For a holomorphic Banach algebra-valued function, put $M_r(f)={\sup }_{|z|=r}r(f(z))$. Apply the Hadamard three-circle theorem to the holomorphic powers $f^{2^n}$. The continuous functions $\Vert f(z{)}^{2^n}{\Vert }^{1/2^n}$ decrease to the spectral radius. Supremum convergence for decreasing continuous functions passes the boundary suprema to the limit, proving the inequality with the same logarithmic interpolation parameter. In particular, at the geometric-mean radius, the square of the middle spectral radius is at most the product of the two boundary suprema.

A 4f pulse shaper uses a diffraction grating to disperse an input pulse, a lens to focus its frequencies at different positions in a shared Fourier plane, a spectral mask or spatial light modulator there to modify them, and a second lens and grating to recombine them. In the symmetric idealization the distances are $f,f,f,f$, with the lenses separated by $2f$. Shaping the spatially resolved spectrum implements spectral pulse shaping rather than a direct time-domain mask.

Phylogenetic species concepts use ancestry and diagnostic inherited differences; versions differ over requiring a clade, an exclusive ancestry pattern at genetic loci, or a distinguishable character combination. State which criterion is intended. ^[ref]

For an original operational example, suppose a phylogenetic tree inferred from one genetic locus separates two sampled biological populations. Compare further genetic loci and additional local samples before interpreting that branch as a species boundary. Otherwise the test may merely identify the particular ancestry of the sampled alleles. Adding more sites from the same genetic locus improves that locus's evidence without creating an independent ancestry test.

A morphological species concept distinguishes species using morphology: measurable differences in form or structure. It can be applied to preserved specimens and fossils when mating cannot be observed. Similar appearance need not imply the same species, and different appearance need not imply different species. ^[ref]

An original operational example is to compare a chosen structure in specimens of comparable age, sex and rearing conditions. Suppose two apparent groups differ in size, but one group contains only juveniles. The observed difference does not yet diagnose two species. Specify the measured phenotype, variation within each sample, and whether the distinction persists across comparable specimens. A single striking specimen is a poor substitute for that comparison.

The biological species concept groups naturally interbreeding biological populations with viable, fertile offspring, distinguished from other such groups by reproductive isolation. An operational test examines mating, fertilization, offspring survival and offspring reproduction, rather than only whether two adults can be paired. Sexual reproduction is its scope: fossils and organisms studied only through asexual reproduction cannot supply the same mating evidence. ^[ref]

For an original hypothetical example, imagine two separated biological populations that reproduce successfully after contact. The separation alone did not demonstrate reproductive isolation. Conversely, failure to mate in one unfamiliar chamber could reflect the assay conditions. Field observations of gene flow and tests under suitable conditions address the biological question more directly. A successful laboratory pairing does not establish frequent interbreeding in nature.

A spacetime extension is an isometric embedding of a spacetime as a proper open subset of a larger connected spacetime with a nondegenerate Lorentzian metric. The allowed differentiability class must be specified: analytic, smooth and low-regularity extensions are different questions. Divergent curvature can obstruct a regular extension even when an incomplete geodesic has finite affine length.

A spacetime event is an idealized occurrence at one point of spacetime, such as a light pulse reaching a detector. An inertial frame assigns spatial coordinates and a time coordinate to it. Different frames can assign different coordinates to the same spacetime event; the event itself is not changed by a coordinate transformation.

A complete entry comprising a sort key together with any associated data. In $(2,A)$, the sort key is $2$ and $A$ stands for the associated data or an identity label. Sorting permutation preservation concerns these complete entries, rather than only their extracted sort keys.

The solar wind is the outward plasma flow from the Sun, carrying an interplanetary magnetic field. Its pressure and field compress intrinsic planetary magnetospheres, form magnetotails, and can drive an induced magnetosphere around a planet with no appreciable intrinsic field.

Firn is compacted, recrystallized old snow that has not yet become dense glacier ice. Its porosity affects meltwater infiltration and heat transfer.

