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
51 views

Interesting question regarding elementary functions

I had this question at a test for a job interview and since and I didn't solve it. Some time later i still can't figure it out, so any insight is helpful. You need to write a function $f(x)$ such ...
3
votes
2answers
130 views

Estimate the arc length of the graph of a particular $\mathcal{C}^1$ function from $[0,1]\to [0,1]$.

Let $f:[0,1]\to[0,1]$ be $\mathcal{C}^1$ such that $f(0) = f(1) = 0$ and $f'$ is nonincreasing ($f$ concave). Show that the arc length of the graph is smaller than 3. I have a rather geometric ...
2
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1answer
34 views

The behavior of a solution to a parametrized equation

Show that the equation $$x\left(1+\log(\frac{1}{\epsilon \sqrt{x}}\right) = 1, x>0, \epsilon>0$$ has exactly two solutions if $\epsilon>0$ is small enough. In this case, let $x(\epsilon)$ be ...
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0answers
30 views

Uniqueness of solutions to ODEs for functions of complex domain

Given the ivp $$ \tag{1} \frac{df}{dz}=F(z,f), \hspace{1cm} f(z_0)=f_0 \hspace{1.5cm} f:\Bbb{C}\rightarrow\Bbb{C} $$ where $f$ and $F$ are complex valued and analytic, then does it folow that $(1)$ ...
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1answer
64 views

Question about Logistic Regression - 5

Can somebody give me a clear explanation about logistic regression in bold below? Logistic regression can be binomial or multinomial. Binomial or binary logistic regression deals with situations in ...
2
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1answer
96 views

For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) \sup_{x \in [i]}f(x) )$ is bounded

$\newcommand{\var}{\operatorname{var}}$ Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that $$|f|_{\var} = \sum_{i=1}^{\infty} \var_n f < \infty,$$ where $\var_n f = ...
3
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1answer
646 views

Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus ...
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1answer
87 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
1
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1answer
107 views

A sum for stirling numbers Pi, e.

In this identity $$1-e{}^{2} = \displaystyle \sum _{n=0}^{\infty } \frac{(-1)^n(\pi )^{2 n}} {(2 n)!}\sum _{k=0}^{2 n} (-1)^{k} S_2(2 n,1-k+2 n),$$ $S_2$ is a Stirling number of the second kind. ...
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1answer
41 views

A question regarding the inverse of continuous mappings.

If there is a continuous mapping $f:\Bbb{R^2}\to\Bbb{R}$, will $f^{-1}$ also be continuous? If there is a differentiable mapping $g:\Bbb{R^2}\to\Bbb{R}$, will $g^{-1}$ be ...
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3answers
1k views

Baby Rudin without knowing multivariate?

I have read Spivak's Calculus and it has went well. I didn't have any problem with the rigorosity of the book at all. Now, I have never had any experience in multivariate. I only have experience with ...
1
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1answer
131 views

Infinitely small functions!

Suppose that $\lim_{x\to a} f(x) =A$ and $\lim_{x\to a} g(x) =B$ and $B \neq 0$. Then $\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)} = \frac{A}{B}$. This is really ...
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0answers
39 views

Integral with function involving decimal expansion

For almost ever $x \in \mathbb{R}$, the decimal expansion $x = a_{0}.a_{1}a_{2}\cdots$ is unique. Let $A_{n}(x) = 1$ if the corresponding $a_{n}$ for $x$ is even and $A_{n}(x) = -1$ if $a_{n}$ is odd. ...
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1answer
42 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
2
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1answer
91 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
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1answer
56 views

Show that the set of 2 continuous functions is closed.

Let $f: \mathbb R \to \mathbb R $ and $g: \mathbb R \to \mathbb R$ be continuous functions. Show the set $ E = \{ x \in\mathbb R: f(x)=g(x)\} $ is closed. My approach A solution I found is the ...
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2answers
53 views

Show that the sequence converges to 0

Given a sequence $\{\frac{(-1)^n}{n}\}$ show directly from the definition that it converges to $0$. Definition of convergence of a sequence is: A sequence $\{p_n\}$ converges if for every ...
3
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1answer
70 views

Show that the limit of $ f(x)$ as $x\rightarrow 0$ is $0$

Show directly from the definition of a limit of a function that lim x->0 (x^(1/3) * sin(1/x)) = 0. The definition is The limit of f as x goes to p is q if ...
3
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3answers
291 views

Limit - Could you help me with it

Can you help me with this limit? What do I have to do? I'm lost. $$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$$ The solution given is $\dfrac{1}{2}$.
2
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1answer
60 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
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1answer
288 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
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4answers
170 views

Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge? [closed]

I need to solve $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$ Does this converge or diverge and why?
5
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1answer
268 views

Using paraboloidal coordinates

I have the 3-dimensional paraboloidal coordinates $$s_{\pm}=\sqrt{x^2+y^2+z^2}\pm z$$ $$\phi=ArcTan(y/x)$$ with the inverse transformation $$x=\sqrt{s_+ \cdot s_-}\cdot cos(\phi)$$ $$y=\sqrt{s_+ ...
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0answers
22 views

Continuity of the solution to a matrix PDE (mapping of a parameter to solution)

I'm considering the following PDE in $\Phi$: $\frac{\partial \Phi(t,s)}{\partial t}$ + $sR\frac{\partial \Phi(t,s)}{\partial s}$ + $\frac{1}{2} s^2 M \frac{\partial^2 \Phi(t,s)}{\partial s^2}$ + ...
1
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1answer
21 views

comparing mean values of a positive functions

Suppose that $D$ is a bounded open set in $\mathbb{R}^{n}$, and $A\subset B\subset D$ measurable sets. Let $f:D\rightarrow [0.+\infty)$ be a measurable function (or even locally integrable) and ...
2
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1answer
52 views

