A semigroup presentation specifies generators $X$ and equalities between nonempty words. The resulting semigroup is the quotient of the free semigroup $X^{+}$ by the smallest congruence containing these equalities. Two words are equal precisely when one can be transformed into the other by finitely many replacements in contexts, using the defining equalities in either direction.

A smooth zero-interior-potential vorticity mode hasDecay for $z\to \infty$ gives $\hat{\psi }=C{e}^{-mz}$. Its penetration depth $1/m$ increases with horizontal wavelength and decreases with buoyancy frequency. Singular neutral interior sheets are a distinct continuous-spectrum class.

For basic shear $U(z)=\Lambda z$ with a resting lower boundary, the rigid-boundary buoyancy condition for quasi-geostrophic waves and the decaying vertical exponential giveThis is real. The edge wave propagates in the same laboratory direction as the basic flow above its resting lower wall. If the basic buoyancy is $M^{2}y+N^{2}z$, then $\Lambda =-M^2/f_0$.

For a forced sphere and a distant equal force-free, torque-free sphere, the leading incident Stokeslet strain is proportional to $(\mathbf{U}\cdot \mathbf{n})(I-3\mathbf{nn})$. Its reflected stresslet is axisymmetric about the line of centers and has zero vorticity on that line. The nominal fifth-order self-spin coefficient therefore vanishes. Finite-size incident-strain corrections and higher force-free scattering multipoles first supply a generic seventh-order angular velocity. Purely longitudinal forcing has zero spin exactly by axial symmetry. The leading translation correction is fourth order for a generic orientation, but sixth order for purely transverse forcing. These cancellations matter when using the method of reflections for Stokes flow to infer orders from far-field decay alone.

For an oriented closed surface smoothly embedded in an oriented four-manifold, a transverse section of the normal bundle gives a homologous small push-off. Its signed zeros are exactly the signed intersections with the original surface. This identifies the evaluation of the Euler class with the self-intersection number.

With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge dt$, ${ϵ}_{123}=1$, and temporal gauge $A_t=0$, the mixed gauge curvature is $F_{ti}={\partial }_t{A}_i$. The self-dual Yang-Mills equations then have the displayed form. Placing $dt$ first in the orientation reverses the sign, so the metric alone does not specify it.

For a strictly feasible primal-dual conic optimization problem and a logarithmically homogeneous barrier, the central path consists of solutionsThe primal-dual gap is $⟨s,y⟩=\nu \mu$. Existence requires appropriate feasibility and boundedness hypotheses, rather than merely a full-rank constraint matrix. Linearizing these equations gives a central-path Newton system.

For a nondegenerate self-concordant barrier, the open local Hessian ballis contained in its barrier domain. This is the Dikin ellipsoid. It supplies a quantitative way to ensure that a damped Newton method stays interior, without relying on Euclidean distance to a possibly curved boundary.

In front of the contact line in the sediment-lubricated viscous mountain model, $h=d$ and $d_x=3{\mu }_{s}U(d-d_{\infty })/(\rho g{d}^3)$. For $\delta =d/d_{\infty }$ and $\ell =\rho g{d}_{\infty }^3/(3{\mu }_{s}U)$, integration gives $x/\ell =P(\delta )-P(3)$, with $P(\delta )={\delta }^3/3+{\delta }^2/2+\delta +\ln (\delta -1)$. The branch $1<\delta \le 3$ is unique and tends exponentially to the uniform far-field depth.

The difference between the angular momentum of circular coplanar orbits at fixed semi-major axes and the system's actual angular momentum along the reference normal. With circular momenta $L_j=m_j\sqrt{G{M}_{\star }{a}_j}$ it is ${\sum }_j{L}_j(1-\sqrt{1-e_j^2}\cos I_j)$. For small planar orbital eccentricities its quadratic part is $\frac{1}{2}{\sum }_j{L}_j|z_j{|}^2$. Its conservation in $\dot{z}=iAz$ requires $L_j{A}_{jk}=L_k{A}_{kj}$.

