Publications

2021

Caginalp, Gunduz

  • Caginalp, C., Caginalp, G., & Swigon, D. (2021). Stochastic Asset Flow Equations: Interdependence of Trend and Volatility. Physica A, 574.

Ermentrout, Bard

  • Riman, N., Victor, J.D., Boie, S.D., & Ermentrout, B. (2021). The Dynamics of Bilateral Olfactory Seach and Navigation. SIAM Review 63 (1), 100-120.
  • Orr, D., & Ermentrout, G.B. (2021). Noise-Driven Oscillations in Couples Excitable Systems. SIAM Journal on Applied Dynamical Systems 20 (2), 826-852.
  • Ding, Y., & Ermentrout, B. (2021). Traveling waves in non-local pulse-coupled networks. Journal of Mathematical Biology 82 (3), 1-20.
  • Savinov, M., Swigon, D., & Ermentrout, B. (2021). Synchronization and locking in oscillators with flexible periods. Chaos: An Interdisciplinary Journal of Nonlinear Science 31 (3), 033143.
  • Hengenius, J.B., Connor, E.G., Crimaldi, J.P., Urban, N.N., & Ermentrout, G.B. (2021). Olfactory navigation in the real world: Simple local search strategies for turbulent environments. Journal of Theoretical Biology 516, 110607.
  • Landsittle, J.A, Ermentrout, G.B., & Stiefel, K.M. (2021). A computational model of the shrimp-goby escape and communication system. Journal of Computational Neuroscience, 1-11.
  • Miyamae, T., Hashimoto, T., Abraham, M., Kawabata, R., Koshikizawa, S., ... & Ermentrout, B. (2021). Kcns3 deficiency disrupts Parvalbumin neuron physiology in mouse prefrontal cortex: Implications for the pathophysiology of schizophrenia. Neurobiology of Disease 155, 105382.
  • Mochan, E., Sego, T.J., Gaona, L., Rial, E., & Ermentrout, G.B. (2021). Compartmental Model Suggests Importance of Innate Immune Response to COVID-19 Infection in Rhesus Macaques. Bulletin of mathematical biology 83 (7), 1-26.

Hajlasz, Piotr

  • Esmayli, B., & Hajłasz, P. (2021). The Coarea Inequality. Annales Fennici Mathematici 46, 965-991.

Kaveh, Kiumars

  • Kaveh, K., Manon, C., Murata, T. (2021). Generic Tropical Initial Ideals of Cohen-Macaulay Algebras. J. Pure Appl. Algebra 225, No. 11, 106713, 11 pages.

Manfredi, Juan

  • Manfredi, J., & Stroffolini, B. (2021). Convergence of the Natural p-means for the p-Laplacian. ESAIM: Control, Optimization and Calculus of Variations 27, 33 pages.

Neilan, Michael

  • Neilan, M., & Otus, M.B. (2021). Divergence-free Scott-Vogelius elements on curved domains. SIAM Journal on Numerical Analysis, 59(2), 1090–1116.

Rubin, Jonathan

  • Rubin, J.E., Signerska-Rynkowska, J., Touboul, J. (2021). Type III Neuronal Dynamics in a Hybrid Nonlinear Model. SIAM J. Appl. Dyn. Syst.,  20(2), 953-980.

Schikorra, Armin

  • Schikorra, A. (2021). Limits of conformal immersions under a bound on a fractional normal curvature quantity. Vietnam Journal of Mathemathics 49, No. 2, 281-317.
  • Antil, H., Rautenberg, C. N., Schikorra, A. (2021). On a fractional version of a Murat compactness result and applications. SIAM Jounal of Mathematical Analysis 53, No. 3, 3158-3187.
  • Mengasha, T., Schikorra, A., Yeepo S. (2021). Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel. Advanced Math, 383(64), 107692.

