Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called "geometric part" is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by A.Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems.Comment: 16 page

On the constructivity of the variational approach to Arnold's Diffusion

The aim of this paper is to discuss the constructivity of the method originally introduced by U. Bessi to approach the phenomenon of topological instability commonly known as Arnold's Diffusion. By adapting results and proofs from existing works and introducing additional tools where necessary, it is shown how, at least for a (well known) paradigmatic model, it is possible to obtain a rigorous proof on a suitable discrete space, which can be fully implemented on a computer. A selection of explicitly constructed diffusing trajectories for the system at hand is presented in the final section.Comment: 37 pages, 3 figures, affiliation information adde

A Kolmogorov theorem for nearly-integrable Poisson systems with asymptotically decaying time-dependent perturbation

The aim of this paper is to prove the Kolmogorov theorem of persistence of Diophantine flows for nearly-integrable Poisson systems associated to a real analytic Hamiltonian with aperiodic time dependence, provided that the perturbation is asymptotically vanishing. The paper is an extension of an analogous result by the same authors for canonical Hamiltonian systems; the flexibility of the Lie series method developed by A. Giorgilli et al., is profitably used in the present generalisation.Comment: 10 page

Normal forms á la moser for aperiodically time-dependent hamiltonians in the vicinity of a hyperbolic equilibrium

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay

Travelling waves over an arbitrary bathymetry:A local stability result

The problem of a travelling wave over an arbitrary quasi-flat bathymetry in a semi infinite channel is studied in the shallow-water formulation. It is shown how the streamfunction can be cast, in the vicinity of an elliptic equilibrium for the fluid flow, in the form of a nearly-integrable non-autonomous Hamiltonian with aperiodic time dependence. The proofs use the tools of perturbation theory in the real-analytic setting. The obtained Hamiltonian provides a natural example in the context of the aperiodically time-dependent Hamiltonian systems studied in Fortunati and Wiggins (2016). Some key properties of the system at hand, such as the stability, can be addressed as a consequence of this theory

Transient Invariant and Quasi-Invariant Structures in an Example of an Aperiodically Time Dependent Fluid Flow

Starting from the concept of invariant KAM tori for nearly-integrable Hamiltonian systems with periodic or quasi-periodic nonautonomous perturbation, the paper analyzes the “analogue” of this class of invariant objects when the dependence on time is aperiodic. The investigation is carried out in a model motivated by the problem of a traveling wave in a channel over a smooth, quasi- and asymptotically flat (from which the “transient” feature) bathymetry, representing a case in which the described structures play the role of barriers to fluid transport in phase space. The paper provides computational evidence for the existence of transient structures also for “large” values of the perturbation size, as a complement to the rigorous results already proven by the first author for real-analytic bathymetry functions. </jats:p

Controlling Dynamic DNA Reactions at the Surface of Single-Walled Carbon Nanotube Electrodes to Design Hybridization Platforms with a Specific Amperometric Readout

: Carbon nanotube (CNT)-based electrodes are cheap, highly performing, and robust platforms for the fabrication of electrochemical sensors. Engineering programmable DNA nanotechnologies on the CNT surface can support the construction of new electrochemical DNA sensors providing an amperometric output in response to biomolecular recognition. This is a significant challenge, since it requires gaining control of specific hybridization processes and functional DNA systems at the interface, while limiting DNA physisorption on the electrode surface, which contributes to nonspecific signal. In this study, we provide design rules to program dynamic DNA structures at the surface of single-walled carbon nanotubes electrodes, showing that specific DNA interactions can be monitored through measurement of the current signal provided by redox-tagged DNA strands. We propose the use of pyrene as a backfilling agent to reduce nonspecific adsorption of reporter DNA strands and demonstrate the controlled formation of DNA duplexes on the electrode surface, which we then apply in the design and conduction of programmable DNA strand displacement reactions. Expanding on this aspect, we report the development of novel amperometric hybridization platforms based on artificial DNA structures templated by the small molecule melamine. These platforms enable dynamic strand exchange reactions orthogonal to conventional toehold-mediated strand displacement and may support new strategies in electrochemical sensing of biomolecular targets, combining the physicochemical properties of nanostructured carbon-based materials with programmable nucleic acid hybridization

Design of specific nucleic acid‐based biosensors for protein binding activity

Nucleic acid-based biosensors for the detection of specific proteins combine the typical programmability of synthetic DNA systems with artificially controlled DNA-protein communication. The high-affinity interaction between a target protein and a specific ligand, such as an aptamer sequence, or a double stranded DNA domain, or a small peptide, is paired with a nature-mimicking molecular mechanism allowing for probing, processing, and translating protein binding activity into a measurable signal. In this Review, two main strategies developed in the context of protein-responsive nucleic acid-based biosensors are discussed. One is the design of proximity-based assays harnessing the spatial colocalization of functional probes within the volume of a multivalent protein. The other is the engineering of dynamic DNA structures that undergo a controlled conformational or structural change upon protein binding. Examples of applications from optical and electrochemical detection of antibodies in biofluids to fluorescence imaging of transcription factors in living cells are reported, and suggestions along with possible future directions in the field are discussed