Showing papers in "Transactions of the American Mathematical Society in 2017"
TL;DR: In this article, a proof of linear inviscid damping for the 2D Euler equations around general monotone shear flows in the framework of Sobolev regularity is presented.
Abstract: Recently there has been much interest in damping phenomena for kinetic equations following the seminal works of Mouhot-Villani on Landau damping and of Bedrossian-Masmoudi on inviscid damping around Couette flow. In this talk I present a proof of linear inviscid damping for the 2D Euler equations around general monotone shear flows in the framework of Sobolev regularity. Here I consider both the settings of an infinite periodic and a finite periodic channel with impermeable walls. In the latter case I explain the non-negligible effect of boundary conditions on the attainable regularity and stability results. ∗Speaker sciencesconf.org:equadiff2015:64596
97 citations
TL;DR: In this paper, it was shown that the silting reduction T /thickP of a triangulated category T with respect to a presilting subcategory P can be realized as a certain subfactor category of T, and that there is a one-to-one correspondence between the set of (pre)silting subcategories of T containing P and the subset of T/thickp containing P. This is analogous to a result for Calabi-Yau reduction.
Abstract: It is shown that the silting reduction T /thickP of a triangulated category T with respect to a presilting subcategory P can be realised as a certain subfactor category of T , and that there is a one-to-one correspondence between the set of (pre)silting subcategories of T containing P and the set of (pre)silting subcategories of T /thickP . This is analogous to a result for Calabi–Yau reduction. This result is applied to show that the Amiot–Guo–Keller construction of d-Calabi–Yau triangulated categories with d-cluster-tilting objects takes silting reduction to Calabi–Yau reduction, and conversely, Calabi–Yau reduction lifts to silting reduction.
88 citations
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TL;DR: In this paper, the authors conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler.
Abstract: We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.
84 citations
TL;DR: In this paper, the authors classify solutions of finite Morse index of the fractional Lane-Emden equation (−∆) s u = |u| p−1 u in R n.
Abstract: We classify solutions of finite Morse index of the fractional Lane-Emden equation (−∆) s u = |u| p−1 u in R n .
74 citations
TL;DR: In this article, almost sure invariance principles for non-stationary time-series arising as observations on dynamical systems are established, which is a strong form of approximation by Brownian motion.
Abstract: We establish almost sure invariance principles, a strong form of approximation
by Brownian motion, for non-stationary time-series arising as observations
on dynamical systems. Our examples include observations on sequential
expanding maps, perturbed dynamical systems, non-stationary sequences of
functions on hyperbolic systems as well as applications to the shrinking target
problem in expanding systems.
74 citations
TL;DR: In this article, a stochastic optimal control problem with an autonomous forward process was studied and the dynamic programming principle was rigorously proved for V. The DPP is important in obtaining a characterization of the value function as a solution of a non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures.
Abstract: We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function V admits a nonlinear Feynman-Kac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for V. The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for V. The DPP is important in obtaining a characterization of the value function as a solution of a non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the Feynman-Kac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists in formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function V of the original control problem.
71 citations
TL;DR: In this paper, a chaotic operator was constructed which is not frequently hypercyclic and thus not frequently chaotic, and it was shown that every chaotic operator is reiteratively hypercyclical.
Abstract: We answer one of the main current questions in Linear Dynamics by constructing a chaotic operator on $\ell^1$ which is not $\mathcal{U}$-frequently hypercyclic and thus not frequently hypercyclic. This operator also gives us an example of a chaotic operator which is not distributionally chaotic. We complement this result by showing that every chaotic operator is reiteratively hypercyclic.
70 citations
TL;DR: In this paper, the authors introduced the first Aeppli-chern class for complex manifolds and showed that the $(1,1)$-component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class.
Abstract: In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $S^{2n-1}\times S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifolds such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.
61 citations
TL;DR: In this paper, it was shown that bilinear fractional integral integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed lebesgue spaces.
Abstract: We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.
60 citations
TL;DR: In this paper, a generalization of the Lambert W function is presented, which involves some special polynomials, and the generalization is used to solve relativistic equations and molecular physics problems.
Abstract: The Lambert W function, giving the solutions of a simple transcendental equation, has become a famous function and arises in many applications in combinatorics, physics, or population dyamics just to mention a few. In the last decade it turned out that some specific relativistic equations and molecular physics problems need solutions of more general transcendental equations. Thus a generalization of the Lambert function is necessary. In this paper we construct this generalization which involves some special polynomials.
58 citations
TL;DR: In this article, the Schrodinger equation with smooth potentials and magnetic type terms with controlled growth at infinity was studied, and it was shown that if the perturbation belongs to a class of unbounded symbols including smooth potential, then the system is reducible.
