Invariant Theory
Book: H. Derksen and G. Kemper, “Computational Invariant Theory”, Encyclopaedia of Mathematical Sciences 130, Springer, Heidelberg, second edition, 2015 (first edition 2002).
H. Derksen, I. Klep, V. Makam, J. Volčič, “Ranks of linear matrix pencils separate simultaneous similarity orbits”, Advances in Mathematics 415, p. 108888, 2023.
H. Derksen, V. Makam, “Polystability in positive characteristic and degree lower bounds for invariant rings”, Journal of Combinatorial Algebra 6 (3), p. 353-405, 2022.
H. Derksen and V. Makam, “Algorithms for orbit closure separation for invariants and semi-invariants of matrices”, Algebra & Number Theory 14 (10), p. 2791-2813, 2020.
H. Derksen and V. Makam, “Weyl’s polarization theorem in positive characteristic”, Transformation Groups, p. 1-20, 2020.
H. Derksen and V. Makam, “The regularity lemma is false over small fields”, Communications in Algebra 48 (3), p. 1333-1339, 2020.
H. Derksen and V. Makam, “Polynomial degree bounds for matrix semi-invariants”, Adv. Math. 310, p. 44-63, 2017.
H. Derksen and V. Makam, “Generating invariant rings of quivers in arbitrary characteristic”, J. Algebra 489, p. 435-445, 2017.
H. Derksen and V. Makam, “Degree bounds for semi-invariant rings of quivers”, J. Pure Appl. Algebra 222 (10), p. 3282-3298, 2018.
H. Derksen and G. Kemper, “Computing invariants of algebraic groups in arbitrary characteristic”, Adv. Math.217 (5), p. 2089-2129, 2008.
H. Derksen, E. Jeandel and P. Koiran, “Quantum automata and algebraic groups”, J. Symbolic Comput. 39 (3-4), p. 357-371, 2005.
H. Derksen, “Degree bounds for syzygies of invariants”, Adv. Math. 185 (2), p. 207-214, 2004.
H. Derksen and G. Kemper, “On global degree bounds for invariants”, In Invariant Theory in all characteristics, CRM Proc. Lecture Notes 35, p. 37-41, Amer. Math. Soc., Providence, RI, 2004.
H. Derksen, “Constructive invariant theory”, In Invariant Theory in all characteristics, CRM Proc. Lecture Notes 35, p. 11-36, Amer. Math. Soc., Providence, RI, 2004.
H. Derksen, “Restrictions of algebraic group representations to finite subgroups”, Bull. London Math. Soc. 34 (2), p. 165-173, 2002.
H. Derksen, “Polynomial bounds for rings of invariants”, Proc. Amer. Math. Soc. 129 (4), p. 955-963, 2001.
H. Derksen, “Computation of invariants for reductive groups”, Adv. Math. 141 (2), p. 366-384, 1999.
H. Derksen and H. Kraft, “Constructive invariant theory”, In Algèbre non-commutative, groups quantiques et invariants (Reims 19995), Sémin. Congr. 2, p. 221-244, Soc. Math. France, Paris, 1997.
Quiver Representations
Book: H. Derksen and J. Weyman, “An Introduction to Quiver Representations”, Graduate Studies in Mathematics 184, American Mathematical Society, 2007.
H. Derksen and J. Fei, “General presentations of algebras”, Adv. Math. 278, p. 210-237, 2015.
H. Derksen, B. Huisgen-Zimmermann and J. Weyman, “Top-stable degenerations of finite-dimensional algebras II”, Adv. Math. 259, p. 730-765, 2014.
H. Derksen and J. Weyman, “The combinatorics of quiver representations”, Ann. Inst. Fourier (Grenoble) 61 (3), p. 1061-1131, 2011.
H. Derksen, J. Weyman and A. Zelevinsky, “Quivers with potentials and their representations II: applications to cluster algebras”, J. Amer. Math. Soc. 23 (3), p. 749-790, 2010.
