The peptide backbone dihedral angles are $\varphi =\angle (C_{i-1}^{\prime },N_i,C_{\alpha i},C_i^{\prime })$ and $\psi =\angle (N_i,C_{\alpha i},C_i^{\prime },N_{i+1})$. Rotation is restricted by ring closure, peptide geometry and steric contacts.

A lithographic mask that controls optical phase as well as transmission to improve useful interference contrast. Benefits and phase conflicts depend on the pattern.

Illumination using noncentral pupil directions to place useful mask diffraction orders inside a projection lens. Grating orientation, coherence and process window affect the benefit.

Pattern-dependent modification of mask edges or assist features to compensate the finite imaging and pattern-transfer process. It is assessed by the resulting printed dimensions, not by similarity of the mask to the desired layout.

Splitting a final pattern between separate lithographic steps to reduce the feature density that each exposure must resolve. The steps need a suitable intervening fixation, transfer or spacer process. Minimum single-feature width, pattern decomposition and overlay remain limitations.

A testing strategy that pools samples to determine which individuals satisfy a binary condition. A negative pool rules out every member, whereas a positive pool requires more information to identify the affected individuals. Conditional probabilities and dynamic programming can determine an efficient adaptive strategy under a specified model.

Poissonization replaces a fixed number of independent observations by an independent Poisson count. It often turns dependent category counts into independent ones. Let labels have probabilities $p_1,\dots ,p_m$ and draw $N\sim \mathrm{Poisson}(t)$ independent of those labels. The category counts $C_i$ then satisfyIndeed their joint Probability generating function is $\mathbb{E}({\sum }_i{p}_i{z}_i{)}^N=\exp [t({\sum }_i{p}_i{z}_i-1)]={\prod }_i\exp [t{p}_i(z_i-1)]$. This proves both the marginal laws and independence. Conditioning their sum back to $N=n$ recovers the Multinomial distribution for $n$ observations.

A rate-$\lambda$ Poisson process gives a dynamic version. Independently marking its arrivals by labels splits it into independent Poisson processes of rates $\lambda p_i$. The first arrival of each label is an independent exponential random variable, which simplifies coupon-collection calculations. Returning to a fixed-sample problem requires conditioning or a separate comparison argument; merely changing a clock does not preserve every probability. For an integrable stopping count determined by the labels and independent clock intervals, Wald's equation gives expected elapsed time equal to the expected count divided by $\lambda$.

Random number generation produces values intended to follow a prescribed probability distribution. A physical source uses a measured uncertain process; a pseudorandom number generator deterministically expands a seed into a reproducible sequence. For a uniform integer on $\{0,\dots ,m-1\}$, reducing a uniform $b$-bit integer modulo $m$ is biased unless $m$ divides $2^b$. Rejection sampling fixes this by accepting only values below $m\lfloor 2^b/m\rfloor$ before taking the remainder. For $b=3,m=3$, accept $0,\dots ,5$: each residue then has exactly two preimages.

A Fibonacci primitive root modulo an odd prime $p$ is a primitive root modulo n $g$ satisfying $g^2\equiv g+1(\mathrm{mod}p)$. Its powers obey $g^{n+2}=g^{n+1}+g^n$, so the same multiplication that generates all nonzero residues also follows a Fibonacci recurrence.

For example, $3$ works modulo five: $3^2\equiv 3+1$ and its powers $1,3,4,2$ exhaust the nonzero residues. Solving the quadratic condition alone is insufficient; the multiplicative order must also equal $p-1$. Away from $p=5$, the quadratic has roots only when its discriminant five is a quadratic residue modulo $p$.

A prime number for which every cyclic rotation of its numerical digits is also a prime number. In decimal representation, $197$ is an example because $197,971,719$ are all prime numbers. A multi-digit circular prime can contain only $1,3,7,9$: any other digit can be rotated into the units position and force divisibility by $2$ or $5$.

A prime number that remains prime after successively deleting digits from a specified end of its positional notation. Left truncation, right truncation, and requiring both are distinct conditions. For example, $3797$ is truncatable from both ends: $3797,797,97,7$ and $3797,379,37,3$ consist entirely of prime numbers.

A prime number expressible as $(x^3-y^3)/(x-y)=x^2+xy+y^2$ for specified consecutive or nearly consecutive positive integers. The first kind has $x=y+1$ and therefore the form $3y^2+3y+1$; the second kind has $x=y+2$ and therefore the form $3y^2+6y+4$. The first kind arises when a difference of two cubes is itself prime: the factorization $x^3-y^3=(x-y)(x^2+xy+y^2)$ forces $x-y=1$ for $x>y>0$.

Two primes whose difference is $6$, for example $5$ and $11$. The name refers to the Latin word for six. The primes need not be consecutive.

A prime quadruplet is a set of four primes $p,p+2,p+6,p+8$. For example, $5,7,11,13$ form a prime quadruplet. Testing candidates uses integer congruences to discard starting values for which one of the four entries has a small prime factor.

A prime constellation is a pattern of nearby primes specified by fixed offsets, such as $\{p,p+2\}$ for a twin primes or $\{p,p+2,p+6,p+8\}$ for a prime quadruplet. Some authors reserve the term for especially densely packed admissible patterns, so that convention should be stated.

A finite offset set is admissible if it does not occupy every residue class modulo any prime. Otherwise every translated pattern has a member divisible by that prime, obstructing prime values except for small exceptions. For instance, the offsets $0,2,4$ cover all classes modulo three, explaining why three primes in that pattern can only be $3,5,7$. Admissibility is necessary, but does not itself prove infinitely many realizations.

The prime gap $g_n$ is the difference $p_{n+1}-p_n$ between consecutive prime numbers. For the primes $2,3,5,7,11$, the first gaps are $1,2,2,4$. Every gap after the first is even because both endpoint primes are odd.

Prime gaps are unbounded: for any $m\ge 2$, the consecutive integers $m!+2,\dots ,m!+m$ are all composite, since each is divisible by its corresponding summand. Thus there are arbitrarily long prime-free intervals. This elementary argument proves existence without determining how large gaps typically are near a specified bound.

A prime counting algorithm that evaluates $\pi (x)$ by recursively counting integers not divisible by the first few primes and correcting for products of larger primes. A small number-theoretic sieve and cached recursive values allow it to handle inputs much larger than the precomputed table.

Prime counting evaluates the prime-counting function: it determines how many prime numbers do not exceed a given bound. A number-theoretic sieve can construct the whole table up to a moderate bound, while methods such as the Lehmer prime-counting algorithm group whole classes of integers to evaluate much larger arguments without listing every prime number.

The prime index of a prime number $p$ is its position in the increasing sequence $p_1=2,p_2=3,p_3=5,\dots$. Thus the index of $11$ is $5$, and the prime-counting function satisfies $\pi (p_n)=n$.

Indexing begins at one here. An implementation using zero-based array positions must subtract one. The index describes ordering among primes, whereas a prime gap measures the numerical distance between neighboring primes. For nonprime $x$, $\pi (x)$ still counts primes up to $x$, but $x$ itself has no prime index.