Research

Sze-Man Ngai

Department of Mathematical Sciences 
Georgia Southern University 
Statesboro, GA 30460-8093 
USA

E-mails: smngai@geoergiasouthern.edu

My main research interests are fractal geometry and the theory of multifractal measures. 
I am also interested in their interactions with the theories of wavelets, self-affine tilings
and fractal differential equations.


Publications

 ·        D.-W. Deng and S.-M. Ngai, Eigenvalue estimates for Laplacians on measure J. Funct. Anal., to appear.

 ·        J. F.-C. Chan, S.-M. Ngai, and A. Teplyaev, One-dimensional wave equations defined by fractal Laplacians, J. Anal. Math., to appear.

 ·         S.-M. Ngai and J. Tong, Infinite iterated function systems with overlaps, Ergodic Theory Dynam. Systems, to appear.

  ·         G. Deng, C. Liu, and Sze-Man Ngai, Dimensions of the boundary of a graph-directed self-similar set with overlaps, Houston J. Math., to appear.

   ·        D.-W. Deng and S.-M. Ngai, Fractal tiles and quasidisks, Math. Z. 279, 2015, 359 – 387.

    ·        Q.-R. Deng and S.-M. Ngai, Dimensions of fractals generated by bi-Lipschitz maps, Abst.  Appl. Anal. 2014, Article ID 549741, http://dx.doi.org/10.1155/2014/549741

     ·        K.-S. Lau and S.-M. Ngai, Boundary theory on the Hata tree, Nonlinear Anal. 95 (2014), 292–307.

     ·         Q. Deng, K.-S. Lau, and S.-M. Ngai, Separation conditions for iterated function systems with overlaps, Contemp. Math. 600 (2013), 1–20.

·         D.-W. Deng, T. Jiang, and S.-M. NgaiStructure of planar integral self-affine tilingsMath. Nachr285 (2012), 447-475.

·         S.-M. NgaiSingularity and L2-dimension of self-similar measuresChaos Solitons Fractals45 (2012), 256-265.

·         K.-S. Lau and S.-M. NgaiMartin boundary and exit space on the Sierpinski gasketSci. China Math.  55 (2012), 475 - 494.

·         S.-M. NgaiSpectral asymptotics of Laplacians associated to one-dimensional iterated function systems with overlapsCanad. J. Math. 63(2011), 648-688.

·         Q. Deng and S.-M. NgaiConformal iterated function systems with overlapsDyn. Syst. 26 (2011), 103 - 123.

·         S.-M. Ngai, F. Wang, and X. Dong, Graph-directed iterated function systems satisfying the generalized finite type condition, Nonlinearity 23 (2010), 2333 - 2350.

·         J. Chen and S.-M. NgaiEigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlapsJ. Math. Anal. Appl. 364 (2010), 222 - 241.

·         Q. Deng and S.-M. NgaiMultifractal formalism for self-affine measures with overlapsArch. Math. 92 (2009), 514 - 625.

·         K.-S. Lau, S.-M. Ngai, and X.-Y. Wang, Separation conditions for conformal iterated function systems,  Monatsh. Math. 156 (2009), no. 4, 325 - 355.

·         S.-M. NgaiMultifractal structure of noncompactly supported measuresFractals,  16 (2008), 209 - 226.

·         K.-S. Lau and S.-M. NgaiA generalized finite type condition for iterated function systemsAdv. Math. 208 (2007), 647 - 671.

·         J. Hu, K-S. Lau and S.-M. NgaiLaplace operators related to self-similar measures on Rd J. Funct. Anal. 239 (2006), 542 - 565.

·         D.-W. Deng and S.-M. NgaiVertices of self-similar tiles Illinois J. Math. 49 (2005), no. 3, 857 - 872.

·         F. Jordan and S.-M. NgaiReptiles with holesProc. Edinb. Math. Soc. (2) 48 (2005), issue 03, 651 - 671.

·         S.-M. Ngai and Y. Wang, Self-similar measures associated with IFS with non-uniform contraction ratios Asian J. Math. (2005), no. 2, 227 - 244.

·         S.-M. Ngai and T.-M. Tang, Topology of connected self-similar tiles in the plane with disconnected interiors,Topology Appl. 150 (2005), no. 1-3, 139 - 155.

·         S.-M. Ngai and T.-M. Tang, A technique in the topology of connected self-similar tiles, Fractals 12, no. 4 (2004), 389 - 403.

·         M. Das and S.-M. NgaiGraph-directed iterated function systems with overlapsIndiana Univ. Math. J. 59 (2004), no. 1, 109-134.

