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It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean analytic spaces rigorously follows from the case of valuations of height one. For non-archimedean spaces this result reduces the problem to studying local structure of smooth Berkovich spaces. [ABSTRACT FROM AUTHOR]

In this paper, we obtain global small solutions and decay estimates for the two-dimensional MHD boundary layer in Gevrey space without any structural assumptions, generalizing the results of [20] in analytic space. The proof method is mainly inspired by [14] and [26] , using new auxiliary functions and finer structural analysis to overcome the difficulty of the loss of derivatives and then we obtain the global well-posedness of the MHD boundary layer in the Gevrey 3 2 space. [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitable decay rates. The whole analysis is divided into two parts: When the initial far-field states are not small, we reformulate the original problem into a small perturbation problem by extracting a suitable background profile which is governed by a heat equation, then we prove a long-time existence of solutions, and the lower bound of lifespan of solutions is given in term of the small parameter of initial perturbation; After establishing the long-time existence, combining the decay properties of far-field state and the properties of solutions to the heat equation, it is shown that both the solution obtained in the first part and the background profile indeed satisfy some smallness requirements when time goes to the lower bound of lifespan of solutions. Then we can show the global existence of solutions after this moment. [ABSTRACT FROM AUTHOR]

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof, of Serre duality on any reduced pure n -dimensional paracompact complex space X . At the core of the paper is the introduction of certain fine sheaves B X n , q of currents on X of bidegree ( n , q ) , such that the Dolbeault complex ( B X n , • , ∂ ¯ ) becomes, in a certain sense, a dualizing complex. In particular, if X is Cohen–Macaulay then ( B X n , • , ∂ ¯ ) is an explicit fine resolution of the Grothendieck dualizing sheaf. [ABSTRACT FROM AUTHOR]

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s > 1 , and suggests that a suitable space for the well-posedness is Gevrey class of order s = 1 , which is exactly the space of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of local isometric embeddings into Euclidean space for analytic Riemannian metrics g , defined on a domain U ⊂ R n , which are singular in the sense that the determinant of the metric tensor is allowed to vanish at an isolated point (say the origin). Specifically, we show that, under suitable technical assumptions, there exists a local analytic isometric embedding u from (U ′ , Π ⁎ g) into Euclidean space E (n 2 + 3 n − 4) / 2 , where Π : U ′ → U \ { 0 } is a finite Riemannian branched cover of a deleted neighborhood of the origin. Our result can thus be thought of as a generalization of the classical Cartan-Janet Theorem to the singular setting in which the metric tensor is degenerate at an isolated point. Our proof uses Leray's ramified Cauchy-Kovalevskaya Theorem for analytic differential systems, in the form obtained by Choquet-Bruhat for non-linear systems. [ABSTRACT FROM AUTHOR]

In this paper, we establish analyticity of solutions to the barotropic compressible Navier-Stokes equations describing the motion of the density ρ and the velocity field u in R 3. We assume that ρ 0 is a small perturbation of 1 and (1 − 1 / ρ 0 , u 0) are analytic in Besov spaces with analyticity radius ω > 0. We show that the corresponding solutions are analytic globally in time when (1 − 1 / ρ 0 , u 0) are sufficiently small. To do this, we introduce the exponential operator e (ω − θ (t)) D acting on (1 − 1 / ρ , u) , where D is the differential operator whose Fourier symbol is given by | ξ | 1 = | ξ 1 | + | ξ 2 | + | ξ 3 | and θ (t) is chosen to satisfy θ (t) < ω globally in time. [ABSTRACT FROM AUTHOR]

The study of singularities of algebraic or analytic spaces over the field of complex numbers is a traditional subject, but, in contrast, the parallel theory over arbitrary algebraically closed fields is still poor and there are lots of interesting questions to be answered. Our main concern here is the study of the Milnor number μ of an isolated hypersurface singularity which is defined as the codimension of the ideal generated by the partial derivatives of a power series that represents locally the hypersurface. This is an important topological invariant of the singularity over the complex numbers, but its behavior changes dramatically when the base field has positive characteristic, in which case it may be infinite and depends upon the local equation of the hypersurface, not being an intrinsic invariant. In this paper we will study the variation of the Milnor number in terms of the local equations, showing that its minimum value has an intrinsic meaning and give necessary and sufficient conditions for its invariance. We also relate the smoothness of the generic fiber of an isolated hypersurface singularity deformation with the finiteness of this number, connecting it to a Bertini type result. At the end, we will show how, in arbitrary characteristic, one defines the sequence μ ⁎ of Milnor numbers of sections of a hypersurface singularity by general linear spaces of increasing codimension. [ABSTRACT FROM AUTHOR]

Systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by Walsh and Szego¨, and in full generality by Bergman. In principle, these are very interesting objects, allowing application to analytic continuation that is not restricted (as Weierstrassian continuation via power series) either by circular geometry or considerations of locality. However, few explicit examples are known, and in general one does not know even gross qualitative features of such systems. The main contribution of the present paper is to prove qualitative results in a quite general situation.It is by now very well known that the phenomenon of “double orthogonality” is not restricted to Bergman spaces of analytic functions, nor even indeed has it any intrinsic relation to analyticity; its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a Hilbert space of functions on some set, to a subset thereof, provided this restriction is injective and compact. However, in this paper only Hilbert spaces of analytic functions are considered, especially Bergman spaces. In the case of the Hardy spaces Fisher and Micchelli discovered remarkable qualitative features of doubly orthogonal systems, and we have shown how, based on the classical potential-theoretic notion of balayage, and its modern generalizations, one can deduce analogous results in the Bergman space set-up, but with restrictions imposed on the geometry of the considered domains and measures; these were not needed in the Fisher–Micchelli analysis, but are necessary here as shown by examples.From a more constructive point of view we study the Bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible (as an inversion followed by a logarithmic derivative) in terms of the polynomial equation of the boundary of the inner domain. [Copyright &y& Elsevier]

We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions. [ABSTRACT FROM AUTHOR]

In this paper we extend vessel theory, or equivalently, the theory of overdetermined 2 D systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces. [ABSTRACT FROM AUTHOR]

In this paper local monomialization theorems are proven for morphisms of complex and real analytic spaces. This gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in [16] and [18] to analytic spaces. [ABSTRACT FROM AUTHOR]

A 3-fold terminal quotient singularity X = C 3 / G admits the economic resolution Y → X , which is “close to being crepant”. This paper proves that the economic resolution Y is isomorphic to a distinguished component of a fine moduli space of θ -stable G -constellations for a suitable GIT parameter θ . [ABSTRACT FROM AUTHOR]

In this paper, we consider Toeplitz operators on a class of analytic function spaces, including standard Bergman spaces on the unit disc and Bargman–Fock spaces. We give a complete characterization of those compact operators whose eigenvalues have a prescribed slow decay to zero. [ABSTRACT FROM AUTHOR]

In this paper, we consider Q log space, the analytic version of Q 1 ( ∂ D ) space. We characterize lacunary series in Q log space. Using a characterization of Q log in terms of functions with absolute values, we also give a necessary and sufficient condition for an inner function to be in Q log . [ABSTRACT FROM AUTHOR]

For the almost Mathieu operator ( H λ , α , θ u ) ( n ) = u ( n + 1 ) + u ( n − 1 ) + 2 λ cos ⁡ 2 π ( θ + n α ) u ( n ) , Avila and Jitomirskaya conjecture that for every phase θ ∈ R ≜ { θ ∈ R | 2 θ + α Z ∈ Z } , H λ , α , θ satisfies Anderson localization if | λ | > e 2 β . In the present paper, we verify the conjecture for | λ | > e 7 β . [ABSTRACT FROM AUTHOR]

In this paper we introduce the family of spaces R M (p , q) , 1 ≤ p , q ≤ + ∞. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when p = ∞) and Bergman spaces (when p = q). We characterize the inclusion between R M (p 1 , q 1) and R M (p 2 , q 2) depending on the parameters. For 1 < p , q < ∞ , our main result provides a characterization of the dual spaces of R M (p , q) by means of the boundedness of the Bergman projection. We show that R M (p , q) is separable if and only if q < + ∞. In fact, we provide a method to build isomorphic copies of ℓ ∞ in R M (p , ∞). [ABSTRACT FROM AUTHOR]

Abstract: This paper analyzes the computational power of dynamical systems robust to infinitesimal perturbations. Previous work on the subject has delved on very specific types of systems. Here we obtain results for broader classes of dynamical systems (including those systems defined by Lipschitz/analytic functions). In particular we show that systems robust to infinitesimal perturbations only recognize recursive languages. We also show the converse direction: every recursive language can be robustly recognized by a computable system. By other words we show that robustness is equivalent to decidability. [Copyright &y& Elsevier]

Abstract: Let be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology to stronger ones in the frame of . First we show that the Mackey dual of a space is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space . This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient is analytic. Several questions, remarks and examples are included. [Copyright &y& Elsevier]

Abstract: In this paper we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of the entire solutions to the higher-dimensional Cauchy–Riemann system in . This allows us to determine the exact value for the growth order of the maximum modulus for all entire monogenic functions without knowing the precise value for the maximum modulus. Furthermore, it enables us to construct easily examples of entire monogenic functions that exhibit growth behavior of order ρ for any real value . [Copyright &y& Elsevier]

Abstract: A famous unsolved conjecture of P. Erdős and J.L. Selfridge states that there does not exist a covering system with the moduli odd, distinct and greater than one. In this paper we show that if such a covering system exists with all square-free, then the least common multiple of has at least 22 prime divisors. [Copyright &y& Elsevier]

We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed α > 0 and every conformal automorphism φ of the disc, f → f ∘ φ (φ ′) α defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the "unique" Hilbert space satisfying this property for a given α > 0. We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions g with the property that the integration operator f → ∫ 0 z f (t) g ′ (t) d t is bounded on a Banach space satisfying the weighted conformal invariance property. [ABSTRACT FROM AUTHOR]

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toën-Vaquié theory of schemes over F 1 , i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (e.g. polydisks) are recovered as a base change of analytic spaces over F 1. We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors. [ABSTRACT FROM AUTHOR]