Newtheorem and theoremstyle test

Newtheorem and theoremstyle test Michael Downesupdated by Barbara Beeton none

1 1 §1 <tag close=" ">1</tag>Test of standard theorem styles

Ahlfors’ Lemma gives the principal criterion for obtaining lower bounds on the Kobayashi metric.

Ahlfors’ Lemma Ahlfors’ Lemma <tag><text font="bold">Ahlfors’ Lemma</text></tag><text font="bold">.</text>

Let $= ⁢ d s 2 ⁢ h (z) | ⁢ d z | 2$ be a Hermitian pseudo-metric on $D r$ , $∈ h ⁢ C 2 (D r)$ , with $ω$ the associated $(1, 1)$ -form. If $≥ Ric ω ω$ on $D r$ , then $≤ ω ω r$ on all of $D r$ (or equivalently, $≤ ⁢ d s 2 ⁢ d s r 2$ ).

Lemma 1.1 1.1 Lemma 1.1 <tag><text font="bold">Lemma 1.1</text></tag><text font="bold"> </text>(negatively curved families)<text font="bold">.</text>

Let ${⁢ d s 1 2, …, ⁢ d s k 2}$ be a negatively curved family of metrics on $D r$ , with associated forms $ω 1$ , …, $ω k$ . Then $≤ ω i ω r$ for all $i$ .

Then our main theorem:

Theorem 1.2 1.2 Theorem 1.2 <tag><text font="bold">Theorem 1.2</text></tag><text font="bold">.</text>

Let $d max$ and $d min$ be the maximum, resp. minimum distance between any two adjacent vertices of a quadrilateral $Q$ . Let $σ$ be the diagonal pigspan of a pig $P$ with four legs. Then $P$ is capable of standing on the corners of $Q$ iff

(1) 1

≥ σ + d max 2 d min 2 .

Corollary 1.3 1.3 Corollary 1.3 <tag><text font="bold">Corollary 1.3</text></tag><text font="bold">.</text>

Admitting reflection and rotation, a three-legged pig $P$ is capable of standing on the corners of a triangle $T$ iff () holds.

Remark Remark <tag><text font="italic">Remark</text></tag><text font="italic">.</text>

As two-legged pigs generally fall over, the case of a polygon of order $2$ is uninteresting.

2 2 §2 <tag close=" ">2</tag>Custom theorem styles Exercise 1 1 Exercise 1 <tag><text font="bold">Exercise 1</text></tag><text font="bold">.</text>

Generalize Theorem to three and four dimensions.

Note 1 1 Note 1 <tag><text font="italic">Note 1</text></tag><text font="italic">:</text>

This is a test of the custom theorem style ‘note’. It is supposed to have variant fonts and other differences.

B-Theorem 1 1 B-Theorem 1 <tag><text font="bold">B-Theorem 1</text></tag><text font="bold">.</text>

Test of the ‘linebreak’ style of theorem heading.

This is a test of a citing theorem to cite a theorem from some other source.

Theorem 3.6 in <cite class="ltx_citemacro_cite">[<bibref bibrefs="thatone" separator="," yyseparator=","/>]</cite>.

No hyperlinking available here yet … but that’s not a bad idea for the future.

3 3 §3 <tag close=" ">3</tag>The proof environment Proof.

Here is a test of the proof environment. ∎

Proof of Theorem <ref labelref="LABEL:pigspan"/>.

And another test. ∎

Proof <text font="upright">(</text>necessity<text font="upright">)</text>.

And another. ∎

Proof <text font="upright">(</text>sufficiency<text font="upright">)</text>.

And another, ending with a display:

= + 11 2 . ∎

4 4 §4 <tag close=" ">4</tag>Test of number-swapping

This is a repeat of the first section but with numbers in theorem heads swapped to the left.

Ahlfors’ Lemma gives the principal criterion for obtaining lower bounds on the Kobayashi metric.

Ahlfors’ Lemma Ahlfors’ Lemma <tag><text font="bold">Ahlfors’ Lemma</text></tag><text font="bold">.</text>

4.1 Lemma 4.1 Lemma 4.1 <tag><text font="bold">4.1 Lemma</text></tag><text font="bold"> </text>(negatively curved families)<text font="bold">.</text>

Let ${⁢ d s 1 2, …, ⁢ d s k 2}$ be a negatively curved family of metrics on $D r$ , with associated forms $ω 1$ , …, $ω k$ . Then $≤ ω i ω r$ for all $i$ .

Then our main theorem:

4.2 Theorem 4.2 Theorem 4.2 <tag><text font="bold">4.2 Theorem</text></tag><text font="bold">.</text>

(2) 2

≥ σ + d max 2 d min 2 .

4.3 Corollary 4.3 Corollary 4.3 <tag><text font="bold">4.3 Corollary</text></tag><text font="bold">.</text>

Admitting reflection and rotation, a three-legged pig $P$ is capable of standing on the corners of a triangle $T$ iff () holds.

References [1] 1 Dummy entry.