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
signal is generated inside the magnetron, with zero
current at the boundary. The response signal outside the magnetron is a
radiowave in the sense that it solves Maxwell's
equation in vacuum, but is
continuous with the matter wave inside. This is the work of 73,74,75.
The
dimensions of the
spherical cavity can be chosen to have a radius of about 0.05m, and
the signal outside the magnetron can be picked up
by an RLC circuit and then amplified. The
signal will resonate with radii of other dimensions, which can be
calculated
using some code to find the zeros of a Bessel
function.) 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,
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. (There is a mistake
in the paper as it currently
stands. In electrolysis, the amount of substance produced should only
depend on the endpoint of the reaction path and the
initial amounts, as no substance is formed in a loop. The amount of
energy expended in producing a given amount of substance
however does depend on the path. The reaction rate formula I gave is
wrong, and I'm currently correcting it) (in progress)
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). It could also be used to construct a
diffusion for a pair of potentials (V,A) in the Lorentz gauge
corresponding to given electric and magnetic fields (E,B) in vacuum,
noting that (V/c^2,A) satisfies the continuity equation, and supporting
the photon model of light.)
62. Electrolysis, Acids and pH (pdf)
(in progress) (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)). The paper supports the
idea of a method of creating
plasmas with a high concentration of H+ ions
through electrolysis of water, converting into steam,
by adapting the designs in 60. This is a possible prototype. The plasma can be
contained in a sphere of appropriate radius
in order to resonate with the
design in 58. The paper gives an approximate estimate for the charging
time of a hydrogen
fuel cell or flow battery, fitting into a car, if we alter the voltage
according to the lines in (60) while cooling the cell to a room
temperature of 20 degrees centigrade, with initial temperatures of
50,75,100 degrees centigrade, as 5 minutes, 2 minutes
40 seconds and 2 minutes. It takes abut 30 hours to recharge a
conventional electric car battery.
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)
78. The Relativity of Rotating Frames
(pdf) (in progress)
A
book I wrote on nonstandard
analysis.
A paper I wrote on aesthetics and geometry;
79. 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.