6,594 research outputs found
Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey
In this survey paper, we focus on the necessary and sufficient conditions for
the unique solvability and unsolvability of the absolute value equations (AVEs)
during the last twenty years (2004 to 2023). We discussed unique solvability
conditions for various types of AVEs like standard absolute value equation
(AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE)
and a class of NGAVE based on interval matrix, P-matrix, singular value
conditions, spectral radius and -property. Based on the unique
solution of AVEs, we also discussed unique solvability conditions for linear
complementarity problems (LCP) and horizontal linear complementarity problems
(HLCP)
Generalized Perron Roots and Solvability of the Absolute Value Equation
19 pages, 2 figuresLet be a real matrix. The piecewise linear equation system is called an absolute value equation (AVE). It is well-known to be equivalent to the linear complementarity problem. Unique solvability of the AVE is known to be characterized in terms of a generalized Perron root called the sign-real spectral radius of . For mere, possibly non-unique, solvability no such characterization exists. We close this gap in the theory. That is, we define the concept of the aligned spectrum of and prove, under some mild genericity assumptions on , that the mapping degree of the piecewise linear function is congruent to , where is the number of aligned values of which are larger than . We also derive an exact -- but more technical -- formula for the degree of in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP
Solitons and kinks in a general car-following model
We study a car-following model of traffic flow which assumes only that a
car's acceleration depends on its own speed, the headway ahead of it, and the
rate of change of headway, with only minimal assumptions about the functional
form of that dependence. The velocity of uniform steady flow is found
implicitly from the acceleration function, and its linear stability criterion
can be expressed simply in terms of it. Crucially, unlike in previously
analyzed car-following models, the threshold of absolute stability does not
generally coincide with an inflection point in the steady velocity function.
The Burgers and KdV equations can be derived under the usual assumptions, but
the mKdV equation arises only when absolute stability does coincide with an
inflection point. Otherwise, the KdV equation applies near absolute stability,
while near the inflection point one obtains the mKdV equation plus an extra,
quadratic term. Corrections to the KdV equation "select" a single member of the
one-parameter set of soliton solutions. In previous models this has always
marked the threshold of a finite- amplitude instability of steady flow, but
here it can alternatively be a stable, small-amplitude jam. That is, there can
be a forward bifurcation from steady flow. The new, augmented mKdV equation
which holds near an inflection point admits a continuous family of kink
solutions, like the mKdV equation, and we derive the selection criterion
arising from the corrections to this equation.Comment: 25 page
On Computing the Translations Norm in the Epipolar Graph
This paper deals with the problem of recovering the unknown norm of relative
translations between cameras based on the knowledge of relative rotations and
translation directions. We provide theoretical conditions for the solvability
of such a problem, and we propose a two-stage method to solve it. First, a
cycle basis for the epipolar graph is computed, then all the scaling factors
are recovered simultaneously by solving a homogeneous linear system. We
demonstrate the accuracy of our solution by means of synthetic and real
experiments.Comment: Accepted at 3DV 201
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