Paper Title - Indian Journal of Scientific Research

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31 Οκτ 2013 (πριν από 4 χρόνια και 9 μέρες)

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Climbing R
obot



Akash K Singh, PhD

IBM Corporation

Sacramento, USA

akashs@us.ibm.com



Abstract

Climbing is a current challenge in robotics.

Observation of vertebrate climbing
proves

that in arboreal

environment like on flat ground locomotion is carried out as

well
by the trunk as by the limbs. Thus to elucidate one

extreme a
trunk
-
driven concept was chosen as base for our

design process.
Often climbing robots are highly special
ized

machines. They only
are able to move on one or two

different substrates. And they
often are expensive solutions,

built for a special application. The
climbing robot we report

about is designed as a modular system.
The modular idea

influences all
developmental areas of a
mechatronic system



the mechanical, the electronic as well as the
software part.

By using different modules, the robot is capable to
climb on

different substrates made of a wide variety of materials.
The

mechanical demands of clim
bing are strong and rule the

design. Substrate contact has to be established and broken

actively. A sensor concept was developed, which allows for

choosing relevant sensor ranges, depending on the desired

type of
control. The configuration described in thi
s paper is

composed of
only two different module types. Due to the

features of these
modules, the robot is capable to climb on

pipe
-
like structures.
The speed of the robot depends on the

gait chosen.


Keywords
-

robotics, service robotics, bionics, climbing
,

biomechatronics
.



I.


I
NTRODUCTION


Importance of automatic inspection is increasing. In

buildings,
cables and pipes have to be inspected

periodically to prevent
or to detect damages. In historic

buildings infrastructure has
grown over many years. Often

maps showing the exact course
of supply lines do not

exist. In new buildings, passable supply
routes are

installed for human beings performing inspection
tasks.

But these routes consume a lot of expensive space. Thus

our main idea is to build small robots,

which can move in

also
small canals along linear structures.

Also in industrial
environments robot system are

needed for moving on flat
ferromagnetic structures. Such

robots use active or passive
magnetic systems to lock on

the surface. Other robotic
syst
ems are automatically

cleaning window areas. Therefore
often vacuum

technologies are chosen to achieve a climbing
ability.

Common goals for mobile robots are locomotion,

idiomotion, manipulation, navigation, orientation,

imitation
and cooperation [2], [22]
. For a climbing robot,

the challenge
in locomotion is high locomotion ability.

The robot has to
move horizontally and vertically,

ascending and descending.
The environment is

unstructured. Often a special infrastructure
for the robot

system cannot be inst
alled, so the robot system
has to use

the available environmental conditions for its
locomotion




cables and pipes with diameters of 10 mm to 80 mm,

ferromagnetic flat surfaces, windows, and walls.


II.

CLIMBING ROBOTS
PRINCIPLE


In the last decades, a lot of
climbing robots have been

developed. We can classify current climbing robots by

application, by technical parameters like size, payload,

and
power consumption. As described before, climbing

robots deal
with the generation of mot ion and with the

substrate c
ontact.
Substrate contact plays an important

role. In most cases, the
robot will be destroyed, if

substrate contact fails. For our
systematic construction

process, classifications by substrate
contact and by

locomotion principle were realized.

In the last
decades, robots often owned stiff backbones.

Due to the space
requirements of control hardware, energy

source and
communicat ion devices, rigid body

constructions were made.
But zoological studies show that

the locomotion of vertebrates
and non
-
vertebrates
is

mainly driven by a flexible trunk
[1],[4],[7],[8],[22].

The work of Fischer and Lehmann [5]
illustrates that

the main
locomotors

organ of vertebrates is the
trunk,

vertebral column. Furthermore, small, ancestral
mammals

obtain up to 50% of the spatial
gain per movement
cycle

by trunk motion.


A.

Classification by Substrate Contact

There are common techniques to generate adhesion to

the
substrate, classified by the nature of forces. In reality,

there is
a cross
-
coupling of adhesion techniques; often a

clear

differentiation is impossible.

• pneumatic


forces generated by low pressure,

• magnetic


magnetic force generated by permanent or

electro
magnets on ferromagnetic surfaces

• electrostatic


forces generated by a electrostatic field

between the robot an
d the surface (e.g. SRI wallclimbing

robot),

• chemical


by using chemical substances to generate

adhesive forces (e.g. Stickybot [9]),

• positive locking


by using the geometrical structures

of the
surface (e.g Spinybot [19], RatNic), and

• hybrid


by
coupling at least two of the above listed

methods
of force generation


This list is only a short summary of adhesion

technologies. A
more complete overview is given in [11].

Another important
attribute is the need of external energy

to hold up the contact
to the substrate. In this point, we

can simply differ into active
and passive contact. Active

means in this case, that we
continuously need power to

keep the connection. Passive
means that after the

connection is established, there is no need
of any power

for maintaining the connection. In the same way,
we can

classify the process of establishment and release of

locking.


B.

Classification by Characteristics of Locomotion

There are a lot of locomotion methods in robotics. Most of

them are also used for climbin
g robots. If we neglect

substrate
contact, we find the same principles like in

terrestrial
locomotion. Only one additional method is

shown in [11]. In
comparison to [10], the methods can be

differed into

• walking


using limb
-
like construction for walking

on

the
substrate

• frame walking


using at least two rigid frames in an

alternating way

• sliding


sliding on the surface of the substrate

• wheeled


using wheels to drive on a mostly flat

substrate

• tracked


using chains or chain
-
like constructions

• brachiation


using at least two arms to brachiate


The degree of freedom (DOF) of trunk motion called

level of
mobility is important for the design process. This

is not in
every case the number of joints integrated in the

robot. Robots
climbing on a sin
gle straight pipe like

shown in [2] have a
level of mobility of one. Robots

walking on flat surfaces have
a level of mobility of three,

if they can walk in two direct ions
and turn around one

axis. A level of mobility of six means full
climbing

mobility.


C.

Properties of Modular Robots


Locomotion on pipe
-
like substrates needs variation in

the
DOF of locomotion from 2D to 6D. Due to this, a

modular
design is chosen. In most cases, these modules

are functional
units. By combination of these single units,

systems with a
new functional quality may be created. In

most cases, one
module is unable to move. By connecting

a lot of modules, we
get chained systems, which are able

to realize complex
movements. One big advantage is that

we can combine
modules in diff
erent ways, so we can

create walking, crawling
or swimming systems. Most

important properties are [3],[21]:

• status of homogeneity,

• status of self
-
configurability,

• torques or forces acting via the module

• power consumption,

• and communication.

If al
l elements of the system are identical, there will be

homogeneity of the modules. If the structure consists of

several, non
-
identical modules, it is called heterogeneous.

In
both cases, the position of the module in the robot

defines its
function. A modula
r robot will be selfconfigurable,

if the robot
identifies its structure by itself.

In this case, the modules could
be put together in different

ways to achieve an adaptable
behavior without any effort

for reconfiguration.


III.

