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Orthogonal projections of an isotropic Gaussian vector are independent. If $P$ is an orthogonal projection of rank $r$ and $Z\sim N(0,I)$, then

Nonparametric regression estimates a regression function without restricting it to a fixed finite-dimensional family.

Density estimation constructs an estimate of an unknown probability density function from sampled observations.

This local phase change is a gauge transformation of the wave function. Multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

For a radial function $u\in H^1({\mathbb{R}}^d)$,This supplies spatial decay without requiring a weighted moment.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable tail errors.

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

For sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

In the convention used here, a Brzozowski NFA has one final state, every state can reach it, and each word labels a path to it from exactly one starting state.

The powerset construction turns an NFA with state set $Q$ into a DFA with state set $\mathcal{P}(Q)$, transitionand accepting subsets that meet the NFA final set.

Without epsilon transitions, defineA word is accepted when the set reached from the initial state meets the final set.

In $R=\mathbb{Z}[\sqrt{-5}]$, the idealhas additive index two. If $I=(a+b\sqrt{-5})$ were principal, multiplication by its generator would givewhich has no solution in integers. Thus $I$ is not principal.

The global phase symmetry of a Dirac field gives the conserved Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

Spacetime translations give the canonical stress-energy tensor

For a point mass and $r>0$, writeThe vacuum scalar-curvature equation gives $r{f}^{\prime }+3f=0$, so $f=C/r^3$. The remaining vacuum equations fix $C=2GM$ in the corresponding Schwarzschild gauge.

For a planar velocity field in polar coordinates,

Shear stress is the tangential component of traction on a surface. For leading unidirectional flow $u(y)$ in a Newtonian fluid, $\tau =\mu du/dy$.

A negative-temperature system is hotter than every positive-temperature system. If heat $\delta Q$ passes from it to a positive-temperature body, the total entropy change is