diff options
| author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2025-08-18 21:47:37 -0300 |
|---|---|---|
| committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2025-08-18 21:47:37 -0300 |
| commit | 337c72cb0c71b50cad612327f292032ce8ff5933 (patch) | |
| tree | 73fefbb1d608a4ad1abcc8491d37c1066d12b76c | |
| download | zif-337c72cb0c71b50cad612327f292032ce8ff5933.tar.gz zif-337c72cb0c71b50cad612327f292032ce8ff5933.tar.bz2 zif-337c72cb0c71b50cad612327f292032ce8ff5933.zip | |
Initial commit
| -rw-r--r-- | .gitignore | 15 | ||||
| -rw-r--r-- | figs/Football_Pallo_valmiina-cropped.jpg | bin | 0 -> 166898 bytes | |||
| -rw-r--r-- | figs/Simons-Foundation-Logo_blue.png | bin | 0 -> 8698 bytes | |||
| -rw-r--r-- | figs/complex.pdf | bin | 0 -> 111232 bytes | |||
| -rw-r--r-- | figs/complexity_3.pdf | bin | 0 -> 11622 bytes | |||
| -rw-r--r-- | figs/complexity_34.pdf | bin | 0 -> 12412 bytes | |||
| -rw-r--r-- | figs/complexity_34_marginal.pdf | bin | 0 -> 13317 bytes | |||
| -rw-r--r-- | figs/complexity_3_marginal.pdf | bin | 0 -> 11684 bytes | |||
| -rw-r--r-- | figs/connected.pdf | bin | 0 -> 78031 bytes | |||
| -rw-r--r-- | figs/cow.png | bin | 0 -> 127593 bytes | |||
| -rw-r--r-- | figs/cow_edge.png | bin | 0 -> 314609 bytes | |||
| -rw-r--r-- | figs/cow_face.png | bin | 0 -> 307202 bytes | |||
| -rw-r--r-- | figs/cow_vert.png | bin | 0 -> 191441 bytes | |||
| -rw-r--r-- | figs/energy_arrow.pdf | bin | 0 -> 4574 bytes | |||
| -rw-r--r-- | figs/energy_arrow_mixed.pdf | bin | 0 -> 4735 bytes | |||
| -rw-r--r-- | figs/energy_arrow_o1.pdf | bin | 0 -> 7030 bytes | |||
| -rw-r--r-- | figs/energy_arrow_question.pdf | bin | 0 -> 4752 bytes | |||
| -rw-r--r-- | figs/fapesp.png | bin | 0 -> 8426 bytes | |||
| -rw-r--r-- | figs/function-0.png | bin | 0 -> 140522 bytes | |||
| -rw-r--r-- | figs/function-1.png | bin | 0 -> 167979 bytes | |||
| -rw-r--r-- | figs/function-2.png | bin | 0 -> 156331 bytes | |||
| -rw-r--r-- | figs/function_1.png | bin | 0 -> 130874 bytes | |||
| -rw-r--r-- | figs/function_2.png | bin | 0 -> 151038 bytes | |||
| -rw-r--r-- | figs/function_3.png | bin | 0 -> 141157 bytes | |||
| -rw-r--r-- | figs/genus.png | bin | 0 -> 442024 bytes | |||
| -rw-r--r-- | figs/gone.pdf | bin | 0 -> 155551 bytes | |||
| -rw-r--r-- | figs/ift-unesp.png | bin | 0 -> 28064 bytes | |||
| -rw-r--r-- | figs/intersections_1.pdf | bin | 0 -> 103052 bytes | |||
| -rw-r--r-- | figs/intersections_2.pdf | bin | 0 -> 113137 bytes | |||
| -rw-r--r-- | figs/intersections_3.pdf | bin | 0 -> 123718 bytes | |||
| -rw-r--r-- | figs/intersections_4.pdf | bin | 0 -> 176822 bytes | |||
| -rw-r--r-- | figs/land_0.75.pdf | bin | 0 -> 44601 bytes | |||
| -rw-r--r-- | figs/land_1.5.pdf | bin | 0 -> 38663 bytes | |||
| -rw-r--r-- | figs/land_2.25.pdf | bin | 0 -> 25046 bytes | |||
| -rw-r--r-- | figs/land_2.pdf | bin | 0 -> 29477 bytes | |||
| -rw-r--r-- | figs/land_3.5.pdf | bin | 0 -> 16994 bytes | |||
| -rw-r--r-- | figs/large_deviation.pdf | bin | 0 -> 11785 bytes | |||
| -rw-r--r-- | figs/logo-ictp-saifr.jpg | bin | 0 -> 7622 bytes | |||
| -rw-r--r-- | figs/middle.pdf | bin | 0 -> 106443 bytes | |||
| -rw-r--r-- | figs/most_squares_complex.pdf | bin | 0 -> 12160 bytes | |||
| -rw-r--r-- | figs/most_squares_complexity.pdf | bin | 0 -> 51036 bytes | |||
| -rw-r--r-- | figs/most_squares_nonzoom.pdf | bin | 0 -> 30121 bytes | |||
| -rw-r--r-- | figs/most_squares_stability.pdf | bin | 0 -> 41384 bytes | |||
| -rw-r--r-- | figs/most_squares_zoom.pdf | bin | 0 -> 27962 bytes | |||
| -rw-r--r-- | figs/most_squares_zoom_2.pdf | bin | 0 -> 28514 bytes | |||
| -rw-r--r-- | figs/msg_marg_spectra.pdf | bin | 0 -> 22957 bytes | |||
| -rw-r--r-- | figs/mu_0.75.pdf | bin | 0 -> 8416 bytes | |||
| -rw-r--r-- | figs/mu_1.5.pdf | bin | 0 -> 8435 bytes | |||
| -rw-r--r-- | figs/mu_2.25.pdf | bin | 0 -> 8412 bytes | |||
| -rw-r--r-- | figs/mu_2.pdf | bin | 0 -> 8406 bytes | |||
| -rw-r--r-- | figs/mu_3.5.pdf | bin | 0 -> 8139 bytes | |||
| -rw-r--r-- | figs/other_sphere.png | bin | 0 -> 66763 bytes | |||
| -rw-r--r-- | figs/other_sphere_all.png | bin | 0 -> 204002 bytes | |||
| -rw-r--r-- | figs/other_sphere_crit.png | bin | 0 -> 75286 bytes | |||
| -rw-r--r-- | figs/other_sphere_edge.png | bin | 0 -> 62588 bytes | |||
| -rw-r--r-- | figs/other_sphere_face.png | bin | 0 -> 216703 bytes | |||
| -rw-r--r-- | figs/other_sphere_vert.png | bin | 0 -> 75126 bytes | |||
| -rw-r--r-- | figs/phases.png | bin | 0 -> 87811 bytes | |||
