?? 24 semi-simple comm ring structures on {a,b,c,d} ...
f2 X f2 vs f4 ....
?? depict f4 structure as for example "ab", meaning a=0 and b=1 ...
?? depict f2Xf2 structure as ... ??? ... for example, "ab", meaning a = (0,0) and b = (1,1) .... ?? ...
(?? so some of that 2+2=2X2 stuff is dimly visible here but ok to soft-pedal, i guess .... ??? ....)
?? but need way to distinguish the 2 types .... ?? 4ab and 2ab ... ??? ....
?? 2^24 as already somewhat annoyingly big ... ?? ....
?? try working with euler summand ...... ???? .... ???? .....
Monday, April 9, 2012
?? 24 semi-simple comm ring structures on {a,b,c,d} ...
?? halving ab/cd ...
12
3ac
4db
ab
c32
d14
?? "halving into ab/cd corresponds to factoring into 12/34" ... ??? ... ?? and vice versa ?? .... ??? this really is that weird thing that i noticed in high school ??? .... ?? still don't quite get all of it though ... the vice versa part .... ???? .... ?? outer automorphism of automorphism group of wreath product here ??? .... ??? _is_ this connected nicely to solution of quartic somehow ??? ....
?? 4! as resulting from making outer inner ??? .... ?? having trouble getting arithmetic to work here .... ???? ....
wreath product here as 8-elt gp ??? .... outer automorphism of order 2, evidently .... ???? .....
?? well, so maybe the semi-direct product here really is some weird 16-elt gp .... ??? ....
?? "element" vs "cross section" of split 2-elt set ....
?? "unordered dual pair of split 4-elt sets" .... ??? ....
?? number of 2+2 decompositions same as number of 2X2 decomposition because of ... outer automorphism here .... ??? .... ????? ....
?? isomorphism between subgroup of 4! preserving a 2+2 decomposition and subgroup preserving a 2X2 decomposition .... ??? .... ???? extending to ..... ???? .... ????? ..... ?? actually, extending to identity automorphism .... ??? ... ???? confusion ... ???? .....
?? is there duality between 6 halves of one 4-elt set and 6 halves of .... ??? .... ??? doesn't seem to quite make sense ??? ...
?? halving ab/cd ...
12
3ac
4db
ab
c32
d14
?? "halving into ab/cd corresponds to factoring into 12/34" ... ??? ... ?? and vice versa ?? .... ??? this really is that weird thing that i noticed in high school ??? .... ?? still don't quite get all of it though ... the vice versa part .... ???? .... ?? outer automorphism of automorphism group of wreath product here ??? .... ??? _is_ this connected nicely to solution of quartic somehow ??? ....
?? 4! as resulting from making outer inner ??? .... ?? having trouble getting arithmetic to work here .... ???? ....
wreath product here as 8-elt gp ??? .... outer automorphism of order 2, evidently .... ???? .....
?? well, so maybe the semi-direct product here really is some weird 16-elt gp .... ??? ....
?? "element" vs "cross section" of split 2-elt set ....
?? "unordered dual pair of split 4-elt sets" .... ??? ....
?? number of 2+2 decompositions same as number of 2X2 decomposition because of ... outer automorphism here .... ??? .... ????? ....
?? isomorphism between subgroup of 4! preserving a 2+2 decomposition and subgroup preserving a 2X2 decomposition .... ??? .... ???? extending to ..... ???? .... ????? ..... ?? actually, extending to identity automorphism .... ??? ... ???? confusion ... ???? .....
?? is there duality between 6 halves of one 4-elt set and 6 halves of .... ??? .... ??? doesn't seem to quite make sense ??? ...
?? "can you define multiplication of natural numbers in terms of "relatively prime" ?" as maybe good example ... ???
?? also defining anything in terms of successor ... ??? ...
?? all possible ways of decorating triangle .... ??? "special" such ... ???? hmm, confusion ... ??? .... ?? taking seriously idea of "frame-x orientations as x's" .... ?? hmmm, yet another horrible confusion here ?? ... "active vs passive" ... hard to explain without good pictures here .... ??? ....
??? each of the three frame-point orientations has a very different look when portrayed in the "collection of all frames exhibiting that orientation to given point p" style .... ??? .... ?? to what extent does this generalize ?? .... ???? .....
