Singular Hyperplanes Research Articles

Global well–posedness of a class of singular hyperbolic Cauchy problems

The goal of this paper is to establish a global well–posedness, cone condition and loss of regularity for singular hyperbolic equations with coefficients in $$L^1((0,T];C^\infty (\mathbb {R}^n)) \cap C^1((0,T];C^\infty (\mathbb {R}^n))$$ and Cauchy data in an appropriate Sobolev space tailored to a metric on the phase space. The coefficients are unbounded near the singular hyperplane $$t=0$$ and polynomially growing as $$|x| \rightarrow \infty .$$ The singular behavior is characterized by the blow–up rate of the coefficients and their first t-derivatives near $$t=0.$$ In order to study the interplay of the singularity in t and unboundedness in x, we consider a class of metrics on the phase space. Our methodology relies on the use of the Planck function associated to the metric to subdivide the extended phase and to define an infinite order pseudodifferential operator for the conjugation. We also give some counterexamples.

Hybrid Newton-Successive Substitution Method for Multiphase Rachford-Rice Equations.

In multiphase (≥3) equilibrium calculations, when the Newton method is used to solve the material balance (Rachford-Rice) equations, poorly conditioned Jacobian can lead to false convergence. We present a robust successive substitution method that solves the multiphase Rachford-Rice equations sequentially using the method of bi-section while considering the monotonicity of the equations and the locations of singular hyperplanes. Although this method is slower than Newton solution, as it does not rely on Jacobians that can become poorly conditioned, it can be inserted into Newton iterations upon the detection of a poorly conditioned Jacobian. Testing shows that embedded successive substitution steps effectively improved the robustness. The benefit of the Newton method in the speed of convergence is maintained.

A characterization of a class of hyperplanes of [formula omitted

A hyperplane of the symplectic dual polar space DW(2n−1,F), n≥2, is said to be of subspace-type if it consists of all maximal singular subspaces of W(2n−1,F) meeting a given (n−1)-dimensional subspace of PG(2n−1,F). We show that a hyperplane of DW(2n−1,F) is of subspace-type if and only if every hex F of DW(2n−1,F) intersects it in either F, a singular hyperplane of F or the extension of a full subgrid of a quad. In the case F is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane H of DW(2n−1,F) is of subspace-type or arises from the spin-embedding of DW(2n−1,F)≅DQ(2n,F) if and only if every hex F intersects it in either F, a singular hyperplane of F, a hexagonal hyperplane of F or the extension of a full subgrid of a quad.

Desingularization of complex multiple zeta-functions

We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding suitable rigorous meaning of the values of multiple zeta-functions at non-positive integer points. We reveal that multiple zeta-functions (which are known to be meromorphic in the whole space with infinitely many singular hyperplanes) turn to be entire on the whole space after taking the desingularization. The desingularized function is given by a suitable finite `linear' combination of multiple zeta-functions with some arguments shifted. It is shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones. We also discuss twisted multiple zeta-functions, which can be continued to entire functions, and their special values at non-positive integer points can be explicitly calculated.

Geometry of Grushin spaces

We compare the Grushin geometry to Euclidean geometry, through quasisymmetric parametrization, bilipschitz parametrization and bilipschitz embedding, highlighting the role of the exponents and the fractal nature of the singular hyperplanes in Grushin geometry. Consider in Rn a system of diagonal vector fields Xj = λj(x) ∂ ∂xj , j = 1, . . . , n,

Highest weight modules and polarized embeddings of shadow spaces

The present paper was inspired by the work on polarized embeddings by Cardinali et al. (J. Algebr. Comb. 25(1):7---23, 2007) although some of our results in it date back to 1999. They study polarized embeddings of certain dual polar spaces, and identify the minimal polarized embeddings for several such geometries. We extend some of their results to arbitrary shadow spaces of spherical buildings, and make a connection to work of Burgoyne, Wong, Verma, and Humphreys on highest weight representations for Chevalley groups. Let Δ be a spherical Moufang building with diagram M over some index set I, whose strongly transitive automorphism group is a Chevalley group $G(\mathbb{F})$ over the field $\mathbb{F}$ . For any non-empty set K?I let Γ be the K-shadow space of Δ. Extending the notion in to this situation, we say that an embedding of Γ is polarized if it induces all singular hyperplanes. Here a singular hyperplane is the collection of points of Γ not opposite to a point of the dual geometry Γ ?, which is the shadow geometry of type opp I (K) opposite to K. We prove a number of results on polarized embeddings, among others the existence of (relatively) minimal polarized embeddings. We assume that $G(\mathbb{F})$ is untwisted. In that case, the point-line geometry Γ has an embedding e K into the Weyl module $V(\lambda_{K})_{\mathbb{F}}^{0}$ of highest weight ? K =? k?K ? k . We show that this embedding is polarized in the sense described above. We then prove that the minimal polarized embedding relative to e K exists and equals the unique irreducible $G(\mathbb{F})$ -module L(? K ) of highest weight ? K . More precisely we show that the polar radical of e K (the intersection of all singular hyperplanes) coincides with the radical of the contravariant bilinear form considered by Wong to obtain the irreducible (restricted) representations of $G(\mathbb{F})$ in positive characteristic. This viewpoint allows us to recognize the irreducible $G(\mathbb{F})$ -modules of highest weight ? K geometrically as minimal polarized embeddings of the appropriate shadow space.

