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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

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On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
Elias Costa's user avatar
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39 votes
0 answers
992 views

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms. It suffices to show that the terms of the sequence $$\,b_n=\mathrm{e}^...
Yiorgos S. Smyrlis's user avatar
35 votes
0 answers
1k views

Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
Mason's user avatar
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32 votes
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993 views

Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
Torsten Schoeneberg's user avatar
28 votes
0 answers
1k views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

Update 24 Nov 2015: It is solved. Please refer to this arXiv paper. For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids ...
sobasu's user avatar
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25 votes
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2k views

Theorem 6.17 in Baby Rudin, 3rd ed: $\int_a^b f \,d\alpha = \int_a^b f(x) \alpha^\prime(x) \,dx$

Here is Theorem 6.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Assume $\alpha$ increases monotonically and $\alpha^\prime \in \mathscr{R}$ on $[a, b]$. Let $f$ be ...
Saaqib Mahmood's user avatar
23 votes
1 answer
387 views

Supremum of $\int_{-\infty}^{\infty}\frac{p'(x)^2}{p(x)^2+p'(x)^2}\,\mathrm{d}x$

Question. Let $P_d = \{ p \in \mathbb{R}[x] : \deg p = d\}$ denote the set of all degree $d$ polynomials with real coefficients. Also, for $p \in \mathbb{R}[x]$, define $$ I(p) = \frac{1}{\pi} \int_{-\...
Sangchul Lee's user avatar
23 votes
0 answers
404 views

Extension of Vector Field in the $\mathcal{C}^r$ topology.

Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^...
Matheus Manzatto's user avatar
22 votes
0 answers
1k views

Refinement of a famous inequality

I refine a famous inequality this is the following : Let $x,y>0$ then we have : $$x^n+y^n\leq \Big(\frac{x^n+y^n}{x^{n-1}+y^{n-1}} \Big)^n+\Big(\frac{x+y}{2}\Big)^n$$ It's equivalent to : $$x^n+1\...
user avatar
22 votes
0 answers
622 views

A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that $\...
MT_'s user avatar
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22 votes
0 answers
541 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
Dawid C.'s user avatar
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21 votes
0 answers
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Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
jacopoburelli's user avatar
21 votes
1 answer
2k views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
Ufuk Can Bicici's user avatar
18 votes
0 answers
185 views

How do permutations of $\Bbb N$ affect series?

Let $G$ be the group of all permutations of $\mathbb{N}$ and $\sum a_n$ a conditionally convergent series of reals. What do we know about how $G$ "acts" on this series? We can partition $G$ ...
anon's user avatar
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18 votes
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662 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
char's user avatar
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18 votes
0 answers
525 views

Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x) = x$ and $f_{n+1}(x) = f_n(x) - f_n'(x) x (1-x)$. These polynomials have some special properties, for example $f_n(0) = 0$, $f_n(1) = 1$, and all $n+1$...
TomH's user avatar
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17 votes
0 answers
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Other approaches to $\int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \text{d}x$

Let $K(x) = \int_0^{\pi/2}\frac{1}{\sqrt{1-x^2 \sin^2 \theta}}d\theta$ be the complete elliptic integral of first kind. It could be shown that $$ \int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \...
Setness Ramesory's user avatar
17 votes
1 answer
757 views

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset\Bbb{R} $ intersect every uncountable $\mathcal{F}_{\sigma}$ set. $B\subset \Bbb{R}$ is said to have the property of Baire if $B=U\triangle M$ where $U$ is open and $M$ is meager. Does ...
Ussesjskskns's user avatar
17 votes
0 answers
255 views

Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
msm's user avatar
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16 votes
0 answers
3k views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega \...
user119615's user avatar
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16 votes
0 answers
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Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ g:\...
Alexander Sokol's user avatar
15 votes
0 answers
581 views

How to prove this lemma related to Rolle's theorem

For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...
Coolwater's user avatar
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15 votes
0 answers
501 views

How to construct examples of functions in the Spaces of type $\mathcal{S}$

There are $3$ $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. They are defined by: $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{...
creative's user avatar
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15 votes
0 answers
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Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
Julian Lee's user avatar
15 votes
1 answer
2k views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
fbg's user avatar
  • 407
15 votes
0 answers
454 views

Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
combcereal's user avatar
15 votes
1 answer
389 views

Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
afshi7n's user avatar
  • 257
14 votes
0 answers
200 views

Does there exist any $p >0$ such that $\frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty$?

Does there exist any $p >0$ such that \begin{equation*} \frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty \;? \end{equation*} If there is one, what's the infimum of those $p$? Is it also a minimum? I ...
Bob's user avatar
  • 5,755
14 votes
0 answers
575 views

Is there any condition that makes a measure zero set necessarily countable?

