On an assertion about Nash--Moser applications

By an example we show that Olaf Mueller's assertion about his new theorems being able to give anew some classical results previously obtained via applications of Nash--Moser type theorems is unfounded. We also give another example indicating some limitations in possible applications of related new inverse function theorems.Comment: 3 pages, AmS-LaTeX; v2: some clarification of wording

An inverse function theorem for Colombeau tame Frolicher-Kriegl maps

For k=1,2,... infty and a Frolicher-Kriegl order k Lipschitz differentiable map f:E supseteq U to E having derivative at x_0 in U a linear homeomorphism E to E and satisfying a Colombeau type tameness condition, we prove that x_0 has a neighborhood V subseteq U with f|V a local order k Lipschitz diffeomorphism. As a corollary we obtain a similar result for Keller C_c^{\infty} maps with E in a class including Frechet and Silva spaces. We also indicate a procedure for verifying the tameness condition for maps of the type x mapsto varphi circ [id,x] and spaces E=C^{\infty}(Q) when Q is compact by considering the case Q=[0,1]. Our considerations are motivated by the wish to try to retain something valuable in an interesting but defective treatment of integrability of Lie algebras by J. Leslie.Comment: AmSLaTeX, 8 pages; v2: scope of Corollary 9 extended, misprints corrected; v3: inaccuracies in 3 Def:s, misprints corrected, reorganization of proofs suggested in the former footnote; v4: minor specifications added, misprints corrected; v5: a forgotten detail added in the proof of 10 Prop., minor rewording in the proof of 8 Th

Seip's differentiability concepts as a particular case of the Bertram--Gloeckner--Neeb construction

From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the "compactly generated" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof for the assertion concerning Seip's theory. We also give an example indicating that the premises in Seip's various inverse and implicit function theorems may be too strong in order for these theorems to have much practical value. Also included is a presentation of the BGN--setting reformulated so as to be consistent with the Kelley--Morse--Godel--Bernays--von Neumann type approach to set theory, as well as a treatment of the function space constructions and development of their basic properties needed in the proof of the main result.Comment: Comments: 32 pages, AmSLaTeX; versions 2--7: correction of misprints and minor mistakes, improvement of unhappy wordings, Proposition "C_{Se0} is BGN" added on page 22 in v2, (7) "locality" added to Proposition 47(=v1:46) in v

A holomorphic map in infinite dimensions

We prove holomorphy E sqcap C(I,varPi) to C(I,varPi) of the map (x,y) mapsto x circ [id,y] where [id,y]:I owns t mapsto (t,y(t)) for a real compact interval I, and where varPi is a complex Banach space and E is a certain locally convex space of continuous functions x:I times varPi to varPi for which x(t,.) is holomorphic for all t in I. We also discuss application of this result to establishing a holomorphic solution map (xi,varphi) mapsto y for functions y:I to varPi satisfying the ordinary differential equation y' = varphi circ [id,y] with initial condition y(t_0) = xi .Comment: 7 pages, LaTeX; v2: a misprint corrected (p. 1, `x' added); v3(="v4"): " if we fix xi=xi^0, " added on p.

The Frolicher--Kriegl differentiabilities as a particular case of the Bertram--Glockner--Neeb construction

We prove that the order $k$ differentiability classes for $k=0,1,...\infty$ in the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl can be obtained as particular cases of the general construction by Bertram, Glockner and Neeb leading to $C^k$ differentiabilities from a given $C^0$ concept.Comment: 8 pages, AmS-LaTe

On the definition of some Banach spaces over bounded domains with irregular boundary

This note aims to clarify the interrelations of certain inequivalently defined Banach spaces denoted by C^i(\bar\Omega) for a natural number i and a bounded open set \Omega. We give some sufficient conditions for the equality of these spaces, and present examples to show that the spaces indeed can be unequal for \Omega having irregular boundary.Comment: 4 pages, AmSTeX; v2: Added: 'surjection' in Exa 1, subscript 'H\"o' in Exa 3. Removed: use of AC in Prop 2; v3: a reference updated, two misprints correcte

Real analyticity of composition is shy

Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\mathbb S^{\,1.}\to\mathbb S^{\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic Lie group since the group multiplication is not real analytic. The authors considered this result "surprising" for the applied concept of infinite-dimensional real analyticity for maps $E\to F$, defined by the property that locally a holomorphic extension $E_{\mathbb C}\to F_{\mathbb C}$ exist. In this note we show that this type of real analyticity is quite rare for composition maps ${\rm f\,}\varphi:x\mapsto\varphi\circ x$ when $\varphi$ is real analytic. Specifically, we show that the smooth Fr\'echet space map ${\rm f\,}\varphi:C\,(\mathbb R)\to C\,(\mathbb R)$ for real analytic $\varphi:\mathbb R\to\mathbb R$ is real analytic in the above sense only if $\varphi$ is the restriction to $\mathbb R$ of some entire function $\mathbb C\to\mathbb C$. We also discuss the possibility of proving that the set of these "admissible" functions $\varphi$ be "small" in the space $A\,(\mathbb R)$ of real analytic functions either in the Baire categorical sense, or in the measure theoretic sense of shyness.Comment: 5 pages, AmS-LaTeX, v2: added Prop. 6: Every inf-dim Silva space is shy in itsel

On Yamamuro's inverse and implicit function theorems in terms of calibrations

For the Frechet space E=C^{\infty}(S^1) and for a smooth \phi: R to R, we prove that the associated map E to E given by x mapsto\phi\circ x satisfies the continuous B\Gamma--differentiability condition in Yamamuro's inverse function theorem only if \phi is affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps.Comment: Comments: 9 pages, AmSLaTeX; versions 2--5: correction of minor mistake

Maximal tripartite entanglement between singlet-triplet qubits in quantum dots

Singlet-triplet states in double quantum dots are promising realizations of qubits, and capacitive coupling can be used to create entanglement between these qubits. We propose an entangling three-qubit gate of singlet-triplet qubits in a triangular setup. Our simulations using a realistic microscopic model show that a maximally entangled Greenberger-Horne-Zeilinger state can be generated as the qubits are evolved under exchange. Furthermore, our analysis for the gate operation can be used to extract the actual experimental pulse sequence needed to realize this

Capacitative coupling of singlet-triplet qubits in different inter-qubit geometries

In the singlet-triplet qubit architecture, the two-qubit interactions required in universal quantum computing can be implemented by capacitative coupling, by exploiting the charge distribution differences of the singlet and triplet states. The efficiency of this scheme is limited by decoherence, that can be mitigated by stronger coupling between the qubits. In this paper, we study the capacitative coupling of singlet-triplet qubits in different geometries of the two-qubit system. The effects of the qubit-qubit distance and the relative orientation of the qubits on the capacitative coupling strength are discussed using an accurate microscopic model and exact diagonalization of it. We find that the trapezoidal quantum dot formations allow strong coupling with low charge distribution differences between the singlet and triplet states. The analysis of geometry on the capacitative coupling is also extended to the many-qubit case and the creation of cluster states