Hypercube Research Articles

Lollipop and cubic weight functions for graph pebbling

Given a configuration of pebbles on the vertices of a graph G, a pebbling move removes two pebbles from a vertex and puts one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest number of pebbles required such that, given an arbitrary initial configuration of pebbles, one pebble can be moved to any vertex of G through some sequence of pebbling moves. Through constructing a non-tree weight function for Q4, we improve the weight function technique, introduced by Hurlbert and extended by Cranston et al., that gives an upper bound for the pebbling number of graphs. Then, we propose a conjecture on weight functions for the n-dimensional cube. We also construct a set of valid weight functions for variations of lollipop graphs, extending previously known constructions.

Tracial oscillation zero and [formula omitted]-stability

Let A be a (not necessarily unital) separable non-elementary simple amenable C⁎-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies a certain condition (C). We show that A is Z-stable if and only if A has strict comparison for positive elements. Extremal boundaries of simplexes which satisfy condition (C) may contain countable disjoint unions of n-dimensional cubes (n∈N) as a subset.

Data Centre Efficiency Enhancement by Metrics Oriented Approach to Revamp Green Cloud Computing Concept

Cloud computing inherits sharing of data from pool of resources existing in data centres when ever demanded. The imminent requirement for this purpose is proficiency of the data centre for fulfilment of this coveted objective. The pursuit of energy-efficient peak performance level is challenged by a simultaneous hike of energy consumption. The energy-efficient metrics contribute a major role for attainment of desired objective of safeguarding the environment. These metrics address the enhancement of the system’s proficiency. An increased energy-efficiency results into reduced consumption of energy resources since these energy resources are mostly non-renewable in nature and are the main source of carbon and heat emissions from operational data centres. As a matter of fact, any individual metric is not capable of achieving enhanced energy-efficient performance in a data centre. Therefore a collective utilization of selected metrics pertaining to power, performance and network traffic can improve the energy-efficient capability of data centre communication systems. The testing platform for such metrics is based on certain architectures which include D Cell, B Cube, Hyper Cube and Fat tree three-tier architectures.

Self-Adaptive Spherical Search With a Low-Precision Projection Matrix for Real-World Optimization.

Since the last three decades, numerous search strategies have been introduced within the framework of different evolutionary algorithms (EAs). Most of the popular search strategies operate on the hypercube (HC) search model, and search models based on other hypershapes, such as hyper-spherical (HS), are not investigated well yet. The recently developed spherical search (SS) algorithm utilizing the HS search model has been shown to perform very well for the bound-constrained and constrained optimization problems compared to several state-of-the-art algorithms. Nevertheless, the computational burdens for generating an HS locus are higher than that for an HC locus. We propose an efficient technique to construct an HS locus by approximating the orthogonal projection matrix to resolve this issue. As per our empirical experiments, this technique significantly improves the performance of the original SS with less computational effort. Moreover, to enhance SS's search capability, we put forth a self-adaptation technique for choosing the effective values of the control parameters dynamically during the optimization process. We validate the proposed algorithm's performance on a plethora of real-world and benchmark optimization problems with and without constraints. Experimental results suggest that the proposed algorithm remains better than or at least comparable to the best-known state-of-the-art algorithms on a wide spectrum of problems.

Application of Multiobjective Optimization to Provide Operational Guidance for Allocating Supply among Multiple Sources

