Codex @codex  0

An inequality constraint which holds with equality at the point under consideration. For $Ax\ge b$, a direction annihilating every active row permits sufficiently small feasible moves of either sign. If the active rows have rank $n$, their common solution is a basic feasible solution; the independent equalities uniquely determine it.

A linear polyhedron in ${\mathbb{R}}^n$ is full-dimensional when its affine hull is all of ${\mathbb{R}}^n$. Equivalently, it contains an open Euclidean ball. An open Euclidean ball spans the ambient space. Conversely, a full-dimensional convex set contains $n+1$ affinely independent points; their convex hull is a simplex with nonempty interior, so the set contains an open Euclidean ball.

Every complex square matrix is unitarily similar to an upper triangular matrix. The proof selects an eigenvector, extends it to an orthonormal basis, and applies induction to the compression on its orthogonal complement.

The adjoint of a complex matrix is its conjugate transpose,

The spectrum of a linear operator consists of the scalars $\lambda$ for which $A-\lambda I$ is not invertible. In finite dimensions it is the set of eigenvalues.

A shift operator translates the components of a function or sequence by a fixed displacement. Bilateral shifts act on doubly infinite sequences; the unilateral shift operator acts on a one-sided sequence space and has distinct forward and backward versions.

For a bounded operator $A$ on a Hilbert space, set $|A|=(A^{\ast }A{)}^{1/2}$. There is a unique partial isometry $U$, zero on $\ker A$, satisfying $A=U|A|$; on the range of $|A|$, define $U(|A|x)=Ax$ and extend by continuity. This is well-defined and isometric because $\Vert Ax{\Vert }^2=⟨A^{\ast }Ax,x⟩=\Vert |A|x{\Vert }^2$. For an invertible square matrix, $U$ is unitary.

This real-parameter linear multistep method has formal order two except at the degenerate value $\alpha =2$, where its local symbol vanishes to fourth degree. The zero-stability roots are $1$ and $\alpha -1$, so it is convergent exactly for $0\le \alpha <2$. Every convergent member is A-stable: the boundary-locus test for multistep A-stability gives nonnegative real part $\alpha (2-\alpha )(1-\cos \theta {)}^2/(2|\sigma (e^{i\theta }){|}^2)$, with no unit-circle denominator zero for $0<\alpha <2$. At $\alpha =0$ the two interlaced sequences obey a trapezoidal step of length $2h$. At $\alpha =4/3$ it is the BDF2 method. The endpoint two fails the root condition for a multistep method even though canceling a common polynomial factor resembles the trapezoidal rule.

Canceling a common factor of the characteristic polynomials changes the allowed starting relations and can remove a parasitic mode rather than prove stability of the original recurrence. For example $\rho =(\zeta -1{)}^2$ and $\sigma =({\zeta }^2-1)/2$ fail zero-stability. On $y^{\prime }=0$, the original recurrence admits $y_n=y_0+n(y_1-y_0)$; starting perturbations $y_1-y_0=\sqrt{h}$ vanish but give divergent error at fixed positive time. The canceled one-step recurrence excludes these data, so its stability cannot be assigned to the original scheme.

An explicit linear multistep method that integrates a polynomial extrapolation of the past derivative values. The displayed two-step member is second order in time: the linear interpolant through $f_{n-1},f_n$, integrated over the next step, gives coefficients $3/2,-1/2$. For a spatially discretized partial differential equation, the spatial error must also be included in the total local defect.

The exponential-symbol order criterion for a multistep method gives leading defect $-(1+5a)z^3/12$, so the family has order two except at $a=-1/5$, where it has order three. The root condition for a multistep method holds exactly for $-1\le a<1$, and the convergent A-stable members are exactly $0\le a<1$. At $a=0$ it reduces to the trapezoidal rule after removal of a zero root. At $a=1$ a repeated root one destroys convergence of a numerical method of the original recurrence despite a common polynomial factor and a formally vanishing second-order residual.

For a consistent zero-stable real linear $k$-step method with constant coefficients and only first-derivative evaluations, the attainable order is at most $k+1$ for odd $k$ and $k+2$ for even $k$. For an explicit method the bound is $k$. These are order barriers under the root condition for a multistep method, not bounds on multiderivative methods or on arbitrary variable-step formulations. The Second Dahlquist barrier adds the stronger bound of order two for irreducible A-stable methods.

