Pearl Bipin

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• Bachelor of Technology
• Engineer at National Institute of Technology Goa

We introduce a two-variable analytic family ζ J (s), representing a deformation of the Riemann zeta-function ζ(s) constructed using an Imaginary Mellin Transform (ImMT) indexed by complex parameters J derived from recursive imaginary units. For each fixed J with ℜ(J) > 0, ζ J (s) is defined via the integral representation I(s, J) = ∞ 0 t s+J−2 e t...

We revisit the classical "diagonal" case of Fermat's equation, a n + b n = c n , n ≥ 2, under the linear constraints b = k a. Using only elementary irrationality and infinite-descent arguments, we show that for each fixed integer k ≥ 1, no nontrivial positive-integer solutions exist. Along the way we illustrate the methods with explicit small-expon...

We propose a simple yet effective pair of greedy heuristics for the Euclidean Traveling Salesman Problem (TSP) that adapt to the vehi-cle's load state. Case 1 (burdened vehicle) employs a classical nearest-neighbor rule to drop load early and reduce travel cost under heavy load. Case 2 (unburdened vehicle) uses a farthest-first start followed by ne...

Complex numbers, particularly the imaginary unit i, are fundamental to analyzing linear time-invariant (LTI) systems, quantum mechanics, and signal processing. However, implementing complex-valued parameters in hardware or optimizing complex functions with real-valued algorithms can be challenging. This paper introduces a principled method for deri...

The detection of sarcasm in text is an inherently difficult task due to its reliance on context, tone, and often implied meaning. This research proposes a novel approach to sarcasm detection that does not rely on complex neural networks but instead utilizes a combination of lexical analysis, sentiment analysis, and semantic word embeddings. Specifi...

We propose a rigorous mathematical framework for complex-valued probability amplitudes, generalizing classical probability through the use of L2 spaces. By modeling events using complex amplitudes and defining probabilities as squared magnitudes of integrals, we introduce a clean method of analyzing interference, superposition, and stochastic evolu...

This paper presents a step-by-step solution for a modified harmonic os-cillator equation incorporating a nonstandard recursive imaginary damping term. An extension of the classical Laplace transform, the Imaginary Laplace Transform (ImLT), is introduced and used to derive the time-domain behavior of the system. We develop a mathematical foundation...

We present a regime-based approach to stock market forecasting that bypasses the need for neural networks. Using polynomial ridge regression within a dynamic sliding window, we segment long-term historical S&P 500 data into interpretable trend regimes. From each segment, we extract statistically engineered features such as slope, R 2 , and volatili...

This paper presents a non-neural network approach for the classification of elec-trocardiogram (ECG) signals. By employing both standard and weighted statistical feature extraction techniques on ECG signals-each comprising 140 data points-and leveraging dimensionality reduction via Principal Component Analysis (PCA), we build an effective classific...

Classical Fourier transforms are a cornerstone in the analysis of signals and systems. However, their applicability is limited when dealing with systems exhibiting recursive oscillatory behavior governed by re-cursive imaginary units. In this paper, we introduce the Recursive Fourier Transform (RFT), a novel extension of the classical Fourier trans...

The Imaginary Mellin Transform (ImMT) is a newly introduced analytical tool that extends the classical Mellin Transform into the domain of recursive imaginary units. This new transform is designed to handle systems exhibiting recursive, multi-layered scaling behavior. The paper defines the ImMT, explores its key properties, and discusses its potent...

This paper presents a lightweight, traditional machine learning pipeline designed to classify phonetically similar spoken words-specifically, "Barbie" and "Puppy". Despite a small initial dataset, the system achieves 92.37% accuracy through a combination of handcrafted MFCC-based statistical features, custom audio augmentation, and optimized suppor...

This paper presents a lightweight, traditional machine learning pipeline designed to classify phonetically similar spoken words-specifically, "Barbie" and "Puppy". Despite a small initial dataset, the system achieves 92.37% accuracy through a combination of handcrafted MFCC-based statistical features, custom audio augmentation, and optimized suppor...

