Thomas Peter Witelski
Professor in the Department of Mathematics
My primary area of expertise is the solution of nonlinear ordinary and partial differential equations for models of physical systems. Using asymptotics along with a mixture of other applied mathematical techniques in analysis and scientific computing I study a broad range of applications in engineering and applied science. Focuses of my work include problems in viscous fluid flow, dynamical systems, and industrial applications. Approaches for mathematical modelling to formulate reduced systems of mathematical equations corresponding to the physical problems is another significant component of my work.
Current Appointments & Affiliations
 Professor in the Department of Mathematics, Mathematics, Trinity College of Arts & Sciences 2011
 Professor in the Department of Mechanical Engineering and Materials Science, Pratt School of Engineering, Duke University 2016
Contact Information
 295 Physics Bldg, Box 90320, Durham, NC 277080320
 Box 90320, Durham, NC 277080320
 witelski@math.duke.edu (919) 6602841
 Google Scholar
 MathSciNet: 357178
 Methods of Mathematical Modeling textbook
 Primary webpage at math.duke.edu
 ResearcherID: G53892016
 Background

Education, Training, & Certifications
 Ph.D., California Institute of Technology 1995
 B.S.E., Cooper Union 1991

Duke Appointment History
 Professor in the Department of Mechanical Engineering and Materials Science, Mechanical Engineering and Materials Science, Pratt School of Engineering 2012  2015
 Associate Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2005  2011
 Assistant Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 1998  2005
 Recognition

Awards & Honors
 Expertise

Subject Headings
 Publications & Artistic Works

Selected Publications

Books
 Witelski, T, and Bowen, M. Methods of Mathematical Modelling: Continuous Systems and Differential Equations. September 18, 2015. Full Text

Conference Papers
 Peterson, E, Shearer, M, Witelski, TP, and Levy, R. "Stability of traveling waves in thin liquid films driven by gravity and surfactant." 2009. Link to Item
 Witelski, TP. "Computing finitetime singularities in interfacial flows." 2002. Link to Item

