The pulmonary trunk is the common artery leaving the right ventricle through the pulmonary valve before dividing into the vessels supplying the two lungs. It is the initial shared segment of the pulmonary artery route in the pulmonary circulation. The name artery describes flow away from the heart; the blood in the pulmonary trunk has not yet acquired oxygen in the lungs.^[ref]

At a junction with negligible storage, conservation of mass givesA constructed total blood flow of $5$ L/min could therefore divide into $3$ and $2$ L/min. Measuring the total does not establish equal flow to the two lungs. In a simple steady model with a common upstream and downstream pressure, branch blood flow rates depend on their respective vascular resistance values; the shared inlet does not force the resistances to be equal.

The pulmonary trunk also supplies an important distinction between two measurements. Right ventricular ejection fraction describes change in a chamber volume, whereas forward blood flow in the pulmonary trunk describes transport through a vessel. They agree through stroke volume and heart rate only under the relevant balance assumptions, including negligible backward flow. An anatomical connection is therefore the starting point for a flow model, rather than a substitute for its assumptions.

Pseudomonas syringae pv. tomato DC3000 is the name used for a well-studied bacterial strain in experimental interactions with tomato and Arabidopsis thaliana. The abbreviation pv. denotes pathovar, a classification related to pathogenic characteristics, while DC3000 identifies the particular strain. Those parts of the label carry more information than the species name Pseudomonas syringae alone.^[ref]

The 2003 genome report described about 6.5 megabases of DNA, comprising one circular chromosome and two plasmids, with 5763 predicted open reading frames. The assembly and annotation created a reference for comparing molecular observations, rather than merely attaching a name to a culture.^[ref] A predicted open reading frame is not automatically an experimentally established protein function.

In a 2011 plant-community experiment, this strain supplied a common challenge against which different bacterial colonists were compared. Both pathogen population and host disease symptoms were measured.^[ref] Holding the challenge strain fixed makes differences between colonist treatments easier to interpret. It does not make the experiment representative of every Pseudomonas syringae strain or every host.

The real projective line ${\mathbb{RP}}^1$ consists of one-dimensional vector subspaces of ${\mathbb{R}}^2$, written $[x:y]$ with $(x,y)\ne (0,0)$ and $[x:y]=[tx:ty]$ for every nonzero real $t$. The chart $y\ne 0$ gives the affine coordinate $x/y$, while $[1:0]$ is one added point at infinity.

A unit vector $(\cos \theta ,\sin \theta )$ and its negative determine the same line, so the angular coordinate is taken modulo $\pi$. This quotient is itself a circle: the map sending that line to $(\cos 2\theta ,\sin 2\theta )$ is well defined and gives a homeomorphism with $S^1$.

Hence the real projective line is compact and can be viewed as the one-point compactification of the real line. Both $+\infty$ and $-\infty$ approach the same added projective point, unlike the two distinct endpoints of an extended closed real interval.

A projective subspace of $\mathbb{P}(V)$ is $\mathbb{P}(W)$ for a nonzero vector subspace $W\subseteq V$. Its projective dimension is $\dim W-1$. Thus one-dimensional vector subspaces represent projective points, while two-dimensional vector subspaces give projective lines.

For example, in ${\mathbb{P}}^3(K)$ the linear equation $x_3=0$ defines a projective plane. Adding the independent equation $x_2=0$ leaves a projective line. A consistent homogeneous linear system therefore describes a projective subspace through the nonzero vectors in its linear solution space.

If two corresponding vector subspaces have zero intersection, their projectivizations are disjoint; otherwise their intersection is $\mathbb{P}(W_1\cap W_2)$. The vector-space dimension formula explains why two projective lines in a plane always meet, whereas two lines in projective three-space can be skew. Nonlinear homogeneous equations instead define more general projective varieties.