For a specified finite prime factor base $B$, a nonzero integer is $B$-smooth if every prime dividing its absolute value belongs to $B$. Negative residues are handled by including the sign $-1$ in their exponent vectors. This notation refers to a finite prime set, rather than the alternative convention in which a numerical smoothness bound is called $B$.

Fix $\eta =1/k$ with $k\ge 2$. Choose $k$ primes in $[H^{\eta -{\eta }^2},H^{\eta }]$, repetitions allowed, and an integer cofactor $m$ so that $m{p}_1\cdots p_k\le H$. The product is between $H^{1-\eta }$ and $H$, so $m\le H^{\eta }$ and all resulting integers are $H^{\eta }$-smooth. The Mertens theorem for reciprocal primes makes the reciprocal prime sum tend to $-\log (1-\eta )>0$. Since $\lfloor H/(p_1\cdots p_k)\rfloor \ge H/(2p_1\cdots p_k)$, there are ${\gg }_{\eta }H$ ordered representations. Each integer has only $O_{\eta }(1)$ prime factors in that interval, counted with multiplicity, so representation multiplicities are bounded by a constant depending on $k$. This proves the density bound. For arbitrary fixed positive $\eta$, choose a smaller reciprocal-integer exponent.

// Target: number-theory.bigb

The six pentads are the six one-factorizations of the complete graph on six points. Permuting the ground-set points permutes these factorizations. The action sends a transposition to three disjoint transpositions and a $3$-cycle to two disjoint $3$-cycles. Its restriction to $A_6$ is faithful by simplicity, and the full action of $S_6$ is faithful because an order-two kernel would be central. It is consequently an automorphism of $S_6$. Its restriction preserves $A_6$ and changes $3$-cycle type, so it cannot be induced on $A_6$ by any conjugation from $S_6$.

A tag SNP represents other nearby single-nucleotide polymorphisms whose alleles are strongly correlated through linkage disequilibrium. Tagging reduces the number of markers needed for association mapping; it does not establish that the tag itself causes the phenotype.

// Target: biology.bigb

For an ideal open ion channel with constant electrical conductance $g$, membrane potential $V$ and reversal potential $E_{\mathrm{rev}}$, define outward electric current as positive:This is an Ohm's law model of the open state, not a universal channel law or a rule for opening probability. In the synthetic figure, $g=10\mathrm{pS}$ and $E_{\mathrm{rev}}=0\mathrm{mV}$. Because $1\mathrm{pS}\cdot 1\mathrm{mV}=10^{-3}\mathrm{pA}$, the open-state electric currents at $-80$ and $-40\mathrm{mV}$ are $-0.8$ and $-0.4\mathrm{pA}$. Zero electric current at the reversal potential does not imply a closed ion channel. Real ion channels can have nonlinear responses, so the straight line is an explicit idealization.

A conical pendulum is a pendulum whose bob executes uniform horizontal circular motion beneath a fixed pivot, with its string at a constant angle $\theta$ to the downward vertical. The string sweeps a right circular conical surface; the bob follows its circular rim, without any physical cone supporting it. The ideal model has a point mass $m>0$, a taut, massless, inextensible string of length $\ell >0$, uniform gravitational acceleration of magnitude $g>0$, and negligible drag force and pivot friction. A nonzero circular motion requires $0<\theta <\pi /2$. This geometry and the corresponding forces are described in Richard Fitzpatrick, The conical pendulum, equations 262–267.

The right triangle formed by the pivot, the orbit centre, and the bob gives $r=\ell \sin \theta$ and $h=\ell \cos \theta$. Here $r$ is the orbit radius and $h$ is the vertical drop from the pivot to the bob's horizontal plane, not the bob's height above a table. Write $T$ for tension, $P$ for the oscillation period of one revolution, and $\omega =2\pi /P$ for the angular speed; then the bob's speed is $v=\omega r$. A side view compresses the horizontal circle to a line segment; the top view reveals the circle and its tangent velocity.

