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After testing nonnegative edge coordinates and any explicit box or budget inequalities, treat those coordinates as lengths. A shortest path of length below one supplies a violated path inequality; a minimum length at least one certifies all path inequalities. If no source-to-target path exists, all those inequalities are vacuous. Thus a polynomial time shortest path problem calculation separates an exponentially constrained linear program.

A heat equation with a nonlinear reaction term depending on the value of its solution.

If $N$ is an elementary abelian normal subgroup and $|G/N|$ is prime to its characteristic, the extension splits and its complements are conjugate by $N$. In additive notation an extension cocycle satisfies $f(h,k)+f(hk,t)=hf(k,t)+f(h,kt)$. Averaging over $t$ gives $f(h,k)=b(h)+hb(k)-b(hk)$, where $b(h)=|G/N{|}^{-1}{\sum }_{t}f(h,t)$. Changing the section by $b$ removes the cocycle. For two complements the difference cocycle has $d(hk)=d(h)+hd(k)$; averaging gives $d(h)=b-hb$, which is conjugation by an element of $N$. These arguments use invertibility of the group order on $N$ and underlie Hall conjugacy in soluble groups.

If $N=⟨A\mid R⟩$ and $H=⟨B\mid S⟩$ have finite group presentations and $\varphi :H\to \mathrm{Aut}(N)$ is an action, choose a word $w_{b,a}(A)$ representing $\varphi (b)(a)$. Then $N{⋊}_{\varphi }H$ has the finite presentationThe action automorphisms also let these relations rewrite conjugation by inverse generators. Every word becomes an $A$-word followed by a $B$-word. The natural map to the semidirect product is injective because its triviality forces the $B$-word to be trivial using $S$ and then the $A$-word to be trivial using $R$.

Let an abelian group $V$ carry an action $\rho$ of a group $H$ by automorphisms, and use multiplication $(h,v)(k,w)=(hk,v+\rho (h)w)$. Comparing $(e,v)(h,w)$ and $(h,w)(e,v)$ shows that a translation is central precisely when its vector is fixed by the entire action. If $H$ is itself abelian, the whole semidirect product is abelian exactly when the action is trivial. For $H=(\mathbb{R},+)$, $V={\mathbb{R}}^2$, and determinant-one linear actions fixing the first coordinate vector, $\rho (x)=\left(\begin{array}{cc}1& r(x)\\ 0& 1\end{array}\right)$, where $r$ is an additive function. Without a regularity assumption, it need not be $r(x)=cx$.

For a finitely generated module $V$ on the right over a semilocal ring, the displayed condition makes the induced module endomorphism on $V/VJ$ injective. This quotient is an Artinian module, so Artinian modules are co-Hopfian makes that map surjective. The remaining quotient module $V/\alpha (V)$ equals its product with $J$, and Nakayama lemma makes it zero.

A finite family of contracting similarities $S_1,\dots ,S_k$ satisfies the open set condition if there is a nonempty bounded open set $O$ with $S_i(O)\subseteq O$ and $S_i(O)\cap S_j(O)=\varnothing$ for $i\ne j$. This permits the closures to meet. For the invariant compact set, this separation makes its Hausdorff dimension equal to the similarity exponent $s$ defined by ${\sum }_i{c}_i^s=1$, where $c_i$ are the contraction ratios. The level-$n$ cylinder sets supply the upper dimension bound; the separated cylinder masses with weights $c_i^s$ supply the matching mass-distribution lower bound. For the 24 maps of ratio $1/5$ retaining all but the central square, $O=(0,1{)}^2$ works and $s=\log 24/\log 5$.

A contracting Euclidean similarity multiplies every Euclidean distance by a constant $c<1$; equivalently it has the form $cRx+b$ for an orthogonal matrix $R$ and translation vector $b$. Finite families of such maps generate compact sets with self-similarity.

For finitely many contracting Euclidean similarities, the map $\mathcal{T}$ on nonempty compact subsets of ${\mathbb{R}}^n$ is a contraction in the Hausdorff metric, with contraction constant ${\max }_i{c}_i<1$. The contraction mapping theorem gives a unique nonempty compact invariant set. If the open set condition holds, its Hausdorff dimension is the unique $s$ satisfying ${\sum }_i{c}_i^s=1$.

For an energy-conserving strong spherical shock wave expanding into mass density $C{r}^{-\beta }$, $0\le \beta <3$, dimensional analysis gives the displayed radius law. Neglect ambient pressure, radiation losses and gravity in this blast-dominated approximation. Write $u=\dot{R}U(\xi )$, $\rho =C{R}^{-\beta }D(\xi )$ and $p=C{R}^{-\beta }{\dot{R}}^{2}P(\xi )$, with $\xi =r/R$. The radial continuity equation, momentum equation and adiabatic pressure equation reduce toThe Strong-shock Rankine-Hugoniot conditions give $U(1)=P(1)=2/(\gamma +1)$ and $D(1)=(\gamma +1)/(\gamma -1)$. The explosion energy fixes the radius coefficient through $E=4\pi C{R}^{3-\beta }{\dot{R}}^2{\int }_0^1[D{U}^2/2+P/(\gamma -1)]{\xi }^{2}d\xi$.

