Current-source swapping in a resistance ratio measurement exchanges two excitation sources between the sensor and reference resistor, allowing an unequal-current contribution to be identified or canceled. The combining rule depends on the circuit topology. In the separate-branch model of branch-current mismatch in a resistance ratio measurement, the exact combination is a geometric mean.

Let $q=R_T/R_{\mathrm{ref}}>0$ and let the two positive source currents have a fixed ratio $k=I_A/I_B$. With source A on the sensor and B on the reference, the measured potential ratio is $r_A=qk$. After exchanging their assignments, $r_B=q/k$. HenceThis assumes that the resistances, current outputs, potential-sensing gains and relevant thermal conditions are unchanged across the two acquisitions. It cancels the multiplicative source ratio, not an additive readout error or an inaccurate assigned reference resistance.

For an original example, let $q=0.2$ and $k=1.1$. The acquisitions give $r_A=0.22$ and $r_B=0.181818\dots$. Their geometric mean is exactly $0.2$. Their arithmetic mean is $0.200909\dots$, high by approximately $0.454545\%$. In general,The arithmetic mean suppresses a small mismatch to second order here, but is not the exact inverse-pair cancellation.

The primary RTD guide uses source interchange in a different three-wire lead-cancellation circuit. Its two readings are averaged according to that circuit's equations; they must not be replaced automatically by the geometric-mean rule above.^[ref]

A drift residual after current-source swapping remains if the source ratios in the two acquisitions are $k_1$ and $k_2$. For unchanged $q$, the same calculation gives $\hat{q}/q=\sqrt{k_1/k_2}$. A $0.2\%$ increase from $k_1$ to $k_2$ therefore makes the combined estimate low by $1-1/\sqrt{1.002}\approx 0.0998502\%$. Sensor self-heating in a ratiometric resistance measurement can also change $q$ between source assignments.

A proposed apparatus check switches two characterized current sources between stable sensing resistors, waits for electrical settling, and compares both individual readings and their product. Reversing the order of the two acquisitions helps distinguish a repeatable source mismatch from drift during the sequence. This tests the stated circuit model without presuming that all source-swapping circuits use the same estimator.

The pure braid group $P_n$ is the kernel of the endpoint-permutation homomorphism from the braid group $B_n$ to the symmetric group $S_n$:Each strand of a pure braid returns to its own original endpoint. It may still wind around other strands, so trivial endpoint permutation does not make the braid trivial.

For two strands, $B_2\cong \mathbb{Z}$ is generated by one crossing ${\sigma }_1$, and $P_2$ consists of its even powers. Thus $P_2\cong \mathbb{Z}$ but has index two in $B_2$. In general, the endpoint map is onto, so $[B_n:P_n]=n!$.

Geometrically, $P_n$ is the fundamental group of the configuration space of $n$ ordered distinct points in the plane. The ordering records which moving point must return to which starting point, matching the pure-strand condition.

Pressure gauge temperature error is the change in a gauge's indication error associated with a change from its specified temperature conditions, while the pressure comparison is otherwise controlled. It belongs to the complete Bourdon tube pressure gauge response, not merely the number of scale divisions. Distinguish an instrument-temperature effect from a real change in the pressure being measured.

A concrete manufacturer specification is WIKA's June2025 US data sheet, which gives an additional error bound of $\pm 0.4\%$ of span per $10\mathrm{K}$ departure from the $20{}^{\circ }\mathrm{C}$ reference.^[ref] This is a version-specific bound; it is not a measured signed correction for an individual gauge.

For an original gauge with span $P_{\mathrm{span}}=1\mathrm{MPa}$ at $40{}^{\circ }\mathrm{C}$, the specified form givesAt a pressure reading of $100\mathrm{kPa}$, that bound is $8\%$ of the reading. The percentage is referenced to span, so the same absolute bound represents a larger relative contribution near the bottom of the range. Without the sign and an appropriate measurement calibration, subtracting $8\mathrm{kPa}$ would not be justified.

For an original controlled comparison, hold reference pressure at $400\mathrm{kPa}$ and allow the gauge assembly to equilibrate first at $20$ and then at $40{}^{\circ }\mathrm{C}$. Suppose indications are $401$ and $407\mathrm{kPa}$. Their errors relative to the reference are $1$ and $7\mathrm{kPa}$, so the observed temperature-associated change is $6\mathrm{kPa}$, or a local average $0.3\mathrm{kPa}{\mathrm{K}}^{-1}$. These are illustrative readings, not manufacturer test results. Repeat at other pressures and on return to the reference temperature to check reversibility and hysteresis.

