Post by Ross FinlaysonPost by Ross FinlaysonPost by Ross FinlaysonPost by Ross FinlaysonPodcasts: Ross Finlayson's study
(Not audio only, but all audio.)
About Philosophy
http://youtu.be/6BNDx-FUwKM
http://youtu.be/C0jIsXfuUKM
http://youtu.be/GgoRuwa2Zcs
http://youtu.be/uLoEv9p16iw
About Mathematics
Standard and Non-Standard Calculus
http://youtu.be/sRr1gBLEmo0
http://youtu.be/HsGOZp7jrEY
Differential Calculus
http://youtu.be/RtdXHM6k07Y
http://youtu.be/dnClOA-lp20
http://youtu.be/7g3UAFxr4S8
Analysis and Methods
http://youtu.be/njSqmjkj0gQ
http://youtu.be/mgreCqD2gqo
http://youtu.be/U7KbayajRaA
About Physics
Non-Linear Field and Wave Physics
http://youtu.be/v82ZeL_-hy0
http://youtu.be/5_vmEhWl9oE
http://youtu.be/VtWy8uwvA3g
http://youtu.be/UkZGZ6FRpS0
http://youtu.be/GIMKbXBEQno
http://youtu.be/3iB85GtOduc
http://youtu.be/7PY3QK8pyMY
http://youtu.be/h5fp3De0SfI
http://youtu.be/dz585sC5dKQ
http://youtu.be/gMBcectYDws
http://youtu.be/fK8KDlwobtI
http://youtu.be/3LnC4srnwtY
http://youtu.be/f3el3DayFSU
http://youtu.be/xIcZim7Y53U
Quantum Mechanics
http://youtu.be/tynLKPjpjjs
http://youtu.be/zs_mr-_VlXY
http://youtu.be/20IVJlUpbHo
http://youtu.be/gxqrBM_s_EA
http://youtu.be/lQxyQboZw2k
http://youtu.be/GL1yqoRf6HM
http://youtu.be/7rVEBVT1kwE
http://youtu.be/MFwNvNxjwxI
http://youtu.be/PnidFex3Nm4
http://youtu.be/1ml1SSifcSU
Relativity
http://youtu.be/qHVOLO1ryGQ
First recorded: http://youtu.be/axl4czl5Bus
Reading from Einstein: mass, charge, energy, and fundaments
http://youtu.be/LRD4Knsoycc
Einstein and theory, decomposition of elements, total field theory, local and global,
GR before SR, SR and locality, GR and non-locality, energy and mass-energy equivalence,
Einstein's formula, numerical derivations of series and approximations in truncating terms,
dimensional analysis in the orders of terms in physical quantities, thermodynamic energy
and kinetics, nuclear reactions, entropy and organization, open and closed, restitution/oscillation
and dissipation/attenuation, Olbers' paradox and cosmic background, reactions and transitions,
separations and decompositions of theories into fundamental theories and complementary
theories of fundamental fields, Maxwell and classical fields, the lettered fields of
electromagnetism, factors of the Parameterized-Post-Newtonian and Modified Newtonian,
theories and the empirical and fundamentally theoretical, theories of fundamental objects
and higher level organization, gravity and space-time, gravity in theory, gravity in relativity,
gravity and the centrifugal, Einstein defends relativity, Einstein defends Newton.
Reading from Einstein: model of a physicist's philosophy of science
http://youtu.be/8DlW0kkDcCI
Einstein and Oppenheimer, Einstein the image, reason and sense, ideals and explanations,
philosophical and physicists' principles, rules of theories, laws of nature, object and context,
sense and object-sense, pure theory, about classical mechanics and motion, field theory and
total field theory, de Broglie and Schroedinger, Einstein about Born and the statistical ensemble,
continuity laws, desiderata of future theory.
Reading from Einstein: tea on the train, the train and the "time", space and "space"
http://youtu.be/ZpWi_nRBmWY
Metaphor, the thought experiment, gedanken, Einstein's train gedanken, tea on the train,
maximums and boundaries, Mach and the acoustic, relativistic dynamics, dynamics and the
cosmic observatory, statics and the terrestrial frame, energy and mass and charge, rotational
kinematics, Des Cartes and Des Cartes' Laplacian Euclidean, the Laplacian and harmonic function
theory vis-a-vis theories of potential, Einstein's model of a working physicist, Einstein's physicist
as a sensitive man, local and global definitions of time, space and "quasi-rigid", theories of
space contraction, Einstein's definitions of local time and global time and "the time",
white holes as kinematic singularities, Einstein on the classical motion in classical space
and classical time, Einstein's strategy.
