An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect...

in neural networks. The adjoint state space is chosen to simplify the physical interpretation of equation constraints. Adjoint state techniques allow the...

costate equation is related to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It...

differential equation (1) is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear homogenous ordinary differential equations can be...

^{\mu }\gamma ^{0},} one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by γ 0...

matrix, related to its inverse Adjoint equation The upper and lower adjoints of a Galois connection in order theory The adjoint of a differential operator...

this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics...

The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system.: 1–2  Its...

{\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle A^{*}} on that space according to the...

specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier...

can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, fluid...

t'}(x\mid x')\,dx.} From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of L {\displaystyle {\mathcal {L}}} , L...

In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved...

In mathematics, a self-adjoint operator on a complex vector space V with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is a linear...

of the adjoint: ( col ⁡ M ) ⊥ = ker ⁡ M ∗ . {\displaystyle (\operatorname {col} M)^{\bot }=\ker M^{*}.} Fredholm's theorem for integral equations is expressed...

The Kolmogorov backward equation (KBE) and its adjoint, the Kolmogorov forward equation, are partial differential equations (PDE) that arise in the theory...

operator, see below. A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system. This definition...

classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though...

classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid...

are scientific computing algorithms used in solving time dependent adjoint equations, as well as reverse mode automatic differentiation. D. K. Pradhan...

Complex Numbers". Applied Linear Algebra and Differential Equations. LibreTexts. "Adjoint matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]...

when working with Fourier transforms. Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality...

This defines another bounded linear operator A* : H2 → H1, the adjoint of A. The adjoint satisfies A** = A. When the Riesz representation theorem is used...

differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal...

the bar above ψ μ {\displaystyle \psi _{\mu }} denotes the Dirac adjoint. This equation controls the propagation of the wave function of composite objects...

evolution equations such as the Kolmogorov backward equation, which describes the evolution of statistics of the process; its L2 Hermitian adjoint is used...

self-adjoint operator is an operator equal to its own (formal) adjoint. If Ω is a domain in Rn, and P a differential operator on Ω, then the adjoint of...

given by a section of a vector bundle (specifically, the adjoint bundle). These equations are D A ∗ F A + ∗ [ Φ , D A Φ ] = 0 , D A ∗ D A Φ = 0 {\displaystyle...

differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. A differential...

{\displaystyle S_{+{\frac {1}{2}}}} satisfies the adjoint equation, that can be obtained from the above equation by replacing θ {\displaystyle \theta } with...