This document discusses groupoids, local systems, and their relationships to differential equations on manifolds. Some key points:
1) Groupoids generalize groups by allowing multiple objects and isomorphisms between them. Representations of groupoids correspond to local systems on manifolds.
2) Local systems on a manifold X are sheaves of vector spaces that are locally isomorphic to a constant sheaf. They correspond to representations of the fundamental groupoid of X.
3) Vector bundles with connections on a Riemann surface B are equivalent to local systems on B. Global sections of bundles generate differential equations, whose solutions can be studied via the bundle's local system or groupoid representation.
Hand-outs of a lecture on Aho-Corarick string matching algorithm on Biosequence Algorithms course at University of Eastern Finland, Kuopio, in Spring 2012
Inversion Theorem for Generalized Fractional Hilbert Transforminventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Hand-outs of a lecture on Aho-Corarick string matching algorithm on Biosequence Algorithms course at University of Eastern Finland, Kuopio, in Spring 2012
Inversion Theorem for Generalized Fractional Hilbert Transforminventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Edhole School provides best Information about Schools in India, Delhi, Noida, Gurgaon. Here you will get about the school, contact, career, etc. Edhole Provides best study material for school students."
Characterizing the Distortion of Some Simple Euclidean EmbeddingsDon Sheehy
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Edhole School provides best Information about Schools in India, Delhi, Noida, Gurgaon. Here you will get about the school, contact, career, etc. Edhole Provides best study material for school students."
Characterizing the Distortion of Some Simple Euclidean EmbeddingsDon Sheehy
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
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Latency is a key indicator of service quality, and important to measure and track. However, measuring latency correctly is not easy. In contrast to familiar metrics like CPU utilization or request counts, the "latency" of a service is not easily expressed in numbers. Percentile metrics have become a popular means to measure the request latency, but have several shortcomings, especially when it comes to aggregation. The situation is particularly dire if we want to use them to specify Service Level Objectives (SLOs) that quantify the performance over a longer time horizons. In the talk we will explain these pitfalls, and suggest three practical methods how to implement effective Latency SLOs.
Monitoring systems will get smarter in order to keep up with the demands of tomorrow's IT architectures. Features like anomaly detection, root cause analysis, and forecasting tools will be critical components of this next level of monitoring. At the same time, the data that monitoring systems ingest is ever increasing in amount and velocity.
This session covers architectural models for advanced online analytics. We argue that stateful online computations provide a means to realize machine learning on high-velocity data. We show how alerting systems, event engines, stream aggregators, and time-series databases interact to support smart, scalable, and resilient monitoring solutions.
Heinrich Hartmann is the Chief Data Scientist at Circonus. He is driving the development of analytics methods that transform monitoring data into actionable information as part of the Circonus monitoring platform. In his prior life, Heinrich pursued an academic career as a mathematician (PhD in Bonn, Oxford). Later he transitioned into computer science and worked as consultant for a number of different companies and research institutions.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Cusps of the Kähler moduli space and stability conditions on K3 surfacesHeinrich Hartmann
Presentation about the paper with the same title http://arxiv.org/abs/1012.3121
Abstract:
In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the K\"ahler moduli space. The K\"ahler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions σ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of σ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of σ-stable objects.
An appendix is devoted to the group of auto-equivalences of the derived category which respect the component Stab†(X) of the stability manifold
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Toxic effects of heavy metals : Lead and Arsenicsanjana502982
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Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
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Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
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What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
2. nition 1.1. A groupoid G is a small category where all arrows are
isomorphisms. In other words G consists of two sets Ob(G) := V;Mor(G) :=
A (vertices/arrows), two maps s; t : A ! V (source/target), a map id : V !
A and an associative composition law
A V A = f(y; x) 2 A Aj s(y) = t(x)g ! A; (y; x)7! y x
satisfying the condition that idp is an identity for all morphisms starting or
ending at p, and any arrow has a two-sided inverse.
Remark 1.2. Let G be a groupoid, and pick vertices pi 2 V in any isomor-phism
class i 2 V= =
. Denote by 0(G; fpig) G the full sub category with
vertices pi.
This inclusion i is an equivalence of categories.
Let us analyse this a bit more closely, we claim that there is a functor
r : G ! 0(G; fpig) such that r i =
id0(G) and i r =idG. We can
arrange r in such a way the
3. rst isomorphism is the identity, but the second
one involves the choice of arrows xp : p ! pi for all p into some representative
pi.
If G is connected, i.e. all objects of G are isomorphic, there is only one
object pi = p to choose and 0(G; p) will be a group. So everyhting we do
with the goupoid will have an equivalent in group theory.
Nevertheless, working with groupoids we avoid choices (pi; p ! pi) which
will turn out to be very handy for our considerations.
