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Equivalently, is conjugate to in if and only if and satisfy the Cauchy – Riemann equations in As an immediate consequence of the latter equivalent definition, if is any harmonic function on the function is conjugate to, for then the Cauchy – Riemann equations are just and the symmetry of the mixed second order derivatives, Therefore an harmonic function admits a conjugated harmonic function if and only if the holomorphic function has a primitive in in which case a conjugate of is, of course, So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain.
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Equivalently and is
With this definition, it is necessary to consider the direction of p ( pointed clockwise or counter-clockwise ) to figure out the sign of L. Equivalently:
Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers.
Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.
Equivalently, the dyne is defined as " the force required to accelerate a mass of one gram at a rate of one centimetre per second squared ":
Equivalently, the DFT is often thought of as a matched filter: when looking for a frequency of + 1, one correlates the incoming signal with a frequency of − 1.
Equivalently, inverting is an involution.
Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable.
Equivalently, it is the lowest wage at which workers may sell their labor.
A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.
Equivalently, a perfect number is a number that is half the sum of all of its positive divisors ( including itself ) i. e. σ < sub > 1 </ sub >( n ) = 2n.
Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.
Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups.
Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors ( the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice ).
Equivalently, an ideal of R is a sub-R-bimodule of R.
Equivalently, a right ideal of is a right-submodule of.
Equivalently, a left ideal of is a left-submodule of.
The row rank of a matrix A is the maximum number of linearly independent row vectors of A. Equivalently, the column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A.
Equivalently and if
Equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with.
Equivalently, a problem is # P-complete if and only if it is in # P, and for any non-deterministic Turing machine (" NP machine "), the problem of computing its number of accepting paths can be reduced to this problem.
Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.
Equivalently, a set is bounded if it is contained in some open ball of finite radius.
Equivalently they will be equal if their coordinates are equal.
Equivalently, if a particle travels in a closed loop, the net work done ( the sum of the force acting along the path multiplied by the distance travelled ) by a conservative force is zero.
b. Equivalently, if a ≠ b, then f ( a ) ≠ f ( b ).
Equivalently, a set is closed if and only if it contains all of its limit points.
Equivalently, M is a maximal submodule if and only if the quotient module A / M is a simple module.
More formally, a set R X Y is called a ( combinatorial ) rectangle if whenever R and R then R. Equivalently, R can also be viewed as a submatrix of the input matrix A such that R = M N where M X and N Y.
Equivalently, if X is a locally compact metric space, then ƒ is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X.
Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals ; that is, given any chain:
Equivalently, a family of closed sets forms a base for the closed sets if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.
Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal.
Equivalently, a density operator ρ is a pure state if and only if
Equivalently and only
Equivalently, are there only finitely many finite groups with m generators of exponent n, up to isomorphism?
Equivalently, a set is recursively enumerable if and only if it is the range of some computable function.
Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.
Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.
Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, ... of elements of X such that x < sub > n + 1 </ sub > R x < sub > n </ sub > for every natural number n.
Equivalently, a directed graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint directed cycles and all of its vertices with nonzero degree belong to a single strongly connected component.
Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and:
Equivalently, an action of a discrete group G on a topological space X is properly discontinuous if and only if any two points x and y have neighborhoods U < sub > x </ sub > and U < sub > y </ sub > such that there are only a finite number of group elements g with g ( U < sub > x </ sub >) meeting U < sub > y </ sub >.
Equivalently, a geodesic metric space M is a real tree if and only if M is a δ-hyperbolic space with δ = 0.
Equivalently, a point x in S is an isolated point of S if and only if it is not a limit point of S.
Equivalently, the extension E / F is Galois if and only if it is algebraic, and the field fixed by the automorphism group Aut ( E / F ) is precisely the base field F. ( See the article Galois group for definitions of some of these terms and some examples.
Equivalently, F ( n, k ) is the number of ways of writing n − 1 as an ordered sum involving only 1 and 2, so that 1 is used exactly k times.
( Equivalently, f ≤ g if and only if f ⊆ g where f and g are identified with their respective graphs.
Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.
A nonempty family Δ of finite subsets of a universal set S is an abstract simplicial complex if, for every set X in Δ, and every subset Y ⊂ X, Y also belongs to Δ. Equivalently, it is an abstract simplicial complex if and only if there do not exist two sets Y ⊂ X such that X belongs to Δ but Y does not.
If s is a sequence starting from s < sub > 0 </ sub > and s ′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that, then A ( s ) is defined if and only if A ( s ′) is defined, and Equivalently, whenever for all n, then
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