A randomized complete block design places every treatment in every block, with a common replication pattern, and uses independent within-block randomization. The usual basic version has one occurrence of each treatment per block; equal repeated occurrences give a replicated complete-block version. Such equal-proportion designs are orthogonal block designs.

A row-column design has two crossed partitions of experimental units into rows and columns. Equal treatment proportions in every row and every column make treatment contrasts orthogonal to both centered block spaces. A repeated three-treatment arrangement in a six-by-six array can give a balanced example.

For two voters and at least three alternatives, pairwise unanimity, pairwise independence and transitivity force the binary social ordering from strategyproof choice to follow one fixed voter. If voter one wins a conflict $a$ against $b$, use profiles $(a>b>c,b>c>a)$ and $(c>a>b,b>c>a)$ to show it also wins $a$ against $c$ and $c$ against $b$. Repeating this implication yields both directions of every pair. If the initial conflict instead follows voter two, exchange the roles. The social choice, being the top of that ordering, is a dictatorship in social choice. This proves the two-voter case of the Gibbard-Satterthwaite theorem.

For a bi-invariant Riemannian metric on a Lie group, the induced inner product on its Lie algebra makes every adjoint operator skew-adjoint. The Levi-Civita connection of a bi-invariant metric gives $R(X,Y)Z=-\frac{1}{4}[[X,Y],Z]$. Thus $K(X,Y)=\frac{1}{4}\Vert [X,Y]{\Vert }^2$ for orthonormal $X,Y$, andThe nullspace of this quadratic form is exactly the center of a Lie algebra. A zero center therefore gives positive Ricci curvature, uniformly bounded below by a positive constant times the metric through left invariance and compactness of the unit sphere in the Lie algebra.

Repeated coordinates contribute zero. Sorting a distinct tuple and subtracting the staircase gives a partition of $m$, and its $m!$ orderings have identical squared summands. The beta-set hook-product identity turns each summand into $(\dim S^{\lambda }/m!{)}^2$. Summing and using the Artin–Wedderburn theorem for $\mathbb{C}{S}_m$ gives $m!{\sum }_{\lambda ⊢m}(\dim S^{\lambda }{)}^2/(m!{)}^2=1$. This is a normalized group algebra dimension identity.

For $A,B>0$, the nonzero zero of the renormalization-group beta function has slope ${\beta }^{\prime }(g_{\ast })=-2B^2/A<0$ and is ultraviolet attractive. The origin is infrared attractive. A fixed-point field-renormalization anomalous dimension gives a power-law quantum field theory propagator factor $d(z)\sim C{z}^{{\gamma }_{\varphi }(g_{\ast })}$. A weak fixed point is needed for control by a perturbative truncation.

For a smooth solution of the heat equation, backward Euler with the centered second difference has residual divided by $k$ equal to $-(k/2+d^2/12)u_{xxxx}+O(k^2+k{d}^2+d^4)$. Both leading terms have the same sign. Under parabolic mesh refinement their coefficient is $-d^2(r/2+1/12)$, which cannot vanish for a positive diffusion Courant number. Consistency of an update $U^{n+1}-U^n=\alpha (r){\delta }^2{U}^{n+1}$ forces $\alpha (r)=r$ for every fixed positive $r$.

For the degree-two Cardinal B-spline basis on knots $1,\dots ,n+3$, sampling at $x_i=i+2$ gives the upper-bidiagonal B-spline collocation matrix $A=(I+S)/2$, where $S$ is the one-step upper shift. Since $S^n=0$, $A^{-1}=2{\sum }_{r=0}^{n-1}(-S{)}^r$ and its maximum absolute row sum is $2n$. The B-spline interpolation operator norm estimate with the order-three stability constant $d_3=3$ gives the displayed linear upper and lower bounds. The lower bound proves failure of uniform boundedness.

Photoionization changes the ion population, often removing bound-state coolants at temperatures where collisional equilibrium would retain them. It can therefore suppress primordial line-cooling peaks. It also supplies photoionization heating. The net loss is cooling minus heating and depends on radiation spectrum, intensity, shielding and gas density. A universal density-independent function of temperature alone is not generally sufficient for irradiated gas.

