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Manifolds with Density: Fuller References

SELECTED PUBLICATIONS IN THE HISTORY OF MANIFOLDS WITH DENSITY:

[1959] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Indo. Control 2 (1959), 101-112, Eqn. 2.3. Gives a version of Gaussian log-Sobolev inequality, used by Perelman, often attributed to Gross [1975] or sometimes Federbush [1969].

[1966] E. Nelson, A quartic interaction in two dimensions, mathematical Theory of Elementary Particles (Goodman, R. and Segal, I., eds.), MIT Press, 1966, 69-73. Gross [G] says that the entire subject of logarithmic Sobolev inequalities and contractivity properties of semigroups was started in this paper.

[1966] Harper, L. H. Optimal numberings and isoperimetric problems on graphs. J. Combinatorial Theory 1 1966 385-393. Apparently uses measure and metric, cited by Ledoux-Talagrand [1991], both cited by [Ros, §1.4, p. 182].

[1969] Paul Federbush, A partially alternate derivation of a result of Nelson, J. Math. Phys. 10 (1969), 50-52. Gives Gaussian log-Sobolev inequality, used by Perelman, often attributed to Gross [1975], actually probably due to Stam [1959].

[1970] André Lichnerowicz, Variétés riemanniennes a tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A650-A653. Studies Ric – Hess log density to prove splitting theorems.

[1973] E Nelson, The free Markov field, J. Funct. Analy. 12, 211-227. Gross survey ([G] below) says equivalent form of Gaussian log-Sobolev inequality.

[1975] Christer Borell, The Brunn-Minkowski inequality in Gauss Space, Invent. Math. 30 (1975) 207-216. Also: V. N. Sudakov and B. S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures, J. Soviet Math. (1978), 9-18 (1974 in Russian). Proof of Gaussian isoperimetric inequality.

[1975] Gross, Leonard, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083.  Proves Gaussian log-Sobolev inequality, used by Perelman. Not yet realized that it was trivial consequence of Gaussian isoperimetric inequality via analytic version which [Ros,  §3.4] traces back to Ehrhard [E, 1984] and Bobkov [B7, 1997], first observed by Ledoux ’94 and Beckner ’96 (published [1999]) (see Morgan Blog and email from Milman). Precursors are Stam [1959] and Federbush [1969] (see Gross survey [G]).

[1977] R. R. Coifman and G. Weiss, Extensions of Hardy Spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. First appearance (as I learned from Cheeger) of metric-measure spaces (singular manifolds  with density) called “homogeneous spaces.”

[1981]  M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. Metric measure spaces; see Shioya below.

[1982] Edward Witten. Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661–692 (1983). Pointed out to me by Doan.  d_t = e^-ht d e^ht   The main idea seems to be the same as Perelman’s: changing the density does not change the topology but it can help with the analysis.

[1984] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., 59 (1984) 335-395.

[1985] D. Bakry and M. Émery, Diffusions Hypercontractive, Séminaire de Probabilités XIX 1983/4, Lecture Notes Math. 1123, Springer, 1985, 177-206. Generalized Ricci curvatures. They mention the earlier classical Lichnerowicz-Bochner-Weitzenböck formula right after their statement of their Proposition 3, as pointed out to me by Houdré.

[1989] Kenji Fukaya. Collapsing Riemannian manifolds to ones with lower dimension. I, II. J. Math. Soc. Japan 41 (1989), 333–356. Preceding Cheeger and Colding, 1997.

[1991] Michel Ledoux and Michel Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. Springer-Verlag, New York, 1991. First text on topic.

[1991] Isaac Chavel and Edgar A. Feldman, Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J. 64 (1991), 473-499. Modified isoperimetric and heat diffusion, weighted Laplacian. “…we introduce weighted Riemannian spaces and graphs, for we will require this expanded category to prove Theorem 2-4.” p. 478.

[1996] D. Bakry and M. Ledoux, Lévy-Gromov isoperimetric inequality for an infinite-dimensional diffusion generator, Inv. Math. 123 (1996), 259-281.

[1997] Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. I, II, III,  J. Differential Geom. 46 (1997), 406-480, 54 (2000), 13-35, 37-74.

[1999] William Beckner, Geometric asymptotics and the logarithmic Sobolev inequality. Forum Math. 11 (1999), no. 1, 105-137.

[2000] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214:3, (2000), 547–563. Additional density on Gauss space log concave implies volume-preserving Lipschitz image of Gauss (implies isoperimetrically worse than Gauss, hence Levy-Gromov for densities on Euclidean space).

[2003] Misha Gromov, Isoperimetry of waists and concentration of maps, Geom. Func. Anal. 13 (2003), 178-215.

[2003] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org (2003). Conceptual development via manifolds with density; specific use of Gross’s Gaussian log-Sobolev inequality. Sect 3.3:  “Logarithmic Sobolev inequalities is a vast area of research; see [G] for a survey and bibliography up to the year 1992; the influence of the curvature was discussed by Bakry-Emery [B-Em]. In the context of geometric evolution equations, the logarithmic Sobolev inequality occurs in Ecker [E1].”

[2004] Vincent Bayle, Propriétés de concavité du profil isopérimétrique et applicationes, graduate thesis, Institut Fourier, Universite Joseph-Fourier, Grenoble I, 2004. Beautiful survey plus second variation version of Bakry-Emery and more.

[2005] Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858. Survey and simple derivation of weighted Levy-Gromov with uniqueness. Revised as Chapter 18 of 4th edition of Geometric Measure Theory: a Beginner’s Guide [2009].

[2006] Alexander Grigor’yan, Heat kernels on weighted manifolds and applications. The ubiquitous heat kernel, 93-191, Contemp. Math., 398, Amer. Math. Soc., Providence, RI, 2006. Credits Chavel-Feldman 1991 and Davies 1992 for weighted Laplacian.

“The notion of a weighted Laplacian was introduced by I. Chavel and E. Feldman [1991] and by E. B. Davies [D, 1992]. Many facts from the analysis on weighted manifolds are similar to those on Riemannian manifolds. However, in the former setting one has an added flexibility of changing the measure without changing the underlying Riemannian structure, which happens to be a powerful technical tool, as was earlier observed by E. B. Davies and B. Simon [1984]. A natural setup for this approach would be a metric measure space with an energy form in the spirit of [FOT, 1994], but this would bring additional technical complications, caused by the singularity of the space.” Ex. 2.1. Scaling metric by a(x) and measure by b(x) yields new Laplacian (1/b)div((b/a)grad). 2.4. Laplacian on model manifolds (surfaces with density of revolution). Lapµ = ∂2/∂r2 + m(r) ∂/∂r + ψ-2 Lap_theta m(r) = (log S(r))’ = (n-1)(log g(r))’ + 2ψ’.

[2006] Karl-Theodor Sturm. On the geometry of metric measure spaces, I, II. Acta Math. 196 (2006), 65-131 and 133-177.

[2006] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009), 903–991; arXiv.org (2006).

[2006] Sun-Yung A. Chang, Matthew J. Gursky, and Paul Yang, Conformal invariants associated to a measure, Proc. Natl. Acad. Sci. USA 103 (2006), 2535-2540.

[2007] Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Émery Ricci tensor, arXiv.org (2007). And see references therein.

[2009] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide. Academic Press, fourth edition, 2009.

RECENT:

[2007] Peter W. Michor and David Mumford. An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal., 23(1):74–113, 2007.

[2008] Edward M. Fan, Topology of three-manifolds with positive P-scalar curvature. Proc. Amer. Math. Soc. 136 (2008), 3255–3261. [2009] Ruan, Qi-hua, Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 6, 71–74.

Summary: In this paper, we generalize Cheng’s maximal diameter theorem and Bishop’s volume comparison theorem to manifolds with an m-Bakry-Émery Ricci curvature. As applications, we obtain some rigidity theorems on the warped product.”

[2009] Y. Ollivier, Ricci Curvature of Markov Chains on Metric Spaces, J. Funct. Anal. 256 (2009), 810-864. http://arxiv.org/abs/math/0701886

[2010] Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons [constant positive generalized Ricci curvature], J. Diff. Geom. 85 (2010), 175-185. Generalized Bishop volume comparison theorem. Cites Wei-Wylie 2007.

[2010] Ma, Li; Du, Sheng-Hua, Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. C. R. Math. Acad. Sci. Paris 348 (2010), 1203–1206.

[2010] Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran, Isoperimetric regions in the plane with density r^p, New York J. Math. 16 (2010), 31-51. http://nyjm.albany.edu/j/2010/16-4.html

[2010] Kolesnikov and Zhdanov, On isoperimetric sets of radially symmetric measures, arXiv.org, 2010. Corollary 6.17. If V >= 0 and convex, then e^V dx satisfies the classical isoperimetric inequality. And much more.

[2010] Pak Tung Ho,The structure of ϕ-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature. Math. Ann. 348 (2010), 319–332.

[2010] Emil Saucan, Curvature based triangulation of metric measure spaces. http://arxiv.org/abs/1002.0007v1

[2010] Kevin Brighton, A Liouville-type theorem for smooth metric measure spaces. http://arxiv.org/abs/1006.0751

[2011] Doan The Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys. 61 (2011), 1625-1629.

[2011] Ovidiu Munteanu and Jiaping Wang, Smooth metric measure spaces with non-negative curvature, arXiv.org, 2011. “As an application, we conclude steady Ricci solitons must be connected at infinity.” http://arxiv.org/abs/1103.0746

[2011] Emil Saucan, A simple sampling method for metric measure spaces. http://arxiv.org/abs/1103.3843  Cites Corwin et al. as well as my MwD.

[2011] Jürgen Jost and Shiping Liu, Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs. http://arxiv.org/abs/1103.4037

[2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, http://arxiv.org/abs/1106.2090

[2011] Sean Howe, The Log-Convex Density Conjecture and vertical surface area in warped products, preprint (2011), http://arxiv.org/abs/1107.4402.

[2011] Tapio Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces. http://arXiv.org/abs/1107.4842v1

[2011] David Bate and Gareth Speight, Differentiability, porosity and doubling in metric measure spaces, http://arxiv.org/abs/1108.0318

[2011] Emanuel Milman, Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition, http://arxiv.org/abs/1108.4609v3, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041–1078. [2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, http://arxiv.org/abs/1109.0222

[2011] Martin Bauer, Philipp Harms, and Peter W. Michor, Sobolev metrics on shape space II: weighted sobolev metrics and almost local metries, arXiv:1109.0404v1 [math.DG] 2 Sep 2011. See also references therein, including [2007, Michor and Mumford] above.

[2011] Jasun Gong, Differentiability of Lipschitz functions on doubling metric measure spaces, http://arxiv.org/abs/1110.4279.

[2011] Siddhartha Gadgi and Manjunath Krishnapur, Lipschitz correspondence between metric measure spaces and random distance matrices, http://arxiv.org/abs/1110.6333.

