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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2answers
67 views

Lagrange multiplier - find minima of a function satisfying a condition

I am supposed to find the points (x,y,z) satisfying the condition $x^2+2y^2-z^2-1=0$ that are the closest to origin (0,0,0). So basically, the idea was to find the minima of $$\Lambda(x,y,z,\lambda) = ...
2
votes
1answer
78 views

Hilbert space question

Let $\{x_n\}$ be a sequence of pairwise orthogonal vectors in a Hilbert space $H$. Prove that the following are equivalent: a) $\displaystyle\sum_{n=1}^\infty \|x_n\|^2<\infty$ b) ...
2
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2answers
461 views

Intuitive explanation for Jacobian matrix having max. rank

As part of a longer definition I came across the following: $f: X\subseteq\mathbb{R}^m \rightarrow \mathbb{R}^n$ ($X$ open, $m<n$) with $rank(Df_{x}) = m$ for all $x \in X$. My question is now if ...
2
votes
2answers
83 views

analysis question

Suppose $f(x) \in \mathbb{R}[x]$ is such that $\operatorname{deg}{f(x)} = 2011$, then $\exists \: c \in \mathbb{R}$ such that $f(c) = f'(c)$. How can I prove/disprove the above statement. Any hints?
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0answers
167 views

A question about weak lower semicontinuity

Let $\Omega \subset \mathbb{R^{n}}$ be a bounded domain and $u , u_j \in H^{1}(\Omega)$ such that $u_j \rightharpoonup u$ in $H^{1}(\Omega)$ $$ F_{1}(u) = \int_{\{ u > 0\}} \dfrac{1}{2}\langle A_1 ...
5
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3answers
358 views

Continuous bijection from $\mathbb{R}^{2} \to \mathbb{R}$

Can anyone give an example of a continuous bijection from $\mathbb{R}^{2} \to \mathbb{R}$
0
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1answer
53 views

Few questions about the following sequence .

If i have a sequence $u_n=u(x+n)$ , $u_n\in C_c^\infty R$, is it bounded in $W^{1,p} (R)$ ? and is it true that for no $q\ge1$ does there exist a subsequence converging strongly in $L^q(R)$ . As ...
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0answers
51 views

Continuous only at rational pts [duplicate]

Possible Duplicate: Set of continuity points of a real function I'm studying continuous in analysis class. I have a question. That's simple! Is there a function that is continuous only at ...
2
votes
1answer
176 views

Proving that $\sum_ {n=1}^{\infty}{\frac{(-1)^n}{\sqrt{n}}} $ converges

Prove that the following series converges: $$\sum_ {n=1}^{\infty}{\frac{(-1)^n}{\sqrt{n}}} $$ $$\frac{1}{\sqrt{n+1}} > \frac{1}{\sqrt{n}}$$ $$\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0$$ ...
14
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2answers
701 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere. If there exists $c \in \mathbb{R}$ ...
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3answers
174 views

Circle coordinate system using two perpendicular vectors

If we have two vectors $A$ and $B$ such that the inner product is $(A,B)=0$ (i.e. they are orthogonal in $R^2$). How can we describe any point along a circle created by rotating these vectors? ...
0
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1answer
713 views

What does it mean to say a boundary is $C^k$?

I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please. And also need some explanation on how to straighten boundary ?
4
votes
1answer
84 views

For a reflexive Banach space, we have $\left\Vert x-y\right\Vert =\left\Vert x+F\right\Vert _{E/F}$

The problem is: Let E be a Banach space and $F\subset E$ be a closed linear subspace. Prove that for every $x \in E$ there exists $y \in F$ such that $\left\Vert x-y\right\Vert =\inf\left\{ ...
0
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1answer
104 views

Does such a bijective mapping exist?

It is a problem I encountered when working on shape optimization(mech eng, not math one). Consider two connected sets $A$ and $B$ in $\mathbb{R}^d$ (it would be nice if $d$ can be chosen arbitrarily, ...
1
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1answer
155 views

Positive Linear Functionals on Von Neumann Algebras

Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal ...
1
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1answer
356 views

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different ...
3
votes
2answers
87 views

Is this $\left|\left(\frac{a}{b}\right)^n-\left(\frac{a}{b}\right)^{n-1}\right|$ bounded?

Let $0.5<a<1$ and let $b=1-a$. Let $n\in \mathbb{N}$. $\left|\left(\frac{a}{b}\right)^n-\left(\frac{a}{b}\right)^{n-1}\right|\le C$. Is ...
0
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2answers
89 views

Analyze the convergence behavior of $\sum_{k=0}^{\infty}\frac{x^{2k}}{2^{2k}}-\frac{x^{2k+1}}{3^{2k+1}}.$

Analyze the convergence behavior of the following series: $$\sum_{k=0}^{\infty}\frac{x^{2k}}{2^{2k}}-\frac{x^{2k+1}}{3^{2k+1}}.$$ I came across this problem as I was preparing for an exam. It is ...
5
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2answers
136 views

$L^p$ space question

Assume $(X,\mathcal{M},\mu)$ is a measure space and for some $1\leq p<\infty$, $1\leq q<\infty$, $L^p(\mu)\subset L^q(\mu)$. Prove there is a constant $C>0$ so that $\|f\|_q\leq C\|f\|_p$ ...
2
votes
1answer
1k views

Radius of convergence of Power Series!