Journal Articles
 Gao, Y, Ji, H, Liu, JG, and Witelski, TP. "Global existence of solutions to a tear film model with locally elevated evaporation rates." Physica D: Nonlinear Phenomena 350 (July 2017): 1325. Full Text
 Ji, H, and Witelski, TP. "Finitetime thin film rupture driven by modified evaporative loss." Physica D: Nonlinear Phenomena 342 (March 2017): 115. Full Text
 George, C, Virgin, LN, and Witelski, T. "Experimental study of regular and chaotic transients in a nonsmooth system." International Journal of NonLinear Mechanics 81 (May 2016): 5564. Full Text
 Sanaei, P, Richardson, GW, Witelski, T, and Cummings, LJ. "Flow and fouling in a pleated membrane filter." Journal of Fluid Mechanics 795 (May 2016): 3659. Full Text
 Smolka, LB, McLaughlin, CK, and Witelski, TP. "Oil capture from a water surface by a falling sphere." Colloids and Surfaces A: Physicochemical and Engineering Aspects 497 (May 2016): 126132. Full Text
 Dijksman, JA, Mukhopadhyay, S, Gaebler, C, Witelski, TP, and Behringer, RP. "Obtaining selfsimilar scalings in focusing flows." Physical review. E, Statistical, nonlinear, and soft matter physics 92, no. 4 (October 22, 2015): 043016. Full Text
 Witelski, TP. "Preface to the special issue on “Thin films and fluid interfaces”." Journal of Engineering Mathematics 94, no. 1 (October 2015): 13. Full Text
 Hall Taylor, N, Hewitt, I, Ockendon, J, and Witelski, T. "A new model for disturbance waves." International Journal of Multiphase Flow 66 (November 2014): 3845. Full Text
 Witelski, T, Virgin, LN, and George, C. "A driven system of impacting pendulums: Experiments and simulations." Journal of Sound and Vibration 333, no. 6 (March 17, 2014): 17341753. Full Text
 Smolka, LB, and Witelski, TP. "Biaxial extensional motion of an inertially driven radially expanding liquid sheet." PHYSICS OF FLUIDS 25, no. 6 (June 2013). Full Text Link to Item
 Chapman, SJ, Trinh, PH, and Witelski, TP. "Exponential Asymptotics for Thin Film Rupture." SIAM Journal of Applied Mathematics 73 (2013): 232253. Full Text
 Witelski, T, Virgin, LN, and George, C. "A driven system of impacting pendulums: Experiments and simulations." Journal of Sound and Vibration (2013).
 Huang, Y, Witelski, TP, and Bertozzi, AL. "Anomalous exponents of selfsimilar blowup solutions to an aggregation equation in odd dimensions." Applied Mathematics Letters 25, no. 12 (2012): 23172321. Full Text
 Wiebe, R, Virgin, LN, and Witelski, TP. "A parametrically forced nonlinear system with reversible equilibria." International Journal of Bifurcation and Chaos 22, no. 6 (2012). Full Text
 Witelski, T, Ambrose, D, Bertozzi, A, Layton, A, Li, Z, and Minion, M. "Preface: Special issue on fluid dynamics, analysis and numerics." Discrete and Continuous Dynamical Systems  Series B 17, no. 4 (2012): iii. Full Text
 Aydemir, E, Breward, CJW, and Witelski, TP. "The effect of polar lipids on tear film dynamics." Bull Math Biol 73, no. 6 (June 2011): 11711201. Full Text Link to Item
 Aguareles, M, Chapman, SJ, and Witelski, T. "Motion of spiral waves in the complex GinzburgLandau equation." Physica D: Nonlinear Phenomena 239, no. 7 (2010): 348365. Full Text
 Bernoff, AJ, and Witelski, TP. "Stability and dynamics of selfsimilarity in evolution equations." Journal of Engineering Mathematics 66, no. 1 (2010): 1131. Full Text
 Witelski, TP. "The subtle art of blowing bubbles (News and Views: Fluid Dynamics)." Nature Physics 5 (May 2009): 315316. (Academic Article) Link to Item
 Gratton, MB, and Witelski, TP. "Transient and selfsimilar dynamics in thin film coarsening." Physica D: Nonlinear Phenomena 238, no. 2324 (2009): 23802394. Full Text
 Hwang, HJ, and Witelski, TP. "Shorttime pattern formation in thin film equations." Discrete and Continuous Dynamical Systems 23, no. 3 (2009): 867885. Full Text
 Smolka, LB, and Witelski, TP. "On the planar extensional motion of an inertially driven liquid sheet." Physics of Fluids 21, no. 4 (2009). Full Text
 Witelski, TP, and Bowen, M. "Singular perturbation theory." Scholarpedia 4 (2009): 39513951. Full Text
 Witelski, TP. "Fluid dynamics: The subtle art of blowing bubbles." Nature Physics 5, no. 5 (2009): 315316. Full Text
 Aguareles, M, Chapman, SJ, and Witelski, T. "Interaction of spiral waves in the complex GinzburgLandau equation." Physical Review Letters 101, no. 22 (2008). Full Text
 Aguareles, M, Chapman, SJ, and Witelski, TP. "Interaction of spiral waves in the Complex GinzburgLandau equation." Physical Review Letters 101, no. 224101 (2008). (Academic Article) Link to Item
 Catllá, AJ, Schaeffer, DG, Witelski, TP, Monson, EE, and Lin, AL. "On spiking models for synaptic activity and impulsive differential equations." SIAM Review 50, no. 3 (2008): 553569. Full Text
 DiCarlo, DA, Juanes, R, LaForce, T, and Witelski, TP. "Nonmonotonic traveling wave solutions of infiltration into porous media." Water Resources Research 44, no. 2 (2008). Full Text
