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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Typo in Paper? (Simple Calculation)

I have a question about obtaining Eq. (17) of this paper. Context: In Theorem 3, they show that for a Gaussian probability path $p_{t}(x\vert x_{1}) = \mathcal N(x\vert \mu_{t}(x_{1}), \sigma^2_{t}(x_{1})I)$, the conditional vector field (VF)…
Hermi
  • 692
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After removing zero section of tautological line bundle, what is the right formulation for integration by part on this non-compact space?

Consider the total space $\mathcal{O}(-1)\rightarrow\mathbb{C}\mathbb{P}^1$, it is simply the blow-up $Bl_0\mathbb{C}^2$, now we remove the zero section $E$, we naturally obtain a biholomorphism from $Bl_0\mathbb{C}^2-E$ to $\mathbb{C}^2-\{0\}$. It…
Maths007
  • 23
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Functional equation $f\left( x^{2}\right) =2f\left( x\right)$and the Cauchy problem $f\left( xy\right) =f\left( x\right) +f\left( y\right)$

Is it possible to show that if a continuous function $f:\mathbb{R} _{+}\rightarrow \mathbb{R}$ verifies the equation $$f\left( x^{2}\right) =2f\left( x\right)$$ then $f$ also satisfies the Cauchy equation $$f\left( xy\right) =f\left( x\right)…
user472228
  • 151
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Is Wikipedia's example of a function not in a tensor product correct?

https://en.wikipedia.org/wiki/Nuclear_space#Motivations_from_geometry explains that we need to take the completed tensor product for $\mathcal{C}^{\infty}(\mathbb{R})\otimes\mathcal{C}^{\infty}(\mathbb{R}) = \mathcal{C}^{\infty}(\mathbb{R}^2)$ to…
oggledog
  • 300
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Find a bijective function between rationals.

Give an explicit example of a bijective function $f:\mathbb Q \to \mathbb Q$ such that $f(x)>x^3$ for each x. My approach := Consider $f(x)=x^3+1$. Then $f$ is one-one but not onto. Since 3 has no pre-image. Now I am unable to construct.
SUJAN DAS
  • 302
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Inequality $|x|^p - |y|^p > p|y|^{p-2} \cdot \langle y, x - y \rangle$; $p>1$

Let $x, y \in \mathbb{R}^N$ be distinct and $p > 1$. According to the paper $\bullet$ Lindqvist, Peter; On the equation $\mbox{div}(|\nabla u|^{p−2}\nabla u) + \lambda|u|^{p−2}u=0$. Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164. the inequality…
Santos
  • 783
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How to show $a_n = (1 + \frac{2}{n} )^n$ is bounded above?

Can someone help me about this question? I have proved $a_n = (1 + \frac{2}{n})^n$ is increasing, but I have been confused by how to show it is bounded above. I have show that for $1 \leq k \leq n$ ${n \choose k} \frac{2^k}{n^k} = \frac{2^k}{k!}…
murtabak
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$\mathbb{Q}\times(\mathbb{R}-\mathbb{Q})\cup(\mathbb{R}-\mathbb{Q})\times\mathbb{Q}$ in $\mathbb{R}^2$

For $(a, b) \in \mathbb{R}^2,$ say $(a, b) \in A \subset \mathbb{R}^2$ if $a$ and $b$ are both rational numbers, or $a$ and $b$ are both irrational numbers; and $(a, b) \in B=\mathbb{R}^2\setminus A \subset \mathbb{R}^2$ otherwise. Given any two…
2512490741
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integral $\int_{0}^{1}t{}^{q}e^{-iXt}dt$

For given $q$ it is easy to compute this integral using integration by parts. For general integer(even) $q>0$, Mathematica gives the formula: $$-Ei[-q,iX]+(ix)^{-1-q}q!$$ where $Ei$ is exponential integral. Is exponential integral related with…
Katja
  • 515
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show that $e^{x^2} / x$ converges to infinity for $x \rightarrow \infty$

I've been given functions $f(x)=e^{x^2}$ and $F(x) = \int_{0}^{x} f(t) \,dt$. I have to show that $$\frac{f(x)}{x} \rightarrow \infty \quad \text{and} \quad F(x) \rightarrow \infty \quad \text{for} \quad x \rightarrow \infty$$ So far, all I have is…
Fjeeds Arcade
  • 39
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How to linearize the equation

I have an equation that I read from a book as $$A = R^2 \arcsin{\left(\frac{a}{R}\right)} - a\left(R-b\right) (1)$$ And it says: compared with R, b is very small, thus, the equation above can be further linearized about $$\frac{a}{R}$$ as: $$A = ab…
AlenKen
  • 23
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The Fundamental Theorem of Calculus, removing hypothesis

I am trying to understand why the Fundamental Theorem of Calculus is written the way it is done in this Wikipedia page. So, I go removing hypothesis and check if anything breaks. First check: are there functions $f:[a,b]→ℝ$ and $F:[a,b]→ℝ$ such that…
SolutionExists
  • 326
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About the principal nth root

[$n_{th}$ root] A difficulty with this choice is that, for a negative real number and an odd index, the principal nth root is not the real one. For example, ${\displaystyle -8}$ has three cube roots, ${\displaystyle -2}, {\displaystyle 1+i{\sqrt…
Andrew Li
  • 431
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1 answer

Can the limit of a sequence be $+\infty$?

The proposition comes from the textbook of Tao Analysis: Proposition 6.4.12. Let $(a_n)_{n=m}^∞ $ be a sequence of real numbers, let $L^+$ be the limit superior of this sequence, and let $L^-$ be the limit inferior of this sequence (thus both $L^+$…
Andrew Li
  • 431
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1 answer

Definition of limits

We have $\lim_{x\to\infty} x^2 = \infty$. What would be the precise definitions of $\lim_{x\to c} f(x) = \infty, \lim_{x\to\infty}f(x) = L, \lim_{x\to\infty}f(x) = \infty$ ? How to go about these? I'm not sure where to even start.
user1127565
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