In a topological space, a set is closed if and only if it coincides with its closure.Equivalently, a set is closed if and only if it contains all of its limit points.This is not to be confused with a closed manifold.…

In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases its number of connected components.Equivalently, an edge is a bridge if and only if it is not contained in any cycle.A graph is said to be bridgeless or isthmus-free if it contains no bridges.…

Equivalently, elements of the extension field can be written as ordered pairs of real numbers.

Equivalently, X is perfectly normal if and only if every closed set is a zero set.Every perfectly normal space is automatically completely normal.…

Take the statement "All red objects have color."This can be equivalently expressed as "If an object is red, then it has color."In other words, the contrapositive is logically equivalent to a given conditional statement, though not sufficient for a biconditional.…

Open interest in a derivative is the sum of all contracts that have not expired, been exercised or physically delivered.Moreover, the open interest is the number of long positions or, equivalently, the number of short positions.…

Such sites may have separate entries if they are considered to be more notable than the formation as a whole.In contrast, extensive formations associated with large areas may be equivalently represented at many locations.Such formations may be listed either without a site or with a site that represents the type locality.…

Equivalently, f is continuous if the inverse image of every open set is open.This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function.…

He closed his eyes, and his thoughts came and went like soft light and shade in a garden close; his happiness was a part of himself, as fragrance is inherent in the summer time.The evil of the last days had fallen from him, and the reaction was equivalently violent.…Cited from A Mere Accident, by George Moore

Selection can be presented as conditioning on an event defined by the data and may affect the coverage probability of a CI for a single parameter.Equivalently, the problem of selection changes the basic sense of P-values.FCR procedures consider that the goal of conditional coverage following any selection rule for any set of (unknown) values for the parameters is impossible to achieve.…

In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common.Equivalently, every edge in such a graph belongs to at most one simple cycle.Equivalently, every block (maximal subgraph without a cut-vertex) is an edge or a cycle.…

Equivalently, the same name has two different numerical values depending on whether it is being used in the long or short scale.Each scale has a logical justification to explain the use of each such differing numerical name and value within that scale.…

Equivalently, we may also use the following definition.

Two vectors are said to be equal if they have the same magnitude and direction.Equivalently they will be equal if their coordinates are equal.…

Then a free-falling observer could distinguish free fall from true absence of forces, because a charged particle in a free-falling laboratory would begin to be pulled relative to the neutral parts of the laboratory, even though no obvious electric fields were present.Equivalently, we can think about a charged particle at rest in a laboratory on the surface of the earth.Since we know the earth's gravitational field of 1 g is equivalent to being accelerated constantly upward at 1 g, and we know a charged particle accelerated upward at 1 g would radiate, why don't we see radiation from charged particles at rest in the laboratory?…

Both are important for the full description of the probability distribution.Or, equivalently, we need to fully define the space on which we want to define f.…

We can ask whether those are the same topologies; equivalently, if they give rise to the same completions.

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers.Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers.The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.…

It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time.Equivalently, it says that time average and average over the statistical ensemble are the same.It has been shown that not only classical systems but also quantum mechanical ones tend to maximize their entropy over time.…

In the sense used here, given any two conditions, the first is a consequence of the second if and only if the second is a sufficient condition for the first.Equivalently, being a consequence of a given condition is coextensive with being a necessary condition for it.The relational verb 'implies' is frequently used for the converse of the relational verb phrase 'is a consequence of'.…