The screening parameter $S$ in $Z_{\mathrm{eff}}=Z-S$. Different atomic orbitals can have different shielding parameters.

Probability density within the inner electron shells. Greater penetration can reduce screening and lower an orbital energy; this helps distinguish many-electron s and p levels with the same principal quantum number.

A trait affected by divergent ecological selection that also contributes to assortative mating. Distinguish one pleiotropic trait from tightly linked separate loci.

Divergence of sensory systems and signals associated with the physical or ecological environment, potentially generating mating barriers.

The box-ball system is a discrete cellular automaton with a row of boxes, each either empty or holding one ball. Consider finitely many balls and unlimited empty boxes to the right. During one update, process the original balls from left to right and move each exactly once to the nearest currently empty box on its right. The ordering and the instruction to move each ball only once are part of the evolution rule.

An isolated block of three balls in boxes $\{0,1,2\}$ moves to $\{3,4,5\}$ in one update. More generally, an isolated block of $k$ balls moves $k$ boxes per step, preserving its shape. Interacting patterns have a soliton interpretation: localized structures re-emerge with conserved identities, and the system has further conserved quantities.^[ref] This gives a particularly concrete setting in which a simple integer update rule produces organized wave-like motion.

The forward orbit of a state is the collection of states reached as time advances. For a discrete dynamical system $F:X\to X$,For a flow map ${\Phi }_t$, the analogous set is $\{{\Phi }_t(x_0):t\ge 0\}$ over the times for which the solution exists. The initial state is included; its future accumulation points instead form the omega-limit set.

For example, $F(x)=x/2$ with $x_0=1$ gives $1,1/2,1/4,\dots$. Zero is approached but is not one of those iterates, so the forward orbit differs from its closure. Forward evolution also need not determine a unique past: $F(x)=x^2$ sends both $1$ and $-1$ to $1$. The distinction matters when discussing attraction or invariance for a map that cannot be inverted.

A semiconjugacy from $f:X\to X$ to $g:Y\to Y$ is an intertwining map $h:X\to Y$:In a topological factor-map convention, $h$ is also continuous and onto. Authors using the term for an arbitrary intertwining map should state that weaker convention. If $h$ is a homeomorphism, the relation is a topological conjugacy.

The identity transports every forward orbit. Induction gives $h(f^n(x))=g^n(h(x))$, so iterating in the larger space and then projecting agrees with projecting first and then iterating. The map $h$ can merge different states, so the projected dynamics need not retain all information about the original orbit.

For example, let $f(\theta )=2\theta (\mathrm{mod}1)$ on the circle, $g(y)=4y(1-y)$ on $[0,1]$, and $h(\theta )={\sin }^2(\pi \theta )$. ThenThus the doubling map maps onto the parameter-four logistic map by a continuous semiconjugacy. Since $h(\theta )=h(1-\theta )$, it is not one-to-one and does not give a topological conjugacy by this map.

Count multisets of allowed positive coin values whose sum is a target. Initialize $w(0)=1$ and $w(s)=0$ for $s>0$. For each coin $c$, update $w(s)\leftarrow w(s)+w(s-c)$ in increasing order of $s\ge c$. The outer loop over coins counts each multiset once while allowing repeated use of a coin.

A dynamic programming method that advances a boundary across a structured object and records only the information needed on that boundary. For tilings by pieces of bounded height, a bit mask recording occupied cells in a few frontier rows can determine every legal continuation. Its state space is typically exponential in the width but independent of the total height.

Count or sum integers by processing their numerical digits and merging prefixes that have the same possible continuations. Typical state records include the current position, whether the prefix still equals an upper limit, a remainder, or a bit mask of digits already used. The final answer comes from accepting exactly those states satisfying the desired digit constraints.

A method for counting or weighting configurations by processing successive slices and recording only the information that crosses the current boundary. Compatible boundary states define a matrix of transitions, so repeated slices can be handled by matrix powers. It is especially useful when the boundary is much smaller than the whole configuration.

The coin change problem asks how denominations can form a target amount. Common versions minimize the number of coins, count unordered combinations, or count ordered sequences; these are different problems.

For positive integer denominations, the minimum-count recurrence is $M(0)=0$ and $M(t)=1+{\min }_{c_i\le t}M(t-c_i)$, using infinity for unreachable amounts. For denominations $1,3,4$ and target six, the optimum is $3+3$, whereas a largest-first greedy rule gives $4+1+1$.

For unordered counting with unlimited coins, process denominations outside the increasing-amount loop. Reversing those loops counts ordered sequences because each final coin becomes a separate last-step choice. Stating the counting convention is therefore essential.

An algorithm that uses dynamic programming: it computes and stores solutions to overlapping subproblems, then combines them through a recurrence relation.

Interval dynamic programming stores answers for contiguous index intervals $[i,j]$ and combines shorter intervals. It applies when an optimal object can be split at an internal index while keeping subproblems contiguous.

For matrix-chain multiplication, with matrix dimensions $d_{i-1}\times d_i$, define $C(i,i)=0$ andThe final term is the scalar multiplication cost of combining the two products. Filling intervals in increasing length makes all dependencies available, giving $O(n^3)$ time and $O(n^2)$ storage. The recurrence optimizes parenthesization while retaining the original matrix order.

Frontier dynamic programming processes a graph or grid in an order and remembers only the processed vertices that still meet unprocessed edges, together with the information needed across that frontier. Its complexity is controlled by frontier width rather than merely by the total vertex count.

For counting paths on a strip, the state may need both vertex degrees and which frontier endpoints are connected through the processed region. Two configurations with the same occupied frontier vertices can differ in whether the next edge closes a forbidden cycle. Recording a canonical connectivity partition prevents this loss of information. This is a graph-oriented form of state compression and transfer dynamic programming.

Transfer dynamic programming advances through repeated layers while storing a state on their interface. If $v_j(s)$ counts partial configurations with boundary state $s$, transitions give $v_{j+1}(t)={\sum }_s{T}_{ts}{v}_j(s)$. With identical layers this is repeated multiplication by a transfer matrix.

For binary strings with no consecutive ones, store whether the last digit is zero or one. Appending a zero is allowed from either state; appending a one is allowed only from the zero state. The resulting two-state recurrence gives Fibonacci numbers. The state must retain all information that a later layer can detect, including connectivity when local occupancies alone do not suffice.

Subset DP is dynamic programming indexed by subsets of a finite ground set, usually encoded by bit masks. A recurrence builds a subset from smaller subsets or removes one chosen element. There are $2^n$ subsets of an $n$-element set, so many such methods require exponential space or time despite avoiding factorial enumeration.

For a shortest path visiting a given set, $D(S,v)$ stores the best cost of a path visiting exactly $S$ and ending at $v$. The recurrence $D(S,v)={\min }_{u\in S\setminus \{v\}}(D(S\setminus \{v\},u)+c_{uv})$ retains the endpoint because it controls the next edge cost. Initial states and prohibited edges must be specified separately.