Riemann Operator Research Articles

Basics on growth orders of polymonogenic functions

In this paper, we introduce generalizations of the classical growth order and the growth type of analytic functions in the context of polymonogenic functions. Polymonogenic functions are null-solutions of higher integer order iterates of a generalized higher dimensional Cauchy–Riemann operator. One of the main goals is to prove generalizations of the famous Lindelöf–Pringsheim theorem linking explicitly these growth orders and growth types with the Taylor series coefficients in the context of this function class.

$$L^2$$ L 2 estimates for the $$\bar{\partial }$$ ∂ ¯ operator

We present the theory of twisted $$L^2$$ estimates for the Cauchy–Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa–Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the $$\bar{\partial }$$ -Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.

The $$L^2$$ L 2 -cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff

We give an example of a pseudoconvex domain in a complex manifold whose \(L^2\)-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in \({\mathbb {C}}^2\), hence Stein. This implies that for \(q>0\), the usual Dolbeault cohomology with respect to smooth forms vanishes in degree \((p,q)\). But the \(L^2\)-Cauchy–Riemann operator on the domain does not have closed range on \((2,1)\)-forms and consequently its \(L^2\)-Dolbeault cohomology is not Hausdorff.

Statistical properties of spectral fluctuations ofNinteracting bosons in a harmonic trap

Spectral fluctuations of a system of N weakly interacting bosons in an isotropic harmonic trap are studied, with the focus on the deviations from Poisson spectral statistics, typical of a quantum integrable systems. We have utilized the ideas formulated by Makino et al. [Phys. Rev. E 67, 066205 (2003)1063-651X10.1103/PhysRevE.67.066205] who have extended the approach of Berry and Robnik [J. Phys. A 17, 2413 (1984)JPHAC50305-447010.1088/0305-4470/17/12/013]. Earlier investigations of the Berry conjecture [Proc. R. Soc. London, Ser. A 356, 375 (1977)1364-502110.1098/rspa.1977.0140] of Poisson spectral statistics mainly considered quantum systems whose classical counterparts are integrable. However, the system of N weakly interacting bosons in the external trap has no classical counterpart. Also, it is a realistic and experimentally achievable system with close relation to Bose-Einstein condensation. Thus, a stringent analysis of the applicability of the Berry conjecture to this kind of systems is indeed required. We observe that for small boson number, the system is close to integrability and the nearest-neighbor level spacing distribution and the level number variance exhibit deviations from Poisson statistics similar to those of rational rectangular billiards.

On Clifford analysis for holomorphic mappings

Abstract In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

A Robin boundary value problem in the upper half plane for the Bitsadze equation

In this article we give the solvability conditions and an integral representation of the solution of a Robin problem for the Bitsadze equation in the upper half plane. In order to do that, we use classical results of complex analysis and carry out the composition of two Robin problems for the Cauchy Riemann operator.

Generalized monogenic functions satisfying differential equations with anti-monogenic right-hand sides on Clifford algebras depending on parameters

The equation Du = F(x, u) in Clifford algebras depending on parameters is considered, where D is the Cauchy–Riemann operator. A sufficient condition for the right-hand side of the equation to be anti-monogenic is given. The given criterion enables the construction of anti-monogenic functions in Clifford algebras depending on parameters and yields interior L p -estimates as well. The results provide generalizations of those in Tutschke and Yüksel (1999) [W. Tutschke and U. Yüksel, Generalized Monogenic Functions Satisfying Differential Equations with Anti-Monogenic Right-Hand Sides, ISAAC Series, Vol. 6, Kluwer Academic Publishers, Dordrecht, 1999, pp. 263–270].

Displacement energy of unit disk cotangent bundles

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle \(D^*M\) in a cotangent bundle \(T^*M\), when the base manifold \(M\) is an open Riemannian manifold. Our main result is that the displacement energy is not greater than \(C r(M)\), where \(r(M)\) is the inner radius of \(M\), and \(C\) is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.

Left Cauchy–Riemann operator and Dolbeault–Grothendieck lemma on the group of Heisenberg type

We extend to differential forms the system of partial differential equations defined for functions as followsThis leads to the cohomology, and then we will prove the corresponding Dolbeault–Grothendieck lemma.

Another proof of the Liouville theorem

We provide another proof of the Liouville theorem that conformal mappings in the dimensions at least three are Mobius transformations under the assumption that the mapping is 1-quasiconformal. Our method employs the Ahlfors Cauchy–Riemann operator.

