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).

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2
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1answer
31 views

positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove$\displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent

When I do my homework ,I met this problem: If positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove:$ \displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent. I ...
3
votes
1answer
20 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
0
votes
1answer
13 views

If $A^k$ consistently approximates $\nabla^2f(x^k)$ with $x^k\to x^*$ and $\nabla^2f(x^*)$ regular, then the $A^k$ are regular, too

Let's call $\left\{A^k\right\}\subseteq\mathbb R^{n\times n}$ a consistent approximation of $\left\{B^k\right\}\subseteq\mathbb R^{n\times n}$ iff ...
1
vote
1answer
78 views
+300

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...
0
votes
2answers
31 views

If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...
0
votes
0answers
47 views

rudin's definition of a compact set

Here are some definitions given in my book: Definition 2.31 By an open cover of a set $E$ in a metric space $X$ we mean a collection $\{G_\alpha \}$ of open subsets of $X$ such that $E \subset ...
-1
votes
1answer
26 views

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$. Solution: Let $X_n=[x_0,x_1,\dots]$; define ...
1
vote
1answer
44 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
0
votes
2answers
36 views

Let $X$ and $Y$ be finite sets. Then $X \cup Y$ is finite and $| X \cup Y| \leq |X| + |Y|$.

Let $X$ and $Y$ be finite sets. Let us assume that they are distinct at least, for otherwise $X \cup Y = X$ and $X$ is finite. Also let us assume that $X$ has cardinality $n$ and $Y$ has cardinality ...
1
vote
1answer
27 views

Solution to the wave equation in $\mathbb{R}^{3}$ with certain initial data

Suppose $f$ is a smooth function satisfying $f(0) = f'(0) = 0$. The question I am working on is to determine the solution $u$ to $u_{tt} - \Delta u = 0$ in $\mathbb{R}^{3}$ with $u(x, 0) = f(|x|)/|x|$ ...
4
votes
2answers
155 views

How do we know ternary expansions with only $0$'s and $2$'s are unique?

Let $c \in [0,1]$ and consider one of its ternary expansions $\sum_{n \ge 1} c_n / 3^n$ s.t. each $c_n = 0$, $1$, or $2$. This ternary expansion needn't be unique. For example: $$ 0.0222222\ldots ...
6
votes
2answers
221 views

Is it possible to develop Analysis solely from Peano's axioms

...and a few definitions on the way? When I studied Calculus using Spivak's book It was clearly shown that, in order to prove some fundamental theorems (intermediate value theorem being one of them), ...
0
votes
0answers
26 views

Jacobian from $\mathbb{R}^n \to \mathbb{R}^m$

Consider a diffeomorphism $F: V \subset \mathbb{R}^N \to U \subset \mathbb{R}^n$, where $U$ and $V$ are open, $N>n$. Moreover, suppose that $f:=F|_M: M \subset V \to U' \subset U$ is also a ...
2
votes
0answers
13 views

Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
0
votes
1answer
19 views

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$.

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$. I'm having difficulty showing the above equalities. I ...
13
votes
5answers
2k views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
0
votes
0answers
10 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
0
votes
1answer
8 views

When we say f(x)->l as x->c then how c becomes a limit point of the domain of defination of f.

I think that if c be an isolated point of the domain of f then continuity of c does not imply existence of the limit of f at c.Is it the only cause?
1
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0answers
15 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...
2
votes
3answers
52 views

Why is 1 + 1 = 0 when we make the addition table for F = {0, 1} (F = field)

In Analysis, I learned that any number system satisfying all the axioms (commutativity, associativity, identity elements, invertibility, distributivity) is called a field. Then the professor mentioned ...
5
votes
1answer
39 views

Showing that $\sup_{(x,y)}f(x,y)=\sup_x\sup_yf(x,y)=\sup_y\sup_xf(x,y)$

Can anyone help me prove this: Let $X$ and $Y$ be nonempty sets and $f:X\times Y\to\Bbb R$ such that $f(X\times Y)$ is bounded. Prove the following statement: ...
1
vote
2answers
39 views

$y'=\frac{y^2}{2x(y-x)}$

I'm trying to solve the following differential equation: $$y'=\frac{y^2}{2x(y-x)}$$ It is supposed to have a relatively easy general solution, but I can't find it. I've tried several things, the ...
2
votes
2answers
67 views

How do I find the exact solution to the boundary value problem $y'' = 4y' + y + 2 − 8x − x^ {2}$ , $y(0) = 0$ and $ y(4) = 16$?

