scholarly journals CARTESIAN PRODUCT PARTITIONING OF MULTI-DIMENSIONAL REACHABLE STATE SPACES

2016 ◽  
Vol 30 (3) ◽  
pp. 413-430 ◽  
Author(s):  
Tuǧrul Dayar ◽  
M. Can Orhan
Keyword(s):  
State Space ◽  
Cartesian Product ◽  
Model Systems ◽  
Dimensional System ◽  
Reachable State ◽  
State Spaces ◽  
Cartesian Products ◽  
Higher Dimensional ◽  
The Matrix ◽  
Minimum Number

Markov chains (MCs) are widely used to model systems which evolve by visiting the states in their state spaces following the available transitions. When such systems are composed of interacting subsystems, they can be mapped to a multi-dimensional MC in which each subsystem normally corresponds to a different dimension. Usually the reachable state space of the multi-dimensional MC is a proper subset of its product state space, that is, Cartesian product of its subsystem state spaces. Compact storage of the matrix underlying such a MC and efficient implementation of analysis methods using Kronecker operations require the set of reachable states to be represented as a union of Cartesian products of subsets of subsystem state spaces. The problem of partitioning the reachable state space of a three or higher dimensional system with a minimum number of partitions into Cartesian products of subsets of subsystem state spaces is shown to be NP-complete. Two algorithms, one merge based the other refinement based, that yield possibly non-optimal partitionings are presented. Results of experiments on a set of problems from the literature and those that are randomly generated indicate that, although it may be more time and memory consuming, the refinement based algorithm almost always computes partitionings with a smaller number of partitions than the merge-based algorithm. The refinement based algorithm is insensitive to the order in which the states in the reachable state space are processed, and in many cases it computes partitionings that are optimal.

scholarly journals More Results on Italian Domination in Cn□Cm

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 465 ◽  
Author(s):  
Hong Gao ◽  
Penghui Wang ◽  
Enmao Liu ◽  
Yuansheng Yang
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Rainbow Domination ◽  
Two Cities ◽  
Minimum Number ◽  
Graph Class ◽  
Domination In Graphs ◽  
Appl Math

Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In such an assignment, the minimum number of troops is the Italian domination number of the empire/graph is denoted as γ I . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained γ I ( C n □ C 3 ) and γ I ( C n □ C 4 ) (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75–85). Stȩpień et al. obtained γ I ( C n □ C 5 ) = 2 n (2-Rainbow domination number of C n □ C 5 , Discret. Appl. Math. 170 (2014): 113–116). In this paper, we study the Italian domination number of the Cartesian products of cycles C n □ C m for m ≥ 6 . For n ≡ 0 ( mod 3 ) , m ≡ 0 ( mod 3 ) , we obtain γ I ( C n □ C m ) = m n 3 . For other C n □ C m , we present a bound of γ I ( C n □ C m ) . Since for n = 6 k , m = 3 l or n = 3 k , m = 6 l ( k , l ≥ 1 ) , γ r 2 ( C n □ C m ) = m n 3 , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668–674), it follows in this case that C n □ C m is an example of a graph class for which γ I = γ r 2 , which can partially answer the question presented by Brešar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394–2400.

scholarly journals Martin boundaries of cartesian products of markov chains

1992 ◽  
Vol 128 ◽  
pp. 153-169 ◽  
Author(s):  
Massimo A. Picardello ◽  
Wolfgang Woess
Keyword(s):  
Markov Chains ◽  
Harmonic Functions ◽  
Cartesian Product ◽  
Martin Boundary ◽  
State Spaces ◽  
Cartesian Products ◽  
Discrete State ◽  
Stochastic Transition ◽  
Transition Operators ◽  
Martin Boundaries

Let P and Q be the stochastic transition operators of two time-homogeneous, irreducible Markov chains with countable, discrete state spaces X and Y, respectively. On the Cartesian product Z = X x Y, define a transition operator of the form Ra = a·P + (1 — a) · Q, 0 < a < 1, where P is considered to act on the first variable and Q on the second. The principal purpose of this paper is to describe the minimal Martin boundary of Ra (consisting of the minimal positive eigenfunctions of Ra with respect to some eigenvalue t, also called t-harmonic functions) in terms of the minimal Martin boundaries of P and Q.

Impact of control representations on efficiency of local nonholonomic motion planning

2011 ◽  
Vol 59 (2) ◽  
pp. 213-218 ◽  
Author(s):  
I. Duleba
Keyword(s):  
Motion Planning ◽  
State Space ◽  
Three Dimensional ◽  
Nonholonomic Systems ◽  
State Spaces ◽  
Controlled Systems ◽  
Higher Dimensional ◽  
Dimensional State Space ◽  
Optimal Motion Planning ◽  
Nonholonomic Motion Planning

Impact of control representations on efficiency of local nonholonomic motion planning In this paper various control representations selected from a family of harmonic controls were examined for the task of locally optimal motion planning of nonholonomic systems. To avoid dependence of results either on a particular system or a current point in a state space, considerations were carried out in a sub-space of a formal Lie algebra associated with a family of controlled systems. Analytical and simulation results are presented for two inputs and three dimensional state space and some hints for higher dimensional state spaces were given. Results of the paper are important for designers of motion planning algorithms not only to preserve controllability of the systems but also to optimize their motion.

