A submodule of a filtered module inherits the displayed filtration by intersection. It is also called the induced filtration.

The quotient filtration records images of the filtered pieces in a quotient module. Together with the subspace filtration, it gives exactness of associated graded modules for induced filtrations.

An ideal two-sided line plume has half-width $b=\alpha z$, constant velocity $w_{\ast }=(\mathcal{B}/(2\alpha ){)}^{1/3}$ and volume flux $q=2\alpha w_{\ast }z$. In a closed box of width $2R$, neglecting the plume's occupied area gives $2R\dot{h}=-q(h)$ for the filling-box first front, initially at ceiling height $H$. With negligible diffusion the front is a material interface; its ambient buoyancy jump retains the initial ceiling-discharge value $\mathcal{B}/q(H)=w_{\ast }^2/H$. The contemporaneous plume reduced gravity $w_{\ast }^2/h(t)$ is a different quantity. The upper ambient generally becomes stratified rather than remaining one uniformly mixed layer.

A system call for requesting persistence of a file's changed contents and associated file metadata. Its scope, successful completion and separate file-system directory synchronization matter to file-system transaction durability.

The fundamental group of the figure-eight knot complement has the displayed two-generator one-relator presentation. It follows from the Artin automorphism of a braid for $({\sigma }_1{\sigma }_2^{-1}{)}^2$ by eliminating the third meridian from its fixed-generator presentation.

A K-isomorphism between extensions of $K$ is a field isomorphism fixing $K$ pointwise. Simple algebraic extensions $K(\alpha )$ and $K(\beta )$ admit one sending $\alpha$ to $\beta$ exactly when their minimal polynomials agree: both then identify with the same quotient $K[X]/(m)$.

Two finite Galois extensions of the rational number field with coprime field discriminants have trivial intersection. Any ramified prime in the intersection would ramify in both fields. The intersection is therefore everywhere unramified at finite primes and has field discriminant of absolute value one. The Minkowski lower bound for a number-field discriminant excludes this in degree greater than one. The Galois correspondence then gives the product degree for their compositum. The normality hypothesis is used in this last intersection-to-degree argument.

With the standard Fibonacci numbers $F_0=0$, $F_1=1$, this identity holds for $n\ge 0$, $k\ge 1$. At $k=1$ it is immediate. If it holds for $k$ at every $n$, apply it at $n+1$ and use $F_{n+2}=F_{n+1}+F_n$ to obtain $F_{n+k+1}=F_{k+1}{F}_{n+1}+F_k{F}_n$, proving the assertion by mathematical induction. Taking $k=n$ gives $F_{2n}=F_n(F_{n+1}+F_{n-1})$ for $n\ge 1$, linking the doubling formula to the Lucas numbers.

The vector $(F_{n+1},F_n{)}^T$ advances by multiplication by $A$. Matrix powers therefore prove modular recurrence identities uniformly in $n$. Explicitly, $A^3\equiv I(\mathrm{mod}2)$, $A^8\equiv I(\mathrm{mod}3)$, and $A^5\equiv 3I(\mathrm{mod}5)$. The last congruence gives $F_{n+5}\equiv 3F_n(\mathrm{mod}5)$, hence $5\mid F_{5j}$ for every nonnegative $j$.

For a surjective function $f:B\to A$ and a function $g:B\to C$, there is a unique function $h:A\to C$ with $g=h\circ f$ if and only if $g$ is constant on each fiber of a function of $f$. Define $h(a)$ to be the common value on $f^{-1}(a)$; it is well-defined and unique because that fiber is nonempty. This uniquely specified value requires no simultaneous arbitrary choices.

For a concrete example, restrict the codomain of the finite function above to its image of a function $I=\{0,1,2\}$, making $f:A\to I$ surjective. Define $g(a)=g(b)=5$, $g(c)=7$ and $g(d)=9$. Then $h(y)=2y+5$ on $I$ gives $g=h\circ f$: knowing the output of $f$ is enough to compute $g$. If instead $g(a)\ne g(b)$, no such $h$ can exist, because $h(f(a))=h(f(b))$ must be the same value.

The surjectivity assumption supplies uniqueness on the full target. If the original unused target value three were retained, the factorization would impose no condition on $h(3)$, and different extensions could agree on every actual input. Thus factorization through a surjection formalizes when a compressed description retains all information needed for a later calculation.

// Target: combinatorics.bigb

With interaction density $\mu {\varphi }^3/3!-\lambda {\varphi }^4/4!$, the Dyson series contains $i{\mathcal{L}}_{\mathrm{int}}$ and hence cubic vertex $i\mu$ and quartic vertex $-i\lambda$. Wick contractions into the labelled slots cancel each factorial. Position integration gives the four-momentum delta function, and equivalent unlabelled contractions leave the Feynman-diagram symmetry factor. The free line carries $i/(p^2-m^2+i0)$ in the mostly-minus convention.

