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Boolean differential calculus (2,199 words) [view diff] no match in snippet view article find links to article

Boolean differential calculus (BDC) (German: Boolescher Differentialkalkül (BDK)) is a subject field of Boolean algebra discussing changes of Boolean variables

looks more like the lowercase Latin ⟨d⟩, a mirrored numeral 6 or a partial derivative symbol ⟨∂⟩. Southern (Serbian, Bulgarian, Macedonian) typography may

identification is often made between a coordinate basis vector eα and the partial derivative operator ∂/∂xα, under the interpretation of vectors as operators acting

T}}\right)_{P}=VT\alpha \left({\frac {\partial S}{\partial V}}\right)_{T}\,} The partial derivative ( ∂ S ∂ V ) T {\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}}

energy, U, is only a function of the temperature T; therefore the partial derivative of heat capacity with respect to T is identically the same as the

time on the outcomes we hope to achieve". More formally, it is the partial derivative of the total expected value with respect to time. Cost of Delay combines

derivative. Note that the equation with a partial derivative is not covariant, since the partial derivative of a tensor is not a tensor, and is only valid

_{\geq 0}^{n}} Where D α {\displaystyle D^{\alpha }} denotes the partial derivative of order α {\displaystyle \alpha } (see multi-index notation). When

The partial derivative of the density with respect to time need not vanish to ensure incompressible flow. When we speak of the partial derivative of the

C_{P}=\left({\frac {\partial H}{\partial T}}\right)_{P}} which is the partial derivative of the enthalpy with respect to temperature while holding pressure

a partial derivative into a total derivative, then: M ( x ) ( y ′ + P ( x ) y ⏟ ) partial derivative {\displaystyle M(x){\underset {\text{partial

_{j}u_{i}\right)} where ∂ i {\displaystyle \partial _{i}} denotes partial derivative with respect to position coordinate x i {\displaystyle x_{i}} . The

Đồng sign ⅆ : Unicode symbol for d used as derivative symbol ∂ : the partial derivative symbol, ∂ {\displaystyle \partial } The Latin letters ⟨D⟩ and ⟨d⟩

exact by the implicit function theorem: if f(x0, y0, z0) = 0, and the partial derivative in z of f is not zero at (x0, y0, z0), then there exists a differentiable

side of the relation and the ⅃ will give the other. Note that the partial derivative is taken along the vertical stem of L (and ⅃) while the last corner

the unity in A ( R ) {\displaystyle A(\mathbb {R} )} , There is a partial derivative operator ∂ {\displaystyle \partial } on A ( R ) {\displaystyle A(\mathbb

unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation

{\displaystyle (\mu _{j})_{j}} are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good x j k {\displaystyle x_{j}^{k}}

{x_{i}}{1+e^{\frac {2\mu x_{i}}{\sigma ^{2}}}}}} . By equating the first partial derivative of the log-likelihood to zero, we obtain a nice relationship ∑ i =

averse. It can be shown that the partial derivative of v with respect to μw is positive, and the partial derivative of v with respect to σw is negative;

relativity, we have to use the covariant derivative instead of the partial derivative in d'Alembert's equation, so we get: 0 = ( x α ) ; β ; γ g β γ = (

to this nonlinear edge is an edge weight that is the second-order partial derivative of the nonlinear node in relation to its predecessors. This nonlinear

^{2}-x_{c}^{2}\right)\cdot \Im _{w}\right)\end{aligned}}} for the first order partial derivative V ′ = ∂ V ∂ x {\displaystyle V'={\frac {\partial V}{\partial x}}}

