scholarly journals Frameworks for Two-Dimensional Keller Maps

10.37236/9210 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Alexander Borisov
Keyword(s):  
Affine Space ◽  
Affine Plane ◽  
Combinatorial Problem ◽  
Large System ◽  
Jacobian Conjecture ◽  
System Of Equations ◽  
Two Dimensional ◽  
Picard Groups ◽  
Keller Map

The classical Jacobian Conjecture asserts that every locally invertible polynomial self-map of the complex affine space is globally invertible. A Keller map is a (hypothetical) counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a map between the Picard groups of suitable compactifications of the affine plane, that satisfy a complicated set of conditions. This is essentially a combinatorial problem. Several solutions to it ("frameworks") are described in detail. Each framework corresponds to a large system of equations, whose solution would lead to a Keller map. 

An ideal-theoretic approach to Keller maps

2019 ◽  
Vol 62 (4) ◽  
pp. 1033-1044
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Affine Space ◽  
Residue Function ◽  
Jacobian Conjecture ◽  
Jacobian Determinant ◽  
Theoretic Approach ◽  
Finite Intersection ◽  
Maximal Ideals ◽  
Keller Map

AbstractA self-map F of an affine space ${\bf A}_k^n $ over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in $k\setminus \{0\}$. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals A PRECONDITIONED METHOD FOR THE SOLUTION OF THE ROBBINS PROBLEM FOR THE HELMHOLTZ EQUATION

2010 ◽  
Vol 52 (1) ◽  
pp. 87-100 ◽  
Author(s):  
JIANG LE ◽  
HUANG JIN ◽  
XIAO-GUANG LV ◽  
QING-SONG CHENG
Keyword(s):  
Linear System ◽  
Helmholtz Equation ◽  
Analytical Formula ◽  
System Of Equations ◽  
Two Dimensional ◽  
Sparse System ◽  
Sine Transform ◽  
Minimum Residual ◽  
Preconditioned Matrix ◽  
Preconditioned Iterative

AbstractA preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz equation leads to a sparse system of equations which is too large to solve directly. The approach taken in this paper is to precondition this linear system with a sine transform based preconditioner and then solve it using the generalized minimum residual method (GMRES). An analytical formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except for some outliers. Numerical results are reported to demonstrate the effectiveness of the proposed method.

Axial flow in a two-dimensional microchannel induced by a travelling temperature wave imposed at the bottom wall

2018 ◽  
Vol 848 ◽  
pp. 1040-1072 ◽  
Author(s):  
Chenguang Zhang ◽  
Harris Wong ◽  
Krishnaswamy Nandakumar
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axial Flow ◽  
Temperature Wave ◽  
Bottom Wall ◽  
Moving Frame ◽  
Global Maximum ◽  
System Of Equations ◽  
Two Dimensional ◽  
Perturbation Solutions

Fluid flow in microchannels has wide industrial and scientific applications. Hence, it is important to explore different driving mechanisms. In this paper, we study the net transport or fluid pumping in a two-dimensional channel induced by a travelling temperature wave applied at the bottom wall. The Navier–Stokes equations with the Boussinesq approximation and the convection–diffusion heat equation are made dimensionless by the height of the channel and a velocity scale obtained by a dominant balance between buoyancy and viscous resistance in the momentum equation. The system of equations is transformed to an axial coordinate that moves with the travelling temperature wave, and we seek steady solutions in this moving frame. Four dimensionless numbers emerge from the governing equations and boundary conditions: the Reynolds number $Re$, a Reynolds number $Rc$ based on the wave speed, the Prandtl number $Pr$ and the dimensionless wavenumber $K$. The system of equations is solved by a finite-volume method and by a perturbation method in the limit $Re\rightarrow 0$. Surprisingly, the leading and first-order perturbation solutions agree well with the computed axial flow for $Re\leqslant 10^{3}$. Thus, the analytic perturbation solutions are used to study systematically the effects of $Re$, $Rc$, $Pr$ and $K$ on the dimensionless induced axial flow $Q$. We find that $Q$ varies linearly with $Re$, and $Q/Re$ versus any of the three remaining dimensionless groups always exhibits a maximum. The global maximum of $Q/Re$ in the parameter space is subsequently determined for the first time. This induced axial flow is driven solely by the Reynolds stress.

An effective study of polynomial maps

2016 ◽  
Vol 16 (08) ◽  
pp. 1750141 ◽  
Author(s):  
Elżbieta Adamus ◽  
Paweł Bogdan ◽  
Teresa Crespo ◽  
Zbigniew Hajto
Keyword(s):  
Affine Space ◽  
Finite Sequence ◽  
Jacobian Conjecture ◽  
Equivalent Condition ◽  
Effective Algorithm ◽  
Polynomial Maps ◽  
Dimension Formula ◽  
Polynomial Map ◽  
Equivalent Statement

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map [Formula: see text] on an affine space of dimension [Formula: see text] over a field of characteristic [Formula: see text], we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of [Formula: see text] as well as a formula for [Formula: see text] in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

scholarly journals Capillary-induced deformations of a thin elastic sheet

2016 ◽  
Vol 374 (2066) ◽  
pp. 20150169 ◽  
Author(s):  
N. D. Brubaker ◽  
J. Lega
Keyword(s):  
Finite Thickness ◽  
Three Dimensional ◽  
Two Dimensions ◽  
System Of Equations ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Drop Volume ◽  
Exactly Solvable ◽  
Encapsulation Process

We develop a three-dimensional model for capillary origami systems in which a rectangular plate has finite thickness, is allowed to stretch and undergoes small deflections. This latter constraint limits our description of the encapsulation process to its initial folding phase. We first simplify the resulting system of equations to two dimensions by assuming that the plate has infinite aspect ratio, which allows us to compare our approach to known two-dimensional capillary origami models for inextensible plates. Moreover, as this two-dimensional model is exactly solvable, we give an expression for its solution in terms of its parameters. We then turn to the full three-dimensional model in the limit of small drop volume and provide numerical simulations showing how the plate and the drop deform due to the effect of capillary forces.

Re-examination of vorticity transfer theory

1977 ◽  
Vol 354 (1676) ◽  
pp. 1-8 ◽  
Keyword(s):  
Momentum Transfer ◽  
Turbulent Diffusion ◽  
Constant Coefficient ◽  
Shear Flows ◽  
Turbulent Shear ◽  
System Of Equations ◽  
Two Dimensional ◽  
Exchange Coefficient ◽  
Loss Of Information ◽  
Transfer Theory

The averaging procedure required to generate the Reynolds equations for turbulent shear flows results in a loss of information. The system of equations is no longer closed, and additional assumptions are required. The classical closure assumptions are based on Taylor’s (1915, 1932) vorticity transfer theory and Prandtl’s (1925, 1942) momentum transfer theory. In this paper we show that the former is in a very fundamental sense incorrect in that only for very special forms of the eddy exchange coefficient (including a constant coefficient) does turbulent ‘diffusion’ of vorticity also conserve momentum. For expository purposes much of the discussion is centred on an idealized simple two dimensional wake in which momentum and vorticity are conserved transferable properties.

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