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Bertotti-Steinmetz Magnetic Core Hysteresis and Eddy Loss Formula
Separate transformer and inductor core loss density into hysteresis k_h*f*B^beta, classical eddy-current k_e*(f*B)^2, and excess domain-wall k_exc*(f*B)^1.5 components.
Read the guide →ScienceBET Adsorption Isotherm: Monolayer Capacity, Constant C & Surface Area
Calculate Brunauer-Emmett-Teller (BET) energetic constant C = 1 + (s / i), monolayer capacity n_m = 1 / (s + i), and N2 specific surface area S_BET (m^2/g) from linearized isotherm slope and intercept.
Read the guide →Engineering & ScienceBethe-Bloch Mass Stopping Power (-dE/rho*dx) & Proton Bragg Peak Formula
Calculate collisional mass stopping power S_mass = 0.307075 * (Z/A) * (z^2 / beta^2) * (LN((2 * m_e * c^2 * beta^2) / ((1 - beta^2) * I_eV)) - beta^2) in MeV*cm^2/g for protons and alpha particles in tissue/water.
Read the guide →Engineering & ScienceBethe-Weizsäcker Semi-Empirical Nuclear Binding Energy (Liquid-Drop SEMF)
Calculate total nuclear binding energy B(A, Z) and binding energy per nucleon B / A (MeV) from Volume, Surface, Coulomb, Asymmetry, and ISEVEN/ISODD Pairing terms across the chart of nuclides.
Read the guide →Rock Mechanics, Tunneling & Underground Mining, Bieniawski RMR89 ClassificationBieniawski Rock Mass Rating (RMR89) & Tunnel Load Formula
Calculate Basic RMR89 (0–100), Strike/Dip Orientation-Adjusted RMR89, Serafim-Pereira Deformation Modulus Em (GPa), and Unal Tunnel Roof Support Load Pressure (kPa).
Read the guide →Utility-Scale Solar Photovoltaics, IEC 60904-1-2 Bifacial Modules & Yield ModelingBifacial Solar PV Albedo Rear Irradiance & Gain Formula
Calculate Unshaded Rear-Side Irradiance G_rear (W/m²), IEC 60904-1-2 Equivalent Effective Irradiance G_eff (W/m²), Bifacial Energy Gain BG (%), and Array DC Output (kW).
Read the guide →Dynamic ArraysBiggest gainers and losers formula: extract Top N & Bottom N movers (SORTBY & TAKE)
Rank the largest and smallest percentage changes without reordering the source table. Distinguish top/bottom rankings from positive-only gainers and negative-only losers.
Read the guide →Lookup & ReferenceBilling Lookup with Standard vs. Partner Client Rates (XLOOKUP + IF)
Calculate billable timecard line totals by selecting between Standard and Retainer/Partner hourly rate columns in a role rate card via XLOOKUP and multiplying by hours logged.
Read the guide →Engineering & ScienceBinary Phase Diagram Lever Rule: Liquid & Solid Equilibrium Mass Fractions
Calculate equilibrium liquid mass fraction w_L = (C0 - C_S) / (C_L - C_S) and solid mass fraction w_S = (C_L - C0) / (C_L - C_S) along a horizontal isothermal tie-line in binary alloy phase diagrams.
Read the guide →Statistical AnalysisBinomial Acceptance Sampling & Critical Defect Cutoff (BINOMDIST & CRITBINOM)
Calculate exact binomial defect probabilities, cumulative lot-acceptance curves, producer’s risk, and the smallest critical defect threshold (CRITBINOM) using exact factorial combinatorics.
Read the guide →StatisticalBinomial probability formula: exact k successes in n independent trials
Calculate exact binomial probability P(X = k), expected successes n*p, and binomial standard deviation SQRT(n*p*(1-p)) using FACT.
Read the guide →Clinical Medicine & PhysiologyBioelectrical Impedance (BIA) Formula: Height^2 / R Index, Kushner TBW & Lukaski FFM
Calculate Resistive Impedance Index H^2 / R_50 (cm^2/Ohm), Kushner-Schoeller Total Body Water (TBW, L), and Fat-Free Mass (FFM = TBW / 0.732) from 50-kHz tetrapolar Bioelectrical Impedance Analysis.
Read the guide →Healthcare Corporate Finance, Royalty Monetization & Structured Biotech DebtBiopharma Synthetic Royalty Hard Cap & Post-Cap Tail Formula
Calculate Non-Dilutive Biopharma Synthetic Royalty Hard-Cap Trigger Sales ($M), Cumulative Royalty + Post-Cap Tail Payout ($M), Biotech Issuer Cap Savings vs Uncapped Royalty, and Royalty Investor Realized Multiple (MOIC).
