Git versioning (why and how) for Mathematicians

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]]>We will look mostly at finite dimensional (not neccesarily associative) division algebras over the real numbers.

Instead of these rather loose notes, you may prefer to look at my texed notes in this pdf file.

There are several classes of algebras over a field which we will look at. By "R-algebra" we mean a not necessarily commutative, not necessarily associative, not necessarily unital ring which at the same time is a module over the ring R, so that the ring multiplication is R-linear. We focus on fields R=k, so the word "module" becomes "vector space".

A *division algebra* over a field is a -algebra which allows division, i.e. for any you have unique elements and , unless . The notation means that is an element such that .

An *associative algebra* over a field is a -algebra in which the multiplication is associative (what else?). An *associative division algebra* can be defined to be an associative algebra with a unit element distinct from such that every element has a unique inverse (right- as well as left-inverse).

An example for an associative algebra which is not a division algebra: the split complex numbers with multiplication and consequently . This algebra has zero divisors: .

If the field is finite, the finite dimensional division algebras over it are obviously also finite. In the other direction, if a division algebra over a field is finite, Wedderburn's little theorem shows that and must be finite fields.

The *Cayley-Dickson construction* is a process to take an associative division algebra and produce a new algebra of twice the dimension. It actually works like the construction that you use to produce the complex numbers from the real numbers: define a multiplication on the direct sum which is again a division algebra. Starting from , the Cayley-Dickson construction produces . Starting from , we get the quaternions and from that the octonions . The next step is no longer a division algebra (the sedenions, of dimension 16).

If we look at the field of real numbers , Frobenius' theorem states that the only finite dimensional associative division algebras over , up to isomorphism, are .

Over any algebraically closed field such as , there are no finite dimensional associative division algebras except itself.

There are infinite dimensional associative algebras: take a Banach space and the linear endomorphisms under composition , then that is a unital associative -algebra (with complete norm, thus a Banach algebra). The only Banach division algebras are , by a theorem of I.Gelfand and S.Mazur.

There are infinite dimensional associative divison algebras, but they don't admit a norm (otherwise they would embed into their completion, which by the previous theorem is finite dimensional). An example was found by Hilbert: take formal Laurent series in one variable over the field and define a new product by extending the rule to all Laurent series. The result is not a field, but an associative division algebra over .

A *central simple algebra* over a field is a finite dimensional associative algebra which is simple and has as center only . Every simple algebra is a central simple algebra over its center. One should probably say something about Brauer groups here, but leaving the name "Brauer" here should suffice to point anyone in the right direction.

A *composition algebra* is a unital algebra equiped with a nondegenerate quadratic form (often called the norm) that allows composition, i.e. .

One can show that composition algebras can only occur in dimensions , by an analogue of the Cayley-Dickson procedure, which takes a subalgebra and returns a subalgebra of twice the dimension. One can show that a proper subalgebra is always associative; The doubling procedure, started from the one-dimensional subalgebra generated by the neutral element, gives at some point a non-associative 8-dimensional subalgebra, which must be the whole thing (if the doubling didn't end earlier on).

Over the real numbers, not only appear, but also things like the split complex numbers, split quaternions, etc. In a composition algebra, an element has an inverse if and only if the norm is nonzero.

Now if one wants to classify the finite dimensional division algebras over without the extra structure of a norm or quadratic form or whatever - you have to do algebraic topology. The solution comes in the form of Adams' solution of the Hopf Invariant One problem, which shows that only the classical Hopf fibrations for have Hopf invariant one, and therefore only the euclidean norm-1-spheres in these -vector spaces admit a multiplication (even up to homotopy). Since any division algebra induces such a multiplication, we know there can't be any other. If you want, a reason behind this is seen in Atiyah's proof, which exploits the 8-fold Bott periodicity, which in turn comes from a mod-8 periodicity in the classification of Clifford algebras (so in fact a very algebraic reason).

The reason I'm interested in brushing up on division algebras? Exceptional groups. In the classification of semisimple Lie algebras (or compact Lie groups, or reductive linear algebraic groups), the exceptional cases (other than the "classical" groups) are rather hard to come by. Unless you have the quaternions and octonions. The magic is best seen in Freudenthal-Tits magic squares, but that can be read over at Wikipedia. Enjoy your algebras.

]]>Continue 2,4,6,8,...

To which the solution is often supposed to be 10,12,14, and so on.

**The problem, as any mathematician knows**, is that there is not the one solution. In fact, given any set of finite numbers one could just talk about the sequence that starts like that and continues with zeroes only, that's a perfectly valid sequence. Of course, one should really figure out the rule behind the finitely many numbers, but there are always many possible choices. The game is not to find any sensible rule, but to find the rule the designer had in mind. It's more about testing your knowledge of culture than an honest test of mathematical ability (or anything else).

**But there is a way to fix this**.

If you look at the sequence above, you could continue with 12,14,16,18,22, and so on - the sequence of all even numbers which are not divisible by 10. This is a valid continuation, of course, and it even follows a sensible rule, just like the sequence of all even numbers. But there is a difference: one description is longer.

While you can always take a finite sequence of numbers and any infinite integer sequence, and craft the rule "the first few numbers are given by the finite sequence, then follows this infinite sequence", this will always be longer than the rule for the infinite sequence alone.

The technical concept that captures the length of descriptions appropriately is *Kolmogorov complexity*, relative to a description language. A description language can be formalized as a Turing machine M that takes descriptions D of integer sequences in some syntax (a list of letters and whitespace, for example) together with a natural number N, and spits out the integer $M(D,N) = a_N$ in the sequence $(a_n)_{n \in \mathbb{N}}$ which is described by D.

One may describe the computation of an integer sequence more or less efficiently, so it makes sense to define the (Kolmogorov) complexity of a sequence relative to a description language as the minimum length of a description in that language. Note that length of words is a natural number, hence the infimum length is always attained and the complexity is a natural number, too.

If we take some description language and bake a new one by requiring everything to start with the symbols "###" and then the same stuff as in the old language, then we have always longer descriptions in the new language. That's why we have to take the complexity relative to a description language. There may be other unexpected behaviour: if you have a single symbol just for the sequence of all even numbers, the description gets very short for this particular sequence (not necessarily for the others). One can compare description languages by looking at the length of the programs that compute the sequences from descriptions. Surely, in the program that interprets the single symbol for the sequence of all even numbers, the computation description for that sequence is hidden. Putting such comparison factors into the Kolmogorov complexity gives a more robust measure of the descriptive complexity. We don't need to go into more details here, as you might as well take as description language "Fortran programs" or "Haskell programs" and be happy with it.

**The proposed fix** to the problem of non-unique solutions to sequences consists now of modifying the question in two steps:

1) Given a finite sequence of integers, what is the shortest description of any rule that produces this sequence (and continues infinitely)? In other words: what is a description whose length attains the Kolmogorov complexity?

2) Since there might be several distinct rules with the same description length, the real question is: What is the minimal Kolmogorov complexity of an infinite sequence that starts with the prescribed values?

