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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3
votes
0answers
43 views

Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...
4
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1answer
131 views

Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator. How can I compute the ...
2
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0answers
79 views

A question about how to use the convexity condition?

At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say that since $p$ is convex, we can deduce that $$ \...
1
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1answer
97 views

Hermite Transform of Tanh

I would like to understand $L^2(\mathbb{R},\mu)$ approximation by polynomials of $\tanh$ up to degree $n$ where $\mu$ is the standard Gaussian distribution. This leads to considering the Hermite ...
4
votes
1answer
112 views

A continuous bi-Lipschitz shrinking of a domain into a compact subset

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is: (1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...
0
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1answer
188 views

Does there exist a continuous nonconstant function $f$ that maps almost all irrationals to rationals? [closed]

Let $f$ be a continuous nonconstant function on the reals. Could it map almost all irrationals to rationals? This is impossible if $f$ maps all irrationals to rationals, by a well known result. This ...
1
vote
2answers
121 views

Global estimate to an L1 function whose Laplacian is a bounded measure

Pretty simple question: Suppose that $u \in L^1(\mathbb{R}^N)$ is such that $\Delta u \in \mathcal{M}(\mathbb{R}^N)$ (i.e., $\Delta u$ is a bounded Radon measure). Does $\nabla u \in L^1(\mathbb{R}^N)$...
35
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12answers
5k views

Examples where existence is harder than evaluation

In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, ...
1
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2answers
91 views

Non-asymptotic upper bound of right tail of Gamma function

I'm wondering if there is any non-asymptotic upper bound for the following Gamma function: $$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$ for $x>0,a>0$? Something like $x^a\exp(-x)$?
43
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7answers
4k views

Two (probably) equal real numbers which are not proved to be equal?

Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal? I never really understood the assertion that "the reals do not have decidable equality"...
2
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0answers
83 views

Discrete Sobolev embedding

It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$ Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
1
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0answers
64 views

Choosing the weight in a particular definition of Besov spaces

Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
2
votes
1answer
222 views

Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$

Let $f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$ $g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
1
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1answer
298 views

Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere

Let $f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$ $g:=\ln f$ (and assume $g'$ is Lipschitz continuous) $n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...
5
votes
2answers
222 views

Bounded weak derivative

Let $f \in L^{\infty}$ be a function such that $f$ and the weak derivatives $D^{\alpha}f\in L^{\infty}$ exist for all $\vert \alpha\vert\ge 2$. Does this imply that also $D^{\alpha}f$ with $\vert \...
1
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0answers
48 views

Dependency of the Wasserstein metric on its parameters

Let the population on some region $\Omega\subset\mathbb R^d$ be modeled by a density function $\rho:\Omega\to (0,+\infty)$. Provided $n\ge 1$ food trucks labeled by their capacity $p_1,\ldots, p_n\in (...
1
vote
1answer
178 views

If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set

If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
3
votes
1answer
177 views

Rademacher, maxima, convex hulls

Let $F\subset \mathbb{R}^n$ be a finite set and $\sigma$ be uniformly distributed over $\{-1,1\}^n$. The usual Rademacher average of $F$ (modulo normalizing factors) is $$ R_n(F)=\mathbb{E}_\sigma \...
4
votes
0answers
147 views

Signed variant of the Flint Hills series

I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one: $$ \sum_{n = 1}^\infty \...
2
votes
1answer
68 views

Integral substitution involving the length and angle of two vectors

Let $F\colon\mathbb R^3\to\mathbb R$ be a compactly supported smooth function. I want to compute $$ \int_{\mathbb R^n}\int_{\mathbb R^n} F(\lVert x\rVert^2,\lVert y\rVert^2,\langle x,y\rangle)~\...
1
vote
1answer
79 views

Good upper bound for $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where $a,b \in (0, 1)$ and $N \ge 1$

Let $a,b \in (0, 1)$ and $N \ge 1$, and consider the incomplete gamma function $x \mapsto \Gamma(1-a,x)$. Question Is there a simple bound (involving 'simple function's) for the expression $\Gamma(1-...
2
votes
2answers
142 views

Lower bound Renyi divergence between two discrete probability distributions

I am trying to understand the proof of Lemma 1 in this paper (Section 9.2). The proof shows that given a discrete probability distribution $P=(p_1,p_2,...,p_k)$ where $p_1 \geq p_2 \geq ... \geq p_k$,...
7
votes
1answer
207 views

Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$

It may be better to move this to a separate question. Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...
6
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1answer
352 views

Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?

Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
5
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0answers
194 views

A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. ...
3
votes
0answers
42 views

Controlling a Schwartz kernel near the diagonal

Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
1
vote
2answers
76 views

Asymptotic Behaviour of Solutions to a Riccati-type ODE with Small Forcing Term

Consider the following ODE: $$f'(r) = f^2(r) + O \left( \frac{1}{r^4} \right)$$ as $r$ goes to infinity. The initial conditions are $f(1) = C <0$. What is the behaviour of a solution $f$ at ...
0
votes
1answer
64 views

Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]

Let $f:[0,1] \to \mathbb{R}, G = graph(f)$. If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1 $ then $H^s(G) < \infty$ What technique can I ...
0
votes
0answers
18 views

Bounds on the Hausdorff dimension for an IFS satisfying the OSC

Let $([a,b],F)$ be an IFS where $F$ satisfies the open set condition for an open interval $G$. Furthermore, assume that each $f_i \in F$ satisfies: $$\forall x,x' \in \stackrel{-}{G}.q_i |x-x'| \le |...
2
votes
0answers
67 views

How we can do the derivative for this equation w.r.t.to time t>0

Let $x\in[0,L]$ and consider the following equation, $$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
2
votes
2answers
271 views

Uniqueness of the limit sequence of discrete probability measures

Let $N_n$ be a sequence of natural numbers increasing to infinity, and suppose we have a sequence of finite sets of distinct points $X_n = \{x_1^{n},x_2^{n},\ldots,x^{n}_{N_n}\} \subset[0,1] \subset \...
1
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0answers
78 views

An oscillatory integral estimate

Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
2
votes
1answer
88 views

Weak Lebesgue spaces and an estimate for BV functions

Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that $$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$ holds for a.e. $...
0
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0answers
59 views

Convergence of Schur-Hadamard multipliers

Picking up on the question asked here: Convergence of sequence of images of Schur multipliers I was wondering if the same holds in the following setting: Let $K$ be an integral operator acting on $...
5
votes
2answers
182 views

Density character of a metric space is an Ulam number

I am reading this paper and I came across the following sentence: Throughout the paper we silently assume [...] that the density character (i.e. the minimum cardinality of a dense subset) of ...
1
vote
1answer
66 views

Independent identical distribution sequence

given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $. I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...
1
vote
1answer
101 views

How to prove the following Whittaker formula

I am a theoretical physicist and I need help in proving the alternate Whittaker formula $W _ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M _ { k , ...
1
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0answers
156 views

Extending Green's theorem from very special regions to more general regions

Green's theorem Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
4
votes
3answers
383 views

Covariant derivative of determinant of the metric tensor

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
5
votes
0answers
123 views

Initial data and heat equation

We assume all solutions to be bounded here! Let $y_{+},y_{-} \in C_c^{\infty}$ be two positive functions. If we then consider the heat equation $$\partial_t u(t,x) = \Delta u(t,x)$$ for two ...
3
votes
1answer
275 views

Oscillatory integrals

Consider the integrals $$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$ I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...
1
vote
1answer
50 views

Limit of doubly indexed functions

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $f_{n,j}$ be a doubly indexed sequence of positive functions in $L^p(\Omega),$ $1<p<\infty.$ Suppose $f_{n,j}$ converges pointwise a.e. ...
1
vote
1answer
55 views

Can a sequence of degree one maps converges to a constant map in $W^{1,2}$ norm?

Can a sequence of degree one maps on, say, the unit circle, converges to a constant map in $W^{1,2}$ norm? If the answer is yes, would you please provide an explicit example?
0
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0answers
75 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
0
votes
1answer
186 views

Evaluation of a double definite integral with a singularity

How to compute the $$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy.$$ Is it possible to compute the integral analytically upto some terms. I believe it should involve ...
1
vote
1answer
82 views

BV function with absolutely continuous divergence

Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...
1
vote
1answer
116 views

Derivative and Jacobian determinant of solution of ODE [closed]

Let $\Phi$ be the unique solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$ where we have assumed $f$ smooth. ...
2
votes
1answer
103 views

Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)

Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation. Question 1. How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$? Question ...
0
votes
1answer
48 views

Boundary behavior of Greens functions on smooth bounded (planar) domains

It is well known that for any smooth bounded (connected) domain $\Omega\subset\mathbb R^d$ with $d\ge2$, we can define a Green's function $G:\Omega\times\Omega\to\mathbb R$ in $\Omega$ which is smooth ...
2
votes
1answer
73 views

About the continuity of a function on BV

For a fixed $u \in BV(\mathbb{R}^N)$, consider the function $h:(0,+\infty) \to BV(\mathbb{R}^N)$, given by $h(t) = u (tx)$. Is $h$ continuous?