Reading from Einstein: classical mechanics and continuum mechanics
http://youtu.be/P08kIRUSkhE
Einstein's "Out of My Later Years", Einstein's theories, the classical mechanics and classical force,
surface mechanics, material points, atomism, success of theories, Newton and Galileo, continuous
media, thermodynamics and the success of discretization, potential theories, theories arising
from natural deduction, Mach and Mill, configuration space and paucity of terms, surface domains
and the applied, the empirical, mathematics and models of material points.
Reading from Einstein: fields, and electromagnetism
1 of 2 http://youtu.be/yJuFhlclPP0
2 of 2 http://youtu.be/FonnPLtX51Y
Fields in mathematics, arithmetic and field operations, alternative derivations of arithmetic,
complementary duals, fields in physics, space-time and space and time, a ray of time or the
origin of time, fields in academia, Maxwell and Faraday, classical fields and potential fields,
classical fields and interfaces, surfaces and material points, intuition and superclassical fields,
complementary duals, magnetism, Meissner, original analysis, Laplace and the differential,
Einstein and time ordering, continuum mechanics, deconstructive accounts of field fundamentals.
Reading from Einstein: relativity and gravitational field
http://youtu.be/yutLelN_t_Y
Cosmology, sky survey, Doppler, standard candles, hydrogen spectroscopy,
LaGrange and Laplace, classical connections, Fitzgerald and Lorentz,
complementary duals, computing the geodesy, "total differential equations",
"in the space", dynamical models and time, linear and non-linear and singular
and non-singular duals, gravitational field equations, Riemannian metric and
covariance, bases of analytical freedom, the quasi-Euclidean, potential theory,
central symmetries, deconstructive/reconstructive accounts.
Reading from Einstein: the field, the time, and quantum probability
http://youtu.be/btAiSlW1eX4
Field equations, differential singularities, modern field theory, quantum theory,
probabilistic quantum theory, Heisenberg and Dirac and Schroedinger and de Broglie
and Bohm, discretization, Planck and running constants, geometry and a deconstructive
account of non-standard analysis, infinities and infinitesimals in mathematics and physics,
quantum spin, particle/wave duality, real wave function, locality and non-locality,
differential equations and the time.
Reading from Einstein: continuity, instinct and intuition
http://youtu.be/C4eTUKaFE7U
Field mechanics, fields and forces, central symmetries, nuclear forces,
relativistic effect, Loch Ness monster, space contraction, ballet dance,
relativistic nanogyroscopes, quantum fields, quantum interpretations,
extra-local action, causality, requirements of theory, definition of a
model physicist, continuity and atomism, a Planck square, Heisenberg
uncertainty and skew, nuclear theory, light and transmutation,
mass and charge and light and matter.
Reading from Einstein: field theory and continuity, mathematical
1 of 2 http://youtu.be/4p3LJEBS68s
2 of 2 http://youtu.be/A8zX5PukCW4
Antique physics, theoretical physics, Anaximander, Heraclitus, Leucippus, Democritus,
Zeno and Aristotle, concluding "Out of My Later Years" Chapter 13: Physics and Reality,
central symmetries and bridges, central symmetries and Einstein's teacup, the rotational
symmetry's "cube wall", Einstein summarizes relativity theory in field theory in physics
and begifts his model physicist.
Reading from Einstein: philosophy and foundations
1 of 2 http://youtu.be/OrK9KzsK9po
2 of 2 http://youtu.be/mbKTJa6SBJM
Central symmetries and relativistic dynamics, history of modern atomism
and chemistry, Dalton's law of constant proportions, running constants,
atomic chemistry and molecular chemistry, fundamentals and foundations,
fundamentals of foundations, philosophy of foundations, rhetoric.
Reading from Einstein: state of the field
http://youtu.be/BpGYn2n1GQk
Atomism, law of constant proportions, object-sense, running constants,
atomic theory and nuclear theory, ultraviolet catastrophe, mass and charge
and light and colour, general relativity and quantum mechanics, field theory
and the statistical ensemble, universals in Einstein's theory, reconciling field
theory and discretization, functional freedom and degrees of freedom,
infrared catastrophe, 20'th century foundations and 21'st century foundations.
