Analytic Function
Simplistically, a mathematical expression that is (at least locally) representable as a convergent power series on an open interval (in the Real Domain – the issue is more complex in the…Complex…Domain). Being expressed as a convergent power series means that the function is infinitely differentiable, therefore its value in the region around any given point... read more
Array
An array is a sequential list1 of values, where position in the sequence denotes some particular relevance to the user. Arrays may take any order. We allow degenerate forms to exist, so that scalars and vectors can also be considered arrays. When the positions in the listing require exactly two numbers2 for access, an array is often referred... read more
Basis
A set of orthogonal (mutually independent) vectors (or functions) that can be used to unambiguously identify exactly one region in an abstract space of interest. When used in the context of technical requirements: the list of topics (some or all of which might be functions) which, when we are talking about any one of them,... read more
Closed Form
Broadly, an expression solving a mathematical problem that is composed entirely of analytic functions, or other “well-behaved”, continuous mathematical functions. The solution is “exact” on one or more continuous ranges, involving no approximations (other than those applicable to measurement of the independent variables). Its applicability is complete within each interval of the underlying function(s). This... read more
Complex Number
A number comprised of two parts: a real part, and an imaginary part. An unpreferred number that is hard for non-SME’s to understand, usually the result of a complex calculation.
Constraint
In the legacy System Engineering context, any technical requirement that is not a Performance Requirement. Sometimes referred to as “Design Constraint”, but this is a misnomer. In the context of optimization, a limit on what values may be obtained by some measure contributing to, or a necessary correlative design feature of, an Objective Function.
Degree of Freedom (DoF)
In Systems Engineering, a $10 expression referring to the number of system characteristics that can take values without regard to interactions with those taken by any other system characteristics, i.e., “independent” variables. This concept is important in System Engineering, because it is the key to limiting the number of Performance Requirements we write. Without it,... read more
Dendrogram
A connected, acyclic graph. “Connected” means there are no loose parts of the graph. “Acyclic” means that any given node can be reached from any other node by exactly one path. These two properties mean that such graphs look like trees (“Dendron” is the Greek root word for “tree”). Many people use the word “taxonomy”... read more
Differential Algebraic Equation (DAE)
A DAE is a system of differential equations where not all of the terms share any single, continuous independent variable and/or derivatives with respect to it, and where some of the terms are purely algebraic (as opposed to differential). This last constraint means that the Jacobian of the system is singular, so the system of... read more
Domain
In Functional Analysis (mathematics): the set of permissible inputs for a function. In the internet: a cluster of internet addresses, all subordinate to some common node of autonomy, authority, or control.
Evaluate
In the System Engineering context, the process of specifying the actual collection of points for a value, rather than the criteria for their selection. The abstraction is borrowed from database design theory.
Function
1) In mathematics: a relation between a set of inputs and outputs. 2) In System Engineering: A means to impose a particular system state1. To be useful as an Engineering abstraction, a function has to relate to the way the system’s state variables are changed (or, held constant; see also Performance Requirement). The state... read more
Functional Decomposition
A decomposition process executed in the functional domain. The completeness criterion is to show that the sub-functions (“child functions”) effect the state change (or maintenance) that spans the function being decomposed (“parent function”). Re-parameterization of the system’s state vector is an acceptable means of demonstrating completeness, provided that the re-parameterization is at-least-locally1 deterministic2 in both... read more
Game Theory
Game Theory is a mathematical construct that attempts to model how rational interactions (e.g., competition, cooperation) should proceed. Use of the word “game” does not suggest that the context is frivolous; only that the people (and organizations) involved can have differing information and ignorance, goals and aversions, advantages and disadvantages, rewards and penalties influencing their courses of action.... read more
Graph Theory
A mathematical framework for characterizing properties of how dots are connected. Since connecting the dots is a primary focus of System Integration, such a framework can come in pretty handy when you’re trying to understand the implications of dot-to-dot connectedness. If this sounds a bit too esoteric, consider the concept of a Specification Tree and... read more
If and only If (IFF)
A condition possessed by a set of criteria which, in and of themselves, constitute the entire necessary and sufficient conditions for something to be either true or false.
Imaginary Number
Two distinct definitions exist: Any number involving the square root of -1. Typically used in numerically intensive, analytically oriented things like signal processing, control theory, and electromagnetism to assist in (for example) stability analysis, but also sometimes used just to make the math work out right (which was, in fact, the original application). Schedule dates... read more
Jacobian Matrix
The matrix formed by all first-order partial derivatives of a vector-valued function. The Jacobian can be interpreted as a “map” of how a system will respond to small changes in the value of its independent variables: a matrix of “influence coefficients” or “sensitivity coefficients”. If time is an independent variable, the Jacobean provides limited insight... read more
Making Everything Right (The Orthogonality Problem)
Most of the time, System Engineers want to ensure that they're dealing with exactly one thing. The concept of "orthogonality" is borrowed from the abstract mathematics of topology for use as an organizational artifice. Being an abstraction, however, the pervasiveness of the idea isn't always immediately apparent. Several practical usage examples are discussed. read more.
Mathematics
Patterns of manipulating quantities or representations thereof. The subject patterns may be either concrete or abstract. read more.
