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An achromatic doublet is an achromatic lens with two optical lens elements. In the contact thin lens model, their optical powers add and their primary colour changes cancel according to achromatic doublet power balance. The conventional positive achromatic doublet uses a converging crown glass element and a weaker diverging flint glass element; these names describe a common choice, not a universal ordering of every modern optical glass composition.

For a finite Markov chain in which an absorbing state is accessible from every state, order the nonabsorbing states first and writeHere $I$ is the identity matrix and $N$ is the matrix inverse of $I-Q$. With $T$ the first hitting time of the absorbing set,Thus $N$ gives the expected occupation count before absorption, including the initial visit; $B_{ia}$ is the absorption probability of state $a$; and $t_i$ is the expected hitting time of the absorbing set. The vector $1$ has every entry one. See Grinstead and Snell, section 11.2, theorems 11.4-11.6.

For the toy routine in absorbing state, direct matrix multiplication givesThe first row of $N$ says that starting at $u$ gives $2.4$ visits to $u$ and $0.8$ to $v$ before termination. Their sum is the mean duration $3.2$. The identities $(I-Q)N=I$, $B=QB+R$, and $t=1+Qt$ independently verify the inverse, outcome, and duration calculations. The absorbing set must be reachable from every state for these finite formulas: applying them to all nonabsorbing states in the closed-cycle example would leave a persistent cycle in $Q$ and make $I-Q$ singular.

An absolutely summing operator is an absolutely p-summing operator with $p=1$. The identity on the infinite-dimensional l2 sequence space is not absolutely summing: the first $n$ coordinate vectors have sum of norms $n$, while their weak 1-norm is $\sqrt{n}$.

For an absolutely p-summing operator $T:E\to F$, there is a probability measure $\mu$ on the dual unit ball with its weak-star topology such that $\Vert Tx\Vert \le {\pi }_p(T)(\int |\varphi (x){|}^{p}d\mu (\varphi ){)}^{1/p}$. Thus $T$ factors through the closed span of the evaluation functions in $L^p(\mu )$, with the final operator having norm at most ${\pi }_p(T)$. Conversely this factorization implies the summing inequality by summing under the integral. Existence of $\mu$ follows from finite-dimensional Hahn-Banach separation theorem: failure for finitely many vectors would contradict the summing inequality for a nonnegative weighted family. Compactness of probability measures then combines the finite constraints. The final operator need only be defined on the indicated closed subspace, not on all of $L^p(\mu )$.

A bounded linear operator $T:E\to F$ is 2-summing if ${\sum }_j\Vert T{x}_j{\Vert }^2\le C^2{\sup }_{\varphi \in B_{E^{\ast }}}{\sum }_j|\varphi (x_j){|}^2$ for every finite family; ${\pi }_2(T)$ is the least $C$. For operators between Hilbert spaces, this is equivalent to being a Hilbert-Schmidt operator, with ${\pi }_2(T)=\Vert T{\Vert }_{\mathrm{HS}}$. The upper bound follows by applying the Hilbert-Schmidt norm ideal inequality to the column operator of the family; testing finite orthonormal families gives the reverse bound.

The Abelian Ward splitting by Laurent series gives this gauge potential through the Cauchy integral formula, with $f_{,B}=\partial f/\partial {\omega }^B$ evaluated on the twistor line. The integrand has total projective degree zero. Holding the contour fixed locally and differentiating gives $F_{A{A}^{\prime }B{B}^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}(2\pi i{)}^{-1}{\oint }_{\Gamma }{f}_{,AB}{\pi }_{D^{\prime }}d{\pi }^{D^{\prime }}$, using ${\pi }_{A^{\prime }}{\iota }_{B^{\prime }}-{\pi }_{B^{\prime }}{\iota }_{A^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}({\iota }_{C^{\prime }}{\pi }^{C^{\prime }})$. Its unprimed two-component spinor is symmetric, so the primed gauge curvature vanishes and the field obeys the anti-self-dual Maxwell equations. These signs use ${ϵ}_{0^{\prime }{1}^{\prime }}=1$, ${ϵ}^{0^{\prime }{1}^{\prime }}=-1$ and raising by left multiplication; other consistent raising conventions can reverse both displayed overall signs.

