arXiv subjects

Gurinder Singh

Publications and source records attributed to Gurinder Singh.

14 recordsLinked to original sources

arXiv2026

Rademacher-type formula and higher order Tur\'{a}n inequalities for $\ell$-regular overpartitions

For $\ell\geq 2$, let $\overline{A}_\ell(n)$ count the number of overpartitions of $n$ with no parts divisible by $\ell$. In this article, we employ the circle method to derive a Rademacher-type formula for $\overline{A}_\ell(n)$, when $\ell$ is a squarefree odd integer. As an application, we derive higher order Tu\'{r}an inequalities for the $\ell$-regular overpartition function using a result of Griffin, Ono, Rolen, and Zagier.

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arXiv2026

On the G\"ollnitz-Gordon-Andrews identities via commutative algebra

The G\"ollnitz-Gordon-Andrews identities generalize the classical partition identities discovered independently by H. G\"ollnitz and B. Gordon. These are Rogers-Ramanujan-type identities involving generating functions of partitions satisfying certain kinds of difference conditions on the one hand and infinite periodic products on the other. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on Afsharijoo's approach, we investigate the G\"ollnitz-Gordon-Andrews identities using techniques from commutative algebra. More generally, we establish a broader family of identities, of which the G\"ollnitz-Gordon-Andrews identities arise as special cases. Our approach interprets the associated generating functions in terms of Hilbert-Poincar\'e series of suitably constructed graded algebras, providing the first commutative algebra framework for these identities.

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arXiv2025

Domain-Aware Quantum Circuit for QML

Designing parameterized quantum circuits (PQCs) that are expressive, trainable, and robust to hardware noise is a central challenge for quantum machine learning (QML) on noisy intermediate-scale quantum (NISQ) devices. We present a Domain-Aware Quantum Circuit (DAQC) that leverages image priors to guide locality-preserving encoding and entanglement via non-overlapping DCT-style zigzag windows. The design employs interleaved encode-entangle-train cycles, where entanglement is applied among qubits hosting neighboring pixels, aligned to device connectivity. This staged, locality-preserving information flow expands the effective receptive field without deep global mixing, enabling efficient use of limited depth and qubits. The design concentrates representational capacity on short-range correlations, reduces long-range two-qubit operations, and encourages stable optimization, thereby mitigating depth-induced and globally entangled barren-plateau effects. We evaluate DAQC on MNIST, FashionMNIST, and PneumoniaMNIST datasets. On quantum hardware, DAQC achieves performance competitive with strong classical baselines (e.g., ResNet-18/50, DenseNet-121, EfficientNet-B0) and substantially outperforming Quantum Circuit Search (QCS) baselines. To the best of our knowledge, DAQC, which uses a quantum feature extractor with only a linear classical readout (no deep classical backbone), currently achieves the best reported performance on real quantum hardware for QML-based image classification tasks. Code and pretrained models are available at: https://github.com/gurinder-hub/DAQC.

quant-ph↗

arXiv2025

Hook length inequalities for $t$-regular partitions in the $t$-aspect

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of $k$. We prove that for any $t\geq2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq0$. We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n>3$, and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq0$. Finally, we state some problems for future works.

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arXiv2025

QFGN: A Quantum Approach to High-Fidelity Implicit Neural Representations

Implicit neural representations have shown potential in various applications. However, accurately reconstructing the image or providing clear details via image super-resolution remains challenging. This paper introduces Quantum Fourier Gaussian Network (QFGN), a quantum-based machine learning model for better signal representations. The frequency spectrum is well balanced by penalizing the low-frequency components, leading to the improved expressivity of quantum circuits. The results demonstrate that with minimal parameters, QFGN outperforms the current state-of-the-art (SOTA) models. Despite noise on hardware, the model achieves accuracy comparable to that of SIREN, highlighting the potential applications of quantum machine learning in this field.

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arXiv2025

Benchmarking MedMNIST dataset on real quantum hardware

Quantum machine learning (QML) has emerged as a promising domain to leverage the computational capabilities of quantum systems to solve complex classification tasks. In this work, we present the first comprehensive QML study by benchmarking the MedMNIST-a diverse collection of medical imaging datasets on a 127-qubit real IBM quantum hardware, to evaluate the feasibility and performance of quantum models (without any classical neural networks) in practical applications. This study explores recent advancements in quantum computing such as device-aware quantum circuits, error suppression, and mitigation for medical image classification. Our methodology is comprised of three stages: preprocessing, generation of noise-resilient and hardware-efficient quantum circuits, optimizing/training of quantum circuits on classical hardware, and inference on real IBM quantum hardware. Firstly, we process all input images in the preprocessing stage to reduce the spatial dimension due to quantum hardware limitations. We generate hardware-efficient quantum circuits using backend properties expressible to learn complex patterns for medical image classification. After classical optimization of QML models, we perform inference on real quantum hardware. We also incorporate advanced error suppression and mitigation techniques in our QML workflow, including dynamical decoupling (DD), gate twirling, and matrix-free measurement mitigation (M3) to mitigate the effects of noise and improve classification performance. The experimental results showcase the potential of quantum computing for medical imaging and establish a benchmark for future advancements in QML applied to healthcare.

