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-15800 | Ring of Polynomial Forms over Field is Vector Space/Corollary | Let $S \subseteq F \sqbrk X$ denote the subset of $F \sqbrk X$ defined as:
:$S = \set {\mathbf x \in F \sqbrk X: \map \deg {\mathbf x} < d}$
for some $d \in \Z_{>0}$.
Then $S$ is a vector space over $F$. | From Ring of Polynomial Forms over Field is Vector Space we note that $\struct {F, +, \times}$ is a vector space over $F$.
The remaining question is that $S$ remains closed under polynomial addition and scalar multiplication.
Let $\mathbf x, \mathbf y \in S$ such that $\map \deg {\mathbf x} = m$ and $\map \deg {\mathbf... | Let $S \subseteq F \sqbrk X$ denote the [[Definition:Subset|subset]] of $F \sqbrk X$ defined as:
:$S = \set {\mathbf x \in F \sqbrk X: \map \deg {\mathbf x} < d}$
for some $d \in \Z_{>0}$.
Then $S$ is a [[Definition:Vector Space|vector space over $F$]]. | From [[Ring of Polynomial Forms over Field is Vector Space]] we note that $\struct {F, +, \times}$ is a [[Definition:Vector Space|vector space over $F$]].
The remaining question is that $S$ remains [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Polynomial Addition|polynomial addition]] and [[Defin... | Ring of Polynomial Forms over Field is Vector Space/Corollary | https://proofwiki.org/wiki/Ring_of_Polynomial_Forms_over_Field_is_Vector_Space/Corollary | https://proofwiki.org/wiki/Ring_of_Polynomial_Forms_over_Field_is_Vector_Space/Corollary | [
"Examples of Vector Spaces"
] | [
"Definition:Subset",
"Definition:Vector Space"
] | [
"Ring of Polynomial Forms over Field is Vector Space",
"Definition:Vector Space",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Polynomial Addition",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Polyn... |
proofwiki-15801 | P-adic Norm of p-adic Number is Power of p | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $x \in \Q_p: x \ne 0$.
Then:
:$\exists v \in \Z: \norm x_p = p^{-v}$
== Lemma ==
{{:P-adic Norm of p-adic Number is Power of p/Lemma}}{{qed|lemma}} | From Rational Numbers are Dense Subfield of P-adic Numbers $\Q$ is dense in $\Q_p$.
By the definition of a dense subset then $\map \cl \Q = \Q_p$.
By Closure of Subset of Metric Space by Convergent Sequence then:
:there exists a sequence $\sequence {x_n} \subseteq \Q$ that converges to $x$.
That is:
:$\ds \lim_{n \math... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $x \in \Q_p: x \ne 0$.
Then:
:$\exists v \in \Z: \norm x_p = p^{-v}$
== [[P-adic Norm of p-adic Number is Power of p/Lemma|Lemma]] ==
{{:P-adi... | From [[Rational Numbers are Dense Subfield of P-adic Numbers]] $\Q$ is [[Definition:Everywhere Dense|dense]] in $\Q_p$.
By the definition of a [[Definition:Everywhere Dense|dense subset]] then $\map \cl \Q = \Q_p$.
By [[Closure of Subset of Metric Space by Convergent Sequence]] then:
:there exists a [[Definition:Rati... | P-adic Norm of p-adic Number is Power of p/Proof 1 | https://proofwiki.org/wiki/P-adic_Norm_of_p-adic_Number_is_Power_of_p | https://proofwiki.org/wiki/P-adic_Norm_of_p-adic_Number_is_Power_of_p/Proof_1 | [
"P-adic Number Theory",
"P-adic Norm of p-adic Number is Power of p"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"P-adic Norm of p-adic Number is Power of p/Lemma"
] | [
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense",
"Closure of Subset of Metric Space by Convergent Sequence",
"Definition:Rational Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Modulus of Limit",
"Convergent Sequ... |
proofwiki-15802 | P-adic Norm of p-adic Number is Power of p | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $x \in \Q_p: x \ne 0$.
Then:
:$\exists v \in \Z: \norm x_p = p^{-v}$
== Lemma ==
{{:P-adic Norm of p-adic Number is Power of p/Lemma}}{{qed|lemma}} | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where:
:$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$
:$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Then $x$ is... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $x \in \Q_p: x \ne 0$.
Then:
:$\exists v \in \Z: \norm x_p = p^{-v}$
== [[P-adic Norm of p-adic Number is Power of p/Lemma|Lemma]] ==
{{:P-adi... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
That is, $\Q_p$ is the [[Definition:Quotient Ring|quotient ring]] $\CC \, \big / \NN$ where:
:$\CC$ denotes the [[Definition:Ring of Cauchy Sequences|commutative ring of Cauchy sequences]] over $\struct {\Q... | P-adic Norm of p-adic Number is Power of p/Proof 2 | https://proofwiki.org/wiki/P-adic_Norm_of_p-adic_Number_is_Power_of_p | https://proofwiki.org/wiki/P-adic_Norm_of_p-adic_Number_is_Power_of_p/Proof_2 | [
"P-adic Number Theory",
"P-adic Norm of p-adic Number is Power of p"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"P-adic Norm of p-adic Number is Power of p/Lemma"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Quotient Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence",
"Definition:Coset/Left Coset",
"Definition:Cauchy Sequence/Normed Division Ring",
"P-adic Norm of p-adic Number is Power of p/Lemma",
"Definit... |
proofwiki-15803 | Fisher-Tippett-Gnedenko Theorem | In general, for iid $\set {X_i} _{i\mathop = 1}^n$ with cdf $\map F x$ we have $\mathbb P \sqbrk {\max_{i \mathop = 1}^n \set {X_i} > x} = \map {F^n} x$.
Let $x_* := \sup \set {x: \map F x < 1}$.
Then as $n \to \infty$:
:$\ds \max_{i \mathop = 1, \mathop \ldots, n} \set {X_i} \stackrel {prob} {\to} x_*$
We normalize it... | Proof under construction
{{Namedfor|Ronald Aylmer Fisher|name2 = Leonard Henry Caleb Tippett|name3 = Boris Vladimirovich Gnedenko|cat = Fisher|cat2 = Tippett|cat3 = Gnedenko}}
Category:Probability Theory
27jovmcthna40dhk0b2mepp5s8klk6x | In general, for iid $\set {X_i} _{i\mathop = 1}^n$ with cdf $\map F x$ we have $\mathbb P \sqbrk {\max_{i \mathop = 1}^n \set {X_i} > x} = \map {F^n} x$.
Let $x_* := \sup \set {x: \map F x < 1}$.
Then as $n \to \infty$:
:$\ds \max_{i \mathop = 1, \mathop \ldots, n} \set {X_i} \stackrel {prob} {\to} x_*$
We normalize... | Proof under construction
{{Namedfor|Ronald Aylmer Fisher|name2 = Leonard Henry Caleb Tippett|name3 = Boris Vladimirovich Gnedenko|cat = Fisher|cat2 = Tippett|cat3 = Gnedenko}}
[[Category:Probability Theory]]
27jovmcthna40dhk0b2mepp5s8klk6x | Fisher-Tippett-Gnedenko Theorem | https://proofwiki.org/wiki/Fisher-Tippett-Gnedenko_Theorem | https://proofwiki.org/wiki/Fisher-Tippett-Gnedenko_Theorem | [
"Probability Theory"
] | [] | [
"Category:Probability Theory"
] |
proofwiki-15804 | Borell-TIS Inequality | Let $T$ be a topological space.
Let $\sequence {f_t}_{t \mathop \in T}$ be a centred (i.e. mean zero) Gaussian process on $T$, such that:
:$\norm f_T := \sup_{t \mathop \in T} \size {f_t}$
is almost surely finite.
Let:
:$\sigma_T^2 := \sup_{t \mathop \in T} \operatorname E \size {f_t}^2$
Then $\map {\operatorname E} {\... | {{ProofWanted}}
{{Namedfor|Christer Borell|name2 = Boris Tsirelson|name3 = Ildar Ibragimov|name4 = and Vladimir Sudakov|cat=Borell|cat2 = Tsirelson|cat3 = Ibragimov|cat4 = Sudakov}}
67cs9lqph3ka28980ky2qedoa9km23p | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $\sequence {f_t}_{t \mathop \in T}$ be a centred (i.e. mean zero) [[Definition:Gaussian Process|Gaussian process]] on $T$, such that:
:$\norm f_T := \sup_{t \mathop \in T} \size {f_t}$
is [[Definition:Almost Surely|almost surely]] finite.
Let:
:$... | {{ProofWanted}}
{{Namedfor|Christer Borell|name2 = Boris Tsirelson|name3 = Ildar Ibragimov|name4 = and Vladimir Sudakov|cat=Borell|cat2 = Tsirelson|cat3 = Ibragimov|cat4 = Sudakov}}
67cs9lqph3ka28980ky2qedoa9km23p | Borell-TIS Inequality | https://proofwiki.org/wiki/Borell-TIS_Inequality | https://proofwiki.org/wiki/Borell-TIS_Inequality | [] | [
"Definition:Topological Space",
"Definition:Gaussian Process",
"Definition:Almost Everywhere"
] | [] |
proofwiki-15805 | Gaussian Isoperimetric Inequality | Let $A$ be a measurable subset of $\R^n$ endowed with the standard Gaussian measure $\gamma^n$ with the density $\dfrac {\map \exp {-\norm x^2 / 2} } {\paren {2 \pi}^{n/2} }$
Denote by:
:$A_\epsilon = \set {x \in \R^n: \map d {x, A} \le \epsilon}$
the $\epsilon$-extension of $A$.
The '''Gaussian isoperimetric inequalit... | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|cat = Gauss}}
Category:Measure Theory
fj42xkxsm5x5ffftwoxwyrazp4sw01t | Let $A$ be a [[measurable]] subset of $\R^n$ endowed with the standard Gaussian measure $\gamma^n$ with the density $\dfrac {\map \exp {-\norm x^2 / 2} } {\paren {2 \pi}^{n/2} }$
Denote by:
:$A_\epsilon = \set {x \in \R^n: \map d {x, A} \le \epsilon}$
the $\epsilon$-extension of $A$.
The '''Gaussian isoperimetric i... | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|cat = Gauss}}
[[Category:Measure Theory]]
fj42xkxsm5x5ffftwoxwyrazp4sw01t | Gaussian Isoperimetric Inequality | https://proofwiki.org/wiki/Gaussian_Isoperimetric_Inequality | https://proofwiki.org/wiki/Gaussian_Isoperimetric_Inequality | [
"Measure Theory"
] | [
"measurable"
] | [
"Category:Measure Theory"
] |
proofwiki-15806 | Negation of Propositional Function in Two Variables | Let $\map P {x, y}$ be a propositional function of two Variables.
Then:
:$\neg \forall x: \exists y: \map P {x, y} \iff \exists x: \forall y: \neg \map P {x, y}$
That is:
:''It is not the case that for all $x$ a value of $y$ can be found to satisfy $\map P {x, y}$''
means the same thing as:
:''There exists at least one... | {{begin-eqn}}
{{eqn | q = \neg \forall x
| l = \exists y
| o = :
| r = \map P {x, y}
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x
| l = \neg \exists y
| o = :
| r = \map P {x, y}
| c = Denial of Universality
}}
{{eqn | ll= \leadstoandfrom
| q = \exist... | Let $\map P {x, y}$ be a [[Definition:Propositional Function|propositional function]] of two [[Definition:Variable|Variables]].
Then:
:$\neg \forall x: \exists y: \map P {x, y} \iff \exists x: \forall y: \neg \map P {x, y}$
That is:
:''It is not the case that for all $x$ a [[Definition:Value of Variable|value]] of $... | {{begin-eqn}}
{{eqn | q = \neg \forall x
| l = \exists y
| o = :
| r = \map P {x, y}
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x
| l = \neg \exists y
| o = :
| r = \map P {x, y}
| c = [[Denial of Universality]]
}}
{{eqn | ll= \leadstoandfrom
| q = \e... | Negation of Propositional Function in Two Variables | https://proofwiki.org/wiki/Negation_of_Propositional_Function_in_Two_Variables | https://proofwiki.org/wiki/Negation_of_Propositional_Function_in_Two_Variables | [
"Universal Quantifier",
"Existential Quantifier"
] | [
"Definition:Propositional Function",
"Definition:Variable",
"Definition:Variable/Value",
"Definition:Variable/Satisfaction",
"Definition:Variable/Value",
"Definition:Variable/Satisfaction"
] | [
"De Morgan's Laws (Predicate Logic)/Denial of Universality",
"De Morgan's Laws (Predicate Logic)/Denial of Existence"
] |
proofwiki-15807 | Valuation Ideal of P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Then the valuation ideal induced by norm $\norm {\,\cdot\,}_p$ is the principal ideal:
:$p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$
where $\Z_p$ denotes the $p$-adic integers. | From P-adic Integers is Local Ring, $\Z_p$ is a local ring.
From Principal Ideal from Element in Center of Ring, $p \Z_p$ is a principal ideal.
Now:
{{begin-eqn}}
{{eqn | l = \norm x_p
| o = <
| r = 1
}}
{{eqn | ll= \leadstoandfrom
| l = \norm x_p
| o = \le
| r = \dfrac 1 p
| c = P-... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Then the [[Definition:Valuation Ideal Induced by Non-Archimedean Norm|valuation ideal induced by norm]] $\norm {\,\cdot\,}_p$ is the [[Definition:Pri... | From [[P-adic Integers is Local Ring]], $\Z_p$ is a [[Definition:Local Ring|local ring]].
From [[Principal Ideal from Element in Center of Ring]], $p \Z_p$ is a [[Definition:Principal Ideal of Ring|principal ideal]].
Now:
{{begin-eqn}}
{{eqn | l = \norm x_p
| o = <
| r = 1
}}
{{eqn | ll= \leadstoandfrom ... | Valuation Ideal of P-adic Numbers | https://proofwiki.org/wiki/Valuation_Ideal_of_P-adic_Numbers | https://proofwiki.org/wiki/Valuation_Ideal_of_P-adic_Numbers | [
"P-adic Numbers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Valuation Ideal Induced by Non-Archimedean Norm",
"Definition:Principal Ideal of Ring",
"Definition:P-adic Integer"
] | [
"P-adic Integers is Valuation Ring Induced by P-adic Norm/Corollary",
"Definition:Local Ring",
"Principal Ideal from Element in Center of Ring",
"Definition:Principal Ideal of Ring",
"P-adic Norm of p-adic Number is Power of p",
"Properties of Norm on Division Ring/Norm of Quotient"
] |
proofwiki-15808 | Integers are Arbitrarily Close to P-adic Integers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $x \in \Z_p$.
Then for $n \in \N$ there exists unique $\alpha \in \Z$:
:$(1): \quad 0 \le \alpha \le p^n - 1$
:$(2): \quad \norm { x -\alpha}_p \le p^{-n}$ | Let $n \in \N$.
From Rational Numbers are Dense Subfield of P-adic Numbers:
:the rational numbers are dense in $\Q_p$.
So there exists:
:$\dfrac a b \in \Q: \norm {x - \dfrac a b}_p \le p^{-n}$
From Unique Integer Close to Rational in Valuation Ring of P-adic Norm, there exists unique $\alpha \in \Z$ such that:
:$\no... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $x \in \Z_p$.
Then for $n \in \N$ there exists [[Definition:Unique|unique]] $... | Let $n \in \N$.
From [[Rational Numbers are Dense Subfield of P-adic Numbers]]:
:the [[Definition:Rational Number|rational numbers]] are [[Definition:Everywhere Dense|dense]] in $\Q_p$.
So there exists:
:$\dfrac a b \in \Q: \norm {x - \dfrac a b}_p \le p^{-n}$
From [[Unique Integer Close to Rational in Valuation Ri... | Integers are Arbitrarily Close to P-adic Integers | https://proofwiki.org/wiki/Integers_are_Arbitrarily_Close_to_P-adic_Integers | https://proofwiki.org/wiki/Integers_are_Arbitrarily_Close_to_P-adic_Integers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Unique"
] | [
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Definition:Rational Number",
"Definition:Everywhere Dense",
"Unique Integer Close to Rational in Valuation Ring of P-adic Norm",
"Definition:Unique",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ri... |
proofwiki-15809 | Integers are Dense in P-adic Integers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$ restricted to the $p$-adic integers.
The integers $\Z$ are dense in the metric space $\struct{\Z_p, d_p}$.
=== Corollary ==... | From Open Ball Characterization of Denseness it is sufficient to show that every open ball of $\struct {\Z_p, d_p}$ contains an element of $\Z$.
Let $x \in \Z_p$ and $\epsilon \in \R_{>0}$.
By definition the open ball $\map {B_\epsilon} x$ is:
:$\map {B_\epsilon} x = \set {y \in \Z_p: \norm y_p < \epsilon}$
From Sequen... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $d_p$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induc... | From [[Open Ball Characterization of Denseness]] it is sufficient to show that every [[Definition:Open Ball|open ball]] of $\struct {\Z_p, d_p}$ contains an [[Definition:Element|element]] of $\Z$.
Let $x \in \Z_p$ and $\epsilon \in \R_{>0}$.
By definition the [[Definition:Open Ball|open ball]] $\map {B_\epsilon} x$ ... | Integers are Dense in P-adic Integers | https://proofwiki.org/wiki/Integers_are_Dense_in_P-adic_Integers | https://proofwiki.org/wiki/Integers_are_Dense_in_P-adic_Integers | [
"P-adic Number Theory",
"Integers are Dense in P-adic Integers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Restriction/Mapping",
"Definition:P-adic Integer",
"Definition:Integer",
"Definition:Everywhere Dense",
"Definition:Metric Space",
... | [
"Open Set Characterization of Denseness/Open Ball",
"Definition:Open Ball",
"Definition:Element",
"Definition:Open Ball",
"Sequence of Powers of Number less than One",
"Definition:Open Ball",
"Integers are Arbitrarily Close to P-adic Integers",
"Definition:Open Ball",
"Definition:Element",
"Open S... |
proofwiki-15810 | P-adic Integer is Limit of Unique Coherent Sequence of Integers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $x \in \Z_p$.
Then there exists a unique coherent sequence $\sequence {\alpha_n}$:
:$\ds \lim_{n \mathop \to \infty} \alpha_n = x$ | By the definition of a coherent sequence it needs to be proved that there exists a unique integer sequence $\sequence {\alpha_n}$:
:$(1): \quad \forall n \in \N: \alpha_n \in \Z$ and $0 \le \alpha_n \le p^{n + 1} - 1$
:$(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$
:$(3): \quad \ds \lim... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $x \in \Z_p$.
Then there exists a [[Definition:Unique|unique]] [[Definition:... | By the definition of a [[Definition:P-adically Coherent Sequence|coherent sequence]] it needs to be proved that there exists a [[Definition:Unique|unique]] [[Definition:Integer Sequence|integer sequence]] $\sequence {\alpha_n}$:
:$(1): \quad \forall n \in \N: \alpha_n \in \Z$ and $0 \le \alpha_n \le p^{n + 1} - 1$
:$(2... | P-adic Integer is Limit of Unique Coherent Sequence of Integers | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers | [
"P-adic Number Theory",
"P-adic Integer is Limit of Unique Coherent Sequence of Integers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Unique",
"Definition:P-adically Coherent Sequence"
] | [
"Definition:P-adically Coherent Sequence",
"Definition:Unique",
"Definition:Integer Sequence",
"Integers are Arbitrarily Close to P-adic Integers",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Sequence"
] |
proofwiki-15811 | Dimension of Vector Space on Cartesian Product | Let $\struct {K, +, \circ}$ be a division ring.
Let $n \in \N_{>0}$.
Let $\mathbf V := \struct {K^n, +, \times}_K$ be the '''$K$-vector space $K^n$'''.
Then the dimension of $\mathbf V$ is $n$. | Let the unity of $K$ be $1$, and the zero of $K$ be $0$.
Consider the vectors:
{{begin-eqn}}
{{eqn | l = \mathbf e_1
| o = :=
| r = \underbrace {\tuple {1, 0, \ldots, 0} }_{n \text { coordinates} }
| c =
}}
{{eqn | l = \mathbf e_2
| o = :=
| r = \underbrace {\tuple {0, 1, \ldots, 0} }_{n ... | Let $\struct {K, +, \circ}$ be a [[Definition:Division Ring|division ring]].
Let $n \in \N_{>0}$.
Let $\mathbf V := \struct {K^n, +, \times}_K$ be the '''[[Definition:Vector Space on Cartesian Product|$K$-vector space $K^n$]]'''.