(?? vs ... "collection of all points exhibiting that orientation to given frame f" style ..... ????? ..... ?? hmm, that style as unhelpful (?? ...) here because whole point is that concept of frame is here (???? .....) coming equipped with canonical (??? ....) way of drawing it ..... ??? ... os ???? .... ?? hmm, hint tha answer to "to what extent does this generalize ?" above is something like ... ??? to the general context where the base carriers are plain old (?? finite ?? ...) sets, so that concept of frame has somewhat good canonical pictorial representation ... ??? or beyond that, to general context where _you've already settled on a good way to draw a frame_, and you want to use it to get good ways of drawing other structures .... ???? ...... ??? then ... ?? general idea of getting picture in which the symmetry group acts "fixed-point-free" on some orbit, so that marking single point on that orbit gives good decoration corresponding to "frame" ... ??? ...)
3 looks : "hypotenuse-to-hypotenuse", "short-side-to-short-side", "touching at only one point" .... ??? .... any nice way to single out one of these ??? ..... ???? ....
?? trying to get straight "classification of the equivalence classes of special decorations that all others can be reduced to" .... ???? nice way of getting exactly one representative of each symmetry class .... ???? ....... if that can be done ... nicely ... .... ???? .....
?? idea of having the special representative being giving each coset its own color ???? ..... color choice issue though .... ??? ....
(??no ??) paradox / confusion concerning .... ??? "distinctive" nature of "orientation classes of frames towards given x" vs "indistinctive" nature of "orientation classes of x's towards given frame" ??? .... ?????
?? also defining anything in terms of successor ... ??? ...
?? all possible ways of decorating triangle .... ??? "special" such ... ???? hmm, confusion ... ??? .... ?? taking seriously idea of "frame-x orientations as x's" .... ?? hmmm, yet another horrible confusion here ?? ... "active vs passive" ... hard to explain without good pictures here .... ??? ....
??? each of the three frame-point orientations has a very different look when portrayed in the "collection of all frames exhibiting that orientation to given point p" style .... ??? .... ?? to what extent does this generalize ?? .... ???? .....
(?? vs ... "collection of all points exhibiting that orientation to given frame f" style ..... ????? ..... ?? hmm, that style as unhelpful (?? ...) here because whole point is that concept of frame is here (???? .....) coming equipped with canonical (??? ....) way of drawing it ..... ??? ... os ???? .... ?? hmm, hint tha answer to "to what extent does this generalize ?" above is something like ... ??? to the general context where the base carriers are plain old (?? finite ?? ...) sets, so that concept of frame has somewhat good canonical pictorial representation ... ??? or beyond that, to general context where _you've already settled on a good way to draw a frame_, and you want to use it to get good ways of drawing other structures .... ???? ...... ??? then ... ?? general idea of getting picture in which the symmetry group acts "fixed-point-free" on some orbit, so that marking single point on that orbit gives good decoration corresponding to "frame" ... ??? ...)
3 looks : "hypotenuse-to-hypotenuse", "short-side-to-short-side", "touching at only one point" .... ??? .... any nice way to single out one of these ??? ..... ???? ....
?? trying to get straight "classification of the equivalence classes of special decorations that all others can be reduced to" .... ???? nice way of getting exactly one representative of each symmetry class .... ???? ....... if that can be done ... nicely ... .... ???? .....
?? idea of having the special representative being giving each coset its own color ???? ..... color choice issue though .... ??? ....
(??no ??) paradox / confusion concerning .... ??? "distinctive" nature of "orientation classes of frames towards given x" vs "indistinctive" nature of "orientation classes of x's towards given frame" ??? .... ?????
?? categorified multiplication and categorified comultiplication for structure types ... ???
"from structure type on pairs of sets, get structure type on sets" ... ??? multiplication .... ??? ... ?? special case where structure type on pairs of sets involves "no correlation" between the two sets .... ???? ....
"from structure type on sets, get structure type on pairs of sets" ... ??? comultiplication .... ????? "new structure on (a,b) as old structure on a+b" .... ??? ....
??? structure type t tw .... ??? nice morphism from comult(t) to ... ???? ..... "no correlation" structure type ........ ???? .....
?? clear up .... ???? .....
?? morphism from comult(t) to t#t ???? ..... ?????? ..... ?????? ......
