id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-12500 | Lucas Number 2n in terms of Square of Lucas Number n | Let $L_n$ denote the $n$th Lucas number.
Then:
:$L_{2 n} = {L_n}^2 + 2 \left({-1}\right)^n$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Note that we have:
{{begin-eqn}}
{{eqn | l = \phi \hat \phi
| r = \dfrac {1 + \sqrt 5} 2 \dfrac {1 - \sqrt 5} 2
| c =
}}
{{eqn | r = \dfrac {1 - 5} 4
| c = Difference of Two Squares
}}
{{eqn | r = -1
| c =
}}
{{end-eqn... | Let $L_n$ denote the $n$th [[Definition:Lucas Number|Lucas number]].
Then:
:$L_{2 n} = {L_n}^2 + 2 \left({-1}\right)^n$ | Let:
:$\phi = \dfrac {1 + \sqrt 5} 2$
:$\hat \phi = \dfrac {1 - \sqrt 5} 2$
Note that we have:
{{begin-eqn}}
{{eqn | l = \phi \hat \phi
| r = \dfrac {1 + \sqrt 5} 2 \dfrac {1 - \sqrt 5} 2
| c =
}}
{{eqn | r = \dfrac {1 - 5} 4
| c = [[Difference of Two Squares]]
}}
{{eqn | r = -1
| c =
}}
{{e... | Lucas Number 2n in terms of Square of Lucas Number n | https://proofwiki.org/wiki/Lucas_Number_2n_in_terms_of_Square_of_Lucas_Number_n | https://proofwiki.org/wiki/Lucas_Number_2n_in_terms_of_Square_of_Lucas_Number_n | [
"Lucas Numbers"
] | [
"Definition:Lucas Number"
] | [
"Difference of Two Squares",
"Closed Form for Lucas Numbers",
"Closed Form for Lucas Numbers"
] |
proofwiki-12501 | 12 Pentominoes | There exist $12$ distinct free pentominoes:
:600px | {{ProofWanted|Work to be done yet to establish method of creation}} | There exist $12$ [[Definition:Distinct|distinct]] [[Definition:Free Polyomino|free]] [[Definition:Pentomino|pentominoes]]:
:[[File:12Pentominoes.png|600px]] | {{ProofWanted|Work to be done yet to establish method of creation}} | 12 Pentominoes | https://proofwiki.org/wiki/12_Pentominoes | https://proofwiki.org/wiki/12_Pentominoes | [
"Pentominoes",
"12"
] | [
"Definition:Distinct",
"Definition:Polyomino/Free",
"Definition:Pentomino",
"File:12Pentominoes.png"
] | [] |
proofwiki-12502 | 18 Fixed Pentominoes | There exist $18$ distinct fixed pentominoes:
:600px | {{ProofWanted|Work to be done yet to establish method of creation}} | There exist $18$ [[Definition:Distinct|distinct]] [[Definition:Fixed Polyomino|fixed]] [[Definition:Pentomino|pentominoes]]:
:[[File:18Pentominoes.png|600px]] | {{ProofWanted|Work to be done yet to establish method of creation}} | 18 Fixed Pentominoes | https://proofwiki.org/wiki/18_Fixed_Pentominoes | https://proofwiki.org/wiki/18_Fixed_Pentominoes | [
"Pentominoes",
"18"
] | [
"Definition:Distinct",
"Definition:Polyomino/Fixed",
"Definition:Pentomino",
"File:18Pentominoes.png"
] | [] |
proofwiki-12503 | Differential of Differentiable Functional is Unique | The differential of a differentiable functional is unique. | === Lemma ===
{{:Differential of Differentiable Functional is Unique/Lemma}}{{qed|lemma}}
Let $J \sqbrk y$ be a differentiable functional.
Suppose the differential of $J \sqbrk y$ is not uniquely defined.
Then at least $2$ different forms of this exist:
:$\Delta J \sqbrk {y; h} = \phi_1 \sqbrk {y; h} + \epsilon_1 \siz... | The [[Definition:Differential of Functional|differential]] of a [[Definition:Differentiable Functional|differentiable functional]] is [[Definition:Unique|unique]]. | === [[Differential of Differentiable Functional is Unique/Lemma|Lemma]] ===
{{:Differential of Differentiable Functional is Unique/Lemma}}{{qed|lemma}}
Let $J \sqbrk y$ be a [[Definition:Differentiable Functional|differentiable functional]].
Suppose the [[Definition:Differential of Functional|differential]] of $J \s... | Differential of Differentiable Functional is Unique | https://proofwiki.org/wiki/Differential_of_Differentiable_Functional_is_Unique | https://proofwiki.org/wiki/Differential_of_Differentiable_Functional_is_Unique | [
"Calculus of Variations"
] | [
"Definition:Differential of Mapping/Functional",
"Definition:Differentiable Functional",
"Definition:Unique"
] | [
"Differential of Differentiable Functional is Unique/Lemma",
"Definition:Differentiable Functional",
"Definition:Differential of Mapping/Functional",
"Definition:Unique",
"Definition:Linear Functional",
"Definition:Infinitesimal",
"Definition:Infinitesimal/Order",
"Definition:Infinitesimal",
"Defini... |
proofwiki-12504 | Ordered Subset of Ordered Set is Ordered Set | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $\struct {S', \preceq'}$ be an ordered subset of $L$.
Then $\struct {S', \preceq'}$ is an ordered set. | By definition of ordered subset:
:$S' \subseteq S$ | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {S', \preceq'}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$.
Then $\struct {S', \preceq'}$ is an [[Definition:Ordered Set|ordered set]]. | By definition of [[Definition:Ordered Subset|ordered subset]]:
:$S' \subseteq S$ | Ordered Subset of Ordered Set is Ordered Set | https://proofwiki.org/wiki/Ordered_Subset_of_Ordered_Set_is_Ordered_Set | https://proofwiki.org/wiki/Ordered_Subset_of_Ordered_Set_is_Ordered_Set | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Ordered Subset",
"Definition:Ordered Set"
] | [
"Definition:Ordered Subset",
"Definition:Ordered Subset",
"Definition:Ordered Subset",
"Definition:Ordered Subset",
"Definition:Ordered Subset"
] |
proofwiki-12505 | Finite Subsets form Ideal | Let $X$ be a set.
Let $\map {\operatorname {Fin} } X$ be the set of all finite subsets of $X$.
Then $\map {\operatorname {Fin} } X$ is ideal in $\struct {\powerset X, \subseteq}$
where $\powerset X$ denotes the power set of $X$. | === Non-Empty ===
By Empty Set is Subset of All Sets:
:$\O \subseteq X$ and $\O$ is finite.
By definition of $\operatorname {Fin}$:
:$\O \in \map {\operatorname {Fin} } X$
Thus by definition:
:$\map {\operatorname {Fin} } X$ is non-empty.
{{qed|lemma}} | Let $X$ be a [[Definition:Set|set]].
Let $\map {\operatorname {Fin} } X$ be the [[Definition:Set of Sets|set]] of all [[Definition:Finite Subset|finite subsets]] of $X$.
Then $\map {\operatorname {Fin} } X$ is [[Definition:Ideal (Order Theory)|ideal]] in $\struct {\powerset X, \subseteq}$
where $\powerset X$ denote... | === Non-Empty ===
By [[Empty Set is Subset of All Sets]]:
:$\O \subseteq X$ and $\O$ is [[Definition:Finite Set|finite]].
By definition of $\operatorname {Fin}$:
:$\O \in \map {\operatorname {Fin} } X$
Thus by definition:
:$\map {\operatorname {Fin} } X$ is [[Definition:Non-Empty Set|non-empty]].
{{qed|lemma}} | Finite Subsets form Ideal | https://proofwiki.org/wiki/Finite_Subsets_form_Ideal | https://proofwiki.org/wiki/Finite_Subsets_form_Ideal | [
"Preorder Theory",
"Power Set"
] | [
"Definition:Set",
"Definition:Set of Sets",
"Definition:Finite Subset",
"Definition:Ideal (Order Theory)",
"Definition:Power Set"
] | [
"Empty Set is Subset of All Sets",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Finite Set"
] |
proofwiki-12506 | Product of Proper Divisors | Let $n$ be an integer such that $n \ge 1$.
Let $\map P n$ denote the product of the proper divisors of $n$.
Then:
:$\map P n = n^{\map {\sigma_0} n / 2 - 1}$
where $\map {\sigma_0} n$ denotes the divisor count Function of $n$. | Let $\map D n$ denote the product of '''all''' the divisors of $n$.
From Product of Divisors:
:$\map D n = n^{\map {\sigma_0} n / 2}$
The proper divisors of $n$ are defined as being the divisors of $n$ excluding $n$ itself.
Thus:
:$\map P n = \dfrac {\map D n} n = \dfrac {n^{\map {\sigma_0} n / 2} } n = n^{\map {\sigma... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Let $\map P n$ denote the [[Definition:Integer Multiplication|product]] of the [[Definition:Proper Divisor of Integer|proper divisors]] of $n$.
Then:
:$\map P n = n^{\map {\sigma_0} n / 2 - 1}$
where $\map {\sigma_0} n$ denotes the [[Definition:Divis... | Let $\map D n$ denote the [[Definition:Integer Multiplication|product]] of '''all''' the [[Definition:Divisor of Integer|divisors]] of $n$.
From [[Product of Divisors]]:
:$\map D n = n^{\map {\sigma_0} n / 2}$
The [[Definition:Proper Divisor of Integer|proper divisors]] of $n$ are defined as being the [[Definition:Di... | Product of Proper Divisors | https://proofwiki.org/wiki/Product_of_Proper_Divisors | https://proofwiki.org/wiki/Product_of_Proper_Divisors | [
"Product of Divisors",
"Divisor Count Function",
"Proper Divisors"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Proper Divisor/Integer",
"Definition:Divisor Count Function"
] | [
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer",
"Product of Divisors",
"Definition:Proper Divisor/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12507 | Product of Divisors | Let $n$ be an integer such that $n \ge 1$.
Let $\map D n$ denote the product of the divisors of $n$.
Then:
:$\map D n = n^{\map {\sigma_0} n / 2}$
where $\map {\sigma_0} n$ denotes the divisor count function of $n$. | We have by definition that:
:$\map D n = \ds \prod_{d \mathop \divides n} d$
Also by definition, $\map {\sigma_0} n$ is the number of divisors of $n$.
Suppose $n$ is not a square number.
Let $p \divides n$, where $\divides$ denotes divisibility.
Then:
:$\exists q \divides n : p q = n$
Thus the divisors of $n$ come in p... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Let $\map D n$ denote the [[Definition:Integer Multiplication|product]] of the [[Definition:Divisor of Integer|divisors]] of $n$.
Then:
:$\map D n = n^{\map {\sigma_0} n / 2}$
where $\map {\sigma_0} n$ denotes the [[Definition:Divisor Count Function|d... | We have by definition that:
:$\map D n = \ds \prod_{d \mathop \divides n} d$
Also by definition, $\map {\sigma_0} n$ is the number of [[Definition:Divisor of Integer|divisors]] of $n$.
Suppose $n$ is not a [[Definition:Square Number|square number]].
Let $p \divides n$, where $\divides$ denotes [[Definition:Divisor ... | Product of Divisors/Proof 1 | https://proofwiki.org/wiki/Product_of_Divisors | https://proofwiki.org/wiki/Product_of_Divisors/Proof_1 | [
"Product of Divisors",
"Divisor Count Function",
"Divisors"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Count Function"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Square Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Divisor Count Function is Odd Iff Argument is Square",
"Definition:Even Integer",
"Definition:Integer",
"Definiti... |
proofwiki-12508 | Product of Divisors | Let $n$ be an integer such that $n \ge 1$.
Let $\map D n$ denote the product of the divisors of $n$.
Then:
:$\map D n = n^{\map {\sigma_0} n / 2}$
where $\map {\sigma_0} n$ denotes the divisor count function of $n$. | {{begin-eqn}}
{{eqn | l = \map D n^2
| r = \paren {\prod_{d \mathop \divides n} d}^2
| c = by definition
}}
{{eqn | r = \paren {\prod_{d \mathop \divides n} d} \paren {\prod_{d \mathop \divides n} \dfrac n d}
| c = Sum Over Divisors Equals Sum Over Quotients
}}
{{eqn | r = \prod_{d \mathop \divides n}... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 1$.
Let $\map D n$ denote the [[Definition:Integer Multiplication|product]] of the [[Definition:Divisor of Integer|divisors]] of $n$.
Then:
:$\map D n = n^{\map {\sigma_0} n / 2}$
where $\map {\sigma_0} n$ denotes the [[Definition:Divisor Count Function|d... | {{begin-eqn}}
{{eqn | l = \map D n^2
| r = \paren {\prod_{d \mathop \divides n} d}^2
| c = by definition
}}
{{eqn | r = \paren {\prod_{d \mathop \divides n} d} \paren {\prod_{d \mathop \divides n} \dfrac n d}
| c = [[Sum Over Divisors Equals Sum Over Quotients]]
}}
{{eqn | r = \prod_{d \mathop \divide... | Product of Divisors/Proof 2 | https://proofwiki.org/wiki/Product_of_Divisors | https://proofwiki.org/wiki/Product_of_Divisors/Proof_2 | [
"Product of Divisors",
"Divisor Count Function",
"Divisors"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Count Function"
] | [
"Sum Over Divisors Equals Sum Over Quotients",
"Definition:Square Root/Positive"
] |
proofwiki-12509 | Square of Reversal of Small-Digit Number | Let $n$ be an integer whose decimal representation consists of sufficiently small digits.
Then the reversal of the square of $n$ is the square of the reversal of $n$. | {{ProofWanted|Needs to be formulated properly.}} | Let $n$ be an [[Definition:Integer|integer]] whose [[Definition:Decimal Notation|decimal representation]] consists of sufficiently small [[Definition:Digit|digits]].
Then the [[Definition:Reversal|reversal]] of the [[Definition:Square (Algebra)|square]] of $n$ is the [[Definition:Square (Algebra)|square]] of the [[Def... | {{ProofWanted|Needs to be formulated properly.}} | Square of Reversal of Small-Digit Number | https://proofwiki.org/wiki/Square_of_Reversal_of_Small-Digit_Number | https://proofwiki.org/wiki/Square_of_Reversal_of_Small-Digit_Number | [
"Recreational Mathematics",
"Square of Reversal of Small-Digit Number",
"Square Numbers",
"Reversals"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Digit",
"Definition:Reversal",
"Definition:Square/Function",
"Definition:Square/Function",
"Definition:Reversal"
] | [] |
proofwiki-12510 | 12 times Divisor Sum of 12 equals 14 times Divisor Sum of 14 | $x = 12$ and $y = 14$ are solutions to the indeterminate equation:
:$x \, \map {\sigma_1} x = y \, \map {\sigma_1} y$
where $\sigma_1$ denotes the divisor sum function. | {{begin-eqn}}
{{eqn | l = 12 \, \map {\sigma_1} {12}
| r = 12 \times 28
| c = {{DSFLink|12}}
}}
{{eqn | r = 12 \times 2 \times 14
| c =
}}
{{eqn | r = 14 \times 24
| c =
}}
{{eqn | r = 14 \, \map {\sigma_1} {14}
| c = {{DSFLink|14}}
}}
{{end-eqn}}
{{qed}} | $x = 12$ and $y = 14$ are solutions to the [[Definition:Indeterminate Equation|indeterminate equation]]:
:$x \, \map {\sigma_1} x = y \, \map {\sigma_1} y$
where $\sigma_1$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | {{begin-eqn}}
{{eqn | l = 12 \, \map {\sigma_1} {12}
| r = 12 \times 28
| c = {{DSFLink|12}}
}}
{{eqn | r = 12 \times 2 \times 14
| c =
}}
{{eqn | r = 14 \times 24
| c =
}}
{{eqn | r = 14 \, \map {\sigma_1} {14}
| c = {{DSFLink|14}}
}}
{{end-eqn}}
{{qed}} | 12 times Divisor Sum of 12 equals 14 times Divisor Sum of 14 | https://proofwiki.org/wiki/12_times_Divisor_Sum_of_12_equals_14_times_Divisor_Sum_of_14 | https://proofwiki.org/wiki/12_times_Divisor_Sum_of_12_equals_14_times_Divisor_Sum_of_14 | [
"12",
"14",
"Divisor Sum Function"
] | [
"Definition:Indeterminate Equation",
"Definition:Divisor Sum Function"
] | [] |
proofwiki-12511 | 12 Knights to Attack or Occupy All Squares on Chessboard | On a standard chessboard, a minimum of $12$ knights are needed to ensure all squares are either occupied or under attack. | First, we show that fewer than $12$ knights are not enough to occupy or attack each square.
Consider the $12$ squares $\text a 1$, $\text a 2$, $\text b 2$, $\text a 8$, $\text b 8$, $\text b 7$, $\text h 8$, $\text h 7$, $\text g 7$, $\text h 1$, $\text g 1$ and $\text g 2$.
{{ChessDiagram|
|
|xx|xx| | | | | |xx ... | On a standard [[Definition:Chessboard|chessboard]], a minimum of $12$ [[Definition:Chess Knight|knights]] are needed to ensure all [[Definition:Square of Chessboard|squares]] are either occupied or [[Definition:Under Attack|under attack]]. | First, we show that fewer than $12$ [[Definition:Chess Knight|knights]] are not enough to occupy or [[Definition:Under Attack|attack]] each [[Definition:Square of Chessboard|square]].
Consider the $12$ [[Definition:Square of Chessboard|squares]] $\text a 1$, $\text a 2$, $\text b 2$, $\text a 8$, $\text b 8$, $\text b... | 12 Knights to Attack or Occupy All Squares on Chessboard | https://proofwiki.org/wiki/12_Knights_to_Attack_or_Occupy_All_Squares_on_Chessboard | https://proofwiki.org/wiki/12_Knights_to_Attack_or_Occupy_All_Squares_on_Chessboard | [
"12",
"Recreational Chess"
] | [
"Definition:Chess/Chessboard",
"Definition:Chess/Piece/Knight",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Rules/Under Attack"
] | [
"Definition:Chess/Piece/Knight",
"Definition:Chess/Rules/Under Attack",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Piece/Knight",
"Definition:Chess/Rules/Under Attack",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Piece/Knight",
"Definiti... |
proofwiki-12512 | Prime Number is Deficient | Let $p$ be a prime number.
Then $p$ is deficient. | A specific instance of Power of Prime is Deficient.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Then $p$ is [[Definition:Deficient Number|deficient]]. | A specific instance of [[Power of Prime is Deficient]].
{{qed}} | Prime Number is Deficient/Proof 1 | https://proofwiki.org/wiki/Prime_Number_is_Deficient | https://proofwiki.org/wiki/Prime_Number_is_Deficient/Proof_1 | [
"Prime Numbers",
"Deficient Numbers",
"Prime Number is Deficient"
] | [
"Definition:Prime Number",
"Definition:Deficient Number"
] | [
"Power of Prime is Deficient"
] |
proofwiki-12513 | Prime Number is Deficient | Let $p$ be a prime number.
Then $p$ is deficient. | Let $p$ be a prime number.
From Divisor Sum of Prime Number:
:$\map {\sigma_1} p = p + 1$
and so:
:$\dfrac {\map {\sigma_1} p} p = \dfrac {p + 1} p = 1 + \dfrac 1 p$
As $p > 1$ it follows that $\dfrac 1 p < 1$.
Hence:
:$\dfrac {\map {\sigma_1} p} p < 2$
The result follows by definition of deficient.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Then $p$ is [[Definition:Deficient Number|deficient]]. | Let $p$ be a [[Definition:Prime Number|prime number]].
From [[Divisor Sum of Prime Number]]:
:$\map {\sigma_1} p = p + 1$
and so:
:$\dfrac {\map {\sigma_1} p} p = \dfrac {p + 1} p = 1 + \dfrac 1 p$
As $p > 1$ it follows that $\dfrac 1 p < 1$.
Hence:
:$\dfrac {\map {\sigma_1} p} p < 2$
The result follows by definit... | Prime Number is Deficient/Proof 2 | https://proofwiki.org/wiki/Prime_Number_is_Deficient | https://proofwiki.org/wiki/Prime_Number_is_Deficient/Proof_2 | [
"Prime Numbers",
"Deficient Numbers",
"Prime Number is Deficient"
] | [
"Definition:Prime Number",
"Definition:Deficient Number"
] | [
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Definition:Deficient Number"
] |
proofwiki-12514 | Ordering on Closure Operators iff Images are Including | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $f, g: S \to S$ be closure operators on $L$.
Then:
:$f \preceq g$ {{iff}} $g \sqbrk S \subseteq f \sqbrk S$
where
:$\preceq$ denotes the ordering on mappings
:$f \sqbrk S$ denotes the image of $f$. | === Sufficient Condition ===
Let $f \preceq g$.
Let $x \in g \sqbrk S$
By definition of image of mapping:
:$\exists y \in S: \map g y = x$
By definition of closure operator/idempotent:
:$\map g {\map g y} = \map g y$
By definition of ordering on mappings:
:$\map f {\map g y} \preceq \map g {\map g y}$
By definition of ... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $f, g: S \to S$ be [[Definition:Closure Operator|closure operators]] on $L$.
