Матеріал з Вікіпедії — вільної енциклопедії.
Подстраница "Користувач Галактион:Нотація Ландау" создана для того, чтобы перенести информацию из раздела "Обговорення" статьи "Нотація Ландау ". Галактион 19:17, 5 березня 2010 (UTC)
Дополнение к статье "Нотацiя Ландау"
[ ред. | ред. код ]
d
⊢
X
1
×
Y
1
⊆
R
2
∧
f
:
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1
↦
Y
1
∧
X
2
×
Y
2
⊆
R
2
∧
g
:
X
2
↦
Y
2
∧
{\displaystyle ~d\vdash \ X_{1}\times Y_{1}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {f} :X_{1}\mapsto Y_{1}\quad \land \quad X_{2}\times Y_{2}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {g} :X_{2}\mapsto Y_{2}\quad \land }
∃
a
∈
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(
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a
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∞
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⊆
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∩
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∀
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∈
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a
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∞
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(
g
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x
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≠
0
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)
→
{\displaystyle ~\exists _{a\ \in \ \mathbb {R} }\ (\ (a,\infty )\subseteq X_{1}\cap X_{2}\ \land \ \forall _{x\ \in \ (a,\infty )}\ (g(x)\neq 0)\ )\quad \to }
(
∀
x
∈
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∞
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f
∈
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∃
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∀
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∞
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f
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x
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≤
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g
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x
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in O(\mathrm {g} ))\quad \leftrightarrow \quad \exists _{c\ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,\infty )}\ \forall _{x\ \in \ (a',\infty )}\ (|f(x)|\ \leq \ c\cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
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˙
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∞
)
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f
∈
o
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g
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↔
∀
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∈
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∈
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∀
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<
ε
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g
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in o(\mathrm {g} ))\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,\infty )}\ \forall _{x\ \in \ (a',\infty )}\ (|f(x)|\ <\ \varepsilon \cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
∞
)
(
f
≍
g
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↔
∀
ε
∈
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0
,
∞
)
∃
a
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∈
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a
,
∞
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∀
x
∈
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∞
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f
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x
)
−
g
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x
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<
ε
⋅
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g
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x
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \asymp \mathrm {g} )\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,\infty )}\ \forall _{x\ \in \ (a',\infty )}\ (|f(x)-g(x)|\ <\ \varepsilon \cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
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∞
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f
∈
Ω
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g
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∃
c
∈
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∞
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a
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∈
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a
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∀
x
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∞
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f
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x
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≥
c
⋅
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g
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x
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in \Omega (\mathrm {g} ))\quad \leftrightarrow \quad \exists _{c\ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,\infty )}\ \forall _{x\ \in \ (a',\infty )}\ (|f(x)|\ \geq \ c\cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
∞
)
(
f
∈
ω
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g
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)
↔
∀
ε
∈
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∞
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∃
a
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∈
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a
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∞
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∀
x
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∞
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>
ε
⋅
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in \omega (\mathrm {g} ))\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,\infty )}\ \forall _{x\ \in \ (a',\infty )}\ (|f(x)|\ >\ \varepsilon \cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
∞
)
(
f
∈
Θ
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g
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↔
∃
{
c
,
C
}
⊆
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0
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∞
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∃
a
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∈
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∀
x
∈
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c
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g
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≤
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f
