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Gaurav Bhatnagar

Publications and source records attributed to Gaurav Bhatnagar.

35 records · Page 2Linked to original sources
arXiv2020

Prime number conjectures from the Shapiro class structure

The height $H(n)$ of $n$, introduced by Pillai in 1929, is the smallest positive integer $i$ such that the $i$th iterate of Euler's totient function at $n$ is $1$. H. N. Shapiro (1943) studied the structure of the set of all numbers at a height. We state a formula for the height function due to Shapiro and use it to list steps to generate numbers at any height. This turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $π(n)$ and the $n$th prime $p_n$. We conjecture formulas for $π(n)$ and $p_n$ in terms of the height function.

math.NT↗
arXiv2019

A partial theta function Borwein conjecture

We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.

math.CO↗
arXiv2019

On Entry II.16.12: A continued fraction of Ramanujan

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt's volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanujan's Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynomials.

math.CA↗
arXiv2018

Automatic Rendering of Building Floor Plan Images from Textual Descriptions in English

Human beings understand natural language description and could able to imagine a corresponding visual for the same. For example, given a description of the interior of a house, we could imagine its structure and arrangements of furniture. Automatic synthesis of real-world images from text descriptions has been explored in the computer vision community. However, there is no such attempt in the area of document images, like floor plans. Floor plan synthesis from sketches, as well as data-driven models, were proposed earlier. Ours is the first attempt to render building floor plan images from textual description automatically. Here, the input is a natural language description of the internal structure and furniture arrangements within a house, and the output is the 2D floor plan image of the same. We have experimented on publicly available benchmark floor plan datasets. We were able to render realistic synthesized floor plan images from the description written in English.

cs.CV↗
arXiv2018

SUGAMAN: Describing Floor Plans for Visually Impaired by Annotation Learning and Proximity based Grammar

In this paper, we propose SUGAMAN (Supervised and Unified framework using Grammar and Annotation Model for Access and Navigation). SUGAMAN is a Hindi word meaning "easy passage from one place to another". SUGAMAN synthesizes textual description from a given floor plan image for the visually impaired. A visually impaired person can navigate in an indoor environment using the textual description generated by SUGAMAN. With the help of a text reader software, the target user can understand the rooms within the building and arrangement of furniture to navigate. SUGAMAN is the first framework for describing a floor plan and giving direction for obstacle-free movement within a building. We learn $5$ classes of room categories from $1355$ room image samples under a supervised learning paradigm. These learned annotations are fed into a description synthesis framework to yield a holistic description of a floor plan image. We demonstrate the performance of various supervised classifiers on room learning. We also provide a comparative analysis of system generated and human written descriptions. SUGAMAN gives state of the art performance on challenging, real-world floor plan images. This work can be applied to areas like understanding floor plans of historical monuments, stability analysis of buildings, and retrieval.

cs.CV↗
arXiv2018

Heine's method and $A_n$ to $A_m$ transformation formulas

We apply Heine's method---the key idea Heine used in 1846 to derive his famous transformation formula for $_2ϕ_1$ series---to multiple basic series over the root system of type $A$. In the classical case, this leads to a bibasic extension of Heine's formula, which was implicit in a paper of Andrews which he wrote in 1966. As special cases, we recover extensions of many of Ramanujan's $_2ϕ_1$ transformations. In addition, we extend previous work of the author regarding a bibasic extension of Andrews' $q$-Lauricella function, and show how to obtain very general transformation formulas of this type. The results obtained include transformations of an $n$-fold sum into an $m$-fold sum.

math.CA↗
arXiv2018

The Determinant of an Elliptic Sylvesteresque Matrix

We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaev's $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.

math.CA↗
arXiv2018

Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems

We list $A_n$, $C_n$ and $D_n$ extensions of the elliptic WP Bailey transform and lemma, given for $n=1$ by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced and very-well-poised ${}_{10}V_9$ elliptic hypergeometric summation formula due to Rosengren, and Rosengren and Schlosser. In our study, we discover two new $A_n$ ${}_{12}V_{11}$ transformation formulas, that reduce to two new $A_n$ extensions of Bailey's $_{10}ϕ_9$ transformation formulas when the nome $p$ is $0$, and two multiple series extensions of Frenkel and Turaev's sum.

math.CA↗
arXiv2017

Spiral determinants

We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves right, up, left, or down along the spiral.

math.CO↗
arXiv2016

A bibasic Heine transformation formula and Ramanujan's $_2ϕ_1$ transformations

We study Andrews and Berndt's organization of Ramanujan's transformation formulas in Chapter 1 of their book Ramanujan's Lost Notebook, Part II. In the process, we rediscover a bibasic Heine's transformation, which follows from a Fundamental Lemma given by Andrews in 1966, and obtain identities proximal to Ramanujan's entries. We also provide a multibasic generalization of Andrews' 1972 theorem concerning a $q$-analogue of the Lauricella function. Our results only require the $q$-binomial theorem, and are an application of what Andrews and Berndt call 'Heine's Method'.

math.CO↗
arXiv2016

Analogues of a Fibonacci-Lucas Identity

Sury's 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn's 2003 book: {\em Proofs that count: The art of combinatorial proof}) has excited a lot of comment. We give an alternate, telescoping, proof of this---and associated---identities and generalize them. We also give analogous identities for other sequences that satisfy a three-term recurrence relation.

math.CO↗
arXiv2016

A formula for the convergents of a continued fraction of Ramanujan

In Entry 16, Chapter 16 of his notebooks, Ramanujan himself gave a formula for the convergents of the famous Rogers-Ramanujan continued fraction. We provide a similar formula for the convergents of a more general continued fraction, namely Entry 15 of Chapter 16.

math.CA↗
arXiv2011

In Praise of an Elementary Identity of Euler

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8ϕ_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

math.CO↗
arXiv2009

Robust Watermarking in Multiresolution Walsh-Hadamard Transform

In this paper, a newer version of Walsh-Hadamard Transform namely multiresolution Walsh-Hadamard Transform (MR-WHT) is proposed for images. Further, a robust watermarking scheme is proposed for copyright protection using MRWHT and singular value decomposition. The core idea of the proposed scheme is to decompose an image using MR-WHT and then middle singular values of high frequency sub-band at the coarsest and the finest level are modified with the singular values of the watermark. Finally, a reliable watermark extraction scheme is developed for the extraction of the watermark from the distorted image. The experimental results show better visual imperceptibility and resiliency of the proposed scheme against intentional or un-intentional variety of attacks.

cs.CR↗
arXiv2008

A new Contrast Based Image Fusion using Wavelet Packets

Image Fusion, a technique which combines complimentary information from different images of the same scene so that the fused image is more suitable for segmentation, feature extraction, object recognition and Human Visual System. In this paper, a simple yet efficient algorithm is presented based on contrast using wavelet packet decomposition. First, all the source images are decomposed into low and high frequency sub-bands and then fusion of high frequency sub-bands is done by the means of Directive Contrast. Now, inverse wavelet packet transform is performed to reconstruct the fused image. The performance of the algorithm is carried out by the comparison made between proposed and existing algorithm.

cs.IT↗