ArgumentError: Differential w.r.t. multiple variables Any[0.030,....] are not allowed.

[DanielePessina]

Describe the bug 🐞

I am trying to model a chromatography column using ModelingToolkit and MethodOfLines, and I can't get past this error message.

From my testing the problematic term is - (1 - ε_column)*(3/rp)*Deff*dz(cₚ(t, z, rp)) - commenting it out discretizes and solves the model. To be clear this should be the the partial derivative evaluated at rp, I'm not sure if this is how it should be coded.

I don't understand how to fix it - previous reports of the same error were due to mixed derivatives, but I don't think I have them do I?

To be clear: the numbers in the Any[...] vector belong to the z domain I think, changing zL changes them too.

I'd be very grateful for some help!

Minimal Reproducible Example 👇

@parameters t, z, r
@variables c(..), cₚ(..)

zL = 0.1 # [m] - length of the column
dp = 85e-6 # [m] diameter of resin particle
rp = dp/2 # [m] radius of resin particle
ε_column = 0.36 # [-] void fraction of column
ε_particle = 0.88 # [-] void fraction of resin particle
kfit = 0.083 # [-] fitting parameter taking resin particle pore diameter, antibody molecule diameter and pore tortuosity into account
qmax = 5.47 # [mol m-3] maximum binding capacity of affinity resin
kd = 7.7e-4 # [mol m-3] equilibrium desorption constant

Dm = 5.2154833353911946e-11 # [m2 s-1]  Molecular diffusivity
Dc = 0.008 # [m]       Internal diameter of the column
Lc = 0.1 # [m]       Length of the column
Ac = pi * (Dc/2)^2 # [m2] Cross sectional area of the column
M = 145000 # [g mol-1] Molecular weight of mAb
cin = 2.27586207e-02 # [mol m-3]
Qin = 1.05000000e-08 # [m3 s-1] inlet volumetric flowrate RT:  8  0.63 mL/min


u = Qin/(Ac*ε_column) # [m s-1] Liquid interstitial velocity
Re = 997*dp*u/0.001002 # [-] Reynold's number of the flow (using water dynamic viscosity and density)
Dax = dp*u*ε_column/(0.339 + 0.033*Re^0.48) # [m2 s-1] Axial dispersion coefficient
Deff = kfit*ε_particle*Dm # [m2 s-1] Effective pore diffusion coefficient
k_film = 3/(rp) # [m-1] film mass transfer coefficient

dt = Differential(t)
dz = Differential(z)
dr = Differential(r)

dz_sq = Differential(z)^2
dr_sq = Differential(r)^2


eqs = [

    ε_column*dt(c(t, z)) ~ -u*ε_column* dz(c(t, z)) + ε_column*Dax*dz_sq(c(t, z)) - (1 - ε_column)*(3/rp)*Deff*dz(cₚ(t, z, rp)), #### this last term is the problem
    ε_particle*dt(cₚ(t, z, r)) ~ Deff*(dr_sq(cₚ(t, z, r)) + 2/(r)*dr(cₚ(t, z, r)))- (1 - ε_particle)*kd*qmax*c(t, z)/(kd + c(t, z))^2,
]

bcs = [
    # boundary conditions
    c(t,0) ~ cin + (Dax/u) * dz(c(t,0)),
    dz(c(t,zL)) ~ 0,

    cₚ(t, z, rp) ~ c(t, z),
    dr(cₚ(t,z, 0)) ~ 0,

    # Initial conditions
    c(0, z) ~ 0,
    cₚ(0, z, r) ~ 0,
]

domain = [z ∈ (0.0, zL), r ∈ (0.0, rp), t ∈ (0.0, 400)]

@named HPLC = PDESystem(eqs, bcs, domain, [t, z, r], [c(t, z), cₚ(t, z, r)])

# prob = discretize(HPLC, MOLFiniteDifference([z=>40, r=>40], t), advection_scheme = WENOScheme())
prob = discretize(HPLC, MOLFiniteDifference([z=>40, r=>40], t), advection_scheme = UpwindScheme())

sol = solve(prob, Tsit5(), saveat = 1)

Error & Stacktrace ⚠️

The system of equations is:

Discretization failed, please post an issue on https://github.com/SciML/MethodOfLines.jl with the failing code and system at low point count.

