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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23 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
0
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1answer
32 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
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2answers
57 views

If f is differentiable with a continuous derivative function, then the set of critical points of f is closed.

If f is differentiable with a continuous derivative function, then the set of critical points of f is closed. Is this a true statement? I'm kinda lost.
2
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1answer
59 views

Is sinus an unique function?

On $\mathbb{R}$, is sinus the unique $C^{\infty}$ function f with all is derivate and itself between -1 and 1 and also $ \frac{df}{dx}(0)=1 $ ?
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1answer
30 views

If a function has asymptote and the derivative does not, then its second derivative is not bounded

Let $f\colon\mathbb{R} \rightarrow \mathbb{R}$ be a function with second derivative everywhere in is domain. Prove that if $\lim_{x\rightarrow\infty}f(x)=b \in \mathbb{R}$ and ...
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1answer
58 views

2048 Tournament Word Problem [closed]

Problem: There was a 2048 tournament. And after the results were counted and were announced, winners got candy. The 1st place got 2 less than a third of candy, 2nd place got 4 less than half of the ...
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2answers
43 views

Prove ${x:d(x,p) < d(x,q)}$ is open in metric space $X$

$X$ is a metric space and $p \neq q$ $\in X$. I want to prove that $E=$ $\{x:d(x,p) < d(x,q) \}$ is open in metric space $X$. I think I can directly prove this by showing every point $x \in E$ ...
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1answer
24 views

$\left\Vert J(x)^{-1}\right\Vert<2\left\Vert J(x^*)^{-1}\right\Vert. $?

Could you please help me to prove this theorem: Suppose $J:{\bf {\rm R}}^m\rightarrow{\bf {\rm R}}^{n\times n}$ is a continuous matrix-valued function. If J(x*) is nonsingular, then there exists ...
3
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1answer
44 views

How to proove that a bijective transformation is NOT continous

I am having this transformation $f: \mathbb R \to \mathbb R$ $$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$ I've already prooved ...
1
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2answers
34 views

Convex function and expectation

I was wondering: if f is a convex function and X a random variable, what does E(f(X)) = f(E(X)) implies? Thanks a lot, David
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9 views

$0$ is an unstable equilibrium if $f$ is Lipschitz with certain conditions

Consider the following system: $$x'=-x^3-xy^2+2x^2y^2$$ $$y'=-2y+x^2y-3x^3y$$ There are two questions: The first one is to show that $(0,0)$ is uniformly asymptotically stable. The second question ...
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0answers
32 views

Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
0
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1answer
40 views

A compact set, which is not closed.

I'm looking for a compact set, which is not closed. I read somewhere that $Z^+$ are compact and not closed, but I don't understand why. Are there any other examples of compact sets that are not ...
3
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2answers
68 views

Suppose that f is integrable on $[a,b]$. Prove there is a number $x$ in $[a,b]$ such that $\int_a^x f = \int_x^b f$

Also, show by example that it is not always possible to choose $x$ in $(a,b)$ I've proven the first part (in the title), but I can't seem to think of a scenario for the second part. Perhaps my brain ...
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0answers
38 views

analyses exam bad translation [closed]

where is the function droves $\frac {x^3} {8} + \frac {3k}{8}x^2$, $k>0$ , for every $k>0$, the graph of the function $f(k)$ in the $1$st quadrant adjacent to the turning point has also a high ...
3
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1answer
41 views

Prove the energy is constant in a PDE?

