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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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Bounded solutions of nonlinear third-order ODEs

I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, ...
Zhang Yuhan's user avatar
1 vote
1 answer
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Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$

Let $\Omega=[0,1]^d$ for some $d\ge 1$, and let $w:\Omega \to (0,\infty)$ be a continuous function. Is is true that $$\Vert w f \Vert_{L^2(\Omega)}^2+ \left\Vert \frac{1}{w}g \right\Vert_{L^2(\Omega)}^...
Tulip's user avatar
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How are concrete (non-abstract) notions of a set defined, such as $\{ 1,2,3 \}$?

I have learnt the axioms of ZFC from Tao's Analysis I, which define what a set abstractly is. However, I was wondering how we define concrete notions of a set, such as $\{ 1,2,3 \}$- how is this ...
Princess Mia's user avatar
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1 answer
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What is the limit of the alternating series $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ as $z\to1$ for a sequence $T_n\sim cn$?

Let $(T_n)_{n>0}$ be an increasing sequence of positive integers and $c>0$ a positive number such that $\lim_{n\to\infty}\frac{T_n}{n}=c$. Write $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ for a real ...
Mathew's user avatar
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In this proof, how does the Fourier transform work? [closed]

I recently read the article 'E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86 (2006), 271–291'. The Theorem 2.1. confused me....
M.A.D.M.A.N's user avatar
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18 views

Density of a Lattice.

For an integer $N \ge 1$ denote by $K_N$ the number of pairs of integers $(m, n)$ with $|m|, |n| \le N$ such that the straight line from the origin to $(m, n)$ does not intersect any other lattice ...
MathematicalPhysicist's user avatar
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0 answers
9 views

Finding an upper bound for / computing a $d$-dimensional integral

I want to know if there is a clever way of computing this $d$-dimensional integral, or if and how it is possible to use polar coordinates? \begin{align} \int_{|m|\leq k_F} d^d m \left( ...
putti.123's user avatar
  • 339
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22 views

The fourier transform of cut-off function of $\mathcal S$ is in $L^1$?

Let $\mathcal S(\mathbb R)$ be Schwartz class and $f\in\mathcal S(\mathbb R)$, and $I:=[-1,1]$. Define $\chi_I f$ by $\chi_I f(x):=\chi_I(x) f(x)=\begin{cases}f(x) &\text{if }x\in I\\ 0&\text{...
daㅤ's user avatar
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What is the reduction formula for $\tan^n(x)\tan(mx)$ [closed]

I started with making $\tan^n(x)$ become $\tan^{n-2}(x) \tan^2x$ I am stuck in the second integration by parts.
Oluwabunmi Sunmola's user avatar
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16 views

Legendre transformation is a continuous map

I was reading about an article about the Legendre transformation in convex analysis. It was defined like this : Let $\varphi$ be a continuous function on $[0,+\infty]$ and convex on $\mathbb{R}_{>...
MoinsUnPuissanceN's user avatar
4 votes
1 answer
90 views

Can anyone explain the constant in this relation

When the following function is plotted in desmos, $$y^x = \frac{(xy)^y}{k},$$ Graph where $k = 1$, i.e. $y^x = (xy)^y$. As k approaches the number $\approx 2.89473713041139$ ($1/k \approx 0....
Nathan Michael's user avatar
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1 answer
43 views

If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$? [duplicate]

Let $a$ and $b$ be any real numbers such that $a < b$, and let $S$ be a (nonempty) subset of the closed bounded interval $[a, b]$ such that $S$ has measure $0$. Now let $f \colon [a, b] \...
Saaqib Mahmood's user avatar
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0 answers
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Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]

Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
Saaqib Mahmood's user avatar
0 votes
1 answer
28 views

Bump function with integral $1$ and value $1$ at zero

How can i contruct a smooth bump function $F$ on $\Bbb{R}^n$ such that $F(0)=1$ and with integral $1$? I have tried to manipulate the function $f(x)=e^{-\frac{1}{x^2}}$ if $x>0$ and $f(x)=0$ if $x \...
Marios Gretsas's user avatar
1 vote
0 answers
41 views

If $f$ is discontinuous but continuous for each component,is there a continuous function $g$ that makes $f \circ g$ discontinuous for some component?

