Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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7 views

a question about sequence and series. prove $ \lim_{n \to \infty}( nlnn)a_{n}=0$?

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} (nlnn)a_{n}=0$$ I totally ...
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7 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
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7 views

quasi-convexity of a function

I would like to know whether the following function is at least quasi-convex. Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plot ...
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1answer
32 views

Does every inner product space has an orthogonal basis?

It is proved that every inner product space has a basis $W$, but I am not sure if every inner product space has an orthogonal basis? It is known that every inner space has a maximal orthogonal set ...
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1answer
32 views

Proving the divergence of a sequence [on hold]

Let $\{a_n\}, \{b_n\}$ be sequences of positive numbers. Set $$c_n = \frac{a_n b_n}{b_{n+1}} - a_{n+1}$$ Suppose that $\sum_{n=1}^\infty \frac{1}{a_n}$ is divergent and $\limsup_n c_n < 0$. Prove ...
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0answers
10 views

How do you use R to find the box counting dimension of a two dimensional set of data? [on hold]

Here's what I have tried, but not sure if it is accurate fd.estim.boxcount(cbind(Sham_INV_1_1712$MeanArtPress,Sham_INV_1_1712$RBF),plot.loglog=TRUE, plot.allpoints=TRUE, nlags="auto") any help on ...
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21 views

Two measures having the same moments

Let $\mu_{1}$ and $\mu_{2}$ be two finite Borel measures supported on $[0, 1]$. Suppose $\int_{\mathbb{R}}x^{k}\, d\mu_{1}(x) = \int_{\mathbb{R}}x^{k}\, d\mu_{2}(x)$ for all $k = 0, 1, 2, \ldots$. ...
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1answer
32 views

Showing that a set is not infinite in measure

Suppose $f_n \geq 0$ for all $n \geq 1$, $f_n \to f$ a.e. on $[0, \infty)$ and there exists a constant $M>0$ such that $$ \sup\limits_{n} \int_{E} f_n(x)dx \leq M \mu(E)$$ for each measurable ...
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1answer
23 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
2
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1answer
27 views

Uniformly bounded derivative implies uniform convergence

Let $f_n$ be a sequence of differentiable functions on $[a, b] \subset \mathbb{R}$. Suppose $\lim_{n \rightarrow \infty} f_n(x) = f(x)$ exists for all $x \in [a, b]$, and the derivatives $|f_n'(x)| ...
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0answers
23 views

Applying the duality between infimum and supremum

Studying the textbook 'Methods of Modern Mathematical Physics' by Reed & Simon, I encountered the following problem - the details are not important but I did not see how to otherwise introduce my ...
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1answer
41 views

Compute limit of a function

Compute: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$ WolframAlpha tells me it's 1/6. Any nice idea how to rewrite that expression? Thanks!
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1answer
44 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
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0answers
54 views

$\int_0^\infty f(x) \; dx < \infty$ implies $\lim_{x \rightarrow \infty} x f(x) = 0$. [duplicate]

Let $f$ be non-negative, monotone decreasing such that $$\int_0^\infty f(x) \; dx < \infty$$ Show that $$\lim_{x \rightarrow \infty} x f(x) = 0.$$ I have the following solution, but wonder if ...
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1answer
36 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in ...
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0answers
14 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
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50 views

Bound for this integral

Using the fact that $$\sqrt{(1+y^2)} - \sqrt{(1+x^2)} \geq \frac{x}{\sqrt{1+x^2}}(y-x)$$ for each $x,y\in \mathbb{R}$. We need to show that $$L(k)- L(h) \geq \int_a^b \frac{h'}{\sqrt{1+{h'}^2}} ...
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1answer
17 views

L2 convergence, subsequence

I'm sai. I have a question about $L^{2}$ convergence. Let $U \subset \mathbb{R}^{d}$ be open. Suppose $u_{n} \in C_{0}^{\infty}(U), n\in \mathbb{N} $ satisfies the following assertion: For any $v ...
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1answer
28 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
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21 views

fix $b \gt 1$. Prove the following statements : (Rudin page 22, question no 6)

Fix $ b \gt 1$. (a) If $m$, $n$, $p$, and $q$ are integers, $n \gt 0$, $q \gt 0$ and $r=\dfrac{m}{n}=\dfrac{p}{q}$, prove that $\{b^{m}\}^{\dfrac{1}{n}}=\{b^{p}\}^{\dfrac{1}{q}}$. Hence it makes ...
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37 views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Let $f,g : \mathbb{R} → \mathbb{R}$ be $L_1$-functions. Set $h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y).$ Prove that $h(x)$ is defined a.e., $h ∈ L_1(\mathbb{R})$ and $∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1.$ So I ...
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0answers
14 views

Boundedness of a certain function defined on a closed bounded real interval

Let $I:=[a,b]$ be a closed bounded real interval , $f: I \to \mathbb R$ be a function such that for every $x \in I$ , $\exists \delta_x>0$ such that $f(x)$ is bounded $ \forall x \in ...
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1answer
48 views

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$. Prove that lim $a_n/a_{n+1} = z_0.$ [duplicate]

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$ of a function which is analytic in $\mathbb{C}$ \ ${z_0}$, $z_0\neq 0$ and has only a simple pole at $z_0.$ Prove that $lim_{n ...
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1answer
47 views

Does $f$ necessarily have infinte oscillations?