CONCEPT OF THE MODUL
AR SYSTEM


As

shown before, for climbing the main focus is on one

hand
on the substrate contact and one the other hand on

the
generation of motion. The idea is, to implement the

different
technologies of adhesion on one robot. The

adequate
technology of adhesion depend
s on the substrate

properties.
These properties could be:

• material

• stiffness

• surface roughness

• ferromagnetic properties

• shape

Also the needs of motion generation depend on the

substrate
properties. A robot doesn’t need a high degree of

freedom
cl
imbing on a flat surface. A robot climbing on

cylindrical
substrates (brackets, wire, pipes) needs a

higher degree of
freedom. Furthermore some adhesion

technologies need a
special way of motion for the

establishment or for the
releasing of substrate conta
ct. So,

the whole mechanical
system of the robot has to be varied

for different tasks.
Because of this, an overall modular

concept is chosen
. It

consists of active modules and passive modules. Passive

means in this case, that they are electronically passiv
e. So,

a
self
-
identification of the system is not possible at this

state.

The passive modules are connectors. They are

necessary to
connect the motion und adhesive modules, so

a power
transmission could be achieved. The mounting

modules are
another type of

modules. They are necessary

to fix equipment
on the system like electronic devices and

accumulator.
Important is a common mechanical

interface. By this, an
intermodular connectivity and very

flexible robotic system are
achieved.



A.

Modular Electronic Syste
m

The robot system is a modular one. Hence, the

electronics has
to be designed in such a way, too. Because

the mechanical
structure isn’t framed, the electrical system

has to be
adaptable. At the current state, the robot should be remote

controlled.
Therefore, a common game pad is used. A

Host PC
with an I²C
-
Converter is used to communicate

with an A/D
-
Converter and a motor controller. The motor

controller is able
to control 21 servo motors. If a higher

number of drives will
be necessary, a second con
troller

can be integrated. Sensor
signal can be read by an A/Dconverter

connected to the I²C
-
Bus, too. The sensor

signals must have a voltage of ±10 V.

The power supply is made by a 7.4V LiPo
-
Cell with

a
capacity of 2 Ah. The maximum current is limited to
20

A,
due to the recommendation made by the manufacturer.

For the
design of modules four main restrictions have

been made:

• Actuators have to be controlled by a special PWMsignal

• Sensor signals has to be conditioned to a 10V
-
level

• The voltage of power

supply is limited to 7,4V

• The overall maximum current is limited to 20A.

By considering this, different modules can be designed

and
integrated in the system with any other

recommendations to
the electronic system.


B.


Modular Mechanical System

To assure m
echanical compatibility of different

modules, a
mechanical interface is designed. The modules

must
have four
tap holes in a square
.
The modules are fixed to the connectors
by four screws

of the size M3x8. The modules should have the
shape of a

cube. The ed
ge length has to be 56mm. The
montage has

to be done by a user. A self
-
organizat ion and
selfassembling

is not planned.


C.

Designed Modules

As described before, restrictions to the design of the

modules
have been done due to the modular idea for the

system. A
t the
current state different types of modules

have been developed.


Adhesion Modules


As described before, there are a lot of different

substrates. And
there is no adhesion technology working

on every substrate.
Hence, different adhesion modules are

design for the modular
system.

The

recommendations are similar to all adhesive
modules. For

the design process a locking force of 50 N and a
locking

torque of 1 Nm.

• magnetic module

• adaptive gripper module

• gripper module for a special diameter of subs
trate

• vacuum module


Motion Modules

Motion could be generated in a translatory or
rotator
way. At
the current state only rotational modules are

realized. The
drives integrated as actuators have a torque

of 3.0 Nm
. The
modules are

• rotational module wit
h DOF = 1

• rotational module with DOF = 2



IV.

CONFIGURATIONS FOR C
LIMBING ON PIPE
-
LIKE
STRUCTURES


A.

Examples of Kinematical Configurations

Possible configuration could be robot structures with

serial,
parallel or mixed structures. At the beginning, only

serial
robot configurations are considered. Two gripper

modules are
chosen, one at each end of the robot. Level of

mobility is
defined by the number and the arrangement of

locomotion
modules between the grippers. The structure

of the robot
should be adapte
d to the climbing task.

The level of

mobility
is one. The locomotion module is a rotational

one. To achieve
a linear motion, three locomotion

modules have to be
combined. An inchworm
-
like motion

pattern is achieved.

The
second example is a configuration fo
r climbing on

a straight
pipe and for changing from this pipe to a parallel

one. For this
task, the level of mobility is two. The two

motions are
translational. To achieve the two translations,

we have to
combine only three locomotion modules.



B.

Reference
System for Climbing on Straight Pipe
-
like

Structure


Configuration

The modular system is suitable for the design of

climbing
robots. For the task “climbing on a straight pipelike

structure”
a simple serial configuration is chosen,

consisting of three
rotational motion modules and two

adaptive gripper modules.
The modules are connected. The configuration is completed
by the

motor controller and the power supply.


Kinematics


The kinemat ics of the robots is very simple. The tool

centre
point (TCP) has to

be controlled in its three

mechanical
freedoms. Therefore, inverse kinemat ical

calculations have to
been done.



For the three coordinates of the TCP we get the

following

equations:

The coordinates of the TCP can be combined to

and
the coordinates of the
drives can be generalized to

The
Jacobian
-

Matrix is used for the inverse kinemat ics

A change
of the angular position of the rotational drive is

calculated by


The calculation is implemented in the control software
.



We consider the following
anycast
field equations defined
over an open bounded piece of
network

and /or feature space
. They describe the dynamics of the mean
anycast
of
each of
n
ode

populations.




We give an
interpretation of the various parameters and
functions that appear in (1),

is finite piece of
nodes

and/or
feature space and is represented as an open bounded set of
. The vector

and

represent points in
. The
function

is the normalized sigmoid function:





It describes the relation between the
input

rate

of
population

as a function of the
packets

potential, for
example,

We note

the

dimensional vector
The

function

represent the initial conditions, see below. We
note

the


dimensional vector

The

function

represent external
factors

from
other
network

areas. We note

the

dimensional
vector
The

matrix of functions

represents the connectivity between
populations

and

see below. The

real values

determine the threshold of activity for eac
h
population, that is, the value of the
nodes

potential
corresponding to 50% of the maximal activity.

The
real
positive values

determine the slopes of the
sigmoids at the origin. Finally the
real positive values

determine the speed at which each
anycast
node

potential decreases exponentially toward its real value.

We also introduce the function

defined by

and the
diagonal

mat rix
I
s the intrinsic
dynamics of the population given by the linear response of
data transfer
.

is replaced by

to use the
alpha function response. We use

for simplicity
although our analysis applies to more general intrinsic
dynamics. For the sake, of generality, the propagation delays
are not assumed to be identical for all populations,

hence they
are described by a matrix

whose element
is
the propagation delay between population

at

and
population

at

The reason for this assumption is that it is
still unclear from
anycast

if propagation delays are
independent of the populations. We assume for technical
reasons that

is continuous, that is

Moreover
packet

data indicate that

is not a symmetric
function i.e.,

thus no assumption is
made about this symmetry unless otherwise stated.

In order to
compute the
righthand s
ide of (1), we need to know the
node
potential factor


on interval

The value of

is
obtained by considering the maximal delay:




Hence we choose


C.

