From e04dac50a699a8bfc36adfc611d1108497d6ad92 Mon Sep 17 00:00:00 2001 From: Indigo5684 <159226326+Indigo5684@users.noreply.github.com> Date: Tue, 30 Sep 2025 13:19:35 -0500 Subject: [PATCH] Definition Punctuation --- docs/math/abstract-algebra/16-rings.md | 2 +- docs/math/abstract-algebra/17-polynomial-rings.md | 2 +- docs/math/abstract-algebra/18-integral-domains.md | 2 +- docs/math/abstract-algebra/DF-10-modules.md | 4 ++-- docs/math/abstract-algebra/DF-12-modules-pids.md | 2 +- docs/math/diffeq/2-1st-order.md | 2 +- docs/math/real-analysis/2-reals.md | 6 +++--- docs/physics/mechanics/2-projectiles-charged-particles.md | 2 +- 8 files changed, 11 insertions(+), 11 deletions(-) diff --git a/docs/math/abstract-algebra/16-rings.md b/docs/math/abstract-algebra/16-rings.md index 779a57a..f19f331 100644 --- a/docs/math/abstract-algebra/16-rings.md +++ b/docs/math/abstract-algebra/16-rings.md @@ -56,7 +56,7 @@ Note that some books impose the condition that $1 \neq 0$. If $1 = 0$, we can sh ## Section 16.3 - Ring Homomorphisms and Ideals -**Definition** Given rings $R$ and $S$, and a mapping $\varphi: R \rightarrow S$, we say that $\varphi$ is a *ring homomorphism* if the following are satisfied for all elements of $R$: +**Definition**. Given rings $R$ and $S$, and a mapping $\varphi: R \rightarrow S$, we say that $\varphi$ is a *ring homomorphism* if the following are satisfied for all elements of $R$: $$ \begin{align} diff --git a/docs/math/abstract-algebra/17-polynomial-rings.md b/docs/math/abstract-algebra/17-polynomial-rings.md index b27492e..aaa8e29 100644 --- a/docs/math/abstract-algebra/17-polynomial-rings.md +++ b/docs/math/abstract-algebra/17-polynomial-rings.md @@ -46,7 +46,7 @@ where either $\deg r(x) < \deg g(x)$ or $r(x)$ is the zero polynomial. ## Section 17.3 Irreducible Polynomials -**Definition** A non-constant polynomial $f(x) \ in F[x]$ is *irreducible* over a field $F$ if it cannot be expressed as the product of two non-identity polynomials $g(x)$ and $h(x)$ in $F[x]$, with the degree of both polynomials strictly less than the degree of $f(x)$. +**Definition**. A non-constant polynomial $f(x) \ in F[x]$ is *irreducible* over a field $F$ if it cannot be expressed as the product of two non-identity polynomials $g(x)$ and $h(x)$ in $F[x]$, with the degree of both polynomials strictly less than the degree of $f(x)$. **Lemma**. Let $p(x) \in \mathbb{Q}[x]$. Then, with $r, s \in \mathbb{Z}, a(x) \in \mathbb{N}[x]$, we can write $p(x) = \frac{r}{s} a(x)$. diff --git a/docs/math/abstract-algebra/18-integral-domains.md b/docs/math/abstract-algebra/18-integral-domains.md index ed145cd..de33e46 100644 --- a/docs/math/abstract-algebra/18-integral-domains.md +++ b/docs/math/abstract-algebra/18-integral-domains.md @@ -57,7 +57,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, **Theorem**. 18.15: Every PID is a UFD. Note that the converse is not true. -**Corollary** 18.16: Let $F$ be a field. Then, $F[x]$ is a UFD. +**Corollary**. 18.16: Let $F$ be a field. Then, $F[x]$ is a UFD. --- diff --git a/docs/math/abstract-algebra/DF-10-modules.md b/docs/math/abstract-algebra/DF-10-modules.md index 35e0f67..18b540e 100644 --- a/docs/math/abstract-algebra/DF-10-modules.md +++ b/docs/math/abstract-algebra/DF-10-modules.md @@ -15,7 +15,7 @@ **Remark**. Modules over a field $F$ and vector spaces over $F$ are identical. -**Definition** An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule. +**Definition**. An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule. **Remark**. If $F$ is a field, submodules are equivalent to subspaces. @@ -124,7 +124,7 @@ This direct product is in itself an $R$-module. **Corollary**. If $F$ is a free $R$-module with basis $A$, then $F \cong