Submanifolds of Fixed Degree in Graded Manifolds for Perceptual Completion

2021 ◽  
pp. 47-55
Author(s):  
Giovanna Citti ◽  
Gianmarco Giovannardi ◽  
Manuel Ritoré ◽  
Alessandro Sarti
Keyword(s):  
Fixed Degree ◽  
Perceptual Completion ◽  
Graded Manifolds

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Thue equations that simultaneously fail the Hasse principle

2021 ◽  
pp. 1-10
Author(s):  
PALOMA BENGOECHEA
Keyword(s):  
Positive Integer ◽  
Hasse Principle ◽  
Fixed Degree ◽  
Binary Forms ◽  
Absolute Value ◽  
Positive Proportion ◽  
The Absolute

Abstract We refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x, y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree ≥ 3 and are ordered by the absolute value of the maximum of the coefficients.

scholarly journals Counting points of fixed degree and given height over function fields

2012 ◽  
Vol 45 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Jeffrey Lin Thunder ◽  
Martin Widmer
Keyword(s):  
Function Fields ◽  
Fixed Degree ◽  
Counting Points

scholarly journals The Hardness of Speeding-up Knapsack

1998 ◽  
Vol 5 (14) ◽  
Author(s):  
Sandeep Sen
Keyword(s):  
Decision Tree ◽  
Knapsack Problem ◽  
Decision Tree Model ◽  
Fixed Degree ◽  
Tree Model ◽  
Pram Model ◽  
Speed Up ◽  
Decision Version ◽  
Polynomial Class ◽  
Strongly Polynomial

We show that it is not possible to speed-up the Knapsack problem efficiently in the parallel algebraic decision tree model. More specifically, we prove that any parallel algorithm in the fixed degree algebraic decision tree model that solves the decision version of the Knapsack problem requires  Omega(sqrt(n)) rounds even by using 2^sqrt(n) processors. We extend the result to the PRAM model without bit-operations. These results are consistent with Mulmuley's recent result on the separation of the strongly-polynomial class and the corresponding NC class in the arithmetic PRAM model.<br />Keywords lower-bounds, parallel algorithms, algebraic decision tree

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