{\vec {n}},t)} , where ∂ 1 G {\displaystyle \partial _{1}G} is the partial derivative of G {\displaystyle G} with respect to the first argument. The divergence

elastici", which contains his theorem for computing displacement as partial derivative of the strain energy. This theorem includes the method of least work

a partial derivative, the multivariable chain rule should be used. Hence, to avoid ambiguity, the function arguments of any term inside of a partial derivative

second order partial derivative of the function f with respect to x. Thus, p t {\displaystyle p_{t}} is the partial derivative of the density p ( t

the derivative is fully known. In fact, the family of arithmetic partial derivative ∂ ∂ p {\textstyle {\frac {\partial }{\partial p}}} relative to the

_{0}\\{\frac {a^{2}}{\rho _{0}}}&0\end{bmatrix}}} and the index denotes the partial derivative with respect to the corresponding variable (i.e. x or t). The eigenvalues

derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian. Sometimes the box symbol

\textstyle {\frac {\partial P}{\partial x_{i}}}} denotes the formal partial derivative of P with respect to x i . {\displaystyle x_{i}.} A non-homogeneous

L}{\partial y'}}{\frac {\Delta y_{m}}{\Delta t}}} Evaluating the partial derivative gives ∂ J ∂ y m = L y ( t m , y m , y m + 1 − y m Δ t ) Δ t + L y

from say coordinates { x α } {\displaystyle \{x^{\alpha }\}} , the partial derivative with respect to a general orthonormal frame { e μ } {\displaystyle

_{1}^{\alpha _{1}}\cdots \partial _{n}^{\alpha _{n}}u} denotes the partial derivative of order α i {\displaystyle \alpha _{i}} in x i {\displaystyle x_{i}}

{\displaystyle A} 's Poisson bracket with the Hamiltonian equals minus its partial derivative with respect to time ∂ A ∂ t = − { A , H } . {\displaystyle {\frac

in each of the N {\displaystyle N} gradients. Now assume that each partial derivative provides as much information for our surrogate as a single primal

power is the rate of energy transferred in an electric circuit. As a partial derivative, it is expressed as the change of work over time: P = ∂ W ∂ t = ∂

transfer onto the normal to the wall of the container. Since the partial derivative ( ∂ u ∂ T ) V {\displaystyle \left({\frac {\partial u}{\partial T}}\right)_{V}}

where L a a ( x ) {\displaystyle L_{aa}(\mathbf {x} )} is second partial derivative in the a {\displaystyle a} direction and L a b ( x ) {\displaystyle

method provides rapid convergence; however, it requires the first partial derivative of the option's theoretical value with respect to volatility; i.e

}}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0} where the partial derivative ∂ ← μ {\displaystyle {\overleftarrow {\partial }}_{\mu }} acts from

bcf}\,} where Einstein's sum convention is implied, Φ,a denotes the partial derivative of the function Φ by the coordinate vectorfield ∂/∂xa which are all

The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule

the perturbation, ∂ μ {\displaystyle \partial _{\mu }} denotes the partial derivative with respect to the x μ {\displaystyle x^{\mu }} coordinate of spacetime

describe its derivative as where f t {\displaystyle f_{t}} denotes the partial derivative of f {\displaystyle f} with respect to t {\displaystyle t} . Namely

{\displaystyle {T_{a}}^{b}} is the stress–energy tensor, the comma indicates a partial derivative and the semicolon indicates a covariant derivative. The terms involving

\partial _{a}={\frac {\partial }{\partial x^{a}}}} means taking the partial derivative with respect to the coordinate x a {\displaystyle x^{a}} . Alternatively

_{1})^{\alpha _{1}}\cdots (-i\partial _{n})^{\alpha _{n}}} is an iterated partial derivative, where ∂j means differentiation with respect to the j-th variable

{\displaystyle \gamma } , its optimal value could be calculated by setting the partial derivative of the objective against γ {\displaystyle \gamma } to 0. The objective

bidifferential operator BΓ( f, g) defined as follows. For each edge there is a partial derivative on the symbol of the target vertex. It is contracted with the corresponding