Mathematical Framework

A convenient functional setting for the non
-
delayed
packet

field equations is to use the space

which is a
Hilbert space endowed with the usual inner product:




To give a meaning to (1), we defined the history space

with

which is the Banach phase space associated with equation (3).
Using the notation

we
writ
e (1) as



Where




Is the linear continuous operator satisfying

Notice that most of the papers on this
subject assume

infinite, hence requiring




Proposition 1.0


If the following assumptions are satisfied.

1.



2.

The external current


3.



Then for any

there exists a unique solution

to (3)

Notice that this result gives existence on

finite
-
time
explosion is impossible for this delayed differential equation.
Nevertheless, a particular solution could grow indefinitely, we
now prove that this cannot happen.


D.

Boundedness of Solutions

A valid model of neural networks should only feature b
ounded
packet node

potentials.


Theorem 1.0

All the trajectories are ultimately bounded by
the same constant

if


Proof

:Let us defined

as


We note





Thus,

i
f





Let us show that the open
route

of

of center 0 and radius

is stable under the dynamics of equation. We know
that

is defined for all

and that

on

the boundary of
. We consider three cases for the initial
condition
If

and set

Suppose that

then

is defined and belongs to

the closure of

because
is closed, in effect to

we also have

because

Thus we deduce that for

and small enough,

which contradicts the definition of T. Thus

and
is stable.

Because f<0 on

implies that
. Finally we consider the case
. Suppose that

then

thus

is monotonically
decreasing and reaches the value of R in finite t
ime when

reaches

This contradicts our assumption. Thus



Proposition

1.1

:
Let

and


be measured simple functions
on

for

define




Then

is a measure on
.



Proof :
If

and if

are disjoint members of
whose union is

the countable additivity of

shows that



Also,

so that

is not identically
.

Next, let

be as before, let

be the distinct values
of t,and let

If

the


a
nd

Thus (2)
holds with

in place of
. Since
is the disjoint union
of the sets

the first half of our
proposition implies that (2) holds.



Theorem

1.1
:
If

is a compact set in the plane whose
complement is connected, if

is a continuous complex
function on

which is holomorphic in the interior of , and if

then there exists a polynomial

such that

for all
.

If the interior of
is
empty, then part of the hypothesis is vacuously satisfied, and
the conclusion holds for every
. Note that
need
to be connected.

Proof:
By Tietze’s theorem,

can be extended to a
continuous function in the plane, with compact support. We
fix one such extension and denote it again by
.

For any

let

be the sup
remum of the numbers


Where

and

are subject to the
condition
. Since

is uniformly continous, we
have


F
rom now on,

will be
fixed. We shall prove that there is a polynomial

such that






By (1),


this proves the theorem.

Our first objective is the
construction of a funct
ion

such that for all




And



Where

is the set of all points in the support of

whose
distance from the complement of

does not
. (Thus
contains no point which is “far within”
.) We construct
as the convolution of

with a smoothing function A. Put

if
put




And define



For all complex
. I
t is clear that
. We claim that




The constants are so adjusted in (6) that (8) holds. (Compute
the integral in polar coordinates), (9) holds simply because

has compact support. To compute (10), express

in polar
coordinates, and note that




Now define



Since

and

have compact support, so does
. Since



And

if

(3) follows from (8). The
difference quotients of

converge boundedly to the
corresponding partial derivatives, since
. Hence
the last expression in (11) may be differentiated under the
integral sign, and we obtain



The last equality depends on (9). Now (10) and (13) give (4).
If we write (13) with

and

in place of

we see
that

has continuous partial derivatives,
if we can show that

in

where

is the set of all

whose
distance from the complement of

exceeds

We shall do
this by showing that




Note that

in
, since

is holomorphic there. Now
if

then

is in the inte
rior of

for all

with

The mean value property for harmonic functions
therefore gives, by the first equation in (11),



For all

, we have now proved (3), (4), and (5)

The
definit ion of

shows that
is compact and that

can be
covered by finitely many open discs

of radius

whose centers are not in

Since

is
connected, the center of each

can be joined to

by a
polygonal path in
. It follows that each
contains a
compact connected set

of diameter at least

so that

is connected a
nd so that

with
. There are functions

and constants

so that the inequalities.




Hold for

and

if



Let

be the complement of

Then
is an
open set which contains

Put

and

for


Define



And



Since,



(18) shows that

is a finite linear combination of the
functions

and
. Hence

By (20), (4), and
(5) we have



Observe that the inequalities (16) and (17) are valid with

in
place of

if

and
Now fix
, put

and

estimate the integrand in (22) by (16) if

by (17) if

The integral in (22) is then
seen to be less than the sum of



And



Hence (22) yields



Since

and

is connected,
Runge’s theorem shows that

can be uniformly
approximated on

by polynomials. Hence (3) and (25) show
that (2) can be satisfied.

This completes the proof.


Lemma

1.0

:
Suppose

the space of all
continuously differentiable functions in the plane, with
compact support. Put



Then the following “Cauchy formula” holds:



Proof:

This may be deduced from Green’s theorem. However,
here is a simple direct proof:

Put

real


If

the chain rule gives



The right side of (2) is therefore equal to the limit, as

of






For each

is periodic in

with peri
od
. The
integral of

is therefore 0, and (4) becomes



As

uniformly. This gives (2)


If

and
, then

, and so

satisfies the condition
.
Conversely,



and so if

satisfies
, then the subspace generated by the
monomials
, is an ideal. The proposition gives a
classification of the monomial ideals in
: they
are in one to one correspondence with the subsets

of

satisfying
. For example, the monomial ideals in

are exactly the ideals
, and the zero ideal
(corresponding to the empty set
). We write


for the ideal corresponding to

(subspace generated by the
).


LEMMA
1.1
. Let

be a subset of
. The the ideal

generated by

is the monomial ideal
corresponding to




Thus, a monomial is in

if and only if it is divisible by one
of the

PROOF. Clearly

satisfies
, and
.
Conversely, if
, then

for some
,
and
. The last statement follows from
the fact that
.

Let

satisfy
. From the geometry of
, it is clear that there is
a finite se
t of elements

of
such that


(The

are the corners of
) Moreover,

is
generated by the monomials
.


DEFINITION
1.0
.

For a nonzero ideal

in
, we let

be the ideal generated by





LEMMA
1.2

Let

be a nonzero ideal in
;
then
is a monomial ideal, and it equals

for some
.

PROOF. Since

can also be described as the ideal
gener
ated by the leading monomials (
rather than the leading
terms) of elements of
.


THEOREM
1.2
.

Every
ideal

in
is finitely
generated; more

precisely,

where
are any elements of

whose leading terms generate


PROOF.

Let
. On applying the division algorithm,
we find

,
where either

or no monomial occurring in it is divisible
by any
. But
, and therefore
, implies that
every monomial occurring in

is divisible by one in
. Thus
, and
.


DEFINITION
1.1
.

A finite subset

of an

ideal

is a standard (
bases for

if
. In other words, S is a
standard basis if the leading term of every element of
is
divisible by at least one of the leading terms of the
.


THEOREM 1.
3

The ring

is Noetherian i.e.,
every ideal is finitely generated.


PROOF.

For


is a principal ideal domain,
which means that every ideal is generated by single element.
We shall prove the theorem by induction on
. Note that the
obvious map


is an
isomorphis m


this simply says that every polynomial

in

variables

can be expressed uniquely as a
polynomial in

with coefficients in
:



Thus the next lemma will complete the proof


LEMMA 1.3.