In finite projective space $\mathrm{PG}(3,q)$ with $q>2$, an ovoid is a set of $q^2+1$ points with no three collinear. At each of its points there is a unique tangent plane meeting the set only at that point. It is a finite-geometric analogue of a strictly convex surface, although a definition using real convexity does not make sense over a finite field.^[ref]

The elliptic quadric supplies examples. Over ${\mathbb{F}}_3$, the equation $x_0^2+x_1^2+x_2^2=x_3^2$ defines ten projective points. In the chart $x_3=1$, exactly one of the first three coordinates is nonzero, giving six points. At infinity all three are nonzero, giving four more after quotienting by simultaneous sign. Thus the count is $6+4=3^2+1$. This example links the incidence condition to a concrete homogeneous polynomial equation.

Over a finite field ${\mathbb{F}}_q$, an elliptic quadric $Q^{-}(2n+1,q)$ is the zero set in projective space $\mathrm{PG}(2n+1,q)$ of a nonsingular quadratic form on a $(2n+2)$-dimensional vector space whose Witt index is $n$. Thus the largest vector subspaces on which the form vanishes identically have dimension $n$, one less than for a hyperbolic form in the same dimension. The word elliptic describes this type of quadratic form.

The especially important case $Q^{-}(3,q)$ contains $q^2+1$ points and no projective line. A line therefore meets it in at most two points, making it an example of an ovoid. These properties make elliptic quadrics useful configurations in finite geometry and in the study of incidence ranks.^[ref]

For example, in $\mathrm{PG}(3,3)$ the equation $x_0^2+x_1^2+x_2^2=x_3^2$ gives an elliptic quadric. With $x_3=1$ there are six solutions, since exactly one of the first three coordinates is nonzero. With $x_3=0$ all three must be nonzero; the eight vectors give four projective points after identifying scalar multiples. There are ten points altogether, as $3^2+1$ predicts.

A product-null spectral contrast is a weighted combination of spectral measurements that cancels a modeled product contribution while retaining sensitivity to the carrier of interest. It is useful when carrier consumption and product formation both change the absorbance recorded in an inhibited autoxidation assay.

After background correction and with path length included in the response coefficients, write a measurement vector asHere $\mathbf{d}$ and $\mathbf{p}$ are known carrier and product spectral response columns. A contrast ${\hat{c}}_D={\mathbf{w}}^{\mathsf{T}}\mathbf{A}$ removes the product and has unit carrier response whenIf the columns are proportional, these constraints are incompatible. More measurements do not help if they add no independent spectral information.

An original three-channel example uses scaled coefficient columns $\mathbf{d}=(3,1,0{)}^{\mathsf{T}}$ and $\mathbf{p}=(1,1,1{)}^{\mathsf{T}}$. The weights $\mathbf{w}=(5,-1,-4{)}^{\mathsf{T}}/14$ satisfy both constraints. Stipulated concentrations $(c_D,c_P)=(0.2,0.1)$ give $\mathbf{A}=(0.7,0.3,0.1{)}^{\mathsf{T}}$ and the contrast returns $0.2$. The coefficient and concentration units have been absorbed into the scaling; these are not measured dye coefficients.

For independent equal-variance channel noise ${\sigma }^2$, the example has variance ${\sigma }^2\Vert \mathbf{w}{\Vert }^2=3{\sigma }^2/14$. These are the minimum-variance weights under the stated noise model: project $\mathbf{d}$ perpendicular to $\mathbf{p}$, then normalize that residual to unit carrier response. The Cauchy-Schwarz inequality shows that any other vector with the same constraints has at least as large a norm. A two-channel difference using just the first two measurements also isolates the carrier, but weights $(1/2,-1/2,0)$ give the larger variance ${\sigma }^2/2$.

For general covariance matrix $\Sigma$, the variance is ${\mathbf{w}}^{\mathsf{T}}\Sigma \mathbf{w}$, so correlated or unequal noise changes the optimal weighting.^[ref] The coupled chemical readout remains the one described by the primary signal-carrier method.^[ref] Additional products or changing protonation require additional response columns; a cancellation calibrated for one product does not remove every unknown spectral contribution.

