For independent affected sib pairs with IBD counts $n_0,n_1,n_2$, the likelihood function of the recombination fraction is proportional to $s^{2n_0+n_1}(1-s{)}^{2n_2+n_1}$, with $s=2\theta (1-\theta )$. At fixed sample size, the total shared-copy count $2n_2+n_1$ is a sufficient statistic by the Fisher-Neyman factorization theorem. Thus a one-sided likelihood-ratio test has the same ordering as a test for excess mean IBD sharing. This equivalence depends on the stated fully penetrant recessive model and does not extend to arbitrary disease models.

// Target: biology.bigb

A motivating experiment used isolated membranes from A-431 biological cells, which supplied abundant EGFR. Separate membrane aliquots were incubated at $0{}^{\circ }\mathrm{C}$ with ATP carrying radioactive phosphorus in its terminal phosphate group, plus magnesium or manganese ions. Radioactivity in material recovered by chemical acid-induced precipitation supplied the readout. Adding EGF beforehand enhanced incorporation relative to the unstimulated preparation; this placed a ligand-dependent biochemical response in a cell-free membrane system.^[ref]

The follow-up comparisons separated increased addition from suppressed removal: phosphate removal was rapid but unaffected by EGF, and anti-EGF antibody reversed activation by removing the ligand. These observations support reversible regulation of membrane-associated protein kinase activity.^[ref] Total incorporation alone cannot identify which protein or amino acid residue received the label, prove intrinsic receptor catalysis, or predict an intact biological cell's response.

The same account records an instructive identification error: receptor phosphorylation was initially assigned to the wrong amino acid residue because released modified amino acids migrated together in the electrophoretic electric field. Reanalysis identified phosphotyrosine residues. Retained radioactivity is evidence about the assay's separation and detection, not unique chemical identity.^[ref]

The negative part of a real-valued function $f$ is the nonnegative function $f_{-}(x)=\max (0,-f(x))$. It records the magnitude of the negative values, rather than retaining their sign: if $f(x)=-3$, then $f_{-}(x)=3$. Together with the positive part, it gives $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$. This decomposition defines the Lebesgue integral of a signed function and makes the integrability condition explicit.

The upper and lower unipotent matrices generate SL2R. For $a\ne 0$, let $w(a)=U(a)L(-a^{-1})U(a)$; multiplication gives $w(a)=\left(\begin{array}{cc}0& a\\ -a^{-1}& 0\end{array}\right)$ and $w(a)w(-1)=\mathrm{diag}(a,a^{-1})$. A determinant-one matrix with upper-left entry $a\ne 0$ factors as $L(c/a)\mathrm{diag}(a,a^{-1})U(b/a)$. If that entry vanishes, first add the second row to the first by $U(1)$. Lower unipotents are conjugates of upper unipotents, so the latter also generate the whole group normally.

For an even-dimensional antisymmetric matrix $M$, put $\theta =(u+iv)/\sqrt{2}$ and $\bar{\theta }=(u-iv)/\sqrt{2}$, with independent Grassmann variables $u,v$. Anticommutation cancels the mixed terms, giving ${\bar{\theta }}^{T}M\theta =(u^{T}Mu+v^{T}Mv)/2$. The constant Grassmann change-of-variables formula changes only the normalization. The two real Pfaffians multiply to $\mathrm{Pf}(M{)}^2=\det M$, agreeing with the complex Grassmann Gaussian integral. In odd dimension $\det M=0$ and both top-degree integrals vanish.

A cubic destabilizing term and quintic saturation describe a subcritical stationary pattern bifurcation. A plane wave $A=R{e}^{iQX}$ has squared amplitude $x=R^2$ satisfying $\mu -Q^2+\alpha x-x^2=0$. The nonzero branches meet at $x=\alpha /2$, $\mu =Q^2-{\alpha }^2/4$. The lower branch is unstable to amplitude perturbations; the upper branch can also undergo an Eckhaus instability of its phase.