[2011] Luigi Ambroio, Nicola Gigli, and Giuseppe Savaré, Density of Lipschitz functons and equivalence of weak gradients in metric measure spaces, http://arxiv.org/abs/1111.3730

[2011] Ben Andrews and Lei Ni, Eignvalue comparison on Bakry-Emery manifolds, http://arxiv.org/abs/1111.4967

[2011] Tapio Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditons of Sturm, http://arxiv.org/abs/1111.5526

[2011] Wolfang Löhr, Equivalence of Gromov-Prohorov- and Gromov’s box-metric on the space of metric measure spaces, http://arxiv.org/abs/1111.5837

[2011] Shouhei Honda, A weak second differentiable structure on rectifiable metric measure spaces, http://arxiv.org/abs/1112.0099

[2012] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010); see blog posts 1 and 2.

[2012] Yan-Hui Su and Hui-Chun Zhang. Rigidity of manifolds with Bakry-Emery Ricci curvature bounded below. Geometriae Dedicata 160 (2012), 1–11.

[2012] Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré, Heat flow and calculus on metric measure spaces with Ricci curvature bounded below – the compact case, http://arxiv.org/abs/1205.3288

[2012] Nicola Gigli, On the differential structure of metric measure spaces and applications, http://arxiv.org/abs/1205.6622

[2012] Luigi Ambrosio, Nicola Gigli, Andrea Mondino, Tapio Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with sigma-finite measure, http://arxiv.org/abs/1207.4924

[2012] Karl-Theodor Sturm, The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, http://arxiv.org/abs/1208.0434  “The space X of all metric measure spaces … is proven to have nonnegative curvature in the sense of Alexandrov.” Geodesics are linear interpolations on products.

[2012] David Bate [student of David Preiss at Warwick], Structure of measures in Lipschitz differentiability spaces. http://arxiv.org/abs/1208.1954

[2012] Haizhong Li and Yong Wei, f-minimal surface and manifold with positive m-Bakry-Émery Ricci curvature, http://arxiv.org/abs/1209.0895v1  Generalize to density Choi-Schoen theorem that minimal surfaces of fixed topological type in closed 3-manifold smoothly compact.

[2012] Christian Ketterer [student of Sturm], Ricci curvature bounds for warped products, http://arxiv.org/abs/1209.1325

[2012] Zahra Sinaei,Harmonic maps on smooth metric measure spaces and their convergence, http://arxiv.org/abs/1209.5893  [Good references.]

[2012] Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, http://arxiv.org/abs/1209.5786

[2012] Jose M. Espinar, Weighted stable CMC surfaces in manifolds with density, http://arxiv.org/abs/1209.6162

[2012] Alessio Figalli, Emanuel Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones http://arxiv.org/abs/1210.3113  Degenerate Wulff shape!

[2012] Xu Cheng, Tito Mejia, Detang Zhou, Stability and compactness for complete f-minimal surfaces, http://arxiv.org/abs/1210.8076. Cites Morgan [2005].

[2012] Alexandru Kristály and Shin-ichi Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications, http://arxiv.org/abs/1211.3171v1

[2012] Ovidiu Munteanu and Jiaping Wang, Geometry of manifolds with densities, http://arxiv.org/abs/1211.3996  Bochner, Laplacian, entropy; Euclidean vol growth, splitting.

[2012] Gang Liu, Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor, Comm. Anal. Geom., http://arxiv.org/abs/1211.3770. 2nd varn, splitting.

[2012] Jon Wolfson, Eigenvalue gap theorems for a class of non symmetric elliptic operators on convex domains, http://arxiv.org/abs/1212.1669. “The class of operators includes the Bakry-Emery laplacian.”

[2013] Matheus Vieira, Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature, Arch. Math. 101 (2013), 581–590.

[2013] Qin Huang, Qihua Ruan, Applications of Some Elliptic Equations in Riemannian Manifolds, http://arxiv.org/abs/1301.1117  (Heintze-Karcher for m-Ricci, later by Batista-Cavalcante; thank and cite Ma.)

[2013] Fabio Cavalletti and Martin Huesmann, Existence and uniqueness of optimal transport maps, http://arxiv.org/abs/1301.1782v1

[2013] Debora Impera and Michele Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, http://arxiv.org/abs/1302.6160

[2013] Mingliang Cai, On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature, http://arxiv.org/abs/1303.2728

[2013] Matthias Erbar, Kazumasa Kuwada, Karl-Theodor Sturm, On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner’s Inequality on Metric Measure Spaces, http://arxiv.org/abs/1303.4382  Good overview introduction.

[2013] Giuseppe Savaré, Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces, http://arxiv.org/abs/1304.0643

[2013] Antonio Cañete, César Rosales, Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities, http://arxiv.org/abs/1304.1438 Good references.

[2013] Xavier Cabre, Xavier Ros-Oton, Joaquim Serra, Sharp isoperimetric inequalities via the ABP method, http://arxiv.org/abs/1304.1724  Uses Figalli-Indrei degenerate Wulff shape to handle nonsmooth cones.

[2013] Marcio Batista, Heudson Mirandola [Brazil], A Sobolev-type inequality for submanifolds in weighted Riemannian Manifolds, http://arxiv.org/abs/1304.2271

[2013] Jia-Yong Wu, L^p-Liouville theorems on complete smooth metric measure spaces, http://arxiv.org/abs/1305.0616

[2013] Xu Cheng, Tito Mejia, Detang Zhou, Simons’ type equation for f-minimal hypersurfaces and applications, http://arxiv.org/abs/1305.2379v1

[2013] Xu Cheng, Detang Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces, http://arxiv.org/abs/1305.4116

[2013] Nicola Garofalo, Andrea Mondino, Li-Yau and Harnack type inequalities in RCD^*(K,N) metric measure spaces, http://arxiv.org/abs/1306.0494

[2013] Haizhong Li, Yong Wei, Sharp diameter estimates for compact manifold with boundary, http://arxiv.org/abs/1306.3715  Ruan generalized Cheng to m-Ricci; LW generalize to manifolds with boundary.

[2013] Ye-Lin Ou, On f-biharmonic maps and f-biharmonic submanifolds, http://arxiv.org/abs/1306.3549, after Wei-Jun Lu, On f-Biharmonic maps between Riemannian manifolds , arXiv:1305.5478

[2013] Marcio Batista, Marcos P. Cavalcante, The Heintze-Karcher-Ros Inequality and Eigenvalue Estimates in Weighted Manifolds http://arxiv.org/abs/1306.4874 (Heintze-Karcher for m-Ricci, earlier by Huang Ruan.)

[2013] Aaron Naber, Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces, http://arxiv.org/abs/1306.6512

[2013] Bingyu Song, Guofang Wei, Guoqiang Wu, Monotonicity Formulas for Bakry-Emery Ricci Curvature, http://arxiv.org/abs/1307.0477

[2013] Xu Cheng, Detang Zhou, Stability properties and gap theorem for complete f-minimal hypersurfaces, http://arxiv.org/abs/1307.5099v1

[2013] Matthew McGonagle, John Ross, The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space, http://arxiv.org/abs/1307.7088

[2013] Bobo Hua, Martin Kell, Chao Xia, Harmonic functions on metric measure spaces, http://arxiv.org/abs/1308.3607

[2013] Emanuel Milman and Liran Rotem, Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures, http://arxiv.org/abs/1308.5695. Also negative dimension.

[2013] Luigi Ambrosio, Andrea Mondino, Giuseppe Savaré, On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces, http://arxiv.org/abs/1309.4664

[2013] Maree Jaramillo, Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below, http://arxiv.org/abs/1309.6685

[2013] Shouhei Honda, Cheeger constant, p-Laplacian, and Gromov-Hausdorff convergence, http://arxiv.org/abs/1310.0304

[2013] Bruno Colbois, Ahmad El Soufi, Alessandro Savo, Eigenvalues of the Laplacian on a compact manifold with density, http://arxiv.org/abs/1310.1490

[2013] Alexander V. Kolesnikov, Emanuel Milman, Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary, http://arxiv.org/abs/1310.2526. Unified positive and negative dimension.

[2013] Fabio Cavalletti, Monge problem in metric measure spaces with Riemannian curvature-dimension condition, http://arxiv.org/abs/1310.4036

[2013] Shin-ichi Ohta, (K,N)-convexity and the curvature-dimension condition for negative N, http://arxiv.org/abs/1310.7993

[2013]  William Wylie, Sectional curvature for Riemannian manifolds with density, http://arxiv.org/abs/1311.0267

[2013] Christian Ketterer, Cones over metric measure spaces and the maximal diameter theorem, http://arxiv.org/abs/1311.1307

[2013] Katherine Castro and César Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, http://arxiv.org/abs/1311.1952

[2013] Gregory R. Chambers, Proof of the Log-Convex Density Conjecture, https://arxiv.org/abs/1311.4012, J. Eur. Math. Soc. 21 (2019), 2301–2332.

[2013] M Rupert and E Woolgar, Bakry-Émery black holes, http://arxiv.org/abs/1310.3894 [2013] E Woolgar, Scalar-tensor gravitation and the Bakry-Emery-Ricci tensor, http://arxiv.org/abs/1302.1893 [2013] Nicola Gigli, Andrea Mondino, Giuseppe Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, http://arxiv.org/abs/1311.4907

[2013] Martin Kell, On Interpolation and Curvature via Wasserstein Geodesics, http://arxiv.org/abs/1311.5407 “The proof of the Borell–Brascamp–Lieb (BBL) inequality for Riemannian manifolds by Cordero-Erausquin-McCann-Schmuckenschläger [CEMS01], and later for Finsler manifolds by Ohta [Oht09], led Lott-Villani [LV09, LV07] and Sturm [Stu06a, Stu06b] to a new notion of a lower bound on the generalized Ricci curvature for metric measure spaces, called curvature dimension.”

[2013] Katrin Fässler, Pekka Koskela, Enrico Le Donne. Nonexistence of quasiconformal maps between certain metric measure spaces, http://arxiv.org/abs/1312.1305v1

[2013] Gregory J Galloway and Eric Woolgar, Cosmological singularities in Bakry-Émery spacetimes, http://arxiv.org/abs/1312.3410

[2013] Guy C. David, Bi-Lipschitz Pieces between Manifolds, http://arxiv.org/abs/1312.3911v1

[2013] Jeff Cheeger, Bruce Kleiner, Inverse limit spaces satisfying a Poincare inequality, http://arxiv.org/abs/1312.5227v1

[2013] Jia-Yong Wu, Peng Wu, Heat Kernels on Smooth Metric Measure Spaces with Nonnegative Curvature, http://arxiv.org/abs/1401.6155

[2014] Takashi Shioya, Metric measure limits of spheres and complex projective spaces, http://arxiv.org/abs/1402.0611 [2014]  Yashar Memarian, Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds, http://arxiv.org/abs/1402.4947 [apparently incorrect]

[2014] Christian Ketterer, Tapio Rajala, Failure of topological rigidity results for the measure contraction property, http://arxiv.org/abs/1403.3105

[2014] César Rosales, Isoperimetric and stable sets for log-concave perturbations of Gaussian measures, http://arxiv.org/abs/1403.4510

[2014] Ayato Mitsuishi, Current and measure homologies, http://arxiv.org/abs/1403.5518

[2014] Karl-Theodor Sturm, Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions, http://arxiv.org/abs/1405.0459. “There are two canonical ways to define the heat flow on a mms, either as the gradient flow for the energy or as the gradient flow for the entropy in the Wasserstein space. …in great generality both approaches will coincide.”