Then:
:$f \preceq g$ {{iff}} $g \sqbrk S \subseteq f \sqbrk S$
where
:$\preceq$ denotes the [[Definition:Ordering on Mappings|ordering ... | === Sufficient Condition ===
Let $f \preceq g$.
Let $x \in g \sqbrk S$
By definition of [[Definition:Image of Mapping|image of mapping]]:
:$\exists y \in S: \map g y = x$
By definition of [[Definition:Closure Operator|closure operator/idempotent]]:
:$\map g {\map g y} = \map g y$
By definition of [[Definition:Orde... | Ordering on Closure Operators iff Images are Including | https://proofwiki.org/wiki/Ordering_on_Closure_Operators_iff_Images_are_Including | https://proofwiki.org/wiki/Ordering_on_Closure_Operators_iff_Images_are_Including | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Closure Operator",
"Definition:Ordering on Mappings",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Closure Operator",
"Definition:Ordering on Mappings",
"Definition:Closure Operator",
"Definition:Antisymmetric Relation",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Image (Se... |
proofwiki-12515 | Multiple of Perfect Number is Abundant | Let $n$ be a perfect number.
Let $m$ be a positive integer such that $m > 1$.
Then $m n$ is abundant. | We have by definition of divisor sum function and perfect number that:
:$\dfrac {\map {\sigma_1} n} n = 2$
But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
:$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$
Hence the result by definition of abundant.
{{qed}} | Let $n$ be a [[Definition:Perfect Number|perfect number]].
Let $m$ be a [[Definition:Positive Integer|positive integer]] such that $m > 1$.
Then $m n$ is [[Definition:Abundant Number|abundant]]. | We have by definition of [[Definition:Divisor Sum Function|divisor sum function]] and [[Definition:Perfect Number|perfect number]] that:
:$\dfrac {\map {\sigma_1} n} n = 2$
But from [[Abundancy Index of Product is greater than Abundancy Index of Proper Factors]]:
:$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$
Hence the... | Multiple of Perfect Number is Abundant | https://proofwiki.org/wiki/Multiple_of_Perfect_Number_is_Abundant | https://proofwiki.org/wiki/Multiple_of_Perfect_Number_is_Abundant | [
"Abundant Numbers",
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Positive/Integer",
"Definition:Abundant Number"
] | [
"Definition:Divisor Sum Function",
"Definition:Perfect Number",
"Abundancy Index of Product is greater than Abundancy Index of Proper Factors",
"Definition:Abundant Number"
] |
proofwiki-12516 | Multiple of Abundant Number is Abundant | Let $n$ be an abundant number.
Let $m$ be a positive integer such that $m > 1$.
Then $m n$ is abundant. | We have by definition of divisor sum function and abundant number that:
:$\dfrac {\map {\sigma_1} n} n > 2$
But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
:$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$
Hence the result by definition of abundant.
{{qed}} | Let $n$ be an [[Definition:Abundant Number|abundant number]].
Let $m$ be a [[Definition:Positive Integer|positive integer]] such that $m > 1$.
Then $m n$ is [[Definition:Abundant Number|abundant]]. | We have by definition of [[Definition:Divisor Sum Function|divisor sum function]] and [[Definition:Abundant Number|abundant number]] that:
:$\dfrac {\map {\sigma_1} n} n > 2$
But from [[Abundancy Index of Product is greater than Abundancy Index of Proper Factors]]:
:$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$
Hence t... | Multiple of Abundant Number is Abundant | https://proofwiki.org/wiki/Multiple_of_Abundant_Number_is_Abundant | https://proofwiki.org/wiki/Multiple_of_Abundant_Number_is_Abundant | [
"Abundant Numbers"
] | [
"Definition:Abundant Number",
"Definition:Positive/Integer",
"Definition:Abundant Number"
] | [
"Definition:Divisor Sum Function",
"Definition:Abundant Number",
"Abundancy Index of Product is greater than Abundancy Index of Proper Factors",
"Definition:Abundant Number"
] |
proofwiki-12517 | Divisor of Perfect Number is Deficient | Let $n$ be a perfect number.
Let $n = r s$ where $r$ and $s$ are positive integers such that $r > 1$ and $s > 1$.
Then $r$ and $s$ are both deficient. | {{WLOG}}, consider $r$.
We have by definition of divisor sum function and perfect number that:
:$\dfrac {\map {\sigma_1} {r s} } {r s} = 2$
But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
:$\dfrac {\map {\sigma_1} {r s} } {r s} > \dfrac {\map {\sigma_1} r} r$
That is:
:$\dfrac {\m... | Let $n$ be a [[Definition:Perfect Number|perfect number]].
Let $n = r s$ where $r$ and $s$ are [[Definition:Positive Integer|positive integers]] such that $r > 1$ and $s > 1$.
Then $r$ and $s$ are both [[Definition:Deficient Number|deficient]]. | {{WLOG}}, consider $r$.
We have by definition of [[Definition:Divisor Sum Function|divisor sum function]] and [[Definition:Perfect Number|perfect number]] that:
:$\dfrac {\map {\sigma_1} {r s} } {r s} = 2$
But from [[Abundancy Index of Product is greater than Abundancy Index of Proper Factors]]:
:$\dfrac {\map {\sigm... | Divisor of Perfect Number is Deficient | https://proofwiki.org/wiki/Divisor_of_Perfect_Number_is_Deficient | https://proofwiki.org/wiki/Divisor_of_Perfect_Number_is_Deficient | [
"Deficient Numbers",
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Positive/Integer",
"Definition:Deficient Number"
] | [
"Definition:Divisor Sum Function",
"Definition:Perfect Number",
"Abundancy Index of Product is greater than Abundancy Index of Proper Factors",
"Definition:Deficient Number"
] |
proofwiki-12518 | Divisor of Deficient Number is Deficient | Let $n$ be a perfect number.
Let $n = k d$ where $r$ is a positive integer.
Then $k$ is deficient. | We have by definition of divisor sum function and perfect number that:
:$\dfrac {\map {\sigma_1} {k d} } {k d} < 2$
But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
:$\dfrac {\map {\sigma_1} {k d} } {k d} > \dfrac {\map {\sigma_1} k} k$
That is:
:$\dfrac {\map {\sigma_1} k} k < 2$
... | Let $n$ be a [[Definition:Perfect Number|perfect number]].
Let $n = k d$ where $r$ is a [[Definition:Positive Integer|positive integer]].
Then $k$ is [[Definition:Deficient Number|deficient]]. | We have by definition of [[Definition:Divisor Sum Function|divisor sum function]] and [[Definition:Perfect Number|perfect number]] that:
:$\dfrac {\map {\sigma_1} {k d} } {k d} < 2$
But from [[Abundancy Index of Product is greater than Abundancy Index of Proper Factors]]:
:$\dfrac {\map {\sigma_1} {k d} } {k d} > \dfr... | Divisor of Deficient Number is Deficient | https://proofwiki.org/wiki/Divisor_of_Deficient_Number_is_Deficient | https://proofwiki.org/wiki/Divisor_of_Deficient_Number_is_Deficient | [
"Deficient Numbers"
] | [
"Definition:Perfect Number",
"Definition:Positive/Integer",
"Definition:Deficient Number"
] | [
"Definition:Divisor Sum Function",
"Definition:Perfect Number",
"Abundancy Index of Product is greater than Abundancy Index of Proper Factors",
"Definition:Deficient Number"
] |
proofwiki-12519 | Abundancy Index of Product is greater than Abundancy Index of Proper Factors | Let $n \in \Z_{>0}$ be a composite number such that $n = r s$, where $r, s \in \Z_{>1}$.
Then:
:$\dfrac {\map {\sigma_1} n} n > \dfrac {\map {\sigma_1} r} r$
and consequently also:
:$\dfrac {\map {\sigma_1} n} n > \dfrac {\map {\sigma_1} s} s$
where $\sigma_1$ denotes the divisor sum function.
That is, the abundancy in... | Consider the divisors of $r$.
Let $d \divides r$, where $\divides$ indicates divisibility.
We have that:
:$d \divides n$
and also that:
:$d s \divides n$
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} r
| r = \sum_{d \mathop \divides r} d
| c =
}}
{{eqn | ll= \leadsto
| l = s \map {\sigma_1} r
... | Let $n \in \Z_{>0}$ be a [[Definition:Composite Number|composite number]] such that $n = r s$, where $r, s \in \Z_{>1}$.
Then:
:$\dfrac {\map {\sigma_1} n} n > \dfrac {\map {\sigma_1} r} r$
and consequently also:
:$\dfrac {\map {\sigma_1} n} n > \dfrac {\map {\sigma_1} s} s$
where $\sigma_1$ denotes the [[Definition:D... | Consider the [[Definition:Divisor of Integer|divisors]] of $r$.
Let $d \divides r$, where $\divides$ indicates [[Definition:Divisor of Integer|divisibility]].
We have that:
:$d \divides n$
and also that:
:$d s \divides n$
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} r
| r = \sum_{d \mathop \divides r} d
... | Abundancy Index of Product is greater than Abundancy Index of Proper Factors | https://proofwiki.org/wiki/Abundancy_Index_of_Product_is_greater_than_Abundancy_Index_of_Proper_Factors | https://proofwiki.org/wiki/Abundancy_Index_of_Product_is_greater_than_Abundancy_Index_of_Proper_Factors | [
"Abundancy"
] | [
"Definition:Composite Number",
"Definition:Divisor Sum Function",
"Definition:Abundancy Index",
"Definition:Composite Number",
"Definition:Abundancy Index",
"Definition:Proper Divisor/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Category:Abundancy"
] |
proofwiki-12520 | Condition for Differentiable Functional to have Extremum | Let $S$ be a set of mappings.
Let $y, h \in S: \R \to \R$ be real functions.
Let $J \sqbrk y: S \to \R$ be a differentiable functional.
Then a necessary condition for the differentiable functional $J \sqbrk {y; h}$ to have an extremum for $y = \hat y$ is:
:$\bigvalueat {\delta J \sqbrk {y; h} } {y \mathop = \hat y} = ... | Suppose $J \sqbrk {y; h}$ attains a minimum for $y = \hat y$.
Then:
:$\Delta J \sqbrk {\hat y; h} \ge 0$
By definition of the differentiable functional:
:$\Delta J \sqbrk {y; h} = \delta J \sqbrk {y; h} + \epsilon \size h$
where:
:$\ds \lim_{\size h \mathop \to 0} \epsilon = 0$
Hence, there exists $\size h$ small enoug... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Mapping|mappings]].
Let $y, h \in S: \R \to \R$ be [[Definition:Real Function|real functions]].
Let $J \sqbrk y: S \to \R$ be a [[Definition:Differentiable Functional|differentiable functional]].
Then a [[Definition:Necessary Condition|necessary condition]] for t... | Suppose $J \sqbrk {y; h}$ attains a [[Definition:Minimum Value of Functional|minimum]] for $y = \hat y$.
Then:
:$\Delta J \sqbrk {\hat y; h} \ge 0$
By definition of the [[Definition:Differentiable Functional|differentiable functional]]:
:$\Delta J \sqbrk {y; h} = \delta J \sqbrk {y; h} + \epsilon \size h$
where:
... | Condition for Differentiable Functional to have Extremum | https://proofwiki.org/wiki/Condition_for_Differentiable_Functional_to_have_Extremum | https://proofwiki.org/wiki/Condition_for_Differentiable_Functional_to_have_Extremum | [
"Calculus of Variations"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Real Function",
"Definition:Differentiable Functional",
"Definition:Conditional/Necessary Condition",
"Definition:Differentiable Functional"
] | [
"Definition:Minimum Value of Functional",
"Definition:Differentiable Functional",
"Definition:Contradiction"
] |
proofwiki-12521 | Superabundant Numbers are Infinite in Number | There are infinitely many superabundant numbers. | {{AimForCont}} the set $S$ of superabundant numbers is finite.
Let $m$ be the greatest element of $S$.
By definition of superabundant, $m$ has the largest abundancy index of all the elements of $S$.
Consider the integer $2 m$.
From Abundancy Index of Product is greater than Abundancy Index of Proper Factors, $2 m$ has ... | There are [[Definition:Infinite Set|infinitely many]] [[Definition:Superabundant Number|superabundant numbers]]. | {{AimForCont}} the [[Definition:Set|set]] $S$ of [[Definition:Superabundant Number|superabundant numbers]] is [[Definition:Finite Set|finite]].
Let $m$ be the [[Definition:Greatest Element|greatest element]] of $S$.
By definition of [[Definition:Superabundant Number|superabundant]], $m$ has the largest [[Definition:A... | Superabundant Numbers are Infinite in Number | https://proofwiki.org/wiki/Superabundant_Numbers_are_Infinite_in_Number | https://proofwiki.org/wiki/Superabundant_Numbers_are_Infinite_in_Number | [
"Superabundant Numbers"
] | [
"Definition:Infinite Set",
"Definition:Superabundant Number"
] | [
"Definition:Set",
"Definition:Superabundant Number",
"Definition:Finite Set",
"Definition:Greatest Element",
"Definition:Superabundant Number",
"Definition:Abundancy Index",
"Definition:Element",
"Definition:Integer",
"Abundancy Index of Product is greater than Abundancy Index of Proper Factors",
... |
proofwiki-12522 | Ordered Set of Closure Systems is Ordered Set | Let $L = \left({S, \preceq}\right)$ be an ordered set.
Then $\operatorname{ClSystems}\left({L}\right)$ is an ordered set,
where $\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems. | By definition of ordered set of closure systems:
:$\operatorname{ClSystems}\left({L}\right) = \left({X, \precsim}\right)$
where
:$X$ is the set of all closure systems of $L$,
:dor all closure systems $s_1 = \left({T_1, \preceq_1}\right), s_2 = \left({T_2, \preceq_2}\right)$ of $L$: $s_1 \precsim s_2 \iff T_1 \subseteq ... | Let $L = \left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Then $\operatorname{ClSystems}\left({L}\right)$ is an [[Definition:Ordered Set|ordered set]],
where $\operatorname{ClSystems}\left({L}\right)$ denotes the [[Definition:Ordered Set of Closure Systems|ordered set of closure systems]]. | By definition of [[Definition:Ordered Set of Closure Systems|ordered set of closure systems]]:
:$\operatorname{ClSystems}\left({L}\right) = \left({X, \precsim}\right)$
where
:$X$ is the [[Definition:Set|set]] of all [[Definition:Closure System|closure systems]] of $L$,
:dor all [[Definition:Closure System|closure syste... | Ordered Set of Closure Systems is Ordered Set | https://proofwiki.org/wiki/Ordered_Set_of_Closure_Systems_is_Ordered_Set | https://proofwiki.org/wiki/Ordered_Set_of_Closure_Systems_is_Ordered_Set | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Ordered Set",
"Definition:Ordered Set of Closure Systems"
] | [
"Definition:Ordered Set of Closure Systems",
"Definition:Set",
"Definition:Closure System",
"Definition:Closure System",
"Definition:Closure System",
"Definition:Closure System",
"Definition:Closure System",
"Definition:Ordered Set"
] |
proofwiki-12523 | Necessary Condition for Integral Functional to have Extremum for given function | Let $S$ be a set of real mappings such that:
:$S = \set {\map y x: \paren {y: S_1 \subseteq \R \to S_2 \subseteq \R}, \paren {\map y x \in C^1 \closedint a b}, \paren {\map y a = A, \map y b = B} }$
Let $J \sqbrk y: S \to S_3 \subseteq \R$ be a functional of the form:
:$\ds \int_a^b \map F {x, y, y'} \rd x$
Then a nec... | From Condition for Differentiable Functional to have Extremum we have
:$\delta J \sqbrk {y; h} \bigg \rvert_{y = \hat y} = 0$
The variation exists if $J$ is a differentiable functional.
The endpoints of $\map y x$ are fixed.
Hence:
:$\map h a = 0$
:$\map h b = 0$.
From the definition of increment of a functional:
{{beg... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Real Function|real mappings]] such that:
:$S = \set {\map y x: \paren {y: S_1 \subseteq \R \to S_2 \subseteq \R}, \paren {\map y x \in C^1 \closedint a b}, \paren {\map y a = A, \map y b = B} }$
Let $J \sqbrk y: S \to S_3 \subseteq \R$ be a [[Definition:Real Functio... | From [[Condition for Differentiable Functional to have Extremum]] we have
:$\delta J \sqbrk {y; h} \bigg \rvert_{y = \hat y} = 0$
The variation exists if $J$ is a [[Definition:Differentiable Functional|differentiable functional]].
The endpoints of $\map y x$ are fixed.
Hence:
:$\map h a = 0$
:$\map h b = 0$.
Fr... | Necessary Condition for Integral Functional to have Extremum for given function | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function | [
"Calculus of Variations"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Functional/Real",
"Definition:Conditional/Necessary Condition",
"Definition:Extremum/Functional",
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Condition for Differentiable Functional to have Extremum",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Definition:Functional/Real",
"Definition:Definite Integral",
"Taylor's Theorem",
"Definition:Definite Integral",
"Definition:Definite Integral",
"Definition:Di... |
proofwiki-12524 | Image of Closure Operator Inherits Infima | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $f$ be a closure operator on $L$.
Then $R = \struct {f \sqbrk S, \precsim}$ inherits infima,
where
:$\mathord \precsim = \mathord \preceq \cap \paren {f \sqbrk S \times f \sqbrk S}$
:$f \sqbrk S$ denotes the image of $f$. | Let $X$ be subset of $f \sqbrk S$ such that
:$X$ admits an infimum in $L$.
By Closure Operator does not Change Infimum of Subset of Image:
:$\map f {\inf_L X} = \inf_L X$
By definition of image of mapping:
:$\inf_L X \in f \sqbrk S$
Thus by Infimum in Ordered Subset:
:$X$ admits an infimum in $R$ and $\inf_R X = \inf_L... | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $f$ be a [[Definition:Closure Operator|closure operator]] on $L$.
Then $R = \struct {f \sqbrk S, \precsim}$ [[Definition:Infima Inheriting|inherits infima]],
where
:$\mathord \precsim = \mathord \preceq \cap \paren {f \sqbrk S \times f... | Let $X$ be [[Definition:Subset|subset]] of $f \sqbrk S$ such that
:$X$ admits an [[Definition:Infimum of Set|infimum]] in $L$.
By [[Closure Operator does not Change Infimum of Subset of Image]]:
:$\map f {\inf_L X} = \inf_L X$
By definition of [[Definition:Image of Mapping|image of mapping]]:
:$\inf_L X \in f \sqbrk ... | Image of Closure Operator Inherits Infima | https://proofwiki.org/wiki/Image_of_Closure_Operator_Inherits_Infima | https://proofwiki.org/wiki/Image_of_Closure_Operator_Inherits_Infima | [
"Closure Operators"
] | [
"Definition:Ordered Set",
"Definition:Closure Operator",
"Definition:Infima Inheriting",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Subset",
"Definition:Infimum of Set",
"Closure Operator does not Change Infimum of Subset of Image",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Infimum in Ordered Subset",
"Definition:Infimum of Set"
] |
proofwiki-12525 | Necessary Condition for Integral Functional to have Extremum for given function/Lemma | Let $\map \alpha x, \map \beta x$ be real functions.
Let $\map \alpha x, \map \beta x$ be continuous on $\closedint a b$.
Let:
:$\forall \map h x \in C^1: \ds \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \rd x = 0$
subject to the boundary conditions:
:$\map h a = \map h b = 0$
Then $\map \beta x$... | Using Integration by Parts allows us to factor out $\map h x$:
{{begin-eqn}}
{{eqn | l = \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \rd x
| r = \int_a^b \map \alpha x \map h x \rd x + \int_a^b \map \beta x \rd \map h x
| c = where $\d \map h x = \map {h'} x \rd x$
}}
{{eqn | r = \in... | Let $\map \alpha x, \map \beta x$ be [[Definition:Real Function|real functions]].
Let $\map \alpha x, \map \beta x$ be [[Definition:Continuous Real Function on Interval|continuous]] on $\closedint a b$.
Let:
:$\forall \map h x \in C^1: \ds \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \rd x = 0... | Using [[Integration by Parts]] allows us to factor out $\map h x$:
{{begin-eqn}}
{{eqn | l = \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \rd x
| r = \int_a^b \map \alpha x \map h x \rd x + \int_a^b \map \beta x \rd \map h x
| c = where $\d \map h x = \map {h'} x \rd x$
}}
{{eqn | r ... | Necessary Condition for Integral Functional to have Extremum for given function/Lemma | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function/Lemma | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function/Lemma | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Boundary Condition",
"Definition:Differentiable Mapping/Real-Valued Function"
] | [
"Integration by Parts",
"Integration by Parts",
"If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes",
"Definition:Interval/Ordered Set/Closed"
] |
proofwiki-12526 | If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes | Let $\map \alpha x$ be a continuous real function on the closed real interval $\closedint a b$.
Let $\ds \int_a^b \map \alpha x \map h x \rd x = 0$ for every real function $\map h x \in C^0 \closedint a b$ such that $\map h a = 0$ and $\map h b = 0$.
where $C^0 \closedint a b$ means continuous functions on $\closedint ... | {{AimForCont}} the real function $\map \alpha x$ is nonzero at some point in $\closedint a b$ for some arbitrary $\map h x$.
