Proof diagrams and term rewriting with applications to computational algebra

In this thesis lessons learned from the use of computer algebra systems and machine assisted theorem provers are developed in order to give an insight into both the problems and their solutions. Many algorithms in computational algebra and automated deduction (for example Grobner basis computations and Knuth-Bendix completion) tend to produce redundant facts and can contain more than one proof of any particular fact. This thesis introduces proof diagrams in order to compare and contrast the proofs of facts which such procedures generate. Proof diagrams make it possible to analyse the effect of heuristics which can be used to guide implementations of such algorithms. An extended version of an inference system for Knuth-Bendix completion is introduced. It is possible to see that this extension characterises the applicability of critical pair criteria, which are heuristics used in completion. We investigate a number of executions of a completion procedure by analysing the associated proof diagrams. This leads to a better understanding of the heuristics used to control these examples. Derived rales of inference are also investigated in this thesis. This is done in the formalism of proof diagrams. Rewrite rules for proof diagrams are defined: this is motivated by the notion of a transformation tactic in the Nuprl proof development system. A method to automatically extract 'useful' derived inference rales is also discussed. 'Off the shelf' theorem provers, such as the Larch Prover and Otter, are compared to specialised programs from computational group theory. This analysis makes it possible to see where methods from automated deduction can improve on the tools which group theorists currently use. Problems which can be attacked with theorem provers but not with currently used specialised programs are also indicated. Tietze transformations, from group theory, are discussed. This makes it possible to link ideas used in Knuth-Bendix completion programs and group presentation simplification programs. Tietze transformations provide heuristics for more efficient and effective implementations of these programs

Surface-enhanced spatially-offset raman spectroscopy (SESORS) in tissue analogs

Surface-enhanced, spatially-offset Raman spectroscopy (SESORS) combines the remarkable enhancements in sensitivity afforded by surface-enhanced Raman spectroscopy (SERS) with the non-invasive, sub-surface sampling capabilities of spatially-offset Raman spectroscopy (SORS). Taken together, these techniques show great promise for in vivo Raman measurements. Herein, we present a step forward for this technique, demonstrating SESORS through tissue analogs of six known and varied thickness, with a large number of distinct spatial offsets, in a back-scattering optical geometry. This is accomplished by spin-coating SERS-active nanoparticles (NPs) on glass slides, and monitoring the relative spectral contribution from the NPs and tissue sections, respectively, as a function of both tissue thickness and spatial offset of the collection probe. The results show that SESORS outperforms SERS alone for this purpose, NP signal is attainable at tissue thicknesses in excess of 6.75 mm, and that greater tissue thicknesses require greater spatial offsets to maximize NP signal, all with an optical geometry optimized for utility. This demonstration represents a step forward toward the implementation of SESORS for non-invasive, in vivo analysis

Detection of multiple nitroaromatic explosives via formation of a Janowsky complex and SERS

Military-grade explosives such as 2,4,6-trinitroluene (TNT) are still a major worldwide concern in terms of terror threat and environmental impact. The most common methods currently employed for the detection of explosives involve colorimetric tests, which are known to be rapid and portable; however, they often display false positives and lack sensitivity. Other methods used include ion mobility mass spectrometry, gas chromatography-mass spectrometry (GC-MS), and liquid chromatography-mass spectrometry (LC-MS), which despite producing more reliable results often require large, expensive instrumentation and specially trained staff. Here we demonstrate an alternative approach that utilizes the formation of a colored Janowsky complex with nitroaromatic explosives through reaction of the enolate ion of 3-mercapto-2-butanone. The colored complex is formed rapidly and can then be detected sensitively using surface-enhanced Raman scattering (SERS). We demonstrate that SERS can be used as a quick, sensitive, and selective technique for the detection of 2,4,6-trinitrotoluene (TNT), hexanitrostillbene (HNS), and 2,4,6-trinitrophenylmethylnitramine (tetryl) with a detection limit of 6.81 ng mL -1 achieved for TNT, 17.2 ng mL -1 for tetryl, and 135.1 ng mL -1 for HNS. This method of detection also requires minimal sample preparation, can be done in a solution-based format, and utilizes the same precursor reagents for complex formation with each of the explosives which can then be identified due to the specificity of the unique SERS response obtained. We demonstrate the ability to simultaneously identify three explosive compounds within a total analysis time of 10 min. This method of detection shows promise for the development of rapid and portable SERS-based assays which can be utilized in the field in order to achieve reliable and quantitative detection

