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Indian fact-checking organization
Digital Forensics, Research and Analytics Center (DFRAC) is an Indian non-profit organization founded by Shujaat Ali Quadri and Prashant Tandon in 2021
DFRAC
Matrix of partial derivatives of a vector-valued function
dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Matrix of second derivatives
_{f}={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial
Hessian_matrix
SI unit of electric capacitance
}{\text{kg}}}}={\dfrac {\text{C}}{\text{V}}}={\dfrac {{\text{A}}{\cdot }{\text{s}}}{\text{V}}}={\dfrac {{\text{W}}{\cdot }{\text{s}}}{{\text{V}}^{2}}}={\dfrac
Farad
Model of enzyme kinetics
}+K_{\mathrm {mA} }b+(K_{\mathrm {mB} }+b)a}}={\dfrac {{\dfrac {Vb}{K_{\mathrm {mB} }+b}}\cdot a}{{\dfrac {K_{\mathrm {iA} }K_{\mathrm {mB} }+K_{\mathrm
Michaelis–Menten_kinetics
Linear filter in the time domain
}}\\{\dfrac {1}{T_{s}}}{\dfrac {\sin \left[\pi {\dfrac {t}{T_{s}}}\left(1-\beta \right)\right]+4\beta {\dfrac {t}{T_{s}}}\cos \left[\pi {\dfrac {t}{T_{s}}}\left(1+\beta
Root-raised-cosine_filter
SI unit of inductance
{A} ^{2}}}&&={\dfrac {\mathrm {J} }{\mathrm {A} ^{2}}}&&={\dfrac {\mathrm {kg} {\cdot }\mathrm {m} ^{2}}{\mathrm {C} ^{2}}}&&={\dfrac {\mathrm {s} ^{2}}{\mathrm
Henry_(unit)
Audio companding in communications
F(x)=\operatorname {sgn}(x){\begin{cases}{\dfrac {A|x|}{1+\ln(A)}},&|x|<{\dfrac {1}{A}},\\[1ex]{\dfrac {1+\ln(A|x|)}{1+\ln(A)}},&{\dfrac {1}{A}}\leq |x|\leq 1,\end{cases}}}
A-law_algorithm
Economic formula of productivity
{\dfrac {\partial Y/Y}{\partial {L}/L}}={\dfrac {\partial Y}{\partial L}}{\dfrac {L}{Y}}=\alpha AL^{\alpha -1}K^{\beta }{\dfrac {L}{Y}}=\alpha {\dfrac {AL^{\alpha
Cobb–Douglas production function
Cobb–Douglas_production_function
Coordinate system whose directions vary in space
_{x}={\dfrac {\partial \mathbf {r} }{\partial x}};\;\mathbf {e} _{y}={\dfrac {\partial \mathbf {r} }{\partial y}};\;\mathbf {e} _{z}={\dfrac {\partial
Curvilinear_coordinates
Vector operator in vector calculus
_{i}={\begin{bmatrix}{\dfrac {\partial A_{11}}{\partial x_{1}}}+{\dfrac {\partial A_{12}}{\partial x_{2}}}+{\dfrac {\partial A_{13}}{\partial x_{3}}}\\{\dfrac {\partial
Divergence
Unit of rotational speed
{\begin{array}{rcrcr}1~{\dfrac {\text{rad}}{\text{s}}}&=&{\dfrac {1}{2\pi }}~{\text{Hz}}&=&{\dfrac {60}{2\pi }}~{\text{rpm}}\\2\pi ~{\dfrac
Revolutions_per_minute
Condition when the angle of deviation is minimal in a prism
2 ) sin ( A 2 ) {\displaystyle n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}} This is useful to calculate
Minimum_deviation
Rotationally symmetric stress distribution
_{z}={\dfrac {F}{A}}={\dfrac {Pr^{2}}{(r+t)^{2}-r^{2}}}\ } Though this may be approximated to σ z = P r 2 t {\displaystyle \sigma _{z}={\dfrac {Pr}{2t}}\
