A Bohr set defined by distance of each character phase to the nearest integer. For $0\le ϵ\le 1/2$, it corresponds to the chord-distance radius $2\sin (\pi ϵ)$. Numerical radius constants depend on the convention. In a group of prime order and with $d=|R|\ge 1$, simultaneous torus approximation yields a nonzero element when $ϵ>N^{-1/d}$ and an arithmetic progression of length at least $\min (N,\lceil ϵN^{1/d}\rceil )$.

For a spinless particle without a magnetic vector potential, Bohmian mechanics sets $\dot{\mathbf{X}}=\hslash \nabla S/m$, where $S$ is the dimensionless quantum phase. Equivalently the velocity is the probability current divided by the probability density. This formula is local to nonzero wavefunction regions.

The amplitude-dependent extra potential in the Madelung equations. Taking the gradient of their phase equation and using the guidance equation yields $mD\mathbf{v}/Dt=-\nabla (V+Q)$, where $D/Dt$ is the material derivative. At wavefunction nodes this local expression may be singular.

For the separation of a two-dimensional central-potential eigenstate, the guidance equation gives zero radial velocity and constant-radius circular motion, with circulation $2\pi \hslash k/m$. A real superposition of the degenerate $k$ and $-k$ eigenstates has zero current away from nodes instead. The speed depends on the actual wavefunction, not just its energy.

For $D\subseteq \mathbb{Z}/q\mathbb{Z}$ of subset density $\alpha$, $2D-2D$ contains a Bohr set of rank at most $8{\alpha }^{-2}$ and a fixed positive width, including for composite $q$. With normalized Fourier coefficients on a finite abelian group, retain the frequencies where $|\widehat{1_D}|\ge {\alpha }^{3/2}/\sqrt{8}$. Parseval identity on a finite group bounds their number and the discarded fourth moment. Their nearly constant phases make $1_D\ast 1_D\ast 1_{-D}\ast 1_{-D}$ positive on the Bohr set.

For a set of density $\alpha$ in a finite cyclic group, select frequencies whose normalized Fourier coefficients on a finite abelian group have magnitude at least ${\alpha }^{3/2}/2$. Parseval identity bounds their number by $4{\alpha }^{-2}$. Fourier inversion of the fourfold convolution shows it is positive on the Bohr set in phase-distance convention of radius $1/6$. This is an explicit form of the Bogolyubov lemma.

A bounded invertible bosonic Bogoliubov transformation is unitarily implementable in the usual bosonic Fock space precisely when its creation-mixing part is a Hilbert-Schmidt operator. Without this condition, the transformation still preserves the algebraic canonical commutation relations, but its two vacua need not be vectors in a common Fock space. Equal nonzero squeezing in countably many independent modes is a simple counterexample.

A Nambu spinor groups an annihilation operator with an opposite-momentum creation operator to express pairing in a quadratic Hamiltonian as a matrix problem. Its components are not independent: the spinor at $-k$ is related to conjugated components at $k$. This redundancy must be accounted for when summing energies and normal ordering constants. Component phases are a convention and change the displayed off-diagonal matrix entries.

The projective morphism ${\mathrm{Proj}}_X{⨁}_{j\ge 0}{\mathcal{I}}^j\to X$ associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an isomorphism. Blowing up a smooth point of an $n$-dimensional smooth variety replaces the point by a projective space ${\mathbb{P}}^{n-1}$, the algebraic exceptional divisor.

Group random edges by their larger endpoint. These independent coordinates each change edges incident to only one vertex. Removing that vertex leaves the same graph under any two outcomes, so their chromatic numbers differ by at most one. The McDiarmid inequality with $n$ coordinate ranges of length one gives the displayed concentration bound.

For a fixed probability strictly between zero and one, the largest independent set has leading size $2{\log }_{1/(1-p)}n$, giving a lower bound for colouring. The matching upper bound uses independent sets in every large subset of a dense random graph, then repeatedly colours and removes such sets. A sufficiently strong probability estimate makes the failure contribution negligible for the expectation.

For $p=(1+\epsilon )/n$, the branching survival probability satisfies $1-\rho =(1-p\rho {)}^n$. If $(1+\epsilon {)}^2<n$, comparison with a Poisson branching process and expansion of the logarithm giveFor the upper bound use $\epsilon ={\sum }_{j\ge 1}(1-n{p}^{j+1}){\rho }^j/(j+1)$ and retain its first nonnegative term. For fixed $n>1$, the leading term is instead $2n\epsilon /(n-1)$. In particular $n=2$ gives $\rho =4\epsilon /(1+\epsilon {)}^2$, showing that the uncorrected exact bound $\rho \le 2\epsilon$ is false for small positive $\epsilon$.

