For radial acceleration $-A/r^2-B/r^3$, conserved specific angular momentum $l$ and $u=1/r$ reduce the Binet equation to the displayed linear equation. If $l^2>B$, its angular oscillation frequency is $\sqrt{1-B/l^2}$; attraction proportional to $r^{-3}$ changes the apsidal angle. A zero of $u$ ends the physical positive-radius branch and corresponds to escape, not passage to a negative radius. A spatial self-intersection need not be a periodic return because velocity may differ.

An infinite path is a binary function all of whose finite prefixes belong to the tree. Its path space $[T]$ is closed in Cantor space, since failure is witnessed by one finite prefix outside $T$.

A binary tree of finite strings is computable when membership of a finite binary string is decidable. An infinite computable tree need not have any computable infinite path. Removing all nodes without infinite extensions need not preserve decidability.

A modular-addition quantum oracle for a linear function over the three-element finite field reveals its two coefficients in one query. Prepare the answer qutrit in ${\mathrm{QFT}}_3|2⟩$, whose unit shift eigenvalue is $e^{2\pi i/3}$. Quantum phase kickback produces a product of Fourier states labelled by the coefficients; inverse quantum Fourier transforms read both coefficients exactly.

The Bell basis diagonalizes the commuting Pauli operators $Z\otimes Z$ and $X\otimes X$. Two local anticommutations make the joint operators commute. Their signs encode parity and relative phase. The displayed projectors turn the signed eigenvalues into literal zero-or-one bit eigenvalues: even parity and plus phase have bit zero.

Apply a Hadamard gate to the first qubit, then a controlled-NOT gate from the first to the second. This maps a computational-basis state $|ij⟩$ to the Bell state with phase bit $i$ and parity bit $j$. Its inverse uses the same gates in reverse order because each gate is self-inverse but they do not commute. Applying the forward sequence twice is not generally a decoding operation.

If every model's likelihood is multiplied by the same positive factor independent of its parameter or model index, Bayes theorem cancels that factor from the normalized posterior. Fixed-count Bernoulli sampling and sampling stopped at a specified positive head count give such proportional likelihoods for the same total heads and tails. This conclusion uses their actual sampling likelihoods; it is not a claim that every selection or stopping rule can be ignored.

Let $N$ have the positive-integer geometric distribution with success parameter $p\in (0,1)$, and conditionally toss $N$ independent coins with head probability $\theta \in (0,1)$. Exactly one head has conditional probability $n\theta (1-\theta {)}^{n-1}$. Combining this with the prior by Bayes theorem gives the displayed posterior with $r=(1-p)(1-\theta )$; its normalization uses ${\sum }_{n\ge 1}n{r}^{n-1}=(1-r{)}^{-2}$. For $p=\theta =1/2$, the posterior is $9n/4^{n+1}$. It is also a shifted negative binomial distribution, with $N-1$ equal in law to the number of failures before two successes of probability $1-r$.

If $E(A)\ge \theta |A{|}^3$, a subset $B\subseteq A$ satisfies $|B|\ge c_{\theta }|A|$ and $|B-B|\le K_{\theta }|B|$. A dependent random choice argument in the popular sum graph finds many four-edge paths representing each member of the difference set. The Petridis minimal-growth lemma then bounds all higher iterated sumsets and difference sets.

There are absolute $c,C>0$ such that finite sets $A,B$ of size $n$, a bipartite graph of density at least $\delta$, and a restricted sumset of size at most $Kn$ yield subsets $A^{\prime },B^{\prime }$ with $|A^{\prime }|,|B^{\prime }|\ge c{\delta }^{C}n$ and $|A^{\prime }+B^{\prime }|\le C{\delta }^{-C}{K}^{C}n$. A graph formed by popular representations of sums turns this into the usual large-additive energy form of the Balog-Szemerédi-Gowers theorem.

For distinct increasing knots and $k\ge 2$, the explicit divided difference formula writes an order-$k$ B-spline as a finite sum of truncated power functions of degree $k-1$. It is a piecewise polynomial function and is globally $C^{k-2}$. Below its first knot the order-$k$ divided difference annihilates a degree-$k-1$ polynomial; above its last knot all values vanish. Nonzero first and last pieces give the exact closed support. Order one gives interval indicators instead of continuous splines.

