A Game of Primes

The basis for most of the ideas mentioned in this paper is the theory of cellular automata. A cellular automata contains a regular grid of cells, with each cell having a pre-defined set of finite states. The initial state is determined at time/state zero. At this point all the cells are assigned their respective starting states. The automata is defined by a set of simple rules that decide the subsequent states of the cells. We aim to create a cellular automata of prime numbers and come up with some axioms, theorems and conjectures for the same

Sequences, Series and Uniform distribution of SP Numbers

We defined numbers of the form $p\cdot a^2$ as SP numbers (Square-Prime numbers) ($a\neq1$, $p$ prime) in 'Distribution of Square-Prime Numbers' (arXiv:2109.10238). These numbers are listed in the OEIS as A228056. Some examples of SP numbers: $75=3 \cdot 25; 108=3 \cdot 36; 45= 5 \cdot 9$. This paper explores sequences of these numbers, sequences between 0 and 1 related to these numbers and analyzes the distribution of some of these sequences.Comment: Re-uploaded after corrections. Paper is currently undergoing peer-revie

An Abelian Loop for Non-Composites

We define an abelian loop on a set $S$ consisting of 1 and all odd prime numbers with an operation $\bullet$, where for $a,b$ $\in$ $S$, $a$ $\bullet$ $b$ is the smallest element of $S$ strictly larger than $|a-b|$. We use theorems and conjectures from number theory to prove properties of the loop and state analogous conjectures about the loop.Comment: This paper has been accepted for publication in the Missouri Journal of Mathematical Sciences and is set to appear in 202

Red Arecanut Seed Extract as a Sustainable Corrosion Inhibitor for Aluminum Submerged in Acidic Corrodent: An Experimental Approach Towards Zero Environmental Impact

The effect of the red arecanut seed (RAS) extract on the corrosion of aluminum in 0.5 M hydrochloric acid environment is reported by weight loss, electrochemical (Tafel plot and impedance spectroscopy), scanning electron microscopy and atomic force microscopy studies. The weight loss study indicated that, protection efficiency of the red arecanut seed extract is directly proportional to its concentration and inversely proportional to solution temperature and aluminum contact time in the test solution. Langmuir adsorption isotherm is best fitted model explaining the adsorption of red arecanut seed extract constituents on aluminum surface in 0.5 M HCl system. The results obtained from Tafel curves indicated the mixed inhibition role of red arecanut seed extract. The impedance spectroscopy technique indicated that, red arecanut seed extract reduces the speed of aluminum corrosion by charge transfer process. The scanning electron microscopy and atomic force microscopy images of aluminum specimens clearly give clues about the adsorption of plant constituents on the surface of the aluminum metal

Filtered rays over iterated absolute differences on layers of integers

The dynamical system generated by the iterated calculation of the high order gaps between neighboring terms of a sequence of natural numbers is remarkable and only incidentally characterized at the boundary by the notable Proth-Glibreath Conjecture for prime numbers. We introduce a natural extension of the original triangular arrangement, obtaining a growing hexagonal covering of the plane. This is just the base level of what further becomes an endless discrete helicoidal surface. % Although the repeated calculation of higher-order gaps causes the numbers that generate the helicoidal surface to decrease, there is no guarantee, and most often it does not even happen, that the levels of the helicoid have any regularity, at least at the bottom levels. However, we prove that there exists a large and nontrivial class of sequences with the property that their helicoids have all levels coinciding with their base levels. This class includes in particular many ultimately binary sequences with a special header. % For almost all of these sequences, we additionally show that although the patterns generated by them seem to fall somewhere between ordered and disordered, exhibiting fractal-like and random qualities at the same time, the distribution of zero and non-zero numbers at the base level has uniformity characteristics. Thus, we prove that a multitude of straight lines that traverse the patterns encounter zero and non-zero numbers in almost equal proportions

On Quasi-Periodicity in Proth-Gilbreath Triangles

Let PG be the Proth-Gilbreath operator that transforms a sequence of integers into the sequence of the absolute values of the differences between all pairs of neighbor terms. Consider the infinite tables obtained by successive iterations of PG applied to different initial sequences of integers. We study these tables of higher order differences and characterize those that have near-periodic features. As a biproduct, we also obtain two results on a class of formal power series over the field with two elements F2 that can be expressed as rational functions in several ways

God and Infinity

This article tries to come up with justifications to God and Infinity based on the existence and non-existence of each other

XLDA: Linear Discriminant Analysis for Scaling Continual Learning to Extreme Classification at the Edge

Streaming Linear Discriminant Analysis (LDA) while proven in Class-incremental Learning deployments at the edge with limited classes (upto 1000), has not been proven for deployment in extreme classification scenarios. In this paper, we present: (a) XLDA, a framework for Class-IL in edge deployment where LDA classifier is proven to be equivalent to FC layer including in extreme classification scenarios, and (b) optimizations to enable XLDA-based training and inference for edge deployment where there is a constraint on available compute resources. We show up to 42x speed up using a batched training approach and up to 5x inference speedup with nearest neighbor search on extreme datasets like AliProducts (50k classes) and Google Landmarks V2 (81k classes)Comment: Submitted at ICML 2023: PAC-Bayes Interactive Learning Worksho

Ant species richness and diversity in some selected localities of Western Ghats

We have sampled ants from twelve different localities in the Uttara Kannada district of Karnataka, India using both a package of standardized trapping methods as well as by intensive all-out-search. The relative efficacies of trapping versus all-out-search are compared. Different methods of trapping are also compared among themselves. A check list of all ants seen by all methods put together, identified up to the genus level is provided. Different localities are compared by the similarity and diversity 'of their ant fauna. A weak but statistically significant positive correlation between ant species diversity. and plant species diversity is demonstrated. This report provides the first estimates of the diversity and abundance of ants for any forest habitat of India