The reason why this "fixes" the problem, is the following: suppose someone has a rule for an infinite integer sequence and writes down a few of the first numbers. If you can come up with a rule that starts like that, but has a shorter description, you can say "oh, but if you wanted that more complicated sequence, you should have given more numbers to make it clear!". In some sense, this would mean that the problem was underspecified.

At least this is a much better complaint than just saying "I could continue any finite sequence however I like". And it's a very very hard problem, to find the Kolmogorov complexity explicitly. Even lower bounds are usually impossible. The complexity itself is not a computable number, so it is not possible (in general) to write a computer program to continue a finite sequence as I proposed. It's that hard.

A technical note at the end: Ordinarily, Kolmogorov complexity is defined for finite strings, which we apply to some encoding of a Turing machine that computes an infinite sequence. This excludes any non-computable infinite sequence, but that's sensible for the problem at hand.

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Let be sets, and denote by the set of set-theoretic maps . Then

This is sometimes called Currying or Schönfinkeln, and it really boils down to

An adjunction is a pair of functors , between categories together with a natural transformation of functors

In the baby case above, and is adjoint to .

We call the left adjoint to and the right adjoint to . The notation ⊣ can be quite useful.

You should care about adjunctions, because their existence implies nice properties: is always cocontinuous (preserves colimits) and is always continuous (preserves limits). This comes from the continuity properties of . There is also a theorem (Freyd's AFT) that describes precisely which extra condition a functor has to satisfy, besides continuity, to be a right adjoint.

Another example comes from vector spaces:

where we give the vector space structure via .

In fact, this is basically the definition of the tensor product, if you watch closely.

In homotopy theory, one often uses the adjunction

between suspension and loop space (where the brackets denote the morphisms in the homotopy category). From the definitions, is a quotient of and is a subspace of , and the source of this fundamental adjunction is the baby case above.

To a lot of "forgetful" functors, like from rings with unit to rings not-necessarily-with-unit, from abelian groups to arbitrary groups, from abelian groups to abelian monoids, from compact Hausdorff spaces to arbitrary topological spaces, one can construct left adjoints which give some sort of "free" objects. For example, to any topological space one can assign the Stone-Cech compactification, and to every abelian monoid one can assign the Grothendieck group.

For categories we can form the functor category of functors and consider the diagonal functor which maps every object to the constant functor. If it has a right adjoint, then we call it Limit () and write

If it has a left adjoint, then we call it Colimit () and write

Such a (co)limit functor exists iff all (co)limits exist in the ordinary sense.

There would be much more to write (e.g. about nerves and realizations) but I'll stop here to prevent this from rotting in the draft section.

Except, I couldn't resist [UPDATE 2014-06-12]:

Given a category we can look at all chains of arrows of length , i.e. such that one has a composition . Such a chain of length can be mapped to a chain of length by merging two adjacent arrows into and it can be mapped to a chain of length by inserting identities. All this structure together forms a simplicial set called the nerve, whose -simplices are precisely the chains of arrows of length . This construction if functorial, and it admits a left adjoint that associates to a simplicial set its "homotopy category":

Write for the homotopy category of a category with weak equivalences, then we have a derived version of the lim and colim adjunctions:

Specializing this to the diagrams being (for holim) and (for hocolim) we are talking about homotopy pullbacks and homotopy pushouts. Now plugging in a terminal object for the outer things, and an arbitrary object in the middle, we get diagrams in like , whose holim is called and whose hocolim is called .

At this point you should probably try to see that the ordinary suspension and loop space fulfill these universal properties in the category of topological spaces.

Anyway, writing down explicitly the holim and hocolim adjunctions for these particular diagrams, you see the adjunction immediately. I guess this is called "synthetic homotopy theory" and it's beautiful, isn't it?

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One of the most basic notions of mathematics are sets with subsets and powersets. From these, one can define a binary relation between sets and to be a subset of and then a map from to to be a binary relation between and which is left-total (each element of is related to at least one element of ) and right-unique (aka "functional", each element of is related to at most one element of ).

A binary operation by on is a map . A binary operation of on itself is called an inner binary operation on . A **magma** is a set with an inner binary operation .

Examples of magmas: the natural numbers with addition (but not the natural numbers with subtraction) and the set of words over an alphabet, with concatenation. The former is a special case of the latter if we encode numbers unary, i.e. is the empty word, is just ( is ) and every other number is a word of length consisting only of s. The alphabet is and is essentialy the same as .

Concatenation can be done in more than one dimension, i.e. we can consider not only words over an alphabet (such that ) with concatenation ("horizontal composition"), but 2-dimensional words composed of letters , such that each with fixed and varying is an ordinary word, and each with fixed and varying is a word written vertically. Such a 2-dimensional word has a "rectangular shape". The set of all 2-dimensional words over an alphabet has two magma structures (horizontal composition and vertical composition) and it can be interesting how they interact.

A classical example of a set with two magma structures is the set of loops in a pointed topological magma (continuous paths from the basepoint to the basepoint), with one operation given by the pointwise operation of the topological magma and the other given by concatenation of loops.

Other magmas: the natural numbers with multiplication, the natural numbers with exponentiation , the complex numbers with multiplication, the octonions with multiplication, the sedenions with multiplication. If you haven't seen the octonions or sedenions before, look up the Cayley–Dickson construction in Wikipedia.

A magma homomorphism is a map such that . The unary encoding of natural numbers discussed above is a magma isomorphism .

A magma operation of a magma on a set is a binary operation of on that is compatible with the magma operation , i.e. .

There are some elementary properties one can study for any magma one encounters:

- Associativity: Is ?
- Neutral Elements: ?
- Units: ?
- The dual magma with .
- Commutativity: Is ?
- Solvability of Equations: Does there exist a unique inner binary operation such that ?

A magma with the property that equations of the form and are uniquely solvable is called a quasigroup. A **loop** is a quasigroup that posesses a neutral element.

An associative magma is sometimes called a semigroup, although that word is also used for **monoid**s, which are associative magmas that posess a neutral element, i.e. an element such that and are both equal to the identity map.

**Groups** can be defined as monoids in which every element is a unit, i.e. every has an inverse element which satisfies and . Groups can also be defined as associative loops.

A Loop is a quasigroup with a neutral element, but it needn't be associative or commutative. In a loop, every element has a left-inverse and a right-inverse, since and are uniquely solvable, but unlike a group, these one-sided inverses don't coincide in general.

In a loop , the left-associator of three elements , defined by , measures how non-associative the loop is (together with the right-associator).

A quasigroup has the Moufang property (or satisfies the Moufang identity) if , which is a weak form of associativity. The Moufang property implies that there exists a neutral element (so the quasigroup is a loop, after all): since is uniquely solvable and is uniquely solvable, we have , for all , so is left-neutral. Now take defined by , which can be shown by the same method to be right-neutral, so we conclude with .

A loop that enjoys the Moufang property is called a **Moufang loop**. In a Moufang loop, left-inverses coincide with right-inverses. In particular, the left-associator and the right-associator coincide (but the associator is still non-trivial in general).

