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A uniform big O bound for a family of functions uses one multiplier and one threshold for every allowed parameter:The order of the quantifiers matters. For $f_{\alpha }(n)=\alpha n$, $0\le \alpha \le 10$ admits $C=10$ and $N=1$, while the unbounded family $\alpha >0$ admits no such constant. Each individual member nevertheless satisfies $f_{\alpha }=O(n)$. A uniform bound is useful when an approximation must work across a whole parameter range, rather than separately for each fixed parameter.

At the specified limit, big theta notation means that two positive constants give both-sided bounds:sufficiently near the limit. Equivalently, $f=O(g)$ and $g=O(f)$ in big O notation. In operation-count analysis, $3n^2+7n+4=\Theta (n^2)$ for $n\ge 1$, using $c=3$ and $C=14$. This supplies a tight scale without asserting that the normalized ratio tends to one.

For an example with no limiting ratio, let $f(n)=n(2+(-1{)}^n)$ for positive integers $n$. Then $n\le f(n)\le 3n$, so $f(n)=\Theta (n)$, but $f(n)/n$ alternates between one and three. A tight growth order is therefore weaker than asymptotic equivalence. The constant bounds also do not fix an exact factor by which a function changes when its argument doubles.

Donald E. Knuth's 1976 paper proposes using $\Theta$ for two-sided constant bounds in algorithm analysis. He credits Bob Tarjan and Mike Paterson with independently suggesting the notation. His discussion gives a concrete motivation for distinguishing an eventual upper estimate from a tight comparison. ^[ref]

When $g$ is nonzero sufficiently near the specified limit, little o notation meansThus the numerator becomes negligible relative to the denominator's scale. As $n\to \infty$, $n=o(n^2)$ because $n/n^2=1/n\to 0$; as $h\to 0$, $h^2=o(h)$ because $h^2/h=h\to 0$. The direction of the limit changes the comparison. ^[ref]

Equivalently, for every $\epsilon >0$, the bound $|f|\le \epsilon |g|$ holds sufficiently near that limit, with a threshold allowed to depend on $\epsilon$. Big O notation needs only one fixed multiplier, making it weaker. For example, $2n=O(n)$ but $2n$ is not $o(n)$. The useful relation $f\sim g⟺f-g=o(g)$ turns asymptotic equivalence into a statement about a negligible remainder.

The Bhattacharyya distance is the negative natural logarithm of the Bhattacharyya coefficient, with $-\ln 0=\infty$. It is zero for identical probability distributions and infinite for disjoint probability supports. Despite the name, it need not satisfy the triangle inequality required of a metric.

An antiparallel beta sheet has neighboring beta strands running in opposite N→C sequence directions. This specifies strand orientation rather than whether the beta strands belong to one polypeptide chain or several.

A parallel beta sheet has neighboring beta strands running in the same direction along their amino acid residue sequences, from the N-terminus toward the C-terminus.

A beta strand is an extended segment of protein backbone in the geometry characteristic of a beta sheet. Several beta strands can associate through backbone hydrogen bonds. A beta strand need not be an entire polypeptide chain.

For a causal autoregressive process with mean $m$ and innovations orthogonal to the observed past, define ${\hat{X}}_{T,h}=X_{T+h}$ for $h\le 0$ and replace future innovations by zero. The displayed recursion is its best linear prediction from a finite past when at least the last $p$ observations are available. Future innovation sums are orthogonal to those observations, which proves the projection property. Mere white noise does not guarantee that a linear forecast equals a conditional mean: that stronger interpretation requires martingale-difference or independent innovations.

The regular power series has $J_0(\rho )=1-{\rho }^2/4+{\rho }^4/64-\cdots$. Its quartic truncation has a double zero at $\rho =\sqrt{8}$, giving a rough disc-well energy estimate $4{\hslash }^2/(\mu a^2)$. A truncation zero is not an exact Bessel zero or a rigorous eigenvalue bound. The double-root behavior particularly warns against treating a low-order local expansion as accurate far from its center.

Apply the Bernstein inequality for algebraic polynomials to the central interval $|x|\le \cos (\pi /(2n))$, using $\sin (\pi /(2n))\ge 1/n$. On the remaining regions apply the Chebyshev nodal derivative comparison to $p^{\prime }$. Since $T_n^{\prime }(\cos \theta )=n\sin (n\theta )/\sin \theta$, the identity expressing $\sin (n\theta )/\sin \theta$ as a sum of $n$ unit complex phases gives $|T_n^{\prime }|\le n^2$. The endpoint limit is $T_n^{\prime }(1)=n^2$, so the constant is sharp.

In ideal steady flow, a contraction of a pipe increases the speed and lowers the pressure. For constant mass density, equal body-force potential at the ends and approximately uniform endpoint speeds, conservation of mass and the Bernoulli equation give $\dot{M}=S_1{S}_2\sqrt{2\rho (p_1-p_2)/(S_1^2-S_2^2)}$ for flow from a wider to a narrower section. Elevation changes contribute a body-force potential difference, and nonuniform profiles or viscous losses require additional information.

