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The General System has been described in [Zeigler76] and [ZPK00] with the standpoints to define (1) the time base, (2) the admissible input segments, (3) the system states, (4) the state trajectory with an admissible input segment, (5) the output for a given state.
A Timed Event System defining the state trajectory associated with the current and event segments came from the class of General System to allows non-deterministic behaviors in it [Hwang2012]. Since the behaviors of DEVS can be described by Timed Event System, DEVS and RTDEVS is a sub-class or an equivalent class of Timed Event System.
A timed event system is a structure
Given a timed event system , the set of its behaviors is called its language depending on the observation time length. Let be the observation time length. If , -length observation language of is denoted by , and defined as
We call an event segment a -length behavior of , if .
By sending the observation time length to infinity, we define infinite length observation language of is denoted by , and defined as
We call an event segment an infinite-length behavior of , if .
State Transition System
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable . The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions f, the Laplace transform is the integral
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal-second per cubic metre, or in the MKS system the rayl per square metre, while that of specific acoustic impedance is the pascal-second per metre, or in the MKS system the rayl. There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electric current resulting from a voltage applied to the system.
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
In quantum field theory, the LSZ reduction formula is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.
In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems.
A Dynkin system, named after Eugene Dynkin is a collection of subsets of another universal set satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems or d-system. These set families have applications in measure theory and probability.
In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
In mathematics, a π-system on a set is a collection of certain subsets of such that
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
DEVS abbreviating Discrete Event System Specification is a modular and hierarchical formalism for modeling and analyzing general systems that can be discrete event systems which might be described by state transition tables, and continuous state systems which might be described by differential equations, and hybrid continuous state and discrete event systems. DEVS is a timed event system.
SP-DEVS abbreviating "Schedule-Preserving Discrete Event System Specification" is a formalism for modeling and analyzing discrete event systems in both simulation and verification ways. SP-DEVS also provides modular and hierarchical modeling features which have been inherited from the Classic DEVS.
FD-DEVS is a formalism for modeling and analyzing discrete event dynamic systems in both simulation and verification ways. FD-DEVS also provides modular and hierarchical modeling features which have been inherited from Classic DEVS.
In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
The behavior of a given DEVS model is a set of sequences of timed events including null events, called event segments, which make the model move from one state to another within a set of legal states. To define it this way, the concept of a set of illegal state as well a set of legal states needs to be introduced.
A segment of a system variable in computing shows a homogenous status of system dynamics over a time period. Here, a homogenous status of a variable is a state which can be described by a set of coefficients of a formula. For example, of homogenous statuses, we can bring status of constant and linear. Mathematically, a segment is a function mapping from a set of times which can be defined by a real interval, to the set [Zeigler76],[ZPK00], [Hwang13]. A trajectory of a system variable is a sequence of segments concatenated. We call a trajectory constant if its concatenating segments are constant.
In theoretical computer science, DEVS is closed under coupling [Zeigper84] [ZPK00]. In other words, given a coupled DEVS model , its behavior is described as an atomic DEVS model . For a given coupled DEVS , once we have an equivalent atomic DEVS , behavior of can be referred to behavior of atomic DEVS which is based on Timed Event System.
Quantum stochastic calculus is a generalization of stochastic calculus to noncommuting variables. The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing measurement, as in quantum trajectories. Just as the Lindblad master equation provides a quantum generalization to the Fokker–Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical Langevin equations.
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.