Cylinder_stress
Control loop feedback mechanism
dfrac {\Delta t}{T_{i}}}+{\dfrac {T_{d}}{\Delta t}}\right)e(t_{k})+\left(-1-{\dfrac {2T_{d}}{\Delta t}}\right)e(t_{k-1})+{\dfrac {T_{d}}{\Delta
PID_controller
Wing shape
≤ 1 , {\displaystyle y_{c}={\begin{cases}{\dfrac {m}{p^{2}}}\left(2px-x^{2}\right),&0\leq x\leq p,\\{\dfrac {m}{(1-p)^{2}}}\left((1-2p)+2px-x^{2}\right)
NACA_airfoil
Statement in mathematics
+ d d {\displaystyle \ {\dfrac {a+b}{b}}={\dfrac {c+d}{d}}} , a − b b = c − d d {\displaystyle \ {\dfrac {a-b}{b}}={\dfrac {c-d}{d}}} . If a b = c
Proportion_(mathematics)
SI unit of magnetic field strength
\mathrm {T={\dfrac {N{\cdot }s}{C{\cdot }m}}} .} Expressed in SI base units, 1 tesla is: T = k g A ⋅ s 2 , {\displaystyle \mathrm {T={\dfrac {kg}{A{\cdot
Tesla_(unit)
Average velocity of particles mainly moving randomly
[u]={\dfrac {\text{A}}{{\dfrac {\text{electron}}{{\text{m}}^{3}}}{\cdot }{\text{m}}^{2}\cdot {\dfrac {\text{C}}{\text{electron}}}}}={\dfrac {\dfrac
Drift_velocity
Constant value used in mathematics
square numbers behaves asymptotically as b x log ( x ) . {\displaystyle {\dfrac {bx}{\sqrt {\log(x)}}}.} This constant b was rediscovered in 1913 by Srinivasa
Landau–Ramanujan_constant
Theorem in mathematical analysis
{\displaystyle {\dfrac {1}{p}}={\dfrac {j}{n}}+\theta \left({\dfrac {1}{r}}-{\dfrac {m}{n}}\right)+{\dfrac {1-\theta }{q}},\qquad {\dfrac {j}{m}}\leq \theta
Gagliardo–Nirenberg interpolation inequality
Gagliardo–Nirenberg_interpolation_inequality
Science of air vehicle orientation and control in three dimensions
π 2 ≡ {\displaystyle \lambda ={\dfrac {\mu }{\rho }}{\sqrt {\dfrac {\pi }{2R\theta }}}={\dfrac {M}{Re}}{\sqrt {\dfrac {k\pi }{2}}}\equiv } mean free path
Aircraft_flight_dynamics
American physicist (1922–2009)
=-G\int \left[{\dfrac {[\rho ]}{r^{3}}}+{\dfrac {1}{r^{2}c}}\left[{\dfrac {\partial \rho }{\partial t}}\right]\right]\mathbf {(} r)dV'+{\dfrac {G}{c^{2}}}\int
Oleg_D._Jefimenko
Type of potential in electrodynamics
Lorenz gauge: ◻ φ = ρ ϵ 0 , ◻ A = μ 0 J {\displaystyle \Box \varphi ={\dfrac {\rho }{\epsilon _{0}}}\,,\quad \Box \mathbf {A} =\mu _{0}\mathbf {J} }
Retarded_potential
Mathematical technique
Euler-Bernoulli beam theory ± E I d 2 w d x 2 = M {\displaystyle \pm EI{\dfrac {d^{2}w}{dx^{2}}}=M} Where w {\displaystyle w} is the deflection and M {\displaystyle
Macaulay's_method
Special function in the physical sciences
) d t {\displaystyle \operatorname {Ai} (x)={\dfrac {1}{\pi }}\int _{0}^{\infty }\!\cos \left({\dfrac {t^{3}}{3}}+xt\right)\,dt} , which converges by
Airy_function
Dimensionless number describing oscillating flow mechanisms
= m U 2 k L 2 = U 2 ω 0 2 L 2 {\displaystyle {\dfrac {mU^{2}}{FL}}={\dfrac {mU^{2}}{kL^{2}}}={\dfrac {U^{2}}{\omega _{0}^{2}L^{2}}}} , where, m = mass
Strouhal_number