This paper introduces the Imaginary Laplace Transform (ImLT), a novel mathematical framework for analyzing systems governed by recursive imaginary coefficients. By extending the classical Laplace transform into a higher-order imaginary domain-through units such as √ −i-we propose a structure for capturing deeply nested oscillatory behaviors. These...

The function x to the power of itself presents a blend of exponential and power-like behavior that places it outside the realm of elementary functions. In this note, we systematically explore its higher-order derivatives using Bell polynomials and the Faà di Bruno formula, and develop a differential operator approach that offers a concise perspecti...

This paper delves into the logarithmic properties of both classical and higher-order imaginary numbers, offering a novel exploration of complex logarithms. We extend the concept of the imaginary unit i to recursively defined higher-order imaginary numbers and investigate their logarithmic behavior. Through detailed step-by-step computations, we exp...

This paper presents an pedagogically valuable and elegant representation of the mathematical constant π derived from complex logarithms. Unlike traditional infinite series or trigonometric identities, the formula showcased offers a concise and direct connection between complex analysis and π, showcasing the deep interrelations between elementary fu...

The present invention is a lightning harvesting apparatus designed to capture and store electrical energy from lightning strikes. The apparatus includes an omnidirectional conductive antenna tower connected to an energy distribution network comprising multiple branches with supercapacitors arranged in parallel. The captured energy is directed i...

This short technical note explores the Central Limit Theorem (CLT) and addresses a common query regarding its application. While it is well known that 30 samples are sufficient for the CLT to ensure the sample mean distribution approaches normality, the question remains: how many groups of 30 samples do we need to create a reliable plot of these sa...

The Sleeping Beauty problem has been a topic of significant debate and analysis in probability theory, particularly in the context of conditional probability and Bayesian reasoning. In this experiment, a fair coin is flipped, and the outcome determines the waking schedule of the subject, "Sleeping Beauty." If the coin lands heads, she wakes up once...

The Sleeping Beauty problem has been a topic of significant debate and analysis in probability theory, particularly in the context of conditional probability and Bayesian reasoning. In this experiment, a fair coin is flipped, and the outcome determines the waking schedule of the subject, "Sleeping Beauty." If the coin lands heads, she wakes up once...

Background: Sarcasm detection is a critical task in natural language processing (NLP), with applications ranging from sentiment analysis to conversational AI. Detecting sarcasm is inherently challenging due to its subtle and context-dependent nature. In this paper, we propose a self-contained sarcasm classification methodology that avoids neural ne...

Background: Modern AI systems often rely on large language models (LLMs) for a wide range of tasks, but these models are computationally intensive and may not be efficient for simpler tasks. Traditional machine learning models, while less powerful, offer computational efficiency for less complex tasks. There is a growing need for an AI architecture...

Background: In traditional logic systems, direct current (DC) voltages are represented by the states +1, 0, and-1, while alternating current (AC) is often overlooked in logic representations. The imaginary unit i, known for its oscillatory nature, provides a novel way to represent AC voltages, given its inherent property of cycling between negative...

Background: In traditional logic systems, direct current (DC) voltages are represented by the states +1, 0, and-1, while alternating current (AC) is often overlooked in logic representations. The imaginary unit i, known for its oscillatory nature, provides a novel way to represent AC voltages, given its inherent property of cycling between negative...

Sorry the research paper "The Number of Numbers" had errors, I fixed it here.

Background: The concept of the universe's lifecycle, from its initial singularity to its potential collapse and rebirth, represents a profound area of cosmological inquiry. This theory explores the possibility that the universe undergoes infinite cycles of expansion and contraction, akin to stellar evolution and black hole formation. Objectives: Th...

Background: The concept of the universe's lifecycle, from its initial singularity to its potential collapse and rebirth, represents a profound area of cosmological inquiry. This theory explores the possibility that the universe undergoes infinite cycles of expansion and contraction, akin to stellar evolution and black hole formation. Objectives: Th...

Background: Moore's Law, which predicts the doubling of transistors on a chip every two years, has driven exponential growth in electronics. However, as transistor sizes approach atomic scales, quantum mechanical effects pose significant limitations to this trend. Objectives: This study aims to explore the theoretical constraints imposed by quantum...

