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 |
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proofwiki-20200 | Adapted Stochastic Process is Supermartingale iff Negative is Submartingale/Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a discrete-time filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Then $\sequence {X_n}_{n \ge 0}$ is a supermartingale {{iff}} $\sequence {-X_n}_{n \ge 0}$ is a sub... | Since $\sequence {X_n}_{n \ge 0}$ is a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process:
:$X_n$ is $\FF_n$-measurable for each $n \in \N$.
From Pointwise Scalar Multiple of Measurable Function is Measurable:
:$-X_n$ is $\FF_n$-measurable for each $n \in \N$.
So $\sequence {-X_n}_{n \ge 0}$ is a di... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Discrete Time|discrete-time filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be a [[Definition:Adapted Stochastic Process/Discrete Time|discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochast... | Since $\sequence {X_n}_{n \ge 0}$ is a [[Definition:Adapted Stochastic Process/Discrete Time|discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process]]:
:$X_n$ is [[Definition:Measurable Function|$\FF_n$-measurable]] for each $n \in \N$.
From [[Pointwise Scalar Multiple of Measurable Function is Measura... | Adapted Stochastic Process is Supermartingale iff Negative is Submartingale/Discrete Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Supermartingale_iff_Negative_is_Submartingale/Discrete_Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Supermartingale_iff_Negative_is_Submartingale/Discrete_Time | [
"Adapted Stochastic Process is Supermartingale iff Negative is Submartingale"
] | [
"Definition:Filtered Probability Space/Discrete Time",
"Definition:Adapted Stochastic Process/Discrete Time",
"Definition:Supermartingale/Discrete Time",
"Definition:Submartingale/Discrete Time"
] | [
"Definition:Adapted Stochastic Process/Discrete Time",
"Definition:Measurable Function",
"Pointwise Scalar Multiple of Measurable Function is Measurable",
"Definition:Measurable Function",
"Definition:Adapted Stochastic Process/Discrete Time",
"Definition:Conditional Expectation",
"Conditional Expectati... |
proofwiki-20201 | Stopped Submartingale is Submartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-submartingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\sequence {X_n^T}_{n \ge 0}$ be the stopped process.
Then $\... | From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale, we have:
:$\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
From Stopped Supermartingale is Supermartingale, we have:
:$\sequence {-X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
From ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Submartingale|$\sequence {\FF_n}_{n \ge 0}$-submartingale]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with respe... | From [[Adapted Stochastic Process is Supermartingale iff Negative is Submartingale]], we have:
:$\sequence {-X_n}_{n \ge 0}$ is a [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]].
From [[Stopped Supermartingale is Supermartingale]], we have:
:$\sequence {-X_n^T}_{n \ge 0}$ is a [[Definiti... | Stopped Submartingale is Submartingale | https://proofwiki.org/wiki/Stopped_Submartingale_is_Submartingale | https://proofwiki.org/wiki/Stopped_Submartingale_is_Submartingale | [
"Stopped Submartingale is Submartingale",
"Submartingales",
"Stopped Processes",
"Stopped Submartingale is Submartingale"
] | [
"Definition:Filtered Probability Space",
"Definition:Submartingale",
"Definition:Stopping Time",
"Definition:Stopped Process",
"Definition:Submartingale"
] | [
"Adapted Stochastic Process is Supermartingale iff Negative is Submartingale",
"Definition:Supermartingale",
"Stopped Supermartingale is Supermartingale",
"Definition:Supermartingale",
"Adapted Stochastic Process is Supermartingale iff Negative is Submartingale",
"Definition:Submartingale",
"Category:Su... |
proofwiki-20202 | Stopped Martingale is Martingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-martingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\sequence {X_n^T}_{n \ge 0}$ be the stopped process.
Then $\seq... | From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Stopped Supermartingale is Supermartingale and Stopped Submartingale is Submartingale:
:$\sequence {X... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Martingale|$\sequence {\FF_n}_{n \ge 0}$-martingale]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with respect to ... | From [[Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale]]:
:$\sequence {X_n}_{n \ge 0}$ is a [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]] and [[Definition:Submartingale|$\sequence {\FF_n}_{n \ge 0}$-submartingale]].
From [[Stopped Supermartingale is Superm... | Stopped Martingale is Martingale | https://proofwiki.org/wiki/Stopped_Martingale_is_Martingale | https://proofwiki.org/wiki/Stopped_Martingale_is_Martingale | [
"Stopped Processes",
"Martingales",
"Stopped Martingale is Martingale"
] | [
"Definition:Filtered Probability Space",
"Definition:Martingale",
"Definition:Stopping Time",
"Definition:Stopped Process",
"Definition:Martingale"
] | [
"Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale",
"Definition:Supermartingale",
"Definition:Submartingale",
"Stopped Supermartingale is Supermartingale",
"Stopped Submartingale is Submartingale",
"Definition:Supermartingale",
"Definition:Submartingale",
"Adapted Stochas... |
proofwiki-20203 | Equivalence of Definitions of Martingale in Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be an adapted stochastic process.
{{TFAE|def = Martingale in Discrete Time|view = martingale in discrete time}} | === Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} = X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} = X_n$.
Fix $n \in \Z_{\ge 0}$.
We i... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \mathop \ge 0}$ be an [[Definition:Adapted Stochastic Process|adapted stochastic process]].
{{TFAE|def = Martingale in Discrete Time|view = marti... | === Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is [[Definition:Integrable Random Variable|integrable]] for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} = X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \... | Equivalence of Definitions of Martingale in Discrete Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Martingale_in_Discrete_Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Martingale_in_Discrete_Time | [
"Martingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Adapted Stochastic Process"
] | [
"Definition:Integrable Random Variable",
"Definition:Proposition",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Defini... |
proofwiki-20204 | Equivalence of Definitions of Submartingale in Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an adapted stochastic process.
{{TFAE|def = Submartingale/Discrete Time|submartingale in discrete time}} | === Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \ge X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} \ge X_n$.
Fix $n \in \Z_{\ge 0}$.
... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Adapted Stochastic Process|adapted stochastic process]].
{{TFAE|def = Submartingale/Discrete Time|submartingale in discrete ti... | === Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is [[Definition:Integrable Random Variable|integrable]] for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \ge X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m... | Equivalence of Definitions of Submartingale in Discrete Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Submartingale_in_Discrete_Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Submartingale_in_Discrete_Time | [
"Submartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Adapted Stochastic Process"
] | [
"Definition:Integrable Random Variable",
"Definition:Proposition",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Defini... |
proofwiki-20205 | Equivalence of Definitions of Supermartingale in Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an adapted stochastic process.
{{TFAE|def = Submartingale/Discrete Time|supermartingale in discrete time}} | === Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \le X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} \le X_n$.
Fix $n \in \Z_{\ge 0}$.
... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Adapted Stochastic Process|adapted stochastic process]].
{{TFAE|def = Submartingale/Discrete Time|supermartingale in discrete ... | === Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is [[Definition:Integrable Random Variable|integrable]] for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \le X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m... | Equivalence of Definitions of Supermartingale in Discrete Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Supermartingale_in_Discrete_Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Supermartingale_in_Discrete_Time | [
"Supermartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Adapted Stochastic Process"
] | [
"Definition:Integrable Random Variable",
"Definition:Proposition",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Defini... |
proofwiki-20206 | Expected Value of Supermartingale Less Than or Equal To Initial Expected Value | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be a supermartingale.
Then:
:$\expect {X_n} \le \expect {X_0}$
for each $n \in \Z_{\ge 0}$. | From Definition 2 of a discrete time supermartingale, we have:
:$\expect {X_n \mid \FF_0} \le X_0$ almost surely.
So from Expectation is Monotone:
:$\expect {\expect {X_n \mid \FF_0} } \le \expect {X_0}$
From Expectation of Conditional Expectation, we have:
:$\expect {\expect {X_n \mid \FF_0} } \le \expect {X_n}$
So:
:... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be a [[Definition:Supermartingale|supermartingale]].
Then:
:$\expect {X_n} \le \expect {X_0}$
for each $n \in \Z_{\ge 0}$. | From [[Definition:Supermartingale/Discrete Time/Definition 2|Definition 2 of a discrete time supermartingale]], we have:
:$\expect {X_n \mid \FF_0} \le X_0$ [[Definition:Almost Everywhere|almost surely]].
So from [[Expectation is Monotone]]:
:$\expect {\expect {X_n \mid \FF_0} } \le \expect {X_0}$
From [[Expectatio... | Expected Value of Supermartingale Less Than or Equal To Initial Expected Value | https://proofwiki.org/wiki/Expected_Value_of_Supermartingale_Less_Than_or_Equal_To_Initial_Expected_Value | https://proofwiki.org/wiki/Expected_Value_of_Supermartingale_Less_Than_or_Equal_To_Initial_Expected_Value | [
"Supermartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Supermartingale"
] | [
"Definition:Supermartingale/Discrete Time/Definition 2",
"Definition:Almost Everywhere",
"Expectation is Monotone",
"Expectation of Conditional Expectation",
"Category:Supermartingales"
] |
proofwiki-20207 | Expected Value of Submartingale Greater Than or Equal To Initial Expected Value | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be a submartingale.
Then:
:$\expect {X_n} \ge \expect {X_0}$
for each $n \in \Z_{\ge 0}$. | From Definition 2 of a discrete time submartingale, we have:
:$\expect {X_n \mid \FF_0} \ge X_0$ almost surely.
So from Expectation is Monotone:
:$\expect {\expect {X_n \mid \FF_0} } \ge \expect {X_0}$
From Expectation of Conditional Expectation, we have:
:$\expect {\expect {X_n \mid \FF_0} } \ge \expect {X_n}$
So:
:$\... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be a [[Definition:Submartingale|submartingale]].
Then:
:$\expect {X_n} \ge \expect {X_0}$
for each $n \in \Z_{\ge 0}$. | From [[Definition:Submartingale/Discrete Time/Definition 2|Definition 2 of a discrete time submartingale]], we have:
:$\expect {X_n \mid \FF_0} \ge X_0$ [[Definition:Almost Everywhere|almost surely]].
So from [[Expectation is Monotone]]:
:$\expect {\expect {X_n \mid \FF_0} } \ge \expect {X_0}$
From [[Expectation of... | Expected Value of Submartingale Greater Than or Equal To Initial Expected Value | https://proofwiki.org/wiki/Expected_Value_of_Submartingale_Greater_Than_or_Equal_To_Initial_Expected_Value | https://proofwiki.org/wiki/Expected_Value_of_Submartingale_Greater_Than_or_Equal_To_Initial_Expected_Value | [
"Submartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Submartingale"
] | [
"Definition:Submartingale/Discrete Time/Definition 2",
"Definition:Almost Everywhere",
"Expectation is Monotone",
"Expectation of Conditional Expectation",
"Category:Submartingales"
] |
proofwiki-20208 | Primitive of x by Square of Logarithm of x | :$\ds \int x \ln^2 x \rd x = \dfrac {x^2 \ln^2 x} 2 - \dfrac {x^2 \ln x} 2 + \dfrac {x^2} 4 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = x \cdot \dfrac \d {\d x} \ln x + \ln x \dfrac \d {\d... | :$\ds \int x \ln^2 x \rd x = \dfrac {x^2 \ln^2 x} 2 - \dfrac {x^2 \ln x} 2 + \dfrac {x^2} 4 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = x \cdot \dfrac \d {\d x} \... | Primitive of x by Square of Logarithm of x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Logarithm_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Square_of_Logarithm_of_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Product Rule for Derivatives",
"Derivative of Natural Logarithm Function",
"Derivative of Identity Function",
"Primitive of Logarithm of x",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power/Examples/x"
] |
proofwiki-20209 | Primitive of x squared by Exponential of a x/Examples/x squared by e^-x | :$\ds \int x^2 e^{-x} \rd x = -e^{-x} \paren {x^2 + 2 x + 2} + C$ | From Primitive of $x^2 e^{a x}$:
{{:Primitive of x squared by Exponential of a x}}
So:
{{begin-eqn}}
{{eqn | l = \int x^2 e^{-x} \rd x
| r = \frac {e^{-x} } {-1} \paren {x^2 - \frac {2 x} {-1} + \frac 2 {\paren {-1}^2} } + C
| c = Primitive of $x^2 e^{a x}$: setting $a = -1$
}}
{{eqn | r = -e^{-x} \paren {x... | :$\ds \int x^2 e^{-x} \rd x = -e^{-x} \paren {x^2 + 2 x + 2} + C$ | From [[Primitive of x squared by Exponential of a x|Primitive of $x^2 e^{a x}$]]:
{{:Primitive of x squared by Exponential of a x}}
So:
{{begin-eqn}}
{{eqn | l = \int x^2 e^{-x} \rd x
| r = \frac {e^{-x} } {-1} \paren {x^2 - \frac {2 x} {-1} + \frac 2 {\paren {-1}^2} } + C
| c = [[Primitive of x squared by... | Primitive of x squared by Exponential of a x/Examples/x squared by e^-x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_a_x/Examples/x_squared_by_e^-x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_a_x/Examples/x_squared_by_e^-x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of x squared by Exponential of a x",
"Primitive of x squared by Exponential of a x",
"Category:Primitives involving Cosine Function"
] |
proofwiki-20210 | Doob's Optional Stopping Theorem/Discrete Time/Supermartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \ma... | We first show that if $T$ is finite almost surely, then:
:$\map {X_{n \wedge T} } \omega \to \map {X_T} \omega$
for almost all $\omega \in \Omega$.
Let $\omega \in \Omega$ be such that $\map T \omega = s < \infty$.
Then for $n \ge s$, we have:
:$\map {X_{n \wedge T} } \omega = \map {X_s} \omega$
From Constant Sequen... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with r... | We first show that if $T$ is [[Definition:Finite Extended Real Number|finite]] [[Definition:Almost Surely|almost surely]], then:
:$\map {X_{n \wedge T} } \omega \to \map {X_T} \omega$
for [[Definition:Almost All|almost all]] $\omega \in \Omega$.
Let $\omega \in \Omega$ be such that $\map T \omega = s < \infty$.
T... | Doob's Optional Stopping Theorem/Discrete Time/Supermartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem/Discrete_Time/Supermartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem/Discrete_Time/Supermartingale | [
"Doob's Optional Stopping Theorem",
"Supermartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Supermartingale",
"Definition:Stopping Time",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Finite Extended Real Number",
"Definition:Almost Everywhere",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"De... | [
"Definition:Finite Extended Real Number",
"Definition:Almost Everywhere",
"Definition:Almost All",
"Constant Sequence in Topological Space Converges",
"Definition:Finite Extended Real Number",
"Definition:Almost Everywhere",
"Definition:Almost All",
"Definition:Almost Sure Convergence",
"Stopped Sup... |
proofwiki-20211 | Doob's Optional Stopping Theorem/Discrete Time/Submartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-submartingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map ... | From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale:
:$\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale. | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Submartingale|$\sequence {\FF_n}_{n \ge 0}$-submartingale]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with respe... | From [[Adapted Stochastic Process is Supermartingale iff Negative is Submartingale]]:
:$\sequence {-X_n}_{n \ge 0}$ is a [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]]. | Doob's Optional Stopping Theorem/Discrete Time/Submartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem/Discrete_Time/Submartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem/Discrete_Time/Submartingale | [
"Doob's Optional Stopping Theorem",
"Submartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Submartingale",
"Definition:Stopping Time",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Finite Extended Real Number",
"Definition:Almost Everywhere",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Defi... | [
"Adapted Stochastic Process is Supermartingale iff Negative is Submartingale",
"Definition:Supermartingale"
] |
proofwiki-20212 | Doob's Optional Stopping Theorem/Discrete Time/Martingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-martingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map {\c... | From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Doob's Optional Stopping Theorem: Discrete Time: Supermartingale and Doob's Optional Stopping Theorem... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Martingale|$\sequence {\FF_n}_{n \ge 0}$-martingale]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with respect to ... | From [[Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale]]:
:$\sequence {X_n}_{n \ge 0}$ is a [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]] and [[Definition:Submartingale|$\sequence {\FF_n}_{n \ge 0}$-submartingale]].
From [[Doob's Optional Stopping Theorem/... | Doob's Optional Stopping Theorem/Discrete Time/Martingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem/Discrete_Time/Martingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem/Discrete_Time/Martingale | [
"Doob's Optional Stopping Theorem",
"Martingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Martingale",
"Definition:Stopping Time",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Finite Extended Real Number",
"Definition:Almost Everywhere",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Definit... | [
"Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale",
"Definition:Supermartingale",
"Definition:Submartingale",
"Doob's Optional Stopping Theorem/Discrete Time/Supermartingale",
"Doob's Optional Stopping Theorem/Discrete Time/Submartingale",
"Definition:Integrable Random Variab... |
proofwiki-20213 | Least Time at which Discrete-Time Adapted Stochastic Process equals or exceeds Real Number is Stopping Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Let $\lambda \in \R$.
Let:
:$T = \inf \set {n \in \Z_{\ge 0} : X_n \ge \lambda}$
Then $T$ is a stopping time wit... | Note that for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$, we have:
:$\inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t$
{{iff}}:
:$\map {X_s} \omega \ge \lambda$ for some $s \le t$.
That is, we have:
:$\ds \set {\omega \in \Omega : \inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t} =... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be a [[Definition:Adapted Stochastic Process/Discrete Time|discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process]].
Let $\lambda ... | Note that for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$, we have:
:$\inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t$
{{iff}}:
:$\map {X_s} \omega \ge \lambda$ for some $s \le t$.
That is, we have:
:$\ds \set {\omega \in \Omega : \inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le... | Least Time at which Discrete-Time Adapted Stochastic Process equals or exceeds Real Number is Stopping Time | https://proofwiki.org/wiki/Least_Time_at_which_Discrete-Time_Adapted_Stochastic_Process_equals_or_exceeds_Real_Number_is_Stopping_Time | https://proofwiki.org/wiki/Least_Time_at_which_Discrete-Time_Adapted_Stochastic_Process_equals_or_exceeds_Real_Number_is_Stopping_Time | [
"Adapted Stochastic Processes",
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Adapted Stochastic Process/Discrete Time",
"Definition:Stopping Time"
] | [
"Definition:Adapted Stochastic Process",
"Definition:Filtration of Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Finite Union",
"Definition:Stopping Time",
"Category:Adapted Stochastic Processes",
"Category:Stopping Times"
] |
proofwiki-20214 | Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
Let $S$ and $T$ be bounded stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ and $S \le T$.
Let $\FF_S$ be the stopped $\sigm... | From Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra:
:$X_S$ is $\FF_S$-measurable.
Then from Conditional Expectation of Measurable Random Variable:
:$\expect {X_S \mid \FF_S} = X_S$ almost surely.
Also from Doob's Optional Stopping Theorem: Discrete Time: Supermartingale... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]].
Let $S$ and $T$ be [[Definition:Bounded Real-Valued Function|b... | From [[Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra]]:
:$X_S$ is [[Definition:Measurable Function|$\FF_S$-measurable]].
Then from [[Conditional Expectation of Measurable Random Variable]]:
:$\expect {X_S \mid \FF_S} = X_S$ [[Definition:Almost Everywhere|almost surel... | Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem_for_Stopped_Sigma-Algebra_of_Bounded_Stopping_Time/Discrete_Time/Supermartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem_for_Stopped_Sigma-Algebra_of_Bounded_Stopping_Time/Discrete_Time/Supermartingale | [
"Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time",
"Supermartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Supermartingale",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Stopping Time",
"Definition:Stopped Sigma-Algebra",
"Definition:Adapted Stochastic Process at Stopping Time",
"Definition:Almost Everywhere"
] | [
"Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra",
"Definition:Measurable Function",
"Conditional Expectation of Measurable Random Variable",
"Definition:Almost Everywhere",
"Doob's Optional Stopping Theorem/Discrete Time/Supermartingale",
"Definition:Integr... |
proofwiki-20215 | Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Submartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-submartingale.
