Introductory discrete math. Discrete math definition.

Introductory discrete math

Discrete math definition

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers.

More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.

Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.

Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

In the university curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science support course; its contents was somewhat haphazard at the time. The curriculum has thereafter developed in conjunction to efforts by ACM and MAA into a course that's basically intended to develop mathematical maturity in freshmen; as such it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics it is sometimes seen a preparatory course, not unlike precalculus in this respect.

The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

Analysis of algorithms

In computer science, the analysis of algorithms is the determination of the amount of resources (such as time and storage) necessary to execute them. Most algorithms are designed to work with inputs of arbitrary length. Usually, the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps (time complexity) or storage locations (space complexity).

Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem. These estimates provide an insight into reasonable directions of search for efficient algorithms.

In theoretical analysis of algorithms it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. Big O notation, Big-omega notation and Big-theta notation are used to this end. For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, or in O(log(n)), colloquially "in logarithmic time".

Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency. However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called a hidden constant.

Exact (not asymptotic) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called model of computation.

A model of computation may be defined in terms of an abstract computer, e.g., Turing machine, and/or by postulating that certain operations are executed in unit time. For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log2 n + 1 time units are needed to return an answer.

Circuit design

The process of circuit design can cover systems ranging from complex electronic systems all the way down to the individual transistors within an integrated circuit. For simple circuits the design process can often be done by one person without needing a planned or structured design process, but for more complex designs, teams of designers following a systematic approach with intelligently guided computer simulation are becoming increasingly common.In integrated circuit design automation, the term "circuit design" often refers to the step of the design cycle which outputs the schematics of the integrated circuit. Typically this is the step between logic design and physical design.[1]

Formal circuit design usually involves the following stages:

• sometimes, writing the requirement specification after liaising with the customer

• writing a technical proposal to meet the requirements of the customer specification

• synthesising on paper a schematic circuit diagram, an abstract electrical or electronic circuit that will meet the specifications

• calculating the component values to meet the operating specifications under specified conditions

• performing simulations to verify the correctness of the design• building a breadboard or other prototype version of the design

and testing against specification

• making any alterations to the circuit to achieve compliance

• choosing a method of construction as well as all the parts and materials to be used

• presenting component and layout information to draughtspersons, and layout and mechanical engineers, for prototype production

• testing or type-testing a number of prototypes to ensure compliance with customer requirements

• signing and approving the final manufacturing drawings• post-design services (obsolescence of components etc.)

Unicast

In computer networking, unicast transmission is the sending of messages to a single network destination identified by a unique address.

Multicast

In computer networking, multicast (one-to-many or many-to-many distribution) is group communication where information is addressed to a group of destination computers simultaneously. Multicast should not be confused with physical layer point-to-multipoint communication.

Group communication may either be application layer multicast or network assisted multicast, where the latter makes it possible for the source to efficiently send to the group in a single transmission. Copies are automatically created in other network elements, such as routers, switches and cellular network base stations, but only to network segments that currently contain members of the group.

Network assisted multicast may be implemented at the Internet layer using IP multicast, which is often employed in Internet Protocol (IP) applications of streaming media, such as Internet television scheduled content (but not media-on-demand) and multipoint videoconferencing, but also for ghost distribution of backup disk images to multiple computers simultaneously. In IP multicast the implementation of the multicast concept occurs at the IP routing level, where routers create optimal distribution paths for datagrams sent to a multicast destination address.

Network assisted multicast may also be implemented at the Data Link Layer using one-to-many addressing and switching such as Ethernet multicast addressing, Asynchronous Transfer Mode (ATM) point-to-multipoint virtual circuits (P2MP) or Infiniband multicast.

Router

A router is a networking device that forwards data packets between computer networks. A router is connected to two or more data lines from different networks (as opposed to a network switch, which connects data lines from one single network). When a data packet comes in one of the lines, the router reads the address information in the packet to determine its ultimate destination. Then, using information in its routing table or routing policy, it directs the packet to the next network on its journey. This creates an overlay internetwork. Routers perform the "traffic directing" functions on the Internet. A data packet is typically forwarded from one router to another through the networks that constitute the internetwork until it reaches its destination node.

The most familiar type of routers are home and small office routers that simply pass data, such as web pages, email, IM, and videos between the home computers and the Internet. An example of a router would be the owner's cable or DSL router, which connects to the Internet through an ISP. More sophisticated routers, such as enterprise routers, connect large business or ISP networks up to the powerful core routers that forward data at high speed along the optical fiber lines of the Internet backbone. Though routers are typically dedicated hardware devices, use of software-based routers has grown increasingly common.

Computer security

Computer security is security applied to computing devices such as computers and smartphones, as well as computer networks such as private and public networks, including the whole Internet. The field includes all the processes and mechanisms by which digital equipment, information and services are protected from unintended or unauthorized access, change or destruction, and is of growing importance due to the increasing reliance of computer systems in most societies.[1] It includes physical security to prevent theft of equipment and information security to protect the data on that equipment. It is sometimes referred to as "cyber security" or "IT security". Those terms generally do not refer to physical security, but a common belief among computer security experts is that a physical security breach is one of the worst kinds of security breaches as it generally allows full access to both data and equipment.

Cybersecurity is the process of applying security measures to ensure confidentiality, integrity, and availability of data. Cybersecurity assures protection of assets, which includes data, desktops, servers, buildings, and most importantly, humans. The goal of cybersecurity is to protect data both in transit and at rest. Countermeasures can be put in place in order to ensure security of data. Some of these measures include, but are not limited to, access control, awareness training, audit and accountability, risk assessment, penetration testing, vulnerability management, and security assessment and authorization.

Database

A database is an organized collection of data. The data is typically organized to model aspects of reality in a way that supports processes requiring information. For example, modelling the availability of rooms in hotels in a way that supports finding a hotel with vacancies.

Database

Database management systems are computer software applications that interact with the user, other applications, and the database itself to capture and analyze data. A general-purpose DBMS is designed to allow the definition, creation, querying, update, and administration of databases. Well-known DBMSs include MySQL, PostgreSQL, Microsoft SQL Server, Oracle, Sybase and IBM DB2.

Database

A database is not generally portable across different DBMSs, but different DBMS can interoperate by using standards such as SQL and ODBC or JDBC to allow a single application to work with more than one DBMS.

Database

Database management systems are often classified according to the database model that they support; the most popular database systems since the 1980s have all supported the relational model as represented by the SQL language. Sometimes a DBMS is loosely referred to as a 'database'.

Deadlock

In concurrent programming, a deadlock is a situation in which two or more competing actions are each waiting for the other to finish, and thus neither ever does.In a transactional database, a deadlock happens when two processes each within its own transaction updates two rows of information but in the opposite order. For example, process A updates row 1 then row 2 in the exact timeframe that process B updates row 2 then row 1. Process A can't finish updating row 2 until process B is finished, but process B cannot finish updating row 1 until process A is finished. No matter how much time is allowed to pass, this situation will never resolve itself and because of this database management systems will typically kill the transaction of the process that has done the least amount of work.

In an operating system, a deadlock is a situation which occurs when a process or thread enters a waiting state because a resource requested is being held by another waiting process, which in turn is waiting for another resource held by another waiting process. If a process is unable to change its state indefinitely because the resources requested by it are being used by another waiting process, then the system is said to be in a deadlock.

Deadlock is a common problem in multiprocessing systems, parallel computing and distributed systems, where software and hardware locks are used to handle shared resources and implement process synchronization.In telecommunication systems, deadlocks occur mainly due to lost or corrupt signals instead of resource contention.