Geometric Arithmetic Research Articles

Atom-Bond Connectivity and Geometric Arithmetic Indices of POPAM and Siloxane Dendrimers

Atom-Bond Connectivity and Geometric Arithmetic Indices of POPAM and Siloxane Dendrimers

On Connectivity Indices of an Infinite Family of the Linear Parallelogram of Benzenoid Graph

A topological index of a graph G is a numeric quantity related to G which is describe molecular graph G. In this paper the Atom Bond Connectivity (ABC) and Geometric-Arithmetic (GA) indices of an infinite class of the linear parallelogram of benzenoid graph.

A finite mixture model of geometric distributions for lossless image compression

In this paper, we proposed a new geometric finite mixture model-based adaptive arithmetic coding (AAC) for lossless image compression. Applying AAC for image compression, large compression gains can be achieved only through the use of sophisticated models that provide more accurate probabilistic descriptions of the image. In this work, we proposed to divide the residual image into non-overlapping blocks, and then we model the statistics of each block by a mixture of geometric distributions of parameters estimated through the maximum likelihood estimation using the expectation–maximization algorithm. Moreover, a histogram tail truncation method within each predicted error block is used in order to reduce the number of symbols in the arithmetic coding and therefore to reduce the effect of the zero-occurrence symbols. Experimentally, we showed that using convenient block size and number of mixture components in conjunction with the prediction technique median edge detector, the proposed method outperforms the well known lossless image compressors.

On topological indices of poly oxide, poly silicate, DOX, and DSL networks

Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such as Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study different interconnection networks and derive analytical closed results of the general Randić index (Rα(G)) for α = 1, [Formula: see text], –1, [Formula: see text] only, for dominating oxide network (DOX), dominating silicate network (DSL), and regular triangulene oxide network (RTOX). All of the studied interconnection networks in this paper are motivated by the molecular structure of a chemical compound, SiO4. We also compute the general first Zagreb, ABC, GA, ABC4, and GA5 indices and give closed formulae of these indices for these interconnection networks.

On Connectivity Indices of an Infinite Family of the Linear Parallelogram of Benzenoid Graph

A topological index of a graph G is a numeric quantity related to G which is describe molecular graph G. In this paper the Atom Bond Connectivity (ABC) and Geometric-Arithmetic (GA) indices of an infinite class of the linear parallelogram of benzenoid graph.

On topological indices of fullerenes

Two degree based topological indices, the atom-bond connectivity (ABC) and the geometric–arithmetic (GA) indices of fullerene networks and carbon nanotube networks are studied. Expressions for ABC and GA indices for these important classes of networks are obtained.

On topological indices of certain interconnection networks

In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numeric number. These topological indices correlate certain physico-chemical properties like boiling point, stability, strain energy etc of chemical compounds. Graph theory has found a considerable use in this area of research.The topological properties of certain networks are studied recently in [13] by Hayat and Imran (2014). In this paper, we extend this study to interconnection networks and derive analytical closed results of general Randić index Rα(G) for different values of “α” for butterfly and Benes networks. We also compute first Zagreb, ABC, and GA indices for these important classes of networks. Moreover, we construct two new classes of mesh derived networks by using some basic operations of graphs on m×n mesh networks, and then study certain topological indices for these classes of networks.

Computation of topological indices of certain networks

There are certain types of topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. Among degree based topological indices, the so-called atom-bond connectivity (ABC), geometric–arithmetic (GA) are of vital importance. These topological indices correlate certain physico-chemical properties such as boiling point, stability and strain energy etc. of chemical compounds.In this paper, we compute ABC4 and GA5 indices for certain networks like silicate, chain silicate, hexagonal, oxide and honeycomb networks. The atom-bond connectivity index (ABC index) and geometric–arithmetic index (GA index) for oxide and chain silicate networks are also computed in this paper.

On geometric-arithmetic index

The geometric-arithmetic (GA) index is a newly proposed graph invariant in mathematical chemistry. We give the lower and upper bounds for GA index of molecular graphs using the numbers of vertices and edges. We also determine the n-vertex molecular trees with the minimum, the second and the third minimum, as well as the second and the third maximum GA indices.

