CAG@LOCS#32
I know a couple of decent sources for independent coffee shops online, though i had to contribute most or all of what Blacksburg has to offer (also promoted on my webpage). I have also found a couple of similar directories of vegetarian restaurants and sources/users of local food. (For pretty much all of this, see "Todo el Mundo" on my webpage.) I also have some links to various classifications of "ethical" clothing, plus the National Green Pages and Advomatic.
While these efforts are commendable and the pages very helpful and full of potential (not a large enough proportion of businesses being counted yet), they paradoxically are too disparate from each other due to a common "flaw": They are highly specialized. For instance, Justice Clothing focuses on unionized (non-sweatshop) clothing manufacture but does not have a way to restrict browsing to, say, vegan or undyed items, while HappyCow distinguishes vegan, vegetarian, and either-friendly restaurants but makes no note of how much locally-grown or -raised produce they use. And i can't even find a site that tells me what locally-owned and/or artsy movie theaters exist where.
What i want is an "ethical shopping" directory of pretty much any and all businesses, subject to sortability and restriction by any of a variety of factors. For instance, i might invoke the universal qualifier "vegan", which would highlight restaurants, beauty salons, movie theaters, etc. that use no animal products and omit any that offered no such products at all. I could then invoke "non-chain" with respect to all but the movie theaters, and "local raw materials" only with respect to restaurants. Finally, i could invoke the spectral qualifiers "independent/Democratic" with respect to each business's political party donations (if they make any) and "worker-owned/co-op" with respect to business structure. Once my settings are established, i would be able to browse the business directory leisurely, and search for types of businesses in the usual ways, without concerning myself over what places might appeal to my ethics. (All do to some extent, and the heavier highlights indicate those that do so more.)
The major point is that conscientious shoppers differ on what qualities make a business worth their patronage. A lot of people would fill in just one or two criteria, with "don't care" being the default value for the others. Of course it would make a lot more sense for someone new to a place to simply ask locals, but this requires guts, a feature somewhat lacking in many individuals, myself included. (I sometimes break through, but too often the people i ask have no idea about the factors i'm interested in.) For this kind of directory to succeed, like such others as CafeHunt and Local Harvest, is merely one devotee in any given city or sizable town, with scattered extras contributing further variety and filling in obscure regions.
Something i didn't mention is the helpfulness of spectral criteria, i.e. ratings instead of "check-marks", which would differentiate between cursorily "good" businesses and honestly conscientious ones in popular areas, and permit marginally (but not sufficiently for a "check") "good" ones in areas where that's the best one can expect to find. I have yet to see such a thing invoked by any such online directory with respect to anything other than quality of taste, appearance, speed, and other such superficial qualities.
On to math.
Um, right. Three Macaulay 2 examples are still running on the hobbes server, i need to talk with Mark about my stepwise ideal initialization method, and i still have far insufficient data by which to conjecture anything on the punchcards that these primary decompositions generate. The program is verifiably working, but since it's not in C i apparently can't request System X time. Anyone out there have (access to) a badass processor they wouldn't mind shoving some of my code through?
Although, this seems like a good spot (approximate point in spacetime) to respond to a recent back-and-forth between myself and gaspaheangea about Schubert varieties. While Baez is a league or two beyond my exposure, he gives a plain and effective exposition of finite Schubert cells/varieties that begs no elucidation (especially from me). Affine Schubert varieties are somewhat less tangible beasts, and in addition to the meaningful definition there is a more concrete description that has come in handy in my calculations.