Then the [[Definition:Dimension of Vector Space|dimension]] of $\mathbf V$ is $n$. | Let the [[Definition:Unity of Ring|unity]] of $K$ be $1$, and the [[Definition:Ring Zero|zero]] of $K$ be $0$.
Consider the [[Definition:Vector (Linear Algebra)|vectors]]:
{{begin-eqn}}
{{eqn | l = \mathbf e_1
| o = :=
| r = \underbrace {\tuple {1, 0, \ldots, 0} }_{n \text { coordinates} }
| c =
}}... | Dimension of Vector Space on Cartesian Product | https://proofwiki.org/wiki/Dimension_of_Vector_Space_on_Cartesian_Product | https://proofwiki.org/wiki/Dimension_of_Vector_Space_on_Cartesian_Product | [
"Vector Space on Cartesian Product",
"Dimension of Vector Space"
] | [
"Definition:Division Ring",
"Definition:Vector Space on Cartesian Product",
"Definition:Dimension of Vector Space"
] | [
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Ring Zero",
"Definition:Vector/Linear Algebra",
"Definition:Standard Ordered Basis/Vector Space",
"Standard Ordered Basis is Basis",
"Definition:Basis of Vector Space",
"Basis Theorem"
] |
proofwiki-15812 | Dimension of Vector Space of Polynomial Functions | Let $\struct {F, +, \times}$ be a field whose unity is $1_F$.
Let $F_n \sqbrk X$ be the ring of polynomials over $F$ whose degree is less than $n$.
Then the dimension of the vector space $F_n \sqbrk X$ is $n$. | Let $B$ be the set of all the identity functions $I^k$ on $F_n \sqbrk X$ where $n \in \Z_{\ge 0}$.
By definition, every element of $F_n \sqbrk X$ is a linear combination of $B$.
Suppose:
:$\ds \sum_{k \mathop = 0}^m \alpha_k I^k = 0, \alpha_m \ne 0$
Then by differentiating $m$ times, we obtain from Nth Derivative of Nt... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $F_n \sqbrk X$ be the [[Definition:Ring of Polynomials in Ring Element|ring of polynomials]] over $F$ whose [[Definition:Degree of Polynomial|degree]] is less than $n$.
Then the [[D... | Let $B$ be the [[Definition:Set|set]] of all the [[Definition:Identity Mapping|identity functions]] $I^k$ on $F_n \sqbrk X$ where $n \in \Z_{\ge 0}$.
By definition, every element of $F_n \sqbrk X$ is a [[Definition:Linear Combination|linear combination]] of $B$.
Suppose:
:$\ds \sum_{k \mathop = 0}^m \alpha_k I^k = 0... | Dimension of Vector Space of Polynomial Functions | https://proofwiki.org/wiki/Dimension_of_Vector_Space_of_Polynomial_Functions | https://proofwiki.org/wiki/Dimension_of_Vector_Space_of_Polynomial_Functions | [
"Dimension of Vector Space",
"Polynomial Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Multiplicative Identity",
"Definition:Ring of Polynomials in Ring Element",
"Definition:Degree of Polynomial",
"Definition:Dimension of Vector Space",
"Definition:Vector Space"
] | [
"Definition:Set",
"Definition:Identity Mapping",
"Definition:Linear Combination",
"Definition:Differentiation",
"Nth Derivative of Nth Power",
"Definition:Linearly Independent/Set",
"Definition:Basis of Vector Space",
"Basis Theorem"
] |
proofwiki-15813 | Vector Space over Division Subring is Vector Space | Let $\struct {L, +_L, \times_L}$ be a division ring.
Let $K$ be a division subring of $\struct {L, +_L, \times_L}$.
Let $\struct {G, +_G, \circ}_L$ be a $L$-vector space.
Let $\circ_K$ be the restriction of $\circ$ to $K \times G$.
Hence let $\struct {G, +_G, \circ_K}_K$ be the vector space induced by $K$.
Then $\struc... | A vector space over a division ring $D$ is by definition a unitary module over $D$.
$S$ is a division ring by assumption.
$\struct {R, +, \circ_S}_S$ is a unitary module by Subring Module is Module/Special Case.
{{qed}} | Let $\struct {L, +_L, \times_L}$ be a [[Definition:Division Ring|division ring]].
Let $K$ be a [[Definition:Division Subring|division subring]] of $\struct {L, +_L, \times_L}$.
Let $\struct {G, +_G, \circ}_L$ be a [[Definition:Vector Space|$L$-vector space]].
Let $\circ_K$ be the [[Definition:Restriction of Operati... | A [[Definition:Vector Space over Division Ring|vector space]] over a [[Definition:Division Ring|division ring]] $D$ is by definition a [[Definition:Unitary Module|unitary module]] over $D$.
$S$ is a [[Definition:Division Ring|division ring]] by assumption.
$\struct {R, +, \circ_S}_S$ is a [[Definition:Unitary Modul... | Vector Space over Division Subring is Vector Space | https://proofwiki.org/wiki/Vector_Space_over_Division_Subring_is_Vector_Space | https://proofwiki.org/wiki/Vector_Space_over_Division_Subring_is_Vector_Space | [
"Examples of Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Division Subring",
"Definition:Vector Space",
"Definition:Restriction/Operation",
"Definition:Vector Space over Division Subring",
"Definition:Vector Space/Division Ring"
] | [
"Definition:Vector Space/Division Ring",
"Definition:Division Ring",
"Definition:Unitary Module over Ring",
"Definition:Division Ring",
"Definition:Unitary Module over Ring",
"Subring Module is Module/Special Case"
] |
proofwiki-15814 | Vector Space on Field Extension is Vector Space | Let $\struct {K, +, \times}$ be a field.
Let $L / K$ be a field extension over $K$.
Let $\struct {L, +, \times}_K$ be the a vector space of $L$ over $K$.
Then $\struct {L, +, \times}_K$ is a vector space. | By definition, $L / K$ is a field extension over $K$.
Thus, by definition, $K$ is a subfield of $L$.
Thus, also by definition, $K$ is a division subring of $L$.
The result follows by Vector Space over Division Subring is Vector Space.
{{qed}}
Category:Examples of Vector Spaces
hv8bgifu71tkrwb1ltfyu52uhx2gdy4 | Let $\struct {K, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $L / K$ be a [[Definition:Field Extension|field extension]] over $K$.
Let $\struct {L, +, \times}_K$ be the a [[Definition:Vector Space on Field Extension|vector space of $L$]] over $K$.
Then $\struct {L, +, \times}_K$ is a [[Defin... | By definition, $L / K$ is a [[Definition:Field Extension|field extension]] over $K$.
Thus, by definition, $K$ is a [[Definition:Subfield|subfield]] of $L$.
Thus, also by definition, $K$ is a [[Definition:Division Subring|division subring]] of $L$.
The result follows by [[Vector Space over Division Subring is Vector ... | Vector Space on Field Extension is Vector Space | https://proofwiki.org/wiki/Vector_Space_on_Field_Extension_is_Vector_Space | https://proofwiki.org/wiki/Vector_Space_on_Field_Extension_is_Vector_Space | [
"Examples of Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Extension",
"Definition:Vector Space on Field Extension",
"Definition:Vector Space"
] | [
"Definition:Field Extension",
"Definition:Subfield",
"Definition:Division Subring",
"Vector Space over Division Subring is Vector Space",
"Category:Examples of Vector Spaces"
] |
proofwiki-15815 | Condition for Linear Divisor of Polynomial | Let $\map P x$ be a polynomial in $x$.
Let $a$ be a constant.
Then $x - a$ is a divisor of $\map P x$ {{iff}} $a$ is a root of $P$. | From the Little Bézout Theorem, the remainder of $\map P x$ when divided by $x - a$ is equal to $\map P a$. | Let $\map P x$ be a [[Definition:Polynomial|polynomial]] in $x$.
Let $a$ be a [[Definition:Constant|constant]].
Then $x - a$ is a [[Definition:Divisor of Polynomial|divisor]] of $\map P x$ {{iff}} $a$ is a [[Definition:Root of Polynomial|root]] of $P$. | From the [[Little Bézout Theorem]], the [[Definition:Remainder (Polynomial Long Division)|remainder]] of $\map P x$ when divided by $x - a$ is equal to $\map P a$. | Condition for Linear Divisor of Polynomial | https://proofwiki.org/wiki/Condition_for_Linear_Divisor_of_Polynomial | https://proofwiki.org/wiki/Condition_for_Linear_Divisor_of_Polynomial | [
"Polynomial Theory"
] | [
"Definition:Polynomial",
"Definition:Constant",
"Definition:Divisor of Polynomial",
"Definition:Root of Polynomial"
] | [
"Little Bézout Theorem",
"Polynomial Long Division",
"Little Bézout Theorem",
"Polynomial Long Division",
"Polynomial Long Division",
"Little Bézout Theorem",
"Polynomial Long Division"
] |
proofwiki-15816 | Complex Number is Algebraic over Real Numbers | Let $z \in \C$ be a complex number.
Then $z$ is algebraic over $\R$. | Let $z = a + i b$.
Let $\overline z = a - i b$ be the complex conjugate of $z$.
From Product of Complex Number with Conjugate:
:$z \overline z = a^2 + b^2$
From Sum of Complex Number with Conjugate:
:$z + \overline z = 2 a$
Thus from Viète's Formulas, both $z$ and $\overline z$ are roots of the polynomial:
:$X^2 - 2 a ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then $z$ is [[Definition:Algebraic Number over Field|algebraic over $\R$]]. | Let $z = a + i b$.
Let $\overline z = a - i b$ be the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
From [[Product of Complex Number with Conjugate]]:
:$z \overline z = a^2 + b^2$
From [[Sum of Complex Number with Conjugate]]:
:$z + \overline z = 2 a$
Thus from [[Viète's Formulas]], both $z$ and $\over... | Complex Number is Algebraic over Real Numbers | https://proofwiki.org/wiki/Complex_Number_is_Algebraic_over_Real_Numbers | https://proofwiki.org/wiki/Complex_Number_is_Algebraic_over_Real_Numbers | [
"Algebraic Numbers"
] | [
"Definition:Complex Number",
"Definition:Algebraic Number over Field"
] | [
"Definition:Complex Conjugate",
"Product of Complex Number with Conjugate",
"Sum of Complex Number with Conjugate",
"Viète's Formulas",
"Definition:Root of Polynomial",
"Definition:Algebraic Number over Field"
] |
proofwiki-15817 | Polynomial with Algebraic Number as Root is Multiple of Minimal Polynomial | Let $F$ be a field.
Let $\map P x$ be a polynomial in $F$.
Let $z$ be a root of $\map P x$.
Then $\map P x$ is a multiple of the minimal polynomial $\map m x$ in $z$ over $F$. | For $z$ to be a root of $F$, $z$ must be algebraic over $F$.
Let us write:
:$\map P x = \map m x \, \map q x + \map r x$
where $\map q x$ and $\map r x$ are polynomials in $F$.
Then either $\map r x = 0$ or $\map \deg {\map r x} < \map \deg {\map m x}$.
Then:
:$\map P z = \map m z \, \map q z + \map r z$
But as $z$ is ... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\map P x$ be a [[Definition:Polynomial|polynomial]] in $F$.
Let $z$ be a [[Definition:Root of Polynomial|root]] of $\map P x$.
Then $\map P x$ is a [[Definition:Multiple of Ring Element|multiple]] of the [[Definition:Minimal Polynomial|minimal polynom... | For $z$ to be a [[Definition:Root of Polynomial|root]] of $F$, $z$ must be [[Definition:Algebraic Number over Field|algebraic over $F$]].
Let us write:
:$\map P x = \map m x \, \map q x + \map r x$
where $\map q x$ and $\map r x$ are [[Definition:Polynomial|polynomials]] in $F$.
Then either $\map r x = 0$ or $\map \d... | Polynomial with Algebraic Number as Root is Multiple of Minimal Polynomial | https://proofwiki.org/wiki/Polynomial_with_Algebraic_Number_as_Root_is_Multiple_of_Minimal_Polynomial | https://proofwiki.org/wiki/Polynomial_with_Algebraic_Number_as_Root_is_Multiple_of_Minimal_Polynomial | [
"Minimal Polynomials"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Polynomial",
"Definition:Root of Polynomial",
"Definition:Multiple/Integral Domain",
"Definition:Minimal Polynomial"
] | [
"Definition:Root of Polynomial",
"Definition:Algebraic Number over Field",
"Definition:Polynomial",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Contradiction",
"Definition:Minimal Polynomial",
"Definition:Multiple/Integral Domain"
] |
proofwiki-15818 | Simple Algebraic Field Extension consists of Polynomials in Algebraic Number | Let $F$ be a field.
Let $\theta \in \C$ be algebraic over $F$.
Let $\map F \theta$ be the simple field extension of $F$ by $\theta$.
Then $\map F \theta$ consists of polynomials that can be written in the form $\map f \theta$, where $\map f x$ is a polynomial over $F$. | Let $H$ be the set of all numbers which can be written in the form $\map f \theta$.
We have that:
:$H$ is closed under addition and multiplication.
:$H$ contains $0$ and $1$
:For every element of $H$, $H$ also contains its negative.
Let $\map f \theta \ne 0$.
Then $\theta$ is not a root of $\map f x$.
Hence from Polyno... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\theta \in \C$ be [[Definition:Algebraic Number over Field|algebraic over $F$]].
Let $\map F \theta$ be the [[Definition:Simple Algebraic Field Extension|simple field extension]] of $F$ by $\theta$.
Then $\map F \theta$ consists of [[Definition:Polyno... | Let $H$ be the [[Definition:Set|set]] of all [[Definition:Number|numbers]] which can be written in the form $\map f \theta$.
We have that:
:$H$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Polynomial Addition|addition]] and [[Definition:Polynomial Multiplication|multiplication]].
:$H$ contai... | Simple Algebraic Field Extension consists of Polynomials in Algebraic Number | https://proofwiki.org/wiki/Simple_Algebraic_Field_Extension_consists_of_Polynomials_in_Algebraic_Number | https://proofwiki.org/wiki/Simple_Algebraic_Field_Extension_consists_of_Polynomials_in_Algebraic_Number | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebraic Number over Field",
"Definition:Simple Algebraic Field Extension",
"Definition:Polynomial",
"Definition:Polynomial"
] | [
"Definition:Set",
"Definition:Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Polynomial Addition",
"Definition:Multiplication of Polynomials",
"Definition:Element",
"Definition:Field Negative",
"Definition:Root of Polynomial",
"Polynomial with Algebraic Number as R... |
proofwiki-15819 | Element of Simple Algebraic Field Extension of Degree n is Polynomial in Algebraic Number of Degree Less than n | Let $F$ be a field.
Let $\theta \in \C$ be algebraic over $F$ of degree $n$.
Let $\map F \theta$ be the simple field extension of $F$ by $\theta$.
Then any element of $\map F \theta$ can be written as $\map f \theta$, where $\map f x$ is a polynomial over $F$ of degree at most $n - 1$. | From Simple Algebraic Field Extension consists of Polynomials in Algebraic Number, an arbitrary element of $\map F \theta$ can be written as $\map f \theta$.
But:
:$\map f x = \map m x \, \map q x + \map r x$
where:
:$\map m x$ is minimal polynomial in $\theta$
:$\map q x$ is a polynomial in $\map F \theta$
:$\map r x$... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\theta \in \C$ be [[Definition:Algebraic Number over Field|algebraic over $F$]] of [[Definition:Degree of Algebraic Number over Field|degree $n$]].
Let $\map F \theta$ be the [[Definition:Simple Algebraic Field Extension|simple field extension]] of $F$ ... | From [[Simple Algebraic Field Extension consists of Polynomials in Algebraic Number]], an arbitrary [[Definition:Element|element]] of $\map F \theta$ can be written as $\map f \theta$.
But:
:$\map f x = \map m x \, \map q x + \map r x$
where:
:$\map m x$ is [[Definition:Minimal Polynomial|minimal polynomial]] in $\the... | Element of Simple Algebraic Field Extension of Degree n is Polynomial in Algebraic Number of Degree Less than n | https://proofwiki.org/wiki/Element_of_Simple_Algebraic_Field_Extension_of_Degree_n_is_Polynomial_in_Algebraic_Number_of_Degree_Less_than_n | https://proofwiki.org/wiki/Element_of_Simple_Algebraic_Field_Extension_of_Degree_n_is_Polynomial_in_Algebraic_Number_of_Degree_Less_than_n | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebraic Number over Field",
"Definition:Algebraic Number over Field/Degree",
"Definition:Simple Algebraic Field Extension",
"Definition:Element",
"Definition:Polynomial",
"Definition:Degree of Polynomial"
] | [
"Simple Algebraic Field Extension consists of Polynomials in Algebraic Number",
"Definition:Element",
"Definition:Minimal Polynomial",
"Definition:Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial"
] |
proofwiki-15820 | Degree of Simple Algebraic Field Extension equals Degree of Algebraic Number | Let $F$ be a field.
Let $\theta \in \C$ be algebraic over $F$ of degree $n$.
Let $\map F \theta$ be the simple field extension of $F$ by $\theta$.
Then $\map F \theta$ is a finite extension of $F$ whose degree is:
:$\index {\map F \theta} F = n$ | Considered as a vector space over $F$, $\map F \theta$ is generated by the set $B$, where:
:$B := \set {1, \theta, \theta^2, \ldots, \theta^{n - 1} }$
But $S$ is linearly independent over $F$, because otherwise:
:$c_0 1 + c_1 \theta + c_2 \theta^2 + \dotsb + c_{n - 1} \theta^{n - 1} = 0$
with all the $c$s non-zero.
Tha... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\theta \in \C$ be [[Definition:Algebraic Number over Field|algebraic over $F$]] of [[Definition:Degree of Algebraic Number over Field|degree $n$]].
Let $\map F \theta$ be the [[Definition:Simple Algebraic Field Extension|simple field extension]] of $F$ ... | Considered as a [[Definition:Vector Space on Field Extension|vector space]] over $F$, $\map F \theta$ is [[Definition:Generated Field Extension|generated]] by the [[Definition:Set|set]] $B$, where:
:$B := \set {1, \theta, \theta^2, \ldots, \theta^{n - 1} }$
But $S$ is [[Definition:Linearly Independent Set|linearly ind... | Degree of Simple Algebraic Field Extension equals Degree of Algebraic Number | https://proofwiki.org/wiki/Degree_of_Simple_Algebraic_Field_Extension_equals_Degree_of_Algebraic_Number | https://proofwiki.org/wiki/Degree_of_Simple_Algebraic_Field_Extension_equals_Degree_of_Algebraic_Number | [
"Degrees of Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Algebraic Number over Field",
"Definition:Algebraic Number over Field/Degree",
"Definition:Simple Algebraic Field Extension",
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree"
] | [
"Definition:Vector Space on Field Extension",
"Definition:Generated Field Extension",
"Definition:Set",
"Definition:Linearly Independent/Set",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Minimal Polynomial",
"Definition:Degree of Polynomia... |
proofwiki-15821 | Degree of Element of Finite Field Extension divides Degree of Extension | Let $F$ be a field whose zero is $0$ and whose unity is $1$.
Let $K / F$ be a finite field extension of degree $n$.
Let $\alpha \in K$ be algebraic over $F$.
Then the degree of $\alpha$ is a divisor of $n$. | Let $\alpha \in K$.
The dimension of $K / F$ considered as a vector space is $n$.
Let $S$ be the set defined as:
:$S := \set {1, \alpha, \alpha^2, \ldots, \alpha_n}$
We have that:
:$\card S = n + 1$
From Cardinality of Linearly Independent Set is No Greater than Dimension:
:$S$ is linearly dependent on $F$.
Hence there... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0$ and whose [[Definition:Unity of Field|unity]] is $1$.
Let $K / F$ be a [[Definition:Finite Field Extension|finite field extension]] of [[Definition:Degree of Field Extension|degree]] $n$.
Let $\alpha \in K$ be [[Def... | Let $\alpha \in K$.
The [[Definition:Dimension of Vector Space|dimension]] of $K / F$ considered as a [[Definition:Vector Space on Field Extension|vector space]] is $n$.