[2014] Yu Kitabeppu, Sajjad Lakzian, Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups, http://arxiv.org/abs/1405.0897

[2014] Renjin Jiang, The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces, http://arxiv.org/abs/1405.0684

[2014] Andrea Mondino, Aaron Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I, http://arxiv.org/abs/1405.2222

[2014] Yi Li, Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature, http://arxiv.org/abs/1406.0125

[2014] Jeffrey S. Case, Sun-Yung Alice Chang, On fractional GJMS operators, http://arxiv.org/abs/1406.1846

[2014] Luigi Ambrosio, Andrea Pinamonti, Gareth Speight, Weighted Sobolev Spaces on Metric Measure Spaces, http://arxiv.org/abs/1406.3000

[2014] Jia-Yong Wu, Peng Wu, Heat kernel on smooth metric measure spaces and applications, http://arxiv.org/abs/1406.5801

[2014] Nicola Gigli, Bangxian Han, The continuity equation on metric measure spaces, http://arxiv.org/abs/1406.6350

[2014] Nicola Gigli, Nonsmooth differential geometry – An approach tailored for spaces with Ricci curvature bounded from below, http://arxiv.org/abs/1407.0809

[2014] Frédéric Bernicot, Thierry Coulhon, Dorothee FreyGradient estimates, Poincaré inequalities, De Giorgi property and their consequences, http://arxiv.org/abs/1407.3906

[2014] Nicola Gigli, Bangxian Han, Independence on p of weak upper gradients on RCD spaces, http://arxiv.org/abs/1407.7350

[2014] A. Barros, R. Batista, E. Ribeiro Jr., Bounds on volume growth of geodesic balls for Einstein warped products, http://arxiv.org/abs/1408.1621

[2014] Erlend Grong, Anton Thalmaier, Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Parts I and II, http://arxiv.org/abs/1408.6873, http://arxiv.org/abs/1408.6872

[2014] David Bate, Sean Li, Characterizations of rectifiable metric spaces, http://arxiv.org/abs/1409.4242 [2014] Emanuel Milman, Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension, http://arxiv.org/abs/1409.4109

[2014] Jeffrey S. Case, A notion of the weighted σ_k-curvature for manifolds with density, http://arxiv.org/abs/1409.4455

[2014] Simone Di Marino, Sobolev and BV spaces on metric measure spaces via derivations and integration by parts, http://arxiv.org/abs/1409.5620

[2014] Jing Mao, The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature, http://arxiv.org/abs/1409.5741

[2014] Takashi Shioya, Metric measure geometry, http://arxiv.org/abs/1410.0428

[2014] Lee Kennard, William Wylie, Positive weighted sectional curvature, http://arxiv.org/abs/1410.1558

[2014] Antonio G. Ache, Micah W. Warren, Coarse Ricci curvature and the manifold learning problem, http://arxiv.org/abs/1410.3351

[2014] Shouhei Honda, Elliptic PDEs on compact Ricci limit spaces and applications, http://arxiv.org/abs/1410.3296

[2014] Karl-Theodor Sturm, Gradient Flows for Semiconvex Functions on Metric Measure Spaces – Existence, Uniqueness and Lipschitz Continuity, http://arxiv.org/abs/1410.3966

[2014] Christian Ketterer, Obata’s rigidity theorem for metric measure spaces, http://arxiv.org/abs/1410.5210

[2014] Jia-Yong Wu, Peng Wu, On L^p-Liouville property for smooth metric measure spaces, http://arxiv.org/abs/1410.7305

[2014] Paul W.Y. Lee, On measure contraction property without Ricci curvature lower bound, http://arxiv.org/abs/1412.4345

[2014] Dominique Bakry (IMT), François Bolley (LPMA), Ivan Gentil (ICJ), The Li-Yau inequality and applications under a curvature-dimension condition, http://arxiv.org/abs/1412.5165

[2015] Fabio Cavalletti, Andrea Mondino, Measure rigidity of Ricci curvature lower bounds, http://arxiv.org/abs/1501.03338

[2015] William Wylie, Some curvature pinching results for Riemannian manifolds with density, http://arxiv.org/abs/1501.06079

[2015] Sérgio Mendonça, Splitting, parallel gradient and Bakry-Emery Ricci curvature, http://arxiv.org/abs/1502.00185

[2015] Guy C. David, Lusin-type theorems for Cheeger derivatives on metric measure spaces, http://arxiv.org/abs/1502.00694

[2015] Fabio Cavalletti, Andrea Mondino, Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds, http://arxiv.org/abs/1502.06465. Invent. Math. 208 (2017), no. 3, 803–849. Generalize Milman smooth to non-branching CD.

[2015] Yin Jiang, Hui-Chun Zhang, Sharp Spectral Gaps on Metric Measure Spaces, http://arxiv.org/abs/1503.00203

[2015] Ezequiel Barbosa, Ben Sharp, Yong Wei, Smooth compactness of f-minimal hypersurfaces with bounded f-index, http://arxiv.org/abs/1503.01945

[2015] M. Carmen Domingo-Juan, Vicente Miquel, Reilly’s type inequality for the Laplacian associated to a density related with shrinkers for MCF, http://arxiv.org/abs/1503.01332

[2015] Vicente Miquel, Francisco Viñado-Lereu, The curve shortening problem associated to a density, http://arxiv.org/abs/1503.02429

[2015] M. P. Cavalcante, J. Q. Oliveira, M. S. Santos, Compactness in weighted manifolds and applications, Results Math. (2015).

[2015]  Wyatt Boyer, Bryan Brown, Gregory R. Chambers, Alyssa Loving, Sarah Tammen, Isoperimetric Regions in ℝ^n with density r^p, https://arxiv.org/abs/1504.01720, Anal. Geom. Metr. Spaces 4 (2016), 236–265.

[2015] Homare Tadano, Remark on a diameter bound for complete manifolds with positive Bakry-Émery Ricci curvature, http://arxiv.org/abs/1504.05384

[2015] Fabio Cavalletti, Andrea Mondino, Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds, http://arxiv.org/abs/1505.02061

[2015] Emanuel Milman, Harmonic Measures on the Sphere via Curvature-Dimension, http://arxiv.org/abs/1505.04335v1

[2015] Christian Ketterer, On the geometry of metric measure spaces with variable curvature bounds, http://arxiv.org/abs/1506.03279

[2015] William Wylie, A warped product version of the Cheeger-Gromoll splitting theorem, http://arxiv.org/abs/1506.03800

[2015] Luigi Ambrosio, Matthias Erbar, Giuseppe Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces, http://arxiv.org/abs/1506.05932

[2015] Emanuel Milman, Spectral Estimates, Contractions and Hypercontractivity, https://arxiv.org/abs/1508.00606

[2015] Christian Ketterer, Evolution variational inequality and Wasserstein control in variable curvature context, http://arxiv.org/abs/1509.02178

[2015] Luigi Ambrosio, Andrea Mondino, Giuseppe Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, http://arxiv.org/abs/1509.07273

[2015] Max Fathi, Yan Shu, Curvature and transport inequalities for Markov chains in discrete spaces, http://arxiv.org/abs/1509.07160

[2015] Shouhei Honda, Spectral convergence under bounded Ricci curvature, http://arxiv.org/abs/1510.05349

[2015] Bang-Xian Han, Conformal transformation on metric measure spaces, http://arxiv.org/abs/1511.03115

[2015] Feng Du, Jing Mao, Qiaoling Wang, Chuanxi Wu, The Gagliardo-Nirenberg inequality on metric measure spaces, http://arxiv.org/abs/1511.04696

[2015] Martin Kell, On Cheeger and Sobolev differentials in metric measure spaces, http://arxiv.org/abs/1512.00828

[2015] Yuzhao Wang, Huaiqian Li, Lower Bound Estimates for The First Eigenvalue of The Weighted p-Laplacian on Smooth Metric Measure Space, http://arxiv.org/abs/1512.01031 [2015] Nicola Gigli, Guido de Philippis, From volume cone to metric cone in the non smooth setting, http://arxiv.org/abs/1512.03113

[2015] Renjin Jiang, Huichun Zhang, Hamilton’s Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces, http://arxiv.org/abs/1512.08306

[2016] Tomasz Adamowicz, Michał Gaczkowski, Przemysław Górka, Harmonic functions on metric measure spaces, http://arxiv.org/abs/1601.03919

[2016] Shin-ichi Ohta, A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetric inequality [for Finsler manifolds with density], http://arxiv.org/abs/1602.00390

[2016] Hui-Chun Zhang, Xi-Ping Zhu, Local Li-Yau’s estimates on metric measure spaces, http://arxiv.org/abs/1602.05347

[2016] William Wylie, Dmytro Yeroshkin, On the geometry of Riemannian manifolds with density, dedicated to Frank Morgan, http://arxiv.org/abs/1602.08000

[2016] Qintao Deng, Fernando Galaz-Garcia, Luis Guijarro, Michael Munn, Three-Dimensional Alexandrov spaces with positive or nonnegative Ricci curvature, http://arxiv.org/abs/1602.07724v1. “We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the ∗(K,N) sense.”

[2016] Erbar Matthias, Juillet Nicolas, Smoothing and non-smoothing via a flow tangent to the Ricci flow, http://arxiv.org/abs/1603.00280

[2016] Karl-Theodor Sturm, Super-Ricci Flows for Metric Measure Spaces. I, http://arxiv.org/abs/1603.02193 [2016] Jeffrey S. Case, A weighted renormalized curvature for manifolds with density, http://arxiv.org/abs/1603.02989

[2016] Yu Kitabeppu, A Bishop type inequality on metric measure spaces with Ricci curvature bounded below, http://arxiv.org/abs/1603.04162

[2016] Xian-tao Huang, Noncompact RCD(0,N) spaces with linear volume growth, http://arxiv.org/abs/1603.05221

[2016] Luigi Ambrosio, Federico Stra, Dario Trevisan, Weak and strong convergence of derivations and stability of flows with respect to MGH convergence, http://arxiv.org/abs/1603.05561

[2016] Luigi Ambrosio, Nicola Gigli, Simone Di Marino, Perimeter as relaxed Minkowski content in metric measure spaces, http://arxiv.org/abs/1603.08412 [2016] Fabrice Baudoin, Daniel J. Kelleher, Poincaré Duality, Bakry-Émery Estimates and Isoperimetry on Fractals, http://arxiv.org/abs/1604.02520

[2016] Ronen Eldan, James R. Lee, Joseph Lehec, Transport-entropy inequalities and curvature in discrete-space Markov chains, http://arxiv.org/abs/1604.06859

[2016] Andrea Mondino, Guofang Wei, On the universal cover and the fundamental group of an RCD∗(K,N)-space, http://arxiv.org/abs/1605.02854  Nice intro.

