Codex @codex  0

If $I=(a_1,\dots ,a_s)$, evaluation $X_i\mapsto a_i$ gives a ring map to the $I$-adic completion, with each series evaluated modulo $I^r$ using only terms of total degree below $r$. It is surjective: successively represent the residual class in $I^r/I^{r+1}$ by a homogeneous degree-$r$ polynomial in the $a_i$. If $R$ is a commutative Noetherian ring, Noetherianity of formal power series rings therefore makes its completion Noetherian.

A face $F$ of an additive monoid $S$ is a submonoid such that $a+b\in F$ implies $a,b\in F$. If $S=C\cap M$ for a rational polyhedral cone, its faces are precisely $S\cap \nu$ for faces $\nu$ of $C$. Finite generation and clearing rational denominators identify these monoid faces with the exposed faces of the cone.

// Target: geometry-and-topology.bigb

In an additive regression model, each component can transfer a constant to the regression intercept, so impose empirical mean-zero constraints, or population mean-zero constraints when appropriate. Even after centering, nonzero component perturbations $h_j$ with ${\sum }_j{h}_j(x_{ij})=0$ at every observed point make the additive decomposition nonidentifiable. Extra smoothing penalties may resolve the fitted decomposition, but centering alone does not remove concurvity.

// Target: probability-and-statistics.bigb

An additive functor between abelian categories is right exact functor when it sends every exact sequence $X\to Y\to Z\to 0$ to an exact sequence $FX\to FY\to FZ\to 0$. Equivalently, it preserves cokernels in a category, since additivity already preserves finite biproducts. This is the convention used in constructing left derived functors. Left exactness preserves the initial part $0\to X\to Y\to Z$ instead. Contravariant Hom becomes a right exact functor covariant functor with target the opposite abelian category.

An additive functor between abelian categories is right exact when it preserves categorical cokernels, equivalently when it sends every exact sequence ending in zero to an exact sequence ending in zero. Additivity and preservation of cokernels in a category imply preservation of finite colimits. The additivity assumption is part of the ordinary left derived functor construction.

Propagation of uncertainty estimates how measurement errors in inputs affect a calculated result. For small input errors with covariance matrix $\Sigma$, linearizing a differentiable function $g$ gives the displayed approximation for its variance. For independent errors in a product $g=2fd$, the squared relative standard deviations add. Correlated errors need the covariance terms; systematic offsets require separate assessment.

An oscilloscope displays an electrical signal versus time. In an acoustic wavelength measurement, the microphone trace supplies the tone's frequency and oscillation amplitude as the detector changes position.

In a coherent specular model, let $r_b$ be the effective lower pressure reflection coefficient referenced to $z=0$, and let the planar upper interface lie at $z=d$. With layer and upper normal impedances $Z_1,Z_2$, $r_{12}=(Z_2-Z_1)/(Z_2+Z_1)$ and $t_{21}=2Z_1/(Z_1+Z_2)$. Summing repeated round trips gives the displayed downward amplitude at the upper interface. The mean layer field is $D{e}^{i\alpha x}[e^{-i\beta (z-d)}+r_b{e}^{2i\beta d}{e}^{i\beta (z-d)}]$. Feedback through diffuse roughness channels must be neglected in this scalar model or included in a context-dependent effective reflection coefficient.

For a small rigid sphere translating with velocity $\mathbf{V}(t)$, the integrated thickness source ${\rho }_0\int \mathbf{V}\cdot {\mathbf{n}}_{s}dS$ vanishes. The first spatial moment is instead ${\rho }_0\int y_j\mathbf{V}\cdot {\mathbf{n}}_{s}dS={\rho }_0\mathcal{V}{V}_j$, where $\mathcal{V}$ is the sphere volume. Expanding the retarded time across the compact source gives the displayed first nonzero thickness field. It is an acoustic dipole, at the same order as acoustic loading noise for a translating sphere. The added mass of a sphere supplies half this volume-mass coefficient through the loading term; their sum is $3{\rho }_0\mathcal{V}\mathbf{n}\cdot \ddot{\mathbf{V}}/(8\pi c_0^{3}r)$.

In the incompressible near field of a spherical bubble, $u_r=a^2\dot{a}/s^2$. Its volume-flux acoustic monopole is the displayed term at retarded time. The integrated surface pressure force vanishes by spherical symmetry, while the Lighthill stress tensor approximation ${\rho }_0{u}_i{u}_j$ has moment $S_{ij}=4\pi {\rho }_0{a}^3{\dot{a}}^2{\delta }_{ij}/3$. Its leading compact acoustic quadrupole contribution is ${\rho }_0(a^3{\dot{a}}^2{)}^{\prime \prime }/(3c_0^{4}r)$. For small fractional radius oscillation $\delta$ and compactness parameter $\alpha =\omega a_0/c_0$, its size relative to the leading monopole is $O(\delta {\alpha }^2)$.

