# American Institute of Mathematical Sciences

• Previous Article
Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants
• ERA-MS Home
• This Volume
• Next Article
The gap between near commutativity and almost commutativity in symplectic category
2013, 20: 77-96. doi: 10.3934/era.2013.20.77

 1 Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Received  December 2012 Revised  June 2013 Published  October 2013

We prove a generalization of Gromov's packing inequality to symplectic embeddings of the boundaries of two balls such that the bounded components of the complements of the image spheres are disjoint. Moreover, we define a capacity which measures the size of Weinstein tubular neighborhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Furthermore, we introduce the spherical variant of the relative Gromov radius and prove its finiteness for monotone Lagrangian tori in symplectic vector spaces.
Citation: Kai Zehmisch. The codisc radius capacity. Electronic Research Announcements, 2013, 20: 77-96. doi: 10.3934/era.2013.20.77
##### References:
 [1] C. Abbas, Finite energy surfaces and the chord problem, Duke Math. J., 96 (1999), 241-316. doi: 10.1215/S0012-7094-99-09608-4.  Google Scholar [2] C. Abbas, Introduction to Compactness Results in Symplectic Field Theory, to appear, Springer, 2013. Google Scholar [3] V. I. Arnol'd, The first steps of symplectic topology, Uspekhi Mat. Nauk, 41 (1986), 3-18, 229.  Google Scholar [4] S. Artstein-Avidan and Y. Ostrover, A Brunn-Minkowski inequality for symplectic capacities of convex domains, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn044, 31 pp. doi: 10.1093/imrn/rnn044.  Google Scholar [5] J.-F. Barraud and O. Cornea, Homotopic dynamics in symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 109-148. doi: 10.1007/1-4020-4266-3_03.  Google Scholar [6] J.-F. Barraud and O. Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. (2), 166 (2007), 657-722. doi: 10.4007/annals.2007.166.657.  Google Scholar [7] S. M. Bates, A capacity representation theorem for some non-convex domains, Math. Z., 227 (1998), 571-581. doi: 10.1007/PL00004394.  Google Scholar [8] P. Biran, Symplectic packing in dimension $4$, Geom. Funct. Anal., 7 (1997), 420-437. doi: 10.1007/s000390050014.  Google Scholar [9] P. Biran and O. Cornea, A Lagrangian quantum homology, in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009, 1-44.  Google Scholar [10] P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989. doi: 10.2140/gt.2009.13.2881.  Google Scholar [11] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888. doi: 10.2140/gt.2003.7.799.  Google Scholar [12] M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc., 66 (1960), 74-76. doi: 10.1090/S0002-9904-1960-10400-4.  Google Scholar [13] L. Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, J. Topol. Anal., 2 (2010), 57-75. doi: 10.1142/S1793525310000240.  Google Scholar [14] Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226. doi: 10.1215/S0012-7094-98-09506-0.  Google Scholar [15] K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds, Amer. Math. Soc. Colloq. Publ., 59, American Mathematical Society, Providence, RI, 2012.  Google Scholar [16] K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 1-44. doi: 10.1017/CBO9780511755187.002.  Google Scholar [17] K. Cieliebak and K. Mohnke, Punctured holomorphic curves and Lagrangian embeddings,, in preparation., ().   Google Scholar [18] M. Damian, Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv., 87 (2012), 433-462. doi: 10.4171/CMH/259.  Google Scholar [19] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567. Google Scholar [20] Y. Eliashberg and L. Polterovich, Local Lagrangian $2$-knots are trivial, Ann. of Math. (2), 144 (1996), 61-76. doi: 10.2307/2118583.  Google Scholar [21] U. Frauenfelder, Gromov convergence of pseudoholomorphic disks, J. Fixed Point Theory Appl., 3 (2008), 215-271. doi: 10.1007/s11784-008-0078-1.  Google Scholar [22] H. Geiges, An Introduction to Contact Topology, Cambridge Stud. Adv. Math., 109, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611438.  Google Scholar [23] H. Geiges and K. Zehmisch, Eliashberg's proof of Cerf's theorem, J. Topol. Anal., 2 (2010), 543-579. doi: 10.1142/S1793525310000446.  Google Scholar [24] H. Geiges and K. Zehmisch, Symplectic cobordisms and the strong Weinstein conjecture, Math. Proc. Cambridge Philos. Soc., 153 (2012), 261-279. doi: 10.1017/S0305004112000163.  Google Scholar [25] H. Geiges and K. Zehmisch, How to recognise a 4-ball when you see one, to appear, Münster J. Math., (2013). Google Scholar [26] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.  