ERROR: ArgumentError: Differential w.r.t. multiple variables Any[0.03076923076923077, 0.04358974358974359, 0.09487179487179487, 0.06923076923076923, 0.007692307692307693, 0.04871794871794872, 0.09743589743589744, 0.03333333333333333, 0.046153846153846156, 0.0641025641025641, 0.041025641025641026, 0.09230769230769231, 0.05641025641025641, 0.015384615384615385, t, 0.038461538461538464, 0.01282051282051282, 0.020512820512820513, 0.05897435897435897, 0.05384615384615385, 0.08461538461538462, 0.07692307692307693, 0.07948717948717948, 0.08974358974358974, 0.028205128205128206, 0.07179487179487179, 0.035897435897435895, 0.010256410256410256, 0.017948717948717947, 0.02564102564102564, 0.07435897435897436, 0.002564102564102564, 0.06153846153846154, 0.08205128205128205, 0.023076923076923078, 0.05128205128205128, 0.005128205128205128, 0.06666666666666667, 0.08717948717948718] are not allowed.
Stacktrace:
 [1] check_equations(eqs::Vector{Equation}, iv::SymbolicUtils.BasicSymbolic{Real})
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/3AmNp/src/utils.jl:197
 [2] ODESystem(tag::UInt64, deqs::Vector{…}, iv::SymbolicUtils.BasicSymbolic{…}, dvs::Vector{…}, ps::Vector{…}, tspan::Nothing, var_to_name::Dict{…}, ctrls::Vector{…}, observed::Vector{…}, tgrad::Base.RefValue{…}, jac::Base.RefValue{…}, ctrl_jac::Base.RefValue{…}, Wfact::Base.RefValue{…}, Wfact_t::Base.RefValue{…}, name::Symbol, systems::Vector{…}, defaults::Dict{…}, guesses::Dict{…}, torn_matching::Nothing, initializesystem::Nothing, initialization_eqs::Vector{…}, schedule::Nothing, connector_type::Nothing, preface::Nothing, cevents::Vector{…}, devents::Vector{…}, parameter_dependencies::Nothing, metadata::MethodOfLines.MOLMetadata{…}, gui_metadata::Nothing, tearing_state::Nothing, substitutions::Nothing, complete::Bool, index_cache::Nothing, discrete_subsystems::Nothing, solved_unknowns::Nothing, split_idxs::Nothing, parent::Nothing; checks::Bool)
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/3AmNp/src/systems/diffeqs/odesystem.jl:190
 [3] ODESystem
   @ ~/.julia/packages/ModelingToolkit/3AmNp/src/systems/diffeqs/odesystem.jl:176 [inlined]
 [4] ODESystem(deqs::Vector{…}, iv::Num, dvs::Vector{…}, ps::Vector{…}; controls::Vector{…}, observed::Vector{…}, systems::Vector{…}, tspan::Nothing, name::Symbol, default_u0::Dict{…}, default_p::Dict{…}, defaults::Dict{…}, guesses::Dict{…}, initializesystem::Nothing, initialization_eqs::Vector{…}, schedule::Nothing, connector_type::Nothing, preface::Nothing, continuous_events::Nothing, discrete_events::Nothing, parameter_dependencies::Nothing, checks::Bool, metadata::MethodOfLines.MOLMetadata{…}, gui_metadata::Nothing)
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/3AmNp/src/systems/diffeqs/odesystem.jl:269
 [5] generate_system(disc_state::PDEBase.EquationState, s::MethodOfLines.DiscreteSpace{…}, u0::Vector{…}, tspan::Tuple{…}, metadata::MethodOfLines.MOLMetadata{…}, disc::MOLFiniteDifference{…})
   @ PDEBase ~/.julia/packages/PDEBase/nzap9/src/discretization_state.jl:41
 [6] symbolic_discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…})
   @ PDEBase ~/.julia/packages/PDEBase/nzap9/src/symbolic_discretize.jl:89
 [7] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…}; analytic::Nothing, kwargs::@Kwargs{…})
   @ PDEBase ~/.julia/packages/PDEBase/nzap9/src/discretization_state.jl:58
 [8] top-level scope
   @ ~/ModToolkit/dev_HPLCmodtoolkit.jl:130
Some type information was truncated. Use `show(err)` to see complete types.


