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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2answers
13 views

Let $(x_n)$ be a convergent sequence in a metric space. Show that $\{x_n: n \in \Bbb N\} \cup \lim x_n$ is a compact subset.

Hello I am trying to show the following. Let $(x_n)$ be a convergent sequence in a metric space. Show that $\{x_n: n \in \Bbb N\} \cup \{\lim x_n\}$ is a compact subset. I am proving by Heinel-Borel, ...
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14 views

Interval of which $S_1$ and $S_2$ are equal in volume.

Let the region $R$ lie above the curve, $y= 1 -ax^2$, and below $y = 1$, on $0\le x \le 1$, and $0 < a \le 1$. So $R$ lies entirely in the first quadrant. Let $S_1$ be the solid obtained by ...
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Inequality for a nested family of convex sets.

We consider a family of bounded open convex sets in $\mathbf{R}^n$ which we'll denote $\{G_h\}_{h \in (0,1]}$. They are non-decreasing in the sense $h_1 > h_2$ implies $G_{h_1} \supseteq G_{h_2}$. ...
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1answer
13 views

Uniform & Pointwise convergence of $\frac{\sin(nx)}{n}$ and $\frac{x\cos(nx)}{n}$

Here are some exercises I've found: Prove or disprove it is uniform and/or pointwise convergent for the following: $f_n(x)=\frac{sin(nx)}{n}$ defined on $(0,+\infty)$ $f_n(x)=\frac{x\cos(nx)}{n}$ ...
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30 views

Image of a ball by a diffeomorphism

Let $f:B_r(x_0)\subset \mathbb{R}^n\to\mathbb{R}^n$ a $C^1$ diffeomorphism with $\Vert f'(x)^{-1}\Vert \le M$ for all $x\in B_r (x_0)$ and $\vert f(x_0)\vert <\frac{r}{M}$. I'm wondering if I can ...
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2answers
25 views

Integral beginner proof

Question: Let $f:[0,\pi]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^{\pi} f(t)\sin(t)dt =0,$$ then the equation $f(x)=0$ admits a solution in $[0,\pi].$ What I've ...
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1answer
21 views

A supposedly obvious equation in Hörmander's The Analysis of Linear Partial Differential Operators I

To quote page 8. Another important example of a $C^1$ map is for two Banach spaces $U$ and $V$ the map $f$ taking an invertible element $T \in L(U,V)$ (T is a continuous linear map between U and V), ...
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Analysis - Prove a function that maps the unit Euclidean ball to R with bounded partial derivatives is uniformly continuous

I am stuck with this problem from my textbook and I cannot see the solution. I'm certain the solution is fairly simple and I am just missing the mark somehow. Any help would be appreciated. Suppose ...
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2answers
55 views

Help to sum these two series $\sin(x)-\sin(2x)+\sin(3x)…$

I am reading Euler's paper entitled "Subsidium Calculi Sinuum" and he wrote down some "sums" for these trigonometric series: \begin{align}S &= \sin(x)-\sin(2x)+\sin(3x)-\sin(4x)...
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1answer
37 views

Discontinuous functions in real analysis.

Let $D=\{x_{1},x_{2},...,x_{n},...\}$. Can we find a function $f:\mathbb{R}\rightarrow\mathbb{R}$, which is discontinuous at each point of $D$?
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74 views

Question regarding fixed point (integrals)

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^1 f(t)\,\mathrm{d}t =\frac{1}{2},$$ then $f$ has a fixed point in $[0,1]$, i.e. $\exists x_0\in [0,1]$ such ...
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1answer
19 views

Problem with Mean Value Theorem and Limit of Derivative

Consider the function $f(y)=\sin(y)$. Then on the interval $[\sqrt x,x]$ by MVT $$\exists t\in(\sqrt x,x) \text{ s.t. } f'(t)=\frac {f(x)-f(\sqrt x) } {x-\sqrt x}. $$ Then by taking $x\to \infty \...
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45 views

Prove that AM-GM-HM recursive function converge to the same value.

