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Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
29 views

Sequence convergence problem under series convergence assumption

If positive series $\sum\limits_{n=1}^\infty a_n$ is convergent, then $\lim\limits_{n\to\infty} na_n=0$. Is that true? If not, could you please show me a counterexample? Thanks a lot!
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1answer
13 views

problem in converting hex to binary dividing by 2

if i convert $ABH$ into binary by table simple its value is $1010$ $1011$. but if i convert using dividing by $2$ then why the answer is different ie $0110$ $1110$. $AB/2=55, r=0$ $55/2=27, r=1$ $...
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3answers
45 views

Calculating improper integral $\displaystyle \int_0^1 \dfrac{\arcsin(x)}{\sqrt{1-x^2}}\,dx$

Calculate improper integral $\displaystyle \int_0^1\dfrac{\arcsin(x)}{\sqrt{1-x^2}}\,dx$ We had the following equation to calculate improper integrals (2nd style): Let f in$\left(a,b\right]$ ...
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2answers
36 views

Fourier transform $\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$ unbounded?

I want to prove or disprove that the Fourier transform $\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$ is unbounded, where $\lVert\cdot \rVert_1$ denotes the $...
2
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1answer
24 views

sum of sequences converging to $0$

Let $I$ a countable set of indices, and for any fixed $i\in I$ let $(a^{(n)}_i)_{n\in\mathbb N}$ be a sequence with values in a non-archimedean local field (e.g. $\mathbb Q_p$). Assume that the ...
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2answers
30 views

determine if the statement with limit is true

I want to determine if the statement $$\lim_{x\to\ a} f(x) = \infty \Rightarrow \lim_{x\to\ a}\frac{1}{f(x)} = 0$$ is true or not (by proving it or proving a contradiction). I know that I have a ...
1
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1answer
14 views

Does the following set have Lebesgue measure zero?

Let $F:\mathbb R^n\to\mathbb R$ is a non-constant continuous function. Is it true that $Leb[(x_1,...,x_n)\in\mathbb R^n:F(x_1,...,x_n)=0]=0?$ Here $Leb$ denotes Lebesgue measure. I don't know if this ...
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2answers
22 views

Sequence Convergence under some assumption

Suppose a positive sequence $\{a_n\}_{n=1}^\infty$ has a convergent subsequence, namely, $\lim\limits_{j\to\infty}a_{n_j}=\mu$. If for all $n>0$, $|a_{n+1}-a_n|<\frac{1}{n}$, then $\lim\limits_{...
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0answers
15 views

Closed operator in sobolev spaces

So suppose we look at the operator $S:H_0^1(0,1) \rightarrow L^2(0,1), \ u\mapsto u' $, where $H_0^1(0,1)$ denotes the closure of the infinitely differentiable functions compactly supported in $(0,1)...
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0answers
14 views

A smooth function whose derivatives satisfy smallness condition near a point

Let $f\in C^{\infty}\left(\mathbb{R}\rightarrow \mathbb{R}\right)$. Suppose that, for $n\geq 1$, we have $$\lim_{x\rightarrow 0}\frac{f^{\prime}(x)+a_{1}xf^{\prime\prime}(x)+...+a_{n-1}x^{n-1}f^{(n)}(...
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2answers
33 views

Does this series converge? $\sum_{n=1}^{\infty} \left( \sqrt{x+\sqrt x} - \sqrt{x-\sqrt x} \right)$

So I have following series: $$\sum_{x=1}^{\infty}\left(\sqrt{x+\sqrt x}-\sqrt{x-\sqrt x}\right)$$ and I have to find out if it converges. By Ratio Test it gave me 1, so no answer. Wolfram Alpha ...
0
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1answer
21 views

Archimedean spiral and length

The Archimedean spiral is given as $r=at,\, a>0, \, \text{for}\, t \in [0, \infty).$ I need to calculate the length of the first turn in the third quadrant. I have absolutely no idea how to compute ...
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0answers
17 views

Proof of Schwarz inequality in Baby Rudin (theorem 1.35)

