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28 votes
10 answers
4k views

Why $ \int_0^{2\pi}\frac{\cos t - r}{1 - 2r\cos t + r^2}\,dt=0$ for $r\in[0,1)$

11 votes
4 answers
741 views

If $f : [a,b]\to\Bbb R$ is continuous, are there $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(a)}{b-a} = \tfrac{f(x_1)-f(x_2)}{x_1-x_2}$?

7 votes
1 answer
390 views

Given $A\in\Bbb R^{n\times n}$, is $C_A := \{SAS^{-1} : S\in GL(n,\mathbb R)\}$ connected?

7 votes
1 answer
625 views

Continuity with respect to Hausdorff metric, also with intersection?

7 votes
2 answers
270 views

A fascinating sequence of polynomials

5 votes
1 answer
74 views

Given real angles $\phi_1,\ldots,\phi_N$, there exist infinitely many integers $n$ such that $\cos(n\phi_k) > 0$ for all $k$.

3 votes
1 answer
275 views

Metric Density always zero or one?

3 votes
2 answers
163 views

$A(t)$ measurable. Does there exists measurable $x(t)\in\ker A(t)$?

3 votes
1 answer
85 views

What's the reproducing kernel of $H^1(a,b)$? [closed]

3 votes
1 answer
82 views

Example for trivial intersection of domains

2 votes
0 answers
38 views

Sequence of bump functions with an additional integral condition on the gradient

2 votes
1 answer
108 views

Something similar to Kronecker's Theorem

2 votes
1 answer
25 views

Are transition probabilities always absolutely continuous w.r.t. invariant measure?

2 votes
1 answer
46 views

Sufficient condition for a matrix to have full rank almost surely

2 votes
2 answers
106 views

Example for $A,B$ non-empty, bounded, and closed, $A\cap B = \emptyset$, but $\operatorname{dist}(A,B) = 0$

2 votes
3 answers
136 views

Weird Lagrange multiplier problem

2 votes
1 answer
267 views

When is the image of a convex set under a holomorphic function convex?

2 votes
1 answer
24 views

Constant for Sobolev multiplier

2 votes
1 answer
51 views

Is $\operatorname{dom}(A^*A)$ a core of $A$?

2 votes
1 answer
184 views

Sum of orthogonal projections whose images span the space

2 votes
1 answer
51 views

One matrix identity implies the other

2 votes
2 answers
94 views

Continuous, injective map between annuli, but with a "hole" in the image

2 votes
1 answer
88 views

Non-coprime pairs with coprime $\gcd$'s.

2 votes
3 answers
1k views

Adjoint of multiplication operator

2 votes
1 answer
125 views

Is this set dense in $L^2(0,1)$?

1 vote
0 answers
74 views

Classical derivative exists a.e. and is $L^1$. Is $f$ continuous?

1 vote
1 answer
25 views

Proof of a theorem on Sobolev multipliers on $(0,1)$ without extending to $\mathbb R$

1 vote
1 answer
62 views

$f\in L^1(0,1)$, $f\ge 0$, $f$ continuous on $(0,1)$. Does then $\lim_{x\to 0}xf(x) = 0$?

1 vote
1 answer
88 views

If $\operatorname{ran}A$ is closed, is then $\operatorname{ran}A^2$ closed?

1 vote
0 answers
236 views

Is infimum over all Lipschitz constants attained?