{"id":5711514,"date":"2022-03-23T12:28:40","date_gmt":"2022-03-23T22:28:40","guid":{"rendered":"https:\/\/math.hawaii.edu\/wordpress\/?p=5711514"},"modified":"2026-01-30T14:50:49","modified_gmt":"2026-01-31T00:50:49","slug":"qualifying-exams-in-applied-mathematics","status":"publish","type":"post","link":"https:\/\/math.hawaii.edu\/wordpress\/qualifying-exams-in-applied-mathematics\/","title":{"rendered":"Qualifying Exams in Applied Mathematics"},"content":{"rendered":"<h5><\/h5>\n<h5 id=\"applied\">Applied mathematics exam<\/h5>\n<p>The applied mathematics qualifying exam covers topics in dynamical systems, partial differential equations, and applied linear algebra.<\/p>\n<hr \/>\n<h6>Material<\/h6>\n<p><span style=\"text-decoration: underline\">Linear algebra<\/span><\/p>\n<ul>\n<li>\u00a0Undergraduate linear algebra<\/li>\n<li>Spaces: Normed spaces, inner product spaces, dual spaces, fundamental subspaces, rank-nullity theorem, direct sums<\/li>\n<li>Matrices and operators: linear, bounded, self-adjoint, unitary, normal, positive-definite, and compact operators; projections, adjoints, matrix exponential<\/li>\n<li>Spectra: Eigenvalues, eigenvectors, Rayleigh quotient, eigendecomposition, singular value decomposition, Schur decomposition, generalized eigenvectors<\/li>\n<li>Orthogonality: orthogonal projections and reflections, orthogonal complement, orthonormality, Gram-Schmidt orthogonalization, orthogonality of range and null space of operators and their adjoints<\/li>\n<li>Hilbert spaces: completeness, Riesz representation theorem, Parseval\u2019s identity<\/li>\n<\/ul>\n<p><span style=\"text-decoration: underline\">Ordinary differential equations<\/span><\/p>\n<ul>\n<li>ODE theory: Local existence and uniqueness of solutions to initial-value problems, continuous dependence on initial conditions, continuous dependence on parameters, Sturm-Liouville theory for one-dimensional boundary-value problems<\/li>\n<li>Dynamical systems: continuous and discrete systems, equilibrium states and their stability, flows, orbits, limit sets, invariant sets; linear systems, stable, unstable, center spaces; nonlinear systems, linearization, topological equivalence, center manifold theory, nullclines, Lyapunov function, limit cycles, Poincar\u00e9\u2013Bendixson theory, stability of periodic solutions, Poincar\u00e9 map.<\/li>\n<li>Bifurcation theory: structural stability, local bifurcations of equilibrium states, genericity, codimension, center manifold theory, global bifurcations (homoclinic, heteroclinic), bifurcations in 1-dimensional maps.<\/li>\n<\/ul>\n<p><span style=\"text-decoration: underline\">Partial differential equations<\/span><\/p>\n<ul>\n<li>PDE theory: classification of PDEs, uniqueness of solutions, energy estimates<\/li>\n<li>Initial-value problems: First-order linear and quasilinear PDEs, method of characteristics, heat equation, wave equation, separation of variables, series solutions, Fourier transform<\/li>\n<li>Boundary-value problems: Laplace equation, Poisson equation, Sturm\u2013Liouville theory, Green\u2019s functions, fundamental solutions, Dirichlet and Neumann boundary conditions, variational formulations of boundary-value problems.<\/li>\n<\/ul>\n<hr \/>\n<p><strong>Textbooks. The following textbooks are recommended:<\/strong><\/p>\n<ul>\n<li>\u00a0Friedberg, Insel, Linear algebra<\/li>\n<li>Roman, Advanced Linear algebra<\/li>\n<li>Cheney, Analysis for applied mathematics<\/li>\n<li>Perko, Differential equations and dynamical systems<\/li>\n<li>Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering<\/li>\n<li>Hirsch, Smale, and Devaney, Differential equations, dynamical systems, and an introduction to chaos<\/li>\n<li>Teschle, Ordinary differential equations and dynamical systems<\/li>\n<li>Bleecker, Csordas, Basic partial differential equations<\/li>\n<li>Evans, Partial differential equations<\/li>\n<li>Haberman, Applied partial differential equations with Fourier series and boundary value problems<\/li>\n<li>Olver, Introduction to partial differential equations<\/li>\n<\/ul>\n<hr \/>\n<h6>Sample Exams (new format)<\/h6>\n<ul>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/samplequals.pdf\">Sample Quals (1 &#8211; 4)<\/a><\/li>\n<\/ul>\n<h6>Sample Exams (old format)<br \/>\n<em>These still have good problems to study from although the balance of material is different from the new format.