In cylindrical orthonormal bases the deformation gradient is $F=\left(\begin{array}{ccc}\rho /r& 0& 0\\ 0& r/\rho & \alpha r\\ 0& 0& 1\end{array}\right)$ and has unit determinant. One principal stretch is $\rho /r$; the squares of the other two are the roots of $z^2-[(r/\rho {)}^2+{\alpha }^2{r}^2+1]z+(r/\rho {)}^2=0$. The reference-volume energy per height is $E=2\pi \int \rho Wd\rho$. Virtual work gives end torque $M={\partial }_{\alpha }E$ and inner pressure $p=(2\pi a{)}^{-1}{\partial }_{a}E$ at fixed reference radii and height.

For axial stretch $\lambda$ and twist per reference length $\alpha$, the deformation in local cylindrical orthonormal bases hasThe principal stretches follow from the eigenvalues of $F^{T}F$. For a cylinder of reference radius $A$, stored energy per reference length is $\mathcal{E}=2\pi {\int }_0^{A}W(F)RdR$. Reversible power $M\dot{\alpha }+N\dot{\lambda }=\dot{\mathcal{E}}$ gives torque $M={\mathcal{E}}_{,\alpha }$ and axial force $N={\mathcal{E}}_{,\lambda }$. The relevant derivative holds the reference twist fixed, not the twist per current length.

For an equilibrium with $P_{Ii}=W_{,F_{iI}}+q{\varphi }_{,F_{iI}}$ and dead loads, the constrained second variation on ${\varphi }_{,F}:\delta F=0$ isIndeed, a constraint-preserving perturbation path has ${\varphi }_{,F}:{\delta }^{2}F=-{\varphi }_{,FF}[\delta F,\delta F]$, and stationarity of the total potential supplies the multiplier term. Linearized kinetic energy plus $\mathcal{Q}/2$ is conserved. Positivity gives the usual energy-norm linear stability; control in a specified displacement norm additionally requires the corresponding coercivity and treatment of rigid-motion modes.

For reference radii $a_0,b_0$, current inner radius $a$ and uniform axial stretch $z>0$, incompressibility integrates to $r^2=a^2+(R^2-a_0^2)/z$. The principal stretches are $r_{,R}=1/(z\lambda )$, $\lambda =r/R$ and $z$. Write $\hat{W}(\lambda ,z)=W((z\lambda {)}^{-1},\lambda ,z)$ and $\mathcal{E}=2\pi {\int }_{a_0}^{b_0}\hat{W}RdR$ per reference length. Virtual work gives $p={\mathcal{E}}_{,a}/(2\pi az)$ and the total axial wall load $N_{\mathrm{wall}}={\mathcal{E}}_{,z}$, both derivatives holding the other displayed parameter fixed. For pressurized closed ends the independently applied axial load is $N_{\mathrm{ext}}=N_{\mathrm{wall}}-\pi a^{2}p$. The distinction follows from the axial pressure power $\pi a^{2}p\dot{z}$.

On two copies of the real line with their usual topologies, let $U=(1,2)$ and $V=(2,4)$. The coordinate projection maps giveThe first two open sets are unbounded strips; their set intersection is an open rectangle. The point $(1.4,3)$ passes both coordinate tests, whereas $(2,3)$ fails the first because the endpoint $2$ is excluded.

Every point of an open set in the Euclidean plane has such an open rectangle around it inside that open set. Conversely, every open rectangle contains a small Euclidean ball around each of its points. Thus the product topology here is the usual metric topology of the Euclidean plane. With finitely many factors all coordinates may be restricted; on an infinite Cartesian product a basic cylinder set must leave all but finitely many factors free.

A colour class of a finite colouring $c:X\to \{1,\dots ,r\}$ is a fibre $c^{-1}(i)$. Each colour class is a monochromatic set, and these fibres give a finite set partition of $X$ after empty fibres are omitted.

The colour profile of a length-$N$ block $b_1,\dots ,b_N$ under a finite colouring $c$ is the ordered tuple $(c(b_1),\dots ,c(b_N))$. There are at most $r^N$ profiles when $c$ uses at most $r$ colours. Equally profiled blocks reproduce every colour configuration at the same relative positions; this converts a colouring of points into a finite colouring of blocks in the colour-focusing proof of Van der Waerden theorem.

A financial market model that omits trading frictions such as transaction costs and a difference between buying and selling prices. The exact idealization must still specify permitted holdings, short selling, and borrowing and lending terms; the worked binomial market allows divisible holdings and unrestricted borrowing at the bank growth factor.