Let $S$ be the [[Definition:Set|set]] defined as:
:$S := \set {1, \alpha, \alpha^2, \ldots, \alpha_n}$
We have that:
:$\card S = n + 1$
From [[Ca... | Degree of Element of Finite Field Extension divides Degree of Extension | https://proofwiki.org/wiki/Degree_of_Element_of_Finite_Field_Extension_divides_Degree_of_Extension | https://proofwiki.org/wiki/Degree_of_Element_of_Finite_Field_Extension_divides_Degree_of_Extension | [
"Degrees of Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree",
"Definition:Algebraic Element of Field Extension",
"Definition:Degree of Algebraic Element",
"Definition:Divisor (Algeb... | [
"Definition:Dimension of Vector Space",
"Definition:Vector Space on Field Extension",
"Definition:Set",
"Cardinality of Linearly Independent Set is No Greater than Dimension",
"Definition:Linearly Dependent/Set",
"Definition:Scalar/Vector Space",
"Definition:Field Zero",
"Definition:Polynomial",
"De... |
proofwiki-15822 | Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2 | Let $K / F$ be a finite field extension of degree $2^m$.
Let $\alpha \in K$ be algebraic over $F$ with degree $3$.
Then $\alpha \notin K$. | {{AimForCont}} $\alpha \in K$.
From Degree of Element of Finite Field Extension divides Degree of Extension:
:$\map \deg \alpha \divides \map \deg {K / F}$
But:
:$3 \nmid 2^m$
From this contradiction, it follows that $\alpha \notin K$.
{{qed}} | Let $K / F$ be a [[Definition:Finite Field Extension|finite field extension]] of [[Definition:Degree of Field Extension|degree]] $2^m$.
Let $\alpha \in K$ be [[Definition:Algebraic Element of Field Extension|algebraic]] over $F$ with [[Definition:Degree of Algebraic Element|degree]] $3$.
Then $\alpha \notin K$. | {{AimForCont}} $\alpha \in K$.
From [[Degree of Element of Finite Field Extension divides Degree of Extension]]:
:$\map \deg \alpha \divides \map \deg {K / F}$
But:
:$3 \nmid 2^m$
From this [[Definition:Contradiction|contradiction]], it follows that $\alpha \notin K$.
{{qed}} | Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2 | https://proofwiki.org/wiki/Algebraic_Element_of_Degree_3_is_not_Element_of_Field_Extension_of_Degree_Power_of_2 | https://proofwiki.org/wiki/Algebraic_Element_of_Degree_3_is_not_Element_of_Field_Extension_of_Degree_Power_of_2 | [
"Field Extensions"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree",
"Definition:Algebraic Element of Field Extension",
"Definition:Degree of Algebraic Element"
] | [
"Degree of Element of Finite Field Extension divides Degree of Extension",
"Definition:Contradiction"
] |
proofwiki-15823 | Construction of Point in Cartesian Plane with Rational Coordinates | Let $\CC$ be a Cartesian plane.
Let $P = \tuple {x, y}$ be a rational point in $\CC$.
Then $P$ is constructible using a compass and straightedge construction. | Let $x = \dfrac m n$ where $m, n \in \Z_{\ne 0}$ are non-zero integers.
Let $y = \dfrac r s$ where $r, s \in \Z_{\ne 0}$ are non-zero integers.
Let $O$ denote the point $\tuple {0, 0}$.
Let $A$ denote the point $\tuple {1, 0}$.
Let $M$ denote the point $\tuple {0, m}$.
Let $N$ denote the point $\tuple {0, n}$.
The $x$-... | Let $\CC$ be a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x, y}$ be a [[Definition:Rational Point in Plane|rational point]] in $\CC$.
Then $P$ is [[Definition:Constructible Point in Plane|constructible]] using a [[Definition:Compass and Straightedge Construction|compass and straightedge constru... | Let $x = \dfrac m n$ where $m, n \in \Z_{\ne 0}$ are non-zero [[Definition:Integer|integers]].
Let $y = \dfrac r s$ where $r, s \in \Z_{\ne 0}$ are non-zero [[Definition:Integer|integers]].
Let $O$ denote the [[Definition:Point|point]] $\tuple {0, 0}$.
Let $A$ denote the [[Definition:Point|point]] $\tuple {1, 0}$.
... | Construction of Point in Cartesian Plane with Rational Coordinates | https://proofwiki.org/wiki/Construction_of_Point_in_Cartesian_Plane_with_Rational_Coordinates | https://proofwiki.org/wiki/Construction_of_Point_in_Cartesian_Plane_with_Rational_Coordinates | [
"Analytic Geometry",
"Compass and Straightedge Constructions"
] | [
"Definition:Cartesian Plane",
"Definition:Rational Point in Plane",
"Definition:Constructible Point in Plane",
"Definition:Compass and Straightedge Construction"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Point",
"Definition:Point",
"Definition:Point",
"Definition:Point",
"Definition:Axis/X-Axis",
"Definition:Line/Straight Line",
"Definition:Axis/Y-Axis",
"Definition:Right Angle/Perpendicular",
"Construction of Lattice Point in Cartesian Pla... |
proofwiki-15824 | Construction of Integer Multiple of Line Segment | Let $AB$ be a line segment in the plane.
Let $AC$ be a line segment in the plane through a point $C$
Let $D$ be a point on $AC$ such that $AD = n AB$ for some $n \in \Z$.
Then $AD$ is constructible using a compass and straightedge construction. | Let $AB$ be given.
Let the straight line through $AC$ be constructed.
Let $D_1$ be constructed on $AC$ such that $A D_1 = AB$ by constructing a circle whose center is at $A$ and whose radius is $B$.
Let $D_0$ be identified as the point $A$.
For each $k \in \set {1, 2, \ldots, n - 1}$, construct a circle whose center is... | Let $AB$ be a [[Definition:Line Segment|line segment]] in the [[Definition:Plane|plane]].
Let $AC$ be a [[Definition:Line Segment|line segment]] in the [[Definition:Plane|plane]] through a [[Definition:Point|point]] $C$
Let $D$ be a [[Definition:Point|point]] on $AC$ such that $AD = n AB$ for some $n \in \Z$.
Then... | Let $AB$ be given.
Let the [[Definition:Straight Line|straight line]] through $AC$ be constructed.
Let $D_1$ be constructed on $AC$ such that $A D_1 = AB$ by constructing a [[Definition:Circle|circle]] whose [[Definition:Center of Circle|center]] is at $A$ and whose [[Definition:Radius of Circle|radius]] is $B$.
Let... | Construction of Integer Multiple of Line Segment | https://proofwiki.org/wiki/Construction_of_Integer_Multiple_of_Line_Segment | https://proofwiki.org/wiki/Construction_of_Integer_Multiple_of_Line_Segment | [
"Euclidean Geometry",
"Compass and Straightedge Constructions"
] | [
"Definition:Line/Segment",
"Definition:Plane Surface",
"Definition:Line/Segment",
"Definition:Plane Surface",
"Definition:Point",
"Definition:Point",
"Definition:Constructible Point in Plane",
"Definition:Compass and Straightedge Construction"
] | [
"Definition:Line/Straight Line",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Point",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"File:Construction of Integer Multiple of Line Segment.png",
"Category:Euclidean Geometry",... |
proofwiki-15825 | Point in Plane is Constructible iff Coordinates in Extension of Degree Power of 2 | Let $\CC$ be a Cartesian plane.
Let $S$ be a set of points in $\CC$.
Let $F$ be the smallest field containing all the coordinates of the points in $S$.
Let $P = \tuple {a, b}$ be a point in $\CC$.
Then:
:$P$ is constructible from $S$ using a compass and straightedge construction
{{iff}}:
:the coordinates of $P$ are con... | A point $P$ is constructed in a compass and straightedge construction from one of $3$ basic operations:
:$(1): \quad$ the intersection of $2$ straight lines
:$(2): \quad$ the intersection of a straight line and the circumference of a circle
:$(3): \quad$ the intersection of the circumferences of $2$ circle.
Let $A$, $B... | Let $\CC$ be a [[Definition:Cartesian Plane|Cartesian plane]].
Let $S$ be a [[Definition:Set|set]] of [[Definition:Point|points]] in $\CC$.
Let $F$ be the smallest [[Definition:Field (Abstract Algebra)|field]] containing all the [[Definition:Cartesian Coordinate System|coordinates]] of the [[Definition:Point|points]]... | A [[Definition:Point|point]] $P$ is constructed in a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]] from one of $3$ basic operations:
:$(1): \quad$ the [[Definition:Intersection (Geometry)|intersection]] of $2$ [[Definition:Straight Line|straight lines]]
:$(2): \quad$ the [... | Point in Plane is Constructible iff Coordinates in Extension of Degree Power of 2 | https://proofwiki.org/wiki/Point_in_Plane_is_Constructible_iff_Coordinates_in_Extension_of_Degree_Power_of_2 | https://proofwiki.org/wiki/Point_in_Plane_is_Constructible_iff_Coordinates_in_Extension_of_Degree_Power_of_2 | [
"Analytic Geometry",
"Compass and Straightedge Constructions"
] | [
"Definition:Cartesian Plane",
"Definition:Set",
"Definition:Point",
"Definition:Field (Abstract Algebra)",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Point",
"Definition:Constructible Point in Plane",
"Definition:Compass and Straightedge Construction",
"Definition:Ca... | [
"Definition:Point",
"Definition:Compass and Straightedge Construction",
"Definition:Intersection (Geometry)",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Line/Straight Line",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Intersection (Geom... |
proofwiki-15826 | Construction of Lattice Point in Cartesian Plane | Let $\CC$ be a Cartesian plane.
Let $P = \tuple {a, b}$ be a lattice point in $\CC$.
Then $P$ is constructible using a compass and straightedge construction. | Let $O$ denote the point $\tuple {0, 0}$.
Let $A$ denote the point $\tuple {1, 0}$.
The $x$-axis is identified with the straight line through $O$ and $A$.
The $y$-axis is constructed as the line perpendicular to $OA$ through $O$.
From Construction of Integer Multiple of Line Segment, the point $\tuple {a, 0}$ is constr... | Let $\CC$ be a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {a, b}$ be a [[Definition:Lattice Point of Cartesian Coordinate System|lattice point]] in $\CC$.
Then $P$ is [[Definition:Constructible Point in Plane|constructible]] using a [[Definition:Compass and Straightedge Construction|compass and ... | Let $O$ denote the [[Definition:Point|point]] $\tuple {0, 0}$.
Let $A$ denote the [[Definition:Point|point]] $\tuple {1, 0}$.
The [[Definition:X-Axis|$x$-axis]] is identified with the [[Definition:Straight Line|straight line]] through $O$ and $A$.
The [[Definition:Y-Axis|$y$-axis]] is constructed as the [[Definitio... | Construction of Lattice Point in Cartesian Plane | https://proofwiki.org/wiki/Construction_of_Lattice_Point_in_Cartesian_Plane | https://proofwiki.org/wiki/Construction_of_Lattice_Point_in_Cartesian_Plane | [
"Analytic Geometry",
"Compass and Straightedge Constructions"
] | [
"Definition:Cartesian Plane",
"Definition:Lattice Point/Cartesian Coordinate System",
"Definition:Constructible Point in Plane",
"Definition:Compass and Straightedge Construction"
] | [
"Definition:Point",
"Definition:Point",
"Definition:Axis/X-Axis",
"Definition:Line/Straight Line",
"Definition:Axis/Y-Axis",
"Definition:Right Angle/Perpendicular",
"Construction of Integer Multiple of Line Segment",
"Definition:Point",
"Definition:Circle",
"Definition:Circle/Center",
"Definitio... |
proofwiki-15827 | Even Integers do not form Integral Domain | Let $2 \Z$ be the set of even integers.
Then $\struct {2 \Z, +, \times}$ is not an integral domain. | From Integer Multiples form Commutative Ring, $\struct {2 \Z, +, \times}$ is a commutative ring.
As $2 \ne 1$, we also have from Integer Multiples form Commutative Ring that $\struct {2 \Z, +, \times}$ has no unity.
Hence by definition it is not an integral domain.
{{qed}} | Let $2 \Z$ be the [[Definition:Even Integer|set of even integers]].
Then $\struct {2 \Z, +, \times}$ is not an [[Definition:Integral Domain|integral domain]]. | From [[Integer Multiples form Commutative Ring]], $\struct {2 \Z, +, \times}$ is a [[Definition:Commutative Ring|commutative ring]].
As $2 \ne 1$, we also have from [[Integer Multiples form Commutative Ring]] that $\struct {2 \Z, +, \times}$ has no [[Definition:Unity of Ring|unity]].
Hence by definition it is not an ... | Even Integers do not form Integral Domain | https://proofwiki.org/wiki/Even_Integers_do_not_form_Integral_Domain | https://proofwiki.org/wiki/Even_Integers_do_not_form_Integral_Domain | [
"Integers",
"Integral Domains"
] | [
"Definition:Even Integer",
"Definition:Integral Domain"
] | [
"Integer Multiples form Commutative Ring",
"Definition:Commutative Ring",
"Integer Multiples form Commutative Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Integral Domain"
] |
proofwiki-15828 | Rational Numbers with Denominator Power of Two form Integral Domain | Let $\Q$ denote the set of rational numbers.
Let $S \subseteq \Q$ denote the set of set of rational numbers of the form $\dfrac p q$ where $q$ is a power of $2$:
:$S = \set {\dfrac p q: p \in \Z, q \in \set {2^m: m \in \Z_{\ge 0} } }$
Then $\struct {S, +, \times}$ is an integral domain. | From Rational Numbers form Integral Domain we have that $\struct {\Q, +, \times}$ is an integral domain.
Hence to demonstrate that $\struct {S, +, \times}$ is an integral domain, we can use the Subdomain Test.
We have that the unity of $\struct {\Q, +, \times}$ is $1$.
Then we note:
:$1 = \dfrac 1 1$
and:
:$1 = 2^0$
an... | Let $\Q$ denote the [[Definition:Rational Number|set of rational numbers]].
Let $S \subseteq \Q$ denote the [[Definition:Set|set]] of [[Definition:Rational Number|set of rational numbers]] of the form $\dfrac p q$ where $q$ is a [[Definition:Integer Power|power of $2$]]:
:$S = \set {\dfrac p q: p \in \Z, q \in \set {2... | From [[Rational Numbers form Integral Domain]] we have that $\struct {\Q, +, \times}$ is an [[Definition:Integral Domain|integral domain]].
Hence to demonstrate that $\struct {S, +, \times}$ is an [[Definition:Integral Domain|integral domain]], we can use the [[Subdomain Test]].
We have that the [[Definition:Unity o... | Rational Numbers with Denominator Power of Two form Integral Domain | https://proofwiki.org/wiki/Rational_Numbers_with_Denominator_Power_of_Two_form_Integral_Domain | https://proofwiki.org/wiki/Rational_Numbers_with_Denominator_Power_of_Two_form_Integral_Domain | [
"Rational Numbers",
"Integral Domains"
] | [
"Definition:Rational Number",
"Definition:Set",
"Definition:Rational Number",
"Definition:Power (Algebra)/Integer",
"Definition:Integral Domain"
] | [
"Rational Numbers form Integral Domain",
"Definition:Integral Domain",
"Definition:Integral Domain",
"Subdomain Test",
"Definition:Unity (Abstract Algebra)/Ring",
"Subdomain Test",
"Definition:Subring",
"Subdomain Test",
"Subring Test",
"Subring Test",
"Subring Test",
"Subring Test"
] |
proofwiki-15829 | Quadratic Integers over 3 form Integral Domain | Let $\R$ denote the set of real numbers.
Let $\Z \sqbrk {\sqrt 3} \subseteq \R$ denote the set of quadratic integers over $3$:
:$\Z \sqbrk {\sqrt 3} = \set {a + b \sqrt 3: a, b \in \Z}$
Then $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is an integral domain. | From Real Numbers form Integral Domain we have that $\struct {\R, +, \times}$ is an integral domain.
Hence to demonstrate that $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is an integral domain, we can use the Subdomain Test.
We have that the unity of $\struct {\R, +, \times}$ is $1$.
Then we note:
:$1 = 1 + 0 \times \sq... | Let $\R$ denote the [[Definition:Real Number|set of real numbers]].
Let $\Z \sqbrk {\sqrt 3} \subseteq \R$ denote the [[Definition:Set|set]] of [[Definition:Quadratic Integer|quadratic integers]] over $3$:
:$\Z \sqbrk {\sqrt 3} = \set {a + b \sqrt 3: a, b \in \Z}$
Then $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is ... | From [[Real Numbers form Integral Domain]] we have that $\struct {\R, +, \times}$ is an [[Definition:Integral Domain|integral domain]].
Hence to demonstrate that $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is an [[Definition:Integral Domain|integral domain]], we can use the [[Subdomain Test]].
We have that the [[Defi... | Quadratic Integers over 3 form Integral Domain | https://proofwiki.org/wiki/Quadratic_Integers_over_3_form_Integral_Domain | https://proofwiki.org/wiki/Quadratic_Integers_over_3_form_Integral_Domain | [
"Real Numbers",
"Examples of Integral Domains",
"Quadratic Integers"
] | [
"Definition:Real Number",
"Definition:Set",
"Definition:Algebraic Integer/Quadratic",
"Definition:Integral Domain"
] | [
"Real Numbers form Integral Domain",
"Definition:Integral Domain",
"Definition:Integral Domain",
"Subdomain Test",
"Definition:Unity (Abstract Algebra)/Ring",
"Subdomain Test",
"Definition:Subring",
"Subdomain Test",
"Subring Test",
"Subring Test",
"Subring Test",
"Subring Test"
] |
proofwiki-15830 | Normal Subgroup of Order 25 in Group of Order 100 | Let $G$ be a group of order $100$.
Then $G$ has a normal subgroup of order $25$. | Let $r$ be the number of Sylow $p$-subgroup of order $5^2 = 25$
The First Sylow Theorem guarantees existence, so $r \ge 1$.
From the Fourth Sylow Theorem:
:$r \equiv 1 \pmod 5$
That is:
:$r \in \set {1, 6, 11, 16, \ldots}$
From the Fifth Sylow Theorem:
:$r \divides \dfrac {100} {25} = 4$
from which it follows that $r =... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $100$.
Then $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $25$. | Let $r$ be the number of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of [[Definition:Order of Group|order]] $5^2 = 25$
The [[First Sylow Theorem]] guarantees existence, so $r \ge 1$.
From the [[Fourth Sylow Theorem]]:
:$r \equiv 1 \pmod 5$
That is:
:$r \in \set {1, 6, 11, 16, \ldots}$
From the [[Fifth Sylow ... | Normal Subgroup of Order 25 in Group of Order 100 | https://proofwiki.org/wiki/Normal_Subgroup_of_Order_25_in_Group_of_Order_100 | https://proofwiki.org/wiki/Normal_Subgroup_of_Order_25_in_Group_of_Order_100 | [
"Groups of Order 100"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure"
] | [
"Definition:Sylow p-Subgroup",
"Definition:Order of Structure",
"First Sylow Theorem",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup"
] |
proofwiki-15831 | Normal p-Subgroup contained in All Sylow p-Subgroups | Let $G$ be a finite group.
Let $p$ be a prime number.
Let $H$ be a normal subgroup of $G$ which is a $p$-group.
Then $H$ is a subset of every Sylow $p$-subgroup of $G$. | Let $P$ be a Sylow $p$-subgroup of $G$.
By Second Sylow Theorem, $H$ is a subset of a conjugate of $P$.
Then:
:$\exists g \in G: H \subseteq g P g^{-1}$.
This implies:
{{begin-eqn}}
{{eqn | l = H
| r = g^{-1} H g
| c = Subgroup equals Conjugate iff Normal
}}
{{eqn | o = \subseteq
| r = g^{-1} \paren {... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let $H$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$ which is a [[Definition:P-Group|$p$-group]].
Then $H$ is a [[Definition:Subset|subset]] of every [[Definition:Sylow p-Subgroup|Sylow $p$-su... | Let $P$ be a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G$.
By [[Second Sylow Theorem]], $H$ is a [[Definition:Subset|subset]] of a [[Definition:Conjugate of Group Subset|conjugate]] of $P$.
Then:
:$\exists g \in G: H \subseteq g P g^{-1}$.
This implies:
{{begin-eqn}}
{{eqn | l = H
| r = g^{-1} H... | Normal p-Subgroup contained in All Sylow p-Subgroups | https://proofwiki.org/wiki/Normal_p-Subgroup_contained_in_All_Sylow_p-Subgroups | https://proofwiki.org/wiki/Normal_p-Subgroup_contained_in_All_Sylow_p-Subgroups | [
"P-Groups",
"Sylow p-Subgroups"
] | [
"Definition:Finite Group",
"Definition:Prime Number",
"Definition:Normal Subgroup",
"Definition:P-Group",
"Definition:Subset",
"Definition:Sylow p-Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"Second Sylow Theorem",
"Definition:Subset",
"Definition:Conjugate (Group Theory)/Subset",
"Subgroup equals Conjugate iff Normal",
"Subset Relation is Compatible with Subset Product/Corollary 2",
"Definition:Subset",
"Definition:Sylow p-Subgroup"
] |
proofwiki-15832 | Existence of Subgroup whose Index is Prime Power | Let $G$ be a finite group.
