All Questions
5,702 questions
-1
votes
1
answer
113
views
Lipschitz function which is surjective on subset implies that the subset is dense
Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a Lipschitz-function. Suppose $A \subseteq \mathbb{R}^n$ is an $(n-1)$-connected subset such that $f(A) = \mathbb{R}^n$. I would like to show that $A\subseteq ...
- 261
-1
votes
1
answer
213
views
Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 581
-1
votes
1
answer
142
views
A pathological (?) function involving powers
This is inspired by a recent math.SE question.
Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide ...
- 13.4k
-1
votes
1
answer
168
views
Searching the roots of a self-consistent transcendental equation
I have the equation
$$M = c_1 + c_2M - c_3T\ln\left(\left|\frac{e^{(c_4M + c_5)/T}-1}{e^{(c_6M + c_5)/T}-1}\right|\right)$$
where $c_1, \dots, c_6$ are constants.
I am interested in the roots of $$M\...
-1
votes
1
answer
155
views
Is this recurrent sequence decreasing?
Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
-1
votes
1
answer
80
views
Minimal covering sets of continuous endomorphisms
For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ covers $\text{End}(X)$ if for every $f\in \...
- 48.9k
-1
votes
1
answer
103
views
Sum of $\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}$ [closed]
I want to calculute or estimate of order $O(n^{2-\varepsilon})$, where $\varepsilon>0$, of the following sum for $0<\alpha<1$
$$\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}.$$
-1
votes
1
answer
119
views
Existence of a function with slow growth on derivatives
Does there exist a smooth compactly supported function $$f \in C^{\infty}_c((0,1))$$
such that
$$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$
...
- 4,135
-1
votes
1
answer
81
views
Closed on generic set implies closed set whole set [closed]
Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
- 1,043
-1
votes
1
answer
102
views
Compactness of a special kind of Integral operators
Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{
& K:{L^2}(0,1) \to {L^2}(0,1) \cr
& f: \to (Kf)(x) = \int\limits_0^1 {k(...
- 617
-1
votes
1
answer
83
views
On probabilistic extension for Bernstein polynomials
Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
- 101
-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
votes
1
answer
69
views
Proof of $\lim_{i\to\infty}\lambda_i^{-1}\left|f(\hat{x}+\lambda_ix,u_i) - f(\hat{x},u_i) - D_xf(\hat{x},u_i)(\lambda_ix)\right| = 0$
I am trying to prove or disprove the next statement that seems necessary for the proof of Proposition 2.9 of this book.
Let $U\subset R^k$ be compact and $f:R^n\times U \to R^m$ be twice ...
- 183
-1
votes
1
answer
59
views
Does there exist any subsequence $(u_{n_k})$ converging strongly in $L^q(\mathbb{R})$, for any $1 \le q \le \infty$? [closed]
Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Does there exist any subsequence $(u_{n_k})$ converging ...
- 17
-1
votes
1
answer
226
views
separable BV space for PDE's, Whats stopping us? [closed]
Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
- 698
-1
votes
1
answer
519
views
Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
-1
votes
1
answer
69
views
Glueing smooth functions give a smooth function if reparametrized [closed]
Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and
$$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 (t)...
- 1,759
-1
votes
1
answer
159
views
Question about the derivative of a fuctional
I have this lemma+proof and i dont understand why it follows from $J'(u_n)\rightarrow 0$ that $-\Delta_p u_n- f(x,u_n)\rightarrow 0$ such that
$J(u)=\frac1p\int_{\Omega} |\bigtriangledown u|^p dx-\...
- 277
-1
votes
1
answer
349
views
A question about approximation of Real analytic functions
Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$
for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in ...
- 105
-1
votes
0
answers
132
views
Trig conjecture about square roots and Arcsin
Let $r(a,b)$ be a rational number depending on positive integers $a,b$ and $r(a,b)$ being nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$.
Let $C(b)$ be a squarefree positive ...
- 769
-1
votes
0
answers
51
views
Existence of continuous crossection
Let $G$ be a second countable locally compact Hausdorff groupoid. Then $G= \sqcup_{u\in G^{0}}G^{u}$ where $G^{0}$ is the unit space of $G$ and $G^{u}=r^{-1}(u)$. Here $r:G \to G^{0}, r(x)=xx^{-1}$.
...
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
-1
votes
1
answer
550
views
Lower bound of an expectation
Suppose a random variable $X$ has unit variance i.e. $\sigma^{2} = 1$. Is there a positive constant $c > 0$ such that
$$\mathbb{E}[\ | X - \mathbb{E}[X] | \ ] \ge c $$
My attempt of a solution is ...
- 643
-1
votes
1
answer
74
views
Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail
Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...
- 1
-1
votes
1
answer
236
views
Natural candidates for sub-half-exponential which limit to half-exponential function from below
There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However sub-half-exponentials (functions whose composition grows ...
- 1,826
-1
votes
1
answer
193
views
Limit of the convolution of derivative of Gaussian heat kernel
I'm looking for the following limit:
$$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...
-1
votes
1
answer
126
views
Is there a name for this family of integral?
This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.
- 101
-1
votes
1
answer
180
views
Orthogonal polynomials of the second kind
Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $L(x^n)= s_n, n\ge 0.$ Given a positive ...
