# Tag Info

Accepted

### Baire class $1$ functions and Baire's characterization theorem

I had a similar problem some time ago and I had the impression that a general review on Baire classes is in fact missing, and would be useful. Probably you already saw it, but in this chapter there is ...
1 vote

• 6,344

• 85.3k
Accepted

### Ruling out the existence of a strange polynomial

The polynomial $f(x,y)=(x^2+1)(5y^2+5y+1)\in\mathbb{Z}[x,y]$ is an example. Note that $5y^2+5y+1>0$ for $y\in\mathbb{Z}$, but $5y^2+5y+1<0$ at $y=-\frac{1}{2}$.
• 2,374

### For which Sheaf topoi is Brouwer's fan theorem true?

The classic 1979 paper Fourman & Hyland, Sheaf models for analysis, has a number of results along these lines. They show in particular in Theorem 3.2 that every spatial topos, i.e. topos of ...
• 3,271
Accepted

• 5,357

### Status of the fundamental theorem of algebra for the locale of real numbers

I can see two issues of non-constructivity here, but I feel that they are both noise rather than central to the mathematics: The leading coefficient of the polynomial could fail to be apart from zero, ...
• 6,832
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• 85.3k
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### Entire function with almost periodic boundary condition?

The answer seems to be negative. Suppose that an entire function $f$ satisfies $f(z+v_i)=e^{A_iz}f(z)$, where $v_1$ and $v_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this ...
• 81.8k

### Quantitative analytic continuation estimate for a function small on a set of positive measure

• 85.3k
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### Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$. ...
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### General version of Weyl's lemma

unfortunately Weyl lemma cannot be generalized to any divergence form elliptic operator. The problem is given by the smoothness of the coefficients $a_{ij}$. There is a huge literature on the subject. ...
Accepted

• 81.8k
Accepted

• 85.3k

### Discontinuous functions without removable discontinuities

I claim that one can always get rid of all removable discontinuities of a function to obtain a function without removable discontinuities. For generality, we shall work in the framework of general ...
• 24.6k
Let us drop the assumption $x_j\in[1,2]$, it is not needed. Proving the result by contradiction, denote our function by $f_N$, suppose that $f_N(z_N)=-i$, and $\mathrm{Im}\ z_N= 1/(N^2R_N)$ where \$R_N\...