All Questions
5,909 questions
3
votes
1
answer
354
views
Evaluation of an interesting Integral [duplicate]
Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically.
$$f(n) = \frac{2}{\pi} \int_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \...
- 188k
-1
votes
1
answer
136
views
An elementary question about integration by parts! [closed]
Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
- 1,190
10
votes
3
answers
414
views
Is an open subset of a rigid space rigid?
Let $X$ be a locally compact Hausdorff space. Call $X$ rigid if its only autohomeomorphism is the identity, $\operatorname{Homeo}(X)=\{1\}$.
Questions:
Let $X$ be rigid. Is it true that every open ...
- 7,276
12
votes
1
answer
352
views
A problem involving the Error Function
I am looking at the following function on the domain $x\geq 0$:
$$F(x)=(x+a)e^{x^2}(1-\mathrm{erf}(x))-\frac{b}{\sqrt\pi},$$
where $a>0$, $0<b<1$ are parameters. From plotting this function ...
- 128k
3
votes
1
answer
113
views
maximum likelihood estimation of X is better than that of f(X)?
Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...
- 128k
3
votes
0
answers
3k
views
Derivative of the regularized upper incomplete gamma function
I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function
\begin{eqnarray}
\Gamma(x)&=&\int_0^\...
- 63.9k
3
votes
1
answer
113
views
Asymptotic expansion of nonlinear Gaussian transformation in terms of covariance
I'm reading this paper and on page 8 the authors state without proof an asymptotic expansion of a multivariate Gaussian integral in terms of the covariance obtained by applying what they call the "...
- 128k
14
votes
1
answer
440
views
Inequalities on elementary symmetric polynomials
I have recently come across the following result.
Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $...
- 61.9k
4
votes
2
answers
3k
views
How to find the minimum of the integral?
Suppose $x(t)$ is differentiable on $(0,T)$ and continuous on $[0,T]$. How to find the minimum and the minimal value of the integral $$\int_0^T\|\dot x(t)+x(t)\|^2dt$$ such that $m\le x(t)\le M$ on $[...
- 1,228
1
vote
0
answers
134
views
Convolution integral of series involving the non-trivial zeros of $\zeta(s)$
Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
- 219
5
votes
1
answer
240
views
Are Pointwise conditions studied?, do they make sense?, do they have any applications?
In weakly formulated PDE (or even ODE), we seem to be interested in solutions that satisfy or take desired values at some boundary points of a domain we are interested in. For example, Dirichlet ...
- 3,085
5
votes
0
answers
313
views
Uniqueness of a SDE with non-negativity constraint
I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed):
\begin{equation}\label{sde}%sde
x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...
- 779
5
votes
1
answer
384
views
Asymptotic growth of the of Taylor coefficients of the inverse of a function
Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the ...
- 61
6
votes
1
answer
2k
views
About weak convergence of probability measure
Suppose $\mu_j$ is a sequence of measures on $\mathbb{R}$. By the definition of weak convergence of measures, $\mu_j$ weak converges to $\mu$ means that for any bounded continuous function $f$, there ...
- 29.8k
1
vote
1
answer
348
views
Continuity of implicitly defined function
Consider a function $g(x)$ defined implicitly via
$$\int_x^{x + g(x)} f(\xi) \,d \xi - u(x) = 0. $$
I know that for every $x$ a unique $g(x)$ exists.
Furthermore $f$ is locally integrable and $u$ is ...
CommunityBot
- 1
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
7
votes
2
answers
257
views
$f$ locally (Lebesgue) integrable function on real line, $g(x):= \lim _{r\to \infty} \frac 1r \int_{x-r}^{x+r} f(t) dt$ exists for every real $x$
Let $f : \mathbb R \to \mathbb R$ be a function such that $f \in L^1[-a,a] , \forall a \in (0,\infty)$ and $g(x) : = \lim _{r\to \infty} \dfrac 1r \int_{x-r}^{x+r} f(t) dt$ exists in $\mathbb R$ for ...
- 4,713
12
votes
2
answers
812
views
Inequality in Gaussian space -- possibly provable by rearrangement?
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
- 13.6k
26
votes
3
answers
3k
views
Sum of Gaussian pdfs
I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum_{n\in\mathbb Z}f(x+n)$), the ...
