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WordPress Latex support

AMS TeX Collections: Distribution

Latex documentation

Latex math symbols

Examples:

e^{\i \pi} + 1 = 0

sin{x} + 1 = 0

i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>

 A  \textbf{bold \textit{Hello \LaTeX}} to start!

This is an in-text z=x+y math equation

———-

Generic function notation:

{f : \mathcal X \, \rightarrow \, \mathcal Y} ,

{\mathcal X \ni x \, \mapsto \, f(x )\in \mathcal Y} ,

\mathcal X \, \stackrel {f} {\longrightarrow} \, \mathcal Y ,

{x \, \longmapsto \, f(x)} ,

{{\mathcal X \, \stackrel {f} {\longrightarrow} \, \mathcal Y \:}\atop {\: x \, \longmapsto \, f(x) } } {\,} .

Inverse Fourier transform:

​ f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x} \,d\xi ,

Binary composition:

{\mathcal S \times \mathcal S \ni (x, y) \, \mapsto \, x \ast y \in \mathcal S} ,

z = x^{y^2} ,

{\oplus \atop {x \in \mathcal{P} } } ,

{\bigoplus \atop {x \in \mathcal{P} } } \mathbf {R} ,

\mathrm{supp} f = \{ x \in X \, : \, f(x) \neq 0 \} ,

R \subseteq \mathcal X \times \mathcal X

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x \pmod a
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\varphi ( x \ast y) = \varphi (x) {\ast}' \varphi (y), \, \forall x, y \in \mathcal G

\varphi ( x \ast y) = \varphi (x) {\ast}' \varphi (y), \, \forall x, y \in \mathcal G

Community: C = (A, P)

Activities: A = \{ A_1, A_2, \ldots, A_{n} \}

Participators: P = \{ P_1, P_2, \ldots, P_{m} \}

Possible participator grouping (of an activity A_{i} ):

\mathrm{P_{GI}}(A_{i}) = \sum_{k=1}^m (1-P_{k})(P_{k} \in A_{i}) A_{i}

Actual participator grouping (of an activity A_{i} ):
\mathrm{G_{I}}(A_{i}) = \sum_{j=1}^m \sum_{s \in {\prod_{}^j}m} P_{s}(P_{s} \in A_{i}) A_{i}

Infinite sequence: {(a_n)}_{n=1}^{\infty}

Possible activity grouping (of a participator P_{k} ):

\mathrm{P_{GI}}(P_{k}) = \sum_{i=1}^n (1-A_{i})(A_{i} \in P_{k}) P_{k}

Actual activity grouping (of a participator P_{k} ):

\mathrm{G_{I}}(P_{k}) = \sum_{i=1}^n \sum_{s \in {\prod_{}^i}n} A_{s}(A_{s} \in P_{k}) P_{k}

Cardinality formulas:

Total number of participators:

|P| = \sum_{k=1}^n (-1)^{k-1} \sum_{s \in {\prod_{}^k}n}|\cap P_{A_{s}}| = m

where

\cap P_{A_{(1,2)}} = P_{A_1} \cap P_{A_2}

Total number of activities:

|A| = \sum_{k=1}^m (-1)^{k-1} \sum_{s \in {\prod_{}^k}m}|\cap A_{P_{s}}| = n

where

\cap A_{P_{(1,2)}} = A_{P_1} \cap A_{P_2}

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checking some expressions:

\mathbf N \times \mathbf N \, \stackrel {+} {\longrightarrow} \, \mathbf N {\mathbf N \times \mathbf N \ni (x, y) \, \mapsto \, x + y \in \mathbf N}

The Tension-gradient of a Scenario:

dS_{cenario} = \sum_{T_{ensions}}{\frac{\partial{S_{cenario}}}{\partial{T_{ension}}}} dT_{ension} \ldots dS_{cenario} = \bigoplus_{T_{ensions}}{\frac{\partial{S_{cenario}}}{\partial{T_{ension}}}} dT_{ension} dI_{ssue} = \sum_{T_{ensions}}{\frac{\partial{I_{ssue}}}{\partial{T_{ension}}}} dT_{ension} dI_{ssue} = \bigoplus_{T_{ensions}}{\frac{\partial{I_{ssue}}}{\partial{T_{ension}}}} dT_{ension} \int\limits_{a}^{b} f(x) \, dx \int_a^b f(x) \, dx \, {\sum\limits_{m \in m}^{ \text {abcd} }} \,