If $f:(a,\infty)\to\mathbb{R}$ is differentiable and $\lim_{x\to \infty} f(x)$ does not exists($\neq \pm \infty$),then is it necessary that $f$ have infinitely many oscillations and therefore ...
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0answers
16 views

Let $f\in C[0,1]$. Compute $\lim_{t\rightarrow \infty} \frac{1}{t} \log \int_0^1 \cosh(tf(x)) dx$ [duplicate]

Let $f\in C[0,1]$. Compute $\lim_{t\rightarrow \infty} \frac{1}{t} \log \int_0^1 \cosh(tf(x)) dx$. Can anyone give me a hint for this type of problem? I don't know where to start. Thank you!
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4answers
65 views

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$. Here are some of my ideas: Also by applying Mean Value theorem, we know that ...
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0answers
14 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
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40 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty$, right? Now, ...
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3answers
43 views

Determined or not?

the function $\dfrac {2x}{3x-\sqrt{x} }$ is not derterined for values of $x$ equale or samller than zero, though when I take the limit $ \lim_{x \to 0^+} \dfrac {2x}{3x-\sqrt{x} }$ the output is zero ...
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1answer
19 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
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1answer
64 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
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2answers
48 views

Supremum of $f_1(x)=1$ and $f_2(x)=x$

I'm trying to understand the supremem of a sequence of functions so I came up with a trivial case as follows - Let $(f_n(x))$ be a sequence of functions with $n$ having a value of either $1$ or $2$. ...
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2answers
119 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
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2answers
171 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
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1answer
42 views

Space that can not define an inner product

Normally, we can see lots of example of Hilbert space, I am wondering if there is a space we can't define an inner product on it?
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0answers
47 views

Derivatives of $\exp\left(-\sqrt{x^2 + y^2 + z^2}\right)$

For which $\ell, m, n \in \mathbb{N}$ are the mixed derivatives $$\frac{\partial^{\ell + m + n}}{\partial x^{\ell} \partial y^{m} \partial z^{n}} \exp\left(-\sqrt{x^2 + y^2 + z^2}\right)$$ in the ...
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1answer
9 views

Geometric interpretation of parallelogram law determines the inner product

We know that if we have parallelogram law,we can determine an inner product, I want to know what is geometric interpretation behind it since the inner product gives an geometric structure of the ...
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1answer
52 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
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0answers
58 views

Is the repeating decimal $0.999… \in \,(0, 1)$? [duplicate]

Is the repeating decimal $0.999.... \in (0, 1)$? It seems like it can't be as $0.999...$ is defined as being equal to $1$.
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1answer
26 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
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2answers
38 views

Intuition for sequences of functions?

A sequence $(a_n)$ of real numbers can be thought of as a function that maps $\mathbb{N}$ to $\mathbb{R}$. The supremum of this sequence, if it exists, will be some $k \in \mathbb{R}$. A regular ...
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1answer
38 views

The inner product determines the structure of the space

The Hilbert space employs inner product to determine the geometric structure,e.x. the angle. But I couldn't understand how. For example, the key structure of Euclidean space $\mathbb{R}^2$ is that it ...
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1answer
55 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier stuff. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see ...
0
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2answers
47 views

Maximum of $\frac{x(1-x)y(1-y)}{1-xy}$ over $[0,1] \times [0, 1]$?

I wish to find the maximum of $$\frac{x(1-x)y(1-y)}{1-xy}$$ over $[0,1] \times [0, 1]$, and where the maximum is achieved. I try to compute the gradient and set to zero, but this is not working out. ...
3
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1answer
65 views

Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
2
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0answers
26 views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$ I want to say that the condition is that $E$ is finite. This ...
1
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0answers
24 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
5
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1answer
74 views

sequence $a_n =\sum_{k=0}^{n-1}(-1)^k\frac{(n-k)^k}{k!}$ converge or diverge?

How check that sequence $a_n =\sum_{k=0}^{n-1}(-1)^k\frac{(n-k)^k}{k!}$ is converge or diverge ?
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2answers
68 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
-1
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2answers
28 views

Determine all values of $p \in R$ such that the sequence is in $l^2$. $\left\{\frac{1}{\sqrt{k}(\ln k)^p}\right\}_{k=2}^{\infty}$ [on hold]

Determine all values of $p \in R$ such that the sequence is in $l^2$ $$\left\{\frac{1}{\sqrt{k}(\ln k)^p}\right\}_{k=2}^{\infty}$$