Let $H$ be a normal subgroup of $G$ which has a finite index in $G$.
Let:
:$p^k \divides \index G H$
where:
:$p$ is a prime number
:$k \in \Z_{>0}$ is a (strictly) positive integer
:$\divides$ denotes divisibility.
Then $G$ contains a subgroup $K$ such that:
:$\index K H = p^k$ | The order $\order {G / H}$ of the quotient group $G / H$ is $\index G H$.
Hence $p^k$ divides $\order {G / H}$.
By Group has Subgroups of All Prime Power Factors, $G / H$ has a subgroup of order $p^k$.
By Correspondence Theorem, this subgroup is in the form $K / H$ where $H \le K \le G$.
Hence the result.
{{qed}} | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $H$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$ which has a [[Definition:Finite Index|finite index]] in $G$.
Let:
:$p^k \divides \index G H$
where:
:$p$ is a [[Definition:Prime Number|prime number]]
:$k \in \Z_{>0}$ is a [[Definition:Strictly P... | The [[Definition:Order of Group|order]] $\order {G / H}$ of the [[Definition:Quotient Group|quotient group]] $G / H$ is $\index G H$.
Hence $p^k$ [[Definition:Divisor of Integer|divides]] $\order {G / H}$.
By [[Group has Subgroups of All Prime Power Factors]], $G / H$ has a [[Definition:Subgroup|subgroup]] of order $... | Existence of Subgroup whose Index is Prime Power | https://proofwiki.org/wiki/Existence_of_Subgroup_whose_Index_is_Prime_Power | https://proofwiki.org/wiki/Existence_of_Subgroup_whose_Index_is_Prime_Power | [
"P-Groups",
"Sylow p-Subgroups"
] | [
"Definition:Finite Group",
"Definition:Normal Subgroup",
"Definition:Index of Subgroup/Finite",
"Definition:Prime Number",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup"
] | [
"Definition:Order of Structure",
"Definition:Quotient Group",
"Definition:Divisor (Algebra)/Integer",
"Group has Subgroups of All Prime Power Factors",
"Definition:Subgroup",
"Correspondence Theorem (Group Theory)",
"Definition:Subgroup"
] |
proofwiki-15833 | Group of Order p^2 q is not Simple | Let $p$ and $q$ be prime numbers such that $p \ne q$.
Let $G$ be a group of order $p^2 q$.
Then $G$ is not simple. | From Group of Order $p^2 q$ has Normal Sylow $p$-Subgroup, $G$ has a normal subgroup of order $p^2$.
Hence the result, by definition of simple group.
{{qed}} | Let $p$ and $q$ be [[Definition:Prime Number|prime numbers]] such that $p \ne q$.
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $p^2 q$.
Then $G$ is not [[Definition:Simple Group|simple]]. | From [[Group of Order p^2 q has Normal Sylow p-Subgroup|Group of Order $p^2 q$ has Normal Sylow $p$-Subgroup]], $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order $p^2$]].
Hence the result, by definition of [[Definition:Simple Group|simple group]].
{{qed}} | Group of Order p^2 q is not Simple | https://proofwiki.org/wiki/Group_of_Order_p^2_q_is_not_Simple | https://proofwiki.org/wiki/Group_of_Order_p^2_q_is_not_Simple | [
"Groups of Order p^2 q",
"Simple Groups"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Simple Group"
] | [
"Group of Order p^2 q has Normal Sylow p-Subgroup",
"Definition:Normal Subgroup",
"Definition:Order of Structure",
"Definition:Simple Group"
] |
proofwiki-15834 | Simple Group of Order Less than 60 is Prime | Let $G$ be a simple group.
Let $\order G < 60$, where $\order G$ denotes the order of $G$.
Then $G$ is a prime group. | First it is noted that Prime Group is Simple.
We also note from Alternating Group is Simple except on 4 Letters that the alternating group $A_5$, which is of order $60$, is simple.
Hence the motivation for the result.
It remains to be shown that all groups of composite order such that $\order G < 60$ are not simple.
Le... | Let $G$ be a [[Definition:Simple Group|simple group]].
Let $\order G < 60$, where $\order G$ denotes the [[Definition:Order of Group|order]] of $G$.
Then $G$ is a [[Definition:Prime Group|prime group]]. | First it is noted that [[Prime Group is Simple]].
We also note from [[Alternating Group is Simple except on 4 Letters]] that the [[Definition:Alternating Group|alternating group $A_5$]], which is of [[Definition:Order of Group|order]] $60$, is [[Definition:Simple Group|simple]].
Hence the motivation for the result.
... | Simple Group of Order Less than 60 is Prime | https://proofwiki.org/wiki/Simple_Group_of_Order_Less_than_60_is_Prime | https://proofwiki.org/wiki/Simple_Group_of_Order_Less_than_60_is_Prime | [
"Simple Groups",
"Groups of Order 60"
] | [
"Definition:Simple Group",
"Definition:Order of Structure",
"Definition:Prime Group"
] | [
"Prime Group is Simple",
"Alternating Group is Simple except on 4 Letters",
"Definition:Alternating Group",
"Definition:Order of Structure",
"Definition:Simple Group",
"Definition:Group",
"Definition:Composite Number",
"Definition:Order of Structure",
"Definition:Simple Group",
"Definition:Set",
... |
proofwiki-15835 | Group of Order 42 has Normal Subgroup of Order 7 | Let $G$ be of order $42$.
Then $G$ has a normal subgroup of order $7$. | We have that:
:$42 = 2 \times 3 \times 7$
From the First Sylow Theorem, $G$ has at least one Sylow $7$-subgroup, which is of order $7$.
Let $n_7$ denote the number of Sylow $7$-subgroups of $G$.
From the Fourth Sylow Theorem:
:$n_7 \equiv 1 \pmod 7$
and from the Fifth Sylow Theorem:
:$n_7 \divides 6$
where $\divides$ d... | Let $G$ be of [[Definition:Order of Group|order]] $42$.
Then $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $7$. | We have that:
:$42 = 2 \times 3 \times 7$
From the [[First Sylow Theorem]], $G$ has at least one [[Definition:Sylow p-Subgroup|Sylow $7$-subgroup]], which is of [[Definition:Order of Group|order]] $7$.
Let $n_7$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $7$-subgroups]] of $G$.
From the [[Fourth Sylow... | Group of Order 42 has Normal Subgroup of Order 7 | https://proofwiki.org/wiki/Group_of_Order_42_has_Normal_Subgroup_of_Order_7 | https://proofwiki.org/wiki/Group_of_Order_42_has_Normal_Subgroup_of_Order_7 | [
"Groups of Order 42"
] | [
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure"
] | [
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Sylow p-Subgroup",
"Definition:Normal Su... |
proofwiki-15836 | Group of Order 54 has Normal Subgroup of Order 27 | Let $G$ be of order $54$.
Then $G$ has a normal subgroup of order $27$. | We have that:
:$54 = 2 \times 3^3$
From the First Sylow Theorem, $G$ has at least one Sylow $3$-subgroup, which is of order $3^3 = 27$.
Let $n_3$ denote the number of Sylow $3$-subgroups of $G$.
From the Fourth Sylow Theorem:
:$n_3 \equiv 1 \pmod 3$
and from the Fifth Sylow Theorem:
:$n_3 \divides 2$
where $\divides$ d... | Let $G$ be of [[Definition:Order of Group|order]] $54$.
Then $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $27$. | We have that:
:$54 = 2 \times 3^3$
From the [[First Sylow Theorem]], $G$ has at least one [[Definition:Sylow p-Subgroup|Sylow $3$-subgroup]], which is of [[Definition:Order of Group|order]] $3^3 = 27$.
Let $n_3$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $3$-subgroups]] of $G$.
From the [[Fourth Sylow... | Group of Order 54 has Normal Subgroup of Order 27 | https://proofwiki.org/wiki/Group_of_Order_54_has_Normal_Subgroup_of_Order_27 | https://proofwiki.org/wiki/Group_of_Order_54_has_Normal_Subgroup_of_Order_27 | [
"Groups of Order 54"
] | [
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure"
] | [
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Sylow p-Subgroup",
"Definition:Normal Su... |
proofwiki-15837 | Group of Order 40 has Normal Subgroup of Order 5 | Let $G$ be of order $40$.
Then $G$ has a normal subgroup of order $5$. | We have that:
:$40 = 2^3 \times 5$
From the First Sylow Theorem, $G$ has at least one Sylow $5$-subgroup, which is of order $5$.
Let $n_5$ denote the number of Sylow $5$-subgroups of $G$.
From the Fourth Sylow Theorem:
:$n_5 \equiv 1 \pmod 5$
and from the Fifth Sylow Theorem:
:$n_5 \divides 8$
where $\divides$ denotes ... | Let $G$ be of [[Definition:Order of Group|order]] $40$.
Then $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $5$. | We have that:
:$40 = 2^3 \times 5$
From the [[First Sylow Theorem]], $G$ has at least one [[Definition:Sylow p-Subgroup|Sylow $5$-subgroup]], which is of [[Definition:Order of Group|order]] $5$.
Let $n_5$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $5$-subgroups]] of $G$.
From the [[Fourth Sylow Theore... | Group of Order 40 has Normal Subgroup of Order 5 | https://proofwiki.org/wiki/Group_of_Order_40_has_Normal_Subgroup_of_Order_5 | https://proofwiki.org/wiki/Group_of_Order_40_has_Normal_Subgroup_of_Order_5 | [
"Groups of Order 40"
] | [
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure"
] | [
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Sylow p-Subgroup",
"Definition:Normal Su... |
proofwiki-15838 | Finite Group with One Sylow p-Subgroup per Prime Divisor is Isomorphic to Direct Product | Let $G$ be a finite group whose order is $n$ and whose identity element is $e$.
Let $G$ be such that it has exactly $1$ Sylow $p$-subgroup for each prime divisor of $n$.
Then $G$ is isomorphic to the internal direct product of all its Sylow $p$-subgroups. | If each of the Sylow $p$-subgroups are unique, they are all normal.
As the order of each one is coprime to each of the others, their intersection is $\set e$.
{{finish|It remains to be shown that the direct product is what is is}} | Let $G$ be a [[Definition:Finite Group|finite group]] whose [[Definition:Order of Group|order]] is $n$ and whose [[Definition:Identity Element|identity element]] is $e$.
Let $G$ be such that it has exactly $1$ [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] for each [[Definition:Prime Divisor|prime divisor]] of $n$... | If each of the [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] are [[Definition:Unique|unique]], they are all [[Definition:Normal Subgroup|normal]].
As the [[Definition:Order of Group|order]] of each one is [[Definition:Coprime Integers|coprime]] to each of the others, their [[Definition:Set Intersection|intersect... | Finite Group with One Sylow p-Subgroup per Prime Divisor is Isomorphic to Direct Product | https://proofwiki.org/wiki/Finite_Group_with_One_Sylow_p-Subgroup_per_Prime_Divisor_is_Isomorphic_to_Direct_Product | https://proofwiki.org/wiki/Finite_Group_with_One_Sylow_p-Subgroup_per_Prime_Divisor_is_Isomorphic_to_Direct_Product | [
"Sylow p-Subgroups"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Sylow p-Subgroup",
"Definition:Prime Factor",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Internal Group Direct Product",
"Definition:Sylo... | [
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Order of Structure",
"Definition:Coprime/Integers",
"Definition:Set Intersection"
] |
proofwiki-15839 | Number of Subgroups of Prime Power Order is Congruent to 1 modulo Prime | Let $G$ be a finite group whose order is $n$.
Let $p$ be a prime number such that $p^k$ is a divisor of $n$.
Then the number of subgroups of order $p^k$ is congruent to $1$ modulo $p$. | Let $r_k$ denote the number of subgroups of $G$ of order $p^k$.
From Number of Order p Elements in Group with m Order p Subgroups:
:The number of elements of order $p$ is $r_1 (p - 1)$.
Since there exists $mp$ elements whose order divide $p$, for some $m \in \Z$:
:The number of elements of order $p$ is $mp - 1$.
Thus,
... | Let $G$ be a [[Definition:Finite Group|finite group]] whose [[Definition:Order of Group|order]] is $n$.
Let $p$ be a [[Definition:Prime Number|prime number]] such that $p^k$ is a [[Definition:Divisor of Integer|divisor]] of $n$.
Then the number of [[Definition:Subgroup|subgroups]] of [[Definition:Order of Group|orde... | Let $r_k$ denote the number of [[Definition:Subgroup|subgroups]] of $G$ of [[Definition:Order of Group|order]] $p^k$.
From [[Number of Order p Elements in Group with m Order p Subgroups]]:
:The number of [[Definition:Element|elements]] of [[Definition:Order of Group|order]] $p$ is $r_1 (p - 1)$.
Since there exists $... | Number of Subgroups of Prime Power Order is Congruent to 1 modulo Prime | https://proofwiki.org/wiki/Number_of_Subgroups_of_Prime_Power_Order_is_Congruent_to_1_modulo_Prime | https://proofwiki.org/wiki/Number_of_Subgroups_of_Prime_Power_Order_is_Congruent_to_1_modulo_Prime | [
"Sylow p-Subgroups"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Definition:Subgroup",
"Definition:Order of Structure",
"Number of Order p Elements in Group with m Order p Subgroups",
"Definition:Element",
"Definition:Order of Structure",
"Definition:Element",
"Definition:Order of Structure",
"Definition:Element",
"Definition:Order of Structure",
"Definition:S... |
proofwiki-15840 | Count of Distinct Homomorphisms between Additive Groups of Integers Modulo m | Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
Let $\struct {\Z_m, +}$ denote the additive group of integers modulo $m$.
The number of distinct homomorphisms $\phi: \struct {\Z_m, +} \to \struct {\Z_n, +}$ is $\gcd \set {m, n}$. | {{MissingLinks|Missing a lot of links I think...especially the last step. The result should exist in {{ProofWiki}}, but I don't know where it is.}}
$\Z_m$ is isomorphic to the quotient group $\Z / m\Z$.
By Universal Property of Quotient Group, to give a group homomorphism from $\Z_m$ to $\Z_n$ is equivalent to give a h... | Let $m, n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\struct {\Z_m, +}$ denote the [[Definition:Additive Group of Integers Modulo m|additive group of integers modulo $m$]].
The number of distinct [[Definition:Group Homomorphism|homomorphisms]] $\phi: \struct {\Z_m, +}... | {{MissingLinks|Missing a lot of links I think...especially the last step. The result should exist in {{ProofWiki}}, but I don't know where it is.}}
$\Z_m$ is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Quotient Group|quotient group]] $\Z / m\Z$.
By [[Universal Property of Quotient Group]], to give... | Count of Distinct Homomorphisms between Additive Groups of Integers Modulo m | https://proofwiki.org/wiki/Count_of_Distinct_Homomorphisms_between_Additive_Groups_of_Integers_Modulo_m | https://proofwiki.org/wiki/Count_of_Distinct_Homomorphisms_between_Additive_Groups_of_Integers_Modulo_m | [
"Additive Groups of Integers Modulo m"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Additive Group of Integers Modulo m",
"Definition:Group Homomorphism",
"Definition:Greatest Common Divisor"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Quotient Group",
"Universal Property of Quotient Group",
"Definition:Group Homomorphism",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Subgroup",
"Homomorphism of Generated Group",
"... |
proofwiki-15841 | Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17 | Let $\struct {\Z_{16}, +}$ denote the additive group of integers modulo $16$.
Let $\struct {\Z'_{17}, \times}$ denote the multiplicative group of reduced residues modulo $17$.
Let $\phi: \struct {\Z_{16}, +} \to \struct {\Z'_{17}, \times}$ be the mapping defined as:
:$\forall \eqclass k {16} \in \struct {\Z_{16}, +}: \... | Let $\eqclass x {16}, \eqclass y {16} \in \struct {\Z_{16}, +}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi {\eqclass x {16} } \times \map \phi {\eqclass y {16} }
| r = \map \phi {x + 16 m_1} \times \map \phi {y + 16 m_2}
| c = {{Defof|Residue Class}}: for some representative $m_1, m_2 \in \Z$
}}
{{eqn | r =... | Let $\struct {\Z_{16}, +}$ denote the [[Definition:Additive Group of Integers Modulo m|additive group of integers modulo $16$]].
Let $\struct {\Z'_{17}, \times}$ denote the [[Definition:Multiplicative Group of Reduced Residues|multiplicative group of reduced residues modulo $17$]].
Let $\phi: \struct {\Z_{16}, +} \to... | Let $\eqclass x {16}, \eqclass y {16} \in \struct {\Z_{16}, +}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi {\eqclass x {16} } \times \map \phi {\eqclass y {16} }
| r = \map \phi {x + 16 m_1} \times \map \phi {y + 16 m_2}
| c = {{Defof|Residue Class}}: for some representative $m_1, m_2 \in \Z$
}}
{{eqn | r... | Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17 | https://proofwiki.org/wiki/Isomorphism_between_Additive_Group_Modulo_16_and_Multiplicative_Group_Modulo_17 | https://proofwiki.org/wiki/Isomorphism_between_Additive_Group_Modulo_16_and_Multiplicative_Group_Modulo_17 | [
"Additive Groups of Integers Modulo m",
"Multiplicative Groups of Reduced Residues",
"Groups of Order 16",
"Examples of Group Isomorphisms"
] | [
"Definition:Additive Group of Integers Modulo m",
"Definition:Multiplicative Group of Reduced Residues",
"Definition:Mapping",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Power (Algebra)/Integer/Knuth Notation",
"Exponent Combination Laws/Product of Powers",
"Definition:Group Homomorphism",
"Definition:Bijection",
"Definition:Prime Number",
"Definition:Coprime/Integers",
"Definition:Multiplicative Group of Reduced Residues",
"Definition:Order of Structure",... |
proofwiki-15842 | P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1 | :$\forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$ | For any $n \in \N$ then:
{{begin-eqn}}
{{eqn | l = \norm {\alpha_{n + 1} - \alpha_n }_p
| r = \norm {\paren {\alpha_{n + 1} - x} + \paren {x - \alpha_n } }_p
}}
{{eqn | o = \le
| r = \max \set {\norm {\alpha_{n + 1} - x}_p, \: \norm {x - \alpha_n }_p }
| c = {{NormAxiomNonArch|4}}
}}
{{eqn | o = \le
... | :$\forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$ | For any $n \in \N$ then:
{{begin-eqn}}
{{eqn | l = \norm {\alpha_{n + 1} - \alpha_n }_p
| r = \norm {\paren {\alpha_{n + 1} - x} + \paren {x - \alpha_n } }_p
}}
{{eqn | o = \le
| r = \max \set {\norm {\alpha_{n + 1} - x}_p, \: \norm {x - \alpha_n }_p }
| c = {{NormAxiomNonArch|4}}
}}
{{eqn | o = \le
... | P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1 | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers/Lemma_1 | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers/Lemma_1 | [
"P-adic Integer is Limit of Unique Coherent Sequence of Integers"
] | [] | [
"Properties of Norm on Division Ring/Norm of Negative"
] |
proofwiki-15843 | P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 3 | $\sequence {\alpha_n}$ is a unique sequence satisfying properties $(1)$, $(2)$ and $(3)$ above. | Suppose that there exists a sequence $\sequence {\alpha'_n}$ with:
:$(1'): \quad \forall n \in \N: \alpha'_n \in \Z$ and $0 \le \alpha'_n \le p^{n + 1} - 1$
:$(2'): \quad \forall n \in \N: \alpha'_{n + 1} \equiv \alpha'_n \pmod {p^{n + 1} }$
:$(3'): \quad \ds \lim_{n \mathop \to \infty} \alpha'_n = x$
{{AimForCont}}:
:... | $\sequence {\alpha_n}$ is a [[Definition:Unique|unique]] [[Definition:Sequence|sequence]] satisfying properties $(1)$, $(2)$ and $(3)$ above. | Suppose that there exists a [[Definition:Sequence|sequence]] $\sequence {\alpha'_n}$ with:
:$(1'): \quad \forall n \in \N: \alpha'_n \in \Z$ and $0 \le \alpha'_n \le p^{n + 1} - 1$
:$(2'): \quad \forall n \in \N: \alpha'_{n + 1} \equiv \alpha'_n \pmod {p^{n + 1} }$
:$(3'): \quad \ds \lim_{n \mathop \to \infty} \alpha'_... | P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 3 | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers/Lemma_3 | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers/Lemma_3 | [
"P-adic Integer is Limit of Unique Coherent Sequence of Integers"
] | [
"Definition:Unique",
"Definition:Sequence"
] | [
"Definition:Sequence",
"Initial Segment of Natural Numbers forms Complete Residue System",
"Definition:Limit of Sequence/Normed Division Ring",
"Properties of Norm on Division Ring/Norm of Negative",
"Definition:Contradiction"
] |
proofwiki-15844 | P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2 | :$\ds \lim_{n \mathop \to \infty} \alpha_n = x$ | From Sequence of Powers of Number less than One:
:$\ds \lim_{n \mathop \to \infty} p^{-n} = 0$
From Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} p^{-\paren {n + 1} } = 0$
By the Squeeze Theorem for Real Sequences :
:$\ds \lim_{n \mathop \to \infty} \norm {x - \alpha_n}_p = 0$.
