Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,258
questions
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How should the "measure theoretic" Jacobians of a dynamical map be understood in Lai-Sang Young's "Recurrence Times and Rates of Mixing"
In Young's article: Recurrence Times and Rates of Mixing, she uses multiple times the notation $JF, JF^k, JF^R$ to mean the Jacobian of a dynamical map $F:\Delta\to\Delta$ w.r.t. a given reference ...
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10
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An expansion for 2d Euler equation
Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$:
$$
-\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
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0
answers
54
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How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
0
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0
answers
70
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Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
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45
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Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
3
votes
2
answers
356
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Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
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137
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Fourier series but different waveform
Given a nondegenerate smooth simple closed curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$. Let $f_n: [0,2\pi]\to \mathbb C \setminus \{0\}$ be such that $f_n(...
2
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0
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103
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A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
2
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0
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80
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An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
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46
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Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
2
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2
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146
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Is there a compactly supported differentiable function whose Fourier transform is not in L1?
In my MSE answer here, I discussed the example of compactly supported continuous function
$$g(x)=
\begin{cases}
\dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\
0,&\text{otherwise}
\end{cases}$$
...
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1
answer
175
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Does this property implies Lipschitz continuity?
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be such that, for $x,y,z \in \mathbb{R}^{n}$, we have that
$$|f(z) - f(x)| \leq |f(z) - f(y)| \Rightarrow \|z-x\| \leq \|z-y\|$$
Can I say that this ...
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1
answer
43
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Everywhere existence of marginals
Let $f\in L^1(\mathbb{R}^2)$ be a (joint) probability density function which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$.
What is a necessary and sufficient condition under which the ...
8
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The cardinality of projections of subsets of the Hilbert cube by inner products
I have three related questions.
Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
2
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1
answer
98
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Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
6
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264
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+100
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
2
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1
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97
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On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
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Are these two spaces of functions identical?
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there.
...
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0
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77
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Zero of gradient of a sum of squares of polynomial
Let $f_1,\ldots,f_n\in \mathbb{R}[x_1,\ldots,x_n]$ have non zero constant jacobian determinant.
Then is it true that $f_1^2+\cdots+f_n^2$ has any critical point?
Let me explain a motivation.
If ...
2
votes
0
answers
99
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Infinite multiplicity set of continuous functions
Definitions:
Fix a function $f:\mathbb{R}\to\mathbb{R}$ such that $f$ obtains each value only finite (possibly $0$) number of times. We say $E \subset \mathbb{N}$ is the "multiplicity set" ...
4
votes
2
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331
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Nontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by
$$ v(r,\theta) = u(\frac{1}{r},\theta)$$
is also harmonic for $r>0$. Note that the Kelvin ...
1
vote
1
answer
77
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Sufficient initial conditions for "Non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
0
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0
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26
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Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions
For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
1
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1
answer
101
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A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
0
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0
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128
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Are there "gaps" between Lipschitz functions and $C^1$ functions?
I was wondering about this fact. By Rademacher theorem, it is well known that a Lipschitz function in $\mathbb{R}^d$ is $\mathcal{L}^d$-a.e. differentiable. Of course, a $C^1$ function is everywhere ...
0
votes
1
answer
69
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Triviality of functions integrated against some trigonometrical kernels
Let's say we have a smooth real symmetric function $f\in C^\infty(\mathbb{R}^2)$ satisfying next identity:
$$\int_{\mathbb{R}^2}(e^{-i\xi x}-e^{-i\xi y})f(x,y)\,dx\,dy=0\quad\forall \xi\in\mathbb{R}. $...
1
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0
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57
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Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
1
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1
answer
74
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On the definition of symmetric rearrangement
For a measurable function $u:\mathbb{R}^{n}\to \mathbb{C}$ one usually defines the symmetric rearrangement $u^{*}:\mathbb{R}^{n}\to \mathbb{R}^{+}$ as follows:
\begin{equation*}
u^{*}(x)=\int_{0}^{\...
0
votes
1
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65
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Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$
We fix $p \in [1, \infty)$. We have for every $f \in L^p (\mathbb R^d)$ that $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$. I wonder if there is an estimate of above decay, i.e.,
Is there a ...
0
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0
answers
22
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Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
3
votes
0
answers
54
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About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
-1
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0
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80
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Stochastic dominance for (~)random harmonic series
$\DeclareMathOperator\Pr{Pr}$Consider the series $\sum_n^\prime a_nR_n$, where $a_n=\frac{(-1)^n}{n+c}$ for some constant $c\in(0,1)$ and $\{R_n\}$ denotes a sequence of i.i.d. Bernoulli random ...
2
votes
1
answer
93
views
Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane
Suppose that
$f$ and $g$ are polynomials with nonnegative coefficients,
the degree of $g$ is greater than the degree of $f$,
$g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
-2
votes
1
answer
60
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Partial derivative in terms of Kronecker delta and the Laplacian operator [closed]
How can the following term:
$$ T_{ij} = \partial_i \partial_j \phi$$
be written in terms of Kronecker delta and the Laplacian operator $\mathbin\bigtriangleup = \nabla^2$?
I mean is there a relation:
$...
0
votes
1
answer
129
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About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
-1
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0
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Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$?
Question: Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have:
$$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$
where $r>...
0
votes
2
answers
124
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Cauchy's functional multiplicative equation on the unit interval
This question might be trivial, but I didn't find a clean reference and have not attempted to prove it myself yet: Let $f:[0,1]\rightarrow [0,1]$ be a continuous and monotonic function such that $f(0)=...
2
votes
1
answer
130
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Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
6
votes
0
answers
564
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Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
0
votes
3
answers
266
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A generalisation of Tchebychev inequality
Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$.
Is it true that exists $ u$ any real function, and $a,b$ monotone ...
2
votes
2
answers
186
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$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
2
votes
1
answer
232
views
Decompose a function into a bounded part and a Lipschitz part
Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$
Are there functions $g,h: \mathbb R^d \...
5
votes
0
answers
91
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
3
votes
0
answers
77
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
2
votes
2
answers
176
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
2
votes
1
answer
145
views
Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued ...
1
vote
1
answer
123
views
To find a $2\pi$-periodic function with a property
I recently came across the following question in my research, and I don't know how to proceed this problem.
Question: How to find a function $g(x)$ such that it satisfies
(1) $2\pi$ periodic
(2) odd
(...
4
votes
1
answer
259
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
0
votes
1
answer
117
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
0
votes
0
answers
87
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On polynomial equation of fourth order depending on two parameters and bound on a maximal root
I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:
\begin{eqnarray}
F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...