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This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula., Comment: 48 pages. Reformulation of Theorem 1 for readability. Extra explanation for the mixed terms in section 2.2.3. Reference updates

In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations u(4)+wu′′+a(x)u=f(x,u), where w is a constant, a∈C(R,R) and f∈C(R×R,R). By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well-known (AR)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.

In this note we point out an error in [2]. We show how to repair the proof in dimension 5. The results are true in general as can easily be seen from recent work of Borman, Eliashberg and Murphy [1]. The proof of Lemma 3.4 in [2] is incorrect. Below we will describe the problem with the proof and then show how it can easily be repaired in dimension 5. We then observe that Lemma 3.4, and thus the main results of the paper, is true in all dimensions based on recent work of Borman, Eliashberg and Murphy [1]. However this approach does not give an explicit construction and hence goes against the sprit of the original paper and in addition all the results of [2] follow directly from [1]. Acknowledgement: We thank Yasha Eliashberg for pointing out the error in the proof of Lemma 3.4 in [2]. The first author was partially supported by a grant from the Simons Foundation (#342144) and NSF grant DMS-1309073. 1. Exact Lagrangians, Liouville flows, and the error in the proof of Lemma 3.4 We begin by recalling the statement of Lemma 3.4 from [2]. To state the lemma we first establish some notation (that is slightly different that what was used in [2]). Consider T 2 × [0, 1] with coordinates (θ, φ, r) and the contact structure ξi = kerαi, i = 1, 2, where αi = ki(r) dθ + li(r) dφ. Here we have k1(r) = cos π 2 r and l(r) = sin π 2 r, and for i = 2 we have k2 and l2 agreeing with k1 and l1 near r = 0 and 1, and the curve (k2(r), l2(r)) in R has 5π/2 winding about the origin. In particular notice that ξ2 is obtained from ξ1 by adding Giroux torsion. Lemma 3.4 from [2] now reads as follows. Lemma 1. Let W be a manifold with contact form λ, there is a contact structure on W × [0, 1] × ([0, 1]× T ) such that the following properties are satisfied: (1) near W×{0}× [0, 1]×T 2 and W× [0, 1]×{0, 1}×T 2 the contact structure is contactomorphic to λ+ eα1, and (2) near W × {1} × [0, 1]× T 2 the contact structure is contactomorphic to λ+ eα2. Here t is the coordinate on the first [0, 1] factor. See [2] for details on how the main constructions and theorems of the paper follow from this lemma. The strategy of the proof in [2] was: (1) To construct a contact structure on W×[0, 1]×T 3 that near W×{0}×T 3 is given by λ+eβ0 and near W ×{1}× T 3 is given by λ+ e × β1, where βi is the contact structure on T 3 with Giroux torsion i and we are thinking of T 3 as S×T 2 with the S-factors Legendrian curves. (2) Then cut W × [0, 1]×T 3 along W × [0, 1]× ({θ0, θ1}×T ) so that one of the resulting pieces is as described in the lemma. To try to arrange this let β = p1 dθ1+p1 dθ2 be the Liouville form on T ∗T 2 = R×T 2 with coordinates (p1, p2, θ1, θ2). Notice that α = λ + β is a contact form on W × T ∗T . We will see below that we can arrange the two items above that are needed for our proof if there is a radial vector field v in R centered at a point p whose flow expands dβ (that is, Lvdβ = dβ) and a Lagrangian torus T 2 in a small neighborhood of {q}×T 2 ⊂ T ∗T 2 that is exact with respect to ιvdβ that is isotopic to {q}×T 2 by an isotopy disjoint from {p} × T . One may easily arrange all of this except for either the last 2010 Mathematics Subject Classification. 57R17 (primary), and 53D35(secondary).

In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data. © 2015 Wiley Periodicals, Inc.