[2016] Luigi Ambrosio, Andrea Mondino, Gaussian-type Isoperimetric Inequalities in RCD(K,∞) probability spaces for positive K, http://arxiv.org/abs/1605.02852  After Bakry-Émery.

[2016] Sajjad Lakzian, Zachary Mcguirk, Global Poincaré inequality on Graphs via Conical Curvature-Dimension Conditions, http://arxiv.org/abs/1605.05432

[2016] Luigi Ambrosio, Shouhei Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, http://arxiv.org/abs/1605.07908

[2016] Max Grieshammer, Thomas Rippl, Partial orders on metric measure spaces, http://arxiv.org/abs/1605.08989

[2016] Qi S Zhang, Meng Zhu, New Volume Comparison results and Applications to degeneration of Riemannian metrics, http://arxiv.org/abs/1605.09420

[2016] A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, Some isoperimetric inequalities on ℝN with respect to weights |x|^α, https://arxiv.org/abs/1606.02195v2

[2016] Lingzhong Zeng, The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces, http://arxiv.org/abs/1606.06429

[2016] Martin Kell, Andrea Mondino, On the volume measure of non-smooth spaces with Ricci curvature bounded below, http://arxiv.org/abs/1607.02036

[2016] Nicola Gigli, Enrico Pasqualetto, Behaviour of the reference measure on spaces under charts, http://arxiv.org/abs/1607.05188

[2016] Sylvester Eriksson-Bique, Classifying Poincaré Inequalities and the local geometry of RNP-Differentiability Spaces, http://arxiv.org/abs/1607.07428 [2016] Vicente Miquel, Francisco Viñado-Lereu, Type I singularities in the curve shortening flow associated to a density, http://arxiv.org/abs/1607.08402

[2016] Robert Haslhofer, Aaron Naber, Ricci Curvature and Bochner Formulas for Martingales, http://arxiv.org/abs/1608.04371

[2016] Luis Guijarro, Jaime Santos-Rodríguez, On the isometry group of RCD∗(K,N)-spaces, http://arxiv.org/abs/1608.06467

[2016] Shiping Liu, Florentin Münch, Norbert Peyerimhoff, Bakry-Emery curvature and diameter bounds on graphs, http://arxiv.org/abs/1608.07778

[2016] Fabio Cavalletti, Andrea Mondino, Optimal maps in essentially non-branching spaces, http://arxiv.org/abs/1609.00782

[2016] Gerardo Sosa, The isometry group of an RCD*-space is Lie, http://arxiv.org/abs/1609.02098

[2016] Lukáš Malý, Nageswari Shanmugalingam, Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity, http://arxiv.org/abs/1609.06808

[2016] Li Ma, Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers, https://arxiv.org/abs/1609.09332

[2016] Shouhei Honda, Ricci curvature and Orientability, https://arxiv.org/abs/1610.02932

[2016] Christian Ketterer, Andrea Mondino, Sectional and intermediate Ricci curvature lower bounds via Optimal Transport, https://arxiv.org/abs/1610.03339

[2016] Nguyen Thac Dung, Nguyen Ngoc Khanh, Quoc Anh Ngô, Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces, https://arxiv.org/abs/1610.03199

[2016] Nguyen Thac Dung, Kieu Thi Thuy Linh, Ninh Van Thu, Gradient estimates for some evolution equations on complete smooth metric measure spaces, https://arxiv.org/abs/1610.03198

[2016] Jia-Yong Wu, Comparison geometry for integral Bakry-Émery Ricci tensor bounds, https://arxiv.org/abs/1610.03926

[2016] Jia-Yong Wu, Liouville property for f-harmonic functions with polynomial growth, https://arxiv.org/abs/1610.03923

[2016] Fabio Cavalletti, Andrea Mondino, Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds, https://arxiv.org/abs/1610.05044. For perimeter, not just Minkowski content.

[2016] Guy C. David, Bruce Kleiner, Rectifiability of planes and Alberti representations, https://arxiv.org/abs/1611.05284

[2016] Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Balls Isoperimetric in ℝ^n with Volume and Perimeter Densities r^m and r^p, https://arxiv.org/abs/1610.05830

[2016] Ushio Tanaka, Gromov’s Problem: Bound the Expansion Coefficient from below in terms of the Observable Diameter of a Metric Measure Space, and its Diameter Bounds, https://arxiv.org/abs/1611.05596

[2016] Panu Lahti, Strong approximation of sets of finite perimeter in metric spaces, https://arxiv.org/abs/1611.06162 [2016] Nicola Gigli, Enrico Pasqualetto, Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces, https://arxiv.org/abs/1611.09645

[2016] Panu Lahti, Strict and pointwise convergence of BV functions in metric spaces, https://arxiv.org/abs/1612.06447

[2016] Fabio Cavalletti, Emanuel Milman, The Globalization Theorem for the Curvature Dimension Condition, https://arxiv.org/abs/1612.07623

[2017] Hilário Alencar, Adina Rocha, The f-Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons, https://arxiv.org/abs/1701.00373

[2017] Hui-Chun Zhang, Xi-Ping Zhu, Weyl’s law on RCD∗(K,N) metric measure spaces, https://arxiv.org/abs/1701.01967

[2017] Jia-Cheng Huang, Hui-Chun Zhang, Sharp gradient estimate for heat kernels on RCD∗(K,N) metric measure spaces, https://arxiv.org/abs/1701.01803

[2017] Luigi Ambrosio, Shouhei Honda, David Tewodrose, Short-time behavior of the heat kernel and Weyl’s law on RCD∗(K,N)-spaces, https://arxiv.org/abs/1701.03906

[2017] Bang-Xian Han, Andrea Mondino, Angles between curves in metric measure spaces, https://arxiv.org/abs/1701.05000

[2017] Davide Barilari, Luca Rizzi, Sharp measure contraction property for generalized H-type Carnot groups, https://arxiv.org/abs/1702.04401

[2017] Chang-Yu Guo, Harmonic mappings between singular metric spaces, https://arxiv.org/abs/1702.05086

[2017] Ángel Arroyo, José G. Llorente, A priori Hölder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces, https://arxiv.org/abs/1702.07175

[2017] Luigi Ambrosio, Shouhei Honda, Local spectral convergence in RCD∗(K,N) spaces, https://arxiv.org/abs/1703.04939

[2017] Panu Lahti, Quasiopen sets, bounded variation and lower semicontinuity in metric spaces, https://arxiv.org/abs/1703.04675

[2017] Kohei Suzuki, Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions, https://arxiv.org/abs/1703.07234

[2017] Manuel Ritoré, Jesús Yepes Nicolás, Brunn-Minkowski inequalities in product metric measure spaces, https://arxiv.org/abs/1704.07717

[2017] Nicola Gigli, Chiara Rigoni, Recognizing the flat torus among RCD∗(0,N) spaces via the study of the first cohomology group, https://arxiv.org/abs/1705.04466

[2017] Hiroki Nakajima, The maximum of the 1-measurement of a metric measure space, https://arxiv.org/abs/1706.01258

[2017] Nicola Gigli, Chiara Rigoni, A note about the strong maximum principle on RCD space, https://arxiv.org/abs/1706.01998

[2017] Luigi Ambrosio, Shouhei Honda, Jacobus W. Portegies, Continuity of nonlinear eigenvalues in CD(K,∞) spaces with respect to measured Gromov-Hausdorff convergence, https://arxiv.org/abs/1706.08368

[2017] Jia-Yong Wu, Myers’ type theorem with the Bakry-Émery Ricci tensor, https://arxiv.org/abs/1706.07897

[2017] Lee Kennard, William Wylie, Dmytro Yeroshkin, The weighted connection and sectional curvature for manifolds with density, https://arxiv.org/abs/1707.05376

[2017] Eric Woolgar, William Wylie, Curvature-dimension bounds for Lorentzian splitting theorems, https://arxiv.org/abs/1707.09058

[2017] Guido De Philippis, Nicola Gigli, Non-collapsed spaces with Ricci curvature bounded from below, https://arxiv.org/abs/1708.02060

[2017] Yu Kitabeppu, A sufficient condition to a regular set of positive measure on RCD spaces, https://arxiv.org/abs/1708.04309

[2017] Nicola Gigli, Christian Ketterer, Kazumasa Kuwada, Shin-ichi Ohta, Rigidity for the spectral gap on RCD(K,∞)-spaces, https://arxiv.org/abs/1709.04017

[2017] Kohei Suzuki, Convergence of Non-Symmetric Diffusion Processes on RCD spaces, https://arxiv.org/abs/1709.09536

[2017] Bang-Xian Han, Ricci tensor on smooth metric measure space with boundary, https://arxiv.org/abs/1709.10143

[2017] Xian-Tao Huang, An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth, https://arxiv.org/abs/1710.05830

[2017] Bang-Xian Han, Characterizations of monotonicity of vector fields on metric measure space, https://arxiv.org/abs/1710.07953

[2017] Helmer Hoppe, Jun Masamune, Stefan Neukamm, H-compactness of elliptic operators on weighted Riemannian Manifolds, https://arxiv.org/abs/1710.09352

[2017] Luca Rizzi, A counterexample to gluing theorems for MCP metric measure spaces, https://arxiv.org/abs/1711.04499

[2017] Willian Isao Tokura, Levi Adriano, Changyu Xia, The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces, https://arxiv.org/abs/1711.04836

[2017] Vitali Kapovitch, Christian Ketterer, RCD meets CAT, https://arxiv.org/abs/1712.02839

[2017] Timo Schultz, Existence of optimal transport maps in very strict CD(K,∞) -spaces, https://arxiv.org/abs/1712.03670

[2017] Luigi Ambrosio, Elia Bruè, Dario Trevisan, Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and RCD(K,∞) spaces, https://arxiv.org/abs/1712.06315

[2017] Cong Hung Mai, Rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds, https://arxiv.org/abs/1712.06904

[2017] Raphael Bouyrie, Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry–Ledoux isoperimetric comparison Theorem, https://arxiv.org/abs/1708.07203

[2017] Matthias Erbar, Karl-Theodor Sturm, Rigidity of cones with bounded Ricci curvature, https://arxiv.org/abs/1712.08093

[2018] Hiroki Nakajima, Takashi Shioya, Isoperimetric rigidity and distributions of 1-Lipschitz functions, https://arxiv.org/abs/1801.01302

[2018] Antonio Bueno, Translating solitons of the mean curvature flow in the space ℍ2×ℝ, https://arxiv.org/abs/1803.02783

[2018] Elia Bruè, Daniele Semola, Regularity of Lagrangian flows over RCD∗(K,N) spaces, https://arxiv.org/abs/1803.04387

[2018] Sebastiano Don, Davide Vittone, A compactness result for BV functions in metric spaces, https://arxiv.org/abs/1803.07545

[2018] Debora Impera, Michele Rimoldi, Alessandro Savo, Index and first Betti number of f-minimal hypersurfaces and self-shrinkers, https://arxiv.org/abs/1803.08268v1

[2018] Daisuke Kazukawa, A new condition for convergence of energy functionals and stability of lower Ricci curvature bound, https://arxiv.org/abs/1804.00407

[2018] Elia Bruè, Daniele Semola, Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows, https://arxiv.org/abs/1804.07128

[2018] Shouhei Honda, Bakry-Émery conditions on almost smooth metric measure spaces, https://arxiv.org/abs/1804.07043

[2018] Ilaria Mondello (LAMA), J. Bertrand (IMT), C Ketterer, T. Richard (LAMA), Stratified spaces and synthetic Ricci curvature bounds, https://arxiv.org/abs/1804.08870

[2018] Antoni Kijowski, Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure, https://arxiv.org/abs/1804.10005. “We conclude with a remarkable observation that strongly harmonic functions in R^n possess the mean value property with respect to infinitely many weight functions obtained from a given weight.”