"from structure type on pairs of sets, get structure type on sets" ... ??? multiplication .... ??? ... ?? special case where structure type on pairs of sets involves "no correlation" between the two sets .... ???? ....
"from structure type on sets, get structure type on pairs of sets" ... ??? comultiplication .... ????? "new structure on (a,b) as old structure on a+b" .... ??? ....
??? structure type t tw .... ??? nice morphism from comult(t) to ... ???? ..... "no correlation" structure type ........ ???? .....
?? clear up .... ???? .....
?? morphism from comult(t) to t#t ???? ..... ?????? ..... ?????? ......
?? categorified riemann zeta function ....
?? assign to each finite set set of semi-simple comm ring structures on it ... ?? n! such structures on n .... ???? ....
??? then would like .... ???? corresponding vsp-valued structure type ...... ????? ..... ?? so would give same decategorification .... ??? ....
?? assigining n!-dim vsp to n .... ??? so cardinality k^[n!] .... ???? ....
?? "structure type where structure on n is assignment to each semi-simple comm ring structure on n of map from its components to 3" ..... ??????
?? for example n = 4 .... ???? 12 one-component structures and 12 two-component ones .... ????? .... 3^12 * (3^2)^12 .... ?? whereas i was naively expecting 3^24 ???? ...... ???? just how screwed up are things here ???? .....
??? dirichlet-exponentiate structure type given by ??? field structure augmented by ...... ????? ..... ????? ............. ?????
spec(number ring) X spec(finite field) -> z^k
?? given alg ab gp, hom into it from spectrums of finite fields ... then dirichet-exponetiate that thing .... ??? .... ??? .....
???????
?? ....
?? assign to each finite set set of semi-simple comm ring structures on it ... ?? n! such structures on n .... ???? ....
??? then would like .... ???? corresponding vsp-valued structure type ...... ????? ..... ?? so would give same decategorification .... ??? ....
?? assigining n!-dim vsp to n .... ??? so cardinality k^[n!] .... ???? ....
?? "structure type where structure on n is assignment to each semi-simple comm ring structure on n of map from its components to 3" ..... ??????
?? for example n = 4 .... ???? 12 one-component structures and 12 two-component ones .... ????? .... 3^12 * (3^2)^12 .... ?? whereas i was naively expecting 3^24 ???? ...... ???? just how screwed up are things here ???? .....
??? dirichlet-exponentiate structure type given by ??? field structure augmented by ...... ????? ..... ????? ............. ?????
spec(number ring) X spec(finite field) -> z^k
?? given alg ab gp, hom into it from spectrums of finite fields ... then dirichet-exponetiate that thing .... ??? .... ??? .....
???????
?? ....
Sunday, April 8, 2012
?? alg ab gp given by ....
spec(z[0,1,2], but with "pointwise" multiplication of polynomials instead of usual ocnvolution) .... ????
?? have we ever thought about this multiplication of polynomials before ???? ...
?? comultiplication given by .... ????? ....
?? no, didn't get basis correct yet, i think .... ????? ....
-
10 01 with - = 11
10
00
01
00
00
10
00
01
with 10 =
10
10
and
01 =
01
01
....
?? so - stays multiplicative identity .... ???? ....
1,11,110,111,1100,1110,1101,1111 .... ???? ....
11
11
11
11
0,1,10,11,100,101,110,111
11
11
11
11
10
10
10
10
11
00
11
00
?? linear functionals on functions on subsets of n .... ????? .... ?? that only depend on the restriction of those functions to ... ?? finite subsets ???? ....
?? linear functionals on functions on finite subsets of n .... ????? .... ?? for which there's a finite size such that the functional only depends on the restriction of the functions to ... ?? subsets below that size ???? ....
spec(z[0,1,2], but with "pointwise" multiplication of polynomials instead of usual ocnvolution) .... ????
?? have we ever thought about this multiplication of polynomials before ???? ...
?? comultiplication given by .... ????? ....
?? no, didn't get basis correct yet, i think .... ????? ....
-
10 01 with - = 11
10
00
01
00
00
10
00
01
with 10 =
10
10
and
01 =
01
01
....
?? so - stays multiplicative identity .... ???? ....
1,11,110,111,1100,1110,1101,1111 .... ???? ....
11
11
11
11
0,1,10,11,100,101,110,111
11
11
11
11
10
10
10
10
11
00
11
00
?? linear functionals on functions on subsets of n .... ????? .... ?? that only depend on the restriction of those functions to ... ?? finite subsets ???? ....
