Authors:Pieter C. Allaart Pages: 1 - 39 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Pieter C. Allaart This paper studies a large class of continuous functions f : [ 0 , 1 ] → R d whose range is the attractor of an iterated function system { S 1 , … , S m } consisting of similitudes. This class includes such classical examples as Pólya's space-filling curves, the Riesz–Nagy singular functions and Okamoto's functions. The differentiability of f is completely classified in terms of the contraction ratios of the maps S 1 , … , S m . Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) f is nowhere differentiable; (ii) f is non-differentiable almost everywhere but with uncountably many exceptions; or (iii) f is differentiable almost everywhere but with uncountably many exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and (iii) above is calculated, and more generally, the complete multifractal spectrum of f is determined.
Authors:Serena Dipierro; Aram L. Karakhanyan Pages: 40 - 81 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Serena Dipierro, Aram L. Karakhanyan In this paper we prove the local Lipschitz regularity of the minimizers of the two-phase Bernoulli type free boundary problem arising from the minimization of the functional J ( u ) : = ∫ Ω ∇ u p + λ + p χ { u > 0 } + λ − p χ { u ≤ 0 } , 1 < p < ∞ . Here Ω ⊂ R N is a bounded smooth domain and λ ± are positive constants such that λ + p − λ − p > 0 . Furthermore, we show that for p > 1 the free boundary has locally finite perimeter and the set of non-smooth points of the free boundary is of zero ( N − 1 ) -dimensional Hausdorff measure. For this, our approach is new even for the classical case p = 2 .
Authors:Kurusch Ebrahimi-Fard; Frédéric Patras Pages: 112 - 132 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Kurusch Ebrahimi-Fard, Frédéric Patras The theory of cumulants is revisited in the “Rota way”, that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.
Authors:Dean Baskin; András Vasy; Jared Wunsch Pages: 160 - 216 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Dean Baskin, András Vasy, Jared Wunsch We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation. In two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the Kerr family. In these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero.
Authors:Fei Fang; Zhong Tan Pages: 217 - 247 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Fei Fang, Zhong Tan In this article, we study the heat flow equation for Dirichlet-to-Neumann operator with critical growth. By assuming that the initial value is lower-energy, we obtain the existence, blowup and regularity. On the other hand, a concentration phenomenon of the solution when the time goes to infinity is proved.
Authors:Pere Ara; Matias Lolk Pages: 367 - 435 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Pere Ara, Matias Lolk We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algebras, O ( E , C ) and O r ( E , C ) , of a separated graph ( E , C ) , and of the abelianized Leavitt path algebra L K ab ( E , C ) as well. These algebras are the (reduced) crossed products with respect to the above-mentioned partial actions, and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary D ∞ -saturated subsets of a certain infinite separated graph ( F ∞ , D ∞ ) built from ( E , C ) , called the separated Bratteli diagram of ( E , C ) . We finally use these tools to study simplicity and primeness of the tame separated graph algebras.
Authors:Diego Alonso-Orán; Antonio Córdoba; Ángel D. Martínez Pages: 436 - 445 Abstract: Publication date: 13 April 2018 Source:Advances in Mathematics, Volume 328 Author(s): Diego Alonso-Orán, Antonio Córdoba, Ángel D. Martínez In this paper we provide an integral representation of the fractional Laplace–Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. The first deals with compact manifolds with or without boundary, while the second approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.
Authors:Loukas Grafakos; Danqing He; Petr Honzík Pages: 54 - 78 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Loukas Grafakos, Danqing He, Petr Honzík We study the rough bilinear singular integral, introduced by Coifman and Meyer [8], T Ω ( f , g ) ( x ) = p.v. ∫ R n ∫ R n ( y , z ) − 2 n Ω ( ( y , z ) / ( y , z ) ) f ( x − y ) g ( x − z ) d y d z , when Ω is a function in L q ( S 2 n − 1 ) with vanishing integral and 2 ≤ q ≤ ∞ . When q = ∞ we obtain boundedness for T Ω from L p 1 ( R n ) × L p 2 ( R n ) to L p ( R n ) when 1 < p 1 , p 2 < ∞ and 1 / p = 1 / p 1 + 1 / p 2 . For q = 2 we obtain that T Ω is bounded from L 2 ( R n ) × L 2 ( R n ) to L 1 ( R n ) . For q between 2 and infinity we obtain the analogous boundedness on a set of indices around the point ( 1 / 2 , 1 / 2 , 1 ) . To obtain our results we ... PubDate: 2018-02-05T06:56:50Z DOI: 10.1016/j.aim.2017.12.013 Issue No:Vol. 326 (2018)
Authors:Amin Gholampour; Artan Sheshmani Pages: 79 - 107 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Amin Gholampour, Artan Sheshmani Motivated by the S-duality conjecture, we study the Donaldson–Thomas invariants of the 2-dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular threefold X fibered over a nonsingular curve. In the case where X is a K3 fibration, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 fiber and the Noether–Lefschetz numbers of the fibration. We prove that a certain generating function of these invariants is a vector modular form of weight − 3 / 2 as predicted in S-duality.
