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We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1,\,2)-dimensional real superspace. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. This work is a simplest generalization of a result by Basdouri and Ben Ammar [Cohomology of $\frak {osp}(1|2)$ with coefficients in $\frak{D}_{\lambda,\mu}$.

The present paper deals with the spectral and the oscillation properties of a linear pencil $A-\lambda B$. Here $A$ and $B$ are linear operators generated by the differential expressions $(py")"$ and $-y"+ cry$, respectively. In particular, it is shown that the negative eigenvalues of this problem are simple and the corresponding eigenfunctions $y_{-n}$ have $n-1$ zeros in $(0,1)$.

23 p. 28 cm

A new technique for constructing \"computational molecules\" for linear finite difference operators is developed. The basic approach is one of approximating a two dimensional surface with a geometrically consistent interpolating polynomial of degree four or five. The desired finite differences operator is then developed from the polynomial. the resulting molecules are geometrically consistent and may be used to solve boundary value problems without the use of fictitious points. Molecules for the biharmonic operator with various boundary conditions included are presented in this paper, as well as molecules representing the boundary conditions for shear and moment along the free edge of a plate. The integrity of the molecules presented is proven by comparisons of solutions for flat plate bending problems by finite difference with exact solutions from the literature. Convergence plots for each problem are also presented.

We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{\lambda,\mu})$ ($\lambda, \mu\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{\lambda,\mu}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)

We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{\lambda,\mu})$ ($\lambda, \mu\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{\lambda,\mu}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

Journal of Research of the National Bureau of Standards

Journal of Research of the National Bureau of Standards

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

On the Interchange of the Variables in Certain Linear Differential Operators. Elliott, E Proceedings of the Royal Society of London (1854-1905). 1889-01-01. 46:358–362

We obtain a global resolution for the sheaf of differential operators on smooth geometric quotients of free linear actions of algebraic groups. The terms of our resolution involve symmetric and alternating powers of vector bundles easily constructed geometrically from the algebraic group and the vector space on which it acts.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol vanishing locus. We obtain uniqueness and a priori maximum principle estimates for smooth solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators for partial Dirichlet or Neumann boundary conditions along the complement of the boundary vanishing locus. We also prove weak maximum principles and uniqueness for solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.

The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u \rVert_{L^1} ] is shown to hold if and only if (A(D)) is elliptic and canceling. Here (A(D)) is a homogeneous linear differential operator (A(D)) of order (k) on (\mathbb{R}^n) from a vector space (V) to a vector space (E). The operator (A(D)) is defined to be canceling if [\bigcap_{\xi \in \mathbb{R}^n \setminus {0}} A(\xi)[V]={0}.] This result implies in particular the classical Gagliardo-Nirenberg-Sobolev inequality, the Korn-Sobolev inequality and Hodge-Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator (L(D)) of order (k) on (\mathbb{R}^n) from a vector space (E) to a vector space (F) is introduced. It is proved that (L(D)) is cocanceling if and only if for every (f \in L^1(\mathbb{R}^n; E)) such that (L(D)f=0), one has (f \in \dot{W}^{-1, n/(n-1)}(\mathbb{R}^n; E)). The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.

The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u \rVert_{L^1} ] is shown to hold if and only if (A(D)) is elliptic and canceling. Here (A(D)) is a homogeneous linear differential operator (A(D)) of order (k) on (\mathbb{R}^n) from a vector space (V) to a vector space (E). The operator (A(D)) is defined to be canceling if [\bigcap_{\xi \in \mathbb{R}^n \setminus {0}} A(\xi)[V]={0}.] This result implies in particular the classical Gagliardo-Nirenberg-Sobolev inequality, the Korn-Sobolev inequality and Hodge-Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator (L(D)) of order (k) on (\mathbb{R}^n) from a vector space (E) to a vector space (F) is introduced. It is proved that (L(D)) is cocanceling if and only if for every (f \in L^1(\mathbb{R}^n; E)) such that (L(D)f=0), one has (f \in \dot{W}^{-1, n/(n-1)}(\mathbb{R}^n; E)). The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.

An analog of Gelfand-Levitan-Marchenko integral equations for multi- dimensional Delsarte transmutation operators is constructed by means of studying their differential-geometric structure based on the classical Lagrange identity for a formally conjugated pair of differential operators. An extension of the method for the case of affine pencils of differential operators is suggested.

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $\lambda$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to different values of $\lambda$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each other for every values of $\lambda\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm \frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case $(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to two different values of the degree: $\lambda$ and $\lambda^{\prime}$, are isomorphic if and only if $\lambda+\lambda^{\prime}=1$.

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition.

We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The proposed method does not require solving the algebraic equation and can be applied in the case when its Galois group is not solvable.

We consider a classical problem of Computer Algebra: symbolic solution of PDEs. We transform the famous Darboux theorems on differential transformations of hyperbolic operator into the space of invariants. We introduce a new idea -- $X$- and $Y$-invariants of such operator as solutions of some equations written in terms of the Laplace invariants of this operator. Explicit formula for the changes in the sets of the $X$- and $Y$-invariants under the Darboux transformations are obtained.

General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic formula for fundamental solutions of corresponding homogenous linear differential equation as series of integrals. Furthermore, multiplicative property of causal Green's functions is shown and by which explicit formulas for causal Green's functions of some classes of decomposable linear differential operators are given. A method to find Green's function of general linear differential operator of given degree in one variable with arbitrary boundary condition according to coefficient functions of differential operator is demonstrated.