De
4. nition 1.3. A linear representation of a groupoid G is a functor
F : G ! VectC; p7! Fp; (x : p ! q)7! x : Fp ! Fq:
In other words, we pick a vector space Fp for each vertex p and a linear
map x : Fp ! Fq (necesserily an isomorphism) for each arrow x : p ! q,
satisfying (y x) = y x, and (idp) = idFp . Linear representations form a
category in the obvious way, which we denote by RepC(G). More generally
we can de
5. ne a representation of G in an arbitary category in the same way.
Remark 1.4. Let 0(G; fpig) be a as in 1.2 the inculsion 0(G; fpig) G
induces an equivalence of categories
: RepC(G) ! RepC(0(G; fpig)):
This equivalence is non-canonical, as the inverse functor and the natural
isomorphism ! idRep(G) involve choices.
Date: May 2009.
1
7. nition 1.5. Let X be a topological space. The fundamental groupoid
(X) of X has the points of X as vertices, and as arrows [
] : p ! q (based-)
homotopy classes of paths starting at p and ending at q. The composition
of arrows is de
8. ned as concatenation of paths1
([
] : q ! r; [] : p ! q)7! [
] [] : p ! r:
The identities idp are the constant paths based at the points p, inverses are
provided by reversing the direction of the path.
Remark 1.6. The fundamental groupoid (X) is connected if and only if X
is path connected. In this case we pick a point x 2 X, then the fundamental
group 1(X; x) of X is just 0((X); x).
Example 1.7. To any covering space f : Y ! X there is a canonical Set-
representation of (X), de
9. ned by
p ! Yp = f1(fpg); ([
] : p ! q)7! T[
] : Yp ! Yq
where T
sends a point r in Yp to ~
(1) 2 Yq where ~
is a/the lift of
starting
at r.
2. Local systems
De
10. nition 2.1. A local system L of rank r on a topological space X is
a sheaf of C-vector spaces, locally isomorphic to the constant sheaf Cr.
A morphism of local systems is a morphism of the underlying sheaves of
vector spaces. In this way local systems form category, which we denote
by LocC(X). We can de
11. ne local systems of (free) abelian groups/k-vector
spaces/sets in the same way.
De
12. nition 2.2. Let L be a local system on X and
: p ! q a path in
X. The inverse image
1(L) is a local system on the unit interval [0; 1].
Note that the stalks
1(L)t and L
(t) are canonoically isomorphic and the
restriction maps induce isomorphisms
Lp H0([0; 1];
1(L)) ! Lq
We denote the composition by PTL
: Lp ! Lq and call it the parallel
transport map. If
0 is a homotopic path, then PTL
= PTL
0 . This can be
seen using the fact that local systems on [0; 1] [0; 1]R are trivial.
Proposition 2.3. Let X be locally simply connected, then the functor
PT : LocC(X) ! RepC((X));L7! PTL
where PTL maps p7! Lp and [
]7! PTL
has an inverse
Loc : RepC((X)) ! LocC(X); F7! Loc(F)
where Loc(F)(U)
Q
p2U Fp is f (vp) j 8
: p ! q in U : [
]v(p) = v(q) g,
such that there are canonical (!) isomorphisms: idLoc ! Loc PT and
PT Loc ! idRep.
1This convention diers form the usual one used by topologists; their
is our
. The Advantage of our de
13. nition is, that parallel-transport/monodromy have the
right associativity behaviour, i.e. become honest representations (as opposed to anti
representations).
14. GROUPOIDS, LOCAL SYSTEMS AND DIFFERENTIAL EQUATIONS 3
Proof. It is easy to see that Loc(F) and Rep(L) are in fact local systems,
resp. representation of the fundamental groupoid.
To construct the
15. rst isomorphism, we start with a representation F of the
fundamental groupoid (X). The stalk Loc(F)p is caononical isomorphic
to Fp for any p 2 X.
Indeed, elements of Loc(F)p are represented by (U; (sq)q2U) where U is a
neighbourhood of p in X, and sq 2 Fq staisfy the condition that [
](sq1) =
(sq2) for all paths
: q1 ! q2 contained in U. Two representatives are
equivalent if they agree on some small neighbourhood of p.
As we assume that any open neighbourhood contains a simply connected
one, we can restrict to simply connected neighbourhoods. But on a simply
connected U we can extend vectors sp 2 Fp uniquely to Loc(F)(U), and
hence the restriction Loc(F)(U) ! Fp is an isomorphism.
To get the desired natural isomorphism of representations Rep(Loc(F)) !
F we only need to check that the following diagramm commutes:
Loc(F)p
PTLoc(F)
/ Loc(F)q
Fp
[
]
/ Fq
Dividing
in smaller pices, we can assume
to be contained in a simply
connected neighbourhood. But there the statement is obvious.