Possible solution to the measurement problem
\Lambda (d)={\begin{cases}{\dfrac {6GM^{2}}{5R_{0}\hbar }}\left({\dfrac {5}{3}}\lambda ^{2}-{\dfrac {5}{4}}\lambda ^{3}+{\dfrac {1}{6}}\lambda ^{5}\right)&{\text{when
Diósi–Penrose_model
Statistical model
_{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {x_{i1}-x_{i2}}{2}}'+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}{\dfrac
Fixed_effects_model
Bug in the Intel P5 Pentium floating-point unit
195,835 − 256 3,145,727 {\displaystyle \textstyle {\dfrac {4{,}195{,}579}{3{,}145{,}727}}={\dfrac {4{,}195{,}835-256}{3{,}145{,}727}}} . Thomas Nicely
Pentium_FDIV_bug
Relating coefficients and roots of a polynomial
+r_{n-1}+r_{n}=-{\dfrac {a_{1}}{a_{0}}}\\[1ex](r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{1}r_{n})+(r_{2}r_{3}+r_{2}r_{4}+\cdots +r_{2}r_{n})+\cdots +r_{n-1}r_{n}={\dfrac
Vieta's_formulas
Family of probability distributions
\ln(-\ln p)~~&{\text{for }}~~\xi =0~~{\text{ and }}~~p\in (0,1)\ ,\\\mu +{\dfrac {\sigma }{\xi }}{\Big (}\ (-\ln p)^{-\xi }-1\ {\Big )}~~&{\text{for }}~~\xi
Generalized extreme value distribution
Generalized_extreme_value_distribution
Technique in electrical circuit analysis
dfrac {V_{\text{s3}}}{Z_{\text{s3}}}}\right)Z_{\text{s1Y}}\\[2ex]&V_{\text{s2Y}}=\left({\dfrac {V_{\text{s2}}}{Z_{\text{s2}}}}-{\dfrac
Y-Δ_transform
Notion in statistics
_{m}}}&={\begin{bmatrix}{\dfrac {\partial \mu _{1}}{\partial \theta _{m}}}&{\dfrac {\partial \mu _{2}}{\partial \theta _{m}}}&\cdots &{\dfrac {\partial \mu _{N}}{\partial
Fisher_information
Approximation technique in integral calculus
A visual representation of the area under the curve y = x2 over [0, 2]. Using antiderivatives this area is exactly 8 3 {\textstyle {\dfrac {8}{3}}} .
Riemann_sum
Length in solid geometry
-\left[{\dfrac {(\mathbf {y} -\mathbf {p} )\cdot \mathbf {a} }{\mathbf {a} \cdot \mathbf {a} }}\right]\mathbf {a} =\mathbf {y} -\left[{\dfrac {\mathbf
Distance from a point to a plane
Distance_from_a_point_to_a_plane
Economic principle
{\displaystyle Q={\dfrac {P-4}{-2}}={\dfrac {4-P}{2}}=2-{\dfrac {P}{2}}} ∂ Q ∂ P = − 1 2 {\displaystyle {\dfrac {\partial Q}{\partial P}}=-{\dfrac {1}{2}}} P
Elasticity_(economics)
Distance beyond which all objects can be brought into an acceptable focus
{\begin{array}{crclcl}&{\dfrac {H}{D/2}}&=&{\dfrac {x}{c/2}}\\\therefore &H&=&{\dfrac {Dx}{c}}&=&{\dfrac {D}{c}}{\Big (}f+{\dfrac {cf}{D}}{\Big )}\\&&=&{\dfrac {Df}{c}}+f&=&{\dfrac
Hyperfocal_distance
Fundamental operation on complex numbers
x=\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}}} Imaginary part: y = Im ( z ) = z − z ¯ 2 i {\displaystyle y=\operatorname {Im} (z)={\dfrac {z-{\overline
Complex_conjugate
Sampling algorithm
{\partial H}{\partial p_{i}}}\quad {\text{and}}\quad {\dfrac {{\text{d}}p_{i}}{{\text{d}}t}}=-{\dfrac {\partial H}{\partial x_{i}}}} where x i {\displaystyle
