On the probability simplex, ${\sum }_i{x}_i=1$. The Cauchy-Schwarz inequality gives $1\le \sqrt{n}\Vert x\Vert$, with equality only when all coordinates are equal. Thus the displayed centroid is the unique closest point to the origin. It is also equidistant from all coordinate vertices, since $\Vert x-e_i{\Vert }^2=\Vert x{\Vert }^2-2x_i+1$.

Every abstract subgroup of finite index in a finitely generated pro-p group is open. Fixed-generator commutator collection in nilpotent groups makes the abstract commutator subgroup closed. The resulting abelian quotient is a quotient of a finite power of the p-adic integers, so its subgroup of $p$th powers is closed and of finite index. Thus the algebraic $G^p[G,G]$ is open and every abstract index-$p$ subgroup is open. An arbitrary finite quotient is a finite p-group; induction on its order, using a normal subgroup of index $p$ and the Schreier lemma, gives the assertion. The normal core reduces nonnormal finite-index subgroups to this case.

A finitely generated powerful p-group in the category of pro-p groups is uniform if all successive indices in its lower p-series are equal. Equivalently it is torsion-free. If its generator number is $d$, the successive indices are $p^d$, the power map carries $G^{p^n}$ bijectively onto $G^{p^{n+1}}$, and an ordered minimal generating set gives unique coordinates $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i$ in the p-adic integers. For $p=2$, powerfulness means $[G,G]\le G^4$.

For a finitely generated pro-p group, its profinite Frattini subgroup has finite elementary abelian quotient. Continuous group homomorphisms to ${\mathbb{F}}_p$ are precisely the linear functionals on this quotient. The topological generation criterion identifies a minimal generating set with a basis, giving the displayed equality. Finite generation is essential when generator number means the least cardinality of an unrestricted dense generating family.

For finite $X$, this is the pro-p completion of the free group on $X$. Every function from $X$ to a pro-p group extends uniquely to a continuous group homomorphism. More generally one can require a basis tending to the identity and impose the universal property for maps with that convergence condition; alternatively the pro-$p$ completion of the abstract free group represents unrestricted maps from an arbitrary set. These infinite-rank conventions should not be conflated when interpreting generator cardinalities.

The inverse limit of the quotients of an abstract group by its normal subgroups of finite $p$-power index. Maps to pro-p groups factor uniquely through the canonical dense image in this completion. Unlike profinite completion, only finite p-groups are used.

A pro-p group whose closed commutator subgroup lies in its closed power subgroup $G^{p^{ϵ}}$, with $ϵ=1$ for odd $p$ and $ϵ=2$ for $p=2$. This is the profinite analogue of a powerful p-group. Finitely generated torsion-free examples are exactly the uniform pro-p groups. The additive group of a finite product of p-adic integers is an abelian example.

If a pro-$p$ presentation on $d$ generators and possibly infinitely many relators $r_i$ satisfieswith the sum convergent and $D$ the Magnus degree of a free pro-p group element, then the presented pro-p group is infinite. The Golod-Shafarevich inequality forces its completed group algebra to be infinite-dimensional, which would be impossible for a finite group. This formulation applies to relators whose degrees tend to infinity and is suitable for enumerated power-relation constructions.

Over $k={\mathbb{F}}_p$, the Nottingham group is $\mathcal{N}(k)=\{f(t)=t+{\sum }_{i\ge 2}{a}_i{t}^i:a_i\in k\}$ with composition and the $t$-adic topology. Equivalently it is the group of continuous $k$-automorphisms of the Laurent series field $k((t))$ that act trivially on $(t)/(t^2)$. With ordinary series composition, the isomorphism sends $f$ to the automorphism $h(t)\mapsto h(f^{-1}(t))$; this inverse avoids reversing the group product. It is a pro-p group, recovered from its finite truncations, and contains elements of arbitrarily large finite $p$-power order.

For a pro-p ring $R$, an $R$-analytic pro-p group is a pro-p group equipped with an analytic manifold structure locally modelled on open subsets of a finite free $R$-module, such that multiplication and inversion are given locally by convergent power series. The number of local coordinates is its analytic dimension. The Nottingham group admits no such structure over any pro-$p$ ring: its hereditary just infiniteness, unbounded finite cyclic torsion and sub-base-three derived-series growth exclude both alternatives in the analytic just-infinite structure theorem.

A closed subgroup of a profinite group that is a pro-p group with the induced topology. Its images in finite continuous quotients are p-groups.

For a dilute well-mixed particle current of conserved bulk volume $W$, triangular section $\lambda h^2$, constant particle settling speed $w_s$ and initial reduced gravity $g_0^{\prime }$, put $s=\sqrt{\varphi /{\varphi }_0}$. The balances give $\dot{X}=Bs{X}^{-1/4}$ and $\dot{\varphi }=-2w_s\varphi /h$, where $B=\mathrm{Fr}\sqrt{g_0^{\prime }}(W/\lambda {)}^{1/4}$. Eliminating time gives $s=1-(X/R{)}^{7/4}$ and the displayed limiting runout. It is approached at infinite time. Early spreading has the inertial $t^{4/5}$ exponent, while settling limits the eventual reach.