Read the guide →Engineering & ScienceBioreactor Volumetric Oxygen Transfer (kLa), OTR & Minimum kLa_min
Calculate aerobic fermenter Oxygen Transfer Rate OTR = k_L*a * (C_star - C_L) (mmol O2/(L*h)), cellular Oxygen Uptake Rate OUR = q_O2 * X, and minimum required k_L*a to prevent dissolved oxygen starvation below C_crit.
Read the guide →EngineeringBiot (Bi), Prandtl (Pr) & Nusselt (Nu) Heat Transfer Dimensionless Numbers
Calculate Biot, Prandtl and Nusselt numbers with distinct solid/fluid conductivities and an illustrative small-Biot screening check.
Read the guide →EngineeringBiot Number (Bi), Fourier Number (Fo) & Lumped-Capacitance Cooling
Calculate Biot and Fourier numbers and an approximate lumped transient temperature using a consistent characteristic length. Check inputs and a Bi < 0.1 screening criterion before displaying temperature.
Read the guide →Biopharma Corporate Development, Licensing & rNPV ValuationBiotech Clinical Phase POS & Risk-Adjusted rNPV Formula
Calculate Biopharma Cumulative Likelihood of Approval LOA = P_Ph1 * P_Ph2 * P_Ph3_NDA, Risk-Adjusted Commercial Launch PV, Pipeline Asset Risk-Adjusted Net Present Value (rNPV), and Breakeven Approval Probability.
Read the guide →Engineering & ScienceBirch-Murnaghan 3rd-Order Isothermal Equation of State (P–V GPa)
Calculate high-pressure crystal compression P(V) (GPa) in diamond-anvil cells and planetary mantles using Francis Birch’s Eulerian finite-strain 3rd-order Equation of State.
Read the guide →Healthcare & ClinicalBISAP Score for Acute Pancreatitis Early Severity & Mortality Risk (0–5)
Calculate the 0–5 point Bedside Index for Severity in Acute Pancreatitis (BISAP) within the first 24 hours of admission from BUN, mental status (GCS), SIRS count, age, and pleural effusion.
Read the guide →Civil, Structural & Geotechnical EngineeringBishop Simplified Method of Slices Formula: Circular Slope Factor of Safety & m_alpha
Compute Alan W. Bishop’s (1955) Simplified Method of Slices circular slope stability Factor of Safety FS = SUM(((c'*b + (W - u*b)*TAN(phi')) / m_alpha)) / SUM(W * SIN(alpha)) using vertical force equilibrium.
Read the guide →Statistical & EngineeringBitwise Permission Flag Extraction & Bit-Shift Formula (INT & MOD Powers of 2)
Test individual permission bits inside integer RBAC bitmasks using INT(MOD(mask / 2^bit, 2)) and perform exact left/right bit shifts (mask * 2^s and INT(mask / 2^s)) in any spreadsheet.
Read the guide →Date & TimeBiweekly and semimonthly pay schedule formulas (with weekend adjustment)
Generate 26-period biweekly pay dates (+14 days) and 24-period semimonthly 15th/month-end pay dates that step back onto Friday.
Read the guide →Quantitative Finance & DerivativesBjerksund-Stensland Closed-Form American Option Trigger Boundary
Compute the Bjerksund-Stensland (1993) closed-form flat early-exercise boundary I_flat, characteristic root beta, and smooth-pasting coefficient alpha without iterative root-finding.
Read the guide →Operations, Logistics & EngineeringBjerrum Field Vane Shear Correction Factor (mu) Formula: Plasticity Index Clay Strength
Correct raw in-situ Field Vane Shear strength s_u,FV for strain-rate and anisotropy effects in high-plasticity soft clays using Laurits Bjerrum’s (1972/1973) mobilization factor mu = 1.7 - 0.54 * LOG10(PI) and s_u,design = mu * s_u,FV.
Read the guide →Quantitative Finance & DerivativesBlack-Scholes d1, d2 Log-Moneyness & Put-Call Parity Arbitrage Check Formula
Compute Black-Scholes-Merton standardized log-moneyness factors d1 and d2 using LN, SQRT, and EXP alongside Hans Stoll’s model-free European Put-Call Parity pricing residual.