Record gauge and reference conditions, equilibration and uncertainty in the pressure comparison. Heating a sealed gas can change its actual pressure through the ideal gas law; an uncontrolled change of that kind cannot be assigned entirely to instrument error. The bound or fitted correction remains one contribution to the final measurement uncertainty.

A cluster bootstrap resamples entire observed groups, retaining all observations and their dependence within each selected group. The sampling unit is the group: draw $K$ group labels with replacement from the $K$ observed groups, concatenate their observations, and recompute the desired statistic. This is a form of bootstrapping suited to independent groups whose internal observations need not be independent. The Stata manual implements it with cluster(); idcluster() distinguishes two resampled copies of the same original group when the fitted procedure needs group identifiers.^[ref]

For an original example, three sensors each supply two perfectly shared readings: $A=(0,0)$, $B=(2,2)$, and $C=(4,4)$. The overall sample mean is $2$. A cluster draw $(A,A,C)$ produces readings $(0,0,0,0,4,4)$ and sample mean $4/3$; the two copies of $A$ remain distinct resampling slots. Conditional on these data, the empirical variance of a group mean drawn uniformly from $0,2,4$ is $8/3$. Averaging three independent draws therefore gives bootstrap varianceResampling six individual rows instead givesThe row procedure halves the conditional variance by pretending that two identical within-sensor readings are independent information. These fractions use the empirical drawing law directly, not a Bessel-corrected estimate of an unknown population variance.

With unequal group sizes, drawing the same number of groups can produce a different number of rows in each replicate. A pooled row mean and a mean of group means are different statistics: unequal-cluster weighting in a cluster bootstrap must follow the intended estimand. Many rows from a few sensors do not replace having many independent sensors; arbitrary dependence between sensors, a dominant group, or too few groups can defeat the intended inference.

A microbolometer is a miniaturized bolometer in which absorbed electromagnetic radiation heats a thermally isolated sensing element and the readout detects the resulting change in electrical resistance. Arrays of microbolometers can form an infrared image without cryogenic detector cooling. A pixel commonly has an absorbing membrane suspended above its readout circuit on narrow supports.^[ref]

An elementary small-signal heat transfer model writesHere $C$ is the sensing element's heat capacity, $G$ its thermal conductance to a fixed-temperature substrate $T_b$, $P$ incident radiant power and $\eta$ its absorbed fraction. The model treats the element as having a single temperature and neglects changes in readout self-heating. A step to constant incident power $P_0$ givesThe thermal time constant therefore connects sensitivity to response speed. Lowering $G$ increases the steady temperature rise, but at fixed $C$ also lengthens the response.^[ref]

For an original example, choose $C=10^{-9}\mathrm{J}{\mathrm{K}}^{-1}$, $G=10^{-7}\mathrm{W}{\mathrm{K}}^{-1}$, $\eta =0.8$ and $P_0=1\mathrm{nW}$. Then $\tau =10\mathrm{ms}$ and the final sensing-element rise is $8\mathrm{mK}$; after $30\mathrm{ms}$ it has reached about $95\%$ of that rise. If the initial resistance is $100\mathrm{k\Omega }$ and the temperature coefficient of resistance is $\alpha =-0.02{\mathrm{K}}^{-1}$, the linear approximation $\Delta R\approx \alpha R\theta$ gives $\Delta R=-16\Omega$ at steady state. These stipulated values illustrate the model, not a particular camera's specification.

The sensing element's temperature rise is not the observed object's temperature rise. Inferring object temperature from the measured infrared signal additionally depends on emissivity, reflected radiation, the optical path and measurement calibration. “Uncooled” also permits temperature stabilization of the array; it describes the absence of required cryogenic operation, not the absence of thermal design.

A four-point flexural test bends a specimen using two outer supports and two inner loading contacts. For symmetric loading, the interval between the inner contacts has constant bending moment and zero transverse shear force, approximating pure bending away from contact disturbances. The Illinois laboratory uses adjustable spans and records force and deflection with a mechanical testing frame.^[ref]

Let the support span be $L$, the distance from each support to its nearest loading contact be $a$, and the total downward applied force be $P$. Each loading contact and each support carries force magnitude $P/2$. Static equilibrium givesThe central constant-moment region has length $L-2a$. Specifying whether a force symbol means total force or force at each nose prevents a factor-of-two error.

