Hypercube Research Articles

A new look at random projections of the cube and general product measures

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube [−1/n,+1/n]n onto Rd converges almost surely to a centered d-dimensional Euclidean ball of radius 2/π, as n→∞. We start by providing an alternative proof of this strong law via the Artstein–Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius 2/π, we determine the asymptotic number of vertices and the volume of the part of the cube projected ‘close’ to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let ν⊗n be the n-fold product measure of a Borel probability measure ν on R, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in Rn. It is shown that the sequence of random measures ν⊗n∘(n−1/2I∗)−1, n∈N, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [−1,1]n and the discrete cube {−1,1}n as special cases.

About the construction of probability hypercubes

About the construction of probability hypercubes

Peiffer pairings in multisimplicial groups and crossed n-cubes and applications for bisimplicial groups

Peiffer pairings in multisimplicial groups and crossed n-cubes and applications for bisimplicial groups

Chaotic darwinian particle swarm optimization for real-time hierarchical congestion management of power system integrated with renewable energy sources

The increasing penetration of renewable energy sources that have intermittent outputs challenge the Independent System Operator while managing transmission congestion. A new Real-Time Hierarchical Congestion Management (RHCM) technique is proposed that reschedules generators in two stages based on Available Congestion Clearing Time (ACCT) of the transmission lines in presence of renewable energy sources. Chaotic Darwinian Particle Swarm Optimization (CDPSO) is used for determining the optimal schedules of demand response loads and reschedules of conventional generators to mitigate congestion. The solar and wind energy sources are modelled using Rayleigh and Beta probability density functions; Latin Hyper cube sampling is adopted for sampling. IEEE 39 bus system is simulated for cases of overloads and contingencies and RHCM using CDPSO is tested to ensure the security of the system. The benefit of RHCM incorporating demand response is presented in terms of reduced congestion relieving costs and decreased power loss.

Shortest-path routing for optimal all-to-all personalized-exchange embedding on hierarchical hypercube networks

Hypercube (HC) networks (N=2n) provide efficient communication for parallel-and-distributed computing (PDC) but the HC-based multi-processor (MP) system is costly and not scalable, while hierarchical hypercube (HHC) networks (N=2n, n=2m+m) are less expensive and more scalable. However, the traditional HHC-routing easily conflicts, especially when executing multiple tasks (k > 2m nodes-per-task). In the past, the node-disjoint-path routing could be used to avoid the conflict but that reliable (S, D) routing limited S(source)=0. Later, the parallel N2N (node-to-node) disjoint-path was proposed for reliable routing but limited (m+1)/2 pairs-of-nodes. Therefore, this study proposes the generalized N2N shortest-path (SP) routing (with arbitrary S) and the parallel N2N SP-routing, based on our hypothesis “the shortest-path routing and the proper HHC-partitioning can avoid the HHC-conflict directly”. Next, our innovation and contribution are 1. the GCD (grouping of cross dual-cube) partitioning to solve the HHC-conflict for k≤2m+1 nodes-per-task, and 2. the SP-ATAPE (all-to-all personalized exchange) embedding on the HHC-MPs. The correctness of the SP-routing was proven and the ATAPE communication was experimented to validate our conflict solution. The ATAPE results confirmed that in any group the GCD mapping could make all tasks (k= 2m+1 nodes-per-task), synchronized with the same control, working without the conflict.

On the Volume of Sections of the Cube

Abstract We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1] n . We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝ n onto a k-dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube [−1, 1] n , n ≥ 2.