Hamiltonian_Monte_Carlo
Measurement of the harmonic distortion present in a signal
{\frac {\cot {\dfrac {\pi }{\sqrt {2}}}\cdot \coth ^{2}{\dfrac {\pi }{\sqrt {2}}}-\cot ^{2}{\dfrac {\pi }{\sqrt {2}}}\cdot \coth {\dfrac {\pi }{\sqrt {2}}}-\cot
Total_harmonic_distortion
Function in fluid mathematics
as, L = − u ∗ 3 κ g T Q ρ c p {\displaystyle L=-{\dfrac {u_{*}^{3}}{\kappa {\dfrac {g}{T}}{\dfrac {Q}{\rho c_{p}}}}}} where κ ≈ 0.40 {\displaystyle \kappa
Monin–Obukhov similarity theory
Monin–Obukhov_similarity_theory
Rules to verify computer program correctness
before skip also holds true afterwards. { P } skip { P } {\displaystyle {\dfrac {}{\{P\}{\texttt {skip}}\{P\}}}} The assignment axiom states that, after
Hoare_logic
Approach to feature extraction
F'(x)\approx {\dfrac {F(x+h)-F(x)}{h}}={\dfrac {G(x)-F(x)}{h}}\,} so that h ≈ G ( x ) − F ( x ) F ′ ( x ) {\displaystyle h\approx {\dfrac {G(x)-F(x)}{F'(x)}}\
Kanade–Lucas–Tomasi feature tracker
Kanade–Lucas–Tomasi_feature_tracker
Statistic quantifying the association between two events
00 {\displaystyle {{\rm {SE}}={\sqrt {{\dfrac {1}{n_{11}}}+{\dfrac {1}{n_{10}}}+{\dfrac {1}{n_{01}}}+{\dfrac {1}{n_{00}}}}}}} . This is an asymptotic
Odds_ratio
Two or more natural numbers with a common abundancy index
= 72 30 = 12 5 {\displaystyle {\dfrac {\sigma (30)}{30}}={\dfrac {1+2+3+5+6+10+15+30}{30}}={\dfrac {72}{30}}={\dfrac {12}{5}}} σ ( 140 ) 140 = 1 + 2 +
Friendly_number
Dimensionless parameter to quantify fluid resistance
}} is defined as c d = 2 F d ρ u 2 A {\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}}} where: F d {\displaystyle F_{\mathrm
Drag_coefficient
Antiderivative of the secant function
+ C {\displaystyle \int \sec \theta \,d\theta ={\begin{cases}{\dfrac {1}{2}}\ln {\dfrac {1+\sin \theta }{1-\sin \theta }}+C\\[15mu]\ln {{\bigl |}\sec \theta
Integral of the secant function
Integral_of_the_secant_function
Type of structure in atomic physics
{H}}_{\text{D}}={}&2g_{\text{I}}\mu _{\text{N}}\mu _{\text{B}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {1}{L_{z}}}\sum _{i}{\dfrac {{\hat {\ell }}_{zi}}{r_{i}^{3}}}\mathbf {I}
Hyperfine_structure
Basketball statistic
Rebounds ) {\displaystyle {\text{Rebound Rate}}={\dfrac {100\times {\text{Rebounds}}\times {\dfrac {\text{Team Minutes Played}}{5}}}{{\text{Minutes Played}}\times
Rebound_rate
0 ≤ G L i ≤ 1 {\displaystyle GL_{i}={\dfrac {(X_{i}+M_{i})-\left|X_{i}-M_{i}\right|}{X_{i}+M_{i}}}=1-{\dfrac {\left|X_{i}-M_{i}\right|}{X_{i}+M_{i}}}\qquad
Grubel–Lloyd_index
How many standard deviations apart from the mean an observed datum is
_{i}x_{i}}{n}}\right)={\dfrac {1}{n^{2}}}\operatorname {Var} \left(\sum _{i}x_{i}\right)={\dfrac {n\sigma ^{2}}{n^{2}}}={\dfrac {\sigma ^{2}}{n}}\end{aligned}}}
Standard_score
Function describing the effects of feedback on a control system
( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}} G ( s ) {\displaystyle G(s)} is called the
Closed-loop_transfer_function
Change in apparent direction of light rays due to special relativity
θ s {\displaystyle \cos \theta _{o}={\frac {\cos \theta _{s}-{\dfrac {v}{c}}}{1-{\dfrac {v}{c}}\cos \theta _{s}}}} This expression can also be written
Relativistic_aberration
Two-dimensional laminar boundary layer that forms on a semi-infinite plate
{\displaystyle u{\dfrac {\partial u}{\partial x}}+v{\dfrac {\partial u}{\partial y}}=-{\dfrac {1}{\rho }}{\dfrac {\partial p}{\partial x}}+{\nu }{\dfrac {\partial
Blasius_boundary_layer
Calculation of complex statistical distributions
partial sums: S n ( h ) = 1 n ∑ i = 1 n h ( X i ) {\displaystyle S_{n}(h)={\dfrac {1}{n}}\sum _{i=1}^{n}h(X_{i})} as n goes to infinity. Particularly, we
Markov_chain_Monte_Carlo
Describes the highest power of primes dividing a binomial coefficient
= 3. {\displaystyle \nu _{2}\!{\binom {10}{3}}={\dfrac {S_{2}(3)+S_{2}(7)-S_{2}(10)}{2-1}}={\dfrac {2+3-2}{2-1}}=3.} Kummer's theorem can be generalized
Kummer's_theorem
Game theory concept
{\begin{bmatrix}{\dfrac {dx_{1}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dx_{2}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dy_{2}^{*}}{dp_{1}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial
Strategic_complements
Multivariate derivative (mathematics)
dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial
Gradient
Equation giving the form of a central force
^{2}}}+u={\begin{cases}{\dfrac {r_{s}c^{2}}{2h^{2}}}+{\dfrac {3}{2}}r_{s}u^{2}-{\dfrac {\Lambda c^{2}}{3h^{2}u^{3}}}&{\text{(particle)}}\\[1.2em]{\dfrac
Binet_equation
Mathematical function
{\dfrac {(2\ell +1)\pi r}{m}}=-{\frac {\pi }{m}}\sum _{r=1}^{m-1}{\frac {r\cdot \sin {\dfrac {2\pi r}{m}}}{\cos {\dfrac {2\pi r}{m}}-\cos {\dfrac {(2\ell
Digamma_function
Property of many linear time-invariant (LTI) systems
{\displaystyle T(z)={\dfrac {z-1}{z}}Z[Y(s)]} T ( z ) = z − 1 z Z [ T ( s ) s ] {\displaystyle T(z)={\dfrac {z-1}{z}}Z[{\dfrac {T(s)}{s}}]} The bilinear
Infinite_impulse_response
Extension of the factorial function
) Γ ( 1 − x 2 ) ) } , {\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma
Hadamard's_gamma_function
Geometric formula for finding the ratio in which a line segment is divided by a point
P=\left({\dfrac {x_{1}+x_{2}}{2}},{\dfrac {y_{1}+y_{2}}{2}}\right)} P = ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) {\displaystyle P=\left({\dfrac {mx_{2}+nx_{1}}{m+n}}
Section_formula
{\displaystyle a_{mn}={\begin{cases}1&{\text{if }}n=0,\\\\{\dfrac {1}{n!}}&{\text{if }}m=1,\\\\{\dfrac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1
