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The end-to-end distance of a polymer chain with $N$ fixed-length links of length $a$ cannot exceed its polymer contour length $Na$. A Gaussian chain spring has no such bound, so it eventually becomes an unphysical extrapolation. For a three-dimensional freely jointed chain under force $f$, the mean extension along the force is $Na[\coth x-1/x]$, where $x=fa/(k_{B}T)$. Its small-$x$ limit is $Nf{a}^2/(3k_{B}T)$, and its large-$x$ limit approaches $Na$ from below.

Use the symmetric transfer matrix $T_{s{s}^{\prime }}=\exp [Ks{s}^{\prime }+h(s+s^{\prime })/2]$ with dimensionless field $h$. For blocking length two set $W=T^2$, then write $W_{s{s}^{\prime }}=A\exp [K^{\prime }s{s}^{\prime }+h^{\prime }(s+s^{\prime })/2]$. Matching its three independent entries proves the title relations and gives $A=(W_{++}{W}_{--}{W}_{+-}^2{)}^{1/4}$. Explicitly $W_{\pm \pm }=e^{2K\pm 2h}+e^{-2K}$ and $W_{+-}=2\cosh h$. At zero field the exact chain recursion is recovered; at $K=0$ the independent-spin field stays unchanged. Splitting the field equally between neighboring bonds is essential for this normalization.

A length $\ell$ with impedance ratio $\alpha$ between identical matched lossless lines has multiple internal reflections. Summing their amplitudes gives the displayed power transmission when the propagation speed is the same in all sections. It is unity at half-wave resonances even though either individual interface reflects. Matched remote terminations remove external echoes but not this frequency dependence.

For a symmetric one-dimensional finite square well, nondegenerate bound states alternate even and odd parity, starting with an even ground state. The $n$th state counted from zero has $n$ interior nodes. Outside the well, it decays with inverse length $\sqrt{2m(U-E_n)}/\hslash$. Matching both the wavefunction and its first derivative at a finite step distinguishes it from the infinite-well approximation.

The first fundamental form of a smooth surface $S\subseteq {\mathbb{R}}^3$ is the restriction of the ambient inner product to each tangent space $T_{p}S$. For a regular parametric surface $X(u,v)$, it is

Here $X_u^2$ means $X_u\cdot X_u$, not a componentwise square. The coefficients form the Gram matrix of the two coordinate tangent vectors. Regularity makes this a positive-definite matrix, so $E>0$ and $EG-F^2>0$.

To calculate the length of a curve $\gamma (t)=X(u(t),v(t))$, apply the chain rule and integrate its speed:The same inner product gives the angle $\alpha$ between nonzero tangent vectors $V,W$ through $\cos \alpha =I(V,W)/\sqrt{I(V,V)I(W,W)}$, and gives the area element $dA=\sqrt{EG-F^2}dudv$ as a positive density. One compact expression therefore supplies rulers, protractors, and surface area measurements.

For the lateral surface of a circular cylinder, take the circumferential arc length $u$ rather than the angle:Direct inner products give $E=G=1,F=0$, hence $I=d{u}^2+d{v}^2$. This is exactly the Euclidean metric on the unrolled plane. For instance, a path with $u=3t,v=4t$, $0\le t\le 1$, has length $5$ on either surface. Every corresponding path retains its length, not merely this example, so sufficiently small neighborhoods retain their intrinsic distances. The local isometry preserves angles and surface areas too. Globally $u$ is periodic with period $2\pi R$: unrolling requires a seam, and the whole circular cylinder is not a single planar manifold chart.

For a symmetric attractive well of width $a$ and depth $V_0$, the first odd bound state obeys $-k\cot (ka/2)=q>0$, where $q$ is the inverse tail length. Its lowest branch has $\pi <ka<2\pi$. Since $k^2<2m{V}_0/{\hslash }^2$, existence requires $V_0>V_{\mathrm{odd}}$; it is also sufficient by continuity of the matching equation on that branch. At equality the exterior solution has zero decay rate and is not square-integrable. Thus a well at or below this threshold has only its even ground state.

For a permutation input, the first row of its RSK insertion tableau is the minimum-tail table for increasing subsequences: its $j$th entry is the least attainable final value of a strictly increasing subsequence of length $j$ in the processed prefix. Inserting $z$ replaces the first entry larger than $z$, or appends if none exists. In replacement position $r$, the previous entry in position $r-1$ supplies a subsequence that can be extended by $z$. Shorter minimum tails were already smaller, and a longer subsequence ending at $z$ would require an old tail in position at least $r$ smaller than $z$, contrary to the replacement rule. This proves the invariant inductively. Consequently the first-row length is the longest increasing subsequence length. The row entries need not be a subsequence together: input $2,4,3,1$ gives first row $1,3$, although one occurs after three. They are minima for different lengths, obtained from potentially different subsequences.

The static spherically symmetric electrically charged black hole in five spacetime dimensions has $f(r)=1-2M/r^2+Q^2/r^4$. When $M>Q$ it has outer and inner horizon radii $r_{\pm }^2=M\pm \sqrt{M^2-Q^2}$; for $M=Q$ it is an extremal black hole, while $M<Q$ exposes a naked singularity.

For a tension-dominated cylindrical lipid vesicle of fixed length $L$ and volume, a real axisymmetric sine amplitude $u_q$ has stiffness $K_q=\pi \sigma L[(q{R}_0{)}^2-1]/R_0$. Fixed volume requires a mean-radius shift $-u_q^2/(4R_0)$. Stable modes have $⟨u_q^2⟩=k_{B}T/K_q$ by the equipartition theorem. Modes with $q{R}_0<1$ have the Rayleigh–Plateau instability, rather than a negative equilibrium variance. Bending modifies very short wavelengths and near-marginal modes.

