A topological spectrum $E$ represents the reduced generalized homology theory ${\tilde{E}}_k(X)={\pi }_k(E\wedge {\Sigma }^{\infty }X)$ on based spaces, where the smash product of spectra is derived. It is covariant in $X$.

A population-averaged log-rate model with exposure $P_{it}$ gives an annual rate ratio $e^{{\beta }_1}$. A generalized estimating equation can account for repeated observations within clusters through a working correlation and sandwich covariance matrix. Independent Poisson standard errors do not account for clustering. An exposure-weighted population rate and an equally weighted average of cluster rates are different estimands.

For independent subjects with repeated Bernoulli observations, specify marginal means $m_{ij}={\mathrm{logit}}^{-1}(w_{ij}^T\gamma )$. A generalized estimating equation uses their derivative matrix $D_i$ and working covariance matrix $V_i$ to solve ${\sum }_i{D}_i^T{V}_i^{-1}(Y_i-m_i)=0$. Subject-level sandwich covariance matrices give robust uncertainty when the mean is correctly specified but the working correlation is not. The coefficients describe population odds ratios; this mean specification does not prescribe a complete joint distribution within a subject.

A working correlation matrix $R_i(\alpha )$ is a positive-definite proposed correlation structure used to weight a generalized estimating equation, with $A_i$ the diagonal matrix of marginal variances. It need not equal the actual correlation matrix. Correct marginal means and independent sampling clusters keep the estimating equations unbiased; a cluster-level sandwich covariance matrix estimates their actual large-sample covariance. For three exchangeable observations, positive definiteness requires $-1/2<\rho <1$.

The skeletal filtration of a CW complex gives this spectral sequence, with differential of bidegree $(r,1-r)$ in cohomological grading. For a finite complex the filtration is finite and convergence is bounded. In a multiplicative theory the associated graded product is the ordinary cohomology product with coefficients in the theory's coefficient ring.

A generalized cohomology theory with natural unital cup products compatible with relative groups, suspension, and the cohomology axioms. Its coefficient groups form a graded ring. Thom orientations make the cohomology of a bundle's Thom space a shifted module over the cohomology of the base.

// Target: geometry-and-topology.bigb

A topological spectrum $E$ represents the reduced generalized cohomology theory ${\tilde{E}}^q(X)=[{\Sigma }^{\infty }X,{\Sigma }^{q}E{]}_{\mathrm{st}}$. For an Omega-spectrum this equals $[X,E_q{]}_{\ast }$, taking $E_q={\Omega }^{-q}{E}_0$ for negative $q$. It is contravariant in $X$.

A time-independent Lindblad equation becomes an affine linear equation for the Generalized Bloch representation. If $\mathcal{L}(I)={\sum }_d[V_d,V_d^{†}]$, then $A_{mn}=\mathrm{Tr}({\sigma }_m\mathcal{L}({\sigma }_n))$ and $c_m=\mathrm{Tr}({\sigma }_m\mathcal{L}(I))/N$ for traceless basis indices. The offset vanishes exactly when the dynamics are unital. Equilibria solve $A{s}_{\ast }=-c$, and deviations evolve as $s(t)-s_{\ast }=e^{At}(s(0)-s_{\ast })$.

For a Hilbert-Schmidt orthonormal traceless Hermitian basis, the Generalized Bloch representation gives the displayed identity by orthogonality. Nonnegative eigenvalues of a density operator sum to one, so their squared sum is at most one, with equality precisely for a rank-one projector. For $N>2$, the radius bound alone does not guarantee positivity: $(2/N)I-P$ for a rank-one projector $P$ has trace one and squared norm one, but one negative eigenvalue.

In a Hilbert-Schmidt orthonormal operator basis, the generator $-i[H,\cdot ]$ is represented on the traceless Hermitian subspace by a real skew-symmetric matrix. Its propagator is therefore in the special orthogonal group and preserves the purity of a density operator. For $N>2$, these rotations form the adjoint image of the accessible unitary group, generally a proper subgroup of the full Euclidean rotation group; preservation of the full spectrum of the density operator imposes further constraints.

