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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$C^1$ Lipschitz function linear growth

I would like to know how to prove that a $C^1$ Lipschitz function has linear growth. (Actually I don't even know if it is true, it is a question in my exam – it says prove that, so it means it is true). Thanks in advance.
Minkowski Yaacov
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Conservative Fields and Denseness

Let $\Omega\subset\mathbb{R}^3$ be a open domain and $F=(F_1,F_2,F_3):\Omega\rightarrow\mathbb{R}^3$ a continuous field. Suppose that does not exist $u:\Omega\rightarrow\mathbb{R}$ with $\nabla u=F$. Is it possible to find…
Tomás
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Can the following norm be put on $ C^1(D)$ that can make it into a Banach space?

I know that $C^1([a,b])$ with norm $\|u\|=\|u\|_\infty+\|u'\|_\infty $ is a Banach space. My question is: Can the following norm be put on $ C^1(D), D \subset \mathbb{R}^2$ that can make it into a Banach space? $$\|u\|_1=\|u\|_\infty+\max\left(…
Thu Le
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Is it possible to express accuracy knowing sensitivity and specificity?

I know the value of the sensitivity Se and the value of the specificity of Sp, they are equal to 78.65 and 90.00, respectively. I know nothing but this. Can I somehow of the equations, which in the photo express the value of the accuracy Ac?
Lionell
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uniformly convergence of heat equation

Problem Let $u(x,t) = \frac{e^\frac{-x^2}{4t}}{\sqrt{4 \pi t}} $ for $ t > 0, > x \in \mathbb{R} $. If a > 0, prove that $u(x,t) \rightarrow 0$ as $t > \rightarrow 0+$, uniformly for $x \in [a, \infty ]$ I proved like this, but I didn't understand…
alryosha
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Sequences and series and the alternating series test

Question: Suppose $(q_{n})_{n=1}^{\infty}$ is a sequence of real numbers such that Lim$_{n \rightarrow \infty} q_{n} = + \infty$. Show that we can find a sequence $(a_{n})_{n=1}^{\infty}$ such that $\sum a_{n}$ is convergent but $\sum a_{n}q_{n}$ is…
user653036
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Find asymptotically equivalent function $f\sim g$

I'm trying to find a asymptotically equivalent function $g$ for $f(x)= \sin{(\frac{\pi}{6^x})} - \dfrac{1}{2}$, ie $f \sim g$. Should I use $\sin{x}$ Taylor expansion or something else? Here, $x$ approaches to $1$.
Metso
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Suppose that $x_n$ $\rightarrow$ + $\infty$and $y_n$ $\rightarrow$ a. True or false? If a>0, then $x_ny_n$ $\rightarrow$ + $\infty$

Suppose that $x_n$ $\rightarrow$ + $\infty$and $y_n$ $\rightarrow$ a True or false? If a>0, then $x_ny_n$ $\rightarrow$ + $\infty$ This is what I have: Let $\epsilon$>0 then $n_0$ $\in$ $\mathbb{N}$ for |$y_n$-a|<$\epsilon$. Let $\epsilon$=a/2 then…
Reign T.
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is $2\sin^2(n\pi/2)+cos(n\pi)$ convergent? divergent to $+\infty$ or $-\infty$, bounded but not convergent, or none of these

is $2\sin^2(n\pi/2)+cos(n\pi)$ convergent? divergent to $+\infty$ or $-\infty$, bounded but not convergent, or none of these What I did was graph it and it was a straight horizontal line on $y=1$. Does a straight horizontal line count as bounded?…
user597188
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Prove that {$x_n$} is not bounded.

Suppose $\{x_n\}$ is a sequence such that $x_{n+1}-x_n\geq10^{-6}$ for all $n$. Prove that $\{x_n\}$ is not bounded. This is what I have: $x_{n+1}-x_n\geq10^{-6}$ $x_{n+1}-x_n\geq\frac{1}{10^{6}}$ $x_{n+1}\geq…
user597188
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for a>0and n∈ℕ, define $a^{1/n}$ to be that unique positive number b such that $b^n$=a. Prove that for a>0,$lim_{n\rightarrow\infty}$ $a^{1/n}$=1.

for a>0and n∈ℕ, define $a^{1/n}$ to be that unique positive number b such that $b^n$=a. Prove that for a>0,$lim_{n\rightarrow\infty}$ $a^{1/n}$=1. I know I need to consider both cases a $\leq$ 1 and a>1. This is what I have so far: a>1 |$a^{1/n}$…
user597188
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What should I study before read Analysis on Manifolds written by Munkres?

Is Rudin's PMA enough before studying analysis on manifolds? Please let me know if there is more. I'd like to know if I can understand the book before buy it.
Yun YJ
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$\sin x$ does not satisfy this quadratic equation

Prove that $\sin x$ is not a rational function using the fact that it is not of the form $p(x)/q(x)$ where $p$ and $q$ are polynomials. Then, by using the above proof, prove that $\sin x$ does not satisfy a "quadratic equation" of the form: …
Dick
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Topologies and continuity: continuous iff continuous at every point

Following Pedersen's Analysis Now, it seems that involving topologies in the definition of continuous functions, yet ignoring them in defining point-wise continuity, makes it difficult (impossible?) to show equivalence between the two. Pedersen's…
AlexJ
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Find a majorizing function

Please, could somebody help me find a function $f(x)$ such that $| \frac{1}{n+n^2 \sin(xn^{-2})}| \le f(x)$ for each $n \in (0, \infty)$. $f(x)$ has to be $\ge 0$ for every $x \in (0, \infty)$ and integrable in $(0,\infty)$. I've been trying, but…
Anne
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