TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Routing Problem ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

PERMUTATION ROUTING AND SORTING ON THE RECONFIGURABLE MESH

1998 ◽  
Vol 09 (02) ◽  
pp. 199-211
Author(s):  
SANGUTHEVAR RAJASEKARAN ◽  
THEODORE MCKENDALL
Keyword(s):  
Lower Bound ◽  
Randomized Algorithms ◽  
Linear Array ◽  
Order Term ◽  
Routing Algorithm ◽  
Sorting Algorithm ◽  
Routing Problem ◽  
Reconfigurable Mesh ◽  
Permutation Routing ◽  
Time Bounds

In this paper we demonstrate the power of reconfiguration by presenting efficient randomized algorithms for both packet routing and sorting on a reconfigurable mesh connected computer. The run times of these algorithms are better than the best achievable time bounds on a conventional mesh. Many variations of the reconfigurable mesh can be found in the literature. We define yet another variation which we call as Mr. We also make use of the standard PARBUS model. We show that permutation routing problem can be solved on a linear array Mr of size n in [Formula: see text] steps, whereas n-1 is the best possible run time without reconfiguration. A trivial lower bound for routing on Mr will be [Formula: see text]. On the PARBUS linear array, n is a lower bound and hence any standard n-step routing algorithm will be optimal. We also show that permutation routing on an n×n reconfigurable mesh Mr can be done in time n+o(n) using a randomized algorithm or in time 1.25n+o(n) deterministically. In contrast, 2n-2 is the diameter of a conventional mesh and hence routing and sorting will need at least 2n-2 steps on a conventional mesh. A lower bound of [Formula: see text] is in effect for routing on the 2D mesh Mr as well. On the other hand, n is a lower bound for routing on the PARBUS and our algorithms have the same time bounds on the PARBUS as well. Thus our randomized routing algorithm is optimal upto a lower order term. In addition we show that the problem of sorting can be solved in randomized time n+o(n) on Mr as well as on PARBUS. Clearly, this sorting algorithm will be optimal on the PARBUS model. The time bounds of our randomized algorithms hold with high probability.

The structure of rank-one convex quadratic forms on linear elastic strains

2003 ◽  
Vol 133 (1) ◽  
pp. 213-224 ◽  
Author(s):  
Kewei Zhang
Keyword(s):  
Quadratic Forms ◽  
Morse Index ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Linear Elastic ◽  
Direct Sum ◽  
Higher Dimensional ◽  
Rank One ◽  
Finite Set ◽  
Elastic Strains

We classify the Morse indices for rank-convex quadratic forms defined on the space of linear elastic strains in two- and three-dimensional linear elasticity. For the higher-dimensional case n > 3, we give a universal lower bound of the largest possible Morse index and various upper bound of this index. We show in the three-dimensional case that the Morse index is at most 1, and in this case the nullity cannot exceed 2. Examples are given that show that the estimates can be reached. We apply the results to study the critical points for smooth rank-one convex functions defined on the space of linear strains. We also examine an example and construct a quasiconvex function that vanishes in a finite set in the direct sum of the null subspace and the negative subspace of the rank-one quadratic form.

scholarly journals Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1880
Author(s):  
Artur Kobus ◽  
Jan L. Cieśliński
Keyword(s):  
Building Block ◽  
Differential Calculus ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Main Building ◽  
Higher Dimensional ◽  
Algebraic Operations ◽  
Low Dimensional

The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

The Structure of the Polytope of Hereditary Information

2019 ◽  
Vol 8 (2) ◽  
pp. 7-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Nucleic Acid ◽  
Hydrogen Bonds ◽  
Phosphoric Acid ◽  
High Intensity ◽  
Information Exchange ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Nitrogen Bases ◽  
Higher Dimensional ◽  
Low Dimensional

The representations of the sugar molecule and the residue of phosphoric acid in the form of polytopes of higher dimension are used. Based on these ideas and their simplified three-dimensional images, a three-dimensional image of nucleic acids is constructed. The geometry of the neighborhood of the compound of two nucleic acid helices with nitrogen bases has been investigated in detail. It is proved that this neighborhood is a cross-polytope of dimension 13 (polytope of hereditary information), in the coordinate planes of which there are complementary hydrogen bonds of nitrogenous bases. The structure of this polytope is defined, and its image is given. The total incident flows from the low-dimensional elements to the higher-dimensional elements and vice versa of the hereditary information polytope are calculated equal to each other. High values of these flows indicate a high intensity of information exchange in the polytope of hereditary information that ensures the transfer of this information.