 Gratton, MB, and Witelski, TP. "Coarsening of dewetting thin films subject to gravity." Physical Review E 77, no. 016301 (2008): 111. (Academic Article) Link to Item
 Gratton, MB, and Witelski, TP. "Coarsening of unstable thin films subject to gravity." Physical Review E  Statistical, Nonlinear, and Soft Matter Physics 77, no. 1 (2008). Full Text
 Gratton, MB, and Witelski, TP. "Coarsening of unstable thin films subject to gravity." Phys Rev E Stat Nonlin Soft Matter Phys 77, no. 1 Pt 2 (January 2008): 016301. Full Text Link to Item
 Santillan, ST, Plaut, RH, Witelski, TP, and Virgin, LN. "Large oscillations of beams and columns including selfweight." International Journal of NonLinear Mechanics 43, no. 8 (2008): 761771. Full Text
 Levy, R, Shearer, M, and Witelski, TP. "Gravitydriven thin liquid films with insoluble surfactant: Smooth traveling waves." European Journal of Applied Mathematics 18, no. 6 (2007): 679708. Full Text
 Schaeffer, DG, Shearer, M, and Witelski, TP. "Boundaryvalue problems for hyperbolic equations related to steady granular flow." Mathematics and Mechanics of Solids 12, no. 6 (2007): 665699. Full Text
 Bowen, M, and Witelski, TP. "The linear limit of the dipole problem for the thin film equation." SIAM Journal on Applied Mathematics 66, no. 5 (2006): 17271748. Full Text
 Witelski, TP, Shearer, M, and Levy, R. "Growing surfactant waves in thin liquid films driven by gravity." Applied Mathematics Research eXpress 2006 (2006). Full Text
 Munch, A, Wagner, B, and Witelski, TP. "Lubrication models with small to large slip lengths." Journal of Engineering Mathematics 53, no. 34 (December 2005): 259283. (Academic Article) Link to Item
 Fetzer, R, Jacobs, K, Münch, A, Wagner, B, and Witelski, TP. "New slip regimes and the shape of dewetting thin liquid films." Phys Rev Lett 95, no. 12 (September 16, 2005): 127801. Full Text Link to Item
 Glasner, KB, and Witelski, TP. "Collision versus collapse of droplets in coarsening of dewetting thin films." Physica D: Nonlinear Phenomena 209, no. 14 SPEC. ISS. (2005): 80104. Full Text
 Haskett, RP, Witelski, TP, and Sur, J. "Localized Marangoni forcing in driven thin films." Physica D: Nonlinear Phenomena 209, no. 14 SPEC. ISS. (2005): 117134. Full Text
 Münch, A, Wagner, B, and Witelski, TP. "Lubrication models with small to large slip lengths." Journal of Engineering Mathematics 53, no. 34 (2005): 359383. Full Text
 Witelski, TP, and Rienstra, SW. "Introduction to practical asymptotics III." Journal of Engineering Mathematics 53, no. 34 (2005): 199. Full Text
 Witelski, TP. "Motion of wetting fronts moving into partially prewet soil." Advances in Water Resources 28, no. 10 SPEC. ISS. (2005): 11331141. Full Text
 Sur, J, Witelski, TP, and Behringer, RP. "Steadyprofile fingering flows in Marangoni driven thin films." Phys Rev Lett 93, no. 24 (December 10, 2004): 247803. Full Text Link to Item
 Borucki, LJ, Witelski, T, Please, C, Kramer, PR, and Schwendeman, D. "A theory of pad conditioning for chemicalmechanical polishing." Journal of Engineering Mathematics 50, no. 1 (2004): 124. Full Text
 Smolka, LB, Belmonte, A, Henderson, DM, and Witelski, TP. "Exact solution for the extensional flow of a viscoelastic filament." European Journal of Applied Mathematics 15, no. 6 (2004): 679712. Full Text
 Witelski, TP, Bernoff, AJ, and Bertozzi, AL. "Blowup and dissipation in a criticalcase unstable thin film equation." European Journal of Applied Mathematics 15, no. 2 (2004): 223256. Full Text
 Merdan, H, and Caginalp, G. "Decay of solutions to nonlinear parabolic equations: renormalization and rigorous results." Discrete and Continuous Dynamical Systems  Series B 3, no. 4 (August 2003): 565588. Full Text
 Glasner, KB, and Witelski, TP. "Coarsening dynamics of dewetting films." Phys Rev E Stat Nonlin Soft Matter Phys 67, no. 1 Pt 2 (January 2003): 016302. Full Text Link to Item
 Glasner, KB, and Witelski, TP. "Coarsening dynamics of dewetting films." Physical Review E  Statistical, Nonlinear, and Soft Matter Physics 67, no. 1 2 (2003): 1630211630212.
 Schaeffer, DG, Shearer, M, and Witelski, T. "Onedimensional solutions of an elastoplasticity model of granular material." Math. Models and Methods in Appl. Sciences 13 (2003): 16291671. (Academic Article)
 Shearer, M, Schaeffer, DG, and Witelski, TP. "Stability of shear bands in an elastoplastic model for granular flow: The role of discreteness." Mathematical Models and Methods in Applied Sciences 13, no. 11 (2003): 16291671. Full Text
 Witelski, TP, and Bowen, M. "ADI schemes for higherorder nonlinear diffusion equations." Applied Numerical Mathematics 45, no. 23 (2003): 331351. Full Text
 Witelski, TP. "Intermediate asymptotics for Richards' equation in a finite layer." Journal of Engineering Mathematics 45, no. 34 (2003): 379399. Full Text
 Witelski, TP. "Nonlinear Differential Equations, Mechanics and Bifurcation." Discrete and Continuous Dynamical Systems  Series B 3, no. 4 (2003): i.
