analisis funcional

sup[ 0 [< x=y [< 1 ]( ∬ d[x]d[y] ) = 1 & x=1 & y=1
sup[ 0 [< x=y [< 1 ]( ∬ [xy] d[x]d[y] ) = (1/4) & x=1 & y=1


med[ 0 [< x=y [< 1 ]( ∬ d[x]d[y] ) = (1/2) & x=(1/2^{(1/2)}) & y=(1/2^{(1/2)})
med[ 0 [< x=y [< 1 ]( ∬ [xy] d[x]d[y] ) = (1/8) & x=(1/2^{(1/4)}) & y=(1/2^{(1/4)})


sup[ 0 [< x=y [< 1 ]( ∫ [ (1+(-x)) ] d[y] ) = (1/4) & x=(1/2) & y=(1/2)
med[ 0 [< x=y [< 1 ]( ∫ [ (1+(-x)) ] d[y] ) = (1/8) & x=(1/2)( 1+(1/2)^{(1/2)} ) & y=(1/2)( 1+(1/2)^{(1/2)} )
x^{2}+(-x)+(1/8)=0

especies combinatóries: parts dos

[ m·( {a_{i_{k}},a_{j_{k}}} ) ]


∑ ( ( m·[ k // 2 ] )·x^{k} )


[ m·( {a_{i_{k}},a_{j_{k}}} ) ] + [ n·( {a_{i_{k}},a_{j_{k}}} ) ] = ...
... [ (m+n)·( {a_{i_{k}},a_{j_{k}}} ) ]


∑ ( ( (m+n)·[ k // 2 ] )·x^{k} )


[ m·( {a_{i_{k}},a_{j_{k}}} ) ] [x] [ n·( {a_{i_{k}},a_{j_{k}}} ) ] = ...
... [ (m·n)·( {a_{i_{k}},a_{j_{k}}} [x] {a_{i_{k}},a_{j_{k}}} ) ]


∑ ( ( (m·n)·[ k // 2 ]^{2} )·x^{k} )

especies combinatóries

[ n·( {a_{k},a_{1},a_{2}},...,{a_{k+(-1)},a_{k},a_{1}} ) ]


∑ ( ( n·k )·x^{k} )


[ m·( {a_{k},a_{1},a_{2}},...,{a_{k+(-1)},a_{k},a_{1}} ) ]+...
... [ n·( {a_{k},a_{1},a_{2}},...,{a_{k+(-1)},a_{k},a_{1}} ) ] = ...
... [ (m+n)·( {a_{k},a_{1},a_{2}},...,{a_{k+(-1)},a_{k},a_{1}} ) ]


∑ ( ( (m+n)·k )·x^{k} )
 
f: [n·( {a_{1}},...,{a_{k}} )] ---> [ n·( {a_{k},a_{1},a_{2}},...,{a_{k+(-1)},a_{k},a_{1}} ) ] és bijectiva.


f({a_{j}}) = {a_{j+(-1)},a_{j},a_{j+1}} = {a_{i+(-1),a_{i},a_{i+1}}} = f({a_{i}})
j+(-1) = i+(-1) & j+1 = i+1
j=i

especies combinatóries

[ n·( {a_{k},a_{2}},...,{a_{k+(-1)},a_{1}} ) ]


∑ ( ( n·k )·x^{k} )


[ m·( {a_{k},a_{2}},...,{a_{k+(-1)},a_{1}} ) ]+[ n·( {a_{k},a_{2}},...,{a_{k+(-1)},a_{1}} ) ] = ...
... [ (m+n)·( {a_{k},a_{2}},...,{a_{k+(-1)},a_{1}} ) ]


∑ ( ( (m+n)·k )·x^{k} )
 
f: [n·( {a_{1}},...,{a_{k}} )] ---> [ n·( {a_{k},a_{2}},...,{a_{k+(-1)},a_{1}} ) ] és bijectiva.


f({a_{j}}) = {a_{j+(-1)},a_{j+1}} = {a_{i+(-1),a_{i+1}}} = f({a_{i}})
j+(-1) = i+(-1) & j+1 = i+1
j=i

especies combinatóries: el octopus

[ A_{1},...,A_{j} ]-[ m·( {a_{1}},...,{a_{k}} ) ]


∑ (m·k+j)·x^{k}


[ A_{1},...,A_{j} ]-[ m·( {a_{1}},...,{a_{k}} ) ] + [ B_{1},...,B_{s} ]-[ n·( {a_{1}},...,{a_{k}} ) ] = ...
... [ A_{1},...,A_{j},B_{1},...,B_{s} ]-[ (m+n)·( {a_{1}},...,{a_{k}} ) ]


∑ ( (m+n)·k+(j+s) )·x^{k}


[ A_{1},...,A_{j} ]-[ m·( {a_{1}},...,{a_{k}} ) ] [x] [ B_{1},...,B_{s} ]-[ n·( {a_{1}},...,{a_{k}} ) ] = ...
... [ A_{j} [x] B_{s} ]-[ (ms+nj)·( {a_{1}},...,{a_{k}} ) ]-[ (m·n)·( {a_{k}} [x] {a_{k}} ) ]