With binary assignment or encouragement $Z$, potential treatment receipt $D(z)$ is the treatment a unit would actually receive under assignment $z$. It differs from the assignment itself when there is noncompliance. Only $D=D(Z)$ is observed for that unit, under the usual consistency in causal inference convention.

The pair $(D(0),D(1))$ defines four principal strata: never-takers $(0,0)$, compliers $(0,1)$, defiers $(1,0)$ and always-takers $(1,1)$. A patient who takes a drug only when encouraged is a complier; one who obtains it under either assignment is an always-taker. Randomization of $Z$ does not itself randomize actual treatment receipt. If $D(1)\ge D(0)$ for every unit, defiers are excluded andunder randomized assignment. Thus the effect of encouragement on uptake identifies the complier fraction, not automatically the effect of receiving treatment on outcomes.^[ref]

Pressure-tap position calibration determines the locations assigned to pressure tappings relative to a specified model coordinate system, with corrections and uncertainties. It is a geometrical measurement calibration, distinct from calibrating the pressure transducer. A correct pressure reading cannot compensate for assigning it to the wrong point on a steep pressure coefficient profile.

A proposed practical procedure measures each hole center against visible reference marks, establishes the leading edge and airfoil chord used for normalization, and records the coordinate transformation. A calibrated imaging scale or translation stage can provide the comparison. Record repeated readings and shared datum errors separately, and check that the measured view represents the intended surface coordinates rather than an oblique projection. This is a proposed metrological procedure, not an assertion that a particular historical airfoil experiment performed it.

An affine pressure-tap coordinate calibration has the simple formwhere $r$ is the instrument coordinate, $a$ an offset and $b$ a scale. With an independently established $x_{\mathrm{LE}}=0$, $c=200\mathrm{mm}$, $a=0.20\mathrm{mm}$ and $b=1.001$, a nominal reading $r=100\mathrm{mm}$ gives $x=100.30\mathrm{mm}$ and $\xi =0.5015$, rather than $0.5000$. For a locally linear pressure coefficient slope of $10$, that coordinate discrepancy corresponds to $\delta C_p=0.015$.

Reference sharing matters. If the same affine map determines the tap, leading-edge and trailing-edge coordinates, its common offset and scale cancel inThis cancellation does not remove distortion, uncertain feature identification or non-affine errors. It shows why the references and their covariances must be part of the measurement model, rather than independently attached after normalization.^[ref]

Robert M. Pinkerton's 54-orifice airfoil experiment illustrates why spatially distributed pressure data matter: the experiment compared measured distributions with potential flow predictions.^[ref] A position check tests the coordinate assignment, whereas additional taps address quadrature ambiguity of pressure-tap data; neither task substitutes for the other.

Correlated position errors of pressure taps occur when uncertainty in one tap coordinate is statistically related to uncertainty in another. A shared origin, common scale or common image transformation can couple all inferred positions. Treating every tap as independent can either overestimate or underestimate the uncertainty in a reconstructed pressure load, depending on the sensitivities and their signs.

For fixed reconstruction weights $w_i$ and a differentiable pressure coefficient profile, the linearized position contribution isHere ${\epsilon }_i$ is dimensionless coordinate error and ${\Sigma }_{\xi }$ its covariance matrix. The general covariance propagation law is the same one used for other correlated inputs in a measurement model.^[ref]

An original two-tap example takes equal coordinate standard uncertainties $s=0.002$ and correlation coefficient $r=0.9$. With sensitivities $\mathbf{b}=(1,-1)$,Discarding correlation gives $u_Q=\sqrt{2}s\simeq 0.002828$, which is too large here. For $\mathbf{b}=(1,1)$ instead, $u_Q=\sqrt{2s^2(1+r)}\simeq 0.003899$; the same independence assumption is now too small. Positive correlation alone does not determine the direction of the error in the output uncertainty.