M_{y}} , and N x {\textstyle N_{x}} , where the subscripts denote the partial derivative with respect to the relative variable, be continuous in the region

variable. It is essentially the complex conjugate of the ordinary partial derivative with respect to. It's important in complex analysis and complex differential

elastici", which contains his theorem for computing displacement as partial derivative of the strain energy. This theorem includes the method of 'least work'

where ∂ u ∂ y {\displaystyle {\frac {\partial u}{\partial y}}} is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction

elastici", which contains his theorem for computing displacement as the partial derivative of the strain energy. This theorem includes the method of "least work"

covariant derivative on tensor or spin-tensor fields is simply the partial derivative in flat coordinates. However the gauge covariant derivative may require

^{\dagger }(x)e^{-i\alpha (x)},\qquad U^{\dagger }=U^{-1}.} Now the partial derivative ∂ μ {\displaystyle \partial _{\mu }} transforms, accordingly, as ∂

{\displaystyle i} -th direction and ∂ j {\displaystyle \partial _{j}} is the partial derivative in the j {\displaystyle j} -th direction. Note that: ε j j = ∂ j u

then the joint probability density function can be computed as a partial derivative f ( x ) = ∂ n F ∂ x 1 ⋯ ∂ x n | x {\displaystyle f(x)=\left.{\frac

characteristics of the more general n-dimensional case. Each second partial derivative needs to be approximated similar to the 1D case Δ u ( x , y ) = u

not the velocity of the path element dl and (2) in general, the partial derivative with respect to time cannot be moved outside the integral since the

_{-1/2}^{1/2}H_{d}(F)^{2}\,dF} Step 3: Minimize the mean square error by doing partial derivative of MSE with respect to s [ n ] {\displaystyle s[n]} ∂ MSE ∂ s [ n

\partial f_{[k]}\mid _{x=x_{0}}} is the k t h {\displaystyle k^{th}} partial derivative of f ( x ) {\displaystyle f(x)} with respect to x {\displaystyle x}

shows the set of population sizes at which the rate of change, or partial derivative, for one population in a pair of interacting populations is zero.

the topological dual space contains only the zero functional. The partial derivative of the p {\displaystyle p} -norm is given by ∂ ∂ x k ‖ x ‖ p = x k

y}\mathbf {v} \cdot d{\mathbf {r} }.} By the path independence, its partial derivative with respect to x {\displaystyle x} (for φ {\displaystyle \varphi

S=-k\langle \log P\rangle =-{\frac {\partial F}{\partial T}},} the partial derivative ∂F/∂N is approximately related to chemical potential, although the

the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal

{\displaystyle {\text{BTree}}:={\text{lfp}}(T\mapsto A\times T^{2}+1)} The partial derivative of the type constructor can be computed to be ( T ↦ A × T 2 + 1 )

{\displaystyle F(x)} consists in determining the triangular code of the partial derivative ∂ F ( x ) ∂ y {\displaystyle {\frac {\partial F(x)}{\partial y}}}

variable u i ∗ {\displaystyle u_{i}^{*}} can be interpreted as the partial derivative of the optimal value z ∗ {\displaystyle z^{*}} of the objective function

{\displaystyle D_{1}} under his control, giving the condition that the partial derivative of his profit with respect to D 1 {\displaystyle D_{1}} should be

to t represents an intersection point at infinity. If at least one partial derivative of the polynomial p is not zero at an intersection point, then the

free fermion theory and including an interaction which promotes the partial derivative in the fermion theory to a gauge-covariant derivative. Physics portal

Your Last Chapbook, with Mathias Svalina, New Lights Press. 2014. Partial Derivative of the Unnamable, Troll Thread. 2012. Goodbye, John! On John Baldessari

\epsilon _{ab}} is a function of space-time. This means that the partial derivative of a spinor is no longer a genuine tensor. As usual, one introduces

elastici", which contains his theorem for computing displacement as partial derivative of the strain energy. In 1824, Portland cement was patented by the