I am trying to do this question Let $0<a_1 < b_1 < c_1$ $$ a_n = HM(a_{n-1}, b_{n-1}, c_{n-1})$$$$ b_n = GM(a_{n-1}, b_{n-1}, c_{n-1})$$$$ c_n = AM(a_{n-1}, b_{n-1}, c_{n-1})$$ Prove that ${...
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How to prove the following inequality about vector inner product? [closed]

For any vectors $\boldsymbol{x}(t)$, $\boldsymbol{y}(t)$ with time-varying elements and any positive definite matrix $\boldsymbol{Z}$ of appropriate dimension, the following inequality holds: $$\pm 2\...
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1answer
36 views

proving supremum of sequence is infinity

For $j \in \mathbb{N}$ We have $$\|a_n\| \geqq \ln\frac{j^2+1}{j+1}$$ gives $\displaystyle \sup_n\|a_n\|=\infty.$ Hence $(a_n)_{n=1}^\infty$ does not converge. Q1 How to prove $\displaystyle \sup_n\|...
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2answers
81 views

How do I prove $[0,3)$ is an open set in $([0,\infty),d)$ standard metric?

This is my first analysis class and I don't know if I'm on the right path. I'm thinking about proving if every point in the interval has a neighborhood contained in it, but the $[0,\infty)$ confuses ...
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0answers
43 views

Where does $a_i = \prod_{t=1}^j |t^\alpha - i^\alpha|$ reach its the minimum

Let $j$ be a positive integer, and $(a_i)_{1\leq i\leq j}$ be a sequence such that $$a_i = \prod_{t=1}^j |t^\alpha - i^\alpha|, \quad\, \alpha \geq 2.$$ Find $i$ such that $a_i$ is the minimum. For $\...
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29 views

Showing function is smooth

$f:S^1\times S^1\rightarrow S^1$ such that $f(z, w)=z^mw^n$ for some integer $m$ and $n$. Then why $f$ is smooth? By definition a smooth function is differentiable infinitely. So in order to prove $f$ ...
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1answer
34 views

A function whose argument is a function

I need to find the most precise method to calculate the total fuel that an aircraft uses to fly from A to B at a certain altitude. The rate of fuel consumption fuelRate (kg/km) is a function of the ...
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1answer
24 views

Could this Beta density related integral be simplified

I encounter the following integral, where $F^{-1}(u)$ is the quantile function for $\mathcal{N}(0,1)$ distribution: for $k \in\mathbb{N}$: $$\int\limits_{1/2}^1 F^{-1}(u) u^k(1-u)^k\,\mathrm{d}u$$ ...
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47 views

Show that $f$ is differentiable and that $f′$ is continuous.

Let $\Omega$ be the set of all invertible linear operators on $\mathbb{R}^n$. $\Omega$ is an open subset of $L(\mathbb{R}^n)$ and the mapping $f(A)= A^{-1}$ is continuous on $\Omega$. Show that $f$ is ...
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13 views

Convert from cartesian to polar or spherical coordinates in order to show functions belong to $L^p$ space or Sobolev space

Q1. Let $f(x) = \frac{1}{\log x} $ and $\Omega = (0,\frac{1}{2})$. Show that $f \in L^p (\Omega)$ for any $p \geq 1$ and $f' \in L^p(\Omega)$ only for $p=1$. Q2. Show that $g(x) = \log( \log ( 1 + {|...
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3answers
65 views

Using definition of limits

Let $c∈\mathbb{R}$ and let $f:\mathbb{R}\setminus{c}\rightarrow\mathbb{R}$ be a function such that $f(x)>0$ for all $x∈\mathbb{R}$. Use the definition of limits to prove that $$ \lim_{x\to c}f(x)=\...
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0answers
37 views

Determine at which points $f(x)=[x]$ has a limit

Let $[x]$ denote the largest integer that is less than or equal to x. Determine at which points $f(x)=[x]$ has a limit. I know that I'm suppose to use the epsilon definition of a limit to do but I'm ...
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37 views

a closed subset $F$ of $Q^{C} \cap [0,1]$ has continuous function

CAn we find a closed subset $F$ of $Q^{C} \cap [0,1]$ such that $I_{Q^{C} \cap [0,1]}$ (where the indicator function on the irrationals in $[0,1]$) is continuous on $F$ and $m([0,1]-F) < \epsilon$ ...
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0answers
37 views