The nonzero quaternions with quaternion multiplication form a non-commutative associative loop. The nonzero octonions form a non-associative Moufang loop. In general, the set of invertible elements of an alternative algebra forms a Moufang loop. The integers with subtraction form a loop with neutral element that does not have the Moufang property.

Moufang's theorem states that the subloop generated by any two elements of a Moufang loop is an associative loop, i.e. a group. This theorem makes working with loops a lot more convenient.

A monoid is an associative magma with a neutral element, but it needn't be commutative or have any inverses. A monoid homomorphism is a magma homomorphism that maps neutral elements to neutral elements.

An archetypical monoid is the set of endomorphisms of some thing, since composition of morphisms is required to be associative and there is the identity morphism. In other words, any monoid can be seen as a category with one object and and vice versa. Observe that a group in this formalism is a category with one object such that all morphisms are invertible.

This points to an obvious generalization of monoids: **categories**. The corresponding generalization of groups is called **groupoid**.

To every commutative monoid one can associate an abelian group such that the monoid homomorphisms into any other abelian group are in bijection with the group homomorphisms . This can be done by the construction familiar from building integers out of natural numbers and is called the Grothendieck group. Other prominent examples of this construction lie in the fundamentals of K-Theory.

The Eckmann-Hilton theorem states for a set with two inner binary operations that have a neutral element and distribute over each other, i.e. , that the two operations actually coincide, their neutral elements coincide and that this operation is associative and commutative.

The classical application of the theorem is the case of the monoid of homotopy classes of loops of an H-space, which shows that, up to homotopy, composition of loops is the same as multiplication of loops. As a corollary, all higher homotopy groups of a topological space , for , are abelian.

It is very nice to work out the Eckmann-Hilton argument as an exercise given a few hints: visualize the two operations as horizontal and vertical composition and look at expressions of the form .

]]>- Is the mathematical definition of Kullback-Leibler distance the key to understand different kinds of information?
- Can we talk about the total information content of the universe?
- Is hypercomputation possible?
- Can we tell for a physical system whether it is a Turing machine?
- Given the fact that every system is continually measured, is the concept of a closed quantum system (with unitary time evolution) relevant for real physics?
- Can we create or measure truly random numbers in nature, and how would we recognize that?
- Would it make sense to adapt the notion of real numbers to a limited (but not fixed) amount of memory?
- Can causality be deﬁned without reference to time?
- Should we re-deﬁne “life”, using information-theoretic terms?

What do you think?

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I wrote about motivic cell structures and the Białynicki-Birula decomposition before. Here I'll explain how to compute a motivic cell structure out of the BB-decomposition *explicitly* and how to get an *explicit* BB-decomposition for any smooth complete toric variety. Then I'll do the examples.

As described in the last post about Białynicki-Birula's *Annals* paper from 1972/73, for a -action on a smooth complete variety over a (possibly non-closed) field (of arbitrary characteristics), there is the plus-decomposition, giving us for each fixed point a subscheme which is isomorphic to an affine space . The number is the dimension of the positive weight part (with respect to the induced -action) of the (co)tangent space of at .

There is another paper of Białynicki-Birula, published in 1976 in the *Bulletin de l'académie polonaise des sciences, Série des sciences math.*, where it is proved that the BB-decomposition of smooth *projective* varieties is *filtrable*, which means that one can choose an order on the fixed points and a partition of into blocks (where is the dimension of ) such that for the union of cells of each block, the are closed subschemes of that form a finite decreasing sequence

Here every is a proper inclusion. The proof uses Sumihiro's equivariant completion, which provides an equivariant closed immersion into a projective space with linear torus action, so there one can filter the projective space by the weights of this action and that provides the closed immersions. If one wants to compute something, one can of course figure out a good order of the fixed points by hand, without looking at equivariant embeddings at all. At the moment, I don't know any algorithm other than brute force to do that.

Now we build a motivic cell structure out of this. This is slightly unusual, as the attaching maps arise "the other way around" than one would expect. If it confuses you, the examples below might provide illumination.

The open subscheme has complement isomorphic to a disjoint union of some .

Look at the closed immersion of smooth schemes and its normal bundle . By the homotopy purity theorem of Morel and Voevodsky we have

If we want to handle more cells at once, we just define by

to be a homotopy cofiber sequence.

One more homotopy cofiber construction gives us , an attaching map of into that gives us, as next homotopy cofiber, the space . So this produces inductively a stable motivic cell structure on , since for a cofiber sequence with two stably cellular spaces, a theorem of Dugger and Isaksen shows that the third space is also stably cellular.

That toric varieties have a motivic cell structure (without referring to the BB-decomposition) is already contained in the paper of Dugger and Isaksen *Motivic Cell Structures*.

From a homotopy cofiber sequence we get a distinguished triangle in the category of motives of the reduced motives . Like with reduced and unreduces cohomology theories, , so we can compute motives of varieties from homotopy cofiber sequences.

It is long known for toric varieties, that any torus cocharacter in general position (in the cocharacter lattice) has the same fixed points as the torus. To describe explicit cell decompositions, one needs to know which cocharacter is "in general position" and which cocharacter has a larger fixed point set. A cocharacter fixes an orbit if is inside the linear subspace generated by . The full torus has as fixed points those with of maximal dimension. To pick a good cocharacter, one just has to avoid the hyperplanes spanned by the codimension 1 cones in the fan.

The fan of the projective line consists of the cone and the cones generated by and respectively. The affine variety corresponding to the -cone is and the affine variety corresponding to the -cone is . Their intersection is the affine variety corresponding to the cone , which is the torus .

The torus acts on itself via the group multiplication, and it acts on and trivially, i.e. these are the fixed points of the action. If you prefer homogeneous coordinates, acts on as , so we have and .

The Kähler differentials are and . The induced -action is and , respectively. We see that the positive weight part at is everything, while at it is nothing. Consequently, the orthogonal at is nothing and at it is everything. Under we get an isomorphic preimage of this orthogonal and take the ideal generated by it. This gives us ideals and . They correspond to the cells and .

This is already the BB-decomposition: .

The BB-filtration of is (with ) just .

To get a motivic cell structure, we need attaching maps for the cell to the set of -cells . The stable attaching map is the cofibration , which one can see as the homotopy cofiber of , i.e. the gluing of along to a point .

The homotopy cofiber sequence yields a distinguished triangle

which we can identify as

Now we have a splitting .

Okay, that was kind of stupid, given that we already knew that is a -cell. It was also kind of stupid that we have computed a stable cell structure, while it is also quite easy to describe an unstable cell structure of projective spaces.

We can also take a different -action on , by taking any cocharacter (i.e. group homomorphism) . These are all of the form for some . If , we get the same weight decomposition of (co)tangent spaces (Kähler differentials), hence the same BB-decomposition. If , we get no decomposition because the fixed points are not isolated (they are everything). If , we get the decomposition , which one might also call the minus-decomposition w.r.t. the first action considered.

Here we have a torus acting and it becomes a slightly more interesting question which cocharacter gives which BB-decomposition.

The fixed points of the torus are the orbit closures corresponding to the maximal cones, which are .

