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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37 views

Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
1
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2answers
118 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
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0answers
78 views

Calculate the limit of an integral

Prove that $$\lim_{\lambda\rightarrow\infty}\int_1^2\frac{\cos\lambda t}{t\sqrt{t-1}}\text{d}t=0.$$ I have tried differentiating the integrand w.r.t $\lambda$ but it doesn't look promising.
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1answer
140 views

How to show that if möbius transformation has an inverse, then it is injective?

Let $f(z)$ be möbius transformation. How to show that if möbius transformation has an inverse, then it is injective? I mean why don't you use this definition to show injectivity of möbius ...
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1answer
117 views

Energy of a solution of the wave equation.

Let $f\in\operatorname{C}^2(\mathbb{R})$ and $g\in\operatorname{C}^1(\mathbb{R})$ be function whose support are compact. By considering a solution $u$ of the problem $$ \begin{cases} u_{tt}(x,t) - ...
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2answers
96 views

Prove that the set $A$ is measurable and find its Lebesgue measure.

Let $A ⊂ [0, 1] × [0, 1]$ be the set of points $(x, y)$ with decimal representations $x = 0.x_1x_2 ..., y = 0.y_1y_2 ...$ such that $x_ny_n = 5$ for all $n ∈ \mathbb{N}.$ Prove that the set $A$ is ...
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1answer
79 views

Show that $f_n(x_n) \to f(x).$

Let $f_n$ continuous,so that $f_n \to f$ uniformly.Let $x_n$ be a sequence of real numbers,such that $x_n \to x$.Show that $f_n(x_n) \to f(x).$ $f_n$ continuous and $f_n \to f$ uniformly,so $f$ is ...
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1answer
44 views

Fourier coefficients of a measure and absolute continuity

A relative of a theorem of Peyriere (found in a 1997 paper of Klemes and Reinhold, "Rank One Transformations with Singular Spectral Type") says that if $\mu$ is a Borel probability measure on $S^1$ ...
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1answer
31 views

Asymptotic expansion of $z^{-x}$

Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form ...
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1answer
28 views

Uniform distance

Uniform distance: $|f-g|_A= \sup \{ f(x)-g(x), x \in A\}, f(x)-g(x) \geq 0 \forall x \in A$ Find the uniform distance of $f(x)=x, g(x)=1 \forall x \in \mathbb{R}$ My attempt is to take cases for ...
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1answer
66 views

If a bounded function is integrable on each interval $[a,1]$, then it is integrable on $[0,1]$.

Let $f:[0,1] \to \mathbb{R}$ bounded,such that $\forall a $ with $0<a<1$, $f$ is integrable at the interval $ [a,1]$,show that f is integrable at $[0,1]$. As $f$ is bounded, $\exists ...
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1answer
75 views

A calculus problem

Question: Suppose that $u(x,t)$ is continuous, together with its first and second partial derivatives; suppose that $u$ and its first partial derivatives are periodic in $x$ of period $1,$ and ...
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0answers
46 views

Summability of subsets of a summable index set.

Summable means that for some index set $A$, $\{S_F\}$ is a convergent real net, where $S_F = \sum_{\alpha\in F}x_{\alpha}$ and the $F\subset A$ are finite sets partially ordered by inclusion. I need ...
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1answer
99 views

Fitting noise with Levenberg–Marquardt algorithm

I've got a sample of noise from a microphone, and I'm trying to fit a curve to the data using the Levenberg–Marquardt algorithm. However, I can't seem to find a good starting function. I've tried a ...
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3answers
102 views

Evaluate the limit $\lim_{t\rightarrow\infty}\left(te^t\int_t^{\infty}\frac{e^{-s}}{s}\text{d}s\right)$

$$\lim_{t\rightarrow\infty}\left(te^t\int_t^{\infty}\frac{e^{-s}}{s}\text{d}s\right)$$ I have no idea where to start. Any help will be appreciated!
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1answer
47 views

Why is the antiderivative equal to $F(x)=|x|-1$

We have: $$f(x)=\left\{\begin{matrix} -1, & -1 \leq x \leq 0 \\ 1, & 0<x \leq 1 \end{matrix}\right.$$ Why is the antiderivative equal to $F(x)=|x|-1$? Do we have to find the integral ...
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1answer
30 views

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$ (a) Prove that there exists $y ∈ [0,1]$ such that $m(E^y) ≥ \frac{1}{4}.$ ...
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0answers
139 views

If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function ...
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0answers
68 views

General solution of ODE

please what is the general solution of $$-(p(t)u')'+q(t)u=0$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in L^1((0,+\infty))$ Thank you
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4answers
1k views

Prove that every irrational numbers can be approximated by rational numbers.

