Insufficient vitamin C can produce the clinical syndrome scurvy. ^[ref]

A Krylov subspace method seeks approximations in spaces generated by repeated matrix applications to a starting vector. For a linear system with starting point $x_0$ and residual $r_0=b-A{x}_0$, the affine search space is $x_0+K_k(A,r_0)$. Choosing an optimality or orthogonality condition within it gives different algorithms. The conjugate gradient method minimizes the energy norm of the error for a symmetric positive-definite matrix; other methods use residual minimization or different projections. The growth of the Krylov subspace can stop early when an invariant subspace is reached.

Recovering a derivative from values controlled only in the uniform norm is an ill-posed problem: the differentiation map is not continuous in those norms. On $[0,1]$, take $f_k(x)=\sin (kx)/k$. Then $\Vert f_k{\Vert }_{\infty }\le 1/k\to 0$, while $f_k^{\prime }(x)=\cos (kx)$ has uniform norm one for every $k$. Arbitrarily small value perturbations can therefore conceal nonvanishing derivative changes.

Numerical differentiation can still be useful when additional smoothness bounds, a suitable sampling scale, or a regularization method control the admissible changes. The basic noise amplification by differentiation of a linear interpolant gives a finite-data version: endpoint uncertainty of size $\delta$ produces derivative uncertainty of order $\delta /h$ at spacing $h$.

Bacterial density is the number of bacteria per unit length, area or volume in a spatial model. It is a field variable, distinct from the organisms themselves.

A total family has dense linear span; it is minimal when no vector lies in the closed span of all the others. Equivalently, no proper subfamily remains total. Such a family has unique bounded coordinate functionals making a biorthogonal system. Indeed put $M_i=\overline{\mathrm{span}}\{x_j:j\ne i\}$ and $d_i=\mathrm{dist}(x_i,M_i)>0$. The functional $m+t{x}_i\mapsto t$ on $M_i+\mathbb{R}{x}_i$ is bounded by $1/d_i$, so the Hahn-Banach theorem extends it to $X$. Its norm is exactly $1/d_i$, by the distance infimum. Uniqueness follows because its values on the total family determine it on a dense subspace. For a countable family, minimality and totality alone do not ensure norm-convergent expansions; the uniform partial-sum criterion for a Schauder basis supplies the additional condition.

If $H$ is a subgroup and $K$ a normal subgroup of a group $G$, their product set $HK=\{hk:h\in H,k\in K\}$ is a subgroup. Normality gives $(h_1{k}_1)(h_2{k}_2)=h_1{h}_2(h_2^{-1}{k}_1{h}_2)k_2\in HK$ and $(hk{)}^{-1}=h^{-1}(h{k}^{-1}{h}^{-1})\in HK$. Together with the identity element this proves the subgroup test. For finite subgroups, $|HK|=|H||K|/|H\cap K|$: the pairs representing one product differ exactly by multiplying the first factor by an element of $H\cap K$ and its second factor by its inverse.

A T1 space is perfectly normal when it is normal and every closed subset is a G-delta set. Equivalently every closed subset is the zero set of a continuous function into $[0,1]$, using the closed G-delta set as a zero set theorem. If disjoint closed sets are the exact zero sets of $f,g$, then $f/(f+g)$ has those sets as its exact zero and one levels. Every subspace is normal, although the converse fails.

A space is hereditarily normal when every subspace is a normal topological space. This is stronger than normality: arbitrary subspaces need not be closed. Every perfectly normal space is hereditarily normal.

The stag hunt is a coordination game with a mutually beneficial risky action and a safer action. A representative two-player payoff matrix isBoth mutual stag and mutual hare are Nash equilibria: neither player can improve by deviating alone. Mutual stag improves both players' payoffs relative to mutual hare, but choosing stag against an uncertain partner can be risky. If the partner chooses stag with probability $p$, stag yields $4p$ while hare yields three, so stag is a best response exactly when $p\ge 3/4$. The example separates mutual gain from assurance that the other player will cooperate.

Aspirin is acetylsalicylic acid, a nonsteroidal anti-inflammatory drug that acetylates cyclooxygenase. Its persistent effect on platelet function illustrates the distinction between a drug’s presence in blood and the lifetime of its effect on a cellular target.

Naproxen is a nonsteroidal anti-inflammatory drug that reduces prostaglandin synthesis through cyclooxygenase inhibition. Its pharmacological class describes a shared mechanism, while its exposure time course is a separate pharmacokinetics question.

Ibuprofen is a nonsteroidal anti-inflammatory drug acting through cyclooxygenase inhibition. It illustrates how one pathway can influence pain, fever and inflammation while also participating in normal tissue functions.

For the normalized sine eigenfunctions of a particle in a box on $(0,a)$, multiplying the lowest one by $ϵ\cos (2\pi x/a)$ gives $ϵ({\varphi }_3-{\varphi }_1)/2$. Thus the first-order diagonal shift is $-ϵ/2$, and the only second-order coupling is to level three. With $E_3=9E_1$, its contribution is $({ϵ}^2/4)/(E_1-E_3)=-{ϵ}^2/(32E_1)$, giving the displayed expansion. This simple selection rule makes the full second-order perturbation theory sum a single term.

For nonlinear least squares $f(x)=\frac{1}{2}\Vert R(x){\Vert }_2^2$, with residual Jacobian matrix $J=DR$, the method approximates the Hessian by $J^{T}J$. The exact Hessian is $J^{T}J+{\sum }_j{R}_j{\nabla }^2{R}_j$. Full column rank of $J$ makes the approximation positive definite. At a zero-residual solution with full-rank Jacobian, $J(x_k{)}^{T}J(x_k)\to {\nabla }^{2}f(x_{\ast })$, so any convergent sequence of full Gauss-Newton steps satisfies the Dennis-Moré condition and converges superlinearly. A nonzero residual at the solution leaves a generally nonvanishing discarded Hessian term; the same conclusion does not follow automatically.

Relative to a fixed orthonormal basis, the displayed linear map is an isometry because its basis images are orthonormal. Its adjoint obeys $C^{\ast }|k⟩|\ell ⟩={\delta }_{k\ell }|k⟩$. If vectors are described by coefficient functions, this adjoint maps $|f⟩|g⟩$ to $|fg⟩$, where the product is pointwise. Although $C$ copies every basis quantum state, it does not copy arbitrary superpositions. For example it sends $(|0⟩+|1⟩)/\sqrt{2}$ to $(|00⟩+|11⟩)/\sqrt{2}$, whereas two copies would also contain the cross terms $|01⟩$ and $|10⟩$.

The correction is the vector obtained by solving the Newton system, added to the current point to give $x_{\mathrm{new}}=x+\delta$. The inverse notation specifies the solution; a numerical implementation normally solves a linear system rather than forming a matrix inverse.

A late manifestation of neurosyphilis involving progressive cognitive, behavioral and neurological disturbance. Its historical psychiatric description preceded a secure explanation of its infectious cause, connecting changes in disease classification with microbiological evidence.^[ref]

A late neurological manifestation of syphilis involving posterior sensory pathways of the spinal cord. Loss of position sense can impair coordination despite preserved muscle strength, connecting the affected pathway with the clinical pattern.

Historical category of kidney disease associated with albuminuria and often oedema, named after Richard Bright. It should not be equated with one modern microscopic diagnosis.^[ref]