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{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in \Theta (\mathrm {g} ))\ \leftrightarrow \ \exists _{\{c,\ C\}\ \subseteq \ (0,\infty )}\ \exists _{a'\ \in \ (a,\infty )}\ \forall _{x\ \in \ (a',\infty )}\ (c\cdot |g(x)|\leq |f(x)|\leq C\cdot |g(x)|)\ )}
Примечание
∀
x
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≲
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{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in O(\mathrm {g} ))\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (|f(x)|\lesssim |g(x)|)}
∀
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{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in o(\mathrm {g} ))\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (|f(x)|\ll |g(x)|)}
∀
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∞
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{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \asymp \mathrm {g} )\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (|f(x)-g(x)|\ll |g(x)|)}
∀
x
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∞
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∞
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≲
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∀
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∈
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∞
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g
∈
O
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f
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)
{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in \Omega (\mathrm {g} ))\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (|g(x)|\lesssim |f(x)|)\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {g} \in O(\mathrm {f} ))}
∀
x
∈
U
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∞
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f
∈
ω
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g
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⇔
∀
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∈
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∞
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≪
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f
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∀
x
∈
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˙
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∞
)
(
g
∈
o
(
f
)
)
{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in \omega (\mathrm {g} ))\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (|g(x)|\ll |f(x)|)\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {g} \in o(\mathrm {f} ))}
∀
x
∈
U
˙
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∞
)
(
f
∈
Θ
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g
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)
⇔
∀
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∈
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˙
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∞
)
(
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f
(
x
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≈
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g
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x
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{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in \Theta (\mathrm {g} ))\ \Leftrightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (|f(x)|\approx |g(x)|)}
∀
x
∈
U
˙
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∞
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(
f
∈
o
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g
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⇒
∀
x
∈
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∞
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f
∈
O
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g
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)
{\displaystyle ~\forall _{x\ \in \ {\dot {U}}(\infty )\ }\ (\mathrm {f} \in o(\mathrm {g} ))\ \Rightarrow \ \forall _{x\ \in \ {\dot {U}}(\infty )}\ (\mathrm {f} \in O(\mathrm {g} ))}
d
⊢
X
1
×
Y
1
⊆
R
2
∧
f
:
X
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↦
Y
1
∧
X
2
×
Y
2
⊆
R
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∧
g
:
X
2
↦
Y
2
∧
{\displaystyle ~d\vdash \ X_{1}\times Y_{1}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {f} :X_{1}\mapsto Y_{1}\quad \land \quad X_{2}\times Y_{2}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {g} :X_{2}\mapsto Y_{2}\quad \land }
∃
a
∈
R
(
(
−
∞
,
a
)
⊆
X
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∩
X
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∧
∀
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∈
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∞
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a
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g
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x
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≠
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→
{\displaystyle ~\exists _{a\ \in \ \mathbb {R} }\ (\ (-\infty ,a)\subseteq X_{1}\cap X_{2}\ \land \ \forall _{x\ \in \ (-\infty ,a)}\ (g(x)\neq 0)\ )\quad \to }
(
∀
x
∈
U
˙
(
−
∞
)
(
f
∈
O
(
g
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↔
∃
c
∈
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0
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∞
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∃
a
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∈
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∞
,
a
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∀
x
∈
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≤
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⋅
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(-\infty )}\ (\mathrm {f} \in O(\mathrm {g} ))\quad \leftrightarrow \quad \exists _{c\ \in \ (0,\infty )}\ \exists _{a'\ \in \ (-\infty ,a)}\forall _{x\ \in \ (-\infty ,a')}\ (|f(x)|\ \leq \ c\cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
−
∞
)
(
f
∈
o
(
g
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)
↔
∀
ε
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∞
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a
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∞
,
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∀
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<
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(-\infty )}\ (\mathrm {f} \in o(\mathrm {g} ))\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (-\infty ,a)}\forall _{x\ \in \ (-\infty ,a')}\ (|f(x)|\ <\ \varepsilon \cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
−
∞
)
(
f
≍
g
)
↔
∀
ε
∈
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0
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∞
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∃
a
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∈