We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that $R^{2n+1}$ has at least three distinct contact structures. This version of the paper contains both the texts of the published version of the paper together with an Erratum to the published version appended to the end., Comment: 19 pages, 2 figures, and erratum to the published version of the paper

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals κ, there is an unexpected connection between invariants ae(κ), b(κ) and a certain cardinal invariant md(κ+) on κ+. As a corollary we get for example the following result. For a successor cardinal κ, even assuming that κ

The statement (and the proof) of Theorem 4.1.1 of the paper ‘Mackey formula in type A’ [Proc. London. Math. Soc. (3) 80 (2000) 545–574] is false. In this corrigendum we provide a correct statement (and a correct proof). As a consequence, we see that Corollary 4.1.2 of the paper holds. So all the other statements in the paper are correct (up to minor misprints…).2000 Mathematical Subject Classification: 20G05, 20G40.

In this paper we generalize an equation studied by Mossino and Temam, to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. We prove existence of a $W^{2,p}$-viscosity solution and regularity up to $C^{1,\alpha}(\overline{\Omega})$ for any $\alpha

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties., Accepted to Transactions of the London Mathematical Society

The mod two cohomology of the three connective covering of S3 has the form F2[X2n] ⊗ E(Sq1X2n) where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the map S3 → B2S1 of degree 2, then the mod 2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background K\"ahler metric, to be constant. In this paper we introduce and study dHYM equations with variable K\"ahler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and mirror to Thomas' moment map for special Lagrangians, to the Donaldson-Fujiki picture of scalar curvature as a moment map. As a consequence one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the K\"ahler-Yang-Mills system of \'Alvarez-C\'onsul, Garcia-Fernandez and Garc\'ia-Prada, and the coupled K\"ahler-Einstein equations of Hultgren-Witt Nystr\"om. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, following ideas of Feng and Sz\'ekelyhidi., Comment: 42 pages, some corrections to Section 4

In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill--Noether theory., 38 pages, significantly reorganized final version to appear in J. London Math. Soc

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed mass $\mu(\mathbb{R}^3)=\nu$, when $\mu$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $\mu$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $\nu_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $\mu\geq0$ with $\mu(\mathbb{R}^3), Comment: Final version to appear in Proc. London Math. Soc

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.

Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function., 45 pages

This paper considers the propagation of TE‐modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. Using the Floquet–Bloch theory in negative‐order Sobolev spaces, we characterize the precise number of eigenvalues created by the line defect in terms of the band functions of the original periodic background medium for arbitrarily weak defects.

In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen-Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen-Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero-Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, computing a new unexpected explicit example of a deformed Calogero-Moser system., Comment: 50 pages, some misprints corrected

ith respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p ‐stabilisation in this setting is also of independent interest.

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox., 18 pages. Some theorem, section and reference numbering is significantly changed from previous versions

In this paper we study the relationship between two fundamental regularity properties of an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting on a Borel measure $\mu$ in $\mathbb{R}^d$, with $s\in (0,d)$. In the classical case when $s=d$ and $\mu$ is equal to the Lebesgue measure, Calder\'on and Zygmund showed that if a CZO is bounded in $L^2$ then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower-dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical harmonics. We introduce sharp geometric conditions on $\mu$, in terms of certain scaled transportation distances, which ensure that an extension of the Calder\'{o}n-Zygmund theorem holds. These conditions are necessary and sufficient in the cases of the Riesz transform and the Huovinen transform. Our techniques build upon prior work by Mattila and Verdera, and incorporate the machinery of symmetric measures, introduced to the area by Mattila., Comment: 32 pages

International audience; Orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (nonexhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.

In the present paper, we prove, for a large class of elliptic curves defined over $\mathbb{Q}$, the existence of an explicit infinite family of quadratic twists with analytic rank $0$. In addition, we establish the $2$-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch--Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over $\mathbb{Q}$ without complex multiplication.

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper, we prove general adjoint functor theorems for functors between ∞‐categories. One of our main results is an ∞‐categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable ∞‐categories. We also discuss the comparison between adjunctions of ∞‐categories and homotopy adjunctions, and give a treatment of Brown representability for ∞‐categories based on Heller's purely categorical formulation of the classical Brown representability theorem.

In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.