For a fixed horizon, a risk-free asset has a deterministic terminal payoff. If the price of a unit terminal payoff is $B>0$, its gross return is $R=1/B$. In a finite state market, the sum of the Arrow state prices is $B$. Risk-free does not mean that a longer-maturity bond has a deterministic resale price before its maturity.

A credit derivative transfers exposure to deterioration of credit quality or a specified credit event of a reference borrower without necessarily transferring the underlying loan or bond. Credit default swaps transfer a defined loss exposure; total return swaps transfer a wider asset return. Funded structures such as credit-linked notes combine the exposure with an investor's initial principal. Documentation specifies the reference entity, obligations, credit events and settlement; a hedge can leave counterparty credit risk and mismatch with the original exposure.

The fixed price parameter at which an option permits an underlying trade. A European call option pays $(S_T-K{)}^{+}$ at maturity; a European put option pays $(K-S_T{)}^{+}$. Specifying $K$ therefore determines how terminal prices translate into the financial payoff.

A filter on a set equipped with a topology converges to $x$ if it contains every neighbourhood of $x$. A Hausdorff space gives every convergent filter at most one limit: disjoint neighbourhoods of distinct purported limits would have empty intersection in the filter.

For a function $f:X\to Y$, the pushforward of a filter on a set $X$ consists of the subsets $B\subseteq Y$ with $f^{-1}(B)\in \mathcal{F}$. Preimages preserve inclusions and finite intersections, so this is a proper filter. Equivalently it is generated by $f(A)$ for $A\in \mathcal{F}$. Because preimages preserve complements too, the pushforward of an ultrafilter is an ultrafilter.

For a source in pure plume balance, $Q(z)=C{F}_s^{1/3}(z+z_v{)}^{5/3}$ and $Q_s=Q(0)$. A single ideal exhaust passes the source volume flux. Above the exhaust height, it removes unprocessed lower fluid and the filling-box first front descends at rate $-\pi Q/A_c$. Below it, processed fluid leaves and only entrained volume grows the processed region. The floor is reached in finite time if the opening is at the floor; otherwise the final approach is exponential. A finite-height aperture supporting bidirectional exchange requires a different model.

An acknowledged file-system transaction remains recoverable after a crash under the stated storage assumptions. In the full-data timeline, durable journal payload and a durable journal commit record suffice even before the journal checkpoint. A crash before durable commit can lose the append while still preserving file-system crash consistency. A successful ordinary write may only have reached a buffer cache; an application requiring persistence must use its documented interface and check errors. On Linux kernel-based systems, fsync on a file descriptor persists the associated file contents and file metadata, but persisting a newly created name also requires fsync on the containing file-system directory.^[ref]

In $L={\mathbb{Q}}_p(\theta )$ with ${\theta }^2=p$, the Eisenstein polynomial $X^2-p$ gives $v_L(\theta )=1$ and $v_L(p)=2$. The element $x=\theta /p$ has field trace zero, but valuation $-1$, so is not in the valuation ring. Cancellation in a field trace contrasts with the multiplicative test in unit norm detects units in a local extension.

For an integral element of a finite extension of non-Archimedean local fields, reduce the multiplication endomorphism on ${\mathcal{O}}_L$ modulo a base uniformizer. Its filtration by the powers of the maximal ideal has $e$ successive quotients, each the residue field ${\kappa }_L$. Multiplication by $a$ induces multiplication by $\bar{a}$ on each quotient. Additivity of the matrix trace through the filtration proves the formula. If the residue characteristic divides the ramification index, the trace of every integral element lies in the base maximal ideal; a Galois extension is not required.

For a finite extension of local fields with normalized discrete valuations, the field norm satisfies the displayed equality. The extended absolute values on a field of all conjugates agree, so $|N(x){|}_K=|x{|}_L^{[L:K]}$. Using $[L:K]=e(L/K)f(L/K)$ and $v_L{|}_K=e(L/K)v_K$ gives the equality. Thus the field norm is a unit precisely when the original element is a unit.

Let $A$ be an integrally closed domain with fraction field $K$, and let $L/K$ be finite. If $b\in L$ is an integral element over $A$, its monic minimal polynomial lies in $A[T]$: its roots are integral, so its coefficients are integral elements of $K$, hence belong to $A$. If that polynomial has degree $r$ and constant coefficient $a_0$, then $N_{L/K}(b)=((-1{)}^r{a}_0{)}^{[L:K(b)]}\in A$. This remains true for inseparable extensions. An infinite algebraic extension has no usual determinant field norm.

A field isomorphism is a bijective field homomorphism; its inverse is also a field homomorphism. For fields extending a specified field $K$, an isomorphism may be required to fix $K$ pointwise, a stronger requirement than an abstract field isomorphism.