Original archive: Cambridge FOI 2024/50 released ZIP. The original is the member NanoMPhil Paper 1 2017.pdf, physical pages 1–13.

A Maxwell fluid has a spring and dashpot in series, so its step-stress creep compliance is $J(t)=1/G_0+t/\eta$. The passive microrheology relation then gives the displayed three-dimensional mean squared displacement. The plateau scale $k_{B}T/(\pi a{G}_0)$ measures elasticity, while the long-time slope $k_{B}T/(\pi a\eta )$ measures viscosity. An entropic network with mesh size $\xi$ has the estimate $G_0\sim k_{B}T/{\xi }^3$, giving plateau scale ${\xi }^3/(\pi a)$.

The nonzero zero-lag extrapolation belongs to the idealized massless spring-dashpot response. Actual displacement variance starts at zero; fast solvent dynamics or inertia regularize the short-time behavior. Consequently the plateau formula applies after those transients and before the Maxwell relaxation time, if those scales are well separated. A sampling interval comparable with $\tau$ gives little access to a genuine plateau, even when spatial localization is sufficiently precise.

A ranked partially ordered set has an integer-valued rank function of a partially ordered set that increases by one whenever $y$ covers $x$, meaning $x<y$ with no element strictly between them. For finite orders with minimal elements, one commonly normalizes every minimal element to rank zero. Conventions about requiring all maximal chains to have the same length should be stated separately.

In the Boolean lattice, $\rho (S)=|S|$: a covering step adds one element. In the positive divisors of $12$ ordered by divisibility, $\rho (d)$ is the number of prime factors counted with multiplicity. Rank levels organize an order into layers and allow questions about its Sperner property.

A finite ranked partially ordered set has the Sperner property if its largest antichain has the same size as its largest rank level. Each level is already an antichain, so the property says that mixing different ranks cannot produce a larger one.

The Boolean lattice of all subsets of an $n$-element set, ordered by inclusion and ranked by subset size, has this property by Sperner's theorem. Its largest antichain has size $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$. For $n=3$, the three singletons form a largest antichain. General ranked orders need not have the property, making it a substantive extremal condition.

The zeta transform on a finite partially ordered set sends $f$ to $F(x)={\sum }_{y\le x}f(y)$. The subset zeta transform is the case where the order is inclusion of subsets.

A subset $I$ of a partially ordered set such that $x\in I$ and $y\le x$ imply $y\in I$. Thus membership forces all smaller elements to belong as well. This definition concerns an order, rather than an ideal of a ring.

An FCC partial dislocation with Burgers vector $a⟨112⟩/6$ lying in a close-packed $\{111\}$ plane. Two suitable Shockley vectors can sum to an FCC perfect vector $a⟨110⟩/2$.

A sessile partial junction associated with intersecting stacking faults. A typical FCC resultant has vector $a⟨110⟩/6$. Its Burgers vector and junction line can define a plane outside the ordinary $\{111\}$ glide family.

A parametric curve in the plane, given by two coordinate functions $x(t),y(t)$. Its speed is $\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}$ wherever these derivatives exist, giving its arc length by integration.

A Lissajous curve is a parametric curve with coordinates $x(t)=A\sin (at+\delta )$ and $y(t)=B\sin (bt)$. The frequency ratio and phase shift control its crossings and whether its trajectory closes.

Paramagnetism arising when thermal excitation populates a magnetic manifold above a less magnetic or nonmagnetic ground manifold. Competing thermal population and orientational randomization can produce a maximum in susceptibility at an intermediate temperature.

For independent distinguishable moments with angular momentum $J$, define $y=g{\mu }_{B}B/(k_{B}T)$. The one-site partition function is $Z=\sinh [(J+1/2)y]/\sinh (y/2)$. Its logarithmic derivative gives $⟨{\mu }_z⟩=g{\mu }_B[(J+1/2)\coth ((J+1/2)y)-\frac{1}{2}\coth (y/2)]$. At $J=1/2$ this reduces to $(g{\mu }_B/2)\tanh (y/2)$.

At fixed positive temperature and weak field, independent spin-$J$ moments have total moment $M=N(g{\mu }_B{)}^{2}J(J+1)B/(3k_{B}T)$. This is the Curie law; the corresponding density divides by the sample volume.

For the ratio of resistor-current amplitude to total-current amplitude to exceed $1/\sqrt{2}$, $|\omega C-1/(\omega L)|<1/R$. The exact positive endpoint difference is $\Delta \omega =1/(RC)$ and the quality factor is $Q=R\sqrt{C/L}$. Greater shunt resistance means weaker dissipation, unlike increasing a series resistance.

When $R_s$ is in series with a parallel RLC circuit, the ratio of parallel-network voltage to voltage across $R_s$ is its impedance divided by $R_s$. Its peak is $R/R_s$, and its phase passes from $+\pi /2$ through zero to $-\pi /2$ as angular frequency rises through $1/\sqrt{LC}$. This is not the transfer ratio relative to the total input voltage, which includes the additional series voltage drop and has peak $R/(R+R_s)$. Identifying voltage endpoints is necessary before using the voltage divider formula.

With no external current, Kirchhoff's current law is $C\dot{V}+V/R+I_L=0$, and $L{\dot{I}}_L=V$. Differentiating gives the title equation. For $4R^2>L/C$, its real solutions are $A{e}^{-t/(2RC)}\cos ({\omega }_{d}t+\varphi )$ with ${\omega }_d^2=1/(LC)-1/(4R^2{C}^2)$. The original current balance fixes the compatible initial inductor current. A single complex exponential represents a physical voltage through its real part; general complex solutions require both conjugate characteristic roots.

For uniform density between concentric radii $R_1,R_2$, subtract the missing solid core's central moment of inertia and then shift the axis by $R_2$. Here $M=4\pi \rho (R_2^3-R_1^3)/3$. The thin-shell limit gives $I=5M{R}_2^2/3$, while a full solid sphere gives $7M{R}_2^2/5$.

A Lychrel number is a positive integer whose repeated reverse-and-add process never reaches a palindromic number in the chosen radix. Each step performs digit reversal and adds the reversed integer to the current value. For example, $56+65=121$ reaches a decimal palindrome in one step, so $56$ is not a base-ten Lychrel number.

Failure to reach a palindromic number within a fixed iteration limit establishes only a candidate. It does not prove that every later step also fails. A computation must therefore distinguish a bounded search result from the infinite claim in the definition.

A palindromic number in decimal representation, with no leading zeros.