For an inextensible planar filament with tailward tangent $\mathbf{t}=(X_s,Y_s)$, a travelling shape has material velocity $(V-U)\mathbf{i}-c\mathbf{t}$ in stationary fluid. Thus the relative fluid velocity is the displayed vector. With $\mathbf{n}=(-Y_s,X_s)$ and $q=V-U>0$, a local inertial normal drag $K_N|w_n|w_n\mathbf{n}$ has axial component $-K_N{q}^2|Y_s{|}^3$. A positive tangential slip $w_t=c-q{X}_s$ gives the opposite axial force. A prescribed tangential law $K_T{w}_t^{3/2}h(s)\mathbf{t}$ therefore gives the force-free balance $K_N{q}^2\int |Y_s{|}^{3}ds=K_T\int (c-q{X}_s{)}^{3/2}{X}_{s}h(s)ds$. The weight $h(s)$ is part of the constitutive approximation and must be justified separately from its speed exponent.

For a periodic inextensible filament, let $q=\mathbf{t}\cdot \mathbf{i}$, $\alpha =⟨q⟩=V/c$, $\beta =⟨q^2⟩$ and $\gamma ={\kappa }_T/{\kappa }_N$. The material velocity is $(V-U)\mathbf{i}-c\mathbf{t}$. Applying the linear local drag tensor ${\kappa }_T\mathbf{tt}+{\kappa }_N(I-\mathbf{tt})$ gives thrust $T={\kappa }_{N}L\{V(1-\gamma )(1-\beta )-U[1-(1-\gamma )\beta ]\}$. Equating this to head drag $\delta {\kappa }_{N}LU$ gives the displayed speed ratio. Whole spatial periods, or period averages, are needed for $c\int qds=VL$. Cauchy-Schwarz gives ${\alpha }^2\le \beta$, with equality only for constant axial tangent projection almost everywhere.

For a heavy spherical head with a much thinner long rigid tail, let $P$ be its excess weight, $H=6\pi \mu a$ the head translation resistance, and $K=K_{N}L$ the tail normal translation resistance. Neglect tail mass relative to head mass, sphere rotation resistance relative to tail rotation resistance, and $a/L$ in tail lever arms. In shear $\alpha z{\mathbf{e}}_1$, the stable orientation satisfies $\sin \theta =\alpha /(\beta +\sqrt{{\beta }^2+{\alpha }^2})$, $\cos \theta >0$, with $\beta =3P/[L(K+4H)]$. Resolving resistive-force theory force and torque balance gives the displayed signed vertical velocity. Since $0<\gamma <1$, stronger shear increases $U_z$ towards zero and reduces the downward settling speed. The stable head is below the far tail end. Very large shear can make the neglected sphere-rotation contribution significant.

Four-terminal sensing uses one pair of contacts to carry electric current and a separate pair to sense electric potential difference. The sensing instrument draws negligible electric current, so the voltage drops along its own leads are negligible, and the current-contact drops outside the sensing interval are excluded. A measurement using just two shared leads includes their electrical resistance and contact drops, which can overwhelm the small sample signal in a resistance-versus-temperature experiment for superconductivity. Four contacts do not automatically remove offsets from the thermoelectric effect. ^[ref]

For a stable offset $V_0$, let $I>0$ denote the magnitude of the reversed electric current. The measured electric potential differences are $V(+I)=IR+V_0$ and $V(-I)=-IR+V_0$, so subtraction gives the displayed electrical resistance. For example, with stipulated $I=1\mathrm{mA}$, $R=0.2\mathrm{m\Omega }$ and $V_0=3\mu \mathrm{V}$, the two readings are $3.2$ and $2.8\mu \mathrm{V}$; their difference correctly recovers $R$. If the offset changes between readings, its change leaves a residual measurement error; the sample and measuring circuit must remain sufficiently stable during reversal.

The finite residual consists of all elements killed by every homomorphism to a finite group. A group is a residually finite group precisely when its finite residual is trivial. Intersecting all finite-index subgroups gives the same result: the normal core of each such subgroup is normal of finite index and lies inside it.