Authors:Israel Michael Sigal; Tim Tzaneteas Pages: 108 - 199 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Israel Michael Sigal, Tim Tzaneteas The Ginzburg–Landau equations play a key role in superconductivity and particle physics. They inspired many imitations in other areas of physics. These equations have two remarkable classes of solutions – vortices and (Abrikosov) vortex lattices. For the standard cylindrical geometry, the existence theory for these solutions, as well as the stability theory of vortices are well developed. The latter is done within the context of the time-dependent Ginzburg–Landau equations – the Gorkov–Eliashberg–Schmid equations of superconductivity – and the abelian Higgs model of particle physics. We study stability of Abrikosov vortex lattices under finite energy perturbations satisfying a natural parity condition (both defined precisely in the text) for the dynamics given by the Gorkov–Eliashberg–Schmid equations. For magnetic fields close to the second critical magnetic field and for arbitrary lattice shapes, we prove that there exist two functions on the space of lattices, such that Abrikosov vortex lattice solutions are asymptotically stable, provided the superconductor is of Type II and these functions are positive, and unstable, for superconductors of Type I, or if one of these functions is negative.
Authors:Osamu Iyama; Øyvind Solberg Pages: 200 - 240 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Osamu Iyama, Øyvind Solberg We generalize the notions of n-cluster tilting subcategories and τ-selfinjective algebras into n-precluster tilting subcategories and τ n -selfinjective algebras, where we show that a subcategory naturally associated to n-precluster tilting subcategories has a higher Auslander–Reiten theory. Furthermore, we give a bijection between n-precluster tilting subcategories and n-minimal Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander–Solberg correspondence [8] as well as a Gorenstein analog of n-Auslander correspondence [22]. The Auslander–Reiten theory associated to an n-precluster tilting subcategory is used to classify the n-minimal Auslander–Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander–Solberg.
Authors:David Chataur; Martintxo Saralegi-Aranguren; Daniel Tanré Pages: 314 - 351 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré For having a Poincaré duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure need a coefficient field or an additional hypothesis on the torsion. In this work, by using the classical geometric process of blowing-up, adapted to a simplicial setting, we build a cochain complex which gives a Poincaré duality via a cap product with intersection homology, for any commutative ring of coefficients. We prove also the topological invariance of the blown-up intersection cohomology with compact supports in the case of a paracompact pseudomanifold with no codimension one strata. This work is written with general perversities, defined on each stratum and not only in function of the codimension of strata. It contains also a tame intersection homology, suitable for large perversities.
Authors:Richard Lärkäng; Hossein Raufi; Jean Ruppenthal; Martin Sera Pages: 465 - 489 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Richard Lärkäng, Hossein Raufi, Jean Ruppenthal, Martin Sera We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson–Păun. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.
Authors:Shiwu Yang Pages: 490 - 520 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Shiwu Yang It is known that the Maxwell–Klein–Gordon equations in R 3 + 1 admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of r-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.
Authors:Susanna Dann; Jaegil Kim; Vladyslav Yaskin Pages: 521 - 560 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Susanna Dann, Jaegil Kim, Vladyslav Yaskin Busemann's intersection inequality asserts that the only maximizers of the integral ∫ S n − 1 K ∩ ξ ⊥ n d ξ among all convex bodies of a fixed volume in R n are centered ellipsoids. We study this question in the hyperbolic and spherical spaces, as well as general measure spaces.
Authors:Alexander G. Melnikov; Keng Meng Ng Pages: 864 - 907 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Alexander G. Melnikov, Keng Meng Ng We prove that c.c. torsion abelian groups can be described by a Π 4 0 -predicate that describes the failure of a brute-force diagonalisation attempt on such groups. We show that there is no simpler description since their index set is Π 4 0 -complete. The results can be viewed as a solution to a 60 year-old problem of Mal'cev in the case of torsion abelian groups. We prove that a computable torsion abelian group has one or infinitely many computable copies, up to computable isomorphism. The result confirms a conjecture of Goncharov from the early 1980s for the case of torsion abelian groups.