Take the diagonal corresponding to the weight in the weight lattice. It doesn't hit any linear subspace generated by codimension one cones, so it has the same isolated fixed points, as the original torus. (In contrast, e.g. fixes a whole ).

From analyzing the positive weight subspaces of the cotangent spaces of at these fixed points, we get

and from this the ideals defining the cells

so the cells are

It is pretty obvious now how much influence the choice of a cocharacter has on the cell decomposition. There are only four different cell decompositions, corresponding to the four maximal-dimensional cones in the fan.

The BB-filtration is . The motivic cell structure is built inductively, we start with the cellular space and attach the cellular space to it (to obtain ) via the homotopy cofiber sequence

In the next step we attach to the stably cellular space the cellular space to obtain via the homotopy cofiber sequence

This is also the motivic cell structure you would get as product cell structure from the previously considered cell structure for , and it is all parallel to classical topology, up to homotopy (though classically the gluing maps don't look that strange).

For the sake of completeness, let's compute the motive from this (i.e. let's look at the distinguished triangles):

shows that , as expected from the observation . The next homotopy cofiber sequence gives

and we get , since there are no non-trivial morphisms .

The fixed points of the torus are , and (corresponding to the maximal dimensional cones in the fan, in counter-clockwise order). Denote by the affine toric variety corresponding to the maximal dimensional cone .

The cotangent spaces at the fixed points are

,

,

.

The diagonal cocharacter is no longer good, since lies in the linear subspace generated by a cone of the fan -- it fixes the projective line .

We can choose the cocharacter , which acts with the same isolated fixed points as the whole torus on . For this one we get:

,

,

,

so that we have cells

,

,

.

This decomposition is one of the common decompositions of into and a at infinity, which is decomposed into and .

The corresponding BB-filtration is just .

The homotopy cofiber sequences that give the motivic cell structure are

As before, we have and .

If we pick another cocharacter , this is still inside the cone , but it has a different scalar product with one of the rays, so the decomposition should be different. Indeed, from computations we find that the big cell now is part of and in we have only a -cell.

The fan for a Hirzebruch surface looks similar to the fan of :

The image shows the fan of .

We pick the torus cocharacter of weight again, since it works for all Hirzebruch surfaces.

The cone generated by and as well as the cone generated by and are just as in the situation of , and the corresponding affine toric varieties glue together to a , which is visible in the cell structure. The big cell is corresponding to the cone with faces and .

The cone generated by and corresponds to a fixed point which is a BB-cell itself and the cone generated by and corresponds to a fixed point with attached BB-cell of dimension .

The BB-filtration is with the fixed point which already is a cell and everything except the big cell (big cell = unique open cell). The motivic cell structure is built from the homotopy cofiber sequences

The space is homotopy equivalent to the at the base, but the gluing map really depends on .

The motive is just the same as the motive of .

It is a well-known fact (and not hard to prove) that all complete nonsingular toric surfaces (implicitly assuming normal) are either , a Hirzebruch surface or a blow-up of one of these at torus fixed points, since one can describe such blow-ups with fans. One easily sees that blowing up a fixed point introduces an additional fixed point with "BB-cell" , therefore an additional to the motivic cell structure (the additional ). This introduces an additional to the motive.

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The paper is essentially about algebraic torus actions on varieties and relating the induced action on the tangent space of a fixed point to the variety itself. The most simple torus is just the multiplicative group (think of or ). In classical Morse theory, one considers "Morse functions", which are a particular kind of function , and their gradient flow, which is the flow associated to the gradient vector field. Such a flow is nothing but a -action! Where the Morse-theory people look at smooth manifolds and apply the exponential function from the tangent space (of a critical point of the Morse function, i.e. a fixed point of the flow) to the whole space , an algebraic geometer has to do something else (as the exponential function is not algebraic). This something else is a gimmick invented by Białynicki-Birula. With this gimmick, a -action with isolated fixed points provides a cell decomposition, like the CW decomposition from classical Morse theory.

We work over an algebraically closed field . Let be a quasi-affine algebraic scheme and a nonsingular closed point. We denote by a reductive algebraic group, though in the end only the 1-dimensional torus is relevant.

Given a -action on a scheme with fixed point , the tangent space gets a natural -action. For any vector space with -action there is a decomposition into the weight-graded pieces. I call the action definite if either the minus- or the plus-part vanishes, and fully definite if also the zero-part vanishes.

**Theorem 2.1**: Given a reductive group acting on with a closed irreducible -invariant subscheme containing a closed fixed point nonsingular in and , to any -invariant subspace of the tangent space that contains one can find a closed irreducible -invariant subscheme that contains and has the prescribed tangent space.

(This is what I consider a replacement for the exponential function).

**Proof idea**: The maximal ideal corresponding to maps -equivariant surjective to . Denote by the tangent space of and by the ideal corresponding to . Since is reductive, there exists a -submodule that maps isomorphically to . Then is an ideal in , so the corresponding closed subscheme of has an irreducible component containing . By construction, .

Uniqueness of the subspace is also discussed in Theorem 2.2, in particular we have a **Corollary to Theorem 2.2**: Let act on with fixed point . If is either the positive, the negative, the non-negative, the non-positive or the zero-part of the graded vector space , then there exists exactly one closed, irreducible and reduced -invariant subscheme through such that is non-singular and .

There is a morphism-version of Theorem 2.1, which is slightly weaker. Roughly, to a -isomorphism of some tangent spaces of two -schemes, you get a third scheme with étale maps to the two others, and if you already have a -isomorphism on subschemes, this is taken into account. The precise statement is

**Theorem 2.4**: Given for sequences of closed immersions of -invariant subschemes of quasi-affine algebraic -schemes and a -isomorphism , such that the -modules and are isomorphic, there exists such a sequence and étale -morphisms that map onto an open subscheme of .

**Proof idea**: Inside embed as and apply Theorem 2.1 to get a subscheme that contains and , with . The projections to the factors are étale at , hence over a smaller subscheme that still contains . Denote by the union of the preimages of the in , then and do the job.

The local structure of affine "cells" comes from

**Theorem 2.5**: For any torus acting on such that is a fixed point and the induced action on is definite, there exists a -invariant open neighborhood of which is -isomorphic to , with a finite-dimensional fully definite -module.

**Proof idea**: The arises as the complement of .

WLOG (as one can show) is reduced, irreducible and acts effectively. Apply Theorem 2.4 to , , , , , , then the resulting are not only étale, but also birational (as one can show), hence open immersions. Then contains an open subscheme which is -isomorphic to for some open beighbourhood of in and gives the statement of the theorem.

In the full proof of Theorem 2.5, the notion of a universal domain is frequently used. This is a device to handle generic points without talking about prime ideals, which I explained in this blog posts about points.

Given a -rep one defines the notion of an -fibration , which carries a -action on and Zariski-locally on looks like , with -action induced by . We call the dimension of the -fibration.

One should remark that an -fibration needn't be a vector bundle, since there might be more -equivariant automorphisms of than the linear ones.

The following gives us a uniqueness property for -fibration-structures on maps .