For example $$f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq (0,0)\\ 0 &(x,y)=(0,0)\end{cases}$$ $f$ is discontinuous but continuous for each component, In polar coordinates(Equivalent ...
9sky's user avatar
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2 votes
0 answers
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How to sketch the series $S(x)=\sum_{n=2}^{\infty}\frac{\sin nx}{\ln n}$ near zero?

$$ \mbox{I am confronted with a series}\quad \operatorname{S}\left(x\right) = \sum_{n = 2}^{\infty}\frac{\sin\left(nx\right)}{\ln\left(n\right)} $$ in a book discussing Fourier Series. The book says ...
Andyqian7's user avatar
1 vote
1 answer
67 views

$\mathbb{N}$ is uncountable?

I recently saw a proof that $\mathbb{R}$ is uncountable using Baire's Category Theorem. It goes like this: Suppose $\mathbb{R}$ is countable then $\mathbb{R} = \cup(x_n)$. Since $\mathbb{R}$ is ...
Jackson Smith's user avatar
4 votes
1 answer
166 views

Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as $$ \log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2} $$ where $\arg(z)$ is the standard branch of the ...
Max's user avatar
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4 votes
1 answer
380 views

Is this "continuous" function really continuous?

Let's assume $n \ge 2.$ Suppose $f:\Bbb R^n \times \Bbb R \to \Bbb R^{n+1}$ has the form $$ f(x,t) = (\phi_t(x),t), $$ where $\phi_{t_0}:\Bbb R^n \to \Bbb R^n$ is a continuous function for each fixed $...
BigbearZzz's user avatar
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0 votes
0 answers
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Prove every Cauchy sequence in R is bounded

Here's my attempt to prove every Cauchy sequence in $\mathbb{R}$ is bounded, I would like to see if there are any flaws in it. Proof: Let ${(x_n)}$ be a Cauchy sequence in $\mathbb{R}$. Now let's ...
Jackson Smith's user avatar
1 vote
2 answers
74 views

About solution to homogeneous ODE $u'' + u = 0$

It is known that the solution to $u'' + u = f(x)$ with $u(0) = u'(0) = 0$ is $$u(x) = \int_0^x\sin(x - \xi)f(\xi)d\xi,$$ where $\sin(x-\xi)$ is a solution to $$\frac{d}{dx}R + R = 0, \ \ \ x > \xi,$...
user57's user avatar
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2 votes
0 answers
46 views

Alternate proof to the Extreme Value Theorem

I'm following Spivak's Calculus and was revisiting some of my notes when I think I found a much more straightforward proof for the Extreme Value Theorem, compared to the one given in the book. I was ...
Aryaan's user avatar
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1 vote
1 answer
38 views

Composition of functions, once for all

I need certainties. Consider two functions $f: A \to B$ and $g: C \to D$. For what I am going to ask, it's not a loss of generality if we consider multivariable functions with domains and or codomains ...
J.N.'s user avatar
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3 votes
0 answers
122 views

What is this notion of continuity? Pt. 2

Let $f : \mathbb{Z}_p\to \mathbb{Z}_q$, where $\mathbb{Z}_p$ ($\mathbb{Z}_q$) are the $p$-adics ($q$-adics) for $p\neq q$. I have encountered the class of all $f$ satisfying \begin{equation}\tag{1}\...
Mark Schultz-Wu's user avatar
0 votes
1 answer
30 views

If $(X, A, m)$ is $\sigma$-finite and $B\subseteq A$ is a $\sigma$-sub-algebra, then is $(X, B, \nu)$ $\sigma$-finite?

Here $\nu$ is defined in the following way: $$\nu(E)= \int_E f dm$$ where $f$ is non-negative, $A$-measureable. I don't see why we can assert that $\nu$ is also $\sigma$-finite since the subsets of $B$...
Hyperbolic Cake's user avatar
0 votes
1 answer
37 views

A simple question about bounding a sum

Let $\Lambda_N:=\{-N+1,\dots,N-1\}^2\subset\mathbb{Z}^2$ be a set of lattice points in $\mathbb{Z}^2,$ and let $\gamma=\frac{1}{\sqrt{N}}.$ For $x\in\Lambda_N$ with $\|x\|_2>\gamma^{-1}$(with $\|\...
Chang's user avatar
  • 329
1 vote
1 answer
192 views

What is this notion of continuity?