Three scalar Yukawa interactions on a closed Dirac field loop give two cyclic orientations. Odd gamma matrix traces vanish, while $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$ gives the displayed numerator. For external invariant mass squares $M^2,{\mu }^2,{\mu }^2$, write $D_a=a^2-m^2$ and similarly for the other two lines. The numerator becomes $4m[D_a+D_b+D_c+4m^2-{\mu }^2-M^2/2]$, reducing the fermion loop to three scalar bubbles and a scalar triangle Feynman integral. The closed fermion loop sign must be included. A massive fermion produces a logarithmic local divergence.

For a massive charged vector with interaction $-\kappa (W^{+}J+W^{-}{J}^{†})$, neglect derivatives at momenta small compared with its mass. Completing the square in $m_W^2{W}^{+}{W}^{-}$ leaves $-{\kappa }^2{J}^{†}J/m_W^2$. With $\kappa =g/(2\sqrt{2})$ this yields the Fermi constant relation $G_F/\sqrt{2}=g^2/(8m_W^2)$. Corrections contain extra derivatives and are suppressed by momentum squared over the heavy mass squared.

An electron Fermi gas whose occupation is governed mainly by the Pauli exclusion principle, with thermal energy small compared with its kinetic Fermi energy. At zero temperature its momentum states are filled up to $p_F$. In the ultrarelativistic limit $E/V=3nc{p}_F/4\propto n^{4/3}$, with $n=4\pi g_s{p}_F^3/(3h^3)$. This supplies the pressure supporting a white dwarf until the relativistic mass limit is approached.

The even positive Fejér kernel of total mass one gives ${\sigma }_{n}f-f=\frac{1}{2}\int K_n(t)[f(\cdot +t)-2f+f(\cdot -t)]dt$. The scaling property of the second modulus of smoothness bounds this by ${\omega }_2(f,\delta )(1+{\delta }^{-2}\int t^2{K}_n(t)dt)$. On $[-\pi ,\pi ]$, $\sin (|t|/2)\ge |t|/\pi$ gives $\int t^2{K}_n(t)dt\le {\pi }^2/n$. Choose $\delta =n^{-1/2}$. For twice continuously differentiable functions, the second-difference integral formula makes the error $O(n^{-1})$.

The normalized Fejér kernel satisfies $0\le K_n(t)\le \min \{n/(2\pi ),\pi /(2n{t}^2)\}$ on $0<|t|\le \pi$. Splitting its first absolute moment at $1/n$ gives $\int |t|K_n(t)dt\le 1/(2\pi n)+(\pi /n)\log (\pi n)$. The chaining inequality for the modulus of continuity, $\omega (f,t)\le (1+t/\delta )\omega (f,\delta )$, then bounds the Fejér sum error by $\omega (f,\delta )[1+{\delta }^{-1}\int |t|K_n]$. Taking $\delta =\log n/n$ proves the stated bound for $n\ge 2$ and hence uniform convergence for continuous periodic functions. The logarithm vanishes at $n=1$, so the displayed estimate is not intended at that index.

For a periodic Hölder continuous function of Hölder exponent $0<\alpha <1$ and Hölder seminorm at most $M$, the degree-$n-1$ Fejér sum has the displayed supremum norm error bound. In the half-kernel normalization $H_n=(2n{)}^{-1}{\sin }^2(nt/2)/{\sin }^2(t/2)$, use $H_n(t)\le \min \{n/2,{\pi }^2/(2n{t}^2)\}$. Splitting the convolution error integral at $1/n$ gives the valid constant $C_{\alpha }=1/[\pi (\alpha +1)]+\pi /(1-\alpha )$. At the endpoint $\alpha =1$ this argument instead gives $O(M\log n/n)$.

Odd-address intervals lie in the first image of the central restrictive interval, away from the critical point. The derivative of the Feigenbaum map is bounded above and away from zero there, so the next even interval has comparable length by the mean value theorem. For the wraparound image, $g^{2^n}({\beta }^{n}x)={\beta }^{n}g(x)$ makes its length a fixed fraction $(1-\beta )/2$ of the central interval. Thus the odd and full partition length-power sums differ only by constant factors for each fixed real exponent, and have the same exponential growth rate.

Suppose a Fatou component $U$ satisfies $R(U)\subseteq U$, and some $a\in U$ has $R^{-1}(a)\subseteq U$. Every connected component of $R^{-1}(U)$ maps properly and surjectively to $U$: properness follows by intersecting compact preimages with the relatively closed component; openness and closedness then give surjectivity. Such a component must meet the entire fiber over $a$, which lies in $U$. Since $U$ is itself one inverse-image component, there can be no other. Hence $R^{-1}(U)=U$ and $R(U)=U$, proving complete invariance.

A Siegel disc is a simply connected periodic Fatou component on which the first-return map is holomorphically conjugate to an irrational rotation of the unit disc: $h\circ R^p\circ h^{-1}(w)=e^{2\pi i\theta }w$, with $\theta \notin \mathbb{Q}$. Its return iterates are normal because any sequence of rotation factors has a convergent subsequence in the unit circle, giving locally uniform limits after conjugation. The iterates generally do not converge to a constant. This is a rotation domain, illustrating stability without attraction.