If

is Noetherian, then so also is


PROOF. For a polynomial





is called the degree of
, and

is its leading coefficient.
We call 0 the leading coefficient of the polynomial 0.

Let

be an ideal in
. The leading coefficients
of the polynomials in

form an ideal

in
, and since

is Noetherian,
will

be finitely generated. Let

be elements of

whose leading coefficients generate
, and
let
be the maximum degree of
. Now let

and
suppose

has degree
, say,

Then

, and so we can write



Now

has degree

.
By continuing in this way, we find that

With

a polynomial of
degree
.

For each
, let

be the subset of

consisting of 0 and the leading coefficients of all polynomials
in

of degree

it is again an ideal in
. Let

be polynomials of degree

whose leading
coefficients generate
. Then the same argument as above
shows that any polynomial

in

of degree

can be
written

With

of
degree
. On applying this remark repeatedly we find
that

Hence




an
d so the polynomials

generate



One of the great successes of category theory in computer
science has been the development of a “unified theory” of the
constructions underlying denotational semantics. I
n the
untyped
-
calculus, any term may appear in the function
position of an application. This means that a model D of the
-
calculus must have the property that given a term

whose
interpretation is

Also, the interpretation of a
functional abstraction like
.

is most conveniently
defined as a function from

, which must then

be
regarded as an element of
D
.

Let

be the
function that picks out elements of
D

to represent elements of

and

be the function that
maps elements of
D

to functions of
D.

Since

is
intended to represent the function

as an element of
D,
it
makes sense to require that


that is,



Furthermore, we often want to view every
e
lement of
D

as representing some function from
D to D

and
require that elements representing the same function be equal


that is




The latter condition is called extensionality. These conditions
together imply that

are inverses
---

that is,
D

is
isomorphic to the space of functions from
D to D

that can be
the interpretations of functional abstractions:

.Let us suppose we are working with the untyped
, we need a solution ot the equation

where A is some predetermined
domain containing interpretations for elements of
C.

Each
element of
D

corresponds to either an element of
A
or an
element of

with a tag. This equation can be
solved by finding least fixed points of the function

from domains to domains
---

that
is, finding domains
X

such that

and
such that for any domain
Y

al
so satisfying this equation, there
is an embedding of
X

to
Y

---

a pair of maps



Such that




Where

means that

in some
ordering representing their informat ion content. The key shift
of perspective from the domain
-
theoretic to the more general
category
-
theoretic approach lies in considering
F
not as a
function on domains, but as a
functor

on a category of
domains.
Instead of a least fixed point of the function,
F.


Definition 1.3
: Let
K
be a category and

as a
functor. A fixed point of
F

is a pair (A,a), where A is a
K
-
object
and

is an isomorphis m. A prefixed
point of F is a pair (A,a), where A is a
K
-
object
and a is any
arrow from F(A) to A


Definition 1.4 :
An

in a category
K

is a diagram
of the following form:



Recall that a cocone

of an

is a
K
-
object
X
and a collection of K

arrows

such
that

for all
. We sometimes write

as a reminder of the arrangement of

components

Similarly, a colimit
is a cocone with
the property that if

is also a cocone then there
exists a unique mediating arrow

such that for all
. Colimits of

are sometimes
referred to as
.

Dually, an

in
K
is a diagram of the following form:


A cone

of an

is a
K
-
object

X and a collection of
K
-
arrows

such that for all
. An
-
limit of an


is a cone

with the property that if
is also a cone, then there
exists a unique mediating arrow

such that for
all

. We write

(or just
) for the
distinguish initial object of
K,
when it h
as one, and

for the unique arrow from

to each
K
-
object A. It is also
convenient to write
to denote all of

except

and
. By analogy,

is
. For
the images of

and

under
F

we write

and


We write

for the
i
-
fold iterated composition of
F


that is,

,etc.

With these definitions we can state that every monitonic
function on a complete lattice has a least fixed point:


Lemma

1.4.

Let
K

be
a category with init ial object

and let

be a functor. Define the

by



If both

and
are
colimits,
then (D,d) is an intial F
-
algebra, where


is the mediat ing arrow from


to the
cocone




Theorem 1.
4

Let a DAG G given in which each node is a
random variable, and let a discrete conditional probability
distribution of each node given values of its parents in G be
specified. Then the product of these conditional distributions
yields a joint probability d
istribution P of the variables, and
(G,P) satisfies the Markov condition.


Proof.
Order the nodes according to an ancestral ordering. Let
be the resultant ordering. Next define.



Where
is the set of parents of
of in G and
is the specified conditional probability
distribution. First we show this does indeed yield a joint
probability distribution. Clearly,

for
all values of the variables. Therefore, to show we have a joint
distribution, as the variables range through all their possible
values, is equal to one. To that end, Specified conditional
distributions are the conditional distributi
ons they notationally
represent in the joint distribution.

Finally, we show the
Markov condition is satisfied. To do this, we need show for

that

whenever

Where
is the set of nondescendents of
of in G. Since
, we need only show
. First for a given
, order the
nodes so that all and only nondescendents of
precede
in the ordering. Note that this ordering depends on
, whereas
the ordering in the first part of the proof does not. Clearly then



follows





We define the
cyclotomic field to be the field


Where
is the
cyclotomic
polynomial.


has degree
over
since
has degree
. The roots of

are just the primitive
roots of unity, so the
complex embeddings of
are simply the
maps



being our fixed choice of primitive
root of unity. Note
that
for every
it follows that
for all
relat ively prime to
. In
particular, the images of the
coincide, so
is Galois over
. This means that we can
write
for
without much fear of
ambiguity; we will do so from now on, the identification being
One advantage of this is that one can easily talk
about cyclotomic fields being extensions of one another,or
intersections or compositums; all of
these things take place
considering them as subfield of

We now investigate some
basic properties of cyclotomic fields. The first issue is whether
or not they are all distinct; to determine this, we need to know
which roots of uni
ty lie in
.
Note, for example, that if
is odd, then
is a
root of unity. We will show that
this is the only way in which one can obtain any non
-
roots of unity.


LEMMA 1.
5

If
divides
, then

is contained in

PROOF. Since
we have
so the
result is clear


LEMMA 1.
6

If
and
are relatively prime, then




and



(Recall the

is the compositum of


PROOF. One checks easily that
is a primit ive
root
of unity, so that



Since
this implies that


We know that
has degree

over


, so we must have




and




And thus that


PROPOSITION 1.
2

For any
and



And


here
and
denote the least common multiple and
the greatest common divisor of
and
respectively.


PROOF. Write
where the
are distinct primes. (We allow
to be zero)



An entirely similar computation shows that


Mutual informat ion measures the information t ransferred
when

is sent and

is received, and is defined as


In a noise
-
free channel,
each

is uniquely connected to the
corresponding

, and so they constitute an input

output pair

for which


bits; that is, the
transferred informat ion is equal to the self
-
information that
corresponds to the input

In a very noisy channel, the output
and input
would be completely uncorrelated, and so

and also
that is, there is no
transference of information. In general, a given channel will
operate between these two ext remes.