{\displaystyle u_{x}=v_{y},\quad v_{x}=-u_{y}.} where ux is the first partial derivative of u with respect to x. It follows that u y y = ( − v x ) y = − (

s_{2},s_{1})} for all s1, s2 ∈ X. Here D1 (resp. D2) denotes the partial derivative with respect to the first (resp. second) variable; the dot product

equation, electromagnetic interaction can be added by promoting the partial derivative to gauge covariant derivative: ∂ μ → D μ = ∂ μ − i e A μ {\displaystyle

of gravity and surface area. Correctly balanced in this way, the partial derivative of pitching moment with respect to changes in angle of attack will

{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}},} where the partial derivative with respect to q ˙ {\displaystyle {\dot {q}}} holds q(t + ε) fixed

λ {\displaystyle \lambda } is a Lagrange multiplier, Ci(q) is the partial derivative of C(q) with respect to qi, evaluated at q, and E l a s t i c i t

{\displaystyle \partial /\partial \theta _{i}} means the left or the right partial derivative. These properties define the integral uniquely. Notice that different

{\displaystyle {\hat {D}}} we use the fact that in frequency space, the partial derivative operator can be converted into a number by substituting i k {\displaystyle

) {\displaystyle A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)} be the partial derivative operator ∂ ∂ x k . {\displaystyle {\tfrac {\partial }{\partial x_{k}}}

{h}}_{k}s[n-k]+(\sum _{k=0}^{N-1}{\hat {h}}_{k}s[n-k])^{2}]} Distribute the partial derivative: ∂ E ∂ h ^ i = ∑ n = − ∞ ∞ [ − 2 x [ n ] s [ n − i ] + 2 ( ∑ k = 0

_{0}=\eta _{s}+\eta _{p}} ); R z {\displaystyle R_{z}} indicates the partial derivative ∂ R ∂ z {\displaystyle {\frac {\partial R}{\partial z}}}  ; σ z z

time variable in the right-hand side could be taken outside of the partial derivative, since the latter regards only variable x. It is now possible to remove

^{2}kT}{c^{2}}}.} Taking the logarithm of each side and taking the partial derivative with respect to log ν {\displaystyle \log \,\nu } yields ∂ log ⁡ B

derivative of P i j … {\displaystyle P_{ij\ldots }} is simply the partial derivative with respect to time, and the position vector X {\displaystyle \mathbf

_{i}{x_{i}}.} To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero: ∂ ℓ ∂ α = n α + n ln ⁡ x m − ∑ i =

square brackets, [ ], denote antisymmetrization of indices; ∂α is the partial derivative with respect to the coordinate, xα. In Minkowski space coordinates

holds for non-eigenfunction wave functions which are stationary (partial derivative is zero) for all relevant variables (such as orbital rotations). The

u ∂ t {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}} partial derivative of the velocity field with respect to time ∇ = ( ∂ ∂ x , ∂ ∂ y , ∂

parameters are assumed constant except for the variables contained in the partial derivative, but only U, S, or X are shown. It follows from the properties of

balanced, the droplet accelerates. This acceleration is not simply the partial derivative ∂v/∂t because the fluid in a given volume changes with time. Instead

V ) {\displaystyle (\partial F/\partial T)(T,P,V)} denotes the (partial) derivative of the state equation F {\displaystyle F} with respect to its T {\displaystyle

)&:=\nabla _{\gamma }\log f(V:Z,\gamma )\end{aligned}}} and ∇ represents partial derivative with respect to a row vector. In the case where γ0 is known, the asymptotic

chemical potential, μi, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species

this as the "environment change" term, and denoted both terms using partial derivative notation (∂NS and ∂EC). This concept of environment includes interspecies

