All Questions
5,628 questions
6
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0
answers
223
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Sum of product maximum
For which pairs of integers $(n,m)$ is the maximum of the following function $$f(x)=\sum_{i_1+\dots +i_n=m}\prod_{k=1}^n x^{i_k}_{k},\ \ x=(x_1,\dots,x_n), \|x\|=1$$ attained when $x_1=\dots=x_n$?
(...
- 53.3k
3
votes
1
answer
403
views
Is there a probability density function providing the least expected value?
Fix constant reals $A>1$ and $D>0$. Let $f:\mathbb{R}\to[0,\infty)$ be a probability density function on $\mathbb{R}$, i.e. $\int_{-\infty}^\infty f(x)\, dx=1$, that is continuous almost ...
- 10.3k
4
votes
1
answer
261
views
Minimizing action squared versus action
I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...
- 614
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
CommunityBot
- 1
2
votes
1
answer
465
views
Showing the derivative of this function is equal to $0$ a.e [closed]
Define $f:[0,1]\to [0,1]$ by $f(0)=0$, and $$f(x)=\sum\limits_{r_n\le x} 2^{ -n }$$ with $0\lt x\le 1$ where $[r_n]_{n\in \mathbb{Z^+} } = \mathbb{ Q} \cap (0,1) $.
How to show that the derivative $...
- 19.3k
8
votes
2
answers
2k
views
Do proper Zariski closed sets of algebraic sets have measure zero
This is a question related to another question I asked: here.
Say we induce a probability measure that is absolutely continuous with respect to to Lebesgue measure onto an irreducible real algebraic ...
CommunityBot
- 1
4
votes
1
answer
370
views
Norms for complex measures
I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
CommunityBot
- 1
10
votes
2
answers
766
views
When polynomial f(x^2) can be factored as g(x)·g(-x) ?
In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where $...
CommunityBot
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1
vote
0
answers
416
views
When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 6,210
2
votes
0
answers
890
views
Obtaining a pointwise bound on the convolution of two singular measures
I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids.
We are in Section 7, near equation (34) (pag.16 of the arxiv).
Notations and ...
- 692
0
votes
0
answers
60
views
Relative homology of interlevel set
Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
H_{\...
- 181
7
votes
1
answer
941
views
Kakeya and Nikodym maximal functions
I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more specifically,...
CommunityBot
- 1
2
votes
0
answers
263
views
A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$
Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
CommunityBot
- 1
2
votes
2
answers
2k
views
convergence of the infima of convex functions
Can one give a reference to a result like this:
If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then $\displaystyle\inf_{\mathbb{R}...
- 128k
5
votes
1
answer
345
views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
CommunityBot
- 1
37
votes
1
answer
2k
views
Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?
(This question is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...
CommunityBot
- 1
6
votes
2
answers
720
views
Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 7,514
6
votes
1
answer
152
views
Terminology for sequences/functions that approach each other
What do I call two sequences $a, b$ such that $\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions $f, g$ such that $\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are ...
- 24.8k
1
vote
1
answer
263
views
When can we "displace" an ultrafilter limit with another limit?
Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)...
- 25.8k
8
votes
1
answer
597
views
complete metric space
Hallo, I have the following question:
Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: $\...
1
vote
0
answers
324
views
Linearization of cones
Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
- 6,962
3
votes
1
answer
133
views
A recurrent sequence related to the Brouwer fixed-point theorem
Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} \sum_{j=0}...
- 60.5k
0
votes
0
answers
127
views
A question of the weights $A_\infty$' equvalent condition in Real &Harmonic analysis
I have a question. The question is to prove:
The weight $w \in A_\infty $if and only if
$\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all
...
- 8,842
0
votes
0
answers
94
views
Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
- 56.9k
0
votes
0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91
3
votes
2
answers
291
views
on completeness of R_mn, the set of all rational functions of type (m,n)
It is known from finite dimensionality of $P_r$, the space of all polynomials of degree less than or equal to $r$, that $P_r$ is complete with respect to uniform norm.
Considering $R_{m,n}[a,b]=\{p/...
- 79.9k
4
votes
1
answer
529
views
Find a continuous function with a prescribed continuity set
It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.
In the book "Understanding Analysis" by Abbott is stated in page 128 ...
