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Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as
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Formally and case
Formally, the case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic ( the distribution under the null hypothesis ) does not depend on the parameters whose values have not been specified under the point null hypothesis.
Formally, the situation has the characteristics of a prisoner's dilemma game that is played repeatedly between the same partners, and in this case " defection "-jumping the queue-would not be advantageous.
Formally, assuming our intuition from the finite dimensional case carries over into the infinite dimensional setting, the measure should then be equal to
Formally and if
Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies
Formally, if d is the dimension of the parameter, and n is the number of samples, if as and as, then the model is semi-parametric.
Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.
If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,
Formally, a hypothesis is compared against its opposite or null hypothesis (" if I release this ball, it will not fall to the floor ").
Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.
Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write
Formally, if is an open subset of the complex plane, a point of, and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on.
Formally, two variables are inversely proportional ( or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion ) if one of the variables is directly proportional with the multiplicative inverse ( reciprocal ) of the other, or equivalently if their product is a constant.
Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.
Formally, if we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ, then the h-polynomial of Δ is
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →
Formally and linear
Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.
Formally, the statement that " value decreases over time " is given by defining the linear differential operator as:
Formally, scalar multiplication is a linear map, inducing a map ( send a scalar λ to the corresponding scalar transformation, multiplication by λ ) exhibiting End ( M ) as a R-algebra.
Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.
Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).
Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Formally, a biased graph Ω is a pair ( G, B ) where B is a linear class of circles ; this by definition is a class of circles that satisfies the theta-graph property mentioned above.
Formally, for any dimension, the orientation of the image of an object under a direct isometry with respect to that object is the linear part of that isometry.
Formally and map
Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
Formally, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H → H ( called the antipode ) such that the following diagram commutes:
Formally and is
Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).
More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,
Formally, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).
Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.
* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.
Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.
Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.
Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset