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A function on the complex upper half-plane is modular invariant if it is unchanged by the fractional-linear action of the modular group. This does not imply holomorphicity: a nonholomorphic Eisenstein series is modular invariant, whereas a modular function additionally satisfies meromorphicity and cusp conditions.

A Dirichlet character $\psi$ modulo $N$ specifying the displayed transformation law is the nebentypus of a modular form. The action of $-I$ forces $\psi (-1)=(-1{)}^k$ for a nonzero form. It describes a character eigenspace of the diamond action rather than an additional analytic condition at modular cusps.

The integral representation of the modified Bessel function of the second kind gives $K_0(x)={\int }_0^{\infty }{e}^{-x\cosh t}dt$ for $x>0$. Taking the Fourier transform of the even continuation and setting $u=\sinh t$ gives $2{\int }_0^{\infty }(1+k^2+u^2{)}^{-1}du=\pi /\sqrt{1+k^2}$. The large-wavenumber terms $\pi /|k|-\pi /(2|k{|}^3)$ are also obtained from the local logarithmic singularity $-\log |x|-(x^2/4)\log |x|$ using the Fourier transform of the absolute logarithm.

The positive occupation operators are $N_e={\sum }_s\int d^{3}p{a}_{\mathbf{p}}^{s†}{a}_{\mathbf{p}}^s/(2\pi {)}^3$ and $N_{\bar{e}}={\sum }_s\int d^{3}p{b}_{\mathbf{p}}^{s†}{b}_{\mathbf{p}}^s/(2\pi {)}^3$. Both commute with the free Hamiltonian. The normal-ordered local density integral $\int :{\psi }^{†}\psi :d^{3}x$ is their difference, not their sum. It generates the global phase symmetry. For electron charge $-e$, electric charge is $-eQ$. Pair creation can change total occupation while preserving this net charge.

For any continuous map between the displayed Real projective spaces, the induced map on cohomology sends the degree-one generator to $\alpha x+\beta y$. Its $2n$th power is $\left(\genfrac{}{}{0ex}{}{2n}{n}\right){\alpha }^n{\beta }^n{x}^n{y}^n$, since all other terms exceed the truncation exponent of one factor. The coefficient is zero modulo two because $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)=2\left(\genfrac{}{}{0ex}{}{2n-1}{n-1}\right)$. Naturality of the evaluation pairing then makes the top induced map on homology zero. This obstruction does not apply to $n=0$, when both spaces are points.

The unit sphere bundle of the tautological line is the double cover $S^n\to {\mathbb{RP}}^n$. In its Gysin sequence, multiplication by the mod-two Euler class $a$ is an isomorphism in successive interior degrees, since the sphere has no interior cohomology. Connectedness in degree zero makes multiplication into degree one injective. At the top, vanishing above dimension $n$ makes the transfer from $H^n(S^n;{\mathbb{F}}_2)$ surjective; together with that injection it makes the top group one-dimensional. Thus every $a^j$, $0\le j\le n$, is the nonzero generator and the next power vanishes.

A Möbius transformation with two distinct fixed points is conjugate to $z\mapsto az$. It is loxodromic when $|a|\ne 1$; on three-dimensional hyperbolic space it translates along the axis joining its fixed points and rotates around that axis by $\arg a$. The positive-real multiplier case has no rotation and is also called hyperbolic. When listing elliptic, parabolic, hyperbolic and loxodromic as separate types, loxodromic refers to the remaining case with nonzero rotation. A determinant-one matrix with nonreal trace has reciprocal eigenvalues that are neither real nor of modulus one, so it supplies this latter case.

The Möbius transformation $M(z)=(3z-1)/(3-z)$ fixes the antipodal pair $1,-1$ under stereographic projection. But $M^{\prime }(1)=2$, so it multiplies infinitesimal spherical lengths at that fixed point by two, as follows from the round line element $2|dz|/(1+|z{|}^2)$. It is not an isometry and hence not a sphere rotation. Fixing an antipodal pair is therefore necessary for a nontrivial rotational Möbius transformation, but is not sufficient.

For $|w|<R$, this Möbius transformation maps the radius-$R$ disc to the unit disc, vanishes at $w$ and has modulus one on the boundary. Expanding gives $|R^2-\overline{w}z{|}^2=R^2|z-w{|}^2$ when $|z|=R$; its pole lies outside the disc. Dividing a holomorphic function by factors corresponding to its interior zeros removes those zeros without changing its boundary modulus. This gives a direct factorization proof of Jensen's formula.