**Corollary to Proposition 3.1**: For any torus acting on a nonsingular , two -representations for and -fibrations (respectively), then is equivalent to . If furthermore is a -fixed point and is definite, then .

**Proof idea**: For any closed point , as -modules, , so the are equivalent. Note that the in is uniquely determined, since there is no nonzero -homomorphism . By (the corollary to) Theorem 2.2 there exists exactly one -invariant subscheme with nonsingular and , but and both fulfill these conditions, hence . This shows .

We call a morphism with -action on a -fibration if it is Zariski-locally over an -fibration for some -reps . If the dimensions of the all coincide, we call that number the dimension of the -fibration.

Now, let and any non-singular reduced algebraic -scheme that can be covered by -invariant quasi-affine open subschemes (for example any smooth projective will do, maybe normal quasiprojective suffices, by Sumihiro's equivariant compactification).

**Theorem 4.1**: Let be the decomposition into connected components. For any there exists a unique locally closed -invariant subscheme and a unique morphism such that

- is a retraction, i.e. is a closed subscheme of and is the identity,
- is a -fibration,
- for any closed fixed point , the tangent space is and the dimension of the -fibration is .

**Proof idea**: Let . By Theorem 2.1 there exists a closed -invariant irreducible subscheme with nonsingular and . By Theorem 2.5, there is an open -stable nonsingular subscheme that still contains and is a trivial -fibration. Using the Corollary to Theorem 2.2 and the Corollary to Proposition 3.1 (the uniqueness statements) we know for that and for any third fixed point we have .

Since every is noetherian, we find such that , so is a -invariant, locally closed subscheme of and a -fibration can be uniquely glued together from the .

Actually, there is also a minus-decomposition, where you use instead. The interplay of these two decompositions for the same -action is explained in **Theorem 4.2**: Let act on a quasi-affine . For a rational point , the orbit closure intersects a connected component in a non-empty set iff or . Moreover, for all connected components.

**Proof idea**: The direction is clear. For the other direction apply Theorem 2.4 to , , , , , . For we have . Only one contains and one can show (using again Theorem 2.1 and 2.2) that actually and .

Moreover, from we get .

**Theorem 4.3**: Let act on a complete nonsingular algebraic -scheme , with the decomposition of the fixed points into connected components. Then there exists a unique locally closed -invariant decomposition and -fibrations such that and for any closed fixed point , .

**Proof idea**: Take the plus-decomposition from Theorem 4.1, then what's missing for the statement ( for and ) follows from analyzing orbit closures (that is actually Theorem 4.2 together with Lemma 4.1 which I didn't include in this summary).

From this follows **Theorem 4.4**: If the -action in Theorem 4.3 has isolated fixed points, then any is isomorphic to an affine space .

(This looks like a cell structure!)

The proofs and results have been improved a little bit (Hesselink removed the assumption that is algebraically closed), so that the current level of generality provides the following

**Theorem**:

Let be a smooth projective -variety over a field . Then

1) is a smooth closed subscheme of (Iversen),

2) Given the connected components , there is a filtration and affine fibrations ,

3) The relative dimension of is the dimension of for any .

I want to remark that any generalization of this theorem to quasiprojective or singular situations would be a very impressive result.

The only generalizations I know of are papers of Skowera and Choudhury on Deligne-Mumford stacks and papers of Carrell and Sommese on the Kähler analogue.

**Theorem** (Karpenko):

Let be a smooth projective variety over a field , equipped with a filtration where the are closed subvarieties, and affine fibrations of relative dimension . Then the Chow motive decomposes .

**Corollary** (Brosnan):

Let be a smooth projective -variety over a field . Then , where the are the connected components of the fixed point locus .

From this, Brosnan re-proved

**Theorem** (Chernousov-Gille-Merkurjev):

For a projective homogeneous variety (for a reductive group) over a field , the kernel of the map consists only of nilpotent elements.

Brosnan proved more, in particular how the motive of decomposes, using this method: , where is length and is the set of minimal length coset representatives of , with the set of roots corresponding to and the set of roots that are killed by a non-central cocharacter of the maximal torus (taking care of the possible non-splitness of the maximal torus).

Using the BB-decomposition, one gets a decomposition of the motive. Actually, one gets a bit more, namely a decomposition in the stable A¹-homotopy category. This is even more analogous to CW decompositions coming from Morse theory.

What you need for a cellular decomposition (in the homotopy-theoretic sense), but what's missing in a direct sum decomposition of the motive, are the gluing maps. One has to extract these gluing maps from the BB-decomposition. This was done by Wendt, who used this approach to show stable cellularity of connected split reductive groups and their classifying spaces, as well as stable cellularity of smooth projective spherical varieties under connected split reductive groups.

As this post is already too long, I might explain the motivic cell structures in another post. Actually, you can just take a look at the preprint.

It remains to see how these cell structures look like explicitly!

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- The usual fundamental group of a topological space with basepoint .
- The simplicial fundamental group of a simplicial set with basepoint .
- The fundamental group of an object with basepoint in a Quillen model category.
- The absolute Galois group of a field with separable hull .
- The fundamental group of a linear algebraic group: root lattice modulo weight lattice.
- The étale fundamental group of a scheme .
- The fundamental group scheme of a scheme over a perfect field with rational basepoint .
- The motivic fundamental group of a scheme over a perfect field with rational basepoint .
- The Tannakian fundamental group object of a fiber functor of a Tannakian category.
- The polynomial fundamental group presheaf of a scheme over a field , with rational basepoint .
- The A¹-fundamental group (pre)sheaf of a scheme over a field , with rational basepoint .

- For an object , the set of objects of some type over , i.e. the set of certain maps to , is often classified by something like a fundamental group. In the case of the topological fundamental group, the covering spaces are in 1:1 correspondence with subgroups of the fundamental group. In particular, there is one covering space which has a trivial fundamental group itself (it is then called simply connected) and the automorphisms of this covering map form precisely the fundamental group of the base space. A similar covering space theory holds for the absolute Galois group and field extensions instead of coverings.
- There are operations that convert one notion of fundamental group to another. For example the singular set of a topological space is a simplicial set that has as simplicial fundamental group exactly the topological fundamental group of the original space (and the geometric realization goes the other way around).
- For many fundamental-group concepts, there are higher homotopy groups and long exact sequences that connect the information of with the . For some fundamental-group concepts, there are no "higher groups" (known so far).
- One may often neglect basepoints, if the resulting groups are isomorphic anyway (under some connectedness assumption). This, however, can be dangerous, as one gets an action of the fundamental group at one basepoint on each other homotopy group, and this action might be non-trivial. Another danger comes from fundamental groups that are not just groups, but also carry some other structure (that of an algebraic variety, or a mixed Hodge structure, for example) -- these additional structures may then depend on the basepoint. If one wants to forget about basepoints, there is the safe way of working with fundamental groupoids instead.
- There are more delicate relations between the various fundamental groups. For a connected projective algebraic variety over one may use an embedding to put the submanifold topology of the complex manifold on , call this and compare with . It turns out that the étale fundamental group is the profinite completion of the topological fundamental group. This may be explained by the fact that the étale topology only allows finite covers, so for example is a topological covering that doesn't come from any étale covering of .
- Some of the fundamental groups on the list are special cases of others. For example, both the topological and the simplicial fundamental group can be defined in the context of Quillen model categories. This also clarifies the relation between them, since the singular set functor and the geometric realization form what is called a "Quillen adjunction", which axiomatizes this relationship. On the other hand, the Galois group of a field is just the étale fundamental group of the spectrum of that field -- which has no direct connection to model categories at all.
- Well, actually it is a bit more complicated: the precise relation between étale fundamental groups and Galois groups with respect to basepoints is described in this MO answer by Minhyong Kim.
- Some of these fundamental groups are really not very fundamental-group-like: The algebraic fundamental group of a reductive linear algebraic group, as explained on MO by Brian Conrad here, is not the étale fundamental group and its relation to it (and other fundamental groups) can be very loose.