Prove that every irrational numbers can be approximated by rational numbers. How can I do it? Ok, I admit. I heard it, I thought it is to be true. And I was a kid. Now I when I think about it, I ...
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2answers
312 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
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1answer
77 views

Pseudodifferential operators and amplitudes

I am studying psudodifferential operators on $\mathbb{R}^n$. Let $U\subset \mathbb{R}^n$ an open subset. A function is $b\in C^\infty(U\times U\times U \times \mathbb{R}^n)$ is an amplitude of order ...
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1answer
35 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
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3answers
63 views

Show that series converges

Show that if $ \{ p_n \} $ is a Cauchy sequence, then it has a subsequence $ \{ p_{n_k}\} $ such that the series $ \sum_{k=1}^\infty b_k $ converges, where $ b_k = d(p_{n_k}, \, p_{n_{k+1}}) $. My ...
2
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1answer
125 views

The real numbers as a completion of the rationals

The real numbers are the completion (i.e. Cauchy sequences modulo equivalence) of the rational numbers. I want to define the real numbers this way, but without using uniform spaces. The problem is ...
2
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1answer
47 views

Do I need the First Mean Value Theorem For Integals?

Let $f$ be a continuous function on $[0,1].$ Suppose that $\int_0^1 f(x) g(x) dx = 0$ for every integrable function $g(x)$ on $[0,1].$ Prove that $f(x) \equiv 0$ on $[0,1]$ This proof is easy to ...
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1answer
32 views

The spherical mean of $h(x,y) = x$.

The spherical mean of a function $h : \mathbb R^2 \to \mathbb R$ is given by $$ \frac{1}{2\pi r} \int_0^{2\pi} h(x + r \cos(\theta), y + r \sin(\theta)) d \theta $$ Now I want to compute the ...
2
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1answer
35 views

$\int_{X}e^{c|f|}\, d\mu < \infty$ implies$\|f\|_{L^{p}(X)} \leq Cp$

Suppose $(X, \mu)$ is a measure space with finite measure and $f: X \rightarrow \mathbb{R}$ a measurable function and there exists a $c > 0$ such that $\int_{X}e^{c|f|}\, d\mu < \infty$. Why ...
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1answer
49 views

Big-O Analysis: Max Bounded by the Sum

I have been asked to show that: $$ \mathcal{O}(Max\{ f(n), g(n) \}) = \mathcal{O}(f(n) + g(n)) $$ I have seen explanations of similar problems, but this is the first time I have encountered the ...
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2answers
90 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
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1answer
62 views

Show that series in Cauchy Sequence

Let $a_n = d(p_n, p_n+1)$ for $n = 1, 2,\cdots $. Show that if the series $\displaystyle \sum^{∞}_{n=1} a_n$ converges, then $\{p_n\}$ is a Cauchy sequence. My Approach: I thought of using the ...
1
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1answer
34 views

Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

Let $(X,A,μ)$ be a $σ$-finite measure space with $μ(X) = ∞.$ Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$ I could easily do this if I ...
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1answer
75 views

if $\lim_{n\to\infty}(4a_{n+2}-4a_{n+1}+a_{n})=2014$ prove the $\lim_{n\to\infty}a_{n}$ is exist and find the value

Let sequence $\{a_{n}\}$ such $$\lim_{n\to\infty}(4a_{n+2}-4a_{n+1}+a_{n})=2014$$ show that $$\lim_{n\to \infty}a_{n}$$ exist and find the limit value. Now I use an ugly method to solve this. I use ...
18
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1answer
345 views

Swapping signs in analysis proofs

Under what minimal conditions are the following interchange of operations valid (including a question of existence, if not given explicitly)? \begin{align*} \lim \int f_n&=\int \lim f_n \\ ...
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2answers
80 views

Why is $\{(x, y) \in [a, b] \times [a, b]: f(x) \geq y\}$ a Borel set?