Let $S$ and $T$ be bounded stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ and $S \le T$.
Let $\FF_S$ be the stopped $\sigma$... | From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale:
:$\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
From Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time: Discrete Time: Supermartingale:
:$\expect {-X_T \mid \FF_S} \le -... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Submartingale|$\sequence {\FF_n}_{n \ge 0}$-submartingale]].
Let $S$ and $T$ be [[Definition:Bounded Real-Valued Function|bound... | From [[Adapted Stochastic Process is Supermartingale iff Negative is Submartingale]]:
:$\sequence {-X_n}_{n \ge 0}$ is a [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]].
From [[Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartinga... | Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Submartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem_for_Stopped_Sigma-Algebra_of_Bounded_Stopping_Time/Discrete_Time/Submartingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem_for_Stopped_Sigma-Algebra_of_Bounded_Stopping_Time/Discrete_Time/Submartingale | [
"Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time",
"Submartingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Submartingale",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Stopping Time",
"Definition:Stopped Sigma-Algebra",
"Definition:Adapted Stochastic Process at Stopping Time",
"Definition:Almost Everywhere"
] | [
"Adapted Stochastic Process is Supermartingale iff Negative is Submartingale",
"Definition:Supermartingale",
"Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartingale",
"Definition:Almost Everywhere",
"Expectation is Linear",
"Definition:Almost Ever... |
proofwiki-20216 | Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Martingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-martingale.
Let $S$ and $T$ be bounded stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ and $S \le T$.
Let $\FF_S$ be the stopped $\sigma$-al... | From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time: Discrete Time: S... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Martingale|$\sequence {\FF_n}_{n \ge 0}$-martingale]].
Let $S$ and $T$ be [[Definition:Bounded Real-Valued Function|bounded]] [... | From [[Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale]]:
:$\sequence {X_n}_{n \ge 0}$ is a [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]] and [[Definition:Submartingale|$\sequence {\FF_n}_{n \ge 0}$-submartingale]].
From [[Doob's Optional Stopping Theorem ... | Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Martingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem_for_Stopped_Sigma-Algebra_of_Bounded_Stopping_Time/Discrete_Time/Martingale | https://proofwiki.org/wiki/Doob's_Optional_Stopping_Theorem_for_Stopped_Sigma-Algebra_of_Bounded_Stopping_Time/Discrete_Time/Martingale | [
"Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time",
"Martingales"
] | [
"Definition:Filtered Probability Space",
"Definition:Martingale",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Stopping Time",
"Definition:Stopped Sigma-Algebra",
"Definition:Adapted Stochastic Process at Stopping Time",
"Definition:Almost Everywhere"
] | [
"Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale",
"Definition:Supermartingale",
"Definition:Submartingale",
"Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartingale",
"Doob's Optional Stopping Theorem for Stopped Sigma... |
proofwiki-20217 | Derivative of Product of Operator-Valued Functions | Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$, and $\struct {Z, \norm \cdot_Z}$ normed vector spaces.
Let $\map B {X, Y}$, $\map B {Y, Z}$ and $\map B {X, Z} $ denote the space of bounded linear transformations between $X$ and $Y$, between $Y$ and $Z$, and between $X$ and $Z$, respectively.
Let $A : I ... | Given arbitrary $x, h \in I$ such that $h \ne 0$, $x+h\in I$ we compute:
:$\norm {\dfrac {\map {\paren {A B} } {x + h} - \map {\paren {A B} } x} h - \map {A'} x \map B x - \map A x \map {B'} x}_{\map B {X, Z} }$
to be:
{{begin-eqn}}
{{eqn | o =
| r = \norm {\dfrac {\map A {x + h}\map B {x + h} - \map A x \map B ... | Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$, and $\struct {Z, \norm \cdot_Z}$ [[Definition:Normed Vector Space|normed vector spaces]].
Let $\map B {X, Y}$, $\map B {Y, Z}$ and $\map B {X, Z} $ denote the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations]] b... | Given arbitrary $x, h \in I$ such that $h \ne 0$, $x+h\in I$ we compute:
:$\norm {\dfrac {\map {\paren {A B} } {x + h} - \map {\paren {A B} } x} h - \map {A'} x \map B x - \map A x \map {B'} x}_{\map B {X, Z} }$
to be:
{{begin-eqn}}
{{eqn | o =
| r = \norm {\dfrac {\map A {x + h}\map B {x + h} - \map A x \map B... | Derivative of Product of Operator-Valued Functions | https://proofwiki.org/wiki/Derivative_of_Product_of_Operator-Valued_Functions | https://proofwiki.org/wiki/Derivative_of_Product_of_Operator-Valued_Functions | [
"Product Rule for Derivatives",
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Differentiable Mapping/Function With Values in Normed Space",
"Definition:Mapping",
"Definition:Real Interval",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Bounded Linear Transformation",... | [
"Norm on Bounded Linear Transformation is Submultiplicative",
"Definition:Derivative/Function With Values in Normed Space",
"Definition:Derivative/Function With Values in Normed Space",
"Differentiable Operator-Valued Function is Continuous",
"Reverse Triangle Inequality/Normed Vector Space",
"Definition:... |
proofwiki-20218 | Doob's Maximal Inequality/Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a non-negative $\sequence {\FF_n}_{n \mathop \ge 0}$-submartingale.
Let:
:$\ds X_n^\ast = \max_{0 \mathop \le k \mathop \le n} X_k$
where $\max$ is the pointwise maximum. ... | Let $\lambda > 0 $ and $n \ge 0$.
Let:
:$E := \set {X^\ast _n \ge \lambda}$.
That is, $E$ is a disjoint union:
:$(1):\quad \ds E = \bigsqcup _{0 \mathop \le k \mathop \le n} E_k$
where:
:$\ds E_k := \set {X_k \ge \lambda} \cap \bigcap _{0 \mathop \le j \mathop \le k-1} \set {X_j < \lambda}$
By construction, we have:
:$... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a [[Definition:Positive Measurable Function|non-negative]] [[Definition:Submartingale|$\sequence {\FF_n}_{n \mathop \ge 0}$-subm... | Let $\lambda > 0 $ and $n \ge 0$.
Let:
:$E := \set {X^\ast _n \ge \lambda}$.
That is, $E$ is a [[Definition:Disjoint Union (Set Theory)/Disjoint Sets|disjoint union]]:
:$(1):\quad \ds E = \bigsqcup _{0 \mathop \le k \mathop \le n} E_k$
where:
:$\ds E_k := \set {X_k \ge \lambda} \cap \bigcap _{0 \mathop \le j \mathop ... | Doob's Maximal Inequality/Discrete Time/Proof 2 | https://proofwiki.org/wiki/Doob's_Maximal_Inequality/Discrete_Time | https://proofwiki.org/wiki/Doob's_Maximal_Inequality/Discrete_Time/Proof_2 | [
"Doob's Maximal Inequality"
] | [
"Definition:Filtered Probability Space",
"Definition:Measurable Function/Positive",
"Definition:Submartingale",
"Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions"
] | [
"Definition:Disjoint Union (Set Theory)/Disjoint Sets",
"Rule for Extracting Random Variable from Conditional Expectation of Product",
"Definition:Characteristic Function (Set Theory)",
"Tower Property of Conditional Expectation",
"Expectation is Linear",
"Integral of Characteristic Function"
] |
proofwiki-20219 | Expectation is Monotone | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be integrable random variables such that:
:$\forall \omega \in \Omega: \map X \omega \le \map Y \omega$
{{explain|Should we craft a link to one of the instances of pointwise inequality? We this for general functions, do we need one for random v... | From the definition of expectation we have:
:$\ds \expect X = \int X \rd \Pr$
and:
:$\ds \expect Y = \int Y \rd \Pr$
The result follows directly from Integral of Integrable Function is Monotone.
{{qed}}
Category:Expectation
kgzhf3379pyvuq13d31whi6ighqubtt | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Integrable Random Variable|integrable random variables]] such that:
:$\forall \omega \in \Omega: \map X \omega \le \map Y \omega$
{{explain|Should we craft a link to one of the instances of po... | From the definition of [[Definition:Expectation|expectation]] we have:
:$\ds \expect X = \int X \rd \Pr$
and:
:$\ds \expect Y = \int Y \rd \Pr$
The result follows directly from [[Integral of Integrable Function is Monotone]].
{{qed}}
[[Category:Expectation]]
kgzhf3379pyvuq13d31whi6ighqubtt | Expectation is Monotone | https://proofwiki.org/wiki/Expectation_is_Monotone | https://proofwiki.org/wiki/Expectation_is_Monotone | [
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable"
] | [
"Definition:Expectation",
"Integral of Integrable Function is Monotone",
"Category:Expectation"
] |
proofwiki-20220 | Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.
Let $\struct{\map {CL} {X, Y}, \norm{\, \cdot \,}}$ be the continuous linear transformation space equipped with the supremum operator norm.
Then $\struct {\map {CL} {X, Y}, \norm{\, \cdot \,} }$ is a Banach Space ... | === Necessary Condition ===
Let $Y$ be a Banach space.
Let $\sequence {T_n}_{n \mathop \in \N} \in \map {CL} {X, Y}$ be a Cauchy sequence.
Let $x \in X$.
Let $\norm {\, \cdot \,}$ be the supremum operator norm. | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $\struct{\map {CL} {X, Y}, \norm{\, \cdot \,}}$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]] equipped with the [[Defin... | === Necessary Condition ===
Let $Y$ be a [[Definition:Banach Space|Banach space]].
Let $\sequence {T_n}_{n \mathop \in \N} \in \map {CL} {X, Y}$ be a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence]].
Let $x \in X$.
Let $\norm {\, \cdot \,}$ be the [[Definition:Supremum Operator Norm|supremum op... | Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space | https://proofwiki.org/wiki/Necessary_and_Sufficient_Conditions_for_Continuous_Linear_Transformation_Space_to_be_Banach_Space | https://proofwiki.org/wiki/Necessary_and_Sufficient_Conditions_for_Continuous_Linear_Transformation_Space_to_be_Banach_Space | [
"Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space",
"Normed Vector Spaces",
"Banach Spaces",
"Continuous Linear Transformation Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Supremum Operator Norm",
"Definition:Banach Space",
"Definition:Banach Space"
] | [
"Definition:Banach Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Supremum Operator Norm",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Banach Space",
"Definition... |
proofwiki-20221 | Area under Curve | Let $f: \R \to \R$ be a real function which is defined and (Darboux) integrable on the closed interval $\closedint a b$.
Consider the graph of $f$ embedded in a Cartesian plane.
Let $\AA$ denote the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.
Then:
:$\AA = \ds \int_a^b \... | === Overview of Proof ===
:500px
Let $x \in \closedint a b$.
Let $\delta x$ be an arbitrarily small positive real number such that $x + \delta x \in \closedint a b$.
Consider the small strip between:
:the $x$-axis
:the vertical straight lines through $\tuple {x, 0}$ and $\tuple {x + \delta x, 0}$
:the curve $\map f x$... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which is defined and [[Definition:Darboux Integrable Function|(Darboux) integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Consider the [[Definition:Graph of Mapping|graph]] of $f$ embedded in a [[Definition:Carte... | === Overview of Proof ===
:[[File:Area-under-Curve.png|500px]]
Let $x \in \closedint a b$.
Let $\delta x$ be an arbitrarily small [[Definition:Positive Real Number|positive real number]] such that $x + \delta x \in \closedint a b$.
Consider the small strip between:
:the [[Definition:X-Axis|$x$-axis]]
:the [[Defini... | Area under Curve | https://proofwiki.org/wiki/Area_under_Curve | https://proofwiki.org/wiki/Area_under_Curve | [
"Area under Curve",
"Definite Integrals",
"Analytic Geometry",
"Area Formulas"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Line/Curve",
"Definition:Line/Straight Line",
"Definition:Axis/X-Axis",
"Definition:Definite Integral... | [
"File:Area-under-Curve.png",
"Definition:Positive/Real Number",
"Definition:Axis/X-Axis",
"Definition:Vertical",
"Definition:Line/Straight Line",
"Definition:Line/Curve",
"Definition:Area",
"Definition:Mathematical Model",
"Definition:Darboux Integral/Geometric Interpretation",
"Definition:Area",
... |
proofwiki-20222 | Single Point Characterization of Simple Closed Contour | Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.
Let $t_0 \in \openint a b$ such that $\gamma$ is complex-differentiable at $t_0$.
Let $S \in \set {-1,1}$ and $r \in \R_{>0}$ such that:
:for all $\epsilon \in \openint 0 r$, we have $\map \gamma {t_0} + ... | We show that for all $t_1 \in \openint a b$ where $\gamma$ is complex-differentiable at $t_1$, there exists $r \in \R_{>0}$ such that for all $\epsilon \in \openint 0 r$, it follows that:
:$\map v { t_1, \epsilon} \in \Int C$.
The result then follows by the definitions of positively oriented contour and negatively orie... | Let $C$ be a [[Definition:Simple Contour (Complex Plane)|simple]] [[Definition:Closed Contour (Complex Plane)|closed contour]] in the [[Definition:Complex Plane|complex plane]] $\C$ with [[Definition:Parameterization of Contour (Complex Plane)|parameterization]] $\gamma: \closedint a b \to \C$.
Let $t_0 \in \openint a... | We show that for all $t_1 \in \openint a b$ where $\gamma$ is [[Definition:Complex-Differentiable Function|complex-differentiable]] at $t_1$, there exists $r \in \R_{>0}$ such that for all $\epsilon \in \openint 0 r$, it follows that:
:$\map v { t_1, \epsilon} \in \Int C$.
The result then follows by the definitions o... | Single Point Characterization of Simple Closed Contour | https://proofwiki.org/wiki/Single_Point_Characterization_of_Simple_Closed_Contour | https://proofwiki.org/wiki/Single_Point_Characterization_of_Simple_Closed_Contour | [
"Orientation of Complex Contour",
"Single Point Characterization of Simple Closed Contour"
] | [
"Definition:Contour/Simple/Complex Plane",
"Definition:Contour/Closed/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Differentiable Mapping/Complex Function/Point",
"Definition:Interior of Simple Closed Contour",
"Definition:Or... | [
"Definition:Differentiable Mapping/Complex Function",
"Definition:Orientation of Contour (Complex Plane)/Positive/Simple Closed",
"Definition:Orientation of Contour (Complex Plane)/Negative/Simple Closed",
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Subdivision of Interval",
"Definiti... |
proofwiki-20223 | Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 1 | Let $\mathbb K = \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\mathbb K$.
Let $CL$ be the continuous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\struct {\map {CL} {X, \mathbb K}, \norm {\, \cdot \,} }$ is a Banach space. | :$X' := \map {CL} {X, \mathbb K}$ is the dual space of $X$
So this theorem is the same as Normed Dual Space is Banach Space.
{{improve|proper proof please}}
{{qed}} | Let $\mathbb K = \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\mathbb K$.
Let $CL$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $\norm {\, \cdot \,}$ be the [[Definition:Supremum ... | :$X' := \map {CL} {X, \mathbb K}$ is the dual space of $X$
So this theorem is the same as [[Normed Dual Space is Banach Space]].
{{improve|proper proof please}}
{{qed}} | Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 1 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Conditions_for_Continuous_Linear_Transformation_Space_to_be_Banach_Space/Corollary_1 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Conditions_for_Continuous_Linear_Transformation_Space_to_be_Banach_Space/Corollary_1 | [
"Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Supremum Operator Norm",
"Definition:Banach Space"
] | [
"Normed Dual Space is Banach Space"
] |
proofwiki-20224 | Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 2 | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space.
Let $CL$ be the continuous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\struct {\map {CL} {X, X}, \norm{\, \cdot \,}}$ is a Banach space. | Take $Y = X$ in Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space.
{{qed}} | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]].
Let $CL$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $\norm {\, \cdot \,}$ be the [[Definition:Supremum Operator Norm|supremum operator norm]].
Then $\struct {\map {CL... | Take $Y = X$ in [[Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space]].
{{qed}} | Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 2 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Conditions_for_Continuous_Linear_Transformation_Space_to_be_Banach_Space/Corollary_2 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Conditions_for_Continuous_Linear_Transformation_Space_to_be_Banach_Space/Corollary_2 | [
"Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space"
] | [
"Definition:Banach Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Supremum Operator Norm",
"Definition:Banach Space"
] | [
"Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space"
] |
proofwiki-20225 | Differentiable Operator-Valued Function is Continuous | Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $f : I \to X$ be a map defined on an interval $I$.
Let $x_0 \in I$ such that $f$ is differentiable at $x_0$.
Then $f$ is continuous at $x_0$. | We have {{hypothesis}} that the derivative $\map {f'} {x_0}$ of $f$ at $x_0$ exists.
Hence:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to x_0} \norm {\map f {x} - \map f {x_0} }_X
| r = \lim_{h \mathop \to 0} \norm {\map f {x_0 + h} - \map f {x_0} }_X
| c = re-write limit
}}
{{eqn | r = \lim_{h \mathop \to ... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $f : I \to X$ be a [[Definition:Mapping|map]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $x_0 \in I$ such that $f$ is [[Definition:Differentiable Mapping/Function With Values in Normed Space|differentia... | We have {{hypothesis}} that the [[Definition:Derivative/Function With Values in Normed Space|derivative]] $\map {f'} {x_0}$ of $f$ at $x_0$ exists.
Hence:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to x_0} \norm {\map f {x} - \map f {x_0} }_X
| r = \lim_{h \mathop \to 0} \norm {\map f {x_0 + h} - \map f {x_0} }... | Differentiable Operator-Valued Function is Continuous | https://proofwiki.org/wiki/Differentiable_Operator-Valued_Function_is_Continuous | https://proofwiki.org/wiki/Differentiable_Operator-Valued_Function_is_Continuous | [
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Mapping",
"Definition:Real Interval",
"Definition:Differentiable Mapping/Function With Values in Normed Space",
"Definition:Continuous Mapping (Normed Vector Space)/Point"
] | [
"Definition:Derivative/Function With Values in Normed Space",
"Definition:Differentiable Mapping/Function With Values in Normed Space",
"Definition:Derivative/Function With Values in Normed Space",
"Definition:Continuous Mapping (Metric Space)/Point/Definition 2",
"Metric Induced by Norm is Metric"
] |
proofwiki-20226 | Reciprocal of 451 | :$\dfrac 1 {451} = 0 \cdotp \dot 00221 \, 7294 \dot 9$ | Performing the calculation using long division:
<pre>
0.0022172949002...
-------------------
451)1.0000000000000
902
---
980
902
---
780
451
---
3290
3157
----
1330
902
----
4280
4059
... | :$\dfrac 1 {451} = 0 \cdotp \dot 00221 \, 7294 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0022172949002...
-------------------
451)1.0000000000000
902
---
980
902
---
780
451
---
3290
3157
----
1330
902
----
... | Reciprocal of 451 | https://proofwiki.org/wiki/Reciprocal_of_451 | https://proofwiki.org/wiki/Reciprocal_of_451 | [
"451",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-20227 | Composition of Continuous Mapping on Compact Space Preserves Uniform Convergence | Let $\struct{ X, d_X }$ be a compact metric space.
Let $\struct{ Y, d_Y }, \struct{ Z, d_Z }$ be metric spaces.
Let $\phi : Y \to Z$ be a continuous mapping.
For all $n \in \N$, let $f_n : X \to Y$ be a continuous mapping.
Let $f: X \to Y$ be a mapping such that $\sequence {f_n}_{n \in \N}$ converges to $f$ uniformly o... | {{AimForCont}} that $\sequence {\phi \circ f_n}_{n \in \N}$ do not converge to $\phi \circ f$ uniformly.
This implies that there exists $\epsilon \in \R_{>0}$ such that for all $n \in \N$, there exist $N_n \ge n$ and $x_n \in X$ such that:
:$\map {d_Z}{ \map { \phi \circ f_{N_n} }{ x_n } , \map { \phi \circ f }{ x_n} }... | Let $\struct{ X, d_X }$ be a [[Definition:Compact Metric Space|compact metric space]].