A common-origin pressure-tap position error can be written ${\epsilon }_i=\delta +{\eta }_i$, with an independent shared offset of variance $s_0^2$ and independent residual errors of variances $s_i^2$. ThenThe common contribution cancels to first order when the sensitivities sum to zero. Repeatedly reading the same misplaced coordinate origin does not remove its uncertainty. Pressure-tap position calibration must therefore retain shared reference information, not just a list of per-hole scatters. If the coordinates also determine the quadrature weights or surface geometry, their changes belong in the full sensitivity calculation; the fixed-weight formula alone is incomplete.

Local contact pressure from additional spatial measurements requires more information than total normal force and two first moments. One explicit reconstruction model partitions the contact into cells of known areas $A_i$ and centroids $(x_i,y_i)$, assuming uniform pressure $p_i$ within each cell. Writing $F_i=A_i{p}_i$ givesThese are three linear measurements of the cell loads. They need not determine every $F_i$, as the nonuniqueness argument in pressure distributions with identical resultant force and center of pressure shows. The primary centroid definition specifies precisely which force/moment information is retained.^[ref]

For an original four-cell example, use equal $0.01{\mathrm{m}}^2$ cells centered at successive positions $(-a,-a),(a,-a),(a,a),(-a,a)$ with $a=0.05\mathrm{m}$. Loads $(10,20,30,40)\mathrm{N}$ give $N=100\mathrm{N}$ and centroid $(0,0.020)\mathrm{m}$. Adding $s(1,-1,1,-1)$ to these loads changes neither the sum nor either first moment. Nonnegative loads permit $-10\le s\le 20\mathrm{N}$. The alternative $(15,15,35,35)\mathrm{N}$ therefore gives the same global measurements.

An independently calibrated measurement of the first cell load resolves this one missing degree of freedom under the four-cell model. If it reads $10\mathrm{N}$, the global moment equations recover the original other three loads. Their cell pressures are $(1,2,3,4)\mathrm{kPa}$. Measuring all four cells instead provides redundant data for checking total load and the assumed geometry.

The local sensor output must actually represent a known cell-integrated force, or have a validated response model; a point sample cannot automatically be treated as an area average. Calibration errors, cell cross-coupling and imperfect coverage become additional measurement inputs. Spatial pressure-map resolution below a sensor cell remains limited even when every cell average is known: unmeasured within-cell peaks are not recovered by simply drawing a finer image. Constitutive or smoothness assumptions can add information, but must be identified as assumptions rather than extra measurements.

A diabatic representation describes coupled electronic states using a basis chosen to vary smoothly with nuclear geometry, often reducing the derivative couplings that appear in an adiabatic basis. In exchange, the electronic energy matrix generally has off-diagonal entries. It is a representation of coupled states, not a claim that the states never interact.

For a two-state example with nuclear coordinate $R$,The diagonal diabatic curves cross at $R=0$, while their coupled adiabatic eigenvalues have an avoided crossing when $V\ne 0$. This makes electron transfer and nonadiabatic reaction models easier to interpret. An exactly derivative-coupling-free basis need not exist globally for every multidimensional molecular problem.

A standard set of $N$ Chebyshev nodes on $[-1,1]$ consists of the zeros of the Chebyshev polynomial $T_N$:They are equally spaced in angle before projection onto the interval, so the resulting real nodes cluster near the endpoints. An affine transformation gives nodes on any nondegenerate finite interval.

For $N=3$, the nodes are $\sqrt{3}/2,0,-\sqrt{3}/2$. Their monic nodal polynomial iswhose maximum absolute value on the interval is $2^{1-N}$. This controls the nodal-product factor in the smooth-function polynomial interpolation error formula, explaining why endpoint clustering can be useful.

The related Chebyshev-Lobatto nodes use $\cos (j\pi /N)$, $j=0,\dots ,N$, and include both endpoints. They are a different node set; the zero-node and endpoint-inclusive conventions should not be silently interchanged.

