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Concatenation joins two strings end to end, retaining the order of their symbols. For example, joining $ab$ and $ba$ gives $abba$. It has the associative property: joining $(uv)w$ or $u(vw)$ gives the same word over an alphabet. The empty word is its identity element. All finite words over an alphabet with this binary operation form a free monoid.

For a strictly distance-decreasing map of a finite-dimensional Euclidean space, one bounded forward orbit suffices for existence of a fixed point. Its closure is compact by the Heine-Borel theorem and invariant by continuity, so strict distance decrease on a compact metric space applies. Strict distance decrease gives uniqueness throughout the whole space, not merely the orbit closure.

For $d>1$, the displayed stress-energy tensor is symmetric and conserved on the free massless scalar field equation $\square \varphi =0$. Its improvement is identically divergence-free. Since $\square {\varphi }^2=2(\partial \varphi {)}^2$ on shell, its trace vanishes. Contracting it with a Conformal Killing vector field therefore produces a conserved conformal current. In $d=2$ the improvement coefficient is zero: the canonical tensor already has zero trace.

These stress measures refer force or area to the reference configuration. Different index-order conventions distinguish a first Piola-Kirchhoff stress tensor from its transposed nominal stress tensor. The second Piola-Kirchhoff stress tensor has two material indices.

The Kirchhoff tensor is the Cauchy stress tensor multiplied by the local volume ratio. Its stress power is per reference volume. Pushing forward the second Piola-Kirchhoff stress tensor gives the displayed formula.

With the right Cauchy-Green deformation tensor $C=F^{T}F$ and the second Piola-Kirchhoff stress tensor $S$, this mixed material stress is generally nonsymmetric. In the material-first nominal stress tensor convention, $PF=SC=M^T$. Thus replacing $P{F}^{-T}$ by $PF$ changes the stress measure. The relation follows from $\mathcal{P}=FS$, without any assumption of material isotropy.

A stress measure $T$ is work-conjugate to a strain measure $E$ when their contraction gives the mechanical power per specified volume for every admissible deformation rate. The reference-volume power is $P_{Ii}{\dot{F}}_{iI}=\tau :D$, with Kirchhoff stress tensor $\tau$ and rate-of-strain tensor $D$. A strain constraint can leave the conjugate stress undetermined by a reaction component that does no admissible work; uniqueness should therefore not be assumed on a restricted set of rates.

The infinitesimal strain tensor is the symmetric part of a displacement gradient tensor. It is the linear approximation to a finite strain measure for small displacement gradients. An infinitesimal rigid rotation contributes no strain, so the elastic energy depends on the symmetric part rather than the entire gradient.

A constructible polygon is a polygon whose vertices can be obtained by a compass-and-straightedge construction from the specified initial data. The classical question for a regular polygon asks which numbers of sides permit an exact construction from a given side or circumscribed circle.

A straight line through a point $a$ in nonzero direction $t$ is the set of vectors $r=a+st$ for real parameter $s$. If $|t|=1$, changes in the parameter equal signed distances along the line. Two such lines intersect exactly when their two parameter formulas give a common vector for some choice of their parameters.

A secant line is a straight line passing through two distinct points of a curve. For a graph $y=f(x)$ at $x=a$ and $x=a+h$, its slope is the difference quotient $(f(a+h)-f(a))/h$. As the second point approaches the first, a limiting secant line can give the tangent line.

In Cartesian coordinates, the slope of a nonvertical straight line is its vertical change divided by its horizontal change. The graph $y=mx+c$ has slope $m$; a vertical straight line has no finite slope. For a tangent line to a differentiable function, the slope is the derivative.

Suppose a family $A\subseteq C(K,\mathbb{R})$ on a compact space is closed under pointwise maximum and minimum. If a target $f$ can be approximated arbitrarily closely at each pair of points by a member of $A$, it can be approximated in the supremum norm. For a fixed first point, take finitely many pairwise approximants whose neighborhoods cover $K$ and take their maximum: it lies above $f-\epsilon$ everywhere and below $f+\epsilon$ near that first point. Take finitely many such first-point neighborhoods covering $K$ and take the minimum of their maxima. The result is within $\epsilon$ of $f$ everywhere. Starting with tolerance $\epsilon /2$ ensures a strict uniform approximation error smaller than $\epsilon$.