Authors:Longting Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Longting Wu The purpose of the article is to give a proof of a conjecture of Maulik and Pandharipande for genus 2 and 3. As a result, it gives a way to determine Gromov–Witten invariants of the quintic threefold for genus 2 and 3.
Authors:F.A. Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): F.A. Grünbaum, L. Velázquez Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces. In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized ‘Schur functions’ meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FR-functions satisfy splitting properties which extend useful factorizations of Schur functions. When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refer to as ‘the measure of the Nevanlinna function’. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple ‘Schur algorithm’ for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the ‘Schur algorithm on the real line’. The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions –as first return generating functions– and their splittings –which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.
Authors:Sylvain Cappell; Alexander Lubotzky; Shmuel Weinberger Abstract: Publication date: Available online 19 January 2018 Source:Advances in Mathematics Author(s): Sylvain Cappell, Alexander Lubotzky, Shmuel Weinberger Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel [6] combined with Mostow rigidity imply that there exists a finite group G = G ( M ) such that any finite subgroup of Homeo + ( M ) is isomorphic to a subgroup of G. Borel [6] asked if there exist M's with G ( M ) trivial and if the number of conjugacy classes of finite subgroups of Homeo + ( M ) is finite. We answer both questions: (1) For every finite group G there exist M's with G ( M ) = G , and (2) the number of maximal subgroups of Homeo + ( M ) can be either one, countably many or continuum and we determine (at least for dim M ≠ 4 ) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dim M ≠ 4 ) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.
Authors:Anton Alekseev; Nariya Kawazumi; Yusuke Kuno; Florian Naef Pages: 1 - 53 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Anton Alekseev, Nariya Kawazumi, Yusuke Kuno, Florian Naef In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).
Authors:Xianzhe Dai; Guofang Wei; Zhenlei Zhang Pages: 1 - 33 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Xianzhe Dai, Guofang Wei, Zhenlei Zhang We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L 2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
Authors:Alexander Kiselev; Changhui Tan Pages: 34 - 55 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Alexander Kiselev, Changhui Tan In recent work of Luo and Hou [10], a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot–Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.
Authors:Jacob Winding Pages: 56 - 86 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Jacob Winding We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of S L r ( Z ) -elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.
Authors:O. Jenkinson; M. Pollicott Pages: 87 - 115 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): O. Jenkinson, M. Pollicott We prove that the algorithm of [19] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set E 2 (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.
Authors:Moulay Tahar Benameur; Varghese Mathai Pages: 116 - 164 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Moulay Tahar Benameur, Varghese Mathai Given a constant magnetic field on Euclidean space R p determined by a skew-symmetric ( p × p ) matrix Θ, and a Z p -invariant probability measure μ on the disorder set Σ which is by hypothesis a Cantor set, where the action is assumed to be minimal, the corresponding Integrated Density of States of any self-adjoint operator affiliated to the twisted crossed product algebra C ( Σ ) ⋊ σ Z p , where σ is the multiplier on Z p associated to Θ, takes on values on spectral gaps in the magnetic gap-labelling group. The magnetic frequency group is defined as an explicit countable subgroup of R involving Pfaffians of Θ and its sub-matrices. We conjecture that the magnetic gap labelling group is a subgroup of the magnetic frequency group. We give evidence for the validity of our conjecture in 2D, 3D, the Jordan block diagonal case and the periodic case in all dimensions.
Authors:Jiayu Li; Chuanjing Zhang; Xi Zhang Pages: 165 - 214 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Jiayu Li, Chuanjing Zhang, Xi Zhang In this paper, we study the asymptotic behavior of the Hermitian–Yang–Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder–Narasimhan–Seshadri filtration, this answers a question by Bando and Siu.
Authors:Yinhe Peng; Liuzhen Wu Pages: 215 - 242 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Yinhe Peng, Liuzhen Wu We show that the Lindelöf property of a topological group is not inherited by its square, thus solving an old problem from [2]. Our construction introduces a new method to this area of mathematics and is of independent interest as it may have some further applications. Our method builds on ideas of [11] and techniques from [20] and give us also an optimal result about the partition properties on ω 1 .