Stop new arrivals at time $t$ in an initially empty general-service infinite-server queue. The number remaining $v$ time units later is Poisson with mean $\lambda {\int }_v^{t+v}\mathbb{P}(S>u)du$. Thus the probability that clearance has not occurred is one minus the exponential of the negative of that mean. Letting $t\to \infty$ gives the displayed equilibrium formula, since ${\int }_v^{\infty }\mathbb{P}(S>u)du=\mathbb{E}(S-v{)}_{+}$. The clearance time is zero when no customers remain at closing.

Multiplication by a generator of ${\mathbb{F}}_{q^n}^{\times }$ on the ${\mathbb{F}}_q$-space ${\mathbb{F}}_{q^n}$ gives a cyclic subgroup regular on nonzero vectors. Its normalizer contains the field Frobenius automorphism. In degree three over ${\mathbb{F}}_2$ the full normalizer has order 21.

The maximal subgroups are point stabilizers and plane stabilizers of order 24, and Singer cycle normalizers of order 21. A proper subgroup containing a 7-element has a unique Sylow 7-subgroup, since the other count gives an impossible index-three subgroup of the simple group. A subgroup of order dividing 24 has a fixed point or a three-point orbit on the seven nonzero vectors. A collinear orbit gives an invariant plane; an independent triple gives its fixed vector sum.

Inverse transpose is an automorphism of order at most two. On projective geometry it implements point-hyperplane duality. For $G{L}_3(2)$ it swaps distinct point- and plane-stabilizer classes, so it is outer.

This group preserves a nondegenerate alternating bilinear form on ${\mathbb{F}}_q^{2m}$. Choosing a nonzero vector and a partner of pairing one, then recursing on their orthogonal complement, gives $|S{p}_{2m}(q)|=q^{m^2}{\prod }_{i=1}^m(q^{2i}-1)$. Its center consists of scalars with ${\lambda }^2=1$.

For column vectors with the preserved space first, $A,D$ are invertible and $B$ arbitrary. Thus $P_k\cong {\mathbb{F}}_q^{k(n-k)}⋊(G{L}_k(q)\times G{L}_{n-k}(q))$. Its orbits on $k$-spaces are classified by intersection dimension with the preserved space.

The projective quotient of determinant-one matrices preserving a nondegenerate Hermitian form over ${\mathbb{F}}_{q^2}$, with conjugation $x\mapsto x^q$. In dimension three its natural two-transitive action has $q^3+1$ isotropic points. For $q\ge 3$ its order is $q^3(q^3+1)(q^2-1)/\gcd (3,q+1)$.

The group of invertible linear maps preserving a nondegenerate Hermitian form over a quadratic finite field is, in an orthonormal basis, the displayed matrix group. Here $q^2$ denotes the size of the field on which the matrices act; some authors denote this same group by $U_n(q)$. Imposing determinant one gives the special unitary subgroup, and subsequently quotienting its scalar center gives the projective special unitary group over a finite field.

If $k_n(q)$ counts conjugacy classes in $G{L}_n(q)$, then ${\sum }_{n\ge 0}{k}_n(q)t^n={\prod }_{i\ge 1}(1-t^i)/(1-q{t}^i)$. Associate a partition to each irreducible polynomial other than t. The Euler product over those polynomials counts monic polynomials with nonzero constant term and is $(1-u)/(1-qu)$.

Assume $a$ and $a^b$ algebraic with algebraic irrational $b$. Apply Siegel lemma to construct a small-coefficient polynomial $P$ vanishing at the algebraic pairs $(s+tb,a^s(a^b{)}^t)$ on a large square grid. The function $P(z,e^{\lambda z})$ is a nonzero exponential polynomial. Its many initial zeros make later grid values exponentially tiny, whereas an algebraic norm estimate forbids such tiny nonzero values. Iterating extends the zero grid indefinitely. Its quadratic zero count in disks contradicts the linear zero-count growth of an exponential polynomial.