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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trajectory in polar coordinates

I need to prove that for a given a smooth trajectory $\vec{z}(t)$ in $\mathbb{R}^2$ defined on some interval $I$ and such that $\|\vec{z}(t)\|=1$, there exists a unique twice differentiable $\varphi(t)$ defined on $I$ such that $\varphi(0)=0$…
keplerr
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Can I express this set as a sum of sets?

Let $X, Y_{1}, ..., Y_{p}$ be nonempty subsets of $R^n, n \geq 1$. Can I express the set $$ \left\{x + \sum_{j = 1}^{p}v_j \mid x \in X, v_j \in Y_j, (v_1)_{n} = \dots = (v_p)_{n}\right\} $$ as a sum of sets, i.e $A + B = \{a + b \mid a \in A, b \in…
AMfrn
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Why is log$(\frac{G}{a})+\frac{a}{G} \geq 1$

I am currently trying to understand Alzers proof of the arithmetic mean geometric-mean inequality. (Link) It states that for $a\displaystyle_i, p_i \in \mathbb{R} \; \text{with} \sum p_i =1\text{ and }\mathbf{G}:= \prod a_i^{p_i}$ there exists an…
ruffler
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two inequalities, application of MVT?

$$py^{p-1}(x-y) \leq x^p-y^p \leq px^{p-1}(x-y)$$ Where $0
user9352
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Question on Axiom of replacement

This axiom comes from chapter 3 "set theory" of Tao Analysis I Axiom $3.6$ (Replacement). Let $A$ be a set. For any object $x \in A$ and any object $y$, suppose we have a statement $P(x, y)$ pertaining to $x$ and $y$, such that for each $x\in A$…
Andrew Li
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Limits and continuity.

I need help with these exercises of Analysis about limits and continuity. Construct a set $ A \subset [0,1] \times [0,1]$ such that $A$ has at most one point en each horizontal line and one in each vertical line and $ \partial A = [0,1] \times…
user89935
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Confused on continuity and uniform continuity

On Rudin's book, there is a comparison between continuity and uniform continuity. It says that if $f$ is continuous on $X$, then it is possible to find, for each $\epsilon>0$ and for each point $p$ of $X$, a number $\delta>0$ having the property…
Beginner
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Why can't a $C^1$-class mapping with nonzero derivative fill a square?

Let $f: [0,1] \rightarrow \mathbb R^2$ be of class $C^1$ with $f'(t)\neq (0,0)$. Why can't $f$ be a Peano-type curve, i.e. $f(I) \neq I\times I$?
Alex
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$f:(0,1) \rightarrow \mathbb{R}$, $x_n$ Cauchy and $f \circ x$ cauchy $\implies f$ uniformly continuous.

$f:(0,1) \rightarrow \mathbb{R}$, and for every Cauchy sequence $x_n \in (0,1)$, $(f \circ x)_n$ is Cauchy: does that mean that $f$ is uniformly continuous ? I believe the answer is yes, continuity has already been asked and answered. But what about…
Pastudent
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Is there a mathematical scenario in the game of snooker where deducting points from Player A instead of adding points to Player B would change result?

Context During a game of snooker I was playing in, I committed a foul resulting in 4 points being added to my opponents score. However, the friendly but inexperienced scorekeeper deducted 4 points from my own score. This led to an unresolved…
Venture2099
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Analysis On Manifolds: Munkres, James R, confusion about Lemma 25,2

By the theorem 16.3 in this book, I know that $A$ as a union of open sets $A_V$, there exists a partition of unity on it. I denote it as $\{\phi_i\}$. Then for each $x\in A$, there exists an open neighborhood $U$ of it so that there's only finitely…
M_k
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Proving certain properties for Babylonian method

The Babylonian method for approximating square roots is divided into three steps: Guess an initial approximation $a$ of $ {\sqrt N}$, where $a$ and $N$ are postive rational numebrs and $N$ is not the square of any rational numbers. Let…
user19170731
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Volume between $z=x^2+y^2$ and $z=x+y$

I need to find the volume between two functions $ z=x^2+y^2 $ and $ z=x + y $. I converted the functions into polar coordinates and ended up with next integral: \begin{cases} z=z \newline x=r\cos(\phi) \newline y=r\sin(\phi) \newline…
Oswald Gross
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Show that a space E is a closed linear subspace of $C[a,b]$

Consider the set $E := \{f \in C[a,b] , \int_a^b f(t) dt = 0 \}$ Show that E is a closed linear subspace of $C[a,b]$. I know that a subset $E$ is closed if $\overline{E} \subseteq E$ (where $\overline{E}$ is the closure of $E$). I don't know how I…
binaryBigInt
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Natural numbers =accumulation values in a sequence

Is there such a sequence where exactly the natural numbers are accumulation values? Because I think not, but not sure how to prove it.
user1117620
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