Isotropic Stability And Resonant Vibration

 The more isotropically stable that a given arbitrary cohesive set of discrete energy quanta is to be, of which is here to work to be displaying a recursively smooth Ricci Flow -- the deeper that its resonant vibration will consequently tend to be. To Be Counted! Sincerely, Samuel David Roach. (1989).

A Little As to When A Discrete Quantum Of Energy Is Gauge-Invariant

 In the general procedural course of "events," in which a given arbitrary discrete quantum of energy is to remain gauge-invariant, -- there is thereby to be the eminent resultant given arbitrary physical tendency, in which the directly associated kinematic flow of the respective local sequential series, in which one is here to be considering the "plucking" action of those directly applicable gauge-bosons, as these are here to be made Gliosis upon the topological surface of those directly corresponding second-order light-cone-gauge eigenstates, to where these said second-order eigenstates, that are here to work to make-up the holonomic substrate, of what basically amounts to being the proximal local presence of the wave-based nature of discrete energy impedance, -- will thence tend to occur in such a general manner, to where this inferred flow of a resultant formation of a set of Schwinger-Indices, that are here to externally propagate, in so as to interact with their infringed upon environment, in such a manner, to be able to form the forces of nature, to where this will therefore tend to work to form, in this particular genus of a case, a homeomorphic flow of force-related eigenstates, over a directly corresponding period of time. Sincerely, Samuel David Roach. (1989).

Indices Of Eminent Dispensation Of Energy Quanta (My Take)

 Let us consider the following perception: Initially: Let's first stipulate, that we will here treat the individually taken physical forces of nature, as respective "causes"; and that we will also treat the main influences, that are here to be formed by such implied forces, as the respective "effects." Let us next say; arbitrarily, that the multiplicity of the individually taken physical forces of nature, are here to act, upon the topological manifolds, of that physical stratum, which is of the correlative respective cohesive sets of each of such inferred discrete energy quanta. That general phenomenology, which is here to be appertaining to those continually expended indices, that are of the eminent dispensation of the metrical Lagrangian-Based flow, that is of the multiplicity of the respective individually taken cohesive sets of discrete energy quanta, works to help at forming those earlier mentioned "main influences," that are here to work to display the exhibition of those physical attributes, that are here to work to appertain to the demonstrable expression of such said "forces," as taken over the course, of a given arbitrary duration of correlative time. Sam Roach.

Conclusion To Derivation -- Importance -- Conservation Of Homotopic Residue

 So, to conclude; Suppose a cohesive mass-bearing set of discrete energy quanta, is to alter in its direction and/or in its relative rate of wave-tug, when in its relation to the motion of electromagnetic energy. This works to help in causing a change in the dimensional-related pulsation, that is of the inferred set of discrete energy. (Thereby working at causing it to have, at each locus in which such an alteration is to occur, the proximal local presence of a metric-related Chern-Simons singularity.) The residual externalized dispensation, that is here to be incurred, at the moment of each of such changes in dimensional-related pulsation, is to work to form the need for an alteration, in the relative differential geometrical-positioning/quantity, that is of those composite partition-based discrepancies, that are here to be delineated in its distribution, along the correlative topological surface, of the directly corresponding superstrings/counter strings of discrete energy permittivity, that are here to work to make-up, the said cohesive mass-bearing set of discrete energy quanta in question. Yet, in order for Casimir Invariance to be maintained, (which is necessary, in order for the net webbing of superstring-related  phenomenology, to be interconnected by field), the total number of such partition-based discrepancies, per each Noether-based mass-bearing cohesive set of discrete energy quanta, for the specific genus of holonomic substrate, that such an inferred type of a Hamiltonian Operator is here to be exhibiting, at a theoretical rest-mass, must thereby tend to mathematically work to bear a certain theoretical Hodge-Index of discrete energy, at such a stipulated rest-based condition. The faster that such an inferred Hamiltonian Operator is to be traveling, in relation to light, the Less Partition-Based Discrepancies that the correlative Individually Taken Mass-Bearing Strings of such an Operator, will tend to have, to where the More Mass-Bearing Strings that such an Operator will thereby tend to need, from within the Ward-Neumman bounds, of its cohesive set of discrete energy quanta. Therefore and consequentially; This works to help in causing, for a change in pulse and/or in geometric delineation, per alteration in mass-related velocity, as is partially described of by relativity, in so as to help to allow for the general process, of "the conservation of homotopic residue." To Be Continued! Sincerely, Samuel David Roach. 

