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 .

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Annutiy choices question

If an annuity pays 4800 annually with a 2% increase per year or has an option of 6000 annually, how many years will the total amount paid is equal in both options Each year is 4800 + 4800 x 1.02 + 4800 x 1.02^2.... vs 6000 + 6000 + 6000...
tom murphy
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proving the existence of a solution to ODE as $t \to \infty$

I am give the following ODE: $x^{'}=\frac{1}{e^{x(t)}+t^2}$ and $x(t_0)=x_0$. I proved that for all $t\geq t_0$, the solution $x(t)$ is bounded. I also proved the existence and uniquence of the solution $x(t)$ to the above ODE when $t\geq t_0$. I am…
User267845467
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Supremum and Infimum with proof

Let $x\in (0,1)$ Compute with careful proof: The greatest lower bound of $(x^n : n \in N)$ and the least upper bound of $(x^n : n \in N)$ Hint: For the infimum (greatest lower bound), first prove that if the greatest lower bound were strictly…
All About Groups
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Continuously differentiable function

Let $f$ : $(-\displaystyle \frac{\pi}{2}, \frac{\pi}{2})\rightarrow \mathbb{R}$ be a continuously differentiable function such that $f(0)=0$ and $f'(x)\geq 1+(f(x))^{2}$ holds for all $x\displaystyle \in(-\frac{\pi}{2}, \frac{\pi}{2})$ . Show…
Joash
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Proof of seperability of the real/complex Lp Metric spaces

If Lp is the set of sequences of real or complex numbers, such that the infinite series, (where you sum up the moduli of the terms in the sequence to the power of p) converges to a finite value. Let the metric d(x,y) be this sum for the sequence…
user157872
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How prove $\lim_{x\to 0,y\to 0}\frac{2y^2x}{y^4+x^3}=0$

prove or disprove $$\lim_{x\to 0,y\to 0}\dfrac{2y^2x}{y^4+x^3}=0？$$ consider $$x^3+y^4>x^4+y^4,(x,y)\to (0,0)$$ so $$0\le |\dfrac{2y^2x}{x^3+y^4}|\le\dfrac{2y^2x}{x^4+y^4}|\le |\dfrac{2y^2x}{2x^2y^2}|=|\dfrac{1}{x}|$$ then I can't
math110
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Tangential equation by a point on the graph

it might be a silly question but I tried everything and could not find the possible error. I got $$ f(x) = e^x $$ and I have to find all possible boundary points of $f(x)$ with tangent(s), which go through the point $$ P (1/1) $$ Well, I'll just…
David Balan
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Prove that $f(A\cap {B})\subseteq {f(A)\cap {f(B)}}$

Let $f:S\to{T}$ be a function. If $A$ and $B$ are two arbitary subsets of $S$ prove that $f(A\cap {B})\subseteq {f(A)\cap{f(B)}}$
Satvik Mashkaria
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Problem with constructing a smooth function with given properties

I wish to construct a function $f:\mathbb R \rightarrow \mathbb R$ of class $C^\infty (\mathbb R)$ with the folowing properties: $f(x)=0$ for $|x|\leq 1$ $f(x)=x$ for $|x| \geq 2$, $|f(x)| \leq |x|$ for $x \in \mathbb R$.
Alex
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Proving Equalities in Analysis

In measure theory, I saw that while proving some "equalities" - $``a=b"$ - (such as measure of any type of an interval is its length, ...), the argument goes as follows: We prove that $a\leq b$ and $b\leq a$. One of these inequalities is mostly…
Groups
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Problem with finding a suitable partition of unity

Let $\{U_s\}_{s\in S}$ be a family of open subsets of $\mathbb R$ with union $W$ with the following property: for each $x\in U_s$ we have $x+1, x-1\in U_s$. Does there exist a sequence $\phi_1,\phi_2, \ldots$ of $1$-periodic, smooth, compactly…
A.B
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Formal construction of $\mathbb Q$: interpretation and equality of elements

Formally the rational numbers are defined as $\mathbb Z \times \mathbb Z / \{0\}$, where $(m_1, n_1)$ and $(m_2, n_2)$ being equivalent if $m_1n_2 = m_2 n_1$. This set equipped with $+$ and $\times$ as defined in the Wiki…
Shuzheng
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A direct application of inverse function theorem

Let $f:U\longrightarrow \mathbb{R}^n$ a function with $U\subset \mathbb{R}^n$ open, $f$ injective of class $C^1$ (i.e. continuous with the first derivate continuous) such that $\forall x\in U$ the derivate $f´(x)=D f(x)$ is an isomorphism. Show that…
Irene
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A problem with the density of sin (N)

Actually I can prove the fact that $\sin(\mathbb{Z})$ is dense in $[-1,1]$ using the result that "any non trivial subgroup of the additive group of $\mathbb{R}$ is either cyclic or is dense in $\mathbb{R}$." But my problem is to prove that…
Adimathematica
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Basic question about interior sets property

Reading a textbook of mine, I've encountered with a simple property and I couldn't prove it is true, I would like if someone could show me why the statement holds so I'll textually copy it. Statement "Let $\Omega$ be any subset of $\mathbb C$ and…
user156441
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