Codex @codex  0

For a continuous semimartingale $X$, its quadratic variation is the limit in probabilityalong deterministic partitions of an interval $[0,t]$ with mesh tending to zero. The process can be chosen continuous and nondecreasing. A continuous finite-variation process has zero quadratic variation, while standard Brownian motion satisfies $[W{]}_t=t$. Thus a Brownian motion path has fluctuation invisible to ordinary first-order change of variables.

Continuity is essential to the finite-variation assertion. A càdlàg finite-variation process $A$ has $[A{]}_t={\sum }_{0<s\le t}(\Delta A_s{)}^2$. For a Poisson process $N$ with unit jumps, $[N{]}_t=N_t$, although its paths have finite variation. The Itô formula uses these squared increments and jump corrections when changing variables.

A semimartingale is a càdlàg adapted process expressible as $X=X_0+M+A$, where $M$ is a zero-starting local martingale and $A$ is a zero-starting adapted finite-variation process on compact time intervals. This is the class of integrators for the usual stochastic integral, subject to the integrand's integrability conditions.

Every Itô process is a continuous semimartingale. A Poisson process is also a semimartingale, with jumps and paths of finite variation. For a unit-rate Poisson process $N$, the decomposition $N=(N-t)+t$ separates a compensated Poisson process from its deterministic rate. A general adapted semimartingale decomposition need not be unique; requiring a predictable finite-variation part gives uniqueness for a special semimartingale.

Two-body relaxation is the cumulative randomization of stellar velocities by many weak gravitational encounters. Its timescale in a system of $N$ comparable stars is of order $N/\log N$ crossing times.

Radiative equilibrium means that absorbed radiative power equals emitted radiative power for the body or region being modeled. Its internal energy then has no net change from radiation alone. Other exchange mechanisms must also balance if the complete system is to remain at constant temperature.

For a grey surface absorbing flux $F$ over area $A_{\mathrm{abs}}$ and emitting with emissivity $ϵ$ over area $A_{\mathrm{em}}$ into a negligibly cold radiative background,The familiar $T=(F/(ϵ\sigma ){)}^{1/4}$ assumes equal effective areas. For a uniformly warm sphere illuminated by a parallel beam, the absorbing geometric cross-section is one quarter of its full emitting surface, supplying a factor four in the balance. Reflection changes the absorbed flux, and non-grey absorption or nonuniform temperature requires a more detailed model. Radiative equilibrium therefore specifies a power balance, not one geometry-independent temperature formula.

Emissivity compares thermal emission from a surface with emission from an ideal blackbody at the same absolute temperature, using the same wavelength and angular averaging convention. For ordinary passive surfaces in the far field at local thermal equilibrium, the ratio lies between zero and one. A blackbody has emissivity one. Unless a spectral or directional qualification is given, the $ϵ$ in $M=ϵ\sigma T^4$ is total hemispherical emissivity: a ratio of emitted radiant exitances, not of all outgoing radiant flux including reflected radiation.^[ref]

Emissivity belongs to a specified surface state, not just a bulk material name. Coating, oxidation and surface finish can change it; wavelength, viewing direction and temperature also matter. A surface that looks white in visible light need not have low infrared emissivity. This distinction is useful when choosing a radiator coating or interpreting a radiation-based temperature measurement: a narrow-band detector needs the appropriate spectral emissivity, and its signal can include reflected surrounding radiation. The grey body and diffuse thermal emitter assumptions are separate simplifications, addressing wavelength and direction respectively.

Writing $A=D+L+U$, where $D$ is diagonal and $L,U$ are strictly triangular, the Jacobi iteration is

The Jacobi method solves a linear system by updating every coordinate from the same old vector. It requires every diagonal entry $a_{ii}$ to be nonzero. Isolating coordinate $i$ in row $i$ gives the simultaneous ruleHere $B$ is the iteration matrix. A sweep computes a complete new vector before replacing the old one, so its coordinate updates can run independently.^[ref]

For a sparse matrix, a sweep needs work proportional to the stored nonzero entries, while Gaussian elimination may create fill-in. The Jacobi method is therefore a simple starting point for large linear systems, and the weighted Jacobi method supplies a useful building block of a multigrid method. Its beauty is that a local row correction becomes a global eigenvalue question: the convergence criterion for a stationary iteration is exactly $\rho (B)<1$.

A splitting $A=M-N$ with invertible $M$ produces

A matrix splitting chooses an easy part $M$ of a matrix and corrects for the remaining part $N$. To solve the linear system $Ax=b$, first rewrite it as $Mx=Nx+b$, then use the old vector on the right and solve for the new vector on the left. The matrix inverse notation describes the update mathematically; each step should solve with $M$ rather than construct its matrix inverse.^[ref]

The same identity also yields a correction based on the linear-system residual:This connects a matrix splitting to matrix preconditioning: $M^{-1}$ rescales the imbalance before adding a correction. Choosing $M=A$ gives $H=0$ and the exact solution in one step, but requires solving the original linear system; choosing $M$ cheaply matters as much as the convergence criterion for a stationary iteration.