A Turing pattern is a spatially structured state produced by interactions between local reactions and diffusion. A Turing instability explains linear pattern onset when a homogeneous equilibrium is stable without diffusion but unstable to a nonzero spatial mode.

For positive concentrations and positive parameters, this reaction–diffusion system has homogeneous equilibrium $(u_{\ast },v_{\ast })=(1/b,1/b^2)$ and reaction Jacobian matrix $\left(\begin{array}{cc}b& -b^2\\ 2/b& -1\end{array}\right)$. Its trace is $b-1$ and determinant is $b$, making the equilibrium reaction-stable exactly when $0<b<1$. Stable complex eigenvalues permit damped oscillations, which must be distinguished from sustained periodic oscillations.

An activator in an activator-inhibitor reaction–diffusion system is a species whose local feedback enhances activation. Near a homogeneous equilibrium, positive self-feedback is reflected in the corresponding diagonal entry of the reaction Jacobian matrix. Cross-coupling to an inhibitor and differing diffusion rates can generate a Turing instability.

An inhibitor in an activator-inhibitor reaction–diffusion system suppresses the activator. Its diffusion and reaction feedback can stabilize a spatially uniform state while allowing some nonzero wavenumbers to grow through a Turing instability.

A quadratic activator can have both basal production and activation inhibited by a second species. A normalized example has reaction rates $f(u,v)=1+r{u}^2/v-u$ and $g(u,v)=q(u^2-v)$, with $r,q>0$ and positive concentrations. At its spatially homogeneous equilibrium, $v=u^2$, so $u_{\ast }=1+r$ and $v_{\ast }=(1+r{)}^2$. Writing $a=(r-1)/(r+1)$, its reaction Jacobian matrix is $J=\left(\begin{array}{cc}a& -r/(1+r{)}^2\\ 2q(1+r)& -q\end{array}\right)$. Thus $\mathrm{tr}J=a-q$ and $\det J=q>0$, giving linear asymptotic stability when $q>a$ and instability when $q<a$. Basal production makes $a$ change sign at $r=1$; a positive activation feedback needed for a conventional Turing instability requires $r>1$.

The scalar equation $u_t=D{u}_{xx}+u(1-u)(u-r)$ with $D>0$ and $0<r<1$ has two stable homogeneous phases $u=0,1$ and an unstable intervening equilibrium $u=r$. Indeed the reaction derivatives at these three equilibria are $-r$, $-(1-r)$ and $r(1-r)$, respectively. A travelling wave can connect the two stable phases. At the equal-depth value $r=1/2$, its decreasing heteroclinic orbit is stationary. Away from this value the phases have unequal potential depths and the front propagates.

Take upward-closed set families of subsets of $[n]$ of size at most $\ell$, closed under the forcing rule in Razborov closure. Normalize any family containing a set of size at most one to the full family, since its clique predicate is identically true. Intersection is the meet in a lattice; closure of the union is the join in a lattice. A family represents the graphs containing a clique on at least one of its members. The approximate meet can lose positive inputs and the approximate join can acquire negative inputs. The Razborov gate-by-gate approximation lemma charges output errors to those local errors.

For a real symmetric matrix $A$ and nonzero real vector $x$, the quotient rule and the quadratic form gradient give the displayed gradient. It vanishes exactly when $Ax=R_A(x)x$, so the stationary points are eigenvectors. Equivalently, the Lagrange multiplier condition for optimizing $x^{T}Ax$ on the unit sphere is $Ax=\lambda x$. The smallest and largest eigenvalues give the global extrema; other eigenvectors can be stationary without being extrema, as a direction mixing larger and smaller eigenvalues changes the quotient in opposite ways.

