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In a longitudinal plane wave, displacement is parallel to the wavevector and the curl vanishes.

The orthogonal projection $P_F$ onto a closed subspace of a Hilbert space $F$ sends $x$ to its component in the orthogonal decompositionThe residual $r$ is orthogonal to every permitted direction. For any $z\in F$, the Pythagorean theorem givesThus $P_{F}x$ uniquely minimizes the distance to $F$. It is a linear map, is idempotent, and is nonexpansive: $\Vert P_{F}x-P_{F}y\Vert \le \Vert x-y\Vert$; taking $y=0$ gives $\Vert P_{F}x\Vert \le \Vert x\Vert$.

In ${\mathbb{R}}^2$, project $v=(1,3)$ onto the line spanned by $u=(2,1)$. The inner product givesThe check $⟨r,u⟩=-2+2=0$ confirms orthogonality, and $\Vert v{\Vert }^2=10=5+5=\Vert P_{F}v{\Vert }^2+\Vert r{\Vert }^2$. Projecting again changes nothing, since the result already lies on the line.

For a finite orthonormal basis $e_1,\dots ,e_m$ of $F$, the same operation is $P_{F}x={\sum }_j⟨x,e_j⟩e_j$, using the convention linear in the first argument. Replacing coordinate vectors by functions gives exactly the same approximation principle. For $f(x)=x^2$ in the real L2 space on $[-1,1]$ and $F=\mathrm{span}\{1,x\}$, the directions are orthogonal, with squared norms $2$ and $2/3$. Their inner products with $f$ are $2/3$ and zero, soDirect integration gives the squared error $\Vert x^2-1/3{\Vert }^2=8/45$. For any competing affine polynomial, the orthogonal residual yieldsEvery deviation from $a=1/3$, $b=0$ adds a nonnegative error. The best function need not pass through any selected data point: it minimizes an integrated error across the whole interval.

Colored noise has nonzero temporal correlation away from equal times. An Ornstein-Uhlenbeck noise has exponential covariance and approaches white noise when its correlation time tends to zero at fixed integrated covariance.

A multivariate Ornstein-Uhlenbeck process obeys $dX=-AXdt+bdW$. If every eigenvalue of $A$ has positive real part, its stationary covariance $\Sigma$ is the unique solution of $A\Sigma +\Sigma A^T=b{b}^T$.

Let $Y$ be the mapping cylinder of the orientation double cover $T^2\to K$ with its free end omitted. If a Hausdorff space contains $Y$ as an open subset, van Kampen extends the orientation quotient of ${\pi }_1(K)$ to a surjection of the ambient fundamental group onto $C_2$. The ambient space is therefore not simply connected.

The null twist is ${\hat{\omega }}_{ab}={\hat{B}}_{[ab]}$. It vanishes for null generators orthogonal to a hypersurface by the Frobenius theorem.

The null shear is the symmetric trace-free part ${\hat{\sigma }}_{ab}={\hat{B}}_{(ab)}-\theta P_{ab}/(D-2)$. It changes the shape of a transverse beam while preserving its area to first order.

The null expansion is $\theta =P^{ab}{\hat{B}}_{ab}$. It is the logarithmic derivative of an infinitesimal transverse area: $\theta =A^{-1}dA/d\lambda$.

If a nonconstant holomorphic function had a local maximum of its modulus, its open image near that point would contain values of larger modulus. Hence no such local maximum exists.

If $P_t$ denotes a path of length $t$, thenfor $s\ge 2$ and $t\ge 1$.

A western boundary current is a narrow, intense return flow that closes a broad wind-driven interior circulation. The beta effect selects the western side for this frictional boundary layer.

Sverdrup balance equates meridional advection of planetary vorticity to wind-stress curl in the weakly frictional ocean interior.

The upper-convected derivative of a contravariant tensor is $\stackrel{▽}{\mathbf{C}}=D\mathbf{C}/Dt-(\nabla \mathbf{u})\mathbf{C}-\mathbf{C}(\nabla \mathbf{u}{)}^T$.

For a connected degree-$n$ cover of a closed orientable surface ${\Sigma }_g$, normality is forced when $n=1$, when $n=2$, or when $g=1$. For $g=0$, existence itself forces $n=1$. For every $g\ge 2$ and $n\ge 3$, nonnormal connected degree-$n$ covers exist.

A universal covering has simply connected total space, so its associated subgroup of the base fundamental group is trivial. The trivial subgroup is normal; hence every universal covering map is a normal covering map.

For $\varphi \in H^1(\mathbb{R})$ with $\Vert \varphi {\Vert }_2=1$, the dilationpreserves the $L^2$ norm and satisfies $\Vert u_R^{\prime }{\Vert }_2^2=R^{-2}\Vert {\varphi }^{\prime }{\Vert }_2^2$. Hence normalized functions on the line have Dirichlet-energy infimum zero, which no nonzero $L^2$ function attains.

A bounded sequence can fail to have a strongly convergent subsequence because its mass escapes to infinity even when its local regularity is uniformly controlled.

Lifting the hyperbolic-scaling action to the universal cover of ${\mathbb{C}}^{\ast }$ preserves the two inseparable limiting orbits. The covering surface is biholomorphic to $\mathbb{C}$, so a simply connected Riemann surface can still have a non-Hausdorff quotient by a covering space action of homeomorphisms.

A Popescu–Rohrlich box is a bipartite no-signalling box attaining the algebraic maximum of the CHSH inequality.

For the negative binomial distribution with positive integer target $r$ and success probability $0<p<1$, the $r$th success occurs after exactly $x$ failures only if the final trial is a success, while the preceding $x+r-1$ trials contain $x$ failures and $r-1$ successes. Choosing the failure locations gives the binomial coefficient $\left(\genfrac{}{}{0ex}{}{x+r-1}{x}\right)$. Each such stopped sequence has probability $(1-p{)}^x{p}^r$ by independence, and the sequences describe mutually exclusive events, so their probabilities add. This proves the probability mass function. Counting arrangements among all $x+r$ positions would wrongly include sequences that already reached the target earlier. The same argument is used for three heads in MIT 18.05, spring 2022, problem set 9 solutions, problem 2(b).

A short route to normalization and moments is to split the failures count into successive waiting blocks:where $F_i$ counts failures after success $i-1$ and before success $i$, with $F_1$ starting at the first trial. Fresh independent Bernoulli trials make the blocks independent, each with the failures-count geometric distribution. Each block has total mass $p{\sum }_{j=0}^{\infty }(1-p{)}^j=1$ by the geometric series, so their sum also defines a normalized probability distribution. The linearity of expectation and variance additivity for independent random variables give $\mathbb{E}X=r(1-p)/p$ and $\mathrm{Var}(X)=r(1-p)/p^2$. This makes the moments transparent without a long calculation of sums. For $p=1$, every block is zero.

The same stopping rule connects a waiting count to a fixed-length count. Let $N=X+r$ be the trial number of the $r$th success, and let $B_n$ count successes in the first $n$ trials, so $B_n$ has a binomial distribution. For any nonnegative integer $n$,The target is still unmet after $n$ trials exactly when fewer than $r$ successes have occurred. With $r=2$, $p=1/2$ and $n=4$, this gives $\mathbb{P}(N>4)=(1+4)/16=5/16$; equivalently, $\mathbb{P}(X\le 2)=1-5/16=11/16$, matching $1/4+1/4+3/16$. At $p=1$, the event identity still holds, with $N=r$ and $B_n=n$ deterministically.