In characteristic different from two, a field generated by finitely many nonzero square roots is the splitting field of a collection of separable quadratic polynomials. Its Galois group embeds into the sign-vector group $(\mathbb{Z}/2{)}^n$. Every index-two subgroup is the kernel of a nontrivial sign character. Coordinate sign characters span the dual of this subgroup of $(\mathbb{Z}/2{)}^n$, by extending a linear functional to the ambient vector space. A product of the corresponding roots transforms by that character and generates precisely the quadratic fixed field.

A primitive element of ${\mathbb{F}}_q$ generates its multiplicative group. This implies ${\mathbb{F}}_q={\mathbb{F}}_p(\alpha )$, but an element generating the field extension need not generate the multiplicative group. For instance $u\in {\mathbb{F}}_3[u]/(u^2+1)$ generates the quadratic extension but has order four, whereas $1+u$ has order eight and is multiplicatively primitive.

Full-weighting restriction transfers a fine-grid residual to a coarser grid by local averaging. In one dimension its weights are $(1,2,1)/4$; in two dimensions they are the tensor-product stencil $\left(\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right)/16$. For linear or bilinear interpolation $P$ and coarsening by two in every direction, it equals $2^{-d}{P}^T$ in unweighted coordinates. This scaling makes restriction the adjoint of interpolation for mesh-weighted inner products.

A geometric multigrid V-cycle pre-smooths an approximate solution, restricts its residual, recursively solves the coarse error equation once from zero, prolongs and adds that error correction, and post-smooths. The recursion ends in a direct solve on a small coarsest grid. With fixed sweep counts and bounded stencil work, coarsening in $d$ dimensions gives total work $O(N){\sum }_{j\ge 0}{2}^{-dj}=O(N)$ per cycle. A mesh-independent convergence factor also requires compatible coarse approximation and effective smoothing; linear work alone does not prove convergence.

For the one-dimensional Poisson equation, use linear interpolation, full-weighting restriction, an exact coarse solve, and one pre- and one post-sweep of weighted Jacobi with weight $2/3$. On a pair of harmonics $\theta ,\theta +\pi$, put $s={\sin }^2(\theta /2)$ and $c=1-s$. The coarse correction is $C=\left(\begin{array}{cc}s& -c\\ -s& c\end{array}\right)$, and the smoother is $S=\mathrm{diag}(1-4s/3,1-4c/3)$. The rank-one error operator $SCS$ has nonzero eigenvalue $s(1-4s/3{)}^2+c(1-4c/3{)}^2=1/9$. This exact harmonic calculation illustrates mesh-independent two-grid contraction for the constant-coefficient model; it is not an unconditional result for arbitrary operators or transfer choices.

Full multigrid solves first on a coarse mesh, interpolates that approximation to the next finer mesh, and applies a fixed number of geometric multigrid V-cycles before proceeding to the next level. If interpolation introduces only the expected discretization-scale error and the cycle count reduces the inherited error sufficiently at every level, the final algebraic error matches the fine-grid discretization error. The geometric sum of level costs is then $O(N)$, unlike reducing an arbitrary fine-grid initial error by an arbitrarily small tolerance.

The MSSM permits a supersymmetric Higgs mass $\mu$, whose natural size is not fixed by soft supersymmetry breaking. Weak-scale symmetry breaking requires it to be comparable to the soft scale rather than a much larger fundamental scale. Superpotential non-renormalization protects a chosen small value radiatively but does not explain that choice; relating its generation to the breaking mechanism addresses the mu problem.

// Target: supersymmetry.bigb

For the MSSM superfield representations augmented by singlet right-handed neutrino chiral superfields $N_i$, the homogeneous cubic gauge-invariant superpotential contains the four Yukawa structures $Q{H}_{u}U$, $Q{H}_{d}D$, $L{H}_{d}E$, $L{H}_{u}N$ and the additional structures $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$. Weak-doublet pairs contract with the antisymmetric two-index tensor; $UDD$ contracts with the color three-index tensor. Consequently the $LLE$ coefficient is antisymmetric in its two $L$ indices, the $UDD$ coefficient in its two $D$ indices, and the $NNN$ coefficient symmetric. With $L(N)=-1$, the last five structures violate baryon number or lepton number, whereas the first four conserve both. Singlet fields must not be omitted when classifying a general cubic superpotential.