 Bernoff, AJ, and Witelski, TP. "Linear stability of sourcetype similarity solutions of the thin film equation." Applied Mathematics Letters 15, no. 5 (2002): 599606.
 Bertozzi, AL, Grün, G, and Witelski, TP. "Dewetting films: Bifurcations and concentrations." Nonlinearity 14, no. 6 (2001): 15691592. Full Text
 Vaynblat, D, Lister, JR, and Witelski, TP. "Rupture of thin viscous films by van der waals forces: Evolution and selfsimilarity." Physics of Fluids 13, no. 5 (2001): 11301141. Full Text
 Vaynblat, D, Lister, JR, and Witelski, TP. "Symmetry and selfsimilarity in rupture and pinchoff: A geometric bifurcation." European Journal of Applied Mathematics 12, no. 3 (2001): 209232. Full Text
 Witelski, TP, Ono, K, and Kaper, TJ. "Critical wave speeds for a family of scalar reactiondiffusion equations." Applied Mathematics Letters 14, no. 1 (2001): 6573. Full Text
 Witelski, TP, Schaeffer, DG, and Shearer, M. "A discrete model for an illposed nonlinear parabolic PDE." Physica D: Nonlinear Phenomena 160, no. 34 (2001): 189221. Full Text
 Witelski, TP, and Bernoff, AJ. "Dynamics of threedimensional thin film rupture." Physica D: Nonlinear Phenomena 147, no. 12 (2000): 155176. Full Text
 Witelski, TP, and Bernoff, AJ. "Stability of selfsimilar solutions for van der Waals driven thin film rupture." PHYSICS OF FLUIDS 11, no. 9 (September 1999): 24432445. Full Text Link to Item
 Witelski, TP, and Hendriks, F. "Stability of gas bearing sliders for large bearing number: Convective instability of the tapered slider." Tribology Transactions 42, no. 1 (1999): 216222.
 Witelski, TP. "Large bearing number stability analysis for tango class gas bearing sliders©." Tribology Transactions 42, no. 3 (1999): 668674.
 Bernoff, AJ, Bertozzi, AL, and Witelski, TP. "Axisymmetric surface diffusion: Dynamics and stability of selfsimilar pinchoff." Journal of Statistical Physics 93, no. 34 (1998): 725776.
 Brenner, MP, and Witelski, TP. "On spherically symmetric gravitational collapse." Journal of Statistical Physics 93, no. 34 (1998): 863899.
 Witelski, TP, Grosberg, AY, and Tanaka, T. "On the properties of polymer globules in the high density limit." Journal of Chemical Physics 108, no. 21 (1998): 91449149.
 Witelski, TP, and Bernoff, AJ. "Selfsimilar asymptotics for linear and nonlinear diffusion equations." Studies in Applied Mathematics 100, no. 2 (1998): 153193.
 Witelski, TP. "Equilibrium interface solutions of a degenerate singular CahnHilliard equation." Applied Mathematics Letters 11, no. 5 (1998): 127133.
 Witelski, TP. "Horizontal infiltration into wet soil." Water Resources Research 34, no. 7 (1998): 18591863.
 Witelski, TP. "Dynamics of air bearing sliders." Physics of Fluids 10, no. 3 (1998): 698708.
 Witelski, TP. "Similarity solutions of the lubrication equation." Applied Mathematics Letters 10, no. 5 (1997): 107113.
 Witelski, TP. "Perturbation Analysis for Wetting Fronts in Richards' Equation." Transport in Porous Media 27, no. 2 (1997): 121134.
 Witelski, TP. "Segregation and mixing in degenerate diffusion in population dynamics." Journal of Mathematical Biology 35, no. 6 (1997): 695712.
 Cohen, DS, and Witelski, TP. "Inaccessible states in timedependent reaction diffusion." Studies in Applied Mathematics 97, no. 4 (1996): 301319.
 Witelski, TP. "Traveling wave solutions for case II diffusion in polymers." Journal of Polymer Science, Part B: Polymer Physics 34, no. 1 (1996): 141150.
 Witelski, TP. "The structure of internal layers for unstable nonlinear diffusion equations." Studies in Applied Mathematics 97, no. 3 (1996): 277300.
 WITELSKI, TP, and COHEN, DS. "FORBIDDEN REGIONS FOR SHOCK FORMATION IN DIFFUSIVE SYSTEMS." STUDIES IN APPLIED MATHEMATICS 95, no. 3 (October 1995): 297317. Link to Item
 Cohen, DS, Jr, ABW, and Witelski, TP. "Shock formation in a multidimensional viscoelastic diffusive system." SIAM Journal on Applied Mathematics 55, no. 2 (1995): 348368.
 Witelski, TP, and Cohen, DS. "Perturbed reversible systems." Physics Letters A 207, no. 12 (1995): 8386.
 Witelski, TP. "Shocks in nonlinear diffusion." Applied Mathematics Letters 8, no. 5 (1995): 2732. Full Text
 Witelski, TP. "Merging traveling waves for the porousFisher's equation." Applied Mathematics Letters 8, no. 4 (1995): 5762.
 Witelski, TP. "Stopping and merging problems for the porous media equation." IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) 54, no. 3 (1995): 227243. Full Text
 Witelski, TP. "An asymptotic solution for traveling waves of a nonlineardiffusion Fisher's equation." Journal of Mathematical Biology 33, no. 1 (1994): 116. Full Text
 Witelski, T, Ng, P, Ying, J, Jundy, J, and Bove, J. "An application of pattern recognition and infrared spectroscopy to water analysis." International Journal of Environmental Analytical Chemistry 44, no. 2 (1991): 127136. (Academic Article)

