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Rajiv Kumar

Publications and source records attributed to Rajiv Kumar.

At least 19 recordsLinked to original sources
arXiv2026

Simulation of advective accretion flows around black holes under various outer boundary conditions

We simulated various accretion disc structures with viscous hydrodynamic (HD) flow around a black hole (BH). We found that the structure of the accretion disc is significantly influenced by changes in the physical parameters of the initial inflowing gases. These physical parameters can be called outer boundary conditions (OBCs) at the outer-accretion boundary and represented on an OBC plane, which is primarily divided into hot-mode and cold-mode inflowing gases. We found smooth and shocked flows in the simulation, which follow the semi-analytical solutions with their OBCs. Interestingly, we observed that certain types of OBCs can produce shocks with jet-like features in the accretion flow. However, other OBCs can produce smooth or shock-free accretion flow, which may or may not have outflows. The smooth flows can display both the lowest and highest angular momentum distributions among the advective flows, depending on the OBCs. Additionally, the nature of the accretion flows can be either steady or quasi-steady, also influenced by the OBCs. We also observed that solutions corresponding to hot-mode gases have a greater tendency to generate outflows compared to those with the cold-mode. Therefore, this qualitative study of OBCs is crucial for understanding accretion physics, which can aid in modeling accretion discs, similar to the quantitative studies (which involve only changing mass accretion rates) of the inflowing gases. Thus, we assert that an accretion model should be based on both the qualitative and quantitative aspects of the initial inflowing gases.

astro-ph.HE↗
arXiv2026

The Mendez-Pinto-Villarreal Conjecture for some classes of monomial ideals

Characterizing when the symbolic and ordinary powers of an ideal coincide is a central problem in commutative algebra, and ideals satisfying this property are called Simis ideals. In this article, we investigate the Simis property of monomial ideals by studying the recent conjecture of Mendez, Pinto, and Villarreal on monomial ideals with minimal irreducible decomposition. Let $I$ be a monomial ideal, and let $\mathcal{F}(I)$ denote the collection of supports of the minimal generators of $I$. Assuming that $\mathcal{F}(I)=\mathcal{F}(\sqrt{I})$, we prove that if $I$ admits more than one minimal generator with a fixed support, then it is not Simis. Using this reduction, we establish the Mendez-Pinto-Villarreal conjecture for two broad classes of monomial ideals, namely support-$3$ monomial ideals and monomial ideals whose associated simplicial complexes are simplicial forests. Finally, we study the Cohen-Macaulay property of monomial ideals whose associated simplicial complexes are grafted and satisfy $\mathcal{F}(I)=\mathcal{F}(\sqrt{I})$.

math.AC↗
arXiv2026

Purity of extremal rays of Betti cones

Let $R$ be a standard graded algebra over an infinite field $\mathsf k$, and let $\mathbb{B}_{\mathbb{Q}}(R)$ and $\mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$ denote the rational cones spanned by the Betti tables of all finitely generated $R$-modules and of those with pure resolutions, respectively. We establish several necessary conditions for the equality $\mathbb{B}_{\mathbb{Q}}(R) = \mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$. When $\operatorname{edim}(R)\ge 2$, we prove that $\mathsf k$ has a pure resolution if and only if it has a linear resolution, and consequently, if the extremal rays of $\mathbb{B}_{\mathbb{Q}}(R)$ are pure, then $R$ is Koszul and good (in the sense of Roos). We show that if $R$ has depth zero, it must be Artinian for the equality of the two cones to hold. For rings with linear pairs of exact zerodivisors, we show that the equality of the cones implies that the $h$-polynomial has degree at most $2$, and use it to characterize generic Gorenstein Artin algebras satisfying $\mathbb{B}_{\mathbb{Q}}(R) = \mathbb{B}_{\mathbb{Q}}^{\mathrm{pure}}(R)$. We also characterize algebras whose extremal rays are exactly the Betti tables of shifts of $R/\mathfrak m^j$ and of pure modules $M$ with $\operatorname{codim}(M)=\operatorname{pdim}(M)$: apart from polynomial rings, these are precisely Cohen--Macaulay algebras of dimension at most one with minimal multiplicity. In addition, we obtain a characterization of Cohen--Macaulay algebras of minimal multiplicity in terms of the extremal rays of the Betti cone of maximal Cohen--Macaulay modules.

