All Questions
5,909 questions
1
vote
1
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250
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Question about the stationary phase method and the smooth function used
A statement of the stationary phase method I know is the following.
Suppose $\phi(x_0) = \phi'(x_0) = 0$ and $\phi''(x_0) \not = 0$. If $\psi$ is a smooth function supported in a sufficiently small ...
- 188k
1
vote
2
answers
235
views
Can we get smooth parition of unity with uniformity?
Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of ...
- 17.2k
7
votes
2
answers
1k
views
Schauder basis $L^p(\mathbb{R})$
Let $\{e_{n}(x)\}_{n=0}^{\infty}$ be orthnormal basis of Hilbert space $L^2(\mathbb{R})$. If $\{e_{n}(x)\}_{n=0}^{\infty} \subset L^p(\mathbb{R})$ for some $p\geq 1$, is the $\{e_{n}(x)\}_{n=0}^{\...
- 4,878
4
votes
0
answers
141
views
algebraic connectivity of a tree
Suppose that $T$ is a tree with $n$ vertices and $L$ is the Laplacian matrix of $T$ and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are laplacian eigenvalues.
I think the multiplicity of $\mu_2$ can ...
- 3
6
votes
1
answer
380
views
Asymptotic value of a multivariate integral
The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems.
Define
$$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
- 37.7k
1
vote
1
answer
350
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Strong convergence in reflecxive Banach space
Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f_n \rightarrow f$ weakly in $X$, and $\|f_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly ...
CommunityBot
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5
votes
1
answer
493
views
On a.e. approximate differentiability of certain continuous real functions
I have the following question:
If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on $[...
- 1,228
13
votes
1
answer
1k
views
Structure of the Cantor part of the derivative of a BV function
It is well known that an integrable function $u \colon \mathbb R^d \to \mathbb R$ is said to be of bounded variation iff the distributional gradient $Du$ is (representable by) a finite Radon measure, ...
- 980
1
vote
1
answer
105
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Inequality satisfied for $t=1/2$ and Measurability implies Log-Concavity
Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that
$$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$
Why must $f$ be log-concave? (That is, why must
$$\...
- 29.8k
8
votes
1
answer
5k
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Lax's proof of the change of variables theorem
I am teaching a course on Multivariable Calculus for Graduate students. I came across this nice article by Lax where a special case of the change of variables theorem is proved:
Theorem. Let $f:\...
3
votes
1
answer
213
views
A really simple probabilistic inequality on the unit interval
Given a probability distribution on the interval $[0,1]$, is there any relationship between the quantity $$\sup_{S}{\mathbb{E}(X|X\in S)^{2}\Pr(X\in S)}$$ over all measurable subsets $S$, and the ...
- 54.2k
1
vote
2
answers
401
views
Reference request for Stieltjes Transform
I am wondering to use Stieltjes transform to signal processing like Fourier Transform. The Fourier is known to give the frequencies of a signal but not sure what Stieltjes transform gives. I am only ...
- 2,135
3
votes
1
answer
461
views
Bounding the "spikiness" of a probability distribution
Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"?
I ask this question because I am interested in the families of probability distributions $f(x)$ ...
- 8,071
0
votes
1
answer
195
views
Approximating the sum $\sum_{n \leq X} a_n$ with a smooth sum $\sum_{n \geq 1} a_n w(X)$
I have a sequence $a_n$ such that $0 \leq a_n \leq \log n$, and I am considering
$\sum_{n \leq X} a_n$. However, I prefer using smooth weights so I would like to approximate it with $\sum_{n \geq 1} ...
- 211
21
votes
2
answers
981
views
What is the optimal speed to approach a red light?
Suppose from distance $d$, while driving at speed $v_0$, I notice that there's a red traffic light in front of me. Suppose that there are no other vehicles, my vehicle has perfect brakes, my maximum ...
- 1,172
4
votes
2
answers
270
views
Is this function always bounded below?
Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for ...
- 105k
46
votes
4
answers
8k
views
Why could Mertens not prove the prime number theorem?
We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 105k
1
vote
0
answers
94
views
Measure of the boundary of the support of a certain function defined by an expectation
Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : ...
- 111
1
vote
1
answer
357
views
Does CLT hold for joint distribution of two dependent binomial variables?
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
CommunityBot
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1
vote
0
answers
326
views
Approximation of Borel sets
Let $\nu$ be a finite Radon measure on $\mathbb{R}^2$ and denote the Lebesgue measure on $\mathbb{R}^2$ by $\mathcal{L}^2$. Assume that $\nu<<\mathcal{L}^2$.
We denote the boundary of $A\subset\...
- 66.3k
4
votes
0
answers
187
views
Asymptotic formula, polynomial, irrational number and uniformly distribution
Problem 1
Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for:
$$\...
- 697
1
vote
1
answer
540
views
Bound of an oscillatory integral from Stein's Harmonic Analysis book
On Stein's ``Harmonic Analysis Real-variable methods, orthogonality, and oscillatory integrals'' (5.13, page 363) there is the following statement. Let $\phi$ be a real homogeneous polynomial on $\...
- 331
1
vote
0
answers
100
views
Higher order derivative of negative power of cosine function
This is a question I encountered in my own research on Generalized
Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the
basis measure for this family is
$$L\left( \...
- 984
24
votes
11
answers
8k
views
The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
CommunityBot
- 1
1
vote
1
answer
178
views
Bochner integrability within a subspace
Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for ...
- 29.8k
9
votes
3
answers
362
views
Maximum zero converges to $\sqrt{2}$
In my research I came upon a recursively defined sequence, and I'm pretty sure it converges to $\sqrt{2}$ though I can't prove it easily. I don't think it is a difficult question but I'm not sure.
...
CommunityBot
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5
votes
0
answers
240
views
The boundary integral of a harmonic function
Let $\Omega\subset\mathbb{R}^{n}$ be a bounded
domain with smooth boundary and $f$ be a harmonic function on $\Omega.$
It is known that
$$
\limsup_{\varepsilon\rightarrow0^{+}}\intop_{\partial\Omega_{...
- 53
9
votes
5
answers
3k
views
Assessing effectiveness of (epsilon, delta) definitions [closed]
There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
- 16.6k
4
votes
1
answer
700
views
Is $L^1(\Omega)$ continuous embedded in the dual of $H^m(\Omega)$ $(m>\frac{d}{2})$?
Let $\Omega$ be a bounded domain of $R^d$ with Lipschitz boundary. If $m>\frac{d}{2}$, such that $H^m(\Omega)$ is continuously embedded in $L^\infty(\Omega)$. Is $L^1(\Omega)$ continuously embedded ...
- 12.6k
4
votes
1
answer
297
views
Proving bounds on analytic functions using only the Taylor expansion
I wonder if there is a general method for obtaining bounds on an analytic function using only its Taylor expansion (not using its special properties such as satisfying a good differential equation, ...
- 91.8k
1
vote
1
answer
165
views
When a product of cdf and tail distribution is increasing?
My problem is the following. Given $F$ and $G$ cumulative distribution functions, with densities $f,g$ (for example on $[0,1]$), what can I say on the monotonicity of $F(x)(1-G(x))$? More specifically:...
- 128k
4
votes
3
answers
3k
views
Is this inequality involving the Frobenius norm right?
Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm.
Is it true that $||AG||_F \geq c(G) ||...
CommunityBot
- 1
1
vote
0
answers
315
views
Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?
In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given:
The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
- 2,784
5
votes
2
answers
1k
views
Summation of double exponential series
Let $q \in (0,1)$ and consider the following summation:
$$S(q,n) = \sum_{i=1}^n {q^2}^i$$
Is there a closed form expression or upper and lower bounds for $S(q,n)$?
