All Questions
5,858 questions
8
votes
3
answers
1k
views
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found ...
- 2,661
7
votes
3
answers
524
views
Rigorous estimates on roots of function
We consider the function
$$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$
The arguments of the two sines differ by a factor ...
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
2
votes
1
answer
210
views
Argmax of a function of $n$ variables under linear constraint
(I start by saying that the tags are probably not accurate but I didn't know what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit ...
- 697
3
votes
0
answers
125
views
Extracting moments of $\max(X_1,\ldots,X_k)$ from asymptotic behavior of $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$
For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions.
Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{...
- 1,295
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
0
answers
154
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 393
53
votes
7
answers
51k
views
Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
- 541
0
votes
1
answer
92
views
If $f(x,t)=\sum_{n \in \mathbb{Z}} a_n(t) e^{in x}$ is $C^\infty$ in $x$ and all $a_n(t)$ continuous, $x$ derivatives of $f$ are continuous in $t$?
This question seem a bit elementary, but I find it more subtle than its looks. So, I post the question here.
Let $f(x,t) : [0,2\pi] \times [0,1] \to \mathbb{C}$ be a function such that $f(0,t)=f(2\pi,...
- 3,477
1
vote
1
answer
151
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
- 391
0
votes
0
answers
119
views
About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
3
votes
1
answer
135
views
Recover an $L^1$ integrand by partial differentiation
Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by
$$F(x,y) = ...
- 41
16
votes
2
answers
1k
views
Definitions of determinant by unique features
A well-known definition of the determinant is:
The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized.
See e....
1
vote
0
answers
76
views
Is this extension of n-th derivatives to ordinal-indexed derivatives trivial? [duplicate]
Let $f$ be a function defined everywhere on the real line, which is infinitely differentiable everywhere, in other words, $f$ is everywhere smooth. I define the $\omega$-th derivative, where $\omega$ ...
- 2,023
10
votes
2
answers
1k
views
Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field
In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple ...
- 2,624
7
votes
3
answers
547
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,223
16
votes
2
answers
2k
views
Proof that block matrix has determinant $1$
The following real $2 \times 2$ matrix has determinant $1$:
$$\begin{pmatrix}
\sqrt{1+a^2} & a \\
a & \sqrt{1+a^2}
\end{pmatrix}$$
The natural generalisation of this to a real $2 \times 2$ ...
- 263
7
votes
0
answers
220
views
Why are these two determinants equal?
This question is a follow up on Mark Wildon's comment from an earlier MO question.
As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by
$$\binom{n}...
- 43.2k
2
votes
1
answer
170
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
- 393
5
votes
1
answer
534
views
Minimiser of a certain functional
Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \...
- 6,223
3
votes
1
answer
557
views
Is there real or complex analytic function whose positive real zeros are the primes?
Related to this question
Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named ...
- 25.4k
1
vote
0
answers
122
views
Implicit function theorem / Implicit selections when Jacobian not invertible
I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
- 393
1
vote
1
answer
191
views
Will this "tree" cover all rational numbers in a range?
Question
I am making a tree using the following two functions:
$$f(x)=\frac{x}{r},\quad g(x)=\frac{x+b}{r}$$
where $1<r<2$ and $0<b$ are rationals. Everything is a real number here.
The ...
- 433
3
votes
2
answers
2k
views
Expected gradient vs. gradient of expectation
Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in ...
- 205
4
votes
3
answers
543
views
Determinant with factorials is not 0?
Below is a simple determinant. I need to show that it is not 0, so that the corresponding matrix is invertible.
$$
D = \begin{vmatrix}
0! & 1! & 2! & \ldots & x!\\
1! & ...
- 109
6
votes
3
answers
267
views
Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let
$$
F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)
$$
be the ...
- 127
16
votes
1
answer
888
views
Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
- 151k
4
votes
1
answer
209
views
Why is there a $\mathcal{H}^d$-null set in the definition of d-rectifiable set?
Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H}^...
- 697
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 7,127
1
vote
0
answers
59
views
Study of the properties of a non-local ODE
I am studying the following non-local ODE
$$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$
The number $x_0$ can ...
