# Tagged Questions

**3**

votes

**1**answer

239 views

### Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...

**1**

vote

**2**answers

296 views

### Limit with theorem of dominated convergence

Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$)
I have to calculate this limit
$$\lim_{|x-y|\to ...

**0**

votes

**1**answer

127 views

### Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense
$$\lim_{|x-y|\to 0} f(x)$$
? ($y$ is a fixed point)
If i have $f$ in $H^2$ I can say that
$$\lim_{|x-y|\to 0} ...

**2**

votes

**2**answers

379 views

### Relationship between the derivative of a matrix and its eigenvalues

Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing?
...

**1**

vote

**0**answers

291 views

### Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...

**0**

votes

**1**answer

291 views

### Integral inequality

Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...

**0**

votes

**0**answers

239 views

### Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...

**1**

vote

**0**answers

111 views

### A question about smoothness

$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold :
$\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all ...

**7**

votes

**2**answers

1k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

**4**

votes

**2**answers

371 views

### Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...

**0**

votes

**1**answer

380 views

### Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...

**1**

vote

**0**answers

213 views

### Supremum of continuous functions on $\mathbb{R}$ [closed]

If $f(x)$ is a continuous function on all of $\mathbb R$, with the property that $\sup_{x\in\mathbb R}|f(x)|\leq 1$. If this is the case, how I can test if the sup is attained or not? (i.e., if there ...

**3**

votes

**1**answer

228 views

### Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \le 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and
$\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$
is finite.
There are at least two ...

**0**

votes

**0**answers

433 views

### A product sum inequality question

For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$
and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$,
do there always exist $z_{1},z_{2},\cdots z_{6}$ in ...

**0**

votes

**1**answer

788 views

### How to determine whether a multivariate function is bounded or not

Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function ...

**2**

votes

**2**answers

567 views

### Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...

**2**

votes

**1**answer

482 views

### A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
...

**2**

votes

**0**answers

258 views

### functions on intervals with endpoints

Would most analysts say that $(2/3) x^{3/2}$ is an antiderivative of $x^{1/2}$ on $[0,\infty)$, or
just on $(0,\infty)$?
More generally, is there a standard interpretation of the assertion "$F$ is an ...

**10**

votes

**1**answer

1k views

### Functions whose antiderivative behaves like xf(x)

I'm wondering if a classification of analytic functions, $f\,$ (it may be that $C^1$ is enough, but I'm not taking any chances, if you have a reason why I only need to consider a larger class of ...

**7**

votes

**2**answers

492 views

### local behavior of a finite Borel measure

Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I ...

**1**

vote

**2**answers

318 views

### Inf of a mutivariate function

Let $f(x_1,\ldots , x_n) = \frac{x_1}{x_2+x_3} + \frac{x_2}{x_3+x_4} + \cdots + \frac{x_n}{x_1+x_2}$, defined for $x_i>0$.
Is there $(x_1, \ldots ,x_n)\in {\mathbb{R}^*_+}^n$ such that ...

**26**

votes

**18**answers

5k views

### Interesting applications (in pure mathematics) of first-year calculus

What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...

**0**

votes

**2**answers

226 views

### A Jordan Arc in the unit disk

Let D be the open unit disk, and J a Jordan arc (that is a homeomorph of [0, 1]) that lies in D, except J(0) lies on the boundary of D, say J(0)=1. I would like to see that D\J([0, 1]) is a path ...

**16**

votes

**11**answers

5k 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 ...

**-3**

votes

**2**answers

246 views

### On \ell_3 norm in R^2

Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$,
in which $\left\Vert \left(x,y\right)\right\Vert ...

**6**

votes

**1**answer

620 views

### Inverse function theorem for DC-functions

I would like to have an inverse (or/and) implicite function theorem for DC-functions.
It seems that I have right definitions, but I fail to prove it...
Definitions:
Let $h:\mathbb R^n\to\mathbb R$ ...

**2**

votes

**2**answers

828 views

### Examples of deterministic processes of quadratic variation which are of unbounded variation

In [FĂ¶llmer 81] (English translation to be found here) writes: "The class of processes of quadratic variation is clearly larger than the class of semimartingales: Just consider a deterministic process ...

**10**

votes

**2**answers

1k views

### Continuous function from $[0,1]$ to $[0,1]$

Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?