[2018] Angelo Alvino, Friedemann Brock, Francesco Chiacchio, Anna Mercaldo, Maria Rosaria Posteraro, The isoperimetric problem for a class of non-radial weights and applications, https://arxiv.org/abs/1805.02518v1

[2018] Jhovanny Muñoz Posso, A generalization of Sobolev trace inequality and Escobar-Riemann mapping type problem on smooth metric measure spaces, https://arxiv.org/abs/1805.03694 [part of PhD thesis under Fernando Codá´Marques]

[2018] Jeff Cheeger, Wenshuai Jiang, Aaron Naber, Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below, https://arxiv.org/abs/1805.07988

[2018] Ana Hurtado, Vicente Palmer, César Rosales, Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights, https://arxiv.org/abs/1805.10055 [2018] Katrin Fässler, Tuomas Orponen, Metric currents and the Poincaré inequality, https://arxiv.org/abs/1807.02969

[2018] Danka Lučić, Enrico Pasqualetto, Infinitesimal Hilbertianity of weighted Riemannian manifolds, https://arxiv.org/abs/1809.05919

[2018] Chris Connell, Xianzhe Dai, Jesús Núñez-Zimbrón, Raquel Perales, Pablo Suárez-Serrato, Guofang Wei, Maximal volume entropy rigidity for ∗(−(N−1),N) spaces, https://arxiv.org/abs/1809.06909

[2018] Jaime Santos-Rodríguez, Invariant measures and lower Ricci curvature bounds, https://arxiv.org/abs/1810.11327

[2018] Fabio Cavalletti, Flavia Santarcangelo, Isoperimetric inequality under Measure-Contraction property, https://arxiv.org/abs/1810.11289. Generalize Levy-Gromov-Milman-CavallettiMondino to MCP spaces.

[2018] L. M. Chasman, Jeffrey J Langford, On Clamped Plates with Log-Convex Density, https://arxiv.org/abs/1811.06423

[2018] Panu Lahti, A sharp Leibniz rule for BV functions in metric spaces, https://arxiv.org/abs/1811.07713v1

[2018] Kazumasa Kuwada, Xiang-Dong Li, Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces, https://arxiv.org/abs/1811.07228

[2018] Simone Di Marino, Nicola Gigli, Enrico Pasqualetto, Elefterios Soultanis, Infinitesimal Hilbertianity of locally CAT(κ)-spaces, https://arxiv.org/abs/1812.02086

[2018] LI MA, LIOUVILLE THEOREMS, VOLUME GROWTH, AND VOLUME COMPARISON FOR RICCI SHRINKERS, PACIFIC J. MATH. 296 dx.doi.org/10.2140/pjm.2018.296.357

[2019] Vitali Kapovitch, Christian Ketterer, Weakly noncollapsed RCD spaces with upper curvature bounds, https://arxiv.org/abs/1901.06966

[2019] Bang-XIan Han, Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds, https://arxiv.org/abs/1902.00942

[2019] Nicola Gigli, Riemann curvature tensor on spaces and possible applications, https://arxiv.org/abs/1902.02282

[2019] Friedemann Brock, Francesco Chiacchio, Some weighted isoperimetric problems on ℝ^N+ with stable half balls have no solutions, https://arxiv.org/abs/1903.04922

[2019] Adriano Cavalcante Bezerra, Changyu Xia, Sharp Estimates for the First Eigenvalues of the Bi-drifting [squared] Laplacian, https://arxiv.org/abs/1903.06728

[2019] Giulia Luise, Giuseppe Savaré, Contraction and regularizing properties of heat flows in metric measure spaces, https://arxiv.org/abs/1904.09825

[2019] Shin-ichi Ohta, Asuka Takatsu, Equality in the logarithmic Sobolev inequality, https://arxiv.org/abs/1904.09400

[2019] Shouhei Honda, New differential operator and non-collapsed RCD spaces, https://arxiv.org/abs/1905.00123

[2019] Timo Schultz, Equivalent definitions of very strict CD(K,N) -spaces, https://arxiv.org/abs/1906.07693

[2019] Davide Barilari, Luca Rizzi, Bakry-Émery curvature and model spaces in sub-Riemannian geometry, https://arxiv.org/abs/1906.08307

[2019] Mathias Braun, Karen Habermann, Karl-Theodor Sturm, Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds, https://arxiv.org/abs/1906.09186

[2019] Gioacchino Antonelli, Elia Bruè, Daniele Semola, Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces, https://arxiv.org/abs/1907.02735

[2019] Vitali Kapovitch, Andrea Mondino, On the topology and the boundary of N-dimensional RCD(K,N) spaces, https://arxiv.org/abs/1907.02614

[2019] Shouhei Honda, Ilaria Mondello, Sphere theorems for RCD spaces, https://arxiv.org/abs/1907.03482

[2019] Angelo Alvino, Friedemann Brock, Francesco Chiacchio, Anna Mercaldo, Maria Rosaria Posteraro, Some isoperimetric inequalities with respect to monomial weights, https://arxiv.org/abs/1907.03659

[2019] Bang-Xian Han, Karl-Theodor Sturm, Curvature-dimension conditions for diffusions under time change, https://arxiv.org/abs/1907.05761

[2019] Ana Hurtado, Vicente Palmer, César Rosales, Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds, https://arxiv.org/abs/1907.07920

[2019] Batu Güneysu, Max von Renesse, Molecules as metric measure spaces with Kato-bounded Ricci curvature, https://arxiv.org/abs/1907.09566

[2019] Christian Ketterer, The Heintze-Karcher inequality for metric measure spaces, https://arxiv.org/abs/1908.06146

[2019] Vitali Kapovitch, Martin Kell, Christian Ketterer, On the structure of RCD spaces with upper curvature bounds, https://arxiv.org/abs/1908.07036

[2019] Abdolhakim Shouman, Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi‑drifting Laplacian, Annals of Global Analysis and Geometry (2019) 55:805–817 https://doi.org/10.1007/s10455-019-09652-1 1 3

[2020] Adson Meira · Rosivaldo Antonio Gonçalves, On the space of f-minimal surfaces with bounded f-index in weighted smooth metric spaces, manuscripta math. 162, 559–563 (2020)

[2020] Kazuhiro Kuwae, Yohei Sakurai, Comparison geometry of manifolds with boundary under lower N-weighted Ricci curvature bounds with ε-range, https://arxiv.org/abs/2011.03730v1

[2021] Florian Johne, Sobolev inequalities on manifolds with nonnegative Bakry-Émery Ricci curvature, https://arxiv.org/abs/2103.08496

[2021] Homare Tadanoa, Some compactness theorems via m-Bakry–Émery and m-modified Ricci curvatures with negative m, Differential Geometry and its Applications 75 (2021), 101-720

[2021] Allan G. Freitasa, Henrique F. de Lima, Eraldo A. Lima Jr., Márcio S. Santosa, Submanifolds immersed in Riemannian spaces endowed with a Killing vector field: Nonexistence and rigidity, Differential Geometry and its Applications 75 (2021) 101-714

[2021] Fabio Cavalletti, Davide Manini, Isoperimetric inequality in noncompact MCP spaces, https://arxiv.org/abs/2110.07528v1

ADDITIONAL PUBLICATIONS IN THE HISTORY OF MANIFOLDS WITH DENSITY

[B3] F. Barthe, Extremal properties of central-half-spaces for product measures, J. Funct. Anal. 182 (2001), 81-107.

[B4] F. Barthe, Log-concave and spherical models in isoperimetry, Geom. Funct. Anal. 12 (2002), 32-55.

[B5] F. Barthe and B. Maurey, Some remarks on isoperimetry of gaussian type, Ann. Inst. H. Poincaré Prob. Stat. 36, (2000), 419-434.

[B7] Sergei Bobkov, An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space, Ann Probab. 24 (1997), 206-214. analytic form

[B8] Sergei Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Prob. 24, 1996, 35-48.

[B9] Sergei Bobkov, Isoperimetric and analytic inequalities for log-concave probability measures, Ann. Prob. 27, (1999), 1903-1921.

[B10] Sergei Bobkov and Christian Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129, 1997.

[CK] E. A. Carlen and C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. 13 (2001), 1-18.

[D] E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. d’Analyse Math. 58 (1992) 99-119.

[E] A. Ehrhard, Inégalités isopérimetriques et intégrales de Dirichlet gaussienes, Ann. Sci. Ecole Norm. Sup. 17 (1984), 317-332.

[FOT] M. Fukushima, Y. Oshima, M. Takeda, “Dirichlet forms and symmetric Markov processes”, Studies in Mathematics 19, De Gruyter, 1994.

[G] Gross, Leonard, Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992), 54–88, Lecture Notes in Math., 1563, Springer, Berlin, 1993. Section 6: “The Gaussian logarithmic Sobolev inequality can be derived in a number of different ways. Its equivalent form, Cor 4.3, which is E. Nelson’s hypercontractivity bound can also be derived in several essentially distinct ways.”

[LS] Landau, H. J.; Shepp, L. A. On the supremum of a Gaussian process. Sankhya Ser. A 32 1970 369-378. A 0-1 law lemma cited by MR of Borell’s [1975] proof of Gaussian isop. inequality: in Gauss space G^n, if convex C about origin has same volume as halfspace containing origin, then for a > 1, |aC| >= |aH|. Relatively trivial proof just uses Brunn-Minkowski for S^n-1. Result also easily from Gaussian isop. inequality because halfspace minimizes not only perimeter but also x dot unit normal at closest boundary approach to origin and then goes farther in lower density and turns less than C.

[L0] Michel Ledoux, Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes, C. F. Acad. Sci. Paris 206 (1988), 79-82. Houdré tells me this is first observation that some log-Sobolev follows from isoperimetry, though not Gross version.

[L1] Michel Ledoux, The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, 89, Amer. Math. Soc., 2001.

[L2] M. Ledoux. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6), 9(2):305–366, 2000.

[L3] M. Ledoux. A simple analytic proof of an inequality by P. Buser. Proc. Amer. Math. Soc., 121(3):951-959, 1994. First observed that a Gaussian log-Sobolev inequality follows from Gaussian isoperimetric inequality. According to [L2, p. 126], Beckner noticed the easy derivation by plugging f=g^2 into Bobkov (see also [Ros, Thm 3.11]).

