arXiv subjects

Ram Krishna Pandey

Publications and source records attributed to Ram Krishna Pandey.

At least 19 recordsLinked to original sources

arXiv2025

Extended inverse results for restricted h-fold sumset in integers

Let $A$ be a finite set of $k$ integers. For $h \leq k$, the restricted $h$-fold sumset $h^{\wedge} A$ is the set of all sums of $h$ distinct elements of $A$. In additive combinatorics, much of the focus has traditionally been on finite integer sets whose sumsets are unusually small (cf.\ Freiman's theorem and its extensions). More recently, Nathanson posed the inverse problem for the restricted sumset $h^{\wedge} A$ when $|h^{\wedge} A|$ is small. For $h \in \{2, 3, 4\}$, this question has already been studied by Mohan and Pandey. In this article, we study the inverse problems for $h^{\wedge} A$ with arbitrary $h \geq 3$ and characterize all possible sets $A$ for certain cardinalities of $h^{\wedge} A$.

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arXiv2024

Some direct and inverse problems for the Restricted Signed sumset in set of integers

Given a positive integer $h$ and a nonempty finite set of integers $A=\{a_{1},a_{2},\ldots,a_{k}\}$, the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} λ_{i} a_{i}: λ_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left| λ_{i} \right| =h\right\rbrace.$$ The direct problem associated with this sumset is to find the optimal lower bound of $|h^{\wedge}_{\pm}A|$, and the inverse problem associated with this sumset is to determine the structure of the underlying set $A$, when $|h^{\wedge}_{\pm}A|$ attains the optimal lower bound. Bhanja, Komatsu and Pandey studied the direct and inverse problem for the restricted $h$-fold signed sumset for $h=2, 3$, and $k$ and conjectured some direct and inverse results for $h \geq 4$. In this paper, we prove these conjectures for $h=4$. We also prove the direct and inverse theorems for arbitrary $h$ under certain restrictions on the set $A$ which are particular cases of the conjectures. Moreover, we prove these conjectures for arithmetic progressions.

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arXiv2024

Freiman's $(3k-4)$-like results for subset and subsequence sums

For a nonempty finite set $A$ of integers, let $S(A) = \left\{ \sum_{b\in B} b: \emptyset \not= B\subseteq A\right\}$ be the set of all nonempty subset sums of $A$. In 1995, Nathanson determined the minimum cardinality of $S(A)$ in terms of $|A|$ and described the structure of $A$ for which $|S(A)|$ is the minimum. He asked to characterize the underlying set $A$ if $|S(A)|$ is a small increment to its minimum size. Problems of such nature are inspired by the well-known Freiman's $3k-4$ theorem. In this paper, some results in the direction of Freiman's $3k-4$ theorem for the set of subset sums $S(A)$ are proved. Such results are also extended to the set of subsequence sums $S(\mathbb{A}) = \left\{ \sum_{b\in \mathbb{B}} b: \emptyset \not= \mathbb{B} \subseteq \mathbb{A} \right\}$ of sequence $\mathbb{A}$, where the notation $\mathbb{B} \subseteq \mathbb{A} $, is used for $\mathbb{B}$ is a subsequence of $\mathbb{A}$. The results are further generalized to a generalization of subset and subsequence sums. The main idea of the proofs of the results is to write the set of subset sums $S(A)$ and the set of subsequence sums $S(\mathbb{A})$ in terms of the $h$-fold sumset $hA$ and the $h$-fold restricted sumset $h^\wedge A$. Such representation also gives other proof of some of the results of Nathanson and Mistri et al.

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arXiv2024

On additive complement with special structures

Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is said to be an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. This article describes various types of additive complements of the set $A$ such as those additive complement of $A$ that does not intersects $A$, additive complements of the form of the union of disjoint infinite arithmetic progressions, additive complement having various density etc. As an application of this study, we also focus on the structure of sumset of arithmetic progression and geometric progression. Apart from this, for given positive real no. $α\leq 1$ and finite set $A$, we investigate a set $B$ such that it can be written as union of disjoint infinite arithmetic progression and density of $A+B$ is $α$.

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arXiv2024

Generalized H-fold sumset and Subsequence sum

Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $h\in H$ where, the sumset $h^{(r)}A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)}A$, i.e., for $|H^{(r)}A|$ and the structure of the underlying sets $A$ and $H$ when $|H^{(r)}A|$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$, since $H^{(r)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.

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arXiv2024

Extended inverse theorems for $h$-fold sumsets in integers

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\left|hA\right| \leq hk+3h-4$. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible.

