where is the length of the string takes in the discrete formulation the form that for the outermost points and the equation of motion are

This formulation is common in the theory of discrete choice models, and makes it easier to extend to certain more complicated models with multiple, correlated choices, as well as to compare logistic regression to the closely related probit model.

Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil ( based on working out some examples ) was striking and novel.

In the latent variable formulation of the multinomial logit model — common in discrete choice theory — the distribution the errors of the latent variables follows a Gumbel distribution.

Thus, one of the basic concepts in the formulation of quantum mechanics is that of light coming in discrete bundles called photons.

The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing.

This formulation is based on the use of recurrence relations to generate progressively finer discrete samplings of an implicit mother wavelet function ; each resolution is twice that of the previous scale.

In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite.

This formulation of QCD in discrete rather than continuous spacetime naturally introduces a momentum cut off at the order 1 / a, where a is the lattice spacing, which regularizes the theory.

If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations (), () and () or equivalently 200 coupled first order differential equations.

The totient is the discrete Fourier transform of the gcd: ()

The sampling rate, sample rate, or sampling frequency () defines the number of samples per unit of time ( usually seconds ) taken from a continuous signal to make a discrete signal.

Though it originated as a ligature of the letters Omega () and Te ( Т т ), it functions as a discrete letter of the alphabet, placed between х and ц.

A discrete fuzzy measure g on a set X is called k-additive () if its Möbius representation verifies, whenever for any, and there exists a subset F with k elements such that.

By assuming that light actually consisted of discrete energy packets, Einstein wrote an equation for the photoelectric effect that agreed with experimental results.

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation.

If the surfaces are approximated by a finite number of planar patches, each of which is taken to have a constant radiosity B < sub > i </ sub > and reflectivity ρ < sub > i </ sub >, the above equation gives the discrete radiosity equation,

The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton ; in essence, it is the Newton interpolation formula, first published in his Principia Mathematica in 1687, namely the discrete analog of the continuum Taylor expansion,

A discrete function could be defined explicitly by a list, or by a formula for f ( n ) or it could be given implicitly by a recurrence relation or difference equation.

A difference equation is the discrete equivalent of a differential equation and can be used to approximate the latter or studied in its own right.

The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur.

The energy eigenvalues from this equation form a discrete spectrum of values, so mathematically energy must be quantized.

Similarly, if g and h are elements of a finite cyclic group G then a solution x of the equation g < sup > x </ sup > = h is called a discrete logarithm to the base g of h in the group G.

While still a student, Ambartsumian thoroughly studied the theory of atomic structure, the formation of energy levels, and the Schrödinger equation and its properties, and when he mastered the theory of eigenvalues of differential equations, he pointed out the apparent analogy between discrete energy levels and the eigenvalues of differential equations.

A lossless digital waveguide realizes the discrete form of d ' Alembert's solution of the one-dimensional wave equation as the superposition of a right-going wave and a left-going wave,

The solutions of Schrödinger equation in polar coordinates for a particle of mass in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in:

In the Bohr model, the allowed orbits were derived from quantized ( discrete ) values of orbital angular momentum, L according to the equation

A hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior – a system that can both flow ( described by a differential equation ) and jump ( described by a difference equation or control graph ).

While still a student Hambardzumyan was very much interested in quantum physics, and thoroughly studied the theory of atomic structure, the formation of energy levels, and Schroedinger's equation and its properties, and when he mastered the theory of eigenvalues of differential equations, he pointed out the apparent analogy between discrete energy levels and the eigenvalues of differential equations.

That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian ( sum of adjacent vertices minus value of vertex ) – the positive solutions to the homogeneous equation:

* Non-Markovian processes – generalized master equation – continuous time with memory of past events, discrete state space, waiting times of events ( or transitions between states ) discretely occur and have a generalized probability distribution.

* Jump Markov process – master equation – continuous time with no memory of past events, discrete state space, waiting times between events discretely occur and are exponentially distributed.

The replicator equation ( in its continuous and discrete forms ) satisfies the folk theorem of evolutionary game theory which characterizes the stability of equilibria of the equation.

In many cases it is not possible to divide the process into a finite number of discrete stages, since the state of the stream is transformed in a continuous manner through the process.

Each wave state has a single discrete spin ( spin up or spin down ).

* discrete input and output bits, allowing control or detection of the logic state of an individual package pin

To this end, he compares George Orwell's Coming Up for Air with Frederik Pohl and C. M. Kornbluth's The Space Merchants and concludes that the basic building block and distinguishing feature of a science fiction novel is the presence of the novum, a term Darko Suvin adapts from Ernst Bloch and defines as " a discrete piece of information recognizable as not-true, but also as not-unlike-true, not-flatly-( and in the current state of knowledge ) impossible ".

Note that every part of the machine — its state and symbol-collections — and its actions — printing, erasing and tape motion — is finite, discrete and distinguishable ; it is the potentially unlimited amount of tape that gives it an unbounded amount of storage space.

Relatively small changes to the analog signal levels due to manufacturing tolerance, signal attenuation or parasitic noise do not leave the discrete envelope, and as a result are ignored by signal state sensing circuitry.

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state.

In general, proteins fold into discrete units that perform distinct cellular functions, but some proteins are also capable of forming an infectious conformational state known as a prion.

The use of the term in Markov chain Monte Carlo methodology covers cases where the process is in discrete time ( discrete algorithm steps ) with a continuous state space.

The Information Age: Economy, Society and Culture comprehends three sociological dimensions — production, power, and experience — stressing that the organisation of the economy, of the state and its institutions, and the ways that people create meaning in their lives through collective action, are irreducible sources of social dynamics — that must be understood as both discrete and inter-related entities.

If a particle can have some discrete orientations of the spin, say the spin can be aligned with the z axis or against it, then the particle can have any state of the form:

For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:

In synchronous sequential circuits, the state of the device changes only at discrete times in response to a clock signal.

* State space for information about state space with discrete states in computer science.

As the game has an absorbing state 5 the distribution of time to absorption is discrete phase-type distributed.

However, if the system contains only a single spatial dimension, then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry.

Model checking tools were initially developed to reason about the logical correctness of discrete state systems, but have since been extended to deal with real-time and limited forms of hybrid systems.

In the context of modern control theory, any variable structure system, like a system under SMC, may be viewed as a special case of a hybrid dynamical system as the system both flows through a continuous state space but also moves through different discrete control modes.

Wisconsin state law had allowed Milwaukee ’ s annexation of such rural and suburban regions that did not qualify for discrete incorporation per the legal incorporation standards.

It has been argued that digital physics, grounded in the theory of finite state machines and hence discrete mathematics, cannot do justice to a physical theory whose mathematics requires the real numbers, which is the case for all physical theories having any credibility.

In a discrete ( i. e. finite state ) market, the following hold:

For example, the maximum entropy prior on a discrete space, given only that the probability is normalized to 1, is the prior that assigns equal probability to each state.