An affine map combines a linear map with a translation. After choosing coordinate origins, it has the form $T:{\mathbb{R}}^p\to {\mathbb{R}}^q$, $T(x)=Mx+b$, where the $q\times p$ matrix $M$ is its linear part of an affine map and $b$ is a fixed vector. Equivalently, it preserves every affine combination. ^[ref]

An original planar example isIt stretches horizontally, applies a shear mapping, and then translates. The vertices $A=(0,0)$, $B=(2,0)$ and $C=(0,2)$ become $A^{\prime }=(1,-2)$, $B^{\prime }=(5,-2)$ and $C^{\prime }=(3,0)$. A square grid becomes a grid of parallelograms; straight edges and their weighted points still fit together.

This map is not a linear map with respect to the displayed origins, because $T(0,0)=(1,-2)$ rather than zero. It also is not a Euclidean similarity: $AB$ has length $2$ and $A^{\prime }{B}^{\prime }$ has length $4$, while the right angle at $A$ becomes $45^{\circ }$. Its determinant is $2$, so it doubles planar area. These metric changes explain why affine geometry studies straightness, incidence and weighted points rather than treating every transformation as an isometry.

The linear part is an invertible matrix, giving the concise inverse $T^{-1}(u,v)=((u-v-3)/2,v+2)$. More generally an affine map is bijective exactly when its linear part of an affine map is bijective. Between equal finite-dimensional spaces, this means an invertible matrix; the translation cannot repair a loss of rank of a linear map.

The cofactor matrix consists of the signed minors of a matrix. It is the transpose of the adjugate matrix. The inverse expression applies to invertible matrices; the definition by minors also applies to singular matrices. Cofactors transform oriented hypersurface area vectors in a change of variables formula.

A popular sum of a finite set $A$ is an element having at least a specified number $u$ of ordered representations as $a+b$, with $a,b\in A$. Popular sums define a dense bipartite graph when additive energy is large. Since the total representation count is $|A{|}^2$, there are at most $|A{|}^2/u$ popular sums.

For every $\theta >0$, some $\gamma (\theta )>0$ has this property: any nonempty finite $A\subseteq \mathbb{Z}$ with $E(A)\ge \theta |A{|}^3$ has an arithmetic progression of length at least $|A{|}^{\gamma (\theta )}$ in $2A-2A$. The small-difference-set form of the Balog-Szemerédi-Gowers theorem, Ruzsa modelling lemma, cyclic Bogolyubov lemma and nonwrapping progression in a cyclic Bohr set prove it. An order-eight Freiman s-isomorphism suffices to lift the progression because its consecutive second-difference equations expand into equalities of eight-term sums.

For a set of shifts $H$ and a map $\theta$ into a circle group, this additive energy counts quadruples whose shift sums and frequency sums both agree. It is the additive energy of the graph of $\theta$ in the product abelian group.

An accelerated life family consists of survival distributions related by positive time scaling: $T_z$ has the distribution of $c_z{T}_0$. Therefore the survivor function is $S_z(t)=S_0(t/c_z)$ and, for densities, the hazard function is $h_z(t)=c_z^{-1}{h}_0(t/c_z)$. This need not be a constant hazard ratio.

For a minimally coupled scalar field with spatial induced metric $h_{ij}$, the field's derivative along the future unit normal is $\Pi$. If the shift vector enters as $d{x}^i-N^{i}dt$, then $\Pi =(\dot{\varphi }+N^i{\partial }_i\varphi )/N$. The canonical momentum of the density is ${\pi }_{\varphi }=\sqrt{h}\Pi$. Confusing this density with the normal derivative loses a volume factor.

The velocity induced by planar vorticity is a convolution with this kernel, whose sign depends on the stream function and vorticity conventions. Its magnitude is $(2\pi |z|{)}^{-1}$ and its derivative is bounded by $C|z{|}^{-2}$. Splitting the convolution at radius $R$ gives $\Vert K\ast \omega {\Vert }_{\infty }\le R\Vert \omega {\Vert }_{\infty }+\Vert \omega {\Vert }_1/(2\pi R)$. The same singularity yields a log-Lipschitz modulus, enough for a Yudovich characteristic flow.