Hence the limit of ... | :$\ds \lim_{n \mathop \to \infty} \alpha_n = x$ | From [[Sequence of Powers of Number less than One]]:
:$\ds \lim_{n \mathop \to \infty} p^{-n} = 0$
From [[Multiple Rule for Real Sequences]]:
:$\ds \lim_{n \mathop \to \infty} p^{-\paren {n + 1} } = 0$
By the [[Squeeze Theorem for Real Sequences]] :
:$\ds \lim_{n \mathop \to \infty} \norm {x - \alpha_n}_p = 0$.
Henc... | P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2 | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers/Lemma_2 | https://proofwiki.org/wiki/P-adic_Integer_is_Limit_of_Unique_Coherent_Sequence_of_Integers/Lemma_2 | [
"P-adic Integer is Limit of Unique Coherent Sequence of Integers"
] | [] | [
"Sequence of Powers of Number less than One",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Squeeze Theorem/Sequences/Real Numbers",
"Definition:Limit of Sequence/Normed Division Ring"
] |
proofwiki-15845 | Group Types of Order Prime Squared | Let $p$ be a prime number.
Let $G$ be a group of order $p^2$.
Then $G$ is isomorphic either to $\Z_{p^2}$ or to $\Z_p \times \Z_p$, where $\Z_p$ denotes the additive group of integers modulo $p$. | From Group of Order Prime Squared is Abelian, $G$ is an abelian group.
From Abelian Group of Prime-power Order is Product of Cyclic Groups, $G$ is either:
:the cyclic group of order $p^2$
:the direct product of the cyclic group of order $p$ with itself.
The result follows from Finite Cyclic Group is Isomorphic to Inte... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $p^2$.
Then $G$ is [[Definition:Group Isomorphism|isomorphic]] either to $\Z_{p^2}$ or to $\Z_p \times \Z_p$, where $\Z_p$ denotes the [[Definition:Additive Group of Integers Modulo m... | From [[Group of Order Prime Squared is Abelian]], $G$ is an [[Definition:Abelian Group|abelian group]].
From [[Abelian Group of Prime-power Order is Product of Cyclic Groups]], $G$ is either:
:the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $p^2$
:the [[Definition:Group Direct Produ... | Group Types of Order Prime Squared | https://proofwiki.org/wiki/Group_Types_of_Order_Prime_Squared | https://proofwiki.org/wiki/Group_Types_of_Order_Prime_Squared | [
"Groups of Order p^2"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Additive Group of Integers Modulo m"
] | [
"Group of Order Prime Squared is Abelian",
"Definition:Abelian Group",
"Abelian Group of Prime-power Order is Product of Cyclic Groups",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Group Direct Product",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Finite ... |
proofwiki-15846 | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers | $\struct {G, \circ}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$. | To prove $G$ is isomorphic to $\struct {\R, +}$, we need to find a bijective homorphism $\phi: \openint {-1} 1 \to \R$:
:$\forall x, y \in G: \map \phi {x \circ y} = \map \phi x + \map \phi y$
From Group Examples: $\dfrac {x + y} {1 + x y}$:
:the identity element of $G$ is $0$
:the inverse of $x$ in $G$ is $-x$.
This a... | $\struct {G, \circ}$ is [[Definition:Isomorphism|isomorphic]] to the [[Definition:Additive Group of Real Numbers|additive group of real numbers]] $\struct {\R, +}$. | To prove $G$ is [[Definition:Isomorphism|isomorphic]] to $\struct {\R, +}$, we need to find a [[Definition:Bijection|bijective]] [[Definition:Group Homomorphism|homorphism]] $\phi: \openint {-1} 1 \to \R$:
:$\forall x, y \in G: \map \phi {x \circ y} = \map \phi x + \map \phi y$
From [[Group/Examples/x+y over 1+xy|Gro... | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1 | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers/Proof_1 | [
"Examples of Groups/x+y over 1+xy"
] | [
"Definition:Isomorphism",
"Definition:Additive Group of Real Numbers"
] | [
"Definition:Isomorphism",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Group/Examples/x+y over 1+xy",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Odd Function",
"Definition:Real Interval/Open",
"Definition:Inv... |
proofwiki-15847 | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers | $\struct {G, \circ}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$. | To prove $G$ is isomorphic to $\struct {\R, +}$, it is sufficient to find a bijective homorphism $\phi: \to \R \to G$:
:$\forall x, y \in G: \map \phi {x + y} = \map \phi x \circ \map \phi y$
From Group Examples: $\dfrac {x + y} {1 + x y}$:
:the identity element of $G$ is $0$
:the inverse of $x$ in $G$ is $-x$.
This al... | $\struct {G, \circ}$ is [[Definition:Isomorphism|isomorphic]] to the [[Definition:Additive Group of Real Numbers|additive group of real numbers]] $\struct {\R, +}$. | To prove $G$ is [[Definition:Isomorphism|isomorphic]] to $\struct {\R, +}$, it is sufficient to find a [[Definition:Bijection|bijective]] [[Definition:Group Homomorphism|homorphism]] $\phi: \to \R \to G$:
:$\forall x, y \in G: \map \phi {x + y} = \map \phi x \circ \map \phi y$
From [[Group/Examples/x+y over 1+xy|Grou... | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2 | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers/Proof_2 | [
"Examples of Groups/x+y over 1+xy"
] | [
"Definition:Isomorphism",
"Definition:Additive Group of Real Numbers"
] | [
"Definition:Isomorphism",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Group/Examples/x+y over 1+xy",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Odd Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
... |
proofwiki-15848 | Open Set Characterization of Denseness/Analytic Basis | Let $\BB \subseteq \tau$ be an analytic basis for $\tau$.
Then $S$ is (everywhere) dense in $X$ {{iff}} every non-empty open set of $\BB$ contains an element of $S$. | === Necessary Condition ===
Let $S$ be everywhere dense in $X$.
By Open Set Characterization of Denseness then every non-empty open set contains an element of $S$.
Every non-empty set of an analytic basis is an open set by definition.
Hence every non-empty set of non-empty open set of $\BB$ contains an element of $S$.
... | Let $\BB \subseteq \tau$ be an [[Definition:Analytic Basis|analytic basis]] for $\tau$.
Then $S$ is [[Definition:Everywhere Dense|(everywhere) dense]] in $X$ {{iff}} every [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open]] set of $\BB$ contains an [[Definition:Element|element]] of $S$. | === Necessary Condition ===
Let $S$ be [[Definition:Everywhere Dense|everywhere dense]] in $X$.
By [[Open Set Characterization of Denseness]] then every [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open]] set contains an [[Definition:Element|element]] of $S$.
Every [[Definition:Non-Empty S... | Open Set Characterization of Denseness/Analytic Basis | https://proofwiki.org/wiki/Open_Set_Characterization_of_Denseness/Analytic_Basis | https://proofwiki.org/wiki/Open_Set_Characterization_of_Denseness/Analytic_Basis | [
"Denseness"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Everywhere Dense",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Element"
] | [
"Definition:Everywhere Dense",
"Open Set Characterization of Denseness",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Non-Empty Set",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definiti... |
proofwiki-15849 | Open Set Characterization of Denseness/Open Ball | Let $\struct {X, d}$ be a metric space.
Let $\tau_d$ be the topology induced by the metric $d$.
Let $S \subseteq X$.
Then $S$ is (everywhere) dense in $\struct {X, \tau_d}$ {{iff}} every open ball contains an element of $S$. | By Open Balls form Basis for Open Sets of Metric Space, the set of open balls are an analytic basis for the topology $\tau_d$.
By Analytic Basis Characterization of Denseness then:
:$S$ is (everywhere) dense in $\struct {X, \tau_d}$ {{iff}} every open ball contains an element of $S$.
{{qed}}
Category:Denseness
chjswen0... | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $\tau_d$ be the [[Definition:Topology Induced by Metric|topology induced]] by the [[Definition:Metric Space|metric]] $d$.
Let $S \subseteq X$.
Then $S$ is [[Definition:Everywhere Dense|(everywhere) dense]] in $\struct {X, \tau_d}$ {{iff}} every... | By [[Open Balls form Basis for Open Sets of Metric Space]], the [[Definition:Set|set]] of [[Definition:Open Ball of Metric Space|open balls]] are an [[Definition:Analytic Basis|analytic basis]] for the [[Definition:Topology|topology]] $\tau_d$.
By [[Analytic Basis Characterization of Denseness|Analytic Basis Character... | Open Set Characterization of Denseness/Open Ball | https://proofwiki.org/wiki/Open_Set_Characterization_of_Denseness/Open_Ball | https://proofwiki.org/wiki/Open_Set_Characterization_of_Denseness/Open_Ball | [
"Denseness"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Metric Space",
"Definition:Everywhere Dense",
"Definition:Open Ball",
"Definition:Element"
] | [
"Open Balls form Basis for Open Sets of Metric Space",
"Definition:Set",
"Definition:Open Ball",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Topology",
"Open Set Characterization of Denseness/Analytic Basis",
"Definition:Everywhere Dense",
"Definition:Open Ball",
"Definition:Element",
... |
proofwiki-15850 | P-adic Valuation Extends to P-adic Numbers | Let $p$ be a prime number.
Let $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on the set of rational numbers.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ be defined by:
:<nowiki>$\forall x \in \Q_p : \map {\nu_p} x = \begin {cases... | It needs to be shown that $\nu_p$:
:$(1): \quad \nu_p$ is a mapping into $\Z \cup \set {+\infty}$
:$(2): \quad \nu_p$ satisfies the valuation axioms $\text V 1$, $\text V 2$ and $\text V 3$
:$(3): \quad \nu_p$ extends $\nu_p^\Q$.
Let $x, y \in \Q_p$. | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ be the [[Definition:P-adic Valuation on Rational Numbers|$p$-adic valuation]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Defi... | It needs to be shown that $\nu_p$:
:$(1): \quad \nu_p$ is a [[Definition:Mapping|mapping]] into $\Z \cup \set {+\infty}$
:$(2): \quad \nu_p$ satisfies the [[Axiom:Valuation Axioms|valuation axioms $\text V 1$, $\text V 2$ and $\text V 3$]]
:$(3): \quad \nu_p$ [[Definition:Extension of Mapping|extends]] $\nu_p^\Q$.
Le... | P-adic Valuation Extends to P-adic Numbers | https://proofwiki.org/wiki/P-adic_Valuation_Extends_to_P-adic_Numbers | https://proofwiki.org/wiki/P-adic_Valuation_Extends_to_P-adic_Numbers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Valuation/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Valuation",
"Definition:Extension of Mapping"
] | [
"Definition:Mapping",
"Axiom:Valuation Axioms",
"Definition:Extension of Mapping",
"Definition:Mapping"
] |
proofwiki-15851 | P-adic Integers is Metric Completion of Integers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $d$ be the subspace metric of the $p$-adic metric on the $p$-adic integers $\Z_p$.
Then $\struct {\Z_p, d}$ is the metric completion of the integers $\Z$. | The integers $\Z$ are a subring of the $p$-adic integers $Z_p$ by Integers form Subring of P-adic Integers.
Hence $\Z \subseteq \Z_p$.
The set of $p$-adic integers $\Z_p$ is closed in the $p$-adic metric by Set of P-adic Integers is Clopen in P-adic Numbers.
By Closure of Subset of Closed Set of Metric Space is Subset ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $d$ be the [[Definition:Metric Subspace|subspace metric]] of the [[Definition:P-adic Metric on P-adic Numbers|$p$-adic metric]] on the [[Definitio... | The [[Definition:Integer|integers]] $\Z$ are a [[Definition:Subring|subring]] of the [[Definition:P-adic Integer|$p$-adic integers]] $Z_p$ by [[Integers form Subring of P-adic Integers]].
Hence $\Z \subseteq \Z_p$.
The [[Definition:Set|set]] of [[Definition:P-adic Integer|$p$-adic integers]] $\Z_p$ is [[Definition:Cl... | P-adic Integers is Metric Completion of Integers | https://proofwiki.org/wiki/P-adic_Integers_is_Metric_Completion_of_Integers | https://proofwiki.org/wiki/P-adic_Integers_is_Metric_Completion_of_Integers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Metric Subspace",
"Definition:P-adic Metric/P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Completion (Metric Space)",
"Definition:Integer"
] | [
"Definition:Integer",
"Definition:Subring",
"Definition:P-adic Integer",
"Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1",
"Definition:Set",
"Definition:P-adic Integer",
"Definition:Closed Set/Metric Space",
"Definition:P-adic Metric/P-adic Numbers",
"Valuation Ring of Non-A... |
proofwiki-15852 | P-adic Metric on P-adic Numbers is Non-Archimedean Metric | Let $p \in \N$ be a prime.
Let $\norm{\,\cdot\,}_p: \Q_p \to \R_{\ge 0}$ be the $p$-adic norm on the $p$-adic numbers $\Q_p$.
Let $d_p$ be the $p$-adic metric on $\Q_p$:
:$\forall x, y \in \Q_p: \map {d_p} {x, y} = \norm{x - y}_p$
Then $d_p$ is a non-Archimedean metric that extends the $p$-adic metric on the rationals ... | The $p$-adic metric on $\Q_p$ is defined as the metric induced by the $p$-adic norm on $\Q_p$.
It follows from Metric Induced by Norm is Metric that $d_p$ is a metric.
By definition of the $p$-adic norm on $\Q_p$, $\norm{\,\cdot\,}_p$ is a non-Archimedean norm.
From Non-Archimedean Norm iff Non-Archimedean Metric, then... | Let $p \in \N$ be a [[Definition:Prime Number|prime]].
Let $\norm{\,\cdot\,}_p: \Q_p \to \R_{\ge 0}$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] $\Q_p$.
Let $d_p$ be the [[Definition:P-adic Metric on P-adic Numbers|$p$-adic metric]] on $\Q_p$:... | The [[Definition:P-adic Metric on P-adic Numbers|$p$-adic metric]] on $\Q_p$ is defined as the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:P-adic Norm on P-adic Numbers|$p$-adic norm]] on $\Q_p$.
It follows from [[Metric Induced by Norm is Metric]] that $d_p$ is a [[Definition:Metric|metri... | P-adic Metric on P-adic Numbers is Non-Archimedean Metric | https://proofwiki.org/wiki/P-adic_Metric_on_P-adic_Numbers_is_Non-Archimedean_Metric | https://proofwiki.org/wiki/P-adic_Metric_on_P-adic_Numbers_is_Non-Archimedean_Metric | [
"P-adic Metrics"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Metric/P-adic Numbers",
"Definition:Non-Archimedean/Metric",
"Definition:Extension of Mapping",
"Definition:P-adic Metric",
"Definition:Rational Number"
] | [
"Definition:P-adic Metric/P-adic Numbers",
"Definition:Metric Induced by Norm",
"Definition:P-adic Norm/P-adic Numbers",
"Metric Induced by Norm is Metric",
"Definition:Metric Space/Metric",
"Definition:P-adic Norm/P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Non-Archimedean No... |
proofwiki-15853 | Borel-Cantelli Lemma 10 to Kochen-Stone Theorem | Let $A_n$ be a sequence of events with $\ds \sum \map \Pr {A_n} = \infty$ and:
:$\ds \liminf_{k \mathop \to \infty} \frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} < \infty$
Then there is a positive probability that $A_n$ occur infinit... | Fix $\ell < k$.
Let $\ds X = \sum_{n \mathop = \ell}^k 1_{A_n}$.
It follows that:
:$\ds \expect X = \sum_{n \mathop = \ell}^k \Pr(A_n)$
and:
:$\ds \expect {X^2} = \sum_{\ell \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m}$
Using the Paley-Zygmund Inequality for $\theta = 0$, we obtain:
{{begin-eqn}}
{{eqn | l =... | Let $A_n$ be a sequence of events with $\ds \sum \map \Pr {A_n} = \infty$ and:
:$\ds \liminf_{k \mathop \to \infty} \frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} < \infty$
Then there is a positive probability that $A_n$ occur infini... | Fix $\ell < k$.
Let $\ds X = \sum_{n \mathop = \ell}^k 1_{A_n}$.
It follows that:
:$\ds \expect X = \sum_{n \mathop = \ell}^k \Pr(A_n)$
and:
:$\ds \expect {X^2} = \sum_{\ell \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m}$
Using the [[Paley-Zygmund Inequality]] for $\theta = 0$, we obtain:
{{begin-eqn}}
{{... | Borel-Cantelli Lemma 10 to Kochen-Stone Theorem | https://proofwiki.org/wiki/Borel-Cantelli_Lemma_10_to_Kochen-Stone_Theorem | https://proofwiki.org/wiki/Borel-Cantelli_Lemma_10_to_Kochen-Stone_Theorem | [
"Kochen-Stone Theorem"
] | [] | [
"Paley-Zygmund Inequality",
"Category:Kochen-Stone Theorem"
] |
proofwiki-15854 | Order of Automorphism Group of Cyclic Group | Let $C_n$ denote the cyclic group of order $n$.
Let $\Aut {C_n}$ denote the automorphism group of $C_n$.
Then:
:$\order {\Aut {C_n} } = \map \phi n$
where:
:$\order {\, \cdot \,}$ denotes the order of a group
:$\map \phi n$ denotes the Euler $\phi$ function. | Let $g$ be a generator of $C_n$.
Let $\varphi$ be an automorphism on $C_n$.
By Homomorphic Image of Cyclic Group is Cyclic Group, $\map \varphi g$ is a generator of $C_n$.
By Homomorphism of Generated Group, $\varphi$ is uniquely determined by $\map \varphi g$.
By Finite Cyclic Group has Euler Phi Generators, there are... | Let $C_n$ denote the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $n$.
Let $\Aut {C_n}$ denote the [[Definition:Automorphism Group|automorphism group]] of $C_n$.
Then:
:$\order {\Aut {C_n} } = \map \phi n$
where:
:$\order {\, \cdot \,}$ denotes the [[Definition:Order of Group|order]... | Let $g$ be a [[Definition:Generator of Cyclic Group|generator]] of $C_n$.
Let $\varphi$ be an [[Definition:Group Automorphism|automorphism]] on $C_n$.
By [[Homomorphic Image of Cyclic Group is Cyclic Group]], $\map \varphi g$ is a [[Definition:Generator of Cyclic Group|generator]] of $C_n$.
By [[Homomorphism of Gen... | Order of Automorphism Group of Cyclic Group | https://proofwiki.org/wiki/Order_of_Automorphism_Group_of_Cyclic_Group | https://proofwiki.org/wiki/Order_of_Automorphism_Group_of_Cyclic_Group | [
"Automorphism Groups",
"Cyclic Groups"
] | [
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Automorphism Group",
"Definition:Order of Structure",
"Definition:Group",
"Definition:Euler Phi Function"
] | [
"Definition:Cyclic Group/Generator",
"Definition:Group Automorphism",
"Homomorphic Image of Cyclic Group is Cyclic Group",
"Definition:Cyclic Group/Generator",
"Homomorphism of Generated Group",
"Finite Cyclic Group has Euler Phi Generators",
"Definition:Group Automorphism"
] |
proofwiki-15855 | Kernel of Group Homomorphism is not Empty | Let $G$ and $H$ be groups whose identity elements are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a homomorphism from $G$ to $H$.
Let $\map \ker \phi$ denote the kernel of $\phi$.
Then:
:$\map \ker \phi \ne \O$
where $\O$ denotes the empty set. | From Identity is in Kernel of Group Homomorphism we have that:
:$e_G \in \map \ker \phi$
Hence the result.
{{Qed}} | Let $G$ and $H$ be [[Definition:Group|groups]] whose [[Definition:Identity Element|identity elements]] are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a [[Definition:Group Homomorphism|homomorphism]] from $G$ to $H$.
Let $\map \ker \phi$ denote the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$... | From [[Identity is in Kernel of Group Homomorphism]] we have that:
:$e_G \in \map \ker \phi$
Hence the result.
