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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6 views

Need Help Creating Symbols [migrated]

I am sorry I need to ask this, but how in the world can I insert mathematical symbols into my questions and answers such as lim, integrals, sums, etc. (I'm new)
2
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1answer
73 views

Real Analysis Delta Epsilon proof

Suppose $f: D \to \mathbb{R}$ and $x_0$ is a limit point of $D$. Prove that $\lim_{x\to x_0} = L$ if and only if for every $\varepsilon>0$, there exists a $\delta>0$ such that if $x$ is in $D$ ...
0
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1answer
37 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
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33 views

a real analysis question,I need to prove whether two sequences are equidistributed or not,really need some help!

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
2
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1answer
35 views

Extracting Bernoulli polynomials from their generating function

The generating function for Bernoulli polynomials is $$ \frac{te^{tx}}{e^t-1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}$$ The only way that I know of to get the coefficients out of this is to use ...
2
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1answer
19 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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0answers
36 views

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
2
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1answer
21 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
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0answers
38 views

Showing that the sequence of functions is not Cauchy

I need to show that $ g_n(x)=x^{1/(2n-1)} $ is not a Cauchy sequence in $C[-1,1] $ w.r.t. supremum norm. I tried to find the maximum of the difference of $g_n$ and $g_m$ by just differentiating but ...
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1answer
24 views

Prove a sequence of integrals converges to 0

Let $E$ be a set of finite Lebesgue measure in ${\bf R}$ and $\{a_n\}_{n \in {\bf N}}$ be a sequence of real numbers. Show that $\int_E \cos(nx + a_n) dx $ goes to 0 as $n \to \infty$. I tried ...
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2answers
45 views

If $\sum_{k=0}^{n}\binom nk=2^n$ then how is $2(\binom n0+\binom n2+\binom n4+…)=2^n$ [duplicate]

$$\sum_{k=0}^{n}\binom nk=2^n$$ then how is $2(\binom n0+\binom n2+\binom n4+...)=2^n$ ?? I don't think it could be because half of the members of the sum are chosen, that seems a bit intuitively ...
3
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0answers
28 views

Prove that the solution of an ODE can be prolonged to $\infty$

I need an help understanding some general techniques in ordinary differential equations. I've never attended a course on ODE, so I'm quite confused on the argument, but I'm trying to improve my ...
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1answer
24 views

Find a solution of the Laplace equation $-\Delta u=1$ with boundary condition $u=0$ on a spherical shell

Let $n\ge 2$ $B_\varepsilon$ and $\overline{B}_\varepsilon$ be the open and closed ball around $0$ with radius $\varepsilon>0$ in $\mathbb{R}^n$, respectively $R>0$, $\rho\in (0,R)$ and ...
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0answers
19 views

Weak convergence in $D^{1,p}(\mathbb{R}^n)$ and in $\mathcal{M}(\mathbb{R}^n)$

What it means: $$u_n\rightharpoonup u ~\text{weakly in }~ D^{1,p}(\mathbb{R}^n)\\ |\nabla(u_n-u)|^p\rightharpoonup\mu ~\text{weakly in}~ \mathcal{M}(\mathbb{R}^n)\\|u_n-u|^{p^{*}}\rightharpoonup\nu ...
9
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2answers
69 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
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0answers
17 views

On the hypothesis of the change of variable theorem

I´m studying the change of variable theorem for a function $f:\mathbb R^n \to \mathbb R$ and my teacher gave us the theorem as follows: Theorem: Let $f:A\subset \mathbb R^n \to \mathbb R$ be ...
2
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2answers
59 views

equation for the beta function

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function, proof the term: $(x + y)B(x + 1, y) = xB(x, y) \space\space \forall x, y > 0$ . Thanks in advance! ...
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2answers
25 views

In differentiability

Let $f(x,y)$ and $g(x,y)$ are differentiable functions in $x$ and $y$. Suppose $f(x,y) = F(g(x,y))$.My question, Is $F$ differentiable function?!.
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1answer
22 views

Convergence and value of improper integral

Show, that the integral $\int_0^\infty e^{-x^a}dx$ exists for all $a > 0$, and show that it's value is $\frac{1}{a}\Gamma(\frac{1}{a})$ where $\Gamma(x)$ is the gamma function. I've tried ...
2
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2answers
26 views

$\overline S = \{ 1-s: s \in S \}$. Prove that $\overline S$ is bounded below.