Authors:Sheng Meng; De-Qi Zhang Pages: 243 - 273 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sheng Meng, De-Qi Zhang An endomorphism f of a projective variety X is polarized (resp. quasi-polarized) if f ⁎ H ∼ q H (linear equivalence) for some ample (resp. nef and big) Cartier divisor H and integer q > 1 . First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-étale quotient of an abelian variety). Finally, we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one. As a consequence, the building blocks of polarized endomorphisms are those of Q-abelian varieties and those of Fano varieties of Picard number one. Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb ( X ) of X, and that the pullback of a power of f acts as a scalar multiplication on the Néron–Severi group of X (modulo torsion) when X is smooth and rationally connected. Partial answers about X being of Calabi–Yau type, or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
Authors:Kei Yuen Chan Pages: 274 - 311 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Kei Yuen Chan This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch–Ciubotaru–Trapa and Ciubotaru. We apply our result to compute the Dirac cohomology of ladder representations for type A n . For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.
Authors:Pedro Resende Pages: 312 - 374 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Pedro Resende We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle π : E → G on an étale groupoid G with G 0 locally compact Hausdorff, equipped with a suitable completion C*-algebra A of its convolution algebra, we obtain a map of involutive quantales p : Max A → Ω ( G ) , where Max A consists of the closed linear subspaces of A and Ω ( G ) is the topology of G. We study various properties of p which mimick, to various degrees, those of open maps of topological spaces. These are closely related to properties of G, π, and A, such as G being Hausdorff, principal, or topological principal, or π being a line bundle. Under suitable conditions, which include G being Hausdorff, but without requiring saturation of the Fell bundle, A is an algebra of sections of the bundle if and only if it is the reduced C*-algebra C r ⁎ ( G , E ) . We also prove that Max A is stably Gelfand. This implies the existence of a pseudogroup I B and of an étale groupoid B associated canonically to any sub-C*-algebra B ⊂ A . We study a correspondence between Fell bundles and sub-C*-algebras based on these constructions, and compare it to the construction of Weyl groupoids from Cartan subalgebras.
Authors:Michel Broué; Ruth Corran; Jean Michel Pages: 375 - 458 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Michel Broué, Ruth Corran, Jean Michel We generalize the definition and properties of root systems to complex reflection groups — roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes.
Authors:Sergii Myroshnychenko; Dmitry Ryabogin Pages: 482 - 504 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sergii Myroshnychenko, Dmitry Ryabogin Let 2 ≤ k ≤ d − 1 and let P and Q be two convex polytopes in E d . Assume that their projections, P H , Q H , onto every k-dimensional subspace H, are congruent. In this paper we show that P and Q or P and −Q are translates of each other. We also prove an analogous result for sections by showing that P = Q or P = − Q , provided the polytopes contain the origin in their interior and their sections, P ∩ H , Q ∩ H , by every k-dimensional subspace H, are congruent.
Authors:Kathrin Bringmann; Chris Jennings-Shaffer; Karl Mahlburg Pages: 505 - 532 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Kathrin Bringmann, Chris Jennings-Shaffer, Karl Mahlburg We analyze the mock modular behavior of P ¯ ω ( q ) , a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is ω ( q ) . We find that the modular completion of P ¯ ω ( q ) is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.
Authors:Avy Soffer; Minh-Binh Tran Pages: 533 - 607 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Avy Soffer, Minh-Binh Tran The system that describes the dynamics of a Bose–Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C 12 and C 22 , which describe collisions of the condensate and the non-condensate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collision operator similar to C 22 , which possesses a blow-up positive radial solution with respect to the L ∞ norm (cf. [29]). On the other hand, at the very low temperature regime (only a portion of the transition temperature T B E C ), the system can be simplified into an equation of C 12 , with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L 1 norm (cf. [3]). In our model, we first decouple the QB, which contains C 12 + C 22 , and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [29], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [29]. A new theory is then supplied.
Authors:Wei He; Yeneng Sun Pages: 608 - 639 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Wei He, Yeneng Sun We prove some regularity properties (convexity, closedness, compactness and preservation of upper hemicontinuity) for distribution and regular conditional distribution of correspondences under the nowhere equivalence condition. We show the necessity of such a condition for any of these properties to hold. As an application, we demonstrate that the nowhere equivalence condition is satisfied on the underlying agent space if and only if pure-strategy Nash equilibria exist in general large games with any fixed uncountable compact action space.
