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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How prove this inequality $[f(x)]^2+[f'(x)]^2\le \max{(A,B)}$

let $f(x)$ be two derivable on $R$,give the two postive numbers $A,B$ and such $$[f(x)]^2\le A$$ $$[f'(x)]^2+[f''(x)]^2\le B$$ show that $$[f(x)]^2+[f'(x)]^2\le \max{(A,B)},\forall x\in R$$ I think maybe well know inequality：…
math110
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Shrinking Map and Fixed Point via Iteration Method

Let $T$ be a map from a compact metric space $X$ into itself satisfying $ d(Tx,Ty) < d(x,y)$ for all distinct $x,y$ in $X$. It is true that $T$ has a unique fixed point. Fix $x_0 \in X$ any point, and define the recursive sequence $\{ x_n \}$ by…
Peter
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Locally Bounded Functional Equation $f(x+y) = f(x) + f(y)$ and Continuity

Let $f$ be a real-valued function on $\mathbb{R}$ s.t. $f(x+y) = f(x) + f(y)$ for all $x,y$ reals. Suppose there are reals $c$ and $M$ s.t. $|f(x)| \leq M $ for all $x$ in $[-c,c]$. Show that $f$ is continuous. I am able to show that $f$ must take…
Peter
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Mathematical Analysis: What is the Velocity of the Falling object through each point in its path?

I have been working through the book called "Mathematics" written by A.D. Aleksandrov, A.n. Kolmogorov and M.A. Lavrent'ev recently and have had some difficulty with understanding Examples given by the authors regarding Mathematical Analysis which…
xAly
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How to construct a smooth function with compact support satisfying $f(x)+f(x^{-1})=1$

How to construct a smooth function with compact support satisfying $$ f(x)+f(x^{-1})=1 $$ For example, let $$ g(x)=\left\{\begin{array}{ll} 0,&\mbox{if $x\leq 0$},\\ \frac{1}{1+e^{\frac{1}{x}-\frac{1}{1-x}}},&\mbox{if $0
Sun
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Investigate whether there exists a function $f : [a,b]\rightarrow R$ that is continuous and that takes exactly twice each of its values.

Let a < b. Investigate whether there exists a function $f : [a,b]\rightarrow R$ that is continuous and that takes exactly twice each of its values.
a1bcdef
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Supremums of measurable functions

According to my textbook, supremums of measurable functions exist and are measurable. But what about the sequence of functions $f_n: [0, 1] \to \mathbb{R}$ given by $f_n = n$? I don't think this sequence has a supremum but I do think all those…
badatmath
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Functional disequality

Let $f \in C^{2}([a,b]) \ $, $f(a)= f(b) = 0 \ $, $f(x) > 0 \ \forall x \in (a,b) \ $, $f(x) + f(x)''>0 \ $. Then $b-a \ge \pi $. Any hint?
WLOG
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Changes in the unit ball

Given the open unit ball in $\mathbb{R}^2$ (or for any other $\mathbb{R}^n$ as well), if I use the function $f:B^2 \rightarrow \mathbb{R}^2$ that does the following: $f(x,y)= (\frac{x}{2}, \frac{y}{2} )$, will the result on the entire ball, i.e.…
Zhan I.s.
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Dual Radon transform: different conventions?

I am having a hard time trying to understand apparently two different definitions of the dual Radon Transform. I am reading simultaneously the book "Mathematics of computerized tomography", by Frank Natterer, and "Radon transform on homogeneous…
Qwertuy
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In such way some condition implies nonexistence of derivative?

Assume that a function $f:[0,1] \rightarrow \mathbb{R}$ is continuous. In what way condition $$\forall_{n\in \mathbb{N}} \forall_{0\leq x\leq 1-\frac{1}{n}} \exists_{0n$$ implies nonexistence of the…
A.B
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Proof verification (Analysis)

Proposition: If $A \subset \mathbb{Z}$, and if there exists $n \in \mathbb{Z}$ such that $n \leq a$ for every $a \in A$ then $A$ has a minimum. If $m \in \mathbb{Z}$ such that $a \leq m$ for every $a \in A$ then $A$ has a maximum. Proof: We know…
Charles Carmichael
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For $A=\{\sin (2n\pi/7) \mid n \in \mathbb{N}\},$ how do I find $\sup(A)$ and $\min(A)$?

I'm kinda new at this and I know what $\sup$ and $\min$ mean, but the problem is when calculating them like the example above. Can you enlighten me please?
Tébina1
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How to prove that if $\|f_n-f\|_p \rightarrow 0$ then $\|F_n-F\|_p\rightarrow 0$

Let $f_n \in C_c^\infty(0,\infty)$ for $n\in \mathbb N$, $f: (0,\infty) \in L^p(0,\infty)$, where $1
Alex
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Requirement of closed and bounded set $[a,b]$ in the Ascoli theorem

In Wikipedia, the Ascoli theorem requires the functions to be continuous on the closed and bounded interval $[a,b]$. However, in the proof given in the book "Theory of Ordinary Differential Equations" by Coddington and Levinson (Tata MaGraw-Hill…
jpv
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