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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.

For a full-row-rank $k\times p$ matrix $R$ and the null hypothesis $R\theta =r$, putUnder the null, asymptotic normality and Slutsky theorem give $W_{n,R}\stackrel{d}{\to }{\chi }_k^2$.

For nonzero $\lambda$, the ordinal ${\omega }^{\lambda }$ is multiplicatively closed exactly when $\lambda$ is additively closed. One direction follows fromFor the other, the leading exponent of an ordinal product shows that products of ordinals below ${\omega }^{\lambda }$ still have leading exponent below $\lambda$.

A long cylinder of radius $a$ and length $L$ settling axially at speed $U$ inside a coaxial container forces displaced liquid back through a narrow annular gap $h\ll a$. For an incompressible Newtonian fluid with dynamic viscosity $\mu$, the displaced volume flux is approximately $\pi a^{2}U$. The pressure-driven part of Couette-Poiseuille flow in a thin gap carries volume flux $2\pi a{h}^3|p_z|/(12\mu )$. Equating these leading terms gives the excess pressure dropThe direct wall-motion contribution to the volume flux is smaller by order $h/a$. The resulting pressure drag is $\pi a^2\Delta p$, larger than side-wall viscous drag by order $a/h$. Balancing it with the buoyancy-corrected weight $\pi a^{2}L\Delta \rho g$, where $\Delta \rho$ is cylinder minus fluid mass density, givesThis leading lubrication theory result assumes slow laminar flow, a nearly uniform gap and negligible end-region corrections. The cubic dependence on gap thickness explains the strong confinement-induced slowing.

The eigenfunctions satisfying $X(0)=0$ and $X^{\prime }(L)=0$ are

For a family with a monotone likelihood ratio in $T$, a level-$\alpha$ upper-tail test in $T$ is a uniformly most powerful test for the corresponding one-sided hypothesis.

Rotational carbon-monoxide line emission traces cold molecular clouds because abundant molecular hydrogen is difficult to observe directly. A conversion factor $X_{\mathrm{CO}}$ relates integrated CO intensity to molecular-hydrogen column density.

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds $c_j$ and angles ${\theta }_j$ from the normal satisfy

An accessible automaton over one letter consists of a directed tail entering one directed cycle. Its minimal quotient is obtained by merging positions having the same future binary acceptance sequence.

A solvability condition in the method of multiple scales is the compatibility requirement allowing a bounded periodic correction at a given perturbation order. Forwith $2\pi$-periodic $F$ in the fast variable, the Fredholm solvability condition for a self-adjoint operator requiresNecessity follows by integration by parts against the two periodic homogeneous solutions. For smooth periodic $F$, sufficiency follows by Fourier series: all coefficients outside modes $\pm 1$ can be divided by $1-n^2$. The forbidden modes would instead generate linearly growing secular terms.

The forcing contains slow derivatives of lower-order amplitudes, so these two conditions become amplitude-phase equations. For a general non-self-adjoint operator, project onto its adjoint null space instead of assuming the same homogeneous modes work.^[ref]

The invariant sigma-algebra of $T$ consists, modulo null sets, of measurable sets $A$ satisfying $T^{-1}A=A$.

Replacing the chosen unit normal $N$ by $-N$ replaces the shape operator $S$ by $-S$. On a surface, Gaussian curvature $K=\det S$ is unchanged, while mean curvature $H=\mathrm{tr}(S)/2$ changes sign.

A minimal surface has vanishing mean curvature vector. For a smooth surface in three-dimensional Euclidean space, this means $H=0$, equivalently $k_1+k_2=0$. Thus Gaussian curvature is $K=-k_1^2\le 0$, but neither principal curvature needs to vanish. A plane and a catenoid both satisfy the condition, with very different Gaussian curvatures.

By the first variation of area formula, a minimal surface is a stationary point of surface area for every smooth variation that fixes its boundary, or has compact support in its interior. The word “minimal” specifies this differential equation; it does not guarantee a global minimum of surface area, or even a local minimum. A stable minimal surface imposes a further condition on the second variation.

For $v,w$ in a real normed space, start with the points at distance $\Vert v-w\Vert /2$ from both endpoints and repeatedly retain those within half the previous diameter of every retained point. Reflection about $(v+w)/2$ preserves every stage, the midpoint belongs to every stage, and the diameters decrease by a factor at least two. Their intersection is therefore exactly the midpoint.

The maximal cyclotomic extension of ${\mathbb{F}}_p$ is ${\overline{\mathbb{F}}}_p$. A primitive $(p^m-1)$th root of unity generates ${\mathbb{F}}_{p^m}$ over ${\mathbb{F}}_p$, and these finite fields exhaust the algebraic closure.