A positive divisor $d$ of a positive integer $n$ is bi-unitary when the greatest common unitary divisor of $d$ and $n/d$ is $1$. The comparison uses unitary divisors, so it differs from requiring the ordinary greatest common divisor to be one.

For $n=p^e$ with $p$ a prime number and $e\ge 1$, consider $d=p^a$, where $0\le a\le e$. The only possible nontrivial common unitary divisor of $p^a$ and $p^{e-a}$ occurs when $a=e-a>0$. Consequently every divisor is bi-unitary except $p^{e/2}$ when $e$ is even.

For example, the bi-unitary divisors of $64=2^6$ are $1,2,4,16,32,64$; only $8$ is excluded. Its unitary divisors are just $1$ and $64$, showing that these two notions differ substantially. For a general positive integer, apply the prime-power condition independently to each prime factor in its prime factorization.

The positive divisors of a positive integer $n$, ordered by divisibility, form a lattice. The greatest lower bound of $a,b\mid n$ is $\gcd (a,b)$, their greatest common divisor; the least upper bound is $\mathrm{lcm}(a,b)$, their least common multiple. Both are again divisors of $n$.

For example, the positive divisors of $12$ are $1,2,3,4,6,12$. The divisors $4$ and $6$ are incomparable under divisibility, but their greatest lower bound is $2$ and their least upper bound is $12$. The bottom and top elements of the whole lattice are $1$ and $n$, respectively. Under prime factorization, the two operations take the coordinatewise minimum and maximum of the prime exponents.

The average shortest path length averages graph distances over distinct ordered pairs of vertices. A nodal value averages distances from one specified vertex to all others. Disconnected pairs require a separate convention.

The rate of accumulation of displacements per atom. Equal final displacement dose reached at different rates can produce different defect populations because diffusion and recovery continue in elapsed time.

Displacing a one-dimensional quantum harmonic oscillator produces a linear force term and a scalar shift relative to the unshifted Hamiltonian. The scalar adds only a common phase. The linear term couples number states separated by one excitation. Arbitrary linear driving of the ground state creates a coherent state; a first-order expansion of that state contains only the vacuum and first excited state.^[ref]

The lattice translation represented by closure failure of a circuit around a dislocation, with a consistent circuit convention. It is a vector, so dislocation reactions conserve its direction as well as its magnitude.

For line tangent $\mathbf{t}$, a screw dislocation has Burgers vector parallel to $\mathbf{t}$, an edge dislocation has it perpendicular, and a mixed dislocation has both components.

Conservative motion within a plane containing both the dislocation line and its Burgers vector. For a nonscrew line their cross product determines the plane normal; a screw line can belong to more than one such plane.

The energy criterion associated with Frederick Charles Frank states that, for dislocations of comparable length with approximately common line-energy factors, a reaction conserving Burgers vector is energetically favourable when the displayed inequality holds. The rule is an energy estimate, not a proof that the geometry or kinetic barriers permit the reaction. Edge/screw character, elastic anisotropy and changed lengths refine it.

A line segment formed by interaction of dislocations from different slip planes. A junction can impede further dislocation glide. Energetic favourability alone does not establish whether the resulting segment is mobile on a particular plane.

A closed dislocation line. Vacancy or interstitial clustering can form a platelet bounded by a loop; its Burgers vector, habit plane and possible stacking fault specify its crystallographic character.

Total dislocation line length per material volume, with units ${\mathrm{m}}^{-2}$. Circular loops with common radius $R$ and number density $N_{\ell }$ contribute $2\pi R{N}_{\ell }$.

Motion of an edge component perpendicular to its glide plane by absorbing or emitting point defects. Unlike conservative dislocation glide, it changes the extra-plane content locally and is coupled to defect transport. Irradiation can drive climb by creating nonequilibrium vacancies and self-interstitials.

A dislocation that accommodates some lattice mismatch at an interface. An initially coherent epitaxial layer or island can instead store elastic deformation; introducing a misfit dislocation trades some strain energy for the defect's formation energy. A projected image must be indexed before a particular spacing discrepancy is assigned to misfit strain.

An order-preserving total dislocation in an ordered crystal, often dissociated into smaller dislocations separated by an antiphase boundary. Two matching $a⟨110⟩/2$ dislocations can restore order in the gamma prime phase.

A dislocation whose Burgers vector is not a full translation of the perfect crystal, so its glide can leave a planar stacking fault. Vector conservation and the actual slip plane constrain reactions and dissociation.

A curved dislocation can be approximated as a line with tension $T_{\ell }$ of order $G{b}^2$. The coefficient depends on elastic constants, character and logarithmic length scales. The simple estimate $T_{\ell }\simeq G{b}^2/2$ gives a pinning bypass stress $Gb/\ell$ for a semicircular segment of free span $\ell$.

A number-theoretic transform is a discrete Fourier transform computed in modular arithmetic. In a finite field with $p$ elements, choose a primitive root of unity $\omega$ of order $L$ dividing $p-1$. The transform above is inverted using ${\omega }^{-1}$ and $L^{-1}$. With sufficient zero padding, it computes polynomial products exactly modulo $p$, avoiding floating-point arithmetic rounding.

cellobiose consists of two glucose units joined by a beta-1,4 linkage. It is a characteristic soluble product in cellulose degradation.

Let $G$ vanish on the boundary of a domain and satisfy $\Delta G(\cdot ,y)={\delta }_y$. Green's second identity givesThus the Poisson kernel is $+{\partial }_{n}G$ for this convention. If instead $-\Delta K=\delta$, then $K=-G$ and the same kernel is $-{\partial }_{n}K$. At the boundary of $x>0$, the outward normal is $-e_x$, so the first convention gives $P=-{\partial }_{x}G$. Changing the delta-source sign or replacing the outward derivative by an inward coordinate derivative changes a sign; changing both leaves the physical kernel unchanged.