Jacobian Multipliers Research Articles

Vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety

Abstract In this paper, we establish several results related to vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffiths-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather–Jacobian multiplier ideals.

Differential Galoisian approach to Jacobi integrability of general analytic dynamical systems and its application

The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction. In this paper, we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems. The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity component. In addition, we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.

On the inverse problems of the polynomial vector fields in ℝ3

In this paper, we deduce the normal forms of polynomial differential systems in ℝ3 having, Fermat cubic surface, Clebsch surface, Goursat surface, Racor surface and 2-dimensional tours as invariant objects. We prove if we have an odd number of polynomial differential equations with the determined inverse Jacobian multipliers then we can make a new polynomial differential system of the first systems so that it’s inverse Jacobian multiplier can be determined by the inverse Jacobian multipliers of the first systems. If new system which is made by them has a limit cycle then we determine it’s position.

Liouvillian Integrability Versus Darboux Polynomials

In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Lienard polynomial differential system \(\dot{x}=y,\, \dot{y}=-f(x)y-g(x)\) with \(\deg f>\deg g\) is not Liouvillian integrable.

Integrability of vector fields versus inverse Jacobian multipliers and normalizers

In this paper we provide characterization of integrablity of a system of vector fields via inverse Jacobian multipliers (matrix) and normalizers of smooth (or holomorphic) vector fields. These results improve and extend some well known ones, including the classical holomorphic Frobenius integrability theorem. Here we obtain the exact expression of an integrable system of vector fields acting on a smooth function via their known common first integrals. Moreover we characterize the relations between the integrability and the existence of normalizers for a system of vector fields. In the case of integrability of a system of vector fields we not only prove the existence of normalizers but also provide their exact expressions.

Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

Poisson and integrable systems are orbitally equivalent through the Nambu bracket. Namely, we show that every completely integrable system of differential equations may be expressed into the Poisson-Hamiltonian formalism by means of the Nambu-Hamilton equations of motion and a reparametrisation related by the Jacobian multiplier. The equations of motion provide a natural way for finding the Jacobian multiplier. As a consequence, we partially give an alternative proof of a recent theorem in [13]. We complete this work presenting some features associated to Hamiltonian maximally superintegrable systems.

The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Let $$\dot{x}=f(x)$$ be a $$C^k$$ autonomous differential system with $$k\in {\mathbb {N}}\cup \{\infty ,\omega \}$$ defined in an open subset $$\varOmega $$ of $${\mathbb {R}}^n$$ . Assume that the system $$\dot{x}=f(x)$$ is $$C^r$$ completely integrable, i.e., there exist $$n-1$$ functionally independent first integrals of class $$C^r$$ with $$2\le r\le k$$ . As we shall see, we can assume without loss of generality that the divergence of the system $$\dot{x}=f(x)$$ is not zero in a full Lebesgue measure subset of $$\varOmega $$ . Then, any Jacobian multiplier is functionally independent of the $$n-1$$ first integrals. Moreover, the system $$\dot{x}=f(x)$$ is $$C^{r-1}$$ orbitally equivalent to the linear differential system $$\dot{y}=y$$ in a full Lebesgue measure subset of $$\varOmega $$ . Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.

Liouvillian integrability of polynomial differential systems

M.F. Singer (Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673–688) proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential systems. In this paper we will extend Singer’s result to any finite dimensional polynomial differential systems. We prove that if an n n –dimensional polynomial differential system has n − 1 n-1 functionally independent Darboux Jacobian multipliers, then it has n − 1 n-1 functionally independent Liouvillian first integrals. Conversely if the system is Liouvillian integrable, then it has a Darboux Jacobian multiplier.

Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

In this paper we consider a class of higher dimensional differential systems in Rn which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or C∞ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.

Geometric nullstellensatz and symbolic powers on arbitrary varieties

In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather–Jacobian multiplier ideal, has been developed independently by Ein–Ishii–Mustata and de Fernex–Docampo. With this new tool, we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms. In this paper, we first extend a result of geometric nullstellensatz due to Ein–Lazarsfeld in nonsingular case to any projective varieties. Then we prove a result on comparison of symbolic powers with ordinary powers on any varieties, which extends results of Ein–Lazarsfeld–Smith and Hochster–Huneke.

Using a Demand System to Estimate Extended Input–Output Multipliers

The interface between household income and expenditure has always been considered to be a key component in the construction of input–output models. However, it can be argued that households are too often treated as if they were just another in dustry in the input–output table. In this paper, we seek to address this problem by developing a new modelling framework in which a micro demand system is used to estim ate the relationship between income and expenditure. This demand system is conjoined with an input–output table for the UK economy, and the system as a whole is solved as a computable general equilibrium model. Comparisons are made between the Jacobian multipliers generated by this model and those derived from a more traditional input–output model in which the income-expenditure linkage is estimated using static coefficients.

Macroeconomic structure and computable general equilibrium models

Economy-wide analysis can be undertaken within three different accounting frameworks: (1) the national income and product accounts, which focus on balance among macroaggregates; (2) the input-output accounts, which focus on intermediate flows and on the sectoral composition of production and demand; and (3) the social accounting matrix (SAM), which provides a framework incorporating both national income and product as well as input-output information. All three of these frameworks have been used to develop fixed coefficient, linear multiplier models. Keynesian multipliers at the macro level have multisectoral counterparts in the input-output model and “multi-institutional” counterparts in SAM-based models. These linear multiplier models tend to be completely demand driven and do not incorporate supply constrains or substitution possibilities. Computable general equilibrium (CGE) models have been developed which capture nonlinear substitution possibilities and multisectoral supply-demand interactions, and also incorporate macro variables and mechanisms for achieving balance among aggregates. “Marginal” multipliers can be derived for CGL models using the Jacobian matrix of partial derivatives at a given equilibrium. This paper develops an approach to analyzing Jacobian multipliers from a CGE model that decomposes macro, sectoral, and institutional linkages in a SAM framework. We give an illustration of SAM-multiplier decomposition using a small CGE model of the United States.