Due to belonging to $C^0$ it is also nonzero in some interval $\closedint {x_1} {x_2}$ contained in $\closedint a b$.
Let us choose $\map h x$ to be of a specific form, while still satisfying th... | Let $\map \alpha x$ be a [[Definition:Continuous Real Function on Interval|continuous real function]] on the [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Let $\ds \int_a^b \map \alpha x \map h x \rd x = 0$ for every [[Definition:Real Function|real function]] $\map h x \in C^0 \closedint a... | {{AimForCont}} the [[Definition:Real Function|real function]] $\map \alpha x$ is nonzero at some point in $\closedint a b$ for some arbitrary $\map h x$.
Due to belonging to $C^0$ it is also nonzero in some interval $\closedint {x_1} {x_2}$ contained in $\closedint a b$.
Let us choose $\map h x$ to be of a specific... | If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes | https://proofwiki.org/wiki/If_Definite_Integral_of_a(x)h(x)_vanishes_for_any_C^0_h(x)_then_C^0_a(x)_vanishes | https://proofwiki.org/wiki/If_Definite_Integral_of_a(x)h(x)_vanishes_for_any_C^0_h(x)_then_C^0_a(x)_vanishes | [
"Calculus of Variations"
] | [
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Real Function"
] | [
"Definition:Real Function",
"Proof by Contradiction"
] |
proofwiki-12527 | Closure Operator does not Change Infimum of Subset of Image | Let $L = \left({S, \preceq}\right)$ be an ordered set.
Let $c: S \to S$ be a closure operator on $L$.
Let $X$ be a subset of $c\left[{S}\right]$ such that
:$X$ admits an infimum,
where $c\left[{S}\right]$ denotes the image of $c$.
Then $\inf X = c\left({\inf X}\right)$ | We will prove that
:$c\left({\inf X}\right)$ is lower bound for $X$.
Let $x \in X$.
By definition of subset:
:$x \in c\left[{S}\right]$
By definition of image of mapping:
:$\exists y \in S: x = c\left({y}\right)$
By definition of closure operator/idempotent:
:$x = c\left({x}\right)$
By definition of infimum:
:$\inf X$ ... | Let $L = \left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $c: S \to S$ be a [[Definition:Closure Operator|closure operator]] on $L$.
Let $X$ be a [[Definition:Subset|subset]] of $c\left[{S}\right]$ such that
:$X$ admits an [[Definition:Infimum of Set|infimum]],
where $c\left[{S}\right]$ de... | We will prove that
:$c\left({\inf X}\right)$ is [[Definition:Lower Bound of Set|lower bound]] for $X$.
Let $x \in X$.
By definition of [[Definition:Subset|subset]]:
:$x \in c\left[{S}\right]$
By definition of [[Definition:Image of Mapping|image of mapping]]:
:$\exists y \in S: x = c\left({y}\right)$
By definition o... | Closure Operator does not Change Infimum of Subset of Image | https://proofwiki.org/wiki/Closure_Operator_does_not_Change_Infimum_of_Subset_of_Image | https://proofwiki.org/wiki/Closure_Operator_does_not_Change_Infimum_of_Subset_of_Image | [
"Closure Operators"
] | [
"Definition:Ordered Set",
"Definition:Closure Operator",
"Definition:Subset",
"Definition:Infimum of Set",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Lower Bound of Set",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Closure Operator",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Closure Operator",
"Definition:Infimum of Set",
"Definiti... |
proofwiki-12528 | Infimum in Ordered Subset | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $R = \struct {T, \preceq'}$ be an ordered subset of $L$.
Let $X \subseteq T$ such that
:$X$ admits an infimum in $L$.
Then $\inf_L X \in T$ {{iff}}
:$X$ admits an infimum in $R$ and $\inf_R X = \inf_L X$ | By definition of ordered subset:
:$T \subseteq S$
and
:$\forall x, y \in T: x \preceq' y \iff x \preceq y$ | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $R = \struct {T, \preceq'}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$.
Let $X \subseteq T$ such that
:$X$ admits an [[Definition:Infimum of Set|infimum]] in $L$.
Then $\inf_L X \in T$ {{iff}}
:$X$ admits an [[Definition... | By definition of [[Definition:Ordered Subset|ordered subset]]:
:$T \subseteq S$
and
:$\forall x, y \in T: x \preceq' y \iff x \preceq y$ | Infimum in Ordered Subset | https://proofwiki.org/wiki/Infimum_in_Ordered_Subset | https://proofwiki.org/wiki/Infimum_in_Ordered_Subset | [
"Ordered Sets",
"Infima"
] | [
"Definition:Ordered Set",
"Definition:Ordered Subset",
"Definition:Infimum of Set",
"Definition:Infimum of Set"
] | [
"Definition:Ordered Subset"
] |
proofwiki-12529 | Operator Generated by Closure System is Closure Operator | Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice.
Let $S = \left({T, \precsim}\right)$ be a closure system of $L$.
Then $\operatorname{operator}\left({S}\right)$ is closure operator,
where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$. | Define $f = \operatorname{operator}\left({S}\right)$. | Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
Let $S = \left({T, \precsim}\right)$ be a [[Definition:Closure System|closure system]] of $L$.
Then $\operatorname{operator}\left({S}\right)$ is [[Definition:Closure Operator|closure operator]],
where $\operator... | Define $f = \operatorname{operator}\left({S}\right)$. | Operator Generated by Closure System is Closure Operator | https://proofwiki.org/wiki/Operator_Generated_by_Closure_System_is_Closure_Operator | https://proofwiki.org/wiki/Operator_Generated_by_Closure_System_is_Closure_Operator | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Closure System",
"Definition:Closure Operator",
"Definition:Operator Generated by Ordered Subset"
] | [] |
proofwiki-12530 | Image of Operator Generated by Closure System is Set of Closure System | Let $L = \struct {X, \vee, \wedge, \preceq}$ be a complete lattice.
Let $S = \struct {T, \precsim}$ be a closure system of $L$.
Then $\map {\operatorname {operator} } S \sqbrk X = T$
where $\map {\operatorname {operator} } S$ denotes the operator generated by $S$. | Define $f = \map {\operatorname {operator} } S$. | Let $L = \struct {X, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $S = \struct {T, \precsim}$ be a [[Definition:Closure System|closure system]] of $L$.
Then $\map {\operatorname {operator} } S \sqbrk X = T$
where $\map {\operatorname {operator} } S$ denotes the [[Definition:Ope... | Define $f = \map {\operatorname {operator} } S$. | Image of Operator Generated by Closure System is Set of Closure System | https://proofwiki.org/wiki/Image_of_Operator_Generated_by_Closure_System_is_Set_of_Closure_System | https://proofwiki.org/wiki/Image_of_Operator_Generated_by_Closure_System_is_Set_of_Closure_System | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Closure System",
"Definition:Operator Generated by Ordered Subset"
] | [] |
proofwiki-12531 | Regular Icosahedron as Pentagonal Antiprism with Pyramidal Endcaps | The regular icosahedron can be considered as a regular pentagonal antiprism with two regular pentagonal pyramid as end caps.
{{stub|Need to construct an icosahedron.}} | {{ProofWanted|Probably get away with a drawing here.}} | The [[Definition:Regular Icosahedron|regular icosahedron]] can be considered as a [[Definition:Regular Pentagon|regular pentagonal]] [[Definition:Antiprism|antiprism]] with two [[Definition:Regular Pentagon|regular pentagonal]] [[Definition:Pyramid|pyramid]] as end caps.
{{stub|Need to construct an icosahedron.}} | {{ProofWanted|Probably get away with a drawing here.}} | Regular Icosahedron as Pentagonal Antiprism with Pyramidal Endcaps | https://proofwiki.org/wiki/Regular_Icosahedron_as_Pentagonal_Antiprism_with_Pyramidal_Endcaps | https://proofwiki.org/wiki/Regular_Icosahedron_as_Pentagonal_Antiprism_with_Pyramidal_Endcaps | [
"Regular Icosahedra"
] | [
"Definition:Icosahedron/Regular",
"Definition:Pentagon/Regular",
"Definition:Antiprism",
"Definition:Pentagon/Regular",
"Definition:Pyramid"
] | [] |
proofwiki-12532 | Construction of Rhombic Dodecahedron | The rhombic dodecahedron can be constructed as follows:
Take a cube $K$ embedded in $3$-dimensional space.
Place $6$ more cubes, each congruent with $K$, so that one face of each coincides with a different face of $K$.
Join the vertices of $K$ to the centers of the adjacent cubes to describe square pyramids whose apice... | {{ProofWanted|Needs a diagram}} | The [[Definition:Rhombic Dodecahedron|rhombic dodecahedron]] can be constructed as follows:
Take a [[Definition:Cube (Geometry)|cube]] $K$ embedded in [[Definition:Ordinary Space|$3$-dimensional space]].
Place $6$ more [[Definition:Cube (Geometry)|cubes]], each [[Definition:Congruent Polyhedra|congruent]] with $K$, s... | {{ProofWanted|Needs a diagram}} | Construction of Rhombic Dodecahedron | https://proofwiki.org/wiki/Construction_of_Rhombic_Dodecahedron | https://proofwiki.org/wiki/Construction_of_Rhombic_Dodecahedron | [
"Rhombic Dodecahedra"
] | [
"Definition:Rhombic Dodecahedron",
"Definition:Cube/Geometry",
"Definition:Ordinary Space",
"Definition:Cube/Geometry",
"Definition:Congruent Polyhedra",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Vertex",
"Definition:Cube/Geometry",
"Definition:Square Pyram... | [] |
proofwiki-12533 | Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative | Let $\map y x:\R \to \R$ be a real function.
Let $\map F {x, y, y'}:\R^3 \to \R$ be a real function.
Suppose $\map F {x, y, y'}$ has continuous first and second derivatives {{WRT|Differentiation}} all its arguments.
Suppose $y$ has a continuous first derivative and satisfies Euler's equation:
:$F_y - \dfrac \d {\d x} ... | Consider the difference
{{begin-eqn}}
{{eqn | l = \Delta F_{y'}
| r = \map F {x + \Delta x, y + \Delta y, y' + \Delta y'} - \map F {x, y, y'}
| c =
}}
{{eqn | r = \Delta x \overline F_{y' x} + \Delta y \overline F_{y'y} + \Delta y' \overline F_{y'y'}
| c = Multivariate Mean Value Theorem
}}
{{end-eqn... | Let $\map y x:\R \to \R$ be a [[Definition:Real Function|real function]].
Let $\map F {x, y, y'}:\R^3 \to \R$ be a [[Definition:Real Function|real function]].
Suppose $\map F {x, y, y'}$ has [[Definition:Continuous Real Function|continuous]] [[Definition:First Derivative|first]] and [[Definition:Second Derivative|sec... | Consider the difference
{{begin-eqn}}
{{eqn | l = \Delta F_{y'}
| r = \map F {x + \Delta x, y + \Delta y, y' + \Delta y'} - \map F {x, y, y'}
| c =
}}
{{eqn | r = \Delta x \overline F_{y' x} + \Delta y \overline F_{y'y} + \Delta y' \overline F_{y'y'}
| c = Multivariate [[Mean Value Theorem]]
}}
{{en... | Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative | https://proofwiki.org/wiki/Conditions_for_C^1_Smooth_Solution_of_Euler's_Equation_to_have_Second_Derivative | https://proofwiki.org/wiki/Conditions_for_C^1_Smooth_Solution_of_Euler's_Equation_to_have_Second_Derivative | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Preimage/Mapping/Element",
"Definition:Continuously Differentiable",
"D... | [
"Mean Value Theorem",
"Combination Theorem for Limits of Functions/Real/Product Rule"
] |
proofwiki-12534 | Recurring Parts of Multiples of One Thirteenth | The multiples of $\dfrac 1 {13}$ from $\dfrac 1 {13}$ to $\dfrac {12} {13}$ can be divided into two sets of equal size:
:one where the digits of the recurring part consists of a cyclic permutation of $076923$
:one where the digits of the recurring part consists of a cyclic permutation of $153846$.
:300px | {{begin-eqn}}
{{eqn | l = \dfrac 1 {13}
| r = 0 \cdotp 076923 \, 076923 \ldots
| c =
}}
{{eqn | l = \dfrac 2 {13}
| r = 0 \cdotp 153846 \, 153846 \ldots
| c =
}}
{{eqn | l = \dfrac 3 {13}
| r = 0 \cdotp 230796 \, 230796 \ldots
| c =
}}
{{eqn | l = \dfrac 4 {13}
| r = 0 \cdot... | The [[Definition:Rational Multiplication|multiples]] of $\dfrac 1 {13}$ from $\dfrac 1 {13}$ to $\dfrac {12} {13}$ can be divided into two [[Definition:Set|sets]] of equal [[Definition:Cardinality|size]]:
:one where the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] consists of a [[Def... | {{begin-eqn}}
{{eqn | l = \dfrac 1 {13}
| r = 0 \cdotp 076923 \, 076923 \ldots
| c =
}}
{{eqn | l = \dfrac 2 {13}
| r = 0 \cdotp 153846 \, 153846 \ldots
| c =
}}
{{eqn | l = \dfrac 3 {13}
| r = 0 \cdotp 230796 \, 230796 \ldots
| c =
}}
{{eqn | l = \dfrac 4 {13}
| r = 0 \cdot... | Recurring Parts of Multiples of One Thirteenth | https://proofwiki.org/wiki/Recurring_Parts_of_Multiples_of_One_Thirteenth | https://proofwiki.org/wiki/Recurring_Parts_of_Multiples_of_One_Thirteenth | [
"Number Theory",
"13",
"One Thirteenth",
"Examples of Reciprocals"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Cardinality",
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Cyclic Permut... | [] |
proofwiki-12535 | Twelve Factorial plus One is divisible by 13 Squared | :$12! + 1$ is divisible by $13^2$. | By calculuation:
{{begin-eqn}}
{{eqn | l = 12! + 1
| r = 479 \, 001 \, 601
| c =
}}
{{eqn | r = 2 \, 834 \, 329 \times 13 \times 13
| c =
}}
{{end-eqn}}
{{qed}} | :$12! + 1$ is [[Definition:Divisor of Integer|divisible]] by $13^2$. | By calculuation:
{{begin-eqn}}
{{eqn | l = 12! + 1
| r = 479 \, 001 \, 601
| c =
}}
{{eqn | r = 2 \, 834 \, 329 \times 13 \times 13
| c =
}}
{{end-eqn}}
{{qed}} | Twelve Factorial plus One is divisible by 13 Squared | https://proofwiki.org/wiki/Twelve_Factorial_plus_One_is_divisible_by_13_Squared | https://proofwiki.org/wiki/Twelve_Factorial_plus_One_is_divisible_by_13_Squared | [
"12",
"13",
"Factorials"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-12536 | Torus can be cut into 13 Pieces with 3 Plane Cuts | A torus can be cut into as many as $13$ separate pieces by $3$ plane cuts. | {{ProofWanted|Lots of background work needed.}} | A [[Definition:Torus (Geometry)|torus]] can be cut into as many as $13$ separate pieces by $3$ [[Definition:Plane|plane]] cuts. | {{ProofWanted|Lots of background work needed.}} | Torus can be cut into 13 Pieces with 3 Plane Cuts | https://proofwiki.org/wiki/Torus_can_be_cut_into_13_Pieces_with_3_Plane_Cuts | https://proofwiki.org/wiki/Torus_can_be_cut_into_13_Pieces_with_3_Plane_Cuts | [
"12",
"13",
"Factorials"
] | [
"Definition:Torus (Geometry)",
"Definition:Plane Surface"
] | [] |
proofwiki-12537 | Operator Generated by Image of Closure Operator is Closure Operator | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $c: S \to S$ be a closure operator on $L$.
Then $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } = c$
where
:$\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times c \sqbrk S}$
:$\map {\operatorname {operator} } {\s... | Let $x \in S$.
By definition of closure operator/inflationary:
:$x \preceq \map c x$
By definition of upper closure of element:
:$\map c x \in x^\succeq$
By definition of image of mapping:
:$\map c x \in c \sqbrk S$
By definition of intersection:
:$\map c x \in x^\succeq \cap c \sqbrk S$
By definitions of infimum and l... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $c: S \to S$ be a [[Definition:Closure Operator|closure operator]] on $L$.
Then $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } = c$
where
:$\mathord \precsim = \mathord \preceq \cap \paren {c... | Let $x \in S$.
By definition of [[Definition:Closure Operator|closure operator/inflationary]]:
:$x \preceq \map c x$
By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
:$\map c x \in x^\succeq$
By definition of [[Definition:Image of Mapping|image of mapping]]:
:$\map c x \in c \sqbrk ... | Operator Generated by Image of Closure Operator is Closure Operator | https://proofwiki.org/wiki/Operator_Generated_by_Image_of_Closure_Operator_is_Closure_Operator | https://proofwiki.org/wiki/Operator_Generated_by_Image_of_Closure_Operator_is_Closure_Operator | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Closure Operator",
"Definition:Operator Generated by Ordered Subset"
] | [
"Definition:Closure Operator",
"Definition:Upper Closure/Element",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Set Intersection",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Set Intersection",
"Definition:Image (Set Theo... |
proofwiki-12538 | If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes | Let $\map \alpha {x, y}$, $\map h {x, y}$ be functions in $\R$.
{{mistake|should that be $\R^2$?}}
Let $\alpha \in C^0$ in a closed region $R$ whose boundary is $\Gamma$.
Let $h \in C^2$ in $R$ and $h = 0$ on $\Gamma$.
Let:
:$\ds \iint_R \map \alpha {x, y} \map h {x, y} \rd x \rd y = 0$
Then $\map \alpha {x, y}$ vanish... | {{AimForCont}} $\map \alpha {x, y}$ is nonzero at some point in $R$.
Then $\map \alpha {x, y}$ is also nonzero in some disk $D$ such that:
:$\paren {x - x_0}^2 + \paren {y - y_0}^2 \le \epsilon^2$
Suppose:
:$\map h {x, y} = \map \sgn {\map \alpha {x, y} } \paren {\epsilon^2 - \paren {x - x_0}^2 + \paren {y - y_0}^2}^3$... | Let $\map \alpha {x, y}$, $\map h {x, y}$ be functions in $\R$.
{{mistake|should that be $\R^2$?}}
Let $\alpha \in C^0$ in a closed region $R$ whose boundary is $\Gamma$.
Let $h \in C^2$ in $R$ and $h = 0$ on $\Gamma$.
Let:
:$\ds \iint_R \map \alpha {x, y} \map h {x, y} \rd x \rd y = 0$
Then $\map \alpha {x, y}$ ... | {{AimForCont}} $\map \alpha {x, y}$ is nonzero at some point in $R$.
Then $\map \alpha {x, y}$ is also nonzero in some [[Definition:Disk|disk]] $D$ such that:
:$\paren {x - x_0}^2 + \paren {y - y_0}^2 \le \epsilon^2$
Suppose:
:$\map h {x, y} = \map \sgn {\map \alpha {x, y} } \paren {\epsilon^2 - \paren {x - x_0}^2 ... | If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes | https://proofwiki.org/wiki/If_Double_Integral_of_a(x,_y)h(x,_y)_vanishes_for_any_C^2_h(x,_y)_then_C^0_a(x,_y)_vanishes | https://proofwiki.org/wiki/If_Double_Integral_of_a(x,_y)h(x,_y)_vanishes_for_any_C^2_h(x,_y)_then_C^0_a(x,_y)_vanishes | [
"Calculus of Variations"
] | [] | [
"Definition:Disk",
"Proof by Contradiction"
] |
proofwiki-12539 | Pell's Equation/Examples/13 | :$x^2 - 13 y^2 = 1$
has the smallest positive integral solution:
:$x = 649$
:$y = 180$ | From Continued Fraction Expansion of $\sqrt {13}$:
:$\sqrt {13} = \sqbrk {3, \sequence {1, 1, 1, 1, 6} }$
The cycle is of length $5$.
By Solution of Pell's Equation, the only solutions of $x^2 - 13 y^2 = 1$ are:
:${p_{5 r} }^2 - 13 {q_{5 r} }^2 = \paren {-1}^{5 r}$
for $r = 1, 2, 3, \ldots$
When $r = 1$ this gives:
:${... | :$x^2 - 13 y^2 = 1$
has the smallest [[Definition:Positive Integer|positive integral]] solution:
:$x = 649$
:$y = 180$ | From [[Continued Fraction Expansion of Irrational Square Root/Examples/13|Continued Fraction Expansion of $\sqrt {13}$]]:
:$\sqrt {13} = \sqbrk {3, \sequence {1, 1, 1, 1, 6} }$
The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fraction|length]] $5$.