H. Derksen, J. Weyman and A. Zelevinsky, “Quivers with potentials and their representations I: Mutations”, Selecta Math. (N.S.) 14 (1), p. 59-119, 2008.
H. Derksen and T. Owen, “New graphs of finite mutation type”, Electron. J. Combin. 15 (1): Research Paper 139, 15, 2008.
H. Derksen, A. Schofield and J. Weyman, “On the number of subrepresentations of a general quiver representation”, J. London Math. Soc. (2) 76 (1), p. 135-147, 2007.
C. Chindris, H. Derksen and J. Weyman, “Counterexamples to Okounkov’s log-concavity conjecture”, Compos. Math. 143 (6), p. 1545-1557, 2007.
H. Derksen and J. Weyman, “Quiver representations”, Notices Amer. Math. Soc. 52 (2), p. 200-206, 2005.
H. Derksen and J. Weyman, “Semi-invariants for quivers with relations”, J. Algebra 258 (1), p. 216-227, 2002.
H. Derksen and J. Weyman, “On the Littlewood-Richardson polynomials”, J. Algebra 255 (2), p. 247-257, 2002.
H. Derksen and J. Weyman, “On the canonical decomposition of quiver representations”, Compositio Math. 133 (3), p. 245-265, 2002.
H. Derksen and J. Weyman, “Generalized quivers associated to reductive groups”, Colloq. Math. 94 (2), p. 151-183, 2002.
H. Derksen and J. Weyman, “Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients”, J. Amer. Math. Soc. 13(3), p. 457-479, 2000.
Tensors
H. Derksen, V. Makam and J. Zuiddam, “Subrank and optimal reduction of scalar multiplications to generic tensors”, arXiv preprint arXiv:2205.15168, 2022.
J. Gryak, K. Najarian and H. Derksen, “Tensor denoising via amplification and stable rank methods”, arXiv preprint arXiv:2301.03761.
N. Tokcan, J. Gryak, K. Najarian and H. Derksen, “Algebraic methods for tensor data”, SIAM Journal on Applied Algebra and Geometry 5 (1), 1-27, 2021.
H. Derksen, “The G-stable rank for tensors and the cap set problem”, Algebra & Number Theory 16 (5), p. 1071-1097, 2022.
H. Derksen and V. Makam, “Highly entangled tensors”, Linear and Multilinear Algebra, 1-14, 2020.
H. Derksen, “A general theory of singular values with applications to signal denoising“, SIAM J. on Applied Algebra and Geometry 2 (4), 535-596, 2018.
H. Derksen and V. Makam, “Explicit tensors of border rank 2d-2 in K2oK2oK2 in arbitrary characteristic“, Linear and Multilinear Algebra, p. 1-13, 2018.
H. Derksen and V.~Makam, “Invariant theory and wheeled PROPs”, arXiv preprint arXiv:1909.00443, p. 1-28, 2019.
H. Derksen and V. Makam, “On non-commutative rank and tensor rank”, Linear and Multilinear Algebra 66 (6), p. 1069–1084, 2018.
H. Derksen, “On the equivalence between low-rank matrix completion and tensor rank”, Linear and Multilinear Algebra 66 (4), p. 645-667, 2018.
H. Derksen, “On the nuclear norm and the singular value decomposition of tensors”, Found. Comput. Math. 16 (3), p. 779-811, 2016.
H. Derksen, “Matrix completion and tensor rank”, Linear and Multilinear Algebra 64 (4), p. 680-685, 2016.
H. Derksen and Z. Teitler, “Lower bound for ranks of invariant forms”, J. Pure Appl. Algebra 219 (12), p. 5429-5441, 2015.
H. Derksen, “Kruskal’s uniqueness inequality is sharp”, Linear Algebra Appl. 438 (2), p. 708-712, 2013.
Commutative Algebra
H. Derksen, R. Eggermont and A. Snowden, “Topological Noetherianity for cubic polynomials”, Algebra & Number Theory 11 (9), p. 2197-2212, 2017.