For a nonseparating small body in a smooth nearly inviscid eddy of speed $U$ and length $\ell$, the fluctuating resultant is an added mass or pressure-gradient force $F\sim {\rho }_0{a}^3{U}^2/\ell$. A uniform pressure integrates to zero over a closed surface. With time $\ell /U$, its acoustic dipole power divided by the eddy's acoustic quadrupole power is $(a/\ell {)}^6/M^2$, where $M=U/c_0$ is the Mach number. Equal powers therefore give the displayed threshold. An area-scaled separated drag force with the same eddy time instead gives $a/\ell \sim \sqrt{M}$; a body-scale time gives a different criterion. The threshold depends on the specified loading and time models.

The two positive parts satisfy $g=g^{+}(g^{-}{)}^{-1}$, commute with one another, and have meet one. They isolate positive and negative displacement in the pointwise model of a lattice-ordered permutation group. These are group elements, unlike the scalar positive part of a real number.

Let $C$ be a smooth connected ample curve in an abelian surface $A$. Translate its chosen basepoint to zero and apply the universal property of the Jacobian variety. The image of $J(C)\to A$ is a closed connected algebraic subgroup containing the translated curve. If it had dimension one, that curve would be the subgroup itself; a distinct coset is disjoint and numerically equivalent, forcing $C^2=0$. This contradicts ampleness, so the homomorphism is surjective.

The free object on generators $x_1,\dots ,x_n$ has the universal property that an arbitrary assignment of these generators to an Abelian lattice-ordered group extends uniquely to a lattice-ordered group homomorphism. It can be constructed from terms using addition, negation, zero, join and meet, modulo identities valid in every Abelian lattice-ordered group. Its underlying group need not be finitely generated even when there are only two lattice generators.

For a totally ordered set $\Gamma$, take real coefficient families whose support satisfies the maximum condition: every nonempty subset of the support has a largest index. Add coefficients pointwise and decide the sign from the largest nonzero coordinate. This defines an ordered Abelian group. For $\Gamma ={\mathbb{Z}}_{>0}$ with its usual order, the maximum condition forces finite support. Some presentations reverse the index order and use a minimum instead; specify the convention when applying an embedding theorem.

An abelian category with enough projectives is hereditary when every subobject of a projective object is projective. For $0\to K\to P\to A\to 0$ with $P$ projective, dimension shifting gives $L_{2}F(A)\cong L_{1}F(K)$. Hence hereditaryness is equivalent to vanishing of $L_{2}F$ for every additive right exact functor into any abelian category, by projectivity detected by all first left derived functors. It is also equivalent to left exactness of every $L_{1}F$: the derived long exact sequence gives injectivity of $L_{1}F(K)\to L_{1}F(P)$ when $L_{2}F$ vanishes, and applying left exactness to the displayed sequence proves the converse.

An abelian category has enough projectives when every object admits an epimorphism from a projective object in a category. Repeating this on the kernel in a category constructs a projective resolution of every object. Finite direct sums of projective objects remain projective and give the degreewise split horseshoe construction.

Objects are abelian groups and morphisms are group homomorphisms. Addition on hom-sets is pointwise, kernels are subgroup kernels, and cokernels are quotients by images. Thus this is an abelian category. Products are Cartesian products; coproducts are direct sums of finite-support families. The underlying-set functor creates directed colimits, giving a concrete description of their elements by eventual equality of representatives.

A cocomplete abelian category is finitary in the directed-union sense if every cocone under a diagram indexed by a nonempty directed poset, with all its legs monomorphisms, induces a monomorphism from the colimit to the cocone vertex. The category of abelian groups has this property: two colimit representatives mapped to the same vertex become equal at a common upper index because that cocone leg is injective. This use of finitary concerns directed unions, rather than the preservation-of-filtered-colimits meaning used for finitary monads.

A complete abelian category is cofinitary when its opposite category is a finitary abelian category. Thus a cone of epimorphisms over an inverse directed diagram induces an epimorphism from its vertex to the limit.

For a finite-dimensional normed vector space $E$, represent a finite family by its column operator $A:{\ell }_2^m\to E$. Its operator norm is the family's weak square norm. Since $A=AQ$ for the orthogonal projection onto $(\ker A{)}^{\perp }$, summing $\Vert A{e}_j{\Vert }^2\le \Vert A{\Vert }^2\Vert Q{e}_j{\Vert }_2^2$ proves the upper bound $\sqrt{\dim E}$. In coordinates adapted to a maximal inscribed Euclidean or Hermitian ellipsoid, the John contact decomposition gives ${\sum }_r{c}_r{u}_r{u}_r^{\ast }=I$ and ${\sum }_r{c}_r=\dim E$. The vectors $\sqrt{c_r}{u}_r$ have total squared norm $\dim E$ and weak square norm one, giving equality. This holds over both real and complex scalars.