A hybrid approach of simulation and metaheuristic for the polyhedra packing problem

This document presents a simulation-based method for the polyhedra packing problem (PPP). This problem refers to packing a set of irregular polyhedra (convex and concave) into a cuboid with the objective of minimizing the cuboid’s volume, considering non-overlapping and containment constraints. The PPP has applications in additive manufacturing and packing situations where volume is at a premium. The proposed approach uses Unity® as the simulation environment and considers nine intensification and two diversification movements. The intensification movements induce the items within the cuboid to form packing patterns allowing the cuboid to decrease its size with the help of gravity-like accelerations. On the other hand, the diversification movements are classic transition operators such as removal and filling of pieces and enlargement of the container, which allow searching on different solution neighborhoods. All simulated movements were hybridized with a probabilistic tabu search. The proposed methodology (with and without the hybridization) was compared by benchmarking with all previous works solving the PPP with irregular items. Results show that satisfactory solutions were reached in a short time; even a few published results were improved.

Limit theorems for some skew products with mixing base maps

We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$ together with a $T$-invariant measure whose base map $\unicode[STIX]{x1D703}$ satisfies certain topological and mixing conditions and the maps $T_{\unicode[STIX]{x1D714}}$ on the fibers are certain non-singular distance-expanding maps. Our results hold true when $\unicode[STIX]{x1D703}$ is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by $T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields162 (2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of $T$ (with respect to the invariant measure) does not seem to have a spectral gap.

Cascade calculation with schematic interactions

In previous works we considered schematic Hamiltonians represented by simplified matrices. We defined two transition operators and calculated transition strengths from the ground state to all excited states. In many cases the strengths decreased nearly exponentially with the excitation energy. Now we do the reverse. We start with the highest energy state and calculate the cascade of transitions until the ground state is reached. On a log plot we show the average transition strength as a function of the number of energy intervals that were crossed. We give an analytic proof of exponential behavior for transition strength in the weak coupling limit for the [Formula: see text] transition operator.

Domain-Dependent and Domain-Independent Problem Solving Techniques

Heuristic search is a general problem-solving method. Some heuristic search algorithms, like the well-known A* algorithm, are domain-independent, in the sense that their knowledge of the problem at-hand is limited to the (1) initial state, (2) state transition operators and their costs, (3) goal-test function, and (4) black-box heuristic function that estimates the value of a state. Prominent examples are A* and Weighted A*. Other heuristic search algorithms are domain-dependent, that is, customized to solve problems from a specific domain. A well-known example is conflict-directed A*, which is specifically designed to solve model-based diagnosis problems. In this paper, we review our recent advancements in both domain-independent and domain-dependent heuristic search, and outline several challenging open questions.

Towards Non-Saturating Recurrent Units for Modelling Long-Term Dependencies

Modelling long-term dependencies is a challenge for recurrent neural networks. This is primarily due to the fact that gradients vanish during training, as the sequence length increases. Gradients can be attenuated by transition operators and are attenuated or dropped by activation functions. Canonical architectures like LSTM alleviate this issue by skipping information through a memory mechanism. We propose a new recurrent architecture (Non-saturating Recurrent Unit; NRU) that relies on a memory mechanism but forgoes both saturating activation functions and saturating gates, in order to further alleviate vanishing gradients. In a series of synthetic and real world tasks, we demonstrate that the proposed model is the only model that performs among the top 2 models across all tasks with and without long-term dependencies, when compared against a range of other architectures.

Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

Dynamical properties of random walks

In this paper, we study dynamical properties such as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreducible Markov chains. As particular cases, we consider simple random walks on [Formula: see text] and [Formula: see text].

Optimal observer-based feedback control for linear fractional-order systems with periodic coefficients

This paper proposes a new technique to design an optimal observer-based feedback control for linear fractional-order systems with constant or periodic coefficients. The proposed observer-based feedback control assures the fastest convergence of the closed-loop system’s states. For this purpose, a state-transition operator is defined in a Banach space and approximated using the fractional Chebyshev collocation method. It is shown that periodic gains of the controller and observer can be independently tuned by minimizing the spectral radius of their associated state-transition operators. The validity and efficiency of the proposed method are demonstrated through two illustrative examples.