The mutua
l informat ion
is defined between the input and the output of a given channel.
An average of the calculat ion of the mutual informat ion for all
input
-
output pairs of a given channel is the average mutual
information:


bits per
symbo
l
.
This calculation is done over the input and output
alphabets. The average mutual information.

The following
expressions are useful for modifying the mutual informat ion
expression:


Then


Where

is
usually called the equivocation.

In a sense, the equivocation
can be seen as the informat ion lost in the noisy channel, and is
a function of the backward conditional probability. The
observation of an output symbol
provides

bits of information. This difference is the
mutual information of the channel.

Mutual Information:
Properties

Since


The mutual information fits the condition


And by interchanging input and output it is also true that


Where


This last entropy is usually called the noise entropy.

Thus, the
informat ion transferred through the channel is the difference
betw
een the output entropy and the noise entropy.
Alternatively, it can be said that the channel mutual
informat ion is the difference between the number of bits
needed for determining a given input symbol before knowing
the corresponding output symbol, and the

number of bits
needed for determining a given input symbol after knowing
the corresponding output symbol

As the channel mutual information expression is a difference
between two quantities, it seems that this parameter can adopt

negative values. However, and is spite of the fact that for some

can be larger than
, this is not
possible for the average value calculated over all the outputs:


Then


Because this expression is of the form


The above expression can be applied due to the factor
which is the product of two probabilit ies, so
that it behaves as the quantity
, which in this expression is
a dummy variable that fits the condition
. It can be
concluded that the average mutual information is a non
-
negative number. It can also be equal to zero, when the input
and the
output are independent of each other. A related
entropy called the joint entropy is defined as




Theorem 1.5:
Entropies of

the binary erasure channel (BEC)

The BEC is defined with an alphabet of two inputs and three
outputs, with

symbol probabilities.

and transition
probabilities




Lemma
1
.7
.
Given an arbitrary restricted t ime
-
discrete,
amplitude
-
continuous channel whose restrictions are
determined by sets
and whose density functions exhibit no
dependence on the state
, let
be a fixed positive integer,
and
an arbitrary probability density function on
Euclidean
n
-
space.
for the density
and
.
For any real
number a, let


Then for each positive integer
, there is a code
suc
h that


Where


Proof: A sequence
such that


Choose the decoding set
to be
. Having chosen
and
, select
such that



Set
, If the process does not terminate
in a finite number of steps, then the sequences
and
decoding sets
form the desired code. Thus
assume that the process terminates after

steps. (Conceivably
). We will show

by showing that
. We proceed as
follows.

Let



E.

Algorithms

Let A be a ring. Recall that an
ideal

a
in A is a subset such
that

a is subgroup of A regarded as a group under addition;





The ideal generated by a subset S
of A is the intersection of all
ideals A containing a
-----

it is easy to verify that this is in fact
an ideal, and that it consist of all finite sums of the form

with

. When
, we
shall write
for the ideal it generates.

Let a and b be ideals in A. The set

is
an ideal, denoted by
. The ideal generated by
is denoted by
.
Note that
. Clearly
consists of all finite sums

with

and
, and if

and
, then
.Let

be an ideal of A. The set of cosets of
in A forms a ring
, and

is a homomorphism
.
The map

is a one to one correspondence
between the ideals of

and the ideals of

containing
An ideal

if
prime

if

and

or
. Thus

is prime if and only if

is nonzero and
has the property that

i.e.,
is an integral domain. An ideal

is
maximal
if

and there does not exist an ideal

contained strictly
between
and
. Thus
is maximal if and only if

has no proper nonzero ideals, and so is a field. Note that

maximal

prime. The ideals of

are all of the
form
, with

and

ideals in

and
. To see this,
note that if

is an ideal in

and
, then

and
. This
shows that

with



and




Let

be a ring. An
-
algebra is a ring

together with a
homomorphism
. A
homomorphism

of
-
algebra

is a homomorphism of rings

such that

for all
.

An
-
algebra
is said
to be
finitely generated

( or of
finite
-
type

over A) if there exist
elements

such that every element of
can be
expressed as a polynomial in the

with coefficients in
, i.e., such that the homomorphis m

sending

to
is surjective. A ring homomorphis m

is
finite,
and

is finitely generated as an A
-
module. Let

be a field, and let
be a
-
algebra. If

in
, then the map

is injective, we can identify
with its image, i.e., we can regard
as a subring of

. If 1=0
in a ring R, the R is the zero ring, i.e.,
.
Polynomi al
rings.

Let


be a field. A
monomial

in

is an
expression of the form

. The
total
degree
of the monomial is
. We somet imes abbreviate it
by
.
The elements of the
polynomial ring

are finite sums



With the obvious notions of equality, addition and
multiplicat ion. Thus the monomials from basis for

as a
-
vector space. The ring
is an integral domain, and the only units in i
t
are the nonzero constant polynomials. A polynomial

is
irreducible
if it is nonconstant and has only
the obvious factorizations, i.e.,

or

is
constant.
Di vision in
. The division algorithm allows
us to divide a nonzero polynomial into another: let

and

be polynomials in
with

then there ex
ist unique
polynomials

such that

with either

or deg

< deg
. Moreover, there is an algorithm for
deciding whether
, namely, find
and check
whether it is zero. Moreover, the Euclidean algorithm allows
to pass from finite set of generators for an ideal in
to a
single generator by successively r
eplacing each pair of
generators with their greatest common divisor.



(
Pure)
lexicographic
ordering (lex
).

Here monomials are
ordered by lexicographic(dictionary) order. More precisely, let

and

be two elements of
;
then

and

(lexicographic ordering) if, in
the vector difference
, the left most nonzero entry
is positive. For example,


. Note that this isn’t
quite how the dictionary would order them: it would put

after
.
Graded reverse
lexicographic order (grevlex).
Here monomials are ordered by
total degree, w
ith ties broken by reverse lexicographic
ordering. Thus,

if
, or

and in

the right most nonzero entry is negative. For
example:


(total degree greater)

.


Orderings

on

.
Fix an ordering on the
monomials in
.
Then we can write an element

of

in a
canonical fashion, by re
-
ordering its
elements in decreasing order. For example, we would write



as



or



Let

, in decreasing order:



Then we define.



The
multidegree
of


to be multdeg
(
)=
;



The

leading coefficient of
to be

LC(
)=
;



The

leading monomial of
to be

LM(
) =
;



The

leading term of
to be

LT(
) =


For the polynomial

the multidegree is
(1,2,1), the leading coefficient is 4, the leading monomial is
, and the leading term is
.
The di vision
algorithm i n

.
Fix a monomial ordering in
. Suppose given a polynomial

and an ordered set

of polynomials; the division algorithm then
constructs polynomials

and

such that

Where either

or no
monomial in

is divisible by any of

Step 1:
If
, divide

into

to get

If
, repeat the process until


(different
) with

not divisible by
. Now divide

into
, and so on, until

With

not divisible by
any

Step 2:
Rewrite
,
and repeat Step 1 with

for
:

(different

)
Monomi al i deals.
In general, an ideal

will contain a
polynomial without containing the individual terms of the
polynomial; for example, the ideal

contains
but not

or
.


DEFINITION
1.5
. An ideal

is
monomial

if


all

with
.