Low-dimensional relativistic degeneracy in quantum plasmas

2013 ◽  
Vol 79 (6) ◽  
pp. 1081-1087 ◽  
Author(s):  
M. AKBARI-MOGHANJOUGHI ◽  
A. ESFANDYARI-KALEJAHI
Keyword(s):  
Three Dimensional ◽  
Thomson Scattering ◽  
Dimensional Case ◽  
Polytropic Index ◽  
Diffraction Effect ◽  
Two Dimensional ◽  
Degenerate Electron ◽  
Relativistic Degeneracy ◽  
Quantum Plasmas ◽  
Low Dimensional

AbstractIn this work we investigate the effect of relativistic degeneracy on different properties of low-dimensional quantum plasmas. Using the dielectric response from the conventional quantum hydrodynamic model, including the quantum diffraction effect (Bohm potential) on free electrons, we explore the existence of the Shukla–Eliasson attractive screening and possibility of the ion structure formation in low-dimensional, completely degenerate electron–ion plasmas. A generalized degeneracy pressure expression for arbitrary relativity parameter in two-dimensional case is derived, indicating that change in the polytropic index (change in the equation of state) for the two-dimensional quantum fluid takes place at the electron number-density of n0 ≃ 1.1 × 1020cm−2 whereas this is known to occur for the three-dimensional case in the electron density of n0 ≃ 5.9 × 1029cm−3. Also, a generalized dielectric function valid for all dimensionalities and densities of a degenerate electron gas is calculated, and distinct properties of electron–ion plasmas, such as static screening, structure factor and Thomson scattering, are investigated in terms of plasma dimensionality.

CORRELATION AND SPECTRAL PROPERTIES OF MULTIDIMENSIONAL THUE–MORSE SEQUENCES

2007 ◽  
Vol 17 (04) ◽  
pp. 1265-1303 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Spectral Properties ◽  
Three Dimensional ◽  
Number System ◽  
Dimensional Case ◽  
Binary Number ◽  
Singular Continuous Spectrum ◽  
One Dimensional ◽  
Number Systems ◽  
Higher Dimensional ◽  
The One

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in ℤm instead of in ℕ. It is shown that the corresponding ±1-valued Thue–Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.

Lower Bounds to the Nth Eigenvalue of the Helmholz Equation Over Three-Dimensional Regions of Arbitrary Shape

1974 ◽  
Vol 41 (3) ◽  
pp. 819-820
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Helmholtz Equation ◽  
Lower Bounds ◽  
Arbitrary Shape ◽  
Three Dimensional ◽  
Helmholz Equation

A method is presented to compute a lower bound to the nth eigenvalue of the Helmholtz equation over three-dimensional regions. The shape of the regions is arbitrary and only their volume need be known.

Computation of Lower Bounds for a Multiple Depot, Multiple Vehicle Routing Problem With Motion Constraints

2015 ◽  
Vol 137 (9) ◽  
Author(s):  
Satyanarayana G. Manyam ◽  
Sivakumar Rathinam ◽  
Swaroop Darbha
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Vehicle Routing Problem ◽  
Optimal Solution ◽  
Stopping Criterion ◽  
Routing Problem ◽  
Motion Constraints ◽  
Multiple Depot ◽  
Simulation Results ◽  
Path Lengths

This paper considers the problem of planning paths for a collection of identical vehicles visiting a given set of targets, such that the total lengths of their paths are minimum. Each vehicle starts at a specified location (called a depot) and it is required that each target to be on the path of at least one vehicle. The path of every vehicle must satisfy the motion constraints of every vehicle. In this paper, we develop a method to compute lower bound to the minimum total path lengths by relaxing some of the constraints and posing it as a standard multiple traveling salesmen problem (MTSP). A lower bound is often important to ascertain suboptimality bounds for heuristics and for developing stopping criterion for algorithms computing an optimal solution. Simulation results are presented to show that the proposed method can be used to improve the lower bounds of instances with four vehicles and 40 targets by approximately 39%.

scholarly journals Lower Bounds for Partial Matchings in Regular Bipartite Graphs and Applications to the Monomer–Dimer Entropy

2008 ◽  
Vol 17 (3) ◽  
pp. 347-361 ◽  
Author(s):  
SHMUEL FRIEDLAND ◽  
LEONID GURVITS
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Bipartite Graphs ◽  
Hyperbolic Polynomials

We derive here the Friedland–Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in r-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three-dimensional monomer–dimer entropy.

On the ergodicity of hyperbolic Sinaĭ–Ruelle–Bowen measures II: the low-dimensional case

2017 ◽  
Vol 38 (8) ◽  
pp. 3042-3061
Author(s):  
MICHIHIRO HIRAYAMA ◽  
NAOYA SUMI
Keyword(s):  
Probability Measure ◽  
Closed Manifold ◽  
Dimensional Case ◽  
Anosov Diffeomorphism ◽  
Stable And Unstable Manifolds ◽  
Higher Dimensional ◽  
Unstable Manifolds ◽  
Low Dimensional

In this paper, we consider diffeomorphisms on a closed manifold $M$ preserving a hyperbolic Sinaĭ–Ruelle–Bowen probability measure $\unicode[STIX]{x1D707}$ having intersections for almost every pair of stable and unstable manifolds. In this context, we show the ergodicity of $\unicode[STIX]{x1D707}$ when the dimension of $M$ is at most three. If $\unicode[STIX]{x1D707}$ is smooth, then it is ergodic when the dimension of $M$ is at most four. As a byproduct of our arguments, we obtain sufficient (topological) conditions which guarantee that there exists at most one hyperbolic ergodic Sinaĭ–Ruelle–Bowen probability measure. Even in higher dimensional cases, we show that every transitive topological Anosov diffeomorphism admits at most one hyperbolic Sinaĭ–Ruelle–Bowen probability measure.

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