We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic, and hence a computational theory of non-linear partial differential equations and their solutions based on variational calculus.

We investigate images of higher-order differential operators of polynomial algebras over a field $k$. We show that, when $\operatorname{char}k>0$, the image of the set of differential operators $\{\unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D70F}_{i}\mid i=1,2,\ldots ,n\}$ of the polynomial algebra $k[\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n},z_{1},\ldots ,z_{n}]$ is a Mathieu subspace, where $\unicode[STIX]{x1D70F}_{i}\in k[\unicode[STIX]{x2202}_{z_{1}},\ldots ,\unicode[STIX]{x2202}_{z_{n}}]$ for $i=1,2,\ldots ,n$. We also show that, when $\operatorname{char}k=0$, the same conclusion holds for $n=1$. The problem concerning images of differential operators arose from the study of the Jacobian conjecture.

The problem of resolution of singularities in positive characteristic can be reformulated as follows: fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps. The first consists of the transformation of $X$ to a hypersurface with $n$-fold points in the so-called monomial case. The second step consists of the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to present a notion of strong monomial case which parallels that of monomial case in characteristic zero: if a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.

In this paper we consider the discreteness of spectrum for higher-order differential operators in weighted function spaces. Using the method of embedding theorems of weighted Sobolev spaces Hnp in weighted spaces Ls,r, we obtain a new sufficient and necessary condition to ensure that the spectrum is discrete, which can be easily used to judge the discreteness of some differential operators.

We give a characterization of contact manifolds in terms of symplectic Lie–Rinehart–Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

In some recent investigations involving certain differential operators for a general family of Lagrange polynomials, Chan el al. encountered and proved a certain summation identity for the Lagrange polynomials in several variables. In the present paper, we derive some generalizations of this summation identity for the Chan-Chyan-Srivastava polynomials in several variables. We also discuss a number of interesting corollaries and consequences of our main results.

Asymptotic approximations to the Green's functions of Sturm–Liouville boundary-value problems on graphs are obtained. These approximations are used to study the regularized traces of the differential operators associated with these boundary-value problems. Various inverse spectral problems for Sturm–Liouville boundary-value problems on graphs resembling those considered in Halberg and Kramer's ‘A generalization of the trace concept' (Duke Mathematics Journal27 (1960), 607–617), for Sturm–Liouville problems, and Pielichowski's ‘An inverse spectral problem for linear elliptic differential operators' (Universitatis Iagellonicae Acta Mathematica27 (1988), 239–246), for elliptic boundary-value problems, are solved.

We prove Pursell–Shanks type results for the Lie algebra $\mathcal{D}(M)$ of all linear differential operators of a smooth manifold M, for its Lie subalgebra $\mathcal{D}^1(M)$ of all linear first-order differential operators of M and for the Poisson algebra S(M) = Pol(T*M) of all polynomial functions on T*M, the symbols of the operators in $\mathcal{D}(M)$. Chiefly, however, we provide explicit formulas completely describing the automorphisms of the Lie algebras $\mathcal{D}^1(M)$, S(M) and $\mathcal{D}(M)$.

We define a 4-parameter family of generically irreducible and inequivalent representations of the Witt Lie algebra on which the infinitesimal rotation operator acts semisimply with infinite-dimensional eigenspaces. They are deformations of the (generically indecomposable) representations on spaces of polynomial differential operators between two spaces of tensor densities on S1, which are constructed by composing each such differential operator with the action of a rotation by a fixed angle.

Let M be a smooth manifold, ${\cal S}$ the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on ${\cal S}$. This cohomology space is closely related to the Vect(M)-modules, ${\cal D}$λ(M), of linear differential operators on the space of tensor densities on M of degree λ.

The purpose of this paper is to define functional realizations of the Khizhnik–Zamolochikov–Bernard (KZB) connection on the bundle of conformal blocks over the moduli space of curves.

The Novikov-Landweber algebra and the Steenrod algebra are set up in terms of the primitive differential operators $D_k=\sum_i x_i^{k+1}\frac{\partial}{\partial x_i}$ acting in the usual way on the integral polynomial ring ${\bf Z}[x_1,\ldots ,x_n,\ldots]$. A commutative wedge product $\vee$ for differential operators is introduced and it is shown that the iterated wedge product $D_k^{\vee r}$ is divisible by $r!$ as an integral operator. The divided differential operator algebra $\cal D$ is generated over the integers by the divided operators $\frac{D_k^{\vee r}}{r!}$ under the wedge product. $\cal D$ is additively isomorphic to the abelian group of symmetric functions in the variables $x_i$. Furthermore $\cal D$ is closed under composition of operators and admits a natural coproduct which makes it a Hopf algebra in two ways, with respect to the composition and wedge products. Under composition $\cal D$ is isomorphic to the Landweber-Novikov algebra. A Hopf sub-algebra is generated under composition by the integral Steenrod squares $SQ^r = \frac{D_1^{\vee r}}{r!}$ and reduces mod 2 to the Steenrod algebra. An explicit product formula for two wedge expressions is developed and used to derive Milnor's product formula for his basis elements in the Steenrod algebra. The hit problem in the Steenrod algebra is reformulated in terms of partial differential operators.

1991 Mathematics Subject Classification 55S10.