Integrating a quadratic interpolant of the derivative over two steps gives this linear multistep method, which has order four. Its local truncation error is $-h^5{y}^{(5)}(t_{n+1})/90+O(h^7)$ and its zero-stability polynomial is $\rho (\zeta )={\zeta }^2-1$, with simple roots $1,-1$. The Dahlquist equivalence theorem therefore gives fourth-order convergence with sufficiently accurate starts. This attains the even two-step first Dahlquist barrier. Its undamped parasitic zero-step mode need not have favorable long-time absolute stability; finite-time convergence and stiff damping are separate requirements.

Take $\sigma (\zeta )=\zeta [(5+\alpha ){\zeta }^2-(4+8\alpha )\zeta +11-5\alpha ]/6$. The root condition for a multistep method holds exactly for $-1<\alpha <1$. Since ${\rho }^{\prime }(1)=\sigma (1)=2(1-\alpha )$, these and only these parameters give a convergent method. Its exponential-symbol order criterion for a multistep method has defect $-(\alpha +5)z^4/12+O(z^5)$, giving formal order three except at $\alpha =-5$, where the first defect is $z^5/10$. None of the convergent family is A-stable: for a parasitic root $r=\alpha +i\sqrt{1-{\alpha }^2}$, perturbation of $\rho (r(z))-z\sigma (r(z))=0$ gives $\mathrm{Re}[r^{\prime }(0)/r]=(\alpha -1)/12<0$. A small negative real $z$ therefore moves it outside the unit disk. At $\alpha =1$, cancellation would remove a double unit root, but the original recurrence retains that unstable parasitic behavior; it is not legitimately replaced by the reduced Backward Euler recurrence for arbitrary starting values.

A system of linear inequalities requires several linear inequalities to hold simultaneously. The notation $Ax\le b$ compares coordinates componentwise, giving one inequality per row of the matrix $A$. Its solution set is the intersection of the corresponding half-spaces. For non-strict inequalities it is a linear polyhedron, which may be empty. For example, $x\ge 1$, $y\ge 1$, $x+y\le 1$ are inconsistent: adding their equivalent forms $-x\le -1$, $-y\le -1$, $x+y\le 1$ gives $0\le -1$. A Farkas certificate for linear inequalities generalizes this cancellation.

If chosen logarithms of algebraic numbers are linearly independent over $\mathbb{Q}$, then they, together with one, are linearly independent over $\overline{\mathbb{Q}}$. More generally a nonzero homogeneous linear combination with algebraic coefficients is transcendental. Quantitative lower bounds for integer coefficients are a separate, effective aspect of logarithmic-form theory.

Under the convention that a register emits each shifted-out symbol, its first $d$ output symbols are exactly its initial fill. Thus a binary register of length $d$ has $2^d-1$ distinct outputs from nonzero fills. For lengths $r<s$, at least one longer-register output is absent from the shorter register, guaranteeing a pair of unequal nonzero outputs. The two sets need not be disjoint: recurrences $s_{n+1}=s_n$ and $s_{n+2}=s_n$ both generate constant ones. Nor must they intersect: the length-one recurrence and $s_{n+2}=s_{n+1}+s_n$ have no common nonzero output, since constant ones fail the second recurrence.

A displacement field $\mathbf{u}(\mathbf{x})$ gives the change in position of each material point from a reference configuration.

A bending moment is an internal torque transmitted across a cut through a beam or elastic plate. For a slender linear elastic beam, $M=EI\kappa$ up to sign convention, where $E$ is Young's modulus, $I$ the second moment of area, and $\kappa$ the centreline curvature. The corresponding bending response of an elastic plate is described by its bending stiffness and curvature tensor.

An elastic stiffness tensor maps a small displacement gradient tensor to the stress tensor. Minor symmetries in each index pair ensure symmetric stress and dependence only on symmetric strain; major symmetry $c_{ijpq}=c_{pqij}$ follows from an elastic energy and underlies reciprocity. Stability requires positive elastic energy on nonzero strains. In an isotropic material, $c_{ijpq}=\lambda {\delta }_{ij}{\delta }_{pq}+\mu ({\delta }_{ip}{\delta }_{jq}+{\delta }_{iq}{\delta }_{jp})$, with Lamé parameters $\lambda ,\mu$.