∑ ( (m·n)·k^{2}+(ms+nj)·k+(j·s) )·x^{k}

especies combinatóries

[ n·( {a_{1}},...,{a_{k}} ) ]


∑ ( ( n·k )·x^{k} )


[ m·( {a_{1}},...,{a_{k}} ) ] + [ n·( {a_{1}},...,{a_{k}} ) ] = [ (m+n)·( {a_{1}},...,{a_{k}} ) ]


∑ ( ( (m+n)·k )·x^{k} )


[ m·( {a_{1}},...,{a_{k}} ) ] [x] [ n·( {a_{1}},...,{a_{k}} ) ] = ...
... [ (m·n)·( {a_{k}} [x] {a_{k}} ) ]


∑ ( ( (m·n)·k^{2} )·x^{k} )

química colesterol nitruro

colesterol nitruro:
H_{2}(OH)_{3}-Hex[ -N- | -C=C-N-C=C- ]-O-Hex[ -N- | -C=C-N-C=C- ]-H_{2}(OH)_{3}
4 para-destructores + 2 para-constructores = 2 para-destructores


gras-insulina nitrura:
H_{2}(OH)_{3}-Hex[ -N- | -C=C=C-N-C=C=C- ]-O-Hex[ -N- | -C=C=C-N-C=C=C- ]-H_{2}(OH)_{3}
4 para-destructores + 6 para-constructores = 2 para-constructores

química azúcar nitruro

azúcar nitruro:
H_{2}(OH)_{4}-Hex[ -C- | -C=C-N-C=C- ]-O-Hex[ -C- | -C=C-N-C=C- ]-H_{2}(OH)_{4}
4 para-destructores + 2 para-constructores = 2 para-destructores


insulina nitrura:
H_{2}(OH)_{4}-Hex[ -C- | -C=C=C-N-C=C=C- ]-O-Hex[ -C- | -C=C=C-N-C=C=C- ]-H_{2}(OH)_{4}
4 para-destructores + 6 para-constructores = 2 para-constructores

química: colesterol-fosfuro

colesterol fosfuro:
(OH)-Hex[ =P= | =C=C=P=C=C= ]-Hex[ =P= | =C=C=P=C=C= ]-(OH)
8 para-destructores + 4 para-constructores = 4 para-destructores


gras-insulina fosfura:
(OH)-Hex[ =P= | =C=C=C=P=C=C=C= ]-Hex[ =P= | =C=C=C=P=C=C=C= ]-(OH)
8 para-destructores + 12 para-constructores = 4 para-constructores

química: azúcar fosfuro

azúcar fosfuro:
Hex[ =C= | =C=C=P=C=C= ]-Hex[ =C= | =C=C=P=C=C= ]
8 para-destructores + 4 para-constructores = 4 para-destructores


insulina fosfura:
Hex[ =C= | =C=C=C=P=C=C=C= ]-Hex[ =C= | =C=C=C=P=C=C=C= ]
8 para-destructores + 12 para-constructores = 4 para-constructores

química colesterol y gras-insulina

colesterol:
(OH)_{2}-Hex[ -O- | -C=C-O-C=C- ]-O_{6}-Hex[ -O- | -C=C-O-C=C- ]-(OH)_{2}
4 para-destructores + 2 para-constructores = 2 para-destructores


gras-insulina:
(OH)_{2}-Hex[ -O- | -C=C=C-O-C=C=C- ]-O_{6}-Hex[ -O- | -C=C=C-O-C=C=C- ]-(OH)_{2}
4 para-destructores + 6 para-constructores = 2 para-constructores

química azúcar y insulina

azúcar:
(OH)_{4}-Hex[ -C- | -C=C-O-C=C- ]-O_{6}-Hex[ -C- | -C=C-O-C=C- ]-(OH)_{4}
4 para-destructores + 2 para-constructores = 2 para-destructores


insulina:
(OH)_{4}-Hex[ -C- | -C=C=C-O-C=C=C- ]-O_{6}-Hex[ -C- | -C=C=C-O-C=C=C- ]-(OH)_{4}
4 para-destructores + 6 para-constructores = 2 para-constructores

dual-románico

románico-italiano
je vare cantare
tú vare cantare


il vare cantare
ella vare cantare


románico-portuguesh
je varesh cantaresh
tú varesh cantaresh


il varesh cantaresh
ella varesh cantaresh


románico-françé
je vare-dom cantare-dom
tú vare-dom cantare-dom


il vare-dom cantare-dom
ella vare-dom cantare-dom

dual-románico

románico-italiano
je havere cantato
tú havere cantato


il havere cantato
ella havere cantato


románico-portuguesh
je haveresh cantadu
tú haveresh cantadu


il haveresh cantadu
ella haveresh cantadu


románico-françé
je havere-dom cantatu-dom
tú havere-dom cantatu-dom


il havere-dom cantatu-dom
ella havere-dom cantatu-dom

dual-románico

románico-italiano
je cantare
tú cantare


il cantare
ella cantare


románico-portuguesh
je cantaresh
tú cantaresh


il cantaresh
ella cantaresh


románico-françé
je cantare-dom
tú cantare-dom


il cantare-dom
ella cantare-dom

combinacions racionals

( (3m+2n)/(6k) ) = (m/(2k))+(n/(3k))