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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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9 views

What's the relationship of $f$ and $h$ in ${\mathbb{E}_i \left[ f(g(x,i)) \right] = h(\mathbb{E}_i \left[ g(x,i) \right] )}$

Let there be a function $f$ and $h$ satisfying ${\mathbb{E}_i \left[ f(g(x,i)) \right] = h(\mathbb{E}_i \left[ g(x,i) \right] )}$ for an arbitrary 2-variable function $g : \mathbb{R}^n \times \...
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0answers
9 views

development of a function and set of convergence

I want to find the development of the function $ e^{x^2}$ and the set of convergence. Considering that: $e^x= \sum_{k=0}^n \frac {x^k}{k!} + R_n(x,0)$ and substituting $x$ with $x^2$ I can write $e^...
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0answers
29 views

Equivalence condition of Absolute Continuity

$f:I\to\mathbb R$ is continuous. Take finite number of subintervals $[x_1,y_1]\cup [x_2,y_2],...,\cup[x_n,y_n]=\mathcal K\subseteq I$, where $\sum_k\mu([x_k,y_k])=K$, i.e. the Lebesgue measure of all ...
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2answers
14 views

Show that $\Lambda(t)\le-\frac C2\Lambda'(t)$ if and only if $e^{2t/C}\Lambda(t)$ is nonincreasing

Let $\Lambda\in C^1([0,\infty))$ and $C>0$. Why does $$\Lambda(t)\le-\frac C2\Lambda'(t)\;\;\;\text{for all }t>0$$ hold if and only if $e^{2t/C}\Lambda(t)$ is nonincreasing in $t$? Is this just ...
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1answer
22 views

General rule for converting between sigma and pi notation

What is the general rule for converting from $\Pi$ to $\sum$ notation?
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0answers
28 views

Proof of Osgood's Uniqueness Theorem

I am working on a proof of Osgood's Uniqueness Theorem. It is a somewhat guided exercise in which I am given intermediate steps to prove that should eventually result in a proof of the greater theorem....
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3answers
35 views

Re-writing pi notation using exponentials

How might one show that the following is true by re-arranging the term on the left: $$\Pi_{r=0}^{n-1} e^\frac{2ri\pi}{n}=(-1)^{n-1}$$
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0answers
28 views

Re-write exponent using sums [on hold]

What are all the possible ways one might re-write the following using the property $e^ae^b=e^{a+b}$: $$e^ \frac{2ri\pi}{n}$$
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1answer
24 views

Sequence with convergent subnets but no convergent subsequences

We can regard a sequence as a special kind of net. But the definition of "subnet" is more flexible than that of "subsequence", so it's easy to find subnets of a sequence that aren't subsequences. In ...
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2answers
36 views

Multiplication proof by induction

Prove by induction that $$\sum_{r=0}^{n-1} r = \frac{1}{2}n(n-1)$$ For all $n\in\mathbb{N}$. This is straight forward. But how can it be used to show the following: $$\Pi_{r=0}^{n-1} e^\frac{2ri\...
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1answer
13 views

Is it possible to express accuracy knowing sensitivity and specificity?

I know the value of the sensitivity Se and the value of the specificity of Sp, they are equal to 78.65 and 90.00, respectively. I know nothing but this. Can I somehow of the equations, which in the ...
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1answer
25 views

Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}

Here we use infinity norm as metrics for $C[0,1]$. The professor claims that this set is compact. I can show this set is relative compact by Arzela Ascoli, i.e., for each subsequence there is a ...
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2answers
48 views

Continuity of $\cos(x), \sin(x)$ and $\exp(x)$

If we wish to prove that $\cos(x), \sin(x)$ and $\exp(x)$ are continuous, can we shorten the argument by saying that each one of them has a power series and all polynomials are continuous, provided ...
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0answers
35 views

Courant's “Introduction to Calculus and Analysis” vs “Differential and Integral Calculus”

I'm considering buying Courant's books, but I have seen two series of textbooks written by him on calculus and analysis. What are the differences between these two? If I want to have a deeper and ...
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2answers
27 views

Convergence of alternating sequence (Proof)

$S_{2n+1} \ -\ S_{2n-1} < 0 $ is what I'm struggling to understand I don't understand where the inequality comes from. I've tried hard to understand it but it's not coming to me unfortunately. ...
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0answers
24 views

How far can the mean of a normal distribution be from its arithmetic mean?