[Lo] J. Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comm. Math. Helv. 78, 2003, 865-883.

[McK] H. P. McKean, Geometry of differential space, Ann. Prob. 1 (1973), 197-206.

[N] E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973), 211-227.

[O] Shin-ichi Ohta,Optimal transport and Ricci curvature in Finsler geometry, Advanced Studies in Pure Mathematics  (2010), 1–20.

[Q1] Qian, Zhongmin, On conservation of probability and the Feller property. Ann. Probab. 24 (1996), no. 1, 280-292. “Motivated by the classical Lichnerowicz-Bochner-Weitzenböck formula, Bakry and Emery [3] introduced a bilinear map Gamma_2, the “curvature” operator of the diffusion operator L, taking the place of the Ricci curvature (which corresponds to the case L = Delta). …” (Intro., pp. 281-282]).

[Q2] Zhongmin Qian, Estimates for weighted volumes and applications, Quart. J. Math. 48 (1997), 235-242.

[Stam] Stam, A. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2 (1959), 101–112. According to [V, Ch 51, Bib. Notes], “At the end of the fifties, Stam established an inequality which can be recast (after simple changes of functions) as the usual logarithmic Sobolev inequality, found fifteen years later by Gross [1975].

[Ros] Antonio Ros, The isoperimetric problem, David Hoffman, editor, Global Theory of Minimal Surfaces (Proc. Clay Math. Inst. 2001 Summer School, MSRI), Amer. Math. Soc., 2005, 175-209.

[RBCM] César Rosales, Vincent Bayle, Antonio Cañete, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Varn. PDE (2007).

[V] Cedric Villani, Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. ISBN: 978-3-540-71049-3

One Comment

1. Frank Morgan » Blog Archive » Manifolds with Density:

[…] Newton’s Principia Manifolds with Density: Fuller References […]


text:
# OEGetXLogP¶

bool OEGetXLogP(const OEChem::OEMolBase &mol, float &xlogp, float *atomxlogps=0)

Returns the XLogP for a given molecule as described in the LogP section. The returned value will be equal to the sum on the individual atom contributions plus the linear regression constant -0.127.

The atomxlogps parameter can be used to retrieve the contribution of each atom to the total XLogP as shown in Listing 2. See example in Figure: Example of depicting the atom contributions of the XLogP.

Listing 2: Example of retrieving individual atom contributions to XLogP

  // declare an array for atom values
float * atomXLogP = new float[mol.GetMaxAtomIdx()];
float xlogp;
bool rc = OEGetXLogP(mol, xlogp, atomXLogP);

if (!rc)
std::cout << "XLogP failed for molecule" << std::endl;
else
{
std::cout << "XLogP = " << xlogp << std::endl;
for (OEIter<OEAtomBase> atom=mol.GetAtoms();atom;++atom)
{
unsigned int idx = atom->GetIdx();
std::cout << idx << " " << atomXLogP[idx] << std::endl;
}
}

// clean up
delete [] atomXLogP;

Example of depicting the atom contributions of the XLogP

Hint

XLogP should be calculated on the neutral form of the molecule. Therefore, calling the OERemoveFormalCharge function of Quacpac TK is highly recommended prior to the XLogP calculation.

format: plain


text:
# Check for pangrams in elisp

I'm trying to find pangrams ie sentences that contain all the letters of the alphabet, for instance "The quick brown fox jumps over the lazy dog".

I'm an absolute beginner in elisp, and it ended up being a really difficult exercice for me. My idea what to put all the letters in a set check if all the letters of the alphabet are in this set.

I don't think it was a very good idea because elisp doesn't seem to really have sets, so I ended up using a hashmap, which seems very clunky.

Another things that really bothers me is the two if in is-pangram. I feel that they make the function really difficult to reason about, although it's doing something very basic.

(defun add-to-hashmap (hashmap item)
"Insert ITEM into HASHMAP and return HASHMAP."
(puthash item t hashmap)
hashmap)

(defun iter-hashmap-keys (hashmap)
"Return a sequence of the keys in HASHMAP"
(let ((keys '()))
(maphash
(lambda (key _) (setq keys (cons key keys)))
hashmap)
keys))

(defun get-letters (s)
"Return the set of letters in S.
The set is a hashmap where keys are ascii characters and values are t."
(setq letters (make-hash-table :test 'equal))
(seq-reduce
(lambda (hashmap letter) (add-to-hashmap hashmap (downcase letter)))
(split-string s "")
letters))

(defvar ALPHABET
(let* ((letters (get-letters "abcdefghijklmnopqrstuvwxyz"))
(_ (remhash "" letters)))
letters))

(defun is-pangram (s)
"Return t is S is a pangram, nil otherwise."
(setq letters (get-letters s))
(if (seq-some
;; this function returns returns nil if the given letter is
;; *present* in the set, and t otherwise. That way, seq-some
;; returns a non-nil value when there's a letter that is *not* in
;; the set.
(lambda (letter) (if (gethash letter letters) nil t))
(iter-hashmap-keys ALPHABET))
nil
t))

(provide 'pangram)
$$$$

(setq letters (get-letters s))

You are setting the same letters variable from different functions; in Emacs Lisp, that sets a global variable which is bad style: (1) that pollutes the global namespace, (2) that could overwrite an existing variable, and (3) the functions are not reentrant and could lead to bad suprises with side-effects. Better introduce variables with let.

Instead of doing (remhash "" letters), you can pass T for OMIT-NULLS in split-string.

Another approach is to transform the input text so as to downcase all letters, remove spaces, remove duplicate letters and check if it corresponds to the expected alphabet. For example:

(equalp (delete-duplicates
(sort (coerce
(downcase
(remove ?\s "The quick brown fox jumps over the lazy dog"))
'list)
'<))
(coerce "abcdefghijklmnopqrstuvwxyz" 'list))
`


text:
The 39th International Symposium on Lattice Field Theory (Lattice 2022)

Aug 8 – 13, 2022
Hörsaalzentrum Poppelsdorf
Europe/Berlin timezone

Noether supercurrent operator mixing from lattice perturbation theory

Aug 8, 2022, 3:00 PM
20m
CP1-HSZ/Ground-0.001 - HS3 (CP1-HSZ)

CP1-HSZ/Ground-0.001 - HS3

CP1-HSZ

50
Oral Presentation Particle physics beyond the Standard Model

Speaker

Dr Marios Costa (University of Cyprus)

Description

In this work we present perturbative results for the renormalization of the supercurrent operator, $S_\mu$, in ${\cal{N}}=1$ Supersymmetric Yang-Mills theory. At the quantum level, this operator mixes with both gauge invariant and noninvariant operators, which have the same global transformation properties. In total, there are 13 linearly independent mixing operators of the same and lower dimensionality. We determine, via lattice perturbation theory, the first two rows of the mixing matrix, which refer to the renormalizations of $S_\mu$, and of the gauge invariant mixing operator, $T_\mu$. To extract these mixing coefficients in the $\overline{MS}$ renormalization scheme and at one-loop order, we compute the relevant two-point and three-point Green’s functions of $S_\mu$ and $T_\mu$ in two regularizations: dimensional and lattice. On the lattice, we employ the plaquette gluonic action and for the gluinos we use the fermionic Wilson action with clover improvement.

Primary authors

Dr Georg Bergner Dr Marios Costa (University of Cyprus) Prof. Haralambos Panagopoulos (University of Cyprus) Ivan Soler (University of Jena) Gregoris Spanoudes (The Cyprus Institute)


text:
I have an engine cycle in mind looking for it's name

1. Jul 10, 2011

shanesworld

This is a simple question I have because I am looking at a particular engine cycle, and I can't seem to think of its name...thus it is very difficult to search for discussions of something you don't know the name of. I'll describe the cycle, if any one knows the name of the cycle please let me know, or remind me of what it is called. This is neither the Carnot cycle, nor the Otto Cycle,( aguable even more simlpe minded)

In four steps on a Pressure-Volume Diagram it goes

1) Gas in piston (or the likes) expands at a constant pressure P1
2) Heat is allowed to escape, but volume is held fixed at V1
3) Gas in piston contracts at constant lower pressure P2
4) Heat is added while volume is held fixed at V2. (V2<V1)

It looks like a square drawn on the P-V diagram. What is it's name?

2. Jul 11, 2011

EWH

Sounds similar to the Ericsson cycle.
isothermal compression - isobaric heat addition - isothermal expansion - isobaric heat rejection
so it's a P-V square.

3. Jul 11, 2011

shanesworld

...well, I don't actually think that is a P-V square though, but it's interesting in itself.

Last edited: Jul 11, 2011
4. Jul 11, 2011

EWH

If all the phases are isobaric or isothermal, then it can be a square on the P-V diagram. If any phase is other than isobaric or isothermal, it can't be a square on the P-V diagram.

5. Jul 11, 2011

shanesworld

Isothermal usuall does not generally correspond to a straight line on P-V diagram. For instance for an ideal gas nRT=PV. At a constant temperature (i.e isothermal) this means that PV=constant; if the volume changes, the pressure must also change. So in a step where pressure is changed and the volume is held fixed, this would not be isobaric in general... there is at least one major case where your statement is not true. That is why I said the cycle you mentioned is not a square on the P-V diagram.

Last edited: Jul 11, 2011
6. Jul 11, 2011

jack action

I don't think someone bothered naming such a cycle as its existence is pointless.

The problem lies in your isobaric expansion phase followed by a isochoric cooling phase. The only way to achieve the isobaric expansion is by constantly increasing the temperature (like in a brayton or a diesel cycle). And right after that, you are cooling the gas ... without a «work retrieving» phase in between! You absolutely need an adiabatic phase, worst case scenario, an isothermal phase to convert the heat you added into work.

The same goes for your isobaric compression where you need to constantly drop the temperature in order to achieve that, hence removing heat ... followed immediately by a heat addition!

If I didn't make any mistake, the maximum efficiency ($\eta$) you can get with the cycle you described is:

$\eta = \frac{k-1}{k}$

where k is the specific heat ratio for the gas. For air, the max efficiency turns out be 29%. And that is assuming infinite pressure and compression ratios!

7. Jul 12, 2011

EWH

"Isothermal usuall does not generally correspond to a straight line on P-V diagram. "

Quite right, my mistake.

8. Jul 18, 2011

shanesworld

Wow, hadn't checked on this thread in a while but thanks for the post. ...appreciate it.


text:
# How much time after will two oppositely charged particles collide for both gravitational force and electrostatic force?

Suppose two point objects charged with opposite charges $$q_1$$ and $$q_2$$ at a distance $$r$$ in a vaccum.

So, the net electrostatic force on both objects $$= F_c = \frac {q_1q_2}{4π\epsilon_0r²}$$ [$$\epsilon_0$$ is vaccum permittivity]

There should be also gravitational force working on those objects. Suppose, the masses of two objects is $$m_1$$ and $$m_2$$

Then, the gravitational force $$= F_g = \frac {Gm_1m_2} {r²}$$

So, the net force working on the objects $$= F_{net} = \frac {4π\epsilon_0Gm_1m_2 + q_1q_2} {4π\epsilon_0r²}$$

I tried to calculate the time taken by the two objects to collide with each other with the net force but failed. I want to find out the equation. So can anyone help me to find out the period of collision in such a situation mentioned above?