Let $\struct{ Y, d_Y }, \struct{ Z, d_Z }$ be [[Definition:Metric Space|metric spaces]].
Let $\phi : Y \to Z$ be a [[Definition:Continuous Mapping (Metric Space)|continuous mapping]].
For all $n \in \N$, let $f_n : X \to Y$ be a [... | {{AimForCont}} that $\sequence {\phi \circ f_n}_{n \in \N}$ do not [[Definition:Uniform Convergence on Metric Space|converge to $\phi \circ f$ uniformly]].
This implies that there exists $\epsilon \in \R_{>0}$ such that for all $n \in \N$, there exist $N_n \ge n$ and $x_n \in X$ such that:
:$\map {d_Z}{ \map { \phi \... | Composition of Continuous Mapping on Compact Space Preserves Uniform Convergence | https://proofwiki.org/wiki/Composition_of_Continuous_Mapping_on_Compact_Space_Preserves_Uniform_Convergence | https://proofwiki.org/wiki/Composition_of_Continuous_Mapping_on_Compact_Space_Preserves_Uniform_Convergence | [
"Uniform Convergence",
"Sequentially Compact Spaces"
] | [
"Definition:Compact Space/Metric Space",
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Mapping",
"Definition:Uniform Convergence/Metric Space",
"Definition:Sequence",
"Definition:Uniform Convergence/Metric Space"
... | [
"Definition:Uniform Convergence/Metric Space",
"Compact Subspace of Metric Space is Sequentially Compact in Itself",
"Definition:Sequentially Compact Space/In Itself",
"Definition:Subsequence",
"Metric is Continous Mapping",
"Definition:Continuous Real Function/Point",
"Uniformly Convergent Sequence Eva... |
proofwiki-20228 | Primitive of Reciprocal of Root of x squared plus Constant | :$\ds \int \frac {\d x} {\sqrt {x^2 + k} } = \ln \size {x + \sqrt {x^2 + k} } + C$ | === Positive Constant ===
Let $k > 0$.
Then $k = a^2$ for some $a \in \R$.
Hence from Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form:
{{:Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form}}
from which the result follows.
{{qed|lemma}} | :$\ds \int \frac {\d x} {\sqrt {x^2 + k} } = \ln \size {x + \sqrt {x^2 + k} } + C$ | === Positive Constant ===
Let $k > 0$.
Then $k = a^2$ for some $a \in \R$.
Hence from [[Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form|Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form]]:
{{:Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form}}
from whic... | Primitive of Reciprocal of Root of x squared plus Constant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_Constant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_Constant | [
"Primitive of Reciprocal of Root of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-20229 | Unique Tree of Order 2 | There exists exactly one tree of order $2$:
:200px | By definition, a graph of order $2$ has exactly $2$ nodes.
By Finite Connected Simple Graph is Tree iff Size is One Less than Order, such a tree has one edge.
There is trivially only one way to connect $2$ nodes with one edge.
Hence the result.
{{qed}}
Category:Tree Theory
tsygc5fl4tlqba2xegmme4svi2p7uuk | There exists [[Definition:Unique|exactly one]] [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order $2$]]:
:[[File:Tree-Order-2.png|200px]] | By definition, a [[Definition:Graph (Graph Theory)|graph]] of [[Definition:Order of Graph|order $2$]] has exactly $2$ [[Definition:Node of Tree|nodes]].
By [[Finite Connected Simple Graph is Tree iff Size is One Less than Order]], such a [[Definition:Tree (Graph Theory)|tree]] has one [[Definition:Edge of Graph|edge]]... | Unique Tree of Order 2 | https://proofwiki.org/wiki/Unique_Tree_of_Order_2 | https://proofwiki.org/wiki/Unique_Tree_of_Order_2 | [
"Tree Theory"
] | [
"Definition:Unique",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"File:Tree-Order-2.png"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Tree (Graph Theory)/Node",
"Finite Connected Simple Graph is Tree iff Size is One Less than Order",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Tree (Graph Theory)/Node",
"... |
proofwiki-20230 | Path Homotopy is Equivalence Relation | Let $X$ be a topological space.
Let $p, q \in X$.
Then path homotopy on the set of all paths in $X$ from $p$ to $q$ is an equivalence relation. | Suppose $X$ is a topological space.
Let $p, q \in X$.
Let $\sim$ denote the path homotopy on the set of all paths in $X$ from $p$ to $q$.
{{ProofWanted}}
Checking in turn each of the criteria for equivalence: | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $p, q \in X$.
Then [[Definition:Path Homotopy|path homotopy]] on the [[Definition:Set|set]] of all [[Definition:Path (Topology)|paths]] in $X$ from $p$ to $q$ is an [[Definition:Equivalence Relation|equivalence relation]]. | Suppose $X$ is a [[Definition:Topological Space|topological space]].
Let $p, q \in X$.
Let $\sim$ denote the [[Definition:Path Homotopy|path homotopy]] on the [[Definition:Set|set]] of all [[Definition:Path (Topology)|paths]] in $X$ from $p$ to $q$.
{{ProofWanted}}
Checking in turn each of the criteria for [[Definit... | Path Homotopy is Equivalence Relation | https://proofwiki.org/wiki/Path_Homotopy_is_Equivalence_Relation | https://proofwiki.org/wiki/Path_Homotopy_is_Equivalence_Relation | [
"Homotopy Theory",
"Examples of Equivalence Relations"
] | [
"Definition:Topological Space",
"Definition:Homotopy/Path/Path Homotopy",
"Definition:Set",
"Definition:Path (Topology)",
"Definition:Equivalence Relation"
] | [
"Definition:Topological Space",
"Definition:Homotopy/Path/Path Homotopy",
"Definition:Set",
"Definition:Path (Topology)",
"Definition:Equivalence Relation"
] |
proofwiki-20231 | Mertens' Convergence Theorem | Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be real or complex sequences.
Let:
:$\ds \sum_{n \mathop = 0}^\infty a_n$ converge to $A$
:$\ds \sum_{n \mathop = 0}^\infty b_n$ converge to $B$.
Let either $\ds \sum_{n \mathop = 0}^\infty a_n$ or $\ds \sum_{n \mathop = 0}^\infty b_n$ con... | {{ProofWanted}}
{{Namedfor|Franz Mertens|cat = Mertens}} | Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be [[Definition:Real Sequence|real]] or [[Definition:Complex Sequence|complex]] [[Definition:Sequence|sequences]].
Let:
:$\ds \sum_{n \mathop = 0}^\infty a_n$ [[Definition:Convergent Sequence|converge]] to $A$
:$\ds \sum_{n \mathop = 0}^... | {{ProofWanted}}
{{Namedfor|Franz Mertens|cat = Mertens}} | Mertens' Convergence Theorem | https://proofwiki.org/wiki/Mertens'_Convergence_Theorem | https://proofwiki.org/wiki/Mertens'_Convergence_Theorem | [
"Convergence"
] | [
"Definition:Real Sequence",
"Definition:Complex Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Convergent Sequence",
"Definition:Absolute Convergence",
"Definition:Absolute Convergence",
"Definition:Cauchy Product"
] | [] |
proofwiki-20232 | Gauss's Hypergeometric Theorem | :$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$ | Let $x, y, n \in \C$ be complex numbers such that $\map \Re {x + y + n + 1} > 0$.
Let $u \in \C$ be a complex number such that $\cmod u < 1$.
Expanding the product of $\paren {1 + u}^{y + n}$ and $\paren {\dfrac {1 + u} u}^x$:
{{begin-eqn}}
{{eqn | l = \paren {1 + u}^{y + n}
| r = \sum_{k \mathop = 0}^\infty \bin... | :$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$ | Let $x, y, n \in \C$ be [[Definition:Complex Number|complex numbers]] such that $\map \Re {x + y + n + 1} > 0$.
Let $u \in \C$ be a [[Definition:Complex Number|complex number]] such that $\cmod u < 1$.
Expanding the product of $\paren {1 + u}^{y + n}$ and $\paren {\dfrac {1 + u} u}^x$:
{{begin-eqn}}
{{eqn | l = \par... | Gauss's Hypergeometric Theorem/Proof 1 | https://proofwiki.org/wiki/Gauss's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Gauss's_Hypergeometric_Theorem/Proof_1 | [
"Gauss's Hypergeometric Theorem",
"Gaussian Hypergeometric Function",
"Gamma Function"
] | [] | [
"Definition:Complex Number",
"Definition:Complex Number",
"Binomial Theorem/Extended",
"Binomial Theorem/Extended",
"Exponent Combination Laws/Power of Product",
"Definition:Coefficient",
"Definition:Series/Number Field",
"Definition:Series/Number Field",
"Negated Upper Index of Binomial Coefficient... |
proofwiki-20233 | Gauss's Hypergeometric Theorem | :$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$ | From Euler's Integral Representation of Hypergeometric Function, we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$
Where $a, b, c \in \C$.
and $\size x < 1$
and $\map \Re c > \map \Re b > 0$.
Sinc... | :$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$ | From [[Euler's Integral Representation of Hypergeometric Function]], we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$
Where $a, b, c \in \C$.
and $\size x < 1$
and $\map \Re c > \map \Re b > 0... | Gauss's Hypergeometric Theorem/Proof 2 | https://proofwiki.org/wiki/Gauss's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Gauss's_Hypergeometric_Theorem/Proof_2 | [
"Gauss's Hypergeometric Theorem",
"Gaussian Hypergeometric Function",
"Gamma Function"
] | [] | [
"Euler's Integral Representation of Hypergeometric Function",
"Euler's Integral Representation of Hypergeometric Function",
"Definition:Limit of Real Function",
"Definition:Definite Integral",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-20234 | Closed Ball is Convex Set | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\map { {B_r}^-} x$ be a closed ball in $X$ with radius $r \in \R_{>0}$ and center $x \in X$.
Then $\map { {B_r}^-} x$ is convex. | Let $y \in \map { {B_1}^-} {\mathbf 0}$.
From {{NormAxiomVector|2}}, it follows that:
:$\norm {r y} = r \norm y$
It follows that:
:$y \in \map { {B_1}^-} {\mathbf 0}$, {{iff}} $r y \in \map { {B_r}^-} {\mathbf 0}$
As $\norm {r y - \mathbf 0} = \norm {\paren {r y + x} - x}$, it follows that:
:$r y \in \map { {B_r}^-} {\... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map { {B_r}^-} x$ be a [[Definition:Closed Ball in Normed Vector Space|closed ball]] in $X$ with [[Definition:Radius of Closed Ball|radius]] $r \in \R_{>0}$ and [[Definition:Center of Closed Ball|center]] $x \in ... | Let $y \in \map { {B_1}^-} {\mathbf 0}$.
From {{NormAxiomVector|2}}, it follows that:
:$\norm {r y} = r \norm y$
It follows that:
:$y \in \map { {B_1}^-} {\mathbf 0}$, {{iff}} $r y \in \map { {B_r}^-} {\mathbf 0}$
As $\norm {r y - \mathbf 0} = \norm {\paren {r y + x} - x}$, it follows that:
:$r y \in \map { {B_r}^... | Closed Ball is Convex Set | https://proofwiki.org/wiki/Closed_Ball_is_Convex_Set | https://proofwiki.org/wiki/Closed_Ball_is_Convex_Set | [
"Closed Balls",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Closed Ball/Metric Space/Radius",
"Definition:Closed Ball/Metric Space/Center",
"Definition:Convex Set (Vector Space)"
] | [
"Closed Unit Ball is Convex Set",
"Definition:Convex Set (Vector Space)",
"Dilation of Convex Set in Vector Space is Convex",
"Definition:Convex Set (Vector Space)",
"Translation of Convex Set in Vector Space is Convex",
"Definition:Convex Set (Vector Space)"
] |
proofwiki-20235 | Paths in Interior and Exterior of Jordan Curve | Let $\gamma : \closedint 0 1 \to \R^2$ be a Jordan curve.
Let $t_1 , t_2 \in \closedint 0 1$ with $t_1 < t_2$.
Let $h \in \R_{>0}$.
Let $\Img \gamma$, $\Int \gamma$ and $\Ext \gamma$ denote the image, interior, and exterior of $\gamma$.
Then there exists $r \in \R_{>0}$ such that $r \le h$, and for all $p_1 \in \map {B... | Let $\phi : \R^2 \to \R^2$ be the homeomorphism defined in the Jordan-Schönflies Theorem such that:
{{begin-eqn}}
{{eqn | l = \phi \sqbrk {\Img \gamma}
| r = \mathbb S_1
}}
{{eqn | l = \phi \sqbrk {\Int \gamma}
| r = \map {B_1}{ \mathbf 0 }
}}
{{eqn | l = \phi \sqbrk {\Ext \gamma}
| r = \R^2 \setmi... | Let $\gamma : \closedint 0 1 \to \R^2$ be a [[Definition:Jordan Curve|Jordan curve]].
Let $t_1 , t_2 \in \closedint 0 1$ with $t_1 < t_2$.
Let $h \in \R_{>0}$.
Let $\Img \gamma$, $\Int \gamma$ and $\Ext \gamma$ denote the [[Definition:Image of Mapping|image]], [[Definition:Interior of Jordan Curve|interior]], and [[... | Let $\phi : \R^2 \to \R^2$ be the [[Definition:Homeomorphic Metric Spaces|homeomorphism]] defined in the [[Jordan-Schönflies Theorem]] such that:
{{begin-eqn}}
{{eqn | l = \phi \sqbrk {\Img \gamma}
| r = \mathbb S_1
}}
{{eqn | l = \phi \sqbrk {\Int \gamma}
| r = \map {B_1}{ \mathbf 0 }
}}
{{eqn | l = \ph... | Paths in Interior and Exterior of Jordan Curve | https://proofwiki.org/wiki/Paths_in_Interior_and_Exterior_of_Jordan_Curve | https://proofwiki.org/wiki/Paths_in_Interior_and_Exterior_of_Jordan_Curve | [
"Jordan Curves"
] | [
"Definition:Jordan Curve",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Jordan Curve/Interior",
"Definition:Jordan Curve/Exterior",
"Definition:Path (Topology)",
"Definition:Path (Topology)",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Radius",
"Definition:Open ... | [
"Definition:Homeomorphism/Metric Spaces",
"Jordan-Schönflies Theorem",
"Definition:Unit Circle",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Path (Topology)",
"Definition:Jordan Curve/Interior",
"Definition:Path (Topology)",
"Definition:Concatenation of Paths",
"Definition:Path (Topolo... |
proofwiki-20236 | Interior and Exterior of Partially Disjoint Jordan Curves | Let $\gamma_0 , \gamma_1 , \gamma_2 : \closedint 0 1 \to \R^2$ be Jordan arcs such that:
:$ \map {\gamma_0} 0 = \map {\gamma_1} 1 = \map {\gamma_2} 1$
:$\map {\gamma_0} 1 = \map {\gamma_1} 0 = \map {\gamma_2} 0$
Let $\gamma_0 \sqbrk { \openint 0 1 }$ and $\gamma_1 \sqbrk { \openint 0 1 }$ be disjoint sets, and let $\ga... | By definition of Jordan curve, it follows that $\gamma_0 * \gamma_1$ and $\gamma_0 * \gamma_2$ are Jordan curves. | Let $\gamma_0 , \gamma_1 , \gamma_2 : \closedint 0 1 \to \R^2$ be [[Definition:Jordan Arc|Jordan arcs]] such that:
:$ \map {\gamma_0} 0 = \map {\gamma_1} 1 = \map {\gamma_2} 1$
:$\map {\gamma_0} 1 = \map {\gamma_1} 0 = \map {\gamma_2} 0$
Let $\gamma_0 \sqbrk { \openint 0 1 }$ and $\gamma_1 \sqbrk { \openint 0 1 }$ b... | By definition of [[Definition:Jordan Curve|Jordan curve]], it follows that $\gamma_0 * \gamma_1$ and $\gamma_0 * \gamma_2$ are [[Definition:Jordan Curve|Jordan curves]]. | Interior and Exterior of Partially Disjoint Jordan Curves | https://proofwiki.org/wiki/Interior_and_Exterior_of_Partially_Disjoint_Jordan_Curves | https://proofwiki.org/wiki/Interior_and_Exterior_of_Partially_Disjoint_Jordan_Curves | [
"Jordan Curves"
] | [
"Definition:Jordan Arc",
"Definition:Disjoint Sets",
"Definition:Disjoint Sets",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Jordan Curve/Interior",
"Definition:Jordan Curve/Ext... | [
"Definition:Jordan Curve",
"Definition:Jordan Curve"
] |
proofwiki-20237 | Sequences of Projections in 2-Sequence Space Converges in Strong Operator Topology | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $p$-sequence normed vector space.
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.
For $n \in \N$ let $P_n \in \map {CL} {\ell^2}$ be the projection operator over $\ell^2$.
Let $\sequence {P_n}_{n \mathop \in \N}$ ... | Let $\mathbf a = \tuple {a_1, a_2, \ldots} \in \ell^2$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {I \mathbf a - P_n \mathbf a}^2_2
| r = \norm {\ldots, 0, a_{n+1}, a_{n + 2}, \ldots}^2_2
| c = {{Defof|Projection Operator over 2-Sequence Spaces}}
}}
{{eqn | r = \sum_{k \mathop = n + 1}^\infty \size {a_k}^2
}}
{... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|$p$-sequence normed vector space]].
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
For $n \in \N$ l... | Let $\mathbf a = \tuple {a_1, a_2, \ldots} \in \ell^2$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {I \mathbf a - P_n \mathbf a}^2_2
| r = \norm {\ldots, 0, a_{n+1}, a_{n + 2}, \ldots}^2_2
| c = {{Defof|Projection Operator over 2-Sequence Spaces}}
}}
{{eqn | r = \sum_{k \mathop = n + 1}^\infty \size {a_k}^2
}}... | Sequences of Projections in 2-Sequence Space Converges in Strong Operator Topology | https://proofwiki.org/wiki/Sequences_of_Projections_in_2-Sequence_Space_Converges_in_Strong_Operator_Topology | https://proofwiki.org/wiki/Sequences_of_Projections_in_2-Sequence_Space_Converges_in_Strong_Operator_Topology | [
"Convergence"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Projection Operator over 2-Sequence Spaces",
"Definition:Sequence",
"Definition:Identity Mapping",
"Definition:Convergent Sequence in Strong Operator Topology",
"Definition:Strong O... | [
"Tail of Convergent Series tends to Zero",
"Definition:Convergent Sequence in Strong Operator Topology",
"Definition:Strong Operator Topology"
] |
proofwiki-20238 | Dixon's Hypergeometric Theorem | :$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map... | From Gauss's Hypergeometric Theorem, we have:
:$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$
Therefore:
{{begin-eqn}}
{{eqn | l = \map F {b + n, c + n; 1 + a + 2 n; 1}
| r = \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {\paren {1 + a + 2 n} - \... | :$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map... | From [[Gauss's Hypergeometric Theorem]], we have:
:$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$
Therefore:
{{begin-eqn}}
{{eqn | l = \map F {b + n, c + n; 1 + a + 2 n; 1}
| r = \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {\paren {1 + a + 2 n... | Dixon's Hypergeometric Theorem/Proof 2 | https://proofwiki.org/wiki/Dixon's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Dixon's_Hypergeometric_Theorem/Proof_2 | [
"Dixon's Hypergeometric Theorem",
"Hypergeometric Functions",
"Gamma Function"
] | [] | [
"Gauss's Hypergeometric Theorem",
"Gauss's Hypergeometric Theorem",
"Rising Factorial as Quotient of Factorials",
"Rising Factorial as Quotient of Factorials",
"Product of Absolutely Convergent Series",
"Rising Factorial in terms of Falling Factorial of Negative",
"Rising Factorial as Quotient of Factor... |
proofwiki-20239 | Sequences of Projections in 2-Sequence Space do not Converge in Uniform Operator Topology | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $p$-sequence normed vector space.
