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Measuring: from geometry to spectra to probability

The descriptions of modern physics (2012) are floating on a language which virtually has no valid structure. The experts are authorities of complicated vagueness. This is a design of (or strategy used by) the propaganda system.
In modern physics entropy is modeled as an arbitrary way of partitioning an energy representation space (in the context of function spaces and their dual function spaces) with less motivation than Bohr's H-atom had, ie energy density is based only on the assumption that a discrete "energy space" should be partitioned in a way which is more consistently related to its continuous structure (whereas Bohr used "continuous well defined geometric structure" to construct a discrete structure).
The spectra of atoms is vaguely based on 1/r terms, and a spherical symmetry which only exists for the two-body atoms (in 3-space), ie only for the H-atom.
The model for a general nucleus is a box-potential energy. Particle-physics is supposed to describe the properties of the general nuclei, but particle-physics is too complicated to ever be able to describe the general nucleus.
For such stable systems as atoms and nuclei, why not consider (why not speculate about) a model of these systems in terms of stable, linear, geometrically separable, controllable math structures (since these fundamental physical structures are stable and (apparently) they are contained within some highly controllable context, of a measurable existence [since they form into stable systems with quite definitive (discrete) measurable properties])?
But for the authorities of complicated vagueness this is not an option (ie simple is not an option).
The stable many-body solar system has only an "unworkable non-linear model," and this is the only type of model which the experts are allowed to consider.
Because the experts are only allowed to consider complicated math patterns for descriptions of stable physical systems, but in this context, this causes the idea of "what is being measured" to not be clear.
For example, the spectra of stable quantum systems depends on the solutions to the radial equation (in the separable energy operator for atoms, really only applicable to the 2-body H-atom) which has a 1/r term in it, and in this context, only ever more complicated math-methods for solution are considered, ie no physical constructs are considered (it is an absolute law of both quantum and particle physics that measuring at this small size scale is not allowed, yet the quantum system's properties are measurable, and they are quite stable and quite precise in their observed values). Thus, (in the realm of abstract complications, to which the professional physicist must adhere) the measuring process (or the context of measuring) is interpreted to be about finding mathematical abstractness which will yield a math solution. That is, there is no clear way in which to choose a discrete spectral representation of the singular (spectral) term 1/r, so to determine a quantum system's spectra one must consider patterns which are very complicated.
Whereas 1/r is related in a discrete manner to the sectional curvature properties of the very stable discrete hyperbolic shapes.
It needs to be explicitly stated that the authority of "science and math" is based on descriptive structures which are: abstract, overly complicated, and built on set structures which are not valid. Within this intellectual realm of abstract complicatedness, math patterns are used to describe the measured properties which are being observed, but these math structures and physical interpretations (done within a context of unnecessary abstractions) do not even come close to describing the observed patterns.
The general systems of: nuclei, atoms, molecules, crystals, and the solar system, do not have adequate descriptions (neither sufficiently accurate [to a valid level of precision, which should be achieved for an overwhelming majority of these general spectral-orbital systems], nor are they useful descriptions in the practical sense [what is being measured is not all that clear (the point of this paragraph), and the descriptions are probabilistic in relation to systems which have relatively few components, and they are not geometric descriptions]).


Measured quantities in the physical realm, ie measurements done within metric-invariant coordinate metric-spaces, seem to be of three different types:
(1) measures of geometric properties (including motions),
(2) measures of spectral or oscillating properties defined in a context of a wave-phase (xp-Et), or (xk-ft), for different wave-functions, Aexp[i((xp-Et)] (see below for definitions), placed within the context of an eigenvalue differential equation, where a system's energy and its oscillations are to be determined by the eigenvalues, or the spectra, of the system, an energy which one is supposed to expect is to be related to the frequencies of the waves and the geometries and material interactions of the waves which, together, compose the system.
In fact, the determination of a general quantum system's spectra is best related to simple discrete shapes in a many-dimensional context (a context which the arrogant, and apparently ignorant, math community ignores).
(3) determination of random events for elementary event spaces which are composed of a finite set of stable, well defined (easily distinguishable) elementary events, ie counting distinguishable random events.

Note: Non-linear "geometric" methods are simply not solvable, and they only have limited validity in the context of feedback systems, where their critical points and subsequent limit cycle properties can sometimes be used in a context of a feedback-system.
It is not even clear if the deformation of general metric-functions has a valid relation to abstract, ridged mathematical geometric properties, in regard to Perelman's deformations of non-linear geometries into the simple, stable shapes which were proposed by Thurston.

That is, it might be better to assume linear, metric-invariance, and geometrically separable shapes on spaces which have non-positive constant curvature with metric-functions which only have constant coefficients as a basis for mathematical stability and quantitative consistency, and to ignore the vast realms of unstable math structures in regard to too great of a form of math generality, and that the set structure of math (also) needs to not be "too big," also for the purpose of ensuring a stable language for math (so its words mean what they are "supposed to mean."