Let the arithmetic mean of the first $n$ terms of the sequence $S_n$ be $\mu_n$. Further assume that $0 < S_n < a$ for all $n \ge 1$. If $S_n$ is normally distributed with a mean $\mu_d$ and ...
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0answers
42 views

Replace the epsilon and delta in the definition of continuity, what kind of continuity do we get?

1) $f : X \to Y$ is a map. For all $x_0\in X$, if for every $\delta > 0$, there is $\varepsilon > 0$ such that $d_X(x_0, x) < \delta$ whenever $d_Y(f(x_0), f(x)) < \varepsilon$. 2)$f : X ...
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0answers
22 views

A difficulty in understanding multivariable Fermat theorem proof.

The theorem and its proof is given below : But I do not understand : 1- why clearly "$F$ is differentiable in this interval"? 2- why $F$ attains its extreme value at $x=a$? 3- why $F^{'}(a_{1})...
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1answer
20 views

Unitary Representations and the Peter-Weyl Theorem

Consider part II of the Peter-Weyl Theorem (see this wikipedia page for more information): Let $\rho$ be a unitary representation of a compact $G$ on a complex Hilbert space $H$. Then $H$ splits ...
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2answers
36 views

Evaluation of $\lim_{x\to\infty} \frac{1}{x}\int_1^x \frac{\log(1+s)}{(1+\log(s))\sqrt{1+s^2}}ds$ [on hold]

I really don't know how to start. I need an hint to get started. Thank you.
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0answers
17 views

What exactly is a partition, $P$, of a grid, $G,$ on a region, $R$, of $\mathbb{R}^{n}$?

I'm studying Jordan regions in analysis but because I don't have the text (I couldn't afford the book and can't find the pdf anywhere) I'm having trouble grasping one of the concepts. We learned about ...
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1answer
54 views

Calculate $\lim_{n\to\infty}\int_0^n\frac{\mathrm{d}x}{n+n^2\sin\frac{x}{n^2}}$

Calculate $$\lim_{n\to\infty}\int_0^n\frac{\mathrm{d}x}{n+n^2\sin\frac{x}{n^2}}$$, well first of all the function isn't monotonic, so I cant use the Lebesgue therem to go with the limit under the ...
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1answer
24 views

Confusion about convolution. Are the integrals equivalent?

The formula for a convolution is given by $$f*g(t,x)=\int_{\mathbb{R}}f(t-u)g(u) \ du. \tag1$$ My question is, is the following equally correct? $$f*g(t,x)=\int_{\mathbb{R}}f(u)g(t-u) du.\tag2$$ I'...
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1answer
12 views

uniformly convergence of heat equation

Problem Let $u(x,t) = \frac{e^\frac{-x^2}{4t}}{\sqrt{4 \pi t}} $ for $ t > 0, > x \in \mathbb{R} $. If a > 0, prove that $u(x,t) \rightarrow 0$ as $t > \rightarrow 0+$, uniformly for $x \...
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3answers
18 views

A difficulty in understanding a step in the proof of Thm. 11.5.6 in Petrovic.