Reading from Einstein: language and science, Einstein's word
http://youtu.be/xxqetiIzanw
Language and communication, language and the space of words, the scientific method
and scientific discourse, generalization and universals, science and ethics, ethics and
morals, science and truth, logic and truth, mass-energy equivalency, Einstein's second
formula of mass-energy equivalence in the centrally symmetrical or rotational.
This is about the last of the reading from Einstein's Out of My Later Years, as it's
the summary of chapters in the middle that convey his scientific message.
That it ends with a particular definition of the mass-energy equivalency that
essentially reflects the setup of the centrally symmetrical i.e. the rotational,
really makes clear that a theory of space-contraction is compatible with
Einstein's theory, final, of relativity.
Analysis and Methods: arithmetization and models
http://youtu.be/ZpWi_nRBmWY
Arithmetic, decomposition of arithmetic, function theory and types, first-order and infinitary
operations, models of arithmetic, bounds and the effective, powers and roots, algebra, real-valued
spaces, complex arithmetic, conjugates and reflections, nine-point theorem, number theory,
congruences, casting out 9's and digit summation congruence, quadratic sieve, numerical models
of data structures, arithmetizations and algebraizations and geometrizations.
Descriptive differential dynamics: vector fields and bundles, geometry and analysis
http://youtu.be/NJ1D2pgTZb0
Methods and analysis, the geometer and the analyst, foundations and Poincare,
Poincare and Dirichlet, Hermann's mathematical physics, continuous domains,
non-linear interactions, principle of minimal interactions, non-linear Lagrangians
and linear Laplacians, a derivation of isomorphisms in duals, direct sums,
Euler-Lagrange differential operators.
Descriptive differential dynamics: a heat equation
http://youtu.be/1fUkrN70BK0
Fourier analysis, Fourier and the heat problem, 1'st and 2'nd law thermodynamics,
restitution/oscillation and dissipation/attenuation, differential equations, integral
equations, setup/ansaetze of capacity and conductivity, boundary value problems,
the differential tool-kit grab-bag, implicits and functions of functions, separation
of variables, quantities, the centrality of the exponential function in differential
analysis, a posteriori checks, usual constants in systems of differential equations
in Fourier analysis.
Descriptive differential dynamics: Fourier series and Cauchy and Riemann's definite integral
http://youtu.be/ftP0fT83n_8
Linearity and superposition, linearity and vector spaces, infinite series and infinite sums,
uniqueness of the complete ordered field, non-standard field operations of (-1,1), shrinking
the quadrant to the unit square, bounded regions and infinite series, orthogonal functions
and coefficients in analytical character, shrinking 1/x and summing into n/x, quadrature and
periodic functions, even and odd functions, Fourier and Darboux and Cauchy and Riemann
and Dirichlet, the definite integral.
Descriptive differential dynamics: x = y, y = x, x = 1/y, y = 1/x
1 of 2 http://youtu.be/WayejS2X4nk
2 of 2 http://youtu.be/w-6Cdm9SBvQ
Numbers and semiotics, perspective and projective infinite, the x-y coordinate quadrants,
positive numbers the first quadrant, x = y, functions and pairs of functions symmetrical
about x = y, 3/4 pi rotation, 1/x and sawtooth wave, shrinking the quadrant to the unit square,
plane curves, tangents and normals, sin and cos and root two over two and x = y, trajectories,
orthogonal functions, scalar infinite, infinite series.
Descriptive differential dynamics: complementary dual plane curves and diff. E.Q.'s
http://youtu.be/O063JzPH2C0
Plane curves, about "an elementary treatise on differential equations", differential analysis
before and after Bourbaki, orbital manifolds and trajectory manifolds, standard and non-standard
infinite series and closed forms, the derivative and the differential, rotations, positive numbers
and x = y and y = x, justification in quantities, dimensional analysis and quantititized analysis,
"off by 1" and quantities, the approach of plane-curve level-lines and their differential function
and differential functions and their plane-curve level-lines their solutions, orthogonal functions
and trajectories, features of parabolas, complementary duals of isoclines, gradient.