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arXiv2023

CoreDeep: Improving Crack Detection Algorithms Using Width Stochasticity

Automatically detecting or segmenting cracks in images can help in reducing the cost of maintenance or operations. Detecting, measuring and quantifying cracks for distress analysis in challenging background scenarios is a difficult task as there is no clear boundary that separates cracks from the background. Developed algorithms should handle the inherent challenges associated with data. Some of the perceptually noted challenges are color, intensity, depth, blur, motion-blur, orientation, different region of interest (ROI) for the defect, scale, illumination, complex and challenging background, etc. These variations occur across (crack inter class) and within images (crack intra-class variabilities). Overall, there is significant background (inter) and foreground (intra-class) variability. In this work, we have attempted to reduce the effect of these variations in challenging background scenarios. We have proposed a stochastic width (SW) approach to reduce the effect of these variations. Our proposed approach improves detectability and significantly reduces false positives and negatives. We have measured the performance of our algorithm objectively in terms of mean IoU, false positives and negatives and subjectively in terms of perceptual quality.

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arXiv2023

Revealing the Underlying Patterns: Investigating Dataset Similarity, Performance, and Generalization

Supervised deep learning models require significant amount of labeled data to achieve an acceptable performance on a specific task. However, when tested on unseen data, the models may not perform well. Therefore, the models need to be trained with additional and varying labeled data to improve the generalization. In this work, our goal is to understand the models, their performance and generalization. We establish image-image, dataset-dataset, and image-dataset distances to gain insights into the model's behavior. Our proposed distance metric when combined with model performance can help in selecting an appropriate model/architecture from a pool of candidate architectures. We have shown that the generalization of these models can be improved by only adding a small number of unseen images (say 1, 3 or 7) into the training set. Our proposed approach reduces training and annotation costs while providing an estimate of model performance on unseen data in dynamic environments.

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arXiv2023

TrueDeep: A systematic approach of crack detection with less data

Supervised and semi-supervised semantic segmentation algorithms require significant amount of annotated data to achieve a good performance. In many situations, the data is either not available or the annotation is expensive. The objective of this work is to show that by incorporating domain knowledge along with deep learning architectures, we can achieve similar performance with less data. We have used publicly available crack segmentation datasets and shown that selecting the input images using knowledge can significantly boost the performance of deep-learning based architectures. Our proposed approaches have many fold advantages such as low annotation and training cost, and less energy consumption. We have measured the performance of our algorithm quantitatively in terms of mean intersection over union (mIoU) and F score. Our algorithms, developed with 23% of the overall data; have a similar performance on the test data and significantly better performance on multiple blind datasets.

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arXiv2022

On the minimum size of subset and subsequence sums in integers

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $α\leq rk$, let $Σ_α (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $α$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $Σ_α (A)$ for $Σ_α (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $Σ_α (A)$ and $Σ_α (\mathcal{A})$. As special cases, we also obtain some already known results in this study.

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arXiv2022

Maximal density and the kappa values for the families $\{a,a+1,2a+1,n\}$ and $\{a,a+1,2a+1,3a+1,n\}$

Let $M$ be a set of positive integers. We study the maximal density $μ(M)$ of the sets of nonnegative integers $S$ whose elements do not differ by an element in $M$. In 1973, Cantor and Gordon established a formula for $μ(M)$ for $|M|\leq 2$. Since then, many researchers have worked upon the problem and found several partial results in the case $|M|\geq 3$, including some results in the case, $M$ is an infinite set. In this paper, we study the maximal density problem for the families $M=\{a,a+1,2a+1,n\}$ and $M=\{a,a+1,2a+1,3a+1,n\}$, where $a$ and $n$ are positive integers. In most of the cases, we find bounds for the parameter \textit{kappa}, denoted by $κ(M)$, which actually serves as a lower bound for $μ(M)$. The parameter $κ(M)$ has already got its importance due to its rich connection with the problems such as the "lonely runner conjecture" in Diophantine approximations and coloring parameters such as "circular coloring" and "fractional coloring" in graph theory.

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arXiv2019

Additive Complements for a given Asymptotic Density

{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that the sumset $A+B:=\{a+b~;a\in A,b\in B\}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:=\{x_1+\cdots+x_k: x_i\in A, i=1,\dots,k\}$ for an integer $k\geq2.$