Note that the construction did not involve any choices.
For the other isomorphism, we start with a local system L. We assign
to it the representation Rep(L) = (p7! Lp; [
]7! PTL
). Let v 2 L(U)
be a section of L. The images in the stalks vq 2 Lq for q 2 U satisfy
vq1 = vq2 for any path
: q1 ! q2 in U since we can pullback the section
to H0([0; 1];
1L) which restrics appropiately at the endpoints.
In this way we constructed a morphism idLoc ! Loc Rep. It is obvi-ous,
that this morpism is an isomorphism at the level of stalks, hence it
is an isomorphism. Note again, that we did not make any choices in the
construction.
Corollary 2.4. Let X be path connected, and locally simply connected, then
there are equivalences of categories:
Loc(X) ! Rep((X)) ! Rep(1(X; x0)):
This, more precise verison of the well known equivalence Loc(X) =
Rep(1(X; x0)), allows us to explicitly construct an inverse Functor:
Rep(1(X; xo)) ! Loc(X):
Namely, choose paths
p : x0 ! p to all points p 2 X, and use them to induce
from a given representation V of 1(X; x0) a representation (X) ! V ectC:
Set p7! Vp := V , and [
] := ([
q]1 [
] [
p]) : Vp ! Vq where [
] : p ! q
is a path and hence [
q]1 [
] [
p] : x0 ! x0 lies in the fundamental gruop.
Then use the equivalence of Rep((X)) to Loc(X) which we described in
detail above.
16. 4 HEINRICH HARTMANN
Remark 2.5. The Functors Loc and Rep are compatible with tensored prod-ucts
and inner Homs.
3. Vector bundles with Connection
There is jet another equivalence of categroies which will be important for
us. Let B be a Riemann surface.
De
17. nition 3.1. Recall that a connection on a holomorphic vector bundle
E over B is a C-linear map of sheaves
r : E !
B
E
satisfying the Leibnitz rule
8f 2 OB; s 2 E : r(fs) = df
s + f r(s)
Note that r is automatically
at, since we are on a Riemann surface. A
at morphism : (E;r) ! (E0;r0) between two vector bundles with con-nections
is a OB-linear map : E ! E0 commuting with the connections,
i.e.
r0 = (id
) r:
The category objects (E;r) and
at morphisms is denoted by FlatVect(B).
Proposition 3.2. There is a canonical equivalence of categories
FlatVect(B) ! LocC(B):
This equivalence is given by the functors:
FlatSect : (E;r)7! kerr E
and
L7! (OB
C L; r : f
s7! df
s):
Moreover the obvious natural isomorphisms
OB
C FlatSect ! idFlatVect; idLoc ! FlatSect(OB
C )
do not depend on choices.
De
18. nition 3.3. Let E be a vector bundle of rank E with connection r. A
global section 2 H0(B;E) is called locally cyclic vector at p if are locally
de
20. elds 1; : : : ; r1 2 Tp such that the derivatives
; r1; r2r1; : : : ;rr1 : : :r1
restrict to a basis on the
21. ber E(p) = Ep=mpEp. We call a cyclic vector if
it is locally cyclic everywhere.
Remark 3.4. As we are on a riemann surface this condition is equivalent to:
For all 2 T , with (p)6= 0, the iterated derivatives
; r; r2
; : : : ;rr1
restrict to a basis of the
22. ber E(p).
If we are given a global section 2 H0(B;E), we can ask:
Where is cyclic?
25. ne i := ri
. The set where sections
0; : : : ; k are lineary dependent is closed, and there will be a minimal k
rk(E) such that they are lineary dependent everywhere.
We would like to conclude, that there is a global linear relation between
these sections. But there are in general too few global holomorphic functions
for this to hold true. So we should at least allow meromorphic coecients.
Let M be the sheaf of meromorphic functions on B. This is a (non-constant)
sheaf of
26. elds, which contains the structure sheaf OB. Assume
that M
E is trivial, i.e. E is generated by meromorphic sections.2
Then H0(M
E) is a vector space of dimension rkE over the
27. eldM(B)
of global meromorphic functions. And there will be a relation
Xk
(1) aii = 0
i=0
with coecients in ai 2M(B).
Let U be the dense open subet of B where all ai are have no poles and
ak6= 0. A priori it is still possible that 0; : : : ; k1 are lineary dependent
at some points in U. The special choice of the sections allows us to say a
bit more. Linear independence means, that
Wr() = 0 ^ ^ k1 2 k(E)
does not vanish. There is an induced connection on k(E) which we can
apply to this section. Using the relation (1) we compute
rWr() = (r0) ^ ^ k1 + + 0 ^ ^ (rk1)
= 0 ^ ^ k2 ^ k
= (ak1=ak)Wr():
Now on the set V U where 6= 0 there is a dual-one form 2
B(V )
satisfying () = 1 and we can form the connection
r0 := r (ak1=ak)
id
of k(E). The equation above just says Wr() is parallel for r0. But if a
parallel section vanishes somwhere, it has to be zero. As Wr() is non-zero
at generic points, we conclude it is non-zero everywhere on V . We thus
proved the following proposition.