This study is motivated by multipleobjective optimization in short-term water management for a regional water utility. Although an increasing application of multi-objective evolutionary algorithm (MOEA) has been reported in the literature, we are not aware of its use for short-term water management by water utilities with diverse supply sources. This study presents an innovative practice for determining monthly resource allocation from multiple water supply sources that consider multiple objectives, including deviation from budgeted production, under or overutilization of a given portfolio of resources, and total cost of water production. This method is comprised of a simulation model, namely a production allocation model (PAM) and a MOEA. The decision variables of the MOEA optimization problem are monthly groundwater production from two groundwater wellfields. TheMOEA is used to search for Pareto optimal solutions across different objectives and the PAM uses MOEA output and considers operational constraints to determine water production from the other four supply sources in the decision horizon. Stochastic demand and supply realizations were generated to capture a wide range of uncertainties which were then sampled by a Latin Hyper Cube to make the computation tractable. A parallel computing environment was used to implement this near real-time decision support tool, providing timely guidance for water resources managers. One major difference between this study and many reported in the literature is that the MOEA was used to find Pareto solutions for each demand-supply realization rather than the entire ensemble. This setup allows water resources managers to explicitly explore Pareto solutions based on different supply and demand outlooks. The application of the innovative practice is demonstrated for a regional wholesale water supply utility, Tampa Bay Water, on the west coast of Florida in the United States. One additional advantage of MOEA-assisted planning is that it allows water managers to combine expert judgments and institutional knowledge in identifying solutions. A comparison between MOEA-assisted monthly production planning and heuristic planning reveals that the potential impact of short-term operations, e.g., deviation from budgeted production, is fully considered in a systematic approach. The proposed method can be applied to other regions with similar challenges in water resources management.

Disjoint Placement Probability of Line Segments via Geometry

We have shown that when any finite number n, of line segments with total combined length less than one, have their centers placed randomly inside the unit interval , the probability of obtaining a mutually disjoint placement of the segments within , is given by the expression where , and denotes the length of the k-th segment, Lk . The result is established by a careful analysis of the geometry of the event, “all segments disjoint and contained within [0,1],” considered as a subset of the uniform probability space of n centers, each of which is in ; that is to say, the unit n-cube of . This event has an interesting geometric structure consisting of disjoint, congruent, (up to a mirror image) polytopes within the unit n-cube. It is shown these event polytopes fit together perfectly to form, except for a set of measure zero, a partition of an n-dimensional cube with common edge length , and hence an n-volume given by the formula. In the case of n = 3 segments, the polytopes form one of the known tetrahedral partitions of the cube as discussed, for example in [4]. In fact for all n > 0, the polytopes comprise a partition of the n-dimensional hypercube, and are therefore n-dimensional space filling.

Determination of metal(oids) in different traditional flat breads distributed in Isfahan city, Iran: Health risk assessment study by latin hypercube sampling.

This survey was conducted to assess the metal(oids) content in 93 samples of bread, including barbari, lavash, and tafton, using inductive couple plasma/optical emission spectroscopy (ICP-OES) method and atomic absorption spectroscopy (AAS). The amounts of measured element were compared with the permissible limit set for bread by FAO/WHO and Iranian National Standardization Organization (INSO). The limit of detection (LOD) was ranged from 6.6 × 10-5 to 2.1 × 10-2 mg l-1 with recoveries ranged from 92% to 102%. The average concentrations of aluminum (Al), arsenic (As), boron (B), cadmium (Cd), iron (Fe), mercury (Hg), magnesium (Mg), sodium (Na), lead (Pb), and zinc (Zn) in bread were 29.88 ± 8, 0.03 ± 0.004, 12.77 ± 3.70, 0.01 ± 0.006, 34.16 ± 8.95, 0.01 ± 0.008, 346.07 ± 36.08, 3314.81 ± 317.19, 0.24 ± 0.11, and 19.65 ± 4.66 mg Kg-1, respectively. Amounts of As, Cd, Hg, Mg, Pb, and Zn were lower, and those of Al, Fe, and Na were higher than the permissible limits defined by FAO/WHO. The Latin Hyper Cube (LHC) sampling results revealed that children were exposed to higher non-carcinogenic risk and adults were more threatened by carcinogenic risk. It is recommended to control the entrance of metals in bread in the farm-to-fork chain in order to prevent probable future health challenges.