Trenchea, Catalin

  • Bukač, M., & Trenchea, C. (2021). Boundary update via resolvent for fluid-structure interaction. Journal of Numerical Mathematics 29, No. 1, 1-22.
  • Bukač, M., Seboldt, A., & Trenchea, C. (2021). Refactorization of Cauchy's method: a second-order partitioned method for fluid-thick structure interaction problems. J. Math. Fluid Mech. 23, no. 3, Paper No. 64.

Wang-Erickson, Carl

  • Wake, P., & Wang-Erickson, C. (2021). The Eisenstein ideal with squarefree level. Adv. Math. 380, Paper No. 107543, 62 pp.

2020

Caginalp, Gunduz

  • Caginalp, G., & DeSantis, M. (2020). Nonlinear price dynamics of S&P 100 stocks. Physica A: Statistical Mechanics and its Applications, 547, 122067.
  • Caginalp, C., & Caginalp, G. (2020). Derivation of non-classical stochastic price dynamics equations. Physica A: Statistical Mechanics and its Applications, 560, 125118.

Chadam, John

  • Chen, X., Saunders, D., & Chadam, J. (2020). Analysis of an optimal stopping problem arising from hedge fund investing. Journal of Mathematical Analysis and Applications, 483(1), 123559.

Chen, Xinfu

  • Carrillo, J. A., Chen, X., Wang, Q., Wang, Z., & Zhang, L. (2020). Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion. SIAM Journal on Applied Mathematics, 80(1), 232-261.
  • Chen, X., Saunders, D., & Chadam, J. (2020). Analysis of an optimal stopping problem arising from hedge fund investing. Journal of Mathematical Analysis and Applications, 483(1), 123559.

DeBlois, Jason

  • DeBlois, J. (2020). Bounds for several-disk packings of hyperbolic surfaces. Journal of Topology and Analysis, 12(01), 131-167.
  • DeBlois, J., Miller, N., & Patel, P. (2020). Effective virtual and residual properties of some arithmetic hyperbolic 3–manifolds. Transactions of the American Mathematical Society, 373(11), 8219-8257.
  • DeBlois, J., & Romanelli, K. (2020). The maximal injectivity radius of hyperbolic surfaces with geodesic boundary. Geometriae Dedicata, 1-27.

Ermentrout, Bard

  • Gumaste, A., Coronas-Samano, G., Hengenius, J., Axman, R., Connor, E. G., Baker, K. L., ... & Verhagen, J. V. (2020). A Comparison between Mouse, In Silico, and Robot Odor Plume Navigation Reveals Advantages of Mouse Odor Tracking. Eneuro, 7(1).
  • Orr, D., & Ermentrout, B. (2020). Oscillations via Excitable Cells.
  • Liu, A., Papale, A. E., Hengenius, J., Patel, K., Ermentrout, B., & Urban, N. N. (2020). Mouse navigation strategies for odor source localization. Frontiers in neuroscience, 14, 218.
  • Touboul, J. D., Piette, C., Venance, L., & Ermentrout, G. B. (2020). Noise-Induced Synchronization and Antiresonance in Interacting Excitable Systems: Applications to Deep Brain Stimulation in Parkinson’s Disease. Physical Review X, 10(1), 011073.
  • Heitmann, S., Ermentrout, G.B. (2020). Direction-Selective Motion Discrimination by Traveling Waves in a Visual Cortex. PLoS Computational Biology 16 (9), e1008164.
  • Ermentrout, B., Park, Y., & Wilson, D. (2020). Correction to ‘Recent advances in coupled oscillator theory’. Philosophical Transactions of the Royal Society A, 378(2170), 20200121.
  • Ermentrout, B., Park, Y., & Wilson, D. (2019). Recent advances in coupled oscillator theory. Philosophical Transactions of the Royal Society A, 377(2160), 20190092.

Fedorov, Roman

  • Fedorov, R., Soibelman, A., & Soibelman, Y. (2020). Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 16, 070.

Gartside, Paul

  • Espinoza, B., Gartside, P., & Pitz, M. (2020). Graph‐like compacta: Characterizations and Eulerian loops. Journal of Graph Theory.