List of integrals of exponential functions
List_of_integrals_of_exponential_functions
Abstract grammatical formalism
{\displaystyle {\dfrac {{\dfrac {{\dfrac {{\dfrac {{\dfrac {\text{the}}{NP/N}}{\dfrac {\text{dog}}{N}}\qquad }{NP}}>}{S/(S\backslash NP)}}T_{>}\qquad {\dfrac
Combinatory categorial grammar
Combinatory_categorial_grammar
expressed in the following form: d x d t = N v ( x ( p ) , p ) {\displaystyle {\dfrac {\bf {dx}}{dt}}={\bf {N}}{\bf {v}}({\bf {x}}(p),p)} The notation for the
Biochemical_systems_equation
Range to estimate an unknown parameter
{\displaystyle {\bar {X}}\pm {\begin{cases}{\dfrac {|X_{1}-X_{2}|}{2}}&{\text{if }}|X_{1}-X_{2}|<1/2\\[8pt]{\dfrac {1-|X_{1}-X_{2}|}{2}}&{\text{if }}|X_{1}-X_{2}|\geq
Confidence_interval
Electron-many photon scattering
{\displaystyle {\dfrac {dN}{dt}}={\dfrac {\sqrt {3}}{2\pi }}{\dfrac {q^{2}mc}{\hbar ^{2}}}{\dfrac {\chi }{\gamma }}\int _{0}^{\chi }{\dfrac {F(\chi ,\eta
Non-linear inverse Compton scattering
Non-linear_inverse_Compton_scattering
Branch of statistical computational learning theory
{\mathcal {F}}}{\dfrac {1}{n}}\left|\sum _{i=1}^{n}f(X_{i})-\mathbb {E} f(Y_{i})\right|\leq \mathbb {E} _{Y}\sup _{f\in {\mathcal {F}}}{\dfrac {1}{n}}\left|\sum
Vapnik–Chervonenkis_theory
Effects of pore diffusion on Rate of heterogeneous chemical reaction
criterion is N W − P = R R p 2 C s D e f f ≤ 3 β {\displaystyle N_{W-P}={\dfrac {{\mathfrak {R}}R_{p}^{2}}{C_{s}D_{eff}}}\leq 3\beta } Where R {\displaystyle
Weisz–Prater_criterion
Conglomeration of discrete solid, macroscopic particles
2 Γ 3 ⟨ ε ⟩ 2 {\displaystyle {\dfrac {d\left\langle \varepsilon ^{2}\right\rangle }{dt}}=lim_{dt\rightarrow 0}{\dfrac {\left\langle \varepsilon ^{2}(t+dt)\right\rangle
Granular_material
Measure of the relative size of firms
I − 1 N ) 1 − 1 N {\textstyle HHI^{*}={\cfrac {\left(HHI-{\dfrac {1}{N}}\right)}{1-{\dfrac {1}{N}}}}} for N > 1 and H H I ∗ = 1 {\displaystyle HHI^{*}=1}
Herfindahl–Hirschman_index
Decimal representation of a number whose digits are periodic
=3\\&\#\mathbf {A} =0&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {13-1}{9}}&={\dfrac {12}{9}}&={\dfrac {4}{3}}\\\\0.3789789\ldots &=0.3{\overline
Repeating_decimal
Theorem used in structural analysis
{\displaystyle {\dfrac {PR}{RB'}}={\dfrac {SQ}{B'S}},} From (1), (2), and (3), Δ B − Δ A + P A ′ L 1 = Δ C − Δ B − Q C ′ L 2 {\displaystyle {\dfrac {\Delta B-\Delta
Theorem_of_three_moments
SI derived unit of magnetic flux
{\displaystyle \mathrm {Wb} =\Omega {\cdot }{\text{C}}={\dfrac {\mathrm {J} }{\mathrm {A} }}={\dfrac {\mathrm {N} {\cdot }\mathrm {m} }{\mathrm {A} }},} where
Weber_(unit)
Formula in number theory
)={\begin{cases}-{\dfrac {\pi }{q^{3/2}}}\sum _{m=1}^{q-1}m\left({\dfrac {m}{q}}\right),&q\equiv 3\mod 4;\\-{\dfrac {1}{2q^{1/2}}}\sum _{m=1}^{q-1}\left({\dfrac {m}{q}}\right)\ln