• Use Newton's second law and solve the resulting differential equation. You'll need two initial conditions. May 22 at 9:48
• @kricheli How do I use F = ma for Coulomb's law? May 22 at 14:46
• This is called a two-body problem. And you'll be hard-pressed not to find an answer to this... physics.stackexchange.com/a/156748/297348 May 22 at 15:07
• May 22 at 15:07
• That is a two body problem with vanishing angular momentum... May 22 at 15:26

In this video by Flammable Maths, the solution to a similar problem is given.

The only difference is that we just need to include the electrostatic force, besides that the process is exactly the same.

Let's say we have two objects $$1$$ and $$2$$ with mass $$m_1,m_2$$ and charge $$q_1,q_2$$ respectivey separated by distance $$R$$ then-

$$\textstyle\displaystyle{F=F_C+F_G=\frac{Gm_1m_2+kq_1q_2}{R^2}}$$

Where $$G$$ is the Newtonian constant of gravitation and $$\textstyle\displaystyle{k=\frac{1}{4\pi\epsilon_0}}$$

By newton's third law we have $$F_{12}=-F_{21}$$ so

$$\textstyle\displaystyle{F_{12}=\frac{Gm_1m_2+kq_1q_2}{(r_2-r_1)^2}=m_1\frac{d^2r_1}{dt^2}}$$

$$\textstyle\displaystyle{F_{21}=-\frac{Gm_1m_2+kq_1q_2}{(r_2-r_1)^2}=m_2\frac{d^2r_2}{dt^2}}$$

Where $$R=r_2-r_1$$

$$\therefore\textstyle\displaystyle{\frac{d^2r_2}{dt^2}-\frac{d^2r_1}{dt^2}}$$

$$\textstyle\displaystyle{=-\frac{Gm_1m_2+kq_1q_2}{(r_2-r_1)^2}\bigg(\frac{1}{m_1}+\frac{1}{m_2}\bigg)}$$

$$\implies\textstyle\displaystyle{\frac{d^2R}{dt^2}=-\frac{\kappa}{R^2}}$$

Now we just need to solve this differential equation-

$$\textstyle\displaystyle{\frac{dv}{dt}=-\frac{\kappa}{R^2}=\frac{dv}{dR}\frac{dR}{dt}}$$

$$\implies\textstyle\displaystyle{-\frac{\kappa}{R^2}=v\frac{dv}{dR}}$$

$$\implies\textstyle\displaystyle{-\kappa\int\frac{1}{R^2}dR=\int vdv}$$

At $$t=0$$, $$R(0)=R_i$$ [The initial radius] $$v(0)=0$$ [velocity at the beginning]

$$\therefore\textstyle\displaystyle{\int_{0}^{v(t)}vdv=-\kappa\int_{R_i}^{R(t)}R^{-2}dR}$$

$$\implies\textstyle\displaystyle{\frac{v^2}{2}=\kappa\bigg(\frac{1}{R}-\frac{1}{R_i}\bigg)}$$

$$\implies\textstyle\displaystyle{v=\frac{dR}{dt}=\pm\sqrt{2\kappa\bigg(\frac{R_i-R}{R_iR}\bigg)}}$$

$$\implies\textstyle\displaystyle{\int_{0}^{T_c}dt=\pm\int_{R_i}^{0}\frac{1}{\sqrt{2\kappa\bigg(\frac{R_i-R}{R_iR}\bigg)}}dR}$$

$$\implies\textstyle\displaystyle{T_c=\pm\sqrt{\frac{R_i}{2\kappa}}\int_{R_i}^{0}\sqrt{\frac{R}{R_i-R}}dR}$$

Solving the integral is simple, if you would like to see the steps then see here. Noting that time can't be negative, we have-

$$\textstyle\displaystyle{T_c=\frac{\pi}{2}\sqrt{\frac{R^3}{2\kappa}}}$$

Now simply substituting the value for $$\kappa$$ and $$k$$ gives us less cleaner formula-

$$\textstyle\displaystyle{T_c=\sqrt{\frac{\pi^3\epsilon_0m_1m_2R^3}{2(m_1+m_2)(4\pi\epsilon_0Gm_1m_2+q_1q_2)}}}$$

Start by writing the problem in the form $$F = -\frac{ C }{ r}$$ for the force between the two particles separated by the distance $$r$$ where $$C$$ is easily identified (note that generally, the gravitational force is much weaker than the Coulomb force and is usually ignored, as well, I have included a $$-ve$$ sign because you are considering opposite charges and the particles attract each other).

Suppose that you hold only particle and let the second particle be released from rest at an initial distance $$r_0$$ from the first. Because the force is central this is really a one-dimensional problem.

Rather than solve the system by applying the equation $$F = m_2 \frac{d^2 x}{d t^2}$$ (where for convenience $$x$$ is the separation of the second particle from the first), it is best to consider the conservation of energy. Then the equation you have to consider is $$-\frac{ C }{ x } + \frac{1}{2} m_2 \left( \frac{ d x }{ d t } \right)^2 = \mbox{ constant}$$ where you can determine the constant given that the second particle is initially at rest and separated by the distance $$x = r_0$$.

From the above you can easily find $$d x / d t$$ as a (rather) complicated function of $$x$$ that may possibly be solved analytically for $$x$$ (see the link in Sarka's answer).

So, what you should find is that when you release the second particle it will move towards the first, increasing in speed, getting closer to the fixed particle, the force increases, ensuring its speed increases further. In fact, the speed will increase without limit (in practice, rather than dealing with point particles you would need to take into account the structure of the "particles").

However there is one important issue that has been ignored, namely that as charges accelerate you will need to take electromagnetic radiation into account (as charges accelerate electromagnetic radiation is emitted by the charge), that is a non-trivial problem to include.

• What if the masses of those two point objects become larger as sun? May 22 at 18:39
• @DebanjanBiswas if the masses are big enough then, yes, you need to include them.
– jim
May 22 at 18:47
• @DebanjanBiswas whether you include gravity or not isn't too important, the potential is still of the form $- C/r$.
– jim
May 23 at 10:08
• What is C in this equation? May 23 at 11:47


text:
## Video From 17w5116: Mostly Maximum Principle

Monday, April 3, 2017 16:09 - 16:50
A priori LIpschitz estimates for unbounded solutions of local and nonlocal Hamilton-Jacobi viscous equations with Ornstein-Uhlenbeck Operator


text:
## Doppler effect problem!

1. The problem statement, all variables and given/known data
Assuming the speed of sound is 338 m/s:

A bat moving at 6.2 m/s is chasing a flying insect. The bat emits a 53 KHz chirp and receives an echo at 53.6 KHz. At what speed is the bat catching up on its prey?

2. Relevant equations

$$f^{'}$$=[ ($$V_{Air}$$ + $$V_{observer}$$) $$/$$ ($$V_{Air}$$ - $$V_{source}$$ ] * f

3. The attempt at a solution

I assumed that since the objects are moving in the same direction I said that the signs are different, but I don't know which one should be positive or negative because normally the signs are the same because Doppler effect usually comes into play when the objects are moving towards or away from each other, but these are moving in the same direction, while the bat is moving quicker and accelerating towards the prey....I just plugged in the numbers, but I don't know which way to work with it....Help?

My answers for Vo (+) and Vs (-) was...Vo = 2.4 m/s
With the Vo (-) and Vs (+) I arrived at...Vo = 10.096
 PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study

Recognitions:
Homework Help
Hi jrrodri7,

 Quote by jrrodri7 1. The problem statement, all variables and given/known data Assuming the speed of sound is 338 m/s: A bat moving at 6.2 m/s is chasing a flying insect. The bat emits a 53 KHz chirp and receives an echo at 53.6 KHz. At what speed is the bat catching up on its prey? 2. Relevant equations $$f^{'}$$=[ ($$V_{Air}$$ + $$V_{observer}$$) $$/$$ ($$V_{Air}$$ - $$V_{source}$$ ] * f 3. The attempt at a solution I assumed that since the objects are moving in the same direction I said that the signs are different, but I don't know which one should be positive or negative because normally the signs are the same because Doppler effect usually comes into play when the objects are moving towards or away from each other, but these are moving in the same direction, while the bat is moving quicker and accelerating towards the prey....I just plugged in the numbers, but I don't know which way to work with it....Help?
I don't believe that is correct. If they are both moving towards each other, then the signs in numerator and denominator should be different.

When the observer is moving towards the source, the observed frequency is greater, so in that case the numerator has a positive sign.

When the source is moving towards the observer, the observed frequency is greater, so the denominator has the negative sign.

So in this problem, which would it be?

Also, they are talking about the reflected wave, so you'll need to use the formula twice. What do you get?

 My answers for Vo (+) and Vs (-) was...Vo = 2.4 m/s With the Vo (-) and Vs (+) I arrived at...Vo = 10.096


text:
## anonymous 5 years ago SR (600y^23)

1. anonymous

$\sqrt{600y ^{23}}$ is the actual equation

2. bahrom7893

$(600y^{23})^{1/2}$ = $10 (y^{22/2})\sqrt{6*y}$

3. bahrom7893

$10y^{11}\sqrt{68y}$

4. bahrom7893

woops sorry: $10y^{11}\sqrt{6*y}$

5. bahrom7893

Please click on become a fan if I helped, I really want to get to the next level!! Thanks =)

6. anonymous

any questions?

7. anonymous

yeah I am still lost as can be...

8. anonymous

your in alegebra 2 right? im might be in the same chapter as you.

9. anonymous

essentially it is algebra 2

10. anonymous

okay soo first see in you can find a sqrt of 600.

11. anonymous

it comes out as a decimal so I know that I have to simplify first and to me it would make the most sense to have it be $\sqrt{100}\sqrt{6}$

12. anonymous

being as 100 is a perfect square

13. anonymous

what my teacher taught me was to write it like Sqrt(100x6) then break those down to more simple numbers. Sqrt(50x2x2x3) [50x2=100] and 2x3=6

14. anonymous

Do you guys want to go through the problem

15. anonymous

?

16. anonymous

i am going throught the problem...

17. anonymous

the sqrt will eventually look like this (5x5x2x2x2x3)

18. anonymous

you have 2 5's and 2 2's. take both and multiple 5 and 2 and put their product on the outside of the squareroot.