I claim that there are two or three major approaches to fundamental groups. The first is via Quillen model categories, which captures the intution of loops at some basepoint modulo basepointed homotopy, and which directly gives us higher homotopy groups and long exact sequences. The second is via Tannakian categories, which captures the intuition of being an automorphism group, and which may directly give us some extra structure on the fundamental group.

Grothendieck's Galois Theory of Topoi captures the intuition of monodromy, but essentially is a variation of the Tannakian point of view (in my eyes).

The brief definition of the fundamental group of an object in a (base-pointed) Quillen model category: Take a fibrant replacement and then where the brackets mean base-pointed homotopy, i.e. with the (left)homotopy relation. For that to make sense, you obviously need an object in the model category; this can be constructed once you have some kind of interval, for example if the model category is in fact a simplicial model category. Alternatively, without an interval you can still define the loop space via the path space fibration and then take where is the (left)homotopy relation.

- The usual fundamental group of a topological space comes from the model structure with homotopy equivalences as weak equivalences and Hurewicz fibrations as fibrations (or any other Quillen equivalent model structure).
- The simplicial fundamental group comes from the Kan model structure, with Kan fibrations as fibrations (and the weak equivalences can be defined without reference to fundamental groups).
- The polynomial fundamental group presheaf is a presheaf of simplicial fundamental groups of singular resolutions.
- The A¹-fundamental group (pre)sheaf is the group (pre)sheaf coming out of the Morel-Voevodsky model category.
- Étale fundamental groups may be defined in terms of an étale homotopy category as well, though this is not widely used.

The brief definition of the fundamental group of a fiber functor of a Tannakian category over a field is just , the -automorphisms of the functor (which are invertible natural transformations from to itself). This is, in general, a pro-algebraic group scheme over . If the Tannakian category is -generated by a single object , then .

- The topological fundamental group of a space with basepoint is the Automorphism group of the fiber functor from the covering category. The covering category is the Tannakian category of covering spaces with lift of the basepoint, and the fiber functor forgets . If there is a single generator of the Tannakian category, i.e. a universal covering, then the Automorphism group of the fiber functor is just the automorphism group of this single covering (where we're talking about automorphisms of the morphism, not just of the space).
- The absolute Galois group of a field comes out of the category of all separable algebraic field extensions , with the fiber functor , since .
- The motivic fundamental group of Tate motives over a scheme is given by the category of mixed Tate motives and the Betti realization. Here, can be constructed as subcategory of cut out by a t-structure if is a number field. For rings of integers of number fields, one can give an ad-hoc modification of that gives the right answer. There are also models from rational homotopy theory (see Deligne-Goncharov, Esnault-Levine).

One might even ask, as Harry Gindi did on MO, whether one could define higher homotopy groups in the Tannakian setting. Obviously, one would need higher Tannakian categories.

Given a Topos of sheaves on some category C, and a point p, we can look at the functor that assigns to each sheaf its stalk at that point. If we take only locally finite, locally constant sheaves (aka local systems), we get a functor to finite sets

which results in an equivalence of to a category of modules under a group, the corresponding Galois group.

See this MO question of Lars Kindler for the relation to the Tannakian POV.

Two more fundamental groups:

- The geometric fundamental group of a scheme over a field is given as the kernel of the morphism . This morphism can be seen to come from a morphism . If is a number field, the geometric fundamental group is the profinite completion of the usual topological fundamental group of the associated complex space.
- The motivic fundamental group of a scheme is similarly given as the kernel of the morphism . There has been some recent research around this object, especially the special case of motivic of . The reason is a deep connection with multiple Zeta values.

Once you have any definition of a fundamental group, it is interesting whether you can find a sheaf topos such that the local systems correspond 1:1 to the representations of this fundamental group. That would be in the spirit of Caramello's unification of mathematics via topos theory.

]]>

A PDF/A file is a document that probably ends in .pdf, complies to the PDF 1.4 standard (not more or less), has ORCed text in the background layer to allow for full-text search, has valid metadata in XMP format (yay!), and the compression is Mixed Raster Compression (MRC) which allows quite small documents (though DJVU is still slightly smaller in my experience). Actually that was more-or-less PDF/A-1b, the basic version. Now there is also PDF/A-2, where you can use better compression (JPEG2000), transparencies and layers, since it's based on PDF 1.7. The "A" in PDF/A stands for "archive-able".

In Debian or Ubuntu GNU/Linux, if you like graphical user interfaces:

`sudo apt-get install scantailor`

will bring you all you need. Under the hood works a command-line tool:

`sudo apt-get install unpaper`

and you can get ScanTailor and unpaper also from their websites.

To find good hardware for scanning that is linux-compatible, just compare technical specifications and prices as usual, and once you have a list of 1--5 devices you might buy, check against the list of linux-supported devices here.

An alternative to the usual flatbed-scanner setup is to construct something yourself, like an open-source book scanner, another open-source book scanner, or a slide-scanner made from a camera.

I use a recent Canon model (LiDe 210) that works without quirks in Ubuntu Linux 12.10. I use scanimage on the command-line and the GUI of XSane (though it looks a bit old-fashioned) so let me tell you about the available options on XSane. Using other scanning software on linux most probably means using another UI to the Sane library, so the options are the same.

For OCR, the best mode is "Gray" or "Color", but not lineart. The resolution should be 300 or 600 DPI, more is usually not necessary and slows down the post-processing. If you're low on memory, high DPI values might even make the post-processing impossible. There is a green-blueish button for automatic gamma, brightness and contrast, which makes sense after acquiring a preview of your scan; I recommend the default enhancement values (1,0,0) since we can post-process later in the proper tools. Some post-processing tools have problems with 16-bit images, so I recommend to use 8-bit (in the "Bit depth"/"Standard Options" window of XSane). For most post-processing tools, it is convenient to have the scans in TIFF or PNG format. With TIFF you have to make sure that lossless compression is activated in the Sane configuration (see also these scanning tips).

Speckles and black borders on a document can make it really hard for OCR software, so you should try to get your scan as clean as possible. It may help to acquire a preview and crop manually.

To make sure the end-result contains all available relevant metadata, I recommend taking as much information as possible into your filename already, like some date attached to the scanned piece (if it is a letter or a photo) and some context. This will later make it easy to move this information into the PDF, especially if you intend to scan many pieces at once.