This very simple paper expresses a much better math context within which to consider the stable spectral structures of general quantum systems, and as to from whence these stable spectral structures come.

This paper is mostly about the math curriculum for 7 and 8 year olds, and the relation that math has to measuring, operators, numbers, functions, linearity, equations, variables, inverses, and stable shapes; so that the descriptive context stays stable and simple, ie the operator structure is diagonal.
However, E Witten, S Hawking, W Thurston, G Perelman, R Penrose, S Weinberg, M Gell-Mann, M Freedman, etc may all profit from going back to the third grade, and follow the simple geometric patterns described by D Coxeter in regard to discrete hyperbolic shapes, placed into the following simple math context.

Simple math context to both accurately and geometrically describe existence (materialism is a subset):

The simple structure [but, apparently, a new context] to be used for the description of (material) existence, is that there are layers of metric-spaces (with various metric-function signatures) which increase by one-dimension (eg in the spatial subspace) with each layer.
That these metric-spaces (mostly hyperbolic and Euclidean, or of other signatures) possess discrete shapes, which, in turn, can contain (the) lower dimensional discrete shapes. The lower dimensional discrete shapes are models of material, but they are also closed metric-spaces.
The models of material interactions are to be based on both the shape of "discrete Euclidean shapes" (of the dimension of the material containing metric-space) which determine the interaction's 2-form force-fields, and the geometric relation that these 2-forms have with the Euclidean fiber group (or it can be placed in either a spin context, or a symplectic context, or a unitary context), so that the interaction is effectively a connection (general derivative, see below), associated to this geometric interaction structure.
The geometric structure of the material interactions both hide the higher dimensions from the lower dimensions (especially if one focuses on the material contained in a particular metric-space one cannot see the higher dimensions, but the material in the higher dimensional spaces will be of a different size than is the size of material within any given metric-space dimensional level, ie it will be the size of the shape of the material containing metric-space) and for small (atomic) material components this interaction structure (which involves both action-at-a-distance and time intervals determined by the rates of the spin-rotations of metric-space states [see below]) creates Brownian motion which is, essentially, the same (or has the same effect) as quantum randomness, which is a (quantum) randomness of small components which fill the space between macroscopic geometry (of classical physics).
The allowed spectral properties which stable discrete hyperbolic shapes can possess is related to the spectral structures (or contents) of the over-all high dimension 11-dimensional hyperbolic metric-space, where the spectral content (or over-all spectral set for all existing metric-space [or material] shapes within the 11-dimension containing hyperbolic metric-space) is related to all the stable shapes which the lower dimensional hyperbolic metric-spaces possess (where the metric-spaces possess very stable discrete hyperbolic shapes, which in turn possess definitive spectral properties), and it is such a large, but finite, set of spectra which the 11-dimensional containing space actually contains, and this spectral set determines (all) the spectral properties... , which all closed metric-spaces, ie all material-spectral-orbital systems, which exist within this 11-dimensional containing space... , can possess.
In this context one counts, "11 chose n," for n<11, to determine the number of independent metric-spaces of dimension-n. However, the spectra also depends on what any particular dimensional level metric-space contains within itself as material components, which also possess spectral properties.
Except for the material interactions, these math structures... , which are to be used for the description of (physical) existence... , are all linear, metric-invariant, "parallelizable and orthogonal," math structures, and these math patterns are (all) closed spaces of non-positive constant curvature (both bounded and unbounded) and they are simple geometric spaces which possess metric-functions that have constant coefficients, ie the main math structures are linear and diagonal, and these simple math patterns are associated to "cubical" simplexes of various dimensions, with various values for their genus (ie number of holes which these discrete hyperbolic shapes possess, ie the number of "cubes" which are attached at vertices to determine the discrete hyperbolic shapes) that is the manifold structure is mostly a global coordinate structure which is linear, geometrically separable, and metric-invariant (the distinguished point, associated to all the vertices of all the cubes, may have some special attributes associated to it, where these special attributes of shape (or angle) exist at the vertices between adjacent cubes (of the discrete shape's fundamental domain).
Furthermore, each metric-space possess a pair of opposite metric-space states, about which spin-rotations of metric-space states are defined. These metric-space states are best placed in complex coordinates, where the states are distinguished by sets of bounded regions in the R(n) and iR(n) subsets of C(n), and thus the fiber groups of the complex coordinates would change from isometry (or spin) Lie groups (for real metric-spaces), to unitary Lie groups.
A Euclidean space's metric-space physical properties are associated to the inertial properties of spatial displacement symmetries, where the two metric-space states are stationary star frames vs. rotating star frames.
A hyperbolic space's physical properties are associated to energy and the positive and negative direction for time flow identifies the two metric-space states for hyperbolic space, etc where further metric-spaces are identified for various new metric-function signatures for the higher dimensional spaces, ie R(s,t) where s+t=n and s is the dimension of the metric-space's spatial subspace and t is the dimension of the temporal subspace, where these new metric-function signatures in the higher dimensions are related to new properties which discrete shapes might possess in higher dimensions.
Note: Life is likely a 5-dimensional or a 7-dimensional discrete hyperbolic shapes which have an odd-genus, and subsequently can have an imbalanced charge on their shapes so as to cause the shape to oscillate and generate its own energy. Mind is related to the fiber group's maximal tori. These life-materials would have a new metric-function signature associated to themselves.
This is all extremely simple and very stable types of math structures.
Furthermore, in this new very simple math context, the idea of what is being measured is much more clear, ie spectra are related to the very stable discrete hyperbolic shapes which exist in the various subspaces of the high-dimensional containment space (see below, for a more definitive math structure within which entropy can be defined).
The hope for a finite set to be used to describe fairly complicated structure for existence is a possibility in this new context (ie the fundamental attributes of existence, namely the spectral set which defines a metric-space structure, is a finite set).