Let a strong shock wave of radius $R$ and speed $V$ sweep a uniform mass density ${\rho }_0$ into a shell of mass $4\pi {\rho }_0{R}^3/3$ and speed $2V/(\gamma +1)$. If the cavity pressure is $\alpha$ times the immediate post-shock pressure, shell momentum balance gives $R\dot{V}=3(\alpha -1)V^2$. Constant total energy gives $R^3{V}^2=\mathrm{constant}$ and hence $\alpha =1/2$. Including shell internal and kinetic energy, and treating the cavity volume as $4\pi R^3/3$ to thin-shell accuracy, yields $R=\xi (E/{\rho }_0{)}^{1/5}{t}^{2/5}$ with ${\xi }^5=75(\gamma -1)(\gamma +1{)}^2/[16\pi (5\gamma -3)]$. This coefficient is approximate; resolving the shell's finite volume changes it.

An inductor is an electrical circuit component designed to provide self-inductance, often a coil such as a solenoid. An ideal linear inductor has constant $L>0$ and zero electrical resistance. Its stored magnetic energy, with zero chosen at zero electric current, isFor self-inductance $0.5\mathrm{H}$ and electric current $3\mathrm{A}$, this gives $U=2.25\mathrm{J}$. Unlike Joule heating, ideal storage is recoverable: the inductor can return energy to an electrical circuit when its electric current decreases. ^[ref].

The short derivation uses the terminal convention in self-inductance. The power absorbed by an ideal inductor isStarting from zero electric current, the integral of this power givesPositive $P_L$ increases stored magnetic energy; negative $P_L$ returns energy. At constant electric current $3\mathrm{A}$ in the example, $P_L=0$ and induced electromotive force is zero, but $U=2.25\mathrm{J}$ still resides in the magnetic field. The battery in the steady RL circuit supplies Joule heating in the electrical resistance, rather than further increasing the inductor's magnetic energy.

During the closed-loop release in RL circuit, stored magnetic energy plus accumulated Joule heating stays $2.25\mathrm{J}$. A real coil also has electrical resistance, so its Joule heating must be included separately from ideal inductor storage. The constant-$L$ formulas assume a fixed linear magnetic response.

Formal verification uses machine-checked formal proofs to establish functional correctness: the C implementation of verified seL4 configurations conforms to its formal specification. Binary correctness connects compiled machine code to that C behaviour where supported. Separate security formal proofs cover integrity, resource-access availability and confidentiality under their configuration requirements; these properties are not all available in every configuration. For example, the documented confidentiality theorem separates scheduler security domains rather than arbitrary pairs of software threads. ^[ref]

The documented assumptions include correct hardware, selected handwritten machine-interface code, correct boot state, translation lookaside buffer and memory cache management, and noninterfering or trusted DMA. The confidentiality result covers information channels in its model; the assumptions page explicitly excludes timing channels from that coverage. ^[ref]

Return to READ(7, 9, 0, 4). Correct IPC delivery does not establish that the service looked up the right file, returned LIME, or ever replied. A service can produce an incorrect payload while the operating-system kernel exactly follows its formal specification. The formal proofs therefore do not automatically prove all user-space services, nor turn the resource-access availability theorem into a guarantee that this file read completes. The verified foundation supports further algorithm correctness and system reasoning; the service protocol and its recovery obligations still need their own justification.

// Destination: computer-science.bigb

The exterior of an unknot is a solid torus. Inclusion of its boundary kills exactly the primitive Seifert longitude, so every essential simple closed curve bounding a disk in that exterior is isotopic, as an unoriented curve, to that longitude. The bounding disk need not avoid other link components.

A thermocouple uses two different electrical conductors and the Seebeck effect to infer a measuring-junction temperature from a voltage and a known reference temperature. The reference material and the reference temperature are distinct choices: B specifies the comparison Seebeck coefficient, while $T_c$ specifies the lower limit of the integral. The two-material experiment in Seebeck effect gives the geometry, sign convention, and calibration limitations.

The useful measurement is a pair: the raw terminal electric potential difference and the thermocouple reference junction temperature. For the A-positive terminal order and homogeneous-wire assumptions of Seebeck effect, write the open-circuit electromotive force asHere $T_h$ is the thermocouple measuring junction temperature and $T_c$ is the common temperature of the reference connections. The subscript $h$ names the measuring end; it need not be hotter. A voltmeter senses the complete connected thermoelectric path through the wires and its leads, in the electrochemical potential convention already described under Seebeck effect. It does not measure an absolute temperature, the individual Seebeck coefficient of A, or an isolated contact electric potential.