{{Qed}} | Kernel of Group Homomorphism is not Empty | https://proofwiki.org/wiki/Kernel_of_Group_Homomorphism_is_not_Empty | https://proofwiki.org/wiki/Kernel_of_Group_Homomorphism_is_not_Empty | [
"Kernels of Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Empty Set"
] | [
"Identity is in Kernel of Group Homomorphism"
] |
proofwiki-15856 | Valuation Ring of P-adic Norm on Rationals/Corollary 1 | The set of integers $\Z$ is a subring of $\OO$. | By Valuation Ring of P-adic Norm on Rationals, the induced valuation ring $\OO$ is the set:
:$\OO = \Z_{\paren p} = \set {\dfrac a b \in \Q : p \nmid b}$
Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \OO$.
Hence $\Z \subseteq \OO$.
By Valuation Ring of Non-Archimedean Division Ring is Subring then $\OO... | The [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is a [[Definition:Subring|subring]] of $\OO$. | By [[Valuation Ring of P-adic Norm on Rationals]], the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|induced valuation ring]] $\OO$ is the [[Definition:Set|set]]:
:$\OO = \Z_{\paren p} = \set {\dfrac a b \in \Q : p \nmid b}$
Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \OO$.
Hence $\Z ... | Valuation Ring of P-adic Norm on Rationals/Corollary 1 | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_on_Rationals/Corollary_1 | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_on_Rationals/Corollary_1 | [
"Valuation Ring of P-adic Norm on Rationals"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Subring"
] | [
"Valuation Ring of P-adic Norm on Rationals",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Set",
"Valuation Ring of Non-Archimedean Division Ring is Subring",
"Definition:Subring",
"Integers form Subdomain of Rationals",
"Definition:Subring",
"Intersection of Subrings is La... |
proofwiki-15857 | Valuation Ring of P-adic Norm is Subring of P-adic Integers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $\Z_{\ideal p}$ be the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$.
Then:
:$(1): \quad \Z_{\ideal p} = \Q \cap \Z_p$.
:$(2): \quad \Z_{\ideal p}$ is a subring of $\Z_p$... | The $p$-adic integers is defined as:
:$\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$
The induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$ is defined as:
:$\Z_{\ideal p} = \set {x \in \Q: \norm x_p \le 1}$
From Rational Numbers are Dense Subfield of P-adic Numbers:
:the $p$-adic norm $\norm {\,\cdot\,}_p$ on $\... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Let $\Z_{\ideal p}$ be the [[Definition:Valuation Ring Induced by Non-Archimedean ... | The [[Definition:P-adic Integer|$p$-adic integers]] is defined as:
:$\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$
The [[Definition:Valuation Ring Induced by Non-Archimedean Norm|induced valuation ring]] on $\struct {\Q,\norm {\,\cdot\,}_p}$ is defined as:
:$\Z_{\ideal p} = \set {x \in \Q: \norm x_p \le 1}$
From [[Ratio... | Valuation Ring of P-adic Norm is Subring of P-adic Integers | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_is_Subring_of_P-adic_Integers | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_is_Subring_of_P-adic_Integers | [
"P-adic Number Theory",
"Valuation Ring of P-adic Norm is Subring of P-adic Integers"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Subring"
] | [
"Definition:P-adic Integer",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Definition:P-adic Norm/P-adic Numbers",
"Definition:Extension of Mapping",
"Definition:P-adic Norm",
"Valuation Ring of Non-Archimedean Division Ring is Sub... |
proofwiki-15858 | Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1 | The set of integers $\Z$ is a subring of $\Z_p$. | Let $\Z_{\paren p}$ be the valuation ring induced by $\norm {\,\cdot\,}_p$ on $\Q$.
By Integers form Subring of Valuation Ring of P-adic Norm on Rationals then:
:$\Z$ is a subring of $\Z_{\paren p}$
By Valuation Ring of P-adic Norm is Subring of P-adic Integers then:
:$\Z_{\paren p}$ is a subring of $\Z_p$
The result f... | The [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is a [[Definition:Subring|subring]] of $\Z_p$. | Let $\Z_{\paren p}$ be the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]] by $\norm {\,\cdot\,}_p$ on $\Q$.
By [[Integers form Subring of Valuation Ring of P-adic Norm on Rationals]] then:
:$\Z$ is a [[Definition:Subring|subring]] of $\Z_{\paren p}$
By [[Valuation Ring of P-adic... | Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1 | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_is_Subring_of_P-adic_Integers/Corollary_1 | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_is_Subring_of_P-adic_Integers/Corollary_1 | [
"Valuation Ring of P-adic Norm is Subring of P-adic Integers"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Subring"
] | [
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Valuation Ring of P-adic Norm on Rationals/Corollary 1",
"Definition:Subring",
"Valuation Ring of P-adic Norm is Subring of P-adic Integers",
"Definition:Subring"
] |
proofwiki-15859 | Laplace Transform of 1 | Let $f: \R \to \R$ be the function defined as:
:$\forall t \in \R: \map f t = 1$
Then the Laplace transform of $\map f t$ is given by:
:$\laptrans {\map f t} = \dfrac 1 s$
for $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \laptrans 1
| c = Definition of $\map f t$
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r =... | Let $f: \R \to \R$ be the [[Definition:Real Function|function]] defined as:
:$\forall t \in \R: \map f t = 1$
Then the [[Definition:Laplace Transform|Laplace transform]] of $\map f t$ is given by:
:$\laptrans {\map f t} = \dfrac 1 s$
for $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \laptrans 1
| c = Definition of $\map f t$
}}
{{eqn | r = \int_0^{\to +\infty} e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r =... | Laplace Transform of 1/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_1 | https://proofwiki.org/wiki/Laplace_Transform_of_1/Proof_1 | [
"Laplace Transform of 1",
"Examples of Laplace Transforms"
] | [
"Definition:Real Function",
"Definition:Laplace Transform"
] | [
"Primitive of Exponential Function",
"Exponential of Zero",
"Complex Exponential Tends to Zero"
] |
proofwiki-15860 | Laplace Transform Exists if Function Piecewise Continuous and of Exponential Order | Let $f$ be a real function which is:
:piecewise continuous in every closed interval $\closedint 0 N$
:of exponential order $\gamma$ for $t > N$.
Then the Laplace transform $\map F s$ of $\map f t$ exists for all $s > \gamma$. | For all $N \in \Z_{>0}$:
:$\ds \int_0^\infty e^{-s t} \map f t \rd t = \int_0^N e^{-s t} \map f t \rd t + \int_N^\infty e^{-s t} \map f t \rd t$
We have that $f$ is piecewise continuous in every closed interval $\closedint 0 N$.
Hence the first of the integrals on the {{RHS}} exists.
Also, as $\map f t$ is of exponenti... | Let $f$ be a [[Definition:Real Function|real function]] which is:
:[[Definition:Piecewise Continuous Function with One-Sided Limits|piecewise continuous]] in every [[Definition:Closed Real Interval|closed interval]] $\closedint 0 N$
:of [[Definition:Exponential Order to Real Index|exponential order $\gamma$]] for $t > ... | For all $N \in \Z_{>0}$:
:$\ds \int_0^\infty e^{-s t} \map f t \rd t = \int_0^N e^{-s t} \map f t \rd t + \int_N^\infty e^{-s t} \map f t \rd t$
We have that $f$ is [[Definition:Piecewise Continuous Function with One-Sided Limits|piecewise continuous]] in every [[Definition:Closed Real Interval|closed interval]] $\cl... | Laplace Transform Exists if Function Piecewise Continuous and of Exponential Order | https://proofwiki.org/wiki/Laplace_Transform_Exists_if_Function_Piecewise_Continuous_and_of_Exponential_Order | https://proofwiki.org/wiki/Laplace_Transform_Exists_if_Function_Piecewise_Continuous_and_of_Exponential_Order | [
"Laplace Transforms"
] | [
"Definition:Real Function",
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Definition:Real Interval/Closed",
"Definition:Exponential Order/Real Index",
"Definition:Laplace Transform"
] | [
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Definition:Real Interval/Closed",
"Definition:Exponential Order/Real Index",
"Laplace Transform of Exponential",
"Definition:Laplace Transform"
] |
proofwiki-15861 | Laplace Transform of Derivative/Discontinuity at t = 0 | Let $f$ fail to be continuous at $t = 0$, but let:
:$\ds \lim_{t \mathop \to 0} \map f t = \map f {0^+}$
exist.
Then $\laptrans f$ exists for $\map \Re s > a$, and:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f {0^+}$ | See Laplace Transform of Derivative/Discontinuity at t = a and use $a = 0$ and $\map f {0^-} = 0$. | Let $f$ fail to be [[Definition:Continuous Mapping|continuous]] at $t = 0$, but let:
:$\ds \lim_{t \mathop \to 0} \map f t = \map f {0^+}$
exist.
Then $\laptrans f$ exists for $\map \Re s > a$, and:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f {0^+}$ | See [[Laplace Transform of Derivative/Discontinuity at t = a]] and use $a = 0$ and $\map f {0^-} = 0$. | Laplace Transform of Derivative/Discontinuity at t = 0 | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative/Discontinuity_at_t_=_0 | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative/Discontinuity_at_t_=_0 | [
"Laplace Transforms of Derivatives"
] | [
"Definition:Continuous Mapping"
] | [
"Laplace Transform of Derivative/Discontinuity at t = a"
] |
proofwiki-15862 | Laplace Transform of Derivative/Discontinuity at t = a | Let $f$ have a jump discontinuity at $t = a$.
Then:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0 - e^{-a s} \paren {\map f {a^+} - \map f {a^-} }$ | See Laplace Transform of Derivative with Finite Discontinuities and use $n = 1$ and $a_1 = a$. | Let $f$ have a [[Definition:Jump Discontinuity|jump discontinuity]] at $t = a$.
Then:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0 - e^{-a s} \paren {\map f {a^+} - \map f {a^-} }$ | See [[Laplace Transform of Derivative with Finite Discontinuities]] and use $n = 1$ and $a_1 = a$. | Laplace Transform of Derivative/Discontinuity at t = a | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative/Discontinuity_at_t_=_a | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative/Discontinuity_at_t_=_a | [
"Laplace Transforms of Derivatives"
] | [
"Definition:Discontinuity (Real Analysis)/Jump"
] | [
"Laplace Transform of Derivative with Finite Discontinuities"
] |
proofwiki-15863 | Laplace Transform of Integral | :$\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$
wherever $\laptrans f$ exists. | Let $\map g t = \ds \int_0^t \map f u \rd u$.
Then:
:$\map {g'} t = \map f t$
and:
:$\map g 0 = 0$
Thus:
{{begin-eqn}}
{{eqn | l = \laptrans {\map {g'} t}
| r = s \laptrans {\map g t} - \map g 0
| c = Laplace Transform of Derivative
}}
{{eqn | r = s \laptrans {\map g t}
| c =
}}
{{eqn | r = \map F s
... | :$\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$
wherever $\laptrans f$ exists. | Let $\map g t = \ds \int_0^t \map f u \rd u$.
Then:
:$\map {g'} t = \map f t$
and:
:$\map g 0 = 0$
Thus:
{{begin-eqn}}
{{eqn | l = \laptrans {\map {g'} t}
| r = s \laptrans {\map g t} - \map g 0
| c = [[Laplace Transform of Derivative]]
}}
{{eqn | r = s \laptrans {\map g t}
| c =
}}
{{eqn | r = \... | Laplace Transform of Integral | https://proofwiki.org/wiki/Laplace_Transform_of_Integral | https://proofwiki.org/wiki/Laplace_Transform_of_Integral | [
"Laplace Transforms of Integrals",
"Laplace Transforms",
"Integral Calculus"
] | [] | [
"Laplace Transform of Derivative"
] |
proofwiki-15864 | Integral of Laplace Transform | :$\ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
wherever $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist. | Let $\map g t := \dfrac {\map f t} t$.
Then:
{{begin-eqn}}
{{eqn | l = \map f t
| r = t \map g t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {\map f t}
| r = \laptrans {t \map g t}
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {\map f t}
| r = -\dfrac \d {\d s} \laptrans {\... | :$\ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
wherever $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist. | Let $\map g t := \dfrac {\map f t} t$.
Then:
{{begin-eqn}}
{{eqn | l = \map f t
| r = t \map g t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {\map f t}
| r = \laptrans {t \map g t}
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {\map f t}
| r = -\dfrac \d {\d s} \laptrans ... | Integral of Laplace Transform | https://proofwiki.org/wiki/Integral_of_Laplace_Transform | https://proofwiki.org/wiki/Integral_of_Laplace_Transform | [
"Integrals of Laplace Transforms",
"Laplace Transforms",
"Integral Calculus"
] | [] | [
"Derivative of Laplace Transform"
] |
proofwiki-15865 | Inclusion Mapping on Subring is Homomorphism | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +{\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$.
Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$.
Then ${i_S}$ is a ring homomorphism. | Let $x, y \in S$.
Then:
{{begin-eqn}}
{{eqn | l = \map {i_S} x + \map {i_S} y
| r = x + y
| c = {{Defof|Inclusion Mapping}}
}}
{{eqn | r = x \mathbin{ + {\restriction_S} } y
| c = as $x, y \in S$
}}
{{eqn | r = \map {i_S} {x \mathbin{ + {\restriction_S} } y}
| c = as $x \mathbin{ + {\restriction... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {S, +{\restriction_S}, \circ {\restriction_S}}$ be a [[Definition:Subring|subring]] of $R$.
Let $i_S: S \to R$ be the [[Definition:Inclusion Mapping|inclusion mapping]] from $S$ to $R$.
Then ${i_S}$ is a [[Definition:Ring Hom... | Let $x, y \in S$.
Then:
{{begin-eqn}}
{{eqn | l = \map {i_S} x + \map {i_S} y
| r = x + y
| c = {{Defof|Inclusion Mapping}}
}}
{{eqn | r = x \mathbin{ + {\restriction_S} } y
| c = as $x, y \in S$
}}
{{eqn | r = \map {i_S} {x \mathbin{ + {\restriction_S} } y}
| c = as $x \mathbin{ + {\restricti... | Inclusion Mapping on Subring is Homomorphism | https://proofwiki.org/wiki/Inclusion_Mapping_on_Subring_is_Homomorphism | https://proofwiki.org/wiki/Inclusion_Mapping_on_Subring_is_Homomorphism | [
"Ring Homomorphisms",
"Inclusion Mappings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Subring",
"Definition:Inclusion Mapping",
"Definition:Ring Homomorphism"
] | [
"Definition:Ring Homomorphism",
"Category:Ring Homomorphisms",
"Category:Inclusion Mappings"
] |
proofwiki-15866 | Inclusion Mapping on Subring is Monomorphism | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +{\restriction_S}, \circ {\restriction_S} }$ be a subring of $R$.
Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$.
Then $i_S$ is a ring monomorphism. | By Inclusion Mapping on Subring is Homomorphism, $i_S$ is a ring homomorphism.
By Inclusion Mapping is Injection, $i_S$ is an injection.
The result follows by definition of (ring) monomorphism.
{{qed}}
Category:Group Monomorphisms
Category:Inclusion Mappings
h5ohqc6l5mbb3zoxs5vxcw9butsp61s | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {S, +{\restriction_S}, \circ {\restriction_S} }$ be a [[Definition:Subring|subring]] of $R$.
Let $i_S: S \to R$ be the [[Definition:Inclusion Mapping|inclusion mapping]] from $S$ to $R$.
Then $i_S$ is a [[Definition:Ring Mono... | By [[Inclusion Mapping on Subring is Homomorphism]], $i_S$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
By [[Inclusion Mapping is Injection]], $i_S$ is an [[Definition:Injection|injection]].
The result follows by definition of [[Definition:Ring Monomorphism|(ring) monomorphism]].
{{qed}}
[[Category:Group... | Inclusion Mapping on Subring is Monomorphism | https://proofwiki.org/wiki/Inclusion_Mapping_on_Subring_is_Monomorphism | https://proofwiki.org/wiki/Inclusion_Mapping_on_Subring_is_Monomorphism | [
"Group Monomorphisms",
"Inclusion Mappings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Subring",
"Definition:Inclusion Mapping",
"Definition:Ring Monomorphism"
] | [
"Inclusion Mapping on Subring is Homomorphism",
"Definition:Ring Homomorphism",
"Inclusion Mapping is Injection",
"Definition:Injection",
"Definition:Ring Monomorphism",
"Category:Group Monomorphisms",
"Category:Inclusion Mappings"
] |
proofwiki-15867 | Negative of Subring is Negative of Ring | Let $\struct {R, +, \circ}$ be a ring.
For each $x \in R$ let $-x$ denote the ring negative of $x$ in $R$.
Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$.
For each $x \in S$ let $\mathbin \sim x$ denote the ring negative of $x$ in $S$.
Then:
:$\forall x \in S: \mathbin \sim x = -x$ | Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$.
By Inclusion Mapping on Subring is Homomorphism, $i_S$ is a ring homomorphism.
By Ring Homomorphism of Addition is Group Homomorphism, $i_S$ is a group homomorphism on ring addition $+$.
Let $x \in S$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbin \sim x
| r... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
For each $x \in R$ let $-x$ denote the [[Definition:Ring Negative|ring negative]] of $x$ in $R$.
Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a [[Definition:Subring|subring]] of $R$.
For each $x \in S$ let $\mathbin \si... | Let $i_S: S \to R$ be the [[Definition:Inclusion Mapping|inclusion mapping]] from $S$ to $R$.
By [[Inclusion Mapping on Subring is Homomorphism]], $i_S$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
By [[Ring Homomorphism of Addition is Group Homomorphism]], $i_S$ is a [[Definition:Group Homomorphism|group... | Negative of Subring is Negative of Ring | https://proofwiki.org/wiki/Negative_of_Subring_is_Negative_of_Ring | https://proofwiki.org/wiki/Negative_of_Subring_is_Negative_of_Ring | [
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Negative",
"Definition:Subring",
"Definition:Ring Negative"
] | [
"Definition:Inclusion Mapping",
"Inclusion Mapping on Subring is Homomorphism",
"Definition:Ring Homomorphism",
"Ring Homomorphism of Addition is Group Homomorphism",
"Definition:Group Homomorphism",
"Definition:Ring (Abstract Algebra)/Addition",
"Group Homomorphism Preserves Inverses",
"Category:Ring... |
proofwiki-15868 | Subtraction of Subring is Subtraction of Ring | Let $\struct {R, +, \circ}$ be an ring.
For each $x, y \in R$ let $x - y$ denote the subtraction of $x$ and $y$ in $R$.
Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$.
For each $x, y \in S$ let $x \sim y$ denote the subtraction of $x$ and $y$ in $S$.
Then:
:$\forall x, y \in S: x \sim... | Let $x, y \in S$.
Let $-x$ denote the ring negative of $x$ in $R$.
Let $\mathbin \sim x$ denote the ring negative of $x$ in $S$.
Then:
{{begin-eqn}}
{{eqn | l = x \sim y
| r = x \mathbin {+ {\restriction_S} } \paren {\mathbin \sim y}
| c = {{Defof|Ring Subtraction}}
}}
{{eqn | r = x + \paren {\mathbin \sim ... | Let $\struct {R, +, \circ}$ be an [[Definition:Ring (Abstract Algebra)|ring]].
For each $x, y \in R$ let $x - y$ denote the [[Definition:Ring Subtraction|subtraction]] of $x$ and $y$ in $R$.
Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a [[Definition:Subring|subring]] of $R$.
For each $x, y \in ... | Let $x, y \in S$.
Let $-x$ denote the [[Definition:Ring Negative|ring negative]] of $x$ in $R$.
Let $\mathbin \sim x$ denote the [[Definition:Ring Negative|ring negative]] of $x$ in $S$.
Then:
{{begin-eqn}}
{{eqn | l = x \sim y
| r = x \mathbin {+ {\restriction_S} } \paren {\mathbin \sim y}
| c = {{Defof... | Subtraction of Subring is Subtraction of Ring | https://proofwiki.org/wiki/Subtraction_of_Subring_is_Subtraction_of_Ring | https://proofwiki.org/wiki/Subtraction_of_Subring_is_Subtraction_of_Ring | [
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Subtraction/Ring",
"Definition:Subring",
"Definition:Subtraction/Ring"
] | [
"Definition:Ring Negative",
"Definition:Ring Negative",
"Negative of Subring is Negative of Ring",
"Category:Ring Theory"
] |
proofwiki-15869 | Initial Value Theorem of Laplace Transform | Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$.
Then:
:$\ds \lim_{t \mathop \to 0} \map f t = \lim_{s \mathop \to \infty} s \, \map F s$
if those limits exist. | We have that $\map {f'} t$ is piecewise continuous with one-sided limits and of exponential order.