End of describing the simple context of physical measuring of physical systems

(1) In the linear, metric-invariant, geometrically separable context, differential equations based on geometry are solvable, and this is the context of classical physics and it is the basis for virtually all of our society's technical development (TV's, computers, Micro-chips, aero-space, including classical thermal and statistical physics, where classical statistical physics is based on a finite elementary event space, based on a fixed number of N particles contained in a closed system [see (3)] etc).
(2) In the linear, metric-invariant, geometrically separable context, for mechanical or electromagnetic waves, or for a few quantum systems, the equations for these systems are also solvable, and are either very useful in classical descriptions, or slightly useful in a limited (probabilistic) context of quantum descriptions. But, this context is not common for the set of general, many-component quantum systems, and determining the spectra of quantum systems based on a more abstract math context, eg many components placed within a context of 1/r potential energy (but where spherical symmetry can no longer be assumed to exist [since there are many components] and geometry is no longer valid [and then there is the quantum-foam which is a part of particle-physics renormalization techniques, etc]) has not been useful. Yet this method, along with the "even worse methods of particle-physics" are, essentially, the only math methods allowed, in regard to determining the spectra of general quantum systems.
(3) In professional math communities there is the belief that indefinably random events which are (nonetheless) distinguishable "elementary" events, can be consistently described in a quantitatively consistent context, but this belief seems to only lead to failure. [This is because such improperly defined events are believed to be consistently related to a stable counting process (because they are distinguishable events) which (it is believed) allows probabilities of these unstable events to be determined, but this is not true (counting is not well defined for events which can change as time changes).]
Even the simple democrat vs. republican statistics, are based on events which are unstable (ie people can change their minds) and thus they cannot be a basis for a reliable as statistics.

Thus if the statistics about democrats and republicans possess any form of reliability, this reliability is related to some other stable property of the society, such as the reliability of a highly controlled propaganda system, where only a limited set of ideas are allowed to be expressed (where it is not clear if any ideas are being expressed by the propaganda system, other than the expression of the inequality of people, and that people should be frightened so that they need to trust the experts, whom are to protect them (the public) and look-out for them [or defraud them with impunity]).

By using the complication scenario of science and math, so that the experts are supposed to be "clearly demonstrated" to be justifiably authoritative (where ordinary people cannot achieve this level of intellectual authority, exactly the same social position as the moral [intellectual] authority of the pope), and thus experts need to be trusted and believed by the public.
This has become one of the main contexts (or models) for propaganda. But science and math have come to fail, and the (so called) math experts, dabbling in economic complications, have come to "not provide information which leads to great wealth," but rather to supply a "new authoritative mechanism" through which to lie, mislead, and steal money, based on the public trusting them.
The system is so corrupt that now the social-economic mess seems to only be controllable by using extreme violence and by the extermination of the public (begun in the 1960's, ie a [recent] consequence of the militarization of the US society which began in the 1950's).
The current political-social system is so corrupt that it is useless, with the exception that it is quite useful to the villains who control it, but these villains are only leading the society to greater ruin, due to narrow monopolistic exploitative basis for their social-economic power (which is supported by a justice system, based on property rights).

The claim that the point of having... ,
"a massively huge government (as crafted by conservatives, despite their rhetoric about opposing big government) is for the security of the public"
... , can be clearly seen to be a bad hoax, since the people who caused the economic collapse of 9-16-08, due to fraud and theft... ,
(though it is claimed to be about complicated financials which only the experts can understand, and fraudulent mortgages were caused by the amoral poor, and their democratic supporters [one might notice "the joke" of having the advantage of possessing the only one-voice in the land, ie controlling the propaganda system by the owners of society, but key to the propaganda strategy is that the "intellectual" authorities of physics and math are a part of that one-voice, ie that voice is carrying the truth, as today's "best" intellectuals understand it to be the truth.])
... , are much worse and more dangerous people than the gangsters who caused the destruction of 9-11-01.

