Schizosaccharomyces pombe is a yeast species used as a model organism. It supplies the candidate genes in the yeast cross-species complementation experiment; the recipient remains Saccharomyces cerevisiae.

A Villain model replaces the cosine Boltzmann factor of the XY model by a periodic Gaussian weight. The Poisson summation formula makes its bond weight equal to ${\sum }_{p\in \mathbb{Z}}\exp [-\beta (\psi -2\pi p{)}^2/2]$. The periodic Gaussian retains integer winding defects and permits an explicit separation into spin waves and vortices in two dimensions. Its replacement of the cosine model is an approximation; subsequent transformations within the Villain model can be exact.

For a symmetric covariant two-tensor, the geodesic transform integrates its evaluation on the unit velocity along boundary-to-boundary or closed geodesics, as appropriate to the manifold. A potential symmetric 2-tensor gives a total derivative along each geodesic and therefore integrates to zero on closed geodesics; boundary data require the corresponding boundary potential to vanish.

The magnetic transform integrates functions or velocity-evaluated one-forms along magnetic trajectories, rather than ordinary geodesics. For closed trajectories, every exact one-form integrates to zero. Hyperbolic dynamics can make these the entire degree-one kernel.

The transform integrates a base function and a velocity-evaluated one-form along the closed trajectories of a surface thermostat. Every exact differential form of degree one integrates to zero, because $F(f\circ \pi )=df(v)$. The relevant kernel theorem concerns these functions of velocity degree zero and one, rather than arbitrary functions on the unit tangent bundle.

Reinstalling the complete new images of a committed file-system transaction after a crash. An uncommitted journal payload is ignored. Rewriting the same image has idempotence: repeating it gives the same contents, unlike performing "append four more bytes" again. If recovery itself crashes, retaining the journal until the journal checkpoint is durable allows the same recovery to restart.

The second diagram exhausts the small full-data model. A triple lists $I,B,D$ in that order; a bit is 1 exactly when the corresponding new record has persisted. Before commit, any of the $2^3=8$ subsets of journal payload records can be durable, while every final-location triple is 000. After commit, the journal triple is 111, while any of the eight final-location triples can be durable. Retirement adds one state: the live journal is gone and the final-location triple is 111. Thus there are $8+8+1=17$ distinct abstract crash states. Waiting for a storage flush without another record reaching persistence does not create an additional state. Retirement is modeled as an atomic change of live-log status: old payload bytes may remain physically present but are no longer part of the live journal.

The short correctness argument has three cases. Before commit, final locations are unchanged. With commit, the complete journal payload restores 111 regardless of the current final-location subset. After retirement, 111 is already durable. Each of the six partial final-location triples is repairable because its journal is retained. The enumeration is exhaustive for these records and phase rules; it excludes torn records, dishonest flush completion, media corruption, journal reuse across several file-system transactions and failures of the recovery code. It is a model check, not a measured crash test of ext4.

Installing committed journal payload at final locations and making those copies durable. Safe retirement follows this completion, not merely submission of the copies. Discarding the journal while some copies remain only in a volatile memory cache can leave an unrecoverable mixture after a crash. Commit establishes a recoverable decision; a journal checkpoint permits reclamation of its journal space.

A marker identifying a complete committed file-system transaction. A marker alone is insufficient if its journal payload is missing or belongs to another group. The simple model prevents this through ordering and fresh record identities. Real journal formats also validate record identities and integrity: the ext4 documentation says replay discards a group lacking its commit or having mismatched integrity checks.^[ref]

The protected new contents and destination information for a file-system transaction. In the diagrams, the three complete journal payload records correspond to $I,B,D$. Records can reach storage in any order within this phase. The journal commit record cannot become durable until all three are durable.

A strongly stretched inextensible worm-like chain has two small transverse tangent components. Its quadratic energy is $\frac{1}{2}\int [A|{\partial }_s{\mathbf{t}}_{\perp }{|}^2+f|{\mathbf{t}}_{\perp }{|}^2]ds-fL$. For a long bulk chain, the equipartition theorem assigns each Fourier mode covariance $k_{B}T/(A{q}^2+f)$. Integrating over $q$ and summing both components gives $⟨|{\mathbf{t}}_{\perp }{|}^2⟩=k_{B}T/\sqrt{Af}$, hence the extension deficit is half that value. With persistence length $L_p=A/(k_{B}T)$, the deficit is $\sqrt{k_{B}T/(4L_{p}f)}$. It requires small slopes and negligible stretching of the backbone.

The Kuhn length is the link length of an equivalent long ideal chain with the same contour length and mean squared end-to-end distance. A three-dimensional worm-like chain satisfies $⟨R^2⟩\sim 2l_{p}L$ for $L\gg l_p$, hence $b_K=2l_p$.

