Systems of Equations over Finite Semigroups and the #CSP Dichotomy Conjecture

AbstractPrevious studies showed that the dynamic equations for a porous fluid-saturated solid may lose hyperbolicity and thus render the boundary-value problem ill-posed while the equations for the same but dry solid remain hyperbolic. This paper presents sufficient conditions for hyperbolicity in both dry and saturated states. Fluid-saturated solids are described by two different systems of equations depending on whether the permeability is zero or nonzero (locally undrained and drained conditions, respectively). The paper also introduces a notion of wave speed consistency between the two systems as a necessary condition which must be satisfied in order for the solution in the locally drained case to tend to the undrained solution as the permeability tends to zero. It is shown that the symmetry and positive definiteness of the acoustic tensor of the skeleton guarantee both hyperbolicity and the wave speed consistency of the equations.

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Equations in finite semigroups: explicit enumeration and asymptotics of solution numbers

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2003.12.001 ◽

2004 ◽

Vol 105 (2) ◽

pp. 291-334

Author(s):

C. Krattenthaler ◽

T.W. Müller

Keyword(s):

Finite Semigroups ◽

Asymptotics Of Solution ◽

Explicit Enumeration

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Families of quasigroup operations satisfying the generalized distributive law

Discrete Mathematics and Applications ◽

10.1515/dma-2020-0018 ◽

2020 ◽

Vol 30 (3) ◽

pp. 187-202

Author(s):

Sergey V. Polin

Keyword(s):

System Of Equations ◽

Systems Of Equations ◽

The Family ◽

Distributive Law ◽

Elimination Of Variables

AbstractThe previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations.

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Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

Numerical Algorithms ◽

10.1007/s11075-020-01060-y ◽

2021 ◽

Author(s):

Florian Mannel

Keyword(s):

Linear Independence ◽

Affine Subspace ◽

Systems Of Equations ◽

Nonlinear Systems Of Equations ◽

Convergence Properties ◽

Initial Matrix ◽

First Time ◽

Special Case ◽

Dimensional Mapping ◽

Finite Number Of Iterations

AbstractWe consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

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