- 452
9
votes
0
answers
522
views
Does the intersection of middle third and middle half Cantor sets contain an irrational number?
Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$.
Likewise let $C_\frac{1}{2}$...
- 2,934
3
votes
1
answer
96
views
Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Let $0 < t_1 < t_2 &...
- 657
27
votes
4
answers
8k
views
Proofs of Young's inequality for convolution
For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
- 3,425
4
votes
1
answer
643
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 697
6
votes
2
answers
622
views
Forcing the uniqueness of a solution of an ODE
For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and
$$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$
Let $y_n$ be the ...
- 449
4
votes
1
answer
288
views
A lower bound for the $L^1$ norm of real trigonometric polynomials
This question is somewhat similar to Minimizing the L1 norm of odd-term trigonometric polynomial. The context of the question is based on the paper Hardy's Inequality and the $L^1$ norm of Exponential ...
- 270
1
vote
2
answers
163
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
- 515
7
votes
2
answers
537
views
How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?
I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any ...
- 3,477
11
votes
1
answer
657
views
Does every differentiable a.e. function admit a maximally differentiable representative?
For $f: \mathbb R \to \mathbb R$ a measurable function, we say $g: \mathbb R \to \mathbb R$ is a modification of $f$ if $f = g$ a.e.
Suppose $f$ Is a measurable function that is differentiable a.e.
We ...
- 6,223
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
39
votes
8
answers
13k
views
Can Cantor set be the zero set of a continuous function?
More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth?
Some days ago I discovered that in this proof ...
- 5,339
2
votes
2
answers
132
views
What are the bounds of $xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a$ for $0 \le x \le 1$ and $a > 0$?
Posting from MSE since it was unanswered in MSE.
Let $0 \le x,y \le 1$ and $a$ be a real and let
$$
f(x,y,a) = xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \tag 1
$$
For a fixed $a$, the graph of the ...
- 5,470
3
votes
2
answers
717
views
Do two ways to differentiate Lipschitz functions coincide?
Let $f\colon \Omega \to \mathbb{R}$ be a Lipschitz function on an open subset $\Omega\subset \mathbb{R}^n$.
By the Rademacher theorem $f$ has first derivative almost everywhere. We denote it $\nabla f$...
- 21.8k
5
votes
2
answers
788
views
Do non-zero derivatives imply tangent lines (and vice versa)?
Let $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ be any continuous function, with image given by $C_\gamma$.
We can say that $\gamma$ has an image tangent at $t \in \mathbb{R}$ if there exists $\...
- 700
4
votes
2
answers
305
views
Generalization of van der Corput's estimate on oscillatory integrals
Question: Given exponents $0<\alpha<\beta$ and an interval
$[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
$\lambda_1,\lambda_2\in\mathbb{R}$,
$$\left|\int_a^be(\...
- 1,701
1
vote
2
answers
180
views
An inequality for a real function
Let $$f(z)=(1+z)^{3/4}-\left(\frac{3}{8}+\frac{\sqrt{3}}{4}\right)^{1/4}-\frac{\left(3 z+\sqrt{6} \sqrt{-1+z^2}\right)^{3/4}}{\left(2 \left(2+\sqrt{3}\right)\right)^{3/4}}.$$ Is there a simple proof ...
- 37
1
vote
0
answers
114
views
Computing sine of gamma function [closed]
In the sense of bit complexity, how difficult is it to compute $$\sin(a\Gamma(x))$$ where $a$ is a constant and $x>1$? Is it possible to avoid the computation of $\Gamma$ as first step?
Is there a ...
- 67
9
votes
2
answers
1k
views
Density of restrictions of harmonic functions inside a ball
Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
- 4,153
5
votes
0
answers
162
views
Closed formula for series $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$
What can be said about $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$ (for $|x|>1$ and $|y|>1$ and $x\neq y$)?
Is there a kind of closed formula for this?
By comparing to the geometric series, this sum ...
- 139
5
votes
1
answer
258
views
On the continuity of a Set-Valued function (correspondence) [closed]
Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued function defined by
\begin{equation*}
f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left(
x\right) ^{T}y\...
- 61