Polynomial in Hilbert space definition

I ve been looking around but haven't really found anything. How does one define polynomials in a Hilbert space $\mathcal{H}$. So a function $F:\mathcal{H}\rightarrow \mathcal{H}$ such that $F$ is a &...
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14 views

Subsequence converging to any k in N

How can I construct an example of a sequence {${x_n}$} such that there exists a subsequent converging to k for any k in N? Can someone give me some hints of an informal construction of this sequence?
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2answers
50 views

How to find the Taylor series of function $f(x)=\frac{a\left(1-x^2\right)+b\sqrt{x}\left(1+x\right)}{2\left(1-x\right)^2}$, at $x=0$

How to find the Taylor series of function $f(x)=\frac{a\left(1-x^2\right)+b\sqrt{x}\left(1+x\right)}{2\left(1-x\right)^2}$, at $x=0$. I know that the Taylor series of the functions: $\frac{t}{1-t}=\...
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0answers
19 views

Confusion in study of derivative of a piecewise function

I have some confusion about the study of the derivatives in $x=0$ of this piecewise function $$f(x)=\begin{cases} \exp\left(-\frac{1}{x^2}\right), \ \text{if} \ x>0 \\ 0, \ \text{if} \ x \leq 0\end{...
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1answer
24 views

There is a metric on N that is not equivalent to the discrete metric

Show that there is a metric on N that is not equivalent to the discrete metric. Any hint or answer would be appreciated. Thank you in advance.
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1answer
19 views

Is the supremum of a Càdlàg function equal to the supremum over a dense set? [closed]

suppose that f is a Cadlag function on [0,1]. Let D be a dense subset of [0,1] is it true that the supremum over D is equal to the supremum over [0,1]? It seems like if I remove the points {0} and {1} ...
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0answers
35 views

Uniform convergence over composition functions

Good evening; A friend who is studying real analysis asked me for help in this problem, I would like any clue of solution or complete solution, I also accept bibliographical references. Let $f: [0,1] \...
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0answers
9 views

Possible generalization of averages

I was thinking if it's possible to generalize the idea of averages, like we have already the arithmetic, geometric and harmonic averages and I realized that they share some properties and I'd like to ...
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3answers
34 views

Proving that $\frac{\text{d}\vec{v}}{\text{dt}}=\frac{\text{d}|\vec{v}|}{\text{dt}}\hat{v}+\frac{\text{d}\hat{v}}{\text{dt}}|\vec{v}|$.

We were taught the following equation on a physics lecture: $$\frac{\text{d}\vec{v}}{\text{d}t}=\frac{\text{d}|\vec{v}|}{\text{d}t}\hat{v}+\frac{\text{d}\hat{v}}{\text{d}t}|\vec{v}|$$ where $\vec{v}$ ...
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1answer
38 views

Bound on Truncated Alternating Harmonic Series

I'm trying to prove the following inequality $$\sum _{n=1}^{2N}\frac{(-1)^n}{n}+\log (2)<\frac{1}{4N+1}.$$ Most of the traditional inequalities I've seen for harmonic numbers aren't tight enough to ...
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2answers
50 views

inequality of $(ax+by+cz)^n$

The Cauchy Swartz inequality says $$ (ax+by+cz)^2 = (a^2+b^2+c^2)(x^2+y^2+z^2). $$ I found that there is a kind of analogous inequality for $(ax+by+cz)^n$ $$ (x+y+z)^n \leq 3^n(x^n+y^n +z^n). $$ if I ...
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2answers
39 views

Limit ($x \to \infty$) of integral of positive and continuous function (in $\mathbb{R}$ and centered at $0$) always finite?

For all functions $f(x)$ that are positive and continuous everywhere in the real numbers. If $\lim\nolimits_{x\to \infty}f(x)=0$, is $\lim\nolimits_{x\to \infty}F(x)$ convergent for all possible $f(x)$...
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1answer
48 views

Experimental Analysis of an Algorithm - How to prove that the graph is $O(n\log n)$?

