Search

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

We prove optimal radially weighted L-2-norm inequalities for the Fourier extension operator associated to the unit sphere in R-n. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_n$ and the corresponding primitive $F_n$. The pointwise convergence of the integrands $f_n$ to some $f$ and the equiintegrability of the functions $f_n$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_n$ converge uniformly to $F$. In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers E. Cabral and P. Y. Lee (2001/2002) is revisited. Under the assumption of pointwise convergence of the integrands $f_n$, the three uniform integrability properties, namely equiintegrability and the two versions of the uniform double Lusin condition, are all equivalent. The first version of the double Lusin condition and its corresponding uniform double Lusin convergence theorem are also extended into the division space.

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant., Comment: 28 pages, 1 figure(with one sentence added)

We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly K\"ahler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.

We study the uniqueness question of transcendental entire functions sharing one finite nonzero value with their derivatives in some angular domains instead of the whole complex plane. The results in the present paper improve and extend the corresponding results from Chang-Fang~[2] and extend Theorem 3 from Zheng~[15]. An example is provided to show that the results in this paper, in a sense, are best possible.

We show that the Teichmuller space of a surface without boundary and with punctures, equipped with the Thurston metric, is the limit in an appropriate sense of Teichmuller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances of mapping classes for their actions on Teichmuller spaces equipped with the Thurston metric. In this paper, we show that the arc metrics on the Teichmuller space of surfaces with boundary limit to the Thurston metric on the Teichmuller space of a surface without boundary, by making the boundary lengths tend to zero. We use this to prove a result on the translation distances for mapping classes. We introduce some notation before stating precisely the results. In all this paper, S = Sg,p,n is a connected orientable surface of finite type, of genus g with p punctures and n boundary components. We assume that S has negative Euler characteristic, i.e., χ(S) = 2 − 2g − p − n 0, we denote by ∂S the boundary of S. A hyperbolic structure on S is a complete metric of constant curvature −1 such that (i) each puncture has a neighborhood isometric to a cusp, i.e., to the quotient {z = x + iy ∈ H 2 | y > a}/hz 7→z + 1i

International audience; This paper is about surfaces of infinite topological type. Unlike the case of surfaces of finite type, there are several deformation spaces associated with a surface of infinite topological type. Such spaces depend on the choice of a basepoint (that is, the choice of a fixed conformal structure or hyperbolic structure on the surface) and they also depend on the choice of a distance on the set of equivalence classes of marked hyperbolic structures. We address the question of the comparison between two deformation spaces, namely, the quasiconformal Teichmüller space and the length-spectrum Teichmüller space. There is a natural inclusion map of the quasiconformal space into the length-spectrum space, which is not always surjective.We work under the hypothesis that the basepoint (a hyperbolic surface) satisfies a condition we call ``upper-boundedness''. This means that this surface admits a pants decomposition defined by curves whose lengths are bounded above. The theory under this upper-boundedness hypothesis shows a dichotomy. On the one hand there are surfaces satisfying what we call Shiga's condition, i.e. they admit a pants decomposition defined by curves whose lengths are bounded above and below. If the base point satisfies Shiga's condition, then the inclusion of the quasiconformal space into the length-spectrum space is surjective, and it is a homeomorphism. In this paper we concentrate on the other kind of upper-bounded surfaces, which we call ``upper-bounded with short interior curves''. This means that the corresponding hyperbolic surface admits a pants decomposition defined by curves whose lengths are bounded above, and such that the lenghts of some interior curves approach zero. We show that in this case the behavior is completely different. Under this hypothesis, the image of the inclusion between the two Teichmüller spaces is nowhere dense in the length-spectrum space. As a corollary of the methods used, we obtain an explicit parametrization of the length-spectrum Teichmüller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmüller space is path-connected.

In the earlier paper (Itoh and Kiyohara, Manuscr Math 114:247–264, 2004), we showed that the cut locus of a general point on two-dimensional ellipsoid is a segment of a curvature line and proved “Jacobi’s last geometric statement” on the singularities of the conjugate locus. In the present paper, we show that a wider class of Liouville surfaces possess such simple cut loci and conjugate loci. The results include the determination of cut loci and the set of poles on two-sheeted hyperboloids and elliptic paraboloids.

