A pycnometer makes the distinction between mass and mass density experimentally tangible. NIST's 1994 report describes a glass vessel with filling capillaries, wire plugs and a covered overflow bowl in a controlled water bath. The operators soaked a pure-water filling to establish the bath temperature, dried the outside, then left it overnight in the balance enclosure to reach thermal equilibrium before weighing. Empty and full force-restoration balance readings, together with known water mass density, calibrated its capacity at the bath temperature. A tin-oxide coating suppressed stray electric charge; an electrometer detected none during use. The authors distinguish bath and weighing temperatures and correct for buoyancy and changes of displacement volume. Capacity and water-reference uncertainties carry into later liquid mass density determinations. Their pycnometer data were not yet analyzed, so this account provides no measured density or achieved precision.^[ref]

The ideal accounting behind this measurement calibration can be derived without numerical observations. Let $m_0,m_w,m_x$ denote corrected masses of the empty vessel, water-filled vessel and sample-filled vessel. At the same filling temperature, with known reference water mass density ${\rho }_w$ and reproducibly filled volume $V_c$,The empty-vessel subtraction removes vessel mass; the water filling sets volume. Doubling $V_c$ for the same uniform sample doubles the sample mass increment but preserves ${\rho }_x$. Uncorrected balance indications, trapped bubbles, incomplete filling or unequal filling temperatures break the simple identification with corrected mass and a common $V_c$. This explains why replication is a statement about comparable states rather than arbitrary instrument readings.

Consider two hypothetical parcels at the same temperature $300\mathrm{K}$, kept separated with negligible partition mass and excluded container volume. Parcel A has mass $2\mathrm{kg}$ and volume $0.002{\mathrm{m}}^3$; parcel B has mass $3\mathrm{kg}$ and volume $0.006{\mathrm{m}}^3$. They may be different materials. Their individual mass densities and the composite mass density areThe composite mass density is total material mass divided by total occupied volume; it need not be the local mass density anywhere. Equal temperature does not make these parcels replicas. In particular, the unweighted average $750\mathrm{kg}{\mathrm{m}}^{-3}$ is wrong. The volume fractions are $1/4$ and $3/4$, givingWith mass fractions $2/5$ and $3/5$, the equivalent rule is a weighted harmonic mean:This follows directly by writing $V_i=m_i/{\rho }_i$ and dividing total volume by total mass; it is also the mass-weighted average of specific volumes. An unweighted average of mass densities is valid when the parcel volume values are equal, not merely when their temperatures are equal.

If a partition is removed and different substances mix, the final volume need not equal the sum of initial volume values. A new temperature, thermodynamic phase or composition may also change the local mass density. Use the actual final volume in $\rho =m/V$; the pre-mixing weighted identity cannot supply it. Even unchanged temperature alone does not guarantee unchanged volume.

For $t$ tetrahedra, count the seven disk types in each tetrahedron. Across each glued face the counts of each of its three normal arc types must agree. These are homogeneous linear equations with integer coefficients. An admissible solution also has at most one nonzero quadrilateral type in each tetrahedron. Matching the arcs in their boundary order reconstructs the normal surface up to an isotopy preserving the tetrahedra.

A rotating rod can draw a liquid with appreciable viscoelasticity inward and raise its free surface around the rod. Tensile normal stresses along curved flow lines produce an inward hoop force, competing with inertia and gravity. It is a manifestation of elastic normal-stress differences, rather than merely a changed scalar shear viscosity.

A polymeric extrudate can expand sideways after leaving a die as the normal-stress differences and stored deformation generated upstream relax. Axial elastic recovery contributes to transverse swelling in an incompressible flow. A Newtonian extrudate can also swell through rearrangement of its velocity profile, so observing swelling alone does not isolate the elastic contribution.

A chief series is a strict normal series of a group with no proper normal-series refinement. Its factors are minimal normal subgroups of the appropriate quotient groups. A chief factor of a finite soluble group is elementary abelian, whereas a composition factor of a finite soluble group is cyclic of prime order. A chief factor can therefore have order larger than a prime.

For conditionally independent annual averages $X_j\mid \theta \sim N(\theta ,v/m_j)$ and prior $\theta \sim N(\mu ,{\sigma }^2)$, the posterior mean is $Z{\overline{X}}_w+(1-Z)\mu$, with ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$. Completing the quadratic likelihood gives posterior precision ${\sigma }^{-2}+W/v$. This is an exact Bayesian credibility version of the Bühlmann–Straub credibility estimate. Increasing the between-risk prior variance increases the experience weight; increasing process variance decreases it. The next total pure premium is $m_{n+1}$ times the estimated per-life mean.

In an additive category with finite categorical limits and colimits in which every epimorphism is a conormal epimorphism, closure of normal monomorphisms under composition forces every monomorphism to be normal. To prove this, write a monomorphism $m$ as $kg$, where $k=\ker \mathrm{coker}m$, and put $l=\ker \mathrm{coker}g$. If $kl=\ker t$ were normal, $tm=0$ would force $tk=0$, hence $l$ would be an isomorphism. This makes $\mathrm{coker}g=0$, so $g$ is epic as well as monic. An epic monomorphism which is a cokernel is invertible, forcing $m$ to be normal. Contrapositively, any failure of normality provides two normal monomorphisms whose composite is not normal.

