Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,295
questions
4
votes
2
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647
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Hypercontractivity of two simple random variables, $E[XY]^s \le E[X^s]E[Y^s].$
For $\alpha,\beta\ge 0$, let $X\in\{1,\alpha\}$ and $Y\in\{1,\beta\}$ be two random variables such that $$XY = \begin{cases}
\alpha\beta \quad & \text{with probability} \quad p_{11}\\
\alpha \quad ...
2
votes
0
answers
113
views
A variant of the optimal transport
Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows:
$$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$
where the inf is taken ...
2
votes
0
answers
403
views
Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]
Is that correct $R^2\cong R$ as measurable spaces?
If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?
2
votes
1
answer
253
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Schwartz space on $\bigcup_{n=1}^CR^n$
I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
5
votes
2
answers
231
views
Can we stay invertible while approximating linear maps in Sobolev spaces?
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.
Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
5
votes
0
answers
618
views
Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)
Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...
1
vote
1
answer
189
views
Giving Uniform Bound on Differences of Sums of Converging Polynomials
The title does not quite capture the essence of the difficulty, please allow me to be more explicit here.
I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
4
votes
0
answers
162
views
Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
1
vote
0
answers
74
views
Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
11
votes
1
answer
942
views
Proof of Green's formula for rectifiable Jordan curves
$\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
10
votes
1
answer
561
views
Nonlinear Schrödinger equation with discrete Laplacian
In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
5
votes
2
answers
413
views
Existence of Solution, System of Equations
Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations ...
13
votes
2
answers
310
views
Semigroup of differentiable functions on real line
Let $D(\mathbb R) $ be the set of all differentiable functions $f: \mathbb R \to \mathbb R$. Then obviously $D(\mathbb R)$ forms a semigroup under usual function composition. Can we characterize (up ...
-1
votes
1
answer
148
views
How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]
It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$
Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
3
votes
0
answers
102
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Dependency of the Wasserstein distance on the parameter: a differential perspective
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below:
$$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
19
votes
1
answer
549
views
Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?
Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$?
Remark 1. The answer to the ...
17
votes
2
answers
3k
views
The Riemann hypothesis as a problem in analysis
The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the ...
1
vote
1
answer
234
views
Reference request for weak solutions of an Elliptic PDE
Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one.
I want to find weak, non trivial, continuous, solutions of $$\...
4
votes
2
answers
140
views
A kind of exponential concavity for polynomials?
Is there $C > 0$ such that the inequality
$$
\prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right)
$$
holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...
3
votes
1
answer
137
views
PDE satisfied by projection of a function onto a subspace
Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$}...
1
vote
1
answer
585
views
Applications of the Calderon-Zygmund theory to PDE's
I am planning to build a PDE topics course focussing on the Calderon-Zygmund theory. I know some important applications of the Calderon-Zygmund theory to elliptic PDEs, but I don't know enough to get ...
31
votes
2
answers
3k
views
Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$
The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned ...
7
votes
2
answers
632
views
Non-separable metric probability space
Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
...
1
vote
0
answers
110
views
Higher Order Partial Derivatives Test
For a nonconstant analytic function $ℝ→ℝ$, a point is a local minimum iff at that point, the order of the first nonzero derivative is even and that derivative is positive. Is there an analogous test ...
41
votes
6
answers
9k
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"Long-standing conjectures in analysis ... often turn out to be false"
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
2
votes
0
answers
189
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
1
answer
217
views
Computing minimum / maximum of strange two variable funcion
I want to compute $$\max_{\frac{1}{5 \theta }\leq \alpha \leq \frac{1}{2}} \left(\frac{\alpha\log (\alpha)}{1-\alpha} + \log \left( 1 - \alpha\right) + \frac{1}{1-\alpha} \cdot \left( f\left(\frac{...
0
votes
2
answers
127
views
Dirichlet problem for capillary equation over convex domain
Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary.
Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function.
Let $L$ be a quasilinear elliptic ...
22
votes
1
answer
2k
views
Differential equation changing sign almost everywhere
Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is $f$ necessarily a constant function?
