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Elements of model theory
Published in Uri Abraham, Models for Concurrency, 2020
y is a free variable, x has three occurrences: it is bound in the second, and free in the third (the one with A(x)). The first occurrence is a formal modifier of the universal quantifier. A formula says something about its free variables, but nothing at all about its bound variables. For example ∃y(2y = x) says that x is an even number, but doesn’t say anything about y. (A formal definition of free and bound occurrences can be found in any logic textbook.) If ϕ is a formula with free variables x1, y3 and z for example, then we shall express this by writing ϕ(x1, y3, z). A sentence is a formula with no free variables.
Introduction to Expert Systems
Published in Chris Nikolopoulos, Expert Systems, 1997
Given a quantified formula ∀X(p), all occurrences of the variable X in p are said to be in the scope of the universal quantifier. A variable not in the scope of any quantifier is a free variable. A quantified variable is bound to its closest quantifier. For example, in the formula ∀X(p(X,Y)⇒∃Zq(Z))
Tuples and Vectors
Published in Jeff Suzuki, Linear Algebra, 2021
Note that in the vector equation of the line and the plane, the parameters can take on any value. We say they are “free variables,” in the sense that there are no constraints on them. On the other hand, the coordinate values x,y,z,… are constrained: we compute their values from the free variables.
Differential flatness based design of robust controllers using polynomial chaos for linear systems
Published in International Journal of Control, 2023
Oladapo Ogunbodede, Tarunraj Singh
To examine the effect of the variation in the number of terms in the parameterisation of the flat output on the solutions for the chance constraint formulation with regards to the number of free variable () in the parameterisation, different solutions from 1 to 5 free variables for a 3rd order polynomial chaos expansion were compared. Figure 5 shows a plot of the position and velocity at the final time for all the realizations over the uncertainty domain using both the deterministic control design and the robust control design. The bounding box shows a visual representation of the bounds of all the realisation. As shown in Figure 5, as the free variable () is increased, improvements in the robustness to parameter variations is observed. This means that there is a strong correlation between the performance of the controller and the number of free variable in the parameterisation of the flat output of the surrogate model. Table 2 shows the values of the bounding box area with increasing number of free variables, which illustrates the decrease in the area of the bounding box which is a proxy for robustness.
Dwell time-based stabilisation of switched delay systems using free-weighting matrices
Published in International Journal of Control, 2018
Ahmet Taha Koru, Akın Delibaşı, Hitay Özbay
We define a new free variable t to bound the cost function: The parameters μi and λi are related with the eigenvalues of Pi, Qi, Zi and Wi as in Equations (10) and (14). We define pi, qi and zi to define maximum eigenvalues of Pi, Qi and Zi, respectively. So, the inequality (18) can be re-written as .
Real-time simulations of human tongue movements with a reduced order model of a non-linear dynamic biomechanical model
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2020
M. Calka, P. Perrier, J. Ohayon, C. Grivot Boichon, M. Rochette, Y. Payan
Figure 1 superimposes the trajectories generated by the ROM and by the full model. To better analyse where most of the errors of the ROM are located, Figure 2 plots the 3 D error map all around the tongue surface. The highest errors are located around the tip and dorsum. This is not surprising since these regions are the most affected ones by SG and GGP activations. This is consistent with the difference in complexity of the ROM when one or two muscles are activated. For single activation, no free variable is needed whereas one free variable is added for double activation. As could be expected on a theoretical level, more non-linearity induces more complexity in the ROM.