For planar vectors $u=(u_x,u_y)$ and $v=(v_x,v_y)$, the scalar $u\times v=u_x{v}_y-u_y{v}_x$ is the determinant of their coordinate columns. It equals the perpendicular component of the three-dimensional cross product after embedding the plane in three dimensions. Its sign records orientation, and its absolute value is twice the area of the triangle spanned by $u,v$.

Transfer of offspring between providers to separate offspring origin from rearing environment. Egg composition, prenatal influences and handling require additional controls.

A systematic difference associated with sample processing groups rather than the biological contrast of interest. Balanced allocation and recorded metadata help separate these effects; statistical adjustment cannot fully recover a contrast perfectly confounded with batch.

A before-after control-impact design compares temporal change at an affected site with change at one or more control sites. In a simple two-period description, its treatment contrast isThis is a difference-in-differences comparison, or a site-type-by-period interaction term. For example, responses falling from $12$ to $8$ at the affected site and from $10$ to $9$ at the control site give $\Delta =-3$: the affected site fell three units more than the comparison site.

The design addresses both a pre-existing site difference and a common temporal change. Its causal interpretation still requires a defensible comparison for how the affected site would have changed without the disturbance. Another site-specific event coinciding with the intervention can confound the contrast. Multiple control sites and repeated sampling before and after the disturbance help characterize natural variation, while the analysis must account for dependence through time.^[ref]

Several observations within one affected site do not become several independent impact-site replicates. Distinguishing the experimental unit from its observational units prevents pseudoreplication; a large within-site sample does not repair missing independent site replication.

Relevance of a test’s cues, actions and setting to the processes of interest outside that test. A task can be controlled yet poorly suited to a species’ sensory or motivational capacities.

Whether an operational measurement supports the intended theoretical interpretation. Successful hidden-mark localization supports a narrower claim than a general claim about consciousness.

A control reproducing handling or other incidental aspects of an intervention while omitting its intended active feature. A sham mark should match tactile and olfactory properties while differing in visibility.

Withholding treatment or hypothesis information from participants, handlers or scorers where appropriate to reduce expectation-dependent differences. What was blinded and which cues remained available should be reported.

A dietary control in which a comparison group receives an amount of food matched to the intake of another group. The timing, energy density and method of matching determine which intake-related explanations it addresses.

A negative control is a comparison expected to lack the response attributed to the tested cause. In a generic analytical measurement, a blank without the target substance reveals background signal or contamination introduced by reagents and handling. A vehicle control can isolate effects of the carrier used to deliver an intervention. A clean negative control supports interpretation but cannot prove that every unwanted influence is absent; the control must share the relevant conditions with the test samples.^[ref]

A positive control is a comparison expected to produce a known response under the measurement or experimental system. It checks that the system can detect the effect it is meant to measure. For a generic assay, a reference sample known to contain the target should yield a signal; failure makes a negative test-sample result difficult to interpret. A positive control and a negative control answer complementary questions: whether the system can respond, and whether a response appears without the target cause.^[ref]

A dermal skeleton organized into relatively large plates or bones, contrasting with a covering dominated by small separate scales or odontodes.

For opposite equal pump wave vectors, a conjugate mixing source has wave vector $-{\mathbf{k}}_3$. In an isotropic nondispersive medium this exactly matches a free backward wave at the generated frequency only in the degenerate case. Away from degeneracy the longitudinal mismatch has magnitude $2n|{\omega }_P-{\omega }_3|/c$; a finite interaction length supplies a tolerance, not exact vector matching.

Because one wave travels backward, its incident boundary condition belongs at the opposite end of the medium. With $u(0)=a$ and $v(L)=0$, differentiating once gives $u^{\prime \prime }=-{\kappa }^{2}u$ and the solutionThe invariant $|u{|}^2+|v{|}^2$ is constant along $z$, while the input/output powers obey transmission minus reflection equal to the incident power in this normalization. A proposed solution must satisfy both first-order equations, not just their second derivatives.

For fully dissociated electrolytes at the same temperature and solvent permittivity, linear screening theory gives ${\lambda }_D^{-2}\propto {\sum }_i{c}_i{z}_i^2=2I$. A divalent symmetric salt of formula concentration $c$ has $I=4c$, not $c$ or $2c$. ion association and nonideal concentrated-solution effects limit this comparison.

antidamping is a velocity-proportional term that supplies rather than removes mechanical energy. For the displayed damped harmonic oscillator with negative damping coefficient, $d[({\dot{x}}^2+{\omega }_0^2{x}^2)/2]/dt=-2\lambda {\dot{x}}^2\ge 0$. causality of a driven solution and its long-time stability are different properties.

For nearest-neighbor spin-singlet pairing on a square lattice, opposite signs of the pairing amplitude on the two bond directions give the displayed momentum-dependent superconducting gap function. It changes sign under a quarter-turn and has nodes on the zone diagonals. This sign structure is distinct from a nonnegative gap whose magnitude is merely anisotropic. A real pure $d_{x^2-y^2}$ pattern does not by itself require broken time reversal.

A superconductor whose pairing order parameter has the angular or crystal-symmetry structure of d-wave superconductivity. The sign of the gap depends on momentum direction and nodes can permit low-energy fermionic Bogoliubov quasiparticles.