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For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas.
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For and nonzero
For given nonzero integers a and b there is a nonzero integer of minimal absolute value among all those of the form ax + by with x and y integers ; one can assume d > 0 by changing the signs of both s and t if necessary.
*( EF2 ) For all nonzero a and b in R,.
* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.
* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.
For another example, the nonzero complex numbers form a group under the operation of multiplication, as do the nonzero real numbers.
For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.
For example, if an apple is sitting in a glass elevator that is descending, an outside observer looking into the elevator sees the apple moving, so to that observer the apple has a nonzero momentum.
For example, suppose ( generalizing to nonzero is straightforward ).
For nonzero Q there exists a canonical linear isomorphism between Λ ( V ) and Cℓ ( V, Q ) whenever the ground field K does not have characteristic two.
# Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S ; there can be no partial operations.
For a general n × n invertible matrix A, i. e., one with nonzero determinant, A < sup >− 1 </ sup > can thus be written as an ( n − 1 )- th order polynomial expression in A: As indicated, the Cayley – Hamilton theorem amounts to the identity
For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.
For instance, it is possible to attempt to " repair " the proof by supposing that a and b have a definite nonzero value to begin with, for instance, at the outset one can suppose that a and b are both equal to one:
For functions of a single variable, the theorem states that if ƒ is a continuously differentiable function with nonzero derivative at the point a, then ƒ is invertible in a neighborhood of a, the inverse is continuously differentiable, and
For a curve of degree d, the weight of any control point is only nonzero in d + 1 intervals of the parameter space.
For each nonzero countable ordinal α there are classes,, and.
For a slower than light particle, a particle with a nonzero rest mass, the formula becomes
For example, when one is subtracting ten thousand minus 4, 679, the leftmost three digits of 4, 679 — 4, 6 and 7 -- are subtracted from 9, and the rightmost nonzero digit — that is, 9 -- is subtracted from 10, yielding the solution: 5, 321.
For real nonzero values of x, the exponential integral Ei ( x ) is defined as
For and vector
For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.
For each picture frame, the 6502 writes graphics commands for the DVG into a defined area of RAM ( the vector RAM ), and then asks the DVG to draw the corresponding vector image on the screen.
For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.
For a vector with linear addressing, the element with index i is located at the address B + c · i, where B is a fixed base address and c a fixed constant, sometimes called the address increment or stride.
For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:
For any system, the following restrictions on measurement results apply, where is the reduced Planck constant and is any direction vector such as x, y, or z:
For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:
For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
For 2D vector animations, the rendering process is the key frame illustration process, while tweened frames are rendered as needed.
For example, binary string 1011 is represented by the vector ( 1, 0, 1, 1 ).
For the purposes of this article, we call this constructed vector the transmitted vector.
For a vector expressed in cylindrical coordinates as
For twisting rotation to exist, it is necessary to tilt the body sideways after takeoff, so that there is now a small component of this horizontal angular momentum vector along the body's long axis.
Malaria Foundation International states, " The outcome of the treaty is arguably better than the status quo going into the negotiations ... For the first time, there is now an insecticide which is restricted to vector control only, meaning that the selection of resistant mosquitoes will be slower than before.
For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:
For a flow, the vector field Φ ( x ) is a linear function of the position in the phase space, that is,
For a more useful solution, we utilize vector identities, which work for any vector, as follows:
For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.
For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.
For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.
For and finite
For finite sets X, the axiom of choice follows from the other axioms of set theory.
For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.
* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).
For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.
For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.
For a finite group, the derived series terminates in a perfect group, which may or may not be trivial.
For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.
For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.
For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.
* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.
For example, Graham's number, though finite, is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes ' number and Moser's number.
For some finite n-valued logics, there is an analogous law called the law of excluded n + 1th.
For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.
Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.
For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.
For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite.
For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits.
For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.
For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.
For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.
For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.
1.019 seconds.