CommunityBot
- 1
2
votes
0
answers
119
views
question about sequences and series (complex analysis may be only elementary real analysis)
I would like to have your help about the proof of the following statement:
If the sequences of complex numbers $\{F_N\}_{N \in \mathbb{N}},\{G_N\}_{N \in \mathbb{N}}$ have the following properties:
(...
- 51
4
votes
0
answers
459
views
Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?
Is there any closed form expression for the following serie?
$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$
Or at least a proof that it is an irrational number. The ...
5
votes
2
answers
1k
views
On the embedding of a function space $X$ into $L^2\cap L^4$
It is well-known that if $\Omega\in \mathbb{R}^n$ is a bounded domain, then we have the embedding
$$
L^4({\Omega})\subset L^2({\Omega})
$$
since $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ ...
CommunityBot
- 1
5
votes
1
answer
226
views
If $f:[0,1]\to\mathbb{R}$ has continuous approximate deriv., is it $C^1[0,1]$?
For $f:[0,1]\to\mathbb{R}$, let $f'_{app}(x)$ denote the approximate derivative (that is, the derivative calculated along some set with density $1$ at $x$, if such a thing exists). Assume that $f'_{...
- 109k
5
votes
1
answer
294
views
Fourier transform of $f$ and $|f|$?
What is the relationship between the Fourier transform of an $L^1$ function $f: \mathbb{R}^d \to \mathbb{C}$ and the Fourier transform of $|f|$?
In other words, what is the relationship between
$$
\...
- 96.4k
2
votes
0
answers
46
views
increasing inter-class distances results in decreasing linear regression error
Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set.
Define $\mathbf{...
3
votes
0
answers
189
views
Discussing results that follow from : $\sum_{n=1}^{\infty}\frac{1}{x_n}$ converges if and only if $\sum_{n=1}^{\infty}\frac{1}{1+x_n}$ converges [closed]
The following result is a simple result but I think it reveals some interesting results as consequences.
Given a sequence $\{x_n\}$ of positive integers then the $\sum_{n=1}^{\infty}\frac{1}{x_n}$ ...
- 850
5
votes
1
answer
1k
views
The spectrum of the discrete Laplacian
Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we ...
- 24.2k
0
votes
2
answers
224
views
Solutions or bounds for $x^{(1-\epsilon)/2}=1-x$
Is there a known expression for the solution of the following simple equation, or at least good bounds on the solution? Assume $\epsilon \in [0,1)$ is a given parameter and $x \in (0,1)$.
$$x^{(1-\...
- 6,734
7
votes
1
answer
387
views
A game-theoretical question in a political economy model
My research question in a dynamic model of political competition boils down to the following conjecture. I am confident that it holds (all simulations work), but I have not been able to prove it yet. ...
- 116
3
votes
0
answers
137
views
Can monoids of "continuous words" be realized as initial monoid objects?
Whenever $X$ is a set, write $X^*$ for the monoid freely generated by $X$. The elements of $X$ are, of course, words in the letters $X$. When $X$ is finite, there also seems to be a great many ...
- 3,793
4
votes
0
answers
125
views
Properties of solution to Schrödinger equation
Given a Schrödinger equation with, let's say continuous, periodic potential
$$-y''(x)+V(x)y(x)=\lambda y(x)$$
where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
- 24.2k
2
votes
1
answer
165
views
If $Z$ is standard normal and $f$ is analytic. Is $g(t)= E[ f(Z-t)]$ analytic?
Let $Z$ be a standard normal.
Now define
\begin{align}
g(t)= E[ f(Z-t)]
\end{align}
where $f(x)$ is a real-analytic function and $|f(x)| \le x^4$.
Question:
Is it true that $g(t)$ is also a real ...
CommunityBot
- 1
7
votes
1
answer
304
views
Argument principle for matrices
Let $f,g$ be entire functions, then the argument principle teaches us that
$$\frac{1}{2\pi i}\int_{\mathbb{C}} g(z) \frac{f'(z)}{f(z)} dz$$
is equal to $g$ evaluated at the zeros of $f.$
Now, let ...
- 52.3k
2
votes
1
answer
104
views
Limits of a quasiperiodic function with two pseudoperiods
Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
...
- 378
2
votes
1
answer
244
views
Growth rate of Lipschitz constants for derivatives of $C^\infty$ functions
Let $f\in C^\infty$ have bounded derivatives, i.e.
$$ \sup_{x\in\mathbb{R}}|f^{(p)}(x)| = B_p < \infty$$
for every $p\ge 1$.