?? linear functionals on functions on finite subsets of n .... ????? .... ?? for which there's a finite size such that the functional only depends on the restriction of the functions to ... ?? subsets below that size ???? ....
??mapping from spec(finite comm ring) to exponential spec(z^n)^[number ring] .... ????
spec(finite comm ring) X [number ring] -> spec(z^n) .... ??? ....
?? number of components that number ring splits into when tensored with finite comm ring .... ????? .....
?? for example number ring = gaussian integers ... ?? .... ?? and n = 3, say ... ?? ...
?? finite comm rings of size one ... ?? ....
?? zero components ??? ... ?? one map ?? ....
?? size two ?? ...
?? is it just one component ?? ... ??? because of ramification ??? ... so 3 maps ?? ....
?? size three ... ??? ...
?? one component .... ?? so 3 maps ??? .....
?? size 4 .... ??? 2 iso classes of semi-simple comm rings here .... ??? ... f_4 and f_2 X f_2 .... ???? .....
?? ...
spec(finite comm ring) X [number ring] -> spec(z^n) .... ??? ....
?? number of components that number ring splits into when tensored with finite comm ring .... ????? .....
?? for example number ring = gaussian integers ... ?? .... ?? and n = 3, say ... ?? ...
?? finite comm rings of size one ... ?? ....
?? zero components ??? ... ?? one map ?? ....
?? size two ?? ...
?? is it just one component ?? ... ??? because of ramification ??? ... so 3 maps ?? ....
?? size three ... ??? ...
?? one component .... ?? so 3 maps ??? .....
?? size 4 .... ??? 2 iso classes of semi-simple comm rings here .... ??? ... f_4 and f_2 X f_2 .... ???? .....
?? ...
Saturday, April 7, 2012
?? using lagrange extrapolation to construct alg ab gp structure on spec(xXz) for x quadratic number ring ..... .... .... ??? ...
?? exponentiableness and "flatness" ??? ... affine line ^ spec(r) .... ??? to what extent this gives good "algebraic model" of r ... ???? ..... ?? exponentiableness and geometric colimits (= algebraic limits ...) ... preservation of them by cartesian (tensor ...) product with exponent e .... ?? modification of colimits and getting more objects to be exponentiable .... ???? ...
?? power series ring .... ??? ....
?? doctrine in which concept "field" can be nicely expressed .... ???? ......
?? tag (??) theory for d-building .... ???? ....
?? spec(gaussian integers) as exponent ... ??? "for each original generator g, generators g_1 and g_i ... for each original relator r, relators r_1 and r_i ..." ...
?? bit about "underlying real variety of complex variety" .... ?????? ....
?? walking idempotent ... r = [gg=g] ... g_1=h, g_r=j ... (h+ji)(h+ji)=h+ji ...
hh-jj = h
jh+hj = j
????
?? measure vs function .... ????? .....
2hj = j
2hj-j = 0
(2h-1)j = 0 ...
?? .... 1/4 - jj = 1/2
jj - 1/4 = -1/2
jj = 1/4 - 1/2
jj = -1/4 ....
j =?= plus or minus i/2
?? h+ji = 1/2 + (i/2)i = 0 .....
?? what in the world is going on here ???? .....
?? spec(r^2) ^ spec(c) ..... ????? ?? over r ..... ????? .....
?? alg ab gp vs motive .... ???? .... ?? ab cat of alg ab gps ?? ... ???? ...
??? "imaginary numbers drag real numbers down so complex field has no real spectrum" .... ???? ....
?? some algebra/geometry contravariance confusion here, but ... ???? ...
?? mystical arrow / dominance / superscript confusion ??? ..... ....
hh-jj = h
2hj = j
(2h-1)j = 0
?? "h = 1/2 or [j=0 and consequently hh=h]" .... ????? ...
?? components of the exponential here ... ?? "real" vs ... ?? "unreal" (????) such .... ???? ...... actual morphism ... "occupied" .... ??? .... ???? ....
?? relationship to exponentiation in group action topos ?? ... ?? "preservation of exponentiation by forgetful functor" .... ???? ....
?? (1+1)^2 = 1+2+1 ..... ????? ....