I am approaching this question by trying to guess the general solution to the boundary value problem. However I haven't come up with one. Can someone explain how to solve this question please?
1
vote
1answer
22 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
3
votes
2answers
130 views

How to apply the Gronwall lemma

Consider $x'=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2x_2,-2x_1-x_2)$. Show that for two solutions $x(t)$ and $y(t)$ of the above differential equation, we have: $$\lVert x(t)-y(t)\rVert \leq ...
2
votes
1answer
64 views

What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
1
vote
1answer
33 views

Orthonormal basis of $L^2(T)$

Why is $\{e_n\mid n\in\mathbb{Z}\}$ an orthonormal basis of $L^2(T)$, where $T=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_T f(z)\,dz:=\int_0^1f(e^{2\pi i t})\,dt$? My try: If $n=m$, ...
4
votes
1answer
115 views

Soft question: what are some elementary motivations of using functional analysis to study probability theory?

Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic ...
7
votes
1answer
291 views

References on the Nash-Moser implicit function theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
2
votes
1answer
60 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
1
vote
0answers
22 views

Find the $\epsilon - \delta$ values for the continuous function - modified step function defined on $[0,1]$

Let the modified step function be defined on $[0,1]$ by : $f(x) = \begin{cases} \bigg( \dfrac {2^n+1}{2}\bigg )x - \dfrac {2^n-1}{2^n} ; & n \in \mathbb N~~ , \dfrac {2} {2^n+1} ...
1
vote
1answer
66 views

Implications of inner products vs normed spaces vs metric spaces

Is it true that: -an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality -a norm can be induced by a metric if and only if the metric ...
1
vote
0answers
56 views

How to construct $\mathbb{R}^N$ where $N$ is a random variable?

How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$? Would someone be so kind ...
88
votes
10answers
5k views

How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
0
votes
0answers
24 views

In the geometrical interpretation for integration how lower and upper rectangular approximation are functions of natural number?

I've attempted to prove this in the following manner. Let Q be a subset of $P[a,b]$ which contains partitions of each order exactly once. Now, if we consider mappings $F:N \to Q$ defined by $F(n)=p$ ...
4
votes
2answers
129 views

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
2
votes
1answer
20 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
0
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0answers
25 views
-1
votes
1answer
38 views

Reposting Question about Schroder-Bernstein

Assume there exists a $1$-$1$ function $f:X\to Y$ and another $1$-$1$ function $g:Y\to X$. Follow the steps to show that there exists a $1$-$1$, onto function $h:X\to Y$ and hence $X\sim Y$. a) The ...
0
votes
1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
0
votes
2answers
41 views

Decide what is the number of roots of the equation

Decide what is the number of roots of the equation $2^x=100x$. I know I can draw a sketch and then check but maybe there is a better method to do that? It's an exam question, thus it must require ...
1
vote
1answer
24 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
1
vote
3answers
1k views

The Least Upper Bound and The Greatest Lower Bound

I am taking math class, and I am not sure about LUB and GLB. I need someone to give this dummy a short explanation about them.... On interval (0,10), 0 is a lower bound and 10 is upper bound, but ...
1
vote
2answers
40 views

Convergence of $\int_2^{\infty}f(x)\,dx$ with a given condition

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
-1
votes
0answers
34 views

On the continuity of Li's numbers. [on hold]

Consider Li's numbers defined by $\lambda_n = \sum_{\rho} \left(1-\left(1-\dfrac{1}{\rho}\right)^n\right)$ where $n$ is a nonnegative integer and the $\rho$ are the nontrivial zeros of the Riemann ...
0
votes
3answers
68 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$ [on hold]

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
0
votes
1answer
13 views

Find the distance such that the angle will be the gratest

Rectangle shaped screen in a cinema is 8m high. It is place on a wall in such a manner that the upper edge of the screen is 12m above the floor. Find the distance between the viewer and the wall where ...
21
votes
2answers
1k views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
-1
votes
2answers
73 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
1
vote
0answers
16 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...