A bounded gravitational system in statistical equilibrium satisfies the time-averaged virial theorem. For inverse-distance gravity with negligible boundary stresses, $2K+W=0$. This relation does not require static particles or a homogeneous density profile, and reaching it does not erase the system's dynamical history.

Two nonzero elements with no common nonunit irreducible factor. In a unique factorization domain this is equivalent to their greatest common divisor being a unit. Coprimality of numerator and denominator makes a fraction reduced, so simultaneous restrictions on its denominator can force it to be a regular function.

A trigonometric function defined where $\cos x\ne 0$. It has real simple poles at $(n+1/2)\pi$, and $\tan ((n+1/2)\pi +\delta )=-\cot \delta$. This local pole identity generates the large roots near tangent poles.

This is the travelling salesman problem on a complete graph with nonnegative symmetric distances satisfying the triangle inequality. Repeated visits can be shortcut without increasing cost. Assigning distance one to edges of an input graph and two to its nonedges creates a metric instance with a tour of cost at most the number of vertices exactly when the graph has a Hamiltonian cycle. Thus the metric problem remains NP-hard.

The best mean-square linear predictor from a semi-infinite past is the orthogonal projection onto the closed linear span of past observations in the random-variable Hilbert space. For a causal invertible ARMA representation, past observations and past innovations generate the same closed span. The next linear innovation is orthogonal to it and is therefore the prediction error.

The best mean-square linear predictor is orthogonal projection onto the span of the available centered observations, plus the known mean. Orthogonal innovations give a convenient basis of that span. Prediction at a future horizon must use only actually observed variables; it need not be a one-step predictor.

For a differentiable normalized eigenstate of a differentiable self-adjoint Hamiltonian, differentiate $H\psi =E\psi$ and pair with $\psi$. The terms involving ${\psi }^{\prime }$ cancel because $H$ is self-adjoint, leaving the stated formula. It turns energy derivatives into observable expectations. A smoothly chosen eigenbranch and suitable operator-domain regularity are required, particularly at degeneracies or crossings.

Writing the wavefunction in modulus and dimensionless quantum phase gives a probability continuity equation and a Hamilton-Jacobi equation with a quantum potential:The phase equations hold locally away from wavefunction nodes. They follow by equating real and imaginary parts of the Time-dependent Schrödinger equation.

Represent each person's successive eligible episodes by separate left-open, right-closed intervals, retaining a common patient identifier and event-order information. A recurrence ends an interval and can be followed by another one; administrative censoring ends the final interval without an event. The row's entry time determines risk set membership. Multiple rows remain one patient history, with possible within-patient recurrent-event dependence.

A gap-time model uses time since the previous recurrence as the episode clock, instead of calendar time since treatment or study entry. Subtract the previous event time from both interval endpoints. Eligibility restrictions become delayed gap entry, and patient/episode labels preserve the actual order of recurrent events. Risk sets on this clock compare eligible episodes of a common gap age, not patients at one common calendar time.

Repeated episodes from one person share susceptibility, treatment and history, so row-wise independence is generally inappropriate. Patient-clustered sandwich covariance matrices estimate uncertainty using whole-patient score contributions under a suitable working model. Shared frailty models instead model persistent latent heterogeneity. These methods require independent sampling units at the patient level and do not necessarily estimate the same marginal versus conditional effect.

A survival concordance index measures agreement between predicted risk ordering and observed event ordering over comparable subject pairs, with half credit for prediction ties. A pair is ordinarily comparable when the earlier observed follow-up ends in an event. Censoring-adjusted versions can reduce dependence on follow-up patterns. Good discrimination does not imply good survival prediction calibration or correct proportional hazards.