For an $n\times n$ matrix $A$, Ryser's formula for the permanent isExpanding a product chooses one column for each row. The alternating subset sum cancels each choice that fails to use every column, by inclusion-exclusion; the surviving choices are exactly the permutations in the permanent.

For a two-by-two matrix, the formula becomes $(a+b)(c+d)-ac-bd=ad+bc$. Maintaining row sums while enumerating subsets in Gray-code order gives an $O(n{2}^n)$ arithmetic algorithm, much smaller than enumerating $n!$ permutations, although still exponential in $n$.

Relative reluctance of a heavier main-group ns electron pair to participate in the higher oxidation state. Retained pairs can nevertheless influence structure through mixing and stereochemical activity.

A conventional measure of atomic extent. Covalent, metallic, ionic and van der Waals definitions differ, so one universal hard-sphere radius should not be assumed.

A circular elastic layer of thickness $h\ll a$ bonded between rigid faces develops shear strain $r\varphi /h$ under a relative twist $\varphi$. Integrating the shear stress times its moment arm gives $K_V=G{J}_p/h$. A small relative tilt produces longitudinal strain $y\theta /h$; when lateral deformation is constrained, the corresponding normal traction gives $K_H=MI/h$. The polar second moment of area is $J_p=\pi a^4/2$ and the diametric second moment of area is $I=\pi a^4/4$. Edge relaxation limits this thin-layer approximation, especially in a nearly incompressible material.

The dianion ${\mathrm{C}}_8{\mathrm{H}}_6^{2-}$, with ten pi electrons over the fused framework. The two overlapping five-membered rings must not be counted as two independent cyclopentadienyl donors.

The monoanionic ${\mathrm{C}}_8{\mathrm{H}}_7^{-}$ coordination ligand obtained by removing one proton from dihydropentalene. An aromatic five-membered portion can bind a metal in an eta-five mode while the fused ring retains an uncoordinated alkene.

A ${\mathrm{C}}_8{\mathrm{H}}_8$ hydrocarbon with two more hydrogen atoms than pentalene; double-bond isomers are possible. One and two deprotonations give chemically distinct hydropentalenide and pentalenide charge states.

Two metal fragments attached to the fused ring system are syn when they lie on the same face and anti when they lie on opposite faces. A direct metal–metal bond between adjacent fragments generally requires the syn arrangement.

For a small, rapid horizontal displacement $a\cos \omega t$, the leading fast angular excursion is $\xi =-(a/l)\cos \Theta \cos \omega t$. Correlating that excursion with the driving torque gives the averaged force $\ddot{\Theta }=-(g/l)\sin \Theta +(a^2{\omega }^2/2l^2)\sin \Theta \cos \Theta$. Its effective potential is $-mgl\cos \Theta +m{a}^2{\omega }^2{\cos }^2\Theta /4$. The downward state loses linear stability above $a\omega =\sqrt{2gl}$ and two tilted minima appear. Small $a/l$ and slow dynamics compared with $\omega$ are required; horizontal and vertical support vibration produce different effective potentials.

The Diophantine equation $x^2-D{y}^2=-1$, usually with a nonsquare positive integer $D$ and integer unknowns. It need not have a solution. For $D=2$, multiplying $u+v\sqrt{2}$ by $3+2\sqrt{2}$ preserves norm $-1$ and gives $(u,v)\mapsto (3u+4v,2u+3v)$.

For $x^2-D{y}^2=1$ with nonsquare positive integer $D$, the solution with $x>1$, $y>0$ and smallest $x$ is the fundamental solution. It generates other positive solutions through powers of $x+y\sqrt{D}$ and can be found using continued fraction convergents of $\sqrt{D}$.

The temperature at which a defined primer duplex is half dissociated under specified conditions. Sequence, salt and concentration matter; a simple composition rule is an approximation rather than a complete assay design.

When the radiofrequency drive is much faster than the Paul trap secular motion and Paul trap micromotion is small, averaging the oscillatory force produces the displayed effective potential. The electric field here is its peak oscillating amplitude. It is an energy, not a static electrostatic potential, so confinement does not contradict the static electrostatic no-trapping theorem. Excess Paul trap micromotion from displacement from the radiofrequency null is distinct from the intended slow Paul trap secular motion.