For $\delta X=-\overline{ϵ}\psi$ and $\delta \psi ={\rho }^{\alpha }{\partial }_{\alpha }Xϵ$, with constant Grassmann-odd Majorana parameters, the commutator on $X$ is the displayed translation. Set $M={ϵ}_1{\overline{ϵ}}_2-{ϵ}_2{\overline{ϵ}}_1$. In two dimensions $\{{\rho }^{\alpha },M\}=2{\overline{ϵ}}_1{\rho }^{\alpha }{ϵ}_2$ times the identity, so the fermion commutator is the translation minus $M{\rho }^{\alpha }{\partial }_{\alpha }\psi$. The remainder vanishes on the fermion field equation. This distinguishes on-shell closure from the off-shell total-derivative invariance of the action.

In flat noncompact spacetime a fixed circular loop's cylindrical worldsheet is not a stationary point of the Nambu–Goto action. A compact coordinate changes the possibilities. The displayed embedding in ${\mathbb{R}}^{2,1}\times S_R^1$ has unit winding, satisfies the wave equation, and obeys $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. Its projection is a circle with static shape, while its motion in the compact direction supplies the second chiral null tangent. The induced worldsheet metric is $2R^2\mathrm{diag}(-1,1)$, so the full worldsheet is nondegenerate.

For a fixed force $\mathbf{F}$ and endpoint displacement $\Delta \mathbf{r}$, the line integral reduces to a dot product:Here $\theta$ is the angle between the force and the displacement. For straight motion, $d$ is also the distance traveled. A $3\mathrm{N}$ force acting along a $4\mathrm{m}$ displacement does $12\mathrm{J}$ of work. Turning the same force perpendicular to the displacement gives $\cos (\pi /2)=0$ and zero work; reversing it gives $\cos \pi =-1$ and $-12\mathrm{J}$. At $\theta =\pi /3$, the work is $6\mathrm{J}$. These signs describe energy transferred to or from the moving body. ^[ref]

For smooth positive $K$ on an interval with regular Dirichlet boundary conditions, there is no turning-point phase shift. Combining conjugate WKB approximation branches to vanish at the first endpoint makes the total phase at the second an integer multiple of $\pi$. The first correction is the integrated WKB phase correction for a slowly varying oscillator. For $K=x+1$ on $[0,1]$ the resulting eigenvalues are $4{\pi }^2{n}^2/9-3/16+O(n^{-2})$.

For smooth $K=|k|>0$ and fixed amplitude flux, use $\sigma =K+{ϵ}^2{\sigma }_1+\cdots$ and $A_0\propto K^{-1/2}$. The exact amplitude-phase equation gives ${\sigma }_1=A_0^{\prime \prime }/(2K{A}_0)$, which is the displayed formula. The first phase correction is order ${ϵ}^2$ in the local wavenumber, hence order $ϵ$ in its integrated exponential phase.

For $\nabla \cdot (k_0(X{)}^{-2}\nabla p)+p=0$ in a slowly varying rigid duct, a real local transverse normal mode $F$ with squared norm $N(X)$ has longitudinal wavenumber $k(X)$. The WKB approximation is $p=A(X)F(y,X)\exp [-i{ϵ}^{-1}{\int }^{X}k(\xi )d\xi ]$. The moving-wall Neumann boundary condition and next-order solvability give $2kN{k}_0^{-2}{A}_X+(kN{k}_0^{-2}{)}_{X}A=0$, hence the displayed amplitude. For $F=\cos (n\pi y/R)$, $N=R$ for $n\ge 1$ and $N=2R$ for $n=0$. The factor $k_0$ is essential when the coefficient in the divergence-form equation varies. This approximation requires a mode separated from cutoff and from other modes.

For $k\ge 1$, the rational middle intersection pairing induces an isomorphism ${\Omega }_{4k}^{\mathrm{Witt}}\cong W(\mathbb{Q})$. The restriction $k>0$ matters: the degree-zero group is $\mathbb{Z}$. In positive dimension congruent to zero modulo four, a space is Witt null-bordant precisely when its middle intersection form is metabolic.

Every odd-dimensional closed oriented Witt space bounds its closed cone: the new vertex has odd-dimensional link, so introduces no Witt obstruction. For a closed oriented two-dimensional pseudomanifold, normalize to a surface, fill each surface component by a handlebody, and attach a collar of the original space using the finite normalization map. Along the singular collar arcs the links are unions of circles; at their interior starting points the links are wedges of two-spheres. These links have zero middle intersection homology, so the construction is a three-dimensional Witt null-bordism.

Allocate each remaining item to a bidder with greatest current marginal gain. For nonnegative monotone submodular set functions, this uses $O(m{n}^2)$ value queries and achieves the displayed guarantee. To prove it by induction, let the first allocation give item $j$ to bidder $r$, and set $w=v_r(\{j\})$. Contract that item into bidder $r$'s value by replacing it with $v_r^{\prime }(S)=v_r(S\cup \{j\})-w$ and leave the other bidders' values unchanged. The remaining greedy run is identical, and $A(v)=w+A(v^{\prime })$. Move $j$ from its owner in an optimal allocation to $r$. Submodularity bounds the owner's loss by its initial singleton marginal, which is no larger than the selected greedy marginal and hence no larger than $w$. The receiver's value cannot decrease, so $\mathrm{Opt}(v^{\prime })\ge \mathrm{Opt}(v)-2w$. Induction proves the guarantee, with the zero-item case immediate even for nonzero empty-bundle values.