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
For $n \in \N$ let $P_n \in \map {CL} {\ell^2}$ be the projection operator... | {{AimForCont}} $\sequence {P_n}_{n \mathop \in \N}$ converges to $I$ in the uniform operator topology.
By definition:
:$\forall \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \norm {I - P_n} < \epsilon$
Suppose $\mathbf e_{N + 1} \in \ell^2$ is such that:
:$\mathbf e_{N + 1} = \tuple {\und... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|$p$-sequence normed vector space]].
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $\norm {\, \... | {{AimForCont}} $\sequence {P_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Uniform Operator Topology|converges]] to $I$ in the [[Definition:Uniform Operator Topology|uniform operator topology]].
By definition:
:$\forall \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \norm {I... | Sequences of Projections in 2-Sequence Space do not Converge in Uniform Operator Topology | https://proofwiki.org/wiki/Sequences_of_Projections_in_2-Sequence_Space_do_not_Converge_in_Uniform_Operator_Topology | https://proofwiki.org/wiki/Sequences_of_Projections_in_2-Sequence_Space_do_not_Converge_in_Uniform_Operator_Topology | [
"Convergence"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Supremum Operator Norm",
"Definition:Projection Operator over 2-Sequence Spaces",
"Definition:Sequence",
"Definition:Identity Mapping",
"Definition:Convergent Sequence in Uniform Op... | [
"Definition:Convergent Sequence in Uniform Operator Topology",
"Definition:Uniform Operator Topology",
"Supremum Operator Norm as Universal Upper Bound",
"Definition:Contradiction",
"Definition:Convergent Sequence in Uniform Operator Topology",
"Definition:Uniform Operator Topology"
] |
proofwiki-20240 | Addition of Numbers is not Distributive over Multiplication | Addition of numbers is not distributive over multiplication.
That is, for numbers $a$, $b$ and $c$ it is not necessarily the case that $a + \paren {b \times c} = \paren {a + b} \times \paren {a + c}$. | Proof by Counterexample:
{{begin-eqn}}
{{eqn | l = 2 + \paren {3 \times 6}
| r = 2 + 18
| c =
}}
{{eqn | r = 20
| c =
}}
{{eqn | l = \paren {2 + 3} \times \paren {2 + 6}
| r = 6 + 8
| c =
}}
{{eqn | r = 48
| c =
}}
{{eqn | o = \ne
| r = 2 + \paren {3 \times 6}
| c =
... | [[Definition:Addition|Addition]] of [[Definition:Number|numbers]] is not [[Definition:Distributive Operation|distributive]] over [[Definition:Multiplication|multiplication]].
That is, for [[Definition:Number|numbers]] $a$, $b$ and $c$ it is not necessarily the case that $a + \paren {b \times c} = \paren {a + b} \times... | [[Proof by Counterexample]]:
{{begin-eqn}}
{{eqn | l = 2 + \paren {3 \times 6}
| r = 2 + 18
| c =
}}
{{eqn | r = 20
| c =
}}
{{eqn | l = \paren {2 + 3} \times \paren {2 + 6}
| r = 6 + 8
| c =
}}
{{eqn | r = 48
| c =
}}
{{eqn | o = \ne
| r = 2 + \paren {3 \times 6}
| ... | Addition of Numbers is not Distributive over Multiplication | https://proofwiki.org/wiki/Addition_of_Numbers_is_not_Distributive_over_Multiplication | https://proofwiki.org/wiki/Addition_of_Numbers_is_not_Distributive_over_Multiplication | [
"Examples of Distributive Operations"
] | [
"Definition:Addition",
"Definition:Number",
"Definition:Distributive Operation",
"Definition:Multiplication",
"Definition:Number"
] | [
"Proof by Counterexample"
] |
proofwiki-20241 | Number of Sides of Polygon equals Number of Interior Angles | Let $A$ be a polygon.
The number of sides of $A$ is equal to the number of interior angles of $A$. | {{ProofWanted|Obvious, which makes it difficult}} | Let $A$ be a [[Definition:Polygon|polygon]].
The number of [[Definition:Side of Polygon|sides]] of $A$ is equal to the number of [[Definition:Interior Angle of Polygon|interior angles]] of $A$. | {{ProofWanted|Obvious, which makes it difficult}} | Number of Sides of Polygon equals Number of Interior Angles | https://proofwiki.org/wiki/Number_of_Sides_of_Polygon_equals_Number_of_Interior_Angles | https://proofwiki.org/wiki/Number_of_Sides_of_Polygon_equals_Number_of_Interior_Angles | [
"Internal Angles"
] | [
"Definition:Polygon",
"Definition:Polygon/Side",
"Definition:Polygon/Internal Angle"
] | [] |
proofwiki-20242 | Equilateral Polygon is not necessarily Equiangular | Let $P$ be an equilateral polygon with more than $3$ sides.
Then it is not necessarily the case that $P$ is also equiangular. | ;Proof by Counterexample
We take as an example the rhombus:
{{:Definition:Rhombus}}{{qed}} | Let $P$ be an [[Definition:Equilateral Polygon|equilateral polygon]] with more than $3$ [[Definition:Side of Polygon|sides]].
Then it is not necessarily the case that $P$ is also [[Definition:Equiangular Polygon|equiangular]]. | ;[[Proof by Counterexample]]
We take as an example the [[Definition:Rhombus|rhombus]]:
{{:Definition:Rhombus}}{{qed}} | Equilateral Polygon is not necessarily Equiangular | https://proofwiki.org/wiki/Equilateral_Polygon_is_not_necessarily_Equiangular | https://proofwiki.org/wiki/Equilateral_Polygon_is_not_necessarily_Equiangular | [
"Polygons"
] | [
"Definition:Polygon/Equilateral",
"Definition:Polygon/Side",
"Definition:Polygon/Equiangular"
] | [
"Proof by Counterexample",
"Definition:Quadrilateral/Rhombus"
] |
proofwiki-20243 | Equiangular Polygon is not necessarily Equilateral | Let $P$ be an equiangular polygon with more than $3$ sides.
Then it is not necessarily the case that $P$ is also equilateral. | ;Proof by Counterexample
We take as an example the rectangle:
{{:Definition:Rectangle}}{{qed}} | Let $P$ be an [[Definition:Equiangular Polygon|equiangular polygon]] with more than $3$ [[Definition:Side of Polygon|sides]].
Then it is not necessarily the case that $P$ is also [[Definition:Equilateral Polygon|equilateral]]. | ;[[Proof by Counterexample]]
We take as an example the [[Definition:Rectangle|rectangle]]:
{{:Definition:Rectangle}}{{qed}} | Equiangular Polygon is not necessarily Equilateral | https://proofwiki.org/wiki/Equiangular_Polygon_is_not_necessarily_Equilateral | https://proofwiki.org/wiki/Equiangular_Polygon_is_not_necessarily_Equilateral | [
"Polygons"
] | [
"Definition:Polygon/Equiangular",
"Definition:Polygon/Side",
"Definition:Polygon/Equilateral"
] | [
"Proof by Counterexample",
"Definition:Quadrilateral/Rectangle"
] |
proofwiki-20244 | Exterior Angle of Regular Polygon | Let $P$ be a regular polygon with $n$ sides.
Then each of the exterior angles of $P$ is equal to $\dfrac {360 \degrees} n$. | From Sum of External Angles of Polygon equals Four Right Angles, the sum of all $n$ exterior angles of $P$ equals $360 \degrees$.
We have {{afortiori}} that the interior angles of $P$ are equal.
Hence the exterior angles of $P$ are also equal.
Hence each exterior angle of $P$ equals $\dfrac {360 \degrees} n$.
{{qed}} | Let $P$ be a [[Definition:Regular Polygon|regular polygon]] with $n$ [[Definition:Side of Polygon|sides]].
Then each of the [[Definition:Exterior Angle of Polygon|exterior angles]] of $P$ is equal to $\dfrac {360 \degrees} n$. | From [[Sum of External Angles of Polygon equals Four Right Angles]], the [[Definition:Addition|sum]] of all $n$ [[Definition:Exterior Angle of Polygon|exterior angles]] of $P$ equals $360 \degrees$.
We have {{afortiori}} that the [[Definition:Interior Angle of Polygon|interior angles]] of $P$ are equal.
Hence the [[D... | Exterior Angle of Regular Polygon | https://proofwiki.org/wiki/Exterior_Angle_of_Regular_Polygon | https://proofwiki.org/wiki/Exterior_Angle_of_Regular_Polygon | [
"External Angles"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Polygon/External Angle"
] | [
"Sum of External Angles of Polygon equals Four Right Angles",
"Definition:Addition",
"Definition:Polygon/External Angle",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/External Angle",
"Definition:Polygon/External Angle"
] |
proofwiki-20245 | Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic | Let $Q$ be a quadrilateral whose sides are $a$, $b$, $c$ and $d$.
Let $\AA$ be the area of $Q$.
Then:
:$Q$ is a cyclic quadrilateral
{{iff}}:
:$\AA$ is the greatest area possible for one with sides $a$, $b$, $c$ and $d$. | From Bretschneider's Formula:
:$\AA = \sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} - a b c d \map {\cos^2} {\dfrac {\alpha + \gamma} 2} }$
where $s$ is the semiperimeter of $Q$.
Hence $\AA$ is greatest exactly when:
:$\map {\cos^2} {\dfrac {\alpha + \gamma} 2} = 0$
That is, when:
:$\map \cos {\dfr... | Let $Q$ be a [[Definition:Quadrilateral|quadrilateral]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$, $c$ and $d$.
Let $\AA$ be the [[Definition:Area|area]] of $Q$.
Then:
:$Q$ is a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]]
{{iff}}:
:$\AA$ is the greatest [[Definition:Area|area]] possible fo... | From [[Bretschneider's Formula]]:
:$\AA = \sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} - a b c d \map {\cos^2} {\dfrac {\alpha + \gamma} 2} }$
where $s$ is the [[Definition:Semiperimeter|semiperimeter]] of $Q$.
Hence $\AA$ is greatest exactly when:
:$\map {\cos^2} {\dfrac {\alpha + \gamma} 2} =... | Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic | https://proofwiki.org/wiki/Area_of_Quadrilateral_with_Given_Sides_is_Greatest_when_Quadrilateral_is_Cyclic | https://proofwiki.org/wiki/Area_of_Quadrilateral_with_Given_Sides_is_Greatest_when_Quadrilateral_is_Cyclic | [
"Cyclic Quadrilaterals"
] | [
"Definition:Quadrilateral",
"Definition:Polygon/Side",
"Definition:Area",
"Definition:Cyclic Quadrilateral",
"Definition:Area",
"Definition:Polygon/Side"
] | [
"Bretschneider's Formula",
"Definition:Semiperimeter",
"Brahmagupta's Formula",
"Definition:Area",
"Definition:Cyclic Quadrilateral",
"Definition:Polygon/Side",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
"Category:Cyclic Quadrilaterals"
] |
proofwiki-20246 | Regular Polygon is Cyclic | Let $P$ be a regular polygon.
Then $P$ is a cyclic polygon. | Let $P$ be a regular polygon.
{{AimForCont}} $P$ is not cyclic.
Let $AB$, $BC$ and $CD$ be sides of $P$ such that $A$, $B$ and $C$ are on the circumference of a circle $K$ such that $D$ is not on that circumference.
This has to be possible, or all vertices of $P$ would lie on $K$.
That would make $P$ cyclic which contr... | Let $P$ be a [[Definition:Regular Polygon|regular polygon]].
Then $P$ is a [[Definition:Cyclic Polygon|cyclic polygon]]. | Let $P$ be a [[Definition:Regular Polygon|regular polygon]].
{{AimForCont}} $P$ is not [[Definition:Cyclic Polygon|cyclic]].
Let $AB$, $BC$ and $CD$ be [[Definition:Side of Polygon|sides]] of $P$ such that $A$, $B$ and $C$ are on the [[Definition:Circumference of Circle|circumference]] of a [[Definition:Circle|circle... | Regular Polygon is Cyclic | https://proofwiki.org/wiki/Regular_Polygon_is_Cyclic | https://proofwiki.org/wiki/Regular_Polygon_is_Cyclic | [
"Regular Polygons"
] | [
"Definition:Polygon/Regular",
"Definition:Cyclic Polygon"
] | [
"Definition:Polygon/Regular",
"Definition:Cyclic Polygon",
"Definition:Polygon/Side",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Circle/Circumference",
"Definition:Polygon/Vertex",
"Definition:Cyclic Polygon",
"Definition:Contradiction",
"Definition:Circle/Center",
"Defi... |
proofwiki-20247 | Long Radius of Regular Polygon equals Radius of Circumcircle | Let $P$ be a regular polygon.
Let $C$ be the circumcircle of $P$.
The long radius of $P$ is equal to the radius of $C$. | By definition of circumcircle, $C$ is the circle such that all vertices of $P$ are on $C$.
By definition, the center of $P$ is defined as the center of $C$.
Hence:
:$OA$ is the radius of $C$
and also:
:$OA$ is the long radius of $P$.
Hence the result.
{{qed}} | Let $P$ be a [[Definition:Regular Polygon|regular polygon]].
Let $C$ be the [[Definition:Circumcircle|circumcircle]] of $P$.
The [[Definition:Long Radius of Regular Polygon|long radius]] of $P$ is equal to the [[Definition:Radius of Circle|radius]] of $C$. | By definition of [[Definition:Circumcircle|circumcircle]], $C$ is the [[Definition:Circle|circle]] such that all [[Definition:Vertex of Polygon|vertices]] of $P$ are on $C$.
By definition, the [[Definition:Center of Regular Polygon|center]] of $P$ is defined as the [[Definition:Center of Circle|center]] of $C$.
Hence... | Long Radius of Regular Polygon equals Radius of Circumcircle | https://proofwiki.org/wiki/Long_Radius_of_Regular_Polygon_equals_Radius_of_Circumcircle | https://proofwiki.org/wiki/Long_Radius_of_Regular_Polygon_equals_Radius_of_Circumcircle | [
"Regular Polygons"
] | [
"Definition:Polygon/Regular",
"Definition:Circumcircle",
"Definition:Polygon/Regular/Long Radius",
"Definition:Circle/Radius"
] | [
"Definition:Circumcircle",
"Definition:Circle",
"Definition:Polygon/Vertex",
"Definition:Polygon/Regular/Center",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Polygon/Regular/Long Radius"
] |
proofwiki-20248 | Apothem of Regular Polygon equals Radius of Incircle | Let $P$ be a regular polygon.
Let $C$ be the incircle of $P$.
The apothem of $P$ is equal to the radius of $C$. | By definition of incircle, $C$ is the circle such that all sides of $P$ are tangent to $C$.
From Regular Polygon can be Circumscribed around Circle, it is established that the center of $P$ and the center of $C$ are the same point.
Hence:
:the radius of $C$
is the same thing as:
:the perpendicular distance from $C$ of ... | Let $P$ be a [[Definition:Regular Polygon|regular polygon]].
Let $C$ be the [[Definition:Incircle|incircle]] of $P$.
The [[Definition:Apothem|apothem]] of $P$ is equal to the [[Definition:Radius of Circle|radius]] of $C$. | By definition of [[Definition:Incircle|incircle]], $C$ is the [[Definition:Circle|circle]] such that all [[Definition:Side of Polygon|sides]] of $P$ are [[Definition:Tangent to Circle|tangent]] to $C$.
From [[Regular Polygon can be Circumscribed around Circle]], it is established that the [[Definition:Center of Regula... | Apothem of Regular Polygon equals Radius of Incircle | https://proofwiki.org/wiki/Apothem_of_Regular_Polygon_equals_Radius_of_Incircle | https://proofwiki.org/wiki/Apothem_of_Regular_Polygon_equals_Radius_of_Incircle | [
"Regular Polygons"
] | [
"Definition:Polygon/Regular",
"Definition:Incircle",
"Definition:Regular Polygon/Apothem",
"Definition:Circle/Radius"
] | [
"Definition:Incircle",
"Definition:Circle",
"Definition:Polygon/Side",
"Definition:Tangent Line/Circle",
"Regular Polygon can be Circumscribed around Circle",
"Definition:Polygon/Regular/Center",
"Definition:Circle/Center",
"Definition:Point",
"Definition:Circle/Radius",
"Definition:Perpendicular ... |
proofwiki-20249 | Regular Polygon can be Circumscribed around Circle | Let $P$ be a regular polygon.
Then it is possible to circumscribe $P$ around a circle $C$, whose center is the same as the center of $P$. | Let $P$ be a regular polygon.
{{AimForCont}} it is not possible to circumscribe $P$ around a circle $C$, whose center is the same as the center of $P$.
Let $AB$, $BC$ and $CD$ be sides of $P$ such that $AB$ and $BC$ are on the tangent to a circle $K$ such that $CD$ is not tangent to that circumference.
There can only b... | Let $P$ be a [[Definition:Regular Polygon|regular polygon]].
Then it is possible to [[Definition:Polygon Circumscribed around Circle|circumscribe]] $P$ around a [[Definition:Circle|circle]] $C$, whose [[Definition:Center of Circle|center]] is the same as the [[Definition:Center of Regular Polygon|center]] of $P$. | Let $P$ be a [[Definition:Regular Polygon|regular polygon]].
{{AimForCont}} it is not possible to [[Definition:Polygon Circumscribed around Circle|circumscribe]] $P$ around a [[Definition:Circle|circle]] $C$, whose [[Definition:Center of Circle|center]] is the same as the [[Definition:Center of Regular Polygon|center]... | Regular Polygon can be Circumscribed around Circle | https://proofwiki.org/wiki/Regular_Polygon_can_be_Circumscribed_around_Circle | https://proofwiki.org/wiki/Regular_Polygon_can_be_Circumscribed_around_Circle | [
"Regular Polygons"
] | [
"Definition:Polygon/Regular",
"Definition:Circumscribe/Polygon around Circle",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Polygon/Regular/Center"
] | [
"Definition:Polygon/Regular",
"Definition:Circumscribe/Polygon around Circle",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Polygon/Regular/Center",
"Definition:Polygon/Side",
"Definition:Tangent Line/Circle",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Definition:Circl... |
proofwiki-20250 | Dougall's Hypergeometric Theorem | :$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } {\map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map ... | By definition of the generalized hypergeometric function
:$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\ove... | :$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } {\map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map ... | By definition of the [[Definition:Generalized Hypergeometric Function|generalized hypergeometric function]]
:$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {\dfrac n 2 ... | Dougall's Hypergeometric Theorem | https://proofwiki.org/wiki/Dougall's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Dougall's_Hypergeometric_Theorem | [
"Dougall's Hypergeometric Theorem",
"Hypergeometric Functions",
"Gamma Function"
] | [] | [
"Definition:Hypergeometric Function/Generalized"
] |
proofwiki-20251 | Kummer's Hypergeometric Theorem | :$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$ | First we note the definition of Gaussian hypergeometric function:
:$\map F {n, -x; x + n + 1; -1} = \ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} } {\paren {x + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}$
where $x^{\overline k}$ denotes the $k$th rising factorial power of $x... | :$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$ | First we note the definition of [[Definition:Gaussian Hypergeometric Function|Gaussian hypergeometric function]]:
:$\map F {n, -x; x + n + 1; -1} = \ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} } {\paren {x + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}$
where $x^{\overline ... | Kummer's Hypergeometric Theorem/Proof 1 | https://proofwiki.org/wiki/Kummer's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Kummer's_Hypergeometric_Theorem/Proof_1 | [
"Kummer's Hypergeometric Theorem",
"Hypergeometric Functions",
"Gamma Function"
] | [] | [
"Definition:Hypergeometric Function/Gaussian",
"Definition:Rising Factorial",
"Definition:Lemma",
"Kummer's Hypergeometric Theorem/Lemma 1",
"Kummer's Hypergeometric Theorem/Lemma 2",
"Dixon's Hypergeometric Theorem",
"Definition:Hypergeometric Function/Generalized",
"Dixon's Hypergeometric Theorem",
... |
proofwiki-20252 | Kummer's Hypergeometric Theorem | :$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$ | From Euler's Integral Representation of Hypergeometric Function, we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$
Where $a, b, c \in \C$.
and $\size x < 1$
and $\map \Re c > \map \Re b > 0$.
