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A first-order model class axiomatized by a theory is closed under extensions exactly when it has an existential axiomatization. If $A$ satisfies all existential consequences of $T$, then $T$ is jointly satisfiable with the universal theory of $A$: any finite obstruction would negate a universal conjunction and produce a violated existential consequence. Embed a model of $T$ into a model elementarily equivalent to $A$, apply extension closure, and transfer truth back to $A$.

To simulate $g$ on the virtual input $f(x)$, store the simulated input-head position and rerun the logspace transducer for $f$ whenever an input symbol is requested. Count emitted symbols until the requested position is reached. The intermediate output has polynomial length, so its positions take $O(\log n)$ bits, and the work space of $g$ is also $O(\log n)$. Storage used by successive recomputations is reused. This proves closure of logspace-computable functions under composition and transitivity of logspace many-one reductions.

A logical matrix specifies truth values, designated values, and operations interpreting the primitive connectives. To prove nonderivability from an axiom set, it suffices that every axiom instance is designated and every inference rule preserves designation, while the proposed conclusion is not designated under some valuation. The primitive connective symbols need not receive their intended classical meanings: preservation by the displayed rules is what makes the syntactic obstruction valid. Rules with discharged assumptions, such as a deduction theorem for propositional logic, must also preserve consequence for arbitrary assumption sets.

Planar Brownian motion started away from zero almost surely avoids zero. On every finite time interval its continuous path therefore has positive minimum modulus, making the logarithmic stochastic integral well-defined. Since complex quadratic covariation satisfies $d[Z,Z]=0$, analytic Itô calculus gives the displayed exponential identity without a drift correction. Its real part is the change in logarithmic radius and its imaginary part is the continuous accumulated argument, including every winding. This is a pathwise logarithmic lift, not a single-valued logarithm on the entire punctured plane.

Suppose strictly feasible stationary points of $f(x)-{\mu }_k{\sum }_j\log c_j(x)$ form a bounded sequence. Every cluster point satisfying the linear independence constraint qualification is a KKT point. Inactive multipliers tend to zero. The active gradients near the limit have uniformly bounded dual vectors, so pairing barrier stationarity with those vectors bounds the active multipliers. A convergent multiplier subsequence then gives stationarity, nonnegative multipliers and complementary slackness, since ${\lambda }_{k,j}{c}_j(x_k)={\mu }_k\to 0$. A stationarity residual tending to zero is also allowed.

This conclusion is stationarity, not global optimality. If the subproblems are globally minimized and strictly feasible points are dense in the feasible set, any cluster point is instead a global minimizer. Compare each barrier minimum with a fixed strictly feasible point; boundedness near the cluster point bounds $\log c_j(x_k)$ above. Taking the limit gives the objective comparison on strictly feasible points, then continuity and density extend it to all feasible points.

For symmetric $A$, every nonzero stationary point of this barrier lies in a negative-eigenvalue eigenspace: $Ax=\lambda x$, $\lambda <0$, $\Vert x{\Vert }^2=1+\mu /\lambda$. The origin is also stationary for every positive $\mu$ and must be considered separately. On a fixed nonzero branch, the Hessian has radial eigenvalue $4{\lambda }^2\Vert x{\Vert }^2/\mu$ and transverse eigenvalues $2({\lambda }_i-\lambda )$. A simple lowest negative eigenvalue gives a minimizing branch with a condition number growing as $1/\mu$; repeated lowest eigenvalues produce zero transverse curvature instead. Nevertheless the minimizing radius tends to one and the limiting vectors solve the constrained quadratic problem. The condition number measures numerical difficulty, not failure of the approximation principle.

For a finite-dimensional complex semisimple Lie algebra with a fixed triangular decomposition of a Lie algebra, category $\mathcal{O}$ consists of finitely generated universal enveloping algebra modules that are direct sums of Cartan weight spaces and are locally finite under the positive nilpotent subalgebra. Their weight spaces are finite-dimensional and their weights lie in a finite union of downward positive-root cones. Verma modules, finite-dimensional modules, submodules and quotients belong to this category. Closure under submodules uses the noetherian property of the enveloping algebra, obtained from the Poincaré-Birkhoff-Witt theorem. The category condition does not require the module to be finite-dimensional.

The generating function holds for $|h|<1$ and fixed $x$. From Rodrigues' formula, $L_n(x)={\sum }_{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)(-x{)}^j/j!$. Sum first over $n\ge j$, using ${\sum }_{n\ge j}\left(\genfrac{}{}{0ex}{}{n}{j}\right)h^n=h^j/(1-h{)}^{j+1}$, and then over $j$ to obtain the exponential function. The absolute-value double series is finite for $|h|<1$, justifying the rearrangement. Alternatively, the weighted inner product and the Laplace transform of a Laguerre polynomial extract the coefficient $h^n$.

The ordinary Laguerre polynomials, normalized by $L_n(0)=1$, are orthonormal for the weighted inner product displayed above. Their coefficient formula is $L_n(x)={\sum }_{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})x^k/k!$. To prove the relation, use Rodrigues' formula $e^{-x}{L}_n=(n!{)}^{-1}(x^n{e}^{-x}{)}^{(n)}$ and integrate by parts $n$ times against $L_m$. For $n>m$ its $n$th derivative vanishes. At zero every derivative of $x^n{e}^{-x}$ of order less than $n$ vanishes, and at infinity the exponential kills every polynomial boundary term. For $n=m$, $L_n^{(n)}=(-1{)}^n$, so the integral is $(n!{)}^{-1}{\int }_0^{\infty }{x}^n{e}^{-x}dx=1$. Symmetry covers $m>n$. Thus these are orthonormal without an extra normalization factor.

