Searcharxiv⌕ Search

arXiv subjects

Manohar Kumar

Publications and source records attributed to Manohar Kumar.

At least 19 recordsLinked to original sources
arXiv2026

Regularity of symbolic powers of complementary edge ideals

Let \(G\) be a finite simple graph on \(n\) vertices, and let \(I_c(G)\) be its complementary edge ideal. We determine \(\reg(I_c(G)^{(t)})\) for every \(t\geq1\) in terms of the number \(c(G)\) of nontrivial connected components of \(G\) and the presence of a \(K_2\)-component. Consequently, \(\reg(I_c(G)^{(t)})\leq \reg(I_c(G)^t) \) for all \(t\geq1\), with equality for every \(t\) if and only if either \(c(G)=1\), or \(c(G)=2\) and \(G\) has a \(K_2\)-component. We also characterize Serre's condition \((S_2)\) for \(R/I_c(G)\), and classify the graphs for which \(R/I_c(G)^{(t)}\) is Cohen-Macaulay for every \(t\geq1\).

math.AC↗
arXiv2026

Homological shift ideals of weighted oriented graphs

In this paper, we study the homological shift ideals of edge ideals associated with weighted oriented graphs. For a weighted oriented graph $D$, let $HS_k(I(D))$ denote the $k^{th}$ homological shift ideal of its edge ideal $I(D)$. If $D$ is vertex-splittable, then we characterize that $HS_1(I(D))$ has linear quotients if and only if $D_6$, $D_7$, and $D_8$ are not induced subgraphs of $D$. Furthermore, we show that if $I(D)$ has linear quotients, then $\sqrt{HS_k(I(D))} = HS_k(I(G))$, for all $k\geq 1$, where $G$ is the underlying simple graph of $D$. We show that if $I(D)$ has homological linear quotients, then $I(G)$ also has homological linear quotients. If $D$ is a tree, then we establish the following characterization: \begin{align*} HS_k(I(D)) \text{ has linear quotients for all } k\geq 0 \iff ~ &G~ \text{is} \text{ a star graph or a broom graph} \\ &\text{ and }~ D \text{ is $D_i$-free, for } i=1,2,5,6,8. \end{align*}

math.AC↗
arXiv2026

$\mathrm{v}$-number of Lov\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs

In this paper, we introduce a new framework for computing certain localized $\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lov\'asz-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\mathrm{v}$-number of the LSS ideal $L_G^{\mathbb{K}}(d)$, denoted by $\mathrm{v}_{\mathfrak{p}_{\emptyset}(G)}(L_G^{\mathbb{K}}(d))$, for all $d \geq 2$, where $\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that $\mathrm{v}(L_G^{\mathbb{K}}(d)) \leq \mathrm{reg}(R/L_G^{\mathbb{K}}(d)),$ where $\mathrm{v}(L_G^{\mathbb{K}}(d))$ and $\mathrm{reg}(R/L_G^{\mathbb{K}}(d))$ denote the $\mathrm{v}$-number of $L_G^{\mathbb{K}}(d)$ and the Castelnuovo-Mumford regularity of $R/L_G^{\mathbb{K}}(d)$, respectively. Also, we give an upper bound for $\mathrm{v}_{I_{K_{n}}}(L_{G}^{\mathbb{R}}(2))$, where $\mathbb{R}$ is field of real numbers. Furthermore, we provide combinatorial descriptions of the localized $\mathrm{v}$-number of parity binomial edge ideals, denoted by $\mathrm{v}_{\mathfrak{p}^{+}(G)}(\mathcal{I}_{G})$, and, as an application, show that $\mathrm{v}(\mathcal{I}_G) \leq \mathrm{reg}(R/\mathcal{I}_G)$ for several classes of non-bipartite graphs. Finally, we prove that $\mathrm{v}(J_G^k)\leq \mathrm{reg}(R/J_G^k)$ for all powers of binomial edge ideals of closed graphs $G$.

math.AC↗
arXiv2026

Detecting crossed Andreev reflection in a quantum Hall interferometer with a superconducting beam splitter

We study time-domain electron interferometry in a Hong-Ou-Mandel (HOM) geometry, where a thin superconductor between two quantum Hall systems acts as the beam splitter. By comparing the measurable current cross correlations at the interferometer outputs with those of a normal-conducting electronic HOM setup, we show that Andreev processes strongly affect the HOM dip. Using a combination of scattering theory and numerical tight-binding simulations for a graphene quantum Hall bar, we show that the change of charge cross correlations can be used to experimentally detect and characterize local and crossed Andreev processes.