By... | Pell's Equation/Examples/13 | https://proofwiki.org/wiki/Pell's_Equation/Examples/13 | https://proofwiki.org/wiki/Pell's_Equation/Examples/13 | [
"Pell's Equation",
"13"
] | [
"Definition:Positive/Integer"
] | [
"Continued Fraction Expansion of Irrational Square Root/Examples/13",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Solution to Pell's Equation",
"Continued Fraction Expansion of Irrational Square Root/Examples/13/Convergents"
] |
proofwiki-12540 | Pell's Equation/Examples/29 | :$x^2 - 29 y^2 = 1$
has the smallest positive integral solution:
:$x = 9801$
:$y = 1820$ | From Continued Fraction Expansion of $\sqrt {29}$:
:$\sqrt {29} = \sqbrk {5, \sequence {2, 1, 1, 2, 10} }$
The cycle is of length is $5$.
By Solution of Pell's Equation, the only solutions of $x^2 - 29 y^2 = 1$ are:
:${p_{5 r} }^2 - 29 {q_{5 r} }^2 = \paren {-1}^{5 r}$
for $r = 1, 2, 3, \ldots$
When $r = 1$ this gives:... | :$x^2 - 29 y^2 = 1$
has the smallest [[Definition:Positive Integer|positive integral]] solution:
:$x = 9801$
:$y = 1820$ | From [[Continued Fraction Expansion of Irrational Square Root/Examples/29|Continued Fraction Expansion of $\sqrt {29}$]]:
:$\sqrt {29} = \sqbrk {5, \sequence {2, 1, 1, 2, 10} }$
The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fraction|length]] is $5$.... | Pell's Equation/Examples/29 | https://proofwiki.org/wiki/Pell's_Equation/Examples/29 | https://proofwiki.org/wiki/Pell's_Equation/Examples/29 | [
"Pell's Equation",
"29"
] | [
"Definition:Positive/Integer"
] | [
"Continued Fraction Expansion of Irrational Square Root/Examples/29",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Solution to Pell's Equation",
"Continued Fraction Expansion of Irrational Square Root/Examples/29/Convergents"
] |
proofwiki-12541 | Simple Variable End Point Problem | Let $y$ and $F$ be mappings.
{{explain|Define their domain and codomain}}
Suppose the endpoints of $y$ lie on two given vertical lines $x = a$ and $x = b$.
Suppose $J$ is a functional of the form
:$(1): \quad J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$
and has an extremum for a certain function $\hat y$.
Then $y... | From Condition for Differentiable Functional to have Extremum we have
:$\bigvalueat {\delta J \sqbrk {y; h} } {y \mathop = \hat y} = 0$
The variation exists if $J$ is a differentiable functional.
We will start from the increment of a functional:
{{explain|make the above link point to a page dedicated to the appropriate... | Let $y$ and $F$ be [[Definition:Mapping|mappings]].
{{explain|Define their domain and codomain}}
Suppose the endpoints of $y$ lie on two given vertical lines $x = a$ and $x = b$.
Suppose $J$ is a [[Definition:Real Functional|functional]] of the form
:$(1): \quad J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$
a... | From [[Condition for Differentiable Functional to have Extremum]] we have
:$\bigvalueat {\delta J \sqbrk {y; h} } {y \mathop = \hat y} = 0$
The variation exists if $J$ is a [[Definition:Differentiable Functional|differentiable functional]].
We will start from the [[Definition:Differentiable Functional|increment of a... | Simple Variable End Point Problem | https://proofwiki.org/wiki/Simple_Variable_End_Point_Problem | https://proofwiki.org/wiki/Simple_Variable_End_Point_Problem | [
"Calculus of Variations"
] | [
"Definition:Mapping",
"Definition:Functional/Real"
] | [
"Condition for Differentiable Functional to have Extremum",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Integration by Parts",
"Condition for Differentiable Functional to have Extremum"
] |
proofwiki-12542 | Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.
Then $\operatorname{Closure}\left({L}\right)$ and $\operatorname{ClSystems}\left({L}\right)^{-1}$ are order isomorphic
where
:$\operatorname{Closure}\left({L}\right)$ denotes the ordered set of closure operators of $L$,
:$\operatorname{ClSystems}\... | By definition of ordered set of closure operators:
:$\operatorname{Closure}\left({L}\right) = \left({X, \preceq'}\right)$
where
:$X$ is the set of all closure operators on $L$,
:for all closure operators $f, g$ on $L$: $f \preceq' g \iff f \preceq g$
:$\preceq$ denotes the ordering on mappings.
By definition of ordered... | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
Then $\operatorname{Closure}\left({L}\right)$ and $\operatorname{ClSystems}\left({L}\right)^{-1}$ are [[Definition:Order Isomorphism|order isomorphic]]
where
:$\operatorname{Closure}\left({L}\right)$ denotes the ... | By definition of [[Definition:Ordered Set of Closure Operators|ordered set of closure operators]]:
:$\operatorname{Closure}\left({L}\right) = \left({X, \preceq'}\right)$
where
:$X$ is the [[Definition:Set|set]] of all [[Definition:Closure Operator|closure operators]] on $L$,
:for all [[Definition:Closure Operator|closu... | Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic | https://proofwiki.org/wiki/Ordered_Set_of_Closure_Operators_and_Dual_Ordered_Set_of_Closure_Systems_are_Isomorphic | https://proofwiki.org/wiki/Ordered_Set_of_Closure_Operators_and_Dual_Ordered_Set_of_Closure_Systems_are_Isomorphic | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Order Isomorphism",
"Definition:Ordered Set of Closure Operators",
"Definition:Ordered Set of Closure Systems",
"Definition:Dual Ordering/Dual Ordered Set"
] | [
"Definition:Ordered Set of Closure Operators",
"Definition:Set",
"Definition:Closure Operator",
"Definition:Closure Operator",
"Definition:Ordering on Mappings",
"Definition:Ordered Set of Closure Systems",
"Definition:Set",
"Definition:Closure System",
"Definition:Closure System",
"Image of Closu... |
proofwiki-12543 | Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation | Let $\map y x$ be a real function such that $\map y a = A$ and $\map y b = B$.
Let $J \sqbrk y$ be a functional of the form:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Then:
:$\dfrac {\delta J} {\delta y} = 0 \implies F_y - \dfrac \d {\d x} F_{y'} = 0$ | The method of finite differences will be used here.
Consider a closed real interval $\closedint a b$, which is divided in $n + 1$ equal parts.
Choose its subdivision to be normal:
:$a = x_0 < x_1 < \cdots < x_n < x_{n + 1} = b$
such that for $i \in set {0, 1, \ldots, n - 1, n}$ we have $x_{i + 1} - x_i = \Delta x$.
App... | Let $\map y x$ be a [[Definition:Real Function|real function]] such that $\map y a = A$ and $\map y b = B$.
Let $J \sqbrk y$ be a functional of the form:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Then:
:$\dfrac {\delta J} {\delta y} = 0 \implies F_y - \dfrac \d {\d x} F_{y'} = 0$ | The method of finite differences will be used here.
Consider a [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$, which is divided in $n + 1$ equal parts.
Choose its [[Definition:Subdivision of Interval|subdivision]] to be [[Definition:Normal Subdivision|normal:]]
:$a = x_0 < x_1 < \cdots < x... | Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation | https://proofwiki.org/wiki/Vanishing_First_Variational_Derivative_implies_Euler's_Equation_for_Vanishing_Variation | https://proofwiki.org/wiki/Vanishing_First_Variational_Derivative_implies_Euler's_Equation_for_Vanishing_Variation | [
"Calculus of Variations"
] | [
"Definition:Real Function"
] | [
"Definition:Real Interval/Closed",
"Definition:Subdivision of Interval",
"Definition:Subdivision of Interval/Normal Subdivision",
"Definition:Kronecker Delta",
"Definition:Variational Derivative"
] |
proofwiki-12544 | Supremum in Ordered Subset | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $R = \struct {T, \preceq'}$ be an ordered subset of $L$.
Let $X \subseteq T$ such that
:$X$ admits an supremum in $L$.
Then $\sup_L X \in T$ {{iff}}
:$X$ admits an supremum in $R$ and $\sup_R X = \sup_L X$ | This follows by {{mutatis}} of the proof of Infimum in Ordered Subset.
{{qed}} | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $R = \struct {T, \preceq'}$ be an [[Definition:Ordered Subset|ordered subset]] of $L$.
Let $X \subseteq T$ such that
:$X$ admits an [[Definition:Supremum of Set|supremum]] in $L$.
Then $\sup_L X \in T$ {{iff}}
:$X$ admits an [[Definiti... | This follows by {{mutatis}} of the proof of [[Infimum in Ordered Subset]].
{{qed}} | Supremum in Ordered Subset | https://proofwiki.org/wiki/Supremum_in_Ordered_Subset | https://proofwiki.org/wiki/Supremum_in_Ordered_Subset | [
"Ordered Sets",
"Suprema"
] | [
"Definition:Ordered Set",
"Definition:Ordered Subset",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Infimum in Ordered Subset"
] |
proofwiki-12545 | Partial Denominators of Continued Fraction Expansion of Irrational Square Root | Let $n \in \Z$ such that $n$ is not a square.
Let the continued fraction expansion of $\sqrt n$ be expressed as:
:$\sqbrk {a_0, a_1, a_2, \ldots}$
Then the partial denominators of this continued fraction expansion can be calculated as:
:$a_r = \floor {\dfrac {\floor {\sqrt n} + P_r} {Q_r} }$
where:
:<nowiki>$P_r = \beg... | The proof proceeds by strong induction.
For all $r \in \Z_{\ge 0}$, let $\map P r$ be the proposition:
:$a_r = \floor {\dfrac {\floor {\sqrt n} + P_r} {Q_r} }$
where:
:<nowiki>$P_r = \begin{cases} 0 & : r = 0 \\
a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\
\end{cases}$</nowiki>
:<nowiki>$Q_r = \begin{cases} 1 & : r = 0... | Let $n \in \Z$ such that $n$ is not a [[Definition:Square Number|square]].
Let the [[Definition:Continued Fraction Expansion of Irrational Number|continued fraction expansion]] of $\sqrt n$ be expressed as:
:$\sqbrk {a_0, a_1, a_2, \ldots}$
Then the [[Definition:Partial Denominator|partial denominators]] of this [[D... | The proof proceeds by [[Second Principle of Mathematical Induction|strong induction]].
For all $r \in \Z_{\ge 0}$, let $\map P r$ be the [[Definition:Proposition|proposition]]:
:$a_r = \floor {\dfrac {\floor {\sqrt n} + P_r} {Q_r} }$
where:
:<nowiki>$P_r = \begin{cases} 0 & : r = 0 \\
a_{r - 1} Q_{r - 1} - P_{r - 1}... | Partial Denominators of Continued Fraction Expansion of Irrational Square Root | https://proofwiki.org/wiki/Partial_Denominators_of_Continued_Fraction_Expansion_of_Irrational_Square_Root | https://proofwiki.org/wiki/Partial_Denominators_of_Continued_Fraction_Expansion_of_Irrational_Square_Root | [
"Continued Fractions"
] | [
"Definition:Square Number",
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Partial Denominator",
"Definition:Continued Fraction Expansion/Real Number"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Second Principle of Mathematical Induction"
] |
proofwiki-12546 | Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema | Let $L = \struct {X, \vee, \wedge, \preceq}$ be a complete lattice.
Let $S = \struct {Y, \precsim}$ be a closure system on $L$.
Then $\map {\operatorname {operator} } S$ preserves directed suprema
{{iff}} $S$ inherits directed suprema.
where $\map {\operatorname {operator} } S$ denotes the operator generated by $S$. | === Sufficient Condition ===
Assume that $\map {\operatorname {operator} } S$ preserves directed suprema.
Let $Z$ be directed subset of $Y$ such that
:$Z$ admits a supremum in $L$.
By Image of Operator Generated by Closure System is Set of Closure System:
:$\map {\operatorname {operator} } S \sqbrk X = Y$
By Operator G... | Let $L = \struct {X, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $S = \struct {Y, \precsim}$ be a [[Definition:Closure System|closure system]] on $L$.
Then $\map {\operatorname {operator} } S$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]]
{{iff}}... | === Sufficient Condition ===
Assume that $\map {\operatorname {operator} } S$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
Let $Z$ be [[Definition:Directed Subset|directed subset]] of $Y$ such that
:$Z$ admits a [[Definition:Supremum of Set|supremum]] in $L$.
By [[Image of Operator ... | Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema | https://proofwiki.org/wiki/Operator_Generated_by_Closure_System_Preserves_Directed_Suprema_iff_Closure_System_Inherits_Directed_Suprema | https://proofwiki.org/wiki/Operator_Generated_by_Closure_System_Preserves_Directed_Suprema_iff_Closure_System_Inherits_Directed_Suprema | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Closure System",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Directed Suprema Inheriting",
"Definition:Operator Generated by Ordered Subset"
] | [
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Image of Operator Generated by Closure System is Set of Closure System",
"Operator Generated by Closure System is Closure Operator",
"Definition:Closure Operator",
"Definition:Closure Operator"... |
proofwiki-12547 | Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations | Euler's Equation for Vanishing Variation is invariant under coordinate transformations. | Let $J \sqbrk y$ be an integral functional:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Suppose, we introduce new curvilinear coordinates $u,v$ such that:
:$x = \map x {u, v}$
:$y = \map y {u, v}$
such that:
:$\begin{vmatrix}
\map {\dfrac {\partial x} {\partial u} } {u, v} & \map {\dfrac {\partial x} {\partial... | [[Definition:Euler's Equation for Vanishing Variation|Euler's Equation for Vanishing Variation]] is invariant under coordinate transformations. | Let $J \sqbrk y$ be an integral functional:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
Suppose, we introduce new curvilinear coordinates $u,v$ such that:
:$x = \map x {u, v}$
:$y = \map y {u, v}$
such that:
:$\begin{vmatrix}
\map {\dfrac {\partial x} {\partial u} } {u, v} & \map {\dfrac {\partial x} {\par... | Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations | https://proofwiki.org/wiki/Euler's_Equation_for_Vanishing_Variation_is_Invariant_under_Coordinate_Transformations | https://proofwiki.org/wiki/Euler's_Equation_for_Vanishing_Variation_is_Invariant_under_Coordinate_Transformations | [
"Calculus of Variations"
] | [
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation",
"Definition:Variational Derivative"
] |
proofwiki-12548 | Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $h: S \to S$ be a closure operator on $L$.
Then $h$ preserves directed suprema
{{iff}} $\struct {h \sqbrk S, \precsim}$ inherits directed suprema.
where
:$h \sqbrk S$ denotes the image of $h$,
:$\mathord\precsim = \mathord\preceq \cap \paren {h \sq... | By Operator Generated by Image of Closure Operator is Closure Operator:
:$\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} } = h$
where $\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} }$ denotes the operator generated by $\struct {h \sqbrk S, \precsim}$
Hence the result holds by Operator... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $h: S \to S$ be a [[Definition:Closure Operator|closure operator]] on $L$.
Then $h$ [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]]
{{iff}} $\struct {h \sqbrk S, \precsim}$ [[Definiti... | By [[Operator Generated by Image of Closure Operator is Closure Operator]]:
:$\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} } = h$
where $\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} }$ denotes the [[Definition:Operator Generated by Ordered Subset|operator generated by]] $\struct {h... | Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema | https://proofwiki.org/wiki/Closure_Operator_Preserves_Directed_Suprema_iff_Image_of_Closure_Operator_Inherits_Directed_Suprema | https://proofwiki.org/wiki/Closure_Operator_Preserves_Directed_Suprema_iff_Image_of_Closure_Operator_Inherits_Directed_Suprema | [
"Closure Operators"
] | [
"Definition:Complete Lattice",
"Definition:Closure Operator",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Directed Suprema Inheriting",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Operator Generated by Image of Closure Operator is Closure Operator",
"Definition:Operator Generated by Ordered Subset",
"Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema"
] |
proofwiki-12549 | Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions | Let $\mathbf y$ be an $n$-dimensional real vector.
Let $J \sqbrk {\mathbf y}$ be a functional of the form:
$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let:
:$\mathbf y \in C^1 \closedint a b$
where $C^1 \closedint a b$ denotes that $\mathbf y$ is continuously differentiable in $\closed... | From Condition for Differentiable Functional of N Functions to have Extremum:
:$\ds \bigvalueat {\delta J \sqbrk {\mathbf y; \mathbf h} } {\mathbf y \mathop = \hat{\mathbf y} } = 0$
For the variation to exist it has to satisfy the requirement for a differentiable functional.
Note that the endpoints of $\map {\mathbf y}... | Let $\mathbf y$ be an $n$-dimensional real vector.
Let $J \sqbrk {\mathbf y}$ be a functional of the form:
$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let:
:$\mathbf y \in C^1 \closedint a b$
where $C^1 \closedint a b$ denotes that $\mathbf y$ is [[Definition:Continuously Differen... | From [[Condition for Differentiable Functional of N Functions to have Extremum]]:
:$\ds \bigvalueat {\delta J \sqbrk {\mathbf y; \mathbf h} } {\mathbf y \mathop = \hat{\mathbf y} } = 0$
For the variation to exist it has to satisfy the requirement for a [[Definition:Differentiable Functional|differentiable functional]... | Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function/Dependent_on_N_Functions | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function/Dependent_on_N_Functions | [
"Calculus of Variations"
] | [
"Definition:Continuously Differentiable"
] | [
"Condition for Differentiable Functional of N Functions to have Extremum",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional"
] |
proofwiki-12550 | Infimum Precedes Coarser Infimum | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.
Let $X, Y$ be subsets of $S$ such that
:$Y$ is coarser than $X$.
Then $\inf X \preceq \inf Y$
where $\inf X$ denotes the infimum of $X$. | We will prove that
:$\inf X$ is lower bound for $Y$.
Let $x \in Y$.
By definition of coarser subset:
:$\exists y \in X: y \preceq x$
By definitions of infimum and lower bound:
:$\inf X \preceq y$
Thus by definition of transitivity:
:$\inf X \preceq x$
{{qed|lemma}}
Hence by definition of infimum:
:$\inf X \preceq \inf ... | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
Let $X, Y$ be [[Definition:Subset|subsets]] of $S$ such that
:$Y$ is [[Definition:Coarser Subset (Order Theory)|coarser than]] $X$.
Then $\inf X \preceq \inf Y$
where $\inf X$ denotes the [[Definition:Infimum of... | We will prove that
:$\inf X$ is [[Definition:Lower Bound of Set|lower bound]] for $Y$.
Let $x \in Y$.
By definition of [[Definition:Coarser Subset (Order Theory)|coarser subset]]:
:$\exists y \in X: y \preceq x$
By definitions of [[Definition:Infimum of Set|infimum]] and [[Definition:Lower Bound of Set|lower bound]]... | Infimum Precedes Coarser Infimum | https://proofwiki.org/wiki/Infimum_Precedes_Coarser_Infimum | https://proofwiki.org/wiki/Infimum_Precedes_Coarser_Infimum | [
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Subset",
"Definition:Coarser Subset (Order Theory)",
"Definition:Infimum of Set"
] | [
"Definition:Lower Bound of Set",
"Definition:Coarser Subset (Order Theory)",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Transitive",
"Definition:Infimum of Set"
] |
proofwiki-12551 | Uniform Prism is Semiregular Polyhedron | Let $\PP$ be a uniform prism whose bases are specifically not square.
Then $\PP$ is a semiregular polyhedron. | {{Recall|Semiregular Polyhedron}}
{{:Definition:Semiregular Polyhedron}}
Let $\PP$ be a uniform prism.
By definition, $\PP$ is regular.
Hence the bases of $\PP$ are regular polygons.
Also by definition, the lateral faces of $\PP$ are square so {{afortiori}} regular polygons.
Suppose the bases of $\PP$ are square.
Then ... | Let $\PP$ be a [[Definition:Uniform Prism|uniform prism]] whose [[Definition:Base of Prism|bases]] are specifically not [[Definition:Square|square]].
Then $\PP$ is a [[Definition:Semiregular Polyhedron|semiregular polyhedron]]. | {{Recall|Semiregular Polyhedron}}
{{:Definition:Semiregular Polyhedron}}
Let $\PP$ be a [[Definition:Uniform Prism|uniform prism]].
By definition, $\PP$ is [[Definition:Regular|regular]].
Hence the [[Definition:Base of Pyramid|bases]] of $\PP$ are [[Definition:Regular Polygon|regular polygons]].
Also by definition,... | Uniform Prism is Semiregular Polyhedron | https://proofwiki.org/wiki/Uniform_Prism_is_Semiregular_Polyhedron | https://proofwiki.org/wiki/Uniform_Prism_is_Semiregular_Polyhedron | [
"Uniform Prisms",
"Semiregular Polyhedra"
] | [
"Definition:Uniform Prism",
"Definition:Prism/Base",
"Definition:Square",
"Definition:Semiregular Polyhedron"
] | [
"Definition:Uniform Prism",
"Definition:Regular",
"Definition:Pyramid/Base",
"Definition:Polygon/Regular",
"Definition:Prism/Lateral Face",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Regular",
"Definition:Pyramid/Base",
"Definition:Quadrilateral/Square",
"Definition:Polyhedron/Face",
... |
proofwiki-12552 | Uniform Antiprism is Semiregular Polyhedron | Let $\PP$ be a uniform antiprism whose bases are specifically not (equilateral) triangles.
Then $\PP$ is a semiregular polyhedron. | {{Recall|Semiregular Polyhedron}}
{{:Definition:Semiregular Polyhedron}}
Let $\PP$ be a uniform antiprism.