In this paper we establish square-function estimates on the double and single layer potentials with rough inputs for divergence form elliptic operators, of arbitrary even order 2m, with variable t-independent coefficients in the upper half-space., 44 pages, to appear in the Proceedings of the London Mathematical Society

In this paper, we generalize all the results obtained on para‐Kahler Lie algebras in [3] to para‐Kahler Lie algebroids. In particular, we study exact para‐Kahler Lie algebroids as a generalization of exact para‐Kahler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra (A,.), the orbits of the action Φ of (A,+) on A∗ given by Φ(a,μ)=exp(La∗)(μ) are pseudo‐Hessian manifolds, where La(b)=a.b. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links., 25 pages, 15 figures. V2: Minor changes. Added reference, fixed typos, and clarified proof of Theorem 5.1

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.

A powerful method pioneered by Swinnerton-Dyer allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by Skorobogatov and Swinnerton-Dyer in 2005, can be applied to study rational points on Kummer varieties. In this paper we extend the method to include an additional step of second descent. Assuming finiteness of the relevant Tate-Shafarevich groups, we use the extended method to show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle on Kummer varieties associated to abelian varieties with all rational 2-torsion, under mild additional hypotheses.

Lattice Gauge theories have been studied in the physics literature as discrete approximations to quantum Yang-Mills theory for a long time. Primary statistics of interest in these models are expectations of the so called "Wilson loop variables". In this article we continue the program initiated by Chatterjee (2015) to understand Wilson loop expectations in Lattice Gauge theories in a certain limit through gauge-string duality. The objective in this paper is to better understand the underlying combinatorics in the strong coupling regime, by giving a more geometric picture of string trajectories involving correspondence to objects such as decorated trees and non-crossing partitions. Using connections with Free Probability theory, we provide an elaborate description of loop expectations in the planar setting, which provides certain insights about structures of higher dimensional trajectories as well. Exploiting this, we construct an example showing that in any dimension, the Wilson loop area law lower bound does not hold in full generality.

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear non-local equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an $H^{5/2}$-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parameterized circular configuration, then global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as $t\rightarrow +\infty$. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions.

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson–Schensted bijection, which is a variant of the type C Robinson–Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson–Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of theexotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space LM(X) (resp LM0(X)) is uniformly noncreasy if and only if the space LM(X) (resp LM0(X)) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.

Given a basic closed 1‐form on a Lie groupoid G, the Morse–Novikov cohomology groups Hθn(G) are defined in this paper. They coincide with the usual de Rham cohomology groups HdRn(G) when θ is exact and with the usual Morse–Novikov cohomology groups Hθn(M) when G is the unit groupoid M⇉M generated by a smooth manifold M. We prove that the Morse–Novikov cohomology groups are invariant under Morita equivalences of Lie groupoids. On orbifold groupoids, we show that these groups are isomorphic to sheaf cohomology groups. Finally, when θ is not exact, we extend a vanishing theorem from smooth manifolds to orbifold groupoids.

Let Pk(n,p) be the set of all real polynomial map germs f=(f1,⋯,fp):(Rn,0)→(Rp,0) with degree of f1,⋯,fp less than or equal to k∈N. The main result of this paper shows that the set of equivalence classes of Pk(n,p), with respect to multi‐K‐bi‐Lipschitz equivalence, is finite.

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents Fp(·),q(·)s(·),ϕ(Rn). These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents Fp(·),q(·)s(·)(Rn) as well as the classical Triebel–Lizorkin spaces Fp,qs(Rn), even the case when p=∞. Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces Fp,qs,τ(Rn) with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.

The purpose of this paper is to provide a detailed description of the spaces that can be specified as $L^2$ domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an analysis of the nature of the minimal domain and of a complementary space in the maximal domain of each of the operators. The key technical result is the nondegeneracy of a certain pairing of cohomology classes associated with the indicial complex. It is further proved that the set of choices of domains leading to Hilbert complexes in the sense of Br\"uning and Lesch form a variety, as well as a theorem establishing a necessary and sufficient condition for the operator in a given degree to map its maximal domain into the minimal domain of the next operator.

Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity radius bounded away from zero. The space parametrizing quasiconformal deformations of Schottky groups of a fixed rank $g \geq 1$ is the marked Schottky space ${\mathcal M}{\mathcal S}_{g}$; this being a complex manifold of dimension $3(g-1)$ for $g \geq 2$ and being isomorphic to the punctured unit disc for $g=1$. In this paper we provide a complete description of the real structures of ${\mathcal M}{\mathcal S}_{g}$, up to holomorphic automorphisms, together their real part.

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

In this paper, we consider a generalized Fisher equation with exponential diffusion from the point of view of the theory of symmetry reductions in partial differential equations. The generalized Fisher-type equation arises in the theory of population dynamics. These types of equations have appeared in many fields of study such as in the reaction-diffusion equations, in heat transfer problems, in biology, and in chemical kinetics. By using the symmetry classification, simplified by equivalence transformations, for a special family of Fisher equations, all the reductions are derived fromthe optimal systemof subalgebras and symmetry reductions are used to obtain exact solutions.

It is known that the weights of a complex weighted homogeneous polynomial $f$ with isolated singularity are analytic invariants of $(\mathbb C^d,f^{-1}(0))$. When $d=2,3$ this result holds by assuming merely the topological type instead of the analytic one. G. Fichou and T. Fukui recently proved the following real counterpart: the blow-Nash type of a real singular non-degenerate convenient weighted homogeneous polynomial in three variables determines its weights. The aim of this paper is to generalize the above-cited result with no condition on the number of variables. We work with a characterization of the blow-Nash equivalence called the arc-analytic equivalence. It is an equivalence relation on Nash function germs with no continuous moduli which may be seen as a semialgebraic version of the blow-analytic equivalence of T.-C. Kuo.

Let Gr(d; n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n. A submanifold X Gr(d; n) gives rise to a differential system ⊂(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such wellknown examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebansky's heavenly equations, and so on. In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3; 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as • the method of hydrodynamic reductions, • the method of dispersionless Lax pairs, • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution, • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2;R) structure induced on a fourfold X ⊂ Gr(3; 5). All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is 6-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3; 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P4 99K Gr(3; 5)). The fourfolds corresponding to `generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kahler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to explore the link between the {\it class} ${\cal C}$ property and the closed positive currents of bidegree $(1,\,1)$ that the manifold supports, a fact leading to the study of semi-continuity properties under deformations of the complex structure of the dual cones of cohomology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly non-Kahler, manifolds defined by the condition that every Gauduchon metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be small in a certain sense. We term them sGG manifolds and find numerical characterisations of them in terms of certain relations between various cohomology theories (De Rham, Dolbeault, Bott-Chern, Aeppli). We also produce several concrete examples of nilmanifolds demonstrating the differences between the sGG class and well-established classes of complex manifolds. We conclude that sGG manifolds enjoy good stability properties under deformations and modifications.

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = \rho g$ with $\rho >0$ constant. We show that if $(M,g)$ satisfies $\delta \mathcal W (\cdot, \cdot, \nabla f) = 0$, then $(M, g)$ is Einstein. Here $\mathcal W$ denotes the Weyl curvature tensor. In the case of noncompact, if $M$ is complete and satisfies the same condition, then $M$ is rigid in the sense that $M$ is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in \cite{l-r}, \cite{m-s} and \cite{p-w3}., Comment: 15pages

Jordan operator algebras are norm-closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras., 35 pages. Corrected final version. To appear Math Nachrichten. Some of this material was described in an OTOA2016 Conference workshop lecture in ISIBANG Bangalore, December 2016. We thank the organizers of that conference for the invitation and support

For the superelliptic curves of the form with , , a prime and for , we show that Here denotes the interval , where is the least prime greater than or equal to . Bennett and Siksek obtained a similar bound for in a recent paper.

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous., Comment: 12 pages

In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds., 15 pages, minor changes

We propose a new mathematical model for the spread of Zika virus. Special attention is paid to the transmission of microcephaly. Numerical simulations show the accuracy of the model with respect to the Zika outbreak occurred in Brazil., Comment: This is a preprint of a paper whose final and definite form is with 'Mathematical Methods in the Applied Sciences', ISSN 0170-4214. Submitted Aug 10, 2017; Revised Nov 13, 2017; accepted for publication Nov 14, 2017