Real and complex singularities (original) (raw)
M M.Sc.-On Some Nonstandard Studies Of Analyticity Near Singularity abstract+intro+ref.
Salahaddin University-Erbil, 2016
In this thesis, we presented some Nonstandard concepts to study the analyticity near the singularity. We analyzed and proved the Existence and Uniqueness Theorems for first order ordinary differential equations in the subset of the monad of the initial standard point. Then the solutions of second order ordinary differential equation (Legendre Equation) are found around the singularity in the monad of zero by using power series method with suitable transformations for singular points. Additionally, we wrote the Legendre polynomial on the form of Mehler-Dirichlet Integral formula to find its solutions near the singularity. Furthermore, Nonstandard analysis tools are successfully applied to find a Nonstandard analytic solution for the first order differential equation near singularity in the following cases: The differential equation with one of the coefficients is unlimited. The general first order ODE in this case were of the form: dy dx = f (x, y) = 1 P(x, y) , where P(x, y) is a polynomial in x and y such that P(x, y) = a1,0x + a0,1y + a1,1xy + a0,2x2 + + aw1,w2xw1yw2, where ai,j is limited for each i, j 0 and w1, w2 are unlimited with 1 wp /2 z Microhal(0) and z is an infinitesimal, for p The differential coefficients are infinitesimals or unlimited and the first order differential equation have the form: dy ω1,ω2 i j xdx = f (x, y) = αx + βy + ∑ i,j=0 i+j≥2 ai,jx y , where ai,j is limited for each i, j ≥ 0 and ω₁, ω₂ are unlimited with 1 , 1 ∈ (ζ − Microhal(0))ᶜ. ω₁ ω₂ • The differential equation have irreducible differential Form. M(x, y)dx + N(x, y)dy = 0, where M(x, y) and N(x, y) is a power series in x and y. Therefore, it is of the form: ω ,ω j=0 2 ai,jxiyj)dx + (aˆx + + ω3,ω 4 j , i,j=0 i+j≥2 where 4 is unlimited with 1 / Microhal 0 and the ωk coefficients are limited. ∏ k=1 ∈ ζ − ( ) • Analyticity of system of differential equation in the monad of singu- larity. We considered the following system: dx₁ φ₁ dx₂ = φ₂ dxn = · · · = , where each φ₁, φ₂, · · · , φn is a power series in x₁, x₂, · · · , xn and n is standard natural number. Finally, we studied in details the idea of the proof of Painleve´’s Theorem in Nonstandard analysis. VII
2007
This paper is an attempt to classify finite-time singularities of PDEs. Most of the problems considered describe free-surface flows, which are easily observed experimentally. We consider problems where the singularity occurs at a point, and where typical scales of the solution shrink to zero as the singularity is approached. Upon a similarity transformation, exact self-similar behaviour is mapped to the
This is a reniew article on the multiplication table associated to the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients to the Gauss-Manin system (Theorem 2.7). We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discnminant and the bifurcation set (Proposition 3. 3). As applications, we establtsh signature formulae for Euler characteristics of real hypersurfaces (Theorem 4.2) by means of these fields. Saito and James William Bruce [2] for the case of hypersurface singularities (i.e. k=1k=1k=1 ).
Taming the movable singularities
The ANZIAM Journal, 2002
We have finally obtained for each of the 6 Painlevés an expression of z, w, w′ that behaves as 1/(z − Z0) + O(1) at each kind of movable singular point. This expression is polynomial in w′ (at most quadratic), and rational in w and z. After it is integrated and exponentiated it yields a function that has a simple zero at each of the singular points.
Asymptotically analytic and other plurisubharmonic singularities
2009
We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge-Ampère masses. They are characterized in terms of Green functions for Demailly's approximations, relative types, and valuations. Furthermore, the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals.
2016
As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a "canonical singularity" and a "log canonical singularity" for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.
On ᵠ ̛ ᵖ Singularities of solutions of Complex Vector Fields
2004
We study the singularities of solutions in 𝙱𝙼𝙾ᵠ ᵖ of an complex vector field L= X + iY. Necessary and sufficient conditions are established in the plane when ℋ𝜑(Σ) = 0, where Σ is the set where X, Y are linearly dependent and ℋ𝜑 is the Hausdorff measure defined by 𝜑.
Isosingular loci and the Cartesian product structure of complex analytic singularities
Transactions of the American Mathematical Society, 1978
Let X be a (not necessarily reduced) complex analytic space, and let F be a germ of an analytic space. The locus of points q in X at which the germ Xq is complex analytically isomorphic to F is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities. 0. Introduction. Let X be a complex analytic space. For q E X, Xq will denote the germ of X at q. In this paper I will study the isosingular loci defined by Definition 0.1. Forp G X let lso{X,p) = {qEX\Xq = Xp). (¡a here and elsewhere will mean complex analytically isomorphic.) It will be shown that: Theorem 0.2. For any p E X, lso{X,p) is a {possibly 0-dimensional) complex submanifold of some open subset of X. Moreover, for any q E Iso(A", p) there is an open neighbornood U of q, and an analytic space Y such that U at Y X {U n lso{X,p)). (X is the cartesian product in the category of analytic spaces.) This result is used to introduce what is, in effect, a partial ordering of complex analytic singularities in terms of their complexity. This, in turn, is used to study the ways in which a germ of an analytic space may be written as the cartesian product of other germs of analytic spaces. Let F be a germ of an analytic space (V not the reduced point). By a decomposition of V of length