The Seebeck coefficient $S(T)$ quantifies the thermoelectric voltage response per small temperature difference at zero electric current. Use $S=-dV/dT$ along a homogeneous material, with voltage understood through electrochemical potential. Its units are volts per kelvin, commonly $\mu V/K$. The word thermopower is also used, despite this being a voltage response rather than power; some authors use that word for $|S|$, so check their convention. ^[ref].

For the A-minus-B terminal order in the Seebeck effect experiment,Thus a small temperature difference around $T_0$ gives $V_{AB}\approx [S_A(T_0)-S_B(T_0)]\Delta T$. The finite-interval ratio $V_{AB}/\Delta T$ is an average of the coefficient difference, not necessarily its value at either endpoint. Ordinary two-material measurements determine a relative Seebeck coefficient: to recover $S_A$ from the measured difference, add a separately known $S_B$. Neither the choice of a zero for voltage nor the use of a familiar reference material makes $S_B$ identically zero.

A concrete research example is Martin and colleagues' 2021 certification experiment. They measured small temperature differences and voltages on six silicon–germanium specimens at 13 base temperatures between $295$ and $900\mathrm{K}$. At each setting they acquired 40 voltage-and-temperature-difference pairs and used linear regression with a free intercept, so a constant voltage offset did not become part of the slope. Platinum voltage leads required a reference-material correction; their terminal order gave $S_{\mathrm{measured}}=S_{\mathrm{Pt}}-S_{\mathrm{specimen}}$, hence $S_{\mathrm{specimen}}=S_{\mathrm{Pt}}-S_{\mathrm{measured}}$. This is an actual measurement protocol, unlike the toy plot above. ^[ref].

Microscopically, a temperature gradient changes charge carrier transport until an opposing electrical response balances it at zero net electric current. In the usual simple semiconductor picture, electrons and electron holes give opposite Seebeck coefficient signs, and their contributions are weighted by their electrical conductivities. The Seebeck coefficient is a transport property: density of states alone does not determine it, since carrier motion and scattering also matter. Its usefulness extends beyond temperature sensing to diagnosing which carriers transport electric charge. ^[ref].

For downward settling velocity magnitude $W_s>0$ and eddy diffusivity ${\kappa }_T=Kz$, the total upward particle flux is $J=-W_s\varphi -Kz{\varphi }_z$. A stationary profile tending to zero at infinity has $J=0$, giving the displayed power law. The exponent compares settling with turbulent mixing. It describes a region above the bed; entrainment from the bed and viscous near-wall effects determine the reference concentration separately.

Second-order comprehension is a scheme asserting the existence of a second-order relation variable whose members are exactly the tuples satisfying a permitted formula:Here $R$ is not free in $\phi$, and the displayed individual and relation parameters are universally closed in each instance. Under full semantics for second-order logic, the selected external subset of $D^k$ is always available. Under Henkin semantics, the selected subset must lie in ${\mathcal{H}}_k$ for every allowed assignment of parameters. Different systems permit different formula classes: for example, arithmetical comprehension theory restricts comprehension to formulas without bound set variables, while still permitting set parameters. Consequently, the exact formula and parameter policy must accompany any claim that a restricted range satisfies second-order comprehension.

An existential second-order logic sentence has the formwhere each $R_i$ is a second-order relation variable of fixed finite arity $k_i$ and $\phi$ is a first-order sentence over the input first-order signature expanded by the $R_i$. The first-order sentence may contain both universal quantification and existential quantification over individuals; only the outer quantification over second-order relation variables is required to be existential. Under full semantics for second-order logic, truth means that some expansion by witness relations satisfies $\phi$.

For a fixed existential second-order logic sentence on a domain of cardinality $n$, the witnesses need ${\sum }_{i=1}^m{n}^{k_i}$ membership bits. The arities and the number of witnesses are constants because the sentence is fixed. After guessing those bits, a verifier evaluates the fixed first-order sentence by enumerating assignments to its finitely many individual variables, taking polynomial time. In the second-order encoding of three-colorability, the three witnesses require $3n$ bits, and the edge constraints can be checked over at most $n^2$ ordered vertex pairs. This is the logical form of the NP principle that a positive answer has a short complexity certificate.

The fixed-sentence qualification matters: it is the input first-order structure that varies, rather than an arbitrarily large formula supplied with each input. Fagin's theorem states the converse as well: every isomorphism-invariant NP property of finite relational first-order structures has such a description. The coloring example uses only unary second-order relation variables; the general theorem allows any fixed finite arities.

A second-order relation variable of arity $k\ge 1$ can be assigned a subset of the $k$-fold Cartesian product $D^k$; its application $R(x_1,\dots ,x_k)$ is true exactly when the tuple $(x_1,\dots ,x_k)$ belongs to that subset. Arity is the number of arguments, so a unary second-order relation variable is a predicate on individual objects. For a finite set $D$ of cardinality $n$, there are $n^k$ possible tuples and $2^{n^k}$ possible relations under full semantics for second-order logic. A relation is encoded by $n^k$ membership bits once the objects have been numbered; the number of possible witnesses can be exponential even when each witness has a short encoding.