Let $f : \mathbb{Z}_p\to \mathbb{R}$, where $\mathbb{Z}_p$ are the $p$-adics. I have encountered the class of all $f$ satisfying $$ \forall x\in\mathbb{Z}_p, f(x) = \lim_{n\to\infty}f(x\bmod p^n) $$ ...
Mark Schultz-Wu's user avatar
0 votes
0 answers
68 views

Theorem 7.48 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: Lebesgue's Criterion for Riemann Integrability [closed]

Here is Theorem 7.48 (Lebesgue's Criterion for Riemann Integrability) in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition: Let $f$ be defined and ...
Saaqib Mahmood's user avatar
1 vote
2 answers
118 views

Prove that $T$ is not a compact operator.

Let $T:\ell_{2}(\mathbb{Z})\rightarrow \ell_{2}(\mathbb{Z})$ be the operator defined by, $$T((x_i)_{i\in \mathbb{Z}})=((y_i)_{i\in \mathbb{Z}}).$$ where $$ y_{j}=\frac{x_{j}+x_{-j}}{2}, \quad j \in \...
Ricci Ten's user avatar
  • 520
-1 votes
0 answers
31 views

Example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$

I am trying to find an example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$. Let $T_F(x) := sup\{ \sum_{i=1}^n |F(x_j) - F(x_{j-1})| :n \in \mathbb{N}, -\infty<...
someeed's user avatar
  • 469
0 votes
1 answer
44 views

Is my formula for this projection correct?

Let $\phi \in L^{2}(\mathbb{R}^{d})$ be fixed. Denote by $P$ the orthogonal projection onto the subspace orthogonal to $\text{span}\{\phi\}$. In other words, for $f \in L^{2}(\mathbb{R}^{d})$ set: $$(...
InMathweTrust's user avatar
3 votes
0 answers
82 views

When is $\max_a \max_b =\max_b \max_a $ [duplicate]

Lets consider a real valued and continuous function $f$ which takes 2 objects as input. If $f$ is symmetric: $f(a,b) = f(b,a)$ then I can obviously say: $$\max\limits_{a\,\in A} \max\limits_{b\, \in B}...
v.tralala's user avatar
  • 299
0 votes
0 answers
88 views

Excercise on Lebesgue-Integral

I have a question regarding Lebesgue-Integration. I am working with the book by Ziemer (Modern Real Analysis, https://www.math.purdue.edu/~torresm/pubs/Modern-real-analysis.pdf) and I am trying to ...
RobRTex's user avatar
  • 49
0 votes
1 answer
38 views

Compactess of a set in $\mathbb{R}^d$ defined as the union of compact sets

Let $f:[a,b]\to \mathbb{R}^d$ be a function of class $C^1$. Let us consider the following set: $$ \mathcal{A}:=\bigcup_{x\in [a,b]}\{y\in \mathbb{R}^d, \quad ||y- f(x)||\leq 1/2 \} $$ I think that ...
hanava331's user avatar
  • 109
2 votes
1 answer
37 views

Stochastic convergence with and without rate

I have a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$. If I know that there exists a sequence of strictly positive real numbers $(\epsilon_n)_{n \in \mathbb{N}}$ which is such that $\...
Akurishen's user avatar
3 votes
1 answer
41 views

Distance of a real number to a discrete set of scaled sine values

Let $M>0$ be an integer, $c\in(0,\frac{1}{2})$ a real number, $$ a_{m,n}:=\frac{2n}{\pi}\sin\frac{m\pi}{2n}, $$ $$ A_n:=\left\{a_{m,n}:~m=1,\ldots,n-1\right\}, \text{ and} $$ $$ d_n:=\operatorname{...
Hui Zhang's user avatar
  • 594
0 votes
1 answer
52 views

$C^1[a,b]$ non-linear function, such that $f'' =0$ a.e.

I've been trying to come up with an example of such a function, but haven't succeeded. I was thinking of using a cantor function, since it is continuous, and its derivative is zero almost everywhere......
Hyperbolic Cake's user avatar
-1 votes
0 answers
72 views

Show that $\liminf(x_k \cdot y_k) = x \cdot \liminf(y_k)$ [closed]

We have a sequence $(x_k)$ in which $\lim(x_k)= x$ and $x > 0$. We also have that $\exists k_0$ such that $y_k > 0 \space \forall k \geq k_0$, then show that $\liminf(x_k y_k) = x \liminf(y_k)$. ...
cardinalcat27's user avatar
3 votes
1 answer
42 views

Why do we need the right-continuity of $F$ at $a$ to prove $V_F[a,c]=\lim_{b\to a^+}V_F[b,c]$ where $F$ is of finite variation?