H. Derksen, “Hilbert series of subspace arrangements”, J. Pure Appl. Algebra 209 (1), p. 91-98, 2007.
H. Derksen, “Universal denominators of Hilbert series”, J. Algebra 285 (2), p. 586-607, 2005.
H. Derksen and J. Sidman, “Castelnuovo-Mumford regularity by approximation”, Adv. Math 188 (1), p. 104-123, 2004.
H. Derksen, “The fundamental theorem of algebra and linear algebra”, Amer. Math. Monthly 110 (7), p. 620-623, 2003.
H. Derksen and J. Sidman, ”A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements”, Adv. Math. 172 (2), p. 151-157, 2002.
Theoretical Computer Science
H. Derksen, E. Viola, “Quasirandom groups enjoy interleaved mixing”, arXiv preprint arXiv:2206.10603, 2022.
H. Derksen, E. Viola, “Fooling polynomials using invariant theory”, 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), p. 399-406, 2022.
Algebraic Statistics
C. Chindris, H. Derksen, “The capacity of quiver representations and the Anantharam-Jog-Nair inequality”, arXiv preprint arXiv:2306.06790, p. 1-17, 2023.
H. Derksen, V. Makam and M. Walter, “Maximum likelihood estimation for tensor normal models via castling transforms”, Forum of Mathematics, Sigma 10, p. e50, 2022.
H. Derksen and V. Makam, “Maximum likelihood estimation for matrix normal models via quiver representations”,SIAM Journal on Applied Algebra and Geometry 5 (2), p. 338-365, 2021.
C. Chindris and H. Derksen, “The capacity of quiver representations and Brascamp-Lieb constants”, International Mathematics Research Notices 2022 (24), p. 19399-19430, 2022. Erratum.
Automorphisms of Affine Space
H. Derksen, A. van den Essen and W. Zhao, “The Gaussian moments conjecture and the Jacobian conjecture”, Israel J. Math. 219 (2), p. 917-928, 2017.
H. Derksen, A. van den Essen, D. R. Finston and S. Maubach, “Unipotent group actions on affine varieties”, J. Algebra 336, p. 200-208, 2011.
H. Derksen. O. Hadas and L. Makar-Limanov, “Newton polytopes of invariants of additive group actions”, J. Pure Appl. Algebra 156 (2-3), p. 187-197, 2001.
H. Derksen, F. Kutzschebauch and J. Winkelmann, “Subvarieties of Cn with non-extendable automorphisms”, J. Reine Angew. Math. 509, p. 213-235, 1999.
Derksen and F. Kutzschebauch, “Global holomorphic linearization of actions of compact Lie groups on Cn”, In Complex geometric analysis in Pohang (1997), Contemp. Math. 22, p. 201-210, Amer. Math. Soc., Providence, RI, 1999.
H. Derksen and F. Kutzschebauch, “Nonlinearizable holomorphic group actions”, Math. Ann. 311 (1), p. 41-53, 1998.
H. Derksen, “Quotients of algebraic group actions”, In Automorphisms of affine spaces (Curaçao, 1994), p. 191-200, Kluwer Acad. Publ., Dordrecht, 1995.
H. Derksen, “Inverse degrees and the Jacobian conjecture”, Comm. Algebra 22 (12), p. 4793-4794, 1994.
H. Derksen, “The kernel of a derivation”, J. Pure Appl. Algebra 84 (1), p. 13-16, 1993.
Combinatorics
H. Derksen and A. Fink, “Valuative invariants for polymatroids”, In 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Nancy, 2010.
H. Derksen, “The graph isomorphism problem and approximate categories”, J. Symbolic Comput. 59, p. 81-112, 2013.
H. Derksen and A. Fink, “Valuative invariants for polymatroids”, Adv. Math. 225 (4), p. 1840-1892, 2010.
H. Derksen, “Symmetric and quasi-symmetric functions associated to polymatroids”, J. Algebraic Combin. 30 (1), p. 43-86, 2009.