Playing with Pythagoras and geometric arithmetic

I just couldn't see it; that was my difficulty with many mathematical abstractions: if I couldn't visually process a maths concept I would resort to rule, hoping that some day it would make sense.Take the Pythagorean Theorem. It has been around for two millenniums plus, readily understood by all academically exposed to the liberal arts by the time they reach puberty; but me, I knew it, a2 + b2 = c2 and the picture of three little squares huddled around the perimeter of a right triangle, that was no help. Why not three equilateral triangles around the perimeter Of course a2 √3/4 + b2 √3/4 = c2 √3/4 is a bit ungainly, but it would work.

Performance evaluation in image processing with GAPP array processor

The application of parallelism considering the field of image processing is an alternative to implement real time image processing. The data parallelism found in array processors simplify the mapping of this kind of problem as each processing element works on part of the image. The GAPP board (Geometric Arithmetic Parallel Processor) is a near-neighbor mesh architecture with 144 processors interconnected as a 12 × 12 bidimensional array. This work analyzes the performance of the GAPP board regarding image processing. The implementation of two image convolution algorithms and the analyses of the obtained results, as well as the utilization of the GAPP board in this kind of application and the performance achieved are discussed. The results found in this work are also compared to the results presented in [11]. Some ways to achieve a better performance of the GAPP array in this kind of application are also presented.

A fast algorithm for gaussian elimination over GF(2) and its implementation on the GAPP

A fast algorithm for Gaussian elimination over GF(2) is proposed. The proposed algorithm employs binary search technique to locate is along the columns of the large binary matrix being triangularized. The algorithm requires 2 m 2 + m log 2 n - 2 m bit operations to triangularize an n × m matrix on a bit-array with n processors and m + 2 + [log 2 n] bits of vertical memory per processor. Details of an implementation on the Geometric Arithmetic Parallel Processor are also presented.

A preliminary evaluation of a massively parallel processor: GAPP

The Geometric Arithmetic Parallel Processor (GAPP) is a systolic planar array processor with 72 (6 × 12) processing elements arranged in a mesh network. It runs at a clock speed of 10 MHz. The greatest advantage of such a design is that as many processor chips as are needed can be implemented without the loss of processing speed as interprocessor communication has been designed into the hardware architecture. It is therefore particularly suitable for processing of large image files with SIMD type of program execution. In this paper we report our study of the GAPP's performance in comparison with a number of other array processor systems, including: CLIP4, ICL DAP and Goodyear MPP. We also compare GAPP with some DSPs such as TMS32020, INMOS Transputer. We have found that GAPP's performance is in the same order of magnitude as these processors.

A Neural Net Associative Memory for Real-Time Applications

A parallel hardware implementation of the associative memory neural network introduced by Hopfield is described. The design utilizes the Geometric Arithmetic Parallel Processor (GAPP), a commercially available single-chip VLSI general-purpose array processor consisting of 72 processing elements. The ability to cascade these chips allows large arrays of processors to be easily constructed and used to implement the Hopfield network. The memory requirements and processing times of such arrays are analyzed based on the number of nodes in the network and the number of exemplar patterns. Compared with other digital implementations, this design yields significant improvements in runtime performance and offers the capability of using large neural network associative memories in real-time applications.

SPL - A different high level language for systolic processors

The Systolic Processors Language, SPL, is a C-based high-level language for systolic processors. This language and its development environment, are designed especially, but not solely, for both the European Wafer Scale Integration Systolic Array Processor (ASAP: ESPRIT-824B), and the NCR Geometric Arithmetic Parallel Processor (GAPP). SPL enhances the software development capability for programmable systolic processors, and aims at simplifying program writing without decreasing the run-time performance. The contribution of SPL, presented in this paper, is the novel memory allocation for each Processing Element (PE), along with a balanced support of data—types, instruction—set, and a rich library of subroutines embedded in it. Altogether, these features provide an efficient back-end code.

Hough transform algorithms for mesh-connected SIMD parallel processors

Hough transform techniques for straight line detection play a key role in the road following algorithms developed by the University of Maryland for the DARPA Autonomous Land Vehicle Project. This report compares several methods of Hough transform computation suitable for implementation on a mesh-connected SIMD parallel processor, such as Goddard Space Flight Center's Massively Parallel Processor (MPP) or Martin Marietta Corp.'s Geometric Arithmetic Parallel Processor (GAPP).