| -rw-r--r-- | figs/phases_1.pdf | bin | 0 -> 9935 bytes | |||
| -rw-r--r-- | figs/phases_12_0.pdf | bin | 0 -> 9738 bytes | |||
| -rw-r--r-- | figs/phases_12_1.pdf | bin | 0 -> 94369 bytes | |||
| -rw-r--r-- | figs/phases_12_2.pdf | bin | 0 -> 105142 bytes | |||
| -rw-r--r-- | figs/phases_12_3.pdf | bin | 0 -> 101052 bytes | |||
| -rw-r--r-- | figs/phases_12_4.pdf | bin | 0 -> 94406 bytes | |||
| -rw-r--r-- | figs/phases_12_5.pdf | bin | 0 -> 10958 bytes | |||
| -rw-r--r-- | figs/phases_2.pdf | bin | 0 -> 12544 bytes | |||
| -rw-r--r-- | figs/phases_3.pdf | bin | 0 -> 12094 bytes | |||
| -rw-r--r-- | figs/phases_4.pdf | bin | 0 -> 11856 bytes | |||
| -rw-r--r-- | figs/shattered.pdf | bin | 0 -> 102104 bytes | |||
| -rw-r--r-- | figs/spectrum_eq.pdf | bin | 0 -> 6593 bytes | |||
| -rw-r--r-- | figs/spectrum_less.pdf | bin | 0 -> 8355 bytes | |||
| -rw-r--r-- | figs/spectrum_more.pdf | bin | 0 -> 6789 bytes | |||
| -rw-r--r-- | zif.bib | 85 | ||||
| -rw-r--r-- | zif.tex | 601 |
74 files changed, 701 insertions, 0 deletions
diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..3418b2f --- /dev/null +++ b/.gitignore @@ -0,0 +1,15 @@ +missfont.log +zif.aux +zif.bbl +zif.bcf +zif.blg +zif.fdb_latexmk +zif.fls +zif.log +zif.nav +zif.out +zif.pdf +zif.run.xml +zif.snm +zif.synctex.gz +zif.toc diff --git a/figs/Football_Pallo_valmiina-cropped.jpg b/figs/Football_Pallo_valmiina-cropped.jpg Binary files differnew file mode 100644 index 0000000..0fa264c --- /dev/null +++ b/figs/Football_Pallo_valmiina-cropped.jpg diff --git a/figs/Simons-Foundation-Logo_blue.png b/figs/Simons-Foundation-Logo_blue.png Binary files differnew file mode 100644 index 0000000..65d55d7 --- /dev/null +++ b/figs/Simons-Foundation-Logo_blue.png diff --git a/figs/complex.pdf b/figs/complex.pdf Binary files differnew file mode 100644 index 0000000..0f70f49 --- /dev/null +++ b/figs/complex.pdf diff --git a/figs/complexity_3.pdf b/figs/complexity_3.pdf Binary files differnew file mode 100644 index 0000000..5c2702d --- /dev/null +++ b/figs/complexity_3.pdf diff --git a/figs/complexity_34.pdf b/figs/complexity_34.pdf Binary files differnew file mode 100644 index 0000000..c39ffef --- /dev/null +++ b/figs/complexity_34.pdf diff --git a/figs/complexity_34_marginal.pdf b/figs/complexity_34_marginal.pdf Binary files differnew file mode 100644 index 0000000..373cb71 --- /dev/null +++ b/figs/complexity_34_marginal.pdf diff --git a/figs/complexity_3_marginal.pdf b/figs/complexity_3_marginal.pdf Binary files differnew file mode 100644 index 0000000..c2db693 --- /dev/null +++ b/figs/complexity_3_marginal.pdf diff --git a/figs/connected.pdf b/figs/connected.pdf Binary files differnew file mode 100644 index 0000000..996dadb --- /dev/null +++ b/figs/connected.pdf diff --git a/figs/cow.png b/figs/cow.png Binary files differnew file mode 100644 index 0000000..ec5456d --- /dev/null +++ b/figs/cow.png diff --git a/figs/cow_edge.png b/figs/cow_edge.png Binary files differnew file mode 100644 index 0000000..0475471 --- /dev/null +++ b/figs/cow_edge.png diff --git a/figs/cow_face.png b/figs/cow_face.png Binary files differnew file mode 100644 index 0000000..0e9dcd9 --- /dev/null +++ b/figs/cow_face.png diff --git a/figs/cow_vert.png b/figs/cow_vert.png Binary files differnew file mode 100644 index 0000000..af79ac9 --- /dev/null +++ b/figs/cow_vert.png diff --git a/figs/energy_arrow.pdf b/figs/energy_arrow.pdf Binary files differnew file mode 100644 index 0000000..1ce8c98 --- /dev/null +++ b/figs/energy_arrow.pdf diff --git a/figs/energy_arrow_mixed.pdf b/figs/energy_arrow_mixed.pdf Binary files differnew file mode 100644 index 0000000..7740085 --- /dev/null +++ b/figs/energy_arrow_mixed.pdf diff --git a/figs/energy_arrow_o1.pdf b/figs/energy_arrow_o1.pdf Binary files differnew file mode 100644 index 0000000..fd14e7a --- /dev/null +++ b/figs/energy_arrow_o1.pdf diff --git a/figs/energy_arrow_question.pdf b/figs/energy_arrow_question.pdf Binary files differnew file mode 100644 index 0000000..576255c --- /dev/null +++ b/figs/energy_arrow_question.pdf diff --git a/figs/fapesp.png b/figs/fapesp.png Binary files differnew file mode 100644 index 0000000..36fe13b --- /dev/null +++ b/figs/fapesp.png diff --git a/figs/function-0.png b/figs/function-0.png Binary files differnew file mode 100644 index 0000000..55c0b52 --- /dev/null +++ b/figs/function-0.png diff --git a/figs/function-1.png b/figs/function-1.png Binary files differnew file mode 100644 index 0000000..99931f7 --- /dev/null +++ b/figs/function-1.png diff --git a/figs/function-2.png b/figs/function-2.png Binary files differnew file mode 100644 index 0000000..c358b5c --- /dev/null +++ b/figs/function-2.png diff --git a/figs/function_1.png b/figs/function_1.png Binary files differnew file mode 100644 index 0000000..c0b87ef --- /dev/null +++ b/figs/function_1.png diff --git a/figs/function_2.png b/figs/function_2.png Binary files differnew file mode 100644 index 0000000..a65296b --- /dev/null +++ b/figs/function_2.png diff --git a/figs/function_3.png b/figs/function_3.png Binary files differnew file mode 100644 index 0000000..44d0473 --- /dev/null +++ b/figs/function_3.png diff --git a/figs/genus.png b/figs/genus.png Binary files differnew file mode 100644 index 0000000..d83d529 --- /dev/null +++ b/figs/genus.png diff --git a/figs/gone.pdf b/figs/gone.pdf Binary files differnew file mode 100644 index 0000000..b2fb052 --- /dev/null +++ b/figs/gone.pdf diff --git a/figs/ift-unesp.png b/figs/ift-unesp.png Binary files differnew file mode 100644 index 0000000..d4c84e0 --- /dev/null +++ b/figs/ift-unesp.png diff --git a/figs/intersections_1.pdf b/figs/intersections_1.pdf Binary files differnew file mode 100644 index 0000000..4161bfa --- /dev/null +++ b/figs/intersections_1.pdf diff --git a/figs/intersections_2.pdf b/figs/intersections_2.pdf Binary files differnew file mode 100644 index 0000000..b423c77 --- /dev/null +++ b/figs/intersections_2.pdf diff --git a/figs/intersections_3.pdf b/figs/intersections_3.pdf Binary files differnew file mode 100644 index 0000000..4d60431 --- /dev/null +++ b/figs/intersections_3.pdf diff --git a/figs/intersections_4.pdf b/figs/intersections_4.pdf Binary files differnew file mode 100644 index 0000000..c92d746 --- /dev/null +++ b/figs/intersections_4.pdf diff --git a/figs/land_0.75.pdf b/figs/land_0.75.pdf Binary files differnew file mode 100644 index 0000000..f3d480b --- /dev/null +++ b/figs/land_0.75.pdf diff --git a/figs/land_1.5.pdf b/figs/land_1.5.pdf Binary files differnew file mode 100644 index 0000000..4584c8e --- /dev/null +++ b/figs/land_1.5.pdf diff --git a/figs/land_2.25.pdf b/figs/land_2.25.pdf Binary files differnew file mode 100644 index 0000000..2bd2254 --- /dev/null +++ b/figs/land_2.25.pdf diff --git a/figs/land_2.pdf b/figs/land_2.pdf Binary files differnew file mode 100644 index 0000000..14d66d8 --- /dev/null +++ b/figs/land_2.pdf diff --git a/figs/land_3.5.pdf b/figs/land_3.5.pdf Binary files differnew file mode 100644 index 0000000..3de33d1 --- /dev/null +++ b/figs/land_3.5.pdf diff --git a/figs/large_deviation.pdf b/figs/large_deviation.pdf Binary files differnew file mode 100644 index 0000000..73e0aeb --- /dev/null +++ b/figs/large_deviation.pdf diff --git a/figs/logo-ictp-saifr.jpg b/figs/logo-ictp-saifr.jpg Binary files differnew file mode 100644 index 0000000..64b51d2 --- /dev/null +++ b/figs/logo-ictp-saifr.jpg diff --git a/figs/middle.pdf b/figs/middle.pdf Binary files differnew file mode 100644 index 0000000..7d55318 --- /dev/null +++ b/figs/middle.pdf diff --git a/figs/most_squares_complex.pdf b/figs/most_squares_complex.pdf Binary files differnew file mode 100644 index 0000000..dbfc896 --- /dev/null +++ b/figs/most_squares_complex.pdf diff --git a/figs/most_squares_complexity.pdf b/figs/most_squares_complexity.pdf Binary files differnew file mode 100644 index 0000000..7e07630 --- /dev/null +++ b/figs/most_squares_complexity.pdf diff --git a/figs/most_squares_nonzoom.pdf b/figs/most_squares_nonzoom.pdf Binary files differnew file mode 100644 index 0000000..77b3b5f --- /dev/null +++ b/figs/most_squares_nonzoom.pdf diff --git a/figs/most_squares_stability.pdf b/figs/most_squares_stability.pdf Binary files differnew file mode 100644 index 0000000..a96b62e --- /dev/null +++ b/figs/most_squares_stability.pdf diff --git a/figs/most_squares_zoom.pdf b/figs/most_squares_zoom.pdf Binary files differnew file mode 100644 index 0000000..8fc914c --- /dev/null +++ b/figs/most_squares_zoom.pdf diff --git a/figs/most_squares_zoom_2.pdf b/figs/most_squares_zoom_2.pdf Binary files differnew file mode 100644 index 0000000..4403a06 --- /dev/null +++ b/figs/most_squares_zoom_2.pdf diff --git a/figs/msg_marg_spectra.pdf b/figs/msg_marg_spectra.pdf Binary files differnew file mode 100644 index 0000000..40b3060 --- /dev/null +++ b/figs/msg_marg_spectra.pdf diff --git a/figs/mu_0.75.pdf b/figs/mu_0.75.pdf Binary files differnew file mode 100644 index 0000000..f37980d --- /dev/null +++ b/figs/mu_0.75.pdf diff --git a/figs/mu_1.5.pdf b/figs/mu_1.5.pdf