A Property of Isometric Mappings Between Dual Polar Spaces of Type $${DQ (2n, {\mathbb K})}$$

Let f be an isometric embedding of the dual polar space $${\Delta = DQ(2n, {\mathbb K})}$$ into $${\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}$$ . Let P denote the point-set of Δ and let $${e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})}$$ denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that $${f(H) = f(P) \cap H^\prime}$$ . We use this to show that there exists a subgeometry $${\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})}$$ of Σ′ such that: (i) $${e^\prime \circ f (x) \in \Sigma}$$ for every point x of $${\Delta; ({\rm ii})\,e := e^\prime \circ f}$$ defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.

Two classes of hyperplanes of dual polar spaces without subquadrangular quads

Let Π be one of the following polar spaces: (i) a nondegenerate polar space of rank n − 1 ⩾ 2 which is embedded as a hyperplane in Q ( 2 n , K ) ; (ii) a nondegenerate polar space of rank n ⩾ 2 which contains Q ( 2 n , K ) as a hyperplane. Let Δ and DQ ( 2 n , K ) denote the dual polar spaces associated with Π and Q ( 2 n , K ) , respectively. We show that every locally singular hyperplane of DQ ( 2 n , K ) gives rise to a hyperplane of Δ without subquadrangular quads. Suppose Π is associated with a nonsingular quadric Q − ( 2 n + ϵ , K ) of PG ( 2 n + ϵ , K ) , ϵ ∈ { − 1 , 1 } , described by a quadratic form of Witt-index n + ϵ − 1 2 , which becomes a quadratic form of Witt-index n + ϵ + 1 2 when regarded over a quadratic Galois extension of K . Then we show that the constructed hyperplanes of Δ arise from embedding.

On the hyperplanes of the half-spin geometries and the dual polar spaces [formula omitted

We describe relationships between locally singular hyperplanes of the dual polar space DQ ( 2 n , K ) , n ⩾ 2 , and hyperplanes of the half-spin geometries HS ( 2 n − 1 , K ) and HS ( 2 n + 1 , K ) for the respective hyperbolic quadrics Q + ( 2 n − 1 , K ) and Q + ( 2 n + 1 , K ) . We use these relationships to classify all hyperplanes of HS ( 9 , K ) and to provide a method for constructing locally singular hyperplanes of DQ ( 2 n + 2 , K ) from locally singular hyperplanes of DQ ( 2 n , K ) . Along our way, we also obtain a new proof for the fact that all hyperplanes of the half-spin geometries arise from embeddings.

The hyperplanes of [formula omitted] and [formula omitted] which arise from their spin-embeddings

We characterize the hyperplanes of the dual polar spaces DQ ( 2 n , K ) and DQ − ( 2 n + 1 , q ) which arise from their respective spin-embeddings. The hyperplanes of DQ ( 2 n , K ) which arise from its spin-embedding are precisely the locally singular hyperplanes of DQ ( 2 n , K ) . The hyperplanes of DQ − ( 2 n + 1 , q ) which arise from its spin-embedding are precisely the hyperplanes H of DQ − ( 2 n + 1 , q ) which satisfy the following property: if Q is an ovoidal quad, then Q ∩ H is a classical ovoid of Q.

A characterization of the SDPS-hyperplanes of dual polar spaces

In [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211], we introduced the class of the SDPS-valuations of dual polar spaces. We showed that these valuations and all their extensions give rise to hyperplanes of dual polar spaces. We call these hyperplanes SDPS-hyperplanes. In the present paper, we show that a hyperplane H of a thick dual polar space is an SDPS-hyperplane if and only if every hex A not contained in H intersects H in either a singular hyperplane or the extension of an ovoid.

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study ( i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ ( 2 n , K ) , then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ ( 8 , K ) and show that every locally singular hyperplane of DQ ( 8 , K ) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.