CommunityBot
- 1
9
votes
2
answers
519
views
The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-...
- 61.9k
1
vote
1
answer
715
views
Importance of Denjoy-Carleman classes as a class.
Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\...
- 25.3k
3
votes
1
answer
379
views
Lipschitz map of the circle onto a triangle
Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane.
- 31.2k
3
votes
1
answer
643
views
Is a Cauchy principal value invariant under a "change of variables"?
Let $f \in C^{\gamma}_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere ...
- 16.2k
1
vote
1
answer
1k
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A question about "nice" functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ''...
- 3,270
3
votes
1
answer
240
views
Lipschitz map of the ellipse
Is there a L-Lipschitz homeomorphism of the Elipse $x^2/4+y^2=1$ onto the unit circle $x^2+y^2=1$ such that $L<1$?
- 7,633
0
votes
1
answer
208
views
The pth power of a distance function is twice continuously differentiable, for $p>2$?
Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...
- 9
1
vote
0
answers
305
views
Adjoint operator in sobolev space
Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta u\|...
- 51
2
votes
1
answer
579
views
Does the Border (Boundary) Points of a convex body make a concave function?
Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis ...
- 42.8k
2
votes
1
answer
310
views
Boundedness of an Oscillating Integral
Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded as $\lambda\...
- 621
1
vote
1
answer
547
views
When does the finite union of convex sets have a hole in it?
Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) \...
- 45k
8
votes
2
answers
2k
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Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
CommunityBot
- 1
1
vote
1
answer
158
views
variational characterization of the average of an $L^p$ function
Let $\Omega$ be a measurable set having finite Lebesgue measure. Let $p\geq 1$ and $u\in
L^p(\Omega)$. Is it true that the minimum value of the real function
$$
c\in
\mathbb{R}^n\mapsto\int_\Omega |u-...
- 621
7
votes
1
answer
463
views
Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions
This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f \...
- 176
5
votes
1
answer
903
views
Uncountable Pre-Image
I've been reading about space filling curves, and been asking myself this question.
If $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous open map, is it true that $\forall x \in$ range$(f)$ ...
- 8,071
2
votes
2
answers
2k
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Does a bounded real function have an analytic continuation [closed]
Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where
$f$ is real-analytic on the open interval $(0,1)$
$f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-...
CommunityBot
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5
votes
1
answer
958
views
Does a nonlinear additive function on R imply a Hamel basis of R?
A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
- 39.8k
0
votes
1
answer
525
views
A simple question from mathematical analysis (assumption changed) [closed]
Let $\forall n=0,1,2,\dots$, $\alpha_{n}(x)$ are POLYNOMIALS in $x$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$
has positive radius of convergence. Can one ...
- 2,715
2
votes
1
answer
391
views
Can you prove that Average(f(x)) is not equal to f(average(x)) for non-linear f in more than one variable [closed]
I am seeking a general mathematical proof & a reference for the proof for something I know intuitively to be true, and can demonstrate by example, but would like to prove. Assume a function with 6 ...
CommunityBot
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-4
votes
1
answer
200
views
How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
CommunityBot
- 1
1
vote
1
answer
353
views
Does $h$ have infinitely many isolated zeros?
Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function:
$$h(s₁,s₂,...,s_{r+1})=\prod_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod_{j=1}^{k}s_{j}\right)$$
Also, ...
- 1,197
0
votes
0
answers
490
views
Sufficient conditions for continuity of function $y\mapsto\min_{[x_0,y]}\phi$
Let $\phi:\mathbb{R}\to\mathbb{R}$ a continuous function.
Fix $x_0\in\mathbb{R}$ and consider
$$\psi:\mathbb{R}\to\mathbb{R},\ \psi(y)=\min_{\xi\in[x_0,y]}\phi(\xi)\ .$$
Is $\psi$ a continuous ...
CommunityBot
- 1
0
votes
1
answer
1k
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Global Implicit Function Theorem
Let $F:\mathbb R^2\rightarrow\mathbb R$ be a measurable function. Under what conditions on $F$ does there exist a function $\theta:\mathbb R^2\rightarrow\mathbb R$ such that
$F(x,\theta(z,x))=z$ for ...
- 16.2k