Research papers of Tristram de Piro in mathematics, physics and chemistry;

                                             1. The Geometry of 1-Based Minimal Types  (dvi) or (pdf) (with Byunghan Kim) (published in Transactions of the American
                                                  Mathematical Society, vol 355, No. 10, (2003))
                                             2. Constructing the Hyperdefinable Group from the Group Configuration (dvi) or (pdf) (with Byunghan Kim and Jessica Young)
                                                 (published in Jornal of Mathematical Logic, 6, (2), (2006))
                                             3. Infinitesimals in a Recursively Enumerable Prime Model  (dvi) or (pdf)
                                             4. Zariski Structures and Algebraic Curvesj(dvi) or (pdf) or ohjgg
                                             5. A Non-Standard Bezout Theorem for Curves (dvi) or (pdf) (published in
Birational Geometry, Kaehler-Einstein Metrics
                                                 and Degenerations, Springer Proceedings in Mathematics & Statistics), (2005).                                           
                                             6. A Theory of Branches for Algebraic Curves  (dvi) or (pdf)  (older version)
                                             7. A Theory of Divisors for Algebraic Curves  (dvi) or (new version) (in progress)
                                             8. Some Geometry of Nodal Curves  (dvi)  or (pdf)
                                             9. Some Notes on Non-Standard Differentiation and Integration (dvi) or (pdf)
                                            10. An Interpretation of Newton's Work in Calculus (dvi) or (pdf) , with attached diagrams; (1, 2, 3, 4, 5, 6)
                                            11. A Theory of Duality for Algebraic Curves  (dvi) or (pdf)
                                            12. Flash Geometry of Algebraic Curves (dvi) or (pdf)  (A Proof of Severi's Conjecture)
                                            13. Severi's Conjecture and Single Node Curves (dvi) or (pdf)
                                            14. A Theory of Harmonic Variations (dvi) or (pdf) 
                                            15. The Geometry of Linear Regular Types (pdf)
                                            16. Some Notes on Quantifier Elimination and Model Completeness (dvi) or (pdf)
                                            17.  A Nonstandard Approach to the Theory of Algebraic Curves (pdf)
                                            18.  Applications of Nonstandard Analysis to Probability Theory (pdf)
                                            19.  A Simple Proof of the Fourier Inversion Theorem Using Nonstandard Analysis  (pdf), (submitted to the Journal of
                                                   Fourier Analysis and Applications), (2014)
                                            20. A Simple Proof of the Uniform Convergence of Fourier Series Using Nonstandard Analysis (pdf),
(submitted to the
                                                  Journal of Fourier Analysis and Applications), (2014)