The Jacobian variety of a smooth projective algebraic curve parametrizes degree-zero line bundles. Its dimension is the genus of the algebraic curve. An effective degree-$d$ Cartier divisor defines a point in a degree-$d$ torsor through the Abel map of an algebraic curve. The theta divisor induces an isomorphism with the dual abelian variety.

If $N$ is invertible in the base field, multiplication by $N$ on an abelian variety is finite étale: its differential is $N$ times the identity, and properness turns its zero-dimensional fibres into finite fibres. The Theorem of the Cube gives $[N{]}^{\ast }L\cong L^{N^2}$ for a symmetric ample line bundle. Top intersection products give degree $N^{2\dim A}$. For each prime power ${\ell }^r\mid N$, the kernels killed by ${\ell }^s$ have orders ${\ell }^{2s\dim A}$; the classification of finite abelian groups forces $2\dim A$ cyclic factors of length $r$. Combining the prime powers gives the displayed group.

For a connected abelian Lie group, the Lie exponential map is an additive group homomorphism and a local diffeomorphism at zero. Its image is an open subgroup, hence the whole connected group. Its kernel is a discrete additive subgroup of a real vector space. A basis of that kernel identifies it with ${\mathbb{Z}}^r$ inside ${\mathbb{R}}^n$, giving the displayed classification. The compact factors record periods of one-parameter subgroups.

An odd square is congruent to one modulo eight. Conversely, for $u\equiv 1(\mathrm{mod}8)$, begin with $x_3=1$ and $x_n^2\equiv u(\mathrm{mod}{2}^n)$. For $n\ge 3$, adding ${\epsilon }_n{2}^{n-1}$ changes the square by ${\epsilon }_n{2}^n{x}_n$ modulo $2^{n+1}$. Since $x_n$ is odd, one choice of ${\epsilon }_n\in \{0,1\}$ gives the next congruence. The corrections tend to zero 2-adically, so completeness produces an odd square root. This directly proves the unit square criterion without needing a unit derivative in Hensel lemma.

These are units that lie in every finite-layer norm image, using the p-adic closures of the unit groups. Compactness makes this intersection the image of the norm inverse limit. Local and global universal norm subgroups need not coincide. In the cyclotomic main-conjecture proof, the finite defect of global units from universal norms is related by class field theory to invariants of the unramified module.

Although the local viscous dissipation is $2\nu S_{ij}{S}_{ij}$, increasingly intense velocity gradients can compensate for vanishing kinematic viscosity. A Burgers vortex illustrates this mechanism: its radius shrinks while its excess integrated viscous dissipation remains finite. Such a local model does not by itself prove the zeroth law of turbulence for a whole flow.

Put a contact thermometer $C$ in thermal contact with $A$, wait for thermal equilibrium, and record response $X_A$. Repeat with $B$ under the same operating constraints. If $X_A=X_B$, and the calibrated response is an injective function of temperature on this range, $C$ occupies the same thermal reference state in both comparisons. The zeroth law of thermodynamics then predicts $A\sim B$. Equality of raw numbers from different uncalibrated thermometers supplies no such premise.^[ref]

Use a probe whose participating heat capacity is small relative to each sample, or use maintained heat baths. Otherwise the matching readings describe the disturbed final states. Also hold the probe's immersion and mechanical conditions fixed: its response can depend on more than temperature. Matching readouts only tests agreement within resolution and repeatability, rather than proving exact equality of temperatures.

Agreement within a tolerance is not itself an equivalence relation. With a $0.10{}^{\circ }\mathrm{C}$ tolerance, readings $25.00$ and $25.16{}^{\circ }\mathrm{C}$ each agree with $25.08{}^{\circ }\mathrm{C}$, yet disagree with each other. This is a limitation of the measurement rule, not a failure of the exact transitivity asserted by the zeroth law of thermodynamics.