Binary files differnew file mode 100644 index 0000000..c60fa34 --- /dev/null +++ b/figs/mu_1.5.pdf diff --git a/figs/mu_2.25.pdf b/figs/mu_2.25.pdf Binary files differnew file mode 100644 index 0000000..0180fd1 --- /dev/null +++ b/figs/mu_2.25.pdf diff --git a/figs/mu_2.pdf b/figs/mu_2.pdf Binary files differnew file mode 100644 index 0000000..5f20309 --- /dev/null +++ b/figs/mu_2.pdf diff --git a/figs/mu_3.5.pdf b/figs/mu_3.5.pdf Binary files differnew file mode 100644 index 0000000..f5d8406 --- /dev/null +++ b/figs/mu_3.5.pdf diff --git a/figs/other_sphere.png b/figs/other_sphere.png Binary files differnew file mode 100644 index 0000000..2f1ffd8 --- /dev/null +++ b/figs/other_sphere.png diff --git a/figs/other_sphere_all.png b/figs/other_sphere_all.png Binary files differnew file mode 100644 index 0000000..990370c --- /dev/null +++ b/figs/other_sphere_all.png diff --git a/figs/other_sphere_crit.png b/figs/other_sphere_crit.png Binary files differnew file mode 100644 index 0000000..74decec --- /dev/null +++ b/figs/other_sphere_crit.png diff --git a/figs/other_sphere_edge.png b/figs/other_sphere_edge.png Binary files differnew file mode 100644 index 0000000..5fe9ba6 --- /dev/null +++ b/figs/other_sphere_edge.png diff --git a/figs/other_sphere_face.png b/figs/other_sphere_face.png Binary files differnew file mode 100644 index 0000000..3a44a54 --- /dev/null +++ b/figs/other_sphere_face.png diff --git a/figs/other_sphere_vert.png b/figs/other_sphere_vert.png Binary files differnew file mode 100644 index 0000000..3e1ec4c --- /dev/null +++ b/figs/other_sphere_vert.png diff --git a/figs/phases.png b/figs/phases.png Binary files differnew file mode 100644 index 0000000..9f790d6 --- /dev/null +++ b/figs/phases.png diff --git a/figs/phases_1.pdf b/figs/phases_1.pdf Binary files differnew file mode 100644 index 0000000..64cd569 --- /dev/null +++ b/figs/phases_1.pdf diff --git a/figs/phases_12_0.pdf b/figs/phases_12_0.pdf Binary files differnew file mode 100644 index 0000000..56d13a3 --- /dev/null +++ b/figs/phases_12_0.pdf diff --git a/figs/phases_12_1.pdf b/figs/phases_12_1.pdf Binary files differnew file mode 100644 index 0000000..0f357dc --- /dev/null +++ b/figs/phases_12_1.pdf diff --git a/figs/phases_12_2.pdf b/figs/phases_12_2.pdf Binary files differnew file mode 100644 index 0000000..03da2a0 --- /dev/null +++ b/figs/phases_12_2.pdf diff --git a/figs/phases_12_3.pdf b/figs/phases_12_3.pdf Binary files differnew file mode 100644 index 0000000..cd4fdde --- /dev/null +++ b/figs/phases_12_3.pdf diff --git a/figs/phases_12_4.pdf b/figs/phases_12_4.pdf Binary files differnew file mode 100644 index 0000000..8717369 --- /dev/null +++ b/figs/phases_12_4.pdf diff --git a/figs/phases_12_5.pdf b/figs/phases_12_5.pdf Binary files differnew file mode 100644 index 0000000..4788ef0 --- /dev/null +++ b/figs/phases_12_5.pdf diff --git a/figs/phases_2.pdf b/figs/phases_2.pdf Binary files differnew file mode 100644 index 0000000..c07d9e9 --- /dev/null +++ b/figs/phases_2.pdf diff --git a/figs/phases_3.pdf b/figs/phases_3.pdf Binary files differnew file mode 100644 index 0000000..07383a2 --- /dev/null +++ b/figs/phases_3.pdf diff --git a/figs/phases_4.pdf b/figs/phases_4.pdf Binary files differnew file mode 100644 index 0000000..818c4f4 --- /dev/null +++ b/figs/phases_4.pdf diff --git a/figs/shattered.pdf b/figs/shattered.pdf Binary files differnew file mode 100644 index 0000000..4642054 --- /dev/null +++ b/figs/shattered.pdf diff --git a/figs/spectrum_eq.pdf b/figs/spectrum_eq.pdf Binary files differnew file mode 100644 index 0000000..03f5572 --- /dev/null +++ b/figs/spectrum_eq.pdf diff --git a/figs/spectrum_less.pdf b/figs/spectrum_less.pdf Binary files differnew file mode 100644 index 0000000..8d2944e --- /dev/null +++ b/figs/spectrum_less.pdf diff --git a/figs/spectrum_more.pdf b/figs/spectrum_more.pdf Binary files differnew file mode 100644 index 0000000..f1555a9 --- /dev/null +++ b/figs/spectrum_more.pdf @@ -0,0 +1,85 @@ +@article{Folena_2020_Rethinking, + author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico}, + title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed $p$-Spin Model}, + journal = {Physical Review X}, + publisher = {American Physical Society}, + year = {2020}, + month = {8}, + volume = {10}, + pages = {031045}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.10.031045}, + doi = {10.1103/PhysRevX.10.031045}, + issue = {3}, + numpages = {26} +} + +@article{Folena_2021_Gradient, + author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico}, + title = {Gradient descent dynamics in the mixed $p$-spin spherical model: finite-size simulations and comparison with mean-field integration}, + journal = {Journal of Statistical Mechanics: Theory and Experiment}, + publisher = {IOP Publishing}, + year = {2021}, + month = {3}, + number = {3}, + volume = {2021}, + pages = {033302}, + url = {https://doi.org/10.1088%2F1742-5468%2Fabe29f}, + doi = {10.1088/1742-5468/abe29f} +} + +@article{Kent-Dobias_2023_How, + author = {Kent-Dobias, Jaron and Kurchan, Jorge}, + title = {How to count in hierarchical landscapes: a full solution to mean-field complexity}, + journal = {Physical Review E}, + publisher = {American Physical Society (APS)}, + year = {2023}, + month = {6}, + number = {6}, + volume = {107}, + pages = {064111}, + url = {https://doi.org/10.1103/PhysRevE.107.064111}, + doi = {10.1103/PhysRevE.107.064111} +} + +@unpublished{Kent-Dobias_2024_Algorithm-independent, + author = {Kent-Dobias, Jaron}, + title = {Algorithm-independent bounds on complex optimization through the statistics of marginal optima}, + year = {2024}, + month = {July}, + url = {https://arxiv.org/abs/2407.02092}, + note = {arXiv preprint}, + archiveprefix = {arXiv}, + eprint = {2407.02092}, + primaryclass = {cond-mat.dis-nn} +} + +@article{Kent-Dobias_2024_Conditioning, + author = {Kent-Dobias, Jaron}, + title = {Conditioning the complexity of random landscapes on marginal optima}, + journal = {Physical Review E}, + publisher = {American Physical Society (APS)}, + year = {2024}, + month = {December}, + number = {6}, + volume = {110}, + pages = {064148}, + url = {http://dx.doi.org/10.1103/PhysRevE.110.064148}, + doi = {10.1103/physreve.110.064148}, + issn = {2470-0053} +} + +@article{Kent-Dobias_2025_On, + author = {Kent-Dobias, Jaron}, + title = {On the topology of solutions to random continuous constraint satisfaction problems}, + journal = {SciPost Physics}, + publisher = {Stichting SciPost}, + year = {2025}, + month = {May}, + number = {5}, + volume = {18}, + pages = {158}, + url = {http://dx.doi.org/10.21468/SciPostPhys.18.5.158}, + doi = {10.21468/scipostphys.18.5.158}, + issn = {2542-4653} +} + @@ -0,0 +1,601 @@ +\documentclass[aspectratio=169,dvipsnames]{beamer} + +\setbeamerfont{title}{family=\bf} +\setbeamerfont{frametitle}{family=\bf} +\setbeamerfont{normal text}{family=\rm} +\setbeamertemplate{navigation symbols}{} +\setbeamercolor{titlelike}{parent=structure,fg=cyan} + +\usepackage{enumitem} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{pifont} +\usepackage{graphicx} +\usepackage{xcolor} +\usepackage{tikz} + +\definecolor{ictpblue}{HTML}{0471b9} +\definecolor{ictpgreen}{HTML}{0c8636} + +\definecolor{mb}{HTML}{5e81b5} +\definecolor{my}{HTML}{e19c24} +\definecolor{mg}{HTML}{8fb032} +\definecolor{mr}{HTML}{eb6235} + +\setbeamercolor{titlelike}{parent=structure,fg=ictpblue} +\setbeamercolor{itemize item}{fg=ictpblue} + +\usepackage[ + style=phys, + eprint=true, + maxnames = 100, + terseinits=true +]{biblatex} + + +\addbibresource{zif.bib} + +\title{ + Understanding the flat parts of random landscapes +} +\author{\textbf{Jaron Kent-Dobias}} +\date{9 September 2025} + +\begin{document} + +\begin{frame} + \maketitle + + \vspace{-8pc} + \begin{minipage}[c]{10pc} + \centering + \includegraphics[height=6pc]{figs/ift-unesp.png} + + \vspace{2em} + + \includegraphics[height=1.5pc]{figs/Simons-Foundation-Logo_blue.png} + \end{minipage} + \hfill\begin{minipage}[c]{10pc} + \centering + \includegraphics[height=6pc]{figs/logo-ictp-saifr.jpg} + + \vspace{2em} + + \includegraphics[height=1.5pc]{figs/fapesp.png} + \end{minipage} + \vspace{2pc} +\end{frame} + +\begin{frame} + \frametitle{How to count: Kac--Rice} + + Number of stationary points with $\nabla H(\pmb x)=0$ given by integral + over Kac--Rice measure + \begin{align*} + \#_\text{points} + &=\int_\Omega d\pmb x\,\delta\big(\nabla H(\pmb x)\big)\,\big|\det\operatorname{Hess}H(\pmb x)\big| + \end{align*} + Note absolute value of the determinant: want to account for curvature but not add $-1$ + + \bigskip + + Can specify properties of points by inserting $\delta$-functions: + \begin{align*} + \#_\text{points}\alert<2>{(E)} + &=\int_\Omega d\pmb x\,\delta\big(\nabla H(\pmb x)\big)\,\big|\det\operatorname{Hess}H(\pmb x)\big| + \alert<2>{\,\delta\big(H(\pmb x)-NE\big)} + \end{align*} + + \emph{How do we condition on marginal minima?