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Any and isotropic

Any disturbance created in a sufficiently small region of isotropic space ( or in an isotropic medium ) propagates from that region in all radial directions.

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 linear transformation f: V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T ( V ) to A as indicated by the following commutative diagram:

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 n ( P, Q ) and V n ( 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 algebraic

* Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.

Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y.

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument.

Any monoid ( any algebraic structure with a single associative binary operation and an identity element ) forms a small category with a single object x.

* Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer ; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not.

Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions.

* Any Lie group with an infinite group of components G / G o cannot be realized as an algebraic group ( see identity component ).

Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group.

Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent.

Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = g u g s of commuting unipotent and semisimple elements g u and g s .

Any and group

* Any member of the genus Eunectes, a group of large, aquatic snakes found in South America

Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

Any formal or informal group — a family, a church, a club, a business, a trade union — may be said to have government.

* Any Lie group G defines an associated real Lie algebra.

* Any topologically closed subgroup of a Lie group is a Lie group.

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

" Any attempt to organize the group ... under a single authority would eliminate their independent initiatives, and thus reduce their joint effectiveness to that of the single person directing them from the centre.

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.

However, in the case of a finitely presented group we know that not all the generators can be trivial ( Any individual generator could be, of course ).

* Any subgroup of a free group is free.

Any group of four students may run for office, but there must always be four students.

Any group that managed to find ways of reasoning effectively would reap benefits for all its members, increasing their fitness.

Any time during a triad conversation, group members can switch seats and one of the co-pilots can sit in the pilot ’ s seat.

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.

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