Linear Change Of Variables Research Articles

Fokas method for linear boundary value problems involving mixed spatial derivatives.

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the [formula omitted]-axis

This paper studies the global dynamics for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y-axis. By linear changes of variables, it can be written as four systems. One of them has been characterized in Llibre (2018). In this paper, the remaining three systems are investigated. Firstly, the normal forms of these systems are given by an algebraic classification of their infinite singular points. Then, we classify the global phase portraits of these systems having canonical forms on the Poincaré disk. Moreover, we obtain the bifurcation diagrams for the corresponding global phase portraits. Combining this work with Llibre (2018), we provide the complete classification of global phase portraits for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y-axis.

Center problem for Λ–Ω differential systems

The Λ-Ω systems are the real planar polynomial differential equations of degree mx˙=−y(1+Λ)+xΩ,y˙=x(1+Λ)+yΩ, where Λ=Λ(x,y) and Ω=Ω(x,y) are polynomials of degree at most m−1 such that Λ(0,0)=Ω(0,0)=0. We study the center problem for these Λ-Ω systems. Any planar vector field with linear type center can be written as a Λ-Ω system if and only if the Poincaré-Liapunov first integral is of the form F=12(x2+y2)(1+O(x,y)). These kind of linear type centers are called weak centers, they contain the class of centers studied by Alwash and Lloyd [1], and also contain the uniform isochronous centers, and the holomorphic isochronous centers, but they do not coincide with the all class of isochronous centers.The main objective of this paper is to study the center problem for two particular classes of Λ-Ω systems of degree m.First if Λ=μ(a2x−a1y), and Ω=a1x+a2y+Ωm−1, where μ,a1,a2 are constants and Ωm−1=Ωm−1(x,y) is a homogenous polynomial of degree m−1, then we prove the following results.(i)These Λ-Ω systems have a weak center at the origin if and only if (μ+m−2)(a12+a22)=0, and ∫02πΩm−1(cos⁡t,sin⁡t)dt=0;(ii)If m=2,3,4,5,6 and (μ+m−2)(a12+a22)≠0, then the given Λ–Ω systems have a weak center at the origin if and only if these systems after a linear change of variables (x,y)⟶(X,Y) are invariant under the transformations (X,Y,t)⟶(−X,Y,−t).Second if Λ=a1x+a2y, and Ω=Ωm−1, where a1,a2 are constants and Ωm−1=Ωm−1(x,y) is a homogenous polynomial of degree m−1, then we prove the following results.(i)These Λ-Ω systems have a weak center at the origin if and only if a1=a2=0, and ∫02πΩm−1(cos⁡t,sin⁡t)dt=0;(ii)If m=2,3,4,5 and a12+a22≠0, then the given Λ–Ω systems have a weak center at the origin if and only if these systems after a linear change of variables (x,y)⟶(X,Y) are invariant under the transformations (X,Y,t)⟶(−X,Y,−t).We observe that the main difficulty to prove results (ii) for m>6 is related with the huge computations necessary for proving them.

Nœther Bases and Their Applications

In this paper, we introduce a new involutive division, called D-Nœther division, and the corresponding notion of a Nœther basis. It is shown that an ideal is in Nœther position, if and only if it possesses a finite Nœther basis. We present a deterministic algorithm which, given a homogeneous ideal, finds a linear change of variables so that the ideal after performing this change possesses a finite Nœther basis (and equivalently is in Nœther position). Furthermore, we define the new concept of an ideal of Nœther type and study its connections with Rees decompositions. We have implemented all the algorithms described in this paper in Maple and assess their performance on a number of benchmark examples.

Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

In this paper, we study the global dynamical behavior of the Hamiltonian system [Formula: see text], [Formula: see text] with the rational potential Hamiltonian [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

A Fractional Variational Approach for Modelling Dissipative Mechanical Systems:Continuous and Discrete Settings

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange equations model a dissipative system in both directions of time, summing up to a set of equations that is invariant under time reversal. Finally, we show that the discrete equations are a meaningful discretisation of the continuous ones.

The boundary of the orbit of the 3-by-3 determinant polynomial

We consider the 3×3 determinant polynomial and we describe the limit points of the set of all polynomials obtained from the determinant polynomial by linear change of variables. This answers a question of Joseph M. Landsberg.