The potential, μi, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species

x}\times \mathbb {R} _{\geq 0}\to \mathbb {C} } and the second spatial partial derivative becomes the discrete laplacian ∂ 2 ψ ∂ x 2 → L Z ψ ( j Δ x , t ) Δ

with electromagnetism in a gauge-invariant way. We can replace the (partial) derivative with the gauge-covariant derivative. Under a local U ( 1 ) {\displaystyle

observe the effect of residual blocks on backpropagation, consider the partial derivative of a loss function E {\textstyle {\mathcal {E}}} with respect to some

P}}\right)_{T}=T\left({\frac {\partial S}{\partial P}}\right)_{T}+V} The partial derivative on the left is the isothermal Joule–Thomson coefficient, μ T {\displaystyle

{\displaystyle h\in R} , where ∇ i {\displaystyle \nabla _{i}} denotes the partial derivative with respect to variable x ( i ) {\displaystyle x^{(i)}} . Nesterov

dY=F_{A}dA+F_{K}dK+F_{L}dL} where F i {\displaystyle F_{i}} denotes the partial derivative with respect to factor i, or for the case of capital and labor, the

academic year 1961–1962. Peetre's research deals with ordinary and partial derivative differential equations, operator interpolation spaces, singular integrals

}{\partial \theta }}\int f(x;\theta )\,dx=0} where the integral and partial derivative have been interchanged (justified by the second regularity condition)

u_{y}(z,0)dz} (where u y ( x , y ) {\displaystyle u_{y}(x,y)} is the partial derivative of u ( x , y ) {\displaystyle u(x,y)} with respect to y {\displaystyle

entropy and salinity can be assumed constant. Therefore, taking the partial derivative of this relation with respect to pressure yields: ( ∂ h ∂ p ) S ,

second covariant derivative of a function by a locally defined second partial derivative, it is necessary to control the first derivative of the local representation

digital elevation model, it is approximated using one of the same partial derivative approximation methods developed for slope. Then the aspect is calculated

allows companies to compute the optimal level of their R&D, as the partial derivative of profits with respect to R&D: Ri* = (aγi / (1-δ))1/(1-γi) The goal

network. The activation function for each neuron is defined as a partial derivative of the Lagrangian  with respect to that neuron's activity From the

the marginal product of capital when labor increases by taking the partial derivative of the marginal product of capital with respect to labor, that is

constant of the system can be described by the Hessian matrix – (second partial derivative of potential V): H = [ H i i H i j H j i H j j ] . {\displaystyle

of any constant is 0 {\displaystyle 0} . Hence, we can reduce each partial derivative to: A 3 ( 0 − 0 ) {\displaystyle A_{3}{\big (}0-0{\big )}} and hence

state, is valid. The volume of activation is found by taking the partial derivative of ΔG‡ with respect to pressure (holding temperature constant): Δ

{\mu _{i}^{\ominus }(T)}{RT}}\right).} The chemical potential is a partial derivative: μ i = ∂ F ( T , V , N ) / ∂ N i {\displaystyle \mu _{i}=\partial

interest, in this case the utility function, equal to zero. When the partial derivative of the utility function with respect to health consumption is assumed

derived, and being loose with the mathematical rigor of taking the partial derivative with respect to something that is not a scalar, the relevant equations

(t)\right)\cdot d\mathbf {l} .\end{aligned}}} Applying the definition of a partial derivative to the integrand of the first term, applying Stokes' theorem to the

variable, the conjugate of the original one. The new variable is the partial derivative of the original function with respect to the original variable. The

partial derivatives are derived from a specific equation. The second partial derivative of f ( x , y ) {\displaystyle f(x,y)} with respect to x {\displaystyle

{\displaystyle {\frac {df}{dx}}(x)} δ ′ ( x ) {\displaystyle \delta '(x)} Partial derivative ∂ ∂ x j {\displaystyle {\frac {\partial }{\partial x_{j}}}} 2 π i

found using c = 0 {\displaystyle c=0} . This means we must take the partial derivative (w.r.t. c {\displaystyle c} ) of the usual trial solution in order

i {\textstyle {\frac {\partial g(\mu )}{\partial x_{i}}}} is the partial derivative of g {\displaystyle g} at the mean vector μ {\displaystyle \mu } with

