Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly, this is equivalent to stabilising a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis and the resulting flag < math > 0 <

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, it can be defined as the integer triples ( a, b, c ), associated to 3 < sup > a </ sup > 5 < sup > b </ sup > 7 < sup > c </ sup >, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

Abstractly we can say that this is a projective line in the space of all conics, on which we take

Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

Abstractly it is assumed that the economy consists only of capitalist production and that the capitalist economy is equal to capitalist society.

Abstractly, the method is as follows:

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:

This method, widely used in drug development, is referred to as rNPV ( risk-adjusted NPV ), and similar methods are used to incorporate credit risk in the probability model of CDS valuation.

For example, consider a model which gives the probability density function of observable random variable X as a function of a parameter θ.

Much like the use of probability in optimal coding theory, rate-distortion theory heavily draws on Bayesian estimation and decision theory in order to model perceptual distortion and even aesthetic judgment.

* Statistical model, in applied statistics, a parameterized set of probability distributions

Although and can be any space of functions, many learning algorithms are probabilistic models where takes the form of a conditional probability model, or takes the form of a joint probability model.

For example, naive Bayes and linear discriminant analysis are joint probability models, whereas logistic regression is a conditional probability model.

For the special case where is a joint probability distribution and the loss function is the negative log likelihood a risk minimization algorithm is said to perform generative training, because can be regarded as a generative model that explains how the data were generated.

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties ( such as convexity ) or because it arises naturally from a particular formulation of the problem ( e. g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost ).

Its form depends on the application: for example, in compression it could be related to the mutual information between and, whereas in statistical modeling, it could be related to the posterior probability of the model given the data.

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

This may be obtained from consideration of whether the required prior probability is greater or lesser than a reference probability associated with an urn model or a thought experiment.

In the context of a mathematical model, such as a probability distribution, the distinction between variables and parameters was described by Bard as follows:

Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.

Unlike the earlier Bohr model of the atom, however, the wave model describes electrons as " clouds " moving in orbitals, and their positions are represented by probability distributions rather than discrete points.

* Quantum cognition, using quantum probability theory to model human behavior

In mathematical terms, a statistical model is frequently thought of as a pair where is the set of possible observations and the set of possible probability distributions on.

In all probability, the council will screen and endorse candidates for the Assembly and for Congress, and then strive to put its full weight behind these pre-primary favorites.

There is a well-known relationship between probability and entropy which states that Af, where **zq is the probability that state ( i.e., volume for an ideal gas ) could be reached by chance alone.

Strictly speaking, this means that the probability for each possible outcome of the experiment can be computed by multiplying together the probabilities of the possible outcomes of the single binomial trials.

We shall find a formula for the probability of exactly X successes for given values of P and N.

Rather than viewing the abortive recovery in 1959-60 as a reason for believing we have lost prospects for growth '', he said `` it should be viewed as a lesson well learned which will increase the probability of substantial improvement in this recovery ''.

Mr. Philip Toynbee writes, for example, that `` in terms of probability it is surely as likely as not that mutual fear will lead to accidental war in the near future if the present situation continues.

If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify ( in the protocol for the experiment or observational study ) that the responses be transformed to stabilize the variance.

According to Richard Dawkins, a distinction between agnosticism and atheism is unwieldy and depends on how close to zero we are willing to rate the probability of existence for any given god-like entity.

In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

The idea is that the dealer's second card has a fairly high probability ( nearly one-third ) to be ten-valued, giving the dealer blackjack and disappointment for the player.

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1 / 3 for all instances.

In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

In other words, there is an algorithm for a quantum computer ( a quantum algorithm ) that solves the decision problem with high probability and is guaranteed to run in polynomial time.

In chemistry, physics, and mathematics, the Boltzmann distribution ( also called the Gibbs Distribution ) is a certain distribution function or probability measure for the distribution of the states of a system.

Alternatively, for a single system at a well-defined temperature, it gives the probability that the system is in the specified state.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

The expression is typically interpreted as the probability amplitude for the state ψ to collapse into the state ϕ.

In quantum mechanics the expression is typically interpreted as the probability amplitude for the state to collapse into the state.

Bayesian probability interprets the concept of probability as " an abstract concept, a quantity that we assign theoretically, for the purpose of representing a state of knowledge, or that we calculate from previously assigned probabilities ," in contrast to interpreting it as a frequency or " propensity " of some phenomenon.