This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb{H}$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\mathbb{R}$. In [AEK17a] we qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16b], we present a complete stability analysis of the equation for any $z\in \mathbb{H}$, including the vicinity of the singularities., Format and numbering was adjusted to the publication in Memoirs of the AMS

In [J. Bures, R. Lavicka, V. Soucek, Elements of quaternionic analysis and Radon transform, Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, 2009], the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on R^(n+1) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra R_n is mapping solutions of the generalized Cauchy-Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in R_n and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317–369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic [Formula: see text]-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central [Formula: see text]-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central [Formula: see text]-value. We can relate the square norm with the central value of some [Formula: see text]-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the [Formula: see text]-functions in our concern.

Using the one-to-one correspondence between Kahler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kahler-Norden manifold to be holomorphically recurrent.

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index.¶In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.

We correct a result in "Extremal disks and extremal surfaces of genus three", Kodai Math. J. 28, no. 1 (2005), 111-130. In the paper we have shown that there exist 16 compact Riemann surfaces of genus three up to conformal equivalence in which two extremal disks are isometrically embedded. However we have three more of them up to conformal equivalence. In the present paper we give these three surfaces and show that they are hyperelliptic. We also determine the groups of automorphisms of them.

WOS: 000340737800002, In this paper, we write Seiberg-Witten-like equations on contact metric manifolds of dimension 5. Since any contact metric manifold has a Spin(e)-structure, we use the generalized Tanaka-Webster connection on a Spin(e) spinor bundle of a contact metric manifold to define the Dirac-type operators and write the Dirac equation. The self-duality of 2-forms needed for the curvature equation is defined by using the contact structure. These equations admit a nontrivial solution on 5-dimensional strictly pseudoconvex CR manifolds whose contact distribution has a negative constant scalar curvature., Scientific Research Foundation of Anadolu University [1105F099], This paper was supported by Project 1105F099 of the Scientific Research Foundation of Anadolu University.

The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi-ϕ-asymptotically nonexpansive mappings in Banach space by means of the hybrid method in mathematical programming. The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces., 17 pages

This paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings. Constant boundary conditions are imposed on the single component of the boundary when the domain is convex, or on each of the two components of the boundary when the domain is a convex ring. A function is called quasiconcave if its superlevel sets, defined in a suitable way when the domain is a convex ring, are all convex. In this paper, we prove that the superlevel sets of the solutions do not always inherit the convexity or ring-convexity of the domain. Namely, we give two counterexamples to this quasiconcavity property: the first one for some two-dimensional convex domains and the second one for some convex rings in any dimension.

This paper focuses on numerical evaluation techniques related to fluctuation theory for Levy processes; they can be applied in various domains, e.g., in finance in the pricing of so-called barrier options. More specifically, with $$\bar{X}_t:= \sup _{0\le s\le t} X_s$$ denoting the running maximum of the Levy process $$X_t$$ , the aim is to evaluate $$\mathbb{P }(\bar{X}_t \in \mathrm{d}x)$$ for $$t,x>0$$ . The starting point is the Wiener–Hopf factorization, which yields an expression for the transform $$\mathbb E e^{-\alpha \bar{X}_{e(\vartheta )}}$$ of the running maximum at an exponential epoch (with $$\vartheta ^{-1}$$ the mean of this exponential random variable). This expression is first rewritten in a more convenient form, and then it is pointed out how to use Laplace inversion techniques to numerically evaluate $$\mathbb{P }(\bar{X}_t\in \mathrm{d}x).$$ In our experiments we rely on the efficient and accurate algorithm developed in den Iseger (Probab Eng Inf Sci 20:1–44, 2006). We illustrate the performance of the algorithm with various examples: Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. In models with jumps, we are also able to compute the density of the first time a specific threshold is exceeded, jointly with the corresponding overshoot. The paper is concluded by pointing out how our algorithm can be used in order to analyze the Levy process’ concave majorant.