?? exponentiableness and "flatness" ??? ... affine line ^ spec(r) .... ??? to what extent this gives good "algebraic model" of r ... ???? ..... ?? exponentiableness and geometric colimits (= algebraic limits ...) ... preservation of them by cartesian (tensor ...) product with exponent e .... ?? modification of colimits and getting more objects to be exponentiable .... ???? ...
?? power series ring .... ??? ....
?? doctrine in which concept "field" can be nicely expressed .... ???? ......
?? tag (??) theory for d-building .... ???? ....
?? spec(gaussian integers) as exponent ... ??? "for each original generator g, generators g_1 and g_i ... for each original relator r, relators r_1 and r_i ..." ...
?? bit about "underlying real variety of complex variety" .... ?????? ....
?? walking idempotent ... r = [gg=g] ... g_1=h, g_r=j ... (h+ji)(h+ji)=h+ji ...
hh-jj = h
jh+hj = j
????
?? measure vs function .... ????? .....
2hj = j
2hj-j = 0
(2h-1)j = 0 ...
?? .... 1/4 - jj = 1/2
jj - 1/4 = -1/2
jj = 1/4 - 1/2
jj = -1/4 ....
j =?= plus or minus i/2
?? h+ji = 1/2 + (i/2)i = 0 .....
?? what in the world is going on here ???? .....
?? spec(r^2) ^ spec(c) ..... ????? ?? over r ..... ????? .....
?? alg ab gp vs motive .... ???? .... ?? ab cat of alg ab gps ?? ... ???? ...
??? "imaginary numbers drag real numbers down so complex field has no real spectrum" .... ???? ....
?? some algebra/geometry contravariance confusion here, but ... ???? ...
?? mystical arrow / dominance / superscript confusion ??? ..... ....
hh-jj = h
2hj = j
(2h-1)j = 0
?? "h = 1/2 or [j=0 and consequently hh=h]" .... ????? ...
?? components of the exponential here ... ?? "real" vs ... ?? "unreal" (????) such .... ???? ...... actual morphism ... "occupied" .... ??? .... ???? ....
?? relationship to exponentiation in group action topos ?? ... ?? "preservation of exponentiation by forgetful functor" .... ???? ....
?? (1+1)^2 = 1+2+1 ..... ????? ....
Friday, April 6, 2012
?? gaussian integers .... homming into finite semi-simple comm ring .... ?? ....
0
1 1 - 1 1
2 f_2 1 1 1
3 0
4 f_2 X f_2 1,1 2 f_4 1 2 1
5 f_5 2 1 3 1 2
6 0
7 0
8 f_2 X f_2 X f_2 1,1,1 6 f_2 X f_4 1,1 2 f_8 1 3 1
??? counting points of spectrum of alg comm ring over finite fields .... ??? ...
_comm ring_ -> _comm ring_ .... ???? ... "tensoring with gaussian integers" ....
?? confusion ?? ... ?? level slip ??? .... ??? pretty big algebraic comm ring ... ??? polynomials in 2 variables .... inf-dim ... ??? vs pretty small hopf algs .... ???? .... ??? .... fin-dim .... ???? ...
?? 4th roots of 1 mod square roots of 1 ??? ..... ???? ....
?? 3rd roots of 1 ........ ?????? .....
0
1 1 - 1 1
2 f_2 1 1 1
3 0
4 f_2 X f_2 1,1 2 f_4 1 2 1
5 f_5 2 1 3 1 2
6 0
7 0
8 f_2 X f_2 X f_2 1,1,1 6 f_2 X f_4 1,1 2 f_8 1 3 1
??? counting points of spectrum of alg comm ring over finite fields .... ??? ...
_comm ring_ -> _comm ring_ .... ???? ... "tensoring with gaussian integers" ....
?? confusion ?? ... ?? level slip ??? .... ??? pretty big algebraic comm ring ... ??? polynomials in 2 variables .... inf-dim ... ??? vs pretty small hopf algs .... ???? .... ??? .... fin-dim .... ???? ...
?? 4th roots of 1 mod square roots of 1 ??? ..... ???? ....
?? 3rd roots of 1 ........ ?????? .....
?? reverse engineer hopf alg from galois rep ?? ... ???? ..... ???? ..... ?? generalize to case including context where "galois gp" is more naively (?? ...) "geometric" .... ??? ....