The two actual forces on the bob are tension toward the pivot and gravitational weight vertically downward. The vertical acceleration is zero, while the horizontal centripetal acceleration is $v^2/r={\omega }^{2}r$. Thus Newton's second law givesThe inward net force is the horizontal component of tension, rather than an extra interaction to add to these two forces. Constant speed still requires acceleration because the velocity changes direction. ^[ref]

Dividing the two component equations and using $r=\ell \sin \theta$ gives the useful collection of predictionsIf $\varphi$ is the azimuthal angle, then $\dot{\varphi }=\pm \omega$; either direction of rotation satisfies ${\dot{\varphi }}^2\cos \theta =g/\ell$. The mass cancels from the angular speed and oscillation period, but not from the tension. The motion is a relative equilibrium: in a rotating frame following the bob at angular speed $\omega$, its position is fixed. This is a concrete way to see how rotational symmetry can turn an apparently moving solution into a stationary one.

A signed spatial filter model of a simple cell assigns a weight $K_{ij}$ to each sampled location and takes its dot product with the visual contrast image $C_{ij}$. Positive weights favor bright features; negative weights favor dark features. The signed driveis a linear function of the image, while the rectified linear unit output $R$ is nonlinear. $D$ and $R$ below are arbitrary model units. To obtain a predicted firing rate one would need a calibrated gain, background response and possibly a different output function. The model does not generate a spike train; a linear-nonlinear-Poisson cascade model adds a stochastic event-generation assumption.

For the first diagram, every one of five rows has weightsA bright central column, with $C_{i3}=1$ and zero elsewhere, gives $D=5/5=1$. A bright middle row gives $D=-1/20-1/20+1/5-1/20-1/20=0$; a uniformly bright patch also gives $D=0$, since every row sums to zero. Move the bright column one site right: $D=5(-1/20)=-1/4$, so $R=0$. Make that shifted column dark instead: $D=1/4$ and $R=1/4$. Thus an OFF-like weight can increase output for darkness. A negative weight is a descriptive sensitivity, not evidence for a particular inhibitory synapse.

For a smooth example use the real, even Gabor filterThe Gaussian function localizes the weights; the cosine supplies alternating polarity. Choose ${\sigma }_x=1/2$, ${\sigma }_y=1$ and $q=\pi$ in arbitrary spatial units. The stripes' spatial frequency is $q/(2\pi )=1/2$ cycles per unit. Their axis makes angle $\theta$ with horizontal, while brightness changes across the perpendicular axis. $\phi$ is their spatial phase. Normalize the full-plane integral of $K{C}_{\theta ,\phi }$ by its value at $\theta =\pi /2$, $\phi =0$. The Gaussian integral and the product identity for cosines givewhereAt $\phi =0$, horizontal stripes give $R\approx 0.004159$, $60^{\circ }$ stripes give $R\approx 0.286741$, and vertical stripes give $R=1$. At the same vertical orientation, $\phi =\pi /2$ gives $D=R=0$, while reversing the polarity with $\phi =\pi$ gives $D=-1$ and $R=0$. The preferred axis persists while the response changes dramatically with position. Shifting the vertical stripes by a quarter of their wavelength changes phase by $\pi /2$ up to sign, without rotating them.

These are two different toy filters, not discretizations of the same fitted receptive field. The discrete filter exactly rejects a uniform contrast patch; this Gabor filter need not reject uniform contrast because its weight integral need not vanish. Neither toy includes stimulus history, recurrent interactions, saturation or a measured response baseline. Their value is a transparent prediction: orientation selectivity can emerge from spatial summation, and it does not by itself imply tolerance to phase.

A renovating shear flow holds a planar shear flow fixed for an interval $T$, then chooses a fresh orientation independently of preceding intervals. Each interval acts on a material line by a rotated copy of the displayed deformation gradient. The orientation law must be specified: independent uniformly distributed orientations define the isotropic model, whereas a biased orientation law generally gives another Lyapunov exponent.

A filter $\mathcal{F}$ on a set $X$ is a nonempty family of subsets that excludes the empty set, is closed under finite intersections, and is upward closed under inclusion.

Every nonempty set $A$ has an element $x\in A$ such that $x\cap A=\varnothing$.