Topology of random -dimensional cubical complexes

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

Chaos on the hypercube

We analyze the spectral properties of a d-dimensional HyperCubic (HC) lattice model originally introduced by Parisi. The U(1) gauge links of this model give rise to a magnetic flux of constant magnitude ϕ but random orientation through the faces of the hypercube. The HC model, which also can be written as a model of 2d interacting Majorana fermions, has a spectral flow that is reminiscent of Maldacena-Qi (MQ) model, and its spectrum at ϕ = 0, actually coincides with the coupling term of the MQ model. As was already shown by Parisi, at leading order in 1/d, the spectral density of this model is given by the density function of the Q-Hermite polynomials, which is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model. Parisi demonstrated this by mapping the moments of the HC model to Q-weighted sums on chord diagrams. We point out that the subleading moments of the HC model can also be mapped to weighted sums on chord diagrams, in a manner that descends from the leading moments. The HC model has a magnetic inversion symmetry that depends on both the magnitude and the orientation of the magnetic flux through the faces of the hypercube. The spectrum for fixed quantum number of this symmetry exhibits a transition from regular spectra at ϕ = 0 to chaotic spectra with spectral statistics given by the Gaussian Unitary Ensembles (GUE) for larger values of ϕ. For small magnetic flux, the ground state is gapped and is close to a Thermofield Double (TFD) state.

Application of OLAP-technology in decision support system for enterprise management

The work is devoted to the study of methods for increasing the profitability of an enterprise based on management decisions that are close to optimal. It is shown that the enterprise, as an abstract control object, integrates at least four components: buildings, facilities and utilities, production facilities, financial and material stocks and personnel. Moreover, each of these components of the production system has its own management system, and enterprise management, therefore, is carried out at the second level of the hierarchy. Since the levers of direct control of the production system are managed by the first-level control systems, parametric, structural and organizational changes become the control actions at the second level and the main tasks of management are the competent redistribution of managerial resources between these influences. Relying on the presentation of information in the form of a hyper cube and taking advantage of the fact that all solutions lie on the plane of equal contributions, and then, taking it as the coordinate plane for counting the utility function, one can reasonably find points on this plane with a resource distribution close to optimal by maximizing this feature.

On Dimensionally Orthogonal Diagonal Hypercubes

On Dimensionally Orthogonal Diagonal Hypercubes

An Efficient Dynamic Parallel and Distributed Network with Hybrid Hyper Cube

An Efficient Dynamic Parallel and Distributed Network with Hybrid Hyper Cube

Inferring structure and parameters of dynamic system models simultaneously using swarm intelligence approaches

Inferring dynamic system models from observed time course data is very challenging compared to static system identification tasks. Dynamic system models are complicated to infer due to the underlying large search space and high computational cost for simulation and verification. In this research we aim to infer both the structure and parameters of a dynamic system simultaneously by particle swarm optimization (PSO) improved by efficient stratified sampling approaches. More specifically, we enhance PSO with two modern stratified sampling techniques, i.e., Latin hyper cube sampling (LHS) and Latin hyper cube multi dimensional uniformity (LHSMDU). We propose and evaluate two novel swarm-inspired algorithms, LHS-PSO and LHSMDU-PSO, which can be used particularly to learn the model structure and parameters of complex dynamic systems efficiently. The performance of LHS-PSO and LHSMDU-PSO is further compared with the original PSO and genetic algorithm (GA). We chose real-world cancer biological model called Kinetochores to asses the learning performance of LHSMDU-PSO and LHS-PSO in comparison with GA and the original PSO. The experimental results show that LHSMDU-PSO can find promising models with reasonable parameters and structure, and it outperforms LHS-PSO, PSO, and GA in our experiments.

Improved Upper Bounds for Identifying Codes in n-dimensional q-ary Cubes

Improved Upper Bounds for Identifying Codes in n-dimensional q-ary Cubes

Effect of biaxial stress state on seismic fragility of concrete gravity dams

Dams are important structures for management of water supply for irrigation or drinking, flood control, and electricity generation. In seismic regions, the structural safety of concrete gravity dams is important due to the high potential of life and economic loss if they fail. Therefore, the seismic analysis of existing dams in seismically active regions is crucial for predicting responses of dams to ground motions. In this paper, earthquake response of concrete gravity dams is investigated using the finite element (FE) method. The FE model accounts for dam-water-foundation rock interaction by considering compressible water, flexible foundation effects, and absorptive reservoir bottom materials. Several uncertainties regarding structural attributes of the dam and external actions are considered to obtain the fragility curves of the dam-water-foundation rock system. The structural uncertainties are sampled using the Latin Hypercube Sampling method. The Pine Flat Dam in the Central Valley of Fresno County, California, is selected to demonstrate the methodology for several limit states. The fragility curves for base sliding, and excessive deformation limit states are obtained by performing non-linear time history analyses. Tensile cracking including the complex state of stress that occurs in dams was also considered. Normal, Log-Normal and Weibull distribution types are considered as possible fits for fragility curves. It was found that the effect of the minimum principal stress on tensile strength is insignificant. It is also found that the probability of failure of tensile cracking is higher than that for base sliding of the dam. Furthermore, the loss of reservoir control is unlikely for a moderate earthquake.

