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The Square-root Lasso minimizesIts residual norm is unsquared. For errors $\epsilon =\sigma Z\ne 0$, the magnitude of the loss gradient at the true regression coefficients is governed byThe unknown noise standard deviation $\sigma$ cancels. This makes a penalty calibrated to that score function pivotal with respect to noise scale.^[ref]

For normalized columns and Gaussian noise, a typical theoretical penalty has order $\sqrt{\log (p)/n}$ without an extra factor of $\sigma$. Prediction guarantees still require assumptions on the number of nonzero regression coefficients, the design matrix and the noise. Pivotal tuning does not remove those assumptions. At an exact zero residual the Euclidean norm is nondifferentiable, so its subdifferential must replace the displayed nonzero-residual gradient.

The Dantzig selector minimizes the coefficient ${\ell }^1$ norm subject to an ${\ell }^{\infty }$ bound on residual correlations.

A dark matter halo is a gravitationally bound concentration of dark matter. Its internal motions are largely supported by a distribution of particle orbits, unlike the cooling gas that can settle into a luminous galactic disk. Halo profiles, abundances and spatial correlations provide useful ingredients for modeling nonlinear clustering. The halo model uses these ingredients as an approximate statistical description of matter.^[ref]

Standard perturbation theory in cosmology expands the matter density contrast and peculiar-velocity divergence in powers of a small initial density field. The linear term evolves independent Fourier modes; higher terms couple them and predict departures from Gaussian statistics. The usual fluid treatment assumes pressureless, irrotational, single-stream motion. It is useful before strongly nonlinear evolution and shell crossing invalidate that approximation.^[ref]

An Einstein-de Sitter universe is a spatially flat expanding model containing only pressureless matter, with zero cosmological constant. Conservation of matter gives $\rho \propto a^{-3}$; inserting this into the Friedmann equations gives $a\propto t^{2/3}$ in cosmic time. Since $dt=ad\tau$, it also gives $a\propto {\tau }^2$ and $\mathcal{H}=2/\tau$ in conformal time. These simple powers make the model useful for isolating gravitational growth, rather than representing the full matter content at every epoch of the actual universe.

Cosmological structure formation is the gravitational amplification of initial linear cosmological density perturbations into the cosmic web and bound objects. Overdense regions attract extra matter while surrounding regions become relatively empty. In the cold dark matter picture, small initial particle velocities preserve fluctuations that stronger free streaming would erase. The resulting spectrum of seed fluctuations influences the populations of galaxies and clusters; gas cooling and feedback add further physics to their visible contents.^[ref]

A gap equation determines a physical mass or excitation gap self-consistently from an interaction and a regulated fluctuation integral.

The Laplace transform of a matrix-valued function is obtained by transforming each entry. A constant matrix may be taken outside on the corresponding side of the integral; in particular, $\mathcal{L}\{AB\}=A\mathcal{L}\{B\}$ when $A$ is constant.

Let $T>0$, and let $g$ be a locally integrable function with period $T$ on nonnegative time. For $\mathrm{Re}s>0$, its unilateral Laplace transform isSplit the defining integral into successive periods and put $t=u+jT$. Periodicity makes each piece $e^{-sjT}{\int }_0^T{e}^{-su}g(u)du$; summing the absolutely convergent geometric series proves the formula. Local integrability on one period also controls the corresponding absolute integrals.

For a square pulse equal to one on $[0,T/2)$ and zero on $[T/2,T)$, repeated periodically, the result is $(1-e^{-sT/2})/[s(1-e^{-sT})]=1/[s(1+e^{-sT/2})]$ in that half-plane. At $s=0$ the defining improper integral generally fails; continuing the quotient algebraically does not enlarge the defining region of absolute convergence of a Laplace transform.

For $a\ge 0$, $\mathcal{L}\{f(t-a)H(t-a)\}(s)=e^{-as}\mathcal{L}\{f\}(s)$.

Suppose $f$ is an absolutely continuous function on every finite interval $[0,T]$, both $e^{-st}f(t)$ and $e^{-st}{f}^{\prime }(t)$ are Lebesgue integrable functions, and $e^{-sT}f(T)\to 0$. Then integration by parts on $[0,T]$ followed by a limit givesHere $f(0+)$ is the right one-sided limit. Exponential order bounds on $f$ and $f^{\prime }$ suffice when $\mathrm{Re}s$ exceeds both bounds. For $f(t)=e^{-2t}$, the identity reads $-2/(s+2)=s/(s+2)-1$ on $\mathrm{Re}s>-2$. This is the ordinary derivative on nonnegative time; differentiating a causal extension by zero at its jump instead introduces a Dirac delta function. ^[ref].

For a constant magnetic field,is the symmetric gauge.

A uniform electric field perpendicular to a uniform magnetic field shifts each cyclotron oscillator's guiding centre and makes its energy depend linearly on the conserved momentum. The formerly degenerate Landau levels therefore become tilted bands.

For a spin-one-half particle with the Pauli magnetic term and gyromagnetic factor two, the orbital energies $\hslash {\omega }_c(n+1/2)$ split into $\hslash {\omega }_{c}n$ and $\hslash {\omega }_c(n+1)$. The lowest level has one spin branch; every positive level has two.

In a planar region of area $A$, each spinless Landau level has guiding-centre degeneracyMagnetic periodic boundary conditions require this flux count to be an integer.

A particle of charge $q$ and mass $m$ in a uniform magnetic field of magnitude $B$ has cyclotron angular frequency ${\omega }_c=|qB|/m$.

For a particle of charge $q$ in a vector potential $\mathbf{A}$, the canonical momentum in the free Hamiltonian is replaced by $\mathbf{p}-q\mathbf{A}$, giving

If $b(x)=B{x}^{-1/5}$, balancing the momentum flux and transverse viscosity gives

For a steady incompressible plume with localized vertical body force $b(x)\delta (y)$, decay of velocity and shear at transverse infinity gives

A Lagrangian with no explicit time dependence conserves $E={\sum }_i{\dot{q}}_i\partial L/\partial {\dot{q}}_i-L$.