PROP
OSITI
ON 1
.3. Let
be a
monomial ideal,
and let
. Then
satisfies the condition

And

is the
-
subspace of

generated by the
.
Conversely, of

is a subset of

satisfying
, then the
k
-
subspace

of

generated by
is a monomial ideal.


PROOF. It is clear from its definit ion that a monomial ideal

is the
-
subspace of


generated by the set of monomials it
contains. If


and

.



If a permutation is chosen uniformly and at random from the

possible permutations in

then the counts

of
cycles of length

are dependent random variables. The joint
distribution of

follows from
Cauchy’s formula, and is given by



for
.


Lemma1.
7
For nonnegative integers


Proof.

This can be established directly by exploit ing
cancellation of the form

when

which occurs between the ingredients in Cauchy’s
formula and t
he falling factorials in the moments. Write
. Then, with the first sum indexed by

and the last sum indexed by

via the correspondence

we have



This last sum simplifies to the indicator

corresponding to the fact that if

then

for

and a random permutation in

must have
some cycle structure
.

The moments of

follow immediately as



We note for future reference that (1.4) can also be written in
the form



Where the

are independent Poisson
-
distribution random
variables that satisfy



The marginal distribution of cycle counts

provides a formula
for the joint distribution of the c
ycle counts

we find the
distribution of

using a combinatorial approach combined
with the inclusion
-
exclusion formula.


Lemma 1.8
.
For




Proof.

Consider the set

of all possible cycles of length

formed with elements chosen from

so that
. For each

consider the “pr
operty”

of
having

that is,
is the set of permutations

such that

is one of the cycles of

We then have
since the elements of

not in

must be permuted among themselves.

To use the inclusion
-
exclusion formula we need to calculate the term

which is
the sum of the probabilities of the
-
fold intersection of
properties, summing over all sets of
distinct properties.
There are two cases to consider. If the
properties a
re
indexed by
cycles having no elements in common, then the
intersection specifies how

elements are moved by the
permutation, and there are

permutations
in the intersection. Th
ere are

such intersections.
For the other case, some t wo distinct properties name some
element in common, so no permutation can have both these
properties, and the
-
fold intersection is empty. Thus



Finally, the inclusion
-
exclusion series for the number of
permutations having exactly

properties is



Which simplifies to (1.1
)

Returning to the original hat
-
check
prob
lem, we
substitute j=1 in (1.1
) to obtain the distribution of
the number of fixed points of a random permutation. For




and the moments of

follow from (1.
2
) with

In
particular, for

the mean and variance of
are both
equal to 1.

The joint distribution of

for any

has an expression similar to (1.7); this to
o can be
derived by inclusion
-
exclusion. For any

with




The joint moments of the first

counts

can be
obtained dir
ectly
from (1.2) and (1.3
) by setting



The limit distribution of cycle counts

It follows immediately from Lemma 1.2 that for each fixed

as



So that
converges in distribution to a random variable

having a Poisson distribution with mean

we use the
notation


where


to descri
be
this. Infact, the limit random variables are independent.


Theorem 1.6

The process of cycle counts converges in
distribution to a Poisson process of

with intensity
.
That is, as




Where the

are independent Poisson
-
distributed random variables with



Proof.

To establish the converges in distribution one shows
that for each fi
xed

as






Error rates

The proof of Theorem
says nothing about the rate of
convergence. Elementary analysis can be used to estimate this
rate when
. Using properties of alternating series with
decreasing terms, for





It follows that



Since



We see from (1.11) that the total variation distance between
the distribution

of

and the distribution

of


Establish the asymptotics of

under conditions

and

where


and

as

for some

We start with the expression





and


Where

refers to the quantity derived from
. It
thus follows that

for a constant
, depending on

and the

and computable explicitly
from (1.1)


(1.3), if Conditions

and

are satisfied
and if

from some

since, under these
circumstances, both

and

tend
to zero as

In particular, for polynomials and square
free polynomials, the relative

error in this asymptotic
approximation is of order

if



For

and

with




Where

under Conditions

and

Since, by the Conditioning Relation,


It follows by direct calculation

that



Suppressing the argument

from now on, we thus obtain







The first sum is at most
the third is bound by





Hence we may take




Required order under Conditions

and

if

If not,

can be replaced by
in the above, which has the required order, without the
restriction on the

implied
by
. Examining the
Conditions

and
it is perhaps surprising to
find that

is required instead of just

that is, that
we should need

to hold for some
. A first observation is that a similar problem arises
with the rate of decay of

as well. For this reason,

is
replaced by
. This makes it possible to replace condition

by the weaker pair of conditions
and
in the
eventual assumptions needed for

to be of order

the decay rate requirement of order

is
shifted from

itself to its first difference. This is needed to
obtain the right approximat ion
error for the random mappings
example. However, since all the classical applications make
far more stringent assumptions about the

than are
made in
. The crit ical point of the proof is seen where
the ini
tial estimate of the difference
. The factor

which should be small, contains a far tail element from

of
the form

which is only small if

being otherwise of order

for any

since

is in any case assumed. For

this gives rise
to a contribution of order

in the estimate of the
difference

which, in the
remainder of the proof, is translated into a contribution of
order
for differences of the form

finally leading to a
contribution of order

for any

in

Some improvement would seem to be possible, defining the
function

by

differences that are
of the form

can be directly
estimated, at a cost of only a single contribution of the form

Then, iterating the cycle, in which one
estimate of a difference in point probabilitie
s is improved to
an estimate of smaller order, a bound of the form


for any

could perhaps be attained, leading to a final error
estimate in order
for any
, to
replace

This would be of the ideal order
for large enough

but would still be coarser for
small




With
and

as in the previous section, we wish to show
that



Where

for any

under Condit ions

and
with
.
The proof uses sharper estimates. As before, we begin with the
formula




Now we observe that



We have




The approximation in (1.2
) is further simplified by noting that





and then by observing that



Combining the contributions of (1.
2)

(1.3
), we thus find tha

The quantity
is seen to be of the order claimed
under Conditions

and
, provided that

this supplementary condition can be removed if

is replaced by

in the definit ion of
, has the required order without the restriction on
the

implied by assuming that
Final
ly, a direct
calculation now shows that



Example 1.
0
.
Consider the point
. For
an arbitrary vector
, the coordinates of the point

are equal to the respective coordinates of the vector

and
. The vector r such as
in the example is called the position vector or the radius vector
of the point

. (Or, in greater detail:

is the radius
-
vector of

w.r.t an origin O). Points are frequently specified by their
radius
-
vectors. This presupposes the choice of O as the
“standard origin”. Let us summar
ize. We have considered

and interpreted its elements in t wo ways: as points and as
vectors. Hence we may say that we leading with the two
copies of

= {points},
= {vectors}

Operations with vectors: multiplication by a number, addition.
Operations with points and vectors: adding a vector to a point
(giving a point), subtracting two points (giving a vector).
treated in this

way is called an
n
-
dimensional affine space
.
(
An
“abstract” affine space is a pair of sets , the set of points and
the set of vectors so that the operations as above are defined
axiomat ically). Not ice that vectors in an affine space are also
known as “free vectors”. Intuitively, they are not fixed at
points and “float freely” in space. From
considered as an
affine space we can precede in two opposite directions:

as
an Euclidean space

as an affine s
pace

as a
manifold.Going to the left means introducing some extra
structure which will make the geomet ry richer. Going to the
right means forgetting about part of the affine structure; going
further in t
his direction will lead us to the so
-
called “smooth
(or differentiable) manifolds”. The theory of differential forms
does not require any extra geometry. So our natural direction
is to the right. The Euclidean structure, however, is useful for
examples and

applications. So let us say a few words about it:


Remark 1.
0
.