General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic formula for fundamental solutions of corresponding homogenous linear differential equation as series of integrals. Furthermore, multiplicative property of causal Green's functions is shown and by which explicit formulas for causal Green's functions of some classes of decomposable linear differential operators are given. A method to find Green's function of general linear differential operator of given degree in one variable with arbitrary boundary condition according to coefficient functions of differential operator is demonstrated.

Thesis (M.S. in M.E.)--Naval Postgraduate School, 1970

In the present book, the theory of linear, ordinary, differential operators of arbitrary order is developed. Extensive use is made of the ideas and theorems of functional analysis, especially those of the theory of linear operators in Hilbert space. For the reader’s convenience all the necessary theorems in functional analysis are developed in the text itself, to make the book, as the author hopes, self-contained and accessible to a wide range of readers-. Many topics in the theory oflinear differential operators can also be discussed without using functional analysis, as Titchmarsh’s treatment [112a] shows. However, the author does not consider such a treatment to be appropriate, since it is only by using the ideas and methods of functional analysis that a deeper understanding of the theory, and its most general results, can be achieved. The book is in two parts. In the first part, which may be described as an elementary theory of differential operators, the use of the methods of functional analysis is kept to a minimum. This part presents the theory of differential operators defined on a finite interval, including the case of differential opera­ tors which are not self-adjoint, the theory being developed on the assumption that the coefficients of the operators are sufficiently smooth, i.e., sufficiently differentiable. To understand Part I the reader needs only an elementary know­ ledge of the theory of ordinary differential and integral equations and of the theory of functions. Part II develops the theory of differential operators, using Hilbert space methods. Here, in addition to the subjects already mentioned, the reader will be expected to have some knowledge of the basic results in the theory of the Lebesgue integral.

In these notes, written to accompany 18.06 lectures in Fall 2007, we discuss these mysteries: Fourier series come from taking concepts like eigenvalues and eigenvectors and Hermitian matrices and applying them to functions instead of �nite column vectors. In this way, we see that important properties like orthogonality of the Fourier series arises not by accident, but as a special case of a much more general fact, analogous to the fact that Hermitian matrices have orthogonal eigenvectors.

We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations. We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four. Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also.

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.

We consider two principal bundles of embeddings with total space $Emb(M,N),$ with structure groups $Diff(M)$ and $Diff_+(M),$ where $Diff_+(M)$ is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: $$ B(M,N) = Emb(M,N)/Diff(M) \hbox{ and } B_+(M,N)= Emb(M,N)/Diff_+(M).$$ From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in \cite{OMYK4}. We show that these groups are regular, and develop the necessary properties for applications to the geometry of $ B(M,N) .$ A case of particular interest is $M=S^1,$ where connected components of $B_+(S^1,N)$ are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space $TB_+(S^1,N)$ is a subgroup of some group $GL_{res},$ following the classical notations of \cite{PS}. These constructions suggest some approaches in the spirit of \cite{Ma2006} that could lead to knot invariants through a theory of Chern-Weil forms.

We consider two principal bundles of embeddings with total space $Emb(M,N),$ with structure groups $Diff(M)$ and $Diff_+(M),$ where $Diff_+(M)$ is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: $$ B(M,N) = Emb(M,N)/Diff(M) \hbox{ and } B_+(M,N)= Emb(M,N)/Diff_+(M).$$ From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in \cite{OMYK4}. We show that these groups are regular, and develop the necessary properties for applications to the geometry of $ B(M,N) .$ A case of particular interest is $M=S^1,$ where connected components of $B_+(S^1,N)$ are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space $TB_+(S^1,N)$ is a subgroup of some group $GL_{res},$ following the classical notations of \cite{PS}. These constructions suggest some approaches in the spirit of \cite{Ma2006} that could lead to knot invariants through a theory of Chern-Weil forms.

Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization of linear ordinary differential operators. Possible applications to factorized Groebner bases computations in the commutative and non-commutative cases are discussed, an application to finding criterions of Darboux integrability of nonlinear PDEs is given.

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.

An algebraic approach for factorizing nonlinear partial differential equations (PDEs) and systems of PDEs is provided. In the particular case of second order linear and nonlinear PDEs and systems of PDEs, necessary and sufficient conditions of factorization are given.

Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that ``irreducible'' parametric factorizations may exist only for a few certain types of factorizations. Examples are given of the parametric families for each of the possible types. For the operators of orders two and three, it is shown that any factorization family is parameterized by a single univariate function (which can be a constant function).

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators $L,$ defined, say, in an open set $\Om\subset \RR^n.$ Suppose the principal symbol $p_k$ of $L$ vanishes to second order at $(x_0,\xi_0)\in T^*\Om\setminus 0,$ and denote by $Q_\H$ the Hessian form associated to $p_k$ on $T_{(x_0,\xi_0)}T^*\Om.$ As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of $L$ at $x_0$ is the existence of some $\theta\in\RR$ such that $\Re (e^{i\theta}Q_\H)\ge 0.$

A complete classification of linear differential operators possessing finite-dimensional invariant subspace with a basis of monomials is presented.

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type.

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type.

We suggest method based on the skeleton decomposition of linear operators in order to reduce ill-posed degenerate differential equations to the non-classic initial-value problem enjoying unique solution