Google Scholar [27] D. Hermann, Inner and outer Hamiltonian capacities, Bull. Soc. Math. France, 132 (2004), 509-541.  Google Scholar [28] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563. doi: 10.1007/BF01232679.  Google Scholar [29] H. Hofer, V. Lizan and J.-C. Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal., 7 (1997), 149-159. doi: 10.1007/BF02921708.  Google Scholar [30] H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.  Google Scholar [31] H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.  Google Scholar [32] H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in Analysis, et cetera, Academic Press, Boston, MA, 1990, 405-427.  Google Scholar [33] H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar [34] M.-Y. Jiang, An inequality for symplectic capacity, Bull. London Math. Soc., 31 (1999), 237-240. doi: 10.1112/S0024609398004962.  Google Scholar [35] L. Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal., 10 (2000), 829-862. doi: 10.1007/PL00001640.  Google Scholar [36] B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc., 65 (1959), 59-65. doi: 10.1090/S0002-9904-1959-10274-3.  Google Scholar [37] D. McDuff, Symplectic embeddings and continued fractions: A survey, Jpn. J. Math., 4 (2009), 121-139. doi: 10.1007/s11537-009-0926-9.  Google Scholar [38] D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, With an appendix by Yael Karshon, Invent. Math., 115 (1994), 405-434. doi: 10.1007/BF01231766.  Google Scholar [39] D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998.  Google Scholar [40] D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.  Google Scholar [41] D. McDuff and F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids, Ann. of Math. (2), 175 (2012), 1191-1282. doi: 10.4007/annals.2012.175.3.5.  Google Scholar [42] M. Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc., 66 (1960), 113-115. doi: 10.1090/S0002-9904-1960-10420-X.  Google Scholar [43] L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8299-6.  Google Scholar [44] F. Schlenk, Embedding Problems in Symplectic Geometry, de Gruyter Expositions in Mathematics, 40, Walter de Gruyter GmbH & Co. KG, Berlin, 2005. doi: 10.1515/9783110199697.  Google Scholar [45] J. Swoboda and F. Ziltener, Coisotropic displacement and small subsets of a symplectic manifold, Math. Z., 271 (2012), 415-445. doi: 10.1007/s00209-011-0870-2.  Google Scholar [46] J. Swoboda and F. Ziltener}, A symplectically non-squeezable small set and the regular coisotropic capacity,, preprint, ().   Google Scholar [47] C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431. doi: 10.1090/S0894-0347-00-00328-3.  Google Scholar [48] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.  Google Scholar [49] K. Zehmisch, Singularities and Self-Intersections of Holomorphic Discs, Doktorarbeit, Universität Leipzig, 2008. Google Scholar [50] K. Zehmisch, Lagrangian non-squeezing and a geometric inequality,, preprint, ().   Google Scholar [51] K. Zehmisch and F. Ziltener, Discontinuous capacities,, preprint, ().   Google Scholar

show all references

##### References:
 [1] C. Abbas, Finite energy surfaces and the chord problem, Duke Math. J., 96 (1999), 241-316. doi: 10.1215/S0012-7094-99-09608-4.  Google Scholar [2] C. Abbas, Introduction to Compactness Results in Symplectic Field Theory, to appear, Springer, 2013. Google Scholar [3] V. I. Arnol'd, The first steps of symplectic topology, Uspekhi Mat. Nauk, 41 (1986), 3-18, 229.  Google Scholar [4] S. Artstein-Avidan and Y. Ostrover, A Brunn-Minkowski inequality for symplectic capacities of convex domains, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn044, 31 pp. doi: 10.1093/imrn/rnn044.  Google Scholar [5] J.-F. Barraud and O. Cornea, Homotopic dynamics in symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 109-148. doi: 10.1007/1-4020-4266-3_03.  Google Scholar [6] J.-F. Barraud and O. Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. (2), 166 (2007), 657-722. doi: 10.4007/annals.2007.166.657.  Google Scholar [7] S. M. Bates, A capacity representation theorem for some non-convex domains, Math. Z., 227 (1998), 571-581. doi: 10.1007/PL00004394.  Google Scholar [8] P. Biran, Symplectic packing in dimension $4$, Geom. Funct. Anal., 7 (1997), 420-437. doi: 10.1007/s000390050014.  Google Scholar [9] P. Biran and O. Cornea, A Lagrangian quantum homology, in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009, 1-44.  