ОБ ОДНОЙ ЗАДАЧЕ ПРЕСЛЕДОВАНИЯ ПРИ НАЛИЧИИ СОПРОТИВЛЕНИЯ СРЕДЫ

This paper considers a game pursuit problem, in which the interceptor (pursuer) and target (evader) move in the same plane under the influence of controlled forces directed always perpendicularly to their velocities. The laws of value variation of controlled forces of interceptor and target are determined by first-order controllers. Besides that, each object is influenced by the force of resistance of a medium which is proportional to the squared velocity of the object and is directed to the side which is opposite to its velocity. It is assumed that during the motion of objects, directions of their velocities are little different from an axis passing through their initial positions. It allows linearizing equations of motion of the pursuer and target. As a result of linearization it turns out that the projections of the position of objects on the axis change by a known law. When there is a coincidence of these projections, the time point prescribes the moment of the end of prosecution process. It is expected that the capture has occurred, if at this time point the module of difference of vector projections of object position on a perpendicular axis does not exceed a predetermined number. Eventually, a linear differential game of pursuit-evasion with fixed end time is obtained. Full information on the state of objects at each time point is available for players. With the help of a linear change of variables, the game comes down to a homogeneous one-dimensional differential game, in which the possible values of control belong to the segments which depend on the time. As a result of the research the set of initial conditions is found, under which the capture of target is possible when in any of its allowable motion, and the control of pursuer that will ensure the capture is built

Linearization of non-degenerate analytic centres in

In this paper, we study the linearization problem of a non-degenerate analytic centre on a centre manifold in . We show how the case of constant divergence of the associated vector field can be linearized with the help of inverse Jacobi multipliers. Moreover, we also combine methods of Lie symmetries from a given analytic commutator with linear part without resonances to get explicitly the linearizing change of variables. Finally, we show how the methods work in some examples.

POINCARÉ CODE: A package of open-source implements for normalization and computer algebra reduction near equilibria of coupled ordinary differential equations

The Poincaré code is a Maple project package that aims to gather significant computer algebra normal form (and subsequent reduction) methods for handling nonlinear ordinary differential equations. As a first version, a set of fourteen easy-to-use Maple commands is introduced for symbolic creation of (improved variants of Poincaré’s) normal forms as well as their associated normalizing transformations. The software is the implementation by the authors of carefully studied and followed up selected normal form procedures from the literature, including some authors’ contributions to the subject. As can be seen, joint-normal-form programs involving Lie-point symmetries are of special interest and are published in CPC Program Library for the first time, Hamiltonian variants being also very useful as they lead to encouraging results when applied, for example, to models from computational physics like Hénon–Heiles. Program summaryProgram title: POINCARÉCatalogue identifier: AEPJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 7694No. of bytes in distributed program, including test data, etc.: 597943Distribution format: tar.gzProgramming language: Maple V11.Computer: See specifications for running Maple V11 or above.Operating system: MS Windows.Classification: 4.3, 5, 16.9.Nature of problem:Computing structure-preserving normal forms near the origin for nonlinear vector fields.Solution method:14 Maple commands are designed to compute various normal forms as well as their generating functions and associated normalizing transformations for nonlinear systems of ordinary differential equations, including Hamiltonian ones. Further reduction of these normal forms–in the presence of Lie-point symmetries–is also considered. All algorithms are based on Lie transform, thus leading to the structure-preserving normal forms in the sense that all properties that can be formulated in terms of graded Lie algebras are preserved.Restrictions:The semisimple part of the leading matrix about the origin (resp. the quadratic part in the Hamiltonian case) is assumed to be already taken into diagonal form (resp. canonical form) via a linear change of variables. In other words, the eigenvalues must be explicitly known.Unusual features:Further reduction of Poincaré–Dulac normal form via the joint normal form commands, taking profitably Lie-point symmetries, is perhaps the main feature of the software. Besides, to the best of our knowledge, and except the Hamiltonian case, it seems that there is no CPC programs for computation of (structure-preserving) normal forms in the general case.Running time:Depends on the input data, mainly on system size, type of the leading matrix (semisimple or not), type of resonances, order of normalization and required options. Instantaneous for the provided examples.

The Crepant Resolution Conjecture for Three-Dimensional Flags Modulo an Involution

After fixing a nondegenerate bilinear form on a vector space V, we define a ℤ2-action on the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov–Witten potential function of [F/ℤ2] agrees (up to unstable terms) with the genus zero Gromov–Witten potential function of a crepant resolution Y of the quotient scheme F/ℤ2, after setting a quantum parameter to −1, making a linear change of variables, and analytically continuing coefficients. We explicitly compute several invariants for the orbifold and the resolution, then argue that these determine the others via basic properties of Gromov–Witten invariants.