The theorem and its proof is given below: But I do not understand how the last equality come from the previous one, could anyone explain this for me please?
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0answers
14 views

Convergence in Euler-Maclaurin

In the Euler-Maclaurin formula, we get an asymptotic approximation with the error term arbitrary small. However, I am confused about the convergence. The problem is that the constants in the terms ...
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1answer
43 views

calculate $f(500)=?$ and give an example of a function that meets the given conditions

$f : \mathbb{R} \to \mathbb{R}$ $\forall_{x\in \mathbb{R}} f(x) * f(f(x))=1 (*) \\ f(1000)=999$ calculate $f(500)=?$ So $f(1000)*f(f(1000))=1 \to 999*f(999)=1 \to f(999)= \frac{1}{999}$ Darboux. $\...
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0answers
9 views

Subharmonic function bounded by linear functions

Problem: Let $\Omega = \{(x,y) \in \mathbb{R}^2 : xy > 0\}$ and suppose $u \colon \overline{\Omega} \to \mathbb{R}$ is continuous and subharmonic on $\Omega$ and satisfies $$u(x,y) \leq \sqrt{x^2 + ...
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0answers
60 views

Can this theorem be extended thus?

Consider the theorem: If $f$ is a continuous and differentiable real-function over an interval $I$ and $f'(x)\ge 0$ on $I,$ then $f$ does not decrease over $I.$ Somehow, I expect this to be true ...
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2answers
65 views

What is wrong with the second proof?

Let $f$ be a real-valued function continuous on an interval $I$ and differentiable on every interior point of $I.$ Then if $f'(x)\ge 0$ everywhere inside $I,$ $f$ does not decrease over $I.$ There's ...
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0answers
10 views

Differentiation and Conformity of a Vector-Valued Function

Let $r=n$ and g be a differentiable transformation. Then g is called conformal if there exists a real-valued function $\mu$ such that $\mu (t) > 0$ and $\mu (t) * D$g$($t$)$ is a rotation of $E^n$ ...
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2answers
106 views

Walter Rudin's mathematical analysis: theorem 2.43. Why proof can't work under the perfect set is uncountable.

I found several discussions about this theorem, like this one. I understand the proof adopts contradiction by assuming the perfect set $P$ is countable. My question is if the assumption is $P$ is ...
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1answer
26 views

Let $d_1$ and $d_2$ be two equivalent metrics. If $A$ is an open subset of $(X,d_1)$ is $A$ an open subset of $(X,d_2)$?

Let $d_1$ and $d_2$ be two equivalent metrics. If $A$ is an open subset of $(X,d_1)$, is $A$ an open subset of $(X,d_2)$? Well, I have that if two metrics are equivalent then every sequence in $(X,...
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2answers
25 views

Is there a metric for which the open unit interval is complete?

Let, $I= (0,1)$ It is well known that $I$ is not a complete with respect to the Euclidean metric $(x,y)\mapsto |x-y|$. However, $(I,|\cdot|)$ is separable. Question: Can we find a metric $d: I\...
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2answers
23 views

Sequences and series and the alternating series test

Question: Suppose $(q_{n})_{n=1}^{\infty}$ is a sequence of real numbers such that Lim$_{n \rightarrow \infty} q_{n} = + \infty$. Show that we can find a sequence $(a_{n})_{n=1}^{\infty}$ such that $\...
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1answer
36 views

Area between curves.

I have to calculate area bounded by curves : $(x^3+y^3)^2=x^2+y^2 $ for $ x,y \ge 0 $. I tried to use polar coordinates, but I have : $r^4(\cos^6\alpha +2\sin^3\alpha\cos^3\alpha + \sin^6\alpha)=1$
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0answers
29 views

If $L_{n}(x)$ is Laguerre's polynom, prove extension $ e^\frac{x}{2} = 2\sum_{n=0}^{\infty} (-1)^n \frac{L_{n}(x)}{n!} $

If $L_{n}(x)= e^x \frac{d^n}{dx^n}(x^n e^{-x}) $ is Laguerre's polynom, prove extension $ e^\frac{x}{2} = 2\sum_{n=0}^{\infty} (-1)^n \frac{L_{n}(x)}{n!} $ I really don't know how resolve this. I ...
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1answer
34 views

Absolutely continuous function not differentiable on uncountable set

From real analysis one knows that an absolutely continuous function is differentiable a.e.. Is there a function showing that this statement cannot be made into "every AC function is differentiable ...
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0answers
8 views

Can determinant of Jacobian essentially bound from below implies locally bi-Lipschitz?