\sigma _{xx}(x)}{\partial x}}} goes to zero as well, by definition of partial derivative. A more intuitive representation of the value of approximation σ x

environment temperature T R {\displaystyle T_{R}} , which is the slope or partial derivative of the internal energy with respect to entropy in the environment

called a fundamental solution of the time-dependent problem if: the partial derivative δ U ( t , τ ) δ t {\displaystyle {\frac {\mathrm {\delta } U(t,\tau

Q_{\nu }(a,b)\sim 0.5} as a ↓ b . {\displaystyle a\downarrow b.} The partial derivative of Q ν ( a , b ) {\displaystyle Q_{\nu }(a,b)} with respect to a {\displaystyle

= −Vdp/dV. Here V is the volume, p is pressure, and dp/dV is the partial derivative of pressure with respect to the volume. The bulk modulus test uses

other hand, the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume:

I_{y}^{2}\rangle \end{bmatrix}}} In words, we find the covariance of the partial derivative of the image intensity I {\displaystyle I} with respect to the x {\displaystyle

at constant pressure, c p {\displaystyle c_{p}} is defined as the partial derivative c p = ∂ T h | p {\displaystyle c_{p}=\partial _{T}h|_{p}} . However

w)=\log \left({\frac {f(z)-f(w)}{z-w}}\right),} its second mixed partial derivative is given by ∂ 2 F ( z , w ) ∂ z ∂ w = f ′ ( z ) f ′ ( w ) ( f ( z

axis. The variable t, time, is assumed to be constant, so that the partial derivative of C1 with respect to y is equal to the partial of C1 with respect

{\alpha }}+{\widehat {\beta }}{\bar {x}}\end{aligned}}} Before taking partial derivative with respect to β ^ {\displaystyle {\widehat {\beta }}} , substitute

is a smooth function F : U → ℝ with: S ∩ U = {(x, y, z) ∈ U : F(x, y, z) = 0} at each point of S ∩ U, at least one partial derivative of F is nonzero.

f_{v}} also uniformly on compacta inside a domain D. Also, respective partial derivative of f v {\displaystyle f_{v}} also compactly converges on domain D

{x_{0}} } , as a function of the time t {\textstyle t} , and the partial derivative is taken for a given fluid parcel x 0 {\textstyle \mathbf {x_{0}}

{\displaystyle {\tfrac {\partial T}{\partial z}}.} the vertical partial derivative of the temperature. The vertical heat difference Q {\displaystyle

v\}_{\eta }:=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)} Hence using partial derivative relations and symplectic condition gives: { u , v } η = ( ∇ η u )

Christoffel symbols constructed from γαβ: where the comma denotes partial derivative by the respective coordinate. With the Christoffel symbols eq. 25

the phase spectrum can be evaluated at two nearby times, and the partial derivative with respect to time be approximated as the difference between the

_{k}+\lambda _{l}}},} where ∂ i {\displaystyle \partial _{i}} denotes partial derivative with respect to parameter θ i {\displaystyle \theta _{i}} . The formula

network. The activation function for each neuron is defined as a partial derivative of the Lagrangian with respect to that neuron's activity From the

= q − gy. The right hand side here is independent of x since its partial derivative with respect to x is 0. So h ( x , y ) = ∫ c y q ( a , s ) d s − g

\epsilon _{IJ}} is a function of space-time. This means that the partial derivative of a spinor is no longer a genuine tensor. As usual, one introduces

= q − gy. The right hand side here is independent of x since its partial derivative with respect to x is 0. So h ( x , y ) = ∫ c y q ( a , s ) d s − g

}=A_{\alpha ,\beta }-A_{\beta ,\alpha }} in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation