangle inequality (in $n$-dimensions), and these have the same ...
JonathanZ's user avatar
  • 11.2k
1 vote
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Legendre transformation is a continuous map

Here is a proof that the Legendre transform is convex. Though I am used to a slightly different version of the definition of Legendre transform. With your definition the transform is concave. I will ...
Steen82's user avatar
  • 806
0 votes

Proving that the set of limit points of a set is closed

Let $x \in E'$ be a limit point of $E'$. $\implies \forall$ neighborhoods $U$ of $x$ $ \exists z \in E'$ such that $z \in U$. Since, $U$ is also a neighborhood of $z$ and $z \in E' \implies \exists y ...
Arohan's user avatar
  • 11
1 vote

Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$

The inequality does not hold for any nonconstant $w(x).$ Assume $f$ does not vanish a.e.. For $d=1$ the LHS is equal $$\int\limits_0^1[w^2(x)|f(x)|^2+w^{-2}(x)|g(x)|^2]\,dx\ge 2\int\limits_0^1 |f(x)|\,...
Ryszard Szwarc's user avatar
2 votes
Accepted

Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$

For $f = \frac{1}{w}, g = w$ the left hand side is just $2$, so we only have to pick $w$ such that the right hand side is larger than that. For example when $d = 1$ for any $n \in \mathbb{N}$ we can ...
user23571113's user avatar
  • 1,458
5 votes
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What is the limit of the alternating series $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ as $z\to1$ for a sequence $T_n\sim cn$?

The answer is no. For example, $$ \sum_{n=1}^\infty (-1)^n z^{4n+(-1)^n} = - \sum_{m=1}^\infty z^{4(2m-1)-1} + \sum_{m=1}^\infty z^{4(2m)+1} = -\frac{z^3}{1-z^8} + \frac{z^9}{1-z^8} = -\frac{z^3+z^5+z^...
Greg Martin's user avatar
  • 82.1k
0 votes

Possibility that all lights $\mathbf{X}=(X_1,X_2,\cdots)$ turn off again with every time turn a light with its number $n\sim\text{geom}(\frac{1}{2})$.

First, let's compute the probability $\mathbb{P}(\mathbf{X}_m = 0)$. For parity reasons, this probability is $0$ when $m$ is odd. For $m$ even, it is more interesting. Notice that $\mathbf{X}_{2m}=0$ ...
Saucitom's user avatar
2 votes

Can anyone explain the constant in this relation

Let $f(x,y) = (xy)^y/y^x$. The graph of $z=f(x,y)$ is a surface that has a saddle point near $(x,y)=(4,1.5)$, which is the intersection point in your second graph. The saddle point can (in principle) ...
Greg Martin's user avatar
  • 82.1k
1 vote

If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$?

If $h=0$ almost everywhere, then it has a null integral. (To prove this, try showing it for characteristic functions, and then use the definition of the integral).
Maxime's user avatar
  • 395
2 votes
Accepted

Bump function with integral $1$ and value $1$ at zero

Consider $$g_0(x)=f\left(\frac12 + x\right)f\left(\frac12-x\right)$$ This is a smooth bump function centred at the origin, but $g_0(0)\neq1$. We can fix that with $g_1(x)=\frac{g_0(x)}{g_0(0)}$. Now ...
Arthur's user avatar
  • 201k
4 votes
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$\mathbb{N}$ is uncountable?

A set is nowhere dense if the interior of its closure is the empty set. Under the discrete metric, the closure of $\{x_n\}$ is just $\{x_n\}$. Further, $x_n$ is also an interior point of $\{x_n\}$, ...
Andrew Lys's user avatar
0 votes

Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

Note, this is not a full solution. For any angle $\theta \in (-2\pi, 2\pi)$ the following holds (you can prove it via geometry): $$ 1-e^{i\theta}= \begin{cases} Ae^{i\frac{\theta-\pi}{2}} & \theta\...
zetko's user avatar
  • 319
8 votes
Accepted

Is this "continuous" function really continuous?

No, this is not necessarily true. As an example, let $n = 2$, $\phi_t: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by, $$\phi_t(x, y) = \begin{cases} (t^{|x|}, y) &, \text{ if }t \in (0, 1], x \in [-...
David Gao's user avatar
  • 9,898
0 votes

Rudin Chapter 3 exercise 5.