F(A)$. -**Definition** For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$. +**Definition**. For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$. ## Section 10.4 - Tensor Products of Modules diff --git a/docs/math/abstract-algebra/DF-12-modules-pids.md b/docs/math/abstract-algebra/DF-12-modules-pids.md index 79af638..1d0a65d 100644 --- a/docs/math/abstract-algebra/DF-12-modules-pids.md +++ b/docs/math/abstract-algebra/DF-12-modules-pids.md @@ -30,7 +30,7 @@ $$ This is the *torsion submodule* of $M$. If $\text{Tor}(M)$ is empty, then $M$ is *torsion-free*. -**Definition** Let $R$ be an integral domain and $M$ be an $R$-module. Then, given a submodule $N$, +**Definition**. Let $R$ be an integral domain and $M$ be an $R$-module. Then, given a submodule $N$, $$ \text{Ann}_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \} diff --git a/docs/math/diffeq/2-1st-order.md b/docs/math/diffeq/2-1st-order.md index 0dbeeaf..caa3904 100644 --- a/docs/math/diffeq/2-1st-order.md +++ b/docs/math/diffeq/2-1st-order.md @@ -65,7 +65,7 @@ $$ \frac{dy}{dt} + p(t)y = g(t) $$ -To find a solution to this differential equation, construct the **integrating factor** $\mu(t)$. +To find a solution to this differential equation, construct the **integrating factor**. $\mu(t)$. $$\mu(t) = k \exp(\int p(t) dt)$$ diff --git a/docs/math/real-analysis/2-reals.md b/docs/math/real-analysis/2-reals.md index d4c4aaf..41f528c 100644 --- a/docs/math/real-analysis/2-reals.md +++ b/docs/math/real-analysis/2-reals.md @@ -21,7 +21,7 @@ 2. $\mathbb{N} \subseteq F^+$ 3. If $a \in F^+$, then $\frac{1}{a} \in F^+$ -**Definition** The order relation $a > b$ and $b < a$ is defined by $a - b \in F^+$. +**Definition**. The order relation $a > b$ and $b < a$ is defined by $a - b \in F^+$. **Theorem**. If $a, b, c \in F$, then @@ -76,9 +76,9 @@ 2. $|a - b| = |b - a|$ 3. $|a - c| \leq |a - b| + |b + c|$ -**Definition** A set together with a function satisfying these three properties is known as a *metric space*. +**Definition**. A set together with a function satisfying these three properties is known as a *metric space*. -**Definition** The $\varepsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\varepsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $|x - a| < \varepsilon$. That is, +**Definition**. The $\varepsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\varepsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $|x - a| < \varepsilon$. That is, $$ V_\varepsilon(a) = (a - \varepsilon, a + \varepsilon) diff --git a/docs/physics/mechanics/2-projectiles-charged-particles.md b/docs/physics/mechanics/2-projectiles-charged-particles.md index c9dba7c..4a7717b 100644 --- a/docs/physics/mechanics/2-projectiles-charged-particles.md +++ b/docs/physics/mechanics/2-projectiles-charged-particles.md @@ -2,7 +2,7 @@ ## Section 2.1 - Air Resistance -**Definition**. The *drag*, or resistive force on an object due to the atmosphere, is denoted as $\mathbf{f}$. Note that this is **not** the force density, but the overall force. In most cases, this force directly opposes the direction of motion. If not, the other component is known as *lift*, however this is mostly negligible. +**Definition**. The *drag*, or resistive force on an object due to the atmosphere, is denoted as $\mathbf{f}$. Note that this is **not**. the force density, but the overall force. In most cases, this force directly opposes the direction of motion. If not, the other component is known as *lift*, however this is mostly negligible. We define air resistance as $\mathbf{f} = -f(v) \hat{\mathbf{v}}$. We consider two types in this text: linear, where $f(v) = f_{lin} = bv$, and quadratic, where $f(v) = f_{quad} = cv^2$. Note that often times we consider both, and state that $f(v) = f_{lin} + f_{quad} = bv + cv^2$.