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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).

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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Stone-Weierstrass Theorem

In the proof of Stone-Weierstrass theorem provided in Rudin's Principles of Mathematical Analysis, why do we only need to show that there exists a sequence of polynomials $P_n$ that converges uniformly to the continuous complex function on $[0,1]$?
Paul
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1 answer

Implicit Function Theorem computation problem

Problem 1, page 78 of Munkres (Analysis on Manifolds): Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x,y_1,y_2)$. Assume that $f(3,-1,2) = \mathbf{0}$ and $$ Df(3,-1,2) = \begin{pmatrix} 1 & 2 & 1 \\ 1 & -1 &…
James R.
  • 1,451
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Iterating Arithmetic, Harmonic and Geometric Means

Starting with a data set $X_{0}$, compute its arithmetic, geometric and harmonic means, $A(X_{0}), G(X_{0})$ and $H(X_{0})$ respectively. Let $X_{1} = \{A(X_{0}),G(X_{0}),H(X_{0})\}$, and compute $A(X_{1}), G(X_{1})$ and $H(X_{1})$. Extending this,…
Elchanan Solomon
  • 30,807
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1 answer

How find this function $f(x)\equiv 0,x\in R$?

Let $f:\mathbb R \to \mathbb R$ have the properties (1): for any prime number $p$ and any real number $x$, $$\sum_{j=0}^{p-1}f\left(x+\dfrac{j}{p}\right)=0$$ (2): there exist real numbers $a$ and $b(>a)$ such that: $x\in (a,b) \implies…
math110
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1 answer

How prove this Ratio Test and Its Generalizations problem?

Question: let $\alpha\in (0,1)$,and the postive sequence $\{a_{n}\}$ such $$\lim_{n\to\infty}\inf \left(n^{\alpha}\left(\dfrac{a_{n}}{a_{n+1}}-1\right)\right) =\lambda\in (0,+\infty)$$ show that $$\lim_{n\to\infty}n^k a_{n}=0,k>0$$ maybe this…
math110
  • 93,304
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2 answers

Analysis on Manifolds Munkres Integration

I needed help in showing that the set $R^{n-1} \times 0$ has measure zero in $R^n$. What I have so far: Let $\epsilon > 0$. If $i_1,\dots,i_{n-1}$ are integers, then define $U_{i_1,\dots,i_{n-1}}=[i_1,i_1+1]\times \cdots \times [i_{n-1},i_{n-1}+1]$.…
Buddy Holly
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Is the set $\left\{\frac{1}{n} \mid n = 2,3,4,\ldots\right\}$ a countable infinite set?

I'm trying to answer a homework question and I need to know if the set $E = \left\{\frac{1}{n} \mid n = 2,3,4,\ldots\right\}$ is countable or not. So $E=\left\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4},\ldots\right\}$. I think it is a countable…
user181728
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Derivative of matrix inversion function?

Let's say I have a function $f$ which maps any invertible $n\times n$ matrix to its inverse. How do I calculate the derivative of this function?
user181600
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Am I going about this the right way?

Show that for any $α ∈ R$, there exist infinitely many rational numbers $\frac{m}{n}$ with $|α − \frac{m}{n^2}| < \frac{1}{n}$. So we know that $-1≤\frac{1}{n}≤1$ which implies $\frac{1}{n^2}≤1$. Case $1$: if $m=n$ then $\frac{m}{n^2} = \frac{1}{n}$…
learning
  • 31
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1 answer

Is the number of subsequential limits of a sequence always countable

I know that a sequence can have many different subsequential limits but is the number of subsequential limits always countable? How do we know?
mathjacks
  • 3,624
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Is the sum of quasi concave functions quasi concave

Is the sum of quasi-concave functions a quasi concave function? I presume that's not in the case in general, but under which conditions is this true?
bonanza
  • 441
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1 answer

Rudin Principles of mathematical analysis p307

"For if $ A=\bigcup A^{'}_{n}$ with $A^{'}_{n} \in M_F(\mu)$, write $A_1=A^{'}_{1} $, and $$ A_n=(A^{'}_1\cup ...\cup A^{'}_n)-(A^{'}_n \cup ... \cup A^{'}_{n-1})$$ $(n=2,3,4,...)$. Then $$ A=\bigcup_{n=1}^{\infty}A_n$$ I can't understand why $A_n$…
JamesWang
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Is set with this property is homeomorphic to Cantor set?

(1) $A$ is nonempty subset of $\mathbb{R}$. (2) For all $x
Maddy
  • 1,875
3
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1 answer

How to prove $f(n) := \frac{1}{2}(a+(f(n-1))^2) ≤ 1-\sqrt{1-a}$

$a \in [0,1]\hspace{2em}f(1) := 0,\hspace{1em} f(n + 1) := \frac{1}{2}(a+(f(n))^2),\hspace{2em} n \in \mathbb{N}$ Prove: $f(n) ≤ 1-\sqrt{1-a}$ I assume that I'll need to convert this recursive function to a non-recursive one in order to prove the…
Lenar Hoyt
  • 1,062
3
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2 answers

Question concerning L'Hospital's rule

I know that L'Hospital's rule is applied when $\lim \frac{f'(x)}{g'(x)}$ must exist. Then, is there an example that $\lim \frac{f'(x)}{g'(x)}$ does not exist but other conditions of L'Hospital's rule hold ? i.e. for example) Are there functions $f$…
user128766
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