I'm having trouble understanding the following step in the proof of Theorem 1.35 (Schwarz inequality) in Baby Rudin. The general idea of the proof is clear, however on this line: $\sum{|Ba_j - Cb_j|^...
4
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1answer
55 views

Find $f$ if $f(x)\leq x$ and $f(x+y)\leq f(x)+f(y)$ for all $x,~y\in \mathbb{R}.$

Find the formula of function $f:\mathbb{R}\to \mathbb{R}$ if: $$f(x)\leq x$$ and $$f(x+y)\leq f(x)+f(y)$$ for all $x,~y\in \mathbb{R}.$ Attempt. Identity function $I(x)=x$ satisfies the needed ...
2
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1answer
30 views

Convergence in Lebesgue-Bochner Space $L^{\infty}(0,T,L^1(\Gamma))$

Let $\Gamma$ be a compact $C^2$ manifold and suppose that $f_n$ is a non negative sequence of functions such that ${\vert \vert f_n \vert \vert}_{L^{\infty}(0,T,L^1(\Gamma))} \le C$ I am interested ...
0
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1answer
19 views

Existence of dominating, monotone decreasing square summable series

Suppose $\{a_i\}_{i\in\Bbb N}$ is an $\ell^2$ sequence. Does there necessarily exist a monotone decreasing sequence $\{b_i\}_{i\in\Bbb N}$ in $\ell^2$ with $b_k \geq |a_k|$ for all $k$?
1
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1answer
57 views

(Hint Needed) Real-Analysis Exam

So I'm trying to study for a qualifying exam and need a hint on the following problem. If someone knows of some useful lemmas and techniques, I would like to hear it! Let $(X,\mathcal{A},\mu)$ be a $\...
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0answers
34 views

Proof that $1^\alpha$ is dense on the unit circle

We take an irrational, real $\alpha$. I have got as far to show that $1^{\alpha}=e^{log(1^{\alpha})}=e^{\alpha ln(1)+i\alpha arg(1)}=e^{i\alpha arg(1)}=e^{i\alpha 2\pi k},k\in\mathbb{Z}$. After ...
1
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1answer
32 views

Measure Theory: Simple Measure Invariance Proof

Let M be a metric space, $f: M \rightarrow M$ be a measurable transformation and $\mu$ be a measure on M. Show that $f$ preserves $\mu$ if and only if $\int \phi d\mu =\int \phi \circ f d\mu$ for ...
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1answer
48 views

Does Cauchy second limit theorem work both ways? [duplicate]

I have been told, that Cauchy second limit theorem doesn't always work both ways - if the limit of $n$th root of sequence exist it doesn't immediately mean the limit of $a_{n+1}/a_n$ is equal to nth ...
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3answers
26 views
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3answers
30 views

Does $\partial B(x_0, r) \subseteq \{x \in X : d(x_0,x) = r \}$ hold in an arbitrary metric space?

Let $X$ be a metric space, and let $B(x_0, r)$ denote the open ball of radius $r$ centred at $x_0 \in X$. Does the statement $\partial B(x_0, r) \subseteq \{x \in X : d(x_0,x) = r \}$ hold true ...
1
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1answer
22 views

Pointwise convergence of $\sum\limits_{k=0}^\infty \frac{\sqrt{a+n}}{n^2}$

I need to prove that this series is pointwise convergent for $a>0$, but the ratio test, root test and the convergent minorant $\frac{1}{n^2}$ are inconclusive, so how would I be able to prove this? ...
1
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1answer
20 views

Problem in a step in the proof that a continuous function sends connected subsets into connected subsets.