The smooth sections of a vector bundle form a module over the commutative ring $C^{\infty }(M)$ by pointwise multiplication. A vector bundle morphism over the identity induces a $C^{\infty }(M)$-linear map between these modules. Bundle morphisms from maps of smooth sections gives the converse, without assuming the bundle has a global frame.

If both sections for every coordinate are binary initial segments, then the family itself is a binary initial segment, apart from the possible family $(\mathcal{P}([n-1])\setminus \{[n-1]\})\cup \{\{n\}\}$. Any missing vertex earlier than a present vertex must disagree with it in every coordinate; the two are therefore complementary and consecutive in binary order on the discrete cube, forcing this sole exceptional shape.

The Mathieu equation is the displayed linear differential equation with periodic coefficients. Floquet theory describes its bounded and exponentially growing solutions, while its periodic Mathieu characteristic values mark the edges of parametric resonance gaps. The forcing normalization matters: replacing $t$ by $t/2$ rescales both the constant and periodic coefficients.

For $\gamma \ne 0$ and $x>0$, the change of variables $u=x^{\gamma }$ convertsinto $Y_{uu}+Y=0$, where $y(x)=Y(u)$. The chain rule cancels the first-derivative terms. Thus $\cos (x^{\gamma })$ and $\sin (x^{\gamma })$ form a pair of linearly independent solutions, with Wronskian $\gamma x^{\gamma -1}$. This is a variable-speed reparametrization of the harmonic oscillator equation.

A smooth invertible change of variables in $u_{xx}-f(x)u=0$ introduces a first derivative in time. Rescaling by ${\dot{x}}^{-1/2}$ removes it. Choosing $\dot{x}=f^{-1/2}$ locally for $f>0$ produces the constant leading coefficient used in the WKB approximation; negative $f$ requires a complex branch or the corresponding oscillatory real formulation.

For $t>0$, the change of variables $\tau =1/t$ and $u(t)=v(\tau )/\tau$ converts $t^4{u}^{\prime \prime }+{\lambda }^{2}u=0$ into $v^{\prime \prime }+{\lambda }^{2}v=0$. The chain rule gives $u^{\prime }=v-\tau v^{\prime }$ and $u^{\prime \prime }={\tau }^3{v}^{\prime \prime }$. Thus $u=t[A\cos (\lambda /t)+B\sin (\lambda /t)]$. The prefactor cancels the derivative term introduced by inverting the independent variable.

For $\psi (t)={\sum }_{m\le t}\Lambda (m)$, finite rearrangement and the Von Mangoldt divisor identity giveThe Stirling formula, or integral comparison for the logarithm, gives $T(x)=x\log x-x+O(\log (x+1))$ for real $x\ge 1$.

For chosen inequivalent unitary irreducible representations $\rho ,\sigma$ of a finite group and a $d_{\rho }\times d_{\sigma }$ matrix $M$, the average ${\mathbb{E}}_x\rho (x)M\sigma (x{)}^{\ast }$ is zero when $\rho \ne \sigma$. For $\rho =\sigma$, it is $(\mathrm{tr}M/d_{\rho })I$. This follows from the Schur lemma, since the average intertwines the two group representations. Equivalent representations in different bases require the corresponding intertwiner; the identity-matrix formula assumes literally the same representative.

For the pure state ${\lambda }_0|00⟩+{\lambda }_1|11⟩$ with real normalized Schmidt coefficients, the two-body Pauli correlators form $T=\mathrm{diag}(s,-s,1)$, $s=2{\lambda }_0{\lambda }_1$. Thus arbitrary local axes give $⟨(\mathbf{a}\cdot \sigma )\otimes (\mathbf{b}\cdot \sigma )⟩=s(a_x{b}_x-a_y{b}_y)+a_z{b}_z$. The negative $yy$ component follows from the phases in the Pauli operators.

A quadratic scan statistic maximizes a directional empirical second moment over candidate unit vectors. For a Gaussian random vector sample with a rank-one covariance spike in one of those directions, the matching statistic has its scale multiplied by the spike's eigenvalue. A chi-squared concentration inequality controls each direction.