Given two power series $$\sum_{n=0}^{\infty} a_nx^n, \sum_{n=0}^{\infty} b_nx^n$$ with convergent radius $R_{1}$ and $R_{2}$ respectively. Suppose $R_{1}<R_{2}$,now what about the convergent radius ...
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1answer
101 views

Extra condition (or proof) for $\int{f_{n}}$ converging to a limit, assuming $f_{n}\to f$ pointwise a.e .and $\int{f}<\infty$

I am trying to show given $f_{n}\to f$ pointwise a.e. and $\int{f}<\infty$, it follows the sequence {$\int{f_{n}}$} has a limit. But I am not sure if extra condition is required. Can anyone give me ...
4
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1answer
136 views

From injective map to continuous map

Let $X$ and $Y$ metric spaces, $f$ is an injective from $X$ to $Y$, and $f$ sets every compact set in $X$ to compact set in $Y$. How to prove $f$ is continuous map? Any comments and advice will be ...
2
votes
1answer
259 views

Tensor product of Hilbert Spaces

I am following this link under "definitions" I need to see why the suggested inner product on the pre-Hilbert space $H_1$ tensor $H_2$ is well defined. Recall that the fundamental tensors are a ...
2
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1answer
73 views

Classification of Type 1 factors

In the proof of this theorem, which says all of the type 1 factors (factors with minimal projections) are isomorphic to $B(\ell^2(I))$ for some $I$, I want to know a few things: The supposed ...
1
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1answer
195 views

Calculating the integral $ \int_{e}^{\infty} e^{-\frac{1}{2} (nx)^2 }dx$

Calculate the following improper integrals $ \displaystyle{ \int_{e}^{\infty} e^{-\frac{1}{2} (nx)^2} dx , \quad \int_{e}^{\infty}x^2 e^{-\frac{1}{2} (nx)^2} dx \quad ,\int_{-\infty}^{\infty} ...
1
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2answers
210 views

Numerical integration given a derivative of a function of two dependent variables

I want to solve the following equation of an integral valued function: $Q = \int_{0}^{x_p}f(t_p,x)dx$ for some particular $x_p$ at a fixed time $t_p$, given some known $Q$ and an initial $f(0,x)$. ...
4
votes
1answer
269 views

The behavior of a density function at infinity

Give $f$ the density function of a random variable. Does it follow that $$\lim_{x\rightarrow \pm\infty}xf(x)=0?$$ I really appreciate it if someone can give me a clue.
2
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3answers
161 views

Measure theory limit question

Let $(X, \cal{M},\mu)$ be a finite positive measure space and $f$ a $\mu$-a.e. strictly positive measurable function on $X$. If $E_n\in\mathcal{M}$, for $n=1,2,\ldots $ and $\displaystyle ...
2
votes
1answer
274 views

Fubini theorem question

Let $f$ and $g$ be Lebesgue measurable nonnegative functions on $\mathbb{R}$. Let $A_y=\{x:f(x) \geq y\}$ Let $F(y)=\int_{A_y} g(x)dx$. Prove $\int_{-\infty}^\infty f(x)g(x)dx=\int_0^\infty F(y)dy$. ...
-1
votes
3answers
124 views

how to solve this inequation?

Let $(x,y)\in[0,1)\times[0,1)$ cum $x^2+y^2<1$. Are there any $\mu\geq\lambda>0$ such that ...
1
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2answers
479 views

Rudin's “Principles of Mathematical Analysis” Example 1.1

This probably involves some very easy algebra, but I am stuck and would appreciate some help. Walter Rudin's Example 1.1 on page 2 of Principles of Mathematical Analysis includes the following ...
1
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4answers
129 views

What exactly is a sequence? (Construction of reals)

I am working through an Analysis textbook and came to the construction of the reals using Cauchy Sequences. I understood the proof more or less but far from completely / intuitively. I have no ...
2
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1answer
82 views

Analysis Questions-Schwartz Space

Given Schwartz Space, I'm interested on proving the following: The function $f(x) := P(x)\exp[-\alpha|x-a|^2 ] $ where $P$ is a polynomial in $x_1,\ldots,x_N$, $\Re(\alpha)>0$ , and $a \in ...
8
votes
1answer
313 views

Ideals in $C(X)$

Let $X$ be a Hausdorf Compact topological space. Please help me to show, for the purpose of understanding an example in some of my lecture notes, that the closed ideals in $C(X)$ are of the following ...
0
votes
1answer
133 views

How does one show $\tan(nz)$ converges uniformly to $-i$ in the upper half plane?

How does one show $\tan(nz)$ converges uniformly to $-i$ in the upper half plane on compact sets? I tried writing out $\tan(nz)$ in terms of exponential functions but I got nowhere.
2
votes
1answer
85 views

Numbers such that their sum of $k^{th}$ powers is always zero.

