An equality between sets expressed in terms of arbitrary input sets that holds for every allowed choice of those inputs. A double inclusion proof proves a set identity by proving both set inclusions. Membership equivalences give another proof: the expressions are equal if and only if each element belongs to either both or neither.
A solution to a partial differential equation that factors into functions of separate variables, for example . For the heat equation , substitution yields , so both sides are a common constant wherever division is valid. The resulting ordinary differential equations must be solved together with the original boundary conditions. Individual separated solutions generate more general solutions by superposition when the equation and boundary data are linear.
A nonzero polynomial is factorwise separable if every irreducible factor is a separable polynomial. Unlike the convention that all roots must be distinct, this condition allows repeated irreducible factors. Its splitting field is a separable field extension precisely when the polynomial is factorwise separable: in one direction every root has a separable minimal polynomial; in the other, adjoining finitely many roots with separable minimal polynomials gives a separable extension. Thus the implication from a separable splitting extension to a polynomial with distinct roots needs the extra squarefree hypothesis. The example already distinguishes the conventions.
Consider an implicit centered heat stencil of width weighted by , combined with an explicit centered stencil of width weighted by . With , its amplification factor is . For , contraction in the discrete norm holds exactly under the displayed condition. It is unconditional for . Put : the condition reduces to . Maximizing this concave quadratic proves the threshold; the Parseval identity converts the multiplier bound into contraction.
Choosing uniformly among currently unused neighbours gives a self-avoiding walk with a path-dependent probability: is the number of choices available at step . In the square lattice, a four-step straight path has probability , whereas the path has probability . There is no trapping at that length, and these unequal probabilities disprove uniformity.
When a trapped attempt is restarted, the final law is the one-attempt law conditional on successful completion. This divides successful-path probabilities by a common normalizing constant and does not remove their unequal products. The local uniformity of each move therefore does not imply uniformity over complete paths.
A self-avoiding polygon on a lattice is a closed nearest-neighbour walk whose vertices are distinct except for the final repetition of the starting point. On the square lattice a nondegenerate polygon has even perimeter at least four, since each step changes the parity of the sum of the coordinates. One can generate such a polygon by generating a self-avoiding walk with edges, retaining it only when its endpoint is a neighbour of its origin, and appending the closing edge. The resulting distribution inherits the bias of the walk sampler unless an additional correction is applied.
A nonnegative second variation requires its highest-derivative coefficient to be nonnegative. If it were negative on an interval, a smooth compactly supported factor times would have negative derivative energy of order , while its zero-order energy stays bounded. Large would contradict nonnegativity. This is a local necessary condition; it does not rule out negative directions caused by the lower-order terms.
For fixed-endpoint variations, a second variation is coercively positive if the displayed estimate holds for one constant and every admissible variation. The endpoint Poincaré inequality makes this equivalent to a positive lower bound in the norm. It is stronger than strict positivity on each nonzero variation: is strictly positive, but high-frequency variations make its ratio to tend to zero.
For with continuous coefficients, coercive positivity implies . Indeed, if on an interval, multiply a smooth function supported there by . Its negative highest-derivative contribution has order , while the mixed term has order at most and the remaining term is bounded. This contradicts the displayed inequality for large . In particular coercive positivity implies the Strengthened Legendre condition, whereas mere strict positivity does not.
No cyclic device can have as its sole effect the extraction of heat from one reservoir and its complete conversion into work. If the reservoir loses heat at temperature , its entropy change is while the cyclic device returns to its starting state. A sole-effect conversion would therefore reduce total entropy.
With the square root positive for on the positive imaginary axis, the displayed conformal map takes the upper half-plane bijectively to . Squaring, subtracting from one, taking the positive-real-part square root, and reciprocating identifies each intermediate slit domain. The boundary prevertices correspond to the lower infinite end, the slit tip at one, and the upper infinite end. The inverse is with positive imaginary part.
The upper half-plane joined to the downward strip has reentrant corners at . The Schwarz-Christoffel formula gives , with at upper infinity. Since , integration gives the displayed map. The inverse trigonometric function is continued analytically in the upper half-plane so its corner values are . The origin is the lower channel end and infinity the upper open end. The complex potentials and respectively normalize unit horizontal velocity in the upper region and unit upward velocity in the channel.
The principal square root maps the upper half-plane bijectively onto with the ray removed. The prevertex zero is the slit tip, and represent the two ends at infinity. The inverse is the unique root with positive imaginary part. Pulling back gives the bounded harmonic function with value one on the slit banks and zero on the imaginary axis. The harmonic function has zero finite boundary values but grows at infinity, explaining the role of boundedness.
For a smooth bounded domain and a bounded continuous scalar function , this problem has a weak solution. Let be the zero-Dirichlet inverse of , defined by the Lax-Milgram theorem. The Poincaré inequality gives . The Rellich-Kondrachov compactness theorem makes compact. The Nemytskii operator is continuous and has bounded image, so is a nonlinear compact operator mapping a sufficiently large closed ball into itself. The Schauder fixed-point theorem gives a fixed point, necessarily in , and its defining identity is the desired weak formulation. This proves existence; uniqueness requires additional assumptions.
For continuous and , restrict to the closed supremum-norm ball of radius in . Boundedness on the compact kernel and nonlinearity boxes makes the integral map preserve this ball for sufficiently small positive . Uniform continuity of proves continuity of the map; uniform continuity of gives an equicontinuous bounded image. Arzelà-Ascoli makes that image relatively compact, and Schauder gives a solution. No global Lipschitz hypothesis or uniqueness conclusion is needed.
For a countable minimal total family in a normed space in a Banach space, let be its unique biorthogonal system and let . Uniform boundedness of these projections implies for every : approximate by a finite linear combination , eventually fixed by , and bound the error by . Conversely, if every vector has a norm-convergent expansion in this family, applying each continuous shows that its coefficients must be . Thus , so the projections are pointwise bounded. The Uniform boundedness principle gives their uniform operator-norm bound. This proves the criterion and uniqueness of all expansions.
A historical excreta-management approach using dry earth in receptacles, together with organized removal.
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