For fixed volume $V$ and symmetric spreading, conservation of mass and the cusp flow between touching cylinders equation give width proportional to $t^{-1/8}$ and support length proportional to $t^{3/8}$. The source-type similarity solution has the displayed profile and $z_N=(8Va/\pi {)}^{1/4}(8\gamma t/(21\mu a){)}^{3/8}$. It is a formal outer lubrication solution; its microscopic nose requires a local description when the axial slope becomes large.

For one directed spin-resolved channel, the density of longitudinal states per unit length and wave number is $1/(2\pi )$. Since $v={\hslash }^{-1}dE/dk$, velocity times density per energy is $1/h$. A reservoir occupation imbalance therefore contributes particle flux $\int \delta f(E)dE/h$; in zero-temperature linear response it is $\delta \mu /h$. Multiplying by charge and transmission gives the Landauer current per mode.

Assume a triangular array of straight, singly quantized Abrikosov vortices with nearest-neighbor distance $a$. Here $\overline{B}>0$ is the magnitude of the spatially averaged internal magnetic field, and ${\Phi }_0=h/(2e)$ is the superconducting flux quantum; $e>0$ is the elementary charge. Two lattice vectors of length $a$ meeting at $60^{\circ }$ span area $A_c=\sqrt{3}{a}^2/2$. One flux quantum per cell givesThus multiplying $\overline{B}$ by five decreases $a$ by $\sqrt{5}$; it does not multiply the flux of each vortex by five. Using the exact SI Planck constant and elementary charge, ${\Phi }_0\simeq 2.06783\times 10^{-15}\mathrm{Wb}$.^[ref]

For chosen internal fields $0.20\mathrm{T}$ and $1.0\mathrm{T}$, the spacings are approximately $109\mathrm{nm}$ and $48.9\mathrm{nm}$. These are geometric model calculations, not measurements from a particular sample. The local magnetic field varies within a cell; replacing $\overline{B}$ by the field at a core would not give this counting relation.

For a planar elastic filament under positive constant extension force $F$, the quadratic force-controlled energy isThe higher-order Euler-Lagrange equation is $B{\zeta }^{\prime \prime \prime \prime }-F{\zeta }^{\prime \prime }=B{\kappa }_0^{\prime \prime }$. On a periodic interval or in a bulk Fourier transform treatment, every nonzero mode obeys the displayed transfer relation. The length deficit is $\frac{1}{2}\int ({\zeta }^{\prime }{)}^{2}dx$, so its ensemble value is determined by the power spectrum of the quenched intrinsic curvature. Boundary conditions must be specified for a finite filament.

Independent link orientations make the canonical partition function $Z_1^N$, with the one-link integral defining the Langevin function. Differentiating its logarithm with respect to force gives $L=Nb\mathcal{L}(Fb/(k_{B}T))$. At high extension, $Nb-L\sim N{k}_{B}T/F$, so the force diverges inversely with remaining length deficit. This requires rigid links and excludes axial bond stretching.

Two spin states per wavevector and parabolic dispersion give $n=2v_d{k}_F^d/(2\pi {)}^d$, where $v_d$ is the unit-ball volume. Thus $E_F={\hslash }^2[n{a}_d{]}^{2/d}/(2m)$ with $a_1=\pi /2$, $a_2=2\pi$, $a_3=3{\pi }^2$. Differentiating the cumulative count yields $g(E_F)=dn/(2E_F)$, normalized per length, area or volume as appropriate.

A freely rotating chain has fixed bond length and fixed angle between successive forward bond vectors, with independently uniform torsion angles. It neglects interactions between distant chain segments. The fixed angle introduces short-range orientational correlations even though the torsional rotations are unrestricted. Its characteristic ratio of a freely rotating chain converts those correlations into an end-to-end size; its equivalent long-chain link is the Kuhn length of a freely rotating chain.

For a particle on a circle with arc length $x\sim x+2\pi R$ and periodic boundary conditions, the Hamiltonian operator $P^2/(2m)$ has orthonormal basis ${\varphi }_n=e^{inx/R}/\sqrt{2\pi R}$, $n\in \mathbb{Z}$, and energy eigenvalues $E_n={\hslash }^2{n}^2/(2m{R}^2)$. To see this, periodicity forces the wave number to be $n/R$, and applying the momentum operator $-i\hslash {\partial }_x$ gives $\hslash n/R$. The integral of $|{\varphi }_n{|}^2$ over one turn is one. The zero energy eigenvalue is simple; each positive energy eigenvalue has quantum degeneracy two, from $n$ and $-n$.

The Fried parameter is the atmospheric coherence length for optical phase. In the standard inertial-range model, the phase structure function is $6.88(\rho /r_0{)}^{5/3}$. For unchanged atmospheric turbulence, $r_0\propto {\lambda }^{6/5}$. The NACO manual, section 3.1 describes this scaling and its role in adaptive correction.

A galactic disk is a flattened stellar or gaseous component of a galaxy, commonly supported mainly by ordered rotation. Its radial mass profile determines its galaxy rotation curve; thickness, truncation and external forces also matter. A galactic disk need not be an exponential galactic disk, and its neutral-gas absorption radius need not equal a stellar scale length.

A uniform-radius magnetic arc of length $L$ and radius $R$, two linear tapers of length $s$ to radius $r$, and an unfringed vacuum gap of length $h$ have the displayed gap field. Here ${\mu }_r=\mu /{\mu }_0$ and $B_g={\mu }_0{H}_g$. Tapering concentrates flux, but at fixed winding current the gap-dominated limit is $H_g\simeq NI/h$ regardless of pole area. Saturation and leakage prevent unrestricted field enhancement.