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Kingman's VUT Equation: G/G/1 queueing wait time from variability & utilization
Estimate real-world factory and service queue wait times Wq = ((Ca^2 + Cs^2)/2) * (u / (1 - u)) * ts when arrivals and processing times are NOT exponential.
Read the guide →Mechanical & Structural EngineeringKirchhoff-Love Circular Plate Flexural Rigidity and Diaphragm Deflection
Calculate thin-plate flexural rigidity D = E*h^3 / (12*(1 - nu^2)) and compare clamped vs simply supported circular diaphragm center deflection under uniform pressure.
Read the guide →Quantitative Finance & DerivativesKirk Spread Option Approximation for Nonzero-Strike Crack and Spark Spreads
Price two-asset commodity spread calls MAX(F1 - F2 - K, 0) in Excel and Arte Sheets using Kirk’s discounted-strike denominator weight and composite spread volatility.
Read the guide →Engineering & ScienceKirkbride Equation: Optimal Distillation Feed-Tray Location (NR / NS)
Locate the optimal feed stage in multicomponent Fenske-Underwood-Gilliland distillation design via C. G. Kirkbride’s rectifying-to-stripping stage ratio N_R / N_S = ((B / D) * (x_HK_F / x_LK_F) * (x_LK_B / x_HK_D)^2)^0.206.
Read the guide →EngineeringKirpich Watershed Time of Concentration (tc) & NRCS Basin Lag Time
Calculate stormwater watershed Time of Concentration t_c = 0.01947 * L^0.77 * S^(-0.385) (minutes, SI flow length L in meters and slope S in m/m) and NRCS Unit Hydrograph Lag Time t_lag = 0.60 * t_c.
Read the guide →Statistical AnalysisKish Design Effect (DEFF), Cluster ICC & Effective Sample Size Formula
Deflate nominal survey and cluster-randomized sample size N using Leslie Kish’s (1965) cluster DEFF_c = 1 + (m - 1)*rho and unequal-weighting DEFF_w = 1 + CV(w)^2.
Read the guide →Survey Statistics & Research DesignKish Unequal-Weight Effective Sample Size (n_eff) Formula
Calculate Kish’s unequal-weighting effective sample size n_eff = (SUM w)^2 / SUM(w^2), weighting design effect DEFF_w = 1 + CV_w^2, and efficiency n_eff / N in Excel and Arte Sheets.
Read the guide →Telecommunications & Queueing NetworksKleinrock Square-Root Channel Capacity Delay Formula
Calculate Leonard Kleinrock’s optimal M/M/1 packet-network link capacities C_i* = lambda_i/mu + (C_tot - F_sum)*(SQRT(lambda_i)/(SQRT(lambda_1)+SQRT(lambda_2))) and minimum average network delay.
Read the guide →Pricing Strategy & Industrial OrganizationKlemperer Switching-Cost Bargain-Then-Ripoff Formula
Calculate Paul Klemperer’s (1987, 1995) below-cost Period-1 customer acquisition price P_1* = c_1 + t - delta*s, Period-2 lock-in harvest price P_2* = c_2 + s, and lifecycle customer profit.
Read the guide →Public Economics & Causal InferenceKleven-Saez Kink Bunching Mass and Behavioral Elasticity Formula
Calculate normalized excess bunching mass b = B / h_0, marginal buncher shift Delta_z = b * w_bin, and exact log-retention elasticity e = (Delta_z / z_star) / LN((1 - m_1) / (1 - m_2)).
Read the guide →Petrophysics, Core Analysis & Tight-Gas Reservoir EngineeringKlinkenberg Gas-Slippage Core Permeability Formula
Calculate True Liquid-Equivalent Klinkenberg Permeability k_inf (mD), Low-Pressure Lab Overestimation (%), In-Situ High-Pressure Reservoir Gas Permeability (mD), and Slippage Multiplier.
Read the guide →Industrial Electric Drives & Rotating Machinery, Kloss Asynchronous Motor Torque-Slip LawKloss Induction Motor Torque-Slip & Breakdown Formula
Calculate Synchronous Rotating-Field Speed Ns (RPM), Rotor Slip s (%), Kloss Electromagnetic Shaft Torque T(s) (N·m), and Direct-On-Line (s = 1) Locked-Rotor Starting Torque (N·m).
Read the guide →Rarefied Gas Dynamics, High-Vacuum Plumbing & Molecular FlowKnudsen Molecular-Flow Tube Conductance & Aperture Formula
Calculate Gas Mean-Free-Path Knudsen Number Kn, Long-Tube Knudsen Conductance (L/s, scaling with D^3!), Entrance Orifice Conductance (L/s), and Short-Tube Series Conductance (L/s).
