arXiv subjects

Malik Amir

Publications and source records attributed to Malik Amir.

8 recordsLinked to original sources

arXiv2026

The K-moment problem: A detailed introduction

We present an expanded expository account of the $K$-moment problem for polynomial algebras over $\R^d$, with special emphasis on compact basic closed semialgebraic sets. The central question is to characterize those linear functionals on $\R[x_1,\dots,x_d]$ which admit representation by integration against a positive Radon measure supported on a prescribed set $K\subseteq\R^d$. We begin with the classical background and with Haviland's formulation of the multidimensional moment problem, then explain how real algebraic geometry enters through quadratic modules, preorderings, and Positivstellens\"atze. The compact case is treated in detail from two complementary perspectives. The geometric route through Schm\"udgen's theorem and the operator-theoretic route through a Gelfand--Naimark--Segal construction and the spectral theorem. We also discuss Putinar's refinement, compare the roles of $T(f)$ and $Q(f)$, and explain how Archimedeanity provides the algebraic shadow of compactness. In order to place the subject in a broader context, we survey determinacy and uniqueness questions, the truncated $K$-moment problem and flat extension phenomena, the relation with sums of squares and Hilbert's seventeenth problem, and the special case of algebraic varieties, where positivity modulo an ideal becomes especially transparent.

math.FA↗

arXiv2026

On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry

Classical Fisher-information asymptotics describe the covariance of regular efficient estimators through the local quadratic approximation of the log-likelihood, and thus capture first-order geometry only. In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-constrained parameter spaces, finite-sample behavior can deviate systematically from these predictions. We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a Riemannian manifold $(\Theta,g)$ with Fisher--Rao metric, immersed in $L^2(\mu)$ through the square-root density map. Under suitable regularity and moment assumptions, we derive an $n^{-2}$ correction to the leading $n^{-1}I(\theta)^{-1}$ covariance term for score-root, first-order efficient estimators. The correction is governed by a tensor $P_{ij}$ that decomposes canonically into three parts, an intrinsic Ricci-type contraction of the Fisher--Rao curvature tensor, an extrinsic Gram-type contraction of the second fundamental form, and a Hellinger discrepancy tensor encoding higher-order probabilistic information not determined by immersion geometry alone. The extrinsic term is positive semidefinite, the full correction is invariant under smooth reparameterization, and it vanishes identically for full exponential families. We then extend the picture to singular models, where Fisher information degenerates. Using resolution of singularities under an additive normal crossing assumption, we describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case. This framework also suggests geometric diagnostics of weak identifiability and curvature-aware principles for regularization and optimization.

math.ST↗

arXiv2023

Enabling Quantum Cybersecurity Analytics in Botnet Detection: Stable Architecture and Speed-up through Tree Algorithms

For the first time, we enable the execution of hybrid machine learning methods on real quantum computers with 100 data samples and real-device-based simulations with 5,000 data samples, thereby outperforming the current state of research of Suryotrisongko and Musashi from 2022 who were dealing with 1,000 data samples and quantum simulators (pure software-based emulators) only. Additionally, we beat their reported accuracy of $76.8\%$ by an average accuracy of $91.2\%$, all within a total execution time of 1,687 seconds. We achieve this significant progress through two-step strategy: Firstly, we establish a stable quantum architecture that enables us to execute HQML algorithms on real quantum devices. Secondly, we introduce new hybrid quantum binary classification algorithms based on Hoeffding decision tree algorithms. These algorithms speed up the process via batch-wise execution, reducing the number of shots required on real quantum devices compared to conventional loop-based optimizers. Their incremental nature serves the purpose of online large-scale data streaming for DGA botnet detection, and allows us to apply hybrid quantum machine learning to the field of cybersecurity analytics. We conduct our experiments using the Qiskit library with the Aer quantum simulator, and on three different real quantum devices from Azure Quantum: IonQ, Rigetti, and Quantinuum. This is the first time these tools are combined in this manner.

quant-ph↗

arXiv2022

Machine Learning Class Numbers of Real Quadratic Fields

We implement and interpret various supervised learning experiments involving real quadratic fields with class numbers 1, 2 and 3. We quantify the relative difficulties in separating class numbers of matching/different parity from a data-scientific perspective, apply the methodology of feature analysis and principal component analysis, and use symbolic classification to develop machine-learned formulas for class numbers 1, 2 and 3 that apply to our dataset.

math.NT↗

arXiv2021

A short note on inadmissible coefficients of weight $2$ and $2k+1$ newforms

Let $f(z)=q+\sum_{n\geq 2}a(n)q^n$ be a weight $k$ normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in \cite{AH} for $k=2$ by ruling out or locating all odd prime values $|\ell|<100$ of their Fourier coefficients $a(n)$ when $n$ satisfies some congruences. We also study the case of odd weights $k\geq 1$ newforms where the nebentypus is given by a real quadratic Dirichlet character.

math.NT↗

arXiv2020

On $L$-Functions of Modular Elliptic Curves and Certain $K3$ Surfaces

Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $τ$-function, one may ask whether an odd integer $α$ can be equal to $τ(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $k\geq 4$. We use these methods for weight $2$ and $3$ newforms and apply our results to $L$-functions of modular elliptic curves and certain $K3$ surfaces with Picard number $\ge 19$. In particular, for the complete list of weight $3$ newforms $f_λ(z)=\sum a_λ(n)q^n$ that are $η$-products, and for $N_λ$ the conductor of some elliptic curve $E_λ$, we show that if $|a_λ(n)|<100$ is odd with $n>1$ and $(n,2N_λ)=1$, then \begin{align*} a_λ(n) \in \,& \{-5,9,\pm 11,25, \pm41, \pm 43, -45,\pm47,49, \pm53,55, \pm59, \pm61, \pm 67\}\\ & \,\,\, \cup \, \{-69,\pm 71, \pm 73,75, \pm79,\pm81, \pm 83, \pm89,\pm 93 \pm 97, 99\}. \end{align*} Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving \begin{align*} a_λ(n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{align*}

math.NT↗

arXiv2019

On Traces of Singular Moduli

This article is an overview of Zagier's and Kim's work on traces of singular moduli. We give more detailed or new proofs to some of their results and also describe some algorithms to compute spaces of Jacobi forms and weight $3/2$ modular forms. Also, for $p$ a prime dividing the order of the Monster group, we give an explicit construction of the Hauptmodul $j_p$ of $Γ_0(p)^*$ using Rademacher series and Rademacher sums.

math.NT↗

arXiv2018

Trajectories of two identical particles on a plane in a constant magnetic field subject to non-Coulomb potentials

Study of the classical motion of two identical particles on a plane subject to non-Coulomb potentials in a constant magnetic field presented in polar coordinates. With the rigorous analysis of the potentials and the constants of motion, we are able to give necessary and sufficient conditions for the existence of bounded and periodic orbits. We also show and analyze numerical solutions of Newton equations of motion.

math-ph↗