Decomposing a G/P. Loosely (meaning "as i faultfully recall"), whenever G is a nice (Kac-Moody) linear group and P is a maximal parabolic subgroup (corresponding to the removal of one node in the Dynkin diagram of G). The quotient G/P can be decomposed into Schubert cells indexed by elements of the Weyl group of G modulo that of P. In my case, G is SLn(C((t))), the group of invertible n×n matrices with formal Laurent series as entries and determinant 1. The appropriate P is then SLn(C[[t]]), the subgroup in which only nonnegative powers of t appear in the entries. (Note that matrices of P must have nonzero constant terms along the diagonal.) Their respective Weyl groups are the affine symmetric group, an infinite Coxeter group generated by the adjacent transpositions s0,...,sn-1 under the usual braid relations (with indices modulo n), and the finite symmetric group Sn obtained by omitting the generator s0. Each Schubert cell in the decomposition of G/P (the affine Grassmannian) consists of those matrices (really matrix orbits over P) that row-reduce (using just C-coefficients) to a particular form; and these forms are indexed by cosets in the quotient of the affine symmetric group by the finite. For instance, one such form is diag(t,1,...,1,t-1) + UT (strictly upper-triangular nonsense, i.e. a big asterisk in the upper-right triangle), and it corresponds to the coset s0. Accordingly, there is a faithful representation of the affine symmetric group inside SLn(C((t))) within which right multiplication (action) by non-s0 generators does not affect this row-reduced form.
Stabilizing the infinite Grassmannian. Refer to Week 188 for its matrix images of Schubert cells; i refer as "pivots" to the indices at which "jumps" in dimension occur, i.e. 1's appear in a column. Loosely, the infinite Grassmannian is just the direct limit of all the finite Grassmannians under a collection of embeddings that preserve the "starting index". For instance, Gr2,4 embeds into Gr5,8 as the union of Schubert cells of the latter with first pivot at or after column 2, second pivot at or before column 5, and remaining pivots at columns 6, 7, and 8. (Draw the matrices.) Hope that's illustrative. Anyway, we can talk about a certain subspace of the infinite Grassmannian (consisting of subspaces of CN generated, analogously to subspaces in a finite Gr, by the rows of Z+×Z "matrices" with no pivots sufficiently far to the left and no nonpivotal columns sufficiently far to the right) preserved by the unipotent operation 1+t, where t is the nilpotent map sending a subspace to its "shift to the right by n columns". This operation actually matches the action of t in the previous definition under a specific embedding of the affine Grassmannian in the infinite, whose image is precisely this u-preserved "subGrassmannian" (actually a subvariety). It is via this embedding that i came across a beautiful normal form into which all the matrices we care about can be row-reduced, thereby identifying affine Schubert cells as "cells" (homeomorphic to Ck for some k).
Damn, but some examples would be nice. I'm just paranoid to post them. And in any event, my laptop has almost exhausted its battery, so i'll wrap up here.
While these efforts are commendable and the pages very helpful and full of potential (not a large enough proportion of businesses being counted yet), they paradoxically are too disparate from each other due to a common "flaw": They are highly specialized. For instance, Justice Clothing focuses on unionized (non-sweatshop) clothing manufacture but does not have a way to restrict browsing to, say, vegan or undyed items, while HappyCow distinguishes vegan, vegetarian, and either-friendly restaurants but makes no note of how much locally-grown or -raised produce they use. And i can't even find a site that tells me what locally-owned and/or artsy movie theaters exist where.
What i want is an "ethical shopping" directory of pretty much any and all businesses, subject to sortability and restriction by any of a variety of factors. For instance, i might invoke the universal qualifier "vegan", which would highlight restaurants, beauty salons, movie theaters, etc. that use no animal products and omit any that offered no such products at all. I could then invoke "non-chain" with respect to all but the movie theaters, and "local raw materials" only with respect to restaurants. Finally, i could invoke the spectral qualifiers "independent/Democratic" with respect to each business's political party donations (if they make any) and "worker-owned/co-op" with respect to business structure. Once my settings are established, i would be able to browse the business directory leisurely, and search for types of businesses in the usual ways, without concerning myself over what places might appeal to my ethics. (All do to some extent, and the heavier highlights indicate those that do so more.)
The major point is that conscientious shoppers differ on what qualities make a business worth their patronage. A lot of people would fill in just one or two criteria, with "don't care" being the default value for the others. Of course it would make a lot more sense for someone new to a place to simply ask locals, but this requires guts, a feature somewhat lacking in many individuals, myself included. (I sometimes break through, but too often the people i ask have no idea about the factors i'm interested in.) For this kind of directory to succeed, like such others as CafeHunt and Local Harvest, is merely one devotee in any given city or sizable town, with scattered extras contributing further variety and filling in obscure regions.