} +\end{frame} + +\begin{frame} + \frametitle{Conditioning on the type of minimum: the spherical models} + \begin{columns} + \begin{column}{0.5\textwidth} + In spherical spin glasses, all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts + \[ + \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\pmb x) + \] + + \bigskip + + Condition on marginal minima by inserting + \[ + \delta\big(\operatorname{Tr}\operatorname{Hess}H(\pmb x)-N\mu_\text{m}\big) + \] + + \end{column} + \begin{column}{0.5\textwidth} + \begin{overlayarea}{\textwidth}{14.5em} + \only<1-2>{\includegraphics[width=\columnwidth]{figs/mu_0.75.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_0.75.pdf}} + \only<3>{\includegraphics[width=\columnwidth]{figs/mu_1.5.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_1.5.pdf}} + \only<4>{\includegraphics[width=\columnwidth]{figs/mu_2.25.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_2.25.pdf}} + \only<5>{\includegraphics[width=\columnwidth]{figs/mu_3.5.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_3.5.pdf}} + \only<6>{\includegraphics[width=\columnwidth]{figs/mu_2.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_2.pdf}} + \end{overlayarea} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Less simple mean-field models} + \framesubtitle{The mixed spherical spin glasses} + + \begin{columns} + \begin{column}{0.41\textwidth} + \alert<1-2>{Pure: all procedures reach $E_\text{th}$} + + \medskip + + \alert<3-4>{Mixed: different procedures reach different energies} + + \medskip + + Dynamical endpoints bounded by complexity of marginal minima + + \bigskip + + \tiny + \fullcite{Folena_2020_Rethinking} + + \smallskip + + \fullcite{Folena_2021_Gradient} + + \smallskip + + \fullcite{Kent-Dobias_2023_How} + \end{column} + \begin{column}{0.59\textwidth} + \begin{overlayarea}{\columnwidth}{0.8\textheight} + \only<1>{% + \includegraphics[width=0.8\columnwidth]{figs/complexity_3_marginal.pdf} + }% + \only<2>{% + \bigskip + + \begin{minipage}[b]{0.83\columnwidth} + \centering + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_3.5.pdf} + \end{minipage} + \raisebox{0.75em}{\includegraphics[width=0.15\columnwidth]{figs/energy_arrow.pdf}} + } + \only<3>{% + \includegraphics[width=0.8\columnwidth]{figs/complexity_34_marginal.pdf} + }% + \only<4>{% + \bigskip + + \begin{minipage}[b]{0.83\columnwidth} + \centering + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_3.5.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + \end{minipage} + \raisebox{0.75em}{\includegraphics[width=0.15\columnwidth]{figs/energy_arrow_o1.pdf}} + }% + \end{overlayarea} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Marginal minima in generic models} + \begin{columns} + \begin{column}{0.5\textwidth} + \alert<1>{In spherical spin glasses, all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts} + + \bigskip + + In generic models, spectral density depends on stationarity, energy, etc! + + \bigskip + + \alert<2>{Example: multi-spherical model + \[ + H(\{\boldsymbol s_1,\boldsymbol s_2\}) + =H_p^{(1)}(\boldsymbol s_1)+H_p^{(2)}(\boldsymbol s_2)+\epsilon\boldsymbol s_1\cdot\boldsymbol s_2 + \]} + \hspace{-0.25em}In most models we don't understand the Hessian at all + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\columnwidth]{figs/mu_0.75.pdf} + \onslide<2>\includegraphics[width=0.9\columnwidth]{figs/msg_marg_spectra.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Towards generic marginal complexity} + \begin{columns} + \begin{column}{0.6\textwidth} + \begin{itemize}[leftmargin=4em] + \item[\color{ictpgreen}\bf Trick \#1:] condition stationary points on \emph{value of smallest eigenvalue} + \end{itemize} + { + \small + \begin{align*} + \hspace{-1em}&\delta(\lambda_\text{min}(A)) \\ + &=\lim_{\beta\to\infty}\int + \frac{d\pmb s\,\delta(N-\|\pmb s\|^2)e^{-\beta\pmb s^TA\pmb s}} + {\int d\pmb s'\,\delta(N-\|\pmb s'\|^2)e^{-\beta\pmb s'^TA\pmb s'}} + \delta\left(\frac{\pmb s^TA\pmb s}N\right) + \end{align*} + } + + \bigskip + + Only works if you happen to have the correct shift $\mu$ + + \bigskip + + \tiny + \fullcite{Kent-Dobias_2024_Conditioning} + + \end{column} + \begin{column}{0.4\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\textwidth]{figs/spectrum_less.pdf} + \onslide<2>\includegraphics[width=\textwidth]{figs/spectrum_more.pdf} + \onslide<3>\includegraphics[width=\textwidth]{figs/spectrum_eq.