Read the guide →Mineral Processing, Quarry & Aggregate Comminution, Blake Jaw Crusher MechanicsBlake Jaw Crusher Nip Angle & Critical RPM Formula
Calculate Crushing-Chamber Nip Angle alpha (deg), Gape-to-CSS Reduction Ratio, Rose & English Critical Toggle Flywheel Speed Ncrit (RPM), and Volumetric Wedge Throughput (t/h).
Read the guide →Statistical & EconometricBland-Altman Bias & 95% Limits of Agreement (LoA) Method-Comparison Formula
Calculate mean inter-device bias d̄ = AVERAGE(D2:D7), difference standard deviation s_d = STDEV.S(D2:D7), and 95% Limits of Agreement d̄ ± 1.96 * s_d to validate clinical measurement instruments.
Read the guide →ScienceBlaney-Criddle Crop Evapotranspiration (ETo, ETc) & Net Irrigation
Calculate FAO Blaney-Criddle reference evapotranspiration ETo = p * (0.46 * T_mean + 8) (mm/day), crop water demand ETc = K_c * ETo, and monthly Net Irrigation Requirement MAX(0, 30 * ETc - P_eff).
Read the guide →Robotics, UAV Propulsion & Electric Drivetrain Engineering, BLDC Motor Kv & Kt PhysicsBLDC Motor Kv Constant, Torque Kt & Shaft Power Formula
Calculate Exact SI Torque Constant Kt = 60 / (2*PI()*Kv) (N·m/A), Electromagnetic Torque (N·m), Resistive Voltage-Drop Loaded Speed (RPM), and Mechanical Shaft Power (kW).
Read the guide →Oncology Pharmacology & BiostatisticsBliss Independence & Loewe Combination Index Formula
Calculate Chester Bliss’s Probabilistic Additive Inhibition E_Bliss = E_A + E_B - E_A * E_B, Excess Over Bliss (EOB), and Loewe / Chou-Talalay Isobole Combination Index CI = d_A / D_{x,A} + d_B / D_{x,B}.
Read the guide →Engineering & ScienceBloch-Grüneisen Metal Resistivity: T^5 Cryogenic vs Linear High-T Law
Model pure metal electrical resistivity ρ(T) across the Thirring high-temperature expansion ρ_0 + ρ_Θ*(T/Θ_R)*(1 - Θ_R^2/(18*T^2)) and Bloch’s cryogenic T^5 acoustic-phonon law.
Read the guide →Computer Science, Databases & Storage EnginesBloom Filter False-Positive & Optimal Hash Formula
Size LSM-tree and cache Bloom filters via optimal hash count k* = ROUND((m/n)*LN(2), 0), false-positive rate (1 - EXP(-k*n/m))^k, and required bits per key -LN(p)/(LN(2)^2).
Read the guide →Plastics Packaging, HDPE Blow Molding & Viscoelastic Die SwellBlow Molding Parison Swell, Blow-Up Ratio & Wall Formula
Calculate Swollen Parison Diameter D_p (mm), Swollen Parison Wall Thickness t_p (mm), True Parison Blow-Up Ratio (BUR), and Finished Bottle Wall Thickness t_wall (mm).
Read the guide →Polymer Film & Flexible Packaging, Tubular Blown-Film Extrusion BUR, Layflat & DDRBlown-Film Extrusion BUR, Layflat & Draw-Down Formula
Calculate Transverse Blow-Up Ratio BUR = D_bubble/D_die, Collapsed Nip Layflat Width (mm), Machine-Direction Draw-Down Ratio DDR, and Forming Orientation Ratio DDR/BUR.
Read the guide →Math & TrigBMI calculation formula: metric (kg/cm), imperial (lb/in), and categories
Calculate Body Mass Index from kilograms and centimeters or pounds and inches, and assign adult screening bands with IFS.
Read the guide →Healthcare & Clinical CalculationsBODE Index for COPD 4-Year Survival & Transplant Referral (0–10)
Calculate Bartolome Celli’s 0–10 multidimensional COPD BODE Index from Body-mass index (<= 21), Obstruction (FEV1 % predicted), Dyspnea (mMRC 0–4), and Exercise capacity (6-minute walk distance).
Read the guide →Engineering & Fluid MechanicsBodenstein Number (Bo) & Levenspiel Closed-Vessel RTD Variance Formula
Quantify non-ideal tubular reactor and chromatography column backmixing via the Bodenstein number Bo = u*L / D_ax and Octave Levenspiel’s exact closed-vessel dimensionless residence-time variance sigma_theta² = 2/Bo - (2/Bo²)*(1 - EXP(-Bo)).