A polymer recovery accounting basis specifies the conserved inventory used to compare initial material with recovered fractions. Possible choices include conditioned dry mass, original repeat-unit equivalents or the amount of a particular element. These bases coincide only under suitable chemical and sampling assumptions. A polymer fraction-recovery balance needs that convention before its recovery percentage has a clear meaning.

For fractions $i$ with measured masses $m_i$, suppose $k_i$ converts one unit of that product mass into the chosen original-material equivalent. Recovery on that basis iswhere $B_0$ is the initial inventory in the same units. Taking every $k_i=1$ is a chemical assumption: it requires that the measured masses are directly comparable. Different solvent contents, water incorporated by reaction, salts, fillers or filter residues can invalidate it.

An original polyglycolic acid example makes the distinction visible. Ignoring initial end-group contributions for a repeat-unit bookkeeping comparison, one millimole of ${\mathrm{C}}_2{\mathrm{H}}_2{\mathrm{O}}_2$ repeat inventory represents approximately 58.04 mg. Complete ester hydrolysis to one millimole of glycolic acid, ${\mathrm{C}}_2{\mathrm{H}}_4{\mathrm{O}}_3$, gives approximately 76.05 mg. The roughly 18.01 mg difference comes from incorporated water, not extra recovered original repeat inventory. The manufacturer's structural and decomposition diagrams identify the polymer and hydrolysis product.^[ref]

Dividing the product's 76.05 mg directly by 58.04 mg would report approximately 131.0% recovery on mixed bases. The monomer-to-original-repeat conversion is instead $k=58.04/76.05\approx 0.76318$, restoring 58.04 mg of original repeat equivalents. Exact finite-chain accounting must include the original end groups and count hydrolyzed bonds; the repeat-unit example deliberately omits that small-chain correction. Soluble oligomers contain a different number of end groups per repeat, so the monomer conversion is not universal for an unidentified soluble mixture.

Carbon-basis closure for polymer degradation offers another explicit inventory. Each repeat and each glycolic-acid molecule has two carbon atoms, so the example contains approximately 24.02 mg of carbon before and after the specified hydrolysis. If subsequent chemistry produces other carbon-containing products, those products belong in that inventory too. Carbon introduced by solvent or background contamination must be distinguished from specimen carbon; otherwise elemental closure can look satisfactory for the wrong reason.

An original fraction count begins with 1.00 mmol of repeat equivalents and recovers 0.60 mmol in the solid, 0.30 mmol in identified soluble material and 0.05 mmol in rinses. Recovery is 95%, regardless of whether the corresponding product masses differ through hydration. The remaining 0.05 mmol is unaccounted on this basis; the deficit alone does not establish whether it was chemically transformed, escaped collection or was underestimated analytically.

A useful experiment records conditioning and drying, collects filtrate and rinses, identifies the soluble products and measures blanks. Kureha's published test uses filtered, dried residue after timed exposure of material at 2% solids concentration to deionized water.^[ref] That retained-residue mass is one well-defined observable. Inferring the full chemical recovery requires the other fractions and their conversion factors. Report those assumptions separately from the molar mass distribution of whichever fraction was analyzed.

For a Poisson random measure $N$ with sigma-finite intensity measure $\mu$, a real deterministic simple integrand $f={\sum }_i{a}_i{1}_{A_i}$ on disjoint sets of finite intensity has compensated Poisson integralIts expected value is zero. Independence of disjoint counts and the Poisson variance formula give $\mathbb{E}[I(f{)}^2]=\int f^{2}d\mu$.

This isometry extends the integral by completion to deterministic $f\in L^2(\mu )$. If $f$ is not also integrable, writing two separate finite integrals $\int fdN-\int fd\mu$ can be invalid; the compensated integral is defined through the joint centered limit. For a constant-rate Poisson process, taking $f=1_{[0,t]}$ gives $N_t-\lambda t$ with variance $\lambda t$.^[ref]

For smooth planar domains and a conformal map $\Phi :D\to \Omega$ extending as a $C^1$ boundary diffeomorphism with nonzero derivative, the Poisson kernel as a density against boundary arclength satisfiesThe derivative is evaluated at the boundary point, not at the interior observation point.