Let $x_1, \ldots,x_n$ be complex numbers such that for any $k$, $$ \sum_{i=0}^n x_i^k = 0.$$ I'd like to show that this implies $x_1 = x_2 = \cdots = x_n = 0.$ I was suggested to use this strategy. ...
3
votes
2answers
177 views

convergence of multiple series

I have a following series $$ \sum\frac{1}{n^2+m^2} $$ As far as I understand it converges. I tried Cauchy criteria and it showed divergency, but i may be mistaken. When i calculate it in matlab or ...
1
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1answer
48 views

Prove :If $u(x)\in \displaystyle{H^{l}(R)}$, then $u(x)\in \displaystyle C^{l-1}(R)$

I come across this problem in my functional analysis book.Prove: If $u(x)\in \displaystyle{H^{l}(R)}$, then $u(x)\in \displaystyle C^{l-1}(R)$,$\displaystyle\lim_{x\to|\infty|}D^{\alpha}u=0 ...
4
votes
1answer
237 views

Does $f_{n}(x)=(1+x^{2n})^{1/2n}$ converge uniformly on $\mathbb{R}$.

Does the sequence of functions defined by $f_{n}(x)=(1+x^{2n})^{1/2n}$ converge uniformly on $\mathbb{R}$. For testing uniform convergence i know if the sequence $x_{n} = \sup \: \{ |f_{n}(x)-f(x) | ...
2
votes
3answers
101 views

An inequality problem

I can't work out that inequality problem. If $x\ge0$, $y\ge0$, how to prove that $$ 1+x+y+xy\leq(x+1)\ln(x+1)+e^y? $$ I tried taylor expansions for $$ln(x+1)$$ and $$e^y,$$ I also tried $$ ...
2
votes
1answer
870 views

Proving that a convex function is Lipschitz

I am trying to show that if $f$ is convex in $(a,b)$ it is Lipschitz in $[c,d]$ where $a \lt c \lt d \lt b$. Here's what I have so far: Let $t_1,t_2 \in \mathbb{R}$ such that $a \lt t_2 \lt c \lt d ...
2
votes
1answer
85 views

Bilogarithmic function

I have shown that the power series $$\sum\limits_{i=1}^\infty \frac{x^n}{n^2} = -\int\limits_{0}^x \frac{\log(1-t)}{t}dt$$ for $x\in (-1,1)$. How can I show that this is also true for the boundary: ...
3
votes
0answers
71 views

Calculus on complete fields of positive characteristic

Are there complete fields of positive characteristic with non-trivial absolute value? What does calculus on them looks like? I'm aware that they have to be non-archimedean, and that the bulk of ...
1
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2answers
186 views

The series $\sum_{n=1}^\infty \frac{n}{n+1}$

$$\sum_{n=1}^\infty \frac{n}{n+1}$$ I have $\displaystyle\lim_{n \to{+}\infty}{\frac{n}{n+1}}=\displaystyle\lim_{n \to{+}\infty}{\frac{\frac{n}{n}}{\frac{n}{n}+\frac{1}{n}}}=\displaystyle\lim_{n ...
1
vote
1answer
70 views

Testing for convergence

$$\sum_{n=1}^{\infty} \frac{n^3}{3^n} z^n$$ As a power series in $z$, it has a radius of convergence, which is what I believe you are looking for. Regroup it as $\displaystyle n^3 \left( \dfrac{z}{3} ...
0
votes
3answers
246 views

Investigate the convergence or divergence of the series

Investigate the convergence or divergence of the series $$\sum_{n=1}^\infty (-1)^{n-1}n=1-2+3-4 + \cdots$$ Could anyone help me with this problem?
2
votes
1answer
157 views

Function in $\mathcal{C}(X\times Y)$ can be “approximated” by sum of finite number of functions of the form $g(x)h(y)$

Let $X,Y$ be compact spaces if $f \in \mathcal C(X \times Y)$ and $\varepsilon > 0$ then $ \exists g_1,\dots , g_n \in \mathcal C(X) $ and $ \exists h_1, \dots , h_n \in\mathcal C(Y) $ such that ...
1
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2answers
250 views

normal to the surface of revolution

Please, help me with my homework... Prove that the normal to the surface of revolution $z = f(\sqrt{x^2+y^2})$ intersect the axis of rotation. first, I need to find $z_x$, $z_y$ $z_x$ = ...
1
vote
1answer
238 views

Exponential of formal power series and Bell polynomials

Wikipedia gives here the following formula for the exponential of a formal power series: $\exp \Big[\ \sum_{n=1}^\infty \frac{a_n}{n!} x^n\ \Big] = \sum_{n=0}^\infty \frac{B_n(a_1,\dots,a_n)}{n!} ...
2
votes
2answers
569 views

$\mathcal{H}$ is relatively compact iff every sequence in $\mathcal{H}$ has a convergent subsequence?

I'm trying to prove that Let $(Y,\rho_{Y}),(K,\rho_{K})$ a complete metric space and a compact metric space, respectively. Let, as well, $Z=\mathcal{C}^{0}(K,Y)$ the metric space of continuous ...