Hajlasz, Piotr

  • Bonk, M., Capogna, L., Hajłasz, P., Shanmugalingam, N., & Tyson, J. (2020). Analysis in Metric Spaces. Notices of the American Mathematical Society, 67(2), 253-256.
  • Alvarado, R., Górka, P., & Hajłasz, P. (2020). Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure. Journal of Functional Analysis, 108628.
  • Goldstein, P., Hajłasz, P., & Pankka, P. (2020). Topologically nontrivial counterexamples to Sard’s theorem. International Mathematics Research Notices, 2020(20), 7073-7096.
  • Alvarado, R.,  & Hajłasz, P. (2020). A note on metric-measure spaces supporting Poincare inequalities. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31, 15-23.
  • P. Hajłasz,  & S. Zimmerman (2020). An implicit function theorem for Lipschitz mappings into metric spaces. Indiana Univ. Math. J. 69, 205–228.

Jiang, Huiqiang

  • Chen, X., Jiang, H., & Liu, G. (2020). Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems-A, 40(6), 3253.

Kaveh, Kiumars

  • Kaveh, K., & Khovanskii, A. G. (2020). Intersections of Hypersurfaces and Ring of Conditions of a Spherical Homogeneous Space. Symmetry, Integrability and Geometry: Methods and Applications, 16(0), 16-12.
  • Escobar, L., & Kaveh, K. (2020). Convex polytopes, algebraic geometry, and combinatorics. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY, 67(8).

Layton, William

  • DeCaria, V., Iliescu, T., Layton, W., McLaughlin, M., & Schneier, M. (2020). An artificial compression reduced order model. SIAM Journal on Numerical Analysis, 58(1), 565-589.
  • Layton, W., & Schneier, M. (2020). Diagnostics for eddy viscosity models of turbulence including data-driven/neural network based parameterizations. Results in Applied Mathematics, 100099.
  • Li, Y., Hou, Y., Layton, W., & Zhao, H. (2020). Adaptive partitioned methods for the time-accurate approximation of the evolutionary Stokes–Darcy system. Computer Methods in Applied Mechanics and Engineering, 364, 112923.
  • Layton, W., & McLaughlin, M. (2020). Doubly-adaptive artificial compression methods for incompressible flow. Journal of Numerical Mathematics, 28(3), 175-192.
  • Jiang, N., Layton, W., McLaughlin, M., Rong, Y., & Zhao, H. (2020). On the Foundations of Eddy Viscosity Models of Turbulence. Fluids, 5(4), 167.

Lennard, Chris

  • Japón, M., Lennard, C., & Popescu, R. (2020). A fixed-point characterization of weakly compact sets in L1 (μ) spaces. Journal of Mathematical Analysis and Applications, 124228.
  • Gallagher, T. M., Japón, M., & Lennard, C. (2020). The nonexpansive and mean nonexpansive fixed point properties are equivalent for affine mappings. Journal of Fixed Point Theory and Applications, 22(4), 1-16.

Lewicka, Marta

  • Lewicka, M. (2020). Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models. Archive for Rational Mechanics and Analysis, 1-31.
  • Lewicka, M., & Lučić, D. (2020). Dimension reduction for thin films with transversally varying prestrain: oscillatory and nonoscillatory cases. Communications on Pure and Applied Mathematics, 73(9), 1880-1932.
  • Lewicka, M., & Peres, Y. (2020). Which domains have two-sided supporting unit spheres at every boundary point?. Expositiones Mathematicae, 38(4), 548-558.
  • Lewicka M., Manfredi J. & Ricciotti D. (2020). Random walks and random Tug of War in the Heisenberg group. Mathematische Annalen 377, 797–-846.
  • Lewicka M. (2020 June) A Course on Tug-of-War Games with Random Noise: Introduction and Basic Constructions (Universitext), Springer; 1st ed. 2020 edition, 263pp.
  • Acu, B., Danielli, D., Lewicka, M., Pati, A.N., Saraswathy RV, Teboh-Ewungkem, M.I. (Eds.) (2020). Advances in Mathematical Sciences, Springer; 1st ed. 2020 edition, 369pg.