This judgment (that the bankers are worse than the airplane hijackers) is (should be) consistent with the US justice system, because (according to the purported values of the US justice system, the national security state is all about protecting "private property" so that "dynamic capital can create a perfect society for everyone" on earth)
The destruction of the property of the economic collapse of 9-16-08, was trillions of times greater than the destruction of 9-11-01. Thus according to the values expressed by the justice system and the national security state, it is property and money which the justice system and the national security state claims to be protecting (where the US law is not about protecting: equality, culture, or life [but again one deals with the propaganda system, wherein the people who support the banksters are pro-life] another one of their cruel jokes [since the system which upholds the owners of society, which the helpers "of the owners of society" are supporting, is a system which is about exterminating people who either get in the way of the investments of the owners of society, or who are of no-value in regard to helping the owners of society gain ever more capital [ever more power within a society which worships money, and those who posses great amounts of money]) The US social system is very much opposed to life, rather it is all about protecting and supporting the oligarchs.

The governing-justice-propaganda system of the US is so corrupted and fraudulent that it is a bad joke. A great deal of the corruption is a result of creating the image of the authoritative expert who can understand things which the public cannot possibly understand (a deep belief in inequality), the authoritative experts who work for the few owners of society, ie the king-pins of narrow monopolistic economic interests.

The only cure is to set-up a new Continental Congress, and simply liquidate the government-justice-financial-secret-agent-military-propaganda, systems, since they "stink of so much corruption," and in their place institute a democracy whose law is based on equality, as proclaimed by the Declaration of Independence, and oppose the oligarchies which all of western culture represents (both communism and capitalism).
Equality means the equal right to know and create, where society provides what people need to create in a selfless manner.
But each person's creativity cannot be judged but their ideas can be expressed and discussed. The computer can classify descriptions (discussions) based on the underlying assumptions upon which one wishes to express truth, as a basis for creativity. [Perhaps language only has meaning in regard to its relation to "practical" human creativity.]


On the other hand there are simple math ideas, which focus on "cubical" simplexes, but which also happen to lead to a model of existence which transcends the material world, but within which the material world is a subset, and it is a new context which leads to a new and (most likely) the proper context of human creativity. This new knowledge, this new set of assumptions about how math patterns should be organized, provides a whole new context for creativity.
These claims should make many people interested in such new ideas, about mankind's creative position within existence. Unfortunately, everyone believes the corrupt overly authoritarian propaganda systems (everyone cannot help but be brained-washed by a propaganda system which organizes language and value within society).
It should be pointed-out that the Thurston-Perelman geometrization, as well as the correct interpretation of Godel's incompleteness theorem, are rebukes to the overly general abstractness in regard to math patterns, and are implicitly saying that new math contexts should be considered by using simpler math patterns and simpler math language, (language needs to stay closer to: assumption, interpretation, context, geometry, stability, consistency, set size, and considerations about "what is measuring?").

New knowledge comes from equality, and equal free-inquiry, which is searching for knowledge (true patterns) related to practical creativity, but practical creativity for humans may really be in higher dimensions.

Simple math curriculum

(the structure of numbers, and measuring, the relation that measuring has to stable geometry,
and the associated operations of numbers and of the measuring process)

Quantities (or numbers) are built from uniform units, of a particular type, by means of (are used in) a counting process.
Counting can define both addition and subtraction, which are inverse operations.
Inverse operators can be used to solve (algebraic) equations, (with variables used to formulate the idea of an equation), where the definition of equations (about numbers) depend on equivalent numbers types but which are represented in different ways.
One can change measuring scales by means of the number operation of multiplication.
Multiplication can also be used to define fractions where the unit of measuring needs to be changed to a new fractional measuring entity to be used within a counting process in regard to the new, but also uniformly defined units of any particular type of fraction, ie this is about the invention of fractional numbers by a change of scale which is possible by means of the number operation of multiplication (and subsequently it is possible to add together fractions)].
The number operation of multiplication can also be used to change the "quantitative type" of a number.
The "identity" multiplication number, ie a uniform unit, can be used to identify a multiplicative inverse, where a number multiplied by its multiplicative inverse is (or results in) a unit.
A multiplicative inverse can (again) be used to solve (algebraic) number equations.