math.AC↗
arXiv2026

Topological properties and Majorana Multiplicity in Zigzag Kitaev Chain

We investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between $\phi=\pi$, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case $\phi=0$. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number $\nu = 1$ and $2$, giving rise to distinct topological phases that support two and four MZMs. The $\nu = 0$ gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.

quant-ph↗
arXiv2026

Majorana bound states in a hybrid Kitaev ladder with long-range pairing

We investigate an inter-leg coupled hybrid Kitaev ladder composed of two parallel superconducting chains with distinct pairing interactions. The upper chain of the ladder hosts conventional $p$-wave pairing, while the lower chain exhibits long-range pairing that decays algebraically with distance. We demonstrate that the mutual influence of long-range pairing exponent, chemical potential, and inter-leg coupling strength gives rise to a rich topological phase diagram characterized by multiple Majorana zero modes and massive Dirac modes. In particular, we show that the inter-leg coupling renormalizes the effective energy scales, leading to a systematic shift of the topological phase boundaries and enabling controlled tuning of the Majorana modes. Furthermore, we identify a transition from a two Majorana zero mode phase to a phase encapsulating four Majorana zero modes, as the long-range pairing exponent is varied. This transition is accompanied by a crossover regime in which Majorana zero modes coexist with massive Dirac modes, reflecting hybridization between edge and bulk excitations. This ladder thus provides a minimal and attractive platform for realizing the impact of a long-range pairing on topological phases. Our results highlight the potential of long-range hybrid systems for engineering tunable topological states relevant for quantum information applications.

quant-ph↗
arXiv2025

Hilbert Coefficients and Regularity of Binomial Edge Ideals

Let $G$ be a simple graph on $n$ vertices, and let $J_G$ denotes the corresponding binomial edge ideal in $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, where $\mathbb{K}$ is a field. We show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal, thereby providing a reduction technique for computing Hilbert coefficients. As an application, for any $i\geq 0$ and a pair $(r,s)$ with $r\geq 2, s\in \mathbb{Z}$, we show that there always exists a graph $G$ such that $\mathrm{reg}(S/J_G)=r$ and $e_i(S/J_G)=s$, where $\mathrm{reg}(S/J_G)$ and $e_i(R/J_G)$ denote the Castelnuovo-Mumford regularity and the $i$-th Hilbert coefficient of $S/J_G$, respectively. In particular, this demonstrates that there is no inherent relationship between the regularity and the Hilbert coefficients for the class of binomial edge ideals.

math.AC↗
arXiv2025

Dynamics of Majorana zero modes across hybrid Kitaev chain

The Kitaev chain has been extensively explored in the context of uniform couplings, with studies focusing either on purely nearest-neighbor interactions or on systems dominated by long-range superconducting pairing. Building on these investigations, we introduce a hybrid Kitaev chain in which the lattice is partitioned into two segments: the left segment comprises nearest-neighbor couplings, while the right segment incorporates long-range pairing. To probe the role of the interface, we study two scenarios: a decoupled (suppressed hopping) case, where the segments are isolated, and a coupled case, where they are connected via interface hopping that enables tunneling. Using this setup, we investigate the behavior of Majorana zero modes at the interface between the two segments, finding that in the decoupled case, Majorana zero modes remain sharply localized at the left segment chain edges while massive Dirac modes remain in right segment chain edges, with their energies and localization strongly dependent on the long-range pairing exponent. Introducing a finite interface coupling enables transfer of Majorana zero modes from the edges of the left segment to those of the right segment of the chain. We characterize this dynamics by the fidelity of state transfer, dynamical rotation, and inverse participation ratio. We show the signature of Majorana zero mode transfer across the interface by the spatiotemporal profile of the probability distribution of the time evolved state.

quant-ph↗
arXiv2025

System versus charger in performance optimization of quantum batteries

Quantum batteries provide a platform for investigating energy storage and extraction in quantum many-body systems. Here, we study a charging protocol in which battery and chargoid roles are assigned to different Hamiltonian components of a standalone many-body spin system. By externally controlling the contribution of the intrinsic battery Hamiltonian during charging, we reveal a tunable competition between intrinsic and charging dynamics. We find that suppressing the intrinsic battery contribution can substantially enhance both the maximum stored energy and charging power, with the magnitude of the enhancement determined by the interaction structure and range. We further investigate the protocol in a Markovian open-system setting that incorporates energy relaxation and pure dephasing. While environmental effects generally degrade charging performance, they can instead enhance energy-storage and energy-extraction dynamics in certain interacting systems. The enhancement associated with the controlled suppression of the intrinsic battery dynamics remains robust in the presence of environmental coupling.