If you want to generate PDF/A compliant PDFs, one solution is to use LaTeX, where you just insert your scan(s) as embedded images, and the metadata where it belongs. There is a tutorial for PDF/A compliant PDFs out of LaTeX, though it doesn't touch the issue of embedding scanned images or OCRed text.

UnPaper is a very useful software to remove any paper artifacts from you scans. In principle, this enables you to get printouts of your scan that look like actual re-prints, not photocopies. This is especially useful for the purpose of OCR.

The standard interface for UnPaper is the command-line, but there are also GUIs available. Some of them are still at an early stage of development, like GScan2PDF, others seem to be discontinued, like OCRFeeder, so I recommend using ScanTailor (download!).

ScanTailor has the assumption of scanning books in mind, so it is optimized to scan two pieces at once and later splitting them into two separate pages. This was useful to me when I wanted to scan large amount of photos, 4 at a time, to split them later in ScanTailor.

Warning: with high resolution come large files, so the post-processing that happens in ScanTailor can be slow. If you have a whole book to scan, I would recommend finding out the right parameters by hand and using the command-line UnPaper instead of ScanTailor.

UnPaper and ScanTailor take image files like TIFF or PNG and give back TIFF.

DjVu files are known for their incredible compression. However, the magic ingredient for that is "Mixed Raster Compression" (MRC), which you can also use in PDFs. Since PDF/A is the archive standard, not DJVU/A, and future tools enable MRC in PDF, DjVu will become even less important.

There is already a wonderfully detailed tutorial online on how to digitize books to DjVu, even with a section covering OCR.

As far as I know, this must be done on the command-line, since no free GUI is available.

To convert a bunch of TIFFs to PDF, there is tiff2pdf. You can supply some metadata on the command-line, to be included in the PDF.

Example usage:

`tiff2pdf -o outputfile.pdf -z -u m -p "A4" -F inputfile.tif`

The switch "-z" enables lossless compression, instead you could use "-j -q 95" for 95% quality JPEG compression. The switch "-p "A4"" specifies the paper size, which could also be "letter". The switch "-F" causes the TIFF to fit the entire PDF page, to avoid borders.

Another example:

`tiff2pdf -o outputfile.pdf -z -u m -p "A4" -F -c "tiff2pdf" -a "Author Name" -t "Document Title" -s "Document Subject" -k "keyword1,keyword2,keyword3"`

−e 20130324103000 inputfile.tif

This line will include the given metadata into the resulting PDF.

Between post-processing the scans and compressing them into a PDF, we might want to run OCR on them. I still use tesseract/hocr2pdf to do that, since the Tesseract engine tends to give me the best results, and hocr2pdf is the only solution I know of that can "hide" the scanned text in a layer behind the scanned image, to give you true full-text search without damaging the scan quality at all.

With whatever input data you have, I recommend the following:

`convert -normalize -density 300 -depth 8 "inputfile.ext" "normalized-input.png"`

since tesseract really works best with normalized images at density 300 and bit-depth 8, in PNG format.

Tesseract is language-sensitive. If you do

`tesseract -l deu -psm 1 "normalized-input.png" "output.pdf" hocr`

it will assume german text (deu=deutsch=german), but the switch "-l eng" will change that to english language. There are many other languages available (see "man tesseract"), and you can build your own.

To merge back the hocr data into the PDF, you need to convert the PNG to JPEG and run hocr2pdf:

`convert "normalized-input.png" "normalized-input.jpg"`

hocr2pdf -i "normalized-input.jpg" -s -o "output.pdf" < "output.pdf.html"

To get the metadata right, you might want to use PDFTk and its dump_data,update_info commands. Take a look at the final shell script below for this.

Standards are only good as long as you can validate them. This is possible for PDF/A with JHOVE, the JSTOR/Harvard Object Validation Environment (pronounced "jove"). Though it still has some bugs, it is the only viable free alternative to Adobe's Windows-only Preflight mode (which is still better, I admit).

After extracting the JHOVE files to some directory "jhove", you have to edit the file "jhove/conf/jhove.conf" and change something in "

After you got that right, run

`java -jar jhove/bin/JhoveView.jar`

to get the interactive program. You can change the configuration there as well. Once I had the strange issue that I had to change the directory from the UI tool to make the CLI tool work...

If you prefer to stay on the command-line, to automate your workflow, try

`java -jar jhove/bin/JhoveApp.jar -m PDF-hul "filename.pdf"`

and watch out for the lines beginning with "Status" and "ErrorMessage".

You'll notice that most documents have some errors, but these don't affect reading the documents. It is actually quite hard to get a PDF/A-conforming document!

I did a little survey on my own archive of PDFs, mostly from the arXiv and mathematical journals, in total about 500 PDFs. The errors (also happening in files that seem to be generated from a TeX source and files from JSTOR or Journal homepages) were:

- InfoMessage: Too many fonts to report; some fonts omitted.: Total fonts = ...
- InfoMessage: Outlines contain recursive references.
- ErrorMessage: Improperly formed date
- ErrorMessage: Lexical error
- InfoMessage: File header gives version as 1.4, but catalog dictionary gives version as 1.6
- ErrorMessage: Invalid page dictionary object
- ErrorMessage: Invalid outline dictionary item
- ErrorMessage: Invalid object number in cross-reference stream
- ErrorMessage: Invalid destination object
- ErrorMessage: Invalid Resources Entry in document
- ErrorMessage: Malformed dictionary
- ErrorMessage: Malformed filter
- ErrorMessage: No PDF header
- ErrorMessage: No PDF trailer
- ErrorMessage: Unexpected error in findFonts: java.lang.ClassCastException: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject cannot be cast to edu.harvard.hul.ois.jhove.module.pdf.PdfDictionary
- ErrorMessage: Unexpected error in findFonts: java.lang.ClassCastException: edu.harvard.hul.ois.jhove.module.pdf.PdfStream cannot be cast to edu.harvard.hul.ois.jhove.module.pdf.PdfDictionary

The last two are obviously bugs in JHOVE. The "too many fonts to report" info message came about 100 times. About 100 files (not the same, but with some overlap) out of the total 500 were invalid PDF/A. Nevertheless, all these files are perfectly readable. It is not clear, if they would be readable on other devices, like a Kindle or Android. I also encountered printing errors with malformed PDFs in the past, so I recommend getting rid of these errors at least in the files you produce after scanning.