Sinc... | :$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$ | From [[Euler's Integral Representation of Hypergeometric Function]], we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$
Where $a, b, c \in \C$.
and $\size x < 1$
and $\map \Re c > \map \Re b > 0... | Kummer's Hypergeometric Theorem/Proof 2 | https://proofwiki.org/wiki/Kummer's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Kummer's_Hypergeometric_Theorem/Proof_2 | [
"Kummer's Hypergeometric Theorem",
"Hypergeometric Functions",
"Gamma Function"
] | [] | [
"Euler's Integral Representation of Hypergeometric Function",
"Euler's Integral Representation of Hypergeometric Function",
"Definition:Limit of Real Function",
"Definition:Definite Integral",
"Integration by Substitution",
"Power Rule for Derivatives",
"Definition:Fraction/Numerator",
"Definition:Fra... |
proofwiki-20253 | Kummer's Hypergeometric Theorem | :$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$ | From Kummer's Quadratic Transformation, we have:
:$\ds \map F {a, b; 1 + a - b; z} = \paren {1 - z}^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; \dfrac {-4 z} {\paren {1 - z }^2} }$
Let $z \to -1$ and we have:
:$\ds \map F {a, b; 1 + a - b; -1} = 2^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; ... | :$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$ | From [[Kummer's Quadratic Transformation]], we have:
:$\ds \map F {a, b; 1 + a - b; z} = \paren {1 - z}^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; \dfrac {-4 z} {\paren {1 - z }^2} }$
Let $z \to -1$ and we have:
:$\ds \map F {a, b; 1 + a - b; -1} = 2^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + ... | Kummer's Hypergeometric Theorem/Proof 3 | https://proofwiki.org/wiki/Kummer's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Kummer's_Hypergeometric_Theorem/Proof_3 | [
"Kummer's Hypergeometric Theorem",
"Hypergeometric Functions",
"Gamma Function"
] | [] | [
"Kummer's Quadratic Transformation",
"Gauss's Hypergeometric Theorem",
"Gauss's Hypergeometric Theorem",
"Legendre's Duplication Formula"
] |
proofwiki-20254 | Dougall-Ramanujan Identity | {{begin-eqn}}
{{eqn | l = \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x + y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1}
| r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1}... | By definition of generalized hypergeometric function of $1$:
:$\ds \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x + y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overlin... | {{begin-eqn}}
{{eqn | l = \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x + y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1}
| r = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1}... | By definition of [[Definition:Generalized Hypergeometric Function|generalized hypergeometric function]] of $1$:
:$\ds \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x + y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1}... | Dougall-Ramanujan Identity | https://proofwiki.org/wiki/Dougall-Ramanujan_Identity | https://proofwiki.org/wiki/Dougall-Ramanujan_Identity | [
"Hypergeometric Functions",
"Gamma Function"
] | [] | [
"Definition:Hypergeometric Function/Generalized"
] |
proofwiki-20255 | Existence of Set of Ordinals leads to Contradiction | The existence of the set of all ordinals leads to a contradiction. | Suppose that the collection of all ordinals is a set.
Let this set be denoted as $\On$.
From {{Corollary|Ordinals are Well-Ordered}}, it is seen that $\Epsilon {\restriction_{\On} }$ is a strict well-ordering on $\On$.
By Element of Ordinal is Ordinal, it is seen that $\On$ is transitive.
Hence $\On$ is itself an ordin... | The existence of the [[Definition:Set|set]] of all [[Definition:Ordinal|ordinals]] leads to a [[Definition:Contradiction|contradiction]]. | Suppose that the [[Definition:Collection|collection]] of all [[Definition:Ordinal|ordinals]] is a [[Definition:Set|set]].
Let this [[Definition:Set|set]] be denoted as $\On$.
From {{Corollary|Ordinals are Well-Ordered}}, it is seen that $\Epsilon {\restriction_{\On} }$ is a [[Definition:Strict Well-Ordering|strict w... | Existence of Set of Ordinals leads to Contradiction | https://proofwiki.org/wiki/Existence_of_Set_of_Ordinals_leads_to_Contradiction | https://proofwiki.org/wiki/Existence_of_Set_of_Ordinals_leads_to_Contradiction | [
"Existence of Set of Ordinals leads to Contradiction",
"Ordinals",
"Naive Set Theory"
] | [
"Definition:Set",
"Definition:Ordinal",
"Definition:Contradiction"
] | [
"Definition:Collection",
"Definition:Ordinal",
"Definition:Set",
"Definition:Set",
"Definition:Strict Well-Ordering",
"Element of Ordinal is Ordinal",
"Definition:Transitive Class",
"Definition:Ordinal",
"Definition:Ordinal",
"Definition:Ordinal",
"Ordinal is not Element of Itself",
"Definitio... |
proofwiki-20256 | Essential Singularity may or may not be Isolated | Let $f$ be a complex function with an essential singularity at $z_0 \in \C$.
Then $z_0$ may or may not be an '''isolated singularity'''. | {{ProofWanted|An example of each. Motivation: to remove confusion and establish this basic fact. The author of this page is infodumping an encyclopedia entry and cannot remember the slightest inkling of his Complex Analysis studies except for the fact that nothing is intuitive and everything needs to be investigated wi... | Let $f$ be a [[Definition:Complex Function|complex function]] with an [[Definition:Essential Singularity|essential singularity]] at $z_0 \in \C$.
Then $z_0$ may or may not be an '''[[Definition:Isolated Singularity|isolated singularity]]'''. | {{ProofWanted|An example of each. Motivation: to remove confusion and establish this basic fact. The author of this page is infodumping an encyclopedia entry and cannot remember the slightest inkling of his Complex Analysis studies except for the fact that nothing is intuitive and everything needs to be investigated wi... | Essential Singularity may or may not be Isolated | https://proofwiki.org/wiki/Essential_Singularity_may_or_may_not_be_Isolated | https://proofwiki.org/wiki/Essential_Singularity_may_or_may_not_be_Isolated | [
"Essential Singularities",
"Isolated Singularities"
] | [
"Definition:Complex Function",
"Definition:Essential Singularity",
"Definition:Isolated Singularity"
] | [
"Category:Essential Singularities",
"Category:Isolated Singularities"
] |
proofwiki-20257 | Derivative of Concatenation of Complex Paths | Let $\gamma_0 : \closedint {a_0}{b_0} \to \C$ be an injective complex-differentiable function.
Let $\gamma_1 : \closedint {a_1}{b_1} \to \C$ be a complex-differentiable function such that $\map {\gamma_0}{b_0} = \map {\gamma_1}{a_1}$.
Let $t_0 \in \openint {a_0}{b_0}$.
Define the concatenation $\gamma_0 * \gamma_1 : \c... | Set $s_0 := \dfrac {t_0 - a_0}{2 \paren{ b_0 - a_0 } }$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\gamma_0 * \gamma_1}{s_0}
| r = \map{\gamma_0}{ \dfrac{2 \paren{ t_0 - a_0 } \paren{ b_0 - a_0 } }{2 \paren{ b_0 - a_0 } } + a_0 }
| c = as $t_0 - a_0 < b_0 - a_0$, so $\dfrac {t_0 - a_0}{2 \paren{ b_0 - a_0 } } <... | Let $\gamma_0 : \closedint {a_0}{b_0} \to \C$ be an [[Definition:Injection|injective]] [[Definition:Complex-Differentiable Function|complex-differentiable function]].
Let $\gamma_1 : \closedint {a_1}{b_1} \to \C$ be a [[Definition:Complex-Differentiable Function|complex-differentiable function]] such that $\map {\gamm... | Set $s_0 := \dfrac {t_0 - a_0}{2 \paren{ b_0 - a_0 } }$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\gamma_0 * \gamma_1}{s_0}
| r = \map{\gamma_0}{ \dfrac{2 \paren{ t_0 - a_0 } \paren{ b_0 - a_0 } }{2 \paren{ b_0 - a_0 } } + a_0 }
| c = as $t_0 - a_0 < b_0 - a_0$, so $\dfrac {t_0 - a_0}{2 \paren{ b_0 - a_0 } }... | Derivative of Concatenation of Complex Paths | https://proofwiki.org/wiki/Derivative_of_Concatenation_of_Complex_Paths | https://proofwiki.org/wiki/Derivative_of_Concatenation_of_Complex_Paths | [
"Complex Differential Calculus"
] | [
"Definition:Injection",
"Definition:Differentiable Mapping/Complex Function",
"Definition:Differentiable Mapping/Complex Function",
"Definition:Concatenation of Paths"
] | [
"Derivative of Complex Composite Function",
"Derivative of Complex Polynomial",
"Definition:Real Interval/Closed",
"Category:Complex Differential Calculus"
] |
proofwiki-20258 | Equivalent Norms on Lipschitz Space/Shift of Finite Type | Let $\struct {X, \sigma}$ be a shift of finite type.
Let $F_\theta$ be the Lipschitz space on $X$.
Let $x_0 \in F_\theta$.
Then the following norms on $F_\theta$ are equivalent:
:$(1): \quad$ Lipschitz norm $\norm\cdot_\theta$
:$(2): \quad \forall f \in F_\theta : \norm f '_\theta := \cmod {\map f {x_0} } + \size f_\th... | {{ProofWanted}}
Category:Functional Analysis
7i6a7q2nn2feoe80vrpetro0ifchb1k | Let $\struct {X, \sigma}$ be a [[Definition:Shift of Finite Type|shift of finite type]].
Let $F_\theta$ be the [[Definition:Lipschitz Space|Lipschitz space]] on $X$.
Let $x_0 \in F_\theta$.
Then the following [[Definition:Norm on Vector Space|norms]] on $F_\theta$ are [[Definition:Equivalence of Norms|equivalent]]:... | {{ProofWanted}}
[[Category:Functional Analysis]]
7i6a7q2nn2feoe80vrpetro0ifchb1k | Equivalent Norms on Lipschitz Space/Shift of Finite Type | https://proofwiki.org/wiki/Equivalent_Norms_on_Lipschitz_Space/Shift_of_Finite_Type | https://proofwiki.org/wiki/Equivalent_Norms_on_Lipschitz_Space/Shift_of_Finite_Type | [
"Functional Analysis"
] | [
"Definition:Shift of Finite Type",
"Definition:Lipschitz Space",
"Definition:Norm/Vector Space",
"Definition:Equivalence of Norms",
"Definition:Lipschitz Norm"
] | [
"Category:Functional Analysis"
] |
proofwiki-20259 | Transfer Operator with respect to One-Sided Shift Space of Finite Type is Linear Bounded Operator | Let $\struct {X ^+, \sigma}$ be a one-sided shift of finite type.
Let $\struct {\map C {X ^+, \C}, \norm \cdot_\infty}$ be the continuous mapping space.
Let $f \in \map C {X ^+, \C}$.
Let $\LL_f : \map C {X ^+, \C} \to \map C {X ^+, \C}$ the transfer operator.
Then $\LL_f$ is a bounded linear operator. | The linearity is clear by definition of $\LL_f$ since Summation is Linear.
Let $\mathbf A$ be the $k\times k$ logical matrix of $X ^+$.
Let $g \in \map C {X ^+, \C}$.
For each $x \in X^+$:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f g} x}
| r = \cmod {\sum_{\map \sigma y = x} e^{\map f y} \map g y}
}}
{{eqn | o ... | Let $\struct {X ^+, \sigma}$ be a [[Definition:One-Sided Shift of Finite Type|one-sided shift of finite type]].
Let $\struct {\map C {X ^+, \C}, \norm \cdot_\infty}$ be the [[Definition:Continuous Mapping Space|continuous mapping space]].
Let $f \in \map C {X ^+, \C}$.
Let $\LL_f : \map C {X ^+, \C} \to \map C {X ^+... | The [[Definition:Linear Operator|linearity]] is clear by definition of [[Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type|$\LL_f$]] since [[Summation is Linear]].
Let $\mathbf A$ be the $k\times k$ [[Definition:Logical Matrix|logical]] [[Definition:Matrix|matrix]] of $X ^+$.
Let $g \i... | Transfer Operator with respect to One-Sided Shift Space of Finite Type is Linear Bounded Operator | https://proofwiki.org/wiki/Transfer_Operator_with_respect_to_One-Sided_Shift_Space_of_Finite_Type_is_Linear_Bounded_Operator | https://proofwiki.org/wiki/Transfer_Operator_with_respect_to_One-Sided_Shift_Space_of_Finite_Type_is_Linear_Bounded_Operator | [
"Ergodic Theory",
"Functional Analysis"
] | [
"Definition:One-Sided Shift of Finite Type",
"Definition:Continuous Mapping Space",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"Definition:Bounded Linear Operator/Normed Vector Space"
] | [
"Definition:Linear Operator",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"Summation is Linear",
"Definition:Logical Matrix",
"Definition:Matrix",
"Category:Ergodic Theory",
"Category:Functional Analysis"
] |
proofwiki-20260 | Ruelle-Perron-Frobenius Theorem | Let $\struct {X_\mathbf A ^+, \sigma}$ be a one-sided shift of finite type.
Let $\map C {X_\mathbf A ^+} := \map C {X_\mathbf A ^+, \C}$ be the continuous mapping space.
Let $F_\theta ^+$ be the space of Lipschitz functions on $X_\mathbf A ^+$.
Let $f \in F_\theta ^+$ be a real-valued function.
Let $\LL_f : \map C {X_\... | {{ProofWanted}}
{{Namedfor|David Pierre Ruelle|name2 = Oskar Perron|name3 = Ferdinand Georg Frobenius|cat = Ruelle|cat2 = Perron|cat3 = Frobenius}}
Category:Ergodic Theory
Category:Functional Analysis
3mx6hi111yd4ulawcq0olmbotwsh3ta | Let $\struct {X_\mathbf A ^+, \sigma}$ be a [[Definition:One-Sided Shift of Finite Type|one-sided shift of finite type]].
Let $\map C {X_\mathbf A ^+} := \map C {X_\mathbf A ^+, \C}$ be the [[Definition:Continuous Mapping Space|continuous mapping space]].
Let $F_\theta ^+$ be the [[Definition:Space of Lipschitz Funct... | {{ProofWanted}}
{{Namedfor|David Pierre Ruelle|name2 = Oskar Perron|name3 = Ferdinand Georg Frobenius|cat = Ruelle|cat2 = Perron|cat3 = Frobenius}}
[[Category:Ergodic Theory]]
[[Category:Functional Analysis]]
3mx6hi111yd4ulawcq0olmbotwsh3ta | Ruelle-Perron-Frobenius Theorem | https://proofwiki.org/wiki/Ruelle-Perron-Frobenius_Theorem | https://proofwiki.org/wiki/Ruelle-Perron-Frobenius_Theorem | [
"Ergodic Theory",
"Functional Analysis"
] | [
"Definition:One-Sided Shift of Finite Type",
"Definition:Continuous Mapping Space",
"Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type",
"Definition:Real-Valued Function",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"Definition:Spectral Radius/B... | [
"Category:Ergodic Theory",
"Category:Functional Analysis"
] |
proofwiki-20261 | Primitive of x by Cosine of x | :$\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | From Primitive of $x \cos a x$:
:$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
The result follows on setting $a = 1$.
{{qed}} | :$\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | From [[Primitive of x by Cosine of a x|Primitive of $x \cos a x$]]:
:$\ds \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
The result follows on setting $a = 1$.
{{qed}} | Primitive of x by Cosine of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_x/Proof_1 | [
"Primitive of x by Cosine of x",
"Primitives involving Cosine Function"
] | [] | [
"Primitive of x by Cosine of a x"
] |
proofwiki-20262 | Primitive of x by Cosine of x | :$\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Derivative of Identity Function
}}
{{end-eqn... | :$\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Derivative of Id... | Primitive of x by Cosine of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_x/Proof_2 | [
"Primitive of x by Cosine of x",
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Identity Function",
"Primitive of Cosine Function",
"Integration by Parts",
"Primitive of Sine Function"
] |
proofwiki-20263 | Primitive of x by Cosine of x | :$\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cos x + x \sin x}
| r = \map {\dfrac \d {\d x} } {\cos x} + \map {\dfrac \d {\d x} } {x \sin x}
| c = Sum Rule for Derivatives
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\cos x} + x \map {\dfrac \d {\d x} } {\sin x} + \map {\dfrac \d {\d x} } x \si... | :$\ds \int x \cos x \rd x = \cos x + x \sin x + C$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cos x + x \sin x}
| r = \map {\dfrac \d {\d x} } {\cos x} + \map {\dfrac \d {\d x} } {x \sin x}
| c = [[Sum Rule for Derivatives]]
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\cos x} + x \map {\dfrac \d {\d x} } {\sin x} + \map {\dfrac \d {\d x} } ... | Primitive of x by Cosine of x/Proof 3 | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Cosine_of_x/Proof_3 | [
"Primitive of x by Cosine of x",
"Primitives involving Cosine Function"
] | [] | [
"Sum Rule for Derivatives",
"Product Rule for Derivatives",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Derivative of Identity Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-20264 | Primitive of Power of a x + b/Proof 3 | {{:Primitive of Power of a x + b}}
where $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }
| r = \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}
| c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {a \paren {n + ... | {{:Primitive of Power of a x + b}}
where $n \ne 1$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }
| r = \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}
| c = [[Power Rule for Derivatives]], [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {a \par... | Primitive of Power of a x + b/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_3 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_3 | [
"Primitive of Power of a x + b"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Definition:Primitive (Calculus)"
] |
proofwiki-20265 | Simple Closed Contour has Orientation | Let $C$ be a simple closed contour in the complex plane $\C$.
Then either $C$ is positively oriented, or $C$ is negatively oriented.
Also, $C$ cannot be both positively and negatively oriented. | Let $\gamma: \closedint a b \to \C$ be a parameterization of $C$.
Let $t \in \openint a b$ such that $\gamma$ is complex-differentiable at $t$.
From Normal Vectors Form Space around Simple Complex Contour, it follows that there exists $r \in \R_{>0}$ such that for all $\epsilon \in \openint 0 r$:
:$\map \gamma t + \eps... | Let $C$ be a [[Definition:Simple Contour (Complex Plane)|simple]] [[Definition:Closed Contour (Complex Plane)|closed contour]] in the [[Definition:Complex Plane|complex plane]] $\C$.
Then either $C$ is [[Definition:Positive Orientation of Simple Closed Contour (Complex Plane)|positively oriented]], or $C$ is [[Defini... | Let $\gamma: \closedint a b \to \C$ be a [[Definition:Parameterization of Contour (Complex Plane)|parameterization]] of $C$.
Let $t \in \openint a b$ such that $\gamma$ is [[Definition:Complex-Differentiable at Point|complex-differentiable]] at $t$.
From [[Normal Vectors Form Space around Simple Complex Contour]], it... | Simple Closed Contour has Orientation | https://proofwiki.org/wiki/Simple_Closed_Contour_has_Orientation | https://proofwiki.org/wiki/Simple_Closed_Contour_has_Orientation | [
"Orientation of Complex Contour"
] | [
"Definition:Contour/Simple/Complex Plane",
"Definition:Contour/Closed/Complex Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Orientation of Contour (Complex Plane)/Positive/Simple Closed",
"Definition:Orientation of Contour (Complex Plane)/Negative/Simple Closed",
"Definition:Orientation o... | [
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Differentiable Mapping/Complex Function/Point",
"Normal Vectors Form Space around Simple Complex Contour",
"Definition:Contour/Image/Complex Plane",
"Complex Plane is Homeomorphic to Real Plane",
"Definition:Complex Number/Complex Plane",
... |
proofwiki-20266 | Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale/Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a discrete-time filtered probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a discrete-time $\sequence {\FF_n}_{n \mathop \ge 0}$-adapted stochastic process.