\,\sum\limits_{k = 0}^{n} \, ,

\, \textstyle\sum_{k=0}^{n} \, f = {\sum\limits_{m \in M}^{ \text {} }}{f_m} m \, , \text{ and } \, g = {\sum\limits_{m \in m}^{ \text {} }}{g_m} m \, , R\{X\} \times R\{X\} \ni (f, g) \, \mapsto \, f \ast g \in {R\{X\}\, , \text{where} \, (f \ast g)(m) \stackrel {\mathrm{def}}{=}{\sum\limits_{m'm'' = m}^{ \text {} }f(m')g(m'')}}

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\mathcal F \, \stackrel {\chi_{apple}} {\longrightarrow} \, \mathbf N \, , \, \mathcal F \, \stackrel {\chi_{pear}} {\longrightarrow} \, \mathbf N \, , \, \mathcal F \, \stackrel {\chi_{banana}} {\longrightarrow} \, \mathbf N

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GRADIENTS

A function f from \mathbf{R}^2 to \mathbf{R} can be described by:

{{\mathbf{R}^2 \, \stackrel {f} {\longrightarrow} \, \mathbf{R} \:}\atop {\: (x,y) \, \longmapsto \, f(x,y) } } {\,}.

The differential df of the function f at the point (a,b) \in \mathbf{R}^2 is given by:

df_{(a,b)} = \frac{\partial f}{\partial x}_{(a,b)} dx + \frac{\partial f}{\partial y}_{(a,b)} dy.

The equation of the level curve ( \, f = const \, ) of the function f at the point (a,b) is given by:

f(x,y)=f(a,b).

The equation of the tangent to the level curve of the function f at the point (a,b) is given by:

\frac{\partial f}{\partial x}_{(a,b)} (x-a) + \frac{\partial f}{\partial y}_{(a,b)} (y-b) = 0.

The gradient

The gradient of the function f is the function {\triangledown f} defined by:

{{\mathbf{R}^2 \, \stackrel {\triangledown f} {\longrightarrow} \, \mathbf{R}^2 \:}\atop {\: (x,y) \, \longmapsto \, (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}) } } {\,}.

Hence the value of {\triangledown f} at the point (x,y) is:

{\triangledown f}_{(x,y)} = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}).

The value of {\triangledown f}_{(x,y)} at the point x=a,y=b is obtained by evaluating the function {\triangledown f}_{(x,y)} at the point (a,b):

{\triangledown f}_{(a,b)} = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y})_{(a,b)}.

Hence we see that the gradient {\triangledown f}_{(a,b)} of the function f at the point (a,b) is perpendicular to the tangent of the level curve of the function f at the point (a,b).

Flying carpets and level surfaces

The “flying carpet equation” of the function f is given by:

z = f(x, y).

The level surface equation of the function f is given by:

g(x,y,z) \stackrel {\mathrm{def}}{=} f(x,y) - z = 0.

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\mathrm{supp} f \, \stackrel {\mathrm{def}}{=} \{ x \in X \, : \, f(x) \neq 0 \} \, .
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In the animation below, the “input” function f is given by:

f(x, y) = \frac{1}{3} (x^2 + 2 y^2) + \frac {3}{4}

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\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}_{x(p)} df(x(u))_{f(x(p))} = f'(x(u))_{x(p)} x'(u)_p du = = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}_{x(p)} \begin{pmatrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \end{pmatrix}_p \begin{pmatrix} du_1 \\ du_2 \end{pmatrix}

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/////// INNEHÅLLSFÖRTECKNINGEN I BOKEN “SIFFRORNA I VÅRA LIV”
(försedda med några länkar och kommentarer)

Innehåll

Inledning XX

1. Matematik runt omkring oss XX
Rekommendationer på Netflix XX
Matematik finns överallt XX

2. Skilda världar XX

I detta kapitel diskuterar författaren skilladen mellan den konkreta värld som vi kan uppfatta med sinnena, och den abstrakta idévärld som introducerades av Platon. I sin berömda grottliknelse, betraktar Platon sinnesförnimmelserna som skuggor av de abstrakta, perfekta och eviga begrepp som grekerna införde i matematiken.