Estimates on Green functions of second order differential operators with singular coefficients

We investigate the Green functions G(x,x^{\prime}) of some second order differential operators on R^{d+1} with singular coefficients depending only on one coordinate x_{0}. We express the Green functions by means of the Brownian motion. Applying probabilistic methods we prove that when x=(0,{\bf x}) and x^{\prime}=(0,{\bf x}^{\prime}) (here x_{0}=0) lie on the singular hyperplanes then G(0,{\bf x};0,{\bf x}^{\prime}) is more regular than the Green function of operators with regular coefficients.

Monogenic vector fields on the poincaré manifold

In this paper a study is made of vector valued functions satisfying the equation , where D E is the Euclidean Dirac operator, and is the (anti-Euclidean) inner product. These functions, called hypermonogenic, were introduced by Leutwiler (see [10-13]). In the first part of this paper we explore the relation between these functions and the Poincaré metric on half space . Indeed, up to a scalar correction factor, these functions are monogenic, in the sense that they are solutions of the Hodge Dirac operator. As a result, they are a logical generalisation of classical holomorphic functions on C - clearly the equation considered reduces to the Cauchy-Riemann equation for n= 2 – in a way similar to, but not equal to, the theory of monogenic functions on Euclidean space. Furthermore, a natural isomorphism between hypermonogenic functions in a half space and in a ball (the second model of hyperbolic space) is given. This isomorphism shows that the series expansion of a hyperharmonic function in the neighbourhood of a point of the manifold can be most easily expressed in terms of the spherical harmonics of Euclidean space. Then two important results for these functions are given. First, a result by Leutwiler, stating that a hypermonogenic function near the ‘boundary’ can be described by its values on and its ‘characteristic function’ on , is generalised from the case n = 3 to arbitrary n> 2. Second, it is proved that a hypermonogenic function can be developed in a neighbourhood of a boundary point in a series of cyhndrical waves. Notice that both results are remarkable in that they deal with an inhomogeneous space, because of the singular hyperplane . Finally, we introduce a similar theory for a metric associated with a subspace of lower dimension, and indicate how it may be Hnked with a higher dimensional analogue of the theory of Padé approximation.

Absolutely continuous states of exit measures for super-Brownian motions with branching restricted to a hyperplane

The exit measures of super-Brownian motions with branching mechanism\(\psi (z) = z^\alpha ,1 1 + \frac{2}{{a - 1}}\) they are singular. However, if the branching is restricted to a singular hyperplane, it is proved that they have absolutely continuous states for alld≥1.

Versality for canonical curves and complete intersections

Let X be a smooth subvariety of Pr , over an algebraically closed eld k. In an attempt to understand the geometry of X , one would like to understand the singular hyperplane sections of X , and how the family of all hyperplane sections behave around a singular one. Classically this involved the study of in ection points on curves. If X is arbitrary nothing much can be said, so suppose that X is general, meaning that it belongs to an open subset of the Hilbert scheme of Pr . It is natural to expect the singularities of hyperplane sections are isolated and versally deformed by the nearby hyperplane sections. If this property holds, we shall call X hyperplane versal. Versality is de ned in (1.2), and roughly means that every possible deformation appears. Apart from the fact that versality is an interesting property in its own right, there is often a normal form for a versal deformation, and it is a useful property for enumerative questions. We are able to prove the following results:

The Singular Solutions of The Generalized Poly Axially Symmetric Helmholz Equations

in this article the solution of the generalized poly-axially symmetric Helmholtz etpıation is obtained on the common intersection of the singular hyperplanes. In the case of spherical region the solution on the common intersection of the singularity hyperplanes is given hy a Pois- p son type integral. We also have obtained Solutions of the eguations h [u ]= o and hp S[u]=sf.

Boundary value problems in generalized biaxially symmetric potential theory

Interior boundary value problems are solved for the operator of generalized biaxially symmetric potential theory. The boundary conditions consist of Dirichlet data on the nonsingular part of the boundary and Dirichlet data or growth restrictions on the singular hyperplanes, depending on the values of parameters of the operator. Continuation of solutions beyond the singular hyperplanes is considered, yielding an improvement of a result of Huber. Potential theoretic methods are used for the investigation.

Some Properties of Generalized Biaxially Symmetric Helmholtz Potentials

For an equation which includes the equation of generalized biaxially symmetric potential theory an investigation is made of the behavior of solutions near the intersection of the two singular hyperplanes. The main results consist of a continuation theorem and uniqueness theorems analogous to those of Huber [3] in generalized axially symmetric potential theory.

On the Uniqueness of Generalized Axially Symmetric Potentials

(k denoting an arbitrary real constant) shall be answered. We shall discuss the behavior of these functions in the neighborhood of the singular hyperplane Xn = 0.