Class_number_formula
Special mathematical function defined as sin(x)/x
{\displaystyle {\frac {d}{dx}}\operatorname {sinc} (x)={\begin{cases}{\dfrac {\cos(x)-\operatorname {sinc} (x)}{x}},&x\neq 0\\0,&x=0\end{cases}}.} The
Sinc_function
Characteristic property of holomorphic functions
y)={\begin{bmatrix}{\dfrac {\partial u}{\partial x}}&{\dfrac {\partial u}{\partial y}}\\[5pt]{\dfrac {\partial v}{\partial x}}&{\dfrac {\partial v}{\partial
Cauchy–Riemann_equations
Ordered binary tree of rational numbers
dfrac {a}{b}},{\dfrac {c}{d}},{\dfrac {e}{f}}\right)&&\\&\swarrow &&\searrow &\\\left({\dfrac {a}{b}},{\dfrac {a+c}{b+d}},{\dfrac
Stern–Brocot_tree
Algorithm for Euclidean division of polynomials
a 1 x + a 0 b 4 x 4 − b 3 x 3 − b 2 x 2 − b 1 x − b 0 {\displaystyle {\dfrac {a_{7}x^{7}+a_{6}x^{6}+a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1
Synthetic_division
v e r a g e i n v e n t o r y C O G S / D a y s {\displaystyle DII={\dfrac {average~inventory}{COGS/Days}}} , alternatively expressed as: D I I = I
Days_in_inventory
Classical statement of gravity as force
R {\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}} Newton's
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
Structure defining distance on a manifold
dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial
Metric_tensor
Question in geometric probability
\Theta }(x,\theta )={\begin{cases}{\dfrac {4}{t\pi }}&:\ 0\leq x\leq {\dfrac {t}{2}},\ 0\leq \theta \leq {\dfrac {\pi }{2}}\\[4px]0&:{\text{elsewhere
Buffon's_needle_problem
u s ¯ r − s | ≤ π ∑ r | u r | 2 . {\displaystyle \left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{r}|u_{r}|^{2}
Hilbert's_inequality
Coordinates comprising a distance and two angles
dfrac {x}{r}}&{\dfrac {y}{r}}&{\dfrac {z}{r}}\\\\{\dfrac {xz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac {yz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac
Spherical_coordinate_system
Dimensionless astrophysics equation
− 1 ) {\displaystyle \left({\dfrac {n-3}{n-1}},2{\dfrac {n+1}{n-1}}\right)} n − 5 ± Δ n 2 − 2 n {\displaystyle {\dfrac {n-5\pm \Delta _{n}}{2-2n}}} (
Lane–Emden_equation
Type of prime number
{\displaystyle \left({\dfrac {p-1}{p}}\right)^{\dfrac {p-3}{2}}=\left(1-{\dfrac {1}{p}}\right)^{\dfrac {p-3}{2}}=\left(1-{\dfrac {1}{p}}\right)^{-3/2}\cdot
Regular_prime
29 − 3 3 ⋅ 31 2 3 3 {\displaystyle {\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {3\cdot 31}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {3\cdot 31}}}{2}}}}{3}}}
List_of_numbers
Approximation of a function by a polynomial
quadratic approximation P 2 ( x ) = 1 + x + x 2 2 {\displaystyle P_{2}(x)=1+x+{\dfrac {x^{2}}{2}}} (red) at a = 0 {\textstyle a=0} . Note the improvement in the
Taylor's_theorem
Kind of arithmetic error
