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Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.
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Given and set
: 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 the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.
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 and S
Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.
Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.
Given concerns about the previous programs using nuclear-tipped interceptors, in the 1980s the U. S. Army began studies about the feasibility of hit-to-kill vehicles, i. e. interceptor missiles that would destroy incoming ballistic missiles just by colliding with them head-on.
Given current trends in the UMC — with overseas churches growing, especially in Africa, and U. S. churches collectively losing about 1, 000 members a week — it has been estimated that Africans will make up at least 30 % of the delegates at the 2012 General Conference, and it is also possible that 40 % of the delegates will be from outside the U. S. One Congolese bishop has estimated that typical Sunday attendance of the UMC is higher in his country than in the entire United States.
Given Rickover's single-minded focus on naval nuclear propulsion, design and operations, it came as a surprise to many when in 1982, near the end of his career, he testified before the U. S. Congress that, were it up to him, he " would sink them all.
Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.
Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).
Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition
Given that the U. S. is still home to the world's largest economy, there's no reason it shouldn't have the most vibrant equity markets — unless regulation is holding back the creation of new public companies.
Given and matrices
Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.
Given the factorization in terms of these matrices, one can now write down immediately an equation
Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.
Given two permutations π and σ of m elements and the corresponding permutation matrices P < sub > π </ sub > and P < sub > σ </ sub >
Given the interpretation via linear mappings and direct sums, there is a special type of block matrix that occurs for square matrices ( the case m = n ).
Given a sequence of matrices, we want to find the most efficient way to multiply these matrices together.
Given the number of possible matrices, it's not surprising that the number of audits can get out of hand .".
Given two density matrices describing independent systems A and B, we have.
Given and each
* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.
Given is each element's name, atomic number, year of first report, name of the discoverer, and some notes related to the discovery.
* Named entity recognition ( NER ): Given a stream of text, determine which items in the text map to proper names, such as people or places, and what the type of each such name is ( e. g. person, location, organization ).
* Part-of-speech tagging: Given a sentence, determine the part of speech for each word.
* Topic segmentation and recognition: Given a chunk of text, separate it into segments each of which is devoted to a topic, and identify the topic of the segment.
# Given an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.
Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):
: Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates.
Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.
Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:
Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.
Given the maximum observed frequency ( MOF ) for a mode on each day of the month at a given hour, the MUF is the highest frequency for which an ionospheric communications path is predicted on 50 % of the days of the month.
* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.
* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.
The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2
Given a fixed physical relationship in space between the different individual microphone transducer array elements, simultaneous DSP ( digital signal processor ) processing of the signals from each of the individual microphone array elements can create one or more " virtual " microphones.
Given " mappings " of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of " rules ", each of the form:
BONUS: " Given a pair of resistors, give their equivalent resistance if they were connected in parallel with each other.
* Given two hypothetical point particles each of Planck mass and elementary charge, separated by any length, α is the ratio of their electrostatic repulsive force to their gravitational attractive force.
Given a testing procedure E applied to each prepared system, we obtain a sequence of values
Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping
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