GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersurfaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.     

Inequalities for two type potential operators on differential forms

The purpose of this paper is to derive some strong-type inequalities for convolution type potential operator applied on differential forms. Caccioppoli-type inequalities for integral type potential operator acting on A-harmonic tensor are also obtained.     

On self-adjointness of the product of two second-order differential operators

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the productL ₂ L ₁ of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators. Supported by the Royal Society and the National Natural Science Foundation of China and the Regional Science Foundation of Inner Mongolia     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

Interpolation by hypercyclic functions for differential operators

We prove that, given a sequence of points in a complex domain Ω without accumulation points, there are functions having prescribed values at the points of the sequence and, simultaneously, having dense orbit in the space of holomorphic functions on Ω. The orbit is taken with respect to any fixed nonscalar differential operator generated by an entire function of subexponential type, thereby extending a recent result about MacLane-hypercyclicity due to Costakis, Vlachou and Niess.     

Projective push-forwards in the Witt theory of algebraic varieties

We define push-forwards along projective morphisms in the Witt theory of smooth quasi-projective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pull-backs and projection formulas.     

Conformally invariant systems of differential operators

Kostant's theory of conformally invariant differential operators on certain homogeneous spaces is generalized to cover conformally invariant systems of endomorphism-valued differential operators. In particular, the connection discovered by Kostant between conformally invariant operators and highest weight vectors in generalized Verma modules is extended.     

Projective dimension of the universal modules for the product of a hypersurface and affine t-space

The main result in this paper is that the. projective dimension of the nth order universal module of the. coordinate ring of the product of a reduced hypersurface and an affine t-spa.ee is finite. This question appears in [1]. We have seen this by showing that projective dimension of the nth order universal module of the ring corresponding to a reduced hypersurface is finite. At the end we present some examples related to projective dimension of the universal modules of the product of variaties.     

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

Compact inverses of first-order normal differential operators

In this work, firstly we describe all normal extensions of a minimal operator generated by linear differential-operator expression of first order in the Hilbert space of vector functions in finite interval. Later on, we investigate discreteness of spectrum and asymptotical behavior of s-numbers of the inverses of these normal extensions.     

Eigenvalue problems and fixed point theorems for a class of positive nonlinear operators

We study the eigenvalue problems for a class of positive nonlinear operators defined on a cone in a Banach space. Using projective metric techniques and Schauder’s fixed-point theorem, we establish existence, uniqueness, monotonicity and continuity results for the eigensolutions. Moreover, the method leads to a result on the existence of a unique fixed point of the operator. Applications to nonlinear boundary-value problems, to differential delay equations and to matrix equations are considered.      

State operators on generalizations of fuzzy structures

We enlarge the language of R?-monoids, which are a non-commutative generalizations of both MV algebras and BL algebras, by adding a unary operation that describes algebraic properties of a state (= an analog of probability measures). The resulting algebras are called stateR?-monoids and state-morphismR?-monoids. We present basic properties of such algebras. We describe subdirectly irreducible algebras, some generators of the varieties of state-morphism R?-monoids, and an interplay between states and state operators.     

Some liftings of poisson structures to Weil bundles

We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.     

Projective and inductive limits in locally convex cones

We define and study the projective and inductive limit notions for locally convex cones. We use convex quasiuniform structure method for this purpose. Also we study the barreledness in the locally convex cones and introduce the notion upper-barreled cones and prove that the inductive limit of upper-barreled cones is upper-barreled.     

A note on differential operators of infinite order

A sufficient condition for entire functions f and g to be such that the series ∑_m=0^∞f^(m)(0)g^(m)/m! represents an entire function is established; and in that case, the growth of the resulting function is described.