For polynomial solutions of the Laguerre differential equation, put $D_n=L_n^{\prime }-L_n$. The equation gives $D_n+t{D}_n^{\prime }=-(n+1)L_n$. Integrating $\int t{D}_n(L_m^{\prime }-L_m)e^{-t}$ by parts gives the displayed identity; the factor $t$ kills the endpoint at zero and exponential decay kills the endpoint at infinity. Its left side is symmetric in $n,m$, so subtracting the swapped identities proves orthogonality for distinct degrees. This provides a direct energy proof of orthogonality for the Laguerre polynomials without assuming their norm formula.

For deterministic continuous functions $h_j$ on a finite interval, their stochastic exponentials $M(h_j)$ are square-integrable martingales and satisfy $dM(h_j)=h_{j}M(h_j)dB$. Thus $F={\sum }_j{c}_{j}M(h_j{)}_T$ has mean ${\sum }_j{c}_j$ and stochastic-integral representation with the displayed integrand. Itô isometry proves uniqueness up to time-probability null sets among square-integrable predictable integrands. It does not give pointwise uniqueness, since changing a value at one deterministic time changes no integral.

A principal complete type is isolated by one first-order consistent first-order formula: modulo the first-order theory, that logical formula implies every member of the partial type. For complete types this is exactly an isolated type. The qualifier complete distinguishes this notion from partial types.

For a finite-dimensional complex semisimple Lie algebra, every derivation is an inner derivation of a Lie algebra. Adjoin a derivation $\delta$ as a one-dimensional acting algebra, obtaining $M=L⋊K\delta$ with $[\delta ,l]=\delta (l)$. The semisimple Lie ideal has a commuting ideal complement result supplies a one-dimensional complementary ideal commuting with $L$. Its vector with derivation component $\delta$ is $\delta -x$ for some $x\in L$. Commuting this with arbitrary $l\in L$ gives $\delta (l)=[x,l]$. The zero derivation is represented by $x=0$.

A first-order system $U_t+A(U)U_x=C(U)$ is strictly hyperbolic at a state when $A$ has distinct real eigenvalues. These separate its characteristic families and permit independent compatibility equations from left eigenvectors. At a repeated eigenvalue strict hyperbolicity fails; if the matrix is also defective, even hyperbolicity in the complete-eigenvector sense fails. For $A=\left(\begin{array}{cc}u& 2\\ 2v& u\end{array}\right)$ the eigenvalues $u\pm 2\sqrt{v}$ are distinct precisely when $v>0$.

For real $x,y$, the exponential function definitions give $\sinh (x+iy)=\sinh x\cos y+i\cosh x\sin y$. Taking the squared complex modulus and using ${\cosh }^{2}x-{\sinh }^{2}x=1$ gives the identity. In particular $|\sinh (x+iy)|\ge |\sinh x|$, uniformly in $y$. This controls the vertical sides of rectangular contours in residue theorem arguments with hyperbolic functions.

The living inner parasite layer of an echinococcal hydatid cyst, distinguished from its laminated layer and the surrounding host tissue.

A brood capsule is a small internal structure produced by the germinal tissue of a hydatid cyst, within which protoscolices develop. It distinguishes that internal reproductive organization from the simpler bladder architecture of a coenurus.^[ref]

For real $a_r$, put $h(z)={\sum }_{r=1}^n{a}_r{z}^r$. The quadratic form equals ${\int }_0^{1}h(x{)}^{2}dx$, proving its nonnegativity. Apply the centred logarithmic keyhole identity bound to the entire function $h^2$. Since $|h(e^{i\theta }{)}^2|=|h(e^{i\theta }){|}^2$, orthogonality of the finite Fourier sum gives ${\int }_0^{2\pi }|h(e^{i\theta }){|}^{2}d\theta =2\pi {\sum }_r{a}_r^2$. Multiplication by the contour estimate's factor $1/2$ proves the upper bound. The matrix $(1/(r+s+1))$ is the principal submatrix of a Hilbert matrix obtained by deleting its first row and column.

The reduction of a real square matrix to upper Hessenberg form uses orthogonal similarity transformations. A Householder reflector at column $k$ acts on rows $k+1,\dots ,n$ to annihilate entries below row $k+1$, and its matching right action preserves similarity. Previous zeros are preserved. Rank-one application of each reflector gives total $O(n^3)$ work. Subsequent shifted systems with $H-\lambda I$ can be factored in $O(n^2)$ work each because only one entry per column lies below the diagonal.

When Gaussian elimination without row interchanges is defined, an upper Hessenberg matrix needs at most one elimination per pivot column, because only its first subdiagonal can be nonzero. The LU decomposition has a unit lower bidiagonal matrix $L$ and an upper triangular matrix $U$. If the input matrix is also symmetric, it is a tridiagonal matrix and $U$ is upper bidiagonal.

The nonzero-pivot condition is essential: $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is upper Hessenberg and invertible but has no such unpivoted LU decomposition. A pivoted factorization can still be computed, with its permutations recorded explicitly.