LoadError: ArgumentError: Differential w.r.t. multiple variables Any[0.03076923076923077, 0.04358974358974359, 0.09487179487179487, 0.06923076923076923, 0.007692307692307693, 0.04871794871794872, 0.09743589743589744, 0.03333333333333333, 0.046153846153846156, 0.0641025641025641, 0.041025641025641026, 0.09230769230769231, 0.05641025641025641, 0.015384615384615385, t, 0.038461538461538464, 0.01282051282051282, 0.020512820512820513, 0.05897435897435897, 0.05384615384615385, 0.08461538461538462, 0.07692307692307693, 0.07948717948717948, 0.08974358974358974, 0.028205128205128206, 0.07179487179487179, 0.035897435897435895, 0.010256410256410256, 0.017948717948717947, 0.02564102564102564, 0.07435897435897436, 0.002564102564102564, 0.06153846153846154, 0.08205128205128205, 0.023076923076923078, 0.05128205128205128, 0.005128205128205128, 0.06666666666666667, 0.08717948717948718] are not allowed.
Stacktrace:
  [1] check_equations(eqs::Vector{Equation}, iv::SymbolicUtils.BasicSymbolic{Real})
    @ ModelingToolkit ~/.julia/packages/ModelingToolkit/3AmNp/src/utils.jl:197
  [2] ODESystem(tag::UInt64, deqs::Vector{Equation}, iv::SymbolicUtils.BasicSymbolic{Real}, dvs::Vector{SymbolicUtils.BasicSymbolic{Real}}, ps::Vector{Any}, tspan::Nothing, var_to_name::Dict{Any, Any}, ctrls::Vector{Any}, observed::Vector{Equation}, tgrad::Base.RefValue{Vector{Num}}, jac::Base.RefValue{Any}, ctrl_jac::Base.RefValue{Any}, Wfact::Base.RefValue{Matrix{Num}}, Wfact_t::Base.RefValue{Matrix{Num}}, name::Symbol, systems::Vector{ODESystem}, defaults::Dict{Any, Any}, guesses::Dict{Any, Any}, torn_matching::Nothing, initializesystem::Nothing, initialization_eqs::Vector{Equation}, schedule::Nothing, connector_type::Nothing, preface::Nothing, cevents::Vector{ModelingToolkit.SymbolicContinuousCallback}, devents::Vector{ModelingToolkit.SymbolicDiscreteCallback}, parameter_dependencies::Nothing, metadata::MethodOfLines.MOLMetadata{Val{true}(), MethodOfLines.DiscreteSpace{2, 2, MethodOfLines.CenterAlignedGrid}, MOLFiniteDifference{MethodOfLines.CenterAlignedGrid, MethodOfLines.ScalarizedDiscretization}, PDESystem, Base.RefValue{Any}, Nothing, MethodOfLines.ScalarizedDiscretization}, gui_metadata::Nothing, tearing_state::Nothing, substitutions::Nothing, complete::Bool, index_cache::Nothing, discrete_subsystems::Nothing, solved_unknowns::Nothing, split_idxs::Nothing, parent::Nothing; checks::Bool)
    @ ModelingToolkit ~/.julia/packages/ModelingToolkit/3AmNp/src/systems/diffeqs/odesystem.jl:190
  [3] ODESystem
    @ ~/.julia/packages/ModelingToolkit/3AmNp/src/systems/diffeqs/odesystem.jl:176 [inlined]
  [4] ODESystem(deqs::Vector{Equation}, iv::Num, dvs::Vector{SymbolicUtils.BasicSymbolic{Real}}, ps::Vector{Num}; controls::Vector{Num}, observed::Vector{Equation}, systems::Vector{ODESystem}, tspan::Nothing, name::Symbol, default_u0::Dict{Any, Any}, default_p::Dict{Any, Any}, defaults::Dict{SymbolicUtils.BasicSymbolic{Real}, Float64}, guesses::Dict{Any, Any}, initializesystem::Nothing, initialization_eqs::Vector{Equation}, schedule::Nothing, connector_type::Nothing, preface::Nothing, continuous_events::Nothing, discrete_events::Nothing, parameter_dependencies::Nothing, checks::Bool, metadata::MethodOfLines.MOLMetadata{Val{true}(), MethodOfLines.DiscreteSpace{2, 2, MethodOfLines.CenterAlignedGrid}, MOLFiniteDifference{MethodOfLines.CenterAlignedGrid, MethodOfLines.ScalarizedDiscretization}, PDESystem, Base.RefValue{Any}, Nothing, MethodOfLines.ScalarizedDiscretization}, gui_metadata::Nothing)
