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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Is this equation valid $\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$,?

While reading a script I found this equation: $\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$ and i cannot figure out how the author did this. I'd appreciate a step-by step equation for this equation.
Sim
  • 208
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Compactness and continuity

I am having some trouble with this: Let $f : X \to Y$ be continuous and bijective with $X$ compact then $f^{-1} : Y \to X$ is continuous? I am not quite sure if this is true, if is not I cant find any counter example, but if it is true I am stuck…
user62758
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Convergence in L^2

Define a function $f_\varepsilon: \mathbb{R} \to \mathbb{R}$ as $$\begin{equation} f_\varepsilon(x) = \begin{cases} -1 & \text{if}\; \,x<-\varepsilon \\ \sin \left(\frac{\pi x}{2 \varepsilon}\right) & \text{if}\;\, |x| \le…
Savannah
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Deducing convergence from bounds on average

Assume I have a bound on the summatory function of the form $$\sum_{n \leqslant x} |a_n|^2 \leqslant x^\alpha$$ Can I then deduce something about the convergence of $$\sum \frac{a_n}{n^k} ?$$ I was expecting getting something using Cauchy-Schwarz…
TheStudent
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What happens if we let the wall thickness of a hollow ball go to zero?

Imagen you have a hollow ball $\Omega$: (I am unsure how to write it down.) $$\Omega(r) = B(R) \setminus B(r), \quad R>r$$ whereas $B(\hat{r}):=\{v\in\mathbb R^3 : |v|\leq \hat{r})\}$ So basically you have two balls, both centered at the origin and…
xotix
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If at each point of a closed interval the $m$-th derivative of $f$ is $0$ for $m$ large enough, then $f$ is polynomial

I am working on exercice 9.5.2 of Analysis by Zorich and I am stuck at the question b. a) A set $E\subset X$ of a metric space $(X,d)$ is nowhere dense in X if it is not dense in any ball, that is, if for every ball $B(x,r)$ there is a second ball…
Jiu
  • 1,545
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Countability of a set of subsequences .

Consider a sequence $x_n$ with positive values such that $\sum _{n=1}^{\infty} x_n $ converges . Is the set of the subsequences $ x_{k_n}$ of $x_n$ such that $\sum _{n=1}^{\infty} x_{k_n} =c \in R$ countable ? $c$ is previously fixed so the…
mike moke
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Continuous map from a hausdorff space to itself

Let X be a compact Hausdorff space, and let $f:X \to X$ be a continuous map of X into itself. Prove that there is a non-empty subset A of X such that $f(A)=A$. If I let $A_1 = f(X)$ and inductively $A_{n+1}=f(A_n)$. Let $A=\cap _{n=1}^\infty A_n…
general1597
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Quotient of two $C^{(k)}$ functions is $C^{(k-1)}$

I am stuck at the following exercise in Zorich. Let $f,g\in C^{(k)}(D;\mathbb{R})$, and suppose that $f(x)=0\Rightarrow g(x)=0$ in the domain $D$. Show that if grad $f \neq 0$, then there is a decomposition $g=h\cdot f$ in $D$, where $h\in…
Jiu
  • 1,545
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How many points of discontinuity?

I need to prove that any monotonic function whose domain is an interval $[a;b]$ can have only finite or countable number of discontinuity points... I don't seem to have any insightful ideas. It even raises more questions in my head. What happens if…
Kapitonas
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Locality and Hilbert Curve

I have a hilbert curve index based on this algorithm. I take two to four values (latitude, longitude, time in unix format and an id code) and create a 1-d hilbert curve. I'm looking for a way to use this data to create a bounding box query (i.e.…
John Horus
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Question about exponents

Question from Rudin chapter 1: Let $b \in \mathbb{R}$ and $ b > 0$ . Let $m,n,p,q \in \mathbb{Z}$ with $ n,q>0$ . Assume $r = m/n = p/q$. Prove that $(b^m)^{1/n} = (b^p)^{1/q}$. The attempt: I think I'm supposed to use the theorem that says "for…
user39794
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4 answers

How to prove $\{y\in\mathbb R^n: |x-y|=r \text{ for some }x\in X\}$ is closed for closed $X$ and fixed positive $r$?

Let $X\subset\mathbb R^n$ be a closed set and $r$ a fixed positive real number. Let $Y=\{y\in\mathbb R^n: |x-y|=r \text{ for some }x\in X\}$. Show that $Y$ is closed. I tried to approach this problem with showing $Y^c$ is open, but I am stucked on…
JFK
  • 1,061
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1 answer

Uniform convergence of $\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$

How can i prove uniform convergence on $E=[0, \frac{\pi}{2}]$ ? $$\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$$
Ilya Fedorov
  • 171
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4 answers

Prove that $\sup \{\frac{a}{b}; a \in \mathbb{N}, b \in \mathbb{N}, a < b\}=1$

Consider a subset of rational numbers $S = \{\frac{a}{b}; a \in \mathbb{N}, b \in \mathbb{N}, a < b\}$. I want to prove that $\sup S = 1$. By the definition of supremum, for $\epsilon > 0$, it suffices to show that there exists $\frac{a}{b} \in S$…
Ted
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