This is a script to call from the command-line, to scan and OCR directly to PDF/A.

usage:

`./scan-archive.sh filename.pdf title subject keywords`

example usage:

`konrad@sagebird:~/Documents/scans$ ./scan-archive.sh Letter-20130324-Bankaccount-closing.pdf "Letter from the bank" finances bank,account,closing`

full script (also available on pastebin):

`#!/usr/bin/env bash`

echo "usage: ./scan-archive.sh filename.pdf title subject keywords"

echo "scanning \"$2\" on \"$3\" about \"$4\"... ($1)"

scanimage --mode Color --depth 8 --resolution 600 --format pnm > out.pnm

echo "processing... ($1)"

scantailor-cli --color-mode=black_and_white --despeckle=normal out.pnm ./

rm -rf cache out.pnm

tiff2pdf -o "$1" -z -u m -p "A4" -F -c "scanimage+unpaper+tiff2pdf+pdftk+imagemagick+tesseract+exactimage" -a "Author Name" -t "$2" -s "$3" -k "$4" out.tif

rm -f out.tif

echo "converting to PDF 1.4 ($1)..."

mv "$1" "$1.bak"

pdftk "$1.bak" dump_data > data_dump.info

pdftk "$1.bak" cat output "$1.bk2" flatten

echo "OCR in lang deu... ($1)"

convert -normalize -density 300 -depth 8 "$1.bk2" "$1.png"

tesseract -l deu -psm 1 "$1.png" "$1" hocr

convert "$1.png" "$1.jpg"

hocr2pdf -i "$1.jpg" -s -o "$1.bk2" < "$1.html"
echo "Inserting metadata... ($1)"
pdftk "$1.bk2" update_info data_dump.info output "$1"
rm -f "$1.bak" "$1.bk2" data_dump.info
rm -f "$1.png" "$1.jpg" "$1.html" "$1.pdf"
echo "done. wrote file. ($1)"
echo "validating... ($1)"
java -jar jhove/bin/JhoveApp.jar -m PDF-hul "$1" |egrep "Status|Message"

You should obviously customize "Author Name", and you might want to skip the validation step in the end. In other environments, "A4" might be better replaced with "Letter" or "A3", depending on your scan format. Purists might want to skipt the conversion to JPEG, which I used to get smaller files. In JPEG2000, the same compression technique that powers DjVu (MRC) is possible.

Maybe one should try the suggestions here for other Tesseract UIs, but I'll stick to the command-line for now. Any other suggestions?

]]>

**Measuring the size** of an object is done by comparing it to some other object of known size. The other object of known size is used to define a unit of size (or length, if you prefer). Measuring mass of an object is done by comparing it to some other object of known mass, which are both measured indirectly by measuring the gravitational force influenced by the earth. With very precise instruments or very heavy objects, one could also measure the force without thinking about a third object. In any case, one would actually measure the work over some known length.

**How do we measure time?** We can compare with some known time-frame, like the duration of one full trip of the earth around the sun (also known as "year"), but only if we're ignoring that this duration changes over time. We can take a look at our watches; There, small Quartz crystals vibrate such that the frequency doesn't change over time, with high precision. To get higher precision, one can only take oscillating signals that are even more stable, like atomic clocks that use the frequency of (microwave) photons that are emmited in electron energy-level transitions. These are the best clocks we have on earth and this level of precision is not reached out of sheer scientific curiosity, but it's necessary for GPS to work.

**What exactly does it mean** to compare a time-frame with some frequency of an oscillation? In the easy example of a year, it means that we take a look at the position of the earth around the sun (for example, it could be summer solstice) when we start the measurement, and we observe how this position changes during measurement. When the measurement is done we integrate ("sum") over all the position changes and get the total length of the path the earth has travelled around the sun. This means that a unit of time is essentially specified by an oscillation in space. While this is not directly true for the atomic clocks, it is true indirectly for two reasons: we observe the electron energy-level transitions ultimately as oscillation of the position of something and every oscillation in space specifies just a scalar mutliple of the time frame specified by the microwave oscillation.

**So, to measure time,** we have to observe how things move over some distances, and our units of time are intimately linked to the distance used in the observation - it is impossible to give a precise definition of "a second" without reference to any distances, lengths or sizes if we want to be able to measure "a second".

**How did we measure distances?** By comparing them with known objects, where "known" means that we ultimately use one thing as reference for all the others, somewhat arbitrarily. A century ago, people used the "Ur-meter", a certain object of 1 meter length, for comparisons. Since then, molecules of the Ur-meter were displaced and the length changed measurably, so 1983 the definition of 1 meter was changed to "the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second". We can safely forget about the archaic factor 299,792,458 and observe that length is measured via time now. Didn't we observe before that time is measured via length?

**The definition of length units** in terms of time the light takes has the advantage that light is the same everywhere and available in abundance, so we can compare measurements without having to move or make copies of any other reference object. The same is proposed for the kilogramm - it can be defined via the energy of a photon (by Einstein's famous formula).

**Measuring time can be a very complicated issue,** since we don't want to measure the time at one place, but usually at multiple places, to compare them. For example, our artificial satellites measure their position by measuring how long the light takes to go between a point on the earth and the satellite. For this, they need to have a clock on board. From the change of their position, they can calculate their speed. This is where things become interesting, since a change of speed alters the perception of time (general relativity). Taking general relativity into account, one can use insights from quantum physics to build high-precision atomic clocks and computers and launch them into space, which gives us GPS.

**What do we mean** by "observe the frequency of a photon"? In the end, it means we build a complicated apparatus that transfers information in the form of (other) photons to us, maybe via a computer screen that displays a number. To build that apparatus, we need some model of the reality, some units of length, time and mass. This means that the measurement is always bound by the theory that is used to make the measurement.

**By the way:** this article of Terence Tao about dimensional analysis inspired me to finish this article now, and I can only recommend it! While we're at it: you should also take a look at Longitude, a book about the importance of time measurement for sea navigation and a true story how we got the first usable clocks.

In short:

Is there a Homotopy Theory of Proofs?

Suppose you take two sets of statements A and B, for example A says that some set X and a product on X satisfies the axioms of a group, and B says something about X and the product which is satisfied for any group. There are many ways to prove , though we might be inclined to see many of them (if not all) somewhat equivalent. Suppose two proofs are considered equivalent, can you image that there are two different ways to see that/how they are equivalent? This is, very roughly, what I think should a homotopy theory of proofs be.

Statement sets (or just statements?) would be points (0-cells), proofs the arrows/morphisms (1-cells) and equivalences of proofs (whatever that may be) the 2-morphisms (2-cells) and so on, with higher homotopies.

One way to get that rigorous might lie in the syntax: take as statements syntactic statements and as proofs full syntactic proofs. Then one can at least assign to two proofs the Hamming distance, which gives us the structure of a (pseudo)metric space.

My question is: is there anyone out there already working on such stuff? Is there any hope or can you give an argument why that is inherently nonsensical?

My next steps would be to seek more examples of different proofs for the same elementary implications to get some feeling for how the transformations between proofs might look like.

Another idea was to look at Turing machines instead, and take reductions as 2-morphisms. Can you think of anything that goes in that direction?

Why this question isn't suitable for MO:

I didn't even try to do an extensive search in the literature (as I don't really know where to start, not being a logician), I can give no good answer to the question "what would you expect from such a theory to prove/do?" and I don't know what would qualify as acceptable answer(s). Maybe in 1-2 years I will have thought enough about this topic to distill a MO-level question. Up to then, this will have to suffice.

I'll be happy with any input (and I know there are at least 2 people who also like this idea)!

[UPDATE 2013-03-29] Thank you for the comments so far, also on Google+!

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