Euclidean geometry.

In

considered as
an affine space we can already do a good deal of geometry.
For example, we can consider lines and planes, and quadric
surfaces

like an ellipsoid. However, we cannot discuss such
things as “lengths”, “angles” or “areas” and “volumes”. To be
able to do so, we have to introduce some more definitions,
making
a Euclidean space. Namely, we define the length
of a vector

to be



After that we can also define distances between points as
follows:



One can check that the distance so defi
ned possesses natural
properties that we expect: is it always non
-
negative and equals
zero only for coinciding points; the distance from A to B is the
same as that from B to A (symmetry); also, for three points, A,
B and C, we have

(the
“triangle inequality”). To define angles, we first introduce the
scalar product of two vectors




Thus

. The scalar product is also denote by dot:
, and hence is often
referred to as the “dot
product” . Now, for nonzero vectors, we define the angle
between them by the equality



The angle itself is defined up to an integral mult iple
of

. For this definition to be consistent we have to ensure
that the r.h.s. of (
4
) does not exceed 1 by the absolute value.
This follows from the inequality



known as the Cauchy

Bunyakovsky

Schwarz inequality
(various combinations
of these three names are applied in
different books
). One of the ways of proving (5
) is to consider
the scalar square of the linear combination

where
. As

is a quadratic polyn
omial in

which is never negative, its discriminant must be less or
equal zero. W
rit ing this explicitly yields (5
). The triangle
inequality for distances als
o follows from the inequality (5
).


Example 1.1.
Consider the function


(the i
-
th
coordinate). The linear function

(the differential of

)
applied to an arbitrary vector

is simply
.From these
examples follows that we can rewrite

as



which is the standard form. Once again: the partial
derivatives
in (1
) are just the coefficients (depending on
);

are linear functions giving on an arbitrary vector

its
coordinates

respectively. Hence





Theorem 1.
7
.

Suppose we have a parametrized curve

passing through

at

and with the
velocity vector

Then




Proof.
Indeed, consider a small increment of the parameter
, Where
. On the other hand, we
have

for an
arbitrary vector
, where

when

.
Combining it together, for the increment of


we
obtain



For a certain

such that
when

(we used the linearity of
). By the definit ion, this
means that the derivative of

at

is exactly
. The statement of the theorem can be expressed
by a simple formula:




T
o calculate the value Of

at a point

on a given vector

one can take an arbitrary curve passing Through

at

with

as the velocity vector at
and calculate the usual
derivative of

at
.


Theorem 1.8
.
For functions
,






Proof. Consider an arbitrary point

and an arbitrary vector

stretching from it. Let a curve

be such that

and
.

Hence


at

and



at

Formulae (1) and (2
) then immediately follow from
the corresponding formulae for the usual derivative Now,
almost without change the theory generalizes to functions
taking values in

instead of
. The only difference is
that now the differential of a map

at a point

will be a linear function taking vectors in

to vectors in
(instead of
) . For an arbitrary vector





+


Where

when

. We have

and




In this matrix notation we have to write vectors as vector
-
columns.


Theorem 1.
9
. For an arbitrary parametrized curve

in
, the differential of a

map

(where
) maps the velocity vector

to the velocity
vector of the curve

in





Proof.

By the definition of the velocity vector,



Where

when
. By the definition of the
differential,



Where

when
. we obtain




For some

when
. This precisely means
that

is the velocity vector of
. As every
vec
tor attached to a point can be viewed as the velocity vector
of some curve passing through this point, this theorem gives a
clear geometric picture of

as a linear map on vectors.



Theorem 1.
10

Suppose we have two maps

and

where

(open
domains). Let
. Then the differential of
the composite map

is the composition of the
differentials of

and





Proof.
We can use the description of the differential
.
Consider a curve

in

with the velocity vector
.
Basically, we need to know to which vector in
it is taken
by
. the curve
. By the
same theorem, it equals the image under

o
f the
Anycast
Flow

vector to the curve

in
. Applying the
theorem once again, we see that the velocity vector to the
curve
is the image under
of the vector
.
Hence

for an arbit rary vector

.


Corollary 1.0
.
If we denote coordinates in
by

and in
by
, and write



Then the chain rule can be expressed as follows:



Where

are taken from (
1). In other words, to get

we have to substitute into (
2) the expression for

from (3
). This can also be expressed by the
following matrix formula:





i.e., if

and

are expressed by matrices of part ial
derivatives, then

is expressed by the product of
these matrices. This is often written as



Or



Where it is assumed that the dependence o
f

on

is given by the map
, the dependence of

on

is given by the map

and the dependence of
on
is given by the composition
.


Definiti on 1.6
.
Consider an open domain
. Consider
also another copy of
, denoted for distinction
, with
the standard coordinates
. A system of coordinates
in the open domain

is given by a map

where

is an open domain of
, such that the
following three conditions are satisfied :

(1)


is smooth;

(2)


is invertible;

(3)


is also smooth


The coordinates of a point

in this system are the
standard coordinates of

In other words,



Here the variables

are the “new” coordinates of
the point



Example 1.2
.
Consider a curve in

specified in polar
coordinates as



We can simply use the chain rule. The map

can be
considered as the composition of the maps
. Then, by the chain
rule, we have



Here

and

are scalar coefficients de
pending on
,
whence the partial derivatives

are vectors
depending on point in
. We can compare this with the
formula in the “standard” coordinates:
.
Consider the vectors
. Explicitly we have



From where it follows that these vectors make a basis at all
points except for the origin (where
). It is instructive to
sketch a

picture, drawing vectors corresponding to a point as
starting from that point. Notice that

are,
respectively, the velocity vectors for the curves


and
. We can
conclude that for an arbitrary curve given in polar coordinates
the velocity vector will have components

if as a basis
we take




A characteristic feature of the
basis

is that it is not
“constant” but depends on point. Vectors “stuck to points”
when we consider curvilinear coordinates.


Proposition 1.3
.

The velocity vector has the same
appearance in all coordinate systems.

Proof.


Follows directly from the chain rule and the
transformation law for the basis
.In particular, the elements
of the basis

(originally, a formal notation) can be
understood directly as the velocity vectors of the coordinate
lines

(all coordinates but

are fixed).
Since we now know how to handle velocities in arbitrary
coor
dinates, the best way to treat the differential of a map

is by its action on the velocity vectors. By
definition, we set



Now

is a linear map that takes vectors attached to a
point

to vectors attached to the point




In particular, for the differential of a function we always have



Where

are arbitrary coordinates. The form of the
differential does not change when we perform a change of
coordinates.