Radiation from rotational and vibrational transitions of molecules removes gas thermal energy. In primordial gas below the atomic threshold, molecular hydrogen is a principal coolant; hydrogen deuteride can matter in suitable chemical conditions. Molecular formation, photodissociation, optical depth and density-dependent level populations control the rate. This allows some small systems below the atomic virial temperature threshold to form stars.

If $A$ is diagonalizable, the anticommutator map $B\mapsto AB+BA$ has eigenvalues $2{\lambda }_1$, ${\lambda }_1+{\lambda }_2$ twice, and $2{\lambda }_2$. Their product gives the displayed characteristic polynomial. The density of diagonalizable complex matrices and continuity of polynomial coefficients prove the formula also for defective matrices.

In a complete two-factor design with equal cell replication, centred factor indicator columns for different factors have zero cross-products. Consequently the two main effects in an additive normal linear model are orthogonal, and their extra sums of squares do not depend on the order in which these main effects are entered. An omitted interaction term can still require a separate adequacy check.

An ANOVA stratum is an orthogonal response subspace representing one level of experimental variation, such as between blocks or within blocks. In a covariance model scalar on each such subspace, its eigenvalue supplies the common variance scale for its projected coordinates. Treatment and residual sums of squares in ANOVA must be compared within appropriate strata.

For an orthogonal projection matrix $P$, a sum of squares in ANOVA is $Y^{T}PY=\Vert PY{\Vert }^2$. Mutually orthogonal projections yield an additive decomposition of total squared response magnitude. Removing the grand mean gives the corrected total.

Applying the amplitude damping channel to the Bloch vector form of a qubit density operator gives an affine map. Transverse components contract by $\sqrt{1-p}$; the longitudinal component contracts by $1-p$ and shifts toward the ground-state north pole. Reading the entries of the output density operator proves the map. The displacement distinguishes this channel from centered depolarization.

The Kraus formula for entanglement fidelity applies to any input, including mixed qubit states. For the amplitude damping channel, the diagonal Kraus trace is $[1+s_z+\sqrt{1-p}(1-s_z)]/2$ and the jump Kraus trace has squared modulus $p(s_x^2+s_y^2)/4$. Summing them gives the formula. Ground-state inputs have unit entanglement fidelity; excited-state inputs have fidelity $1-p$.

For a local basis $e_a$ of the tangent bundle, an affine connection has components defined by ${\nabla }_{e_c}{e}_b={\Gamma }^a{}_{bc}{e}_a$. If $V=V^b{e}_b$, then $({\nabla }_{e_c}V{)}^a=e_c(V^a)+{\Gamma }^a{}_{bc}{V}^b$. The connection components depend on the chosen basis and have an inhomogeneous transformation law; they are not themselves a tensor field. In a coordinate basis, a torsion-free connection has symmetry in the two lower indices.

In a commutative diagram of two kernel rows, a monic arrow between the final objects makes the square between their kernels and middle objects a pullback in a category. The kernel universal property factors any categorical cone uniquely. Conversely, if the square between middle and final objects is a pullback, the induced arrow between their kernels in a category is an isomorphism, since both kernels have the same universal property after pullback.

Let $B=0$ with probability $\pi$, and otherwise let $B$ have a Gamma distribution with mean one and variance $\tau$. Given $B$, counts $Y_j$ are conditionally independent Poisson random variables with means $B{\mu }_j$. For $q=1-\pi$, the law of total variance and law of total covariance giveEach component has a zero-inflated negative binomial distribution, but the components are not independent after marginalizing $B$. The zero component is shared by the whole profile. At $\tau =0$, interpret the positive Gamma component as a point mass at one.

For a vertex $(g/c_W){\gamma }^{\mu }(c_V-c_A{\gamma }^5)$ into one massless color-singlet particle-antiparticle pair, the massive-vector spin average and two-body Lorentz-invariant phase space give $\Gamma =g^2{m}_Z(c_V^2+c_A^2)/(12\pi c_W^2)$. With Standard Model neutral-current vector and axial couplings, one neutrino flavor has ${\Gamma }_{\nu }=g^2{m}_Z/(96\pi c_W^2)$ and one charged-lepton flavor has ${\Gamma }_{\ell }={\Gamma }_{\nu }(1-4s_W^2+8s_W^4)$. Chiral projectors already remove inactive neutrino helicities. These are tree-level massless results.