To derive it, use conformal invariance of harmonic measure and change boundary variables: $d{s}_{\Phi (\zeta )}=|{\Phi }^{\prime }(\zeta )|d{s}_{\zeta }$. The two Poisson integrals must give the same harmonic extension for every continuous boundary datum, identifying the densities. Kernels defined against an angular measure instead require the corresponding measure conversion.

For $\Phi (z)=Rz$ from the unit disk to the radius-$R$ disk, the rule gives $P_{R\mathbb{D}}(Rz,R\zeta )=P_{\mathbb{D}}(z,\zeta )/R$. At the center this is $1/(2\pi R)$, whose integral around circumference $2\pi R$ is one. This normalization check detects a misplaced derivative factor.

For functions $f,g:X\to V$ taking values in the same vector space, pointwise addition is defined by $(f+g)(x)=f(x)+g(x)$ for every $x\in X$. For $f(x)=x$ and $g(x)=x^2$, the resulting function is $x\mapsto x+x^2$; at $x=2$ its value is six.

The zero function and pointwise scalar multiplication make the collection of all such functions a vector space. The vector space identities follow by evaluating at an arbitrary input and using the corresponding identities in $V$. This construction also preserves continuity and linearity when the original functions have those properties, and is the addition used in the superposition principle for a linear differential equation.

For functions $f,g:X\to \mathbb{R}$ or $\mathbb{C}$, their pointwise multiplication is the function $fg$ with $(fg)(x)=f(x)g(x)$. Both values are evaluated at the same input before multiplication. If $f(x)=x+1$ and $g(x)=x^2$, then $(fg)(2)=3\cdot 4=12$ and $(fg)(x)=x^3+x^2$.

Together with pointwise addition, this makes the continuous functions on a topological space into an algebra over a field. Closure properties still need checking for other function spaces: two L2 space functions need not have an L2 space product. For example, $f(x)=g(x)=x^{-1/3}$ on $(0,1)$ have integrable squares, but $(fg{)}^2=x^{-4/3}$ is not integrable. The ambient function space matters.

A podocyte foot process is a fine extension of a podocyte that lies against the glomerular basement membrane. Neighboring podocyte foot process structures interdigitate, leaving narrow filtration slits crossed by a slit diaphragm. The podocyte is one whole epithelial cell; its many processes collectively shape part of the glomerular filtration barrier.

An electron microscope section makes the distinction concrete: identify the basement membrane, then the small profiles resting on its outer side, then the gaps between those profiles. A profile is not necessarily a separate cell; multiple profiles can belong to branches of one podocyte. Conversely, the gap between two profiles is not itself a podocyte foot process.

In a 1999 study, immunogold labeling placed nephrin at the slits between these processes. Figure3 includes sections across the slit and a tangential view with a row of gold particles. Combining the label with the local geometry helped locate a protein at a junction rather than merely somewhere in the glomerulus.^[ref]

Wheat refers to cereal grasses of the taxonomic genus Triticum, cultivated for their grain. Bread wheat, Triticum aestivum, exhibits allopolyploidy: its somatic cells contain six chromosome sets, conventionally written AABBDD, totaling $2n=42$. The letters distinguish three related ancestral genomes, each represented twice.^[ref]

This ancestry creates a problem during meiosis. A chromosome must normally pair and exchange DNA with its true homologous chromosome, rather than with a related chromosome from a different ancestral genome. The Ph1 locus helps stabilize that choice. In experiments comparing wheat with and without this locus, researchers used chromosome imaging and markers of chromosome synapsis and chromosomal crossover. Loss of Ph1 delayed synapsis relative to the stage at which chromosome ends cluster, allowing more opportunity for pairing between related chromosomes. The experiments also changed growth conditions and measured resulting crossovers.^[ref]

Polyploidy therefore involves more than carrying extra DNA: successful reproduction requires mechanisms that organize related chromosomes into reliably inherited sets.