Google Scholar [10] P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989. doi: 10.2140/gt.2009.13.2881.  Google Scholar [11] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888. doi: 10.2140/gt.2003.7.799.  Google Scholar [12] M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc., 66 (1960), 74-76. doi: 10.1090/S0002-9904-1960-10400-4.  Google Scholar [13] L. Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, J. Topol. Anal., 2 (2010), 57-75. doi: 10.1142/S1793525310000240.  Google Scholar [14] Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226. doi: 10.1215/S0012-7094-98-09506-0.  Google Scholar [15] K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds, Amer. Math. Soc. Colloq. Publ., 59, American Mathematical Society, Providence, RI, 2012.  Google Scholar [16] K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 1-44. doi: 10.1017/CBO9780511755187.002.  Google Scholar [17] K. Cieliebak and K. Mohnke, Punctured holomorphic curves and Lagrangian embeddings,, in preparation., ().   Google Scholar [18] M. Damian, Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv., 87 (2012), 433-462. doi: 10.4171/CMH/259.  Google Scholar [19] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567. Google Scholar [20] Y. Eliashberg and L. Polterovich, Local Lagrangian $2$-knots are trivial, Ann. of Math. (2), 144 (1996), 61-76. doi: 10.2307/2118583.  Google Scholar [21] U. Frauenfelder, Gromov convergence of pseudoholomorphic disks, J. Fixed Point Theory Appl., 3 (2008), 215-271. doi: 10.1007/s11784-008-0078-1.  Google Scholar [22] H. Geiges, An Introduction to Contact Topology, Cambridge Stud. Adv. Math., 109, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611438.  Google Scholar [23] H. Geiges and K. Zehmisch, Eliashberg's proof of Cerf's theorem, J. Topol. Anal., 2 (2010), 543-579. doi: 10.1142/S1793525310000446.  Google Scholar [24] H. Geiges and K. Zehmisch, Symplectic cobordisms and the strong Weinstein conjecture, Math. Proc. Cambridge Philos. Soc., 153 (2012), 261-279. doi: 10.1017/S0305004112000163.  Google Scholar [25] H. Geiges and K. Zehmisch, How to recognise a 4-ball when you see one, to appear, Münster J. Math., (2013). Google Scholar [26] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.  Google Scholar [27] D. Hermann, Inner and outer Hamiltonian capacities, Bull. Soc. Math. France, 132 (2004), 509-541.  Google Scholar [28] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563. doi: 10.1007/BF01232679.  Google Scholar [29] H. Hofer, V. Lizan and J.-C. Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal., 7 (1997), 149-159. doi: 10.1007/BF02921708.  Google Scholar [30] H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.  Google Scholar [31] H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.  Google Scholar [32] H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in Analysis, et cetera, Academic Press, Boston, MA, 1990, 405-427.  Google Scholar [33] H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar [34] M.-Y. Jiang, An inequality for symplectic capacity, Bull. London Math. Soc., 31 (1999), 237-240. doi: 10.1112/S0024609398004962.  Google Scholar [35] L. Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal., 10 (2000), 829-862. doi: 10.1007/PL00001640.  Google Scholar [36] B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc., 65 (1959), 59-65. doi: 10.1090/S0002-9904-1959-10274-3.  Google Scholar [37] D. McDuff, Symplectic embeddings and continued fractions: A survey, Jpn. J. Math., 4 (2009), 121-139. doi: 10.1007/s11537-009-0926-9.  Google Scholar [38] D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, With an appendix by Yael Karshon, Invent. Math., 115 (1994), 405-434. doi: 10.1007/BF01231766.  Google Scholar [39] D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998.  Google Scholar [40] D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.  Google Scholar [41] D. McDuff and F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids, Ann. of Math. (2), 175 (2012), 1191-1282. doi: 10.4007/annals.2012.175.3.5.  Google Scholar [42] M. Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc., 66 (1960), 113-115. doi: 10.1090/S0002-9904-1960-10420-X.  Google Scholar [43] L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8299-6.  Google Scholar [44] F. Schlenk, Embedding Problems in Symplectic Geometry, de Gruyter Expositions in Mathematics, 40, Walter de Gruyter GmbH & Co. KG, Berlin, 2005. doi: 10.1515/9783110199697.  Google Scholar [45] J. Swoboda and F. Ziltener, Coisotropic displacement and small subsets of a symplectic manifold, Math. Z., 271 (2012), 415-445. doi: 10.1007/s00209-011-0870-2.  