Then $\sequence {X_n}_{n \mathop \ge 0}$ is a $\sequence {\FF_n}_{n \mathop... | For each $n \in \Z_{\ge 0}$, we have:
:$\expect {X_{n + 1} \mid \FF_n} = X_n$ almost surely
{{iff}}:
:$\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost surely
and:
:$\expect {X_{n + 1} \mid \FF_n} \ge X_n$ almost surely.
That is:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \mathop \ge 0}$-martingale {{iff}}... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Discrete Time|discrete-time filtered probability space]].
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a [[Definition:Adapted Stochastic Process/Discrete Time|discrete-time $\sequence {\FF_n}_{n \mathop ... | For each $n \in \Z_{\ge 0}$, we have:
:$\expect {X_{n + 1} \mid \FF_n} = X_n$ [[Definition:Almost Surely|almost surely]]
{{iff}}:
:$\expect {X_{n + 1} \mid \FF_n} \le X_n$ [[Definition:Almost Surely|almost surely]]
and:
:$\expect {X_{n + 1} \mid \FF_n} \ge X_n$ [[Definition:Almost Surely|almost surely]].
That is... | Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale/Discrete Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Martingale_iff_Supermartingale_and_Submartingale/Discrete_Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Martingale_iff_Supermartingale_and_Submartingale/Discrete_Time | [
"Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale"
] | [
"Definition:Filtered Probability Space/Discrete Time",
"Definition:Adapted Stochastic Process/Discrete Time",
"Definition:Martingale/Discrete Time",
"Definition:Supermartingale/Discrete Time",
"Definition:Submartingale/Discrete Time"
] | [
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Martingale/Discrete Time",
"Definition:Supermartingale/Discrete Time",
"Definition:Submartingale/Discrete Time"
] |
proofwiki-20267 | Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale/Continuous Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a continuous-time $\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process.
Then $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale {{iff}} it is a ... | For each $t, s \in \hointr 0 \infty$ with $0 \le s < t$, we have:
:$\expect {X_t \mid \FF_s} = X_s$ almost surely
{{iff}}:
:$\expect {X_t \mid \FF_s} \le X_s$ almost surely
and:
:$\expect {X_t \mid \FF_s} \ge X_s$ almost surely.
That is:
:$\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale {{iff... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Adapted Stochastic Process/Continuous Time|continuous-time $\sequence {\FF_t}_{t \ge 0}$-adapted ... | For each $t, s \in \hointr 0 \infty$ with $0 \le s < t$, we have:
:$\expect {X_t \mid \FF_s} = X_s$ [[Definition:Almost Everywhere|almost surely]]
{{iff}}:
:$\expect {X_t \mid \FF_s} \le X_s$ [[Definition:Almost Everywhere|almost surely]]
and:
:$\expect {X_t \mid \FF_s} \ge X_s$ [[Definition:Almost Everywhere|alm... | Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale/Continuous Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Martingale_iff_Supermartingale_and_Submartingale/Continuous_Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Martingale_iff_Supermartingale_and_Submartingale/Continuous_Time | [
"Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Adapted Stochastic Process/Continuous Time",
"Definition:Martingale/Continuous Time",
"Definition:Supermartingale/Continuous Time",
"Definition:Submartingale/Continuous Time"
] | [
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Martingale/Continuous Time",
"Definition:Supermartingale/Continuous Time",
"Definition:Submartingale/Continuous Time",
"Category:Adapted Stochastic Process is Martingale iff Supermartingale and Su... |
proofwiki-20268 | Adapted Stochastic Process is Supermartingale iff Negative is Submartingale/Continuous Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process.
Then $\sequence {X_t}_{t \ge 0}$ is a supermartingale {{iff}} $\sequence {-X_t}_{t \ge 0}$ is a submartingale. | Since $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process:
:$X_t$ is $\FF_t$-measurable for each $t \in \hointr 0 \infty$.
From Pointwise Scalar Multiple of Measurable Function is Measurable:
:$-X_t$ is $\FF_t$-measurable for each $t \in \hointr 0 \infty$.
So $\sequence {-X_t}_{t \... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Adapted Stochastic Process/Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-adapted stochastic proce... | Since $\sequence {X_t}_{t \ge 0}$ is a [[Definition:Adapted Stochastic Process/Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-adapted stochastic process]]:
:$X_t$ is [[Definition:Measurable Function|$\FF_t$-measurable]] for each $t \in \hointr 0 \infty$.
From [[Pointwise Scalar Multiple of Measurable Function is Measu... | Adapted Stochastic Process is Supermartingale iff Negative is Submartingale/Continuous Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Supermartingale_iff_Negative_is_Submartingale/Continuous_Time | https://proofwiki.org/wiki/Adapted_Stochastic_Process_is_Supermartingale_iff_Negative_is_Submartingale/Continuous_Time | [
"Adapted Stochastic Process is Supermartingale iff Negative is Submartingale"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Adapted Stochastic Process/Continuous Time",
"Definition:Supermartingale/Continuous Time",
"Definition:Submartingale/Continuous Time"
] | [
"Definition:Adapted Stochastic Process/Continuous Time",
"Definition:Measurable Function",
"Pointwise Scalar Multiple of Measurable Function is Measurable",
"Definition:Measurable Function",
"Definition:Adapted Stochastic Process",
"Definition:Conditional Expectation",
"Conditional Expectation is Linear... |
proofwiki-20269 | Euler's Integral Representation of Hypergeometric Function | :$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c} {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{-a} \rd t$ | Letting $\size x < 1$ and expanding the product of $\paren {1 - x t}^{-a}$:
{{begin-eqn}}
{{eqn | l = \paren {1 - x t}^{-a}
| r = \sum_{k \mathop = 0}^\infty \binom {-a} k \paren {-1}^k \paren {x t}^k
| c = Extended Binomial Theorem
}}
{{eqn | ll= \leadsto
| r = \sum_{k \mathop = 0}^\infty \paren {\bi... | :$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c} {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{-a} \rd t$ | Letting $\size x < 1$ and expanding the product of $\paren {1 - x t}^{-a}$:
{{begin-eqn}}
{{eqn | l = \paren {1 - x t}^{-a}
| r = \sum_{k \mathop = 0}^\infty \binom {-a} k \paren {-1}^k \paren {x t}^k
| c = [[Extended Binomial Theorem]]
}}
{{eqn | ll= \leadsto
| r = \sum_{k \mathop = 0}^\infty \paren... | Euler's Integral Representation of Hypergeometric Function | https://proofwiki.org/wiki/Euler's_Integral_Representation_of_Hypergeometric_Function | https://proofwiki.org/wiki/Euler's_Integral_Representation_of_Hypergeometric_Function | [
"Euler's Integral Representation of Hypergeometric Function",
"Gaussian Hypergeometric Function",
"Beta Function",
"Gamma Function"
] | [] | [
"Binomial Theorem/Extended",
"Negated Upper Index of Binomial Coefficient",
"Rising Factorial as Quotient of Factorials",
"Exponent Combination Laws/Product of Powers",
"Rising Factorial as Quotient of Factorials"
] |
proofwiki-20270 | Pfaff's Transformation | :$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$ | From Euler's Integral Representation of Hypergeometric Function, we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - xt}^{- a} \rd t$
Letting $s = \paren {1 - t}$, we now have:
{{begin-eqn}}
{{eqn | l = \map F {a, b;... | :$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$ | From [[Euler's Integral Representation of Hypergeometric Function]], we have:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - xt}^{- a} \rd t$
Letting $s = \paren {1 - t}$, we now have:
{{begin-eqn}}
{{eqn | l = \map F... | Pfaff's Transformation | https://proofwiki.org/wiki/Pfaff's_Transformation | https://proofwiki.org/wiki/Pfaff's_Transformation | [
"Pfaff's Transformation",
"Euler's Integral Representation of Hypergeometric Function",
"Gaussian Hypergeometric Function",
"Gamma Function"
] | [] | [
"Euler's Integral Representation of Hypergeometric Function",
"Integration by Substitution/Definite Integral",
"Binomial Theorem/Extended",
"Negated Upper Index of Binomial Coefficient",
"Rising Factorial as Quotient of Factorials",
"Exponent Combination Laws/Product of Powers",
"Rising Factorial as Quo... |
proofwiki-20271 | Gelfand's Spectral Radius Formula/Bounded Linear Operator | Let $\struct {X, \norm \cdot _X}$ be a Banach space over $\C$.
Let $\map B X$ be the set of bounded linear operators on $X$.
Let $\norm \cdot_{\map B X}$ denote the operator norm on $\map B X$.
Let $T \in \map B X$.
Let $\size {\map \sigma T}$ be the spectral radius of $T$.
Then:
:$\ds \size {\map \sigma T} = \lim_{n \... | Let $z \in \C$ be such that:
:$\ds \cmod z > \inf_{n \mathop \in \N_{>0} } \paren {\norm {T^n}_{\map B X} }^{1/n}$
That is, there exists an $m \in \N_{>0}$ such that:
:$\ds \cmod z > \paren {\norm {T^m}_{\map B X} }^{1/m}$
Then:
{{begin-eqn}}
{{eqn | l = \paren {T - z I}^{-1}
| r = -z^{-1} \paren {I - z^{-1} T}^{... | Let $\struct {X, \norm \cdot _X}$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $\map B X$ be the [[Definition:Set|set]] of [[Definition:Bounded Linear Operator on Normed Vector Space|bounded linear operators]] on $X$.
Let $\norm \cdot_{\map B X}$ denote the [[Definition:Norm on Bounded Linear Transfor... | Let $z \in \C$ be such that:
:$\ds \cmod z > \inf_{n \mathop \in \N_{>0} } \paren {\norm {T^n}_{\map B X} }^{1/n}$
That is, there exists an $m \in \N_{>0}$ such that:
:$\ds \cmod z > \paren {\norm {T^m}_{\map B X} }^{1/m}$
Then:
{{begin-eqn}}
{{eqn | l = \paren {T - z I}^{-1}
| r = -z^{-1} \paren {I - z^{-1} T}... | Gelfand's Spectral Radius Formula/Bounded Linear Operator | https://proofwiki.org/wiki/Gelfand's_Spectral_Radius_Formula/Bounded_Linear_Operator | https://proofwiki.org/wiki/Gelfand's_Spectral_Radius_Formula/Bounded_Linear_Operator | [
"Bounded Linear Operators",
"Spectral Theory",
"Gelfand's Spectral Radius Formula"
] | [
"Definition:Banach Space",
"Definition:Set",
"Definition:Bounded Linear Operator/Normed Vector Space",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Spectral Radius/Bounded Linear Operator"
] | [
"Invertibility of Identity Minus Operator",
"Division Theorem",
"Definition:Spectral Radius/Bounded Linear Operator",
"Definition:Normed Dual Space",
"Definition:Complex Function",
"Resolvent Mapping is Analytic",
"Definition:Analytic Function/Complex Plane",
"Invertibility of Identity Minus Operator"... |
proofwiki-20272 | Right-Limits of Filtration of Sigma-Algebra form Filtration | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\family {\FF_t}_{t \mathop \ge 0}$ be a continuous-time filtration of $\Sigma$.
For each $t \in \hointr 0 \infty$, let $\FF_{t^+}$ be the right-limit of $\family {\FF_t}_{t \mathop \ge 0}$ at $t$.
Then $\family {\FF_{t^+} }_{t \ge 0}$ is a filtration of $\Sigma$. | We need to show that $\FF_{t^+}$ is a sub-$\sigma$-algebra of $\Sigma$ for each $t \in \hointr 0 \infty$, and:
:$\FF_{t^+} \subseteq \FF_{q^+}$
for $t, q \in \hointr 0 \infty$ with $t \le q$.
From the definition of the right-limit, we have:
:$\ds \FF_{t^+} = \bigcap_{s \mathop > t} \FF_s$
for each $t \in \hointr 0 \inf... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\family {\FF_t}_{t \mathop \ge 0}$ be a [[Definition:Filtration of Sigma-Algebra/Continuous Time|continuous-time filtration]] of $\Sigma$.
For each $t \in \hointr 0 \infty$, let $\FF_{t^+}$ be the [[Definition:Right-Limit of Filtra... | We need to show that $\FF_{t^+}$ is a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$ for each $t \in \hointr 0 \infty$, and:
:$\FF_{t^+} \subseteq \FF_{q^+}$
for $t, q \in \hointr 0 \infty$ with $t \le q$.
From the definition of the [[Definition:Right-Limit of Filtration of Sigma-Algebra|right-lim... | Right-Limits of Filtration of Sigma-Algebra form Filtration | https://proofwiki.org/wiki/Right-Limits_of_Filtration_of_Sigma-Algebra_form_Filtration | https://proofwiki.org/wiki/Right-Limits_of_Filtration_of_Sigma-Algebra_form_Filtration | [
"Filtrations of Sigma-Algebras"
] | [
"Definition:Measurable Space",
"Definition:Filtration of Sigma-Algebra/Continuous Time",
"Definition:Right-Limit of Filtration of Sigma-Algebra",
"Definition:Filtration of Sigma-Algebra/Continuous Time"
] | [
"Definition:Sub-Sigma-Algebra",
"Definition:Right-Limit of Filtration of Sigma-Algebra",
"Intersection of Sigma-Algebras",
"Definition:Sub-Sigma-Algebra",
"Intersection of Family is Subset of Intersection of Subset of Family",
"Category:Filtrations of Sigma-Algebras"
] |
proofwiki-20273 | Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration/Discrete Time | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\sequence {\FF_n}_{n \ge 0}$ be a discrete-time filtration of $\Sigma$.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\FF_\infty$ be the limit of the filtration $\sequence {\FF_n}_{n \ge 0}$.
Then:
:$\set {\omega \in \Omega : \map T... | Note that we have:
{{begin-eqn}}
{{eqn | l = \set {\omega \in \Omega : \map T \omega < \infty}
| r = \set {\omega \in \Omega : \map T \omega \in \Z_{\ge 0} }
}}
{{eqn | r = \bigcup_{t \in \Z_{\ge 0} } \set {\omega \in \Omega : \map T \omega = t}
}}
{{end-eqn}}
From the definition of a stopping time, we have:
:$\set ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\sequence {\FF_n}_{n \ge 0}$ be a [[Definition:Filtration of Sigma-Algebra/Discrete Time|discrete-time filtration]] of $\Sigma$.
Let $T$ be a [[Definition:Stopping Time/Discrete Time|stopping time]] with respect to $\sequence {\FF_... | Note that we have:
{{begin-eqn}}
{{eqn | l = \set {\omega \in \Omega : \map T \omega < \infty}
| r = \set {\omega \in \Omega : \map T \omega \in \Z_{\ge 0} }
}}
{{eqn | r = \bigcup_{t \in \Z_{\ge 0} } \set {\omega \in \Omega : \map T \omega = t}
}}
{{end-eqn}}
From the definition of a [[Definition:Stopping Time/Di... | Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration/Discrete Time | https://proofwiki.org/wiki/Event_of_Stopping_Time_Equal_to_Infinity_is_Measurable_in_Limit_of_Filtration/Discrete_Time | https://proofwiki.org/wiki/Event_of_Stopping_Time_Equal_to_Infinity_is_Measurable_in_Limit_of_Filtration/Discrete_Time | [
"Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration"
] | [
"Definition:Measurable Space",
"Definition:Filtration of Sigma-Algebra/Discrete Time",
"Definition:Stopping Time/Discrete Time",
"Definition:Limit of Filtration of Sigma-Algebra/Discrete Time"
] | [
"Definition:Stopping Time/Discrete Time",
"Set is Subset of Union",
"Definition:Sigma-Algebra Generated by Collection of Subsets",
"Definition:Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Countable Union",
"Definition:Closure (Abstract Algebra)/Algeb... |
proofwiki-20274 | Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration/Continuous Time | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\sequence {\FF_t}_{t \ge 0}$ be a continuous-time filtration of $\Sigma$.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\FF_\infty$ be the limit of the filtration $\sequence {\FF_t}_{t \ge 0}$.
Then:
:$\set {\omega \in \Omega : \map... | Note that we have $\map T \omega < \infty$ {{iff}} $\map T \omega \le N$ for some $N \in \N$.
So, we have:
:$\ds \set {\omega \in \Omega : \map T \omega < \infty} = \bigcup_{N \in \N} \set {\omega \in \Omega : \map T \omega \le N}$
Since $T$ is a stopping time, we have:
:$\set {\omega \in \Omega : \map T \omega \le ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\sequence {\FF_t}_{t \ge 0}$ be a [[Definition:Filtration of Sigma-Algebra/Continuous Time|continuous-time filtration]] of $\Sigma$.
Let $T$ be a [[Definition:Stopping Time/Discrete Time|stopping time]] with respect to $\sequence {... | Note that we have $\map T \omega < \infty$ {{iff}} $\map T \omega \le N$ for some $N \in \N$.
So, we have:
:$\ds \set {\omega \in \Omega : \map T \omega < \infty} = \bigcup_{N \in \N} \set {\omega \in \Omega : \map T \omega \le N}$
Since $T$ is a [[Definition:Stopping Time/Continuous Time|stopping time]], we have:... | Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration/Continuous Time | https://proofwiki.org/wiki/Event_of_Stopping_Time_Equal_to_Infinity_is_Measurable_in_Limit_of_Filtration/Continuous_Time | https://proofwiki.org/wiki/Event_of_Stopping_Time_Equal_to_Infinity_is_Measurable_in_Limit_of_Filtration/Continuous_Time | [
"Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration"
] | [
"Definition:Measurable Space",
"Definition:Filtration of Sigma-Algebra/Continuous Time",
"Definition:Stopping Time/Discrete Time",
"Definition:Limit of Filtration of Sigma-Algebra/Continuous Time"
] | [
"Definition:Stopping Time/Continuous Time",
"Set is Subset of Union",
"Definition:Sigma-Algebra Generated by Collection of Subsets",
"Definition:Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Countable Union",
"Definition:Closure (Abstract Algebra)/Alg... |
proofwiki-20275 | Convex Real Function is Pointwise Supremum of Affine Functions | Let $f : \R \to \R$ be a convex real function.
Then there exists a set $\SS \subseteq \R^2$ such that:
:$\ds \map f x = \sup_{\tuple {a, b} \in \SS} \paren {a x + b}$
for each $x \in \R$. | Let $x \in \R$.
We construct an affine function $\phi_x : \R \to \R$ such that:
:$\map f y \ge \map {\phi_x} y$
and $\map {\phi_x} x = \map f x$.
We will then argue that:
:$\map f x = \sup \set {\map {\phi_y} x : y \in \R}$
Define a function $g_x : \R \setminus \set x \to \R$ by:
:$\map {g_x} y = \dfrac {\map f y -... | Let $f : \R \to \R$ be a [[Definition:Convex Real Function|convex real function]].
Then there exists a set $\SS \subseteq \R^2$ such that:
:$\ds \map f x = \sup_{\tuple {a, b} \in \SS} \paren {a x + b}$
for each $x \in \R$. | Let $x \in \R$.
We construct an affine function $\phi_x : \R \to \R$ such that:
:$\map f y \ge \map {\phi_x} y$
and $\map {\phi_x} x = \map f x$.
We will then argue that:
:$\map f x = \sup \set {\map {\phi_y} x : y \in \R}$
Define a function $g_x : \R \setminus \set x \to \R$ by:
:$\map {g_x} y = \dfrac {\ma... | Convex Real Function is Pointwise Supremum of Affine Functions | https://proofwiki.org/wiki/Convex_Real_Function_is_Pointwise_Supremum_of_Affine_Functions | https://proofwiki.org/wiki/Convex_Real_Function_is_Pointwise_Supremum_of_Affine_Functions | [
"Convex Real Functions",
"Convex Real Function is Pointwise Supremum of Affine Functions"
] | [
"Definition:Convex Real Function"
] | [
"Definition:Convex Real Function/Definition 3",
"Definition:Increasing/Real Function",
"Limit of Monotone Real Function/Increasing",
"Category:Convex Real Functions",
"Category:Convex Real Function is Pointwise Supremum of Affine Functions"
] |
proofwiki-20276 | Orthogonal Set is Linearly Independent Set | Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over a real or complex subfield $\mathbb F$.