 Teaching & Mentoring

Recent Courses
 Scholarly, Clinical, & Service Activities

Presentations & Appearances
 Finitetime rupture in thin films driven by nonconservative effects. Making a Splash: Droplets, Jets and Other Singularities Workshop. ICERM. March 21, 2017 2017
 Multiscale dynamics of dewetting fluid films. Applied PDE Seminar. Dept of Applied Math, University of Washington. November 17, 2016 2016
 Directions for Graduate and Undergraduate Modeling Courses. 2016 SIAM Annual Meeting. SIAM. July 13, 2016 2016
 Experimental study of regular and chaotic transients in a nonsmooth system. Dynamics Days US 2016. January 8, 2016 2016
 Nonlinear dynamics of dewetting thin films. May 29, 2014 2014
 Multiscale dynamics of dewetting fluids. July 1, 2013 2013
 Crowddynamicsdriven instabilities of bridges. June 11, 2013 2013
 Basic questions on the design and efficiency of toothbrushes. May 17, 2013 2013
 Modeling studies of the dynamics of read/write heads in computer hard disk drives. July 26, 2012 2012
 The influence of surfactants on transient behaviors in forced film thinning. Workshop on Surfactant Driven Thin Film Flows. Fields Institute. February 23, 2012 2012
 Nonlinear dynamics of driven coupled impacting pendula. January 20, 2012 2012
 Transient and selfsimilar dynamics in thin film coarsening. July 18, 2010 2010
 Transient and selfsimilar dynamics in thin film coarsening. July 18, 2010 2010
 Transient Stability Analysis for an Inertiallydriven Extensional Motion of a Planar Liquid Sheet. May 26, 2010 2010
 Transient Stability Analysis for an Inertiallydriven Extensional Motion of a Planar Liquid Sheet. May 26, 2010 2010
 Transient and selfsimilar dynamics in thin film coarsening. May 24, 2010 2010
 Mean field model of droplet coarsening. April 17, 2010 2010
 Mean field models and transient effects for coarsening dynamics in fluid films. April 13, 2010 2010
 Coarsening: transient and selfsimilar dynamics in 1D. March 12, 2010 2010
 Perturbation analysis for impulsive differential equations. Duke Math Grad/Faculty seminar. February 12, 2010 2010
 Film coarsening: transient and selfsimilar dynamics in 1D. August 11, 2009 2009
 Coarsening of Dewetting Films I: the Influence of Gravity. July 9, 2009 2009
 Perturbation Analysis for Impulsive Differential Equations. July 9, 2009 2009
 On computing solutions of thin film equations. April 4, 2009 2009
 Coarsening: transient and selfsimilar dynamics in 1D. April 1, 2009 2009
 Coarsening: transient and selfsimilar dynamics in 1D. March 23, 2009 2009
 Coarsening dynamics of thin fluid films. November 1, 2008 2008
 Coarsening dynamics of dewetting fluid films. September 6, 2008 2008