Background : Let us consider the Lebesgue measure space $(\Bbb{R}, \mathcal{L}(\Bbb{R}),m) $. Here measurable set means Lebesgue measurable and measure means Lebesgue measure. $\mathcal{S}\subset \...
Ussesjskskns's user avatar
14 votes
1 answer
683 views

About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$

This inequality is due to user RiverLi : Let $0<x\leq 1$ then we have : $$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$ I propose another one wich states : Let $0<x\leq 1$ ...
Miss and Mister cassoulet char's user avatar
14 votes
1 answer
570 views

Function of two sets

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
pi66's user avatar
  • 7,174
14 votes
0 answers
329 views

Putnam 2018 - Exercise A.5 - proof check

The problem statement is as follows. Let $f:\Bbb R \to \Bbb R$ be an infinitely differentiable function satisfying $f(0) = 0$ and $f(1) = 1$, and $f(x) \geq 0$ for all $x \in \Bbb R$. Show that there ...
dfnu's user avatar
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14 votes
0 answers
2k views

Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

Here is part (a) of Theorem 6.12 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$f_1 + ...
Saaqib Mahmood's user avatar
13 votes
0 answers
350 views

Hardy's inequality proof using Doob's inquality

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
mathex's user avatar
  • 642
13 votes
0 answers
194 views

Rearranging series and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. Let $...
Noah Schweber's user avatar
13 votes
0 answers
172 views

How close can polynomials be in magnitude?

If $p$ and $q$ are polynomials and $|p(t)|=|q(t)|$ on the interval $[0,1]$, then in fact $p(t)=\omega q(t)$ for some constant phase $\omega$. I am curious about quantitative strengthenings of this ...
felipeh's user avatar
  • 3,810
13 votes
0 answers
373 views

Existence of function satisfying $f(f'(x))=x$ almost everywhere

My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...
Pascal's user avatar
  • 3,781
13 votes
1 answer
365 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
Rajesh D's user avatar
  • 4,247
13 votes
0 answers
1k views

Schilling's proof of the Feynman-Kac Formula for Brownian motion

This is part of a proof to the Feynman-Kac formula from Schilling's Brownian motion. I need some help understanding the proof to this theorem. Theorem (Kac 1949). Let $(B_t)_{t\ge 0}$ be a $d$-...
nomadicmathematician's user avatar
13 votes
2 answers
373 views

Covering a set $A \subset \Bbb R$ by two families of disjoint intervals taken from given intervals

Let $A \subset \Bbb R$ be a bounded set. Every element $a \in A$ is the center of some given open interval, let's denote it by $I_a=(a-r_a, a+r_a)$. I'm interested in knowing the following: Can we ...
Emolga's user avatar
  • 3,537
13 votes
0 answers
626 views

Fréchet L-Spaces

NOTE: The question has now been posted on MathOverflow: Fréchet L-Spaces According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous ...
Bumblebee's user avatar
  • 18.3k
12 votes
0 answers
350 views

Approximate the sum of a non $C^1(0,1)$ function by its integral

Consider the function $f: [0,1] \to \mathbb{C}$ defined by $$ f(x)=\sum_{n=1}^{9} e^{2\pi n i x}, $$ so that $$ |f(x)|=\bigg|\frac{\sin(9\pi x)}{\sin(\pi x)}\bigg|. $$ I'm interested in approximating $...
Itachi's user avatar
  • 586
12 votes
0 answers
316 views

"Taylor series" is to "Volterra series" as "Padé approximant" is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
Mike Battaglia's user avatar
12 votes
0 answers
373 views

Convergence of a Stochastic Process - Am I missing something obvious?

In the paper On the Convergence of Stochastic Iterative Dynamic Programming Algorithms (Jaakkola et al. 1994) the authors claim that a statement is "easy to show". Now I am starting to suspect that it ...
Felix B.'s user avatar
  • 2,395
12 votes
0 answers
9k views

Rigorous Proof of Slutsky's Theorem

I was hoping to type up my proof of Slutsky's Theorem and get confirmation on the excruciating details being all correct... Statement of Slutsky's Theorem: $$\text{Let }X_n, \ X,\ Y_n,\ Y,\text{ share ...
OGV's user avatar
  • 541
12 votes
0 answers
501 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
Pedro Vaz Pimenta's user avatar
12 votes
0 answers
740 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
michek's user avatar
  • 668
11 votes
0 answers
148 views
+50

When is $ \sum_{k \in \mathbb{Z}}\left(\frac{\sin(k)}{k}\right)^{n}=2 \int_{0}^{\infty}\left(\frac{\sin(x)}{x}\right)^{n}dx$?

Define sequences $$a_n = \sum_{k \in \mathbb{Z}}\left(\dfrac{\sin(k)}{k}\right)^{n}, b_n = \int_{0}^{\infty}\left(\dfrac{\sin(x)}{x}\right)^{n}dx, \quad n \in \mathbb{N}.$$ I am trying to see if ...
Sam's user avatar
  • 3,233
11 votes
0 answers
268 views

Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?

This question was firstly asked in Mathematics Stack Exchange. Getting no answer, I copied it to Math Overflow, as Moishe Kohan commented. For a vector space $V$ over $\mathbb R$, I say a subset $S$ ...
yummy's user avatar
  • 303
11 votes
0 answers
286 views

The fabulous Wenger's Summation

The user Avery Wenger said he came up with this problem randomly and I think it's very interesting! Let $s:\mathbb N\to\mathbb N$ be the sum of digits function. Then, Wenger defines $w:\mathbb N\to\...
Alma Arjuna's user avatar
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