Let $F:\mathbb R^n\to\mathbb R^n$ be a Lipschitz map, so $DF:\mathbb R^n\to\mathbb R^{n\times n}$ is $L^\infty$ map. Suppose there is a $c>0$ such that $|\det DF(x)|>c$ for almost every $x\in\...
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1answer
47 views

A multiple choice question about $e^{-|x|}$

For $f:\mathbb{R}\rightarrow\mathbb{R}$, $f(x) = \exp (-|x|)$ Is the function a) bounded b) differentiable c) $f(\mathbb{R})$ is compact d) $f$ has a minimum According to the question only one ...
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0answers
8 views

The uniform continuity of functions with Banach space values. [duplicate]

Let $f\colon [a,b]\to\mathbb{R}$ be a continuous function. Since $[a,b]$ is compact, then by continuity of $f$ we also have that $f$ is uniformly continuous on $[a,b]$. Suppose now that $F\colon [a,b]\...
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0answers
36 views

Bayesian Statistics exercise?

I am having issues trying to solve this exercise in Bayesian analysis. The waiting time in minutes until being serviced by a phone call center follows an Exponential(λ) model, with E[y|λ] = 1/λ. Out ...
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0answers
31 views

Prove that if $f$ is a linear transformation then $Df(a)(x) = f(x) $

The question and its answer is given below but I do not know why $r = o$, could anyone explain this for me please?
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0answers
35 views

Use the given data to find an approximation for $f(x)$.

I have the following question in the book: But I do not know how to answer it, the only piece of information that my professor said is that a function differentiable means that it can be approximated ...
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2answers
17 views

Prove the equality for the function

could you help me with the problem? Let $f:R \to R $ is differentiable on $R$. Prove that exists $x \in [0;1] $ such that: $$(\frac4\pi)((f(1)-f(0) = (1+x^2)f'(x)$$ I tried to use mean value theorem: ...
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0answers
20 views

pointwise convergence against a continous function

Decide on the function sequences (an) n∈N, (bn) n∈N, (cn) n∈N and (dn) n∈N from R to R if these converge pointwise against a continuous function: $$ \begin{array}{ll}{\text { a) }} & {a_{n}(x) :=\...
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0answers
20 views

Could an arbitrary function in $\Bbb R^n$ get arbitrarily close to zero in an arbitrary neighbourhood?

Is the following the statement true: $$ \text{Let} \quad f:U \subseteq \mathbb{R}^m \rightarrow \mathbb{R}^n$$ be such that $$|f(x)|\gt0 \quad \forall x \in U$$ Then, there is an open neighbourhood ...
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1answer
17 views

exercise on convergent sequences

How can I prove that a sequence in a metric space converges to an element iff every sub-sequence has a sub-sub-sequence convergent to that element. Thank for the help!!
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2answers
44 views

If $f(a)=f(b)=0$ and $|f''(x)|\le M$ prove $|\int_a^bf(x)\mathrm{d}x| \le \frac{M}{12}(b-a)^3$

If $f(a)=f(b)=0$ and $|f''(x)|\le M$. Prove $$|\int_a^bf(x)\mathrm{d}x| \le \frac{M}{12}(b-a)^3$$ I have thought about that since $f(a) = f(b) = 0 $ there is $\xi$ such that $f'(\xi) = 0$. Then when ...
0
votes
1answer
31 views

Empirical Process and Azuma's Inequality

Given independent random variables $X_1,\dots,X_n$ and a family $\mathcal{F}$ of functions $f:\mathbb{R}\to [0,1]$, let $$Z(X_1,\dots,X_n)=\frac{1}{\sqrt{n}}\sup_{f\in \mathcal{F}}\left|\sum_{i=1}^n\...