For an original example, $L=180\mathrm{mm}$, $a=60\mathrm{mm}$ and $P=80\mathrm{N}$ give reactions $40\mathrm{N}$ and central moment $2.4\mathrm{N}\mathrm{m}$. A rectangular specimen with width $b=20\mathrm{mm}$ and thickness $h=5\mathrm{mm}$ has $I=b{h}^3/12=208.33\mathrm{m}{\mathrm{m}}^4$. The nominal maximum bending stress magnitude is $M(h/2)/I=28.8\mathrm{MPa}$.

The apparatus measures force and a chosen displacement; the beam model converts them into stress, strain or flexural modulus. Midpoint displacement and displacement at the inner loading contacts are generally different. Fixture motion can additionally include machine compliance and contact indentation. Record the displacement location and both spans along with specimen dimensions.^[ref]

Unlike a central three-point load, this geometry provides a finite interval of maximum nominal moment. It tests a larger highly stressed region, so flaw distribution and where failure begins can matter when comparing results between fixtures.

Pure bending is a beam loading state with zero axial resultant $N$, zero transverse shear force $V$ and a nonzero constant bending moment $M$ over the interval considered. Equal and opposite end couples provide an ideal realization. Zero net force on a section does not mean zero local stress: opposing longitudinal tensions and compressions can form a couple.

For a homogeneous slender elastic beam, small-deformation kinematics and Hooke's law giveHere $y$ is measured from the centroidal neutral axis, $I$ is the second moment of area, $E$ is Young's modulus and $\kappa$ is curvature. Uniform $E$, $I$ and $M$ produce uniform curvature. The geometric assumptions and moment sign convention are those of the elementary bending stress model.^[ref]

For an original example, let $M=2\mathrm{N}\mathrm{m}$ and $EI=50\mathrm{N}{\mathrm{m}}^2$. Then $\kappa =0.04{\mathrm{m}}^{-1}$. Across a $0.10\mathrm{m}$ interval the tangent direction changes by approximately $\kappa \ell =0.004$ radians. At $1\mathrm{mm}$ from the neutral axis, the strain magnitude is $4\times 10^{-5}$. The small-slope sag relative to the interval's chord is $\kappa {\ell }^2/8=50\mu \mathrm{m}$.

The central region of a symmetric four-point flexural test is an experimental approximation: the external load contacts lie outside a constant-moment interval. Local contact disturbances remain near the loading noses, and transverse shear exists in the outer spans. Calling the central interval pure bending therefore specifies a region and a model; it does not claim that the entire fixture has zero force or shear.

Bending stress is the longitudinal normal stress associated with a bending moment in a beam. In a homogeneous slender elastic beam with small deformation, plane sections remaining plane give longitudinal strain ${\epsilon }_{xx}=-\kappa y$ relative to the neutral axis. Applying Hooke's law gives ${\sigma }_{xx}=-E\kappa y$.^[ref]

Choose the moment convention $M=-{\int }_{A}y{\sigma }_{xx}dA$. ThenHere $I$ is the second moment of area, and the neutral axis passes through the section centroid when axial resultant is zero and $E$ is uniform. The sign tells which side is in tension; the magnitude grows linearly with distance from that axis.

For an original rectangular section with width $b=12\mathrm{mm}$ and thickness $h=6\mathrm{mm}$, $I=b{h}^3/12=216\mathrm{m}{\mathrm{m}}^4$. A moment $M=1.2\mathrm{N}\mathrm{m}$ produces maximum surface magnitudeThe moment is $1200\mathrm{N}\mathrm{mm}$ when using millimetre section dimensions. At fixed $M$ and $b$, doubling $h$ reduces this nominal surface stress fourfold, not merely twofold, because $I$ grows as $h^3$.

These are beam-model stresses away from local load contacts and discontinuities. Added axial force contributes $N/A$ to the longitudinal stress; a notch can produce a stress concentration beyond the nominal distribution. For inhomogeneous sections the neutral-axis position must be obtained from the actual stiffness distribution rather than assumed from geometry alone.