D_S's and Jay Zha's answers rely on a different definition of $\limsup$ than the one given in Rudin so here is an attempt based solely on the content in Rudin. Suppose towards a contradiction that $\...
Elijah's user avatar
  • 1
1 vote

About solution to homogeneous ODE $u'' + u = 0$

Another approach, let $f(x) = u(x)^2+(u'(x))^2$. Then $f(0) = 0$ and $f'(x) = 0$ hence $f(x) = 0$.
copper.hat's user avatar
  • 174k
1 vote
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About solution to homogeneous ODE $u'' + u = 0$

We can check that $u''+u=0$ has the Solution $u=c_1e^{-x}+c_2e^{+x}$ : $u'=-c_1e^{-x}+c_2e^{+x}$ $u''=c_1e^{-x}+c_2e^{+x}$ [[ Solving Characteristic Polynomial $D^2+1=0$ , we get $D=+1,D=-1$ , hence ...
Prem's user avatar
  • 12.3k
1 vote
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Composition of functions, once for all

Let's address your immediate questions: Is it necessary or sufficient to have $g(C) \subseteq A$? It is both necessary and sufficient. Why necessary? If there is some $c \in C$ such that $g(c) \notin ...
Ben Steffan's user avatar
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3 votes
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A simple question about bounding a sum

First, the condition $\|x\|_2>\gamma^{-1}$ leads to $\frac{1}{\|\gamma x\|_2} <1$, which implies that $\frac{\gamma^2}{\|\gamma x\|_2^2}$ dominates the other terms in the sum. Hence, we only ...
alex440's user avatar
  • 556
2 votes
Accepted

If $(X, A, m)$ is $\sigma$-finite and $B\subseteq A$ is a $\sigma$-sub-algebra, then is $(X, B, \nu)$ $\sigma$-finite?

This need not be the case even if $f \equiv 1$. Say, $X = \mathbb{R}$, $A$ being the Borel $\sigma$-algebra, $m$ being the Lebesgue measure, and $B = \{\varnothing, \mathbb{R}\}$. Then $m$ is $\sigma$-...
David Gao's user avatar
  • 9,898
3 votes
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What is this notion of continuity?

Edit #2: My apologies! This argument doesn't actually work at all and your counterexample works over $\mathbb{R}$ as well; if we take $d(x)$ to be the number of nonzero $p$-adic digits then $f(x) = e^{...
Qiaochu Yuan's user avatar
2 votes

Prove that $T$ is not a compact operator.

For $\lambda\neq0$, $(T-\lambda I)x=0$ iff $x(-n)=(2\lambda -1)x(n)$. In particular, for $\lambda=1$ we have that $$N(T-I)=\{x\in\ell_2(\mathbb{Z}): x(-n)=x(n)\}$$ This subspace is not finite ...
Mittens's user avatar
  • 40.8k
5 votes
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Prove that $T$ is not a compact operator.

For any compact operator $T$ if $x_n$ tends to $0$ weakly then $\|Tx_n\|\to 0$ (see page 4). Let $\{e_n\}$ denote the standard orthonormal basis in $\ell^2(\mathbb{Z}).$ By the Bessel inequality $e_n\...
Ryszard Szwarc's user avatar
1 vote

Stochastic convergence with and without rate

Here's a partial answer. Suppose not only $X_n\to 0$ stochastically, but even in $L^1$, i.e. $\mathbb E[\lvert X_n \rvert]\to 0$. Then choosing e.g. $\epsilon_n := \sqrt{\mathbb E[\lvert X_n \rvert]}$ ...
Joseph Expo's user avatar
2 votes
Accepted

Is my formula for this projection correct?

I should first point out the formula you wrote down for $P$ only works if $\phi$ is a unit vector, i.e., if $\int |\phi(x)|^2 \, dx = 1$. Assuming that is the case, the formula you wrote for $P \...
David Gao's user avatar
  • 9,898
1 vote
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Compactess of a set in $\mathbb{R}^d$ defined as the union of compact sets

The Heine-Borel theorem tells us that it is sufficient to check that $\mathcal{A}$ is closed and bounded. Rewrite $\mathcal{A}$ as: $$ \mathcal{A} = \{y \in \mathbb{R}^d: \exists x \in [a, b], \...
Thành Nguyễn's user avatar
2 votes
Accepted

Distance of a real number to a discrete set of scaled sine values

If you want to explore tighter bounds, you could try $$\sin(x) \geq \pi ^{-\frac{\pi ^2}{3}} x \left(\pi ^2-x^2\right)^{\frac{\pi ^2}{6}} \quad \quad\text{for} \quad x\in (0,\pi)$$ the maximum ...
Claude Leibovici's user avatar
3 votes
Accepted

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

The squeeze theorem is not needed here, just the definition of $\lim_{a \to -\infty}$. To simplify the notation, set $f(t) = V_F([t, b])$ and $L = V_F(-\infty,b]$. In the quoted proof it has been ...
Martin R's user avatar
  • 117k
3 votes
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$C^1[a,b]$ non-linear function, such that $f'' =0$ a.e.

Your idea is correct. Let $C$ be the Cantor function, Then the function $$ F(x)=\int_0^x C(t)\, dt $$ Is in $C^1$ with $F’(x)=C(x)$ and $F’’(x)=0$ a.e.
GReyes's user avatar
  • 17k
3 votes

Fundamental solution for Helmholtz equation in higher dimensions

@Manuel Cañizares and @Mark Viola: I found some original work in [1, p231] about the solution steps above, with the scaled homogeneous solution of the corresponding Bessel differential equation, ...
fzotter's user avatar
  • 31

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