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + + +\begin{frame} + \frametitle{Towards generic marginal complexity} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{itemize}[leftmargin=4em] + \item[\color{ictpgreen}\bf Trick \#1:] condition stationary points on \emph{value of smallest eigenvalue} + \end{itemize} + { + \small + \begin{align*} + \hspace{-3em}&\delta(\lambda_\text{min}(A)) \\ + \hspace{-3em}&=\lim_{\beta\to\infty}\int + \frac{d\pmb s\,\delta(N-\|\pmb s\|^2)e^{-\beta\pmb s^TA\pmb s}} + {\int d\pmb s'\,\delta(N-\|\pmb s'\|^2)e^{-\beta\pmb s'^TA\pmb s'}} + \delta\left(\frac{\pmb s^TA\pmb s}N\right) + \end{align*} + } + + \medskip + + \begin{itemize}[leftmargin=4em] + \item[\color{ictpgreen}\bf Trick \#2:] adjust $\mu\propto\operatorname{Tr}\operatorname{Hess}H$ until order-$N$ large deviation breaks + \end{itemize} + + \bigskip + + \tiny + \fullcite{Kent-Dobias_2024_Conditioning} + + \end{column} + \begin{column}{0.5\textwidth} + \hspace{0.9em} + \includegraphics[scale=0.8]{figs/spectrum_less.pdf} + \hspace{-1.6em} + \includegraphics[scale=0.8]{figs/spectrum_eq.pdf} + \hspace{-1.6em} + \includegraphics[scale=0.8]{figs/spectrum_more.pdf} + \\ + \includegraphics[scale=0.8]{figs/large_deviation.pdf} + + \vspace{-1em} + + \small + \begin{align*} + \hspace{-0.5em}G_0(\mu)=\frac 1N\log\Big\langle\delta\big(\lambda_\text{min}(A-\lambda\mu)\big)\Big\rangle_{A\in\text{GOE}(N)} + \end{align*} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Marginal complexity: example} + \begin{columns} + \begin{column}{0.5\textwidth} + Example: model of nonlinear least squares: + \[ + H(\pmb s)=\frac12\sum_{i=1}^{M}V_i(\pmb s)^2 + \] + for spherical $\pmb s\in\mathbb R^N$ and $M=\alpha N$ inhomogeneous random functions $V_i$ + \[ + V_i(\boldsymbol s)=H_2^{(i)}(\boldsymbol s)+H_3^{(i)}(\boldsymbol s) + \] + + \bigskip + + \tiny + \fullcite{Kent-Dobias_2024_Conditioning} + + \smallskip + + \fullcite{Kent-Dobias_2024_Algorithm-independent} + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\vspace{4em}\includegraphics[width=\textwidth]{figs/most_squares_complex.pdf} + \onslide<2>\vspace{-1.75em}\includegraphics[width=\textwidth]{figs/most_squares_complexity.pdf} + + \vspace{-1.95em} + + \hspace{-0.25em}\colorbox{white}{\includegraphics[width=\textwidth]{figs/most_squares_stability.pdf}} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \begin{columns} + \begin{column}{0.5\textwidth} + The Euler characteristic $\chi(\Omega)$ is a topological invariant of a manifold $\Omega$ + + \medskip + + Defined by tiling the manifold, then taking the alternating sum + \begin{align*} + \chi(\Omega_{\text{cow}}) + &= + {\only<2,5->{\color{Red}}\#_\text{vertices}} + &&\hspace{-1em}- + {\only<3,5->{\color{ictpgreen}}\#_\text{edges}} + &&\hspace{-1em}+ + {\only<4,5->{\color{ictpblue}}\#_\text{faces}} + \\ + &\color{White}\only<2->{\color{Black}}= + {\only<2,5->{\color{Red}}2904} + &&\hspace{-1em}\color{White}\only<3->{\color{Black}}- + {\only<3,5->{\color{ictpgreen}}8706} + &&\hspace{-1em}\color{White}\only<4->{\color{Black}}+ + {\only<4,5->{\color{ictpblue}}5804} \\ + &\color{White}\only<5->{\color{Black}}=2 + \end{align*} + \[ + \color{White}\only<6->{\color{Black}}\chi(\Omega_\text{football}) + ={\only<6->{\color{Red}}60}-{\only<6->{\color{ictpgreen}}90}+{\only<6->{\color{ictpblue}}32}=2 + \] + + \color{White}\only<7>{\color{Black}}Cow is homeomorphic to a sphere + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1,5>\includegraphics[width=\textwidth]{figs/cow.png} + \onslide<2>\includegraphics[width=\textwidth]{figs/cow_vert.png} + \onslide<3>\includegraphics[width=\textwidth]{figs/cow_edge.png} + \onslide<4>\includegraphics[width=\textwidth]{figs/cow_face.png} + \onslide<6->\hspace{2em}\includegraphics{figs/Football_Pallo_valmiina-cropped.jpg} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \framesubtitle{Characteristics of the characteristic} + + \begin{columns} + \begin{column}{0.5\textwidth} + For closed, connected 2-dimensional manifolds, related to genus $g$ by + $\chi=2-2g$ + + \medskip + + General properties: + \vspace{-0.5em} + \[ + \chi(\Omega)=0 \text{ for odd-dimensional $\Omega$} + \] + \vspace{-1.6em} + \[ + \chi(S^D)=2\text{ for even }D + \] + \[ + \chi(\Omega_1\sqcup\Omega_2)=\chi(\Omega_1)+\chi(\Omega_2) + \] + \[ + \chi(\Omega_1\times\Omega_2)=\chi(\Omega_1)\times\chi(\Omega_2) + \] + + \smallskip + + Examples: + \vspace{-0.5em} + \[\chi(M\text{ even-$D$ spheres})=2M\] + \vspace{-1.6em} + \[\chi(S^1\times\text{anything})=0\] + \end{column} + \begin{column}{0.5\textwidth} + \includegraphics[width=\textwidth]{figs/genus.png} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \framesubtitle{Computing