Hence:
:$\ds \lim_{s \mathop \to \infty} \int_0^\infty e^{-s t} \map {f'} t \rd t = 0$
Suppose that $f$ is continuous at $t = 0$.
From Laplace Transform of Derivative:
:$(1): \quad \laptrans {\map {f'} t} = s \map F s - \... | Let $\laptrans {\map f t} = \map F s$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\ds \lim_{t \mathop \to 0} \map f t = \lim_{s \mathop \to \infty} s \, \map F s$
if those limits exist. | We have that $\map {f'} t$ is [[Definition:Piecewise Continuous Function with One-Sided Limits|piecewise continuous with one-sided limits]] and of [[Definition:Exponential Order|exponential order]].
Hence:
:$\ds \lim_{s \mathop \to \infty} \int_0^\infty e^{-s t} \map {f'} t \rd t = 0$
Suppose that $f$ is [[Definiti... | Initial Value Theorem of Laplace Transform | https://proofwiki.org/wiki/Initial_Value_Theorem_of_Laplace_Transform | https://proofwiki.org/wiki/Initial_Value_Theorem_of_Laplace_Transform | [
"Initial Value Theorem of Laplace Transform",
"Properties of Laplace Transforms",
"Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Definition:Exponential Order",
"Definition:Continuous Real Function/Point",
"Laplace Transform of Derivative",
"Definition:Limit of Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Point",
"L... |
proofwiki-15870 | Final Value Theorem of Laplace Transform | Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$.
Then:
:$\ds \lim_{t \mathop \to \infty} \map f t = \lim_{s \mathop \to 0} s \, \map F s$
if those limits exist. | From Laplace Transform of Derivative:
:$(1): \quad \laptrans {\map {f'} t} = s \, \map F s - \map f 0$
We have that:
{{begin-eqn}}
{{eqn | l = \lim_{s \mathop \to 0} \laptrans {\map {f'} t}
| r = \lim_{s \mathop \to 0} \int_0^\infty e^{-s t} \map {f'} t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r =... | Let $\laptrans {\map f t} = \map F s$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\ds \lim_{t \mathop \to \infty} \map f t = \lim_{s \mathop \to 0} s \, \map F s$
if those limits exist. | From [[Laplace Transform of Derivative]]:
:$(1): \quad \laptrans {\map {f'} t} = s \, \map F s - \map f 0$
We have that:
{{begin-eqn}}
{{eqn | l = \lim_{s \mathop \to 0} \laptrans {\map {f'} t}
| r = \lim_{s \mathop \to 0} \int_0^\infty e^{-s t} \map {f'} t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{e... | Final Value Theorem of Laplace Transform | https://proofwiki.org/wiki/Final_Value_Theorem_of_Laplace_Transform | https://proofwiki.org/wiki/Final_Value_Theorem_of_Laplace_Transform | [
"Final Value Theorem of Laplace Transform",
"Properties of Laplace Transforms",
"Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Laplace Transform of Derivative",
"Fundamental Theorem of Calculus",
"Definition:Continuous Real Function/Point",
"Laplace Transform of Derivative/Discontinuity at t = 0",
"Fundamental Theorem of Calculus"
] |
proofwiki-15871 | Evaluation of Integral using Laplace Transform | Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$.
Then:
:$\ds \int_0^{\to \infty} \map f t \rd t = \map F 0$
assuming the integral is convergent. | By definition of Laplace transform:
:$\ds \int_0^{\to \infty} e^{-s t} \map f t \rd t = \map F s$
The result follows by taking the limit as $s \to 0$.
{{qed}} | Let $\laptrans {\map f t} = \map F s$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\ds \int_0^{\to \infty} \map f t \rd t = \map F 0$
assuming the integral is [[Definition:Convergent Integral|convergent]]. | By definition of [[Definition:Laplace Transform|Laplace transform]]:
:$\ds \int_0^{\to \infty} e^{-s t} \map f t \rd t = \map F s$
The result follows by taking the [[Definition:Limit of Real Function|limit]] as $s \to 0$.
{{qed}} | Evaluation of Integral using Laplace Transform | https://proofwiki.org/wiki/Evaluation_of_Integral_using_Laplace_Transform | https://proofwiki.org/wiki/Evaluation_of_Integral_using_Laplace_Transform | [
"Laplace Transforms",
"Integral Calculus"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Convergent Integral"
] | [
"Definition:Laplace Transform",
"Definition:Limit of Real Function"
] |
proofwiki-15872 | Area between Smooth Curve and Line is Maximized by Semicircle | Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space.
Let $y$ have a total length of $l$.
Let it be contained in the upper half-plane with an exception of endpoints, which are on the $x$-axis.
Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area.
Then $y$ ... | By Area between Smooth Curve and Line with Fixed Endpoints is Maximized by Arc of Circle the maximizing curve is an arc of a circle.
It is described as follows:
:If $\dfrac l \pi \le \lambda < \infty$ then:
::$y = \sqrt {\lambda^2 - x^2} - \sqrt {\lambda^2 - a^2}$
:where:
::$l = 2 \lambda \, \map \arctan {\dfrac a {\sq... | Let $y$ be a [[Definition:Smooth Curve|smooth curve]], embedded in $2$-[[Definition:Dimension of Vector Space|dimensional]] [[Definition:Real Euclidean Space|Euclidean space]].
Let $y$ have a total [[Definition:Length of Curve|length]] of $l$.
Let it be contained in the upper [[Definition:Half-Plane|half-plane]] with... | By [[Area between Smooth Curve and Line with Fixed Endpoints is Maximized by Arc of Circle]] the maximizing [[Definition:Curve|curve]] is an [[Definition:Arc of Circle|arc of a circle]].
It is described as follows:
:If $\dfrac l \pi \le \lambda < \infty$ then:
::$y = \sqrt {\lambda^2 - x^2} - \sqrt {\lambda^2 - a^2... | Area between Smooth Curve and Line is Maximized by Semicircle | https://proofwiki.org/wiki/Area_between_Smooth_Curve_and_Line_is_Maximized_by_Semicircle | https://proofwiki.org/wiki/Area_between_Smooth_Curve_and_Line_is_Maximized_by_Semicircle | [
"Isoperimetrical Problems"
] | [
"Definition:Smooth Curve",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Arc Length",
"Definition:Half-Plane",
"Definition:Directed Smooth Curve/Endpoints",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Directed Smooth Curve/Endpoints",
"... | [
"Area between Smooth Curve and Line with Fixed Endpoints is Maximized by Arc of Circle",
"Definition:Line/Curve",
"Definition:Circle/Arc",
"Definition:Area",
"Definition:Definite Integral",
"Primitive of Root of a squared minus x squared",
"Definition:Arc Length",
"Definition:Strictly Positive",
"De... |
proofwiki-15873 | Valuation Ring of Non-Archimedean Division Ring is Clopen | Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$.
Let $\OO$ be valuation ring induced by $\norm{\,\cdot\,}$.
Then $\OO$ is a both open and closed in the metric induced by $\norm{\,\cdot\,}$. | The valuation ring $\OO$ Is the open ball $\map {B_1^-} {0_R}$ by definition.
By Open Balls of Non-Archimedean Division Rings are Clopen then $\OO$ is both open and closed in the metric induced by $\norm {\,\cdot\,}$.
{{qed}}
Category:Normed Division Rings
Category:Valuation Ring of Non-Archimedean Division Ring is Clo... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean normed division ring]] with [[Definition:Ring Zero|zero]] $0_R$.
Let $\OO$ be [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]] by $\norm{\,\cdot\,}$.
Then $\OO$ is a both [[D... | The [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring]] $\OO$ Is the [[Definition:Open Ball|open ball]] $\map {B_1^-} {0_R}$ by definition.
By [[Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen|Open Balls of Non-Archimedean Division Rings are Clopen]] then $\OO$ i... | Valuation Ring of Non-Archimedean Division Ring is Clopen | https://proofwiki.org/wiki/Valuation_Ring_of_Non-Archimedean_Division_Ring_is_Clopen | https://proofwiki.org/wiki/Valuation_Ring_of_Non-Archimedean_Division_Ring_is_Clopen | [
"Normed Division Rings",
"Valuation Ring of Non-Archimedean Division Ring is Clopen"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Ring Zero",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Metric Induced by Norm on Division Ring"
] | [
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Open Ball",
"Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Metric Induced by Norm on Division Ring",
"Category... |
proofwiki-15874 | Valuation Ring of Non-Archimedean Division Ring is Clopen/Corollary 1 | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric. | The $p$-adic integers $\Z_p$ is the valuation ring induced by $\norm {\,\cdot\,}_p$ by definition.
By Valuation Ring of Non-Archimedean Division Ring is Clopen then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric.
{{qed}}
Category:Valuation Ring of Non-Archimedean Division Ring is Clopen
iq8... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Then the [[Definition:P-adic Integer|$p$-adic integers]] $\Z_p$ is both [[Definition:Open Set of Metric Space|open]] and [[Definition:Closed Set of M... | The [[Definition:P-adic Integer|$p$-adic integers]] $\Z_p$ is the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]] by $\norm {\,\cdot\,}_p$ by definition.
By [[Valuation Ring of Non-Archimedean Division Ring is Clopen|Valuation Ring of Non-Archimedean Division Ring is Clopen]] then ... | Valuation Ring of Non-Archimedean Division Ring is Clopen/Corollary 1 | https://proofwiki.org/wiki/Valuation_Ring_of_Non-Archimedean_Division_Ring_is_Clopen/Corollary_1 | https://proofwiki.org/wiki/Valuation_Ring_of_Non-Archimedean_Division_Ring_is_Clopen/Corollary_1 | [
"Valuation Ring of Non-Archimedean Division Ring is Clopen"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:P-adic Metric/P-adic Numbers"
] | [
"Definition:P-adic Integer",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Valuation Ring of Non-Archimedean Division Ring is Clopen",
"Definition:P-adic Integer",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:P-adic Metric/P-adic Numbers",
"Cate... |
proofwiki-15875 | Bessel Function of the First Kind of Negative Integer Order | :$\map {J_{-n} } x = \paren {-1}^n \map {J_n} x$ | {{begin-eqn}}
{{eqn | l = \map {J_{-n} } x
| r = \dfrac 1 \pi \int_0^\pi \map \cos {-n \theta - x \sin \theta} \rd \theta
| c = Integral Representation of Bessel Function of the First Kind/Integer Order
}}
{{eqn | r = \dfrac 1 \pi \int_0^\pi \map \cos {-n \paren {\pi - \theta} - x \sin \paren {\pi - \theta... | :$\map {J_{-n} } x = \paren {-1}^n \map {J_n} x$ | {{begin-eqn}}
{{eqn | l = \map {J_{-n} } x
| r = \dfrac 1 \pi \int_0^\pi \map \cos {-n \theta - x \sin \theta} \rd \theta
| c = [[Integral Representation of Bessel Function of the First Kind/Integer Order]]
}}
{{eqn | r = \dfrac 1 \pi \int_0^\pi \map \cos {-n \paren {\pi - \theta} - x \sin \paren {\pi - \t... | Bessel Function of the First Kind of Negative Integer Order | https://proofwiki.org/wiki/Bessel_Function_of_the_First_Kind_of_Negative_Integer_Order | https://proofwiki.org/wiki/Bessel_Function_of_the_First_Kind_of_Negative_Integer_Order | [
"Bessel Functions"
] | [] | [
"Integral Representation of Bessel Function of the First Kind/Integer Order",
"Sine of Supplementary Angle",
"Reversal of Limits of Definite Integral",
"Cosine of Angle plus Integer Multiple of Pi",
"Integral Representation of Bessel Function of the First Kind/Integer Order"
] |
proofwiki-15876 | Series Expansion of Bessel Function of the First Kind | {{begin-eqn}}
{{eqn | l = \map {J_n} x
| r = \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k +... | We employ Frobenius's method to find the solutions to the Bessel's Equation:
:$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$
for $n \ge 0$, in the form:
:$\ds \map y x = \sum_{k \mathop = 0}^\infty A_k x^{k + r}$
defined on $x > 0$, for some constants $r, A_i$, with $A_0 \ne 0$, whic... | {{begin-eqn}}
{{eqn | l = \map {J_n} x
| r = \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k +... | We employ [[Definition:Frobenius's Method|Frobenius's method]] to find the solutions to the [[Definition:Bessel's Equation|Bessel's Equation]]:
:$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$
for $n \ge 0$, in the form:
:$\ds \map y x = \sum_{k \mathop = 0}^\infty A_k x^{k + r}$
... | Series Expansion of Bessel Function of the First Kind | https://proofwiki.org/wiki/Series_Expansion_of_Bessel_Function_of_the_First_Kind | https://proofwiki.org/wiki/Series_Expansion_of_Bessel_Function_of_the_First_Kind | [
"Bessel Functions"
] | [] | [
"Definition:Frobenius's Method",
"Definition:Bessel's Equation",
"Definition:Bessel's Equation",
"Translation of Index Variable of Summation",
"Definition:Coefficient of Polynomial",
"Definition:Recursive Sequence/Recurrence Relation"
] |
proofwiki-15877 | Recurrence Formula for Bessel Function of the First Kind | Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.
Then:
:$\map {J_{n + 1} } x = \dfrac {2 n} x \map {J_n} x - \map {J_{n - 1} } x$
And:
:$\map {J_{n + 1} } x = -2 \map {J_n'} x + \map {J_{n - 1} } x$
{{refactor|page per result|level = basic}} | From Generating Function for Bessel Function of the First Kind of Order n of x we have:
:$\ds \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} } = \sum_{n \mathop = - \infty}^\infty \map {J_n} x t^n$
Differentiating both sides of the equation with respect to $t$:
{{begin-eqn}}
{{eqn | l = \dfrac x 2 \paren {1 + \dfrac 1 {... | Let $\map {J_n} x$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order $n$]].
Then:
:$\map {J_{n + 1} } x = \dfrac {2 n} x \map {J_n} x - \map {J_{n - 1} } x$
And:
:$\map {J_{n + 1} } x = -2 \map {J_n'} x + \map {J_{n - 1} } x$... | From [[Generating Function for Bessel Function of the First Kind of Order n of x]] we have:
:$\ds \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} } = \sum_{n \mathop = - \infty}^\infty \map {J_n} x t^n$
Differentiating both sides of the equation with respect to $t$:
{{begin-eqn}}
{{eqn | l = \dfrac x 2 \paren {1 + \df... | Recurrence Formula for Bessel Function of the First Kind | https://proofwiki.org/wiki/Recurrence_Formula_for_Bessel_Function_of_the_First_Kind | https://proofwiki.org/wiki/Recurrence_Formula_for_Bessel_Function_of_the_First_Kind | [
"Bessel Functions",
"Recurrence Relations"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order"
] | [
"Generating Function for Bessel Function of the First Kind of Order n of x",
"Generating Function for Bessel Function of the First Kind of Order n of x",
"Translation of Index Variable of Summation",
"Translation of Index Variable of Summation"
] |
proofwiki-15878 | Derivative of x^n by Bessel Function of the First Kind of Order n of x | Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.
Then:
:$\map {\dfrac \d {\d x} } {x^n \map {J_n} x} = x^n \map {J_{n - 1} } x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {x^n \map {J_n} x}
| r = n x^{n - 1} \map {J_n} x + x^n \map {J_n'} x
| c = Product Rule for Derivatives
}}
{{eqn | r = n x^{n - 1} \map {J_n} x - x^n \paren {\frac {\map {J_{n + 1} } x - \map {J_{n - 1} } x} 2}
| c = Recurrence Formula for Bessel Func... | Let $\map {J_n} x$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order $n$]].
Then:
:$\map {\dfrac \d {\d x} } {x^n \map {J_n} x} = x^n \map {J_{n - 1} } x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {x^n \map {J_n} x}
| r = n x^{n - 1} \map {J_n} x + x^n \map {J_n'} x
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = n x^{n - 1} \map {J_n} x - x^n \paren {\frac {\map {J_{n + 1} } x - \map {J_{n - 1} } x} 2}
| c = [[Recurrence Formula for Besse... | Derivative of x^n by Bessel Function of the First Kind of Order n of x | https://proofwiki.org/wiki/Derivative_of_x^n_by_Bessel_Function_of_the_First_Kind_of_Order_n_of_x | https://proofwiki.org/wiki/Derivative_of_x^n_by_Bessel_Function_of_the_First_Kind_of_Order_n_of_x | [
"Bessel Functions"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order"
] | [
"Product Rule for Derivatives",
"Recurrence Formula for Bessel Function of the First Kind",
"Recurrence Formula for Bessel Function of the First Kind"
] |
proofwiki-15879 | Series Expansion of Bessel Function of the First Kind/Negative Index | {{begin-eqn}}
{{eqn | l = \map {J_{-n} } x
| r = \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gam... | From Series Expansion of Bessel Function of the First Kind:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {J_n} x
| r = \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}
| c =
}}
{{eqn | r = \sum... | {{begin-eqn}}
{{eqn | l = \map {J_{-n} } x
| r = \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gam... | From [[Series Expansion of Bessel Function of the First Kind]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {J_n} x
| r = \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}
| c =
}}
{{eqn | r =... | Series Expansion of Bessel Function of the First Kind/Negative Index | https://proofwiki.org/wiki/Series_Expansion_of_Bessel_Function_of_the_First_Kind/Negative_Index | https://proofwiki.org/wiki/Series_Expansion_of_Bessel_Function_of_the_First_Kind/Negative_Index | [
"Bessel Functions"
] | [] | [
"Series Expansion of Bessel Function of the First Kind"
] |
proofwiki-15880 | Integral to Infinity of Dirac Delta Function | Let $\map \delta x$ denote the Dirac delta function.
Then:
:$\ds \int_0^{+ \infty} \map \delta x \rd x = 1$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_0^{+ \infty} \map {F_\epsilon} x \rd x
| r... | Let $\map \delta x$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
Then:
:$\ds \int_0^{+ \infty} \map \delta x \rd x = 1$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_0^{+ \infty} \map {F_\epsilon} x \rd x
... | Integral to Infinity of Dirac Delta Function | https://proofwiki.org/wiki/Integral_to_Infinity_of_Dirac_Delta_Function | https://proofwiki.org/wiki/Integral_to_Infinity_of_Dirac_Delta_Function | [
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Function"
] | [
"Definition:Dirac Delta Function/Definition 1",
"Integral of Constant/Definite",
"Definition:Dirac Delta Function/Definition 1"
] |
proofwiki-15881 | Integral to Infinity of Dirac Delta Function by Continuous Function | Let $\map \delta x$ denote the Dirac delta function.
Let $g$ be a continuous real function.
Then:
:$\ds \int_0^{+ \infty} \map \delta x \map g x \rd x = \map g 0$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_0^{+ \infty} \map {F_\epsilon} x \map g x \rd x
... | Let $\map \delta x$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $g$ be a [[Definition:Continuous Real Function|continuous real function]].
Then:
:$\ds \int_0^{+ \infty} \map \delta x \map g x \rd x = \map g 0$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_0^{+ \infty} \map {F_\epsilon} x \map g x \r... | Integral to Infinity of Dirac Delta Function by Continuous Function | https://proofwiki.org/wiki/Integral_to_Infinity_of_Dirac_Delta_Function_by_Continuous_Function | https://proofwiki.org/wiki/Integral_to_Infinity_of_Dirac_Delta_Function_by_Continuous_Function | [
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Function",
"Definition:Continuous Real Function"
] | [
"Integral of Constant/Definite",
"Darboux's Theorem",
"Definition:Maximal/Element",
"Definition:Minimal/Element",
"Squeeze Theorem"
] |
proofwiki-15882 | Integral to Infinity of Shifted Dirac Delta Function by Continuous Function | Let $\map \delta x$ denote the Dirac delta function.
Let $g$ be a continuous real function.
Let $a \in \R_{\ge 0}$ be a positive real number.
Then:
:$\ds \int_0^{+ \infty} \map \delta {x - a} \, \map g x \rd x = \map g a$ | We have that:
:$\map \delta {x - a} = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < a \\ \dfrac 1 \epsilon & : a \le x \le a + \epsilon \\ 0 & : x > a + \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_0^{+ \infty} \map {F_\epsilon} x \... | Let $\map \delta x$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $g$ be a [[Definition:Continuous Real Function|continuous real function]].
Let $a \in \R_{\ge 0}$ be a [[Definition:Positive Real Number|positive real number]].