- 1
-1
votes
1
answer
63
views
Idempotent solutions to the implict function theorem other than the identity?
I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...
- 1
-1
votes
1
answer
173
views
For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$
Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
- 11
-1
votes
1
answer
148
views
Analytic extension of the exterior Newtonian potential into the domain
I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.
Definition of Newtonian ...
-2
votes
1
answer
423
views
Brouwer's theorem 2.0? [closed]
Let $f\in C([0,1]^n,\mathbb R^n) $ with $[0,1]^n \subset f([0,1]^n)$
Is it true that $\exists x \in [0,1]^n, f(x) =x$?
- 4,074
-2
votes
1
answer
175
views
Simple closed form for $\int \lfloor x \rfloor dx$? [closed]
Wolfram Alpha claims there is no closed form in terms of standard funcions
for $\int \lfloor x \rfloor dx$ but we believe we found
simple closed form agreeing with experimental data.
Define $i_1(x)=x -...
- 25.4k
-2
votes
2
answers
487
views
Does function $f(x)=f(2x)$, $f(x)$ - non const, exist? ($f(x)$ - continuous function on real numbers) [closed]
When I tried solve it I had found just answer "No". I spoke with some people but I cannot understand why the answer is exactly it...
Frankly speaking, this function haunts me:
$f(x) = abs((...
-2
votes
1
answer
102
views
Partial derivative in terms of Kronecker delta and the Laplacian operator [closed]
How can the following term:
$$ T_{ij} = \partial_i \partial_j \phi$$
be written in terms of Kronecker delta and the Laplacian operator $\mathbin\bigtriangleup = \nabla^2$?
I mean is there a relation:
$...
- 117
-2
votes
1
answer
169
views
Question about Lipschitz conditions
Let $f$ be a function on some real interval $[a,b]$. Suppose that $\forall x\in [a,b]$, there exists a positive constant $C$ such that
$$ |f(x)-f(y)| \leq C|x-y| $$
for all $y \in [a,b]$.
Does each $x ...
- 419
-2
votes
1
answer
1k
views
Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
- 395
-2
votes
1
answer
99
views
A question on the zeros involving the equation containing exponential factor [closed]
I recently encounter a puzzle that: how to show that for any constant $c_1,c_2,c_3,c_4 \in \mathbb{R}$ the equation
$$c_1 e^t+c_2e^{-t}+c_3 e^{\alpha t}+c_4 e^{-\alpha t}=0$$
has at most only one ...
- 353
-2
votes
1
answer
880
views
a question regarding the interchange the order of finite summation with finite integration [closed]
Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...
- 603
-2
votes
1
answer
283
views
Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?
Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...
-2
votes
1
answer
652
views
Definition and properties of the inverse of the flow of an ODE [closed]
At lesson, the teacher considers a flow $\Phi$ given by the solutions of the ode system for $t\in[0, T]$ and $x\in\mathbb R^d$,
$$
\begin{cases}
y'(s)=b(y(s), s),&s\leq T\\
y(t)=x
\end{cases},\...
- 171
-2
votes
1
answer
180
views
Is there a function $f$ that is a finite sum of functions with finite products of the inputs of $f$ as inputs with this property?
Note: This question aims to be a generalization of Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? and Is it possible to create a polynomial $p(x)$ with this ...
- 265
-2
votes
1
answer
708
views
Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]
Edit: I got rid of my old definitions. Everything should be clear now
Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...
- 63
-2
votes
1
answer
286
views
Why this function is monotonic?
Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...
- 395
-2
votes
1
answer
214
views
About infinite products and Euler Gamma functions [closed]
I am interested in knowing how to calculate infinite products like (or reading any reference about it):
$$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$
Inserting it into ...
-2
votes
1
answer
248
views
Upper and lower limits [closed]
Find the following limits:
(1) $\limsup_{n\to\infty } \sin (n!) $
(2) $\liminf_{n\to\infty } \sin (n!) $
(3) $\limsup_{n\to\infty } \cos (n!) $
(4) $\liminf_{n\to\infty } \cos (n!) .$
- 469
-2
votes
1
answer
93
views
Express the connection between roots [closed]
$\DeclareMathOperator\elim{lim}\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\lmb{lmb}\DeclareMathOperator\Lmb{Lmb}\DeclareMathOperator\mts{mts}$There are two similar functions; they determine the ...
- 55
-2
votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 109
-2
votes
1
answer
100
views
Understanding the performed change of variable in this integration [closed]
I'm stuck on a passage I do not understand, which reads:
$$\int_{r<|y|<1} \bigg| \frac{1}{(|y|^2 - r^2)^s |y|^n} - \frac{1}{|y|^{n+2s}}\bigg|\ \text{d}y$$
$$\int_1^r \bigg| \frac{1}{(t^2 - r^2)^...
- 99
-2
votes
1
answer
475
views
Is a function piecewise continuous if it has a left-limit and a right-limit at every point in its interval domain and equals at least one of these? [closed]
Suppose a real-valued function f, whose domain is an interval, has the property that
at every point in its domain it has a left-limit and a right-limit, and it equals at least one of these. Is it ...