Read the guide →Clinical Medicine & PhysiologyBody Roundness Index (BRI) Calculator: Thomas Waist-Height Ellipse Eccentricity Formula
Compute Diana M. Thomas et al.’s (2013) geometric Body Roundness Index BRI = 364.2 - 365.5 * SQRT(1 - ((WC / (2 * PI()))^2) / ((0.5 * Height)^2)) and sagittal ellipse eccentricity e from Waist and Height.
Read the guide →Statistical & Clinical AnalysisBoey Score Formula: Perforated Peptic Ulcer Operative Mortality Risk (0 to 3)
Stratify emergency surgical mortality and laparoscopic Graham-patch candidacy in perforated peptic ulcer using John Boey’s classic 3-factor score (Preoperative Shock, Severe Medical Illness ASA >= III/IV, Perforation > 24 Hours).
Read the guide →Engineering & ScienceBohm Diffusion Coefficient Formula: Turbulent Cross-Field Plasma Transport
Calculate David Bohm’s anomalous cross-field plasma diffusivity D_B = T_e_eV / (16 * B) (m^2/s) and cylindrical Bessel radial confinement time τ_B = a^2 / (5.783 * D_B) in Hall thrusters and magnetrons.
Read the guide →ScienceBohr Hydrogen Energy Levels & Rydberg Spectral Wavelength Formula
Use a consistent approximate hydrogen Rydberg model for vacuum emission wavelengths, photon energies and shell energies, with positive integer level validation.
Read the guide →Industrial Energy Efficiency, Steam Distribution & Plant UtilitiesBoiler Condensate Return Ratio & Fired-Fuel Savings Formula
Calculate Steam Plant Blended Feedwater Temperature (°C), Sensible Enthalpy Power Saved (kWth), Boiler-Efficiency-Adjusted Fired Fuel Saved (kW), and Cold Makeup Demand (kg/hr).
Read the guide →Quantitative Finance & TradingBollinger Bands Formula: Upper/Lower Envelope, %B Position & Squeeze Bandwidth
Calculate John Bollinger’s adaptive volatility envelope (SMA +- k*sigma), normalized %B position oscillator (Close - Lower)/(Upper - Lower), and relative Bandwidth %.
Read the guide →Mechanical Fasteners, Bolted Joint Assembly & Structural ReliabilityBolt Torque Preload Tension, Nut-Factor K & Proof Formula
Calculate ISO Metric Bolt Tensile Proof Load (kN), Achieved Clamp Preload Tension F_i (kN) from Wrench Torque T = K*D*F_i, Proof-Load Utilization (%), and Anti-Seize Yield Gate.
Read the guide →Mechanical Design & FastenersBolted-Joint Stiffness Factor (C) and Separation Safety Formula
Calculate Shigley’s bolted-joint stiffness constant C = k_b / (k_b + k_m), bolt tension F_b = MAX(P, F_i + C*P), and joint-separation safety factor n_0 = F_i / (P*(1 - C)).
Read the guide →Commercial Real Estate, Workplace Space Planning & BOMA StandardsBOMA Rentable vs Usable Square Feet, Load & Loss Factor Formula
Calculate ANSI/BOMA Z65.1 Billed Rentable Square Feet RSF = USF * R/U_Load_Ratio, Equivalent Market Loss Factor (RSF - USF) / RSF, Annual Base Rent, and True Effective Cost per Carpetable Usable Square Foot.
Read the guide →EngineeringBond (Eotvos) Number & Ohnesorge Number for Droplet & Inkjet Physics
Calculate Bond, Ohnesorge and inverse-Ohnesorge numbers with separate liquid and ambient densities, explicit length conventions and guarded interactive examples.
Read the guide →Mineral Processing & Comminution EngineeringBond Ball-Mill Work Index Grinding Energy Formula
Calculate Fred Bond’s Specific Comminution Energy W = 10 * W_i * (1 / SQRT(P_80) - 1 / SQRT(F_80)) in kWh/t, Size Reduction Ratio F_80 / P_80, Mill Pinion Power, and Drawn Motor Megawatts.
Read the guide →FinancialBond Convexity & Duration-Convexity second-order price change formula
Estimate a bond’s full-price sensitivity with modified duration and convexity. The second-order calculation remains an approximation, especially for large yield changes.
Read the guide →Financial & AccountingBond Coupon Schedule & Actual/Actual Day Counts (EDATE & DATEDIF)
Derive previous coupon date (COUPPCD), next coupon date (COUPNCD), accrued days (COUPDAYBS), remaining days (COUPDAYSNC), period length (COUPDAYS), and remaining coupons (COUPNUM).
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