Manfredi, Juan

  • Brustad, K. K., Lindqvist, P., & Manfredi, J. J. (2020). A Discrete Stochastic Interpretation of the Dominative $ p $-Laplacian. Differential and Integral Equations, 33(9/10), 465-488.
  • Domokos, A., Manfredi, J.J. (2020) $C^{1,α}$-subelliptic regularity on SU(3)SU(3) and compact, semi-simple Lie groups. Anal.Math.Phys. 10, 4, 1-32.
  • del Teso, F., Manfredi, J. J., & Parviainen, M. (2020). Convergence of dynamic programming principles for the p-Laplacian. Advances in Calculus of Variations, 1(ahead-of-print).
  • Domokos, A., & Manfredi, J. J. (2020, March). Hilbert-Haar coordinates and Miranda's theorem in Lie groups. In Bruno Pini Mathematical Analysis Seminar (Vol. 11, No. 1, pp. 94-118).
  • Domokos, A., & Manfredi, J. J. (2020, March). Regularity for quasilinear PDEs in Carnot groups via Riemannian approximation. In Bruno Pini Mathematical Analysis Seminar (Vol. 11, No. 1, pp. 119-142).
  • Lewicka M., Manfredi J., & Ricciotti D. (2020). Random walks and random Tug of War in the Heisenberg group. Mathematische Annalen 377, 797–-846.

Ming, Chen

  • Chen, M., Liang, Z., Wang, D., & Xu, R. (2020). Energy equality in compressible fluids with physical boundaries. SIAM Journal on Mathematical Analysis, 52(2), 1363-1385.
  • Chen, R. M., Hu, J., Wang, D., Wang, T., & Yuan, D. (2020). Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics. Journal of Differential Equations, 269(9), 6899-6940.

Neilan, Michael

  • Fu, G., Guzmán, J., & Neilan, M. (2020). Exact smooth piecewise polynomial sequences on Alfeld splits. Mathematics of Computation, 89(323), 1059-1091.
  • Neilan, M., Salgado, A. J., & Zhang, W. (2020). The Monge–Ampère equation. In Handbook of Numerical Analysis (Vol. 21, pp. 105-219). Elsevier.
  • Guzmán, J., Lischke, A., & Neilan, M. (2020). Exact sequences on Powell–Sabin splits. Calcolo, 57(2), 1-25.
  • Neilan, M. (2020, July). The Stokes complex: A review of exactly divergence–free finiteelement pairsfor incompressibleflows. In 75 Years of Mathematics of Computation: Symposium on Celebrating 75 Years of Mathematics of Computation, November 1-3, 2018, the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island (Vol. 754, p. 141). American Mathematical Soc..
  • Neilan, M., & Zytoon, A. (2020) Connection between Grad-Div Stabilized Stokes Finite Elements and Divergence-Free Stokes Finite Elements. Int. J. Numer. Anal. Model., 17(6), 839–857.
  • Linke, A., Merdon, C., & Neilan, M. (2020). Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem. Electronic Transactions on Numerical Analysis, 52, 281–294.

Popescu, Roxana

  • Japón, M., Lennard, C., & Popescu, R. (2020). A fixed-point characterization of weakly compact sets in L1 (μ) spaces. Journal of Mathematical Analysis and Applications, 124228.

Rabier, Patrick

  • Rabier, P. J. (2020). The generic dimension of spaces of $\mathbf {A} $-harmonic polynomials. Publicacions Matemàtiques, 64(1), 157-182.

