All Questions
5,659 questions
37
votes
12
answers
5k
views
Examples where existence is harder than evaluation
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, ...
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 ...
- 3,017
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
35
votes
19
answers
9k
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 ...
34
votes
1
answer
2k
views
Ruling out the existence of a strange polynomial
Does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that
$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$
and
$$\displaystyle \liminf_{(x,y) \in \mathbb{R}^2} f(x,y) = -\...
- 26.9k
34
votes
2
answers
2k
views
Is it always possible to calculate the limit of an elementary function?
I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...
- 496
33
votes
1
answer
3k
views
About the validity of a new conjecture about a diophantine equation
Let us consider the following conjecture:
Conjecture: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$.
...
- 1,197
33
votes
2
answers
2k
views
What is the smallest set of real continuous functions generating all rational numbers by iteration?
I recently came across this problem from USAMO 2005:
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
- 4,862
33
votes
4
answers
2k
views
Hahn-Banach theorem with convex majorant
At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
- 16.4k
33
votes
1
answer
2k
views
For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
- 43.2k
33
votes
1
answer
2k
views
How quickly can the derivative of an everywhere differentiable function change sign?
Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
- 635
33
votes
5
answers
12k
views
Differentiable functions with discontinuous derivatives
For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...
- 19.7k
32
votes
4
answers
4k
views
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,200
32
votes
4
answers
18k
views
About the Riemann integrability of composite functions
When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions.
For the composite function $f \circ g$, He presented three cases:
1) ...
- 627
31
votes
13
answers
6k
views
Classic applications of Baire category theorem
I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
31
votes
4
answers
8k
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 ...
- 313
31
votes
2
answers
3k
views
A natural construction of real numbers?
Summary
Someone claims $\mathbb{R}$ can be constructed as the following intriguing quotient, which is related to Gromov's bounded cohomology. I want to find out if it is true.
$$\frac{\bigl\{f:\mathbb{...
- 5,230
31
votes
1
answer
3k
views
What did Rolle prove when he proved Rolle's theorem?
Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...
user21349
31
votes
2
answers
3k
views
Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$
The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned ...
31
votes
1
answer
2k
views
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
I ...
user46896
30
votes
4
answers
3k
views
A counterexample for Sard's theorem in $C^1$ regularity
I can't seem to find an example of a function $f \colon \mathbb{R}^2\to \mathbb{R}$ which is $C^1$ and such that the set of its critical values is not of zero measure.
What examples are there?
$...
- 478
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 407
30
votes
2
answers
1k
views
Minimum number of $|\cdot|$ operations necessary to express $\max$
For two variables, their maximum
$\max\{x_1,x_2\}$ can be expressed using one $|\cdot|$ operation:
$$
\max\{x_1,x_2\} = \frac12(x_1+x_2+|x_1-x_2|).
$$
For $3$ variables, it seems fairly clear that ...
- 6,541
30
votes
1
answer
2k
views
Have any numbers been proven to be normal that weren't constructed to be?
It's easy to construct an example of a number that's normal in a given base, but for most given numbers it's notoriously hard to prove that they're normal.
Has any number ever been proven to be normal ...
- 1,311
30
votes
0
answers
899
views
Three real polynomials
Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
- 91.8k
29
votes
2
answers
4k
views
Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
- 43.2k
29
votes
1
answer
1k
views
About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$
I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far.
I ...
- 483
29
votes
3
answers
2k
views
Wanted: Positivity certificate for the AM-GM inequality in low dimension
I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
Can one write the LHS as a sum
$\,\...
- 489
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 19.6k
28
votes
3
answers
3k
views
Expressing the Riemann Zeta function in terms of GCD and LCM
Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
- 5,470
28
votes
3
answers
2k
views
Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
- 1,565
28
votes
7
answers
5k
views
Rolle's theorem in n dimensions
This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
28
votes
4
answers
3k
views
Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$
Let $x>0$ and $n$ be a natural number. Prove that:
$$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$
This question is very similar to many contests problems, but ...
- 1,054
28
votes
4
answers
2k
views
For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?
Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true :
$$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...
- 850
28
votes
4
answers
3k
views
"Converse" of Taylor's theorem
Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a_0,a_1,\ldots,a_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum_{i=0}^k a_i(c)t^i+o(t^k)$ (for ...
- 3,146
28
votes
1
answer
1k
views
Polynomials non-negative on the integers
Let $P$ be a real polynomial of exact degree $2n$ ($n \geq 1$) whose zeros are real numbers and such that
\begin{equation*}
P(j) \geq 0
\quad \text{for any} \quad
j \in \mathbb{Z}.
\end{equation*}
...
28
votes
0
answers
1k
views
Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
- 6,476
27
votes
5
answers
1k
views
Is this a known question about the expression of a function on $\Bbb R^2$ as an infinite sum of products?
The question below was posted on Mathematics Stack Exchange. It received no answer, and I do not expect any direct answer to it here. However, the question seems to me a natural one. Thus I wonder ...
- 2,437
27
votes
2
answers
1k
views
Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent
Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
- 447
27
votes
3
answers
2k
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Integral $\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$
How to evaluate this integral:
$$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$$
I'm making use of the integral ...
- 389
27
votes
1
answer
2k
views
Is every real number in [0,1] a product of three (or more) Cantor set's numbers?
It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
- 60.5k
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
27
votes
2
answers
1k
views
Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
- 28k
27
votes
1
answer
2k
views
Linear combination of sine and cosine
I was explaining to my students the other day why $\cos(2x)$ is not a linear combination of $\sin(x)$ and $\cos(x)$ over $\mathbb{R}$. Besides the canonical method of using special values of sine and ...
- 960
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
26
votes
5
answers
8k
views
Proof that no differentiable space-filling curve exists
Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere ...
- 151k
26
votes
3
answers
16k
views
the dual space of C(X) (X is noncompact metric space)
It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
- 1,706
26
votes
2
answers
9k
views
Maximal ideals in the ring of continuous real-valued functions on ℝ
For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
- 6,734
26
votes
3
answers
3k
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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 ...
- 23.2k
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.7k