For $0<\epsilon \le 1/4$ and $n\epsilon \ge 2$, the uncorrected exact upper bound does hold for the binomial branching process. Here $n{p}^2\le \epsilon$ and $n{p}^3\le 1/4$, so at $x=2\epsilon$ the first two terms of the nonnegative series give $\epsilon (1-n{p}^2)+(4/3){\epsilon }^2(1-n{p}^3)\ge \epsilon$. Monotonicity of that series yields $\rho \le 2\epsilon$. This recovers the intended large-$n$ bracket without claiming it for every finite reproduction law.

In this idealized model, a critically rotating spherical cloud splits into two equal-mass homologous spheres of radius $a/2$ that share the orbital frequency of their touching circular binary. Their combined orbital and spin inertia is $(1+\alpha )M{a}^2/4$. Equating post-fission angular momentum to $\alpha M\sqrt{GM{R}_c}$ gives the displayed separation relation. A compact outcome $a<R_c$ requires $0<\alpha <1/3$. The model is an angular-momentum budget, not a hydrodynamic proof that stellar fission occurs.

Enumerate pairs of strategy supports, solve the equations making each supported action indifferent, then check strict positivity of supported probabilities and all omitted-action payoff inequalities. These conditions suffice for a Nash equilibrium. Under nondegeneracy of a bimatrix game, only pairs of equal support size need be tested.

A probability distribution over joint action profiles implements a convex combination of their two-player payoff vectors. The attainable expected payoffs form the convex hull of the pure joint payoffs. The distribution need not factor into independent mixed strategies, and a freely enforceable lottery need not satisfy the incentive constraints of a Nash equilibrium.

Let $\alpha$ be a badly approximable number, and sum $B={\sum }_{D<d\le 2D}{\sum }_{W<w\le 2W,dw\le X}{a}_d{b}_{w}e(\alpha dw)$ with $|a_d|\le \tau (d)$ and $|b_w|\le 1$. The Cauchy-Schwarz inequality, divisor-square summatory bound and geometric sums give $|B{|}^2\ll D{\log }^{3}X(DW+W{\sum }_{h\le W}\min (D,\Vert h\alpha {\Vert }^{-1}))$. Separated rotations bound the last sum by $O_{\alpha }(W\log (2W))$, giving the displayed estimate. A hyperbolic cutoff remains valid because the allowed first-variable values after expansion form an interval.

If a bounded linear map $P$ reproduces every element of an approximating polynomial space $V$, then $f-Pf=(f-v)-P(f-v)$ for every $v\in V$. The triangle inequality and infimum over $v$ prove the displayed error bound. The same argument works for any linear approximating space, and applies in particular to de la Vallée Poussin sums.

Cosine substitution is an isometry from continuous functions on $[-1,1]$ to even continuous $2\pi$-periodic functions. It satisfies $\omega (\tilde{f},\delta )\le \omega (f,\delta )$ because cosine is Lipschitz continuous with constant one. Even degree-at-most-$n$ trigonometric polynomials correspond exactly to algebraic polynomials via $T_j(\cos \theta )=\cos (j\theta )$. Averaging a periodic approximant with its reflection cannot increase its error against an even target, so the algebraic and periodic best errors are equal.

Starting at time one, the orthogonal residuals satisfy $U_t=X_t-{\lambda }_t{U}_{t-1}$, with ${\lambda }_t={\gamma }_1/{\phi }_{t-1}$ and ${\phi }_t={\gamma }_0-{\gamma }_1{\lambda }_t$. Earlier innovation directions have zero covariance with the new observation. Explicitly, with $S_m={\sum }_{j=0}^m{\theta }^{2j}$, ${\phi }_t={\sigma }^2{S}_t/S_{t-1}$ and ${\lambda }_t=\theta S_{t-2}/S_{t-1}$ for $t\ge 2$. These formulas follow by induction from $S_t=(1+{\theta }^2)S_{t-1}-{\theta }^2{S}_{t-2}$, with $S_0=1$ and $S_1=1+{\theta }^2$. This includes $\theta =\pm 1$, where ${\phi }_t={\sigma }^2(t+1)/t$.

In a centered MA(1) process, $X_{T+2}$ is uncorrelated with every observation through time $T$. Its best linear predictor from that information is therefore zero, with error variance ${\sigma }^2(1+{\theta }^2)$. One-step innovation formulas using $X_{T+1}$ cannot be substituted when that observation is unavailable.