cond-mat.mes-hall↗
arXiv2026

Landau-Level-Resolved Mode Mixing and Shot Noise in Gate-Defined Graphene Quantum Point Contacts

Graphene quantum point contacts (QPCs) in the quantum Hall regime host competing transport mechanisms including chiral edge propagation, valley degeneracy, and gate-induced mode mixing. Their interplay is not visible in conductance alone. Shot noise directly probes the statistics of transmission eigenvalues, revealing microscopic mode partitioning that conductance cannot access. We develop a hybrid framework combining tight-binding simulations of gate-defined graphene QPCs with random matrix theory (RMT) to predict shot noise and Fano factor signatures across different quantum Hall regimes, validated against experimental conductance maps of hBN-encapsulated graphene Hall bars. Three distinct regimes are identified: adiabatic propagation, sharp mode filtering, and multi-mode mixing driven by localized states beneath the split gate. For higher Landau levels ($N_L > 0$), complete mode mixing produces the universal chaotic-cavity limit $F \simeq 1/4$. Strikingly, the zeroth Landau level ($N_L = 0$) converges to $F = 1/3$. This distinct value originates in the sublattice polarization of the $N_L = 0$ edge state: coupling to mixed-sublattice localized states beneath the gate is suppressed, confining transport to an effective single channel ($N = 1$). Complete mixing within this single channel yields a flat transmission eigenvalue distribution and hence exactly $F = 1/3$ from single-channel RMT, numerically coincident with but mechanistically distinct from pseudo-diffusive zero-field graphene transport. The $F = 1/3$ versus $F = 1/4$ crossover is a Landau-level-resolved noise signature absent in conductance, providing a direct discriminator between single-channel and multi-channel chaotic transport in graphene QPCs.

cond-mat.mes-hall↗
arXiv2026

Expected Moral Shortfall for Ethical Competence in Decision-making Models

Moral cognition is a crucial yet underexplored aspect of decision-making in AI models. Regardless of the application domain, it should be a consideration that allows for ethically aligned decision-making. This paper presents a multifaceted contribution to this research space. Firstly, a comparative analysis of techniques to instill ethical competence into AI models has been presented to gauge them on multiple performance metrics. Second, a novel mathematical discretization of morality and a demonstration of its real-life application have been conveyed and tested against other techniques on two datasets. This value is modeled as the risk of loss incurred by the least moral cases, or an Expected Moral Shortfall (EMS), which we direct the AI model to minimize in order to maximize its performance while retaining ethical competence. Lastly, the paper discusses the tradeoff between preliminary AI decision-making metrics such as model performance, complexity, and scale of ethical competence to recognize the true extent of practical social impact.

cs.CY↗
arXiv2025

Componentwise linearity of powers of edge ideals of weighted oriented graphs

In this paper, we study the componentwise linearity of powers of edge ideal of a weighted oriented graph $D$. We give a characterization for componentwise linearity of the edge ideal $I(D)$ in terms of forbidden subgraphs of $D$. If $D$ is house-free or complete $r$-partite, then the following statements are equivalent: (1) $I(D)$ is componentwise linear; (2) $I(D)$ is vertex splittable; (3) $I(D)$ has linear quotient property; (4) both $G$ and $H(I(D)_{(2)})$ are co-chordal and $D_1,D_2,D_3,D_4$ as in Figure 3, are not induced subgraphs of $D$. Furthermore, if $D$ is a complete $r$-partite weighted oriented graph, then we show that: $I(D)^k$ is componentwise linear, for some $k\geq 2 \iff I(D)$ is componentwise linear.

math.AC↗
arXiv2025

ApplE: A Modular Ontology of Applied Ethics and Event Context for Ethical Decision Modeling

Applied ethics applies ethical decision-making to domain-specific contexts using contextual information such as agents, actions, temporal and spatial settings, and theoretical constructs such as utility, virtues, rights, and duties. However, representing an ethical decision is challenging as it may be abstract, context-sensitive, and semantically heterogeneous. Nevertheless, important ethical and contextual factors can be formally modeled to support structured ethical reasoning. Knowledge representation and reasoning provide a mechanism to translate abstract ethical concepts into machine-interpretable conceptual structures in the context of an event. To achieve this, we propose ApplE, an Applied Ethics ontology that models ethical theory and event context within a unified and modular conceptual framework for ethical decision-making. The ontology was developed using a modified version of the Simplified Agile Methodology for Ontology Development (SAMOD), which facilitates iterative refinement of classes and relationships, as well as the participation of a domain expert. The modular development of ApplE combines Ethics Theory with Event Context to capture semantic relationships between ethical principles, agents, actions, consequences, intentions, and domains. Using ApplE, we modeled a use case from the medical domain to demonstrate the ontology's representational expressivity and reasoning capabilities. In addition to ontological reasoning and consistency checks, ApplE is also evaluated using the three-fold testing process of SAMOD. ApplE follows the FAIR principles and is positioned to be used as a reusable semantic and conceptual modeling resource for ethical AI systems and ontology-driven applications.