·         E. J. Bird, S.-M. Ngai and A. TeplyaevFractal Laplacians on the unit interval Ann. Sci. Math. Quebec 27 (2003), no. 2, 135-168.

·         K.-S. Lau and S.-M. NgaiDimensions of the boundaries of self-similar sets Experiment. Math. 12 (2003), 13-26.

·         S.-M. Ngai and N. Nguyen, The Heighway dragon revisited Discrete Comput. Geom. 29 (2003), 603-623.

·         S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps J. London Math. Soc. (2) 63 (2001), no. 3, 655-672.

·         K.-S. Lau, S.-M. Ngai, and H. RaoIterated function systems with overlaps and self-similar measures J. London Math. Soc. (2) 63 (2001), no. 1, 99-116.

·         K.-S. Lau and S.-M. NgaiSecond-order self-similar identities and multifractal decompositions Indiana Univ. Math. J. 49 (2000), 925-972. 

·         H. Fan, S.-M. Ngai and K.-S. Lau, Iterated function systems with overlaps Asian J. Math. 4 (2000), 527-552.

·         S.-M. Ngai, V. Sirvent, J. J. P. Veerman, and  Y. Wang, On 2-reptiles in the plane, Geom. Dedicata 82 (2000), 325-344.

·         K.-S. Lau and S.-M. NgaiMultifractal measures and a weak separation condition, Adv. Math. 141 (1999), 45-96.

·         K.-S. Lan and S.-M. NgaiLq-spectrum of Bernoulli convolutions associated with P.V. numbers, Osaka J. Math. 36 (1999),  993 -1010.

·         K.-S. Lau and S.-M. NgaiLq-spectrum of the Bernoulli convolution associated with the golden ratio, Studia Math. 131 (1998), 225-251.

·         S.-M. NgaiMultifractal decomposition for a family of overlapping  self-similar measures, Fractal frontiers (Denver, CO, 1997), 151--161, World Sci. Publishing, River Edge, NJ, 1997.

·         S-M. Ngai, A dimension result arising from the Lq-spectrum of a measure, Proc. Amer. Math. Soc. 125 (1997), 2943-2951.

For reprints or preprints, please send me an e-mail: smngai@georgiasouthern.edu

Selected Presentations

·         Fractal differential equations defined by iterated function systems with overlaps, Optimal Configurations on the Sphere and Other Manifolds, May 19, 2010, Vanderbilt University, Nashville, TN, U.S.A.

·         Fractal Laplacians defined IFSs with overlaps, Fractal Laplacians Defined by Iterated Function Systems
with Overlaps, 
AMS 2010 Spring Central Section Meeting, April 10, 2010, Macalester College, St Paul, MN, U.S.A.

·         Spectral asymptotics of fractal Laplacians defined by iterated function systems with overlaps, The 29th Annual Southeastern-Atlantic Regional Conference on Differential EquationsOctober 17, 2009, Mercer University, GA, U.S.A.

·         Conformal iterated function systems with overlaps,Workshop on Fractals and Tilings 2009Strobl, Austria, July 6 - 10, 2009

·         Fractal Laplacians defined by iterated function systems with overlaps, Workshop on fractal geometry and related topics, Zhuhai Campus, Zhongshan University, China, December 2008.

·         Fractal Laplacians defined by iterated function systems with overlaps and their spectral asymptoticsWorkshop on fractal geometry and the theory of tilingsZhongshanUniversity, China, July 2008.

·         Spectral dimension of fractal Laplacians defined by iterated function systems with overlaps, Workshop on Functional Analysis and Optimization Theory, The Chinese University of Hong Kong, December 2006.

·         Multifractal structure of noncompactly supported measures, Special Session on Fractals and Tilings, 21st Summer Conference on Topology and its Applications, Georgia Southern Univ. Statesboro, GA,  July 2006.

·         On a class of Laplacians defined by fractal measures. Conference on Differential & Difference Equations and Applications. Florida Institute of Technology, Melbourne, FL. August 2005.

·         Fractal Laplace operators on bounded open subsets of Euclidean spaces. 2nd Conference on Analysis and Probability on Fractals, Cornell University, Ithaca, NY,  May -- June 2005.


 

·         Fractal Laplace operators on open subsets of RdInternational Conference on Applicable Harmonic Analysis, Hangzhou, China, May 2005.


 

·         Iterated function systems of generalized finite type. The Third International Congress  of Chinese Mathematicians (ICCM 2004), December, 2004.