The character of a Young permutation module counts fixed tabloids and is integer-valued. Dominance order on partitions makes its decomposition into Specht modules triangular with diagonal multiplicity one. Descending induction in any order refining dominance therefore expresses each irreducible Specht character as an integer combination of permutation characters, proving integrality.

The graded abelian group ${⨁}_{n\ge 0}R(S_n)$ of ordinary virtual characters has product $\chi \star \psi ={\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}(\chi ⊠\psi )$ for degrees $m,n$. Its unit is the trivial degree-zero character. Associativity follows by transitivity of induced representations through the three-factor Young subgroup; commutativity follows by conjugating the block positions. The Frobenius characteristic map identifies this ring with the ring of symmetric functions, sending irreducible Specht characters to Schur functions. This product is different from pointwise multiplication of characters of the same symmetric group.

For the defining SU(2) action ${\eta }^{\prime }=A\eta$, the conjugate row ${\bar{\eta }}_{\alpha }=({\eta }^{\alpha }{)}^{\ast }$ transforms by $A^{†}=A^{-1}$. Thus ${\bar{\eta }}_{\alpha }{\eta }^{\alpha }$ is invariant. The alternating invariant tensor ${ϵ}_{12}=1$ satisfies $A^TϵA=ϵ$. Choose its inverse ${ϵ}^{12}=-1$; then ${ϵ}^{\alpha \gamma }{ϵ}_{\gamma \beta }={\delta }^{\alpha }{}_{\beta }$. Raising the conjugate index, ${\tilde{\eta }}^{\alpha }={ϵ}^{\alpha \beta }{\bar{\eta }}_{\beta }$, gives another defining spinor. The resulting antilinear intertwiner squares to $-I$, so the defining representation is a pseudoreal representation.

For the block-rotation maximal torus in the special orthogonal group $SO(2m+1)$, the Weyl group acts by permutations and independent inversions of $z_i$. Its invariant Laurent polynomials are the symmetric polynomials in $x_i=z_i+z_i^{-1}$, hence the polynomial ring in their elementary symmetric polynomials $e_1,\dots ,e_m$. Restriction of the group representation ring of a compact group is injective because every group element is conjugate into the torus and characters distinguish virtual representations. For the defining complexified representation $V$, the exterior-power characters satisfy ${\sum }_k{\chi }_{{\Lambda }^{k}V}{t}^k=(1+t){\prod }_i(1+x_{i}t+t^2)$. For $k\le m$, its coefficient is $e_k$ plus an integral linear combination of $e_j$ for $j<k$. Induction puts every $e_k$ in the restriction image, proving equality with the invariant ring and giving polynomial generators $[{\Lambda }^{1}V],\dots ,[{\Lambda }^{m}V]$.

The shortest prefix before the repeating tail of a decimal representation. For a reduced fraction with denominator $2^u{5}^{v}m$, where $m>1$ is coprime to ten, its length is $\max (u,v)$; the tail length is given by decimal period from multiplicative order.

For independent uniform orientations in a renovating shear flow, the incident angle relative to each new shear is uniform even after earlier deformations. One interval stretches a unit material line by $\ell (\theta {)}^2=1+a\sin 2\theta +a^2{\sin }^2\theta$. Write this as $A+B\cos (2\theta -\delta )$, where $A=1+a^2/2$ and $A^2-B^2=1$. The angular logarithmic integral gives $⟨\log \ell ⟩=\frac{1}{2}\log [(A+1)/2]$. Add these independent logarithmic increments to get the displayed rate. At fixed shear rate, small $T$ gives $\mu \sim {\Lambda }^{2}T/8$ and large $T$ gives $\mu \sim \log (|\Lambda |T/2)/T$. The unique positive maximum is at $|\Lambda |T\approx 3.9605826$, obtained from $\log z=2(1-z^{-1})$ with $z=1+a^2/4$.

