Given these changes, the classical terms " system ", " expansion " and " peripheral " no longer have the same connotations.

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given the same bullet and cartridge, a faster burning powder will generally use a smaller charge of powder than required with a slower powder.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

* Coreference resolution: Given a sentence or larger chunk of text, determine which words (" mentions ") refer to the same objects (" entities ").

Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given the same set of resources, IS-IS can support more routers in an area than OSPF.

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

Given that multiple people often consume food from the same source, it is common for more than a single person to be affected simultaneously.

Given a gain of about 100 and a laser efficiency of about 1 %, HiPER produces about the same amount of fusion energy as electrical energy was needed to create it.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

Given the choice between playing a game at an arcade three or four times ( perhaps 15 minutes of play for a typical arcade game ), and renting, at about the same price, exactly the same game — for a video game console — the console became the preferred choice.

: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.

Pope Saint Adeodatus I or Deodatus I ( which is Given by God in Latin, also called Deusdedit, which is God Has Given ; both are now considered variants of the same name ) ( died 8 November 618 ) was Pope from 13 November 615 to his death.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given a set of integers, does some nonempty subset of them sum to 0?

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given a set

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given a specific task to solve, and a class of functions, learning means using a set of observations to find which solves the task in some optimal sense.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

" Given that science continually seeks to adjust its theories structurally to fit the facts, i. e., adjusts its maps to fit the territory, and thus advances more rapidly than any other field, he believed that the key to understanding sanity would be found in the study of the methods of science ( and the study of structure as revealed by science ).

Given these facts, algorithm defined by the following pseudocode:

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesized logic program which entails all the positive and none of the negative examples.

* Given two different minds and the same facts, how different are the products!

Given the facts that solar thermal power is reliable, can deliver peak load and does not cause pollution, a price of US $ 0. 10 per kWh starts to become competitive, although a price of US $ 0. 06 has been claimed With some operational cost a simple target is 1 dollar ( or lower ) investment for 1 kWh production in a year.

Apart from the very rare “ Kirill of Turov ’ ( even when this designation occurs, it tends to be just one of several variant readings ), the headings in ‘ his ’ manuscripts include “ Kirill the monk ,” “ Kirill the philosopher ,” ‘ Saint Kirill ,” “ The Blessed father Kirill ,” “ the blessed monk Kirill ,” “ Kirill the unworthy monk ,” “ the venerable Kirill .” Given this variety of labels, ‘ Kirill ’ s ’ texts invite several candidates for being their more likely authors ( writers whose existence has been substantiated with historical facts ).

Given the established facts, the court determined that a trial was unnecessary and granted summary judgment in favor of CompuServe for all claims.

Given these facts and other considerations, the actions of PNG could be deemed to be valid in that:

Given these facts, waiting more than 48 hours between bouts of training for a particular muscle is a waste of time.

Given the facts in Donovan, who is in the better position to bear the risk?