Authors:Julien Barral; De-Jun Feng Pages: 640 - 718 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Julien Barral, De-Jun Feng Let μ be a planar Mandelbrot measure and π ⁎ μ its orthogonal projection on one of the principal axes. We study the thermodynamic and geometric properties of π ⁎ μ . We first show that π ⁎ μ is exact dimensional, with dim ( π ⁎ μ ) = min ( dim ( μ ) , dim ( ν ) ) , where ν is the Bernoulli product measure obtained as the expectation of π ⁎ μ . We also prove that π ⁎ μ is absolutely continuous with respect to ν if and only if dim ( μ ) > dim ( ν ) . Our results provides a new proof of Dekking–Grimmett–Falconer formula for the Hausdorff and box dimension of the topological support of π ⁎ μ , as well as a new variational interpretation. We obtain the free energy function τ π ⁎ μ of π ⁎ μ on a wide subinterval [ 0 , q c ) of R + . For q ∈ [ 0 , 1 ] , it is given by a variational formula which sometimes yields phase transitions of order larger than 1. For q > 1 , it is given by min ( τ ν , τ μ ) , which can exhibit first order phase transitions. This is in contrast with the analyticity of τ μ over [ 0 , q c ) . Also, we prove the validity of the multifractal formalism for π ⁎ μ at each α ∈ ( τ π ⁎ μ ′ ( q c − ) , τ π ⁎ μ ′ ( 0 + ) ] .
Authors:Alexandru D. Ionescu; Victor Lie Pages: 719 - 769 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Alexandru D. Ionescu, Victor Lie A basic model for describing plasma dynamics is given by the “one-fluid” Euler–Maxwell system, in which a compressible electron fluid interacts with its own self-consistent electromagnetic field. In this paper we prove long-term regularity of solutions of this system in 3 spatial dimensions, in the case of small initial data with nontrivial vorticity. Our main conclusion is that the time of existence of solutions depends only on the size of the vorticity of the initial data, as long as the initial data is sufficiently close to a constant stationary solution.
Authors:Sayan Bagchi Pages: 814 - 823 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sayan Bagchi In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R 2 so that ( Γ , Λ ) becomes a HUP.
Authors:Manuel Ritoré; Jesús Yepes Nicolás Pages: 824 - 863 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Manuel Ritoré, Jesús Yepes Nicolás Given one metric measure space X satisfying a linear Brunn–Minkowski inequality, and a second one Y satisfying a Brunn–Minkowski inequality with exponent p ≥ − 1 , we prove that the product X × Y with the standard product distance and measure satisfies a Brunn–Minkowski inequality of order 1 / ( 1 + p − 1 ) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn–Minkowski inequality is obtained in X × Y when Y satisfies a Prékopa–Leindler inequality. In particular, we show that the classical Brunn–Minkowski inequality holds for any pair of weakly unconditional sets in R n (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn–Minkowski inequality. Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn–Minkowski's inequalities are derived from our results.
Authors:Matthew Hedden; Thomas E. Mark Pages: 1 - 39 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Matthew Hedden, Thomas E. Mark We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book decompositions with connected binding. We prove this by showing that the rank of Floer homology gives bounds for the number of boundary parallel right or left Dehn twists necessary to add to a surface automorphism to guarantee that the associated contact manifold is tight or overtwisted, respectively. By examining branched double covers, we also show that the rank of the Khovanov homology of a link bounds the fractional Dehn twist coefficient of its odd-stranded braid representatives.
Authors:Elizaveta Rebrova; Roman Vershynin Pages: 40 - 83 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Elizaveta Rebrova, Roman Vershynin Can the behavior of a random matrix be improved by modifying a small fraction of its entries' Consider a random matrix A with i.i.d. entries. We show that the operator norm of A can be reduced to the optimal order O ( n ) by zeroing out a small submatrix of A if and only if the entries have zero mean and finite variance. Moreover, we obtain an almost optimal dependence between the size of the removed submatrix and the resulting operator norm. Our approach utilizes the cut norm and Grothendieck–Pietsch factorization for matrices, and it combines the methods developed recently by C. Le and R. Vershynin and by E. Rebrova and K. Tikhomirov.
Authors:Mauro Di Nasso; Lorenzo Luperi Baglini Pages: 84 - 117 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Mauro Di Nasso, Lorenzo Luperi Baglini We present general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. The goal is to contribute to an overall theory of Ramsey properties of (nonlinear) Diophantine equations that encompasses the known results in this area under a unified framework. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters β N , grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers N ⁎ .
Authors:Sun Kim Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sun Kim In 1980, D. M. Bressoud obtained an analytic generalization of the Rogers–Ramanujan–Gordon identities. He then tried to establish a combinatorial interpretation of his identity, which specializes to many well-known Rogers–Ramanujan type identities. He proved that a certain partition identity follows from his identity in a very restrictive case and conjectured that the partition identity holds true in general. In this paper, we prove Bressoud's conjecture for the general case by providing bijective proofs.
Subscription journal
[3175 journals]