Rubin, Jonathan

  • Letson, B., & Rubin, J. E. (2020). LOR for analysis of periodic dynamics: A one-stop shop approach. SIAM Journal on Applied Dynamical Systems, 19(1), 58-84.
  • Letson, B., Rubin, J.E. (2020) Local orthogonal rectification:  Deriving natural coordinates to study flows relative to manifolds. Discrete & Continuous Dynamical Systems-B 25 (9), 3725.
  • Vich, C., Dunovan, K., Verstynen, T., & Rubin, J. (2020). Corticostriatal synaptic weight evolution in a two-alternative forced choice task: a computational study. Communications in Nonlinear Science and Numerical Simulation, 82, 105048.
  • Rubin, J. E., Vich, C., Clapp, M., Noneman, K., & Verstynen, T. (2020). The credit assignment problem in cortico‐basal ganglia‐thalamic networks: A review, a problem and a possible solution. European Journal of Neuroscience.
  • Wang, Y., & Rubin, J. E. (2020). Complex bursting dynamics in an embryonic respiratory neuron model. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(4), 043127.
  • Theis, N., Dash, K., Patel, N., Muldoon, B., Iyengar, S., Rubin, J., ... & Prasad, K. (2020). Multiplex Network Pattern Analysis for Structure-Function Connectivity Coupling in Psychosis Risk. Biological Psychiatry, 87(9), S201-S202.
  • Greer, M. L., Akman, O., Comar, T. D., Hrozencik, D., & Rubin, J. E. (2020). Paying Our Dues: The Role of Professional Societies in the Evolution of Mathematical Biology Education. Bulletin of Mathematical Biology, 82(5), 59-59.
  • Whalen, T.C., Willard, A.M., Rubin, J.E, Gittis, A.H. (2020). Delta oscillations are a robust biomarker of dopamine depletion severity and motor dysfunction in awake mice. J. Neurophysiol. 124:312-329.
  • Codianni, M., Daun, S., Rubin, J.E. (2020) Bottom-up and top-down control of rhythm generation in a central pattern generator unit for stick insect locomotion. Biological Cybernetics, 114: 533-555.
  • Duan, X., Rubin, J. E., & Swigon, D. (2020). Identification of affine dynamical systems from a single trajectory. Inverse Problems. 36:085004
  • Phillips, R. S., Rosner, I., Gittis, A. H., & Rubin, J. E. (2020). The effects of chloride dynamics on substantia nigra pars reticulata responses to pallidal and striatal inputs. Elife, 9, e55592.
  • Codianni, M. G., Daun, S., & Rubin, J. E. (2020). The roles of ascending sensory signals and top-down central control in the entrainment of a locomotor CPG. Biological Cybernetics, 114(6), 533-555.
  • Schwab BC, Kase D, Zimnik A, Rosenbaum R, Codianni MG, Rubin JE, Turner RS. (2020 Oct). Neural activity during a simple reaching task in macaques is counter to gating and rebound in basal ganglia-thalamic communication. PLoS Biol;18(10):e3000829.
  • Whalen TC, Willard AM, Rubin JE, Gittis AH. (2020 Aug). Delta oscillations are a robust biomarker of dopamine depletion severity and motor dysfunction in awake mice. J Neurophysiol;124(2):312-329.

Schikorra, Armin

  • Schikorra, A., & Van Schaftingen, J. (2020). An estimate of the Hopf degree of fractional Sobolev mappings. Proceedings of the American Mathematical Society, 148(7), 2877-2891.
  • Khomrutai, S., & Schikorra, A. (2020). On C1, alpha-regularity for critical points of a geometric obstacle-type problem. Journal de Mathématiques Pures et Appliquées.
  • Lenzmann, E., & Schikorra, A. (2020). Sharp commutator estimates via harmonic extensions. Nonlinear Analysis, 193, 111375.
  • Schikorra, A. (2020). Hölder-topology of the Heisenberg group. Aequationes mathematicae, 94(2), 323-343.
  • Bugiera, L., Lenzmann, E., Schikorra, A., & Sok, J. (2020). On symmetry of traveling solitary waves for dispersion generalized NLS. Nonlinearity, 33(6), 2797.
  • Mazowiecka, K., Rodiac, R., & Schikorra, A. (2020). Epsilon-regularity for p-harmonic maps at a free boundary on a sphere. Analysis & PDE, 13(5), 1301-1331.

Swigon, David

  • Duan, X., Rubin, J. E., & Swigon, D. (2020). Identification of affine dynamical systems from a single trajectory. Inverse Problems.