We continue our previous study of improved Hardy, Rellich and uncertainty principle inequalities on a Riemannian manifold M, started in our earlier paper from 2009. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as an improved version of our uncertainty principle inequalities on a Riemannian manifold M. In particular, we obtain sharp constants for these inequalities on the hyperbolic space ?n. © 2013 American Mathematical Society.

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheimʼs theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen–Koskelaʼs concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with p-integrable upper gradient in n-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey–Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.

The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent $p$ and regularity $s$. Counterexamples are included to show that these results are indeed sharp. In particular, we precisely settle the question of boundedness of Riesz transforms on $L^p$, including the endpoint. The utility of such results in the study of nonlinear PDE is that they allow us to deduce important results, such as the fractional product and chain rules for the Dirichlet Laplacian, directly from the classical Euclidean setting. As an application, we discuss the local well-posedness and stability problems for energy-critical NLS. All the results of this paper play an essential role in the authors' proof of large-data global well-posedness and scattering for the energy-critical NLS in three dimensional exterior domains; see arXiv:1208:4904., 35pp

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension n with a q-dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a 2-parameter subspace of H(1,3) reaching many 3-dimensional geometries, and a 2-parameter family in H(0,n) of left-invariant metrics on n-dimensional compact and non-compact semisimple Lie groups., 30 pages, 11 figures; new version, many references added, section on Ricci solitons removed (to be included in a forthcoming paper)

TThe notion of Lagrangian $H$-umbilical submanifolds of K a hler manifolds introduced in [3, 4] is closely related with several problems in Lagrangian geometry (cf. [7]). The classification of such submanifolds was done in a series of author's papers [3, 4, 5]. On the other hand, the study of Lagrangian submanifolds of para-K a hler manifolds was initiated very recently in [10]. In this paper we study Lagrangian $H$-submanifolds of para-K a hler manifolds. As results we prove several fundemental properties of such submanifolds. Moreover, we are able to classify Lagrangian $H$-umbilical submanifolds of the para-K a hler $n$-plane $({\mathbb E}^{2n}_n,g_0,P)$ for $n\geq 3$.

This paper analyzes and simplifies Abhyankar’s proof of embedded resolution of surface singularities in positive characteristic. Abhyankar’s proof is obtained by combining the results in the papers [2], [4], [5], [6] and the first two chapters of the book [7]. This proof is extremely influential, but because of its length and complexity, is not generally well known and understood. In this article, I have written a report on the proof, hoping to make the main ideas more generally known. I give complete proofs of the essential parts of the proof. Some lemmas, which are given complete self contained proofs in Abhyankar’s work, are merely stated and cited in this paper. Some of these cited results can be proven directly without great difficulty. I have made substantial simplifications in the original proofs, but have made a point of not making simplications which eliminate an original and interesting idea which could possibly have application to resolution in higher dimension. Resolution of singularities in characteristic zero and in all dimensions was first proven by Hironaka [25]. More recently, there have been significant simplifications of this proof, including in [9], [11], [12], [20], [21], [27], [28], [33], [43], [44]. The first proof of resolution of surface singularities in characteristic p > 0 was by Abhyankar [1]. There have been other proofs of resolution of surface singularities in characteristic p > 0 since this time, including the proof analyzed in this paper, and proofs by Hironaka [26], Lipman [36], Hauser [24] and Cossart, Jannsen and Saito [16]. The first proof of resolution of singularities of 3-folds in positive characteristic p > 5 was given by Abhyankar [7], using the embedded resolution theorems for surfaces analyzed in this paper. A greatly simplified proof appears in [18], using Hironaka’s algorithm [26] for embedded resolution of surface singularities. Recently, Cossart and Piltant have proven resolution of singularities of 3-folds in all characteristics [14], [15]. Some of the recent papers attacking resolution in higher dimensions and positive characteristic are [13], [19], [29], [30], [31], [32], [34], [38], [41], and [42]. Abhyankar’s proof of embedded resolution of surface singularities is essentially a generalization of Zariski’s characteristic zero proof [45] of embedded resolution of surface singularities. Zariski’s final global proof, deducing the Theorem of Beppo Levi from local uniformization, extends without much difficulty to characteristic p > 0. The essential point where Zariski’s proof does not extend to characteristic p > 0 is in local uniformization of a particular type of valuation ν which dominates a normal local domain of dimension two. This difficult case occurs when ν is rational nondiscrete. For the most part, for simplicity, we restrict to the analysis of this fundamental case. The global argument used to deduce global resolution from local uniformization does not extend to dimension three, even in characteristic zero, as birational geometry is tremendously more complicated in higher dimensions. Even the algorithm of Beppo Levi fails in dimension three [40].