the Euler characteristic} + \begin{columns} + \begin{column}{0.5\textwidth} + Morse theory: gradient flow on an arbitrary ``height'' function $h$ makes a complex + \begin{align*} + \chi(\Omega) + &= + {\only<2,5>{\color{Red}}\#_\text{vertices}} + - + {\only<3,5>{\color{ictpgreen}}\#_\text{edges}} + + + {\only<4,5>{\color{ictpblue}}\#_\text{faces}} + +\cdots \\ + &= + {\only<6>{\color{ictpblue}}\#_\text{index 0}} + - + {\only<6>{\color{ictpgreen}}\#_\text{index 1}} + + + {\only<6>{\color{Red}}\#_\text{index 2}} + +\cdots \\ + &=\sum_{i=0}^D(-1)^i\#_\text{index i} + \end{align*} + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\textwidth]{figs/other_sphere.png} + \onslide<2>\includegraphics[width=\textwidth]{figs/other_sphere_vert.png} + \onslide<3>\includegraphics[width=\textwidth]{figs/other_sphere_edge.png} + \onslide<4>\includegraphics[width=\textwidth]{figs/other_sphere_face.png} + \onslide<5>\includegraphics[width=\textwidth]{figs/other_sphere_all.png} + \onslide<6>\includegraphics[width=\textwidth]{figs/other_sphere_crit.png} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \framesubtitle{Computing the Euler characteristic} + + \begin{columns} + \begin{column}{0.6\textwidth} + \[ + \Omega=\left\{ + \pmb a\in\mathbb R^N\mid\|\pmb a\|^2=N, V_0=\hat f(J^i\mid\pmb a)\;\forall\;1\leq i\leq M + \right\} + \] + Pick whatever height function $h:\Omega\to\mathbb R$ you like: $h(\pmb a)=\frac1N\pmb + a_0\cdot\pmb a$ for arbitrary $\pmb a_0$ + \[ + \#_{\substack{\text{critical}\\\text{points}}} + =\int_\Omega d\pmb x\, + \delta\big(\nabla h(\pmb x)\big)\, + \big|\det\operatorname{Hess}h(\pmb x)\big| + \] + \[ + \hspace{-1em}\operatorname{sgn}\big(\det\operatorname{Hess}(\pmb x)\big) + = + \operatorname{sgn}\left(\prod_{i=1}^D\lambda_i\right) + =(-1)^{\text{index}} + \] + \[ + \chi(\Omega) + =\sum_{i=0}^D(-1)^i\#_\text{index i} + =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x) + \] + \end{column} + \begin{column}{0.4\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\textwidth]{figs/function_1.png} + \onslide<2>\includegraphics[width=\textwidth]{figs/function_2.png} + \onslide<3>\includegraphics[width=\textwidth]{figs/function_3.png} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{A simple model of nonlinear least squares} + \framesubtitle{Results} + \begin{columns} + \begin{column}{0.5\textwidth} + $M$ data points, $N$ parameters, $\alpha=M/N$ + \[ + V_0=\hat f(J\mid \pmb a) + =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p} + \] + Simplest case: $p=1$, $\Omega$ is intersection of random hyperplanes with the sphere + + \medskip + + Results in $\chi(\Omega)=2$ or $\chi(\Omega)=0$ depending on whether solutions exist + + \medskip + + \hspace{2em}\includegraphics[width=0.33\textwidth]{figs/connected.pdf} + \hfill + \includegraphics[width=0.33\textwidth]{figs/gone.pdf}\hspace{2em} + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\textwidth]{figs/intersections_1.pdf} + \onslide<2>\includegraphics[width=\textwidth]{figs/intersections_2.pdf} + \onslide<3>\includegraphics[width=\textwidth]{figs/intersections_3.pdf} + \onslide<4>\includegraphics[width=\textwidth]{figs/intersections_4.pdf} + \onslide<5>\includegraphics[width=\textwidth]{figs/phases_1.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{A simple model of nonlinear least squares} + \framesubtitle{Results} + \begin{columns} + \begin{column}{0.5\textwidth} + $M$ data points, $N$ parameters, $\alpha=M/N$ + \[ + V_0=\hat f(J\mid \pmb a) + =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p} + \] + + For $p\geq2$, different phases with $|\chi(\Omega)|\gg1$ with varying sign + + \medskip + + \includegraphics[width=0.23\textwidth]{figs/middle.pdf} + \includegraphics[width=0.23\textwidth]{figs/complex.pdf} + \includegraphics[width=0.23\textwidth]{figs/shattered.pdf} + \includegraphics[width=0.23\textwidth]{figs/gone.pdf} + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\centering\includegraphics[width=0.8\textwidth]{figs/middle.pdf}\\$\chi(\Omega)\ll0$ + \onslide<2>\centering\includegraphics[width=0.8\textwidth]{figs/complex.pdf}\\$\chi(\Omega)\ll0$ + \onslide<3>\centering\includegraphics[width=0.8\textwidth]{figs/shattered.pdf}\\$\chi(\Omega)\gg0$ + \onslide<4>\includegraphics[width=\textwidth]{figs/phases_2.pdf} + \onslide<5>\includegraphics[width=\textwidth]{figs/phases_3.pdf} + \onslide<6>\includegraphics[width=\textwidth]{figs/phases_4.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\end{document} |
