For , a genuine two-cycle obeys with . Subtracting its two component equations gives , then substitution gives the displayed factorization. Cancelling loses the antipodal cycle , present for every . The nonzero-sum family instead has and , requiring . Retaining all vanishing factors is essential in polynomial cycle classification.
Find a graph path whose least edge weight is as large as possible. In a connected undirected graph, use the unique route in a maximum spanning tree. Deleting a least-weight edge on that route gives a separating cut of a graph; a heavier crossing edge would improve the tree's total weight, so every competing route contains an edge no heavier than the deleted one. This proves the bottleneck optimum without assuming distinct weights.
An even-length graph path in an edge coloring is repetitively colored when its first half and second half have the same color sequence in the same order. For a path with distinct edges colored independently and uniformly from colors, the probability of a repetition on edges is : after the first colors, the last colors are prescribed. Reversing the path preserves this property, so the corresponding bad events may be indexed by unoriented paths.
On , the midpoint parity operator is . Substitution gives and , so it is a bounded self-adjoint operator and a unitary operator, with . It is the parity operator in the centered coordinate .
For an infinite square well, obeys . Reflection preserves the Dirichlet boundary conditions and commutes with the second derivative, so commutes with the Hamiltonian operator on its operator domain. By contrast, differentiating once gives for the momentum operator. This distinction permits conserved midpoint parity with changing mean momentum.
Adjective referring to the parasympathetic nervous system.
In a complete Non-Archimedean absolute value field, a series converges if and only if its terms tend to zero: the ultrametric inequality bounds every finite tail by its largest term. The displayed sequence tends to zero in . But each block contributes to the real sum of its p-adic absolute values, so that real sum diverges.
For a positive even weight on a symmetric interval, any orthogonal polynomial of degree has the parity of . Induct on . The reflected polynomial has the same leading coefficient. By substitution in the weighted inner product and the inductive parity of all lower polynomials, it is orthogonal to them. Hence has lower degree and is orthogonal to the entire space containing it. Its squared norm is zero, forcing . The base case is a constant. This proof needs neither monic nor unit-norm normalization; it only compares polynomials with the same leading coefficient.
An orthonormal family of orthogonal polynomials also gives each member unit norm in the chosen weighted inner product. Its pairwise inner products are the Kronecker delta. Dividing every nonzero orthogonal polynomial by its norm produces such a family; the sign or complex phase still requires a convention, such as a positive real leading coefficient.
Assume a positive even weight on a symmetric domain has finite polynomial moments. Subtracting the orthogonal projection onto a finite-dimensional subspace of onto the lower-degree polynomials gives the unique monic orthogonal polynomial. Reflection produces another such monic polynomial, , so uniqueness gives the parity identity. Every coefficient with the opposite parity to its degree therefore vanishes.
For the inner product obtained by integrating with weight on , odd moments vanish and the moment of is . The first monic orthogonal polynomials are , and . Positive definiteness follows because the weight is positive almost everywhere and a nonzero polynomial cannot vanish almost everywhere.
A system of ordinary differential equations specifies derivatives of several unknown functions of one independent variable. A vector-valued form bundles its scalar equations. A scalar second-order ordinary differential equation becomes the first-order system by introducing position and velocity variables. Conversely the first equation identifies and the second recovers the original equation. Its initial condition must specify both variables.
Where , its logarithmic derivative changes this nonlinear second-order ordinary differential equation into the first-order linear differential equation . For continuous , initial values and therefore giveDifferentiation proves both the equation and the initial data. The exponential preserves the sign of , so the solution never reaches the singular value zero on any finite interval where is continuous. For and , the expression reduces to .
An autonomous differential equation for can be reduced by one order on a monotone solution branch by using as the new independent variable. Setting and applying the chain rule gives the displayed derivative identities. The absence of explicit dependence makes the resulting equation involve only and derivatives of . A point with is singular for this coordinate change; for example gives on an appropriate branch, so need not remain finite there.
A group element has finite order if some positive integer power equals the identity element. Its order of a group element is the least such positive integer.
For equality-constrained optimization, restrict the Hessian matrix of the optimization Lagrangian to feasible tangent directions. If the columns of form a basis of the nullspace of the constraint Jacobian matrix , the resulting quadratic form has matrix . Positive definiteness gives positive quadratic curvature in every nonzero tangent direction. Nonsingularity of this reduced matrix, together with full row rank of , characterizes nonsingularity of the corresponding equality-constrained saddle-point matrix.
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