The overdamped filament bending equation diagonalizes in the complete free-free bending spectrum. Positive modes decay at rates $A{k}_n^4/\zeta$, while translation and tilt remain fixed. The long-time shape is the affine least-squares projection of the initial height. Free-end boundary values conserve the total height integral and its first spatial moment. For square-integrable initial data the expansion gives the smoothing semigroup solution.

The integral ${\int }_{\mathbb{R}}{e}^{-t^2}(z-t{)}^{-1}dt$ in the lower half-plane extends to the displayed entire function. In the upper half-plane the continued function differs from the directly evaluated real-contour integral by $2\pi i{e}^{-z^2}$. Its algebraic expansion is $\sqrt{\pi }{z}^{-1}{\sum }_{m\ge 0}(2m-1)!!/(2z^2{)}^m$. The real axes are switching lines for the exponentially small term under the Stokes-multiplier convention; the diagonal rays $\arg z=\pi /4+j\pi /2$ are equal-magnitude boundaries for the exponential and algebraic mechanisms. Stating both the residue jump and the sector domains prevents confusion between switching and dominance conventions.

The saddle and endpoint asymptotic expansions of this entire function have algebraic coefficients $C{\omega }^j/z$ in sectors $(2j-1)\pi /3<\arg z<(2j+1)\pi /3$, where $C=\Gamma (1/3)/3$ and $\omega =e^{2\pi i/3}$. Deforming the contour onto the decay ray $\arg (zt)=2j\pi /3$ gives its complete integral $C{\omega }^j$. The endpoint contribution is $-e^{-z^3}(3z^3{)}^{-1}{\sum }_{r\ge 0}(-1{)}^r(2/3{)}_r/z^{3r}$. The algebraic coefficient switches at $\arg z=\pi /3,\pi ,5\pi /3$; the equal-exponential-magnitude anti-Stokes lines are the intervening rays $\pi /6+j\pi /3$.

Two sectorial fundamental solutions of the same linear system differ on their overlap by a constant invertible right factor, since $(Y_j^{-1}{Y}_{j+1}{)}^{\prime }=0$. With a common formal normalization these factors encode the exponentially small ambiguities invisible to the asymptotic expansion. They are called Stokes matrices. Equivalently their inverses act on column vectors of solution coefficients; the convention must be specified when ordering products.

At an unramified irregular singular point with distinct leading eigenvalues, a sectorial fundamental solution is a fundamental matrix whose entries are holomorphic functions asymptotic to a prescribed formal matrix on a sector of the logarithm cover. For competing cubic exponentials, the usual construction uses six overlapping sectors with basic angle $\pi /3$. A formal power series need not converge, and two sectorial realizations may differ by exponentially small terms across a sector boundary.

For Stokes flow without body forces, $\nabla \cdot \mathbf{\sigma }=0$. Symmetry of the Newtonian fluid stress tensor also gives $\nabla \cdot (\mathbf{x}\times \mathbf{\sigma })=0$, interpreted componentwise as the angular-momentum flux. The divergence theorem makes the resultant force and torque the same on any two homologous enclosing surfaces lying in a nonsingular fluid region. In a boundary perturbation of a nearly spherical particle, evaluating each perturbation field on a fixed enclosing surface avoids spurious geometric force terms: the geometric terms from changing the surface normal and area cancel those from evaluating the base stress at the displaced surface.

Extend the velocity rigidly through a moving sphere in an incompressible tube flow. The extended velocity is continuous and divergence-free, so its total flux is independent of height. Quiescent far-field conditions make this flux zero. The sphere contributes $V_z{A}_s$ through a cross-section; the rotational contribution integrates to zero over its circular slice. Thus the fluid flux compensates the solid displacement. A descending sphere generates upward fluid flux in every section cutting its interior. This statement assumes a localized flow with no imposed throughflow and negligible tube-end effects.