The bearing end-stop engagement temperature is the temperature state at which free relative expansion first exhausts the clearance to an axial stop. It must specify the initial setting and how both shaft and housing temperatures change. A non-locating bearing arrangement normally accommodates relative expansion; an exhausted travel limit changes the axial constraint.^[ref]

For an illustrative common temperature increase $\Delta T$, write free relative movement aswhere ${\delta }_0$ is the assembly offset toward the stop. With positive remaining gap and $\beta >0$, engagement occurs atFor hypothetical ${\alpha }_s=12\times 10^{-6}{\mathrm{K}}^{-1}$, ${\alpha }_h=8\times 10^{-6}{\mathrm{K}}^{-1}$ and $L_s=L_h=0.50\mathrm{m}$, $\beta =2\times 10^{-6}m/K$. A0.10mm gap with no initial offset closes after50K common heating. Treating the housing as fixed would instead give16.67K, a different boundary-condition model.

An initial offset ${\delta }_0=0.02\mathrm{mm}$ reduces the engagement increase to40K. With combined axial compliance $C_b+C_s=1.25\times 10^{-7}m/N$, the engaged bearing thermal reaction slope isAt60K increase, the excess20K beyond engagement therefore gives320N of bearing thermal end-stop load in this linear quasistatic model.

If $\beta =0$, common uniform heating does not change the relative gap in this model. If $\beta <0$, heating moves away from the stated stop; cooling or another operating change can be relevant instead. Temperature gradients or independently heated components require their actual relative-displacement model rather than a universal scalar engagement temperature.

A bearing nominal axial setting is the chosen reference position of a shaft/support assembly within its available bearing axial travel interval. It determines how much travel remains in each direction before thermal or assembly changes encounter a limit. The locating support establishes the axial reference, while the non-locating bearing arrangement accommodates the permitted movement.^[ref]

Let physical travel be $[a,b]$, nominal setting $x_0$ and all required changes be $d\in [d_{\min },d_{\max }]$. Feasibility requiresThe interval is nonempty exactly when the available span is at least the required span. This follows directly by requiring $a\le x_0+d_{\min }$ and $x_0+d_{\max }\le b$.

For an original example, physical travel $[-0.30,0.30]\mathrm{mm}$ and required changes $[-0.17,0.23]\mathrm{mm}$ permit settings from $-0.13$ to $+0.07\mathrm{mm}$. Setting $x_0=-0.03\mathrm{mm}$ gives operating endpoints $-0.20$ and $+0.20\mathrm{mm}$, leaving0.10mm at each limit. Setting $x_0=0$ instead leaves0.13mm on one side and0.07mm on the other.

The worst endpoint bearing axial travel margin isIn this symmetric-margin objective, its maximum occurs at $x_0=(a+b-d_{\min }-d_{\max })/2$, with $m_{\max }=[(b-a)-(d_{\max }-d_{\min })]/2$. The example gives0.10mm. A real setting can have additional positioning requirements; balancing these geometric margins alone does not determine every functional constraint.

A posterior median is a median of the posterior distribution after observing data. Writing its cumulative distribution function as $F$, a value $m$ is a posterior median when $F(m^{-})\le 1/2\le F(m)$. This definition permits atoms and non-unique medians.

If the posterior has finite first absolute moment, its medians minimize posterior absolute-error loss $\mathbb{E}[|\theta -a|\mid x]$. For a continuous posterior, differentiating with respect to $a$ gives $2F(a)-1$; the general case uses the corresponding left and right derivatives. Thus a minimizer is exactly a point with at least half the posterior mass on either side.

For a posterior placing probability $0.7$ at zero and $0.3$ at ten, the median is zero and its expected absolute loss is three. The posterior mean is three, whose absolute loss is $0.7(3)+0.3(7)=4.2$. Choosing a Bayes estimator requires specifying the loss, not merely selecting a conventional summary of the posterior.

Quadratic baseline curvature in gated energy integration describes a baseline whose second time derivative is nonzero over the measurement gate. Averaging only its two endpoint values is exact for a linear baseline but generally not a quadratic one. The baseline-drift bias in pulse-energy integration article derives the resulting endpoint-subtraction error.