With constant buoyancy and no settling or entrainment, the box volume is proportional to $h^2\ell$. Thus $h=h_0\sqrt{x_0/\ell }$ for an initial lock length $x_0$. A gravity-current front condition gives $\dot{\ell }=F\sqrt{g^{\prime }{h}_0}(x_0/\ell {)}^{1/4}$, so ${\ell }^{5/4}$ increases linearly with time. This is a late-time integral approximation; it does not reproduce the initial rarefaction or the detailed nose.

Assume independent stationary Compound Poisson process inputs, preemptive-resume priority for voice, first come first served within each class, capacity $C$, a shared buffer of $B$ units of work, packet sizes bounded by $s_{\max }<B$, and voice delay gate $D>0$. Write $B_0=B-s_{\max }$ and $K=CD$. For $n_v,n_d$ flows with per-flow effective bandwidths $a_v,a_d$, the continuous-time exponential workload bound givesprovided $n_v{a}_v({\theta }_T)+n_d{a}_d({\theta }_T)\le C$ and $n_v{a}_v({\theta }_V)\le C$. To prove this, couple with the unthinned infinite-buffer aggregate and voice queues driven by the same offered packets. Dropping work reduces both the total and voice workloads under this service discipline. A buffer-fit failure implies total workload greater than $B_0$, and a voice-gate failure implies voice workload at least $K$. Apply the two bounds and a union bound; Poisson arrivals see time averages converts stationary prearrival probabilities to offered-packet loss fractions. For target losses ${\ell }_d,{\ell }_v$, choose $0<{\eta }_B<\min ({\ell }_d,{\ell }_v)$ and ${\eta }_V={\ell }_v-{\eta }_B$, then setThe two effective bandwidth constraints, together with strict aggregate mean-load stability, define a sufficient integer admission region. Voice loss includes the shared-buffer term, even though data packets have lower service priority.

For a regular curve with positive curvature of a space curve, the Frenet-Serret formulas give $dT/ds=\kappa N$. If the unit tangent vector has constant component along a fixed vector $a$, differentiation of $T\cdot a$ gives $\kappa N\cdot a=0$. Hence every principal normal vector is perpendicular to that direction. Nonzero curvature is needed to define the principal normal.

If two principal moments of inertia coincide, for example $I_1=I_2\ne I_3$, every direction in their two-dimensional eigenspace is a principal axis. The perpendicular axis is determined, but there is no privileged orthogonal pair within that plane. Rotating the pair in the plane leaves the diagonal inertia tensor unchanged. If all three principal moments of inertia coincide, ${\mathbf{I}}_O=I_{0}1$ and every axis through $O$ is principal. This follows from the eigenvalue multiplicity and eigenspace, rather than from a failure of diagonalization. ^[ref]

Equal principal moments of inertia do not prove that a body's outline has continuous rotational symmetry. For example, six equal point masses at $(\pm a,0,0)$, $(0,\pm a,0)$ and $(0,0,\pm a)$ have ${\mathbf{I}}_O=4m{a}^{2}1$. This discrete layout has the same axial moment of inertia in every direction, although its actual mass distribution changes under most rotations. Conversely, even when all three principal moments of inertia are distinct, principal axes are unoriented lines: reversing an eigenvector's sign gives the same axis. Near repeated eigenvalues, small measurement errors can rotate fitted principal axes substantially within the nearly degenerate eigenspace, so quoted directions need an uncertainty assessment as well as quoted principal moments of inertia.

For strictly positive principal moments of inertia and a fixed rotational kinetic energy $K_0>0$, the allowed angular velocity components form an ellipsoid:Its semiaxis lengths $s_i$ have units of angular speed. The longest semiaxis corresponds to the smallest principal moment of inertia: that direction permits the greatest angular speed for the same rotational kinetic energy. This is a surface in angular velocity space, not the physical outline of the body. If one principal moment of inertia vanishes in an ideal line-mass model, this constant-energy surface is unbounded in that direction and is no longer an ellipsoid.

The rotational energy ellipsoid gives the geometry behind Poinsot's ellipsoid, but a constant-energy surface alone does not determine a free-rotation trajectory. A torque-free trajectory must also satisfy the conserved angular momentum constraint and the Euler equations for a torque-free rigid body.

For $N\ge 2$, the effective principal congruence subgroup is torsion-free. Its modular curve cover of $X(1)$ has analytic ramification indices $2,3,N$ above the two elliptic orbits and the modular cusp. If $\mu$ is its projective index, the numbers of points above these three values are $\mu /2,\mu /3,\mu /N$. Applying the Riemann-Hurwitz formula gives $2g-2=\mu (1/6-1/N)$. Here $\mu =6$ for $N=2$ and $\mu =N^3{\prod }_{p\mid N}(1-p^{-2})/2$ for $N\ge 3$; level one is the identity cover.

The population principal components are centered linear compounds along an orthonormal eigenbasis of the covariance matrix, ordered by decreasing eigenvalue. They are uncorrelated and have variances ${\lambda }_j$. Each successive direction maximizes variance subject to unit length and orthogonality to earlier directions. Uncorrelatedness does not imply independence unless additional distributional assumptions hold.

For positive marginal variances, standardizing each variable to variance one converts covariance-based PCA on $Z$ into correlation-based PCA on $X$. This makes the analysis insensitive to the original measurement units but changes the optimization criterion. Standardization gives small-variance variables the same marginal weight as large-variance variables; it is a modelling choice, not an automatic improvement.