Recently, Han and Lu [HL] developed a discrete Littlewood-Paley-Stein analysis and multi-parameter Hardy space theory associated with isotropic flag singular integral operators initially studied by Muller-Ricci-Stein [MRS] and Nagel-Ricci-Stein [NRS]. The purpose of this paper is to carry out the multi-parameter Hardy space theory associated with non-isotropic flag singular integrals. The boundedness of such flag singular integral operator $T$ from $H^p_F$ to $H^p_F$ and from $H^p_F$ to $L^p$ are established in this paper. Discrete Calderon's identity and Min-Max principle derived here are the main tools used to establish the non-isotropic multi-parameter Hardy space theory.

The present paper essentially contains two results that generalize and improve some of the constructions of [arXiv:0801.1493]. First of all, in the case of one derivation, we prove that the parameterized Galois theory for difference equations constructed in [arXiv:0801.1493] can be descended from a differentially closed to an algebraically closed field. In the second part of the paper, we show that the theory can be applied to deformations of q-series, to study the differential dependency with respect to x\frac{d}{dx} and q\frac{d}{dq}. We show that the parameterized difference Galois group (with respect to a convenient derivation defined in the text) of the Jacobi Theta function can be considered as the Galoisian counterpart of the heat equation., 17 pages. To appear in Pacific Journal of Mathematics

This paper deals with coefficient estimates for close-to-convex functions with argument $\beta$ ($-\pi/2, Comment: This paper has been withdrawn by the author. There is no good reason to keep it since the author found a crucial sign error in equation 1

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N u,v in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group S n and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for N u,v represent elements of the Coxeter double Bruhat cell G u,v ⊂GL n . The Cartan subgroup H acts on G u,v by conjugation. The standard Poisson structure on the space of weights of N u,v induces a Poisson structure on G u,v , and hence on the quotient G u,v /H, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for N u,v . The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows. Following the ideas developed in our previous papers, we introduce a cluster algebra $ \mathcal{A} $ on $ {\mathcal{R}_n} $ compatible with the obtained Poisson bracket. Generalized Backlund–Darboux transformations map solutions of one Coxeter–Toda lattice to solutions of another preserving the corresponding Weyl function. Using network representation, we construct generalized Backlund–Darboux transformations as appropriate sequences of cluster transformations in $ \mathcal{A} $ .

Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equal to n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions be singular at all the points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, selfadjoint first-order differential operators, and selfadjoint elliptic pseudodifferential operators. As a result, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a new proof of a well known result of Branson-Gilkey, which was obtained by means of Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of the third author's paper [Po1]. Corrections to that statement are given in Appendix B., Final version. To appear in International Journal of Mathematics. 20 pages

In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting. In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group. The second part of the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert space of holomorphic functions is defined such that the $L^2$-heat transform becomes a unitary isomorphism., 18 pages

In this paper, we show that the composition of the generalized Levy Laplacian of order p (which is gener- ated by the corresponding Cesaro mean) and a certain linear transformation of the domain of the function to which it is applied is proportional to the generalized Levy Laplacian of smaller or larger order; this makes it possible to transform the generalized Levy Laplacian of any order into the classical Levy Laplacian. The con- struction is based on an expression of generalized Cesaro means in terms of ordinary ones, which is also suggested in this paper. By virtue of results of (5), this implies that all results on the classical Levy Laplacian have analogues for the generalized Levy Laplacian. In particular, we consider a relationship between the gen- eralized Levy Laplacian and quantum random pro- cesses.