Let $S$ be a nonempty subset of $\mathbb R$ that is bounded above. Set $\overline S = \{ 1-s: s \in S \}$. Prove that $\overline S$ is bounded below. Prove that $\overline S$ is bounded below. I'm ...
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2answers
23 views

sum approximation of a Lipschitz-continuous function

Let $f: [0, 1] \to \mathbb{R}$ be a Lipschitz continuous function with a Lipschitz constant $L > 0$, meaning: $$|f(x) - f(y)| ≤ L|x - y| \space\space\space \forall x, y \in [0, 1]$$ For the ...
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4answers
71 views

Let $x \in \mathbb Q \setminus \{0\}$ and $y \in \mathbb R\setminus \mathbb Q$. Prove that $\frac{x}{y} \in \mathbb R \setminus \mathbb Q$

Let $x \in \mathbb Q\setminus \{0\}$ and $y \in \mathbb R\setminus \mathbb Q.$ Prove that $\frac{x}{y} \in \mathbb R \setminus \mathbb Q$ I saw this question in a basic analysis test but it confuses ...
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2answers
16 views

positive definite and 2-norm [on hold]

If A is positive definite, then is there always exist positive real number c which is independent of x such that holds? (There was a mistake in question. I changed it. sorry)
5
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2answers
249 views

Lebesgue non-measurable function

Can we give an example of Lebesgue non-measurable function, for which set $\{x: f(x)=C\}~\forall C\in\mathbb{R}$ is measurable? Thanks.
3
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1answer
18 views

Lower semicontinuous energy functional on compact space of Lipschitz functions

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space $A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ ...
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1answer
34 views

Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

I am studying Furstenberg's article Strict ergodicty and transformation of the torus and I'm stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, ...
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0answers
37 views

Limit of integral expression

I would like to evaluate this limit I have a function $A(x)$ such that it possesses a minimum at $x = x_0$ function and is strictly convex in a neighbourhood of a point $x_0$. Given a sufficiently ...
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0answers
11 views

Reconstructing a measure from its (absolutely continuous) marginals

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation ...
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33 views

Question about convergence in $L^2$ (revisited)

Yesterday I asked the folowing question: Question about convergence in $L^2$ which was answered negatively with a counterexample. Here, I wonder if one can find the right set to look at: Assume we ...
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2answers
51 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [on hold]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
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0answers
17 views

Elliptic regulartiy for nonlinear elliptic equations

Here is the question: Let us consider the Schrodinger type equation $$ \left\{ \begin{aligned} &-\Delta u + u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N \\ &u\in H^1(\mathbb{R}^N) \qquad ...
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0answers
23 views

Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
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3answers
47 views

Integral equal to Riemann Zeta Function

As part of a homework problem in Rudin, I need calculate $$ \int_{1}^{N} \frac{[x]}{x^{s+1}} \,dx$$ where $[x]$ is the floor function. Clearly $[x]$ has derivative $0$ everywhere but the integers. ...
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2answers
23 views

Calculate in terms of f and r only

Let $f :\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function and define $u: \mathbb{R}^n \rightarrow \mathbb{R}$ by $u(x) = f(r)$, where $r = \lVert x\rVert_2$, for all $x \in ...
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1answer
38 views

Computing $\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}$ [duplicate]

How do you compute $$\lim_{n\to\infty} \dfrac{(n!)^{1/n}}{n}\;?$$ I know that the answer is $\dfrac{1}{e}$ by plugging it into WolframAlpha, but I have no idea how to get there.
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1answer
31 views