Background: Creating virtual clones of individuals using classical natural language processing (NLP) offers a cost-effective alternative to neural network-based models. This theoretical research explores the development of decentralized chatbots trained on specific corpora to emulate different facets of a person's personality and knowledge without...

Background: Traditional mathematics treats zero and infinity as well-defined concepts, yet their practical attainment remains elusive. Zero represents the convergence point of number lines, while infinity symbolizes divergence. In this paper, we explore the theory that zero and infinity are bounded but unattainable in practical life, forming the co...

Background: Traditional mathematics treats zero and infinity as well-defined concepts, yet their practical attainment remains elusive. Zero represents the convergence point of number lines, while infinity symbolizes divergence. In this paper, we explore the theory that zero and infinity are bounded but unattainable in practical life, forming the co...

Background: Traditional probability theory assumes ideal conditions, such as a fair coin with only two possible outcomes-heads or tails. However, real-world factors, including environmental disturbances (e.g., wind, vibrations), can affect the behavior of a coin during a flip. In this paper, we introduce the concept of "deviant behavior" in probabi...

Background: Ternary logic gates, utilizing the base-1, 0, 1, represent a departure from conventional binary systems, offering expanded states for digital representation. This paper explores the integration of the imaginary unit i within ternary logic, introducing new dimensions to computing paradigms beyond traditional binary logic gates. Objective...

Background: Image classification is a fundamental task in computer vision, crucial for various applications, from medical diagnostics to autonomous driving. In this paper, we propose an image classification methodology that leverages classical machine learning techniques with a focus on weighted statistical features, aiming for simplicity, interpre...

Background: Image classification is a fundamental task in computer vision, crucial for various applications, from medical diagnostics to autonomous driving. In this paper, we propose an image classification methodology that leverages classical machine learning techniques with a focus on weighted statistical features, aiming for simplicity, interpre...

Background: The traditional binary logic gates operate on a binary system with inputs and outputs taking values of 0 and 1. In this paper, we explore the concept of logic gates using a ternary base, where inputs and outputs can take values of -1, 0, and 1. Objectives: Our study aims to demonstrate the implementation and functionality of basic logic...

Background: The traditional binary logic gates operate on a binary system with inputs and outputs taking values of 0 and 1. In this paper, we explore the concept of logic gates using a ternary base, where inputs and outputs can take values of -1, 0, and 1. Objectives: Our study aims to demonstrate the implementation and functionality of basic logic...

Background: Anomaly detection is a critical aspect of data analysis, demanding rigorous methodologies to ensure accuracy and reliability. In this paper, we present an anomaly detection algorithm specifically engineered for detecting anomalies within rotated elliptical clusters. Objectives: Our study aims to bridge the gap between theoretical insigh...

Background: Anomaly detection is a critical aspect of data analysis, demanding rigorous methodologies to ensure accuracy and reliability. In this paper, we present an anomaly detection algorithm specifically engineered for detecting anomalies within rotated elliptical clusters. Objectives: Our study aims to bridge the gap between theoretical insigh...

Background: Anomaly detection is a critical aspect of data analysis, demanding rigorous methodologies to ensure accuracy and reliability. In this paper, we present an anomaly detection algorithm specifically engineered for detecting anomalies within rotated elliptical clusters. Objectives: Our study aims to bridge the gap between theoretical insigh...

Anomaly detection is a critical aspect of data analysis, demanding rigorous methodologies to ensure accuracy and reliability. In this paper, we present an anomaly detection algorithm specifically engineered for the detection of anomalies within rotated elliptical clusters. Our approach leverages precise geometric computations and robust statistical...

Anomaly detection is a critical aspect of data analysis, demanding rigorous methodologies to ensure accuracy and reliability. In this paper, we present an anomaly detection algorithm specifically engineered for the detection of anomalies within rotated elliptical clusters. Our approach leverages precise geometric computations and robust statistical...

Anomaly detection, a fundamental task in data analysis, requires robust mathematical frameworks. We present to you a pioneering anomaly detection algorithm poised to redefine the landscape of data analysis. Within this study, we introduce a novel approach meticulously tailored to the nuanced characteristics of elliptical clusters, showcasing an int...