Google Scholar [46] J. Swoboda and F. Ziltener}, A symplectically non-squeezable small set and the regular coisotropic capacity,, preprint, ().   Google Scholar [47] C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431. doi: 10.1090/S0894-0347-00-00328-3.  Google Scholar [48] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.  Google Scholar [49] K. Zehmisch, Singularities and Self-Intersections of Holomorphic Discs, Doktorarbeit, Universität Leipzig, 2008. Google Scholar [50] K. Zehmisch, Lagrangian non-squeezing and a geometric inequality,, preprint, ().   Google Scholar [51] K. Zehmisch and F. Ziltener, Discontinuous capacities,, preprint, ().   Google Scholar
 [1] Viktor L. Ginzburg and Basak Z. Gurel. The Generalized Weinstein--Moser Theorem. Electronic Research Announcements, 2007, 14: 20-29. doi: 10.3934/era.2007.14.20 [2] Benny Avelin, Tuomo Kuusi, Mikko Parviainen. Variational parabolic capacity. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5665-5688. doi: 10.3934/dcds.2015.35.5665 [3] Lasse Kiviluoto, Patric R. J. Östergård, Vesa P. Vaskelainen. Sperner capacity of small digraphs. Advances in Mathematics of Communications, 2009, 3 (2) : 125-133. doi: 10.3934/amc.2009.3.125 [4] Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253 [5] Jimmy Garnier, FranÇois Hamel, Lionel Roques. Transition fronts and stretching phenomena for a general class of reaction-dispersion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 743-756. doi: 10.3934/dcds.2017031 [6] Joseph E. Paullet, Joseph P. Previte. Analysis of nanofluid flow past a permeable stretching/shrinking sheet. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4119-4126. doi: 10.3934/dcdsb.2020090 [7] P. D. Howell, J. J. Wylie, Huaxiong Huang, Robert M. Miura. Stretching of heated threads with temperature-dependent viscosity: Asymptotic analysis. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 553-572. doi: 10.3934/dcdsb.2007.7.553 [8] Ricardo P. Beausoleil, Rodolfo A. Montejo. A study with neighborhood searches to deal with multiobjective unconstrained permutation problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 193-216. doi: 10.3934/jimo.2009.5.193 [9] Joseph D. Skufca, Erik M. Bollt. Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks. Mathematical Biosciences & Engineering, 2004, 1 (2) : 347-359. doi: 10.3934/mbe.2004.1.347 [10] Alexander Shmyrov, Vasily Shmyrov. The optimal stabilization of orbital motion in a neighborhood of collinear libration point. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 185-189. doi: 10.3934/naco.2017012 [11] Tobias Sutter, David Sutter, John Lygeros. Capacity of random channels with large alphabets. Advances in Mathematics of Communications, 2017, 11 (4) : 813-835. doi: 10.3934/amc.2017060 [12] Jie Xiao. On the variational $p$-capacity problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (3) : 959-968. doi: 10.3934/cpaa.2015.14.959 [13] Najwa Najib, Norfifah Bachok, Norihan Md Arifin, Fadzilah Md Ali. Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 423-431. doi: 10.3934/naco.2019041 [14] Ghulam Rasool, Anum Shafiq, Hülya Durur. Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2497-2515. doi: 10.3934/dcdss.2020399 [15] Adriano Festa, Simone Göttlich, Marion Pfirsching. A model for a network of conveyor belts with discontinuous speed and capacity. Networks & Heterogeneous Media, 2019, 14 (2) : 389-410. doi: 10.3934/nhm.2019016 [16] Jianbin Li, Ruina Yang, Niu Yu. Optimal capacity reservation policy on innovative product. Journal of Industrial & Management Optimization, 2013, 9 (4) : 799-825. doi: 10.3934/jimo.2013.9.799 [17] Qing Yang, Shiji Song, Cheng Wu. Inventory policies for a partially observed supply capacity model. Journal of Industrial & Management Optimization, 2013, 9 (1) : 13-30. doi: 10.3934/jimo.2013.9.13 [18] Yanjun He, Wei Zeng, Minghui Yu, Hongtao Zhou, Delie Ming. Incentives for production capacity improvement in construction supplier development. Journal of Industrial & Management Optimization, 2021, 17 (1) : 409-426. doi: 10.3934/jimo.2019118 [19] Dandan Hu, Zhi-Wei Liu. Location and capacity design of congested intermediate facilities in networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 449-470. doi: 10.3934/jimo.2016.12.449 [20] Rongrong Jin, Guangcun Lu. Representation formula for symmetrical symplectic capacity and applications. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4705-4765. doi: 10.3934/dcds.2020199

2020 Impact Factor: 0.929