                                            21.  A Simple Proof of the Martingale Representation Theorem using Nonstandard Analysis (pdf)
                                            22.  Solving the Heat Equation Using Nonstandard Analysis  (pdf) 0 1 2 3 4 5 6 7  (attached files for option
                                                   pricing; run spectrum.m in MATLAB) (in progress)
                                            23. Decay Rates for Cusp Functions (pdf)
                                            24. A Note on Inflexions of Curves (pdf) (in progress)
                                            25. Non Standard Analysis and Physics (pdf) (in progress)
                                            26. A Simple Proof of the Uniform Convergence of Fourier Series in Solutions to the Wave Equation (pdf)
                                            27. A Note on Convergence of Fourier Series (pdf)
                                            28. A Lemma on Polynomial Roots (pdf)
                                            29. An Inversion Theorem for Laplace Transforms
                                            30. A Topological Note (pdf, required in 14)
                                            31. Electron Bunching  (pdf) (in progress)
                                            32. Solving Schrodinger's Equation Using Nonstandard Analysis (pdf) (in progress)
                                            33. Bounding the Number of Maximal Torsion Cosets on Algebraic Varieties (pdf)
                                            34. Historical Research into Plucker and Laplace (pdf)
                                            35. Historical Research into Laplace (contd) (pdf)
                                            36. Historical Research into Newton (pdf)
                                            37. Historical Research into Piero della Francesca (pdf)
                                            38. A Nonstandard Approach to Solving N'th Order, Linear, Inhomogeneous ODE's with Smooth Function Coefficients (pdf)
                                            39. A Proof of the Ergodic Theorem using Nonstandard Analysis (pdf)
                                            40. Applications of Nonstandard Analysis to Riemann Sums (pdf)
                                            41. Riemann Sums for Returning Points (pdf) (in progress)
                                            42. Notes on the Weil Conjectures for Curves (pdf)
                                            43. A Nonstandard Approach to Equidistribution (pdf)
                                            44.  A Nonstandard Approach to Equidistribution in Ergodic Theory (pdf) (in progress)
                                            45. Results on the Nonstandard Laplacian (pdf, required in 32) (in progress)
                                            46. A Nonstandard Version of the Fokker-Planck Equation (pdf) (in progress)
                                            47. Nonstandard Martingales, Markov Chains and the Heat Equation (pdf)
                                            48. Nonstandard Methods for Solving the Heat Equation (pdf)  (in progress, but still have original version, submitted to JLA)
                                            49. Nonstandard Methods for Solving Schrodinger's Equation (pdf) (in progress)
                                            50. Oscillatory Integrals (pdf) (in progress)
                                            51  Simple Proofs of the Riemann-Lebesgue Lemmas using Nonstandard Analysis (pdf)
                                            52. Computing the Distribution of Velocities of Some Solutions to the Nonstandard Diffusion Equation (pdf) (in progress)
                                            53. Schrodinger's Equation and Related Charge Density (pdf)
                                            54. A Nonstandard Version of Dirichlet's Theorem (pdf)
                                            55. Antiderivatives of Inverse Functions (pdf)
                                            56. A Nonstandard Solution to the Wave Equation (pdf)
                                            57. A Nonstandard Poisson Summation Formula (pdf)
                                            58. Some Arguments for the Wave Equation in Quantum Theory (pdf), published in Open Journal of Mathematical Sciences
                                                  (Volume 5)), (2021) (I've added a few footnotes since the original publication and included a design to generate the
                                                   corresponding electromagnetic signal using a cavity magnetron. The dimensions of the larger spherical cavity have to be
                                                   calculated from the paper, (*)) The paper recovers the proportionality in the spacings of the spectral lines of the Balmer
                                                   series, which can be measured using a hydrogen discharge tube and a diffraction grating.  
                                            59. Some Arguments for the Wave Equation in Quantum Theory 2 (pdf), (published in Open Journal of Mathematical Sciences
                                                   (Volume 6)), (2022) (This provides the theoretical justification for the idea that if an electromagnetic system is in thermal
                                                   equilibrium and doesn't radiate at infinity for all obsevers, then the charge and current obeys the relations given in (58). The
                                                   result can probably be strengthened without the assumption of thermal equilibrium, a case which is considered in (72). It
                                                   seems clear that in the design of (58), as the charge is confined, the configuration shouldn't radiate at infinity, by Rutherford's
                                                   observation for atomic systems. The vanishing of the magnetic field in (58) can be checked by zero induction through a loop.
                                                   The signal can be detected by deflection of electrons; replacing the deflecting coils on a cathode ray tube with a wave guide.)
                                                   I've added a  lemma at the end and filled in a proof since the original publication.
                                            60. Equilibria in Electrochemistry and Maximal Rates of Reaction (pdf), (published in Open Journal of Mathematical Sciences
                                                  (Volume 7)), (2023) I've added a footnote since the original publication. (files for driving a motor to alter the potential
                                                  along a maximal reaction path in electrolysis of water 1 2 (matlab), 3 (Arduino)), Graphs showing the difference between
                                                  equilibrium and maximal reaction lines 1 2 3 4, we currently keep temperature and pressure in  electrolyzers fixed. I've
                                                  included the basic design and a further design more compatible with existing PEM technology. Water can be difficult to
                                                  pressurise, so currently the best strategy for a prototype is to alter the temperature of a 2% sulfuric acid or 1% sodium
                                                  bicarbonate solution, while altering the direct current through a rectifier using the motor, a potentiometer and an L298N
                                                  driver.You can check the potentials with a multimeter. You can get basic control of a motor using Arduino, with the
                                                  following circuit. Here is a table showing the different amounts of gas production from electrolysis, using varying temperature
                                                  and potential, these results are theoretically justified. Here is the motor in action.
                                            61. The Continuity Equation and Particle Motion (pdf) (in progress) (Attempting to support the intuitive idea that current is
                                                  determined by charge and the individual velocity of particles. The result is  used  in (63).) 
                                            62.  Electrolysis, Acids and pH (pdf) (considers Faraday's theory of electrolysis, definitions of acidity, relates pH with cell
                                                   potential in electrolysis of water and includes some notes about error terms used in (60))
                                            63. Some Arguments for the Wave Equation in Quantum Theory 3 (pdf),
(published in Open Journal of Mathematical Sciences),
                                                  (Volume 7), (2023) (I've included another design to generate the corresponding fundamental electromagnetic signal using an
                                                  antenna. The three fundamental modes and one reverse mode, as in the paper, could be simulated with four antennae or
                                                  combined directly in one. The approximation becomes better with the use of an n-gon and large n, instead of a pentagon,
                                                  (**) The field created by the inducting loop can be cancelled by winding the inner approximation like in one side of a transformer.
                                                  The signal can be detected by electromagnetic induction, using a circular loop for the antenna, attached to a radio receiver, 
                                                  or using an oscilloscope to measure the emf.)
                                            64. Some Arguments for the Wave Equation in Quantum Theory 4 (pdf) (in progress) (Still a collection of notes, the main aim
                                                  being to characterise electromagnetic fields which are non-radiating. This would find an application in optical amplifiers.)
                                            65. Non Oscillatory Functions and a Fourier Inversion Theorem for Functions of Very Moderate Decrease (pdf) (This improves
                                                  almost everywhere convergence to everywhere convergence in certain cases of Carlsen's result for functions in L^2(R); these
                                                  cases occurring naturally in Physics), submitted to the Journal of Fourier Analysis and Applications, (2023)
                                            66.  A Note on Polar and Caretesian Derivatives (pdf)
                                            67.  Rate Laws and Collision Theory (pdf) (in progress) (Technically a chemistry paper, but also applies to mathematical finance.
                                                   The idea is to use Brownian motion to determine the mean free path of particles involved in a simple two substance reaction,
                                                    with connections to the Arrhenius equation. Develops the theory of (48).)
                                            68.  Some Arguments for the Wave Equation in Quantum Theory 5: No Radiation of Light  (pdf) (in progress)
                                            69.  Some Arguments for the Wave Equation in Quantum Theory 6: Waves, Current and Charge (pdf)
                                            70.  Some Arguments for the Wave Equation in Quantum Theory 7: The Hyperbolic Method (pdf) (in progress) (Existence of the
                                                   electromagnetic fields defined by the "causal" Jefimenko equations) ((68,69,70) will eventually contain the material from (64)).
                                            71.  Some Results in Biochemistry and Biophysics (pdf) (This paper advoctes the use of sodium bicarbonate for the treatment of
                                                   both acidosis and associated heightened channel potential.)
                                            72.  Some Arguments for the Wave Equation in Quantum Theory 8 (pdf) (2023) (published in Open Journal of Mathematical Sciences),
                                                   (Volume 7) (2023) as "Some Arguments for the Wave Equation in Quantum Theory 4",developing the theory of thermal
                                                   equilibrium in electromagnetism)
                                            73.  Microwave Engineering (pdf) (2023) (implements the ideas from 58,59,61,63,64 in the context of microwave engineering)
                                            74.  Microwave Engineering 2 (pdf) (in progress) (2023)  (computes the impedance of surface current and  potential  in cavity
                                                   magnetrons, with the idea of tuning magnetrons to match the resonant and responsive modes in all directions, uses 73. Once
                                                   the surface impedance is known, one can build an RL or RC circuit to effectively receive the signal generated by the magnetron.)
                                            75.   Microwave Engineering 3 (pdf) (repeats the arguments of 73,74 for the spherical cavity magnetron, uses 58 to find the zero
                                                    current at boundary condition on the charge, with applications to confining plasmas in spheromaks.)
                                            76.   Functions Analytic at Infinity and Normality (pdf) (existence of Fourier transform for normal functions)
                                            77.   A Fourier Inversion Theorem for Normal Functions (pdf)
                                          