Gaussian Process Regression Based Multi-Objective Bayesian Optimization for Power System Design

Within a disruptively changing environment, design of power systems becomes a complex task. Meeting multi-criteria requirements with increasing degrees of freedom in design and simultaneously decreasing technical expertise strengthens the need for multi-objective optimization (MOO) making use of algorithms and virtual prototyping. In this context, we present Gaussian Process Regression based Multi-Objective Bayesian Optimization (GPR-MOBO) with special emphasis on its profound theoretical background. A detailed mathematical framework is provided to derive a GPR-MOBO computer implementable algorithm. We quantify GPR-MOBO effectiveness and efficiency by hypervolume and the number of required computationally expensive simulations to identify Pareto-optimal design solutions, respectively. For validation purposes, we benchmark our GPR-MOBO implementation based on a mathematical test function with analytically known Pareto front and compare results to those of well-known algorithms NSGA-II and pure Latin Hyper Cube Sampling. To rule out effects of randomness, we include statistical evaluations. GPR-MOBO turnes out as an effective and efficient approach with superior character versus state-of-the art approaches and increasing value-add when simulations are computationally expensive and the number of design degrees of freedom is high. Finally, we provide an example of GPR-MOBO based power system design and optimization that demonstrates both the methodology itself and its performance benefits.

An Efficient Dissipation-Preserving Numerical Scheme to Solve a Caputo–Riesz Time-Space-Fractional Nonlinear Wave Equation

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

Robust optimization of the aerodynamic design of a helicopter rotor blade based on multi-fidelity meta-models

Robust optimization (RO) of designs becomes indispensable to deal with inherent manufacturing uncertainties. Yet, RO is still too expensive to solve complex problems such as the design of helicopter rotor blades. A promising way to reduce this cost is to use the multi-fidelity meta-model-based optimization (MFMBO), which combines low fidelity models (LFM), high fidelity models (HFM), meta-models, design of experiment (DoE), and optimization techniques. The field of MFMBO is not been completely explored despite the high number of proposed strategies. This paper contributes to the MFMBO cost-effectiveness improvement by proposing a new strategy. The main feature of the latter is the mutual adaptive refinement of the HFM and LFM DoE sets. A new idea is proposed to generate initial nested DoEs, which is implemented using an ordinary and an optimal Latin Hyper Cube method. These DoEs are refined adaptively and mutually using an improved MaxMin-distance criterion. The expensive HFM is used only to define a correction C of the LFM. C is used to adjust LFM evaluations during MBO. Meta-models are constructed using the Radial Basis Functions technique. The strategy is tested with two RO techniques, namely, the Non-Dominated Sorting Genetic Algorithm (NSGA-II) and the Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D). The implemented strategy is validated on three different mathematical benchmarks, compared with four different MFMBO strategies and applied on a helicopter rotor blade. This validation shows that the proposed MFMBO is competitive and can produce solutions as accurate as the HFM-based MBO while reducing significantly the overall computation time (up to 40%).

A Clustering Algorithm for Evolving Data Streams Using Temporal Spatial Hyper Cube

As applications generate massive amounts of data streams, the requirement for ways to analyze and cluster this data has become a critical field of research for knowledge discovery. Data stream clustering’s primary objective and goal are to acquire insights into incoming data. Recognizing all possible patterns in data streams that enter at variable rates and structures and evolve over time is critical for acquiring insights. Analyzing the data stream has been one of the vital research areas due to the inevitable evolving aspect of the data stream and its vast application domains. Existing algorithms for handling data stream clustering consider adding various data summarization structures starting from grid projection and ending with buffers of Core-Micro and Macro clusters. However, it is found that the static assumption of the data summarization impacts the quality of clustering. To fill this gap, an online clustering algorithm for handling evolving data streams using a tempo-spatial hyper cube called BOCEDS TSHC has been developed in this research. The role of the tempo-spatial hyper cube (TSHC) is to add more dimensions to the data summarization for more degree of freedom. TSHC when added to Buffer-based Online Clustering for Evolving Data Stream (BOCEDS) results in a superior evolving data stream clustering algorithm. Evaluation based on both the real world and synthetic datasets has proven the superiority of the developed BOCEDS TSHC clustering algorithm over the baseline algorithms with respect to most of the clustering metrics.