Extensive Adiabatic Invariants for Nonlinear Chains

We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to exploit the complete resonance, while keeping the extensive nature of the system. Next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear Schrodinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of the system size, for a large set of initial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory, with fluctuation much smaller than expected according to standard statistical estimates.

Integrals and Banach spaces for finite order distributions

Let B c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A n c to be the space of tempered distributions that are the nth distributional derivative of a unique function in B c . Similarly with A n r from B r . A type of integral is defined on distributions in A n c and A n r . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A n c and A n r are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B c and B r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A 1 c is the completion of the L 1 functions in the Alexiewicz norm. The space A 1 r contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.

Improved dimensionally-reduced visual cortical network using stochastic noise modeling

In this paper, we extend our framework for constructing low-dimensional dynamical system models of large-scale neuronal networks of mammalian primary visual cortex. Our dimensional reduction procedure consists of performing a suitable linear change of variables and then systematically truncating the new set of equations. The extended framework includes modeling the effect of neglected modes as a stochastic process. By parametrizing and including stochasticity in one of two ways we show that we can improve the systems-level characterization of our dimensionally reduced neuronal network model. We examined orientation selectivity maps calculated from the firing rate distribution of large-scale simulations and stochastic dimensionally reduced models and found that by using stochastic processes to model the neglected modes, we were able to better reproduce the mean and variance of firing rates in the original large-scale simulations while still accurately predicting the orientation preference distribution.

A Class of Gorenstein Artin Algebras of Embedding Dimension Four

In this article, we study height four graded Gorenstein ideals I in k[x, y, z, w] such that I 2 is of height one and generated by three quadrics. After a suitable linear change of variables, I ∩ k[x, y, z] is either Gorenstein or of type two. The former case was studied by Iarrobino and Srinivasan [8] where they give the structure of the ideal and its resolution. We study the latter case and give the structure of these ideals and their minimal resolution. We also explicitly write the form of the generators of I and the maps in the free resolution of R/I.

Constructing a Poincaré map for a holonomic mechanical system with two degrees of freedom subject to impacts

An approach to the construction of Poincare maps for a nonlinear system with impulsive effect is proposed. The approach is based on linear change of variables that brings the Poincare map into the simplest form

MacWilliams-type identities for fragment and sphere enumerators

Let P = n 1 1 ⊕ ⋯ ⊕ n t 1 be the poset given by the ordinal sum of the antichains n i 1 with n i elements. We derive MacWilliams-type identities for the fragment and sphere enumerators, relating enumerators for the dual C ⊥ of the linear code C on P and those for C on the dual poset P ̌ . The linear changes of variables appearing in the identities are explicit. So we obtain, for example, the P -weight distribution of C ⊥ as the P ̌ -weight distribution times an invertible matrix which is a generalization of the Krawtchouk matrix.

SAGBI bases and degeneration of spherical varieties to toric varieties

Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.

DYNAMIC OUTPUT FEEDBACK SYNTHESIS WITH GENERAL FREQUENCY DOMAIN SPECIFICATIONS

This paper considers a control synthesis problem for linear systems to meet design specifications given in terms of multiple frequency domain inequalities in (semi)finite ranges. Dynamic output feedback controllers of order equal to the plant are considered. A new multiplier expansion is proposed to convert the synthesis condition to a linear matrix inequality (LMI) condition through the linearizing change of variables by Scherer, Masubuchi, de Oliveira et al. In the single objective setting, the LMI condition may or may not be conservative, depending upon the choice of the basis for the multiplier expansion. We provide a qualification for the basis matrix to yield nonconservative LMI conditions. It turns out to be difficult to determine the basis matrix meeting such qualification in general. However, it is shown that qualified bases can be found for some cases, and that the qualification can be used to find reasonable choices of the basis for other cases. Finally, the synthesis method is applied to a multiple objective control problem for an active magnetic bearing to demonstrate its utility.

Towards the complete classification of homogeneous two-component integrable equations

In this article we suggest an improved method for classifying general two-component integrable evolution equations, homogeneous in a given weighting scheme. This method relies on linear changes of variables and on an appropriate splitting of the solution space. To illustrate the method, we implement the classification of coupled KdV-type equations. We show that there are five nontriangular systems possessing higher order generalized symmetries. One of these systems is previously unknown. This seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme, without any restrictions on the form of the main matrix.