Holomorphic function with Taylor coefficients that tend to 0

Suppose $f$ is holomorphic on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$ and continuous on $\overline{\mathbb{D}}$. If we can write $F(z) = \sum_{n = 0}^{\infty}a_{n}z^{n}$ for $z \in ...
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2answers
3k views

Continuity of sin(1/x)

I'm dealing with the continuity of $\sin(\frac{1}{x})$. I think that I have a proof but I'm not sure if it's right! Here is my proof: We take the functions $g(x)=\frac{1}{x}$ and $h(x)=\sin(x)$, now ...
0
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1answer
58 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
0
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1answer
82 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
0
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1answer
71 views

The linearity of $D \beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathcal{L}(\mathbb{E_1} \times \mathbb{E_2},F)$

Let $\mathbb{E_1}, \mathbb{E_2}$ and $\mathbb{F}$ normed spaces of finite dimensions and $\beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathbb{F}$ is one bilinear function. Then $D \beta : ...
2
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1answer
62 views

If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
3
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1answer
159 views

How Find the sum $\sum_{n=2}^{\infty}\left(\sum_{j=1}^{n}\frac{1}{(2j-1)^2} \binom{2n}{n}\frac{1}{2^{2n}(2n+1)}\right)$

Prove or disprove $$I=\sum_{n=2}^{\infty}\left(1+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots+\dfrac{1}{(2n-1)^2}\right) \binom{2n}{n}\dfrac{1}{2^{2n}(2n+1)}=\dfrac{\pi^3}{48}-\dfrac{1}{6}$$ My ...
1
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1answer
70 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
4
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2answers
412 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
4
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0answers
195 views

Differentiability-Related Condition that Implies Continuity

I previously asked a related question here that I did not phrase as I intended. This is a revision of that question: It is a well-known fact that differentiability implies continuity. And, for ...
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0answers
35 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
2
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1answer
60 views

a question about Kolmogorov's Existence Theorem

I (a beginner in probability) have some confusions arising from problem 36.7 of the book "probability and measure" by Billingsley. It says that there is on the unit interval with Lebesgue measure no ...
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3answers
120 views

A diffeomorphism with negative Jacobian swaps the orientation?

Let C be a simple close oriented curve $C^1$ in $\mathbb{R}^2$ and let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a diffeomorphism such that $\forall (x, y) \in C$ it holds that the determinant of the ...
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0answers
134 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
1
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1answer
37 views

What is the difference between 1-dim.harmonic oscillator and 2-dim. harmonic oscillator?

I ask myself what exactly is meant with "2-dimensional harmonic oscillator". I only know the situation of a bob hanging on a bar... is that 1-dimensional or 2-dimensional?
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1answer
46 views

Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...
1
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1answer
96 views

Orthogonality of associated Legendre polynomials

Let $P_n(x)$ be the $n$-th degree Legendre polynomial. Let $k$ be a nonnegative integer less than or equal to both $n,m$. How to prove that $$ \int_{-1}^1 (1-x^2)^k D^kP_n(x) D^kP_m(x)\,dx = ...
3
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2answers
304 views

Why is the domain of convergence of a power series a perfect disk?

I've been going over power series in my Differential Equations class for approximating solutions, and one thing that's been fascinating me is the statement that there is a radius of convergence, ...
5
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1answer
95 views

proving that continuous function smaller than integral is identically zero

$f : [0,1] \to \mathbb{R}$ is continuous and $f \geq 0$. There is $C>0$ with $|f(x)| < C \int_{0}^{x} |f(t)| dt$ for all $x \in [0,1]$. (so $f(0)=0$) Is it true that $f = 0$? or is there any ...
2
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1answer
228 views

Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
2
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3answers
65 views

Show $ f(x,y)= 2x\sin(\frac{1}{\sqrt{x^2 + y^2}}) - x\cos(\frac{1}{\sqrt{x^2 + y^2}}) \sqrt{x^2+y^2}^{-1}$ not continuous at $(0,0)$

$f(x,y) = 2x\sin(\frac{1}{\sqrt{x^2 + y^2}}) - \frac{x\cos(\frac{1}{\sqrt{x^2 + y^2}})}{ \sqrt{x^2+y^2}} $ I'm a bit puzzled. The statement is obviously true if you plot the function. Formal: $\quad ...
1
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0answers
34 views

arbitrary reparametrization

Let $\alpha: (a,b)\rightarrow \mathbb{R}^n$ of class $C^{\infty}$ with $\Vert\alpha^{\prime}\Vert>0 $ then if $\{ k,m,n\} \subset \mathbb{R}_+$ there repametrizacion $\beta: (m,n)\rightarrow ...
0
votes
2answers
53 views

Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
2
votes
0answers
90 views

ODE with constraints

Given the ODE system $$\dot{x} = y \\ \dot{y} = \frac{1}{\alpha} (z - y)$$ where $\alpha > 0$ is a constant. How can I find a bound for $z$ depending on $x$ such that $\forall t ~x(t) \geq 0$ under ...
0
votes
1answer
96 views

An equality from Fritz John's paper

Prove: $\frac{\rho}{(4\pi)^2}\int_{|\xi|=1}d\omega_\xi\int_{|\eta|=1}f(x^0+r\xi+\rho\eta)d\omega_\eta=\int_{|r-\rho|}^{r+\rho}\frac{\lambda}{8\pi ...