I would like to find a proof or a counterexample for the following ...
- 24.2k
5
votes
1
answer
305
views
Expectation of max of Gaussian multiplied by a functional of Gaussian
Let $X \in \mathbb{R}^{d}$ follows the standard Gaussian distribution $N(0, I_d)$. Let $Y = \max_{j\in[d] } X_j$. It is not hard to see that
\begin{align}
\mathbb{E}\left [ Y \cdot X\right] = \sum_{j=...
- 128k
5
votes
2
answers
3k
views
Are there functions which are neither convex nor concave everywhere but are continuous? [closed]
By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(...
- 1,996
1
vote
1
answer
132
views
A question about a continuous curve in $\mathbb{R}^2$
Let $f:[0,1]\longrightarrow\mathbb{R}^2$ be a continuous function such that
$f(1)-f(0)=(1,0).$ Is it right that we can find $0\leq t_1<t_2\leq1$ such that $f(t_2)-f(t_1)=(\pm1,0)$ and for any $t_1\...
- 15.4k
1
vote
0
answers
75
views
Integrability over the closure of a domain [closed]
Let $D$ be a bounded domain in $\mathbb{R}^{N}$ ($N\geq2$) and $E$ a closed subset of $D$ with empty interior. Suppose $f$ is a measurable function defined on $D$ and integrable on $D\setminus E$, i....
- 411
1
vote
0
answers
150
views
Proving the existence of a sequence with recursive growth constraints
Fix $0 < c < 1/3$. Show that there exists a strictly increasing sequence $a_k > 0$, a sequence $b_k \geq 0$, and an infinite set $K \subseteq \mathbb{Z}_{>0}$ such that
\begin{align}\...
- 161
1
vote
2
answers
275
views
A corollary of Gibbs' inequality
Gibbs' inequality is equivalent to:
\begin{equation}
\sum_{i} \ln q_i^{p_i}-\ln p_i^{p_i} \leq 0
\end{equation}
where $p_i,q_i \in [0,1]$ and $\sum_i p_i = \sum_i q_i=1$.
Now, a friend of mine ...
- 3,871
10
votes
1
answer
228
views
Distribution of good diophantine approximations
Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
- 2,217
11
votes
3
answers
618
views
smooth functional to detect whether a function has a zero
Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties:
$F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$.
$F$ is ...
- 1,092
2
votes
1
answer
291
views
Approximation of an injective continuous curve by injective piecewise linear curves
Let $f:[a,b]\longrightarrow\mathbb{R}^2$ be an injective continuous function. For any $d>0$, does there exist a piecewise linear curve: $g:[a,b]\longrightarrow\mathbb{R}^2$ such that $g$ is also ...
- 17.2k
3
votes
1
answer
474
views
A generalized logarithmic function
Consider the function
$$f_\epsilon(x):=\int_0^1 \frac{(1-z)^x-1}{\log(1-\epsilon z)}\, dz,$$
defined for a parameter $\epsilon \in (0,1]$ and $x \geq 0$. When $\epsilon=1$, this is $\log(1+x)$. One ...
- 6,486
4
votes
0
answers
122
views
Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?
I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those ...
CommunityBot
- 1
1
vote
0
answers
585
views
A Lemma on convex domain which is a Lipschitz domain
I am reading the following paper:
https://docs.wixstatic.com/ugd/1de1d9_cd82cb002eaa4eefa9af574eb5efdff2.pdf
I am stuck on the proof of lemma 2.3 on page 6.
I don't see why does the property (i) of ...
- 1,594
2
votes
1
answer
922
views
One side Harnack inequality for Subharmonic function
It is well known that for any non negative Harmonic function w ($\Delta w=0$, $w\geq 0$) in a ball, $B_1(0)$, $\exists$, C>0 such that $\forall y\in B_{1/2}(0)$
$$
Cw(0)\leq w (y)
$$
It is a clear ...
- 17.2k
9
votes
1
answer
260
views
Cover of the positive real numbers by intervals
For which real numbers $x$ and $y$ does the following hold?:
$$
\bigcup_{\frac{a}{b} \in \mathbb{Q}^+}
\left[\frac{a}{b},\frac{a}{b}+\frac{1}{a^x b^y}\right]
\ = \ \mathbb{R}^+
$$
- 28.2k
0
votes
1
answer
218
views
Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
- 17.2k