quant-ph↗
arXiv2025

Support-2 monomial ideals that are Simis

A monomial ideal $I\subseteq \mathbb{K}[x_1,\ldots , x_n]$ is called a Simis ideal if $I^{(s)}=I^s$ for all $s\geq 1$, where $I^{(s)}$ denotes the $s$-th symbolic power of $I$. Let $I$ be a support-2 monomial ideal such that its irreducible primary decomposition is minimal. We prove that $I$ is a Simis ideal if and only if $\sqrt{I}$ is Simis and $I$ has standard linear weights. This result thereby proves a recent conjecture for the class of support-2 monomial ideals proposed by Mendez, Pinto, and Villarreal. Furthermore, we give a complete characterization of the Cohen-Macaulay property for support-2 monomial ideals whose radical is the edge ideal of a whiskered graph. Finally, we classify when these ideals are Simis in degree 2.

math.AC↗
arXiv2025

Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs

Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R/I_3(G))$ is bounded below by $2\nu_3(G)$, where $\nu_3(G)$ denotes the $3$-path induced matching number of $G$. We give a class of graphs, namely, trees for which the lower bound is attained. Also, for a unicyclic graph $G$, we show that $reg(R/I_3(G))\leq 2\nu_3(G)+2$ and provide an example that shows that the given upper bound is sharp.

math.AC↗
arXiv2024

Study of accretion disks around black holes with two types of gas inflows

We utilized steady-state, axisymmetric, viscous hydrodynamic fluid equations around a black hole in a Schwarzschild geometry background. Here, the relativistic Schwarzschild geometry is mimicked by the Paczy{\'n}sky-Wiita potential. We investigated two types of inflowing gases that can generate different kinds of accretion flows around the central objects. The inflowing gases are presented on the local energies ($B_{ob}$) of the gases versus the outermost accretion boundary locations ($r_{ob}$) plane we named it the outermost boundary condition (OBC)-plane. Based on the energies of the inflowing gases we found two types of inflowing gas classified as cold-mode and hot-mode inflowing gases in the OBC-plane. Doing so we have found the initial temperature of the inflowing gases can be a parameter for the study of the accretion process. As it can affect the disk structure and optical depth of the accretion flow, which in turn can impact the radiative emissions observed in many accreting sources.

astro-ph.HE↗
arXiv2024

Shock waves in the magnetic reconnection in the flares on the accretion disk of the SGR~A*

Sgr~A* often shows bright, episodic flares observationally, the mechanism of the flares intermittent brightening is not very clear. Many people believe the flares may formed by the non-thermal particles, which can be a consequence of the magnetic reconnection and shock waves. In this work, we use the larger magnetic loop in the presence of pseudo-Newtonian potential which mimics general relativistic effects. The simulation results show that the reconnection of magnetic field lines passes through a current sheet, which bifurcates into two pairs of slow shocks. We also find the shock waves heat the plasma, especially when the plasma density is low. The shock wave heating effect by the magnetic reconnection is confirmed by the simulation results, and thus the process of instantaneous brightening of the flares on the accretion disk can be explained.

astro-ph.HE↗
arXiv2024

Development and Validation of Fully Automatic Deep Learning-Based Algorithms for Immunohistochemistry Reporting of Invasive Breast Ductal Carcinoma

Immunohistochemistry (IHC) analysis is a well-accepted and widely used method for molecular subtyping, a procedure for prognosis and targeted therapy of breast carcinoma, the most common type of tumor affecting women. There are four molecular biomarkers namely progesterone receptor (PR), estrogen receptor (ER), antigen Ki67, and human epidermal growth factor receptor 2 (HER2) whose assessment is needed under IHC procedure to decide prognosis as well as predictors of response to therapy. However, IHC scoring is based on subjective microscopic examination of tumor morphology and suffers from poor reproducibility, high subjectivity, and often incorrect scoring in low-score cases. In this paper, we present, a deep learning-based semi-supervised trained, fully automatic, decision support system (DSS) for IHC scoring of invasive ductal carcinoma. Our system automatically detects the tumor region removing artifacts and scores based on Allred standard. The system is developed using 3 million pathologist-annotated image patches from 300 slides, fifty thousand in-house cell annotations, and forty thousand pixels marking of HER2 membrane. We have conducted multicentric trials at four centers with three different types of digital scanners in terms of percentage agreement with doctors. And achieved agreements of 95, 92, 88 and 82 percent for Ki67, HER2, ER, and PR stain categories, respectively. In addition to overall accuracy, we found that there is 5 percent of cases where pathologist have changed their score in favor of algorithm score while reviewing with detailed algorithmic analysis. Our approach could improve the accuracy of IHC scoring and subsequent therapy decisions, particularly where specialist expertise is unavailable. Our system is highly modular. The proposed algorithm modules can be used to develop DSS for other cancer types.