A focused correction follows from Simpson's rule. For a quadratic baseline on $[0,T]$,For an original example, take $T=12\mu \mathrm{s}$ and baseline readings $10,11,14\mathrm{mW}$ at the beginning, midpoint and end. The integrated baseline is $0.136\mu \mathrm{J}$. Endpoint averaging would instead subtract $0.144\mu \mathrm{J}$, biasing the inferred signal by $-8\mathrm{nJ}$.

This Simpson baseline integration for pulse-energy correction needs a genuine baseline at the midpoint. A sample taken during the pulse contains signal as well as background. A separate matched blank run or independently characterized reference can supply additional information; treating a contaminated pulse sample as baseline does not improve the correction. If the midpoint estimate is biased by $\Delta b$, the Simpson integral is biased by $2T\Delta b/3$.

Historical NIST measurements used before/after baselines and averaged them for background and thermal-drift compensation.^[ref] The three-point quadratic example is an independent mathematical extension, not a description of that apparatus's processing.

Balloon volumetric compliance is the local change in balloon volume per change in a specified pressure difference, with the mechanical state and other conditions defined:It has units of volume per pressure. It differs from diameter response $\mathrm{d}D/\mathrm{d}p$ in a balloon pressure-diameter curve and from a fractional volume response $(1/V)\mathrm{d}V/\mathrm{d}p$.

For an illustrative cylindrical region whose length can also change,The second term vanishes only when length is held fixed. A diameter-only observation does not otherwise determine the whole volume response.

The Nipro chart gives $2.00$ and $2.16\mathrm{mm}$ for its nominal $2.00\mathrm{mm}$ balloon at labeled rows $6$ and $10\mathrm{atm}$.^[ref] Using those diameters in a chosen fixed-length $10\mathrm{mm}$ cylinder gives the finite-interval estimateThe assumed cylinder is not a measured manufacturer balloon volume. The calculation illustrates why a finite volume change differs from substituting one endpoint diameter into a local derivative.

A balloon compliance fixture correction separates the balloon from the measurement apparatus. In a simple quasistatic model where both experience the same pressure increment, measured injected-volume response can be $C_{\mathrm{measured}}=C_{\mathrm{balloon}}+C_{\mathrm{fixture}}$. A blank or independently characterized fixture is needed before assigning the entire response to the balloon.

Pressure reference, external constraint, loading history, temperature and equilibration also belong to the definition. Hysteresis can give different paths on inflation and deflation. Compliance describes a response in the stated regime; it does not alone supply a rupture pressure or a failure probability.

Coronary angioplasty enlarges a narrowed segment of a coronary artery using an intravascular intervention. In balloon angioplasty, a balloon catheter applies outward pressure within the affected segment and is then deflated. A vascular stent, when used, is a separate retained support rather than the balloon itself.

The historical primary report by A. R. Grüntzig, A. Senning and W. E. Siegenthaler describes a catheter system introduced through a systemic artery under local anaesthesia, with controlled balloon inflation. Its 50-patient series reported 32 technical successes and measured both the stenosis and the coronary pressure difference.^[ref] This connects a mechanical intervention to observations of anatomy and a physiological consequence.

An original series-resistance model explains why improving one segment does not determine the whole circulation's response. Let a narrowed segment have vascular resistance $R_s$, the rest of a represented pathway resistance $R_b$, and assume a fixed total driving pressure difference $\Delta p$. ThenIn consistent illustrative units, take $\Delta p=100$, $R_s=4$ and $R_b=6$, giving $Q=10$. If intervention reduces the local resistance to $1$ while the other assumptions remain fixed, flow becomes $100/7\simeq 14.29$. A fourfold increase in the local segment's conductance has increased total flow only by a factor $10/7$, not fourfold. The pressure loss across that segment changes from $40$ to about $14.29$ in the same chosen units.

Real coronary blood flow also involves pulsatility, deformable vessels and responses of smaller vessels, so this is an explanatory network model rather than a patient prediction. Diameter versus area stenosis adds another caution: a geometric narrowing percentage requires its measurement convention and reference segment before it is converted into an area or a flow-model parameter.

Evaluation separates immediate technical success, persistent openness, symptoms, later interventions and other clinical outcomes. The original authors reported failures and emergency bypasses and called for prospective randomized comparison with surgical and medical management.^[ref] A selected successful-case measurement does not supply the outcome distribution for every attempted procedure or establish comparative benefit by itself.