The practical side of numbers
(in relation to fitting together and coupling subsystems to create a new system)

Measuring is about a measurable property, or a function, which exists within "the measured property's" measurable (or represent-able) containment set (or context), where the containment set is composed of other (different) measuring types, so that the new measurable property, ie the new function, can be represented in the variables of the measured values of the (original) containing coordinate set.
This can be illustrated by the example of measuring "How far?" a toy air-rocket lands from one's feet. The containing space would be the x,y,z coordinates of 3-Euclidean-space, and the measuring function would be {Square-Root [(x-x1)^2 + (y-y1)^2 + (z-z1)^2]}, where (x,y,z) are the coordinates of where the air-rocket landed, and (x1,y1,z1) are the coordinates of one's feet.
That is, measuring is a set of function values which, in turn, are defined on a domain space of measurable, independent, coordinates. This containing set of coordinates are called the function's domain space.
A function's values can be represented by math expressions, by means of math operations acting on the function's containing coordinate space's variables, which are the variables of the function's domain space.


The operations on functions are most naturally the operations of both derivatives, and a derivative's "almost inverse" integral operator.
This is because these operators allow measuring (values) to be consistently related in a linear manner to the values of the numbers of the domain space.
Though the operations of addition and multiplication can be defined for functions [when functions are single valued for each domain element], because functions represent number values [in a single-valued context].
The operators of the derivative and the integral are linear operators.
This allows the relations between the functions-values and the function's domain-values to be consistently related to a well defined uniform unit of measuring defined on either set of numbers, ie on either the domain-space, or on the function-value-space (ie the measured property).

To identify a scale relation between two sets of quantities, the function's values versus the function's domain-space values, one needs to define a linear relation, ie f(x)=y=mx, between the f(x) values (or the values along the y-axis) and the domain values, x (on the x-axis) of the (x,y) rectangular coordinates.

Equations, and differential equations (the natural equations for measuring)

If one wants to identify a scale relation between two sets of quantities, the function's values versus the function's domain space values, so that the function's values are well defined, then one needs to identify a cause for the scale inter-relations (between the function's values and its domain values) to exist, and in this case the same type of values need to be defined in (by) two equivalent, but different, representations of the same type of numbers [both number sets are of the same type], so that the local changes of scale, namely, linear relations between the function value and its domain-space of values (or the space of the function's variables), can be related by an equality (which exists between the two different, but equivalent, linear representations of the same number type), ie Ax=y, where A is linear relation, ie A is a matrix with constant components, x is a local coordinate (column) vector and y is the equivalent linear, but different, representation of the same type of number, eg mass x acceleration = the linear force-field defined at the same spatial point as the acceleration.

For example, the geometry of material (contained in a metric-space coordinate space) causes a material object's locally linear measurable properties to determine an equation (a differential equation), where
Both... ,
the object's locally measurable properties of motion (or the 2nd time derivative of its position), graphed in relation to a domain space of time,
As well as
the material-geometry-object (basis for determining, or) determined force-field (geometric) relations [which are both representations of a property which is the same quantitative type, which can also be defined at the same position as the object's position, in the material containing coordinates]
... , can be set equal to one another so as to define a differential equation.

That is, the force-field is y, in the linear relation Ax=y, so that these two representations... ,
(a linear approximation of a graph of motion whose domain space is time,
as well as
the vector force-field values defined at the position of the moving object)
... , are set to be equal to one another, so as to define a set of differential-equations.

The solution functions to this set of differential equations, ie the 3-components of the local coordinate vector, x, provides the measurable geometric properties (of position, or motion in the 3-dimensional domain space) of the measured material object (the object is contained in the domain space). That is the solution function for the object's position (or motion) is a function of position in 3-space (or of motion in 3-space, ie a 3-component tangent property of an object's path [a path of its position property defined in 3-space]) and can be found as a solution to the linear differential equation, and then used to determine a graph in 3-space, ie the 3-component solution function, (but now defined over a domain space of time).
This linear equation can only be solved, so as to provide controllable descriptive information, if each independent motion component always has the same direction as its associated independent force-field component. This identifies a context for coordinates which are globally parallelizable and orthogonal, ie the differential equation is separable, (also called a geometrically separable [coordinate] shape). This condition allows the local linear measures of the measurable property to be consistent with the local linear geometric measures defined on the domain coordinates. This has to do with the local relation that the domain coordinates have to both geometric measures on the domain space and the global coordinate shape which allows local geometric measures and local linear measures of function values (relative to its domain values) to stay consistent with one another, the local linear changes in scale of both the function values and the geometric measure values are represented by diagonal matrices, ie the math operator A in the linear relation Ax=y is a matrix which has constant coefficients.

Measuring spectral values

Measurable material properties can also be spectral properties, Ax=kx, where k is a row vector and x is a column vector, which are assumed to be associated to both
(1) the properties of sets of (oscillating) functions, the components of the column vector, x,
(where the set of oscillating functions define a function-space [see below]), and
(2) to the sets of operators, A, which act on these functions (a complete set of commuting Hermitian operators, A, acting on sets of functions, x).
These operators, A, which when acting on a function space, also define (systems of) differential equations, whose solution functions (are supposed to) determine averages of material object's positions (or of the object's measurable properties) for a material system's components (or the average positions of the material system's particle-components) where these particle-components also posses the property of particular spectral values, which can be determined when the components are observed, (but this observation causes the oscillating function of the material system to project down to one eigenfunction, of the many eigenfunctions which are summed together to represent the oscillating system. Thus, when a spectral system is observed, the spectral system's wave-function "collapses," [this is action-at-a-distance, but is it a valid context for measuring spectra?]) and apparently the collision-interaction hypothesized to be associated with an observation (measurement), also implies the quantum system's "average form" has been lost, otherwise "would not action-at-a-distance also re-establish the quantum system's wave-form the next instant, after the observation?"

