Asymptotic results for finite superpositions of Ornstein–Uhlenbeck processes

A model of intermittency based on superposition of Lévy driven Ornstein–Uhlenbeck processes is studied in [6 Grahovac, D., Leonenko, N., Sikorskii, A., and Te?niak, I. 2016. Intermittency of superpositions of Ornstein–Uhlenbeck type processes. J. Stat. Phys. 165:390–408.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, as shown in Theorem 5.1 in that paper, finite superpositions obey a (sample path) central limit theorem under suitable hypotheses. In this paper we prove large (and moderate) deviation results associated with this central limit theorem.     

Grouped Functional Time Series Forecasting: An Application to Age-Specific Mortality Rates

Age-specific mortality rates are often disaggregated by different attributes, such as sex, state, and ethnicity. Forecasting age-specific mortality rates at the national and sub-national levels plays an important role in developing social policy. However, independent forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider reconciling forecasts of age-specific mortality rates, extending the methods of Hyndman et al. in 2011 Hyndman, R. J., Ahmed, R. A., Athanasopoulos, G., and Shang, H. L. (2011), “Optimal Combination Forecasts for Hierarchical Time Series,” Computational Statistics and Data Analysis, 55, 2579–2589.[Crossref], [Web of Science ®] , [Google Scholar] to functional time series, where age is considered as a continuum. The grouped functional time series methods are used to produce point forecasts of mortality rates that are aggregated appropriately across different disaggregation factors. For evaluating forecast uncertainty, we propose a bootstrap method for reconciling interval forecasts. Using the regional age-specific mortality rates in Japan, obtained from the Japanese Mortality Database, we investigate the one- to ten-step-ahead point and interval forecast accuracies between the independent and grouped functional time series forecasting methods. The proposed methods are shown to be useful for reconciling forecasts of age-specific mortality rates at the national and sub-national levels. They also enjoy improved forecast accuracy averaged over different disaggregation factors. Supplementary materials for the article are available online.     

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005 Bodnar , M. , Velazquez , J. J. L. ( 2005 ). Derivation of macroscopic equations for individual cell-based models: a formal approach . Math. Methods Appl. Sci. 28 ( 15 ): 1757 – 1779 .[Crossref], [Web of Science ®] , [Google Scholar]; Holm and Putkaradze, 2005 Holm , D. D. , Putkaradze , V. ( 2005 ). Aggregation of finite size particles with variable mobility . Phys. Rev. Lett. 95 : 226106 . [Google Scholar]; Mogilner and Edelstein-Keshet, 1999 Mogilner , A. , Edelstein-Keshet , L. ( 1999 ). A non-local model for a swarm . J. Math. Biol. 38 ( 6 ): 534 – 570 .[Crossref], [Web of Science ®] , [Google Scholar]; Morale et al., 2005 Morale , D. , Capasso , V. , Oelschläger , K. ( 2005 ). An interacting particle system modelling aggregation behavior: from individuals to populations . J. Math. Biol. 50 ( 1 ): 49 – 66 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Topaz and Bertozzi, 2004 Topaz , C. M. , Bertozzi , A. L. ( 2004 ). Swarming patterns in a two-dimensional kinematic model for biological groups . SIAM J. Appl. Math. 65 ( 1 ): 152 – 174 (electronic) .[Crossref], [Web of Science ®] , [Google Scholar]; Topaz et al., 2006 Topaz , C. M. , Bertozzi , A. L. , Lewis , M. A. ( 2006 ). A nonlocal continuum model for biological aggregation . Bull. Math. Biol. 68 ( 7 ): 1601 – 1623 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.     

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ? R ² ? J ? Rⁿ (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R ² such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = G_k = (2) × ? × (2).