Abstract: We study the Schrodinger equation on \({\mathbb{R}}\) with a potential behaving as \({x^{2l}}\) at infinity, \({l \in [1, + \infty)}\) and with a small time quasiperiodic perturbation. We prove that if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible.
TL;DR: In this article, it was shown that any non-positive eigenvalue of a Schrodinger operator -Δ+V in L^2 (R^ν) with complex potential has absolute value at most a constant times for 0 < γ ≤ ν/2 in dimension ν ≥ 2.
Abstract: Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrodinger operator -Δ+V in L^2 (R^ν) with complex potential has absolute value at most a constant times ||V||^(γ+ν/2)/γ)_(γ+ν/2) for 0 < γ ≤ ν/2 in dimension ν ≥ 2. We prove this conjecture for radial potentials if 0 < γ < ν/2 and we ‘almost disprove’ it for general potentials if 1/2 < γ < ν/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.
TL;DR: In this article, Wu and Yau introduced a curvature notion called real bisectional curvature for Hermitian manifolds with negative holomorphic sectional curvature and showed that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitians but has non-positive real curvature.
Abstract: Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds When the metric is Kahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-Kahler, it is slightly stronger than $H$ We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature
TL;DR: In this paper, the matched product decomposition of an arbitrary finite left brace is used to construct new families of finite simple left braces. But the complexity of the decomposition is not discussed.
Abstract: Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to $B$. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to construct new families of finite simple left braces.
TL;DR: In this paper, a new generalization of the Cheeger-Gromoll splitting theorem is presented, where a warped product splitting is obtained under the existence of a line. But the curvature condition in this splitting is a curvature dimension inequality of the form $CD(0, 1).
Abstract: We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is $CD(0,1)$.
TL;DR: In this article, the authors explore the consequences of the structure of the discrete automorphic spectrum of the split orthogonal group SO(5) for holomorphic Siegel modular forms of degree 2.
Abstract: We explore the consequences of the structure of the discrete automorphic spectrum of the split orthogonal group SO(5) for holomorphic Siegel modular forms of degree 2. In particular, the combination of the local and global packet structure with the local paramodular newform theory for GSp(4) leads to a strong multiplicity one theorem for paramodular cusp forms.
TL;DR: In this paper, strongly self-absorbing actions of locally compact groups on C-algebras were studied and an equivariant generalization of a strongly selfabsorbing Calgebra to the setting of Cdynamical systems was introduced.
Abstract: We introduce and study strongly self-absorbing actions of locally compact groups on C-algebras. This is an equivariant generalization of a strongly self-absorbing C-algebra to the setting of Cdynamical systems. The main result is the following equivariant McDufftype absorption theorem: A cocycle action (α, u) : G y A on a separable C-algebra is cocycle conjugate to its tensorial stabilization with a strongly self-absorbing action γ : G y D, if and only if there exists an equivariant and unital ∗-homomorphism from D into the central sequence algebra of A. We also discuss some non-trivial examples of strongly self-absorbing actions.
TL;DR: The irreducible modules for the parafermion vertex operator algebra associated to any finite dimensional Lie algebra and any positive integer are identified, the quantum dimensions are computed and the fusion rules are determined.
Abstract: The irreducible modules for the parafermion vertex operator algebra associated to any finite dimensional Lie algebra and any positive integer are identified, the quantum dimensions are computed and the fusion rules are determined.
TL;DR: In this paper, a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value was obtained and a blow-up method was used to derive curvature estimates for such manifolds.
Abstract: In this paper we obtain first a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value. We also use a blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The major part of this paper is the study of how a property of the conformal infinity influences the geometry of the interior of a conformally compact Einstein manifold. Specifically we are interested in conformally compact Einstein manifolds with conformal infinity whose Yamabe invariant is close to that of the round sphere. Based on the approach initiated by Dutta and Javaheri we present a complete proof of the relative volume inequality
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\left (\frac {Y(\partial X, [\hat {g}])}{Y(\mathbb {S}^{n-1}, [g_{\mathbb {S}}])}\right )^{\frac {n-1}{2}}\leq \frac {Vol(\partial B_{g^+}(p, t))} {Vol(\partial B_{g_{\mathbb {H}}}(0, t))} \leq \frac {Vol(B_{g^+}(p, t))} {Vol(B_{g_{\mathbb {H}}}(0, t))}\leq 1,
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for conformally compact Einstein manifolds. This leads not only to the complete proof of the rigidity theorem for conformally compact Einstein manifolds in arbitrary dimension without spin assumption but also a new curvature pinching estimate for conformally compact Einstein manifolds with conformal infinities having large Yamabe invariant. We also derive curvature estimates for such manifolds.