Target Function without Local Minimum for Systems of Logical Equations with a Unique Solution

Many of the applied algorithms that have been developed for solving a system of logical equations or the Boolean satisfiability problem have solved the problem in the Boolean domain. However, other approaches have recently been developed and improved. One of these developments is the transformation of a system of logical equations to a real continuous domain. The essence of this development is that a system of logical equations is transformed into a system in a real domain and the solution is sought in a real continuous domain. A real continuous domain is a richer domain, as it involves many well-developed algorithms. In this paper, we have constructively transformed the solution of any system of logical equations with a unique solution into an optimization problem for a polylinear target function in a unit n-dimensional cube Kn. The resulting polylinear target function in Kn does not have a local minimum. We proved that only once by calculating the gradient of the polylinear target function at any interior point of the Kn cube, we can determine the solution to the system of logical equations.

Optimal ATAPE task scheduling on reconfigurable and partitionable hierarchical hypercube networks

The hierarchical hypercube (HHC) network (N=2n, n=2m+m, m≥2) is one of scalable networks for the large-scale parallel-computing systems since its implementation cost is lower than the popular hypercube (HC) network. However, the complicated HHC-routing may conflict (due to the reduction-of-edges in the hierarchical construction) and cannot be reconfigured simply for k > 2m nodes-per-task. Thus, a crucial research to fulfill the HHC-network is the conflict-free routing since the conflict is not handled easily during runtime. Recently, the HHC-conflict was solved by the shortest-path routing and the special HHC-partitioning, called the grouping-of-cross dual-cube (GCD) partitioning, for k=2m+1 nodes-per-task. In this study, we propose the reconfigurable-and-partitionable HHC (RP-HHC) network for 2m+2 ≤ k ≤ N/2M-1 nodes-per-task (M=2m-1) by the grouping-of-cross sub-system (GCS) combining (the optimal reconfiguring). Moreover, we present the efficient task-scheduling and processor-allocation, incorporated with the reconfigured binary-tree. Since the first-fit policy is fast but returns low-performance while the best-fit policy yields high-performance but is time-consuming, we propose the efficient best-fit driven-policy to work as fast as the first-fit policy while maintaining the high-performance. The correctness of new reconfiguring and time-complexity of task-allocation were analyzed. In experiments, all-to-all personalized exchange (ATAPE) communication was assessed to confirm no-conflict on the RP-HHC systems.

Polylinear Transformation Method for Solving Systems of Logical Equations

In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit n-dimensional cube Kn into a system of polylinear–polynomial equations in a mathematically constructive way. Secondly, we proved that if we slightly modify the system of logical equations, namely, add no more than one special equation to the system, then the resulting system of logical equations and the corresponding system of polylinear–polynomial equations in Kn+1 is equivalent. The paper proposes an algorithm and proves its correctness. Based on these results, further research plans are developed to adapt the proposed method.

Momentary finite element for elasticity 3D problems

A description of a new 8-node finite element in the form of a hexahedron is given for solving elasticity 3D problems. This finite element has the following features. This is a linear approximation of functions in the element, one point of integration and taking into account the moments of forces in the element. The finite element is based on “rare mesh” FEM schemes—finite element schemes in the form of n-dimensional cubes (square, cube, etc.) with templates in the form of inscribed simplexes (triangle, tetrahedron, etc.). Among the rare mesh schemes, schemes in 3-dimensional and 7-dimensional spaces are successful, in which the simplex can be arranged symmetrically with respect to the center of the n-dimensional cube. The rare mesh FEM schemes have not the hourglass instability due to the fact that the template of the finite element operator has the form of a simplex. Compared to traditional linear finite elements in the form of a simplex, rare mesh schemes are more economical and converge better, since they do not have the effect of overestimated shear stiffness. Moment FEM schemes are constructed by rare mesh schemes higher dimensional projection, respectively, on a two-dimensional or three-dimensional finite element mesh. The resulting finite elements are close to the known polylinear elements and surpass them in efficiency. The schemes contain parameters that allow you to control the convergence of numerical solutions. The possibility of applying this approach to the construction of numerical schemes for solving other problems of mathematical physics is discussed.