Read the guide →Engineering & ScienceKohlrausch's Square-Root Law of Molar Conductivity & Weak-Acid Dissociation (alpha)
Calculate strong-electrolyte molar conductivity Lambda_m = Lambda_m^0 - K * SQRT(c) and weak-electrolyte degree of dissociation alpha = Lambda_m / Lambda_m^0 in electrochemistry.
Read the guide →Malting Grain Chemistry & Brewhouse Process ControlKolbach Index, Lintner Diastatic & Mash Efficiency Formula
Read a Malt Certificate of Analysis (COA) by calculating Kolbach Index % SNR = 100*(Soluble_N / Total_N), converting European Windisch-Kolbach (°WK) to Degrees Lintner (°L) = (WK + 16)/3.5, and auditing Brewhouse Mash Extract Efficiency.
Read the guide →Economics & Market MicrostructureKolm-Pollak Absolute Inequality Index K(kappa) & Translation-Invariant EDE Income
Measure absolute dollar income dispersion using Serge-Christophe Kolm’s (1976) Leftist Inequality Index K(kappa) = (1 / kappa) * LN((1 / N) * SUM(EXP(kappa * (mu - y_i)))) and Equally Distributed Equivalent income mu - K(kappa).
Read the guide →AI Accelerator Efficiency & Physics of ComputingKoomey Compute TOPS/W & Landauer Limit Formula
Calculate AI Accelerator Compute Efficiency (TOPS/W), Energy per Operation (pJ/op), Rolf Landauer’s Minimum Irreversible Bit-Erasure Energy k_B * T * LN(2) in zeptojoules, and Headroom Factor Above Landauer.
Read the guide →Operations, Logistics & EngineeringKoros-Nowak & Madon-Boudart Test Formula: Catalytic Turnover Frequency & Transport Disguise
Perform the definitive experimental test for heat- and mass-transfer artifacts in heterogeneous catalysis (Koros & Nowak, 1967; Madon & Boudart, 1982) by computing log-log slope d(LN R_vol) / d(LN Site_Density) across diluted active-metal loadings (Slope = 1.0 intrinsic vs. 0.5 pore-diffused vs. 0.0 film-limited!).
Read the guide →Civil, Structural & Geotechnical EngineeringKostiakov-Lewis (Modified Kostiakov) Irrigation Infiltration Formula: Z(t) & Intake Rate
Design furrow, border, and basin surface irrigation opportunity times using A. N. Kostiakov (1932) and M. R. Lewis / Mezencev’s Modified Kostiakov-Lewis Cumulative Intake Z(t) = k * (t^a) + f_0 * t and Instantaneous Rate I(t).
Read the guide →Engineering & Fluid MechanicsKoutecký-Levich Equation Formula: Electrocatalyst Kinetic vs. Diffusion RDE Current
Deconvolve intrinsic electrocatalyst kinetic current density j_k from convective mass-transport limiting current j_L on a rotating disk electrode (RDE) via 1/j = 1/j_k + 1/j_L.
Read the guide →Engineering & ScienceKozai-Lidov Mechanism Formula: Maximum Eccentricity & Critical Inclination
Calculate secular Kozai-Lidov conserved vertical angular momentum L_z and peak quadrupole eccentricity e_max = SQRT(1 - (5/3)*cos^2(i_0)) in hierarchical triple systems.
Read the guide →Engineering & ScienceKozeny-Carman Porous Media Permeability Formula (Darcies & m^2)
Calculate intrinsic porous-bed permeability k = ((phi^3) * (dp^2)) / (180 * ((1 - phi)^2)) in Darcies and square micrometers from porosity phi and effective grain diameter dp_mm.
Read the guide →Pulp & Paper Chemical Engineering, Kraft Digesters & Recovery BoilersKraft White Liquor Active/Effective Alkali & Sulfidity Formula
Calculate North American Na2O-Basis Active Alkali AA (g/L), Effective Alkali EA (g/L), White-Liquor Sulfidity (%), and Digester Effective Alkali Charge (% on Oven-Dry Wood).
Read the guide →Development Economics & Personnel TheoryKremer O-Ring Production & Assortative Wage Formula
Calculate Michael Kremer’s (1993) multiplicative O-Ring output, supermodular positive assortative matching surplus Y_max*(q_H - q_L)^2, and convex n-task equilibrium wage ratio (q_H / q_L)^n.