Descriptive differential dynamics: fixed-point and the semi-dimensional
1 of 2 http://youtu.be/8bkTr58108o
2 of 2 http://youtu.be/rQCKS3mexjo
Fixed-point, the origin, definitions over time, modern definition, differential equations / integral equations, elementary theories of mathematical objects, models of integers, models of real numbers, implicits, definitions of curves, restrictions of comprehension and the insensate, restrictions of comprehension and constraints, continuity, continuous motion and uniform continuous motion, definition of tangent and normal, identity as a dimension, the "semi-dimensional", implicits as constants, implicits as constraints, Clairaut's theorem, the identity's integral.
Descriptive differential dynamics: the co-semi-dimensional identity constraint
http://youtu.be/on7RFb_yTxU
Zero and indeterminate forms, triviality and vacuity and discreteness and continuity,
integral equations and prederivatives, coordinates and spaces in the identity dimension,
limit theorems and completeness, nowhere and the middle of nowhere, zero and infinity,
Clairaut and d'Alembert and Lagrange, y = -x + 2y and x = y, singular solutions and
indeterminate forms, roots of the identity function and roots of 0,
mathematical conscientiousness and rigorous formalism.
Descriptive differential dynamics: the co-semi-dimensional identity constraint
http://youtu.be/VnMxLKy4Qkw
Zero and indeterminate forms, triviality and vacuity and discreteness and continuity,
integral equations and prederivatives, coordinates and spaces in the identity dimension,
limit theorems and completeness, nowhere and the middle of nowhere, zero and infinity,
Clairaut and d'Alembert and Lagrange, y = -x + 2y and x = y, singular solutions and
indeterminate forms, roots of the identity function and roots of 0,
mathematical conscientiousness and rigorous formalism.
Descriptive differential dynamics: convergence and continuity, definition and differentiation
http://youtu.be/jUjiY4WlP3U
Systems of numbers and coordinates, the identity dimension, the linear curriculum, proofs and conjectures,
convergence criteria, arithmetic and algebra and geometry and number theory and function theory and
type theory in spaces and topoi, sheaves/symmetries/submersions and sub-fields of analysis, overlapping
definition and disambiguation, definitions and derivations or axioms and theorems, the revisitation of
derivation, mathematical conscientiousness and hypocrisy, mathematical historicism, definitions of
continuity, mathematics the field, mathematical objects and a theory of mathematics, miseries of induction
and complementary duals in analytical deduction, logical paradoxes and non-logical paradoxes.
Descriptive differential dynamics: identity dimension and derivative-stopping
http://youtu.be/-1xmnwDpj_g
Arithmetic and algebra and analysis, operator calculus and the working objects,
identity dimension and quadrature, d'Alembert, 1/x and log x and x(log x-1),
zero-to-velocity and stopping distance as inverses, context of "the envelope"
and the identity dimension, "cube wall", "quadrature wall" and x^2 + x = (a^2 - C^2)/2,
constants and parameters and partials and fullers, constants and abscissae and
ordinates, homogeneous equations and the identity dimension.
Kind of looking at 1887, https://www.jstor.org/stable/pdf/1967647.pdf
1895: https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1895.0014
1903: https://books.google.com/books?id=PB0LAAAAYAAJ&lpg=PA41&ots=qobplVjlvn&dq=%22integral%20equation%22%20envelope%20singular&pg=PR3#v=onepage&q&f=false
1903: Singular points of functions which satisfy partial differential equations of the elliptic type, Boecher
2019: https://www.routledge.com/Singular-Differential-Equations-and-Special-Functions/Campos/p/book/9780367137236
Since starting to survey differential equations first there was getting into ODE's and the usual
or what was from the linear undergraduate curriculum, then about methods of symmetries and
about topology then into the "differential geometry", which now I call "numerical methods in
differential geometry of a subset of continuous functions" , then calling these "sheaves, symmetries,
and submersions".
I was pretty surprised to learn that the usual notion of "differential equations, and their solutions",
was sort of opposite the more old-fashioned "plane curves, and their integral equations".
Then I got into that the identity function or x = y is sort of a dimension itself in the differential,
and basically looking only at curves in the first quadrant of the xy plane and all in positive numbers,
basically for a deconstructive account that then can build out into negative numbers instead of
having to build out into complex numbers.