An empirical temperature labels classes of thermal equilibrium using a reproducible thermometer response. Equal labels express the zeroth law of thermodynamics comparison. A suitable continuous response and calibration let ordinary laboratory states be represented on a numerical scale; the abstract equivalence relation alone does not guarantee a useful real-valued coordinate. An injective function of the label preserves equality, so the zeroth law of thermodynamics alone fixes neither unit nor spacing. Thermometer calibration supplies a chosen response relation; thermodynamic temperature supplies the additional absolute-scale structure.

Two thermodynamic systems are in thermal contact when their boundary permits heat transfer. They need not already be in thermal equilibrium: contact between unequal temperatures is precisely how equilibration can begin. Contact specifies an allowed exchange; equilibrium specifies its balance.

For $m^2>0$, the zero-dimensional scalar field theory with Euclidean action $m^2{x}^2/2+\lambda x^4/4!$ has a Gaussian integral normalization $I(0)=\sqrt{2\pi /m^2}$. Its Feynman rules assign $1/m^2$ to each propagator, $-\lambda$ to each quartic vertex, and the reciprocal Feynman-diagram symmetry factor to each vacuum diagram. The integral itself includes disconnected vacuum diagrams; its logarithm selects connected ones. At second order the three possible numbers of edges joining the two vertices are zero, two and four, with Feynman-diagram symmetry factors $128,16,48$ respectively. Their weights add to $35{\lambda }^2/(384m^8)$.

The Wick contractions give the general coefficient $I(0)(-1{)}^v(4v-1)!!/[v!(4!{)}^v{m}^{4v}]$. Successive absolute coefficients have ratio $(4v+3)(4v+1)/[24(v+1)m^4]$, which tends to infinity. Thus this is an asymptotic expansion as $\lambda \downarrow 0$, rather than a convergent power series. For every fixed truncation the Taylor theorem remainder estimate for $e^{-u}$, $u\ge 0$, can be integrated against the Gaussian distribution to justify the corresponding remainder bound.

The continuously compounded yield of a unit-face zero-coupon bond is the constant rate producing its current discount factor over the remaining maturity. It equals the maturity average of the instantaneous forward rate. Differentiation gives $f(t,T)=R(t,T)+(T-t)R_T(t,T)$, and both rates tend to the short rate as $T\downarrow t$. This definition uses continuous compounding; quoted coupon-bond yields require a different cash-flow equation.

Every proper class is in bijection with the universe. Over class-theoretic Zermelo set theory, with class-parameter separation, this is equivalent to replacement plus global choice. For replacement, a proper functional image of a set would give a surjection from that set onto the universe, contradicted by diagonal separation. The proper class of ordinals supplies the global well-order. Conversely rank followed by uniformly chosen rank-segment well-orders is set-like, and every proper class in that order has order type equal to the entire ordinal class.

For $\mathbf{r}=a[\mathbf{q}+b(t)(p(q_1),0,0)]$, put $J=1+b{p}^{\prime }$. Before shell crossing, mass conservation gives $\rho =\bar{\rho }/J$. The dust background equation $3\ddot{a}/a=-4\pi G\bar{\rho }$ and the growing-mode equation $\ddot{b}+2H\dot{b}=4\pi G\bar{\rho }b$ imply ${\nabla }_r\cdot \ddot{\mathbf{r}}=-4\pi G\bar{\rho }+4\pi G\bar{\rho }b{p}^{\prime }/J=-4\pi G\rho$. Thus planar Zeldovich approximation trajectories satisfy the full nonlinear dust force-divergence equation until the map loses invertibility. All three physical coordinates are included in the divergence; the two unperturbed directions still contribute background acceleration.

A quantum anomaly is the failure of a classical symmetry to survive quantization because the functional measure or regulator cannot preserve it.