Foundation implies that no descending membership sequence exists. Conversely, with dependent choice, failure of foundation gives a nonempty set in which every element contains another element of that set. Successively selecting such members produces a descending membership sequence. Thus the two conditions are equivalent with dependent choice; full choice is sufficient.
For a solvable group, its derived length is the least nonnegative integer at which the derived series reaches the trivial group, with and the commutator subgroup of . A nontrivial abelian group has derived length one; a nonabelian group with nontrivial abelian commutator subgroup has derived length two.
For in a compact positive interval , split the integration region at one. Near zero the smallest exponent gives the displayed bound, whose integral is . Above one the largest exponent gives the bound, integrable by exponential decay. A single integrable function therefore dominates all parameter derivatives locally. Dominated differentiation under the integral sign gives . The same argument with higher powers of gives all higher derivatives.
The right derivative is the displayed one-sided limit, when it exists as a finite number. It describes differentiation at the left endpoint of a closed interval without requiring the function to be defined to the left. An ordinary derivative at an interior point requires matching left and right derivatives. For example, the absolute value function has right derivative one at zero and left derivative minus one, so it has no ordinary derivative there.
A purely imaginary root of has frequency one and first appears at . Roots cross with positive real derivative ; all roots lie in the left half-plane for smaller positive delay, since a hypothetical root with would require both and . A small-amplitude expansion yields and , verifying a nondegenerate Hopf bifurcation. The limiting period is .
For a twice continuously differentiable deformation map, each row of its cofactor matrix has zero divergence in reference coordinates. In three dimensions write . Applying gives terms in which a symmetric pair of second partial derivatives contracts with an antisymmetric Levi-Civita symbol, so each term vanishes. This identity is the cancellation that makes the Piola-Kirchhoff stress tensor transform force balance correctly between configurations.
Two constant deformation gradients are rank-one connected when their difference is an outer product . For a nonzero jump its matrix rank is one; equality is the trivial compatible case. The vector specifies a reference interface normal, and is the jump amplitude. The Hadamard jump condition identifies this algebraic condition with continuity of an affine deformation across a planar interface. The normal can be normalized by rescaling without changing the product.
In the usual first-order Hilbert system, logical implication discharge requires the appropriate generalization restriction: a variable generalized in the derivation must not be free in the assumption being discharged. Equivalently use a proof system whose generalization rule already forbids variables free in open assumptions. One can first discharge a temporary assumption and then generalize if the quantified variable is not free in the remaining assumptions. Ignoring this side condition would allow deriving universal conclusions from a single arbitrary instance as an open hypothesis.
Conditional modus ponens derives from and . It is the step which translates a modus ponens rule inference when discharging a common assumption. In the case-elimination Hilbert calculus, the third scheme first gives ; the second scheme combines the two conditionals to give .
In ZF, an injection yields an injective nonsurjective self-map by shifting to and fixing the complement. Conversely, if an injection omits , the natural-number recursion theorem defines . Any repetition would, after cancellation, put in the image of . Thus is injective. The argument uses no axiom of choice, even when some infinite sets are Dedekind-finite sets.
A database query has the same result when its ambient domain is extended while all input database relations are unchanged.
A relational schema is in Boyce-Codd normal form if the determinant of every non-trivial functional dependency is a superkey.
A relational schema is in fourth normal form if the determinant of every non-trivial multivalued dependency is a superkey. This implies Boyce-Codd normal form because every functional dependency implies a multivalued dependency.
A decomposition of a relational schema whose component relational projections always reconstruct the original database relation by natural join, without extra database tuples.
A decomposition in which the database integrity constraints can be enforced using constraints on individual component database relations, without computing their relational join.
Let map the unit disk conformally to a planar Darcy flow region, with at a point source of flux . With zero velocity potential on the boundary, the logarithmic source potential becomes . Differentiating the inverse map at fixed physical position gives , while . The boundary law therefore gives the displayed equation, for nonzero .
On a -dimensional vector space, a cyclic vector gives a basis . Write and set . Then . Since polynomials in commute with , kills every basis vector, so . No nonzero polynomial of degree less than can kill , by linear independence. Hence the minimal polynomial is . The matrix in this basis is , and the characteristic polynomial of a companion matrix is as well.
The CRC generator polynomial is the agreed divisor in a cyclic redundancy check. Under the basic convention, undetected error polynomials are precisely its nonzero multiples.
If an algebra acts on a vector space only by scalar multiples of the identity, every vector generates at most a one-dimensional submodule. A higher-dimensional such module is therefore not cyclic. For example upper triangular -by- matrices act by the first diagonal entry on their first-row ideal. For this ideal is a noncyclic submodule of the cyclic regular module. Thus a submodule of a cyclic module need not be cyclic, even over a finite-dimensional algebra.
Along the antipodal two-cycle of , the Jacobian matrix is at both points. Its two-step periodic-orbit multipliers are squares of the roots of . The Jury criterion makes the cycle attracting precisely in the displayed interval, and repelling for . This stable family is distinct from the always unstable nonzero-sum cycles.
There are unlisted articles, also show them or only show them.