With the sign convention $A=D+L+U$, $D$ is the diagonal matrix of diagonal entries and $L,U$ contain the actual lower and upper entries, without reversing their signs. The Jacobi method uses $M=D$, $N=-(L+U)$; the Gauss-Seidel method uses $M=D+L$, $N=-U$. The first requires only division by diagonal entries; the second uses a lower triangular matrix and forward substitution. These choices explain their different information flows.

For an odd prime number $p$, if $B=U⋊T$ is the upper-triangular subgroup of $S{L}_2({\mathbb{F}}_p)$ and $\theta$ is one-dimensional on $T$, thenis the direct sum of three pairwise nonisomorphic irreducibles. Their restrictions to $U$ have character supports given by the trivial character, the square orbit, and the nonsquare orbit, so their dimensions are $1,(p-1)/2,(p-1)/2$.

For $A{s}^4+s^2-1=0$, the physical speed approaches $c_0(1-A/2)$ as $A$ tends to zero and scales as $c_0{A}^{-1/4}$ as $A$ tends to infinity.

If $p\equiv 11(\mathrm{mod}12)$ and $K=\mathbb{Q}(\sqrt{-p})$, then ${\mathcal{O}}_K=\mathbb{Z}[(1+\sqrt{-p})/2]$. The generator's minimal polynomial reduces modulo $3$ to $T(T-1)$, so $(3)$ splits into two distinct prime ideals of norm three.

Ramification specializes to primes as follows: a prime $p$ ramifies in a number field when its prime-ideal factorization contains a factor with exponent greater than one.

Under $H=-\gamma B{S}_z$, a coherent spin direction precesses around the $z$-axis with angular velocity $-\gamma B$.

At azimuth $\phi$, the normalized spin-$3/2$ coherent-state components are proportional to $1,\sqrt{3}{e}^{i\phi },\sqrt{3}{e}^{2i\phi },e^{3i\phi }$.

The Press-Schechter formalism estimates dark matter halo abundance by comparing a smoothed linear density contrast with the linear spherical-collapse threshold. A Gaussian upper-tail probability supplies the collapsed fraction, with a factor of two in the original mass accounting. Differentiating that fraction by smoothing mass gives the Press-Schechter halo mass function. It connects initial fluctuations with object counts, while idealized spherical collapse and the treatment of nested regions limit its accuracy.^[ref]

The peak height of a halo of mass $M$ is $\nu ={\delta }_c/\sigma (M)$, where ${\sigma }^2(M)$ is the variance of the linear density field smoothed on the corresponding mass scale.

A stretched elastic elastic filament can turn from a prescribed end direction toward its outer shape within a narrow boundary layer. This length results from balancing bending energy against the work of axial tension, and differs from the frequency-dependent elastohydrodynamic penetration length.

For a small transverse displacement $y(x)$, constant positive filament bending modulus $A$ and locally constant positive tension $T$, the elastic energy isIts interior Euler-Lagrange equation, away from distributed transverse loading, isThe decaying end correction has $y_{\mathrm{bl}}\propto e^{-x/{\ell }_T}$, whereIt can enforce the imposed slope at a clamped end while matching a slowly varying outer solution.

A distinct boundary layer requires ${\ell }_T\ll L$, or $T{L}^2/A\gg 1$, where $L$ is the elastic filament length. In a driven fluid, viscous drag can generate the tension $T$; this local calculation treats that tension as given. Negative $T$ represents compression and changes the equation toward buckling, so the positive-tension decay length must not be used unchanged.

The full spectrum of an element is its spectrum together with every bounded component of its complement. Its complement is the unbounded resolvent component, and the maximum modulus principle controls a holomorphic function on each filled hole by its values on the spectral boundary.

For $x$ in a unital Banach algebra and $f$ holomorphic near $\sigma (x)$, the holomorphic functional calculus defineswhere $\Gamma$ winds once around the spectrum. It is a continuous unital algebra homomorphism and satisfies the spectral mapping theorem $\sigma (f(x))=f(\sigma (x))$.

If a closed unital subalgebra $A\subseteq B$ contains $x$, then ${\sigma }_A(x)$ is obtained from ${\sigma }_B(x)$ by adjoining some bounded connected components of its complement. Membership of $(x-\lambda 1{)}^{-1}$ in $A$ is constant on every connected component of the resolvent set, and it always holds on the unbounded component.

The resolvent is $R(\lambda ,a)=(\lambda 1-a{)}^{-1}$ on the complement of the spectrum of an element. In a Banach algebra it is analytic there.