cs.CY↗
arXiv2024

Half-quantized Hall Plateaus in the Confined Geometry of Graphene

Since the ground-breaking discovery of the quantum Hall effect, half-quantized quantum Hall plateaus have been some of the most studied and sought-after states. Their importance stems not only from the fact that they transcend the composite fermion framework used to explain fractional quantum Hall states (such as Laughlin states). Crucially, they hold promise for hosting non-Abelian excitations, which are essential for developing topological qubits - key components for fault-tolerant quantum computing. In this work, we show that these coveted half-quantized plateaus can appear in more than one unexpected way. We report the observation of fractional states with conductance quantization at $\nu_H = 5/2$ arising due to charge equilibration in the confined region of a quantum point contact in monolayer graphene.

cond-mat.mes-hall↗
arXiv2024

The slope of v-function and Waldschmidt constant

In this paper, we study the asymptotic behaviour of the v-number of a Noetherian graded filtration $\mathcal{I}= \{I_{[k]}\}_{k\geq 0}$ of a Noetherian $\mathbb{N}$-graded domain $R$. Recently, it is shown that $\mathrm{v}(I_{[k]})$ is periodically linear in $k$ for $k \gg 0$. We show that all these linear functions have the same slope, i.e. $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I_{[k]})}{k}$ exists, which is equal to $\displaystyle \lim_{k \rightarrow \infty}\frac{\alpha(I_{[k]})}{k}$, where $\alpha(I)$ denotes the minimum degree of a non-zero element in $I$. In particular, for any Noetherian symbolic filtration $\mathcal{I}= \{I^{(k)}\}_{k\geq 0}$ of $R$, it follows that $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I^{(k)})}{k}=\hat{\alpha}(I)$, the Waldschmidt constant of $I$. Next, for a non-equigenerated square-free monomial ideal $I$, we prove that $\mathrm{v}(I^{(k)}) \leq \mathrm{reg}(R/I^{(k)})$ for $k\gg 0$. Also, for an ideal $I$ having the symbolic strong persistence property, we give a linear upper bound on $\mathrm{v}(I^{(k)})$. As an application, we derive some criteria for a square-free monomial ideal $I$ to satisfy $\mathrm{v}(I^{(k)})\leq \mathrm{reg}(R/I^{(k)})$ for all $k\geq 1$, and provide several examples in support. In addition, for any simple graph $G$, we establish that $\mathrm{v}(J(G)^{(k)}) \leq \mathrm{reg}(R/J(G)^{(k)})$ for all $k \geq 1$, and $\mathrm{v}(J(G)^{(k)}) = \mathrm{reg}(R/J(G)^{(k)})=\alpha(J(G)^{(k)})-1$ for all $k\geq 1$ if and only if $G$ is a Cohen-Macaulay very-well covered graph, where $J(G)$ is the cover ideal of $G$.

math.AC↗
arXiv2023

Componentwise linearity of edge ideals of weighted oriented graphs

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is co-chordal. This is an analogue of Fr\"oberg's theorem for weighted oriented graphs. We give combinatorial characterizations of componentwise linearity of $I(D)$ if $V^+$ are sinks or $\vert V^+ \vert\leq 1$. Furthermore, if $G$ is chordal or bipartite or $V^+$ are sinks or $\vert V^+ \vert\leq 1$, then we show the following equivalence for $I(D)$: $$ \text{Vertex splittable}\,\, \Longleftrightarrow\,\, \text{Linear quotient}\,\, \Longleftrightarrow\,\, \text{Componentwise linear}.$$

math.AC↗
arXiv2023

Regularity of symbolic and ordinary powers of weighted oriented graphs and their upper bounds

In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For any weighted oriented complete graph $K_n$, we show that $\reg(I(K_n)^{(k)})\leq \reg(I(K_n)^k)$ for all $k\geq 1$. Also, we give explicit formulas for $\reg(I(K_n)^{(k)})$ and $\reg(I(K_n)^{k})$, for any $k\geq 1$. As a consequence, we show that $\reg(I(K_n)^{(k)})$ is eventually a linear function of $k$. For any weighted oriented graph $D$, if $V^+$ are sink vertices, then we show that $\reg(I(D)^{(k)}) \leq \reg(I(D)^k)$ with $k=2,3$ and equality cases studied. Furthermore, we give formula for $\reg(I(D)^2)$ in terms of $\reg(I(D)^{(2)})$ and regularity of certain induced subgraphs of $D$. Finally, we compare the regularity of symbolic powers of weighted oriented graphs $D$ and $D'$, where $D'$ is obtained from $D$ by adding a pendant.