Normalization
The mathematical process of transforming a data set into a Gaussian (“normal”) distribution, so that standard statistical concepts (e.g, mean, median, variance, kurtosis) are valid. Unless the data have been verified as conforming or transforming into the Gaussian shape, no parametric statistics should be trusted. Note that this verification process “consumes” degrees of freedom in... read more
Numerical Methods
Numerical methods use numerical approximation rather than symbolic algebra to estimate solutions to Engineering problems to within some practical limit of uncertainty. In this context, “practical” can refer to the limits of calculational precision or to the limit of Experimental Uncertainty during a model-validating test, or to the limit of tolerance in an allocated requirement. Most of these... read more
Numerical Stability
A property of computational processes such that small changes in the input(s) result in small changes of the output(s). “Small” is, of course, relative to the nature of the computation and usage of the result. Issues of numerical stability can produce a highly non-linear component of uncertainty, leading to highly variable tolerances.
Objective Function
In optimization theory, the equation (function) to be maximized or minimized, subject to constraints. If the concept is applied to SE, then the objective function for development is parameterized by the Measures of Requirement as allocated by way the development specification. We have to be careful with this concept, however, because it is tempting to... read more
Ontology
A type of classification scheme identifying the kinds of things that exist and the relationships between them. The “things” can be either concrete or abstract, or a mixture of the two. A real-world instance of a thing can possess the attributes of more than one ontological class, so this concept admits to the notion of... read more
Order
The dimensionality of an array; the number of indices necessary to access exactly one position (a.k.a., “element”). Rank and degree are sometimes used as synonyms, but can have special implications in certain usages. Scalars have zero order, vectors are of order one and matrices are of order two. Arrays can be of any order.
Ordinary Differential Equation (ODE)
In analysis, an ODE is an equation where all of the terms share the same (single) continuous independent variable and/or derivatives with respect to it. The independent variable is often time, but can be any other technically relevant variable for which data are available. Contrast with PDE and DAE.
Parameter
A heavily over-loaded term. The meaning strongly depends on the usage, and no attempt is made here to cover all of them. Typically, some kind of variable in the sense of algebra…but not always!
Partial Differential Equation (PDE)
A PDE is a differential equation having more than one continuous independent variable (and derivatives with respect to thereto). Contrast with ODE and DAE.
Precision
The granularity (fineness) with which a number is quoted, sometimes described to as “the repeatability limits”. Contrast with accuracy, tolerance, and uncertainty.
Range (set theory)
The set from which a value may be selected for some given variable.
Regression Analysis
A statistical technique wherein data are compared to certain pre-defined geometric shapes to produce meaningful descriptive simplifications like “average” and “standard deviation”. When applied to test data, this process is sometimes called “data reduction”, where the statistics are referred to as the “reduced data”. I have observed that many of the non-cognoscenti think that regression... read more
Scalar
Formally, an individual element of a vector space. A zero-th order array. Mostly, that isn’t very helpful. As a practical matter, a scalar is a variable that needs exactly one value to describe it. Contrast with vector.
Simulation
Simulation is a type of analysis proceeding through chains of events or circumstances (that is, as a matter of cause-and-effect). The term is often used when the solution of time-domain ODE’s is required to calculate an end state, or a series of intermediate states. It can, however, be used with respect to other independent variables... read more
Singular (Matrix)
A singular matrix is one that cannot be inverted. This means that solution techniques that use division by any such matrix cannot be employed. Here, “dividing” refers to the notion that because we have (for example) Distance = Rate * Time, we can calculate rate by inverting the equation: Time = Distance / Rate. If... read more
Stable
“Stability” is a condition in which a system’s rates of change with respect to one or more independent variables are bounded. The term is used in many contexts and may seem to have different definitions in each, but they all simplify to something very close to this one.
State
A particular combination of values for a defining set of mutually independent parameters by which some system or circumstance is modeled. The parameters are sometimes referred to as “state variables”. In the sense of how the word “value” is used herein, a state is more accurately defined as some combination of ranges on a set... read more
Taxonomy
A classification scheme identifying the kinds of things that exist and the similarity between them, usually presented in the form of a dendrogram. The classes are mutually exclusive: any given thing can belong to at most one class1. The concept does not admit to the notion of multiple inheritance. Contrast with ontology. Footnotes In a... read more
Tensor-valued
A tensor is an array of mathematical expressions that expresses dependency of the array’s output parameters on the input parameters. In the context of System Engineering, tensor-valued parameters admit to explicit relationships between numbers and circumstances. A “value” can (therefore) then be a recipe1 for generating a number based on circumstances, rather than just some... read more
Uncertainty
Uncertainty expresses the extents of the range within which a value can exist, quoted as a distribution about some nominal value. The magnitude(s) of the uncertainty in any given situation can be influenced by (for example) the calibration accuracy of an instrument, or the numerical precision of a calculation. Unlike tolerance, the concept of uncertainty... read more
Value
(1) In the technical vernacular, a value is the exact amount of some particular property, usually represented as a number. (2) As generally used herein, a value is a non-empty set of points in a space associated with a topical parameter. This notion arises because exact amounts are rarely encountered outside of theoretical constructs1. ... read more
Vector
Formally, an element of a vector space. An array of order 1. Mostly, that doesn’t help very much. As a practical matter, a simple list of scalar-valued data (usually numeric) in which their position in the list denotes some particular meaning of relevance to the user. Different combinations of values (in the various positions in... read more