scholarly journals Unitarity in KK-graviton production, a case study in warped extra-dimensions

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
A. de Giorgi ◽  
S. Vogl
Keyword(s):  
Extra Dimensions ◽  
Sum Rules ◽  
Effective Theory ◽  
Higher Dimensional ◽  
Massive Spin ◽  
The Matrix ◽  
Dimensional Gravity ◽  
Warped Extra Dimensions ◽  
Kaluza Klein

Abstract The Kaluza-Klein (KK) decomposition of higher-dimensional gravity gives rise to a tower of KK-gravitons in the effective four-dimensional (4D) theory. Such massive spin-2 fields are known to be connected with unitarity issues and easily lead to a breakdown of the effective theory well below the naive scale of the interaction. However, the breakdown of the effective 4D theory is expected to be controlled by the parameters of the 5D theory. Working in a simplified Randall-Sundrum model we study the matrix elements for matter annihilations into massive gravitons. We find that truncating the KK-tower leads to an early breakdown of perturbative unitarity. However, by considering the full tower we obtain a set of sum rules for the couplings between the different KK-fields that restore unitarity up to the scale of the 5D theory. We prove analytically that these are fulfilled in the model under consideration and present numerical tests of their convergence. This work complements earlier studies that focused on graviton self-interactions and yields additional sum rules that are required if matter fields are incorporated into warped extra-dimensions.

scholarly journals Connectivity matrix model of quantum circuits and its application to distributed quantum circuit optimization

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Ismail Ghodsollahee ◽  
Zohreh Davarzani ◽  
Mariam Zomorodi ◽  
Paweł Pławiak ◽  
Monireh Houshmand ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Connectivity Matrix ◽  
Circuit Optimization ◽  
Novel Approach ◽  
The Matrix ◽  
Minimum Number ◽  
Two Phases

AbstractAs quantum computation grows, the number of qubits involved in a given quantum computer increases. But due to the physical limitations in the number of qubits of a single quantum device, the computation should be performed in a distributed system. In this paper, a new model of quantum computation based on the matrix representation of quantum circuits is proposed. Then, using this model, we propose a novel approach for reducing the number of teleportations in a distributed quantum circuit. The proposed method consists of two phases: the pre-processing phase and the optimization phase. In the pre-processing phase, it considers the bi-partitioning of quantum circuits by Non-Dominated Sorting Genetic Algorithm (NSGA-III) to minimize the number of global gates and to distribute the quantum circuit into two balanced parts with equal number of qubits and minimum number of global gates. In the optimization phase, two heuristics named Heuristic I and Heuristic II are proposed to optimize the number of teleportations according to the partitioning obtained from the pre-processing phase. Finally, the proposed approach is evaluated on many benchmark quantum circuits. The results of these evaluations show an average of 22.16% improvement in the teleportation cost of the proposed approach compared to the existing works in the literature.

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

scholarly journals Inverse-square law between time and amplitude for crossing tipping thresholds

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180504 ◽  
Author(s):  
Paul Ritchie ◽  
Özkan Karabacak ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Dimensional System ◽  
Stochastic Forcing ◽  
Inverse Square Law ◽  
System A ◽  
Asymptotic Expressions ◽  
Higher Dimensional ◽  
Random Disturbances

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

scholarly journals Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
D. Ya. Khusainov ◽  
J. Diblík ◽  
Z. Svoboda ◽  
Z. Šmarda
Keyword(s):  
Trivial Solution ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Dimensional System ◽  
Global Instability ◽  
Degree Polynomial ◽  
Differential Systems ◽  
Right Hand ◽  
The Matrix ◽  
Full Derivative

The present investigation deals with global instability of a generaln-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

scholarly journals Strong chromatic index of products of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Olivier Togni
Keyword(s):  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Induced Matching ◽  
Colour Class ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

scholarly journals Revealing topology with transformation optics

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Lizhen Lu ◽  
Kun Ding ◽  
Emanuele Galiffi ◽  
Xikui Ma ◽  
Tianyu Dong ◽  
...  
Keyword(s):  
Physical Space ◽  
Real Space ◽  
Virtual Space ◽  
Transformation Optics ◽  
Coordinate Space ◽  
Dimensional System ◽  
Edge States ◽  
And Topology ◽  
Higher Dimensional ◽  
Insight Into

AbstractSymmetry deepens our insight into a physical system and its interplay with topology enables the discovery of topological phases. Symmetry analysis is conventionally performed either in the physical space of interest, or in the corresponding reciprocal space. Here we borrow the concept of virtual space from transformation optics to demonstrate how a certain class of symmetries can be visualised in a transformed, spectrally related coordinate space, illuminating the underlying topological transitions. By projecting a plasmonic system in a higher-dimensional virtual space onto a lower-dimensional system in real space, we show how transformation optics allows us to construct a topologically non-trivial system by inspecting its modes in the virtual space. Interestingly, we find that the topological invariant can be controlled via the singularities in the conformal mapping, enabling the intuitive engineering of edge states. The confluence of transformation optics and topology here can be generalized to other wave realms beyond photonics.

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