Then:
:$\ds \int_0^{+ \infty} \map \delta {x - a} \, \map g x... | We have that:
:$\map \delta {x - a} = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < a \\ \dfrac 1 \epsilon & : a \le x \le a + \epsilon \\ 0 & : x > a + \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_0^{+ \infty} \map {F_\epsilo... | Integral to Infinity of Shifted Dirac Delta Function by Continuous Function | https://proofwiki.org/wiki/Integral_to_Infinity_of_Shifted_Dirac_Delta_Function_by_Continuous_Function | https://proofwiki.org/wiki/Integral_to_Infinity_of_Shifted_Dirac_Delta_Function_by_Continuous_Function | [
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Function",
"Definition:Continuous Real Function",
"Definition:Positive/Real Number"
] | [
"Integral of Constant/Definite",
"Darboux's Theorem",
"Definition:Maximal/Element",
"Definition:Minimal/Element",
"Squeeze Theorem",
"Definition:Dirac Delta Function"
] |
proofwiki-15883 | Function which is Zero except on Countable Set of Points is Null | Let $S \subseteq \R$ be a subset of $\R$ such that $S$ is countable, either finite or countably infinite.
Let $f: \R \to \R$ be a real function such that:
:$\forall x \in \R \setminus S: \map f x = 0$
That is, except perhaps for the elements of $S$, the value of $f$ is zero.
Then $f$ is a null function. | This is an instance of Measurable Function Zero A.E. iff Absolute Value has Zero Integral.
{{qed}} | Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of $\R$ such that $S$ is [[Definition:Countable Set|countable]], either [[Definition:Finite Set|finite]] or [[Definition:Countably Infinite Set|countably infinite]].
Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] such that:
:$\forall x \in \R \... | This is an instance of [[Measurable Function Zero A.E. iff Absolute Value has Zero Integral]].
{{qed}} | Function which is Zero except on Countable Set of Points is Null | https://proofwiki.org/wiki/Function_which_is_Zero_except_on_Countable_Set_of_Points_is_Null | https://proofwiki.org/wiki/Function_which_is_Zero_except_on_Countable_Set_of_Points_is_Null | [
"Null Functions"
] | [
"Definition:Subset",
"Definition:Countable Set",
"Definition:Finite Set",
"Definition:Countably Infinite/Set",
"Definition:Real Function",
"Definition:Element",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Zero (Number)",
"Definition:Null Function"
] | [
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral"
] |
proofwiki-15884 | Laplace Transform of Bessel Function of the First Kind of Order Zero | Let $J_0$ denote the Bessel function of the first kind of order $0$.
Then the Laplace transform of $J_0$ is given as:
:$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$ | From Bessel Function of the First Kind of Order Zero:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \times 6... | Let $J_0$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $0$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_0$ is given as:
:$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$ | From [[Bessel Function of the First Kind of Order Zero]]:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \ti... | Laplace Transform of Bessel Function of the First Kind of Order Zero/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero/Proof_1 | [
"Laplace Transforms of Bessel Functions",
"Laplace Transform of Bessel Function of the First Kind of Order Zero"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Bessel Function of the First Kind/Instances/Order 0",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Positive Integer Power",
"Binomial Theorem/General Binomial Theorem",
"Binomial Coefficient of Minus Half",
"Definition:Summation"
] |
proofwiki-15885 | Laplace Transform of Bessel Function of the First Kind of Order Zero | Let $J_0$ denote the Bessel function of the first kind of order $0$.
Then the Laplace transform of $J_0$ is given as:
:$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$ | From Bessel Function of the First Kind of Order Zero:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \times 6... | Let $J_0$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $0$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_0$ is given as:
:$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$ | From [[Bessel Function of the First Kind of Order Zero]]:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \ti... | Laplace Transform of Bessel Function of the First Kind of Order Zero/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero/Proof_2 | [
"Laplace Transforms of Bessel Functions",
"Laplace Transform of Bessel Function of the First Kind of Order Zero"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Bessel Function of the First Kind/Instances/Order 0",
"Laplace Transform of Positive Integer Power",
"Binomial Theorem/General Binomial Theorem"
] |
proofwiki-15886 | Laplace Transform of Bessel Function of the First Kind of Order Zero | Let $J_0$ denote the Bessel function of the first kind of order $0$.
Then the Laplace transform of $J_0$ is given as:
:$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$ | By definition of Bessel function of the first kind, $\map {J_0} t$ satisfies Bessel's equation:
{{begin-eqn}}
{{eqn | l = t^2 \, \map {\dfrac {\d^2} {\d t^2} } {\map {J_0} t} + t \, \map {\dfrac \d {\d t} } {\map {J_0} t} + \paren {t^2 - 0^2} {\map {J_0} t}
| r = 0
| c =
}}
{{eqn | n = 1
| ll= \leads... | Let $J_0$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $0$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_0$ is given as:
:$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$ | By definition of [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]], $\map {J_0} t$ satisfies [[Definition:Bessel's Equation|Bessel's equation]]:
{{begin-eqn}}
{{eqn | l = t^2 \, \map {\dfrac {\d^2} {\d t^2} } {\map {J_0} t} + t \, \map {\dfrac \d {\d t} } {\map {J_0} t} + \paren {t^2 -... | Laplace Transform of Bessel Function of the First Kind of Order Zero/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero/Proof_3 | [
"Laplace Transforms of Bessel Functions",
"Laplace Transform of Bessel Function of the First Kind of Order Zero"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel's Equation",
"Laplace Transform of Derivative",
"Laplace Transform of Second Derivative",
"Bessel Function of the First Kind/Instances/Order 0",
"Derivative of Bessel Function of the First Kind of Order 0",
"Derivative of Laplace Transform",
... |
proofwiki-15887 | Laplace Transform of Bessel Function of the First Kind | Let $J_n$ denote the Bessel function of the first kind of order $n$.
Then the Laplace transform of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$ | From Bessel Function of the First Kind of Order Zero:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \times 6... | Let $J_n$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $n$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {... | From [[Bessel Function of the First Kind of Order Zero]]:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \ti... | Laplace Transform of Bessel Function of the First Kind of Order Zero/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero/Proof_1 | [
"Laplace Transform of Bessel Function of the First Kind",
"Laplace Transforms of Bessel Functions"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Bessel Function of the First Kind/Instances/Order 0",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Positive Integer Power",
"Binomial Theorem/General Binomial Theorem",
"Binomial Coefficient of Minus Half",
"Definition:Summation"
] |
proofwiki-15888 | Laplace Transform of Bessel Function of the First Kind | Let $J_n$ denote the Bessel function of the first kind of order $n$.
Then the Laplace transform of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$ | From Bessel Function of the First Kind of Order Zero:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \times 6... | Let $J_n$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $n$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {... | From [[Bessel Function of the First Kind of Order Zero]]:
{{begin-eqn}}
{{eqn | l = \map {J_0} t
| r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac t 2}^{2 k}
| c =
}}
{{eqn | r = 1 - \dfrac {t^2} {2^2} + \dfrac {t^4} {2^2 \times 4^2} - \dfrac {t^6} {2^2 \times 4^2 \ti... | Laplace Transform of Bessel Function of the First Kind of Order Zero/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero/Proof_2 | [
"Laplace Transform of Bessel Function of the First Kind",
"Laplace Transforms of Bessel Functions"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Bessel Function of the First Kind/Instances/Order 0",
"Laplace Transform of Positive Integer Power",
"Binomial Theorem/General Binomial Theorem"
] |
proofwiki-15889 | Laplace Transform of Bessel Function of the First Kind | Let $J_n$ denote the Bessel function of the first kind of order $n$.
Then the Laplace transform of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$ | By definition of Bessel function of the first kind, $\map {J_0} t$ satisfies Bessel's equation:
{{begin-eqn}}
{{eqn | l = t^2 \, \map {\dfrac {\d^2} {\d t^2} } {\map {J_0} t} + t \, \map {\dfrac \d {\d t} } {\map {J_0} t} + \paren {t^2 - 0^2} {\map {J_0} t}
| r = 0
| c =
}}
{{eqn | n = 1
| ll= \leads... | Let $J_n$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $n$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {... | By definition of [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]], $\map {J_0} t$ satisfies [[Definition:Bessel's Equation|Bessel's equation]]:
{{begin-eqn}}
{{eqn | l = t^2 \, \map {\dfrac {\d^2} {\d t^2} } {\map {J_0} t} + t \, \map {\dfrac \d {\d t} } {\map {J_0} t} + \paren {t^2 -... | Laplace Transform of Bessel Function of the First Kind of Order Zero/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind_of_Order_Zero/Proof_3 | [
"Laplace Transform of Bessel Function of the First Kind",
"Laplace Transforms of Bessel Functions"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel's Equation",
"Laplace Transform of Derivative",
"Laplace Transform of Second Derivative",
"Bessel Function of the First Kind/Instances/Order 0",
"Derivative of Bessel Function of the First Kind of Order 0",
"Derivative of Laplace Transform",
... |
proofwiki-15890 | Laplace Transform of Bessel Function of the First Kind | Let $J_n$ denote the Bessel function of the first kind of order $n$.
Then the Laplace transform of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$ | {{begin-eqn}}
{{eqn | l = \laptrans {t^2 \frac {\d^2 x} {\d t^2} + t \frac {\d x} {\d t} + \paren {t^2 - \alpha^2} x} s
| r = 0
| c = Laplace Transform of Bessel's Equation
}}
{{eqn | l = \frac {\d^2} {\d s^2} \laptrans {x' '} - \frac \d {\d s} \laptrans {x'} + \frac {\d^2} {\d s^2} \laptrans x - \alpha^2... | Let $J_n$ denote the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order]] $n$.
Then the [[Definition:Laplace Transform|Laplace transform]] of $J_n$ is given as:
:$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {... | {{begin-eqn}}
{{eqn | l = \laptrans {t^2 \frac {\d^2 x} {\d t^2} + t \frac {\d x} {\d t} + \paren {t^2 - \alpha^2} x} s
| r = 0
| c = [[Definition:Laplace Transform|Laplace Transform]] of [[Definition:Bessel's Equation|Bessel's Equation]]
}}
{{eqn | l = \frac {\d^2} {\d s^2} \laptrans {x' '} - \frac \d {\... | Laplace Transform of Bessel Function of the First Kind/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind | https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind/Proof_2 | [
"Laplace Transform of Bessel Function of the First Kind",
"Laplace Transforms of Bessel Functions"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order",
"Definition:Laplace Transform"
] | [
"Definition:Laplace Transform",
"Definition:Bessel's Equation",
"Definition:Arbitrary Constant",
"Final Value Theorem of Laplace Transform"
] |
proofwiki-15891 | Laplace Transform of Sine of Root | :$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the Laplace transform of the function $f$. | {{begin-eqn}}
{{eqn | l = \sin \sqrt t
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1}!}
| c = {{Defof|Real Sine Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}!} t^{n + \frac 1 2}
| c =
}}
{{eqn | ll=... | :$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$. | {{begin-eqn}}
{{eqn | l = \sin \sqrt t
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1}!}
| c = {{Defof|Real Sine Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}!} t^{n + \frac 1 2}
| c =
}}
{{eqn | ll=... | Laplace Transform of Sine of Root/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_Root | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_Root/Proof_1 | [
"Laplace Transform of Sine of Root",
"Laplace Transforms involving Sine Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Laplace Transform of Power",
"Linear Combination of Laplace Transforms",
"Gamma Difference Equation",
"Gamma Function of Positive Half-Integer"
] |
proofwiki-15892 | Laplace Transform of Sine of Root | :$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the Laplace transform of the function $f$. | Let $\map y t := \sin \sqrt t$.
Differentiating twice {{WRT|Differentiation}} $t$, we get:
:$(1): \quad 4 t y' ' + 2 y' ' + y = 0$
Let $\map Y s = \laptrans {\map t y}$ be the Laplace transform of $y$.
Then taking the Laplace transform of $(1)$:
{{begin-eqn}}
{{eqn | l = -4 \map {\dfrac \d {\d s} } {\laptrans {\map {y'... | :$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$. | Let $\map y t := \sin \sqrt t$.
[[Definition:Differentiation|Differentiating]] twice {{WRT|Differentiation}} $t$, we get:
:$(1): \quad 4 t y' ' + 2 y' ' + y = 0$
Let $\map Y s = \laptrans {\map t y}$ be the [[Definition:Laplace Transform|Laplace transform]] of $y$.
Then taking the [[Definition:Laplace Transform|Lap... | Laplace Transform of Sine of Root/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_Root | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_Root/Proof_2 | [
"Laplace Transform of Sine of Root",
"Laplace Transforms involving Sine Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Definition:Differentiation",
"Definition:Laplace Transform",
"Definition:Laplace Transform",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Laplace Transform of Second Derivative"
] |
proofwiki-15893 | Laplace Transform of Cosine of Root over Root | :$\laptrans {\dfrac {\cos \sqrt t} {\sqrt t} } = \sqrt {\dfrac \pi s} \, \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the Laplace transform of the function $f$. | Let $\map f t = \sin \sqrt t$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\cos \sqrt t} {2 \sqrt t}
| c =
}}
{{eqn | l = \map f 0
| r = 0
| c =
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn | l = \laptrans {\map {f'} t}
| r = \dfrac 1 2 \laptrans {\dfrac {\cos \sqrt t} {\sqrt t} ... | :$\laptrans {\dfrac {\cos \sqrt t} {\sqrt t} } = \sqrt {\dfrac \pi s} \, \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$. | Let $\map f t = \sin \sqrt t$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\cos \sqrt t} {2 \sqrt t}
| c =
}}
{{eqn | l = \map f 0
| r = 0
| c =
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn | l = \laptrans {\map {f'} t}
| r = \dfrac 1 2 \laptrans {\dfrac {\cos \sqrt t} {\sqrt... | Laplace Transform of Cosine of Root over Root/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine_of_Root_over_Root | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine_of_Root_over_Root/Proof_1 | [
"Laplace Transform of Cosine of Root over Root",
"Laplace Transforms involving Cosine Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Laplace Transform of Derivative",
"Laplace Transform of Sine of Root"
] |
proofwiki-15894 | Laplace Transform of Gauss Error Function | :$\laptrans {\map \erf t} = \dfrac 1 s \map \exp {\dfrac {s^2} 4} \map \erfc {\dfrac s 2}$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\erf$ denotes the Gauss error function
:$\erfc$ denotes the complementary error function
:$\exp$ denotes the exponential function. | By Derivative of Gauss Error Function, we have:
:$\ds \map {\frac \d {\d t} } {\map \erf t} = \frac 2 {\sqrt \pi} e^{-t^2}$
By Primitive of Exponential Function, we have:
:$\ds \int e^{-s t} \rd t = -\frac {e^{-s t} } s$
So:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \erf t}
| r = \int_0^\infty e^{-s t} \map \e... | :$\laptrans {\map \erf t} = \dfrac 1 s \map \exp {\dfrac {s^2} 4} \map \erfc {\dfrac s 2}$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\erf$ denotes the [[Definition:Gauss Error Function|Gauss error function]]
:$\erfc$ denotes ... | By [[Derivative of Gauss Error Function]], we have:
:$\ds \map {\frac \d {\d t} } {\map \erf t} = \frac 2 {\sqrt \pi} e^{-t^2}$
By [[Primitive of Exponential Function]], we have:
:$\ds \int e^{-s t} \rd t = -\frac {e^{-s t} } s$
So:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \erf t}
| r = \int_0^\infty e^{... | Laplace Transform of Gauss Error Function | https://proofwiki.org/wiki/Laplace_Transform_of_Gauss_Error_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Gauss_Error_Function | [
"Examples of Laplace Transforms",
"Gauss Error Function"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Gauss Error Function",
"Definition:Complementary Error Function",
"Definition:Exponential Function"
] | [
"Derivative of Gauss Error Function",
"Primitive of Exponential Function",
"Integration by Parts",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Exponential Tends to Zero and Infinity",
"Limit to Infinity of Gauss Error Function",
"Exponential of Zero",
"Definite Integral on Zero I... |
proofwiki-15895 | Laplace Transform of Gauss Error Function | :$\laptrans {\map \erf t} = \dfrac 1 s \map \exp {\dfrac {s^2} 4} \map \erfc {\dfrac s 2}$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\erf$ denotes the Gauss error function
:$\erfc$ denotes the complementary error function
:$\exp$ denotes the exponential function. | {{begin-eqn}}
{{eqn | l = \laptrans {\map \erf {\sqrt t} }
| r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \map \exp {-u^2} \rd u}
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!}... | :$\laptrans {\map \erf t} = \dfrac 1 s \map \exp {\dfrac {s^2} 4} \map \erfc {\dfrac s 2}$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\erf$ denotes the [[Definition:Gauss Error Function|Gauss error function]]
:$\erfc$ denotes ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map \erf {\sqrt t} }
| r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \map \exp {-u^2} \rd u}
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!}... | Laplace Transform of Gauss Error Function of Root/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Gauss_Error_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Gauss_Error_Function_of_Root/Proof_1 | [
"Examples of Laplace Transforms",
"Gauss Error Function"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Gauss Error Function",
"Definition:Complementary Error Function",
"Definition:Exponential Function"
] | [
"Primitive of Power",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Real Power",
"Gamma Difference Equation",
"Gamma Function Extends Factorial",
"Euler's Reflection Formula",
"Sine of Complement equals Cosine",
"Cosine of Integer Multiple of Pi",
"Gamma Function of One Half",
... |
proofwiki-15896 | Laplace Transform of Gauss Error Function of Root | :$\laptrans {\map \erf {\sqrt t} } = \dfrac 1 {s \sqrt {s + 1} }$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\erf$ denotes the Gauss error function | {{begin-eqn}}
{{eqn | l = \laptrans {\map \erf {\sqrt t} }
| r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \map \exp {-u^2} \rd u}
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!}... | :$\laptrans {\map \erf {\sqrt t} } = \dfrac 1 {s \sqrt {s + 1} }$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\erf$ denotes the [[Definition:Gauss Error Function|Gauss error function]] | {{begin-eqn}}
{{eqn | l = \laptrans {\map \erf {\sqrt t} }
| r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \map \exp {-u^2} \rd u}
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!}... | Laplace Transform of Gauss Error Function of Root/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Gauss_Error_Function_of_Root | https://proofwiki.org/wiki/Laplace_Transform_of_Gauss_Error_Function_of_Root/Proof_1 | [
"Laplace Transform of Gauss Error Function of Root",
"Gauss Error Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Gauss Error Function"
] | [
"Primitive of Power",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Real Power",
"Gamma Difference Equation",
"Gamma Function Extends Factorial",
"Euler's Reflection Formula",
"Sine of Complement equals Cosine",
"Cosine of Integer Multiple of Pi",
"Gamma Function of One Half",
... |
proofwiki-15897 | Laplace Transform of Sine Integral Function | :$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Si$ denotes the sine integral function | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\sin t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = \sin t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}... | :$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\Si$ denotes the [[Definition:Sine Integral Function|sine integral function]] | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\sin t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = \sin t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'}... | Laplace Transform of Sine Integral Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function/Proof_1 | [
"Laplace Transform of Sine Integral Function",
"Sine Integral Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Sine Integral Function"
] | [
"Laplace Transform of Sine",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Initial Value Theorem of Laplace Transform",
"Sum of Arctangent and Arccotangent",
"Arctangent of Reciprocal equals Arccotangent"
] |
proofwiki-15898 | Laplace Transform of Sine Integral Function | :$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Si$ denotes the sine integral function | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^t \dfrac 1 u \paren {u - \dfrac {u^3} {3!} + \dfrac {u^5} {5!} - \dfrac {u^... | :$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\Si$ denotes the [[Definition:Sine Integral Function|sine integral function]] | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^t \dfrac 1 u \paren {u - \dfrac {u^3} {3!} + \dfrac {u^5} {5!} - \dfrac ... | Laplace Transform of Sine Integral Function/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function/Proof_3 | [
"Laplace Transform of Sine Integral Function",
"Sine Integral Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Sine Integral Function"
] | [
"Primitive of Power",
"Laplace Transform of Positive Integer Power",
"Power Series Expansion for Real Arctangent Function"
] |
proofwiki-15899 | Laplace Transform of Sine Integral Function | :$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Si$ denotes the sine integral function | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^1 \dfrac {\sin t v} v \rd v
| c = Integration by Substitution $u = t ... | :$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\Si$ denotes the [[Definition:Sine Integral Function|sine integral function]] | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^1 \dfrac {\sin t v} v \rd v
| c = [[Integration by Substitution]] ... | Laplace Transform of Sine Integral Function/Proof 4 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function/Proof_4 | [
"Laplace Transform of Sine Integral Function",
"Sine Integral Function",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Sine Integral Function"
] | [
"Integration by Substitution",
"Laplace Transform of Sine",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
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