Structural analysis and weight optimization of automotive chassis by Latin hypercube sampling using metal matrix composites

Automotive chassis is used for supporting the functioning structure and balancing on the road. Being an initial requirement chassis vehicles like trucks, trailers, and semi-trailers have stepping skeleton (ladder chassis). Automotive chassis comprise of two bars organized corresponding to the longitudinal hub of the casing and a transverse beam arrangement laterally. Such chassis, the side elements of the lead frames generally consist of open channels or other sections. The work involves a ladder type frame of a heavy-duty truck considering the MMCs Al GA 7-230 and Al6092/SiC/17.5p MMC materials respectively using in comparison with ST52E steel. Latin hyper cube scheme of response surface method is used for the optimization purpose on ANSYS 18.1. Results shows that, the mass of chassis is 62.564Kg and 71.502 Kg using Graphite Al GA 7-230 and Al6092/SiC/17.5p MMC respectively whereas the actual mass is 214 Kg in case of conventionally used Structural steel alloy. Therefore, a significant amount of mass is reduced considering a number of other performance factors.

Transformation Method for Solving System of Boolean Algebraic Equations

In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

Applying Infinite Petri Nets to the Cybersecurity of Intelligent Networks, Grids and Clouds

Correctness of networking protocols represents the principal requirement of cybersecurity. Correctness of protocols is established via the procedures of their verification. A classical communication system includes a pair of interacting systems. Recent developments of computing and communication grids for radio broadcasting, cellular networks, communication subsystems of supercomputers, specialized grids for numerical methods and networks on chips require verification of protocols for any number of devices. For analysis of computing and communication grid structures, a new class of infinite Petri nets has been introduced and studied for more than 10 years. Infinite Petri nets were also applied for simulating cellular automata. Rectangular, triangular and hexagonal grids on plane, hyper cube and hyper torus in multidimensional space have been considered. Composing and solving in parametric form infinite Diophantine systems of linear equations allowed us to prove the protocol properties for any grid size and any number of dimensions. Software generators of infinite Petri net models have been developed. Special classes of graphs, such as a graph of packet transmission directions and a graph of blockings, have been introduced and studied. Complex deadlocks have been revealed and classified. In the present paper, infinite Petri nets are divided into two following kinds: a single infinite construct and an infinite set of constructs of specified size (and number of dimensions). Finally, the paper discusses possible future work directions.

Solving software project scheduling problem using grey wolf optimization

In this paper, we will explore the application of grey wolf optimization (GWO) methodology in order to solve the software project scheduling problem (SPSP) to seek an optimum solution via applying different instances from two datasets. We will focus on the effects of the quantity of employees as well as the number of tasks which will be accomplished. We concluded that increasing employee number will decrease the project’s duration, but we could not find any explanation for the cost values for all instances that studied. Also, we concluded that, when increasing the number of the tasks, both the cost and duration will be increased. The results will compare with a max-min ant system hyper cube framework (MMAS-HC), intelligent water drops algorithm (IWD), firefly algorithm (FA), ant colony optimization (ACO), intelligent water drop algorithm standard version (IWDSTD), and intelligent water drop autonomous search (IWDAS). According to these study and comparisons, we would like to say that GWO algorithm is a better optimizing tool for all instances, except one instance that FA is outperform the GWO.

On the average-case complexity of Boolean functions under binomial distribution on their domains

Abstract Given a binomial probability distribution on the n-dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n-place Boolean functions is established.