A book I wrote on nonstandard analysis.

A paper I wrote on aesthetics and geometry;

                                            78.  Cosmati Pavements: The Art of Geometry, (published in Bridges Leeuwarden Proceedings (2008))

 I've included a design to achieve igntion and control the firing rate wirelessly in a car or aircraft engine and a design to achieve controlled nuclear fusion. Both
ideas rely on the telecommunication
designs of (58) and (63), the fusion idea is also related to the papers (60) and (61). These connections might lead to further
integration between these fields, the automotive, aviation and fusion industries.



Some powerpoint presentations for the Newton Project at Culverhay;

                                             1. Integration
                                             2. The Fundamental Theorem of Calculus

                                         
  My PhD adviser, at M.I.T, was Professor Byunghan Kim, who is an expert in the area of simple theories. A simple theory is a structure in which certain
amalgamation properties hold. You can find, here, some geometric pictures of 1-amalgamation and 3-amalgamation (1>2>3), used, particularly, in the
second paper. My PhD thesis was mainly concerned with developing properties of minimal structures in such theories. A minimal structure is, very roughly
speaking, an abstract version of an algebraic curve. Since then, I have been a research fellow at Edinburgh University, The University of Camerino and
 The University of Exeter, specialising in the geometry of such curves. My masters thesis at Manchester University was on the Model Theory of Lie Algebras
(dvi).

The contents of this page are copyrighted.

(*)  An attempt to recreate this design could be dangerous, due to the increase in charge density and current inside the spherical cavity.Note that this design
is not the same as putting a hollow metallic sphere inside a microwave oven; the radiation caused by a cavity magnetron would induce a current through the
metal and cause arcing, due to the changing magnetic field. The radiation generated by this design should have a zero magnetic field, though it can still
accelerate charge due to the electric field. 


(**)  This design is probably safer. It differs from a conventional transmitting antenna in that the current and charge density, at a given time, changes along
different portions of the loop.