This is a continuation of this post. The book claims the following without proof: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. Suppose also that $a<c$ and $F$...
Shenron's user avatar
  • 75
2 votes
3 answers
56 views

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

I am reading the proof of the following result: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. If $b\in\mathbb{R}$, then $$ V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,...
Shenron's user avatar
  • 75
0 votes
0 answers
9 views

Why is studying upper bounds for $|I_\delta(\mathcal P,\mathcal L)|$ useful?

A natural problem in incidence geometry is counting the number of incidences of points and lines. For example, if $\mathcal P$ is a collection of points in $\Bbb R^d$, and $\mathcal L$ is a collection ...
stoic-santiago's user avatar
2 votes
0 answers
18 views

Sum of Dirichlet kernel for angle differences over $n$ angles on unit circle

Let $$D(\theta_i-\theta_j):= \frac{\sin((n+1/2)(\theta_i-\theta_j))}{2\sin\frac{\theta_i-\theta_j}{2}}$$ being the Dirichlet type of kernel of angle difference between $\theta_i$ and $\theta_j$ where $...
chloe's user avatar
  • 1,052
0 votes
0 answers
33 views

lower bound of $\sum \cos^k(\theta_i-\theta_j)$

In this question, Lower bound of sum of cosine of angle difference, the lower bound $\sum_{i,j\in [n]}\cos^2(\theta_i-\theta_j)\geq \frac{n^2}{2}$ is given in the answer. I am wondering, what is the ...
chloe's user avatar
  • 1,052
0 votes
1 answer
33 views

Which metrics (on vector spaces) can be induced?

Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$. I ...
HIH's user avatar
  • 451
1 vote
0 answers
55 views

Radon-Nikodym derivatives with restricted support

Let $\lambda$ be the Lebesque measure on $\mathbb{R}$ and $f$ and $g$ be two PDFs of two random variables. Then we can write: $$f(x) = \frac{d \mu}{d \lambda}(x) \qquad g(x) = \frac{d \nu}{d \lambda}(...
Jan Stuller's user avatar
  • 1,189
0 votes
0 answers
39 views

Reference for integral of an exponential with quartic argument [closed]

I have been trying to estimate an $L_{2}$ norm of some weighted polynomials. In that context an integral like below shows up. In wikipedia https://en.wikipedia.org/wiki/...
Sam Hilary's user avatar
0 votes
0 answers
23 views

Vector space structure for curve linear coordinates

The space ${\Bbb R}^2$ is the space of all 2-tuples $(x_1,x_2)$, where $x_1,x_2 \in {\Bbb R}$. Vector space structure is introduced in ${\Bbb R}^2$ in a straightforward manner. We can visualise this ...
WhyNót's user avatar
  • 43
0 votes
1 answer
25 views

Normal Cones on a Circular Arc

Let $C = \{(x, y) \in \mathbb{R}^2: x^2 + y^2 \leq 1, y \geq 0, y \leq x\}.$ I wish to determine the normal cone $N_C(x_0, y_0)$, for any $(x_0, y_0)$ on the boundary of $C$. There are six cases to ...
V. Elizabeth's user avatar
0 votes
0 answers
38 views

A question about the definition of derivative in different coordinate systems

The definition of the derivative goes like this: If $x$ is an interior point of a set $E \subseteq {\Bbb R}^n$, then a function $f: {\Bbb R}^n \rightarrow {\Bbb R}^m$ is said to be differentiable at $...
WhyNót's user avatar
  • 43
0 votes
1 answer
60 views

Solving $f(x) = f(\frac{a + b x}{c + d x}) = f(\frac{a' + b' x}{c' + d' x})$?

How to solve the equation $$f(x) = f(\frac{a x + b}{c x + d}) = f(\frac{a'x + b'}{c'x + d'})$$ For given real $a,a',b,b',c,c',d,d'$ ? Maybe this system of equations is a bit overdetermined in its ...
mick's user avatar
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