For nonzero angular momentum $L$, put $u=1/r$ and $\lambda =1+2mB/L^2$. The Binet equation is $u_{\theta \theta }+\lambda u=0$. Positive $\lambda$ gives cosine scattering branches, zero gives linear reciprocal radius, and negative gives exponential combinations, with energy distinguishing sinh, exponential and cosh branches. Only intervals with $u>0$ are physical.

Transfer of inherited variation between lineages through hybridization and subsequent backcrossing, followed by a selective benefit in the recipient. Shared ancestry at a region and evidence of its benefit are separate requirements.

A ligand-regulated protein family whose members often act as DNA-binding regulators of transcription. Subcellular distribution and activation mechanisms vary between members.

A projection of the intestinal lining that increases exposed absorptive surface. Mammalian small-intestinal villi must be distinguished from microscopic microvilli on individual epithelial cells and from the normally non-villous colon.

An epithelial invagination housing proliferative populations in the adult mammalian intestinal lining. Crypt structure, supporting cells and developmental timing vary with region and species.

The small intestine is the gut region between stomach and large intestine, divided into duodenum, jejunum and ileum. It performs extensive digestion and absorption. Surface organization and segment-specific transport link its anatomy to the processing of different nutrients.^[ref]

A lysosome-related organelle of the Caenorhabditis elegans intestine. Loss of granule formation can alter a gut-marker assay without eliminating the intestinal founder lineage.

A cell at the crypt base, including actively cycling Lgr5-expressing stem cells interspersed with supporting cells in the mouse small intestine.

A functional description for a cell that can restore intestinal stem-cell output after damage. It does not by itself establish a permanently distinct, uniquely marked quiescent lineage.

An intermetallic compound and an early example of a heavy fermion material identified by low-temperature calorimetry and electrical transport.

Use Alice's CHSH axes $\hat{x},\hat{y}$ and Bob's $(\hat{x}+\hat{y})/\sqrt{2},(-\hat{x}+\hat{y})/\sqrt{2}$, with sign pattern $S=C_{11}-C_{13}+C_{31}+C_{33}$. If Eve measures both singlet particles along planar angles $\alpha ,\beta$, the intercept-resend attack correlation gives $S=-\sqrt{2}{\cos }^2(\alpha -\beta )$. Consequently $-\sqrt{2}\le S\le 0$. Randomly mixing axes independently of the users' secret settings preserves this interval by averaging. The bound is specific to these settings and this attack, rather than a general security theorem.

An integer factorization algorithm that iterates a simple map such as $x\mapsto x^2+c\mathrm{mod}n$ and searches for a nontrivial greatest common divisor of $n$ and the difference of two iterates. A collision modulo an unknown prime factor can reveal that factor before a collision modulo $n$. A cycle detection schedule such as Floyd's cycle-finding algorithm limits the number of stored iterates. Unsuccessful choices of seed or map can be retried.

An ordered factorization of a positive integer $n>1$ is an ordered list of factors greater than one whose product is $n$. For example, $6$, $2\cdot 3$, and $3\cdot 2$ are the three ordered factorizations of $6$. Taking successive partial products gives a bijection with strictly increasing divisor chains from $1$ to $n$.

A general-purpose classical integer factorization algorithm combining polynomial relations, arithmetic in number fields and sparse linear algebra to construct congruent squares modulo the input. Its familiar general-input complexity estimate is heuristic and subexponential in the input's value, roughly $\exp (O(n^{1/3}(\log n{)}^{2/3}))$ for $n$ input bits. This is not polynomial in the bit length. Special input structure, small factors and preprocessing can make other algorithms preferable.

Integer arithmetic in which storage grows with the number of numerical digits needed, instead of imposing a fixed word size. It allows exact addition, multiplication, and exponentiation on large integers, subject to available memory; operation costs grow with operand length.

Roman numerals represent integers with symbols such as $I=1$, $V=5$, $X=10$, $L=50$, $C=100$, $D=500$, and $M=1000$. In a common modern convention, most values are added, with designated subtractive pairs such as $IV=4$ and $IX=9$.

For example, $MCMXLIV=1000+900+40+4=1944$. Unlike positional notation, a symbol does not acquire a power-of-ten weight from its position, and the ordinary system has no zero digit. Historical usage varies, so the modern subtractive convention should not be imposed on every inscription.

A cubefree integer is a positive integer divisible by no cube of a prime. Equivalently, every prime exponent in its prime factorization is at most two.

A Hilbert number is a positive integer congruent to $1$ modulo $4$: the first examples are $1,5,9,13,17$. The Hilbert numbers form a multiplicative set, because a product of two integers congruent to $1$ modulo $4$ again has that residue.

This set inherits multiplication from the integers, but restricting its allowed factors changes divisibility questions. For example, $9$ belongs to the Hilbert numbers while $3$ does not. Its ordinary factorization $9=3\times 3$ is therefore unavailable as a factorization into smaller Hilbert numbers. The squarefree Hilbert number definition uses this restricted collection of factors.

For a positive integer $n={\prod }_p{p}^{e_p}$, its cube-free part is $c={\prod }_p{p}^{e_p\mathrm{mod}3}$. Then $n=a^{3}c$ with $a={\prod }_p{p}^{\lfloor e_p/3\rfloor }$, and no prime cube divides $c$. This decomposition is unique by unique factorization.

For example, $72=2^3\cdot 3^2=2^3\cdot 9$, so its cube-free part is $9$. This is different from the radical of an integer, which keeps each prime only once and gives $6$ here. Removing the cube factor simplifies cube roots and identifies integers whose ratio is a rational cube.