By definition, $\PP$ is regular.
Hence the bases of $\PP$ are regular polygons.
Also by definition, the lateral faces of $\PP$ are equilateral triangles so {{afortiori}} regular polygons.
Suppose the bases of $\PP... | Let $\PP$ be a [[Definition:Uniform Antiprism|uniform antiprism]] whose [[Definition:Base of Antiprism|bases]] are specifically not [[Definition:Equilateral Triangle|(equilateral) triangles]].
Then $\PP$ is a [[Definition:Semiregular Polyhedron|semiregular polyhedron]]. | {{Recall|Semiregular Polyhedron}}
{{:Definition:Semiregular Polyhedron}}
Let $\PP$ be a [[Definition:Uniform Antiprism|uniform antiprism]].
By definition, $\PP$ is [[Definition:Regular|regular]].
Hence the [[Definition:Base of Pyramid|bases]] of $\PP$ are [[Definition:Regular Polygon|regular polygons]].
Also by def... | Uniform Antiprism is Semiregular Polyhedron | https://proofwiki.org/wiki/Uniform_Antiprism_is_Semiregular_Polyhedron | https://proofwiki.org/wiki/Uniform_Antiprism_is_Semiregular_Polyhedron | [
"Uniform Antiprisms",
"Semiregular Polyhedra"
] | [
"Definition:Uniform Antiprism",
"Definition:Antiprism/Base",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Semiregular Polyhedron"
] | [
"Definition:Uniform Antiprism",
"Definition:Regular",
"Definition:Pyramid/Base",
"Definition:Polygon/Regular",
"Definition:Prism/Lateral Face",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Regular",
"Definition:Pyramid/Base",
"Definition:Triangle (Geometry)/Equilateral",
"Defi... |
proofwiki-12553 | Finer Supremum Precedes Supremum | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.
Let $X, Y$ be subsets of $S$ such that
:$X$ is finer than $Y$.
Then $\sup X \preceq \sup Y$
where $\sup X$ denotes the supremum of $X$. | We will prove that
:$\sup Y$ is upper bound for $X$.
Let $x \in X$.
By definition of finer subset:
:$\exists y \in Y: x \preceq y$
By definitions of supremum and upper bound:
:$y \preceq \sup Y$
Thus by definition of transitivity:
:$x \preceq \sup Y$
{{qed|lemma}}
Hence by definition of supremum:
:$\sup X \preceq \sup ... | Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a [[Definition:Complete Lattice|complete lattice]].
Let $X, Y$ be [[Definition:Subset|subsets]] of $S$ such that
:$X$ is [[Definition:Finer Subset (Order Theory)|finer than]] $Y$.
Then $\sup X \preceq \sup Y$
where $\sup X$ denotes the [[Definition:Supremum of Se... | We will prove that
:$\sup Y$ is [[Definition:Upper Bound of Set|upper bound]] for $X$.
Let $x \in X$.
By definition of [[Definition:Finer Subset (Order Theory)|finer subset]]:
:$\exists y \in Y: x \preceq y$
By definitions of [[Definition:Supremum of Set|supremum]] and [[Definition:Upper Bound of Set|upper bound]]:
... | Finer Supremum Precedes Supremum | https://proofwiki.org/wiki/Finer_Supremum_Precedes_Supremum | https://proofwiki.org/wiki/Finer_Supremum_Precedes_Supremum | [
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Subset",
"Definition:Finer Subset (Order Theory)",
"Definition:Supremum of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Finer Subset (Order Theory)",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Transitive",
"Definition:Supremum of Set"
] |
proofwiki-12554 | Thirteen Catalan Polyhedra | There exist exactly $13$ distinct Catalan polyhedra:
:Triakis tetrahedron
:Triakis octahedron
:Disdyakis dodecahedron
:Tetrakis hexahedron
:Triakis icosahedron
:Disdyakis triacontahedron
:Pentakis dodecahedron
:Rhombic dodecahedron
:Rhombic triacontahedron
:Deltoidal icositetrahedron
:Deltoidal hexecontahedron
:Pentago... | By definition, the Catalan polyhedra are the dual polyhedra of the Archimedean polyhedra.
There are $13$ Archimedean polyhedra, and so there are $13$ Catalan polyhedra.
{{ProofWanted|Include links to specific results linking the Archimedean polyhedra with their duals.}} | There exist exactly $13$ [[Definition:Distinct|distinct]] [[Definition:Catalan Polyhedron|Catalan polyhedra]]:
:[[Definition:Triakis Tetrahedron|Triakis tetrahedron]]
:[[Definition:Triakis Octahedron|Triakis octahedron]]
:[[Definition:Disdyakis Dodecahedron|Disdyakis dodecahedron]]
:[[Definition:Tetrakis Hexahedron|Te... | By definition, the [[Definition:Catalan Polyhedron|Catalan polyhedra]] are the [[Definition:Dual Polyhedron|dual polyhedra]] of the [[Definition:Archimedean Polyhedron|Archimedean polyhedra]].
There are $13$ [[Definition:Archimedean Polyhedron|Archimedean polyhedra]], and so there are $13$ [[Definition:Catalan Polyhed... | Thirteen Catalan Polyhedra | https://proofwiki.org/wiki/Thirteen_Catalan_Polyhedra | https://proofwiki.org/wiki/Thirteen_Catalan_Polyhedra | [
"Catalan Polyhedra"
] | [
"Definition:Distinct",
"Definition:Catalan Polyhedron",
"Definition:Triakis Tetrahedron",
"Definition:Triakis Octahedron",
"Definition:Disdyakis Dodecahedron",
"Definition:Tetrakis Hexahedron",
"Definition:Triakis Icosahedron",
"Definition:Disdyakis Triacontahedron",
"Definition:Pentakis Dodecahedro... | [
"Definition:Catalan Polyhedron",
"Definition:Dual Polyhedron",
"Definition:Archimedean Polyhedron",
"Definition:Archimedean Polyhedron",
"Definition:Catalan Polyhedron",
"Definition:Archimedean Polyhedron",
"Definition:Dual Polyhedron"
] |
proofwiki-12555 | Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation | Let $x: \R \to \R$ and $y: \R \to \R$ be real functions.
Let $J \sqbrk {x, y}$ be a functional of the form:
:$\ds J \sqbrk {x, y} = \int_{t_0}^{t_1} \map \Phi {t, x, y, \dot x, \dot y} \rd t$
where $\dot y$ denotes the derivative of $y$ {{WRT|Differentiation}} $t$:
:$\dot y = \dfrac {\d y} {\d t}$
Then $J \sqbrk {x, y}... | In the functional:
:$\ds \JJ \sqbrk y = \int_{x_0}^{x_1} \map F {x, y, y'} \rd x$
let the argument $y$ stand for a curve which is given in a parametric form.
In other words, let the curve be described by the vector $\tuple {\map y t,\map x t}$ rather than $\paren {\map y x, x}$.
Then the functional can be rewritten as... | Let $x: \R \to \R$ and $y: \R \to \R$ be [[Definition:Real Function|real functions]].
Let $J \sqbrk {x, y}$ be a functional of the form:
:$\ds J \sqbrk {x, y} = \int_{t_0}^{t_1} \map \Phi {t, x, y, \dot x, \dot y} \rd t$
where $\dot y$ denotes the [[Definition:Derivative|derivative]] of $y$ {{WRT|Differentiation}} $... | In the functional:
:$\ds \JJ \sqbrk y = \int_{x_0}^{x_1} \map F {x, y, y'} \rd x$
let the argument $y$ stand for a curve which is given in a parametric form.
In other words, let the curve be described by the [[Definition:Vector (Linear Algebra)|vector]] $\tuple {\map y t,\map x t}$ rather than $\paren {\map y x, x}... | Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Integral_Parametric_Functional_to_be_Independent_of_Parametric_Representation | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Integral_Parametric_Functional_to_be_Independent_of_Parametric_Representation | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Derivative",
"Definition:Cartesian Plane",
"Definition:Homogeneous Function"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Cartesian Plane",
"Derivative of Composite Function"
] |
proofwiki-12556 | Intersection of Upper Section with Directed Set is Directed Set | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $A, B$ be subsets of $S$ such that
:$A \cap B \ne \O$
and
:$A$ is an upper set,
:$B$ is a directed set.
Then $A \cap B$ is a directed set. | Let $x, y \in A \cap B$.
By definition of intersection:
:$x, y \in A$ and $x, y \in B$
By definition of directed subset:
:$\exists z \in B: x \preceq z \land y \preceq z$
By definition of upper section:
:$z \in A$
Thus by definition of intersection:
:$z \in A \cap B$
Thus:
:$\exists z \in A \cap B: x \preceq z$ and $y ... | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$ such that
:$A \cap B \ne \O$
and
:$A$ is an [[Definition:Upper Section|upper set]],
:$B$ is a [[Definition:Directed Subset|directed set]].
Then $A \cap B$ is a [[Definition:Directed Subset|... | Let $x, y \in A \cap B$.
By definition of [[Definition:Set Intersection|intersection]]:
:$x, y \in A$ and $x, y \in B$
By definition of [[Definition:Directed Subset|directed subset]]:
:$\exists z \in B: x \preceq z \land y \preceq z$
By definition of [[Definition:Upper Section|upper section]]:
:$z \in A$
Thus by de... | Intersection of Upper Section with Directed Set is Directed Set | https://proofwiki.org/wiki/Intersection_of_Upper_Section_with_Directed_Set_is_Directed_Set | https://proofwiki.org/wiki/Intersection_of_Upper_Section_with_Directed_Set_is_Directed_Set | [
"Upper Sections"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Upper Section",
"Definition:Directed Subset",
"Definition:Directed Subset"
] | [
"Definition:Set Intersection",
"Definition:Directed Subset",
"Definition:Upper Section",
"Definition:Set Intersection"
] |
proofwiki-12557 | Convex Set is Contractible | Let $V$ be a topological vector space over $\R$ or $\C$.
Let $A\subset V$ be a convex subset.
Then $A$ is contractible. | {{WIP|Lots of imprecision and incorrect grammar which needs to be resolved. Too much to do on the fly}}
Let $x_0 \in A$.
Define $H : A \times \closedint 0 1 \to A$ by:
:$\map H {x, t} = t x_0 + \paren {1 - t} x$
Note that by the definition of a convex set, $\Cdm H$ is also a convex set.
{{Proofread|Verify if the codoma... | Let $V$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$ or $\C$.
Let $A\subset V$ be a [[Definition:Convex Set (Vector Space)|convex subset]].
Then $A$ is [[Definition:Contractible Space|contractible]]. | {{WIP|Lots of imprecision and incorrect grammar which needs to be resolved. Too much to do on the fly}}
Let $x_0 \in A$.
Define $H : A \times \closedint 0 1 \to A$ by:
:$\map H {x, t} = t x_0 + \paren {1 - t} x$
Note that by the definition of a [[Definition:Convex Set (Vector Space)/Definition 1|convex set]], $\Cdm ... | Convex Set is Contractible | https://proofwiki.org/wiki/Convex_Set_is_Contractible | https://proofwiki.org/wiki/Convex_Set_is_Contractible | [
"Vector Spaces",
"Topology",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Contractible Space"
] | [
"Definition:Convex Set (Vector Space)/Definition 1",
"Definition:Convex Set (Vector Space)/Definition 1",
"Definition:Identity Mapping",
"Definition:Set",
"Definition:Constant Mapping",
"Definition:Homotopy",
"Definition:Identity Mapping",
"Definition:Constant Mapping",
"Definition:Convex Set (Vecto... |
proofwiki-12558 | Open Subgroup is Closed | Let $G$ be a topological group.
Let $H\leq G$ be an open subgroup.
Then $H$ is closed. | By Right and Left Regular Representations in Topological Group are Homeomorphisms, the left cosets of $H$ are open.
By Left Coset Space forms Partition, the complement of $H$ is a union of left cosets of $H$.
Because the complement of $H$ is open, $H$ is closed.
{{qed}}
Category:Topology
Category:Group Theory
Category:... | Let $G$ be a [[Definition:Topological Group|topological group]].
Let $H\leq G$ be an open [[Definition:Subgroup|subgroup]].
Then $H$ is [[Definition:Closed Set (Topology)|closed]]. | By [[Right and Left Regular Representations in Topological Group are Homeomorphisms]], the [[Definition:Left Coset|left cosets]] of $H$ are open.
By [[Left Coset Space forms Partition]], the [[Definition:Relative Complement|complement]] of $H$ is a [[Definition:Set Union|union]] of [[Definition:Left Coset|left cosets]... | Open Subgroup is Closed | https://proofwiki.org/wiki/Open_Subgroup_is_Closed | https://proofwiki.org/wiki/Open_Subgroup_is_Closed | [
"Topology",
"Group Theory",
"Topological Groups"
] | [
"Definition:Topological Group",
"Definition:Subgroup",
"Definition:Closed Set/Topology"
] | [
"Right and Left Regular Representations in Topological Group are Homeomorphisms",
"Definition:Coset/Left Coset",
"Left Coset Space forms Partition",
"Definition:Relative Complement",
"Definition:Set Union",
"Definition:Coset/Left Coset",
"Definition:Closed Set/Topology",
"Category:Topology",
"Catego... |
proofwiki-12559 | Topological Group is Hausdorff iff Identity is Closed | Let $G$ be a topological group.
Let $e$ be its identity element.
Then $G$ is a $T_2$ (Hausdorff) space {{iff}} $\set e$ is closed in $G$. | === Necessary Condition ===
Suppose $G$ is a $T_2$ (Hausdorff) space.
By $T_2$ Space is $T_1$ Space, $\set e$ is closed.
{{qed|lemma}} | Let $G$ be a [[Definition:Topological Group|topological group]].
Let $e$ be its [[Definition:Identity Element|identity element]].
Then $G$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] {{iff}} $\set e$ is [[Definition:Closed Set (Topology)|closed]] in $G$. | === Necessary Condition ===
Suppose $G$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
By [[T2 Space is T1 Space|$T_2$ Space is $T_1$ Space]], $\set e$ is [[Definition:Closed Set (Topology)|closed]].
{{qed|lemma}} | Topological Group is Hausdorff iff Identity is Closed | https://proofwiki.org/wiki/Topological_Group_is_Hausdorff_iff_Identity_is_Closed | https://proofwiki.org/wiki/Topological_Group_is_Hausdorff_iff_Identity_is_Closed | [
"Topological Groups",
"Hausdorff Spaces"
] | [
"Definition:Topological Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:T2 Space",
"Definition:Closed Set/Topology"
] | [
"Definition:T2 Space",
"T2 Space is T1",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:T2 Space"
] |
proofwiki-12560 | Topological Group is T1 iff T2 | Let $G$ be a topological group.
Then $G$ is a $T_1$ space {{iff}} $G$ is a $T_2$ (Hausdorff) space. | === Necessary Condition ===
Follows directly from $T_2$ Space is $T_1$ Space.
{{qed|lemma}} | Let $G$ be a [[Definition:Topological Group|topological group]].
Then $G$ is a [[Definition:T1 Space|$T_1$ space]] {{iff}} $G$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | === Necessary Condition ===
Follows directly from [[T2 Space is T1 Space|$T_2$ Space is $T_1$ Space]].
{{qed|lemma}} | Topological Group is T1 iff T2 | https://proofwiki.org/wiki/Topological_Group_is_T1_iff_T2 | https://proofwiki.org/wiki/Topological_Group_is_T1_iff_T2 | [
"Topological Groups",
"T1 Spaces",
"Hausdorff Spaces"
] | [
"Definition:Topological Group",
"Definition:T1 Space",
"Definition:T2 Space"
] | [
"T2 Space is T1"
] |
proofwiki-12561 | Jensen's Inequality (Complex Analysis) | Let $D \subset \C$ be an open set with $0 \in D$.
Let $R > 0$ be such that $\map B {0, R} \subset D$.
Let $f: D \to \C$ be analytic with $\map f 0 \ne 0$.
Let $\cmod {\map f z} \le M$ for $\cmod z \le R$.
Let $0 < r <R$.
Then the number of zeroes of $f$, counted with multiplicity, for $\cmod z \le r$, is at most:
:$\df... | {{ProofWanted}}
{{Namedfor|Johan Jensen|cat = Jensen}} | Let $D \subset \C$ be an [[Definition:Open Set (Complex Analysis)|open set]] with $0 \in D$.
Let $R > 0$ be such that $\map B {0, R} \subset D$.
Let $f: D \to \C$ be [[Definition:Analytic Function|analytic]] with $\map f 0 \ne 0$.
Let $\cmod {\map f z} \le M$ for $\cmod z \le R$.
Let $0 < r <R$.
Then the number o... | {{ProofWanted}}
{{Namedfor|Johan Jensen|cat = Jensen}} | Jensen's Inequality (Complex Analysis) | https://proofwiki.org/wiki/Jensen's_Inequality_(Complex_Analysis) | https://proofwiki.org/wiki/Jensen's_Inequality_(Complex_Analysis) | [
"Complex Analysis",
"Jensen's Inequality"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Analytic Function",
"Definition:Multiplicity (Complex Analysis)"
] | [] |
proofwiki-12562 | Complement of Subset with Property (S) is Closed under Directed Suprema | Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $X$ be a subset of $S$ with property (S).
Then $\relcomp S X$ is closed under directed suprema. | Let $D$ be a directed subset of $S$ such that:
:$D \subseteq \relcomp S X$
{{AimForCont}}
:$\sup D \notin \relcomp S X$
By definition of relative complement:
:$\sup D \in X$
By definition of property (S):
:$\exists y \in D: \forall x \in D: y \preceq x \implies x \in X$
By definition of reflexivity:
:$y \in X$
By defin... | Let $L = \struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$ with [[Definition:Property (S)|property (S)]].
Then $\relcomp S X$ is [[Definition:Closed under Directed Suprema|closed under directed suprema]]. | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that:
:$D \subseteq \relcomp S X$
{{AimForCont}}
:$\sup D \notin \relcomp S X$
By definition of [[Definition:Relative Complement|relative complement]]:
:$\sup D \in X$
By definition of [[Definition:Property (S)|property (S)]]:
:$\exists y \in D:... | Complement of Subset with Property (S) is Closed under Directed Suprema | https://proofwiki.org/wiki/Complement_of_Subset_with_Property_(S)_is_Closed_under_Directed_Suprema | https://proofwiki.org/wiki/Complement_of_Subset_with_Property_(S)_is_Closed_under_Directed_Suprema | [
"Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Property (S)",
"Definition:Closed under Directed Suprema"
] | [
"Definition:Directed Subset",
"Definition:Relative Complement",
"Definition:Property (S)",
"Definition:Reflexivity",
"Definition:Set Intersection",
"Definition:Non-Empty Set",
"Empty Intersection iff Subset of Complement",
"Definition:Contradiction"
] |
proofwiki-12563 | Lifting The Exponent Lemma for Sums | Let $x, y \in \Z$ be integers with $x + y \ne 0$.
Let $n \ge 1$ be an odd natural number.
Let $p$ be an odd prime.
Let:
:$p \divides x + y$
and:
:$p \nmid x y$
where $\divides$ and $\nmid$ denote divisibility and non-divisibility respectively.
Then:
:$\map {\nu_p} {x^n + y^n} = \map {\nu_p} {x + y} + \map {\nu_p} n$
wh... | This follows from the Lifting The Exponent Lemma with $y$ replaced by $-y$.
{{qed}} | Let $x, y \in \Z$ be [[Definition:Integer|integers]] with $x + y \ne 0$.
Let $n \ge 1$ be an [[Definition:Odd Integer|odd]] [[Definition:Natural Number|natural number]].
Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let:
:$p \divides x + y$
and:
:$p \nmid x y$
where $\divides$ and $\nmid$ denote [[Definition:Div... | This follows from the [[Lifting The Exponent Lemma]] with $y$ replaced by $-y$.
{{qed}} | Lifting The Exponent Lemma for Sums | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma_for_Sums | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma_for_Sums | [
"Lifting The Exponent Lemma"
] | [
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Natural Numbers",
"Definition:Odd Prime",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:P-adic Valuation"
] | [
"Lifting The Exponent Lemma"
] |
proofwiki-12564 | Lifting The Exponent Lemma for p=2 | Let $x, y \in \Z$ be distinct odd integers.
Let $n \ge 1$ be a natural number.
Let:
:$4 \divides x - y$
where $\divides$ denotes divisibility.
Then
:$\map {\nu_2} {x^n - y^n} = \map {\nu_2} {x - y} + \map {\nu_2} n$
where $\nu_2$ denotes $2$-adic valuation. | Let $k = \map {\nu_2} n$.
Then $n = 2^k m$ with $2 \nmid m$.
By $p$-adic Valuation of Difference of Powers with Coprime Exponent:
:$\map {\nu_2} {x^n - y^n} = \map {\nu_2} {x^{2^k} - y^{2^k} }$
Note that:
:$x^{2^k} - y^{2^k} = \paren {x - y} \cdot \ds \prod_{i \mathop = 0}^{k - 1} \paren {x^{2^i} + y^{2^i} }$
By Square... | Let $x, y \in \Z$ be [[Definition:Distinct|distinct]] [[Definition:Odd Integer|odd]] [[Definition:Integer|integers]].
Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let:
:$4 \divides x - y$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then
:$\map {\nu_2} {x^n - y^n} = ... | Let $k = \map {\nu_2} n$.
Then $n = 2^k m$ with $2 \nmid m$.
By [[P-adic Valuation of Difference of Powers with Coprime Exponent|$p$-adic Valuation of Difference of Powers with Coprime Exponent]]:
:$\map {\nu_2} {x^n - y^n} = \map {\nu_2} {x^{2^k} - y^{2^k} }$
Note that:
:$x^{2^k} - y^{2^k} = \paren {x - y} \cdot \d... | Lifting The Exponent Lemma for p=2 | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma_for_p=2 | https://proofwiki.org/wiki/Lifting_The_Exponent_Lemma_for_p=2 | [
"Lifting The Exponent Lemma"
] | [
"Definition:Distinct",
"Definition:Odd Integer",
"Definition:Integer",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:P-adic Valuation"
] | [
"P-adic Valuation of Difference of Powers with Coprime Exponent",
"Square Modulo 4",
"P-adic Valuation is Valuation",
"Axiom:Valuation Axioms"
] |
proofwiki-12565 | Classification of Convex Polyhedra whose Faces are Regular Polygons | The convex polyhedra whose faces are all regular polygons are as follows:
:The $5$ Platonic solids
:The uniform prisms and uniform antiprisms, countably infinite in number
:The $13$ Archimedean polyhedra
:The $92$ Johnson polyhedra. | The Platonic solids are the convex polyhedra all of whose faces are congruent and whose vertices are regular.
From Five Platonic Solids, there are $5$ of these.
The uniform prisms are made from two regular polygons of an arbitrary number of sides forming the bases, separated by as many squares as there are sides of the... | The [[Definition:Convex Polyhedron|convex polyhedra]] whose [[Definition:Face of Polyhedron|faces]] are all [[Definition:Regular Polygon|regular polygons]] are as follows:
:The $5$ [[Definition:Platonic Solid|Platonic solids]]
:The [[Definition:Uniform Prism|uniform prisms]] and [[Definition:Uniform Antiprism|uniform ... | The [[Definition:Platonic Solid|Platonic solids]] are the [[Definition:Convex Polyhedron|convex polyhedra]] all of whose [[Definition:Face of Polyhedron|faces]] are [[Definition:Congruent Polygons|congruent]] and whose [[Definition:Vertex of Polygon|vertices]] are [[Definition:Regular Vertex|regular]].
From [[Five Pla... | Classification of Convex Polyhedra whose Faces are Regular Polygons | https://proofwiki.org/wiki/Classification_of_Convex_Polyhedra_whose_Faces_are_Regular_Polygons | https://proofwiki.org/wiki/Classification_of_Convex_Polyhedra_whose_Faces_are_Regular_Polygons | [
"Convex Polyhedra"
] | [
"Definition:Convex Polyhedron",
"Definition:Polyhedron/Face",
"Definition:Polygon/Regular",
"Definition:Platonic Solid",
"Definition:Uniform Prism",
"Definition:Uniform Antiprism",
"Definition:Countably Infinite/Set",
"Definition:Archimedean Polyhedron",
"Definition:Johnson Polyhedron"
] | [
"Definition:Platonic Solid",
"Definition:Convex Polyhedron",
"Definition:Polyhedron/Face",
"Definition:Congruence (Geometry)",
"Definition:Polygon/Vertex",
"Definition:Regular Vertex",
"Five Platonic Solids",
"Definition:Uniform Prism",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"D... |
proofwiki-12566 | Chinese Remainder Theorem/General Result | Let $n_1, n_2, \ldots, n_k$ be positive integers.
Let $b_1, b_2, \ldots, b_k$ be integers such that:
:$\forall i \ne j: \gcd \set {n_i, n_j} \divides b_i - b_j$
Then the system of linear congruences:
{{begin-eqn}}
{{eqn | l = x
| o = \equiv
| r = b_1
| rr = \pmod {n_1}
}}
{{eqn | l = x
| o = \eq... | === Existence ===
We prove this by induction on $k$. | Let $n_1, n_2, \ldots, n_k$ be [[Definition:Positive Integer|positive integers]].
Let $b_1, b_2, \ldots, b_k$ be [[Definition:Integer|integers]] such that:
:$\forall i \ne j: \gcd \set {n_i, n_j} \divides b_i - b_j$
Then the [[Definition:Simultaneous Linear Congruences|system of linear congruences]]:
{{begin-eqn}}... | === Existence ===
We prove this by [[Proof by Mathematical Induction|induction]] on $k$. | Chinese Remainder Theorem/General Result | https://proofwiki.org/wiki/Chinese_Remainder_Theorem/General_Result | https://proofwiki.org/wiki/Chinese_Remainder_Theorem/General_Result | [
"Chinese Remainder Theorem"
] | [
"Definition:Positive/Integer",
"Definition:Integer",
"Definition:Simultaneous Congruences/Linear",
"Definition:Congruence (Number Theory)"
] | [
"Principle of Mathematical Induction"
] |
proofwiki-12567 | Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function | Let $\map F {x, y, z_1, \ldots, z_n}$ be a function in differentiability class $C^2$ {{WRT|Differentiation}} all its variables.
Let $y = \map y x \in C^n\openint a b$ such that:
:$\map y a = A_0, \map {y'} a = A_1, \ldots, \map {y^{\paren {n - 1} } } a = A_{n - 1}$
and:
:$\map y b = B_0, \map {y'} b = B_1, \ldots, \map... | From Condition for Differentiable Functional to have Extremum we have
:$\bigvalueat {\delta J \sqbrk {y h} } {y \mathop = \hat y} = 0$
For the variation to exist it has to satisfy the requirement for a differentiable functional.
Note that the endpoints of $\map y x$ are fixed.
$\map h x$ is not allowed to change values... | Let $\map F {x, y, z_1, \ldots, z_n}$ be a function in [[Definition:Differentiability Class|differentiability class $C^2$]] {{WRT|Differentiation}} all its variables.
Let $y = \map y x \in C^n\openint a b$ such that:
:$\map y a = A_0, \map {y'} a = A_1, \ldots, \map {y^{\paren {n - 1} } } a = A_{n - 1}$
and:
:$\map... | From [[Condition for Differentiable Functional to have Extremum]] we have
:$\bigvalueat {\delta J \sqbrk {y h} } {y \mathop = \hat y} = 0$
For the variation to exist it has to satisfy the requirement for a [[Definition:Differentiable Functional|differentiable functional]].
Note that the endpoints of $\map y x$ are f... | Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_Function/Dependent_on_Nth_Derivative_of_Function | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_Function/Dependent_on_Nth_Derivative_of_Function | [
"Calculus of Variations"
] | [
"Definition:Differentiability Class"
] | [
"Condition for Differentiable Functional to have Extremum",
"Definition:Differentiable Functional",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Definition:Differentiable Functional",
"Generalized Integration by... |
proofwiki-12568 | Logarithm of Infinite Product of Complex Numbers | Let $\sequence {z_n}$ be a sequence of nonzero complex numbers.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|The infinite product $\ds \prod_{n \mathop {{=}} 1}^\infty z_n$ converges to $z \in \C_{\ne 0}$.}}
{{item|(2):|The series $\ds \sum_{n \mathop {{=}} 1}^\infty \log z_n$ converges to $\log z + 2 k \pi i$ for some integ... | === $(1)$ implies $(2)$ ===
Suppose $\ds \prod_{n \mathop = 1}^\infty z_n$ converges to $z$.
By Convergence of Series of Complex Numbers by Real and Imaginary Part, it suffices to show that:
:$\ds \sum_{n \mathop = 1}^\infty \Re \log z_n = \Re \log z$
:$\ds \sum_{n \mathop = 1}^\infty \Im \log z_n = \Im \log z + 2 k \p... | Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence]] of nonzero [[Definition:Complex Number|complex numbers]].
{{TFAE}}
{{begin-itemize}}
{{item|(1):|The [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop {{=}} 1}^\infty z_n$ [[Definition:Convergent Product|converges]] to $z \in \C_{\ne 0}$... | === $(1)$ implies $(2)$ ===
Suppose $\ds \prod_{n \mathop = 1}^\infty z_n$ [[Definition:Convergent Product|converges]] to $z$.
By [[Convergence of Series of Complex Numbers by Real and Imaginary Part]], it suffices to show that:
:$\ds \sum_{n \mathop = 1}^\infty \Re \log z_n = \Re \log z$
:$\ds \sum_{n \mathop = 1}^\... | Logarithm of Infinite Product of Complex Numbers | https://proofwiki.org/wiki/Logarithm_of_Infinite_Product_of_Complex_Numbers | https://proofwiki.org/wiki/Logarithm_of_Infinite_Product_of_Complex_Numbers | [
"Infinite Products"
] | [
"Definition:Sequence",
"Definition:Complex Number",
"Definition:Continued Product/Infinite",
"Definition:Convergent Product",
"Definition:Series",
"Definition:Convergent Series/Number Field",
"Definition:Integer"
] | [
"Definition:Convergent Product",
"Convergence of Series of Complex Numbers by Real and Imaginary Part"
] |
proofwiki-12569 | Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers | Let $\sequence {a_n}$ be a sequence of complex numbers.
Let $\log$ denote the complex logarithm.
{{TFAE|def = Absolute Convergence of Product}} | === 1 iff 2 ===
Follows directly from Equivalence of Definitions of Absolute Convergence of Product.
{{qed}} | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Complex Number|complex numbers]].
Let $\log$ denote the [[Definition:Complex Logarithm|complex logarithm]].
{{TFAE|def = Absolute Convergence of Product}} | === 1 iff 2 ===
Follows directly from [[Equivalence of Definitions of Absolute Convergence of Product]].
{{qed}} | Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Absolute_Convergence_of_Product_of_Complex_Numbers | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Absolute_Convergence_of_Product_of_Complex_Numbers | [
"Complex Analysis",
"Infinite Products"
] | [
"Definition:Sequence",
"Definition:Complex Number",
"Definition:Natural Logarithm/Complex"
] | [
"Equivalence of Definitions of Absolute Convergence of Product"
] |
proofwiki-12570 | Square of Quadratic Gauss Sum | Let $p$ be an odd prime.
Let $a$ be an integer coprime to $p$.
Let $\map g {a, p}$ denote the quadratic Gauss sum of $a$ and $p$.
Then:
:$\map g {a, p}^2 = \paren {\dfrac {-1} p} \cdot p$ | {{proof wanted}}
Category:Number Theory
sbtg7mjhsmx6mt58lwwambdm1bzoxea | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $a$ be an [[Definition:Integer|integer]] [[Definition:Coprime Integers|coprime]] to $p$.
Let $\map g {a, p}$ denote the [[Definition:Quadratic Gauss Sum|quadratic Gauss sum]] of $a$ and $p$.
Then:
:$\map g {a, p}^2 = \paren {\dfrac {-1} p} \cdot p$ | {{proof wanted}}
[[Category:Number Theory]]
sbtg7mjhsmx6mt58lwwambdm1bzoxea | Square of Quadratic Gauss Sum | https://proofwiki.org/wiki/Square_of_Quadratic_Gauss_Sum | https://proofwiki.org/wiki/Square_of_Quadratic_Gauss_Sum | [
"Number Theory"
] | [
"Definition:Odd Prime",
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Quadratic Gauss Sum"
] | [
"Category:Number Theory"
] |
proofwiki-12571 | Absolutely Convergent Product is Convergent | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\mathbb K$ be complete.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then it is convergent. | Let $P_n$ and $Q_n$ denote the $n$th partial products of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ and $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ respectively.
We show that $\sequence {P_n}$ is Cauchy.
We have, for $m > n$:
{{begin-eqn}}
{{eqn | l = \norm {P_m - P_n}
| r = \prod_{k \ma... | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a [[Definition:Valued Field|valued field]].
Let $\mathbb K$ be [[Definition:Complete Metric Space|complete]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be [[Definition:Absolute Convergence of Product|ab... | Let $P_n$ and $Q_n$ denote the $n$th [[Definition:Partial Product|partial products]] of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ and $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ respectively.
We show that $\sequence {P_n}$ is [[Definition:Cauchy Sequence|Cauchy]].
We have, for $m > n$:
{{b... | Absolutely Convergent Product is Convergent | https://proofwiki.org/wiki/Absolutely_Convergent_Product_is_Convergent | https://proofwiki.org/wiki/Absolutely_Convergent_Product_is_Convergent | [
"Infinite Products"
] | [
"Definition:Valued Field",
"Definition:Complete Metric Space",
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product",
"Definition:Convergent Product"
] | [
"Definition:Sequence of Partial Products",
"Definition:Cauchy Sequence",
"Definition:Convergent Product",
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Convergent Product",
"Absolutely Convergent Product Does not Diverge to Zero",
"Definition:Convergent Product",
"Category:... |
proofwiki-12572 | Simplest Variational Problem with Subsidiary Conditions | Let $J \sqbrk y$ and $K \sqbrk y$ be (real) functionals, such that
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
:$\ds K \sqbrk y = \int_a^b \map G {x, y, y'} \rd x = l$
where $l$ is a constant.
Let $y = \map y x$ be an extremum of $F \sqbrk y$, and satisfy boundary conditions:
:$\map y a = A$
:$\map y b = B$
Th... | Let $J \sqbrk y$ be a functional, for which $y = \map y x$ is an extremal with the boundary conditions $\map y a = A$ and $\map y b = B$.
Choose two points, $x_1$ and $x_2$ from the interval $\closedint a b$.
Let $\delta_1 \map y x$ and $\delta_2 \map y x$ be functions, different from zero only in the neighbourhood of ... | Let $J \sqbrk y$ and $K \sqbrk y$ be [[Definition:Real Functional|(real) functionals]], such that
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
:$\ds K \sqbrk y = \int_a^b \map G {x, y, y'} \rd x = l$
where $l$ is a constant.
Let $y = \map y x$ be an extremum of $F \sqbrk y$, and satisfy boundary conditions:... | Let $J \sqbrk y$ be a functional, for which $y = \map y x$ is an extremal with the boundary conditions $\map y a = A$ and $\map y b = B$.
Choose two points, $x_1$ and $x_2$ from the interval $\closedint a b$.
Let $\delta_1 \map y x$ and $\delta_2 \map y x$ be [[Definition:Real Function|functions]], different from zer... | Simplest Variational Problem with Subsidiary Conditions | https://proofwiki.org/wiki/Simplest_Variational_Problem_with_Subsidiary_Conditions | https://proofwiki.org/wiki/Simplest_Variational_Problem_with_Subsidiary_Conditions | [
"Calculus of Variations"
] | [
"Definition:Functional/Real"
] | [
"Definition:Real Function",
"Definition:Neighborhood (Real Analysis)",
"Definition:Variational Derivative",
"Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation"
] |
proofwiki-12573 | Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema | Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $X$ be a closed under directed suprema subset of $S$.
Then $\relcomp S X$ is inaccessible by directed suprema. | Let $D$ be a directed subset of $S$ such that
:$\sup D \in \relcomp S X$
By definition of relative complement:
:$\sup D \notin X$
By definition of closed under directed suprema:
:$D \nsubseteq X$
By Complement of Complement:
:$D \nsubseteq \relcomp S {\relcomp S X}$
Thus by Empty Intersection iff Subset of Relative Com... | Let $L = \struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Closed under Directed Suprema|closed under directed suprema]] [[Definition:Subset|subset]] of $S$.
Then $\relcomp S X$ is [[Definition:Inaccessible by Directed Suprema|inacces... | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$\sup D \in \relcomp S X$
By definition of [[Definition:Relative Complement|relative complement]]:
:$\sup D \notin X$
By definition of [[Definition:Closed under Directed Suprema|closed under directed suprema]]:
:$D \nsubseteq X$
By [[Compl... | Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema | https://proofwiki.org/wiki/Complement_of_Closed_under_Directed_Suprema_Subset_is_Inaccessible_by_Directed_Suprema | https://proofwiki.org/wiki/Complement_of_Closed_under_Directed_Suprema_Subset_is_Inaccessible_by_Directed_Suprema | [
"Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Closed under Directed Suprema",
"Definition:Subset",
"Definition:Inaccessible by Directed Suprema"
] | [
"Definition:Directed Subset",
"Definition:Relative Complement",
"Definition:Closed under Directed Suprema",
"Complement of Complement",
"Empty Intersection iff Subset of Relative Complement"
] |
proofwiki-12574 | Complement of Upper Section is Lower Section | Let $\struct {S, \preceq}$ be an ordered set.
Let $L$ be an upper section.
Then $S \setminus L$ is a lower section. | This follows {{mutatis}} from the proof of Complement of Lower Section is Upper Section.
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $L$ be an [[Definition:Upper Section|upper section]].
Then $S \setminus L$ is a [[Definition:Lower Section|lower section]]. | This follows {{mutatis}} from the proof of [[Complement of Lower Section is Upper Section]].
{{qed}} | Complement of Upper Section is Lower Section | https://proofwiki.org/wiki/Complement_of_Upper_Section_is_Lower_Section | https://proofwiki.org/wiki/Complement_of_Upper_Section_is_Lower_Section | [
"Upper Sections",
"Lower Sections"
] | [
"Definition:Ordered Set",
"Definition:Upper Section",
"Definition:Lower Section"
] | [
"Complement of Lower Section is Upper Section"
] |
proofwiki-12575 | Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set | Let $T = \struct {S, \preceq, \tau}$ be a relational structure with Scott topology
where $\struct {S, \preceq}$ is an up-complete ordered set.
Let $A$ be a subset of $S$.
Then $A$ is closed {{iff}} $A$ is lower and closed under directed suprema. | === Sufficient Condition ===
Assume that
:$A$ is closed.
By definition of closed set:
:$\relcomp S A \in \tau$
By definition of Scott topology:
:$\relcomp S A$ is upper and inaccessible by directed suprema.
By Complement of Complement:
:$\relcomp S {\relcomp S A} = A$
Thus by Complement of Upper Section is Lower Sectio... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Relational Structure with Topology|relational structure with]] [[Definition:Scott Topology|Scott topology]]
where $\struct {S, \preceq}$ is an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $A$ be a [[Definition:Subset|subset]]... | === Sufficient Condition ===
Assume that
:$A$ is [[Definition:Closed Set (Topology)|closed]].
By definition of [[Definition:Closed Set (Topology)|closed set]]:
:$\relcomp S A \in \tau$
By definition of [[Definition:Scott Topology|Scott topology]]:
:$\relcomp S A$ is [[Definition:Upper Section|upper]] and [[Definitio... | Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set | https://proofwiki.org/wiki/Closed_Set_iff_Lower_and_Closed_under_Directed_Suprema_in_Scott_Topological_Ordered_Set | https://proofwiki.org/wiki/Closed_Set_iff_Lower_and_Closed_under_Directed_Suprema_in_Scott_Topological_Ordered_Set | [
"Topological Order Theory",
"Closed Sets"
] | [
"Definition:Relational Structure with Topology",
"Definition:Scott Topology",
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Closed Set/Topology",
"Definition:Lower Section",
"Definition:Closed under Directed Suprema"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Scott Topology",
"Definition:Upper Section",
"Definition:Inaccessible by Directed Suprema",
"Complement of Complement",
"Complement of Upper Section is Lower Section",
"Definition:Lower Section",
"Complement of Inaccessi... |
proofwiki-12576 | Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema | Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $X$ be an inaccessible by directed suprema subset of $S$.
Then $\relcomp S X$ is closed under directed suprema. | Let $D$ be a directed subset of $S$ such that
:$D \subseteq \relcomp S X$
By Empty Intersection iff Subset of Relative Complement:
:$D \cap X = \O$
By definition of inaccessible by directed suprema:
:$\sup D \notin X$
Thus by definition of relative complement:
:$\sup D \in \relcomp S X$
{{qed}} | Let $L = \struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $X$ be an [[Definition:Inaccessible by Directed Suprema|inaccessible by directed suprema]] [[Definition:Subset|subset]] of $S$.
Then $\relcomp S X$ is [[Definition:Closed under Directed Suprema|clo... | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$D \subseteq \relcomp S X$
By [[Empty Intersection iff Subset of Relative Complement]]:
:$D \cap X = \O$
By definition of [[Definition:Inaccessible by Directed Suprema|inaccessible by directed suprema]]:
:$\sup D \notin X$
Thus by definiti... | Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema | https://proofwiki.org/wiki/Complement_of_Inaccessible_by_Directed_Suprema_Subset_is_Closed_under_Directed_Suprema | https://proofwiki.org/wiki/Complement_of_Inaccessible_by_Directed_Suprema_Subset_is_Closed_under_Directed_Suprema | [
"Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Inaccessible by Directed Suprema",
"Definition:Subset",
"Definition:Closed under Directed Suprema"
] | [
"Definition:Directed Subset",
"Empty Intersection iff Subset of Relative Complement",
"Definition:Inaccessible by Directed Suprema",
"Definition:Relative Complement"
] |
proofwiki-12577 | Convergence of Complex Sequence in Polar Form | Let $z \ne 0$ be a complex number with modulus $r$ and argument $\theta$.