Measuring a function's value in the context of its "local linear measurable values" is about identifying a tangent line (or set of tangent lines to partial derivatives) [or a tangent linear subspace] to the function's graph (of the function which represents the measured property of a physical system) defined over the function's "coordinate domain-space," so as to be able to relate the function's values to its coordinate-values in a linear relation.
Note: Only linear relations between such different sets can remain quantitatively consistent.
The cause for a measurable property (or function) to be related to its domain space is about the fact that local linear measuring relations exist between equivalent types of different linear representations of the same type of numbers, so that an equality exists between the equivalent types of numbers which have different (linear) representations [of different math relations], but both linear representations (of the same number type) are defined over the same domain space, where the different representations (of the same (equivalent) measured values) are also determined at the same domain point (where the linear measuring values can both be defined), so that the two representations (of the linearly measured values) are equal (or at least proportional, where equality might require multiplying one side of the equation by a constant factor).
This identifies a differential equation, or a system of partial differential equations.

The tangent measure approximations, the derivatives of various orders which are a part of a differential equation, can be inverted (or their order reduced) due to the function-properties involved in determining the slope of a linear (or linear representation of the) tangent approximation of the property's measured-values, ie the linear approximation of the function values by its domain values. The local values of a function, eg f(x) vs. f(x+dx), are linearly related to the domain values by a limit process (limits are determined by the way in which a sequence of function values behaves [in regard to convergence or divergence] where the sequence of function values are defined be evaluating a sequence of domain values, which, in turn, are converging to a point in the domain space, eg x+dx(n) approaches x). Because these linear approximations of a function's values by means of a limit process have such well-defined function properties (which identify a well-defined pattern between (two) different types of functions) this has meant that the solution function of the differential equation can be found with the help of an (almost) inverse operator (determined by the well-defined function properties of a derivative process).
That is, by observing the pattern of the new functions formed when (another) a set of functions are differentiated, allows one to determine the function (formation) pattern of the inverse (or integral) operator.

The linear representation of the (solution function's) linear tangent subspace to the function's graph can be used to determine a representation of the tangent subspace (or the linear representation of a set of tangent lines to the partial derivatives of a partial differential equation can be used to determine a representation of the tangent subspace).
The matrix values (with constant coefficients) which determine the tangent space (or the set of tangent lines) give the linear change of scale relations, A(dx), that exist between the function's values, f(x), and its local (linear) domain values, x+dx (ie local coordinate vectors), f(x+dx) = f(x) + A(dx).
These linear subspaces, eg tangent subspaces to a function's graph, can also be related to local linear geometric measures of the domain space, but now associated (as a factor) to a function's values, ie differential-forms. These differential-forms can be summed (over a partition of the function's domain region) so as to relate the local linear measured function-values to an identifiable function, but which is defined on the boundary of the region in the domain space to which the differential equation applies, so as to find the differential equation's solution function, ie the anti-derivative (function) defined on the region's boundary.
That is, the local geometric measures associated to function values (or differential-forms) [by means of a local linear approximation of the properties of the measured values multiplied by the appropriate local linear geometric measures] invert the local linear differentiation process on the spatial region upon which the solution function is defined, up to its boundary, where the boundary region becomes the domain of the function's anti-derivative, where the anti-derivative is the (regional) function's (almost) inverse function.
This is essentially the relation which can be defined on a cubical region as a domain space for f, where the local linear approximations for f can be defined, and where these local linear approximations are being related, by the inversion operator, to the faces of the cubical region, where these faces become the domain region of the anti-derivative, F, ie dF=f.

There are questions about "if df=0 then when does f=dg?" where d is the exterior derivative of a differential form, where it is known that ddf=0 for any differential-form f.
The property, f=dg if df=0, is true if there are no holes in the domain space of f.
These are questions about the existence of a potential energy function related to a force-field, in turn, associated to a material geometry in the domain space (the geometry of either mass or charge, but if it is charge, then the geometry also deals with charged current-flows).

