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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Action of $ (I - \Delta)^k$ on a product of functions

Is there a formula for $ (I - \Delta)^k (fg) $ where $ \Delta $ is the Laplacian and $k \in Z $ similar to the Newton-Leibnitz formula for the derivatives of a product?
nenad
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Help understanding norms regarding $L^p$ spaces.

I'm studying $L^p$ spaces, and I'm confused on a couple of things, I think I'm either not understanding notation, or missing some fundamental concept. I'm given the definition $$\|f\|_p = \left(\int_X |f|^p\right)^\frac{1}{p}$$ (although, I…
temp_s
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Prequisite for understanding Abel's work

Hi I hope this question is not too soft I am curious about the work of Abel. I have found his work in this link http://www.abelprize.no/nedlastning/litteratur/houzel_the_work.pdf Currently I have some proof based knowledge from highschool calculus,…
user420309
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Continuity of function which is Lipschitz with respect to each variables separately

Let a function $f: I\times J \rightarrow \mathbb R$, where $I,J$ are intervals in $\mathbb R$, be Lipschitz with respect of each variable separately. Is it then $f$ continuous with respect of both variables? Thanks
Richard
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Analysis Question (limits of two sequences)

Suppose that $(s_n)$ and $(t_n)$ are sequences of positive numbers such that $\lim \limits_{n \to \infty} \frac{s_n}{t_n}$=a and that $(s_n)$ diverges to infinity. What can you conclude? This problem has me a little confused. Is it true that $(t_n)$…
mbb254
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Question about higher differentiability of inverse function

Assume that $f:I: \rightarrow J$ is a bijection of class $C^1$ and $f'(x)>0$ for all $x \in I$, where $I$, $J$ are intervals in $\mathbb R$. Then by the known theorem $f^{-1}$ is of class $C^1$ and $(f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}$ for $y\in…
A.B
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Proof with triangle inequality

Find a number $Y$ such that $|x^5 - 3x^4 + 2x^3 - x| \leq Y$ for all $x \in [-10, 9]$. My attempt: By triangle inequality it follows that $|x^5 - 3x^4 + 2x^3 - x| \leq |x^5| + |3x^4| + |2x^3| + |x|$ $= |x|^5 + 3|x|^4 + 2|x^3| + |x|$ It is obvious…
Adagio
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Proofs with absolute values

For real numbers x and y, show that $$\max\{x,y\} = \frac{x+y +|x-y|}{2}$$ My attempt: If $x\geq y$, then $$\frac{x+y+|x-y|}{2} = \frac{x+y+x-y}{2} = \frac{2x}{2} = x = \max\{x,y\}.$$ Similarly, if $x < y$, then $$\frac{x+y+|x-y|}{2} =…
Adagio
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Decide whether they can be written as the difference of two monotonically increasing functions

Decide for the following functions $f : \mathbb{R} \to \mathbb{R}$, whether they can be written as the difference of two monotonically increasing functions: a) $f(x)=sin(x)$ b) $f(x)= \left\{\begin{matrix} x^2sin(\frac{1}{x}) & if &x\neq 0 \\ 0 &…
Melina
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Proving something is uniformly continuous

Given the following function: $$x\sin\bigg(\frac{1}{x}\bigg)$$ How do I show that this is uniformly continuous on $(0, \infty)$? The reason why I am having trouble is that I am use to finding pathologies in functions which causes them to not have a…
emka
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A question about partition of unity

If $Y\subset X$ are open sets in $\mathbb{R}^n$,then $Y$ is said to have a $C^1$ boundary in $X$ if for every boundary point $x_{0}\in X$ of $Y$ one can find a $C^1$ function $\rho$ in a neighborhood $X_{0}$ of $x_{0}$ such that $$ \rho(x_0)=0,\quad…
Tomas
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Show the existence of a maximum in an open interval

Let $f:ℝ → ℝ$, continous, differentiable with $f(a)>0$, $a ∈ ℝ$ $\lim\limits_{x \rightarrow \infty}{f(x)\leq 0} $ & $\lim\limits_{x \rightarrow -\infty}{f(x)\leq 0}$ I want to show, that there exists an Maximum, but my problem is that I don't have a…
Sheosha
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Continuous functions with the same Fourier coefficients

I know that if two continuous $2\pi$ -periodic functions $f,g$ have the same Fourier coefficients then $f=g$. Is the assumption about $2\pi$ periodicity of functions essential? Thanks
Richard
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producing a stock market game based on variable data

I recently launched a stock market game (http://linkdaq.net) based on the top 50,000 sites ordered by the amount of links, it works reasonably well but I'm not 100% happy with the maths behind it. Let me first be upfront and say I've no experience…
Dom Hodgson
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mathematical analysis problem about intermediate value theorem

Suppose $f(x)\in C^2(-\infty ,\infty )$, $|f(x)|\le 1$, and $(f(0))^2+(f'(0))^2=4$. Prove that $\exists \xi $ such that $f(\xi )+f''(\xi )=0$. I think the function $(f(x))^2+(f'(x))^2$ may help but I don't know how to use this function.
user360777
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