Specifically, I am looking for ...
- 2,175
6
votes
1
answer
340
views
Inequality for functions on [0,1], continued
Let $0<a<1,\; \psi_a(x)=\displaystyle \prod_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set
$$f_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi_a(x).$$
Question. Is it true that, ...
CommunityBot
- 1
-3
votes
1
answer
125
views
Does the Hadwiger-Nelson graph have a perfect matching?
The Hadwiger-Nelson graph on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E_n)$ where $$E_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean ...
- 85.6k
6
votes
1
answer
396
views
Well definition of a function
I've edited, just skip the first attempt and go to the second one.
THE FRAMEWORK: let us consider a real topological vector space $V$.
We denote with $\mathscr C_k(V)$ the set of all continous ...
CommunityBot
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10
votes
2
answers
3k
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Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 4,713
5
votes
0
answers
280
views
Proving that a certain function (related to a volume of a region) has a bounded derivative
Let $F$ be a homogeneous form in $n$ variables with integer coefficients.
Let $D$ be a closed box in $\mathbb{R}^n$ (product of closed and bounded intervals). Assume that the partial $\partial F/\...
- 3,625
3
votes
2
answers
632
views
Solution of the functional equation $f(x+1)=g(x)f(x)$
In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X97953439), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$
(where $f,g:\mathbb{R}^+\rightarrow \...
- 128k
1
vote
1
answer
161
views
Continuity of image of resolvent operator with respect to resolvent parameter
Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev ...
- 19.3k
2
votes
1
answer
357
views
Does this limit always exist?
Here is the question; it may seem very simple, but it is difficult (at least for me).
Let $f(x)$ be a continuous function on $R$ that is strictly increasing, and suppose $g(x)=f(x)-x$ is a periodic ...
- 23.2k
11
votes
1
answer
1k
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Proof of the "Neo-classical Inequality", a fractional extension of the binomial theorem
I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$:
$$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\...
- 13.4k
1
vote
1
answer
201
views
Existence of a certain norm on space of measurable functions
Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can ...
- 17k
6
votes
1
answer
402
views
Why are $\sigma$-algebras preferable to $\sigma$-rings?
The following is said without further explanation in Folland's Real Analysis:
Some authors prefer to take the domains of measures to be $\sigma$-rings rather
than $\sigma$-algebras. The reason is ...
- 1,074
2
votes
0
answers
1k
views
Is there an infinite product like this for $\cos x$?
There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
- 2,784
4
votes
1
answer
192
views
When does iteration of an infinite Toeplitz matrix converge?
Consider a Toeplitz matrix $T$, indexed by $\mathbb{N}_0 \times \mathbb{N}_0$. given by the sequence $t_k,k \in \mathbb{Z}$ where $t_k \geq 0,\sum_{k=-\infty}^\infty t_k=1$. By this I mean that $T_{i,...
- 61.9k
3
votes
1
answer
336
views
The generalization of commutative property of orthogonal projectors on a subspace to the whole space
Let for $i\in [n]$, $P_i$ be some orthogonal projectors defined on the Hilbert space $W$ such that they commute on subspace $V < W$ (i.e, for any $i, j \in [n]$ and $v \in V$: $P_iP_j(v) = P_jP_i(v)...
- 24.2k
3
votes
0
answers
240
views
About optimizing decay rate of Fourier transforms?
Suppose we have a density function $f(t)$ of a random variable and $f \in C^1(R)$. If characteristic function of $f$ is $\phi_f(x) \asymp O(x^{-\beta})$ and $f$ satisfies some restrictive conditions ...
- 39.1k
9
votes
1
answer
734
views
Constructive analysis and synthetic differential geometry
I am curious if (any of) the various inequivalent constructions of the real line in constructive mathematics can be used to build a model of Kock and Lawvere's synthetic differential geometry? In ...
- 66.8k