Let $\sequence {z_n}$ be a sequence of nonzero complex numbers.
Let $r_n$ be the modulus of $z_n$ and $\theta_n$ be an argument of $z_n$.
Then $z_n$ converges to $z$ {{iff}} the following hold:
:$(1): \quad r_n$ converges to $r$
:$(2): \quad$ The... | Suppose $r_n \to r$ and $\theta_n + 2 k_n \pi \to \theta$.
We have, by Complex Modulus of Difference of Complex Numbers:
{{begin-eqn}}
{{eqn | l = \cmod {z_n - z}^2
| r = r_n^2 + r^2 - 2 r r_n \, \map \cos {\theta_n + 2 k_n \pi - \theta}
| c =
}}
{{end-eqn}}
Because Cosine Function is Continuous:
:$\map ... | Let $z \ne 0$ be a [[Definition:Complex Number|complex number]] with [[Definition:Complex Modulus|modulus]] $r$ and [[Definition:Argument of Complex Number|argument]] $\theta$.
Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence]] of nonzero [[Definition:Complex Number|complex numbers]].
Let $r_n$ be the [[Defi... | Suppose $r_n \to r$ and $\theta_n + 2 k_n \pi \to \theta$.
We have, by [[Complex Modulus of Difference of Complex Numbers]]:
{{begin-eqn}}
{{eqn | l = \cmod {z_n - z}^2
| r = r_n^2 + r^2 - 2 r r_n \, \map \cos {\theta_n + 2 k_n \pi - \theta}
| c =
}}
{{end-eqn}}
Because [[Cosine Function is Continuous... | Convergence of Complex Sequence in Polar Form | https://proofwiki.org/wiki/Convergence_of_Complex_Sequence_in_Polar_Form | https://proofwiki.org/wiki/Convergence_of_Complex_Sequence_in_Polar_Form | [
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Argument of Complex Number",
"Definition:Sequence",
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Sequence"... | [
"Complex Modulus of Difference of Complex Numbers",
"Cosine Function is Continuous",
"Modulus of Limit",
"Complex Modulus of Difference of Complex Numbers",
"Convergence of Cosine of Sequence",
"Definition:Sequence",
"Definition:Integer",
"Definition:Convergent Sequence/Real Numbers",
"Category:Comp... |
proofwiki-12578 | Complex Modulus of Sum of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively.
Then:
:$\cmod {z_1 + z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 + 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$ | We have:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}^2
| r = \paren {z_1 + z_2} \paren {\overline {z_1} + \overline {z_2} }
| c = Modulus in Terms of Conjugate and Sum of Complex Conjugates
}}
{{eqn | r = z_1 \overline {z_1} + z_2 \overline {z_2} + z_1\overline {z_2} + \overline {z_1} z_2
| c =
}}
{{... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\theta_1$ and $\theta_2$ be [[Definition:Argument of Complex Number|arguments]] of $z_1$ and $z_2$, respectively.
Then:
:$\cmod {z_1 + z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 + 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$ | We have:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}^2
| r = \paren {z_1 + z_2} \paren {\overline {z_1} + \overline {z_2} }
| c = [[Modulus in Terms of Conjugate]] and [[Sum of Complex Conjugates]]
}}
{{eqn | r = z_1 \overline {z_1} + z_2 \overline {z_2} + z_1\overline {z_2} + \overline {z_1} z_2
| c ... | Complex Modulus of Sum of Complex Numbers/Proof 1 | https://proofwiki.org/wiki/Complex_Modulus_of_Sum_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Sum_of_Complex_Numbers/Proof_1 | [
"Complex Modulus",
"Complex Modulus of Sum of Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Argument of Complex Number"
] | [
"Modulus in Terms of Conjugate",
"Sum of Complex Conjugates",
"Modulus in Terms of Conjugate",
"Sum of Complex Number with Conjugate",
"Product of Complex Numbers in Polar Form",
"Argument of Complex Conjugate equals Negative of Argument"
] |
proofwiki-12579 | Complex Modulus of Sum of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively.
Then:
:$\cmod {z_1 + z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 + 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$ | {{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}^2
| r = \paren {\cmod {z_1} \cos \theta_1 + \cmod {z_2} \cos \theta_2}^2 + \paren {\cmod {z_1} \sin \theta_1 + \cmod {z_2} \sin \theta_2}^2
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = 2 \cmod {z_1} \cmod {z_2} \cos \theta_1 \cos \theta_2 + \cmod {z_1}^2 \cos^2 ... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\theta_1$ and $\theta_2$ be [[Definition:Argument of Complex Number|arguments]] of $z_1$ and $z_2$, respectively.
Then:
:$\cmod {z_1 + z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 + 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$ | {{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}^2
| r = \paren {\cmod {z_1} \cos \theta_1 + \cmod {z_2} \cos \theta_2}^2 + \paren {\cmod {z_1} \sin \theta_1 + \cmod {z_2} \sin \theta_2}^2
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = 2 \cmod {z_1} \cmod {z_2} \cos \theta_1 \cos \theta_2 + \cmod {z_1}^2 \cos^2 ... | Complex Modulus of Sum of Complex Numbers/Proof 2 | https://proofwiki.org/wiki/Complex_Modulus_of_Sum_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Sum_of_Complex_Numbers/Proof_2 | [
"Complex Modulus",
"Complex Modulus of Sum of Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Argument of Complex Number"
] | [
"Sum of Squares of Sine and Cosine",
"Cosine of Difference"
] |
proofwiki-12580 | Complex Modulus of Difference of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively.
Then:
:$\cmod {z_1 - z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \map \cos {\theta_1 - \theta_2}$ | By Complex Argument of Additive Inverse, $\theta_2 + \pi$ is an argument of $-z_2$.
We have:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 - z_2}^2
| r = \cmod {z_1}^2 + \cmod {-z_2}^2 + 2 \cmod {z_1} \cmod {-z_2} \map \cos {\theta_1 - \theta_2 - \pi}
| c = Complex Modulus of Sum of Complex Numbers
}}
{{eqn | r = \c... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\theta_1$ and $\theta_2$ be [[Definition:Argument of Complex Number|arguments]] of $z_1$ and $z_2$, respectively.
Then:
:$\cmod {z_1 - z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \map \cos {\theta_1 - \theta_2}$ | By [[Complex Argument of Additive Inverse]], $\theta_2 + \pi$ is an [[Definition:Argument of Complex Number|argument]] of $-z_2$.
We have:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 - z_2}^2
| r = \cmod {z_1}^2 + \cmod {-z_2}^2 + 2 \cmod {z_1} \cmod {-z_2} \map \cos {\theta_1 - \theta_2 - \pi}
| c = [[Complex Mo... | Complex Modulus of Difference of Complex Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Difference_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Difference_of_Complex_Numbers | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Argument of Complex Number"
] | [
"Complex Argument of Additive Inverse",
"Definition:Argument of Complex Number",
"Complex Modulus of Sum of Complex Numbers",
"Complex Modulus of Additive Inverse"
] |
proofwiki-12581 | Uniform Absolute Convergence of Infinite Product of Complex Functions | Let $X$ be a compact metric space.
Let $\sequence {f_n}$ be a sequence of continuous functions $X \to \C$.
Let $\ds \sum_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$.
Then:
:$\map f x = \ds \prod_{n \mathop = 1}^\infty \paren {1 + \map {f_n} x}$ converges uniformly absolutely on $X$
:$f$ is continuo... | === Lemma ===
{{:Uniform Absolute Convergence of Infinite Product of Complex Functions/Lemma}}{{qed|lemma}}
Because $\ds \sum_{n \mathop = 1}^\infty f_n$ converges uniformly, there exists $n_0 > 0$ such that $\size {\map {f_n} x} < \dfrac 1 2$ for $n \ge n_0$ and $x \in X$.
Then:
:$\map \Re {1 + \map {f_n} x} > 0$
By B... | Let $X$ be a [[Definition:Compact Metric Space|compact metric space]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Continuous Mapping (Metric Spaces)|continuous functions]] $X \to \C$.
Let $\ds \sum_{n \mathop = 1}^\infty f_n$ [[Definition:Uniform Absolute Convergence|converge uniforml... | === [[Uniform Absolute Convergence of Infinite Product of Complex Functions/Lemma|Lemma]] ===
{{:Uniform Absolute Convergence of Infinite Product of Complex Functions/Lemma}}{{qed|lemma}}
Because $\ds \sum_{n \mathop = 1}^\infty f_n$ [[Definition:Uniform Convergence|converges uniformly]], there exists $n_0 > 0$ such ... | Uniform Absolute Convergence of Infinite Product of Complex Functions | https://proofwiki.org/wiki/Uniform_Absolute_Convergence_of_Infinite_Product_of_Complex_Functions | https://proofwiki.org/wiki/Uniform_Absolute_Convergence_of_Infinite_Product_of_Complex_Functions | [
"Infinite Products"
] | [
"Definition:Compact Space/Metric Space",
"Definition:Sequence",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Uniform Absolute Convergence",
"Definition:Uniform Absolute Convergence of Product",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Complex Exponential is Uniformly Continuous on Half-Planes/Corollary",
"Definition:Uniform Convergence",
"Bounds for Complex Logarithm",
"Comparison Test for Uniformly Convergent Series",
"Definition:Uniform Absolute Convergence",
"Uniform Limit Theorem",
"Definition:Continuous Mapping",
"Continuous ... |
proofwiki-12582 | Simplest Variational Problem with Subsidiary Conditions for Curve on Surface | Let $J \sqbrk {y, z}$ be a (real) functional of the form:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, z, y', z'} \rd x$
Let there exist admissible curves $y, z$ lying on the surface:
:$\map g {x, y, z} = 0$
which satisfy boundary conditions:
:$\map y a = A_1, \map y b = B_1$
:$\map z a = A_2, \map z b = B_2$
Let $J \sqbr... | Let $J \sqbrk y$ be a functional, for which the curve $y = \map y x, z = \map z x$ is an extremal with the boundary conditions $\map y a = A, \map y b = B$ as well as $\map g {x, y, z} = 0$.
Choose an arbitrary point $x_1$ from the interval $\closedint a b$.
Let $\delta \map y x$ and $\delta \map z x$ be functions, dif... | Let $J \sqbrk {y, z}$ be a [[Definition:Real Functional|(real) functional]] of the form:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, z, y', z'} \rd x$
Let there exist admissible curves $y, z$ lying on the surface:
:$\map g {x, y, z} = 0$
which satisfy boundary conditions:
:$\map y a = A_1, \map y b = B_1$
:$\map z a... | Let $J \sqbrk y$ be a functional, for which the curve $y = \map y x, z = \map z x$ is an extremal with the boundary conditions $\map y a = A, \map y b = B$ as well as $\map g {x, y, z} = 0$.
Choose an arbitrary point $x_1$ from the interval $\closedint a b$.
Let $\delta \map y x$ and $\delta \map z x$ be [[Definition... | Simplest Variational Problem with Subsidiary Conditions for Curve on Surface | https://proofwiki.org/wiki/Simplest_Variational_Problem_with_Subsidiary_Conditions_for_Curve_on_Surface | https://proofwiki.org/wiki/Simplest_Variational_Problem_with_Subsidiary_Conditions_for_Curve_on_Surface | [
"Calculus of Variations"
] | [
"Definition:Functional/Real"
] | [
"Definition:Real Function",
"Definition:Neighborhood (Real Analysis)",
"Definition:Variational Derivative",
"Sum of Infinite Geometric Sequence",
"Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation",
"Vanishing First Variational Derivative implies Euler's Equation for ... |
proofwiki-12583 | Empty Intersection iff Subset of Relative Complement | Let $S$ be a set.
Let $A, B$ be subset of $S$.
Then $A \cap B = \O \iff A \subseteq \relcomp S B$ | :$A \cap B = \O$
{{iff}}
:$\forall x \in S: x \notin A \cap B$ by Empty Set as Subset
{{iff}}
:$\forall x \in S: \neg \paren {x \in A \land x \in B}$ by definition of intersection
{{iff}}
:$\forall x \in S: x \notin A \lor x \notin B$ by De Morgan's Laws (Logic)/Disjunction of Negations
{{iff}}
:$\forall x \in S: x \in... | Let $S$ be a [[Definition:Set|set]].
Let $A, B$ be [[Definition:Subset|subset]] of $S$.
Then $A \cap B = \O \iff A \subseteq \relcomp S B$ | :$A \cap B = \O$
{{iff}}
:$\forall x \in S: x \notin A \cap B$ by [[Empty Set as Subset]]
{{iff}}
:$\forall x \in S: \neg \paren {x \in A \land x \in B}$ by definition of [[Definition:Set Intersection|intersection]]
{{iff}}
:$\forall x \in S: x \notin A \lor x \notin B$ by [[De Morgan's Laws (Logic)/Disjunction of Nega... | Empty Intersection iff Subset of Relative Complement | https://proofwiki.org/wiki/Empty_Intersection_iff_Subset_of_Relative_Complement | https://proofwiki.org/wiki/Empty_Intersection_iff_Subset_of_Relative_Complement | [
"Relative Complement",
"Set Intersection",
"Empty Set",
"Subsets"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Empty Set as Subset",
"Definition:Set Intersection",
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Rule of Material Implication",
"Definition:Relative Complement",
"Subset in Subsets"
] |
proofwiki-12584 | Subset in Subsets | Let $S, B$ be sets.
Let $A$ be subset of $S$.
Then:
:$A \subseteq B \iff \forall x \in S: x \in A \implies x \in B$ | === Sufficient Condition ===
Follows by definition of subset. | Let $S, B$ be [[Definition:Set|sets]].
Let $A$ be [[Definition:Subset|subset]] of $S$.
Then:
:$A \subseteq B \iff \forall x \in S: x \in A \implies x \in B$ | === Sufficient Condition ===
Follows by definition of [[Definition:Subset|subset]]. | Subset in Subsets | https://proofwiki.org/wiki/Subset_in_Subsets | https://proofwiki.org/wiki/Subset_in_Subsets | [
"Subsets"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Subset",
"Definition:Subset"
] |
proofwiki-12585 | Empty Set as Subset | Let $S$ be a set.
Let $A$ be a subset of $S$.
Then:
:$A = \O \iff \forall x \in S: x \notin A$ | Sufficient condition follows by definition of empty set.
For necessary condition assume that:
:$\forall x \in S: x \notin A$
Let $x$ be arbitrary.
{{AimForCont}} that:
:$x \in A$
By definition of subset:
:$x \in S$
By assumption:
:$x \notin A$
Thus this contradicts:
:$x \in A$
{{qed}}
Category:Empty Set
Category:Subset... | Let $S$ be a [[Definition:Set|set]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$A = \O \iff \forall x \in S: x \notin A$ | Sufficient condition follows by definition of [[Definition:Empty Set|empty set]].
For necessary condition assume that:
:$\forall x \in S: x \notin A$
Let $x$ be arbitrary.
{{AimForCont}} that:
:$x \in A$
By definition of [[Definition:Subset|subset]]:
:$x \in S$
By assumption:
:$x \notin A$
Thus this [[Definition:... | Empty Set as Subset | https://proofwiki.org/wiki/Empty_Set_as_Subset | https://proofwiki.org/wiki/Empty_Set_as_Subset | [
"Empty Set",
"Subsets"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Empty Set",
"Definition:Subset",
"Definition:Contradiction",
"Category:Empty Set",
"Category:Subsets"
] |
proofwiki-12586 | Pythagorean Triangles whose Areas are Repdigit Numbers | The following Pythagorean triangles have areas consisting of repdigit numbers: | From Pythagorean Triangle whose Area is Half Perimeter, the area of the $3-4-5$ triangle is $6$, which is trivially repdigit.
The next Pythagorean triangles in area are:
:the $6-8-10$ triangle, which has area $\dfrac {6 \times 8} 2 = 24$
:the $5-2-13$ triangle, which has area $\dfrac {5 \times 12} 2 = 30$
So there are ... | The following [[Definition:Pythagorean Triangle|Pythagorean triangles]] have [[Definition:Area|areas]] consisting of [[Definition:Repdigit Number|repdigit numbers]]: | From [[Pythagorean Triangle whose Area is Half Perimeter]], the [[Definition:Area|area]] of the [[Pythagorean Triangle/Examples/3-4-5|$3-4-5$ triangle]] is $6$, which is trivially [[Definition:Repdigit Number|repdigit]].
The next [[Definition:Pythagorean Triangle|Pythagorean triangles]] in [[Definition:Area|area]] are... | Pythagorean Triangles whose Areas are Repdigit Numbers | https://proofwiki.org/wiki/Pythagorean_Triangles_whose_Areas_are_Repdigit_Numbers | https://proofwiki.org/wiki/Pythagorean_Triangles_whose_Areas_are_Repdigit_Numbers | [
"Pythagorean Triangles",
"693",
"1924",
"2045",
"666,666"
] | [
"Definition:Pythagorean Triangle",
"Definition:Area",
"Definition:Repdigit Number",
"Definition:Area",
"Definition:Area"
] | [
"Pythagorean Triangle whose Area is Half Perimeter",
"Definition:Area",
"Pythagorean Triangle/Examples/3-4-5",
"Definition:Repdigit Number",
"Definition:Pythagorean Triangle",
"Definition:Area",
"Pythagorean Triangle/Examples/6-8-10",
"Definition:Area",
"Pythagorean Triangle/Examples/5-12-13",
"De... |
proofwiki-12587 | Exponential of Series Equals Infinite Product | Let $\sequence {z_n}$ be a sequence of complex numbers.
Suppose $\ds \sum_{n \mathop = 1}^\infty z_n$ converges to $z \in \C$.
Then $\ds \prod_{n \mathop = 1}^\infty \map \exp {z_n}$ converges to $\exp z$. | Let $S_n$ be the $n$th partial sum of $\ds \sum_{n \mathop = 1}^\infty z_n$.
Let $P_n$ be the $n$th partial product of $\ds \prod_{n \mathop = 1}^\infty \map \exp {z_n}$.
By Exponential of Sum, $\map \exp {S_n} = P_n$ for all $n \in \N$.
By Exponential Function is Continuous, $\ds \lim_{n \mathop \to \infty} \map \exp ... | Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Complex Number|complex numbers]].
Suppose $\ds \sum_{n \mathop = 1}^\infty z_n$ [[Definition:Convergent Series of Numbers|converges]] to $z \in \C$.
Then $\ds \prod_{n \mathop = 1}^\infty \map \exp {z_n}$ [[Definition:Convergent Product|conv... | Let $S_n$ be the $n$th [[Definition:Partial Sum|partial sum]] of $\ds \sum_{n \mathop = 1}^\infty z_n$.
Let $P_n$ be the $n$th [[Definition:Partial Product|partial product]] of $\ds \prod_{n \mathop = 1}^\infty \map \exp {z_n}$.
By [[Exponential of Sum]], $\map \exp {S_n} = P_n$ for all $n \in \N$.
By [[Exponential ... | Exponential of Series Equals Infinite Product | https://proofwiki.org/wiki/Exponential_of_Series_Equals_Infinite_Product | https://proofwiki.org/wiki/Exponential_of_Series_Equals_Infinite_Product | [
"Infinite Products"
] | [
"Definition:Sequence",
"Definition:Complex Number",
"Definition:Convergent Series/Number Field",
"Definition:Convergent Product"
] | [
"Definition:Series/Sequence of Partial Sums",
"Definition:Sequence of Partial Products",
"Exponential of Sum",
"Exponential Function is Continuous",
"Image of Complex Exponential Function"
] |
proofwiki-12588 | Convergence of Series of Complex Numbers by Real and Imaginary Part | Let $\sequence {z_n}$ be a sequence of complex numbers.
Then:
:the series $\ds \sum_{n \mathop = 1}^\infty z_n$ converges to $Z \in \C$
{{iff}}:
:the series:
::$\ds \sum_{n \mathop = 1}^\infty \map \Re {z_n}$
:and:
::$\ds \sum_{n \mathop = 1}^\infty \map \Im {z_n}$
:converge to $\map \Re Z$ and $\map \Im Z$ respectivel... | Let:
:the $n$th partial sum of $\sequence {z_n}$ be denoted $Z_n$
:the $n$th partial sum of $\sequence {\map \Re {z_n} }$ be denoted $U_n$
:the $n$th partial sum of $\sequence {\map \Im {z_n} }$ be denoted $V_n$
Then:
:$Z_n = U_n + i V_n$
Let:
:$\ds \lim_{n \mathop \to \infty} U_n = U$
:$\ds \lim_{n \mathop \to \infty}... | Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Complex Number|complex numbers]].