\u00a0<\/em><\/h6>\n<ul>\n<li><a title=\"Applied Mathematics Sample Exam 1\" href=\"http:\/\/math.hawaii.edu\/home\/graduate\/qualifying-exams\/applied-math\/sample_exam1.pdf\" target=\"_blank\" rel=\"noopener\">Sample Exam 1<\/a><\/li>\n<li><a title=\"Applied Mathematics Sample Exam 2\" href=\"http:\/\/math.hawaii.edu\/home\/graduate\/qualifying-exams\/applied-math\/sample_exam2.pdf\" target=\"_blank\" rel=\"noopener\">Sample Exam 2<\/a><\/li>\n<li><a title=\"Applied Mathematics Sample Exam 3\" href=\"http:\/\/math.hawaii.edu\/home\/graduate\/qualifying-exams\/applied-math\/sample_exam3.pdf\" target=\"_blank\" rel=\"noopener\">Sample Exam 3<\/a><\/li>\n<li><a title=\"Applied Mathematics Sample Exam 4\" href=\"http:\/\/math.hawaii.edu\/home\/graduate\/qualifying-exams\/applied-math\/sample_exam4.pdf\" target=\"_blank\" rel=\"noopener\">Sample Exam 4<\/a><\/li>\n<\/ul>\n<hr \/>\n<h6>Previous Exams (current format)<\/h6>\n<ul>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/Applied_math_qual_S26.pdf\">January 2026<\/a><\/li>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/Applied_math_qual_S25-1.pdf\">January 2025<\/a><\/li>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/Fall_2024_Applied_Qual_Problems.pdf\">August 2024<\/a><\/li>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/qual2024spring.pdf\">January 2024<\/a><\/li>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/qual2023fall-new.pdf\">August 2023<\/a><\/li>\n<\/ul>\n<p><!--more--><\/p>\n<p><!--more--><\/p>\n<p>Previous Exam (old format)<\/p>\n<ul>\n<li><a href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/applied_qual_old_f2023.pdf\">August 2023 (old format)<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for January 2023\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/applied_qual_jan_2023.pdf\" target=\"_blank\" rel=\"noopener\">January 2023<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for August 2022\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/03\/applied_qual_aug_2022.pdf\" target=\"_blank\" rel=\"noopener\">August 2022<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for January 2022\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/06\/applied_qual_jan2022.pdf\" target=\"_blank\" rel=\"noopener\">January 2022<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for August 2021\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/05\/applied_qual_aug2021.pdf\" target=\"_blank\" rel=\"noopener\">August 2021<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for August 2020\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/05\/applied_qual_aug2020.pdf\" target=\"_blank\" rel=\"noopener\">August 2020<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for January 2020\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/05\/applied_qual_jan2020.pdf\" target=\"_blank\" rel=\"noopener\">January 2020<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for January 2019\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2022\/05\/applied_qual_jan2019.pdf\" target=\"_blank\" rel=\"noopener\">January 2019<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for Spring 2018\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2016\/01\/AppliedMath_Spring_2018.pdf\" target=\"_blank\" rel=\"noopener\">Spring 2018<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for Fall 2017\" href=\"https:\/\/math.hawaii.edu\/wordpress\/wp-content\/uploads\/2016\/01\/AppliedMath-Fall-2017.pdf\" target=\"_blank\" rel=\"noopener\">Fall 2017<\/a><\/li>\n<li><a title=\"Applied Mathematics Qualifying Exam for Fall 2016\" href=\"http:\/\/math.hawaii.edu\/home\/graduate\/qualifying-exams\/applied-math\/AppliedMath_Fall_2016.pdf\" target=\"_blank\" rel=\"noopener\">Fall 2016<\/a><\/li>\n<\/ul>\n<p><!-- Older folder for previous exams https:\/\/math.hawaii.edu\/home\/graduate\/qualifying-exams\/applied-math\/\" --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Applied mathematics exam The applied mathematics qualifying exam covers topics in dynamical systems, partial differential equations, and applied linear algebra. Material Linear algebra \u00a0Undergraduate linear algebra Spaces: Normed spaces, inner product spaces, dual spaces, fundamental subspaces, rank-nullity theorem, direct sums Matrices and operators: linear, bounded, self-adjoint, unitary, normal, positive-definite, and compact operators; projections, adjoints, matrix &hellip; <a href=\"https:\/\/math.hawaii.edu\/wordpress\/qualifying-exams-in-applied-mathematics\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Qualifying Exams in Applied Mathematics<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":49,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18],"tags":[],"class_list":["post-5711514","post","type-post","status-publish","format-standard","hentry","category-graduate-posts"],"_links":{"self":[{"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/posts\/5711514","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/users\/49"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/comments?post=5711514"}],"version-history":[{"count":13,"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/posts\/5711514\/revisions"}],"predecessor-version":[{"id":5713585,"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/posts\/5711514\/revisions\/5713585"}],"wp:attachment":[{"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/media?parent=5711514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/categories?post=5711514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hawaii.edu\/wordpress\/wp-json\/wp\/v2\/tags?post=5711514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}