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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Application of the Inverse Function Theorem to polar co-ordinate transformation

$U=\{(u,v)\in {R}^2:u>0\}$ Define a function $F:U \rightarrow R^2$ by $F(u,v)= (u\cos(v),u\sin(v))= (x,y)$ a) Show $F$ is an open mapping on $U$. [I've done this.] b) Calculate $\Large \frac{\partial u}{\partial x},\frac{\partial u}{\partial…
user9352
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Does uniform integrability imply a subsequence that convergences almost surely?

Assume $f_n(x)$ is a sequence of uniformly integrable functions on $[0,T]$.Is there a subsequence of $f_n(x)$ that converges to some function $f$ almost surely, i.e. $f_{n_k}(x)$ is a subsequence of $f_n(x)$ such that $f_{n_k}(x) \to f(x)$…
solver
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Sigma-Algebra generated by a partion

Let $ \Omega $ a set and $ \mathcal{A}\subseteq \mathcal{P}(\Omega) $ a finite $ \sigma $-Algebra over $ \Omega $. Show there exists a partition $ \Omega=A_1\cup...\cup A_n $ such that $$ \mathcal{A}=\{A_{k_1}\cup...\cup A_{k_r}:r\in…
hallo007
  • 545
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Prove that a parameter dependent family of ellipses covers the plane

Consider the family of ellipses defined by $\frac{u^2}{(c+\frac{1}{c})^2}+\frac{v^2}{(c-\frac{1}{c})^2}=1$ with $0
dario
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Smooth real valued function on a Manifold

Let M be a smooth manifold . $f: M\rightarrow R$ be a real valued function. Show that if f is smooth at a point p in M , then f is smooth in a neighborhood of p in M.
Infinity
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the existence of a function $f$,satisfying $1\le f^{(n)}(x)+f^{(n+1)}(x)+\cdots+f^{(3n)}(x) \le 3$

I have a problem and the problem is: Determine whether or not there exists an infinitely differentiable function $f(x)$ such that for every real $x$ and for every positive integer $n$ $$1\le f^{(n)}(x)+f^{(n+1)}(x)+\cdots+f^{(3n)}(x) \le 3.$$ where…
fusheng
  • 1,135
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General solutions of the equation $ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0$

What is a general solution of the equation $$ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0 \textrm{ for } x,y \in \mathbb R, h>0, $$ with unknown function $f: \mathbb R^2 \rightarrow \mathbb R$? Functions of the form $f(x,y)=g(x)+h(y)$ are solutions. Are…
Richard
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Formalizing using $\sin h \approx h$ to evaluate a limit.

Assume $f \in C^{\infty}(\mathbb{R}^2)$ satisfies $f(\sin 2t, \sin t)=0$ for all $t\in \mathbb{R}$. I want to evaluate the limit \begin{equation*} \lim_{h \to 0}\frac{f(2h,h)}{h}. \end{equation*} It seems that (from the context from which this…
CriStiaN Alvarez
  • 23
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How prove this $f(0)=f(1)$,

let $f:[0,1]\longrightarrow [0,1]$ be a continuous function such that $f(0)=f(1)$, show that : if the number $l$ is not of the form $\dfrac{1}{n}$ there exsits a function of this form on whose graph one cannot inscrible a horizontal chord of length…
math110
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When do we call a collection of mathematical objects a "space"?

It seems like there's a space for everything, functions, matrices, complex numbers, so on. Because I see this term "space" used so often, I have to ask: when is a set of mathematical objects a "space"?
askquestions2
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Proving that a function is asympotically linear

I have a function: $$ y = \frac{20x^2 + 14x + 2}{7m+11xm-m\sqrt{37+70x+x^2}} $$ Note: In this case, m is simply some parameter When I plot out this graph, it looks asymptotically linear. How can I go about checking: If the function is, in fact,…
Mui
  • 23
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1 answer

$\lim x_0^2 + x_1^2+...+x_n^2$ where $x_n=x_{n-1}-x_{n-1}^2$

So, we are given a sequence $x(n)$ for which $x_{n+1}= x_n-x_n^2$ , $x_0=a$, $0 \le a \le 1$ I was first requested to show that it converges and to find $\lim_{n \to \infty} x_n$. I will post my answer here for you to check if it is right :S so,…
Plom
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If $|\dot{x}(t)|$ exponentially converges to $0$ as $t\to\infty$, can we prove the existence of $\lim_{t\to\infty}x(t)$?

If $\lim_{t \to \infty} \dot{x}(t)=0$ and $|\dot{x}(t)|$ exponentially converges to $0$ (i.e., there exist the positive constants $c_1$ and $c_2$, such that $|\dot{x}(t)|\leq c_1e^{-c_2~t}$, $t\geq 0$), can we prove the existence of…
CUG
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Prove that a unit circle contains its limit points (that it is closed under its analysis definition)

By a unit circle, I just mean $D = S^1$ i.e. a unit disk in $R^2$. My game is to use some arbitrary sequence $x_n = (a_n,b_n)$ such that $b_n = \sqrt{1-(a_n)^2}$ with $0 \leq a_n \leq 1$ and show that it converges to a boundary point. I basically…
Jerry Randle
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Question on connected spaces and continuous functions

Let $f:X \rightarrow Y$ continuous where the metric space $(X,d)$ is connected. On $Y$ we use the discrete metric. I want to show that $f$ must be constant on $X$. My approach: We may assume that $|X| \geq 2$ (otherwise the claim is trivial). Let…
user42761
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