...
3
votes
1
answer
271
views
Function square-integrable
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...
2
votes
3
answers
292
views
Uniqueness of solution depending on constant?
I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation.
I encountered the following integral equation for functions $f:[...
10
votes
1
answer
785
views
Current vs Varifold
I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
1
vote
1
answer
63
views
Generalizations of Pedal Coordinates
I recently "stumbled upon" the article
Pedal coordinates, Dark Kepler and other force problems by Petr Blaschke from 2017; further search about Pedal Coordinates didn't bring up any other ...
-3
votes
1
answer
370
views
A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
1
vote
0
answers
117
views
Estimation of the integral $\int_a^b e^{2\pi i f(x)} dx $
Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that
$$
\min_{x\in [a,b]} |f'(x)|>\lambda
$$
It is ...
18
votes
2
answers
568
views
Existence of an antiderivative function on an arbitrary subset of $\mathbb{R}$
Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $x$ for every $x\in I$ where $I\subset \mathbb R$ could be arbitrary. Does there always exist a function $F:\mathbb{R}\to \mathbb{R}$ differentiable ...
1
vote
1
answer
120
views
Do functions exist and are they dense? Or does it depend on the basis?
Consider an orthonormal basis $(\varphi_n)_{n \in \mathbb N}$ of $L^2(\mathbb R).$
We consider the functionals $\Phi_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...
1
vote
2
answers
211
views
Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?
Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
1
vote
1
answer
195
views
An equation in Gamma function has at most (n-1) positive solutions
I have to prove some result. And for that, I need to prove this new problem.
To prove,
$c_{1}\Gamma(z+b_{1})+c_{2}\Gamma(z+b_{2})+\ldots+c_{n}\Gamma(z+b_{n})=0$
has at most $(n-1)$ real positive ...
3
votes
0
answers
129
views
Function and distance on bounded set [closed]
Does there exist a bounded set $A \subset \mathbb{R}^ n$ (for some $n$) and a function $f:A\to A$ (not necessarily onto) such that $\forall x,y \in A$ then $\|x-y\| < \|f(x)-f(y)\|$?
If such a ...
6
votes
1
answer
182
views
Reference request: A collection of topologies on $\mathbb{N}$ formed via series
First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...
1
vote
1
answer
130
views
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
1
vote
0
answers
87
views
Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
2
votes
2
answers
231
views
Sum of a terms in a divergent series taken along indices the sum of whose reciprocal diverges. Can the sum converge?
Let $\{x_n\}_{n=1}^{\infty}$ be a monotone decreasing sequence of positive real numbers such that $\sum_{n=1}^{\infty} x_n$ diverges. Also let $\{k_n\}_{n=1}^{\infty}$ be a strictly increasing ...
3
votes
1
answer
166
views
On the values of an entire function
Let $0<q<1$ and consider the entire function $f(z)=\displaystyle \sum_{k=0}^\infty q^{k^2}z^k$. For $a>1,$ denote $m_j=f(a^j),\; j=0,1,2,\dots.$
Question: Does there exist an entire function ...
3
votes
1
answer
406
views
Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?
Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$.
Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows:
$$\mathcal{...
2
votes
1
answer
361
views
Description of (completely) bounded operator
I am somewhat a beginner in the field of operator algebras and was wondering about the following:
Let $T$ be a linear map between the space of bounded operators $B(H)$ on some Hilbert space and $S$ a ...
0
votes
2
answers
216
views
An inequality on length of two curves [closed]
I am looking for a proof, reference, comment of an inequality as follows:
If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that:
$f(a)=g(a)$ and $f(b)=g(b)$
$(...
0
votes
1
answer
188
views
Is $f(x)$ is more curvature than $g(x)$ then length of $f(x)$ seem longer than length of $g(x)$?
In my obsevation:
If $f(x)$ and $g(x)$ be two continuous funcions, and they have derivative and second derivative are also continuous in interval $[a, b]$. If $f(x)$ is more curvature than $g(x)$ in ...
1
vote
0
answers
201
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...