Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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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 ...
Iosif Pinelis's user avatar
4 votes
1 answer
326 views

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 ...
aleari1009's user avatar
2 votes
1 answer
291 views

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 ...
Nate River's user avatar
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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 ...
aglearner's user avatar
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8 votes
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281 views

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 ...
TheSimpliFire's user avatar
0 votes
1 answer
183 views

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? ...
Riku's user avatar
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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 ...
jack412's user avatar
  • 63
4 votes
1 answer
265 views

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 ...
HyyFly's user avatar
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3 answers
501 views

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 $...
Ypbor's user avatar
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3 votes
1 answer
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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 ...
HyyFly's user avatar
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1 vote
1 answer
152 views

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,...
Ryo Ken's user avatar
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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}(...
Math's user avatar
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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(...
Ali's user avatar
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5 votes
1 answer
174 views

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 $\...
Nate River's user avatar
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3 votes
0 answers
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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}...
node's user avatar
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0 votes
1 answer
143 views

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 ...
Gateau au fromage's user avatar
0 votes
1 answer
128 views

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)}\...
T. Amdeberhan's user avatar
1 vote
2 answers
371 views

Sum of reciprocals of A086005

Does the sum of reciprocals of terms of A086005 converge?
Daniel Sebald's user avatar
1 vote
0 answers
57 views

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}) \\ &...
G. Panel's user avatar
  • 557
0 votes
0 answers
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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 ...
igorf's user avatar
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3 votes
0 answers
114 views

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}^{\...
Jone Sweden's user avatar
1 vote
0 answers
155 views

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 : $\...
Sourav Ghosh's user avatar
3 votes
1 answer
328 views

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 $...
W.J.'s user avatar
  • 379
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 ...
Oleg Zubelewicz's user avatar
4 votes
1 answer
286 views

Sharpest version of semiclassical Calderon-Vaillancourt theorem

Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
Yonah Borns-Weil's user avatar
1 vote
1 answer
175 views

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{...
G. Panel's user avatar
  • 557
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 $...
edgar314's user avatar
2 votes
0 answers
150 views

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 ...
anyon's user avatar
  • 181
2 votes
0 answers
111 views

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}...
Calculix's user avatar
  • 321
8 votes
1 answer
356 views

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 ...
David E Speyer's user avatar
2 votes
0 answers
73 views

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 ...
user124297's user avatar
4 votes
2 answers
366 views

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 $\partial C$. ...
G. Panel's user avatar
  • 557
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 ...
Shaq155's user avatar
  • 449
2 votes
0 answers
122 views

Multiple integral with diagonal constraint (short-range)

I am looking for an upper bound on the following integral: $$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$ ...
Thomas Kojar's user avatar
  • 4,414
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 ...
Johnny T.'s user avatar
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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 ...
Nate River's user avatar
  • 4,822
9 votes
0 answers
301 views

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 ...
Noah Schweber's user avatar
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 ...
Hermi's user avatar
  • 274
6 votes
1 answer
270 views

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 + ...
Nate River's user avatar
  • 4,822
3 votes
0 answers
66 views

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 $\mathbb{R}\flat$,...
Madeleine Birchfield's user avatar
8 votes
1 answer
440 views

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 ...
Ali's user avatar
  • 4,077
1 vote
2 answers
112 views

$\log(t)$ 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 ...
Fetchinson0234's user avatar
1 vote
0 answers
48 views

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,\...
ABIM's user avatar
  • 5,019
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 ...
Hermi's user avatar
  • 274
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 ...
Madeleine Birchfield's user avatar
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. ...
user avatar
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!...
T. Amdeberhan's user avatar
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(...
Guido Li's user avatar
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 < ...
Po-Hung Yeh's user avatar
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. ...
Hermi's user avatar
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