Prove $f$ differentiable: $\|f(v)\| \leq M \cdot \|v\|^{N+1}$

Let $f:\Bbb R^n\to \Bbb R^m$. Assume that there exists $M\in\Bbb R$ positive such that for all $v \in \mathbb R^n$: $$ \|f(v)\|\leq M\cdot \|v\|^{N+1} $$ for some natural number $N\geq 1$. Prove $f$ ...
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0answers
14 views

Show the following equivalence of the sets

Could someone help me show the following relation: In analysis we know that for E subset of $R^n$ ploygonally connected $\rightarrow$ pathwise connected $\rightarrow$ connected. And I also know that ...
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2answers
48 views

A metric space on which every real-valued function is continuous

Let $(X,d)$ is a metric space such that every arbitrary function $f:X\to\Bbb{R}$ is continuous. Then which option is right? a) $X$ is bounded. b) Every subset of $X$ is closed. c) Every subset of $X$ ...
4
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1answer
68 views

$f$ convex and concave, then $f=ax+b$ [duplicate]

Let $f$ be a real function defined on some interval $I$. Assuming that $f$ both convex and concave on $I$, i.e, for any $x,y\in I$ one has $$f(\lambda x+(1-\lambda)y)=\lambda f(x)+(1-\lambda)f(y),\, ...
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1answer
23 views

existence of an improper integral

Let $f: [1, \infty ) \to \mathbb{C}$ be a continuous function with a bounded antiderivative $F(x)$ on $[1, \infty)$. Show, that the integral $$ \int_1^\infty \frac{f(x)}{x^s} dx$$ exists for each $s ...
2
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1answer
54 views

$f(x)$ is convex $\Leftrightarrow f'$ is monotonically increasing

How can I prove that $f(x)$ is convex on an interval if and only if $f'(x)$ is monotonically increasing ? • Let $\lambda \in (0,1)$ and $x,y \in I$. $f(x)$ is convex on an interval, if the ...
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3answers
50 views

Rudin exercise 8.21

Following is an exercise in Rudin's Principles of Mathematical Analysis (exercise 8.21). Let $$L_n = \frac1{2\pi}\int^\pi_{-\pi}\left|\frac{\sin(n+\frac12)x}{\sin\frac12x}\right|\,\text dx\space ...
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0answers
23 views

Locally Lipschitz complex function [on hold]

I want to study the property of being locally Lipschitz for the following function $$f(z)=\vert z\vert^\gamma z^2$$ with $\gamma\in\mathbb{R}$. Some hints to study this problem?
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1answer
50 views

How is this concluded in the following differential equation?

Question: Solve the differential equation $$t^2x''-2tx'+(t+2)x=t^6$$ Answer:$$x(t)=u(t)v(t); x'=u'v+uv' ; x''=u''v+2u'v'+uv'';$$ $$t^2(u''v+2u'v'+uv'')-2t(u'v+uv')+(t^2+2)uv= t^6 $$ then: ...
2
votes
1answer
46 views

Question about convergence in $L^2$

Assume we have a sequence of functions $\{f_n\}_{n\geq 0}\subset L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$, i.e. $$\lim_{n\to \infty} \int_0^1 |f_n(x)-f(x)|^2dx =0.$$ Is it then true ...
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0answers
61 views

How to solve $1 = \sum_{p \text{ prime}} x^{-p-1}$?

As the title says, I am trying to solve the equation $$1 = \sum_{p \text{ prime}} x^{-p-1}$$ and I'm not really sure where to begin. I got this from an exercise in a book and apparently there is a ...
4
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1answer
60 views

Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant. Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< ...
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0answers
13 views
+50

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + ...
2
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1answer
40 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
0
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1answer
26 views

Congruent Sub-Intervals with Reimann-Integrable Functions

Let $f:[a,b]\to\Bbb R$ be a Riemann-integrable function. Prove that for each $\sigma\gt0$ there exists a partition $\mathcal P$ of $[a,b]$ into congruent sub-intervals(that is, $x_{j}=a+{j(b-a)\over ...