A choice of renormalization conditions and finite counterterms defines a renormalization scheme. The minimal subtraction scheme removes only regulator poles; the modified minimal subtraction scheme also removes conventional universal constants. Different schemes can assign different values to renormalized couplings while describing the same physical observables after parameters are consistently matched.

The bare mass is the mass parameter in a regulated field-theory action before a renormalization condition relates it to a measured mass. For a scalar, loop self-energy corrections change the propagator denominator to $p^2-m_0^2-\Sigma (p^2)$. The pole mass solves $m^2-m_0^2-\Sigma (m^2)=0$ for a stable particle. The bare mass generally depends on the regulator when the pole mass is held fixed.

A bare coupling is an interaction coefficient in a regulated action. It is related to a renormalized coupling through counterterms and any required powers of the renormalization scale. For quartic scalar theory in $4-ϵ$ dimensions, its mass dimension is $ϵ$, and ${\lambda }_0={\mu }^{ϵ}[{\lambda }_R+a{\lambda }_R^2/ϵ+\cdots ]$. At fixed regulator, changing $\mu$ while leaving the theory unchanged keeps ${\lambda }_0$ fixed; the corresponding change of ${\lambda }_R$ defines its renormalization-group beta function.

Alternately routing a rate-$r$ Poisson process produces non-Poisson streams with common interarrival law Erlang$(2,r)$. The odd-numbered stream has an initial exponential delay, while the even-numbered stream is an ordinary renewal process. Their equilibrium delay survivor is $e^{-rt}(1+rt/2)$. The probability of no odd-stream point in $(s,s+t]$ is $e^{-rt}[1+rt(1-e^{-2rs})/2]$; for the even stream replace the minus by a plus. Conditioning on the parity of $N(s)$ proves these formulas.

Alice holds the first two qubits and Bob the third of a three-vertex graph-state wire. Alice measures at $\alpha$ and $(-1{)}^r\beta$, with outcomes $r,s$, and sends these two bits. Bob applies $Z^r{X}^s$, the inverse of the known Pauli frame $X^s{Z}^r$, obtaining $U(\beta )U(\alpha )|+⟩$ up to global phase. Bob need not know the angles. Every branch succeeds, with branch probability $1/4$, and no postselection is needed. Alice's measurements and Bob's corrections constitute local operations and classical communication on the previously shared entangled resource.

For a target pure state $|\psi ⟩$ of a qubit, choose its orthogonal complement so thatMeasuring the sender's half of this singlet state in that basis leaves the receiver in $|\psi ⟩$ when the sender obtains $|{\psi }^{\perp }⟩$. The outcome has probability one half. One classical success flag heralds the exact preparation without conveying the target's classical description.

// Target: quantum-theory.bigb

Prepare independent candidate blocks by one-bit heralding of a remote state preparation block, but send only the first successful block's index. Success eventually occurs with probability one; a prefix code for a rare geometric success uses at most $n+e/(e-1)$ expected bits per block. The expected singlet state consumption is $n{2}^n$. Alternatively, limiting the number of trials to $n{2}^n$ gives failure probability at most $e^{-n}$ and a fixed message length $n+{\log }_{2}n+O(1)$. Both versions approach one classical bit per remotely prepared qubit in their respective expected-length or high-success senses.

// Target: quantum-theory.bigb

Suppose two orthonormal qubit bases have disjoint physical rays. Alice chooses which conjugate basis to measure on her half of a Bell state. A device that perfectly labels Bob's individual member of either basis would reveal Alice's choice without her outcome being communicated. Spacelike operation would therefore violate relativistic causality, even though ordinary projective measurements see the same reduced density matrix $I/2$.

A moving rod can have both ends inside a shorter shed on a single constant-time slice of the shed frame. Lorentz transformation sends the two simultaneous endpoint events to different times in the rod frame; it does not imply simultaneous containment there. Entrance and first collision can be spacelike separated and occur in reversed time order. After impact, length changes must propagate through the rod causally; a perfectly rigid instantaneous stopping model is not relativistically admissible.