Real Analytic Space Research Articles

Structures of sets of solutions to the Hartree-Fock equation

The Hartree-Fock equation which is the Euler-Lagrange equation corresponding to the Hartree-Fock energy functional is used in many-electron problems. Since the Hartree-Fock equation is a system of nonlinear eigenvalue problems, the study of structures of sets of all solutions needs new methods different from that for the set of eigenfunctions of linear operators. In this paper we prove that the sets of all solutions to the Hartree-Fock equation associated with critical values of the Hartree-Fock energy functional less than the first energy threshold are unions of a finite number of compact connected real-analytic spaces. The result would also be a basis for the study of approximation methods to solve the equation.

Incompressible Euler Limit from Boltzmann Equation with Diffuse Boundary Condition for Analytic Data

A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance, we capture the Navier-Stokes equations of $$\begin{aligned} \textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}} \end{aligned}$$satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of the Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative \(L^p\)-\(L^\infty \) estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green’s function approach. The incompressible Euler limit follows as a byproduct of our framework.

Real Analytic Local Well‐Posedness for the Triple Deck

AbstractThe triple‐deck model is a classical high‐order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure‐displacement relation that links pressure to the tangential slip. In order to overcome this, we split the triple‐deck system into two coupled equations: a Prandtl‐type system on ℍ and a Benjamin‐Ono‐type equation on ℝ. This splitting enables us to extract a crucial leading‐order cancellation at the top of the lower deck. We develop a functional framework to subsequently extend this cancellation into the interior of the lower deck, which enables us to prove the local well‐posedness of the model in tangentially real analytic spaces. © 2019 Wiley Periodicals, Inc.

Algebraic intersection spaces

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces [Formula: see text], including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze “local duality obstructions,” which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in [Formula: see text] of a tubular neighborhood of the singular set.

Normalization of Complex Analytic Spaces from a Global Viewpoint

In this work, we study some algebraic and topological properties of the ring of global analytic functions on the normalization of a reduced complex analytic space . If is a Stein space, we characterize in terms of the (topological) completion of the integral closure of the ring of global holomorphic functions on X (inside its total ring of fractions) with respect to the usual Frechet topology of . This shows that not only the Stein space but also its normalization is completely determined by the ring of global analytic functions on X. This result was already proved in 1988 by Hayes–Pourcin when is an irreducible Stein space, whereas in this paper we afford the general case. We also analyze the real underlying structures and of a reduced complex analytic space and its normalization . We prove that the complexification of provides the normalization of the complexification of if and only if is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization.

Local monomialization of analytic maps

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.

Resolution of singularities of the cotangent sheaf of a singular variety

The main problem studied here is resolution of singularities of the cotangent sheaf of a complex- or real-analytic space X0 (or of an algebraic variety X0 over a field of characteristic zero). Given X0, we ask whether there is a global resolution of singularities σ:X→X0 such that the pulled-back cotangent sheaf of X0 is generated by differential monomials in suitable coordinates at every point of X (“Hsiang–Pati coordinates”). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of σ. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang–Pati 1985, Pardon–Stern 2001). Consequences include monomialization of the induced Fubini–Study metric on the smooth part of a complex projective variety X0; there have been important applications of the latter to L2-cohomology.

On globally defined semianalytic sets

In this work we present the concept of C-semianalytic subset of a real analytic manifold and more generally of a real analytic space. C-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan (C-analytic sets for short). More precisely S is a C-semianalytic subset of a real analytic space (X, OX ) if each point of X has a neighborhood U such that S ∩ U is a finite boolean combinations of global analytic equalities and strict inequalities on X. By means of paracompactness C-emianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on X. The family of C-semianalytic sets is closed.

Equisingularity of sections, (𝑡^{𝑟}) condition, and the integral closure of modules

This paper uses the theory of integral closure of modules to study the sections of both real and complex analytic spaces. The stratification conditions, which play a key role, are the ( t r ) (t^{r}) conditions introduced by Thom and Trotman. Our results include an algebraic formulation of the ( t r ) (t^{r}) in terms of the integral closure of modules, and a new simple proof showing how the ( t r ) (t^{r}) conditions improve under Grassmann modification. In the complex analytic case, we characterise ( t r ) (t^{r}) in terms of the genericity of the multiplicity of a certain submodule of the Jacobian module, then use the principle of specialisation of integral dependence for modules to give an equimultiplicity criterion for ( t r ) (t^{r}) . As a consequence we obtain numerical criteria for Verdier equisingularity of families of plane sections in various situations.