But the idea of "holes in space" is best considered in the context of the genus (where genus means the number of holes in a discrete shape) in either discrete Euclidean shapes (which can only have one such hole) or discrete hyperbolic shapes, which can often have many holes in their shapes (or a genus greater than one), where hyperbolic space is equivalent to space-time. Spheres are excluded, since spheres do not have any holes, but spheres are also non-linear, ie the metric-functions of spheres can have variable coefficients, thus working with spheres is quantitatively inconsistent, leading to chaotic behavior when a sphere is deformed.
Discrete hyperbolic shapes are stable and have well defined spectral properties, where these spectral properties are relatable to the sectional curvature, 1/r, for a circle with radius, r. Thus the spectra of a discrete hyperbolic shape would be related to its "circle-space" which is associated to the discrete-hyperbolic-shape's associated cubical simplex.
That is the 1/r term can be related to a set of definite discrete spectra.
A cube in n-space can have n separate circles associated to itself, so as to form an mn-circle space, if there are m cubes composing the "cubical" simplex of the discrete-hyperbolic-shape in n-dimensional hyperbolic-space, ie the genus of the discrete shape is m, ie each cube identifies one (n-1)-dimensional hole in the n-dimensional discrete shape.
The sectional curvature of a discrete hyperbolic shape is proportional to kinetic energy, for mass, m, (or charge related to mass) of material flowing about the (given) circle, in the circle-space associated to a discrete hyperbolic shape.
That is, in a (hyperbolic) metric-space (ie in space-time) the issue of spectral-existence is best related to the stable discrete hyperbolic shapes which compose material systems in hyperbolic space. In this case, the spectra is determined by the various radii of the circles which compose the circle-space associated to the discrete hyperbolic shape's "cubical" simplex structure, where "cubical" simplex can really mean right-rectangular simplex, and thus the circle-space associated to a right rectangular simplex can have a variety of different spectral values associated to itself.
However, there are further conditions in regard to the reflection groups which allow a cubical simplex to become a (space-filling) lattice in hyperbolic n-space, where each dimension, n, has different restrictions on the allowed discrete-hyperbolic-shapes, restrictions due to the properties of the shape's reflection groups (as identified by D Coxeter).
When n is 2 or 3 then there is a wide variety of discrete hyperbolic shapes.

[the two sets of numbers (function values, and coordinate values)]

The stability of the quantitative relations that exist between a function's values and its domain values depend on linear independent changes of scale (which exist between the local function values and local domain values). This means that the matrix operators which represent the local linear measuring relations can always be diagonal (which also means that such linear differential equations are solvable, and their solution functions controllable). The discrete Euclidean shapes and the discrete hyperbolic shapes are both parallelizable and orthogonal, or equivalently geometrically separable, except possibly at one (distinguished) point on the shape where different angular relations might exist between adjacent cubes of the discrete shape's 'cubical" simplex.

The operators on numbers are addition and multiplication along with their inverse operators subtraction and division (respectively), while the operators on measured values, ie the operators on functions, are the derivative and its inverse (the integral operator).
In algebra, the other issues about operations are about "order of operations," in particular the commutative property (in regard to the order of operations), for operators defined on either numbers or on functions. For matrix operators, commutative matrices means that the matrices are diagonal, and this means that the linear math is solvable.
Note: The eigenvalue structure of operators is related to the properties of the diagonal of the (general, ie not a diagonal matrix,) matrix. For example, the sum of the terms along the diagonal of a matrix is invariant to metric-invariant local coordinate transformations. This is considered to be another abstract math pattern which might be useful in limiting the spectral properties of general spectral systems, but it has fairly limited practical application (and has become part of the lore of abstract complications for the math community) and instead the much simpler model of very stable discrete hyperbolic shapes seems to have a wider range for general application in practical, meaningful matters of useful descriptions.

In regard to the matrices defined for local linear approximations of a function's values in regard to its domain values, the property of the (derivative) operators being commutative means the matrix is a diagonal operator.
That is, fx(x+dx,y,z) = fx(x,y,z) + (mx)(dx) (is a diagonal pattern);
fx(x+dx,y,z) = fx(x,y,z) + (mx)dx + (mxy)dy + (mxz)dz, (which is not a diagonal pattern) etc.
and this linear, diagonal (matrix) relation must exist at all points of the geometry of the system contained in the domain space (whose measured property (or measured value) is represented by the function f(x,y,z)).

That is, parallelizable and orthogonal, or (equivalently) geometrically separable, shapes in the domain space are the fundamental math patterns which are quantitatively stable. That is, the linear changes of scale between the function values and the domain values must always remain linearly independent and orthogonal for all coordinate points of the system, perhaps with an exception for a finite number of points (or one point) depending on the either simple or complicated nature of the exceptions.

When describing a system's measurable properties within a linear context of differential equations defined on a function's domain space, which is a coordinate metric-space, then there are
linearly measurable geometric properties which must remain geometrically separable
spectral properties defined by a set of operators which act on a function-space so as to define a system of differential equations, where the domain space for each function in the function space is the same domain space, and that domain space is a metric-invariant metric-space.

For metric-invariant metric-spaces there are three 2nd order linear equations which are metric-invariant. (1) The elliptic equation, (2) the heat equation, (3) the wave-equation.