Service to the Profession
 Faculty Mentor . Graduate Student Mathematical Modeling Camp 2017 (GSMMC). Rensselaer Polytechnic Institute, Dept of Math. June 14, 2017  June 17, 2017 2017
 Main organizer. 2016 Mathematical Problems in Industry (MPI) Workshop. Duke University. June 13, 2016  June 17, 2016 2016
 Member of organizing committee. Dynamics Days 2016 US. Duke University. January 7, 2016  January 10, 2016 2016
 Associate Editor. Journal of Engineering Mathematics. 2016 2016
 Member of the editorial board. MathematicsinIndustry Case Studies. 2016 2016
 Associate Director. Statistics and Applied Mathematical Sciences Institute (SAMSI). SAMSI. July 1, 2014  June 30, 2017 2014  2017
 Member of board of Series Editors. Springer Texts in Applied Mathematics. Springer Publishing. 2014 2014
 Meber of organizing committee. Flows and Patterns: The Physics of Fluids, Granular Materials, and Soft Matter. Duke University. October 12, 2013  October 13, 2013 2013
 Faculty Mentor. Graduate Student Mathematical Modeling Camp 2013 (GSMMC). Rensselaer Polytechnic Institute, Dept of Math. . June 11, 2013  June 14, 2013 2013
 Meber of organizing committee. BIRS workshop 12w5035: Thin liquid films and fluid interfaces: models, experiments and applications. December 9, 2012  December 14, 2012 2012
 Faculty Mentor : Graduate Student Mathematical Modeling Camp 2011 (GSMMC). June 11, 2011  June 12, 2011 2011
 member of organizing committee : Dynamics Days 2011. January 5, 2011  January 8, 2011 2011
 Committee member : Gene Golub SIAM Summer School Committee. Society of Industrial and Applied Mathematics. January 1, 2011  December 31, 2018 2011  2018
 Main guest editor : FAN (Fluid Dynamics, Analysis, Numerics) proceedings/ JT Beale Special issue DCDSB. January 2011  January 2012 2011  2012
 coEditorinChief : Journal of Engineering Mathematics. January 1, 2011  December 31, 2015 2011  2015
 Chair, organizing committee : FAN2010: Fluid dynamics, Analysis and Numerics (A conference in honor of Thomas Beale). June 28, 2010  June 30, 2010 2010
 Faculty mentor/problem presenter : 2nd UK Graduate Modelling Camp. April 6, 2010  April 9, 2010 2010
 member of organizing committee : CTMS Workshop: Theory and Applications of Multiscale Modeling. September 11, 2009  September 12, 2009 2009
 Minisymposium sessions coorganizer : SIAM 2009 Annual meeting, MS65/MS76 Contemporary Applications of Perturbation and Asymptotic Methods I/II. July 9, 2009  July 10, 2009 2009
 Division Editor : Journal of Mathematical Analysis and Applications. July 2009  December 2014 2009  2014
 Faculty Mentor : Graduate Student Mathematical Modeling Camp 2009. June 9, 2009  June 12, 2009 2009
 Associate Editor. Discrete and Continuous Dynamical Systems, series B. 2006  2017 2006  2017
 Associate Editor. European Journal of Applied Mathematics. 2004 2004

Service to Duke
Some information on this profile has been compiled automatically from Duke databases and external sources. (Our About page explains how this works.) If you see a problem with the information, please write to Scholars@Duke and let us know. We will reply promptly.