Something i didn't mention is the helpfulness of spectral criteria, i.e. ratings instead of "check-marks", which would differentiate between cursorily "good" businesses and honestly conscientious ones in popular areas, and permit marginally (but not sufficiently for a "check") "good" ones in areas where that's the best one can expect to find. I have yet to see such a thing invoked by any such online directory with respect to anything other than quality of taste, appearance, speed, and other such superficial qualities.
On to math.
Um, right. Three Macaulay 2 examples are still running on the hobbes server, i need to talk with Mark about my stepwise ideal initialization method, and i still have far insufficient data by which to conjecture anything on the punchcards that these primary decompositions generate. The program is verifiably working, but since it's not in C i apparently can't request System X time. Anyone out there have (access to) a badass processor they wouldn't mind shoving some of my code through?
Although, this seems like a good spot (approximate point in spacetime) to respond to a recent back-and-forth between myself and gaspaheangea about Schubert varieties. While Baez is a league or two beyond my exposure, he gives a plain and effective exposition of finite Schubert cells/varieties that begs no elucidation (especially from me). Affine Schubert varieties are somewhat less tangible beasts, and in addition to the meaningful definition there is a more concrete description that has come in handy in my calculations.
Decomposing a G/P. Loosely (meaning "as i faultfully recall"), whenever G is a nice (Kac-Moody) linear group and P is a maximal parabolic subgroup (corresponding to the removal of one node in the Dynkin diagram of G). The quotient G/P can be decomposed into Schubert cells indexed by elements of the Weyl group of G modulo that of P. In my case, G is SLn(C((t))), the group of invertible n×n matrices with formal Laurent series as entries and determinant 1. The appropriate P is then SLn(C[[t]]), the subgroup in which only nonnegative powers of t appear in the entries. (Note that matrices of P must have nonzero constant terms along the diagonal.) Their respective Weyl groups are the affine symmetric group, an infinite Coxeter group generated by the adjacent transpositions s0,...,sn-1 under the usual braid relations (with indices modulo n), and the finite symmetric group Sn obtained by omitting the generator s0. Each Schubert cell in the decomposition of G/P (the affine Grassmannian) consists of those matrices (really matrix orbits over P) that row-reduce (using just C-coefficients) to a particular form; and these forms are indexed by cosets in the quotient of the affine symmetric group by the finite. For instance, one such form is diag(t,1,...,1,t-1) + UT (strictly upper-triangular nonsense, i.e. a big asterisk in the upper-right triangle), and it corresponds to the coset s0. Accordingly, there is a faithful representation of the affine symmetric group inside SLn(C((t))) within which right multiplication (action) by non-s0 generators does not affect this row-reduced form.
Stabilizing the infinite Grassmannian. Refer to Week 188 for its matrix images of Schubert cells; i refer as "pivots" to the indices at which "jumps" in dimension occur, i.e. 1's appear in a column. Loosely, the infinite Grassmannian is just the direct limit of all the finite Grassmannians under a collection of embeddings that preserve the "starting index". For instance, Gr2,4 embeds into Gr5,8 as the union of Schubert cells of the latter with first pivot at or after column 2, second pivot at or before column 5, and remaining pivots at columns 6, 7, and 8. (Draw the matrices.) Hope that's illustrative. Anyway, we can talk about a certain subspace of the infinite Grassmannian (consisting of subspaces of CN generated, analogously to subspaces in a finite Gr, by the rows of Z+×Z "matrices" with no pivots sufficiently far to the left and no nonpivotal columns sufficiently far to the right) preserved by the unipotent operation 1+t, where t is the nilpotent map sending a subspace to its "shift to the right by n columns". This operation actually matches the action of t in the previous definition under a specific embedding of the affine Grassmannian in the infinite, whose image is precisely this u-preserved "subGrassmannian" (actually a subvariety). It is via this embedding that i came across a beautiful normal form into which all the matrices we care about can be row-reduced, thereby identifying affine Schubert cells as "cells" (homeomorphic to Ck for some k).
Damn, but some examples would be nice. I'm just paranoid to post them. And in any event, my laptop has almost exhausted its battery, so i'll wrap up here.