The field with two elements is ${\mathbb{F}}_2=\mathbb{Z}/2\mathbb{Z}$, with elements $0,1$ and arithmetic modulo two. Its only nonzero element is its own multiplicative inverse; in particular, $1+1=0$ and $-1=1$. These rules satisfy the field axioms.

Interpreting $0,1$ as bits, addition is exclusive-or and multiplication is logical conjunction. For instance, $(1,0,1)+(1,1,0)=(0,1,1)$ in the vector space ${\mathbb{F}}_2^3$. This makes linear algebra over the field with two elements useful for binary linear codes and parity constraints.

For integers $s\ge 3$ and $n\ge 1$, the $n$th $s$-gonal number isThese figurate numbers begin at one, and their successive increments form an arithmetic progression:Summing these increments proves the formula. For $s=3,4,5,6$, it gives triangular numbers, square numbers, pentagonal numbers and hexagonal numbers. For example, the first three pentagonal numbers are $1,5,12$, whose successive increments are four and seven. Increasing the number of polygon sides changes the step between increments while keeping this common quadratic structure.

The $n$th square pyramidal number counts a square-based stack with layers containing $1^2,2^2,\dots ,n^2$ objects:For example, $S_3=1+4+9=14$. To prove the formula, let $F(n)=n(n+1)(2n+1)/6$. Direct algebra gives $F(n)-F(n-1)=n^2$ and $F(0)=0$, so mathematical induction identifies $F(n)$ with the sum of squares.

Differences $S_b-S_{a-1}$ give sums over consecutive square numbers. The leading term $n^3/3$ shows cubic growth of the total, even though the largest layer has only $n^2$ objects.

A transistor whose active semiconductor channel is a thin film on a supporting substrate. Semiconductor purity, channel continuity, gate dielectric and contacts jointly determine switching; a high intrinsic charge carrier mobility alone does not guarantee a uniform transistor array.

With the sum convention, the left factorial of a nonnegative integer $n$ is $!n={\sum }_{k=0}^{n-1}k!$, with $!0=0$. The first values are $0,1,2,4,10,34$. Its recurrence relation $!(n+1)=!n+n!$ follows by separating the last summand. The notation $!n$ is also sometimes used for a derangement count, so the defining convention matters: the sum gives $!4=10$, whereas there are nine derangements of four labels.

The superfactorial of a nonnegative integer $n$ is the product ${\prod }_{k=1}^{n}k!$. This convention differs from other similarly named iterated factorial operations, so the product definition should be stated.

A mathematical expression given by a finite combination of specified operations and standard functions, rather than an unevaluated recurrence relation or an unbounded procedure. Which functions are admitted is part of the convention. For example, the sum of the first $n$ positive integers has the closed-form expression $n(n+1)/2$.

The mean number of planets per surveyed star in a specified parameter bin. It need not equal the fraction of stars hosting at least one such planet. Inferring it from transits requires accounting for geometric transit probability, transit survey completeness and false positives.

A Cambrian fossil-bearing succession in Canada preserving diverse animal anatomy, including Metaspriggina.

An exceptionally preserved early Cambrian biota from China, including Haikouichthys and Myllokunmingia.

Behaviour or control that reduces the success of selfish actions by other group members. Selection on policing depends on its costs, effects on productivity and genetic or direct benefits to the policing party.

Differences between parental and offspring strategies favoured by selection, including allocation and duration of care. Costs imposed on other offspring have different weights in the parties’ fitness accounts.

Competition among offspring over resources, care or other opportunities. It can interact with parent-offspring conflict by redistributing a fixed supply and altering total parental provisioning.

Differences in the outcomes favoured by selection on males and females, for example over mating, fertilization or parental care. Shared benefits from reproduction do not imply identical optima.

Conflict among genetic elements whose transmission or expression favours different outcomes despite residence in one organism. Organismal performance alone is not the relevant fitness currency for every element.

Repeated removal of high-energy trapped atoms while lowering the energy cutoff as the sample cools. Elastic rethermalization replenishes the tail. The useful outcome is increased phase-space density of an atomic gas, not temperature reduction alone.