Let $S \subseteq V$ be an orthogonal set.
Let $\mathbf 0 \notin S$, where $\mathbf 0$ denotes the zero vector of $V$.
Then $S$ is a linearly independent set. | Let $n \in \N$.
Let $\lambda_1, \ldots, \lambda_n \in \mathbb F$, and let $u_1, \ldots, u_n \in S$ such that:
:$\ds \sum_{i \mathop= 1}^n \lambda_i u_1 = \mathbf 0$
To prove that $S$ is a linearly independent set, we must show that $\lambda_k = 0$ for all $k \in \set { 1, \ldots, n}$.
We calculate:
{{begin-eqn}}
{{eqn ... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]] over a [[Definition:Real Subfield|real]] or [[Definition:Complex Subfield|complex subfield]] $\mathbb F$.
Let $S \subseteq V$ be an [[Definition:Orthogonal (Linear Algebra)/Set|orthogonal set]].
Let $\mathbf 0 \noti... | Let $n \in \N$.
Let $\lambda_1, \ldots, \lambda_n \in \mathbb F$, and let $u_1, \ldots, u_n \in S$ such that:
:$\ds \sum_{i \mathop= 1}^n \lambda_i u_1 = \mathbf 0$
To prove that $S$ is a [[Definition:Linearly Independent Set|linearly independent set]], we must show that $\lambda_k = 0$ for all $k \in \set { 1, \ld... | Orthogonal Set is Linearly Independent Set | https://proofwiki.org/wiki/Orthogonal_Set_is_Linearly_Independent_Set | https://proofwiki.org/wiki/Orthogonal_Set_is_Linearly_Independent_Set | [
"Orthogonality (Linear Algebra)",
"Linear Independence"
] | [
"Definition:Inner Product Space",
"Definition:Real Subfield",
"Definition:Complex Subfield",
"Definition:Orthogonal (Linear Algebra)/Set",
"Definition:Zero Vector",
"Definition:Linearly Independent/Set"
] | [
"Definition:Linearly Independent/Set",
"Definition:Inner Product",
"Inner Product with Zero Vector",
"Definition:Inner Product",
"Definition:Linear Transformation",
"Definition:Inner Product",
"Definition:Non-Negative Definite Mapping",
"Field has no Proper Zero Divisors"
] |
proofwiki-20277 | Convex Real Function is Measurable | Let $f : \R \to \R$ be a convex real function.
Then $f$ is measurable. | From Convex Real Function is Continuous, $f$ is continuous.
From Continuous Mapping is Measurable, $f$ is measurable.
{{qed}} | Let $f : \R \to \R$ be a [[Definition:Convex Real Function|convex real function]].
Then $f$ is [[Definition:Measurable Function|measurable]]. | From [[Convex Real Function is Continuous]], $f$ is [[Definition:Continuous Real Function|continuous]].
From [[Continuous Mapping is Measurable]], $f$ is [[Definition:Measurable Function|measurable]].
{{qed}} | Convex Real Function is Measurable/Proof 1 | https://proofwiki.org/wiki/Convex_Real_Function_is_Measurable | https://proofwiki.org/wiki/Convex_Real_Function_is_Measurable/Proof_1 | [
"Measurable Functions",
"Convex Real Functions",
"Convex Real Function is Measurable"
] | [
"Definition:Convex Real Function",
"Definition:Measurable Function"
] | [
"Convex Real Function is Continuous",
"Definition:Continuous Real Function",
"Continuous Mapping is Measurable",
"Definition:Measurable Function"
] |
proofwiki-20278 | Convex Real Function is Measurable | Let $f : \R \to \R$ be a convex real function.
Then $f$ is measurable. | From Convex Real Function is Pointwise Supremum of Affine Functions: Corollary, there exists a countable set $\SS \subseteq \R^2$ such that:
:$\ds \map f x = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a x + b}$
for each $x \in \R$.
From Linear Function is Continuous, the map $x \mapsto a x + b$ is continuous for eac... | Let $f : \R \to \R$ be a [[Definition:Convex Real Function|convex real function]].
Then $f$ is [[Definition:Measurable Function|measurable]]. | From [[Convex Real Function is Pointwise Supremum of Affine Functions/Corollary|Convex Real Function is Pointwise Supremum of Affine Functions: Corollary]], there exists a [[Definition:Countable Set|countable set]] $\SS \subseteq \R^2$ such that:
:$\ds \map f x = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a x + b}$... | Convex Real Function is Measurable/Proof 2 | https://proofwiki.org/wiki/Convex_Real_Function_is_Measurable | https://proofwiki.org/wiki/Convex_Real_Function_is_Measurable/Proof_2 | [
"Measurable Functions",
"Convex Real Functions",
"Convex Real Function is Measurable"
] | [
"Definition:Convex Real Function",
"Definition:Measurable Function"
] | [
"Convex Real Function is Pointwise Supremum of Affine Functions/Corollary",
"Definition:Countable Set",
"Linear Function is Continuous",
"Definition:Continuous Real Function",
"Continuous Mapping is Measurable",
"Definition:Measurable Function",
"Definition:Countable Set",
"Pointwise Supremum of Measu... |
proofwiki-20279 | Conditional Jensen's Inequality | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG$ be a sub-$\sigma$-algebra of $\Sigma$.
Let $X$ be an integrable random variable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $f : \R \to \R$ be a convex function such that $\map f X$ is integrable.
... | Note that from Convex Real Function is Measurable and Composition of Measurable Mappings is Measurable, $\map f X$ is $\Sigma$-measurable, so the hypotheses make sense.
By Convex Real Function is Pointwise Supremum of Affine Functions: Corollary, there exists a countable set $\SS \subseteq \R^2$ such that:
:$\ds \map... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$.
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]].
Let $\expect {X \mid \GG}$ be a version of the [[Definiti... | Note that from [[Convex Real Function is Measurable]] and [[Composition of Measurable Mappings is Measurable]], $\map f X$ is [[Definition:Measurable Function|$\Sigma$-measurable]], so the hypotheses make sense.
By [[Convex Real Function is Pointwise Supremum of Affine Functions/Corollary|Convex Real Function is Poin... | Conditional Jensen's Inequality | https://proofwiki.org/wiki/Conditional_Jensen's_Inequality | https://proofwiki.org/wiki/Conditional_Jensen's_Inequality | [
"Conditional Expectation",
"Convex Real Functions",
"Jensen's Inequality"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Convex Real Function",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Alg... | [
"Convex Real Function is Measurable",
"Composition of Measurable Mappings is Measurable",
"Definition:Measurable Function",
"Convex Real Function is Pointwise Supremum of Affine Functions/Corollary",
"Definition:Countable Set",
"Conditional Expectation is Monotone",
"Conditional Expectation is Linear",
... |
proofwiki-20280 | Martingale Composed with Convex Function is Submartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-martingale.
Let $f : \R \to \R$ be a convex function such that $\map f {X_t}$ is integrable for each $t \in \hointr 0 \infty$.
Then $\sequ... | Since $\sequence {X_t}_{t \ge 0}$ is a martingale, we have:
:$X_t$ is $\FF_t$-measurable
for each $t \in \hointr 0 \infty$.
From Convex Real Function is Measurable and Composition of Measurable Mappings is Measurable:
:$\map f {X_t}$ is $\FF_t$-measurable
for each $t \in \hointr 0 \infty$.
So $\sequence {\map f {X_t}... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Martingale in Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-martingale]].
Let $f : \R \to \R$ be... | Since $\sequence {X_t}_{t \ge 0}$ is a [[Definition:Martingale in Continuous Time|martingale]], we have:
:$X_t$ is [[Definition:Measurable Function|$\FF_t$-measurable]]
for each $t \in \hointr 0 \infty$.
From [[Convex Real Function is Measurable]] and [[Composition of Measurable Mappings is Measurable]]:
:$\map f... | Martingale Composed with Convex Function is Submartingale | https://proofwiki.org/wiki/Martingale_Composed_with_Convex_Function_is_Submartingale | https://proofwiki.org/wiki/Martingale_Composed_with_Convex_Function_is_Submartingale | [
"Submartingales",
"Martingales",
"Convex Real Functions"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Martingale/Continuous Time",
"Definition:Convex Real Function",
"Definition:Integrable Random Variable",
"Definition:Submartingale/Continuous Time"
] | [
"Definition:Martingale/Continuous Time",
"Definition:Measurable Function",
"Convex Real Function is Measurable",
"Composition of Measurable Mappings is Measurable",
"Definition:Measurable Function",
"Definition:Adapted Stochastic Process",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",... |
proofwiki-20281 | Submartingale Composed with Increasing Convex Function is Submartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-submartingale.
Let $f : \R \to \R$ be an increasing convex function such that $\map f {X_t}$ is integrable for each $t \in \hointr 0 \infty... | Since $\sequence {X_t}_{t \ge 0}$ is a martingale, we have:
:$X_t$ is $\FF_t$-measurable
for each $t \in \hointr 0 \infty$.
From Convex Real Function is Measurable and Composition of Measurable Mappings is Measurable:
:$\map f {X_t}$ is $\FF_t$-measurable
for each $t \in \hointr 0 \infty$.
So $\sequence {\map f {X_t}... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Submartingale/Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-submartingale]].
Let $f : \R \to \R$... | Since $\sequence {X_t}_{t \ge 0}$ is a [[Definition:Martingale in Continuous Time|martingale]], we have:
:$X_t$ is [[Definition:Measurable Function|$\FF_t$-measurable]]
for each $t \in \hointr 0 \infty$.
From [[Convex Real Function is Measurable]] and [[Composition of Measurable Mappings is Measurable]]:
:$\map f... | Submartingale Composed with Increasing Convex Function is Submartingale | https://proofwiki.org/wiki/Submartingale_Composed_with_Increasing_Convex_Function_is_Submartingale | https://proofwiki.org/wiki/Submartingale_Composed_with_Increasing_Convex_Function_is_Submartingale | [
"Submartingales",
"Convex Real Functions"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Submartingale/Continuous Time",
"Definition:Increasing/Real Function",
"Definition:Convex Real Function",
"Definition:Integrable Random Variable",
"Definition:Submartingale/Continuous Time"
] | [
"Definition:Martingale/Continuous Time",
"Definition:Measurable Function",
"Convex Real Function is Measurable",
"Composition of Measurable Mappings is Measurable",
"Definition:Measurable Function",
"Definition:Adapted Stochastic Process",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",... |
proofwiki-20282 | Absolute Value of Martingale is Submartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-martingale.
Then $\sequence {\size {X_t} }_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-submartingale. | From Characterization of Integrable Functions:
:$\size {X_t}$ is integrable for each $t \in \hointr 0 \infty$.
From Absolute Value Function is Convex, $x \mapsto \size x$ is a convex function.
From Martingale Composed with Convex Function is Submartingale, we have:
:$\sequence {\size {X_t} }_{t \ge 0}$ is a $\sequenc... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Martingale in Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-martingale]].
Then $\sequence {\siz... | From [[Characterization of Integrable Functions]]:
:$\size {X_t}$ is [[Definition:Integrable Random Variable|integrable]] for each $t \in \hointr 0 \infty$.
From [[Absolute Value Function is Convex]], $x \mapsto \size x$ is a [[Definition:Convex Real Function|convex function]].
From [[Martingale Composed with Conve... | Absolute Value of Martingale is Submartingale | https://proofwiki.org/wiki/Absolute_Value_of_Martingale_is_Submartingale | https://proofwiki.org/wiki/Absolute_Value_of_Martingale_is_Submartingale | [
"Martingales",
"Submartingales"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Martingale/Continuous Time",
"Definition:Submartingale/Continuous Time"
] | [
"Characterization of Integrable Functions",
"Definition:Integrable Random Variable",
"Absolute Value Function is Convex",
"Definition:Convex Real Function",
"Martingale Composed with Convex Function is Submartingale",
"Definition:Submartingale/Continuous Time"
] |
proofwiki-20283 | Euler's Transformation | :$\ds \map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$ | First, we observe:
{{begin-eqn}}
{{eqn | l = \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} - 1}
| r = \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} - 1} \times \dfrac {x - 1} {x - 1}
| c = multiplying top and bottom by $x - 1$
}}
{{eqn | r = \dfrac x {x - \paren {x - 1} }
| c =
}}
{{eqn | n = 1
... | :$\ds \map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$ | First, we observe:
{{begin-eqn}}
{{eqn | l = \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} - 1}
| r = \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} - 1} \times \dfrac {x - 1} {x - 1}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $x - 1$
}}
{{eqn | r = \dfrac x {x - \... | Euler's Transformation | https://proofwiki.org/wiki/Euler's_Transformation | https://proofwiki.org/wiki/Euler's_Transformation | [
"Gaussian Hypergeometric Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Pfaff's Transformation",
"Pfaff's Transformation",
"Pfaff's Transformation",
"Exponent Combination Laws/Product of Powers",
"Pfaff's Transformation"
] |
proofwiki-20284 | Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a continuous-time martingale such that $\size {X_t}^2$ is integrable for each $t \in \hointr 0 \infty$.
Let $s, t \in \hointr 0 \infty$ be such that $0 \le s < t$.
Let:
:... | We have, for $i < j$:
{{begin-eqn}}
{{eqn | l = \expect {\paren {X_{t_j} - X_{t_i} }^2 \mid \FF_{t_i} }
| r = \expect {X_{t_j}^2 - 2 X_{t_j} X_{t_i} + X_{t_i}^2 \mid \FF_{t_i} }
}}
{{eqn | r = \expect {X_{t_j}^2 \mid \FF_{t_i} } - 2 \expect {X_{t_j} X_{t_i} \mid \FF_{t_i} } + \expect {X_{t_i}^2 \mid \FF_{t_i} }
| ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Martingale in Continuous Time|continuous-time martingale]] such that $\size {X_t}^2$ is [[Definit... | We have, for $i < j$:
{{begin-eqn}}
{{eqn | l = \expect {\paren {X_{t_j} - X_{t_i} }^2 \mid \FF_{t_i} }
| r = \expect {X_{t_j}^2 - 2 X_{t_j} X_{t_i} + X_{t_i}^2 \mid \FF_{t_i} }
}}
{{eqn | r = \expect {X_{t_j}^2 \mid \FF_{t_i} } - 2 \expect {X_{t_j} X_{t_i} \mid \FF_{t_i} } + \expect {X_{t_i}^2 \mid \FF_{t_i} }
|... | Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale | https://proofwiki.org/wiki/Conditional_Expectation_of_Sum_of_Squared_Increments_of_Square-Integrable_Martingale | https://proofwiki.org/wiki/Conditional_Expectation_of_Sum_of_Squared_Increments_of_Square-Integrable_Martingale | [
"Conditional Expectation",
"Martingales",
"Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Martingale/Continuous Time",
"Definition:Integrable Random Variable",
"Definition:Subdivision of Interval/Finite",
"Definition:Almost Everywhere"
] | [
"Conditional Expectation is Linear",
"Definition:Martingale/Continuous Time",
"Definition:Measurable Function",
"Rule for Extracting Random Variable from Conditional Expectation of Product",
"Definition:Martingale/Continuous Time",
"Conditional Expectation of Measurable Random Variable",
"Conditional Ex... |
proofwiki-20285 | Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale/Corollary | :$\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2} = \expect {X_t^2 - X_s^2} = \expect {\paren {X_t - X_s}^2}$ | From Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale:
:$\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2 \mid \FF_s} = \expect {X_t^2 - X_s^2 \mid \FF_s} = \expect {\paren {X_t - X_s}^2 \mid \FF_s}$ almost surely.
From Expectation of Conditional Expectation, ... | :$\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2} = \expect {X_t^2 - X_s^2} = \expect {\paren {X_t - X_s}^2}$ | From [[Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale]]:
:$\ds \expect {\sum_{i \mathop = 1}^n \paren {X_{t_i} - X_{t_{i - 1} } }^2 \mid \FF_s} = \expect {X_t^2 - X_s^2 \mid \FF_s} = \expect {\paren {X_t - X_s}^2 \mid \FF_s}$ [[Definition:Almost Everywhere|almost surely]].
From [... | Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale/Corollary | https://proofwiki.org/wiki/Conditional_Expectation_of_Sum_of_Squared_Increments_of_Square-Integrable_Martingale/Corollary | https://proofwiki.org/wiki/Conditional_Expectation_of_Sum_of_Squared_Increments_of_Square-Integrable_Martingale/Corollary | [
"Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale"
] | [] | [
"Conditional Expectation of Sum of Squared Increments of Square-Integrable Martingale",
"Definition:Almost Everywhere",
"Expectation of Conditional Expectation"
] |
proofwiki-20286 | Expected Value of Martingale is Constant in Time/Continuous Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-martingale.
Then:
:$\expect {X_t} = \expect {X_0}$
for each $t \in \hointr 0 \infty$. | From the definition of a continuous-time martingale, we have:
:$\expect {X_t \mid \FF_0} = X_0$ almost surely
for each $t \in \hointr 0 \infty$.
So:
:$\expect {\expect {X_t \mid \FF_0} } = \expect {X_0}$
From Expectation of Conditional Expectation, we have:
:$\expect {\expect {X_t \mid \FF_0} } = \expect {X_t}$
So we... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Martingale in Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-martingale]].
Then:
:$\expect {X_... | From the definition of a [[Definition:Martingale in Continuous Time|continuous-time martingale]], we have:
:$\expect {X_t \mid \FF_0} = X_0$ [[Definition:Almost Everywhere|almost surely]]
for each $t \in \hointr 0 \infty$.
So:
:$\expect {\expect {X_t \mid \FF_0} } = \expect {X_0}$
From [[Expectation of Condition... | Expected Value of Martingale is Constant in Time/Continuous Time | https://proofwiki.org/wiki/Expected_Value_of_Martingale_is_Constant_in_Time/Continuous_Time | https://proofwiki.org/wiki/Expected_Value_of_Martingale_is_Constant_in_Time/Continuous_Time | [
"Expected Value of Martingale is Constant in Time"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Martingale/Continuous Time"
] | [
"Definition:Martingale/Continuous Time",
"Definition:Almost Everywhere",
"Expectation of Conditional Expectation"
] |
proofwiki-20287 | Expected Value of Martingale is Constant in Time/Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a discrete-time filtered probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a martingale.
Then:
:$\expect {X_n} = \expect {X_0}$
for each $n \in \Z_{\ge 0}$. | From Definition 2 of a discrete time martingale, we have:
:$\expect {X_n \mid \FF_0} = X_0$ almost surely.
So:
:$\expect {\expect {X_n \mid \FF_0} } = \expect {X_0}$
{{explain|Why does "almost surely equal" imply directly that expectations are equal? It is intuitively plausible to a person coming new to this page, but ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a [[Definition:Discrete Time Filtered Probability Space|discrete-time filtered probability space]].
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a [[Definition:Martingale in Discrete Time|martingale]].
Then:
:$\expect {X_n} = \expect {X_0}$
... | From [[Definition:Martingale/Discrete Time/Definition 2|Definition 2 of a discrete time martingale]], we have:
:$\expect {X_n \mid \FF_0} = X_0$ [[Definition:Almost Surely|almost surely]].