Commutative Gorenstein rings appeared at first in the paper [3] Gorenstein, D. 1952. An Arithmetic Theory of Adjoint Plane Curves. Trans. AMS., 72: 414–436. [Crossref], [Web of Science ®] , [Google Scholar]. Torsion-free modules over commutative Gorenstein domains were investigated in [1] Bass, H. 1963. On the Ubiquity of Gorenstein Rings. Math. Z., 82(1): 8–27. [Crossref] , [Google Scholar]. Noncommutative Gorenstein orders were considered in [2] Drozd, Yu. A., Kirichenko, V. V. and Roiter, A. V. 1967. On Hereditary and Bass Orders. Izv. Akad. Nauk SSSR Ser. Mat., 31: 1415–1436. Math. USSR – Izvestija, 1967, 1, 1357–1375 [Google Scholar] and [10] Roggenkamp, K. W. 1970. Lattices Over Orders II Berlin, Heidelberg, New York: Springer-Verlag. [Crossref] , [Google Scholar]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5] Kirichenko, V. V. 1978. On Quasi-Frobenius Rings and Gorenstein Orders. Trudy Math. Steklov Inst., 148: 168–174. (in Russian) [Google Scholar]. Arbitrary tiled orders were considered in [4] Jategaonkar, V. A. 1974. Global Dimension of Tiled Orders Over a Discrete Valuation Ring. Trans. AMS., 196: 313–330. [Crossref], [Web of Science ®] , [Google Scholar], 11-14 Simson, D. 1992. Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. Vol. 4, Gordon and Breach Science Publishers. Zavadskij, A. G. 1973. The Structure of Orders with Completely Decomposable Representations. Mat. Zametki, 13: 325–335. (in Russian) Zavadskij, A. G. and Kirichenko, V. V. 1976. Torsion-free Modules over Prime Rinqs. Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 57: 100–116. J. Soviet. Math. 1979, 11, 598–612 Zavadskij, A. G. and Kirichenko, V. V. 1977. Semimaximal Rings of Finite Type. Mat. Sbornik, 103(No. 3): 323–345. Math. USSR Sbornik, 1977, 32 (3), 273–291 .     

ON TWO SIMPLE CRITERIA FOR RECOGNIZING COMPLETE INTERSECTIONS IN CODIMENSION 2

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P ⁿ , n ≥ 3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p ≤ n ? 1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P ⁿ and p ≤ n. Finally, using our characterization, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu ([5] Franco, D., Kleiman, S. L. and Lascu, A. T. Gherardelli Linkage and Complete Intersections Math. AG/0003075 [Google Scholar]) on self-linking and complete intersections in positive characteristic.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.     

THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES

In 1989 Nichols and Zoeller [NZ] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111: 381–385. [Crossref], [Web of Science ®] , [Google Scholar] showed that finite dimensional k-Hopf algebras are free over Hopf subalgebras. An analog result for Yetter Drinfeld Hopf algebras was not known. In this paper the existence of such a basis will be proved. Moreover the existence of a basis in a certain categorial sense cannot be expected.     

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004 Chalub , F. A. C. C. , Markowich , P. A. , Perthame , B. , Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]), Hwang et al. (2005a Hwang , H. J. , Kang , K. , Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . [Google Scholar] b Hwang , H. J. , Kang , K. , Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . [Google Scholar]) using direct and strong dispersive effects.

Here, we use the weak dispersion estimates of Castella and Perthame (1996 Castella , F. , Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. [Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . [Google Scholar]) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.     

Unifying strongly clean power series rings

It is unknown whether a power series ring over a strongly clean ring is, itself, always strongly clean. Although a number of authors have shown that the above statement is true in certain special cases, the problem remains open, in general. In this article, we look at a class of strongly clean rings, which we call the optimally clean rings, over which power series are strongly clean. This condition is motivated by work in [10 Diesl, A. J., Dorsey, T. J., Garg, S., Khurana, D. (2012). A note on completeness and strongly clean rings, preprint. [Google Scholar]] and [11 Diesl, A. J., Dorsey, T. J., Iberkleid, W., LaFuente-Rodriguez, R., McGovern, W (2013). Strongly clean triangular matrices over abelian rings, preprint. [Google Scholar]]. We explore the properties of optimally clean rings and provide many examples, highlighting the role that this new class of rings plays in investigating the question of strongly clean power series.     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].     

