On Symmetry of Complete Graphs over Quadratic and Cubic Residues

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.

Comparison Failure and Successful Methodologies for Diffusion Measurements Undertaken inside Two Different Testing Rooms

The scattering phenomenon is known to be of great importance for the acoustic quality of a performance arts space. The scattering of sound can be achieved in different ways: it can be obtained by the presence of architectural and/or decorating elements inside a room (e.g., columns, statues), by the geometry and roughness of a surface (e.g., Quadratic Residue Diffuser (QRD)) and by the diffraction effect occurring when a sound wave hits the edges of an obstacle. This article deals with the surface scattering effects and the diffusion phenomenon only related to MDF and plywood panels tested by disposing the wells both horizontally and vertically. The test results undertaken inside a semi-reverberant room and inside a large reverberant room have been compared to highlight the success and the failure of the measuring methodologies. In detail, according to the existing standards and regulations (i.e., ISO 17497—Part 2), diffusion measurements have been undertaken on a few selected types of panel: two QRD panels (made of Medium Density Fiberboard (MDF) and plywood) with and without a smooth painted solid wood placed behind the QRD. The panels have been tested inside two rooms of different characteristics: a semi-anechoic chamber (Room A) and a large reverberant room (Room B). The volume size influenced the results that have been analyzed for both chambers, showing an overlap of reflections on panels tested inside Room A and a clear diffusion response for the panels tested inside Room B. In terms of the diffusion coefficient in all the octave bands between 125 Hz and 8 kHz, results should not be considered valid for panels tested in Room A because they were negatively impacted by extraneous reflections, while they are reliable for panels tested in Room B.

Machine learning for decoding linear block codes: case of multi-class logistic regression model

<p>Facing the challenge of enormous data sets variety, several machine learning-based algorithms for prediction (e.g, Support vector machine, multi layer perceptron and logistic regression) have been highly proposed and used over the last years in many fields. Error correcting codes (ECCs) are extensively used in practice to protect data against damaged data storage systems and against random errors due to noise effects. In this paper, we will use machine learning methods, especially multi-class logistic regression combined with the famous syndrome decoding algorithm. The main idea behind our decoding method which we call logistic regression decoder (LRDec) is to use the efficient multi-class logistic regression models to find errors from syndromes in linear codes such as bose, ray-chaudhuri and hocquenghem (BCH), and the quadratic residue (QR). Obtained results of the proposed decoder have a significant benefit in terms of bit error rate (BER) for random binary codes. The comparison of our decoder with many competitors proves its power. The proposed decoder has reached a success percentage of 100% for correctable errors in the studied codes.</p>