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Page "Tensor algebra" ¶ 14
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Any and linear
Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.
:“ Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T ( V ) to A .”
Any linear polarization of light can be written as an equal combination of right-hand ( RHC ) and left-hand circularly ( LHC ) polarized light:
Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:
# Any sequence satisfying the recurrence relation can be written uniquely as a linear combination of solutions constructed in part 1 as λ varies over all distinct roots of p ( t ).
Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ
* Any linear functional L is either trivial ( equal to 0 everywhere ) or surjective onto the scalar field.
Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence ( hence " the " canonical divisor ).
* Any linear function is both concave and convex.
A weight on a Lie algebra g over a field F is a linear map λ: g F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.
Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point ; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.
Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.
Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line.
Any point that is rigidly connected to the body can be used as reference point ( origin of coordinate system L ) to describe the linear motion of the body ( the linear position, velocity and acceleration vectors depend on the choice ).
* Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
Any linear code can be represented as a graph, where there are two sets of nodes-a set representing the transmitted bits and another set representing the constraints that the transmitted bits have to satisfy.
Any linear time-invariant operation on s ( t ) produces a new spectrum of the form H ( f )• S ( f ), which changes the relative magnitudes and / or angles ( phase ) of the non-zero values of S ( f ).
Any system in a large class known as linear, time-invariant ( LTI ) is completely characterized by its impulse response.
Any linear function is homogeneous of degree 1, since by the definition of linearity
Any linear combination
Any other sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U < sub > n </ sub >( P, Q ) and V < sub > n </ sub >( P, Q ).
Any linear system containing only voltage sources, current sources, and other resistors can be converted to a Thévenin equivalent circuit, comprising exactly one voltage source and one resistor representing " internal resistance ".
Any Jacobi field can be represented in a unique way as a sum, where is a linear combination of trivial Jacobi fields and is orthogonal to, for all.

Any and transformation
Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation.
Any such stress-energy pseudotensor can be made to vanish locally by a coordinate transformation.
* Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, ( x, y ) ( n + x, y ), optionally followed by a reflection in either the horizontal axis, ( x, y ) ( x ,− y ), or the vertical axis, ( x, y ) (− x, y ), provided that this axis is chosen through or midway between two dots, or a rotation by 180 °, ( x, y ) (− x ,− y ) ( ditto ).
Any waveform can be disassembled into its spectral components by Fourier analysis or Fourier transformation.
Any segments of the enterprise that are not service-oriented can also be documented in order to consider transformation requirements if a service needs to communicate with the business processes automated by such segments.
Any canonical transformation involving a type-2 generating function G < sub > 2 </ sub >( q, P, t ) leads to the relations
Any transformation semigroup can be turned into a semigroup action by the following construction.
Any two triplets obeying these relationships must be related by a linear transformation ; they represent utility indices differing only by scale and origin.
Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space.

Any and f
Note: Any nonlinear electronic block driven by two signals with frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub > would generate intermodulation ( mixing ) products.
Any frequency component above f < sub > s </ sub >/ 2 is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies.
Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.
* quantification Any morphism f: X Y in a category with pullbacks induces a monotonous map acting by pullbacks ( A monotonous map is a functor if we consider the preorders as categories ).
Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself.
Any covector field α has components in the basis of vector fields f. These are determined by
Any meromorphic function f gives rise to a divisor denoted ( f ) defined as
* AGSP ( Any Given Screen Position ) is the combination of VSP and Linecruncher, f. ex.
f. Any other task assigned by The Attorney General of Canada or a provincial minister
Any non-linear function, f ( a, b ), of two variables, a and b, can be expanded as
A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism from G to a complete graph K < sub > k </ sub >: the vertices of K < sub > k </ sub > correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K < sub > k </ sub > that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K < sub > k </ sub >, and every two distinct vertices of K < sub > k </ sub > are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K < sub > k </ sub >.
* Any word that begins with an " f "
Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an electron, when placed in an external magnetic field B ( measured in teslas ) that is not aligned with its magnetic moment, will precess at a frequency f ( measured in hertz ), that is proportional to the external field:
Any section s of E over B induces a section of f < sup >*</ sup > E, called the pullback section f < sup >*</ sup > s, simply by defining.
* Any two maps f, g: Y X are homotopic.
* Any map f: Y X is null-homotopic.
Any non-vanishing holomorphic function f defined on the strip can be approximated by the ζ-function.

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