Trenchea, Catalin

  • Burkardt, J., & Trenchea, C. (2020). Refactorization of the midpoint rule. Applied Mathematics Letters, 106438.

Vainchtein, Anna

  • Vainchtein, A., Cuevas-Maraver, J., Kevrekidis, P. G., & Xu, H. (2020). Stability of traveling waves in a driven Frenkel-Kontorova model. Communications in Nonlinear Science and Numerical Simulation, 105236.
  • Vainchtein, A. (2020). Rarefactive lattice solitary waves with high-energy sonic limit. Physical Review E, 102(5), 052218.

van Doorn, Floris

  • Sojakova, K., Doorn, F. V., & Rijke, E. (2020, July). Sequential Colimits in Homotopy Type Theory. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (pp. 845-858).
  • van Doorn, F., Ebner, G., and Lewis, R.Y. (2020). Maintaining a Library of Formal Mathematics. CICM 2020: 13th Conference on Intelligent Computer Mathematics.
  • The mathlib Community (2020). The Lean Mathematical Library. CPP 2020: Certified Programs and Proofs.
  • Han, J. M., & van Doorn, F. (2020, January). A formal proof of the independence of the continuum hypothesis. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (pp. 353-366).

Wang, Dehua

  • Li, L. A., Wang, D., & Wang, Y. (2020). Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier–Stokes equations. Communications in Mathematical Physics, 376(1), 353-384.
  • M. Chen, J. Hu, and D. Wang. (2020). Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients. Math. Ann. 376, 863-912..
  • Chen, M., Liang, Z., Wang, D., & Xu, R. (2020). Energy equality in compressible fluids with physical boundaries. SIAM Journal on Mathematical Analysis, 52(2), 1363-1385.
  • Wang, D., Wu, J., & Ye, Z. (2020). Global regularity of the three-dimensional fractional micropolar equations. Journal of Mathematical Fluid Mechanics, 22, 1-36.
  • Chen, R. M., Hu, J., Wang, D., Wang, T., & Yuan, D. (2020). Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics. Journal of Differential Equations, 269(9), 6899-6940.
  • Liu, C. J., Wang, D., Xie, F., & Yang, T. (2020). Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces. Journal of Functional Analysis, 108637.
  • Mu, Y., & Wang, D. (2020). Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows. Calculus of Variations and Partial Differential Equations, 59(4), 1-42..
  • Z. Liang, & D. Wang. (2020). Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows. Math. Models Methods Appl. Sci. 30, no. 12, 2445-2486.

Wang, Linhong

  • Wang, L., & Wang, X. (2020). A note on generic Clifford algebras of binary cubic forms. Algebras and Representation Theory, 23(4), 1797-1806.

Wang-Erickson, Carl

  • Wake, P.,  & Wang-Erickson, C. (2020). The rank of Mazur's Eisenstein ideal. Duke Math. J. 169, no. 1, 31–115.
  • Levin, B.,  & Wang-Erickson, C. (2020). A Harder-Narasimhan theory for Kisin modules. Algebr. Geom. 7, no. 6, 645–695.

Yao, Song

  • Yao, S. (2020). $L^p$ solutions of reflected backward stochastic differential equations with jumps. ESAIM: Probability and Statistics, 24, 935-962.

Yotov, Ivan

  • Ambartsumyan, I., Khattatov, E., Nordbotten, J. M., & Yotov, I. (2020). A multipoint stress mixed finite element method for elasticity on simplicial grids. SIAM Journal on Numerical Analysis, 58(1), 630-656.
  • Ambartsumyan, I., Khattatov, E., Wang, C., & Yotov, I. (2020). Stochastic multiscale flux basis for Stokes-Darcy flows. Journal of Computational Physics, 401, 109011.
  • Caucao, S., Li, T., & Yotov, I. (2020, June). A Cell-Centered Finite Volume Method for the Navier–Stokes/Biot Model. In International Conference on Finite Volumes for Complex Applications (pp. 325-333). Springer, Cham.