Matematiken som en enda stor historia XX
Det fina med elegans XX

3. Ett liv utan tal XX
På en ö långt från pirahã XX
Utan att mäta! XX
Att arbeta med små mängder XX
Jag vet inte exakt! Stora mängder och hjärnan XX
Till och med en kyckling kan känna igen figurer XX
Bidrar matematik med något? XX

4. Matematik för länge, länge sedan XX
Dags att deklarera XX
Hemläxor i Mesopotamien XX
Bröd, öl och tal i Egypten XX
De alltid lika teoretiska grekerna XX
Nördarna i Kina XX

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5. Det förändras hela tiden
Newton mot Leibniz XX
Allt mindre steg XX
Stegräkning XX
Vädret – något mer föränderligt finns inte XX
Integraler och differentialer i byggnadskonst, policyplaner och fysiken XX
Integraler åt alla? XX

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6. Grepp om osäkerheten XX
Spel i matematiken XX
Myntfördelning XX
Två gånger Thomas XX
Vadå spel? Matematik i praktiken XX
Mer data! XX
Vad John Snow faktiskt visste XX
Nicholas Cage och simbassänger XX
Stämmer det verkligen? Förvräng världen med statistik! XX
När man inte vill fråga alla XX

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7. Att vandra i tankar XX
Enkelriktat XX
Googles vandringar genom internet XX
Att se på film med en graf XX
Effektivare cancerbehandling med matematikens hjälp XX
Facebook, vänner och artificiell intelligens XX
Grafer i bakgrunden XX

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8. Nyttan med matematik XX
Fel i matematiken XX
Är allt en slump? XX
Matematik hjälper XX
Även i vardagen XX

KOMMENTAR:

• Vad är matematik? Om metamatematik, matematisk design och matematiska.
• Math Makers versus Math Fakers
• Matteplåster

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\oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\dfrac {\partial M}{\partial x}}-{\dfrac {\partial L}{\partial y}}\right)dx\,dy

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What is Really Behind the Increase of Atmospheric CO2? – Prof. Murry Salby

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E - A = \dfrac{dr}{dt} \;\;\;\;\;\;\;\;\;\;\;\; \text{Net Emission} \newline

E - A = (\underbrace{E_N - A_N}_{\text{\textcolor{green} {Natural}}}) + (\underbrace{E_A - A_A}_{\text{\textcolor{red} {Anthropogenic}}}) = \dfrac{dr}{dt} \;\;\;\;\;\;\;\; \text{Perturbed}

\;\;\;\;\;\;\;\;\;\;\, \simeq f(T) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{Despite 60\% increase of} \; E_A

\;\;\;\;\;\; \implies \;\;\;\;\;\; (E_A - A_A) \, \ll \, (E_N - A_N) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1) \newline

Now

\;\;\;\; E_A \simeq 3.0 \, \text{ppmv/yr} \, > \, E - A \, = \, 1.5 \, \text{ppmv/yr} \;\;\;\;\;\;\;\; (2) \newline

It can be shown that:

(1) and (2) \;\;\implies \;\;\ A_A \, \simeq \, E_A \;\;\;\;\; \text{Strong Offset by Absorption. }

Then

\;\;\;\;\;\;\;\;\;\;\; \underbrace{\dfrac{dr}{dt} - E_A}_{\text{less than 0}} = \underbrace{(E_N - A_N) - A_A}_{\text{\textcolor{green} {Natural}}} \newline

reduces to

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \dfrac{dr}{dt} \, \simeq \, E_N - A_N

CO_2 is therefore controlled by Net

Natural Emission

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= time 37:32 min:sek

OBSERVED EVOLUTION:

ANTHROPOCENTRIC EMISSION

versus

RESPONSE OF CO_2

= time 37:43 min:sek

Cumulative Anthropogenic Emission of CO_2 must obey a conservation law expressed by a continuity equation.

For CO_2 to increase, it must therefore have two components: A component equal to the absorption of CO_2 , which offsets its removal.

This is the Compensated Component of Anthropogenic Emission.

The Non-Compensated component is equivalent to Net Anthropogenic Emission.

\Delta r_A \; \textcolor{blue}{\Delta {r_A}_1} \; \textcolor{red}{\Delta {r_A}_2} \ge 3 - \textcolor{blue}{\Delta {r_A}_1} \;

\;\;\;\;\;\; [ E_A ] = \underbrace{ [ A_A ]}_{\text{Compensated}} + \underbrace{ \Delta r_A}_{\text{Non-compensated}} \;\;\;\;\;\; , \;\;\;\;\; [\;\;] = \int dt

\;\;\;\;\;\;\;\;\;\;\;\;\;\, = \alpha [\,\overline{r_A}\,] + \underbrace{\Delta r_A}_{\text{= Net Emission}} \; , \;\;\; {\alpha}^{-1} \, = \, \text{Absorption time}

//////// Important construction to remember:

\;\;\;\; \underbrace{\underbrace{\Delta r_A}_{\text{Non-Compensated}}}_{\text{= Net Emission}}

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