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Structural Control of Large-Scale Flexibly Automated Manufacturing Systems
Published in Cornelius Leondes, The Design of Manufacturing Systems, 2019
Spyros A. Reveliotis, Mark A. Lawley, Placid M. Ferreira
A formal proof for this theorem is provided in Reference [56] and it essentially combines similar results initially developed in References [30] and [54]. The significance of this result is demonstrated by means of the flexible manufacturing cell depicted in Figure 4.9. This cell presents the typical layout of many contemporary automated manufacturing cells, consisting of a number of workstations, served by a central robotic manipulator. Each workstation possesses an input and an output buffer, while the manipulator can carry only one part at a time. For such a system, it is clear that deadlock-free operation can be established by controlling the allocation of the buffering capacity of the system workstations and treating the manipulator as the enabler of the authorized job transfers. Furthermore, the SU-(sub-)RAS defined by the system workstations meets the specifications 1 and/or 2 prescribed in the conditions of Theorem 4.1 and, therefore, in such a system, the optimal deadlock avoidance can be attained through one-step lookahead on the allocation of the buffering capacity of the system workstations.
Density Functional Theory (DFT): Periodic Advancement and New Challenges
Published in Tanmoy Chakraborty, Lalita Ledwani, Research Methodology in Chemical Sciences, 2017
which offers a formal proof for the second Hohenberg–Kohn theorem. Assuming differentiability of F[ρ]+∫v(r)ρ(r)dr, this theorem requires that the ground-state density satisfies the Euler–Lagrange equations () 0=δδρ(r){F[ρ]+∫ρ(r)v(r)dr−μ(∫ρ(r)dr−N)},
Green’s Function Method
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
If we take R as a large integer, we obtain the second 6-sequence given in (8.210). This 6-sequence actually gives the formal proof of the Fourier transform. In particular, if we multiply (8.213) by J(ξ) and integrate with respect to ξ from minus infinity to plus infinity, we have f(x)=∫-∞∞δ(x-ξ)f(ξ)dξ=12π∫-∞∞∫-∞∞eik(x-ξ)f(ξ)dkdξ $$ f(x) = \mathop \smallint \limits_{{ - \infty }}^{\infty } \delta (x - \xi )f(\xi )d\xi = \frac{1}{{2\pi }}\mathop \smallint \limits_{{ - \infty }}^{\infty } \mathop \smallint \limits_{{ - \infty }}^{\infty } e^{{ik(x - \xi )}} f(\xi )dkd\xi $$
Investigating a Computational Explanation of the Black Hole Illusion
Published in The International Journal of Aerospace Psychology, 2022
Victoria Jakicic, Logan Boyer, Gregory Francis
A related prediction of the algorithm involves the width of the runway in a BHI situation. Perrone (1983) discussed that the dimensions of the runway (i.e. the runway width) would affect a pilot’s perceived angle , and below we prove how different runway widths either increase or decrease the severity with which misestimates . Consider three different widths where each width is greater than the one previous, with the restriction that on the retinal plane. Perrone’s algorithm predicts that the largest of the three widths will produce the best estimate of the approach angle, which should correspond to the best landing performance. A formal proof is given as Theorem A.3 in the Appendix.
Fragments of quasi-Nelson: residuation
Published in Journal of Applied Non-Classical Logics, 2023
It is convenient to leave for later (Proposition 4.19) a formal proof that quasi-Nelson pocrims are actually pocrims in the sense of Blok and Raftery (1997). By definition, quasi-Nelson pocrims form a variety, henceforth denoted by . Using the twist representation, one can easily check that the -reduct of every (quasi-)Nelson algebra is indeed a member of . As in the previous cases, other prominent examples can be found among the intuitionistic algebras. The proof of the following proposition, as well as any subsequent proof which has been omitted in this section, can be found in the Appendix.
Consensus and coordination on groups SO(3) and S 3 over constant and state-dependent communication graphs
Published in Automatika, 2021
Aladin Crnkić, Milojica Jaćimović, Vladimir Jaćimović, Nevena Mijajlović
Notice that on special orthogonal (or unitary) groups the maximum of the trace is achieved for the identity matrix. Hence, the minimum of is achieved when the matrix , where denotes identity matrix. It follows then that consensus configurations are exactly global minima of the function . Formal proof of this statement is given in the next theorem, which is a corollary of Proposition 7 from [2].