Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,269
questions
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About Euclidean distances
$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$.
Let $d$ denote the Euclidean distance in $\R^n$.
Do then ...
4
votes
1
answer
326
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Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
2
votes
1
answer
291
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Matching the integral of a function on smaller open sets
Let $f: [0, 1] \to \mathbb R$ be Lebesgue integrable with $\int_0^1 f \, d \mu = C.$
Question: For every $K$ with $0 < K \leq 1$, does there exist an open subset $U$ of $[0, 1]$ of Lebesgue measure ...
1
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0
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98
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Real analytic map with connected fibers
Let $X,Y$ be compact real analytic varieties. Suppose $Y$ is connected and there is a surjective analytic map $f:X\to Y$ such that each fiber of $f$ is connected. How to prove that $X$ is connected as ...
8
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0
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Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
0
votes
1
answer
183
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Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]
Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
5
votes
2
answers
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Relationship between KL, chi-squared, and Hellinger
There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
4
votes
1
answer
265
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Equidistribution of distances of integer points to a circle
I have noticed in the following graph that the euclidean distance
between points $k \in\mathbb{Z}^2\cap C_7^1$ ($C_7^1:={}$Circle with radius 7 and shell with thickness 1) and the nearest point on the ...
5
votes
3
answers
501
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How to prove this (corollary of) hyperplane separation theorem?
$X$ is a nonempty convex subset of $\mathbb{R}^n$ whose element is $x=\left(x_1,...,x_n\right)$.
The theorem is as follows.
If for each $x\in X$, there is an $i \in \left\{1,...,n\right\}$ such that $...
3
votes
1
answer
196
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Volume of 3-dimensional region
Let $G$ be bounded finitely connected domain in $\mathbb{R}^3$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature.
Each sufficiently small open ...
1
vote
1
answer
152
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Integral involving Bessel and Laguerre function
Is there a formulas for the following integral
$$\int^\infty_0 e^{-ar^2}L^1_k(b r^2)J_1(cr)r^d dr $$
where $L^1_k$ is the Laguerre polynomials of type 1 and $J_1$ is the Bessel function with $a,...
2
votes
1
answer
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Characterization of extendible distributions
I asked this question on Mathematics Stackexchange, but got no answer.
I found the following question which characterize the extension of a distribution in $\mathbb{R}$:
Let $f \in L_{\text{loc}}^{1}(...
1
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0
answers
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Control of solutions to nonlinear elliptic equations away from boundary
Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(...
5
votes
1
answer
174
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Intermediate value property for Sobolev functions
Let $d \geq 2$, and let $f \in W^{1, 1} (\mathbb R^d)$ be a Sobolev function.
Question: For any $a, b \in \mathbb R$ such that $\text{essinf } f \leq a < b \leq \text{esssup } f$, is it true that $\...
3
votes
0
answers
134
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A uniqueness result for the Neumann problem for the Laplace equation
Let $\Omega \subset \mathbb{R}^{3}$ be a $C^{1}$-domain, not necessarily bounded. Consider solutions $\phi : \overline{\Omega} \to \mathbb{R}$, $\phi \in C^{\infty}(\Omega) \cap C^{1}(\overline{\Omega}...
0
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1
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143
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Determine if an integral expression is in $L^2(\mathbb{R})$
Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...
0
votes
1
answer
128
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Seeking an integral formulation for an algebraic function
While working with a generating function for the Catalan numbers, I came across the integral representation
$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...
1
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2
answers
371
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Sum of reciprocals of A086005
Does the sum of reciprocals of terms of A086005 converge?
1
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0
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Identification of a limit point of a sequence of solution of ODE
Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$,
\begin{align*}
& v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\
&...
0
votes
0
answers
78
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An interpolation of $n!$ such that its derivatives have few zeros
The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties:
$\Gamma(n)=(n-1)!$ for $n=1,2,3,...$.
The $k$'th derivative $\Gamma^{(k)}$ has no zeros on $(0,+\infty)$ when $k$ is ...
3
votes
0
answers
114
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Choose a sub series of a random series, such that its expectation can be a given real number
Suppose $a>0$, and we have an infinite series of Bernoulli random variables $B_k$ with
$$\mathbb{Pr}{\large[}B_k=1{\large]} = \frac{1}{1+e^{a\cdot 2^k}}$$
Then
$$\text{E}\left[\sum_{k=-\infty}^{\...
1
vote
0
answers
155
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Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
3
votes
1
answer
328
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Regularity of boundary of a level set of a $C^{1,\alpha}$ function
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$.
What i want to ask is, if $S_C$ is nonempty for some $...
2
votes
2
answers
526
views
A net of lower semicontinuous functions
Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise.
Please is it true that one can extract a countable ...
4
votes
1
answer
286
views
Sharpest version of semiclassical Calderon-Vaillancourt theorem
Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
1
vote
1
answer
175
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Lipschitz aspect of a projection on the boundary of a convex
Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that
\begin{...
0
votes
1
answer
269
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
2
votes
0
answers
150
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Equivalence of implicit function theorem and Peano existence theorem in ODEs?
I was recently reading a book about the implicit function theorem (IFT): The implicit function theorem: history, theory, and applications, and before that I learned that Peano's existence theorem can ...