Characterization of Minkowski measurability in terms of surface area

The r-parallel set to a set A in Euclidean space consists of all points with distance at most r from A. Recently, the asymptotic behaviour of volume and surface area of the parallel sets as r tends to 0 has been studied and some general results regarding their relations have been established. Here we complete the picture regarding the resulting notions of Minkowski content and S-content. In particular, we show that a set is Minkowski measurable if and only if it is S-measurable, i.e. if and only if its S-content is positive and finite, and that positivity and finiteness of the lower and upper Minkowski contents imply the same for the S-contents and vice versa. The results are formulated in the more general setting of Kneser functions. Furthermore, the relations between Minkowski and S-contents are studied for more general gauge functions. The results are applied to simplify the proof of the Modified Weyl–Berry conjecture in dimension one.

On Berry's conjectures about the stable order in PCF

PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types. 2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF. For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models.) And we restrict the syntax to "game terms", with a graphical representation.

On the Riemann Boundary Value Problem for Null Solutions to Iterated Generalized Cauchy–Riemann Operator in Clifford Analysis

In this paper we consider a kind of Riemann boundary value problem (for short RBVP) for null solutions to the iterated generalized Cauchy–Riemann operator and the polynomially generalized Cauchy–Riemann operator, on the sphere of \({\mathbb{R}^{n+1}}\) with Holder-continuous boundary data. Making full use of the poly-Cauchy type integral operator in Clifford analysis, we give explicit integral expressions of solutions to this kind of boundary value problems over the sphere of \({\mathbb{R}^{n+1}}\) . As special cases solutions of the corresponding boundary value problems for the classical poly-analytic and meta-analytic functions are also derived, respectively.

Universal Series on a Riemann Surface

Abstract Every holomorphic function on a compact subset of a Riemann surface can be uniformly approximated by partial sums of a given series of functions. Those functions behave locally like the classical fundamental solutions of the Cauchy–Riemann operator in the plane.

FLUCTUATIONS OF THE RIEMANN STAIRCASE

The one-sided power spectrum of the fluctuations of the Riemann staircase is calculated numerically and shown to be Lorentzian, like the one-sided power spectrum of the fluctuations of the spectral staircase for classically chaotic systems. This result provides further evidence in support of Berry's conjecture that the En's of the enigmatic Riemann zeroes 1/2 + iEn are real because they are the quantized energies of a system which is classically chaotic.

Friedrichs' extension lemma with boundary values and applications in complex analysis

Let Q be a linear first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension Q min (the closure of the graph) and the maximal closed extension Q max (in the sense of distributions) of Q in L p -spaces (1 ≤ p < ∞) coincide. In this article, we show that the same is true for boundary values with respect to Q min and Q max. This gives a useful characterization of weak boundary values, particularly for the Cauchy–Riemann operator. As an application, we derive the Bochner–Martinelli–Koppelman formula for L p -forms with weak -boundary values.

Index formulas for higher order Loewner vector fields

Let ∂ z ¯ be the Cauchy–Riemann operator and f be a C n real-valued function in a neighborhood of 0 in R 2 in which ∂ z ¯ n f ≠ 0 for all z ≠ 0 . In such cases, ∂ z ¯ n f is known as a Loewner vector field due to its connection with Loewner's conjecture that the index of such a vector field is bounded above by n. The n = 2 case of Loewner's conjecture implies Carathéodory's conjecture that any C 2 -immersion of S 2 into R 3 must have at least two umbilics. Recent work of F. Xavier produced a formula for computing the index of Loewner vector fields when n = 2 using data about the Hessian of f. In this paper, we extend this result and establish an index formula for ∂ z ¯ n f for all n ⩾ 2 . Structurally, our index formula provides a defect term, which contains geometric data extracted from Hessian-like objects associated with higher order derivatives of f.

The Cauchy–Riemann equations on product domains

We establish the L 2 theory for the Cauchy–Riemann equations on product domains provided that the Cauchy–Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on (p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $M$ with a base point $p \in M$ whose radial curvature at $p$ is bounded from below by that of a non-compact model surface of revolution $\widetilde {M}$ which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point $\tilde {p} \in \widetilde {M}$. Notice that our model $\widetilde {M}$ does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove the main theorem, we need a new type of the Toponogov comparison theorem. As an application of the main theorem, we present a partial answer to Milnor’s open conjecture on the fundamental group of complete open manifolds.

Floer homology of cotangent bundles and the loop product

We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas–Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy–Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.