More generally, suppose that an experiment consists of a number of independent trials, that each trial results in either a `` success '' or a `` non-success '' ( `` failure '' ), and that the probability of success remains constant from trial to trial.

More generally, in a k-dimensional array, the address of an element with indices i < sub > 1 </ sub >, i < sub > 2 </ sub >, …, i < sub > k </ sub > is

More recently, an abettor is generally known as an accomplice.

More expensive pigments are generally replicated by hues.

More generally, one curve is a curvilinear asymptote of another ( as opposed to a linear asymptote ) if the distance between the two curves tends to zero as they tend to infinity, although usually the term asymptote by itself is reserved for linear asymptotes.

More generally, as noted by A. W.

More generally, a binary relation between two sets A and B is a subset of.

More generally, a new component is said to be backward compatible if it provides all of the functionality of the old component.

More generally, an n-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance from an outer control point on a unit circle, where t is 360 / n degrees, and n > 2.

More generally, every C *- algebra is a Banach algebra.

More generally, a series

More generally, if ƒ has a branch-point near the origin of such a nature that then

More generally, calculus ( plural calculi ) refers to any method or system of calculation guided by the symbolic manipulation of expressions.

More generally, many philosophers and scientists have been unhappy about the difficulty of producing a definition that does not involve circularity or fuzziness.

More broadly, philosophers who do not accept the possibility of zombies generally believe that consciousness is reflected in behavior ( including verbal behavior ), and that we attribute consciousness on the basis of behavior.

Among the teachings of Jesus Christ in the Gospel of Luke and the Gospel of Matthew, the message to his followers that one should " Turn the other cheek " and his example in the story Pericope Adulterae, in which Jesus intervenes in the stoning of an adulteress, are generally accepted as his condemnation of physical retaliation ( though most scholars agree that the latter passage was " certainly not part of the original text of St John's Gospel ") More militant Christians consider Romans 13: 3 – 4 to support the death penalty.

* More generally, compact groups such as an orthogonal group are compact, while groups such as a general linear group are not.

More generally, compact sets can be separated by open sets: if K < sub > 1 </ sub > and K < sub > 2 </ sub > are compact and disjoint, there exist disjoint open sets U < sub > 1 </ sub > and U < sub > 2 </ sub > such that and.

More generally, electric current can be represented as the rate at which charge flows through a given surface as:

More generally, the idea of a contractive mapping can be defined for maps between metric spaces.

More generally, unspecific coercion may be the form taken by disciplinary coercion, and this appears to be in fact the case within the most effective command systems of the modern world.

More generally, judges look at the difficulty and execution of jumps, stunts and tumbling, synchronization, creativity, the sharpness of the motions, showmanship, and overall routine execution.

More generally among Baptists, a variety of parachurch agencies and evangelical educational institutions may be supported generously or not at all, depending entirely upon the local congregation's customs and predilections.

More generally, one can consider finite direct sums of matrix algebras.

More generally, inheritance now is quite different than those in the past.

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).

More generally, Stokes ' theorem applies to oriented manifolds M with boundary.

More recent statements of the theorem are sometimes careful to exclude the equality condition ; that is, the condition is if x ( t ) contains no frequencies higher than or equal to B ; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B.

** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem.

More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface.

More examples of the use of the empty product in mathematics may be found in the binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, proof that e is irrational, prime factor, binomial series, and multiset.

More precisely, the time hierarchy theorem for deterministic Turing machines states that for all time-constructible functions,

More precisely, the theorem states that for any given number of colours c, and any given integers n < sub > 1 </ sub >,..., n < sub > c </ sub >, there is a number, R ( n < sub > 1 </ sub >, ..., n < sub > c </ sub >), such that if the edges of a complete graph of

More precisely, the change of variables formula is stated in the next theorem:

More generally: all uniformly convex Banach spaces are reflexive according to the Milman – Pettis theorem.

More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows.

More generally, while Hurwitz's theorem states that an identity of form,

More generally, one can consider the image, kernel, coimage, and cokernel, which are related by the fundamental theorem of linear algebra.

More generally, Horn clauses are relevant to automated theorem proving by first-order resolution.

More precisely, Gabriel's theorem states that:

More generally, the Chevalley – Shephard – Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring.

More specifically, the primitive element theorem characterizes those finite degree extensions such that there exists with.

More generally, the theorem fails for equipped with any norm () ( Schwartz 1969, p. 20 ).

More precisely, find necessary and sufficient conditions on the tuple ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) and ( y < sub > 1 </ sub >, ..., y < sub > n </ sub >) separately, so that there is an element of R with the property that x < sub > i </ sub >· r = y < sub > i </ sub > for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.

More precisely, their theorem states that there is no apportionment system that has the following properties ( as the example we take the division of seats between parties in a system of proportional representation ):

More generally, the equipartition theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax < sup > 2 </ sup >, where A is a constant, has an average energy of ½k < sub > B </ sub > T in thermal equilibrium.