Suppose $f: I = [a, b] \rightarrow I$ is a Borel function. Why is $\{(x, y) \in [a, b] \times [a, b]: f(x) \geq y\}$ a Borel set? Since $f$ is a Borel function, for any open set $U$, $f^{-1}(U)$ is ...
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3answers
186 views

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
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0answers
66 views

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ [duplicate]

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ Prove that $f = 0$ a.e. Not sure how to start this question. Any ...
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3answers
177 views

Q: $\lim_{n\to \infty}\left(1 + \frac{1}{n}\right)^{n} = e$

I am having difficulty with the proof $$\lim_{n\to \infty}\left(1 + \frac{1}{n}\right)^{n} = e$$ in Rudin's Principles of Mathematics. In particular, the last few steps. The proof is as follows: ...
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1answer
18 views

Problem with equation

I have problem with showing that Left side is equal to right side. $$ \frac {1-e^{2i\pi yKr/2^n}} {1-e^{2i\pi yr /2^n}} = e^{i\pi (K-1)r/2^n}\frac{\sin(\pi yKr/2^n)}{\sin(\pi y r / 2^n)} $$ My only ...
4
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1answer
140 views

What are the $n$th roots of the identity function?

What are all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f^n=I$ where $f^n$ denotes the composition $f\circ f\circ f\dots \circ f$ of $f$ with itself $n$ times, and ...
3
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2answers
82 views

uniform convergence of a functional sequence

Is this sequence of functions $$f_n(x)=n^3x(1-x)^n$$ converges uniformly for $x\in[0,1]$. I need some help on this.
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1answer
113 views

surface integral (curl F n ds)

Let $F$ be a vector field and let $n$ be normal vector of the closed surface $S$. Then show that $$\iint_S \mathrm{curl} \ F \cdot n\ ds=0. $$ I need help on this exercise.
4
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1answer
195 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
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4answers
306 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
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1answer
80 views

Neighbourhood of a matrix

Sometimes I find definitions which say that something happens in a neighbourhood of a matrix. For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if ...
2
votes
1answer
61 views

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Let $f$ and $g$ be integrable functions on a measure space $(X,Σ,μ).$ For each $t ∈ \mathbb{R},$ consider the sets $F_t =\{x∈X :f(x)>t\}, G_t =\{x∈X :g(x)>t\}.$ Prove that $\int|f − g| = ...
1
vote
1answer
31 views

Proof of Cauchy's functional equation for rational arguments

We have thesis that for every $c\in\mathbb{Q}$ every additive function has form of $f(x)=cx$. In the proof we're showing that $f(nx)=nf(x)$. Then we're supposed to replace $nx$ by $\frac{1}{n}x$. Why ...
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0answers
26 views

$\gamma(t)$ is not asymptotically stable unless $\int_0^T \nabla \cdot f(\gamma(t))dt \leq 0$

Let $f \in C^1(E)$ where E is an open subset of $\mathbb{R^n}$ containing a periodic orbit $\gamma(t)$ of $x'=f(x)$ of period $T$. Then $\gamma(t)$ is not asymptotically stable unless $$\int_0^T ...
0
votes
1answer
25 views

Formula to minimise the output of 2 variable

I am working on a problem where I have 2 variables, and I am trying to create a formula that highlights the output if and only if both variables are low. My variables are time and price change. If ...
1
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1answer
51 views

Does the pre image of a open interval is a open interval, if the function is absolutely continuous and non decreasing?

Assume that $u:[0,1]\to \mathbb{R}$ is a absolutely continuous (A.C. for short), non decreasing function. Suppose that $u(0)=\alpha$ and $u(1)=\beta$. Take any open interval $J\subset [\alpha,\beta]$. ...