Let $f: S\rightarrow M$ be a function from a metric space $S$ to another metric space $M$. Let $X$ a connected subset of $S$. If $f$ is continuous on $X$ then $f(X)$ i a connected subset of $M$. ...
1
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0answers
19 views

On a nonlinear regression problem

Consider the function $f\colon \mathbb{R}^2\to \mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data $\{(x_1^{(i)},...
1
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0answers
31 views

Interchange of limit and derivative of bounded convex function on half space

Let $u\in C^2((0, \infty)^2; \mathbb{R})$ be a convex and bounded and let \begin{align} f(y)\colon= \lim_{x\to \infty}u(x, y), ~~~y >0. \end{align} Then, can I guarantee that $f$ is ...
1
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1answer
27 views

Show that $f(x)=0,\;0\leq x<1/2,\; f(x)=1,\;1/2\leq x\leq 1$ is Riemann integrable over $[0,1]$ and find its value.

I want to prove that \begin{align} f:[0&,1]\to \Bbb{R}\\&x\mapsto \begin{cases}0,&0\leq x<1/2,\\\\ 1,&1/2\leq x\leq 1. \end{cases}\end{align} is Riemann integrable over $[0,1]$ ...
2
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2answers
31 views

Proving lipschitz continuity based on bounded gradient

I would like to prove that if $S$ is a non-empty convex subset, and $f:S \xrightarrow{} R $ is a convex and differentiable function, that the following holds true: $ |f(x)-f(y)| \le L\|x-y \|_{2} \...
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0answers
39 views

Showing convexity of a set in $\mathbb{C}^k$

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert Space and $T\in\mathscr{B(\mathcal{H})}$. Suppose $$\mu_k(T):=\left\{ \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ ...
1
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1answer
58 views

Prove or find a counterexample $\forall x>0: f(2x)-f(x)<g(3x)-g(2x)$, given information about $f, g$.

Let $f,g:(0, \infty) \rightarrow \mathbb{R}$ be two functions that satisfy the following for all $x>0$: $g'(x)>f'(x)$ and $f''(x)>0$. Prove or find a counterexample: $$ \forall x>...
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0answers
13 views

Using diagonal argument to prove existence of $P$-a.s. convergent convex combination for a sequence of processes $(X^k_t)_{t \geq 0}$?

There is a famous Lemma by Komlos often used in Probability Theory and Stochastic processes Lemma: For any sequence of nonnegative random variable $(X^n)_{n \geq 0}$ there exists a convex ...
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0answers
18 views

Solving a differential equation with generalized functions

We need to solve the equation $y' = 0$ (for the class of generalized functions where $y = (y,\varphi)$ and $(y',\varphi) = (y,-\varphi')$. The textbook I am currently reading uses the following ...
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0answers
64 views

Analysis with $a_{n+1}=f(a_n).$

$Hello, \;\;teacher...$ $f:R \to R, \;f(x)=ax^3+bx^2+cx+d \;$ $ f'(x) \ge 0\;$ for all x in R. There exists {$a_{n}$} such that $a_{n+1}=f(a_{n}),\;(n=1,2,3,...),\;a_{1}=t$ $g:R \to R, \;g(t)=\...
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2answers
32 views
0
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1answer
27 views

$\frac{d}{dx}:C^1([0,1])\to C([0,1])$ as a closed, unbounded operator

First recall that a (potentially unbounded) operator $T:D(T)\subseteq X\to Y$ is closed whenever $(x_n)_{n=1}^\infty\subset D(T)$ convergent to $x\in X_0$ and $(Tx_n)_{n=1}^\infty$ convergent in $X_1$ ...
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2answers
39 views

Showing that a sequence is convergent, and finding its limit [on hold]

Okay, so I am very confused about these particular types of problems. Now, I have seen analysis before, so I think that I am just making this way more difficult for myself. I need to show that a ...
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0answers
23 views

Showing holomorphy of integral

Let $f \in L^1$ with compact support. I want to show that $\hat f: \mathbb R \to \mathbb C, f(z):=\int_{\mathbb R}f(x)e^{-ixz}d\lambda(x), $ can be extended to a holomorphic function on $\mathbb C$ ...
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1answer
35 views

