Ternary scale theorems: Difference between revisions

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==== Statement (2) ====

~~In case 2, let~~ {{nowrap|''n'' ≥ 3}} and let {{nowrap|~~(2, 1) − (1, 1) {{=}}~~ '''g'''1|~~(1, 2) − (2, 1) {{=}}~~ '''g'''2}} be the two alternants. Let '''g'''3 be the closing generator after stacking alternating '''g'''1 and '''g'''2. Then the generator circle is {{nowrap|('''g'''1 '''g'''2){{floor|''n''/2}}}} '''g'''3. If a step is formed by stacking ''k'' generators, we may assume that ''k'' is odd, and the combinations of alternants corresponding to a step come in exactly 3 sizes:

Let {{nowrap|''n'' ≥ 3}} and let {{nowrap|'''g'''1|'''g'''2}} be the two alternants. Let '''g'''3 be the closing generator after stacking alternating '''g'''1 and '''g'''2. Then the generator circle is {{nowrap|('''g'''1 '''g'''2){{floor|''n''/2}}}} '''g'''3. If a step is formed by stacking ''k'' generators, we may assume that ''k'' is odd, and the combinations of alternants corresponding to a step come in exactly 3 sizes:

# {{nowrap|{{ceil|{{frac|''k''|2}}}}'''g'''1 + {{floor|{{frac|''k''|2}}}}'''g'''2}}

# {{nowrap|{{floor|{{frac|''k''|2}}}}'''g'''1 + {{ceil|{{frac|''k''|2}}}}'''g'''2}}

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 ==== Statement (2) ====
-In case 2, let {{nowrap|''n'' &ge; 3}} and let {{nowrap|(2, 1) − (1, 1) {{=}} '''g'''<sub>1</sub>|(1, 2) − (2, 1) {{=}} '''g'''<sub>2</sub>}} be the two alternants. Let '''g'''<sub>3</sub> be the closing generator after stacking alternating '''g'''<sub>1</sub> and '''g'''<sub>2</sub>. Then the generator circle is {{nowrap|('''g'''<sub>1</sub> '''g'''<sub>2</sub>)<sup>{{floor|''n''/2}}</sup>}} '''g'''<sub>3</sub>. If a step is formed by stacking ''k'' generators, we may assume that ''k'' is odd, and the combinations of alternants corresponding to a step come in exactly 3 sizes:
+Let {{nowrap|''n'' &ge; 3}} and let {{nowrap|'''g'''<sub>1</sub>|'''g'''<sub>2</sub>}} be the two alternants. Let '''g'''<sub>3</sub> be the closing generator after stacking alternating '''g'''<sub>1</sub> and '''g'''<sub>2</sub>. Then the generator circle is {{nowrap|('''g'''<sub>1</sub> '''g'''<sub>2</sub>)<sup>{{floor|''n''/2}}</sup>}} '''g'''<sub>3</sub>. If a step is formed by stacking ''k'' generators, we may assume that ''k'' is odd, and the combinations of alternants corresponding to a step come in exactly 3 sizes:
 # {{nowrap|{{ceil|{{frac|''k''|2}}}}'''g'''<sub>1</sub> + {{floor|{{frac|''k''|2}}}}'''g'''<sub>2</sub>}}
 # {{nowrap|{{floor|{{frac|''k''|2}}}}'''g'''<sub>1</sub> + {{ceil|{{frac|''k''|2}}}}'''g'''<sub>2</sub>}}