Sensitive SERS nanotags for use with 1550 nm (retina-safe) laser excitation

Chalcogenopyrylium nanotags demonstrate an unprecedented SERS performance with a retina safe, 1550 nm laser excitation. These unique nanotags consisting of chalcogenopyrylium dyes and 100 nm gold nanoparticles produce exceptional SERS signals with picomolar detection limits obtained at this extremely red-shifted and eye-safe laser excitation

Through tissue imaging of a live breast cancer tumour model using handheld surface enhanced spatially offset Resonance Raman Spectroscopy

In order to improve patient survival and reduce the amount of unnecessary and traumatic biopsies, non-invasive detection of cancerous tumours is of imperative and urgent need. Multicellular tumour spheroids (MTS) can be used as an ex vivo cancer tumour model, to model in vivo nanoparticle (NP) uptake by the enhanced permeability and retention (EPR) effect. Surface enhanced spatially offset Raman spectroscopy (SESORS) combines both surface enhanced Raman spectroscopy (SERS) and spatially offset Raman spectroscopy (SORS) to yield enhanced Raman signals at much greater sub-surface levels. By utilizing a reporter that has an electronic transition in resonance with the laser frequency, surface enhanced resonance Raman scattering (SERRS) yields even greater enhancement in Raman signal. Using a handheld SORS spectrometer with back scattering optics, we demonstrate the detection of live breast cancer 3D multicellular tumour spheroids (MTS) containing SERRS active NPs through 15 mm of porcine tissue. False color 2D heat intensity maps were used to determine tumour model location. In addition, we demonstrate the tracking of SERRS-active NPs through porcine tissue to depths of up to 25 mm. This unprecedented performance is due to the use of red-shifted chalcogenpyrylium-based Raman reporters to demonstrate the novel technique of surface enhanced spatially offset resonance Raman spectroscopy (SESORRS) for the first time. Our results demonstrate a significant step forward in the ability to detect vibrational fingerprints from a tumour model at depth through tissue. Such an approach offers significant promise for the translation of NPs into clinical applications for non-invasive disease diagnostics based on this new chemical principle of measurement

Sensitive SERS nanotags for use with a hand-held 1064 nm Raman spectrometer

This is the first report of the use of a hand-held 1064 nm Raman spectrometer combined with red shifted surface enhanced Raman scattering (SERS) nanotags to provide an unprecedented performance in the short-wave infrared (SWIR) region. A library consisting of 17 chalcogenopyrylium nanotags produce extraordinary SERS responses with femtomolar detection limits being obtained using the portable instrument. This is well beyond previous SERS detection limits at this far red shifted wavelength and opens up new options for SERS sensors in the SWIR region of the electromagnetic spectrum (between 950-1700 nm)

Resonance Raman detection of antioxidants using an iron oxide nanoparticle catalysed decolourisation assay

Nanozymes are metal nanoparticles with catalytic properties that can be used to oxidise peroxidase substrates giving a colorimetric response which can be detected using UV-vis, and recently, Raman spectroscopy. Due to their ease of synthesis and increased stability, nanozymes are being increasing investigated to replace conventional enzymes for the detection of biomolecules. Here we exploit the catalytic activity of iron oxide (Fe2O3) nanoparticles combined with the substrate 2,2-Azinobis(3-ethyl-benzothiazoline-6-sulfonic acid) (ABTS) in a decolourisation assay for the detection of antioxidants. Fe2O3 nanoparticles were used to catalyse the oxidation of ABTS to its green radical cation which, upon the addition of an antioxidant, resulted in a decolourisation due to the reduction of the radical cation caused by the hydrogen donating antioxidant. The assay was applied for the detection of multiple antioxidants (glutathione, chlorogenic acid and ascorbic acid), and was followed by monitoring the resonance Raman scattering from the ABTS solution using a portable Raman system with 785 nm laser excitation. This novel assay has the potential to be optimised to detect antioxidant activity in body fluid with low limits of detection with point of use monitoring