ADIABATIC DECOMPOSITION OF THE -DETERMINANT OF THE DIRAC LAPLACIAN. I. THE CASE OF AN INVERTIBLE TANGENTIAL OPERATOR

ABSTRACT

We discuss the decomposition of the ζ-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold. The result was announced in the Note Ref. [16]. The proof sketched in the Note was based on results of Brüning and Lesch (see Ref. [4]). In the meantime we have found another proof, more direct and elementary, and closer to the spirit of the original papers which initiated the study of the adiabatic decomposition of the spectral invariants (see Refs. [7] Douglas, R.G. and Wojciechowski, K.P. 1991. Adiabatic Limits of the η-Invariants. The Odd-dimensional Atiyah–Patodi–Singer Problem. Comm. Math. Phys., 142: 139–168. [Crossref], [Web of Science ®] , [Google Scholar] and [21] Singer, I.M. 1988. “The η-Invariant and the Index”. In Mathematical Aspects of String Theory Edited by: Yau, S.-T. pp. 239–258. Singapore: World Scientific Press.  [Google Scholar]). We discuss this proof in detail. We study the general case (non-invertible tangential operator) in forthcoming work (see Refs. [17] Park, J. and Wojciechowski, K.P. 2001. Scattering Theory and Adiabatic Decomposition of the ζ-Determinant of the Dirac Laplacian 01–02. IUPUI Preprint [Google Scholar] and [18] Park, J. and Wojciechowski, K.P. 2001. Adiabatic Decomposition of the ζ-Determinant of the Dirac Laplacian II. The Case of Non-invertible Tangential Operator In preparation [Google Scholar]). In the Appendix we present the computation of the cylinder contribution to the ζ-function of the Dirac Laplacian on a manifold with boundary, which we need in the main body of the paper. This computation is also used to show the vanishing result for the ζ-function on a manifold with boundary.     

Stratifying Systems via Relative Projective Modules

ABSTRACT

In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]), and also the “Comparison method” given in Platzeck and Reiten (2001 Platzeck , M. I. , Reiten , I. ( 2001 ). Modules of finite projective dimension for standardly stratified algebras . Comm. in Algebra 29 : 973 – 986 . [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]) and Proposition 2.1 in Ringel (1991 Ringel , C. M. ( 1991 ). The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences . Math. Z. 208 : 209 – 223 .[Crossref], [Web of Science ®] , [Google Scholar]) follows.

     

Large time behavior of solutions of a dissipative semilnear heat equation

We study the current of the Pauli operator in a strong constant magnetic field. We prove that in the semi-classical limit the persistent current and the current from the interaction of the spin with the magnetic field cancel, in the case where the magnetic field is very strong. Furthermore we calculate the next term in the asymptotics and estimate the error. Finally, we discuss the connection between this work and the semi-classical estimate of the energy in strong magnetic fields proved by Lieb, Solovej and Yngvason [8] Lieb, E., Solovej, J. P. and Yngvason, J. 1994. Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys., 161: 77–124. [Crossref], [Web of Science ®] , [Google Scholar].

     

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.     

Extension of Goulden–Jackson cluster method on pattern occurrences in random sequences and comparison with Régnier–Szpankowski method

The Goulden–Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1 Goulden, I.P. and Jackson, D.M. 1979. An inversion theorem for cluster decompositions of sequences with distinguished subsequences. Journal of London Mathematical Society, Second Series, 20: 567–576. [Crossref], [Web of Science ®] , [Google Scholar]]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In this paper, we elaborate on one of the several extensions in [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. The extended result is then compared with the method of Régnier–Szpankowski [3 Régnier, M. and Szpankowski, W. 1997. On the approximate pattern occurrences in a text. Proceedings of the compression and complexity of sequences 1997, : 253–264.  [Google Scholar]], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier–Szpankowski method can be simplified to the extended Goulden–Jackson method.     

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 ( 5 ): 43 – 49 [CROSSREF] [CSA] [Crossref] , [Google Scholar]) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999 Bogomolny , E. , Gerland , U. , Schmit , C. ( 1999 ). Models of intermediate spectral statistics . Phys. Rev. E 59 : 1315 – 1318 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004 Burq , N. , Zworski , M. ( 2004 ). Geometric control in the presence of a black box . JAMS 17 : 443 – 471 [CROSSREF] [CSA] [Web of Science ®] , [Google Scholar]).     

Generalized Mean Curvature Flow in Carnot Groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4 Chen , Y. , Giga , Y. , Goto , S. ( 1991 ). Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations . J. Diff. Geom. 33 : 749 – 786 .[Crossref], [Web of Science ®] , [Google Scholar]] and [12 Evans , L. C. , Spruck , J. ( 1991 ). Motion of level sets by mean curvature. I . J. Diff. Geom. 33 ( 3 ): 635 – 681 . [Google Scholar]]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.