Typically, a linear projection of the Grassmannian in its Plücker embedding is generically injective, unless the image of the Grassmannian is a linear space. A notable exception are self-adjoint ...linear projections, which have even degree. We consider linear projections of Gr3C6 with low-dimensional centers of projection. When the center has dimension less than five, we show that the projection has degree 1. When the center has dimension five and the projection has degree greater than 1, we show that it is self-adjoint.
•The linearization problem is solved for equations of order 2, 3, and 4 from the Chain.•The Lie pseudo-group of equivalence transformation is found for all orders.•The explicit expression of ...parameter functions under point transformation is found.•Some important equivalence classes under point transformations are exhibited.•Canonical forms are exhibited for common types of parameter functions.
The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations. The Lie pseudo-group of equivalence transformations for equations of arbitrary orders from the chain are then found, together with expressions for the transformed parameter functions. An important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited.
The present paper studies uniqueness properties of the solution of the inverse problem for the Sturm-Liouville equation with discontinuous leading coefficient and the separated boundary conditions. ...It is proved that the considered boundary-value is uniquely reconstructed, i.e. the potential function of the equation and the constants in the boundary conditions are uniquely determined by given Weyl function or by the given spectral data.
This article, we study sufficient conditions for the controllability of second-order impulsive neutral integro-differential evolution systems with an infinite delay in Banach spaces by using the ...theory of cosine families of bounded linear operators and fixed point theorem.
In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated ...combined dynamic derivatives have led to the proposal of improved approximation expressions for computational application. One such expression is the diamond alpha (⋄α) derivative, which is defined as a linear combination of delta and nabla derivatives. Several dynamic equations and inequalities, as well as hybrid dynamic behavior—which does not occur in the real line or on discrete time scales—are analyzed using this combined concept. In this study, we consider a ⋄α Dirac system under boundary conditions on a uniform time scale. We examined some basic spectral properties of the problem we are considering, such as the simplicity, the reality of eigenvalues, orthogonality of eigenfunctions, and self adjointness of the operator. Finally, we construct an expression for the eigenfunction of the ⋄α Dirac boundary value problem (BVP) on a uniform time scale.
Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient ...condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.
In this paper, the previously obtained description of invertible linear ordinary differential operators is generalized to invertible linear discrete-time operators, unimodular matrices, and ...transformations of control systems. The description is based on assigning a model in the form of a set of squares on the plane to each generalized invertible operator. The number of squares of the model is invariant and can be used to classify generalized invertible operators and, in particular, transformations of control systems.
Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal ...works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem Ly=λy, for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L−λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems.
We consider invertible linear ordinary differential operators whose inversions are also differential operators. To each such operator we assign a numerical table. These tables are described in the ...elementary geometrical language. The table does not uniquely determine the operator. To define this operator uniquely some additional information should be added, as it is described in detail in this paper. The possibility of generalization of these results to partial differential operators is also discussed.
The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of ...genus two the set of orbits is infinite. KCI Citation Count: 2
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