{\begin{array}{l}\;\;\;{\dfrac {d}{dx}}{\dfrac {1}{x}}\\={\dfrac {d}{d}}{\dfrac {1}{x^{2}}}\\={\dfrac {\cancel {d}}{\cancel {d}}}{\dfrac {1}{x^{2}}}\\=-{\dfrac {1}{x^{2}}}\end{array}}}
Anomalous_cancellation
Quantity in relativistic physics
{\begin{aligned}\gamma &={\dfrac {1}{\sqrt {1-\beta ^{2}}}}\\[1ex]&=\sum _{n=0}^{\infty }\beta ^{2n}\prod _{k=1}^{n}\left({\dfrac {2k-1}{2k}}\right)\\[1ex]&=1+{\tfrac
Lorentz_factor
Math problem
y)&={\dfrac {1}{1-xyF(x,xy)}}&={\dfrac {1}{1-{\dfrac {xy}{1-x^{2}yF(x,x^{2}y)}}}}&=\cdots &={\dfrac {1}{1-{\dfrac {xy}{1-{\dfrac {x^{2}y}{1-{\dfrac {x^{3}y}{\cdots
Coins_in_a_fountain
Random model in mathematics
\right)\\&={\dfrac {(\alpha +\gamma -1)!}{(\alpha -1)!\,(\gamma -1)!}}{\dfrac {\Gamma (\gamma +n-k)\Gamma (\alpha +k)}{\Gamma (\alpha +\gamma +n)}}\\&={\dfrac {\left(\alpha
Pólya_urn_model
Three-dimensional packing problem
{\displaystyle 1} 1.0000 1 Trivially optimal. Point 2 1 2 {\displaystyle {\dfrac {1}{2}}} 0.5000 0.25 Trivially optimal. Line segment 3 2 3 − 3 {\displaystyle
Sphere_packing_in_a_sphere
Force acting on charged particles in electric and magnetic fields
_{\mathbf {x} }={\hat {x}}{\dfrac {\partial }{\partial x}}+{\hat {y}}{\dfrac {\partial }{\partial y}}+{\hat {z}}{\dfrac {\partial }{\partial z}}} and
Lorentz_force
Logarithm to the base of the mathematical constant e
\exp x} Derivative d d x ln x = 1 x , x > 0 {\displaystyle {\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0} Antiderivative ∫ ln x d x = x ( ln x − 1 ) +
Natural_logarithm
Coding theory algorithm
({\dfrac {1}{2}}).\alpha N\delta \left(({\dfrac {\delta }{2}})-({\dfrac {\lambda }{d}})\right)=\left(({\dfrac {1}{4}}).\alpha N(\delta ^{2}-O({\dfrac {\lambda
Zemor's_decoding_algorithm
Guarantees chords of length 1/n exist for functions satisfying certain conditions
g:\left[a,{\dfrac {b+a}{2}}\right]\to \mathbb {R} } defined by g ( x ) = f ( x + b − a 2 ) − f ( x ) {\displaystyle g(x)=f\left(x+{\dfrac {b-a}{2}}\right)-f(x)}
Universal_chord_theorem
DFRAC
DFRAC
DFRAC
DFRAC
Boy/Male
Muslim/Islamic
Life Soul
Boy/Male
Hindu, Indian
Love to God
Girl/Female
Muslim
Hope, Shining light
Boy/Male
Indian, Sanskrit
Moon
Girl/Female
Latin
Leafy branch.
Girl/Female
Indian
Fem of manar: light-house
Surname or Lastname
English
English : habitational name from Bainbridge in North Yorkshire, named for the Bain river on which it stands (which is named with Old Norse beinn ‘straight’) + Old English brycg ‘bridge’.A family of this name was very prominent in Princeton, NJ, from the mid 17th century.
Girl/Female
Muslim
Lord rams devotees, Daughter of cyprus (Daughter of cyprus)
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Princess of God; Beautification of Paradise
Boy/Male
Tamil
Vajrapani | வஜà¯à®°à®ªà®¾à®¨à¯€
Lord Indra
DFRAC
DFRAC
DFRAC
DFRAC
DFRAC