So:
:$\expect {\expect {X_n \mid \FF_0} } = \expect {X_0}$
{{explain|Why does "almost surely equal" imply directly that expecta... | Expected Value of Martingale is Constant in Time/Discrete Time | https://proofwiki.org/wiki/Expected_Value_of_Martingale_is_Constant_in_Time/Discrete_Time | https://proofwiki.org/wiki/Expected_Value_of_Martingale_is_Constant_in_Time/Discrete_Time | [
"Expected Value of Martingale is Constant in Time"
] | [
"Definition:Filtered Probability Space/Discrete Time",
"Definition:Martingale/Discrete Time"
] | [
"Definition:Martingale/Discrete Time/Definition 2",
"Definition:Almost Everywhere",
"Expectation of Conditional Expectation",
"Category:Expected Value of Martingale is Constant in Time"
] |
proofwiki-20288 | Inradius of Triangle in Terms of Exradii | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Then we have the following relation:
:$\dfrac 1 r = \dfrac 1 {\rho_a} + \dfrac 1 {\rho_b} + \dfrac 1 {\rho_c}$
where:
:$r$ is the inradius
:$\rho_a$, $\rho_b$ and $\rho_c$ are the exradii of $\triangle AB... | The area $\AA$ of $\triangle ABC$ is given by:
{{begin-eqn}}
{{eqn | l = \AA
| r = \rho_a \paren {s - a}
| c = Area of Triangle in Terms of Exradius
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\rho_a}
| r = \frac {s - a} \AA
| c = rearranging
}}
{{end-eqn}}
Similarly:
{{begin-eqn}}
{{eqn | l = ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Then we have the following relation:
:$\dfrac 1 r = \dfrac 1 {\rho... | The [[Definition:Area|area]] $\AA$ of $\triangle ABC$ is given by:
{{begin-eqn}}
{{eqn | l = \AA
| r = \rho_a \paren {s - a}
| c = [[Area of Triangle in Terms of Exradius]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\rho_a}
| r = \frac {s - a} \AA
| c = rearranging
}}
{{end-eqn}}
Similarly:
... | Inradius of Triangle in Terms of Exradii | https://proofwiki.org/wiki/Inradius_of_Triangle_in_Terms_of_Exradii | https://proofwiki.org/wiki/Inradius_of_Triangle_in_Terms_of_Exradii | [
"Incircles of Triangles",
"Excircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Incircle of Triangle/Inradius",
"Definition:Excircle of Triangle/Exradius"
] | [
"Definition:Area",
"Area of Triangle in Terms of Exradius",
"Area of Triangle in Terms of Inradius",
"Category:Incircles of Triangles",
"Category:Excircles of Triangles"
] |
proofwiki-20289 | Expected Value of Supermartingale is Decreasing in Time/Continuous Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-supermartingale.
Let $t, s \in \hointr 0 \infty$ with $0 \le s < t$.
Then, we have:
:$\expect {X_t} \le \expect {X_s}$ | From the definition of a supermartingale, we have:
:$\expect {X_t \mid \FF_s} \le X_s$ almost surely.
From Expectation is Monotone, we have:
:$\expect {\expect {X_t \mid \FF_s} } \le \expect {X_s}$
From Expectation of Conditional Expectation, we have:
:$\expect {X_t} \le \expect {X_s}$
{{qed}} | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Supermartingale/Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-supermartingale]].
Let $t, s \in \... | From the definition of a [[Definition:Supermartingale/Continuous Time|supermartingale]], we have:
:$\expect {X_t \mid \FF_s} \le X_s$ [[Definition:Almost Everywhere|almost surely]].
From [[Expectation is Monotone]], we have:
:$\expect {\expect {X_t \mid \FF_s} } \le \expect {X_s}$
From [[Expectation of Conditiona... | Expected Value of Supermartingale is Decreasing in Time/Continuous Time | https://proofwiki.org/wiki/Expected_Value_of_Supermartingale_is_Decreasing_in_Time/Continuous_Time | https://proofwiki.org/wiki/Expected_Value_of_Supermartingale_is_Decreasing_in_Time/Continuous_Time | [
"Expected Value of Supermartingale is Decreasing in Time"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Supermartingale/Continuous Time"
] | [
"Definition:Supermartingale/Continuous Time",
"Definition:Almost Everywhere",
"Expectation is Monotone",
"Expectation of Conditional Expectation"
] |
proofwiki-20290 | Expected Value of Submartingale is Increasing in Time/Continuous Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-submartingale.
Let $t, s \in \hointr 0 \infty$ with $0 \le s < t$.
Then, we have:
:$\expect {X_s} \le \expect {X_t}$ | From the definition of a submartingale, we have:
:$\expect {X_t \mid \FF_s} \ge X_s$ almost surely.
From Expectation is Monotone, we have:
:$\expect {\expect {X_t \mid \FF_s} } \ge \expect {X_s}$
From Expectation of Conditional Expectation, we have:
:$\expect {X_s} \le \expect {X_t}$
{{qed}} | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $\sequence {X_t}_{t \ge 0}$ be a [[Definition:Submartingale/Continuous Time|$\sequence {\FF_t}_{t \ge 0}$-submartingale]].
Let $t, s \in \hoin... | From the definition of a [[Definition:Submartingale/Continuous Time|submartingale]], we have:
:$\expect {X_t \mid \FF_s} \ge X_s$ [[Definition:Almost Everywhere|almost surely]].
From [[Expectation is Monotone]], we have:
:$\expect {\expect {X_t \mid \FF_s} } \ge \expect {X_s}$
From [[Expectation of Conditional Ex... | Expected Value of Submartingale is Increasing in Time/Continuous Time | https://proofwiki.org/wiki/Expected_Value_of_Submartingale_is_Increasing_in_Time/Continuous_Time | https://proofwiki.org/wiki/Expected_Value_of_Submartingale_is_Increasing_in_Time/Continuous_Time | [
"Expected Value of Submartingale is Increasing in Time"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Submartingale/Continuous Time"
] | [
"Definition:Submartingale/Continuous Time",
"Definition:Almost Everywhere",
"Expectation is Monotone",
"Expectation of Conditional Expectation"
] |
proofwiki-20291 | Conditional Expectations of Integrable Random Variable with respect to Filtration forms Martingale/Continuous Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $Z$ be an integrable random variable.
For each $t \in \hointr 0 \infty$, let $\expect {Z \mid \FF_t}$ be a version of the conditional expectation of $Z$ given $\FF_t$.
For each $t \in \hointr 0 \infty$... | From the definition of the conditional expectation of $Z$ given $\FF_t$, we have that:
:$X_t$ is $\FF_t$-measurable for each $t \in \hointr 0 \infty$.
So $\sequence {X_t}_{t \ge 0}$ is $\sequence {\FF_t}_{t \ge 0}$-adapted.
Now let $s, t \in \hointr 0 \infty$ have $s \le t$.
Then we have:
{{begin-eqn}}
{{eqn | l = ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $Z$ be an [[Definition:Integrable Random Variable|integrable random variable]].
For each $t \in \hointr 0 \infty$, let $\expect {Z \mid \FF_t}... | From the definition of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $Z$ given $\FF_t$]], we have that:
:$X_t$ is [[Definition:Measurable Function|$\FF_t$-measurable]] for each $t \in \hointr 0 \infty$.
So $\sequence {X_t}_{t \ge 0}$ is [[Definition:Adapted Stochastic Process|... | Conditional Expectations of Integrable Random Variable with respect to Filtration forms Martingale/Continuous Time | https://proofwiki.org/wiki/Conditional_Expectations_of_Integrable_Random_Variable_with_respect_to_Filtration_forms_Martingale/Continuous_Time | https://proofwiki.org/wiki/Conditional_Expectations_of_Integrable_Random_Variable_with_respect_to_Filtration_forms_Martingale/Continuous_Time | [
"Conditional Expectations of Integrable Random Variable with respect to Filtration forms Martingale"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Martingale/Continuous Time"
] | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Measurable Function",
"Definition:Adapted Stochastic Process",
"Tower Property of Conditional Expectation",
"Definition:Martingale/Continuous Time"
] |
proofwiki-20292 | Pfaff-Saalschütz Theorem | :$\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } { c^{\overline n} \paren {c - a - b}^{\overline n} }$ | By definition of the generalized hypergeometric function:
:$\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} \paren {-n}^{\overline k} } { c^{\overline k} \paren {1 + a + b - c - n}^{\overline k... | :$\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } { c^{\overline n} \paren {c - a - b}^{\overline n} }$ | By definition of the [[Definition:Generalized Hypergeometric Function|generalized hypergeometric function]]:
:$\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} \paren {-n}^{\overline k} } { c^{\... | Pfaff-Saalschütz Theorem | https://proofwiki.org/wiki/Pfaff-Saalschütz_Theorem | https://proofwiki.org/wiki/Pfaff-Saalschütz_Theorem | [
"Pfaff-Saalschütz Theorem",
"Hypergeometric Functions"
] | [] | [
"Definition:Hypergeometric Function/Generalized",
"Euler's Transformation",
"Euler's Transformation",
"Binomial Theorem/Extended",
"Negated Upper Index of Binomial Coefficient",
"Rising Factorial as Quotient of Factorials",
"Definition:Coefficient",
"Rising Factorial as Quotient of Factorials",
"Fal... |
proofwiki-20293 | Linear Span is Linear Subspace | Let $V$ be a vector space over a division ring $K$.
Let $S \subseteq V$ be a subset of $V$.
Then the linear span $\map \span S$ is a subspace of $V$. | First, suppose that $S = \O$.
By definition of linear combination of empty set, it follows that $\map \span \O = \set \bszero$, where $\bszero$ denotes the zero vector of $V$.
From Zero Subspace is Subspace, it follows that the trivial vector space $\set \bszero$ is a subspace of $V$.
Suppose instead that $S$ is non-em... | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Division Ring|division ring]] $K$.
Let $S \subseteq V$ be a [[Definition:Subset|subset]] of $V$.
Then the [[Definition:Linear Span|linear span]] $\map \span S$ is a [[Definition:Vector Subspace|subspace]] of $V$. | First, suppose that $S = \O$.
By definition of [[Definition:Linear Combination of Empty Set|linear combination of empty set]], it follows that $\map \span \O = \set \bszero$, where $\bszero$ denotes the [[Definition:Zero Vector|zero vector]] of $V$.
From [[Zero Subspace is Subspace]], it follows that the [[Definition... | Linear Span is Linear Subspace/Proof 1 | https://proofwiki.org/wiki/Linear_Span_is_Linear_Subspace | https://proofwiki.org/wiki/Linear_Span_is_Linear_Subspace/Proof_1 | [
"Vector Subspaces",
"Linear Combinations",
"Linear Span is Linear Subspace"
] | [
"Definition:Vector Space",
"Definition:Division Ring",
"Definition:Subset",
"Definition:Generated Submodule/Linear Span",
"Definition:Vector Subspace"
] | [
"Definition:Linear Combination/Empty Set",
"Definition:Zero Vector",
"Zero Subspace is Subspace",
"Definition:Trivial Vector Space",
"Definition:Vector Subspace",
"Definition:Non-Empty Set",
"Two-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Two-Step Vector Subspace Test",
"Definition... |
proofwiki-20294 | Linear Span is Linear Subspace | Let $V$ be a vector space over a division ring $K$.
Let $S \subseteq V$ be a subset of $V$.
Then the linear span $\map \span S$ is a subspace of $V$. | This is a special case of Generated Submodule is Linear Combinations.
As such, the statement follows immediately from that theorem.
{{qed}} | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Division Ring|division ring]] $K$.
Let $S \subseteq V$ be a [[Definition:Subset|subset]] of $V$.
Then the [[Definition:Linear Span|linear span]] $\map \span S$ is a [[Definition:Vector Subspace|subspace]] of $V$. | This is a special case of [[Generated Submodule is Linear Combinations]].
As such, the statement follows immediately from that theorem.
{{qed}} | Linear Span is Linear Subspace/Proof 2 | https://proofwiki.org/wiki/Linear_Span_is_Linear_Subspace | https://proofwiki.org/wiki/Linear_Span_is_Linear_Subspace/Proof_2 | [
"Vector Subspaces",
"Linear Combinations",
"Linear Span is Linear Subspace"
] | [
"Definition:Vector Space",
"Definition:Division Ring",
"Definition:Subset",
"Definition:Generated Submodule/Linear Span",
"Definition:Vector Subspace"
] | [
"Generated Submodule is Linear Combinations"
] |
proofwiki-20295 | Characterization of Stopping Times with respect to Right-Limit Filtration | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $T : \Omega \to \closedint 0 \infty$ be a random variable.
Let $\sequence {\GG_t}_{t \ge 0}$ be the right-limit filtration associated with $\sequence {\FF_t}_{t \ge 0}$.
{{TFAE}}
:$(1) \quad$ $T$ is a s... | === $(1)$ implies $(2)$ ===
Suppose $T$ is a stopping time with respect to $\sequence {\GG_t}_{t \ge 0}$ and let $t \in \hointr 0 \infty$.
Then, for each $n \in \N$ we have:
:$\ds \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \GG_{t - \frac 1 n}$
Since by the definition of the right-limit filtration, ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space/Continuous Time|continuous-time filtered probability space]].
Let $T : \Omega \to \closedint 0 \infty$ be a [[Definition:Random Variable|random variable]].
Let $\sequence {\GG_t}_{t \ge 0}$ be the [[Definitio... | === $(1)$ implies $(2)$ ===
Suppose $T$ is a [[Definition:Stopping Time/Continuous Time|stopping time]] with respect to $\sequence {\GG_t}_{t \ge 0}$ and let $t \in \hointr 0 \infty$.
Then, for each $n \in \N$ we have:
:$\ds \set {\omega \in \Omega : \map T \omega \le t - \frac 1 n} \in \GG_{t - \frac 1 n}$
Since ... | Characterization of Stopping Times with respect to Right-Limit Filtration | https://proofwiki.org/wiki/Characterization_of_Stopping_Times_with_respect_to_Right-Limit_Filtration | https://proofwiki.org/wiki/Characterization_of_Stopping_Times_with_respect_to_Right-Limit_Filtration | [
"Stopping Times",
"Right-Limit Filtrations of Sigma-Algebras"
] | [
"Definition:Filtered Probability Space/Continuous Time",
"Definition:Random Variable",
"Definition:Right-Limit Filtration of Sigma-Algebra",
"Definition:Stopping Time/Continuous Time",
"Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions",
"Definition:Measurable Function"
] | [
"Definition:Stopping Time/Continuous Time",
"Definition:Right-Limit Filtration of Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Countable Union",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Right-Limit Filtration of Sigma-Al... |
proofwiki-20296 | Chu-Vandermonde Identity/Falling Factorial Variant | :$\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\underline k} s^{\underline {n-k} } = \paren {r + s}^{\underline n}$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \dbinom n k r^{\underline k} s^{\underline {n - k} }
| r = \sum_{k \mathop = 0}^n \paren {\dfrac {n!} {k! \paren{n - k}!} } \paren{ \dfrac {r!} {\paren {r - k}!} } \paren{ \dfrac {s!} {\paren {s - \paren {n - k} }!} }
| c = {{Defof|Binomial Coefficient}} and... | :$\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\underline k} s^{\underline {n-k} } = \paren {r + s}^{\underline n}$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \dbinom n k r^{\underline k} s^{\underline {n - k} }
| r = \sum_{k \mathop = 0}^n \paren {\dfrac {n!} {k! \paren{n - k}!} } \paren{ \dfrac {r!} {\paren {r - k}!} } \paren{ \dfrac {s!} {\paren {s - \paren {n - k} }!} }
| c = {{Defof|Binomial Coefficient}} and... | Chu-Vandermonde Identity/Falling Factorial Variant | https://proofwiki.org/wiki/Chu-Vandermonde_Identity/Falling_Factorial_Variant | https://proofwiki.org/wiki/Chu-Vandermonde_Identity/Falling_Factorial_Variant | [
"Chu-Vandermonde Identity",
"Binomial Coefficients",
"Binomial Theorem"
] | [] | [
"Chu-Vandermonde Identity"
] |
proofwiki-20297 | Chu-Vandermonde Identity/Rising Factorial Variant | :$\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\overline k} s^{\overline {n-k} } = \paren {r + s}^{\overline n}$ | From Rising Factorial as Factorial by Binomial Coefficient, we have:
{{begin-eqn}}
{{eqn | l = r^{\overline k}
| r = k! \dbinom {r + k - 1} k
| c =
}}
{{eqn | l = s^{\overline {n - k} }
| r = \paren{n - k}! \dbinom {s + n - k - 1} {n - k}
| c =
}}
{{eqn | l = \paren{r + s}^{\overline n}
... | :$\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\overline k} s^{\overline {n-k} } = \paren {r + s}^{\overline n}$ | From [[Rising Factorial as Factorial by Binomial Coefficient]], we have:
{{begin-eqn}}
{{eqn | l = r^{\overline k}
| r = k! \dbinom {r + k - 1} k
| c =
}}
{{eqn | l = s^{\overline {n - k} }
| r = \paren{n - k}! \dbinom {s + n - k - 1} {n - k}
| c =
}}
{{eqn | l = \paren{r + s}^{\overline n}
... | Chu-Vandermonde Identity/Rising Factorial Variant | https://proofwiki.org/wiki/Chu-Vandermonde_Identity/Rising_Factorial_Variant | https://proofwiki.org/wiki/Chu-Vandermonde_Identity/Rising_Factorial_Variant | [
"Chu-Vandermonde Identity",
"Binomial Coefficients",
"Binomial Theorem"
] | [] | [
"Rising Factorial as Factorial by Binomial Coefficient",
"Negated Upper Index of Binomial Coefficient/Corollary 2",
"Exponent Combination Laws/Product of Powers",
"Chu-Vandermonde Identity",
"Negated Upper Index of Binomial Coefficient",
"Rising Factorial as Factorial by Binomial Coefficient"
] |
proofwiki-20298 | Equivalence of Definitions of Connected Manifold | Let $M$ be a topological manifold.
{{TFAE|def = Connected Manifold}} | === Definition 1 implies Definition 2 ===
Let $M$ be connected.
From Topological Manifold is Locally Path-Connected:
:$M$ is locally path-connected.
From Connected and Locally Path-Connected Implies Path-Connected:
:$M$ is path-connected.
{{qed|lemma}} | Let $M$ be a [[Definition:Topological Manifold|topological manifold]].
{{TFAE|def = Connected Manifold}} | === Definition 1 implies Definition 2 ===
Let $M$ be [[Definition:Connected Topological Space|connected]].
From [[Topological Manifold is Locally Path-Connected]]:
:$M$ is [[Definition:Locally Path-Connected Space|locally path-connected]].
From [[Connected and Locally Path-Connected Implies Path-Connected]]:
:$M$ is... | Equivalence of Definitions of Connected Manifold | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Manifold | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Manifold | [
"Connected Manifolds"
] | [
"Definition:Topological Manifold"
] | [
"Definition:Connected Topological Space",
"Topological Manifold is Locally Path-Connected",
"Definition:Locally Path-Connected Space",
"Connected and Locally Path-Connected Implies Path-Connected",
"Definition:Path-Connected/Topological Space"
] |
proofwiki-20299 | Topological Manifold is Locally Path-Connected | Let $M$ be a topological manifold.
Then $M$ is a locally path-connected space. | By definition of manifold:
:$M$ is a locally Euclidean space
The result follows from Locally Euclidean Space is Locally Path-Connected
{{qed}} | Let $M$ be a [[Definition:Topological Manifold|topological manifold]].
Then $M$ is a [[Definition:Locally Path-Connected Space|locally path-connected space]]. | By definition of [[Definition:Topological Manifold|manifold]]:
:$M$ is a [[Definition:Locally Euclidean Space|locally Euclidean space]]
The result follows from [[Locally Euclidean Space is Locally Path-Connected]]
{{qed}} | Topological Manifold is Locally Path-Connected | https://proofwiki.org/wiki/Topological_Manifold_is_Locally_Path-Connected | https://proofwiki.org/wiki/Topological_Manifold_is_Locally_Path-Connected | [
"Topological Manifolds",
"Locally Path-Connected Spaces"
] | [
"Definition:Topological Manifold",
"Definition:Locally Path-Connected Space"
] | [
"Definition:Topological Manifold",
"Definition:Locally Euclidean Space",
"Locally Euclidean Space is Locally Path-Connected"
] |
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