H. Derksen, “Error-correcting codes and Bh–sequences”, IEEE Trans. Inform. Theory 50 (3), p. 476-485, 2004.
H. Derksen, C. Eggermont and A. van den Essen, “Multimagic squares”, Amer. Math. Monthly 114 (8), p. 703-713, 2007.
Number Theory
H. Derksen and D. Masser, “Linear equations over multiplicative groups, recurrences, and mixing III”, Ergodic Th. Of Dyn. Systems, p. 1-19, 2017, online.
H. Derksen and D. Masser, “Linear equations over multiplicative groups, recurrences, and mixing II”, Indag. Math. (N.S.) 26 (11), p. 113-136, 2015.
H. Derksen and D. Masser, “Linear equations over multiplicative groups, recurrences, and mixing I”, Proc. London Math. Soc. (3), 104 (5), p. 1045-1083, 2012.
H. Derksen, “A Skolem-Mahler-Lech theorem in positive characteristic and finite automata”, Invent. Math. 168 (1), p. 175-224, 2007.
Biomedical Applications
C. Minoccheri, O. Alge, J. Gryak, K. Najarian, H. Derksen, “Quadratic multilinear discriminant analysis for tensorial data classification”, Algorithms 16 (2), p. 104, 2023.
H. Yao, H. Derksen, J. R. Golbus, J. Zhang, K. D. Aaronson, J. Gryak, K. Najarian, “A novel tropical geometry-based interpretable machine learning method: pilot application to delivery of advanced heart failure therapies”, IEEE Journal of Biomedical and Health Informatics 27 (1), p. 239-250, 2022.
N. Reamaroon, M. W. Sjoding, J. Gryak, B. D. Athey, K. Najarian, H. Derksen, “Automated detection of acute respiratory distress syndrome from chest X-Rays using Directionality Measure and deep learning features”, Computers in biology and medicine 134, p. 104463, 2021.
M. R. Mathis, M. Engoren, S. Kheterpal, K. Gunnerson, A. Williams, B. Biesterveld, A. Croteau, K. Ward, H. Alam, H. Derksen, G. Liu, R. Kim, N. Tokcan, K. Najarian, J. Gryak, “Early detection of postoperative deterioration in cardiac surgery patients using electronic health record and waveform data: A machine learning approach”, Anesthesia and Analgesia 132 (5 S_ SUPPL), p. 999-1003, 2021.
Z. Li, H. Derksen, J. Gryak, C. Jiang, Z. Gao, W. Zhang, H. Ghanbari, P. Gunaratne and K. Najarian,“Prediction of cardiac arrhythmia using deterministic automata”, Biomedical Signal Processing and control 63, p. 1-15, 2021.
L. Hernandez, R. Kim, N. Tokcan, H. Derksen, B. E. Biesterveld, A. Croteau, A. M. Williams, M. Mathis, K. Najarian, J. Gryak, “Multimodal tensor-based method for integrative and continuous patient monitoring during postoperative cardiac care”, Artificial Intelligence in Medicine 113, p. 102032, 2021.
M. Bagherian, R. B. Kim, C. Jiang, M. A. Sartor, H. Derksen and K. Najarian,“Coupled matrix–matrix and coupled tensor–matrix completion methods for predicting drug–target interactions”, Briefings in bioinformatics 22 (2), p. 2161-2171, 2021.
N. Reamaroon, M. W. Sjoding, H. Derksen, E. Sabeti, J. Gryak, R. P. Barbaro, B. D. Athey and K. Najarian, “Robust segmentation of lung in chest x-ray: applications in analysis of acute respiratory distress syndrome”, BMC Medical Imaging 20 (1), 1-13, 2020.
Z. Li, H. Derksen, J. Gryak, M. Hooshmand, A.Wood, H. Ghanbari, P. Gunaratne and K. Najarian, “Markov models for detection of ventricular arrhythmia”, 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), p.
1488-1491, 2019.
E. Sabeti, J. Gryak, H. Derksen, C. Biwer, S. Ansari, H. Isenstein, A. Kratz and K. Najarian, “Learning Using Concave and Convex Kernels: applications in predicting quality of sleep and level of fatigue in fibromyalgia”, Entropy 21 (5), 442, p. 1-16, 2019.
Z. Li, H. Derksen, J. Gryak, M. Hooshmand, A. Wood, H. Ghanbari, P. Gunaratne and K. Najarian, “Supraventricular tachycardia detection via machine learning algorithms”, 2018 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), p. 2419-2422, 2018.
Z. Li, H. Derksen, J. Gryak, H. Ghanbari, P. Gunaratne and K. Najarian, “A novel atrial fibrillation prediction algorithm applicable to recordings from portable devices”, 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), p. 4034-4037, 2018.
S. Ansari, N. Farzaneh, M. Heung, K. Oldham, H. Derksen K. R. Ward and K. Najarian, “Real-time detection of intradialytic hypotension using a novel polyvinylidene fluoride based sensor”, In IEEE-EMBS International Conference on Biomedical and Health Informatics (BHI), 2016.
R. Fazlali, N. Karimi, S.M. R. Soroushmehr, S. Samavi, B. Nallamothu, H. Derksen and K. Najarian, “Robust catheter identification and tracking in X-ray angiographic sequences”, Engineering in Medicine and Biology (EMBC), 37th Annual International Conference of the IEEE, p. 7901-7904, 2015.
S. Kazeminia, N. Karimi, S.M.R. Soroushmehr, S. Samavi, H.Derksen and K. Najarian, “Region of interest for lossless compression of bone X-ray images”, Engineering in Medicine and Biology (EMBC), 37th Annual International Conference of the IEEE, p. 3061-3064, 2015.
S.M.R. Soroushmehr, A. Bafna, S. Schlosser, K. Ward, H. Derksen and K. Najarian, “CT image segmentation in traumatic brain injury”, Engineering in Medicine and Biology (EMBC), 37th Annual International Conference of the IEEE, p. 2973-2976, 2015.
C. Biwer, A. Rothberg, H. IglayReger, H. Derksen, C. F. Burant and K. Najarian, “Windowed persistent homology: a topological signal processing algorithm applied to clinical obesity data”, PLOS one 12 (5): e0177696, 2017.
A. Belle, S. Ansari, M. Spadafore, V. Convertino, K. R. Ward, H. Derksen and K. Najarian, “A signal processing approach for detection of hemodynamic instability before decompensation”, PLOS one 11 (2):e014544, 2016.
Computer Vision and Image Processing
P. D. McKee, H. Derksen, J. A. Christian, “View Invariants for Three-Dimensional Points with Constrained Observer Motion”, Journal of Guidance, Control, and Dynamics 46 (2), p. 277-285, 2023.
J.A. Christian, H. Derksen, “Invariant Theory as a tool for spacecraft navigation”, AAS/AIAA Astrodynamics Specialist Conference, 2022.
J.A. Christian, H. Derksen and R. Watkins, “Lunar crater identification in digital images”, The Journal of the Astronautical Sciences 68 (4), 1056-1144, 2021.
M. Nejati, S. Samavi, H. Derksen and K. Najarian, “Denoising by low-rank and sparse representation”, J. Vis. Commun. Imag. R. 36, p 28-39, 2016.
Y. Ma, A. Y. Yang, H. Derksen and R. Fossum, “Estimation of subspace arrangements with applications in modeling and segmenting mixed data”, SIAM Review 50 (3), p. 413-458, 2008.
Y. Ma, H. Derksen, W. Hong and J. Wright, “Segmentation of multivariate mixed data via lossy coding and compression”, IEEE Trans. On Pattern Analysis and Machine Intelligence (PAMI) 29 (9), p. 1546-1562, 2007.
Rao, H. Derksen, R. Fossum, Y. Ma, A. Wagner and A. Yang, “The algebra and statistics of generalized principal component analysis”, In SPIE conference on Visual Communications and Image Processing (VCIP), 2007
J. Wright, W. Hong, Y. Ma and H. Derksen, “Segmentation of multivariate mixed data via lossy coding and compression”, In SPIE conference on Visual Communications and Image Processing (VCIP), 2007.