solid stress is stress carried by solid constituents of a tissue or material. Compressive solid stress, interstitial fluid pressure and elastic stiffness are distinct quantities that can coexist in a tumour.

Contact mechanics studies how touching bodies deform, transmit forces and separate. Even a ball that appears to touch a flat surface at one point develops a finite contact patch: local elastic deformation spreads the load into a nonuniform pressure distribution. Hertzian contact is a useful ideal model when the surfaces are smooth, the contact patch is small and adhesion and friction can be neglected.

The contact patch is part of the solution, rather than a prescribed loaded area. In nonadhesive contact the nonpenetration constraint prevents overlap, and the normal contact force can push but cannot pull; unilateral contact complementarity joins those requirements. Pressing a spherical probe into a specimen while recording force and indentation tests the contact law and can determine a reduced elastic modulus. Surface roughness, plastic deformation and adhesion require additional modeling, especially when interpreting a measured pull-off force.

A helical symmetry combines an axial rotation and a proportional axial translation. A helix with $z=b\theta$ is unchanged as a set by this transformation. If a mechanical Lagrangian is invariant, the Noether conserved quantity for a mechanical point symmetry is $L_z+b{P}_z$, where the two momenta include all moving components. Pure axial rotation and pure axial translation need not separately be symmetries. The helical symmetry of a potential is a related special case in which the constraint is replaced by an invariant potential.^[ref]

An energy balance accounts for the change of stored energy as inputs minus outputs. For mechanical motion, externally supplied power and dissipative losses must be included before imposing a conserved mechanical energy.

Biosecurity comprises measures that reduce introduction or spread of potentially harmful organisms through specified pathways. In ecology, inspecting transported material can reduce introductions, while restricting movement from an invaded area can reduce further spread.

The pathway matters: stopping one route does not establish that a region is protected if other routes remain active. Surveillance must also distinguish true absence from nondetection, as in assessment of species eradication.

Complete removal of a target population from a defined area, requiring sustained surveillance to distinguish absence from nondetection.

Alternation between shorter and longer chemical bonds along a conjugated system. In the Hückel method, unequal nearest-neighbour couplings split the frontier molecular orbitals. An electronic energy gain must be compared at a stated constraint on the mean coupling and against the geometric distortion cost.

Overlap between adjacent p-type atomic orbitals permits delocalized electronic states across more than one chemical bond. A cyclic path with suitable occupancy can support aromaticity; an external lone pair can act as a donor without contributing an extra independent pair to every ring count.

A 1,6-Conjugate addition attacks the delta carbon of an extended conjugated electrophile. The resulting extended enolate can be protonated at different sites; proton transfer or equilibration can favor a product retaining carbonyl–alkene conjugation.

A nucleophile adds reversibly at the beta carbon to form an enolate, which is functionalized before elimination restores the alkene.

A molecular conformation is a spatial arrangement accessible through motions such as rotations around single bonds, without changing the molecule's connectivity. Distinct conformations can have different potential energies and interconvert at different rates.

For ethane, rotation around the carbon-carbon bond connects staggered and eclipsed conformations. For cyclohexane, ring motions connect chair conformations. A drawing therefore specifies more than a molecular formula: the geometry can control steric interactions, orbital alignment, and chemical reactivity. A sufficiently high interconversion barrier can make particular conformations separately observable.

A conformational form distinguished by rotation about a specified bond. Interconversion can be slow when resonance gives that bond partial multiple-bond character.

configuration interaction retaining a reference determinant and its single and double excitations. Conventional matrix-vector iterations have leading $O(o^2{v}^4)$ work for occupied and virtual dimensions $o,v$, hence $O(N^6)$ when both grow with system size. Truncated CI is not generally size extensive.

A spatially modulated spin density, distinct from a charge density wave. Opposite modulations of spin-up and spin-down occupations can give a periodically varying magnetization while their total charge density stays uniform.

A periodic modulation of electronic charge density, often coupled to a lattice distortion. In a one-dimensional metal, a Peierls distortion with $Q=2k_F$ can open a gap at opposite Fermi points, lowering the occupied electron energy at an elastic energy cost. A charge modulation and a spin density wave are distinct: the latter can modulate spin while leaving the total charge nearly uniform.

Project Euler presents problems combining mathematics with the design of efficient algorithms.^[ref]

A choice of $k$ distinct elements from a set of $n$ elements, with order ignored. Equivalently it is a $k$-element subset. The binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ counts such choices; choosing with repetition instead produces a multiset.

A finite connected union of unit squares on the square lattice, joined along whole sides. Counting conventions may identify shapes under translation, rotation, or reflection, so those conventions must be specified.

A cyclic sequence over an alphabet of size $q$ in which every word of length $n$ appears exactly once as a cyclic window. Its length is $q^n$. The De Bruijn graph converts its construction into an Eulerian cycle problem.

A transfer matrix stores the weights of allowed transitions between finitely many boundary states. Its powers sum weights of compatible sequences of transitions. For a cyclic boundary condition, the matrix trace $\mathrm{tr}(T^n)$ counts weighted closed sequences of length $n$.