Vorticity for the Navier-Stokes equations

The definition that I know of is the vorticity $\omega$ is the curl of the velocity $u$. Now I'm reading a note saying $\omega$ is defined to be the $d\times d$ antisymmetric matrix: $$\omega = \frac{...
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0answers
8 views

derivative of surface integral over sphere

let $u$ be harmonic in the domain $U \subset \mathbb{R}^n$ and $B_R(0) \subset U$ and $u(0)=0, u\neq 0$. Let $0<r<R$. Define $a(r):= \frac{1}{r^{n-1}} \int_{\partial B_r(0)} u^2dS, b(r):= \frac{...
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2answers
26 views

Pointwise convergence of Fourier series of $x \mapsto x \sin(\pi x), x\in[-1,1]$

Consider $x \mapsto x \sin(\pi x), x\in[-1,1]$. The task is to comput the Fourier series of this function and determine the pointwise limit of the Fourier series $$c_0+\sum_{k=1}^{\infty}\Big(a_k\...
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0answers
42 views

Is there such a series?

Is there a series of the form $$\sum_{n=0}^{\infty}a_nx^n$$ such that it is convergent on $$-1\le x \le 1$$ and divergent on all other real number $x$?
0
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1answer
34 views

Proof of an equality

Show that $(1+r)(-(1+r)^{-1}-(1+r)^{-2}-...-(1+r)^{-T+1})'_r = \frac {1+r}{r^2} (1 - \frac{Tr+1}{(1+r)^{T}} )$
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2answers
39 views

Game design equation (Two variables(multiple two variables) and one Result (for every case)) how to make the Equation?

I tried to solve this using multiple methods but i couldn't figure it out. i don't know what to use in order to solve this basically i have an input x and x2 Heading when x = 1 and x2 = 1 Y = 45 ...
2
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1answer
88 views

Darboux continuity of the function $f(x) = \limsup_{n \to \infty} \frac{(x_{1}+…x_{n})^{2}}{n^{2}}$

Let $f : [0,1] \to [0,1]$ be a function that assigns to each $x \in [0,1]$ the following value: $$ x = 0/x_{1}x_{2}x_{3}... \hspace{0.3cm} \text{be the binary expansion of }x $$ define $f(x)$ to be : ...
7
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3answers
156 views

Find the sum $\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\sqrt{41+\cdots}}}}}$

Okay so this can be written as $$\sqrt{5+\sqrt{(5+6)+\sqrt{(5+6+8)+\sqrt{(5+6+8+10)+\sqrt{(5+6+8+10+12)\cdots}}}}}$$ Putting it as $y$ and squaring both sides doesn't seem to help, and I don't know ...
0
votes
0answers
24 views

Strong continuity of $\langle Au,v \rangle=\int u^3 v dx$

I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by $$\langle Au,v \rangle=\int u^3 v dx$$ is strongly (weak to ...
0
votes
1answer
40 views

Subset of a metric space is a metric space.

I have a question? Why is it that every subset of a metric space is a metric space? I mean what if the subset is the empty set, then it can't be a metric space, right? because a metric space is by ...
1
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1answer
51 views

Non-zero continuous function $f(x)$ such that $\int_0^1x^k (1-x)^{n-k} f(x)dx=0$ for every $k=0,1,…,n$ and $n$ is a non-negative integer.

Does there exists a non-zero continuous function $f$ such that $\displaystyle \int_0^1 x^k (1-x)^{n-k} f(x)dx=0$ for every $k=0,1,2,...,n$ where $n$ is a non-negative integer. EDIT: The problem is ...
0
votes
1answer
26 views

Question with regards to proof about diameter of a closed ball.

I have a question with regards to my understanding: Consider the following: If a,z $\in$ X and r,s$\in$ $\mathbb{R}^+$ then diam(B[a,r]) $\leq$ 2r. Where B[a,r] is the closed ball and X is a metric ...
1
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4answers
55 views

Trigonometric equation $ \cos{x} + \cos{\sqrt{2}x} = 2$

I can not find a good way to solve this rather simple-looking equation. $ \cos{x} + \cos{\sqrt{2}x} = 2$ I can see that 0 is a solution, but is there a good way of solving it for all the potential ...