Synonym

A synonym is a word or phrase that means exactly or nearly the same as another word or phrase in the same language. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn (σύν) ("with") and onoma (ὄνομα) ("name"). An example of synonyms are the words begin, start, and commence. Words can be synonymous when meant in certain senses, even if they are not synonymous in all of their senses. For example, if we talk about a long time or an extended time, long and extended are synonymous within that context. Synonyms with exact interchangeability share a seme or denotational sememe, whereas those with inexactly similar meanings share a broader denotational or connotational sememe and thus overlap within a semantic field. Some academics call the former type cognitive synonyms to distinguish them from the latter type, which they call near-synonyms.

In the figurative sense, two words are sometimes said to be synonymous if they have the same connotation:

Penalty (gridiron football)

In American football and Canadian football, a penalty is a sanction called against a team for a violation of the rules, called a foul. Officials initially signal penalties by tossing a bright yellow (American Football) or orange (Canadian football) colored "penalty flag" onto the field toward or at the spot of a foul. Many penalties result in moving the football toward the offending team's end zone, usually either 5, 10, or 15 yards, depending on the penalty. Most penalties against the defensive team also result in giving the offense an automatic first down, while a few penalties against the offensive team cause them to automatically lose a down. In some cases, depending on the spot of the foul, the ball is moved half the distance to the goal line rather than the usual number of yards, or the defense scores an automatic safety.

Rationale

Because football is a high-contact sport requiring a balance between offense and defense, many rules exist that regulate equality, safety, contact, and actions of players on each team. It is very difficult to always avoid violating these rules without giving up too much of an advantage. Thus, an elaborate system of fouls and penalties has been developed to "let the punishment fit the crime" and maintain a balance between following the rules and keeping a good flow of the game. Players and coaches are constantly looking for ways to find an advantage that stretches the limitations imposed by the rules. Also, the frequency and severity of fouls can make a large difference in the outcome of a game, so coaches are constantly looking for ways to minimize the number and severity of infractions committed by their players.

Tau-leaping

In probability theory, tau-leaping, or τ-leaping, is an approximate method for the simulation of a stochastic system. It is based on the Gillespie algorithm, performing all reactions for an interval of length tau before updating the propensity functions. By updating the rates less often this sometimes allows for more efficient simulation and thus the consideration of larger systems.

Many variants of the basic algorithm have been considered.

Algorithm

The algorithm is analogous to the Euler method for deterministic systems, but instead of making a fixed change

the change is

where is a Poisson distributed random variable with mean .

Given a state with events occurring at rate and with state change vectors (where indexes the state variables, and indexes the events), the method is as follows:

Initialise the model with initial conditions .

Calculate the event rates .

Choose a time step . This may be fixed, or by some algorithm dependent on the various event rates.

For each event generate , which is the number of times each event occurs during the time interval .