For a finite-dimensional Hermitian matrix $A$, the Rayleigh quotient has precisely the range $[{\lambda }_{\min },{\lambda }_{\max }]$, where the endpoints are its extreme eigenvalues. The finite-dimensional spectral theorem writes a nonzero vector as $x={\sum }_j{c}_j{e}_j$ in an orthonormal eigenbasis, givingThus the quotient is a convex combination of eigenvalues. The bounds follow immediately, and equality at either endpoint means that $x$ lies in the corresponding extreme eigenspace, including any repeated eigenvalue. Every intermediate value occurs: for orthonormal extreme eigenvectors, $x=\sqrt{1-t}{e}_{\min }+\sqrt{t}{e}_{\max }$ gives $R_A(x)=(1-t){\lambda }_{\min }+t{\lambda }_{\max }$ for $0\le t\le 1$. If all eigenvalues are equal, the quotient is constant.

The Hermitian hypothesis matters. The real matrix $B=\left(\begin{array}{cc}1& 4\\ 0& 1\end{array}\right)$ has only the eigenvalue $1$, but $(1,1)B(1,1{)}^T/2=3$. Its quotient is governed by the symmetric part of a matrix $(B+B^T)/2$, not by a real orthonormal eigenbasis of $B$. A nonsymmetric matrix therefore need not obey the displayed eigenvalue bounds.

For a nonzero real solution of ${\varphi }^{\prime \prime }+(C-k^2)\varphi =0$ on a finite interval, integration by parts gives $\mathcal{R}=\int (C{\varphi }^2-{\varphi }^{\prime 2})dz/\int {\varphi }^{2}dz$ in the displayed form. A Neumann boundary condition only at the lower endpoint does not remove the upper boundary term. Equality with $k^2$ additionally requires $\varphi (z_2){\varphi }^{\prime }(z_2)=0$. With the lower condition, nonconstant oscillatory solutions satisfy this when their phase advance is an integer multiple of $\pi /2$; affine solutions must be constant, while nonzero hyperbolic-cosine solutions fail the condition. Infinite intervals additionally require convergence of both integrals.

For gas pressure depending only on radius and initial speed zero, multiplying the Rayleigh equation for an inviscid spherical bubble by $2R^2\dot{R}$ integrates to the displayed identity. A polytropic pressure $p_g=K/R^{3\kappa }$ gives elementary power integrals when $\kappa \ne 1$, and a logarithm when $\kappa =1$. Turning radii are the positive zeros of this first integral. At $\kappa =2$ and $p_g(R_0)=p_{\infty }/2$, they are $R_0$ and $R_0/2^{1/3}$; the inward and outward accelerations at them have opposite signs, giving bounded undamped oscillation in the ideal model.

For spherical expansion in an unbounded incompressible flow, continuity gives $u_r=R^2\dot{R}/r^2$. Integrating the kinetic energy outside the bubble gives $K=2\pi \rho R^3{\dot{R}}^2$. The Rayleigh equation for an inviscid spherical bubble then yields $\dot{K}=(p_b-p_{\infty })\dot{V}$, where $V=4\pi R^3/3$. With $p_b=p_{\infty }{V}_0/V$ and constant $p_{\infty }$, integration gives $K-K_0=p_{\infty }[V_0\log (V/V_0)-V+V_0]$. This neglects viscosity, surface tension and external body-force differences, which contribute additional terms in other bubble models.

A high-Rayleigh number closure $\mathrm{Nu}=C{\mathrm{Ra}}^{1/3}$ gives $q=C{k}_T[g\beta /(\nu \kappa ){]}^{1/3}(\Delta T{)}^{4/3}$, independent of layer depth. It assumes boundary-layer-controlled heat transfer, a well-mixed interior, fixed material properties and a fixed Prandtl number regime. It is a model, not a universal asymptotic law for every turbulent convection regime or for an impulsive initial conductive transient.

In the simply connected homotopy category, a rational space is a simply connected CW complex whose positive homotopy groups are vector spaces over $\mathbb{Q}$. Multiplication by every nonzero integer on these groups is invertible. More generally one uses nilpotent spaces with localized fundamental group and higher homotopy groups; the simply connected version avoids local-coefficient twisting.