Analytic surface germs with minimal Pythagoras number

2003 ◽  
Vol 244 (4) ◽  
pp. 725-752 ◽  
Author(s):  
José F. Fernando
Keyword(s):  
Analytic Surface ◽  
Pythagoras Number

scholarly journals HIGHER PRODUCT PYTHAGORAS NUMBERS OF SKEW FIELDS

2010 ◽  
Vol 03 (01) ◽  
pp. 193-207
Author(s):  
Dejan Velušček
Keyword(s):  
Laurent Series ◽  
Skew Field ◽  
Skew Fields ◽  
Pythagoras Number

We introduce the n–th product Pythagoras number p n(D), the skew field analogue of the n–th Pythagoras number of a field. For a valued skew field (D, v) where v has the property of preserving sums of permuted products of n–th powers when passing to the residue skew field k v and where Newton's lemma holds for polynomials of the form Xn - a, a ∈ 1 + I v , p n(D) is bounded above by either p n( k v ) or p n( k v ) + 1. Spherical completeness of a valued skew field (D, v) implies that the Newton's lemma holds for Xn - a, a ∈ 1 + I v but the lemma does not hold for arbitrary polynomials. Using the above results we deduce that p n (D((G))) = p n(D) for skew fields of generalized Laurent series.

Green's Forms and Meromorphic Functions on Compact Analytic Varieties

1951 ◽  
Vol 3 ◽  
pp. 108-128 ◽  
Author(s):  
Kunihiko Kodaira
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Point ◽  
Positive Definite ◽  
Analytic Surface ◽  
Analytic Variety ◽  
Complex Dimension ◽  
Analytic Varieties ◽  
Complex Analytic Variety ◽  
Zero Points

Let be a compact complex analytic variety of the complex dimension n with a positive definite Kâhlerian metric [4] ; the local analytic coordinates on will be denoted by z = (z 1 z 2, … , zn). Now, suppose a meromorphic function f(z) defined on as given. Then the poles and zero-points of f(z) constitute an analytic surface in consisting of a finite number of irreducible closed analytic surfaces Γ1, Γ2, … , Γk, each of which is a polar or a zero-point variety of f(z).

scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

scholarly journals Complex Clifford analysis and domains of holomorphy

1990 ◽  
Vol 48 (3) ◽  
pp. 413-433 ◽  
Author(s):  
John Ryan
Keyword(s):  
Clifford Analysis ◽  
Integral Formula ◽  
Complex Variables ◽  
Analytic Surface ◽  
Higher Dimension ◽  
Cauchy’S Integral Formula ◽  
Domains Of Holomorphy ◽  
Huygens Principle ◽  
Real Analytic ◽  
Real Analytic Surface

AbstractIntegrals related to Cauchy's integral formula and Huygens' principle are used to establish a link between domains of holomorphy in n complex variables and cells of harmonicity in one higher dimension. These integrals enable us to determine domains to which analytic functions on real analytic surface in Rn+1 may be extended to solutions to a Dirac equation.

Modeling of Long-Range Reverberation Signal for Rough Bottom Consisting of Polygon Facets

2018 ◽  
Vol 26 (04) ◽  
pp. 1850041
Author(s):  
Youngmin Choo ◽  
Woojae Seong
Keyword(s):  
Long Range ◽  
Order Approximation ◽  
Higher Order ◽  
Ocean Bottom ◽  
Analytic Surface ◽  
Surface Integral ◽  
Computational Burden ◽  
Analytic Integration ◽  
The Cost ◽  
Delay Difference

To acquire a stable reverberation signal from an irregular ocean bottom, we derive the analytic surface integral of a scattered signal using Stokes’ theorem while approximating the bottom using a combination of polygon facets. In this approach, the delay difference in the elemental scattering area is considered, while the representative delay is used for the elemental scattering area in the standard reverberation model. Two different reverberation models are applied to a randomly generated rough bottom, which is composed of triangular facets. Their results are compared, and the scheme using analytic integration shows a converged reverberation signal, even with a large elemental scattering area, at the cost of an additional computational burden caused by a higher order approximation in the surface integral of the scattered signals.

From Analytic Surface to Analytic Space

1992 ◽  
Vol 40 (2) ◽  
pp. 381-404 ◽  
Author(s):  
Warren S. Poland
Keyword(s):  
Analytic Space ◽  
Analytic Surface ◽  
Clinical Theory ◽  
Analytic Process ◽  
Clinical Illustration

Utilizing a clinical illustration, the concept of the surface of the patient's mind, which arose early in analytic history, is reexamined in relation to the analytic space, the unique affective and communicative dyadic context of the analytic process. The shift from analytic surface to analytic space reflects in clinical theory the metapsychological shift from early structural views to current appreciation of compromise formation. Also, this approach permits broadening of consideration of active unconscious forces in both the patient and the analyst.

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