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∞
,
a
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∀
x
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∞
,
a
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f
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x
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−
g
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x
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<
ε
⋅
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g
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x
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|
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)
∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(-\infty )}\ (\mathrm {f} \asymp \mathrm {g} )\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (-\infty ,a)}\ \forall _{x\ \in \ (-\infty ,a')}\ (|f(x)-g(x)|\ <\ \varepsilon \cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
−
∞
)
(
f
∈
Ω
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g
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↔
∃
c
∈
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0
,
∞
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∃
a
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∈
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∞
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a
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∀
x
∈
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∞
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a
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f
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≥
c
⋅
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g
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x
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)
∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(-\infty )}\ (\mathrm {f} \in \Omega (\mathrm {g} ))\quad \leftrightarrow \quad \exists _{c\ \in \ (0,\infty )}\ \exists _{a'\ \in \ (-\infty ,a)}\ \forall _{x\ \in \ (-\infty ,a')}\ (|f(x)|\ \geq \ c\cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
−
∞
)
(
f
∈
ω
(
g
)
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↔
∀
ε
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∞
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∃
a
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∈
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∞
,
a
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∀
x
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a
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(
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f
(
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>
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g
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x
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(-\infty )}\ (\mathrm {f} \in \omega (\mathrm {g} ))\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (-\infty ,a)}\ \forall _{x\ \in \ (-\infty ,a')}\ (|f(x)|\ >\ \varepsilon \cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
(
−
∞
)
(
f
∈
Θ
(
g
)
)
↔
∃
{
c
,
C
}
⊆
(
0
,
∞
)
∃
a
′
∈
(
−
∞
,
a
)
∀
x
∈
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−
∞
,
a
′
)
(
c
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g
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x
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≤
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f
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≤
C
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g
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x
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)
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}(-\infty )}\ (\mathrm {f} \in \Theta (\mathrm {g} ))\ \leftrightarrow \ \exists _{\{c,\ C\}\ \subseteq \ (0,\infty )}\ \exists _{a'\ \in \ (-\infty ,a)}\ \forall _{x\ \in \ (-\infty ,a')}\ (c|g(x)|\leq |f(x)|\leq C|g(x)|))}
[ ред. | ред. код ]
d
⊢
X
1
×
Y
1
⊆
R
2
∧
f
:
X
1
↦
Y
1
∧
X
2
×
Y
2
⊆
R
2
∧
g
:
X
2
↦
Y
2
∧
{\displaystyle ~d\vdash \ X_{1}\times Y_{1}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {f} :X_{1}\mapsto Y_{1}\quad \land \ X_{2}\times Y_{2}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {g} :X_{2}\mapsto Y_{2}\quad \land }
∃
{
a
,
b
}
⊆
R
∧
a
<
b
(
(
a
,
b
)
⊆
X
1
∩
X
2
∧
∀
x
∈
(
a
,
b
)
(
g
(
x
)
≠
0
)
)
→
{\displaystyle ~\exists _{\{a,b\}\ \subseteq \ \mathbb {R} \ \land \ a\ <\ b}\ (\ (a,b)\subseteq X_{1}\cap X_{2}\ \land \ \forall _{x\ \in \ (a,b)}\ (g(x)\neq 0)\ )\quad \to }
(
∀
x
∈
U
˙
+
(
a
)
(
f
∈
O
(
g
)
)
↔
∃
c
∈
(
0
,
∞
)
∃
b
′
∈
(
a
,
b
)
∀
x
∈
(
a
,
b
′
)
(
|
f
(
x
)
|
≤
c
⋅
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g
(
x
)
|
)
)
∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}^{+}(a)}\ (\mathrm {f} \in O(\mathrm {g} ))\quad \leftrightarrow \quad \exists _{c\ \in \ (0,\infty )}\ \exists _{b'\ \in \ (a,b)}\ \forall _{x\ \in \ (a,b')}\ (|f(x)|\ \leq \ c\cdot |g(x)|)\ )\quad \land }
(
∀
x
∈
U
˙
+
(
a
)
(
f
∈
o
(
g
)
)
↔
∀
ε
∈
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0
,
∞
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∃
b
′
∈
(
a
,
b
)
∀
x
∈
(
a
,
b
′
)
(
|
f
(
x
)
|
<
ε
⋅
|
g
(
x
)
|
)
)
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}^{+}(a)}\ (\mathrm {f} \in o(\mathrm {g} ))\quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{b'\ \in \ (a,b)}\ \forall _{x\ \in \ (a,b')}\ (|f(x)|\ <\ \varepsilon \cdot |g(x)|)\ )}
[ ред. | ред. код ]
d
⊢
X
1
×
Y
1
⊆
R
2
∧
f
:
X
1
↦
Y
1
∧
X
2
×
Y
2
⊆
R
2
∧
g
:
X
2
↦
Y
2
∧
{\displaystyle ~d\vdash \ X_{1}\times Y_{1}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {f} :X_{1}\mapsto Y_{1}\quad \land \quad X_{2}\times Y_{2}\subseteq \mathbb {R} ^{2}\ \land \ \mathrm {g} :X_{2}\mapsto Y_{2}\quad \land }
∃
{
a
,
b
}
⊆
R
∧
a
<
b
(
(
a
,
b
)
⊆
X
1
∩
X
2
∧
∀
x
∈
(
a
,
b
)
(
g
(
x
)
≠
0
)
)
→
{\displaystyle ~\exists _{\{a,b\}\ \subseteq \ \mathbb {R} \ \land \ a\ <\ b}\ (\ (a,b)\subseteq X_{1}\cap X_{2}\ \land \ \forall _{x\ \in \ (a,b)}\ (g(x)\neq 0)\ )\quad \to }
(
∀
x
∈
U
˙
−
(
b
)
(
f
∈
O
(
g
)
)
↔
∃
c
∈
(
0
,
∞
)
∃
a
′
∈
(
a
,
b
)
∀
x
∈
(
a
′
,
b
)
(
|
f
(
x
)
|
≤
c
⋅
|
g
(
x
)
|
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∧
{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}^{-}(b)}\ (\mathrm {f} \in O(\mathrm {g} ))\quad \leftrightarrow \quad \exists _{c\ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,b)}\ \forall _{x\ \in \ (a',b)}\ (|f(x)|\ \leq \ c\cdot |g(x)|)\ )\quad \land }
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{\displaystyle ~(\forall _{x\ \in \ {\dot {U}}^{-}(b)}\ (\mathrm {f} \in o(\mathrm {g} ))\ \quad \leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{a'\ \in \ (a,b)}\ \forall _{x\ \in \ (a',b)}\ (|f(x)|\ <\ \varepsilon \cdot |g(x)|)\ )}
[ ред. | ред. код ] Галактион 18:08, 13 серпня 2009 (UTC)