It's sort of the locus of all the envelopes of the trivial solutions or the 0 solution, so it ends up
looking sort of like the middle of things like the linear fractional equation, or about these days
the linear fractional transform, and for plugging it into Clairaut and d'Alembert and so on thus far,
about how it's trivial but also it's in the middle of a lot of things in the limit of things.
This gets a lot into the equations where "in this singular section there's that x = y so that lots
of the equations can be solved in terms of one variable, because, it's a condition they're equal".
There's a sort of integration defined about functions symmetric about the, "identity dimension".
Then the reciprocal function or 1/x a hyperbola is also examined, kind of about the cycloids and
their locus, but mostly about a notion of taking the first quadrant and shrinking it to a unit square,
then making some non-standard orthogonal functions about usual approaches afforded those,
a sort of "hat series".
So, my latest idees fixes include of course the whole "continuous domains" bit after my modern foundations,
then this hat series, getting into the identity dimension, integration in the identity dimension, then after
something like "roots of zero" and "roots of the identity dimension" (or, the singular in the differential),
about the identity dimension and how it's got its solutions up for a deconstructive account of the
trivial or 0 solution and making a primary sub-field of differential analysis.
Of course I'm mostly interested in that for complementary duals in harmonic functions,
about theories of potential.
Descriptive differential dynamics: addition formulae in symmetrical differential operators
http://youtu.be/BDiDhiRbAjY
Analysis and differential analysis, the identity dimension and 1/x, indeterminate and singular forms,
fin de siecle and Y2K, differential analysis as a field, descriptive differential dynamics, contrivances,
addition formulas (formulae), operator calculus and orders and ranks in powers, differential operators,
"power-taking addition formulae", "anti-operators", 1/x to ln x and x to 0 as symmetrical anti-operators,
hypergeometric functions, series and combined series, series in growing and vanishing terms, parameters,
distributions and shape and scale parameters, identity dimension and trivial solutions, identity and zero.
Descriptive differential dynamics: identity dimension metric, 0 1 infinity
http://youtu.be/XqbjVnx1TTg
Original analysis, harmonic functions and the upper half-plane, the positive or first quadrant
and its octants as half-plane, coordinate spaces, hypergeometric regular singular points,
Cantor space and square Cantor space, factorial, Legendre polynomials, Rodrigues formula,
Rolle's theorem, power series and increasing denominators, power series of [0,1], power series
of orthogonal functions, closed forms and reduction, substitution and transformation of
variables and coordinates, identity dimension coordinates, identity dimension metric.
Descriptive differential dynamics: classical and extra-classical orthogonal functions
http://youtu.be/LYb8YT-SjFs
Classical orthogonal polynomials, Legendre to Jacobi to Askey, geometry and topology,
dynamics of topology, dynamics of discretization and dynamics of the continuum limit,
induction and deduction in points and spaces, descriptive differential dynamics,
descriptive set theory and axiomatic set theory, function theory and the Cartesian,
non-standard functions and non-Cartesian functions, the exponential and its own anti-derivative,
functions that are their own anti-derivative, probability theory and the uniqueness of distribution
functions, the quadratic sieve and trilateralometry or n-gonometry, multivariate analysis and
higher-order orthogonal functions.
Descriptive differential dynamics: Fourier analysis derivation
1 of 2 http://youtu.be/CUEp6AKqaWY
2 of 2 http://youtu.be/FyvtpN6xmdg
Coordinate settings and transforms, identity dimension, continuous and piece-wise continuous
integrability, Riemann-Lebesgue theorem convergence, "hyperbola hat" series, equations of
wave motion and periodic functions, the semi-circular wave and n-gonometry, Dirichlet-Riemann
convergence, models of Fourier analysis and implementations of integration.
Descriptive differential dynamics: Fourier-style analysis and inch-worm functions
http://youtu.be/nvmrlCw6iw4
Deconstructive Fourier-style analysis, convergence criteria, Lipschitz condition,
windowing and boxing, Dirichlet kernel, periodicity and modularity,
large-angle approximation, analog and digital waveforms,
inch-worm functions and n-gonometry,
Michaelson's harmonic analyzer, Gibbs phenomena,
hysteresis and overshoot and ringing,
complementary sine sinc(x) = sin(x)/x,
approximation and bounds in error terms.