TL;DR: In this paper, it was shown that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S).
Abstract: We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Γ. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups.
TL;DR: In this paper, simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coin-variant algebra are presented.
Abstract: In all finite Coxeter types but I2(12), I2(18) and I2(30), we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which ...
TL;DR: Theorem A for the truncated singular integral immediately implies that the singular integral along the primes T f(x) = X as discussed by the authors : n∈ N ∈ N.
Abstract: (x) : n∈ N.Moreover, one does not need to find a dense class of functions for which the pointwise convergence holdswhich sometimes might be a difficult problem.Variational estimates were the subject of many papers, see [6, 7, 8, 17] and the references therein. Letus notice that Theorem A for the truncated singular integral immediately implies that the singular integralalong the primesT f(x) =X
TL;DR: In this paper, the authors studied the stiff pressure limit of the porous medium equation, where the initial density is a bounded, integrable function with a sufficient decay at infinity.
Abstract: In this paper we study the \"stiff pressure limit\" of the porous medium equation, where the initial density is a bounded, integrable function with a sufficient decay at infinity. Our particular model, introduced by Perthame-Quiros-Vazquez, describes the growth of a tumor zone with a restriction on the maximal cell density. In a general context, this extends previous results of Caffarelli-Vazquez and Kim who restrict the initial data to be the characteristic function of a compact set. In the limit a Hele-Shaw type problem is obtained, where the interface motion law reflects the acceleration effect of the presence of a positive cell density on the expansion of the maximal density (tumor) zone.
TL;DR: In this paper, an explicit cycle map from Z(r)X|D,Nis the Nisnevich motivic complex of the pair (X,D) to KM r,X |D|D|Nis[−r] a shift of the relative Milnor K-sheaf of (X and D) was constructed.
Abstract: Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from Z(r)X|D,Nis the Nisnevich motivic complex of the pair (X,D) to KM r,X|D,Nis[−r] a shift of the relative Milnor K-sheaf of (X,D). We show that this map induces an isomorphism H M,Nis(X|D,Z(r)) ∼= H (XNis,K r,X|D,Nis), for all i ≥ dimX. This generalizes the well-known isomorphism in the case D = 0. We use this to prove a certain Zariski descent property for the motivic cohomology of the pair (Ak, (m + 1){0}).
TL;DR: In this paper, the authors studied the nonlinear stochastic time-fractional diffusion equations in the spatial domain driven by multiplicative space-time white noise and established the existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure.
Abstract: We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain $\\mathbb{R}$, driven by multiplicative space-time white noise. The fractional index $\\beta$ varies continuously from $0$ to $2$. The case $\\beta=1$ (resp. $\\beta=2$) corresponds to the stochastic heat (resp. wave) equation. The cases $\\beta\\in \\:]0,1[\\:$ and $\\beta\\in \\:]1,2[\\:$ are called {\\it slow diffusion equations} and {\\it fast diffusion equations}, respectively. Existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure, are established. Upper bounds on all $p$-th moments $(p\\ge 2)$ are obtained, which are expressed using a kernel function $\\mathcal{K}(t,x)$. The second moment is sharp. We obtain the H\\\"older continuity of the solution for the slow diffusion equations when the initial data is a bounded function. We prove the weak intermittency for both slow and fast diffusion equations. In this study, we introduce a special function, the {\\it two-parameter Mainardi functions}, which are generalizations of the one-parameter Mainardi functions.
TL;DR: In this paper, the authors introduced bridge trisections of knotted surfaces in the four-sphere and proved that every knotted surface in the 4D space admits a bridge splittings of links in the 3D space.
Abstract: We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number.
TL;DR: In this article, the authors constructed infinitely many complete, immersed selfshrinkers with rotational symmetry for each of the following topological types: the sphere, the plane, the cylinder, and the torus.
Abstract: We construct infinitely many complete, immersed self-shrinkers with rotational symmetry for each of the following topological types: the sphere, the plane, the cylinder, and the torus.
TL;DR: In this paper, a weak Gibbs property and a weak form of specification are used to derive level-2 large deviations principles for symbolic systems equipped with a large class of reference measures.
Abstract: We use a weak Gibbs property and a weak form of specification to derive level-2 large deviations principles for symbolic systems equipped with a large class of reference measures. This has applications to a broad class of symbolic systems, including $\beta$-shifts, $S$-gap shifts, and their factors. A crucial step in our approach is to prove a `horseshoe theorem' for these systems.
TL;DR: In this paper, a matroid configuration is defined as a union of complete intersections of a fixed codimension, and its ideals generalize the Stanley-Reisner ideals of matroids.
Abstract: Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial gen- eralization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of Liaison Theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen- Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms.