Cohen-Macaulay Representations

Geoffrey Sparks

Description

Cohen-Macaulay Representations Graham J. Leuschke Roger Wiegand SYRACUSE UNIVERSITY E-mail address: UNIVERSITY OF NEBRASKA–LINCOLN E-mail address:

• geometric mckay correspondence
• maximal cohen
• macaulay modules
• restrictions on modules over a ring restrict
• structure of the ring
• burban-drozd
• complete classification to date
• representation theory
• finite groups

Sujets

• Ausländer
• Wiegand
• Syracuse University
• Restrict
• Finite
• Module
• Theory
• Macaulay
• Complete

Informations

• Publié par: erza0
• Nombre de lectures: 14
• Langue: English

Extrait

Database Modelling in UML
By Geoffrey Sparks, [e-mail] : http://www.sparxsystems.com.au
Originally published in Methods & Tools e-newsletter : http://www.martinig.ch/mt/index.html
Data Profile (as proposed by RationalIntroduction
Software).
Some familiarity with object-orientedWhen it comes to providing reliable,
design, UML and relational databaseflexible and efficient object persistence
modelling is assumed.for software systems, today's designers
and architects are faced with many
choices. From the technological The Class Model
perspective, the choice is usually
between pure Object-Oriented, Object- The Class Model in the UML is the
Relational hybrids, pure Relational and main artefact produced to represent the
custom solutions based on open or logical structure of a software system.
proprietary file formats (eg. XML, It captures the both the data
OLE structured storage). From the requirements and the behaviour of
vendor aspect Oracle, IBM, Microsoft, objects within the model domain. The
POET and others offer similar but techniques for discovering and
often-incompatible solutions. elaborating that model are outside the
scope of this article, so we will assume
This article is about only one of those the existence of a well designed class
choices, that is the layering of an model that requires mapping onto a
object-oriented class model on top of a relational database.
purely relational database. This is not
to imply this is the only, best or The class is the basic logical entity in
simplest solution, but pragmatically it the UML. It defines both the data and
is one of the most common, and one the behaviour of a structural unit. A
that has the potential for the most class is a template or model from
misuse. which instances or objects are created
at run time. When we develop a logical
We will begin with a quick tour of the model such as a structural hierarchy in
two design domains we are trying to UML we explicitly deal with classes.
bridge: firstly the object-oriented class
model as represented in the UML, and When we work with dynamic
secondly the relational database model. diagrams, such as sequence diagrams
and collaborations, we work with
For each domain we look only at the objects or instances of classes and their
main features that will affect our task. inter-actions at run-time.
We will then look at the techniques
and issues involved in mapping from The principal of data hiding or
the class model to the database model, encapsulation is based on localisation
including object persistence, object of effect. A class has internal data
behaviour, relationships between elements that it is responsible for.
objects and object identity. We will Access to these data elements should
conclude with a review of the UML be through the class's exposed
behaviour or interface. Adherence toPerson Person Class attributes :
the encapsulated
- Address: CAddress
data
# Name:
St i# Age: double
+ getAge() : int
+ setAge(n)
+ getName() :
St i+setName(s)
A simple person class
with no state or Class operations:
behaviour shown the behaviour
Attributes and operations define the state of an object
trun-time and the capabilities or behaviour of the
bj t
Figure 1 - Classes, attributes and operations
this principal results in more relationship that is most interesting: for
maintainable code. example an Address class may be
associated with a Person class. The
mapping of this relationship into theBehaviour
relational data space requires some
Behaviour is captured in the class care.
model using the operations that are
defined for the class. Operations may Aggregation is a form of association
be externally visible (public), visible to that implies the collection of one class
children (protected) or hidden of objects within another. Composition
(private). By combining hidden data is a stronger form of aggregation that
with a publicly accessible interface and implies one object is actually
hidden or protected data manipulation, composed of others. Like the
a class designer can create highly association relationship, this implies a
maintainable structural units that complex class attribute that requires
support rather than hinder change. careful consideration in the process of
mapping to the relational domain.
Relationships and Identity
While a class represents the template
Association is a relationship between 2 or model from which many object
classes indicating that at least one side instances may be created, an object at
of the relationship knows about and run time requires some means of
somehow uses or manipulates the other identifying itself such that associated
side. This relationship may by objects may act upon the correct object
functional (do something for me) or instance. In a programming language
structural (be something for me). For like C++, object pointers may be
this article it is the structural passed around and held to allowobjects access to a unique object with which to locate or re-create the
instance. required object.
Often though, an object will be
destroyed and require that it be re- The Relational Model
created as it was during its last active
instance. These objects require a The relational data model has been
storage mechanism to save their around for many years and has a
internal state and associations into and proven track record of providing
to retrieve that state as required. performance and flexibility. It is
essentially set based and has as its
Inheritance provides the class model fundamental unit the 'table', which is
with a means of factoring out common composed of a set of one or more
behaviour into generalised classes that 'columns', each of which contains a
then act as the ancestors of many data element.
variations on a common theme.
Inheritance is a means of managing Tables and Columns
both re-use and complexity. As we will
see, the relational model has no direct A relational table is collection of one
counterpart of inheritance, which or more columns each of which has a
creates a dilemma for the data unique name within the table construct.
modeller mapping an object model Each column is defined to be of a
onto a relational framework. certain basic data type, such as a
number, text or binary data. A table
Navigation from one object at run time definition is a template from which
to another is based on absolute table rows are created, each row being
references. One object has some form an instance of a possible table instance.
of link (a pointer or unique object ID)
A class hierarchy FamilyPerson Aggregation captures the concept
showing a generalised ofcollection or composition between classes
person class from
which other classes
are derived
Parent Child The main relationships we are
interested in are Association,
Aggregation and Inheritance.
2
1..n Thesedescribe the ways classes interactAssociation captures
or relate to each
a having or using
other
relationship between
classes
Figure 2 - UML Class model notationShare
A Person may
own a set of
Shares
A Person is composed of
a strict set of ID
0..n documents (having nA Person may
elements)reside at zero
or more
addresses 0..1
Address Person ID Document
0..n 0..n
1 n
An Address may Three forms of the Aggregation relationship. The weak form is
have zero or depicted with an unfilled diamond head, the strong form
more Persons in (composition) with a filled head.
residence
Figure 3- Aggregation Relationships
Public Data Access Database stored procedures provide a
means of extending database
The relational model only offers a functionality through proprietary
public data access model. All data is language extensions used to construct
equally exposed and open to any functional units (scripts). These
process to update, query or manipulate functional procedures do not map
it. Information hiding is unknown. directly to entities, nor have a logical
relationship to them.
Behaviour
Navigation through relational data sets
The behaviour associated with a table is based on row traversal and table
is usually based on the business or joins. SQL is the primary language
logical rules applied to that entity. used to select rows and locate
instances from a table set.
Constraints may be applied to columns
in the form of uniqueness Relationships and Identity
requirements, relational integrity
constraints to other tables/rows, The primary key of a table provides the
allowable values and data types. unique identifying value for a
particular row. There are two kinds of
Triggers provide some additional primary key that we are interested in:
behaviour that can be associated with firstly the meaningful key, made up of
an entity. Typically this is used to data columns which have a meaning
enforce data integrity before or after within the business domain, and
updates, inserts and deletes. second the abstract unique identifier,
such as a counter value, which have nobusiness meaning but uniquely identify Having looked at the two domains of
a row. We will discuss this and the interest and compared some of the
implications of meaningful keys later. important features of each, we will