Then:
:the [[Definition:Complex Series|series]] $\ds \sum_{n \mathop = 1}^\infty z_n$ [[Definition:Convergent Series of Numbers|converges]] to $Z \in \C$
{{iff}}:
:the [[Definition:Series of Numbers|series]]:... | Let:
:the $n$th [[Definition:Partial Sum|partial sum]] of $\sequence {z_n}$ be denoted $Z_n$
:the $n$th [[Definition:Partial Sum|partial sum]] of $\sequence {\map \Re {z_n} }$ be denoted $U_n$
:the $n$th [[Definition:Partial Sum|partial sum]] of $\sequence {\map \Im {z_n} }$ be denoted $V_n$
Then:
:$Z_n = U_n + i V_n$... | Convergence of Series of Complex Numbers by Real and Imaginary Part | https://proofwiki.org/wiki/Convergence_of_Series_of_Complex_Numbers_by_Real_and_Imaginary_Part | https://proofwiki.org/wiki/Convergence_of_Series_of_Complex_Numbers_by_Real_and_Imaginary_Part | [
"Convergent Complex Sequences",
"Complex Series"
] | [
"Definition:Sequence",
"Definition:Complex Number",
"Definition:Series/Complex",
"Definition:Convergent Series/Number Field",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field"
] | [
"Definition:Series/Sequence of Partial Sums",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series/Sequence of Partial Sums",
"Definition:Convergent Sequence/Complex Numbers"
] |
proofwiki-12589 | Lower Closure of Element is Closed under Directed Suprema | Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $x \in S$.
Then $x^\preceq$ is closed under directed suprema,
where $x^\preceq$ denotes the lower closure of $x$. | Let $D$ be a directed subset of $S$ such that
:$D \subseteq x^\preceq$
By Lower Closure of Element is Ideal:
:$x^\preceq$ is directed.
By definition of up-complete:
:$D$ and $x^\preceq$ admit suprema.
By Supremum of Subset:
:$\sup D \preceq \map \sup {x^\preceq}$
By Supremum of Lower Closure of Element:
:$\sup D \prece... | Let $L = \struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then $x^\preceq$ is [[Definition:Closed under Directed Suprema|closed under directed suprema]],
where $x^\preceq$ denotes the [[Definition:Lower Closure of Element|lower closure]] of $x... | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$D \subseteq x^\preceq$
By [[Lower Closure of Element is Ideal]]:
:$x^\preceq$ is [[Definition:Directed Subset|directed]].
By definition of [[Definition:Up-Complete|up-complete]]:
:$D$ and $x^\preceq$ admit [[Definition:Supremum of Set|supr... | Lower Closure of Element is Closed under Directed Suprema | https://proofwiki.org/wiki/Lower_Closure_of_Element_is_Closed_under_Directed_Suprema | https://proofwiki.org/wiki/Lower_Closure_of_Element_is_Closed_under_Directed_Suprema | [
"Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Closed under Directed Suprema",
"Definition:Lower Closure/Element"
] | [
"Definition:Directed Subset",
"Lower Closure of Element is Ideal",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Supremum of Subset",
"Supremum of Lower Closure of Element",
"Definition:Lower Closure/Element"
] |
proofwiki-12590 | Pythagorean Triangle whose Hypotenuse and Leg differ by 1 | Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T$.
Then:
: the hypotenuse of $P$ is $1$ greater than one of its legs
{{iff}}:
: the generator for $T$ is of the form $G = \tuple {n, n + 1}$ where $n \in \Z_{> 0}$ is a (strictly) positive integer. | We have from Solutions of Pythagorean Equation that the set of all primitive Pythagorean triples is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where $m, n \in \Z$ such that:
:$m, n \in \Z_{>0}$ are (strictly) positive integers
:$m \perp n$, that is, $m$ and $n$ are coprime
:$m$ and $n$ are of opposite parity... | Let $P$ be a [[Definition:Pythagorean Triangle|Pythagorean triangle]] whose [[Definition:Side of Polygon|sides]] correspond to the [[Definition:Pythagorean Triple|Pythagorean triple]] $T$.
Then:
: the [[Definition:Hypotenuse|hypotenuse]] of $P$ is $1$ greater than one of its [[Definition:Leg of Right Triangle|legs]]
{... | We have from [[Solutions of Pythagorean Equation]] that the [[Definition:Set|set]] of all [[Definition:Primitive Pythagorean Triple|primitive Pythagorean triples]] is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where $m, n \in \Z$ such that:
:$m, n \in \Z_{>0}$ are [[Definition:Strictly Positive Integer|(stri... | Pythagorean Triangle whose Hypotenuse and Leg differ by 1 | https://proofwiki.org/wiki/Pythagorean_Triangle_whose_Hypotenuse_and_Leg_differ_by_1 | https://proofwiki.org/wiki/Pythagorean_Triangle_whose_Hypotenuse_and_Leg_differ_by_1 | [
"Pythagorean Triangles"
] | [
"Definition:Pythagorean Triangle",
"Definition:Polygon/Side",
"Definition:Pythagorean Triple",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Generator for Primitive Pythagorean Triple",
"Definition:Strictly Positive/Integer"
] | [
"Solutions of Pythagorean Equation",
"Definition:Set",
"Definition:Pythagorean Triple/Primitive",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Parity of Integer",
"Solutions of Pythagorean Equation",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/In... |
proofwiki-12591 | Generator for Almost Isosceles Pythagorean Triangle | Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T = \tuple {a, b, c}$.
Let the generator for $T$ be $\tuple {m, n}$.
Then:
:$P$ is almost isosceles
{{iff}}
:$\tuple {2 m + n, m}$ is the generator for the Pythagorean triple $T'$ of another almost isosceles Pythagorean triangle $P'$. | By definition of almost isosceles:
:$\size {a - b} = 1$
First note that, from Consecutive Integers are Coprime, an almost isosceles Pythagorean triangle is a primitive Pythagorean triangle.
Hence $T$ and $T'$ are primitive Pythagorean triples.
Thus it is established that by Solutions of Pythagorean Equation, both $P$ a... | Let $P$ be a [[Definition:Pythagorean Triangle|Pythagorean triangle]] whose [[Definition:Side of Polygon|sides]] correspond to the [[Definition:Pythagorean Triple|Pythagorean triple]] $T = \tuple {a, b, c}$.
Let the [[Definition:Generator for Pythagorean Triple|generator]] for $T$ be $\tuple {m, n}$.
Then:
:$P$ is [... | By definition of [[Definition:Almost Isosceles Pythagorean Triangle|almost isosceles]]:
:$\size {a - b} = 1$
First note that, from [[Consecutive Integers are Coprime]], an [[Definition:Almost Isosceles Pythagorean Triangle|almost isosceles Pythagorean triangle]] is a [[Definition:Primitive Pythagorean Triangle|primit... | Generator for Almost Isosceles Pythagorean Triangle | https://proofwiki.org/wiki/Generator_for_Almost_Isosceles_Pythagorean_Triangle | https://proofwiki.org/wiki/Generator_for_Almost_Isosceles_Pythagorean_Triangle | [
"Pythagorean Triangles"
] | [
"Definition:Pythagorean Triangle",
"Definition:Polygon/Side",
"Definition:Pythagorean Triple",
"Definition:Generator for Pythagorean Triple",
"Definition:Almost Isosceles Pythagorean Triangle",
"Definition:Generator for Pythagorean Triple",
"Definition:Pythagorean Triple",
"Definition:Almost Isosceles... | [
"Definition:Almost Isosceles Pythagorean Triangle",
"Consecutive Integers are Coprime",
"Definition:Almost Isosceles Pythagorean Triangle",
"Definition:Primitive Pythagorean Triangle",
"Definition:Pythagorean Triple/Primitive",
"Solutions of Pythagorean Equation",
"Definition:Parity of Integer",
"Defi... |
proofwiki-12592 | General Variation of Integral Functional/Dependent on N Functions | Let $J$ be a (real) functional of the form:
$\ds J \sqbrk {\ldots y_i \ldots} = \int_{x_0}^{x_1} \map F {x, \ldots y_i \ldots, \ldots y_i' \ldots} \rd x, i = \openint 1 n$
Then:
$\ds \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \frac \d {\d x} F_{y_i'} } \map {h_i} x + \intlimits {\sum_{i \matho... | Let $y = \map y x, y = \map {y^*} x$ be smooth real functions.
Let $\map h x = \map {y^*} x - \map y x$
Let the endpoints of the curve $y_i = \map {y_i} x, i = \openint 1 n$ be:
:$P_0 = \tuple {x_0, \dotsc, y_i^0 \dotsc}$
:$P_1 = \tuple {x_1, \dotsc, y_i^1 \dotsc}$
Let the endpoints of the curve $y_i = y_i^* = \map {y_... | Let $J$ be a [[Definition:Real Functional|(real) functional]] of the form:
$\ds J \sqbrk {\ldots y_i \ldots} = \int_{x_0}^{x_1} \map F {x, \ldots y_i \ldots, \ldots y_i' \ldots} \rd x, i = \openint 1 n$
Then:
$\ds \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \frac \d {\d x} F_{y_i'} } \map {h... | Let $y = \map y x, y = \map {y^*} x$ be [[Definition:Smooth Real Function|smooth real functions]].
Let $\map h x = \map {y^*} x - \map y x$
Let the [[Definition:Endpoints of Directed Smooth Curve|endpoints of the curve]] $y_i = \map {y_i} x, i = \openint 1 n$ be:
:$P_0 = \tuple {x_0, \dotsc, y_i^0 \dotsc}$
:$P_1 = \... | General Variation of Integral Functional/Dependent on N Functions | https://proofwiki.org/wiki/General_Variation_of_Integral_Functional/Dependent_on_N_Functions | https://proofwiki.org/wiki/General_Variation_of_Integral_Functional/Dependent_on_N_Functions | [
"Calculus of Variations"
] | [
"Definition:Functional/Real"
] | [
"Definition:Smooth Real Function",
"Definition:Directed Smooth Curve/Endpoints",
"Definition:Directed Smooth Curve/Endpoints",
"Definition:Real Interval",
"Equation of Straight Line in Plane",
"Definition:Tangent Line",
"Definition:Differential of Mapping/Functional",
"Definition:Increment/Functional"... |
proofwiki-12593 | Kusmin-Landau Inequality | Let $I$ be the half-open interval $\hointl a b$.
Let $f: I \to R$ be continuously differentiable.
Let $f'$ be monotonic.
Let $\norm {f'} \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\norm {\, \cdot \,}$ denotes the distance to nearest integer.
Then:
:$\ds \sum_{n \mathop \in I} e^{2 \pi i \map f n} = \map... | {{ProofWanted}}
{{Namedfor|Rodion Osievich Kuzmin|name2 = Edmund Georg Hermann Landau|cat = Kuzmin|cat2 = Landau}} | Let $I$ be the [[Definition:Half-Open Real Interval|half-open interval]] $\hointl a b$.
Let $f: I \to R$ be [[Definition:Continuously Differentiable|continuously differentiable]].
Let $f'$ be [[Definition:Monotone Real Function|monotonic]].
Let $\norm {f'} \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\... | {{ProofWanted}}
{{Namedfor|Rodion Osievich Kuzmin|name2 = Edmund Georg Hermann Landau|cat = Kuzmin|cat2 = Landau}} | Kusmin-Landau Inequality | https://proofwiki.org/wiki/Kusmin-Landau_Inequality | https://proofwiki.org/wiki/Kusmin-Landau_Inequality | [
"Exponential Sums"
] | [
"Definition:Real Interval/Half-Open",
"Definition:Continuously Differentiable",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Distance to Nearest Integer Function",
"Definition:Big-O Notation"
] | [] |
proofwiki-12594 | Closure of Singleton is Lower Closure of Element in Scott Topological Lattice | Let $T = \struct {S, \preceq, \tau}$ be a up-complete topological lattice with the Scott topology.
Let $x \in S$.
Then:
:$\set x^- = x^\preceq$
where
:$\set x^-$ denotes the topological closure of $\set x$
:$x^\preceq$ denotes the lower closure of $x$. | By Lower Closure of Element is Closed under Directed Suprema:
:$x^\preceq$ is closed under directed suprema.
By Lower Closure of Singleton:
:$\set x^\preceq = x^\preceq$
By Lower Closure is Lower Section:
:$x^\preceq$ is a lower section.
By Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ord... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Up-Complete|up-complete]] [[Definition:Topological Lattice|topological lattice]] with the [[Definition:Scott Topology|Scott topology]].
Let $x \in S$.
Then:
:$\set x^- = x^\preceq$
where
:$\set x^-$ denotes the [[Definition:Closure (Topology)|topological closur... | By [[Lower Closure of Element is Closed under Directed Suprema]]:
:$x^\preceq$ is [[Definition:Closed under Directed Suprema|closed under directed suprema]].
By [[Lower Closure of Singleton]]:
:$\set x^\preceq = x^\preceq$
By [[Lower Closure is Lower Section]]:
:$x^\preceq$ is a [[Definition:Lower Section|lower secti... | Closure of Singleton is Lower Closure of Element in Scott Topological Lattice | https://proofwiki.org/wiki/Closure_of_Singleton_is_Lower_Closure_of_Element_in_Scott_Topological_Lattice | https://proofwiki.org/wiki/Closure_of_Singleton_is_Lower_Closure_of_Element_in_Scott_Topological_Lattice | [
"Topological Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Closure (Topology)",
"Definition:Lower Closure/Element"
] | [
"Lower Closure of Element is Closed under Directed Suprema",
"Definition:Closed under Directed Suprema",
"Lower Closure of Singleton",
"Lower Closure is Lower Section",
"Definition:Lower Section",
"Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set",
"Definition:Clos... |
proofwiki-12595 | Pythagorean Triangle cannot be Isosceles | Let $P$ be a Pythagorean triangle.
Then $P$ is not isosceles. | Let $P$ be a Pythagorean triangle.
{{AimForCont}} $P$ is isosceles.
Let the legs of $P$ be of length $a$.
Let the hypotenuse of $P$ be of length $h$.
We have from Pythagoras's Theorem that:
:$2 a^2 = h^2$
and so:
:$\dfrac h a = \sqrt 2$
By definition, $h$ and $a$ are integers.
Hence, by definition, $\sqrt 2$ is a ratio... | Let $P$ be a [[Definition:Pythagorean Triangle|Pythagorean triangle]].
Then $P$ is not [[Definition:Isosceles Triangle|isosceles]]. | Let $P$ be a [[Definition:Pythagorean Triangle|Pythagorean triangle]].
{{AimForCont}} $P$ is [[Definition:Isosceles Triangle|isosceles]].
Let the [[Definition:Leg of Right Triangle|legs]] of $P$ be of [[Definition:Length of Line|length]] $a$.
Let the [[Definition:Hypotenuse|hypotenuse]] of $P$ be of [[Definition:Len... | Pythagorean Triangle cannot be Isosceles | https://proofwiki.org/wiki/Pythagorean_Triangle_cannot_be_Isosceles | https://proofwiki.org/wiki/Pythagorean_Triangle_cannot_be_Isosceles | [
"Isosceles Triangles",
"Pythagorean Triangles"
] | [
"Definition:Pythagorean Triangle",
"Definition:Triangle (Geometry)/Isosceles"
] | [
"Definition:Pythagorean Triangle",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Linear Measure/Length",
"Pythagoras's Theorem",
"Definition:Integ... |
proofwiki-12596 | Brahmagupta-Fibonacci Identity/Corollary | :$\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c - b d}^2 + \paren {a d + b c}^2$ | {{begin-eqn}}
{{eqn | l = \paren {a^2 + b^2} \paren {c^2 + d^2}
| r = \paren {a c + b d}^2 + \paren {a d - b c}^2
| c = Brahmagupta-Fibonacci Identity
}}
{{eqn | ll= \leadsto
| l = \paren {a^2 + \paren {-b}^2} \paren {c^2 + d^2}
| r = \paren {a c + \paren {-b} d}^2 + \paren {a d - \paren {-b} c}... | :$\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c - b d}^2 + \paren {a d + b c}^2$ | {{begin-eqn}}
{{eqn | l = \paren {a^2 + b^2} \paren {c^2 + d^2}
| r = \paren {a c + b d}^2 + \paren {a d - b c}^2
| c = [[Brahmagupta-Fibonacci Identity]]
}}
{{eqn | ll= \leadsto
| l = \paren {a^2 + \paren {-b}^2} \paren {c^2 + d^2}
| r = \paren {a c + \paren {-b} d}^2 + \paren {a d - \paren {-b... | Brahmagupta-Fibonacci Identity/Corollary | https://proofwiki.org/wiki/Brahmagupta-Fibonacci_Identity/Corollary | https://proofwiki.org/wiki/Brahmagupta-Fibonacci_Identity/Corollary | [
"Brahmagupta-Fibonacci Identity"
] | [] | [
"Brahmagupta-Fibonacci Identity"
] |
proofwiki-12597 | Scott Topological Lattice is T0 Space | Let $T = \struct {S, \preceq, \tau}$ be a complete topological lattice with Scott topology.
Then $T$ is a $T_0$ space. | Let $x, y \in S$ such that:
:$x \ne y$
By Closure of Singleton is Lower Closure of Element in Scott Topological Lattice:
:$\set x^- = x^\preceq$ and $\set y^- = y^\preceq$
Thus by Lower Closures are Equal implies Elements are Equal:
:$\set x^- \ne \set y^-$
Hence by Characterization of T0 Space by Distinct Closures of ... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | Let $x, y \in S$ such that:
:$x \ne y$
By [[Closure of Singleton is Lower Closure of Element in Scott Topological Lattice]]:
:$\set x^- = x^\preceq$ and $\set y^- = y^\preceq$
Thus by [[Lower Closures are Equal implies Elements are Equal]]:
:$\set x^- \ne \set y^-$
Hence by [[Characterization of T0 Space by Distinct... | Scott Topological Lattice is T0 Space | https://proofwiki.org/wiki/Scott_Topological_Lattice_is_T0_Space | https://proofwiki.org/wiki/Scott_Topological_Lattice_is_T0_Space | [
"Topological Order Theory",
"Examples of T0 Spaces"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:T0 Space"
] | [
"Closure of Singleton is Lower Closure of Element in Scott Topological Lattice",
"Lower Closures are Equal implies Elements are Equal",
"Characterization of T0 Space by Distinct Closures of Singletons",
"Definition:T0 Space"
] |
proofwiki-12598 | Lower Closures are Equal implies Elements are Equal | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $x, y \in S$ such that
:$x^\preceq = y^\preceq$
where $x^\preceq$ denotes the lower closure of $x$.
Then $x = y$ | By definitions of lower closure of element and reflexivity:
:$x \in x^\preceq$ and $y \in y^\preceq$
By definition of lower closure of element:
:$x \preceq y$ and $y \preceq x$
Thus by definition of antisymmetry:
:$x = y$
{{qed}} | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x^\preceq = y^\preceq$
where $x^\preceq$ denotes the [[Definition:Lower Closure of Element|lower closure]] of $x$.
Then $x = y$ | By definitions of [[Definition:Lower Closure of Element|lower closure of element]] and [[Definition:Reflexivity|reflexivity]]:
:$x \in x^\preceq$ and $y \in y^\preceq$
By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
:$x \preceq y$ and $y \preceq x$
Thus by definition of [[Definition... | Lower Closures are Equal implies Elements are Equal | https://proofwiki.org/wiki/Lower_Closures_are_Equal_implies_Elements_are_Equal | https://proofwiki.org/wiki/Lower_Closures_are_Equal_implies_Elements_are_Equal | [
"Lower Closures"
] | [
"Definition:Ordered Set",
"Definition:Lower Closure/Element"
] | [
"Definition:Lower Closure/Element",
"Definition:Reflexivity",
"Definition:Lower Closure/Element",
"Definition:Antisymmetric Relation"
] |
proofwiki-12599 | Square of Pythagorean Prime is Hypotenuse of Pythagorean Triangle | Let $p$ be a Pythagorean prime.
Then $p^2$ is the hypotenuse of a Pythagorean triangle. | By Fermat's Two Squares Theorem, a Pythagorean prime, $p$ can be expressed in the form:
:$p = m^2 + n^2$
where $m$ and $n$ are (strictly) positive integers.
:$(1): \quad m \ne n$, otherwise $p$ would be of the form $2 m^2$ and so even and therefore not a prime.
:$(2): \quad m \perp n$, otherwise $\exists c \in \Z: p = ... | Let $p$ be a [[Definition:Pythagorean Prime|Pythagorean prime]].
Then $p^2$ is the [[Definition:Hypotenuse|hypotenuse]] of a [[Definition:Pythagorean Triangle|Pythagorean triangle]]. | By [[Fermat's Two Squares Theorem]], a [[Definition:Pythagorean Prime|Pythagorean prime]], $p$ can be expressed in the form:
:$p = m^2 + n^2$
where $m$ and $n$ are [[Definition:Strictly Positive Integer|(strictly) positive integers]].
:$(1): \quad m \ne n$, otherwise $p$ would be of the form $2 m^2$ and so [[Definitio... | Square of Pythagorean Prime is Hypotenuse of Pythagorean Triangle | https://proofwiki.org/wiki/Square_of_Pythagorean_Prime_is_Hypotenuse_of_Pythagorean_Triangle | https://proofwiki.org/wiki/Square_of_Pythagorean_Prime_is_Hypotenuse_of_Pythagorean_Triangle | [
"Pythagorean Triangles"
] | [
"Definition:Pythagorean Prime",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Pythagorean Triangle"
] | [
"Fermat's Two Squares Theorem",
"Definition:Pythagorean Prime",
"Definition:Strictly Positive/Integer",
"Definition:Even Integer",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Parity of Integer",
"Definition:Even Integer",
"Definition:Prime Number",
"Solutions of Pythagorean E... |
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