Mathematics Stack Exchange
Mathematics Stack Exchange

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).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Extending of domain of smooth function of two variables

Let $f: [a,b]\times [c,d] \rightarrow \mathbb R$ be a smooth function of two variable (assuming that in boundary points $f$ has continuous one side partial derivatives). Is a simple way to extend $f$ to a smooth function $F: \mathbb R \times…
R.S
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What are appropriate techniques to aggregate rankings from experts?

I am not sure if this would be considered a programming or a math or another problem. I've come across a situation in my day to day work and am looking for an optimal solution. If you have a list of rankings from experts (let's say players in a…
Reily Bourne
  • 131
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Help with norm definition

I'd like to show that the following defines a norm on $\mathbb C^n$: $||x||=(a_1^2+a_2^2)^{1/2}$ Where $x=(x_1,..,x_n)$, $a_1$ is the maximum of the $|x_i|$'s and $a_2$ the second maximum. But I have problems with the triangle inequality…
tina
  • 33
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Evaluating $\lim_{n \to \infty}\frac{1}{n}(n!)^{\frac{1}{n}}$

I'm trying to find and prove the value of$$\lim_{n \to \infty}\frac{1}{n}(n!)^{\frac{1}{n}}$$ I was thinking that since $$\frac{1}{n}(n!)^{\frac{1}{n}} = \frac{1}{n} \left[ (1)^{\frac{1}{n}}(2)^{\frac{1}{n}}...(n)^{\frac{1}{n}} \right]$$ and we…
Mike
  • 282
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Is the image f(A) of a n-Lebesgue measurable function m-Lebesgue measurable when f is Lipschitz?

Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is Lipschitz, where $n\geq m$. Let $\lambda_n$ and $\lambda_m$ denote the Lebesgue measure on $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively. If $A\subset\mathbb{R}^n$ is $\lambda_n$--measurable, is…
Mittens
  • 39,145
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About Cauchy product of some sequences

Assume that we have two sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$ such that for each $l\in \mathbb N $ the sequence $\left(|n|^l a_n\right)_{n \in \mathbb Z}$ is bounded, there exists $s \in \mathbb N$ such that…
A.B
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The set of points of boundedness of a function is open

I was reading (self study) the book by Thomson Bruckner and Bruckner and ran into one of the exercises that I think I have the proof for by not sure it is right. The statement of the problem goes as follows. A function $f:\mathbb{R} \mapsto…
Ramesh Kadambi
  • 423
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Uncountably many?

Is there any way to show that the set of disjoint translations of the cantor ternary set is countable? That is show that there are countably many disjoint sets of the form $\{x+C: x\in \mathbb{R}\}$??? Thanks
Brandon
  • 81
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Is there a continuous characteristic function on $\mathbb{R}$ ?(help with proof)

I have a question that says: Is there a continuous characteristic function on $\mathbb{R}$? If $A\subset \mathbb{R}$ show that $X_A$ is continuous at each point of $int(A)$. Are there any other points of continuity? In this problem I have…
user197950
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Antenna adjustment: A real-life problem -- solving for a factor when only a change is known

I'm an amateur radio operator, and part of this involves making antennas. One of the simplest is a dipole, which requires two pieces of wire. The total length of these is given by the formula: Total length (in metres) = $143$ / Frequency (in…
philpem
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Prove that if $B(x,r)\subset B(x',r')\Longleftrightarrow d(x,x')\leq r'-r$

For this proof, we're in $\mathbb{R}^n$ and $d$ is the Euclidean metric. So the claim can be written $(|x-y|
HouseOfWinston
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Find the gradient of unimodal function

I am doing the following exercise: given the level lines of the unimodal function $f$ with minimum $x^*$, a point $x_0$, and vectors $v_1$,$v_2$, $v_3$, $v_4$, $v_5$, one of which is equal to $\nabla f(x_0)$, which vector $v_i$ is equal to $\nabla…
cholo14
  • 451
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Can I get better approximation of $\sum_{k=1}^{n} k^k$

Is it possible to get approximation$f(n)$ of $\sum_{k=1}^{n} k^k$ with \begin{align} \lim_{n\to +\infty }\left(f(n)-\sum_{k=1}^{n} k^k\right)=0 \end{align} Thanks for your attention!
Golbez
  • 4,276
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Is $f^{(0)}:=f$ a valid assumption for a proof involving derivatives on induction?

My motivation for this was proving the general Cauchy-Integral formula (in complex analysis) for an arbitrary derivative. Every book I read shows at least the first derivative using a $\delta-\epsilon$ argument, but we already did this…
Squirtle
  • 6,698
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Convergence of $\sum2^{-\sqrt{k}} $

So I am going to determine whether this series converges or not: $$\sum_{k=0}^\infty 2^{-\sqrt{k}} $$ Since this chapter is about the ratio test, I applied that test to this series. I end up with this limit $$\lim_{k \to \infty}…
fejz1234
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