19. anonymous

your equation should look like this. 10sqrt(6y^23) [6=product of remianing numbers in root]

20. anonymous

for just figure out how many times 2 can go into 23 any remaing numbers remain in the root. answer = $10y^{11}\sqrt{6y}$


text:
# nuclear transmutation equation

U.S. Department of Energy, Nuclear Physics and Reactor Theory. split apart. Natural transmutation by stellar nucleosynthesis in the past created most of the heavier chemical elements in the known existing universe, and continues to take place to this day, creating the vast majority of the most common elements in the universe, including helium, oxygen and carbon. Glasstone, Sesonske. 7. Most of absorption reactions result in fission reaction, but a minority results in radiative capture forming 236U. 238U is not capable of undergoing fission reaction after absorbing thermal neutron, on the other hand 238U can be fissioned by fast neutron with energy higher than >1MeV. () equations/solutions. Ceramic targets containing actinides can be bombarded with neutrons to induce transmutation reactions to remove the most difficult long-lived species. In physics, nuclear transmutation is the conversion of one chemical element or an isotope into... Fuel Depletion – Isotopic Changes. The Big Bang also produced small amounts of lithium, beryllium and perhaps boron. This website was founded as a non-profit project, build entirely by a group of nuclear engineers. The cross-section for radiative capture for thermal neutrons is about 270 barns (for 0.025 eV neutron). Insert a brief calculation of doses[citation needed]. Two of them, Tc-99 and I-129, are mobile enough in the environment to be potential dangers, are free or mostly free of mixture with stable isotopes of the same element, and have neutron cross sections that are small but adequate to support transmutation. Which equation represents artificial transmutation. We have step-by-step solutions for your textbooks written by Bartleby experts! The solution, as well as equivalent solutions for three nuclides and the general case, are known as Bateman. Answer Lead-210 is a short lived isotope of lead with a half-life of 22.3 years. Transmutation of transuranium elements (TRUs, i.e. first, it is limited by the costly and cumbersome need to separate LLFP isotopes before they can undergo transmutation. Most natural transmutation on the Earth today is mediated by cosmic rays (such as production of carbon-14) and by the radioactive decay of radioactive primordial nuclides left over from the initial formation of the solar system (such as potassium-40, uranium and thorium), plus the radioactive decay of products of these nuclides (radium, radon, polonium, etc.). More lithium, beryllium and boron were produced later, in a natural nuclear reaction, cosmic ray spallation. transmutation matrix in the calculation of complex radioactive decay chains to differential equations: the Bateman … heat -infrared photon radiation-, which is an advantage for the storage or disposal af radioactive waste), chemical and biological properties of the element. See the answer. If you have, this isn't the first time you've heard of the idea of turning something fairly useless into gold. For fast neutrons its fission cross-section is on the order of barns. As a reactor is... Evolution Equations. Transmutation In 1919, Ernest Rutherford (the man who discovered the nucleus) and his research group became the first people to cause transmutation - the changing of one nucleus into another. ²⁷₁₃ Al + ⁴₂ He → ³⁰₁₅ P + ¹₀ n. Fission. The impossibility of the metallic transmutation had been debated amongst alchemists, philosophers and scientists since the Middle Ages. He bombarded alpha particles on Nitrogen-14 to produce Oxygen-17 with protons. They are also mixed with larger quantities of other isotopes of the same element. This may be hard to believe, but the concept of this fairy tale isn't so fa… Fusion. By the transmutation constant λ i,j we understand probability of the i th nuclide production per time unit from the j th nuclide destruction, as a result of nuclear interaction with the whole spectrum of interacting particles or due to the natural nuclear decay. Given that this length of storage is necessary, the fission products with shorter half-lives can also be stored until they decay. 235/92 U → 231/90 Th + 4/2 He. It can be done in an artificial manner and also occurs naturally. The charge is conserved, or, the sum of charges on the left is equal to the sum of charges in the right. These equations describe the rate of burnup of U-235, the rate of buildup of Pu-239, production of Pu-240 and Pu-241, the buildup of neutron absorbing fission products, and the overall rate of reactivity change in the reactor due to the changing composition of the fuel. Write a balanced nuclear equation that represents the transmutation of two nuclei. For example, at a burnup of 40GWd/tU, about 40% of the total energy released comes from bred plutonium. About 26% of all absorption reactions result in fission. Rutherford bombarded nitrogen atoms with high-speed α particles from a natural radioactive isotope of radium and observed protons resulting from the reaction… The initial fuel load of a new reactor core (so called first core) is entirely fresh fuel, that is, fuel with no plutonium or fission products present. The pseudo-proton is a particle predicted by Santilli [2] based on the Artificial transmutation. However, U-233, which is fissile, will be present in the UNF. This corresponds to a breeding ratio for this fuel burnup of about 0.4 to 0.5. Fission and Fusion. Also, Tc-99 can substitute for U-238 in supplying Doppler broadening for negative feedback for reactor stability. Section 9.2 presented the process for developing the nuclear equation pattern for a radioactive decay reaction, in which a more stable daughter nucleus is generated upon the emission, or release, of … It is useful in the analysis of sedimentation chronology. plutonium 239). Transmutation of all fission products with half-life greater than one year is studied in Grenoble,[20] with varying results. It can occur by the radioactive decay of a nucleus, or the reaction of a nucleus with another particle. Keywords: Pseudo-proton, Low Energy Nuclear Transmutation, Hydrogen Energy Source, Neutron, Pseudo-protonium, Detection, Spectroscopy 1. Trying to figure out the other product from our nuclear equation, I know nucleons are conserved, so if I have 238 nucleons on the left, I need 238 nucleons on the right. Most of absorption reactions result in fission reaction, but a part of reactions result in radiative capture forming 242Pu. [21] Types of nuclear reactions include fission, fusion and. neptunium, americium, and curium, about 0.1wt% each in LWRs' UNF) has the potential to help solve some problems posed by the management of radioactive waste by reducing the proportion of long-lived isotopes it contains. Alchemists pursued the philosopher's stone, capable of chrysopoeia – the transformation of base metals into gold. A transmutation entails a change in the structure of atomic nuclei and hence may be induced by a nuclear reaction (q.v. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1. So 234 plus four gives me a total of … These are a type of red giant that "puffs" off its outer atmosphere, containing some elements from carbon to nickel and iron. 56 2 54 4 26 1Fe H Mn The cross-section for radiative capture for thermal neutrons is about 99 barns (for 0.0253 eV neutron). Textbook solution for Chemistry: Matter and Change 1st Edition Dinah Zike Chapter 24.3 Problem 19PP. [3] While alchemists often understood chrysopoeia as a metaphor for a mystical, or religious process, some practitioners adopted a literal interpretation, and tried to make gold through physical experiment. The half-life of 239U is approximately 23.5 minutes. Most common reactor fuels are composed of either natural or partially enriched uranium. Since uranium is present in MOX, although plutonium will be burnt, second generation plutonium will be produced through the radiative capture of U-238 and the two subsequent beta minus decays. Nuclear particles and radiations. The process of the transmutation of fertile materials to fissile materials is referred to as fuel breeding.eval(ez_write_tag([[300,250],'nuclear_power_net-box-3','ezslot_3',103,'0','0'])); Fertile materials are not capable of undergoing fission reaction after absorbing thermal (slow or low energy) neutrons and these materials are not capable of sustaining a nuclear fission chain reaction. Nuclear transmutation is the conversion of one nuclide into another. Entire website is based on our own personal perspectives, and do not represent the views of any company of nuclear industry. Nuclear transmutation is the conversion of one chemical element or an isotope into another.. A transmutation can be achieved either by nuclear reactions (in which an outside particle reacts with a nucleus) or by radioactive decay (where no outside particle is needed). Special reference: Paul Reuss, Neutron Physics, EDP Sciences, 2008, ISBN: 2759800415. "[6], Rutherford and Soddy were observing natural transmutation as a part of radioactive decay of the alpha decay type. During the cycle, plutonium can be burnt in a power reactor, generating electricity. In 1919, Ernest Rutherford was able to accomplish transmutation of nitrogen into oxygen at the University of Manchester, using alpha particles directed at nitrogen 14 N + α → 17 O + p. This was the first observation of an induced nuclear reaction, that is, a reaction in which particles from one decay are used to transform another atomic nucleus. Well, I have four from my alpha particle, so I need 234 more. ... Write the balanced nuclear equation for the production of the following transuranium elements: berkelium-244, … is a fissile isotope and its fission cross-section for thermal neutrons is about 585 barns (for 0.0253 eV neutron). Approximately 3 – 4% of the heavy nuclei are fissioned. in the atomic nucleus, nuclear transmutation occurs in any process where the number of protons or neutrons in the nucleus is changed. The principle behind partitioning and transmutation is that there is an optimum means of managing each group of elements in spent nuclear fuel. January 1993. For example, the observed light curves of supernova stars such as SN 1987A show them blasting large amounts (comparable to the mass of Earth) of radioactive nickel and cobalt into space. Radioisotopes usages and wastes Lesson 1 – Nuclear Transmutations Nuclear chemistry is the study of changes that occur in the nucleus of atoms. Cockcroft and Walton split lithium with high energy protons April 1932. See decay chain. In the medium term the fission products of highest concern are strontium-90 and caesium-137; both have a half-life of about 30 years. The process of transmutation can be represented in terms of a chemical reaction as it is also a conversion of one element into another. For example: Special reference: W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1. This is called a chain reaction. During the fuel burning, the content of the U-235 continuously decreases and the content of the plutonium increases (up to ~1% of Pu ). The caesium-137 is responsible for the majority of the external gamma dose experienced by workers in nuclear reprocessing plants[18] and, in 2005, to workers at the Chernobyl site. Well, I have four from my alpha particle, so I need 234 more. Nuclear Decay Types of Reactions Alpha Emission Beta Emission Gamma Emission Positron Emission Electron Capture Transmutation I Transmutation II Review Practice Quiz … Trying to figure out the other product from our nuclear equation, I know nucleons are conserved, so if I have 238 nucleons on the left, I need 238 nucleons on the right. Blackett correctly identified the underlying integration process and the identity of the residual nucleus. William Alfred Fowler, Margaret Burbidge, Geoffrey Burbidge, and Fred Hoyle, 'Synthesis of the Elements in Stars', Takibayev A., Saito M., Artisyuk V., and Sagara H., ', Glenn Seaborg produced several thousand atoms of gold from bismuth, List of nuclear waste treatment technologies, http://history.aip.org/history/exhibits/rutherford/sections/atop-physics-wave.html. In the nuclear transmutation represented by^239_94 Pu(^4_2 He, ^1_0 n)?, what is the product? Physics of Nuclear Kinetics. Instead, they should simply be stored until they decay. Which nuclear equation represents a natural transmutation. The artificial transmutation is shown with the number of protons. When irradiated with fast neutrons in a nuclear reactor, these isotopes can undergo nuclear fission, destroying the original actinide isotope and producing a spectrum of radioactive and nonradioactive fission products. For fast neutrons its fission cross-section is on the order of barns. $$\ce{^{9}_{4}Be}$$ + $$\ce{^{1}_{0}n}$$ $$\rightarrow$$ $$\ce{^{10}_{4}Be}$$ Back to top; 9.3: Half-Life; 9.5: Fission and Fusion Write the balanced nuclear equation for the reaction. About 85% of all absorption reactions result in fission. This can be achieved by adding a yttrium deuteride (YD2) moderator. Flux gradually depletes the uranium-235, decreasing core reactivity ( compensated by control rods, chemical shim or burnable ). 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