Example 1
.3
Consider a 1
-
form in

given in the
standard coordinates:



In the polar coordinates we will have
, hence



Substituting into
, we get



Hence

is the formula for

in the polar
coordinates. In particular, we see that this is again a 1
-
form, a
linear combination of the differentials of coordinates with
functions as coefficients. Secondly, in a more conceptual way,
we can define a 1
-
form in a domain

as a linear function on
vectors at every point of
:



If
, where
. Recall that the
differentials of functions were defined as linear funct
ions on
vectors (at every point), and

at every point
.


Theorem 1.9
.
For arbit rary 1
-
form

and path
, the
integral

does not change if we change parametrizat ion of

provide the orientation remains the same.

Proof:
Consider

and


As

=





Let

be a rational prime and let

We write

for

or this section. Recall that

has degree

over

We wish to show that

Note that

is a root of

and thus is an algebraic
integer; since

is a r
ing we have that

We
give a proof without assuming unique factorizat ion of ideals.
We begin with some norm and trace computations. Let

be
an integer. If
is not divisible by

then

is a primit ive

root of unity, and thus its conjugates are

Therefore




If

does divide

then

so it has only the one
conjugate 1, and

By linearity of the
trace, we find that


We also need to compute the norm of
. For this, we use
the factorization




Plugging in

shows that




Since the

are the conjugates of
this shows
that

The key result for determining the
ring of integers

is the following.


LEMMA 1.
9





Proof.
We saw above that

is a mult iple of

in

so the inclusion

is immediate.
Suppose now that the inclusion is strict. Since
is an ideal of

containing

and
is
a maximal ideal of
, we must have

Thus we can write



For some

That is,

is a unit in



COROLLARY 1.1

For any



PROOF. We have



Where the

are the complex embeddings of

(which we
are really viewing as automorphisms of
) with the usual
ordering. Furthermore,

is a multiple of

in

for every

T
hus


Since the trace is also a
rational integer.


PROPOSITION 1.4

Let

be a prime number and let

be the

cyclotomic field. Then


Thus

is an
integral basis for
.

PROOF. Let

and write


With

Then




By the linearity of the trace and our above calculations we find
that

We also have

so

Next consider the
algebraic integer


This is an
algebraic in
teger since

is. The same argument as
above shows that

and continuing in this way we find
that all of the

are in
. This completes the proof.



Example

1.
4

Let
, then the local ring

is simply
the subring of

of rational numbers with denominator
relatively prime to
. Note that this ring
is not the
ring
of
-
adic integers; to get
one must complete
. The usefulness of

comes from the fact that it h
as
a particularly simple ideal structure. Let
be any proper ideal
of

and consider the ideal

of

We claim
that

That is, that

is generated by the
elements of

in

It is clear from the definition of an
ideal that

To prove the other inclusion,
let

be any element of
. Then we can write

where

and

In particular,

(since

and

is an ideal), so

and

so

Since

this implies that

as claimed.We can use this
fact to determine all of the ideals of

Let

be any ideal
of
and consider the ideal factorization of
in

write it as

For some

and some ideal

relat ively prime to

we claim first that

We now find that



Since

Thus every ideal of

has the form

for some

it follows immediately that
is noetherian. It i
s also now
clear that
is the unique non
-
zero prime ideal in
. Furthermore, the inclusion

Since

this map is also surjection, since the
residue class of

(with

and
) is
the image of

in

which makes sense since

is
invertible in

Thus the map is an isomorphis m. In
particular, it is now abundantly clear that every non
-
zero
prime ideal of
is maximal.

To show that
is a
Ded
ekind domain, it remains to show that it is integrally
closed in
. So let

be a root of a polynomial with
coefficients in

write this polynomial as

With

and

Set

Multiplying by

we find that

is the root of a monic polynomial with coefficients in

Thus

since

we have
. Thus
is integrally close in



COROLLARY
1.
2. Let

be a number field of

degree

and let

be in

then


PROOF. We assume a bit more Galois theory than usual for
this proof. Assume first that

is Galois. Let

be an
element of

It is clear that

since

this shows
that
. Taking the product
over all

we have

Since

is
a rational integer and

is a free
-
module of rank



W
ill have order

therefore


This completes the proof. In the general case, let

be the
Galois closure of

and set



F.

Control
for Climbing Motion

The robot can move in two different modes. The first

mode is
a stepping mode. In the stepping mode, the order

of the
grippers is constant. For example, the upper gripper

can be
moved but it will stay above the other gripper. The

second
mode is a flipping mode. For example, the robot is

flipping
and the upper gripper is moved downwards

passing the other
gripper.

The control software is programmed as a state
machine.

Every stepping mode is implemented as state. As

described before, the cal
culation of the kinematics is

different
due to the gripper, which is fixed on the

substrate. When both
grippers are locked on the substrate,

another state is needed.
There are state for a flipping

motion, for a stepping motion, a
state for the two
-
gripperl
ocking,

and an initializing state.


G.

Functional Tests

First qualitative test have been done. The climbing

motion is
very stable. The robot climbs ascending and

descending in a
flipping mode and a stepping mode.

Those modes could be
interpreted as gaits. One

gait is a

stepwise climbing as
describes before. The other gait is a

flipping motion pattern
.

In
the flipping gait the power consumption is about 14

Watts at a
climbing speed of 0.7m/s. The maximum step

length is about
510 mm. The robot is able to overste
p

obstacles with a height
of 100 mm.



V.

CONCLUSION


The modular robot system consists of different

heterogenic
types of modules, passive connector elements,

control
hardware, power supply. Due to the modular

concept it is
possible to configure different set
ups of the

robot.

In this way,
the mechanics of the robot can be adapted

to the requirements
of the climbing task. In comparison to

other climbing robots, a
generalist system is realized. The

system is remote
-
controlled
by the user via game pad. Its

mass d
epends on the
configuration and is in the range of

m = 1...2 kg. A sensory
system is capable of being

integrated for detecting the contact
between robot and

substrate. Safety and robustness of the
locking on the

substrate can be controlled.

A reference sys
tem
is built, capable to climb pipe
-
like

substrates. Servo drives are
suitable for this design of

robots. In future the system will be
enhanced, to be able to

climb not only on pipe
-
like structures,
but also on flat

surfaces. Different modules could be com
bined
to

climbing robots, optimized to different applications. In

this
way, robot configuration could be tested for service

robots, for
industrial robots or for scientific robots.




A.

Authors and Affiliations

Dr Akash Singh is working with IBM Corporation a
s an IT
Architect and has been designing Mission Critical System and
Service Solutions; He has published papers in IEEE and other
International Conferences and Journals.

He joined IBM in Jul 2003 as a IT Architect which
conducts research and design of High

Performance Smart Grid
Services and Systems and design mission critical architecture
for High Performance Computing
Platform

and Computational
Intelligence and High Speed Communication systems. He is a
member of IEEE (Institute for Electrical and Electron
ics
Engineers), the AAAI (Association for the Advancement of
Artificial Intelligence) and the AACR (American Association
for Cancer Research). He is the recipient of numerous awards
from World Congress in Computer Science, Computer
Engineering and Applied
Computing 2010, 2011, and IP
Multimedia System 2008 and Billing and Roaming 2008. He is
active research in the field of Artificial Intelligence and
advancement in Medical Systems. He is in Industry for 18
Years where he performed various
roles

to provide t
he
Leadership in Information Technology and Cutting edge
Technology.

VI.

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