Read the guide →EngineeringKremser Gas Absorption Factor (A) & Theoretical Scrubber Stages
Calculate the absorption factor and ideal equilibrium-stage estimate for a dilute countercurrent absorber, with input validation, a pinch check and stable evaluation near A = 1.
Read the guide →Engineering & ScienceKretschmann Surface Plasmon Resonance (SPR) Biosensor Angle Formula
Calculate the prism-coupled Surface Plasmon Resonance dip angle θ_SPR = DEGREES(ASIN((1 / n_p) * SQRT((|ε_m| * n_d^2) / (|ε_m| - n_d^2)))) for label-free antibody-antigen optical biosensors.
Read the guide →Statistical & Clinical AnalysisKrippendorff’s Alpha (α) Formula: Coincidence Matrix & Finite-Sample Reliability
Calculate Klaus Krippendorff’s content-analysis reliability coefficient α = 1 - D_o / D_e = 1 - (n - 1)*n_01 / (n_0*n_1) from a symmetric 2×2 coincidence matrix.
Read the guide →Statistical & EngineeringKronecker Delta Numeric Equality Indicator Formula (Exact & Tolerance-Safe)
Implement the Kronecker Delta indicator (returning 1 when x = y and 0 otherwise, with strict ISNUMBER type safety and optional floating-point tolerance) plus vector exact-match counting.
Read the guide →International Macroeconomics & Sovereign RiskKrugman Currency Crisis & Shadow Exchange Rate Formula
Calculate Paul Krugman’s (1979) and Flood-Garber’s (1984) shadow floating exchange rate s_tilde(0), speculative attack timing T* = (R_0 - R_min) / mu - alpha, and discrete reserve wipeout alpha * mu.
Read the guide →International Trade & Economic GeographyKrugman Love-of-Variety Gain & Home-Market Effect Formula
Compute Dixit-Stiglitz-Krugman autarky varieties L / (sigma * F), frictionless trade welfare gain ((L_H + L_F)/L_H)^(1/(sigma - 1)), and iceberg Home-Market Effect firm share.
Read the guide →Engineering & ScienceKubelka-Munk Diffuse Reflectance Function F(R) = K/S & Inverse Reflectance
Linearize diffuse reflectance R_inf of opaque paints, textiles, papers, and catalyst powders into absorption-to-scattering ratio F(R_inf) = (1 - R_inf)^2 / (2 * R_inf) = K / S and invert K/S back to R_inf.
Read the guide →Statistical & PsychometricsKuder-Richardson Formula 20 (KR-20) Dichotomous Test Reliability
Calculate internal-consistency reliability KR-20 = (K / (K - 1)) * (1 - SUM(p_i * q_i) / σ_X^2) for binary right/wrong certification exams and diagnostic checklists.
Read the guide →StatisticalKullback-Leibler (KL) Divergence & Population Stability Index (PSI) drift formulas
Measure probability distribution shift D_KL(P || Q) = SUM(P * LN(P / Q)) and symmetric credit-model Population Stability Index (PSI) in spreadsheets.
Read the guide →Engineering & Fluid MechanicsKutateladze Number (Ku) Formula: Countercurrent Gas-Liquid Flooding & Droplet Entrainment
Predict diameter-independent countercurrent gas-liquid flooding and droplet carry-over in large-diameter pipes, nuclear downcomers, and packed columns using Samson S. Kutateladze’s dimensionless gas flux number Ku_g = j_g * SQRT(rho_g) / (g * sigma * (rho_l - rho_g))^0.25.
Read the guide →Mechanical Design & KinematicsKutzbach-Gruebler Planar Mechanism Mobility (DOF) Formula
Calculate planar linkage degrees of freedom M = 3*(L - 1) - 2*J_1 - J_2 + r_red in Excel and Arte Sheets to distinguish 1-DOF mechanisms, trusses, hyperstatic frames, and parallel-crank paradoxes.
Read the guide →Quantitative FinanceKyle Lambda Order-Flow Price Impact & Reciprocal Market Depth Formula (SLOPE)
Estimate Kyle’s (1985) price-impact coefficient lambda = SLOPE(Price_Change, Signed_Order_Flow) and implied book depth 1 / lambda from signed trade-imbalance intervals.
Read the guide →Engineering & Fluid MechanicsKynch & Talmage-Fitch Thickener Sizing Formula: Unit Area (UA) & Clarifier Diameter
Scale 1-liter graduated-cylinder hindered settling tests up to continuous industrial gravity thickeners using G. J. Kynch & Talmage-Fitch Unit Area UA = t_u / (C_0 * H_0) and tank diameter D = SQRT(4 * F * UA / pi).