Multiple failure criteria-based fragility curves for structures equipped with SATMDs

In this paper, a procedure to develop fragility curves of structures equipped with semi-active tuned mass dampers (SATMDs) considering multiple failure criteria has been presented while accounting for the uncertainties of the input excitation, structure and control device parameters. In this procedure, Latin hypercube sampling (LHS) method has been employed to generate 30 random SATMD-structure systems and nonlinear incremental dynamic analysis (IDA) has been conducted under 20 earthquakes to determine the structural responses, where failure probabilities in each intensity level have been evaluated using Monte Carlo simulation (MCS) method. For numerical analysis, an eight-story nonlinear shear building frame with bilinear hysteresis material behavior has been used. Fragility curves for the structure equipped with optimal SATMDs have been developed considering single and multiple failure criteria for different performance levels and compared with that of uncontrolled structure as well as structure controlled using passive tuned mass damper (TMD). Numerical analysis has shown the capability of SATMDs in significant enhancement of the seismic fragility of the nonlinear structure. Also, considering multiple failure criteria has led to increasing the fragility of the structure. Moreover, it is observed that the influence of the uncertainty of input excitation with respect to the other uncertainties is considerable.

Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture

In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the n-dimensional cube graph has maximum degree at least $\sqrt{n}$. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

An extension of golden section algorithm for n-variable functions with MATLAB code

Golden section search method is one of the fastest direct search algorithms to solve single variable optimization problems, in which the search space is reduced from [a, b] to [0,1]. This paper describes an extended golden section search method in order to find the minimum of an n-variable function by transforming its n-dimensional cubic search space to the zero-one n-dimensional cube. The paper also provides a MATLAB code for two-dimensional and three-dimensional golden section search algorithms for a zero-one n-dimensional cube. Numerical results for some benchmark functions up to five dimensions and a comparison of the proposed algorithm with the Neldor Mead Simplex Algorithm is also provided.

An Efficient Exchanged Hyper Cube for Parallel and Distributed Network

Enormously parallel distribution memory designs are accepting and expanding regard to satisfy the expanding need on processing power. Numerous topologies have been projected for interconnecting the processors of distributed computing systems. The hypercube topology has attracted significant consideration because of a significant number of attractive properties. The engaging properties of the hypercube topology, for example, vertex and edge balance, recursive structure, logarithmic diameter, maximally fault-tolerance, simple routing and broadcasting, and the capacity to recreate other interconnection systems with least overhead have made it a brilliant possibility for some parallel processing applications. Numerous varieties of the hypercube topology have been accounted for the literature, mostly to add the computational power of the hypercube. One of the gorgeous versions of the hypercube was introduced for the improvement of the presented Exchanged hypercube. An Exchanged hypercube has the equivalent structural complexities of the hypercube. It protects the gorgeous properties of the hypercube and diameter the communication time by dropping the diameter by a factor of two. This paper presents the fundamental communication and some of the essential operations normally required in parallel computing on the Exchanged hypercube interconnection networks.

On the Perfectness of Minimal Regular Partitions of the Edge Set of the n-Dimensional Cube

On the Perfectness of Minimal Regular Partitions of the Edge Set of the n-Dimensional Cube

Continuous Flattening of the 2-Dimensional Skeleton of the Square Faces in a Hypercube

The surface of a 3-dimensional cube can be continuously flattened onto any of its faces, by moving creases to change the shapes of some faces successively, following Sabitov’s volume preserving theorem. Let \(C_n\) be an n-dimensional cube with \(n \ge 4\), and S be the set of its 2-dimensional faces, i.e., the 2-dimensional skeleton of the square faces in \(C_n\). We show that S can be continuously flattened onto any face F of S, such that the faces of S that are parallel to F, do not have any crease, that is, they are rigid during the motion.