2
votes
0
answers
111
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Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc
Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$:
$$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
8
votes
1
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356
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Two dice yielding uniform distribution, part 2
Since this question is on the front page again, a generalization.
Let $p$ be prime, and let $a$ and $b$ be positive integers with $a+b=p-1$. Is it possible to have two loaded dice, one with sides ...
2
votes
0
answers
73
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Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
4
votes
2
answers
366
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Is this projection on the boundary of a convex Lipschitz?
Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, and $u\in\mathbb{R}^n\setminus\{0\}$ such that $u$ does not belong to the asymptotic cone of $C$ and is nowhere tangent to $\partial C$. ...
2
votes
1
answer
326
views
Property implies finite propagation speed
Let $u(x, t)$ be a (non-negative, bounded) function on $\mathbb{R}^{n}\times [0, +\infty)$ and suppose that $u$ satisfies some time-independent PDE, e.g. $\partial_{t}u=\Delta_{p}u$. Let us assume ...
2
votes
0
answers
122
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Multiple integral with diagonal constraint (short-range)
I am looking for an upper bound on the following integral:
$$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$
...
0
votes
2
answers
507
views
Smooth approximation for non differentiable function
Let $f(t) = \min(\frac{1}{\lvert t\rvert}, 1)$. I would like to find a smooth approximating function $g$ such that $f(t) \leq g(t)$ for all real $t$. Is there a nice function $g$ out there? Any ...
3
votes
1
answer
125
views
Weak convergence of integral averages
Note: This is a refinement of a previous problem.
Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f_n$ by
$$f_n ...
9
votes
0
answers
301
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A game of harmonic series(s)
Given a set $A\subseteq\mathbb{R}_{>0}$, consider the following (two-player, perfect-information, length-$\omega$) game $H_A$:
Players $1$ and $2$ alternately play strictly increasing natural ...
2
votes
1
answer
120
views
Can we say that there exists a measurable function $f$ such that $ \nu=f_{\#}\mu$?
Define a coupling $\pi\in \Pi(\mu,\nu)$ on the product space $(X\times X,\mathcal{F}\times\mathcal{F})$. let $\pi_x$ be the disintegration of $\pi$ with respect to the $\mu$, i.e. there exists a Borel ...
6
votes
1
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270
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Convergence of integral averages in $L^1$
Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f_n$ by
$$f_n (x) := \frac{1}{2g_n (x)} \int_{x - g_n (x)}^{x + ...
3
votes
0
answers
66
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Discreteness of the higher inductive-inductive Cauchy real numbers in real cohesive homotopy type theory
We work in cohesive homotopy type theory with propositional resizing, so that there is only one type of Dedekind real numbers $\mathbb{R}$ up to equivalence, and Mike Shulman's axiom $\mathbb{R}\flat$,...
8
votes
1
answer
440
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Dirichlet-to-Neumann map on Lipschitz domains
Let $\Omega$ be a bounded domain with a Lipschitz boundary. Consider the Dirichlet-to-Neumann map $\Lambda:H^{\frac{1}{2}}(\partial \Omega)\to H^{-\frac{1}{2}}(\partial \Omega)$ defined via
$$ \langle ...
1
vote
2
answers
112
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$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$
Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute.
I'm ...
1
vote
0
answers
48
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Optimal regularity of polynomial interpolators
Definitions
We define the "complexity" of any polynomial function $p:\mathbb{R}^n\rightarrow \mathbb{R}^m$ as $m\binom{n+\deg(p)}{n}$ (i.e the dimension of $\oplus_{i=1}^m\,\mathbb{R}[X_1,\...
1
vote
1
answer
155
views
How to prove that is a consistent estimator?
Let $\hat{\pi}^N$ be an AW-consistent estimator of $\pi$ (i.e., $\hat{\pi}^N$ is a strongly consistent estimator of $\pi$ under adapted (or called nested) Wasserstein distance $AW(\pi, \hat{\pi}^N)\to ...
4
votes
1
answer
286
views
Equivalence of real numbers in terms of Dedekind cuts and Cauchy nets of rational numbers
We work in weakly predicatively constructive mathematics, in that we accept function sets but do not accept power sets or excluded middle. More specifically, we shall assume a sequential universe ...
1
vote
1
answer
70
views
Identification of $\lim_{n\to\infty}f_n^{-1}$ with $f_n:\mathbb R_+\to (0,1]$ strictly decreasing and converging pointwise
Let $f_n: \mathbb R_+\to (0,1]$ be continuous and strictly decreasing for every $n\ge 1$. Assume that the pointwise limit of $(f_n)_{n\ge 1}$ exists, denoted by $f$, and is also strictly decreasing. ...
3
votes
0
answers
315
views
When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
2
votes
0
answers
214
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
2
votes
1
answer
153
views
Does that exponent of (absolute value of derivative) is constrained implies Lipschitz continuity?
Given $C^1([a, b])$ functions $f_n$ that converge to a continuous real-valued function $f_n \to f$ on a closed interval $[a, b] \subset \mathbb R$, suppose
$$
\int_a^b |f_n'(x)|^{1 + \epsilon} dx < ...
1
vote
1
answer
138
views
How to get the estimator?
They introduce a new correlation. For $\pi\in \Pi(\mu,\nu)$ the set of coupling of two probability measures $\mu$ and $\nu$ on a Polish space $(X,d)$. The author introduces a plugin estimator.
...