The object of the present paper is to study certain curva-ture restriction on an LP-Sasakian manifold with a coefficient α . Amongothers it is shown that if an LP-Sasakian manifold with a coefficient α is a manifold of constant curvature, then the manifold is the productmanifold. Also it is proved that a 3-dimensional Ricci semisymmetricLP-Sasakian manifold with a constant coefficient α is a spaceform. 1. IntroductionIn 1989, Matsumoto [6] introduced the notion of LP-Sasakian manifolds.Then Mihai and Rosca [7] introduced the same notion independently and theyobtained several results in this manifold. In a recent paper, De, Shaikh, andSengupta [3] introduced the notion of LP-Sasakian manifolds with a coefficient α which generalizes the notion of LP-Sasakian manifolds. Recently, T. Ikawaand his coauthors [4], [5] studied Sasakian manifolds with Lorentzian metricand obtained several results in this manifold. The object of the present paperis to study certain curvature restriction on an LP-Sasakian manifold with acoefficient

This paper studies $\EPDiff(S^1)$, the Euler-Poincaré equation for diffeomorphisms of $S^1$, with the Weil-Petersson metric on the coset space $\PSL_2(\R)\setminus\Diff(S^1)$. This coset space is known as the universal Teichmüller space. It has another realization as the space of smooth simple closed curves modulo translations and scalings. $\EPDiff(S^1)$ admits a class of solitonlike solutions (teichons) in which the ``momentum'' $m$ is a distribution. The solutions of this equation can also be thought of as paths in the space of simple closed plane curves that minimize a certain energy. In this paper we study the solution in the special case that $m$ is expressed as a sum of four delta functions. We prove the existence of the solution for infinite time and find bounds on its long-term behavior, showing that it is asymptotic to a one-parameter subgroup in $\Diff(S^1)$. We then present a series of numerical experiments on solitons with more delta functions and make some conjectures about these.

A Lagrangian submanifold in a Kaehler manifold is said to be Hamiltonian-stationary (or simply $H$-stationary) if it is a critical point of the area functional restricted to (compactly supported) Hamiltonian variations. In an earlier paper \cite{cg}, $H$-stationary Lagrangian submanifolds of constant curvature in the complex projective 3-space $CP^3$ with positive relative nullity are classified. In this paper we completely classify $H$-stationary Lagrangian submanifolds of constant curvature in the complex hyperbolic 3-space $CH^3$ with positive relative nullity. As an immediate by-product, several explicit new families of $H$-stationary Lagrangian submanifolds in $CH^3$ are obtained.

In [6] we proved Chen's inequality regarded as a problem of constrained maximum. In this paper we introduce a Riemannian invariant obtained from Chen's invariant, replacing the sectional curvature by the Ricci curvature of k-order. This invariant can be estimated, in the case of submanifolds M in space forms $\widetilde{M}(c)$, varying with c and the mean curvature of M in $\widetilde{M}(c)$. A improuvement of this inequality in the Lagrangian case is obtained., The paper is submitted to Rocky Mountain Journal of Mathematics

