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 or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. 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.
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency.
Signal processing is a subfield of mathematics, information and electrical engineering that concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal.
An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in automata theory. Abstraction of computing processes is used in both the computer science and computer engineering disciplines and usually assumes a discrete time paradigm.
Hidden Markov Model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states.
A communication channel or simply channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used to convey an information signal, for example a digital bit stream, from one or several senders to one or several receivers. A channel has a certain capacity for transmitting information, often measured by its bandwidth in Hz or its data rate in bits per second.
In communication systems, signal processing, and electrical engineering, a signal is a function that "conveys information about the behavior or attributes of some phenomenon". In its most common usage, in electronics and telecommunication, this is a time varying voltage, current or electromagnetic wave used to carry information. A signal may also be defined as an "observable change in a quantifiable entity". In the physical world, any quantity exhibiting variation in time or variation in space is potentially a signal that might provide information on the status of a physical system, or convey a message between observers, among other possibilities. The IEEE Transactions on Signal Processing states that the term "signal" includes audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. In a later effort of redefining a signal, anything that is only a function of space, such as an image, is excluded from the category of signals. Also, it is stated that a signal may or may not contain any information.
Computer simulation is the reproduction of the behavior of a system using a computer to simulate the outcomes of a mathematical model associated with said system. Since they allow to check the reliability of chosen mathematical models, computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics, astrophysics, climatology, chemistry, biology and manufacturing, human systems in economics, psychology, social science, health care and engineering. Simulation of a system is represented as the running of the system's model. It can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions.
Network traffic simulation is a process used in telecommunications engineering to measure the efficiency of a communications network.
In systems science, a sampled-data system is a control system in which a continuous-time plant is controlled with a digital device. Under periodic sampling, the sampled-data system is time-varying but also periodic; thus, it may be modeled by a simplified discrete-time system obtained by discretizing the plant. However, this discrete model does not capture the inter-sample behavior of the real system, which may be critical in a number of applications.
In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case, calculus methods can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis. On the other hand, discrete choice analysis examines situations in which the potential outcomes are discrete, such that the optimum is not characterized by standard first-order conditions. Thus, instead of examining “how much” as in problems with continuous choice variables, discrete choice analysis examines “which one.” However, discrete choice analysis can also be used to examine the chosen quantity when only a few distinct quantities must be chosen from, such as the number of vehicles a household chooses to own and the number of minutes of telecommunications service a customer decides to purchase. Techniques such as logistic regression and probit regression can be used for empirical analysis of discrete choice.
Not to be confused with computer engineering.
The Global Environmental Multiscale Model (GEM), often known as the CMC model in North America, is an integrated forecasting and data assimilation system developed in the Recherche en Prévision Numérique (RPN), Meteorological Research Branch (MRB), and the Canadian Meteorological Centre (CMC). Along with the NWS's global forecast system (GFS), which runs out to 16 days, the ECMWF's Integrated Forecast System (IFS), which runs out 10 days, the Naval Research Laboratory Navy Global Environmental Model (NAVGEM), which runs out eight days, and the UK Met Office's Unified Model, which runs out to six days, it is one of the five predominant synoptic scale medium-range models in general use.
The MEMO model is a Eulerian non-hydrostatic prognostic mesoscale model for wind-flow simulation. It was developed by the Aristotle University of Thessaloniki in collaboration with the Universität Karlsruhe. The MEMO Model together with the photochemical dispersion model MARS are the two core models of the European zooming model (EZM). This model belongs to the family of models designed for describing atmospheric transport phenomena in the local-to-regional scale, frequently referred to as mesoscale air pollution models.
In statistics and machine learning, discretization refers to the process of converting or partitioning continuous attributes, features or variables to discretized or nominal attributes/features/variables/intervals. This can be useful when creating probability mass functions – formally, in density estimation. It is a form of discretization in general and also of binning, as in making a histogram. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand.
A limited dependent variable is a variable whose range of
possible values is "restricted in some important way." In econometrics, the term is often used when
estimation of the relationship between the limited dependent variable
of interest and other variables requires methods that take this
restriction into account. For example, this may arise when the variable
of interest is constrained to lie between zero and one, as in
the case of a probability, or is constrained to be positive,
as in the case of wages or hours worked.
A digital signal is a signal that is being used to represent data as a sequence of discrete values; at any given time it can only take on one of a finite number of values. This contrasts with an analog signal, which represents continuous values; at any given time it represents a real number within a continuous range of values.
LIMDEP is an econometric and statistical software package with a variety of estimation tools. In addition to the core econometric tools for analysis of cross sections and time series, LIMDEP supports methods for panel data analysis, frontier and efficiency estimation and discrete choice modeling. The package also provides a programming language to allow the user to specify, estimate and analyze models that are not contained in the built in menus of model forms.
In mathematics and, in particular, mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.