A novel nanozyme assay utilising the catalytic activity of silver nanoparticles and SERRS

Artificial enzymes have become an increasingly interesting area of research due to their many advantages over natural protein enzymes which are expensive, difficult to isolate and unable to stand harsh environments. An important area of this research involves using metal nanoparticles as artificial enzymes, known as nanozymes, which exhibit peroxidase-like activity enabling them to catalyse the oxidation of substrates such as 3,3’,5,5’-tetramethylbenzidine (TMB) in the presence of hydrogen peroxide (H2O2), giving a colorimetric response. Here we exploit the catalytic activity of silver nanoparticles (Ag NPs) in a surface based silver-linked immunosorbent assay (SLISA) to detect human C-reactive protein (CRP), an inflammatory marker. Ag NPs were conjugated to antibodies with specific recognition for the corresponding target antigenic molecule, CRP, and used to catalyse the oxidation of TMB by H2O2. The resulting coloured oxidation product was detected using SERRS. We demonstrate that Ag NPs can replace the enzymes used in a conventional ELISA and a detection limit of 1.09 ng/mL of CRP can be achieved. It indicates the promise for SLISAs for biomarker detection and opens the way for further assays of this nature to be created. This novel assay has the potential to be optimised to detect lower levels of CRP and can be further extended for the sensitive and specific detection of other relevant biomarkers

Through barrier detection of ethanol using handheld Raman spectroscopy — Conventional Raman versus spatially offset Raman spectroscopy (SORS)

Spatially offset Raman spectroscopy (SORS) provides chemical analysis at depth even when obscuring barriers such as plastic or tissue are present. As the collection probe is moved further away from the point of laser excitation, scattered photons from deeper layers begin to dominate the acquired spectra, thus giving rise to through barrier detection. Here, we demonstrate the potential of conventional Raman (CR) and SORS for through barrier detection using handheld spectrometers. We report the collection of Raman signals from an ethanol solution through plastic at thicknesses of up to 21 mm using SORS in combination with multivariate analysis. SORS is compared to CR, where we also demonstrate impressive through barrier detection of ethanol at depths up to 9 mm. We also highlight the advantage of applying multivariate analysis for through barrier detection using CR or SORS, particularly when peaks with similar spectral features are present in both the barrier and analyte spectra. In addition, to the best of our knowledge, this is the first report of the assessment of the maximum level of through barrier detection using handheld CR and SORS instruments with a backscattering geometry

Plasmonic and photothermal properties of silica-capped gold nanoparticle aggregates

Owing to their biocompatibility, gold nanoparticles have many applications in healthcare, notably for targeted drug delivery and the photothermal therapy of tumors. The addition of a silica shell to the nanoparticles can help to minimize the aggregation of the nanoparticles upon exposure to harsh environments and protect any Raman reporters adsorbed onto the metal surface. Here, we report the effects of the addition of a silica shell on the photothermal properties of a series of gold nanostructures, including gold nanoparticle aggregates. The presence of a Raman reporter at the surface of the gold nanoparticles also allows the structures to be evaluated by surface-enhanced Raman scattering (SERS). In this work, we explore the relationship between the degree of aggregation and the position and the extinction of the near-infrared plasmon on the observed SERS intensity and in the increase in bulk temperature upon near-infrared excitation. By tailoring the concentration of the silane and the thickness of the silica shell, it is possible to improve the photothermal heating capabilities of the structures without sacrificing the SERS intensity or changing the optical properties of the gold nanoparticle aggregates