Proposition 3.5. Let E be a vector bundle generated by meromorphic sec-
tions, 2 H0(B;E) and 2 H0(B; TB); 6= 0. Let
Xk
i=0
ai i = 0
be a minimal relation with meromorphic coecients ai 2M(B) beween the
derivatives i = ri
.
Then 0; : : : ; k1 are linear independet over
U = f t 2 B j a0; : : : ; ak have no pole at t, ak(t)6= 0; (t)6= 0: g
2This is always the case in the algebraic situation where B is a smooth, variety of
dimension 1 over C, and E an algebraic vector bundle with connection.
28. 6 HEINRICH HARTMANN
Moreover is a cyclic vector for the vectorbundle F = OU 0; : : : ; k1
spanned by this sections.
4. Differential equations
Let B be a riemann surface.
De
29. nition 4.1. A dierential dierential on B of order n is a C-linear
morphism of sheaves of the from
(2) ai i(f)
D : OB ! OB; f7! n(f)
nX1
i=0
where ai 2 H0(B;OB) are holomorphic functions and 2 H0(B; TB) is
a nowhere-vanishing, holomorphic vector
30. eld, which acts by directional
derivative on functions.
Remark 4.2. If = f for another vector
31. eld , equation (2) changes into
fnn
nX1
i=0
bi i
for some functions bi, which can be determined by iterated use of the Leibitz
rule. We see, that we need to divide by fn in order to recover the form (2),
which is crucial for Cauchy's theorem below. This is why we insist to be
nowhere vanishing.
There are of course more general de
32. nitions of a dierential equation, see
for example [?], for our application B C P1 this generality is sucient.
Proposition 4.3 (Cauchy). The solutions to a dierential equation form a
local system. More precisely if D : OB ! OB is a dierential equation of
order n the kernel
B U7! LD(U) = f f 2 OB(U) j Df = 0 g
is a sheaf of C-vector spaces locally isomorphic to Cn
B.
Remark 4.4. Even if ai are algebraic functions, the solutions of a dierential
equation will not be algebraic. This can be seen at even the simplest example
@tf = f on B = C, which is solved by the exponential function. Hence
working with sheaves of holomorphic functions is crucial.
Remark 4.5. As LD OB parallel transport in LD along a path
de
33. nes
an analytic continuation of a local solution of D. It follows, that solutions
of dierential equations can be extended along arbitary paths.
There is also a
at vector bundle that we can associate to D.
De
34. nition 4.6. Let D = n
Pn1
i=0 ai i be a dierential equation. De
35. ne
ED := OB e0; : : : ; en1 to be the trivial vector bundle of rank n, and
together with the connection on E, de
36. ned by setting
rei :=
(
ei+1 for i n 1 Pn1
i=0 aiei for i = n 1
(3)
The relation between ED and LD is clari
38. GROUPOIDS, LOCAL SYSTEMS AND DIFFERENTIAL EQUATIONS 7
Proposition 4.7. Let D be a dierential equation. The local systems LD
and Loc(ED
_) are canonically isomorphic.
Proof. Given a
at co-section : ED ! OB, the function f = (e0) is a
solution of the dierential equation since
nf = (rn
nX1
e0) = (
i=0
aiei) =
nX1
i=0
ai(ei) =
nX1
i=0
ai(ri
e0) =
nX1
i=0
aiif:
Conversly, we de
39. ne a map
LD ! E_; f7!
nX1
i=0
(if)i
where i is the dual basis of the vector bundle E = OB e0; : : : ; en1 .
Flatness is checked easily:
nX1
r(
i=0
(if)i) =
nX1
i=0
(i+1f)i +
nX1
i=0
(if)(ri)
=
Xn
i=1
(if)i1 +
nX1
i=0
(if)(i1 ain1)
= (nf)n1
nX1
i=0
(if)ain1 = 0:
Given a dierential equation D, the section e0 2 H0(B;ED) is always
cyclic. We can also go in the other direction:
De
40. nition 4.8. Given a
at vector bundle (E;r) of rank n, a cyclic vec-tor
2 H0(B;E) and a nowhere vanishing vector
41. eld . The dierential
equation associated to this datum is
D(E;r; ; ) := n
nX1
i=0
ai i : OB ! OB
where ai are the coecient in the expansion n =
Pn1
i=0 aii, i := ri
.
Question: Does D(E;r; ; ) really depend on ?