I picked up a copy of Lefschetz "Selected Papers", he's my new favorite algebraic geometer,
and as he puts it, for "algebraic GEOMETRY".
Descriptive differential dynamics: fixed-point and the singular in spaces and varieties
1 of 2 http://youtu.be/tr96eumt8c8
2 of 2 http://youtu.be/ziloVYXbU3E
Spaces and points, fixed-point, space-time and the space-time spider, book-keeping
in dimensions, differential equations, definitions and dynamics, Lefschetz on algebraic
geometry, infinity and fixed-point, schools of names, algebraic geometry, Poincare,
reading in English and French, f(x,y) = 0 and x=y and x-y, Analysis Situs, intersection
numbers, translations and models and model theory, deconstructivist and reconstructivist
approach, fixed-point and branches.
Descriptive differential dynamics: effective and virtual, continua and individua
http://youtu.be/yiBHjB2pVso
Algebraic geometry the geometer's or algebraicist's, Banach-Tarski and its original
derivation, arithmetic and simpler arithmetics, geometry and algebra, continua and
individua, Vitali and Hausdorff, Vitali and the non-measurable, sides of points,
Lefschetz and Picard and Savari, effective and virtual, arithmetization and addition
formulae, the analytic, the Fourier analysis, convergence and completeness,
small-angle and large-angle and windows and boxing, Fejer sums, Fejer kernel,
rules and varieties.
Descriptive differential dynamics: completeness in continuity and constructions
http://youtu.be/D3_WBmAfDpQ
Features of differential analysis, curves and integral equations and differential equations
and solutions, completeness, completeness and continuity, completeness and constructions,
circle as infinitely-sided polygon and infinitely-sided polygon as circle, complementary duals
and meeting in the middle, symmetry and invariants and conservation laws, continuity as
central in mathematical objects, algebraic geometry and analysis situs, Fourier-style analysis,
existence and uniqueness, convergence and boundaries, interchangeability of components
of derivations, combinations of transforms in completeness, dynamical modeling and the
descriptive, features of linear and non-linear analysis, impulse and rest, the mechanics of
vibrating strings, continuity and completeness or "Zeno's mirror", Zeno's uniform motion
and acceleration and deceleration, meeting and parting, identity dimension.
Not everything is
cut and dried,
not everything is
black and white,
"not nothing is not everything": "is".
The complementary dual, is a higher level mental construct,
than the opposite.
The dialectic isn't for asymmetry, though it's applied that way,
it's for symmetry.
The difference between "versus" and "vis-a-vis",
is one has a loser and the other's not a game.
The inductive impasse, reflects that there are cases for induction
that never complete, but, in the continuous and in time, they do.
It's the continuous and the uniform in time, infinitely-divisible,
that's about the simplest prototype of a model of change, in
a model of state.
The, inductive impasse either way, is simply reflected in what
are points in a line, how they are drawn, and, points on a line,
and how they divide, what is the drawn-out and what is the divided,
making line-continuity first then besides signal-continuity,
for field-continuity.
The, "meeting in the middle", the "middle of nowhere",
is the center of the square of opposition and the dialectic,
in the complementary duals, about for example the point,
the local, the global, and the total.
The anaphora and cataphora, nouns, the synthetic and analytic, adjectives,
the continuous and discrete, complements, here what's generally put
first is the universals, that work out same as "void".
So, the context of the oscillating and the attenuative, for
the restitutive and dissipating, is that the oscillating radiates
while the attenuative has a floor, in a "default" or "ground" model.
There are others, where the tendencies and propensities reflects
actions or states. This is used to define the thermodynamic and
anything else which is open in physics.
The kinetic, and laws of motion, here is addressed with a dialectic,
about this, for example "Zeno's starter". This is about "what is the
impulse" or the singularity of beginnings, of change, which are
first modeled as perfect inelastic collisions, but physics is an open
system. So anyways this idea of derivative stopping is about powers
and inverse powers and their derivatives, and for a model of addition
formulae or index formulae, about an operator calculus of higher,
and, lower, orders of acceleration, with respect to displacement
and rest, with respect to time (singular). The idea is to work up
that the C^\infty functions who eventually in some order have
a derivative that's zero, that there's a family of functions whose
integrals eventually reflect a constant, these being symmetrical
in positive and negative powers, for "Newton's Zero-eth Laws".