A finite collection of the standard partition-normalized B-splines is nonnegative and has sum at most one on the entire real line. Extend the knot sequence in both directions. The full order-one collection sums to one, and summing the Cox-de Boor recurrence preserves that sum because the two coefficients of each lower-order function add to one. The finite collection is a subset. On the basic knot interval the finite sum equals one, also following from Marsden's identity.

For a linearly independent partition-normalized B-spline basis, equivalence of norms gives a lower coefficient-stability constant. The upper bound follows from the subpartition of unity for B-splines. A knot-independent stability bound can be chosen depending on spline order. This norm comparison transfers matrix estimates to the B-spline interpolation operator norm.

The auxiliary-metric brane action is a quadratic embedding action with an independent worldvolume metric. Its embedding equation is ${\partial }_{\mu }(\sqrt{-\gamma }{\gamma }^{\mu \nu }{\partial }_{\nu }{X}^a)=0$. Its metric equation is $h_{\mu \nu }-{\gamma }_{\mu \nu }({\gamma }^{\rho \sigma }{h}_{\rho \sigma }-(p-1))/2=0$. For $p\ne 1$, tracing determines ${\gamma }^{\rho \sigma }{h}_{\rho \sigma }=p+1$ and hence $\gamma =h$; eliminating the auxiliary field gives twice the invariant worldvolume volume, before including the overall brane tension. The Weyl-invariance exception for the string among branes prevents unique elimination at $p=1$.

In a cyclic group of arbitrary order $q$, a Bohr set with $d$ frequencies, defined by $\Vert rx/q\Vert \le 1/16$, contains an arithmetic progression of distinct residues of length at least $q^{1/(d+1)}/32$. Partition the $d$-dimensional cube into $Q^d$ boxes, with $Q=\lfloor q^{1/(d+1)}\rfloor$. The pigeonhole principle supplies $1\le t\le Q^d$ with $\Vert rt/q\Vert \le 1/Q$ simultaneously. The residues $jt$, $0\le j\le \lfloor Q/16\rfloor$, remain in the Bohr set and do not wrap around the cyclic group. The weaker exponent permits composite moduli without assuming that every nonzero step has large order.

Loss of summer sea ice need not be irreversible at fixed external forcing. Open water can release excess heat in autumn, and thin ice grows faster than thick ice during winter because it offers less resistance to thermal conduction. Continued climate change can nevertheless shift the long-term ice cover downward. A one-summer perturbation and persistent forcing are different physical experiments.

A monomial alternant is the determinant formed from powers with an integer exponent tuple $\ell$. It is a polynomial when the exponents are nonnegative, and otherwise a Laurent polynomial. Equal exponents give equal columns and zero determinant; interchanging exponents changes its sign. Ordered strictly decreasing nonnegative exponents give a basis of alternating polynomials, by grouping monomials into permutation orbits.

For the constant gas-radiation ratio vertical disc structure, the integrated alpha prescription $\bar{\nu }\Sigma =\int \mu dz=(\alpha /\Omega )\int pdz$ gives the displayed mean kinematic viscosity and $\lambda /(1+\lambda )=\alpha H\Omega \kappa \Sigma /(8c)$. Uniform dynamic viscosity and nonuniform pressure are compatible because this prescription is an integrated equality. Imposing the corresponding pointwise equality would not preserve the exact vertical solution.

Over a field $k$, an affine curve parametrized by positive integer powers of one parameter. Its coordinate ring is the subalgebra generated by those powers, and its defining prime ideal is the kernel of $k[x_1,\dots ,x_n]\to k[t]$ sending $x_i$ to $t^{a_i}$. Equal exponent sums yield binomial relations. Studying those relations connects an additive semigroup of exponents to the geometry of the curve.

Mixing two lotteries with the same third lottery and the same positive weight preserves their strict preference order. For affine utility on probability measures, the difference of the two mixed utilities is the original difference multiplied by that weight. At weight zero the strict preference disappears, so positivity is essential.