With convention $t^{-1}at=\varphi (a)$, choose representatives of the right cosets of $A$ and $B$ in the base group, with identity representing each subgroup. Normal strings have the form $g_0{t}^{{ϵ}_1}{r}_1\cdots t^{{ϵ}_n}{r}_n$, where $r_i$ represents a coset of $B$ for ${ϵ}_i=1$ and of $A$ for ${ϵ}_i=-1$, and $r_i\ne 1$ when adjacent exponents are opposite. Left multiplication by base elements or stable letters defines permutations of these strings. They satisfy the HNN relations. Applying the induced action to the empty string distinguishes different normal strings, proving uniqueness without presupposing Britton's lemma.

// Target: geometry-and-topology.bigb

For normal $x$, the closed unital C-star subalgebra generated by $x,x^{\ast }$ is commutative. When the ambient spectrum is real, its complement is connected. Membership of the resolvent in this subalgebra is both open and closed on that complement, and holds at large modulus by the Neumann series. Thus the subalgebra spectrum is the same real set. Its isometric Gelfand transform takes $x$ to a real-valued function and $x^{\ast }$ to its conjugate, proving equality. Normality is necessary: the nonzero matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ has spectrum $\{0\}$ but is not self-adjoint.

The unital continuous functional calculus has image $C^{\ast }(1,x)$, the norm closure of star polynomials including constants. The smallest closed star subalgebra $C^{\ast }(x)$ need not contain the ambient identity: for $x=0$ it is zero. In a nonunital algebra, apply the unital calculus in the C-star unitization and retain functions vanishing at zero to recover $C^{\ast }(x)$ inside the original algebra.

Since a normal C-star element commutes with its adjoint, words in those two generators can be rearranged into the displayed finite sum. Including the constant term generates the unital star algebra; omitting it generates the nonunital star algebra. Under continuous functional calculus, such a sum is the ordinary function ${\sum }_{j,k}{c}_{jk}{z}^j{\overline{z}}^k$ on the spectrum. The Stone-Weierstrass theorem makes these functions uniformly dense.

Choose an orientation of a vector space on $T_{q}N$. The quotient identification for a regular value gives a smoothly oriented normal bundle along $F=f^{-1}(q)$. An oriented normal frame followed by a frame of $TF$ defines the latter's orientation through the ambient orientation. Changing normal lifts adds only tangent components, so the determinant sign is unchanged. No orientation on all of $N$ is required.

A continuity correction shifts a discrete probability boundary by half a lattice spacing before using a continuous normal approximation. For an upper-tail event $T\ge t$ on a unit lattice, the approximation is $1-\Phi ((t-\frac{1}{2}-E(T))/\sqrt{\mathrm{Var}(T)})$. Lower tails use the boundary $t+\frac{1}{2}$. The correction is used in approximate Wilcoxon signed-rank tests and Wilcoxon rank sum tests; tied ranks also require the appropriate variance adjustment.

For a twice continuously differentiable regular curve describing motion, let $v=\Vert r^{\prime }(t)\Vert >0$, $T=r^{\prime }/v$, and let $N$ be the principal normal vector where the curvature $\kappa$ is positive. Differentiating the velocity vector $r^{\prime }=vT$ givesThe tangential component changes the speed; the normal component changes the direction of the velocity vector. A particle can therefore have nonzero acceleration at constant speed, as in circular motion with constant speed. The formula follows from $dT/dt=(dT/ds)(ds/dt)=\kappa vN$ and the product rule. When $\kappa =0$, the normal contribution vanishes even though the principal normal vector is undefined there. ^[ref]

In dimensionless coordinates, take $r(t)=(t,t^2)$. At $t=1$,The resulting vector components areBoth increasing speed and turning contribute to the same acceleration.

For mutually orthogonal projectors summing to the identity, a pinching map discards coherences between their ranges while preserving each diagonal block. It is a quantum channel with Kraus operators $P_i$, and a unital quantum channel. A state is fixed by pinching exactly when it commutes with every projector.

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

Diagonalize two density operators with eigenvalues $r_i,s_j$, and put $q_{ij}=|⟨i|j⟩{|}^2$. This overlap matrix has row and column sums one. The scalar inequality $u\ln (u/v)\ge u-v$ yields $D(\rho \Vert \sigma )\ge {\sum }_{ij}{q}_{ij}(r_i-s_j)=0$ in natural units; division by $\ln 2$ gives bits. Equality forces $r_i=s_j$ whenever $q_{ij}>0$, so $\sigma |i⟩=r_i|i⟩$ and $\rho =\sigma$. Zero eigenvalues use limits, and failure of support inclusion gives infinite quantum relative entropy.

An irreducible Markov chain is recurrent exactly when every nonnegative superharmonic function for a Markov chain is constant. In the recurrent case, absorb the chain at $j$. The inequality $g_i\ge ({\tilde{P}}^{n}g{)}_i\ge g_j{\mathbb{P}}_i({\tau }_j\le n)$ and almost-sure hitting give $g_i\ge g_j$; exchange $i,j$. In the transient case, the hitting probability of a state $j$ is one there and satisfies the superharmonic inequality, but its one-step average at $j$ is the strictly smaller return probability. It is a nonconstant example.

For $\mathrm{Re}s>1$ this Eisenstein series also equals ${\sum }_{\gamma \in {\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})}(\mathrm{Im}\gamma \tau {)}^s$, where ${\Gamma }_{\infty }$ consists of translations and their negatives. It is modular invariant. The nonholomorphic Eisenstein series summed over all nonzero integer pairs instead equals $2\zeta (2s)E(\tau ,s)$: separate each pair into a positive integer times a primitive pair. This normalization matters in Rankin–Selberg unfolding.