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Inhomogeneous Systems of Equations
1983Let aij, 1 ≤ i ≤ k, 1 ≤ j ≤ n be a set of constants, and fix k constants u1,u2,...,ukThe system $$\left\{ \begin{gathered} {a_{11}}{x_1} + {a_{12}}{x_2} + \cdots + {a_{1n}}{x_n} = {u_1}, \hfill \\ {a_{21}}{x_1} + {a_{22}}{x_2} + \cdots + {a_{2n}}{x_n} = {u_2}, \hfill \\ \vdots \hfill \\ {a_{k1}}{x_1} + {a_{k2}}{x_2} + \cdots + {a_{kn}}{x_n} = {u_k},
Thomas Banchoff, John Wermer
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Special solution of the inhomogeneous Bloch equation
Journal of Mathematical Physics, 2003A solution of the inhomogeneous Bloch equation is given for a class of three-dimensional time-varying magnetic fields by finding the fundamental system in terms of a set of the three independent solutions of the homogeneous Bloch equation. This class is distinguished by requiring a nonlinear relation between one of the magnetic field components and the
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Toward solving the inhomogeneous Bloch equation
Journal of Mathematical Physics, 2003The homogeneous Bloch equation reduces to the Riccati equation. By linearizing the Riccati equation, a set of three solutions of the homogeneous Bloch equation is found. The fundamental matrix becomes singular. We clarify the utility and limitation of our approach to solve the homogeneous Bloch equation.
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2019
Let α be an algebraic integer of degree n ≥ 3, \(K={\mathbb Q}(\alpha )\), and let \(0\neq m\in {\mathbb Z}\). In some applications for index form equations in sextic and octic fields (cf. Sects. 11.2.1, 11.2.2, and 14.2.3) we shall need to solve equations of type $$\displaystyle N_{K/{\mathbb Q}}(x+\alpha y +\lambda )=m \;\;\; \mathrm {in ...
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Let α be an algebraic integer of degree n ≥ 3, \(K={\mathbb Q}(\alpha )\), and let \(0\neq m\in {\mathbb Z}\). In some applications for index form equations in sextic and octic fields (cf. Sects. 11.2.1, 11.2.2, and 14.2.3) we shall need to solve equations of type $$\displaystyle N_{K/{\mathbb Q}}(x+\alpha y +\lambda )=m \;\;\; \mathrm {in ...
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The Boltzmann Equation for Inhomogeneous Electric Fields
Proceedings of the Physical Society, 1963A rigorous derivation is given of the conventional Boltzmann equation for a metal in an inhomogeneous electric field E(x, t). It is assumed that the electrons are scattered elastically by static impurities. The equation is shown to be valid provided the following conditions hold: (i) h?
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Maximum principles for inhomogeneous equation
2011 International Conference on Multimedia Technology, 2011A class of inhomogeneous semilinear elliptic equations is considered. The Hopf maximum principles are used to deduce that certain functions defined for solutions of the equation attain a maximum on the boundary of the demain or at a critical point of the solution. Dirichlet problem and Neumann problem are considered also.
null Yafei He, Hailiang Zhang
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On the determination of an inhomogeneity in an elliptic equation
Applicable Analysis, 2006We consider the determination of the subdomain D, , in the boundary value problem (ED ): −Δu + χ Du = 0 in Ω and u = f on ∂Ω. To each admissible vector field V we associate the boundary measurements corresponding to the solution of , where DV = (I + V)D.
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Systems of Inhomogeneous Linear Equations
2010Many problems in physics and especially computational physics involve systems of linear equations, which arise e.g. from linearization of a general nonlinear problem or from discretization of differential equations. If the system is nonsingular and has full rank, a formal solution is given by matrix inversion. If the matrix is singular or the number of
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