I calculated the $$ \begin{align} \frac{dE(t)}{2\,dt} & = \int_\Omega u_tu_{tt}+DuDu_t+u^3u_t\,dx \\ & =\int_\Omega [u_t(u_{tt}-\Delta u)+u^3u_t] \, dx+\int_{\partial \Omega} u_t ...
2
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0answers
54 views

Covariant derivative for a covector field

In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine. Also, we ...
2
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1answer
28 views

Completeness, Compactness, Sequentially Compactness for $X = [0,1] \cap \mathbb{Q}$

$X = [0,1] \cap \mathbb{Q} \subset \mathbb{R}$ a metric space with the metric of $\mathbb{R}$. Show $X$ is not complete, is totally bounded, and is not sequentially compact. For completeness. I ...
0
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1answer
19 views

something about diffeomorphism

Suppose $A$ and $B$ are both open sets, and there is a diffeomorphism $g$ between them. My book says that the chain rule implies that $Dg$ is non-singular. I don't understand. Can anyone tell my why?
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1answer
32 views

calculate weak derivate of $|x-2|^2$

Let $u$ be a function with $u(x):=|x-2|^2$ on $I:=(-1,1)$. I want to test whether $u \in H^2(I) \backslash H^3(I)$. Let $\phi$ be in $C_0^\infty(I)$. Then: $T_u(\phi '') = \int_{-1}^1 |x-2|^2 ...
3
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1answer
32 views

A problem with equality in a inequality for convex function

Let $f:\rightarrow \mathbb R$ be a convex function on a convex subset $D$ of linear space $X$. Assume that for some pairwise disjoit $x_1,x_2,x_3\in D$ and some $t_1,t_2,t_3\in (0,1)$ such that ...
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0answers
21 views

Reference about $p$-homogeneous functions

I'm looking for a book about $p$-homogeneous functions. I am particularly interested in the associated (nonlinear) eigenvalue problems. However, a reference containing most of the known properties of ...
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2answers
29 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
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0answers
16 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
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0answers
12 views

positive integrable part implies downside integrable

Let $A: M\rightarrow GL(d)$ measurable where $(M, \mathcal{B},\mu)$ is a probability space, then are equivalent: $$\log^+\Vert A^{\pm1}(x)\Vert\in L^1(\mu)\Leftrightarrow \log^-\Vert ...
1
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1answer
23 views

Metric Spaces: closure of a set is the set of all limits of sequences in that set

I am studying metric spaces and got confused about many different ways of defining the closure. Let $S$ be a subset of $M.$ Then, the closure of $S$ is $ \{x \in M : \forall \epsilon>0, \ \ ...
7
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1answer
83 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
1
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1answer
35 views

Is function $f$ also uniformly continuous?

I've been thinking on the following problem lately: Let $(X,d)$ be a metric space and $f_1,f_2,...,f_n: X \rightarrow \mathbb{R}$ and $f(x) = \max\{f_1(x),f_2(x),...,f_n(x) \}$,$x\in X$ If the ...
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2answers
43 views
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0answers
52 views

Is $F$ Riemann integrable, then $F'$ Riemann integrable?

I know Newton-Leibnetz theorem: if $f \in \mathcal{R}[a ,b]$($f$ is Riemann integrable on $[a ,b]$), and if exists differentiable function $F$ satisfy $F'=f$ on $[a, b]$, then ...
1
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1answer
42 views

Geodesic equation

Assume that you have a parametrization of a surface $f:\Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3,(u,v) \mapsto f(u,v)$. Now if I have a curve defined by $g(t)=f(0,t)$. The geodesic ...
1
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1answer
36 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
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2answers
35 views

An other definition of continuity

I know that $$f: E\rightarrow F~\text{is continuous}~\Longleftrightarrow \forall V ~\text{open in}~F, f^{-1}(V)~\text{is open in}~ E$$ How to prove that $$f: E\rightarrow F~\text{is ...
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4answers
28 views

Find the limit of function using Taylor series

Good evening, I'm somehow stuck on solving some easy exercises : $$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
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1answer
68 views

Prove that limit goes to inf

Let $f:\mathbb R \to \mathbb R$ be such that $f(x), f'(x)$ and $f''(x)$ are all positive for each $x \in \mathbb R$. Apply the MVT to $f$ on each interval $[n,n+1]$ for $n=1,2, 3,\dots$ and show that ...
1
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1answer
26 views

interpolation properties of analytic paths

Assume we are given $n$ points in $\mathbb{C}^k$ can we find an analytic path $\phi:[0,1]\to \mathbb{C}^k$ passing through these $n$ points?
2
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1answer
22 views

Continuity and the closure

I want to prove that $$f:E\rightarrow F~\text{is continuous}\Rightarrow \overline{f^{-1}(B)}\subset f^{-1}(\overline{B}),\forall B\subset F$$ I say let $B\subset F$ and let $x\in ...
3
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1answer
53 views

Problem 3.14(e) in Baby Rudin

If $\{s_n\}$ be a sequence of complex numbers, define its arithmetic mean $\sigma_n$ by $$\sigma_n \colon= \frac{s_0 + s_1 \cdots + s_n}{n+1} \, \, (n = 0, 1, 2, \ldots). $$ Put $a_n = s_n - s_{n-1}$ ...
-7
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1answer
46 views

Is Infinity the limit? [closed]

Few weeks ago I heard the phrase "Infinity is not the limit!" in a movie. Today I thought about it form the math side. Why the infinity is not the limit? How can I prove it? Thanks.
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22 views

Convergence of a sequence of polynomials

Let $T_{n}(x)$ be the n-th Chebyshev polynomial. Is the sequence $T_{n-1}(x)-\cos(n\alpha)x$ convergent in $[-1,1]$?. With $\alpha \in [0,2\pi]$ fixed value. Weak or Uniformly. Thanks in advance.
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2answers
69 views

How to prove that $\lim_{x \to \infty} x = \infty$

Please refrain from using logic symbols, as I do not understand those. So, this is the question: $$\lim_{x \to \infty} x = \infty$$ Proving this using the actual formal definition of a limit. So ...
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3answers
38 views

No continuous transformation $f([a,b])= ]a,b[$

$ a,b\in\mathbb R$ with $a<b $. Now I want to show that there is NO continuous transformation $f: [a,b] \to \mathbb R $ with $f([a,b])= ]a,b[$ How can I proove that this transformation don't ...
0
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1answer
14 views

Are the roots of a smooth function, a smooth function?

Let $f(x,y)$ be a smooth function. It is given that for every $x$ there exists at least one $y$ such that $f(x,y)=0$. Is this possible to select one such $y$ for every $x$, such that the $y$'s are a ...
0
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1answer
19 views

Definition of limit points and isolated points

Definition of isolated points: If the points $p \in E \subset X$ is not a limit point, then $p$ is called isolated point of $E$. My question is.. Then all the points in open set is called interior ...
0
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0answers
28 views

Differentiation in Banach space

Let $F:\mathcal{B}([0,1],\mathbb{R})\ni f\mapsto||f||^2-f(0)\in \mathbb{R}$. How to prove that $F$ has derivative ? I know that I should use definition of Fréchet derivative, but I don't see what ...
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2answers
71 views

How do I mathematically explain this relationship?

At 40cm, 1.96N was produced At 46cm, 1.47N was produced At 56cm, 0.98N was produced At 80cm, 0.49N was produced. Is it inverse?
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2answers
68 views

Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Let $X$ be the vector space of all complex $n \times n$ matrices and define $T \colon X \to X$ by $Tx \colon= bx$, where $b \in X$ is fixed and $bx$ denotes the usual product of matrices. I know that ...
2
votes
0answers
39 views

Lebesgue measure is separable?

I would like to better understand the following definition: $(M, \mathcal {A}, \mu) $ a probability space is separable if there exists a countable family $ \mathcal {E} \subset \mathcal {A} $ such ...
6
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2answers
330 views

How can I pick up analysis quickly?

I have a 2-3 week recess from university for winter break. In this time, I would like to learn analysis, starting with Walter Rudin's Principles of Mathematical Analysis, and then, if at all possible, ...
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2answers
32 views

Do the properties of cauchy sequences carry over to converging functions?

Let $f$ be a function that is increasing and bounded above. Then $f$ is converging. I know that a converging sequence {$x_n$} is Cauchy. So does $f$ take on the definition of a Cauchy sequence too?