scholarly journals ALGEBRAIC GEOMETRY AND HOFSTADTER TYPE MODEL

2002 ◽  
Vol 16 (14n15) ◽  
pp. 2097-2106
Author(s):  
SHAO-SHIUNG LIN ◽  
SHI-SHYR ROAN
Keyword(s):  
Algebraic Geometry ◽  
Physical Model ◽  
Algebraic Curves ◽  
Spectral Curve ◽  
Rational Curves ◽  
Type Model ◽  
Bethe Equation ◽  
Explicit Solutions ◽  
Polynomial Roots ◽  
High Genus

In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of the polynomial Bethe equation; the relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooperation with the consideration of the physical model.

scholarly journals On Curve Veering and Flutter of Rotating Blades

1994 ◽  
Vol 116 (3) ◽  
pp. 702-708 ◽  
Author(s):  
D. Afolabi ◽  
O. Mehmed
Keyword(s):  
Algebraic Geometry ◽  
Catastrophe Theory ◽  
Algebraic Curves ◽  
Rotation Speed ◽  
Elastic Stability ◽  
Rotating Blades ◽  
Rotating Blade ◽  
Modes Of Vibration ◽  
Curve Veering ◽  
As If

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to “curve veering”), or attract each other (leading to “frequency coalescence”). Our results are reached by using standard arguments from algebraic geometry—the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

scholarly journals Injective linear series of algebraic curves on quadrics

2020 ◽  
Vol 66 (2) ◽  
pp. 231-254
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Algebraic Curves ◽  
Linear Series ◽  
Central Importance ◽  
Arbitrary Characteristic ◽  
Space Curves

Abstract We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).

A Methodology for the Analysis of Physical Model Scale Effects on the Simulation of Wave Propagation up to Wave Breaking: Preliminary Physical Model Results

2008 ◽  
Author(s):  
Conceic¸a˜o Fortes ◽  
Maria da Grac¸a Neves ◽  
Joa˜o Alfredo Santos ◽  
Rui Capita˜o ◽  
Artur Palha ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Physical Model ◽  
Scale Effects ◽  
National Laboratory ◽  
Physical Models ◽  
Type Model ◽  
Research Activity ◽  
Joint Research ◽  
Common Structure ◽  
Constant Slope

This paper describes the experiments performed at the National Laboratory for Civil Engineering (LNEC) aiming at simulating, in a flume, the wave propagation along a constant slope bottom that ends on a sea wall coastal defence structure, a common structure employed in the Portuguese coast. The objective of these tests is to calibrate the parameters of FUNWAVE, a Boussinesq type model, for wave propagation in coastal regions. This is the first step in the validation of a methodology to combine numerical and physical models in the study of the interactions between beaches and structures. This work is performed in the framework of the Composite Modelling of the Interactions between Beaches and Structures (CoMIBBs) project, a joint research activity of the HYDRALAB III European project.

scholarly journals Approximate Implicitization of Parametric Curves Using Cubic Algebraic Splines

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaolei Zhang ◽  
Jinming Wu
Keyword(s):  
Algebraic Curves ◽  
Approximation Error ◽  
Rational Curves ◽  
Parametric Curves ◽  
Approximate Implicitization

This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a globalG2continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is flexible and efficient.

Infinite-dimensional Lie groups and algebraic geometry in soliton theory

1985 ◽  
Vol 315 (1533) ◽  
pp. 393-404 ◽  
Keyword(s):  
Algebraic Geometry ◽  
Trivial Solution ◽  
Explicit Solutions ◽  
Type I ◽  
Solutions To Equations ◽  
Kyoto School ◽  
Soliton Theory ◽  
Infinite Dimensional ◽  
Funct Anal ◽  
Infinite Dimensional Lie Groups

We study several methods of describing ‘explicit’ solutions to equations of Korteweg-de Vries type: (i) the method of algebraic geometry (Krichever, I.M. Usp. mat. Nauk 32, 183-208 (1977)); (ii) the Grassmannian formalism of the Kyoto school (iii) acting on the trivial solution by the ‘group of dressing transformations’ (Zakharov, V. E. & Shabat, A. B. Funct. Anal. Appl. 13 (3), 13-22 (1979)). I show that the three methods are more or less equivalent, and in particular that the ‘ r -functions’ of method (ii) arise very naturally in the context of method (iii).

A comparison between Suzuki invariant code and one-point Hermitian code

2021 ◽  
pp. 2150104
Author(s):  
Abdulla Eid
Keyword(s):  
Algebraic Geometry ◽  
Minimum Distance ◽  
Algebraic Curves ◽  
Automorphism Groups ◽  
Algebraic Geometry Codes ◽  
Hermitian Codes ◽  
Relative Minimum ◽  
Minimum Distances ◽  
Hermitian Code

In this paper we compare the performance of two algebraic geometry codes (Suzuki and Hermitian codes) constructed using maximal algebraic curves over [Formula: see text] with large automorphism groups by choosing specific divisors. We discuss their parameters, compare the rate of these codes as well as their relative minimum distances, and we show that both codes are asymptotically good in terms of the rate which is in contrast to their behavior in terms of the relative minimum distance.

scholarly journals Correspondence theorems via tropicalizations of moduli spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Andreas Gross
Keyword(s):  
Moduli Spaces ◽  
Toric Variety ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Tropical Curves ◽  
Algebraic Tori ◽  
Pass Through ◽  
General Correspondence ◽  
Correspondence Theorem ◽  
Generic Points

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.

scholarly journals Residually finite rationally p groups

2019 ◽  
Vol 22 (03) ◽  
pp. 1950016
Author(s):  
Thomas Koberda ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Large Class ◽  
Algebraic Curves ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residual Property ◽  
Residually Finite ◽  
Torsion Freeness ◽  
Do So

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

scholarly journals On Curve Veering and Flutter of Rotating Blades

1993 ◽  
Author(s):  
Daré Afolabi ◽  
Oral Mehmed
Keyword(s):  
Algebraic Geometry ◽  
Catastrophe Theory ◽  
Algebraic Curves ◽  
Rotation Speed ◽  
Elastic Stability ◽  
Rotating Blades ◽  
Rotating Blade ◽  
Modes Of Vibration ◽  
Curve Veering ◽  
As If

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to “curve veering”), or attract each other (leading to “frequency coalescence”). Our results are reached by using standard arguments from algebraic geometry — the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

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