This question is probably stupid, but I've been trying to figure this out for hours and I still couldn't find anything about it. Probably I'm just too lost. So basically, I'm analysing an algorithm by ...
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1answer
119 views

Prove that $a_n\to 0$ but $\sum a_n = \infty$.

Let $f:[-1,1]\to\mathbb{R}$ be a $C^2-$function with $f(0)=0$, $f^{\prime}(0)=1$ and $f^{\prime}(x)>1$ for all $x>0$. Let $a_1\in (0,1)$ and define recursively $a_{n+1}$ by equation $f(a_{n+1})=...
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1answer
39 views

What is wrong with this theorem about closed and bounded non empty subsets of $\mathbb{R}$?

I am self-studying metric spaces from the book by Satish Shirali. One of the theorems from the chapter on "Topology of Metric Spaces" states: Let $F$ be a nonempty bounded closed subset of $\...
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0answers
12 views

Solving inhomogeneous Airy equation

Hey I have following ODE: $$-i s_0 Re \omega_{2,0}'+A i \beta Re u_{2,0}'+A x_2 i \alpha_0 Re \omega_{2,0}'=\frac{\partial^2 \omega_{2,0}'}{\partial x_2^2}$$ The inhomogenity $u_{2,0}'$ is given as a ...
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0answers
14 views

Extension of divergence free vector fields

Let $ \Omega \subset \mathbb{R}^n $ be a bounded domain with smooth boundary. Let $ v = (v_1,\dots, v_{n}) \in W^{1,1}(\Omega) $ such that $ \text{div}(v) = 0 $ (note that $ v $ is not identically ...
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0answers
55 views

Evaluating $\int_{1}^{\infty} \frac{1}{\lfloor{x}\rfloor!}dx$

I thought the improper integral $\int_{1}^{\infty} \frac{1}{\lfloor{x}\rfloor!}dx$ converge, while the textbook says it's not. Of course, $\lfloor{x}\rfloor$ is the greatest integer function. Here is ...
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1answer
26 views

Finding roots of function with exponential.

Let $f(x)=e^{-x}$ and $g(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{(x-1)^2}{2}}+\frac{1}{\sqrt{2\pi}}e^{-\frac{(x+1)^2}{2}}$. To complete an exercise I had to show there exists $x_1<x_2$ such that $g(x)\le ...
2
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1answer
45 views

Contraction function on $\mathbb{Q}$ that does not have a fixed point

We know the contraction principle holds on the complete metric space. I am thinking about cases where the space is not complete, such as $\mathbb{Q}$. Could anyone help giving an example of a ...
0
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1answer
30 views

Complement of a Vitali set

We know that it holds for the Vitali set that $V\subset [0,1]$ and that Vitali set is unmeasurable. I know that the complement $\mathbb{R}\setminus V$ is also not measurable. However, I was wondering ...
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1answer
38 views

How to conclude that $\sum_{n>e^t} f_t(n)<e^{-t}?$

Given a Poisson process $N_t$. If we set $$f_t(n):=P(N_t=n)=\frac{e^{-t}t^n}{n!}.$$ We know $\sum_{n\ge 0} f_t(n)=1$. My question is that how to conclude that $$\sum_{n>et} f_t(n)<e^{-t}?$$ This ...
3
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1answer
43 views

Bounded operator such that image of ball is $(1-\varepsilon)$-dense is surjective.

This is not a homework problem, I found it in some notes and I am curious on how to prove it. The statement is as follows: Let $T:X\to Y$ be a bounded linear operator between Banach spaces. If there ...
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1answer
45 views

Series expansion of the power 1/4

There exists an inequality that links the expression $$\sum_{k\ge0}(-1)^{k+j} (2\mu)^{k}\binom{1/4}{k+j-1}$$ where $\mu <1/2$ and $j\ge 2$ with the expression $$\sum_{k\ge 0 }(-1)^k (2\mu)^k \binom{...
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1answer
22 views

Regularity of heat equation with inhomogeneous Dirichlet boundary

Does anyone know a reference for the regularity theory of the Heat equation with inhomogeneous Dirichlet boundary condition: $$\begin{cases} u_t(t,y) = u_{yy} (t,y) & (t,y)\in [0,T]\times \mathbb{...

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