math.AC↗
arXiv2021

Strong magnetoresistance in a graphene Corbino disk at low magnetic fields

We have measured magnetoresistance of suspended graphene in the Corbino geometry at magnetic fields up to $B=0.15\,$T, i.e., in a regime uninfluenced by Shubnikov-de Haas oscillations. The low-temperature relative magnetotoresistance $[R(B)-R(0)]/R(0)$ amounts to $4000 B^2\% $ at the Dirac point ($B$ in Tesla), with a quite weak temperature dependence below $30\,$K. A decrease in the relative magnetoresistance by a factor of two is found when charge carrier density is increased to $|n| \simeq 3 \times 10^{-10}$ cm$^{-2}$. The gate dependence of the magnetoresistance allows us to characterize the role of scattering on long-range (Coulomb impurities, ripples) and short-range potential, as well as to separate the bulk resistance from the contact one. Furthermore, we find a shift in the position of the charge neutrality point with increasing magnetic field, which suggests that magnetic field changes the screening of Coulomb impurities around the Dirac point. The current noise of our device amounts to $10^{-23}$ A$^2$/$\sqrt{\textrm{Hz}}$ at $1\,$kHz at $4\,$K, which corresponds to a magnetic field sensitivity of $60$ nT/$\sqrt{\textrm{Hz}}$ in a background field of $0.15\,$T.

cond-mat.mes-hall↗
arXiv2018

Defects in h-BN tunnel barrier for local electrostatic probing of two dimensional materials

Defects in hexagonal boron nitride (h-BN) layer can facilitate tunneling current through thick h-BN tunneling barriers. We have investigated such current-mediating defects as local probes for materials in two dimensional heterostructure stacks. Besides $IV$ characteristics and negative differential conductance, we have characterized the electrical properties of h-BN defects in vertical graphene-h-BN-Cr/Au tunnel junctions in terms of low frequency current noise. Our results indicate a charge sensitivity of 1.5$\times$$ 10^-5$e/$\sqrt Hz$ at 10 $Hz$, which is equal to good metallic single electron transistors. The noise spectra at low frequency are governed by a few two-level fluctuators. For variations in electrochemical potential, we achieve a sensitivity of 0.8$\mu$eV/$\sqrt Hz$.

cond-mat.mes-hall↗
arXiv2018

Breakdown of zero-energy quantum Hall state in graphene in the light of current fluctuations and shot noise

We have investigated the cross-over from Zener tunneling of single charge carriers to avalanche type of bunched electron transport in a suspended graphene Corbino disk in the zeroth Landau level. At low bias, we find a tunneling current that follows the gyrotropic Zener tunneling behavior. At larger bias, we find avalanche type of transport that sets in at a smaller current the larger the magnetic field is. The low-frequency noise indicates strong bunching of the electrons in the avalanches. On the basis of the measured low-frequency switching noise power, we deduce the characteristic switching rates of the avalanche sequence. The simultaneous microwave shot noise measurement also reveals intrinsic correlations within the avalanche pulses and indicate decrease of correlations with increasing bias.

cond-mat.mes-hall↗
arXiv2017

Gyrotropic Zener tunneling and nonlinear IV curves in the zero-energy Landau level of graphene in a strong magnetic field

We have investigated tunneling current through a suspended graphene Corbino disk in high magnetic fields at the Dirac point, i.e. at filling factor $\nu$ = 0. At the onset of the dielectric breakdown the current through the disk grows exponentially before ohmic behaviour, but in a manner distinct from thermal activation. We find that Zener tunneling between Landau sublevels dominates, facilitated by tilting of the source-drain bias potential. According to our analytic modelling, the Zener tunneling is strongly affected by the gyrotropic force (Lorentz force) due to the high magnetic field

cond-mat.mes-hall↗
arXiv2017

Fast and accurate shot noise measurements on atomic-size junctions in the MHz regime

Shot noise measurements on atomic and molecular junctions provide rich information about the quantum transport properties of the junctions and on the inelastic scattering events taking place in the process. Dissipation at the nanoscale, a problem of central interest in nano-electronics, can be studied in its most explicit and simplified form. Here, we describe a measurement technique that permits extending previous noise measurements to a much higher frequency range, and to much higher bias voltage range, while maintaining a high accuracy in noise and conductance. We also demonstrate the advantages of having access to the spectral information for diagnostics.

cond-mat.mes-hall↗