?? the (???) galois action on x (alg gp, or alg ring, or ... ?? ... ??? alg something ?? ....) depends only on the underlying scheme (?? ...) of x .... ??? ...
?? so for example, the (??? ...) galois action on "alg gaussian integers" as trivial .... ???? .... ??? ...
??? some other galois group / action here ??? .... ??? auts of "alg alg integers" ??? .... ??? ... ??? .... ??? ....
?? the (???) galois action on x (alg gp, or alg ring, or ... ?? ... ??? alg something ?? ....) depends only on the underlying scheme (?? ...) of x .... ??? ...
?? so for example, the (??? ...) galois action on "alg gaussian integers" as trivial .... ???? .... ??? ...
??? some other galois group / action here ??? .... ??? auts of "alg alg integers" ??? .... ??? ... ??? .... ??? ....
Thursday, April 5, 2012
?? role of flatness (?? ...) of k in .... ???? getting ... ??? i was going to distinguish between .... getting "underlying set of tensor product with k" to be representable, vs getting it to lift to _comm ring_ .... ?? but the latter seems completely tautological, right ??? ... ???? ....
?? consider ... ??? many different products here .... categorified and decategorified .... dircichlet series ... comm ring ... alg comm ring ... bistable hopf alg ... .... ??? .... .... cartesian, "tensor", "direct sum", ..... ???? .... correspondences / relationship among such ... ???? ....
??? confusion here (?? ...) ... ?? comm ring vs alg comm ring ... ??? ... ??? galois action .... ???? ....
?? consider ... ??? many different products here .... categorified and decategorified .... dircichlet series ... comm ring ... alg comm ring ... bistable hopf alg ... .... ??? .... .... cartesian, "tensor", "direct sum", ..... ???? .... correspondences / relationship among such ... ???? ....
??? confusion here (?? ...) ... ?? comm ring vs alg comm ring ... ??? ... ??? galois action .... ???? ....
?? so _does_ "direct sum" of bistable hopf algs correspond to multiplication of associated categorified zeta functions ??? .... ???? .... ?? work it out ...
?? even if it does, hard to see how could account for characteristically (...) weird coefficients of l-fn ... ??? unless ... ?? decategorification process used is somehow weirder than ... what it seems to be in the zeta function case .... ??? .... ??? .....
?? underlying additive algebraic (abelian) group of algebraic comm ring .... "direct sum" decomposition of that algebraic abelian group .... ??? ...
?? "algebraic direct sum" .... ???? tensor product .... ???? ......
?? hopf alg and fourier duality .... ???? .....
?? categorified zeta function with f_q-vsp coefficients .... ??? relating in some way to "green type" .... ???? .... ???? ..... ??? vague memories about thinking about stuff like ... ?? to what extent idea of substituting something (????) into dirichlet series makes (categorified ... ??? ...) sense .... ???? ....
?? even if it does, hard to see how could account for characteristically (...) weird coefficients of l-fn ... ??? unless ... ?? decategorification process used is somehow weirder than ... what it seems to be in the zeta function case .... ??? .... ??? .....
?? underlying additive algebraic (abelian) group of algebraic comm ring .... "direct sum" decomposition of that algebraic abelian group .... ??? ...
?? "algebraic direct sum" .... ???? tensor product .... ???? ......
?? hopf alg and fourier duality .... ???? .....
?? categorified zeta function with f_q-vsp coefficients .... ??? relating in some way to "green type" .... ???? .... ???? ..... ??? vague memories about thinking about stuff like ... ?? to what extent idea of substituting something (????) into dirichlet series makes (categorified ... ??? ...) sense .... ???? ....
?? ag morphism from theory of z/2-torsor to theory of line object ... ??? .... ?? pull z/2-graded vsp back along z -> z/2 ?? .... ??? ok to be sloppy about fourier dual of z/2 here because pretty galois-stable ??? ... ??? ... ??? ....
?? wait, does this make any sense ?? .... would be surprising if it did, since i think that it was based on a misconception .... ?? and seems like should give paradox .... ??? ... ??
?? go other way by pushing forward along z -> z/2 .... ???? ....
?? wait, does this make any sense ?? .... would be surprising if it did, since i think that it was based on a misconception .... ?? and seems like should give paradox .... ??? ... ??
?? go other way by pushing forward along z -> z/2 .... ???? ....
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