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arXiv2019

Inverse problems for certain subsequence sums in integers

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $α$ ($\leq k$), let \[Σ_α (A):=\{s(B): B \subset A, |B|\geq α\}.\] Now, let $\mathcal{A}=(\underbrace{a_{1},\ldots,a_{1}}_{r_{1}~\text{copies}}, \underbrace{a_{2},\ldots,a_{2}}_{r_{2}~\text{copies}},\ldots, \underbrace{a_{k},\ldots,a_{k}}_{r_{k}~\text{copies}})$ be a finite sequence of integers with $k$ distinct terms, where $r_{i}\geq 1$ for $i=1,2,\ldots,k$. Given a subsequence $\mathcal{B}$ of $\mathcal{A}$, the sum of all terms of $\mathcal{B}$, denoted by $s(\mathcal{B})$, is called the subsequence sum of $\mathcal{B}$. For $0\leq α\leq \sum_{i=1}^{k} r_{i}$, let \[Σ_α (\bar{r},\mathcal{A}):=\left\{s(\mathcal{B}): \mathcal{B}~\text{is a subsequence of}~\mathcal{A}~\text{of length} \geq α\right\},\] where $\bar{r}=(r_{1},r_{2},\ldots,r_{k})$. Very recently, Balandraud obtained the minimum cardinality of $Σ_α (A)$ in finite fields. Motivated by Baladraud's work, we find the minimum cardinality of $Σ_α(A)$ in the group of integers. We also determine the structure of the finite set $A$ of integers for which $|Σ_α (A)|$ is minimal. Furthermore, we generalize these results of subset sums to the subsequence sums $Σ_α (\bar{r},\mathcal{A})$. As special cases of our results we obtain some already known results for the usual subset and subsequence sums.

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arXiv2019

Direct and inverse results on restricted signed sumsets in integers

Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0},λ_{1}, \ldots, λ_{k-1}) \in \{-1,0,1\}^{k},~Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the restricted signed sumset of $A$. The direct problem for the restricted signed sumset $h\hat{}_{\underline{+}}A$ is to find the minimum number of elements in $h\hat{}_{\underline{+}}A$ in terms of $|A|$. The inverse problem for $h\hat{}_{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h\hat{}_{\underline{+}}A|$ is minimal. In this article, we solve some cases of both direct and inverse problems for $h\hat{}_{\underline{+}}A$, when $A$ is a finite set of integers. In this connection, we also pose some questions as conjectures in the remaining cases.

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arXiv2019

Improving Facial Emotion Recognition Systems Using Gradient and Laplacian Images

In this work, we have proposed several enhancements to improve the performance of any facial emotion recognition (FER) system. We believe that the changes in the positions of the fiducial points and the intensities capture the crucial information regarding the emotion of a face image. We propose the use of the gradient and the Laplacian of the input image together with the original input into a convolutional neural network (CNN). These modifications help the network learn additional information from the gradient and Laplacian of the images. However, the plain CNN is not able to extract this information from the raw images. We have performed a number of experiments on two well known datasets KDEF and FERplus. Our approach enhances the already high performance of state-of-the-art FER systems by 3 to 5%.

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arXiv2018

Binary Document Image Super Resolution for Improved Readability and OCR Performance

There is a need for information retrieval from large collections of low-resolution (LR) binary document images, which can be found in digital libraries across the world, where the high-resolution (HR) counterpart is not available. This gives rise to the problem of binary document image super-resolution (BDISR). The objective of this paper is to address the interesting and challenging problem of super resolution of binary Tamil document images for improved readability and better optical character recognition (OCR). We propose multiple deep neural network architectures to address this problem and analyze their performance. The proposed models are all single image super-resolution techniques, which learn a generalized spatial correspondence between the LR and HR binary document images. We employ convolutional layers for feature extraction followed by transposed convolution and sub-pixel convolution layers for upscaling the features. Since the outputs of the neural networks are gray scale, we utilize the advantage of power law transformation as a post-processing technique to improve the character level pixel connectivity. The performance of our models is evaluated by comparing the OCR accuracies and the mean opinion scores given by human evaluators on LR images and the corresponding model-generated HR images.

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arXiv2018

Direct and Inverse Theorems on Signed Sumsets of Integers

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0}, \ldots, λ_{k-1}) \in \mathbb{Z}^{k},~ Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the {\it signed sumset} of $A$. The {\it direct problem} for the signed sumset $h_{\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\underline{+}}A|$ in terms of $|A|$. The {\it inverse problem} for $h_{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h_{\underline{+}}A|$ is minimal. In this article, we solve both the direct and inverse problems for $|h_{\underline{+}}A|$, when $A$ is a finite set of integers.

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arXiv2018

MSCE: An edge preserving robust loss function for improving super-resolution algorithms

With the recent advancement in the deep learning technologies such as CNNs and GANs, there is significant improvement in the quality of the images reconstructed by deep learning based super-resolution (SR) techniques. In this work, we propose a robust loss function based on the preservation of edges obtained by the Canny operator. This loss function, when combined with the existing loss function such as mean square error (MSE), gives better SR reconstruction measured in terms of PSNR and SSIM. Our proposed loss function guarantees improved performance on any existing algorithm using MSE loss function, without any increase in the computational complexity during testing.

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