Crossing one nondegenerate critical point of Morse index $\lambda$ in a compact sublevel interval attaches a handle $D^{\lambda }\times D^{m-\lambda }$. Between critical levels a suitably normalized gradient flow identifies the sublevels. The Morse lemma supplies the local quadratic model for the attachment. Retracting each handle to its core disk gives a CW complex model with one $\lambda$-cell per critical point. Closed manifolds have finite such decompositions; a noncompact manifold requires a proper bounded-below exhaustion for the corresponding statement.

The free chain group in degree $i$ has one oriented generator for each index-$i$ critical point. A cellular chain complex obtained from the Morse handle-attachment theorem computes the integral homology. Under the Morse-Smale gradient flow condition its differential can instead be obtained by signed counts of connecting trajectories. Over a field, if $r_i$ is the rank of ${\partial }_i$, then $c_i=b_i+r_i+r_{i+1}$. This identity implies both weak and strong Morse inequalities.

A Morse function is perfect over a specified field when its numbers of critical points equal the Betti numbers over that field. Over the rationals this forces every integer matrix in its Morse chain complex to have rank zero, hence to vanish. Consequently its integral homology is free of rank $c_i$ in every degree; a manifold with integral homology torsion cannot admit such an integral-critical-count perfect function.

The critical points of $F$ are exactly pairs of critical points of its factors. Its Hessian matrix is their block direct sum, so their Morse indices add and nondegeneracy is preserved. The critical-count polynomial is the product of those of the factors. This construction also gives products of arbitrary smooth functions, with the critical-point count still multiplying even when some Hessians are degenerate.

The incoming and outgoing boundaries are the regular levels zero and one, and all critical points lie in the interior. Near the boundary use a collar coordinate as the function. The Morse handle-attachment theorem produces a relative chain group freely generated by the interior critical points. Thus $\chi (W,M_0)$ is the alternating sum of their Morse indices, regardless of how their critical values are ordered.

A descending gradient-like field decreases $f$ away from its critical points and agrees with a negative gradient in chosen Morse lemma coordinates near those points. It transports attaching spheres and belt spheres across regular levels. Perturbations away from the critical points can arrange the transverse intersections used by the Morse-Smale complex, while preserving the prescribed local models.

A centimorgan is one hundredth of a morgan. For a sufficiently short genetic map interval, one centimorgan corresponds approximately to a $1\%$ recombination fraction, rather than an exact identity for all intervals.

Antithetic variates deliberately couple unbiased estimators of the same expectation with negative covariance. Their average can have smaller variance than an average using independent draws. For example, evaluating a monotone integrand at $U$ and $1-U$, where $U$ has a uniform distribution, gives opposite fluctuations while preserving both marginal distributions.

A control variate is a random variable with known expected value whose centered value is subtracted from a Monte Carlo estimator to reduce its variance. For a mean-zero vector $G$, put $S=\mathrm{Cov}(G)$ and $c=\mathrm{Cov}(G,H)$. When $S$ is invertible, the optimal estimator averages $H-c^T{S}^{-1}G$ and has variance $(\mathrm{Var}(H)-c^T{S}^{-1}c)/N$ for $N$ independent draws. Improvement is strict exactly when $c\ne 0$.

For $n$ observations, taking the mean of their scalar sum returns that sum, not their sample mean. If $a-b\overline{H}$ is unbiased for a target, replacing the mean by a sum gives expected value $a-bn\mathbb{E}H$ and multiplies its variance by $n^2$. A mean-zero control variate changes variance but cannot correct this missing normalization factor.

In the finite lattice approximation for monotone clique circuits, set $m=r=\lfloor n^{1/4}\rfloor$ and $\ell =\lfloor {\log }_{2}n\rfloor$. A nonuniversal output accepts at most ${\sum }_{s=2}^{\ell }s!(r-1{)}^s(m/n{)}^s\le 2(\ell rm/n{)}^2$ of uniform positive cliques. Its missing positives require the circuit size times the positive clique error of a truncated lattice meet to be bounded below. A universal output accepts every negative colouring input; those errors require the circuit size times the negative colouring error of a forced-set closure to be at least one. Both alternatives imply the displayed superpolynomial bound by the Razborov gate-by-gate approximation lemma.

A monotone Boolean circuit uses only AND and OR gates, variables, and optional constants. It computes a monotone Boolean function; negation gates are excluded.