eess.IV↗
arXiv2024

Multi-Stain Multi-Level Convolutional Network for Multi-Tissue Breast Cancer Image Segmentation

Digital pathology and microscopy image analysis are widely employed in the segmentation of digitally scanned IHC slides, primarily to identify cancer and pinpoint regions of interest (ROI) indicative of tumor presence. However, current ROI segmentation models are either stain-specific or suffer from the issues of stain and scanner variance due to different staining protocols or modalities across multiple labs. Also, tissues like Ductal Carcinoma in Situ (DCIS), acini, etc. are often classified as Tumors due to their structural similarities and color compositions. In this paper, we proposed a novel convolutional neural network (CNN) based Multi-class Tissue Segmentation model for histopathology whole-slide Breast slides which classify tumors and segments other tissue regions such as Ducts, acini, DCIS, Squamous epithelium, Blood Vessels, Necrosis, etc. as a separate class. Our unique pixel-aligned non-linear merge across spatial resolutions empowers models with both local and global fields of view for accurate detection of various classes. Our proposed model is also able to separate bad regions such as folds, artifacts, blurry regions, bubbles, etc. from tissue regions using multi-level context from different resolutions of WSI. Multi-phase iterative training with context-aware augmentation and increasing noise was used to efficiently train a multi-stain generic model with partial and noisy annotations from 513 slides. Our training pipeline used 12 million patches generated using context-aware augmentations which made our model stain and scanner invariant across data sources. To extrapolate stain and scanner invariance, our model was evaluated on 23000 patches which were for a completely new stain (Hematoxylin and Eosin) from a completely new scanner (Motic) from a different lab. The mean IOU was 0.72 which is on par with model performance on other data sources and scanners.

cs.CV↗
arXiv2024

Betti cones over fibre products

Let $R$ be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded $R$-modules. As an application of this, we show that the existence of an $R$-module of finite regularity and infinite projective dimension forces $R$ to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded $R$-modules, and show that when $\operatorname{depth}(R)=1$, they are spanned by the Betti tables of pure $R$-modules if and only if $R$ is Cohen-Macaulay with minimal multiplicity.

math.AC↗
arXiv2023

Powers of vertex cover ideals of Simplicial Trees

In $2011$, Herzog, Hibi, and Ohsugi conjectured that if $J$ is the cover ideal of a chordal graph, then $J^s$ is componentwise linear for all $s \ge 1.$ In 2022, H\`a and Tuyl considered objects more general than chordal graphs and posed the following problem: Let $J(\Delta)$ be the cover ideal of a simplicial tree $\Delta.$ Is it true that $J(\Delta)^s$ is componentwise linear for all $s \geq 1$? In this article, we give an affirmative answer to this problem.

math.AC↗
arXiv2023

FPCD: An Open Aerial VHR Dataset for Farm Pond Change Detection

Change detection for aerial imagery involves locating and identifying changes associated with the areas of interest between co-registered bi-temporal or multi-temporal images of a geographical location. Farm ponds are man-made structures belonging to the category of minor irrigation structures used to collect surface run-off water for future irrigation purposes. Detection of farm ponds from aerial imagery and their evolution over time helps in land surveying to analyze the agricultural shifts, policy implementation, seasonal effects and climate changes. In this paper, we introduce a publicly available object detection and instance segmentation (OD/IS) dataset for localizing farm ponds from aerial imagery. We also collected and annotated the bi-temporal data over a time-span of 14 years across 17 villages, resulting in a binary change detection dataset called \textbf{F}arm \textbf{P}ond \textbf{C}hange \textbf{D}etection Dataset (\textbf{FPCD}). We have benchmarked and analyzed the performance of various object detection and instance segmentation methods on our OD/IS dataset and the change detection methods over the FPCD dataset. The datasets are publicly accessible at this page: \textit{\url{https://huggingface.co/datasets/ctundia/FPCD}}

cs.CV↗