// Target: probability-and-statistics.bigb

An integer power sum adds a fixed nonnegative integer power of consecutive integers. Let $N$ be a nonnegative integer and $p$ a nonnegative integer. For $p=0$, use the constant polynomial $x^0=1$, including at zero, so $S_0(N)=N$. Summing the Bernoulli polynomial finite difference identity makes the intermediate terms cancel and givesThe sum is empty when $N=0$. For the endpoints $1,\dots ,N$, the formula valid also at $p=0$ is insteadFor $p\ge 1$, $B_{p+1}(1)=B_{p+1}(0)$, and this becomes the standard positive-endpoint formula. The exception at $p=0$ is essential: $B_1(1)-B_1(0)=1$.^[ref]

For $p\ge 1$, adding $N^p$ to $S_p(N)$ changes the $B_1$ contribution from $-N^p/2$ to $+N^p/2$. Equivalently,where only $B_1^{+}$ differs from our Bernoulli numbers. A quick endpoint check is $p=1$: the sums over $0,\dots ,N-1$ and $1,\dots ,N$ are $N(N-1)/2$ and $N(N+1)/2$, respectively. The latter is a triangular number.

A slowly varying, steady, inviscid channel flow has constant discharge $Q=uh$ per unit width and the displayed Bernoulli equation. The bed-height function of depth has its maximum at $h_c=(Q^2/g{)}^{1/3}$, where the Froude number is one. At a nondegenerate critical crest, a differentiable profile changes between the subcritical flow and supercritical flow branches. With subcritical upstream squared Froude number $F<1$ and upstream depth $h_1$, the shock-free supercritical downstream depth is $h_1(F+\sqrt{F^2+8F})/4$.

In a steady adiabatic inviscid fluid with a conservative body force, the Bernoulli invariant is the displayed specific energy, constant along each streamline. Dotting the Euler equations with velocity and using the specific enthalpy identity gives $\mathbf{u}\cdot \nabla b=0$. In ideal magnetohydrodynamics, magnetic work need not vanish; the planar ideal-MHD Bernoulli invariant includes the corresponding electromagnetic correction.

This one-step operator combines immediate cost with optimized discounted continuation. It is monotone. For nonnegative costs, iterating it from zero produces increasing finite-horizon values bounded above by the infinite-horizon value. If a nonnegative fixed point has a minimizing measurable selector, repeated conditional expectation and dropping the nonnegative terminal value show that the corresponding stationary policy costs no more than that fixed point. Thus the infinite-horizon value, when the Bellman equation holds, is the least nonnegative fixed point.

A Bell inequality is a constraint on observable correlations satisfied by the specified class of local hidden-variable models. In finite-setting scenarios such constraints bound linear combinations of probabilities or expectation values. The CHSH inequality is one example, following from Bell factorization, measurement independence and bounded outcomes. A violation rules out the conjunction of the assumptions defining that model class; it does not alone identify a unique false premise.

A Mach-Zehnder interferometer splits an incident wave into two arms and recombines them at a second beam splitter. A relative quantum phase changes the interference at its output ports. With symmetric lossless 50:50 splitters, the ports can be calibrated to have detection probabilities ${\cos }^2(\theta /2)$ and ${\sin }^2(\theta /2)$.

// Target: optics.bigb

A polarizing beam splitter sends two orthogonal linear polarizations to different output paths. Acting coherently on $\alpha |H⟩+\beta |V⟩$, it correlates path with photon polarization instead of measuring photon polarization unless a path is actually detected or discarded.

For an ideal polarizing beam splitter with one input, choose orthogonal unit electromagnetic polarization vectors ${\mathbf{e}}_H,{\mathbf{e}}_V$ and an incident power-normalized field $a_H{\mathbf{e}}_H+a_V{\mathbf{e}}_V$. If $H$ is transmitted and $V$ reflected, the output powers are $P_{\mathrm{transmitted}}=|a_H{|}^2$ and $P_{\mathrm{reflected}}=|a_V{|}^2$. The associated complex amplitudes can acquire different wave phases; the routing is not an equal-power split for every incident electromagnetic polarization. For linear polarization at angle $\theta$ to $H$, input power $P$ gives $P{\cos }^2\theta$ and $P{\sin }^2\theta$, respectively. At $\theta =30^{\circ }$, a stipulated $4\mathrm{mW}$ input gives $3\mathrm{mW}$ transmitted and $1\mathrm{mW}$ reflected. A $45^{\circ }$ input gives equal powers even though an $H$ input goes entirely into the transmitted output.^[ref]

Real polarizing beam splitters have leakage into the unwanted output and losses. Each linear polarization has its own field transformation and power coefficients. To make an ordinary power wave interference term between the two output contributions, first bring them into the same path with overlapping transverse profiles and a common electromagnetic polarization, for example using a polarizer. Merely placing two orthogonal linear polarizations on a polarization-insensitive photodetector adds their powers. Coherent routing can be reversed by suitable optics before detection; discarding a path loses access to that part of the field but supplies no recorded measurement outcome by itself.

// Target: optics.bigb

Maximum a posteriori estimation selects a parameter value maximizing its posterior density relative to a specified reference measure. Its output is a maximum a posteriori estimate. The maximizing value need not be invariant under a nonlinear reparameterization, because a density transforms with a Jacobian.

For data $x$, a positive prior distribution density $\pi$ and likelihood function $\ell$, Bayes' theorem gives posterior density $h(\theta \mid x)=\pi (\theta )\ell (\theta )/m(x)$. Thus $h/\pi$ reconstructs the likelihood up to a constant. Estimate the posterior density from simulation, divide by the known prior density and maximize over the search region. A posterior mode alone instead estimates a maximum a posteriori estimate; it estimates a likelihood maximizer only when the prior is constant there. Prior support must include the likelihood maximizer of interest.

// Target: biology.bigb