10 citations · 15 across the 12 of their papers we have counts for
15 papers
Graduate Mathematics in the Age of AI: Forming Mathematicians for Original, Independent, and Responsible Inquiry
Bacim Alali
Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticat…
Quantum uncertainty in a macroscopic domain
Othman Q. Malhas, Bacim Alali
We develop a classical model-theoretic representation of partial Boolean algebras and use it to formulate quantum-like uncertainty without abandoning classical propositional logic.…
Discontinuity Analysis and Semi-Analytic Spectral Approximation for the Nonlocal Poisson Equation
Thinh Dang, Bacim Alali, Nathan Albin
We study a nonlocal Poisson problem with discontinuous source term and analyze how the regularity of the integral kernel determines the discontinuity structure of the corresponding…
Fourier Spectral Method for Nonlocal Equations on Bounded Domains
Ilyas Mustapha, Bacim Alali, Nathan Albin
This work introduces efficient and accurate spectral solvers for nonlocal equations on bounded domains. These spectral solvers exploit the fact that integration in the nonlocal for…
Regularity of Solutions for Peridynamics Equilibrium and Evolution Equations on Periodic Distributions
Thinh Dang, Bacim Alali, Nathan Albin
Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator c…
Regularity of Solutions for the Nonlocal Wave Equation on Periodic Distributions
Thinh Dang, Bacim Alali, Nathan Albin
This work addresses the regularity of solutions for a nonlocal wave equation over the space of periodic distributions. The spatial operator for the nonlocal wave equation is given…
Asymptotic Analysis for the Eigenvalues of Peridynamic Operators
Bacim Alali, Nathan Albin, Thinh Dang
Explicit representations of the eigenvalues of the peridynamic operator have been recently derived in [5]. These representations are given in terms of generalized hypergeometric fu…
Regularity of Solutions for the Nonlocal Diffusion Equation on Periodic Distributions
Ilyas Mustapha, Bacim Alali, Nathan Albin
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation…
Linear peridynamics Fourier multipliers and eigenvalues
Bacim Alali, Nathan Albin
A characterization for the Fourier multipliers and eigenvalues of linear peridynamic operators is provided. The analysis is presented for state-based peridynamic operators for isot…
Fourier spectral methods for nonlocal models
Bacim Alali, Nathan Albin
Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace o…
Fourier multipliers for nonlocal Laplace operators
Bacim Alali, Nathan Albin
Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in . The Fourier multipliers are given t…
Peridynamics and Material Interfaces
Bacim Alali, Max Gunzburger
The convergence of a peridynamic model for solid mechanics inside heterogeneous media in the limit of vanishing nonlocality is analyzed. It is shown that the operator of linear per…
A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics
Bacim Alali, Kuo Liu, Max Gunzburger
A nonlocal vector calculus was introduced in [2] that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A generaliza…
Effective Conductivities of Thin-Interphase Composites
Bacim Alali, Graeme W. Milton
A method is presented for approximating the effective conductivity of composite media with thin interphase regions, which is exact to first order in the interphase thickness. The a…
Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation
Bacim Alali, Robert Lipton
A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented her…