Read the guide →Healthcare & ClinicalLACE Index Calculator: 30-Day Unplanned Readmission & Death Risk (0–19)
Compute Carl van Walraven’s LACE Index (0–19 points: Length of stay, Acuity of admission, Charlson comorbidity score, and 6-month ED visits) to target transitional care before hospital discharge.
Read the guide →Public Procurement & Mechanism DesignLaffont-Tirole Procurement Incentive Power Formula
Calculate Jean-Jacques Laffont & Jean Tirole’s (1986) optimal cost-sharing contract slope b* = 1 - (lambda_f/(1 + lambda_f))*gamma_e*(beta - beta_L) between Fixed-Price and Cost-Plus.
Read the guide →Behavioral Economics & Household FinanceLaibson Quasi-Hyperbolic (beta, delta) Discount Formula
Model time-inconsistent preference reversals using Phelps-Pollak and David Laibson’s (1997) quasi-hyperbolic discount sequence {1, beta*delta, beta*delta^2, ...}.
Read the guide →Mechanical Design & Machine ElementsLame Shrink-Fit Contact Pressure and Slip-Free Torque Formula
Calculate interference press/shrink-fit interface pressure p_if = (E * delta_r / (2 * R_if)) * (1 - (R_if / R_out)^2) and slip-free torque capacity T_max in Excel and Arte Sheets.
Read the guide →EngineeringLame Thick-Walled Cylinder Hoop & Radial Stress (Pressure Vessels)
Calculate inner-bore peak tensile hoop stress sigma_theta = P_i * (r_o^2 + r_i^2) / (r_o^2 - r_i^2) and outer-surface hoop stress in hydraulic cylinders, gun barrels, and autoclaves where t / r_i > 0.10.
Read the guide →Microeconomic Consumer Theory & Hedonic PricingLancaster Characteristics Shadow Price & Frontier Formula
Solve Kelvin Lancaster’s (1966) 2x2 Hedonic Attribute Shadow Prices (pi_1, pi_2) from two frontier goods and test whether a third competing SKU expands or falls inside the Characteristics Efficiency Frontier.
Read the guide →Operations Research & StrategyLanchester’s Square & Linear Laws Formula: Concentration Advantage & Remaining Force
Model competitive market-share and operations-research engagements using Frederick Lanchester’s Square Law (alpha * A_0^2 vs beta * B_0^2), parity threshold, and surviving resource capacity.
Read the guide →FinancialLanded Cost per unit formula: allocate ocean freight & customs tariffs across SKUs
Estimate per-unit cost using supplied duty rates and a weight-based allocation of shared freight.
Read the guide →Water Chemistry, Cooling-Water Scaling & Corrosion ControlLangelier Saturation (LSI) & Ryznar Stability (RSI) Formula
Calculate AWWA / ASTM D3739 Calcium Carbonate Saturation pHs, Langelier Saturation Index LSI (pH - pHs), Ryznar Stability Index RSI (2*pHs - pH), and Scale vs Corrosion Regime.
Read the guide →Engineering & ScienceLangevin Function L(x) = coth(x) - 1/x for Superparamagnetic Nanoparticles
Evaluate Paul Langevin’s magnetization curve M / M_sat = coth(x) - 1/x as a function of magnetic-to-thermal energy ratio x = (mu * B) / (k_B * T) for MRI iron-oxide ferrofluids and dielectric dipole alignment.
Read the guide →EngineeringLangmuir & Freundlich Adsorption Isotherm Capacity formulas
Compare equilibrium Langmuir and Freundlich uptake for a hypothetical solute/adsorbent system with explicit concentration and loading units.
Read the guide →Engineering & ScienceLangmuir Probe Plasma Diagnostics: Bohm Speed & Ion Saturation Current Formula
Calculate the Bohm acoustic ion sheath entry speed u_B = SQRT(e * T_e_eV / m_i) in m/s and Langmuir probe ion saturation current I_sat = 0.61 * e * n_e * A_p * u_B in mA for RF etch and thruster plasmas.
Read the guide →Chemical, Process & Catalysis EngineeringLangmuir-Hinshelwood Bimolecular Surface Kinetics Formula & Self-Inhibition Peak
Calculate competitive bimolecular Langmuir-Hinshelwood catalytic reaction rate r = k_s * K_A * P_A * K_B * P_B / (1 + K_A * P_A + K_B * P_B)^2 and optimal reactant partial pressure P_A,opt = (1 + K_B * P_B) / K_A.
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