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

In this paper, we announce a result on the relation of the analytic torsion withthe Laurent expansion of the Ruelle zeta function at s ¼ 0 for odd dimensional noncompacthyperbolic manifolds with cusps.Key words: Analytic torsion; Ruelle zeta function; hyperbolic manifold with cusps.1. Introduction. In his seminal paper [1],Fried derived a formula relating the analytic torsionto the Laurent expansion of the Ruelle zeta functionat s ¼ 0 for compact hyperbolic manifold of odddimension. The corresponding formula for the etainvariant and the value of the odd type Selberg zetafunction at s ¼ 0 had been proved by Millson in [6].Recently in [7] the formula of Millson has beengeneralized to the case of noncompact hyperbolicmanifolds with cusps. Here the eta invariant isdefined by certain regularized trace of odd heatoperator, which is essentially the same as theb-trace of Melrose introduced in [5]. We also appliedthe result for the weighted unipotent orbitalintegral in [4] to compute the contribution fromcusps. Hence it is a natural question whether ageneralization of the formula of Fried for analytictorsion could be obtained employing these methods.In this paper, we announce such a generalization ofthe formula of Fried, the relationship of the analytictorsion with the Laurent expansion of the Ruellezeta function at s ¼ 0 for noncompact hyperbolicmanifolds with cusps. First we follow the idea ofMelrose in [5] to define the analytic torsion for thisnoncompact case, which is explained in Section 3.Recently in [2,3] it is also shown that the Ruellezeta function has the meromorphic extension overC for odd dimensional hyperbolic manifolds withcusps. This is briefly reviewed in Proposition 4.1.The detailed proofs of results announced in thispaper will be given in [8].2. Laplacians over hyperbolic manifoldswith cusps. Let us recall that a ð2n þ1Þ-dimen-sional noncompact hyperbolic manifold with cuspsis given byX ¼ nSO

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

This paper studies the problem of the uniqueness of meromorphic functions sharing three values. The results in this paper improve some theorems given by H. Ueda, H. X. Yi and other authors.

We continue the study of the center problem for the ordinary differential equation $v'=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$ started in our earlier papers. In this paper we present the highlights of the algebraic theory of centers., 35 pages

This is a foundational paper on flows of G_2 Structures. We use local coordinates to describe the four torsion forms of a G_2 Structure and derive the evolution equations for a general flow of a G_2 Structure on a 7-manifold. Specifically, we compute the evolution of the metric, the dual 4-form, and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernandez-Gray. As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G_2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G_2 geometry., 50 pages, no figures. Minor rewrites. References added and corrected. For Version 3: Minor revisions and clarifications. This final version is a longer, more detailed version of the shorter paper to appear in the Quarterly Journal of Mathematics

Cauchy problems for quasi-hyperbolic factorized operator-differential equations of higher even orders with constant domains were considered in [1]. The Cauchy problem for hyperbolic operatordifferential equations with variable domains was investigated in [2, 3] for second-order equations. The present paper deals with the proof of the well-posed solvability in the strong sense of Cauchy problems for some quasi-hyperbolic factorized operator-differential equations of higher even orders with unbounded operator coefficients whose domains vary; mixed problems for some hyperbolic partial differential equations with coefficients in the boundary conditions smoothly depending on time can be reduced to such problems. For the proof, we use modifications and generalizations of the functional method of energy inequalities in [1]. Unlike [1], in the present paper, the derivation of a priori estimates with the use of abstract smoothing operators is generalized to the case of variable domains of variable unbounded operator coefficients; the proof of the solvability by induction, decomposition of operators into operator factors, and the use of the Lemmas 5 and 6 below is a new technique; and a formula for their strong solutions is derived for the first time [see formula (25) below]. This formula generalizes a similar formula for smooth (classical) solutions and shows that, by analogy with smooth solutions, strong solutions of these Cauchy problems can be found in a recursive way on the basis of operator factors. In addition, unlike [1], in the present paper, we do not repeat any steps of the proofs in [4]. In conclusion, we consider an example of new well-posed mixed problems for hyperbolic factorized partial differential equations of even order with coefficients in the boundary conditions depending on t and smooth with respect to t.

This paper continues the study of a class of compact convex hypersurfaces in Euclidean space $R^{n+1}, ~n \geq 1$, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $R^3$ reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are also important in rendering problems in computer graphics. The notion of a focal function for reflectors plays a central role similar to that of the Minkowski support function for convex bodies. In this paper the basic question of when a given function is a focal function of a convex reflector is answered by establishing necessary and sufficient conditions. In addition, some smoothness properties of reflectors and of the associated directrix hypersurfaces are also etablished., 15 pages

In this paper, we construct a family of asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to -6. The manifolds we constructed can be arbitrary close to anti-de Sitter-Schwarzschild manifolds at infinity. Hence, the mass of our manifolds can be very large or very small. The main arguments we used in this paper is gluing methods which was used in \cite{M}., 23pages