 

·         A generalized finite type condition for iterated function systems, 2004 AMS Central Sectional Meeting, Northwestern University, Evanston, IL, October 23, 2004.


 

·         Topological structure of connected self-similar tiles in the plane, Department of Mathematics, the Chinese University of Hong Kong, June 26, 2003.


 

·         Hausdorff dimension of the boundaries of a class of self-similar sets, International Conference on Fractal Geometry & Stochastics III, Friedrichroda, Germany, March 20, 2003.


 

·         Topological structure of reptiles and self-affine tiles, AMS Southeasten Sectional Meeting, University of Central Florida, Orlando, FL, November 10, 2002.

·         Fractal Laplacians on an interval,  Conference on Analysis and Probability on Fractals, Cornell University, Ithaca, N.Y., June 16, 2002.


 

·         Fractal Fourier Series, Department of Mathematics, The Chinese University of Hong Kong, May 15,  2002.


 

·         Eigenvalues and eigenfunctions of a class of fractal Laplacians on the unit interval, Special Session on Dynamic Equations on Time Scales, AMS-MAA Joint Meeting, Atlanta, Georgia, March 10, 2002.


 

·         Hausdorff dimension of boundaries of self-similar sets,  Department of Mathematics, The Chinese University of Hong Kong, December 22,  2001.


 

·         Graph-directed iterated function systems with overlaps, Special Session on Fractals of the American Mathematical Society Meeting, Columbus, OH, September 22, 2001.


 

·         Self-similar measures associated  to iterated function systems with different contraction ratios, Department of Mathematics, The Chinese University of Hong Kong, June 23, 2001.


 

·         Hausdorff dimension of graph self-similar sets with overlaps, Department of Mathematics, The Chinese University of Hong Kong, June 9, 2001.


 

·         Self-similar measures and dilation equations, Colloquium, Department of Mathematics, University of Louisville, November 10, 2000.


 

·         Fractal measures and dilation equations,  Georgia Institute of Technology, March 15, 2000.

·         Laplacians on fractals I, II, III, lecture series given at the Chinese University of Hong Kong, August 9-11, 1999.

·         Absolute Continuity of Self-Similar Measures, Young Analysts Meeting of the Southeast, Furman University, Greenville, SC, July 6-10, 1999.

·         Hausdorff Dimension of Overlapping Self-Similar Sets, Southeastern Analysis Meeting, Vanderbilt Univ., Nashville, TN, May 20-23, 1999.

·         Dilation equations and iterated function systems with overlapsMinisymposium on ``Wavelets and Their Applications", SIAM Annual Meeting, Atlanta, GA, May 9-15, 1999.


 

·          Hausdorff dimension of overlapping self-similar fractals,  Georgia Institute of Technology, January 13, 1999.

·         Second-order self-similar identities, AMS special session meeting on Fractal Geometry and Related Topics, Louisville, KY, March 20-21, 1998.

·         Self-similarity in fractals, wavelets and tilingsCornell University, March 1998.

·         Multifractal formalism for overlapping self-similar measures, Cornell University, February 1998.

·         Some Applications of the Renewal Theorem in Fractal Geometry, Conference on Dynamical Systems and Fractal Geometry, ZhongshanUniv., Dec 23-26, 1997.

·         Using Mathematical Softwares in Education and Research, South China Normal Univ., Dec 22, 1997.

·         Dilation equations with scaling factor equal to the golden ratio, Workshop on Wavelets and Their Applications, Hong Kong, May5 - 9, 1997.

·         Multifractal decomposition for a family of overlapping self-similar measures,


Fractal 97 --- The 4-th International Multidisciplinary Working Conference on Fractals, Denver Colorado, USA, April 8 - 11 1997.

·         Multifractal decomposition for certain iterated function systems with overlaps, Hong Kong Polytechnic University, Dec 1996.

·         Multifractal decomposition for certain overlapping self-similar measuresZhongshan Univ., China, and Guangdong Univ. of Technology, China, Nov 1996.

·         A dimension result arising from the Lq-spectrum of a  measure, The South China Normal University, May 1996.

·         Multifractal formalism and a dimension formula,  Workshop on nonlinear dynamics, chaos and complexity,  Hong Kong Baptist Univ., March 1996.

·         Iterated function systems and multifractal decomposition, Hong Kong University of Science and Technology, April 1995.

·         L^q-dimension of the Bernoulli convolution associated with the P.V. numbers,  Conference in Wavelets and Fractals, University of  Pittsburgh, May 1994.


My Mathematical Geneology



      (Last updated: November 18, 2009.)

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