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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Generalizing integration

Is it possible to generalize the notion of integrals to other sets than the reals? In particular, would it be possible to integrate over a subset of the reals such as the rationals or the irrationals? What about more exotic sets?
haroba
  • 1,155
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Find sup, inf, min, max of the set B

Given the set $$B=\left\{\frac{1}{n}+(-1)^n, n \in \mathbb N\right\}$$ I have to find $\sup B$, $\inf B$, $\max B$, $\min B$.$$$$ For $n=even:$ $$B_{even}=\left\{\frac{1}{2k}+1, k=1,2,...\right\}$$ For $n=odd:$ $$B_{odd}=\left\{\frac{1}{2k+1}-1,…
Mary Star
  • 13,956
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About Borel's lemma

Borel lemma states that for $x_0 \in \mathbf{R}$ and for a real sequence $(a_n)_{n \in \mathbf{N_0}}$ there exists a smooth function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f^{(n)}(x_0)=a_n$ for $n \in \mathbf{N_0}$. However is it true for…
Richard
  • 4,432
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Differentiabilty only at a single point implies the jacobian is singular?

let $f:\mathbb{R}^n \to \mathbb{R}^n$ be a function differentiable at a single point $p \in \mathbb{R}^n$ yet not differentiable at any other point in $\mathbb{R}^n$. The inverse function theorem tells us that if the jacobian of $f$ is non…
user116457
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1 answer

Reconciling the two statements

I had just recently picked up Functional Analysis so my problem may sound trivial. But I appreciate any help. I am having trouble to reconcile the two statements (said to be true in my notes): let $B \subset$ X*, dual space of X and define $B^o$…
user2319
  • 43
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Given $c_{i} \rightarrow c$, prove if $c \ge 0$ then $\limsup c_{i} a_{i} = c \limsup a_{i}$

Let $a_{i}$ be a sequence of real numbers and suppose that $\limsup a_{i}$ is finite. Let $c_{i}$ be another sequence and suppose $c_{i}$ converges to $c$. Prove that if $c \ge 0$, then $\limsup c_{i} a_{i} = c \limsup a_{i}$ . I worked out the…
user113255
  • 161
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How prove that $\sum_{n=1}^{\infty}\frac{\pi{(n)}}{n^2}<\infty$ for some bijective $\pi(n)$

Let $\mathbb{N}=\{1,2,\cdots\}$. Does there exist a bijective function $\pi:\mathbb{N} \to \mathbb{N}$ such that $$\sum_{n=1}^{\infty}\dfrac{\pi{(n)}}{n^2}<\infty ?$$ My try: note $$\sum_{n=1}^{\infty}\dfrac{1}{n^2} $$ is convergent, and then I…
user94270
4
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1 answer

$x^2 f^{''}(x)+4xf^{'}(x)+2f(x)\geq 0$, prove $f(x)\leq 0$

More specifically, suppose $f$ is continuous on $[a,b]$ with $f(a)=f(b)=0$ and $x^2f^{\prime \prime}(x)+4xf^{\prime}(x)+2f(x)\geq 0$ for $x\in (a,b)$. Prove that $f(x)\leq 0$ for $x\in [a,b]$. I'm not sure how to approach this question. Any help…
Mael
  • 741
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Roots of $x^x-\tan (x)$

I conjecture, that the function $f(x)=x^x-\tan x$ has exactly one root in any of the intervals $\left[\dfrac{2n+1}{2}\pi,\dfrac{2n+3}{2}\pi\right]$ , where $n$ is a nonnegative integer. Does anyone know a proof? I tried the trick using the function…
Peter
  • 84,454
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3 answers

Minimum of set $\{\frac{m}{n} + \frac{4n}{m}\}$

We have the following set: $\mathcal{A} = \{ \frac{m}{n} + \frac{4n}{m};\ \ m, n \in \mathbb{N} \} $ Attempting to prove that the set's minimum is 4 yields: $$\frac{m}{n}+\frac{4n}{m} = \frac{m^2 + 4 n ^2}{mn} \geq 4$$ $$m^2 + 4n^2 \geq 4mn$$ $$m^2…
Luka Toni
  • 485
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Trying to Understand Baby Rudin Proof

If p is a limit point of a set E, then every neighborhood of p contains infinitely many points of E Proof: Suppose there is a neighborhood N of p which contains only a finite number of points of E. Let r be the minimum of the distances of these…
Juniper Leaf
  • 409
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Surjective local diffeomorphism not injective

I would like to find a function $F:\mathbb{R}^N\to\mathbb{R}^N$, for $N\geq 2$, which is surjective and a local diffeomorphism, but which is not injective. I can solve the problem by using partition of unity, but I would like to find an explicit…
Tomás
  • 22,559
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Convex function problem and mean value theorem

I dont know how to solve this problem with not convex condition! Let $f \colon A \subseteq \mathbb R^n \to \mathbb R$ differentiable with $A$ convex and suppose $\|\mathop{\rm grad} f(x)\| \le M$ for $x \in A$. Prove $|f(x)-f(y)| \le M\|x-y\|$ for…
user96813
  • 41
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1 answer

Lower bound of sum of cosine of angle difference

I am reading the paper and got stuck at the formula above (5.13): Let $n\geq 4$. (1) why $$\sum_{i\neq j\in[n]}\cos^2(\theta_i-\theta_j)\geq n^2/2$$ holds? Note that this corresponds to the formula $$n^2/2\leq \|X\|_F^2$$ in the paper, where $X$ is…
chloe
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