Microlocal analysis of quantum fields on curved space–times: Analytic wave front sets and Reeh–Schlieder theorems

We show in this article that the Reeh–Schlieder property holds for states of quantum fields on real analytic curved space–times if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e., without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states of the Klein–Gordon field are further investigated in the present work and the (analytic) microlocal spectrum condition is shown to be equivalent to simpler conditions. We also prove that any quasifree ground or KMS state of the Klein–Gordon field on a stationary real analytic space–time fulfills the analytic microlocal spectrum condition.

Relations among analytic functions. I

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let Φ:X→Y be a morphism of real analytic spaces, and let Ψ:𝒢→ℱ be a homomorphism of coherent modules over the induced ring homomorphism Φ * :𝒪 Y →𝒪 X . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ℛ a = Ker Ψ ^ a , a∈X, are upper semi-continuous in the analytic Zariski topology of X. We prove semicontinuity in many cases (e.g. in the algebraic category). Semicontinuity of the “diagram of initial exponents” provides a unified point of view and explicit new techniques which substitute for coherence in both geometric problems on the images of mappings (semianalytic and subanalytic sets) and analytic problems on the singularities of differentiable functions (in particular, the classical division and composition problems).

Topologien auf Schnittmoduln koh�renter komplex-und reell-analytischer Garben

In this paper we consider coherent complex-analytic sheaves F on a complex-analytic space X, and study two canonical topologies, inductive resp. projective locally convex, on F(A,F) for subsets a⊂x. We are interested in conditions on A for which these topologies coincide, and get as a main result that this is the case for real analytic spaces which can be imbedded in some ℝl and have the original X as a complexification. By complexification we apply our results to coherent real-analytic sheaves.

Blowing-up orientato

In this paper we define a new kind of blowing-up, as a functor from the category of real analytic spaces to the category of real semianalytic spaces, in such a way that orientability is preserved. Then we prove an existence theorem for «oriented blowing-ups» of real analytic spaces.

Extension theorems for smooth functions on real analytic spaces and quotients by Lie groups and smooth stability

AbstractExtension theorems are proved for smooth functions on a coherent real analytic space for which local defining functions exist which are finitely determined in the sense of J. Mather, (1968), and for smooth functions invariant under the action of a compact lie groupG. thus providing the main step in the proof that smooth infinitesimal stability implies smooth stability in the appropriate categories. In addition the remaining step for the category ofCxG-manifolds of finite orbit type is filled in.

Triangulation of subanalytic sets and proper light subanalytic maps

The smallest nonempty class ~c4 of subsets of a real analytic space M which is closed under the formation of locally finite unions, intersections, complements, and connected components and which contains Ac~g ~t {0} for any A in ,~ and realvalued function g analytic in a neighborhood of Clos A is the class of semianalytic subsets of M (see [6, w 1] for a list of references, the most basic being [11]). The smallest such class also containing all proper real analytic images of semianalytic sets is the (strictly larger, for dim M > 3 ) class of subanalytic subsets of M ([6, 8, 14, 15]). A continuous function from a subset of real analytic space M into a real analytic space N is called a subanalytic map if its graph is a subanalytic subset of M • N. Here triangulations of subanalytic sets are constructed using only subanalytic maps throughout. A map is proper (respectively, light) if its inverse image preserves compact (respectively, discrete) sets. Our main results are:

Local homological properties of analytic sets

Using Milnor's fibration theorem [5] for analytic hypersurfaces and the “conic structure lemma”, one gets informations on local homological properties of analytic spaces, as also on the topological behaviour of algebraic sets at ∞. A simple proof of D. Sullivan's theorem about the Euler-Poincare characteristic of (X, X-x), for X a real or complex analytic space, as well as some extensions and generalisations are given.