    @ ModelingToolkit ~/.julia/packages/ModelingToolkit/3AmNp/src/systems/diffeqs/odesystem.jl:269
  [5] generate_system(disc_state::PDEBase.EquationState, s::MethodOfLines.DiscreteSpace{2, 2, MethodOfLines.CenterAlignedGrid}, u0::Vector{Pair{SymbolicUtils.BasicSymbolic{Real}, Float64}}, tspan::Tuple{Float64, Float64}, metadata::MethodOfLines.MOLMetadata{Val{true}(), MethodOfLines.DiscreteSpace{2, 2, MethodOfLines.CenterAlignedGrid}, MOLFiniteDifference{MethodOfLines.CenterAlignedGrid, MethodOfLines.ScalarizedDiscretization}, PDESystem, Base.RefValue{Any}, Nothing, MethodOfLines.ScalarizedDiscretization}, disc::MOLFiniteDifference{MethodOfLines.CenterAlignedGrid, MethodOfLines.ScalarizedDiscretization})
    @ PDEBase ~/.julia/packages/PDEBase/nzap9/src/discretization_state.jl:41
  [6] symbolic_discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{MethodOfLines.CenterAlignedGrid, MethodOfLines.ScalarizedDiscretization})
    @ PDEBase ~/.julia/packages/PDEBase/nzap9/src/symbolic_discretize.jl:89
  [7] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{MethodOfLines.CenterAlignedGrid, MethodOfLines.ScalarizedDiscretization}; analytic::Nothing, kwargs::@Kwargs{advection_scheme::UpwindScheme})
    @ PDEBase ~/.julia/packages/PDEBase/nzap9/src/discretization_state.jl:58
  [8] top-level scope
    @ ModToolkit/dev_HPLCmodtoolkit.jl:130
  [9] eval
    @ ./boot.jl:385 [inlined]
 [10] include_string(mapexpr::typeof(identity), mod::Module, code::String, filename::String)
    @ Base ./loading.jl:2076
 [11] include_string(m::Module, txt::String, fname::String)
    @ Base ./loading.jl:2086
 [12] invokelatest(::Any, ::Any, ::Vararg{Any}; kwargs::@Kwargs{})
    @ Base ./essentials.jl:892
 [13] invokelatest(::Any, ::Any, ::Vararg{Any})
    @ Base ./essentials.jl:889
 [14] inlineeval(m::Module, code::String, code_line::Int64, code_column::Int64, file::String; softscope::Bool)
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/eval.jl:271
 [15] (::VSCodeServer.var"#69#74"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/eval.jl:181
 [16] withpath(f::VSCodeServer.var"#69#74"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams}, path::String)
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/repl.jl:276
 [17] (::VSCodeServer.var"#68#73"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/eval.jl:179
 [18] hideprompt(f::VSCodeServer.var"#68#73"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/repl.jl:38
 [19] (::VSCodeServer.var"#67#72"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/eval.jl:150
 [20] with_logstate(f::Function, logstate::Any)
    @ Base.CoreLogging ./logging.jl:515
 [21] with_logger
    @ ./logging.jl:627 [inlined]
 [22] (::VSCodeServer.var"#66#71"{VSCodeServer.ReplRunCodeRequestParams})()
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/eval.jl:263
 [23] #invokelatest#2
    @ ./essentials.jl:892 [inlined]
 [24] invokelatest(::Any)
    @ Base ./essentials.jl:889
 [25] (::VSCodeServer.var"#64#65")()
    @ VSCodeServer ~/.vscode/extensions/julialang.language-julia-1.105.2/scripts/packages/VSCodeServer/src/eval.jl:34
in expression starting at Julia/ModToolkit/dev_HPLCmodtoolkit.jl:130