(1) The elliptic equation defines planetary orbits, as well as box potentials for spectral systems.
(2), the heat equation, identifies the energy eigenvalue equation... ,
where only
(a ) the singular geometric 1/r term, for the potential energy, of a quantum system based on a spherically symmetric 2-body (interacting) system,
(b) the infinite potential-wall box, ie a quantum model of a crystal,
... , have been related to fundamental quantum systems, and they are not all that successful at describing the spectral properties of these systems, while
(3) the wave-equation, deals with light and sound, [and for a few models for the electromagnetic-field vector-potential in regard to various types of charge and charged-current geometries for electromagnetic systems]. The 1st order Dirac equation is based on a relativistic wave-equation, whose "square root" along with the Dirac matrices, identifies the Dirac operator. The Dirac operator is coupled with a non-linear connection (derivative) term to define particle-physics equations.

That is, spectral systems defined within metric-spaces, essentially, depend on the 1/r term, and spectral cut-offs and spectral approximations as the basis for the spectral energy structures of general quantum systems such as atoms and crystals.
The idea which is used to describe the properties of a spectral system is that they are to be defined upon eigenfunctions, exp(ix), which oscillate about a spectral system's average values of particle-position and orbital energy values. The oscillations are defined around the singular geometry of the potential energy of a spherically symmetric, 1/r, term associated to a 2-body interaction.
Measurements of a spectral system relate to both the position and the spectral-orbital energy properties of the components, which interact, and compose the quantum system. But "how does one determine the spectra from the 1/r singular and continuous geometry?" How are the discrete spectra of a general quantum system to be determined?
The solution to the H-atom's radial equation diverges, while an arbitrary truncation of this diverging series provides a model of spectral discreteness, but how does arbitrary truncation translate into a physical process? This same way of modeling a many component quantum system does not work for the 3-body quantum system. This program has not been all that successful at identifying the spectral structures of the general (observed) spectral properties of fundamental quantum (or spectral) physical systems.

However, within the context of discrete hyperbolic shapes there is a new very simple context within which the stable spectra of physical systems at all size scales can be easily understood.

It should be noted that the spectra of electromagnetic systems can be described fairly well in the context of continuity and where the spectral approximations work well for an approximately continuous spectral model, or for a context in which the discrete spectra of the system are known (or are controllable).

These spectral representations essentially deal with the oscillating functions, such as Aexp[i(kx-ft)], where A is the wave-amplitude, k is 1/(wave-length), and f (frequency) is 1/T, where T is the wave's period, and where x is a space measure and t a time measure in the periodic function's domain space.

Such a space of periodic functions can be transformed to a dual space of functions by a Fourier transform.
Note: Transformations of function spaces to their dual function spaces also changes the operator structure so that differential equations associated to sets of differential operators become algebraic equations which are easier to solve and then one can dual transform (the algebraic solution) back to the original function space, so as to get the solution function to the differential equation. (At least that is the idea, but it has not been used successfully to solve "all that many" (only a few) quantum systems (have been so solved))

These dual spaces are often pairs of values which define an "action," eg momentum and position or frequency and energy "action" pairs. Thus it is believed that much information about a system's energy spectra can be determined from working in energy's dual space of frequency (or vise-versa).

The symplectic spaces place action variables into the same metric-space structure, but now the metric-function is an anti-symmetric operator.
If one models symplectic spaces such as x^p (position-momentum space), as having a local (or global) geometric structure of discrete Euclidean and hyperbolic shapes (respectively), then both (macroscopic and microscopic) material interactions as well as the discrete hyperbolic shapes which either are interacting or allow one to model the existence of a discrete-state partition structure of the system's energy state properties.

This gives a definitive geometric-spectral model of the attribute of (apparently) quantum systems being partitioned into discrete energy regions in the energy space, in regard to the energy-time (or energy-frequency) dual representations of quantum systems (but of which there are no geometric-physical models for such an energy partition within the abstract math structures which are being employed to model quantum systems).

The issue is simple
(1) consider function spaces which have no (or only a very limited) relation to any mechanism through which spectral properties of the function space can be determined (1/r, and spectral approximations, often determined in the frequency function space, where spectral approximations are introduced for an algebraic solution process, [as opposed to the algebraic (frequency) space's dual energy function space]),
(2) consider discrete hyperbolic shapes attached to
(a ) a crystal's lattice sites, or to an enclosed box,
(b) attached to a "macroscopic discrete hyperbolic shape" associated to the region within which a fluid is confined and whose "cubical" simplex structure is composed of the many small facial aspects of the atomic-components, as well as the "local" Euclidean interaction structures associated to the component material which compose a fluid, which is confined within some macroscopic spatial region.
Thus again providing a geometry for a model for the partition of energy into relatively low energy states for macroscopic regions, for either a fluid or for a more ridged crystal lattice.