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arXiv2025

On hook length biases in $t$-regular partitions

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$.

math.CO↗

arXiv2024

Towards 6G-V2X: Aggregated RF-VLC for Ultra-Reliable and Low-Latency Autonomous Driving

We are witnessing a transition to a new era where driverless cars will be pervasively connected to deliver significantly improved safety, traffic efficiency, and travel experiences. A diverse set of advanced vehicular use cases including connected autonomous vehicles will be made possible by building upon the emerging sixth-generation (6G) wireless networks. Among many 6G wireless technologies, the principal objective of this paper is to introduce the potential benefits of the hybrid integration of Vehicular Visible Light Communication (V VLC) and Vehicular Radio Frequency (V RF) communication systems by studying the impact of interference as well as various meteorological phenomenon viz. rain, fog and dry snow. In particular, we show that regardless of any meteorological impact, a properly configured link-aggregated hybrid V-VLC/V-RF system is capable of meeting stringent ultra high reliability (>99.999%) and ultra-low latency (<3 ms) requirements, making it a promising candidate for 6G Vehicle-to-Everything (V2X) Communications. To stimulate future research in the hybrid RF-VLC V2X space, we also highlight the potential challenges and research directions.

eess.SP↗

arXiv2024

Hook length biases in ordinary and $t$-regular partitions

In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular partitions. For a positive integer $k$, let $p_{(k)}(n)$ denote the number of hooks of length $k$ in all the partitions of $n$. We prove that $p_{(k)}(n)\geq p_{(k+1)}(n)$ for all $n\geq0$ and $n\ne k+1$; and $p_{(k)}(k+1)- p_{(k+1)}(k+1)=-1$ for $k\geq 2$. For integers $t\geq2$ and $k\geq1$, let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. We find generating functions of $b_{t,k}(n)$ for certain values of $t$ and $k$. Exploring hook length biases for $b_{t,k}(n)$, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that $b_{2,2}(n)\geq b_{2,1}(n)$ for all $n>4$, whereas $b_{2,2}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. We also propose some conjectures on biases among $b_{t,k}(n)$.

math.CO↗

arXiv2023

Arithmetic properties and asymptotic formulae for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let $σ_o\text{mex}(n)$ (resp., $σ_e\text{mex}(n)$) denote the sum of odd (resp., even) minimal excludants over all the partitions of $n$. Recently, Baruah et al. proved a few congruences for these partition functions modulo $4$ and $8$, and asked for asymptotic formulae for the same. In this article, we study the lacunarity of $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$ modulo arbitrary powers of $2$ and also prove some infinite families of congruences for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$ modulo $4$ and $8$. We also obtain Hardy-Ramanujan type asymptotic formulae for both $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$.

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arXiv2023

Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$

Lin introduced the partition function $\text{PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of $n$ with designated summands in which all parts are odd. For $k\geq0$, Lin conjectured congruences for $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+2}$. In this article, we develop a new approach to study these congruences. We study the generating functions of $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+3}$ for certain values of $k$. We also study $\text{PDO}_t(n)$ modulo powers of $2$. We establish infinitely many congruences for $\text{PDO}_t(n)$ modulo $8$ and $32$. We prove several congruences modulo small powers of $2$ and discuss the existence of congruences modulo arbitrary powers of $2$ similar to those in Lin's conjecture. In reference to this, we also pose some problems for future work.

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arXiv2023

Certain Diophantine equations and new parity results for $21$-regular partitions

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd primes $p$ for which the Diophantine equation $8x^2+27y^2=jp$ has primitive solutions for some $j\in\left\lbrace1,4,8\right\rbrace$, and we also prove that the Dirichlet density of such primes is equal to $1/6$. Recently, Yao provided new infinite families of congruences modulo $2$ for $b_{3}(n)$ and those congruences involve every prime $p\geq 5$ based on Newman's results. Following a similar approach, we prove new infinite families of congruences modulo $2$ for $b_{21}(n)$, and these congruences imply that $b_{21}(n)$ is odd infinitely often.

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arXiv2022

Optical IRS Aided B5G V2V Solution for Road Safety Applications

In this work, we showcase the potential benefit of employing optical intelligent reflecting surfaces (O-IRS) for improving safety message dissemination for vehicular visible light communication (V-VLC) systems particularly at the road intersections. Buildings, roadside structures, signboards, and other impediments commonly hinder line-of-sight (LoS) communication between vehicles at urban crossroads scenarios. We propose using O-IRS at road intersection to improve the communication link's reliability. We compare the performance of proposed scheme with baseline scenarios such as optical relay and non line-of-sight (NLOS) road reflection (NRR) aided vehicle-to-vehicle (V2V) communication. From obtained results, it has been shown that O-IRS offers considerable performance enhancement as compared to the baseline scenarios. In particular, O-IRS can achieve longer communication range as compared to the optical relay aided V-VLC systems while ensuring desired quality-of-service (QoS).

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arXiv2022

Arithmetic properties of certain $t$-regular partitions

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\in \{6,10,14,15,18,20,22,26,27,28\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\geq 0$ for which $b_t(An+ B)\equiv 0\pmod 2$ for all $n\geq 0$. We prove that, for any $t\geq 2$ and prime $\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}$. Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions.

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