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

[0ca39b1e] Chairmarks v1.2.1
  [a93c6f00] DataFrames v1.6.1
⌃ [2b5f629d] DiffEqBase v6.151.5
  [0c46a032] DifferentialEquations v7.13.0
  [5b8099bc] DomainSets v0.7.14
  [56d4f2e9] Gridap v0.18.3
  [94925ecb] MethodOfLines v0.11.1
⌃ [961ee093] ModelingToolkit v9.26.0
  [5fb14364] OhMyREPL v0.5.27
  [91a5bcdd] Plots v1.40.5
⌃ [1fd47b50] QuadGK v2.9.4
⌃ [0c5d862f] Symbolics v5.34.0
  [592b5752] Trapz v2.0.3

• Output of using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)

  [47edcb42] ADTypes v1.6.1
  [da404889] ARFFFiles v1.4.1
  [621f4979] AbstractFFTs v1.5.0
  [80f14c24] AbstractMCMC v5.2.0
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  [7d9f7c33] Accessors v0.1.37
  [79e6a3ab] Adapt v4.0.4
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⌅ [8e7c35d0] BlockArrays v0.16.43
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  [792122b4] EarlyStopping v0.3.0
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⌅ [a98d9a8b] Interpolations v0.14.0
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⌃ [033835bb] JLD2 v0.4.50
  [1019f520] JLFzf v0.1.7
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⌃ [98e50ef6] JuliaFormatter v1.0.58
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⌃ [ccbc3e58] JumpProcesses v9.11.1
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⌃ [5078a376] LazyArrays v1.10.0
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⌃ [7ed4a6bd] LinearSolve v2.30.2
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⌅ [a3b82374] MatrixFactorizations v2.2.0
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⌅ [78c3b35d] Mocking v0.7.9
⌃ [961ee093] ModelingToolkit v9.26.0
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  [f09324ee] Netpbm v1.1.1
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⌃ [e7bfaba1] NumericalIntegration v0.2.0
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⌃ [1dea7af3] OrdinaryDiffEq v6.86.0
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⌃ [43a3c2be] PairPlots v2.7.3
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⌃ [1fd47b50] QuadGK v2.9.4
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⌃ [0bca4576] SciMLBase v2.45.1
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⌃ [0a514795] SparseMatrixColorings v0.3.5
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⌃ [9672c7b4] SteadyStateDiffEq v2.2.0
⌃ [789caeaf] StochasticDiffEq v6.66.0
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⌃ [2efcf032] SymbolicIndexingInterface v0.3.26
⌃ [19f23fe9] SymbolicLimits v0.2.1
⌅ [d1185830] SymbolicUtils v2.1.2
⌃ [0c5d862f] Symbolics v5.34.0
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⌅ [8ea1fca8] TermInterface v0.4.1
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⌅ [3bb67fe8] TranscodingStreams v0.10.10
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⌃ [410a4b4d] Tricks v0.1.8
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⌅ [efce3f68] WoodburyMatrices v0.5.6
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⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
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⌅ [88015f11] LERC_jll v3.0.0+1
⌅ [dad2f222] LLVMExtra_jll v0.0.30+0
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⌅ [e9f186c6] Libffi_jll v3.2.2+1
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⌅ [89763e89] Libtiff_jll v4.5.1+1
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⌃ [5ced341a] Lz4_jll v1.9.4+0
  [856f044c] MKL_jll v2024.2.0+0
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⌅ [fb77eaff] Sundials_jll v5.2.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
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  [d1454406] Xorg_libXinerama_jll v1.1.4+4
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  [ea2f1a96] Xorg_libXrender_jll v0.9.11+0
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  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
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⌅ [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
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  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
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  [b27032c2] LibCURL v0.6.4
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  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2

• Output of versioninfo()

Julia Version 1.10.4
Commit 48d4fd48430 (2024-06-04 10:41 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin22.4.0)
  CPU: 8 × Apple M1 Pro
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 6 default, 0 interactive, 3 GC (on 6 virtual cores)
Environment:
  JULIA_EDITOR = code

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