Let $T_b(z)=z+b$ and $I(z)=1/z$ on the Riemann sphere. For $s\ne 0$, direct composition gives the displayed identity. Every nonzero complex scaling $z\mapsto \lambda z$ is obtained by choosing $s^2=-\lambda$. Since translations, scalings and reciprocal inversion generate the Möbius group, translations and reciprocal inversion alone also generate it.

A nonconstant Möbius transformation mapping the real line into the unit circle has the displayed form with $|\lambda |=1$ and $k$ nonreal. Indeed $|ax+b{|}^2=|cx+d{|}^2$ for real $x$ yields $|a|=|c|\ne 0$, $|b/a|=|d/c|$ and $\mathrm{Re}(b/a)=\mathrm{Re}(d/c)$. Thus $d/c$ is $b/a$ or its complex conjugate; the first possibility makes the map constant, so the second is necessary. Conjugate numerator and denominator have equal modulus on the real line, proving the converse as well.

This Möbius transformation maps the upper half-plane onto the unit disk. For $\mathrm{Im}z>0$, the identity $1-|C(z){|}^2=4\mathrm{Im}z/|z+i{|}^2$ puts the image inside the unit disk. Its inverse is $z=i(1+w)/(1-w)$, so the map is a conformal bijection. On the extended real boundary it maps onto the unit circle. It is the scalar complex-analytic version of a Cayley map, rather than the matrix parametrization used in Cayley transform.

A specimen-stage reference bearing a scale of known length, imaged through the microscope objective during microscope scale calibration.

A ruled scale viewed together with the intermediate microscope image. Its intervals acquire specimen length units through microscope scale calibration.

A Mersenne prime is a Mersenne number that is a prime number. It is the prime factor required by the Euclid-Euler theorem; $3,7,31,127$ generate $6,28,496,8128$. The family is named after Marin Mersenne.

A greedy matching starts with the empty matching in a graph, scans all edges in some order, and selects an edge exactly when both endpoints are unmatched vertices in a matching. For a finite graph, a complete scan returns a maximal matching: a rejected edge had a matched endpoint, which stays matched thereafter. Stopping before all edges have been considered need not give maximality. Changing the order can change the result; on P₄, middle-first gives one pair, whereas outer-first gives two. A greedy algorithm of this form never revisits its choices, while an augmenting path in a matching does.

An integer linear combination is a linear combination whose coefficients are integers. The lattice basis definition uses unique integer linear combinations, rather than arbitrary real coefficients. For example, $(1,0)$ is a real linear combination of $(2,0)$ but is not an integer linear combination of that single vector.

A contour plot displays several level curves of a real-valued function of two variables, usually with labels indicating their values. It represents a surface's height on a flat page. Closely spaced level curves indicate a large change per unit distance only when the gaps between their labeled values are comparable.

The 2024 operating instructions for PHYWE's 04361-00 apparatus describe a frit between two gas chambers. A copper-coil heat exchanger first restores inlet temperature after cylinder-regulator cooling. An upstream manometer reads gauge pressure; the outlet stays at atmospheric pressure. Two NiCr-Ni thermocouples sit 2–5 mm from the frit. The digital meter's difference mode reports $0.01\mathrm{K}$ precision.^[ref]

The manual specifies at most $1\mathrm{bar}$ gauge pressure, one hour of apparatus acclimatization, half an hour of meter warm-up, about 1.5 minutes of flow settling, then 10–15 seconds of averaging. It identifies a stagnant-gas zeroing error that shifts the fitted intercept and notes a maximum nitrogen signal of about $0.3\mathrm{K}$ in this setup.^[ref]

Define $x=p_1-p_2>0$ and $y=T_1-T_2$, so a small-drop fit is $y\simeq {\mu }_{\mathrm{JT}}x+c$. A constant sensor offset $c$ leaves the slope unchanged; pressure-dependent drift or heat leakage can alter it. The observable is the differential temperature slope across the plug, not cylinder-outlet cooling. Display precision alone does not establish accuracy or prove an isenthalpic process. This low-pressure, room-temperature apparatus does not map a complete Joule-Thomson inversion curve.