Gentle Reminder, As To The Reason Of Necessity -- Homotopic Residue

 The general physical reason, as to why it is necessary for homotopic residue to be conserved; is on account of the general physical condition, by which it is necessary for the multiplicity of the vast "arena" of discrete energy quanta, to be able to have a directional field interconnection amongst each other, -- in order for individually taken discrete increments of energy, to be able to have any sort of a prolonged spontaneous interdependence of inter-relationship with one another, over the span of any feasible duration of time. Homotopy works to allow for superstring-related phenomenology, to work to bear the multiplicity of an orientable tense of field amongst each other. When the residue of the inter-mingling of such said/inferred fields is conserved, this general type of an eminent procedure, tends to work to allow for a prolonged spontaneous tense, of an interdependent directional orientation amongst the other countless arrays of superstring-related phenomenology. This just inferred general type of an eminent process, works to help allow for the general physical condition to occur, by which superstring-related phenomenology, is thence to work to bear a consequentially resultant interdependent relationship amongst each other, by the reductional means, in which the delineation of the accommodating respective fields, that work to help in guiding the multiplicity of the flow of energy-based inter-relationships amongst each other, may thereby be able to happen, so that discrete energy and therefore energy, may then be able to persist at existing, over any feasible time period, that may thereby be taken into consideration. To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989). 

Perpetual Accommodation -- Correlative Attribute Of Partition-Based Discrepancies

 The prior mentioned perpetual accommodation, as to the proportional alteration, in the respective correlative physical attribute of partition-based discrepancies, by an alteration in both the mass, length, and time, -- in order to accommodate for a change in the correlative "dimensional-related discrepancy," is at the "heart" of the general process, of what I have worked to describe of, as being, "the conservation of homotopic residue." I will continue with the suspense later! Sincerely, SAMUEL DAVID ROACH.

A "Tad-Bit" Of A Straight Forward Interjetction

Since the more anti chiral, that the charges are to be, of two different given arbitrary cohesive sets of discrete energy quanta, when taken towards one another, -- the more highly attractive that these two different inferred sets of energy, are thence to be towards each other -- to where it follows then, that the more attractive that two different cohesive sets of discrete energy quanta, are to be towards one another, the more likely that such a pair of attracted mass-bearing eigenstates, will consequently tend to result in working to display the behavior, of exhibiting the general activity, of operating as an isotropic dual state of a covariant, co-determinable, and co-differentiable entity, that is here to be displaying the general behavior, as exhibiting a tense of group action. Consequently, as a reverse-fractal of what I just mentioned; when there is to be a more macroscopic tense, of a strong set of electrodynamic group-related attractions -- that are here to be bound in such a manner, to where such a said group-related state, is to act as one cohesive entity, on the whole, it may then be said here, as well, that such a herein inferred relatively "macroscopic" phenomenology, of such a given respective case scenario, is here to consequently tend to result in working to display the behavior, of exhibiting the general activity of operating as an isotropic state, by acting as a covariant, co-determieable, and co-differentiable entity, that is here to be displaying the general behavior, as exhibiting a tense of group action. I will continue with the suspense later! (1989). To Be Continued! Sincerely, Sam Roach. 

As To Counter Strings -- Development Of Idea -- Homotopic Residue

 During the general course of each individually taken iteration of group-related instanton, the multiplicity of those Noether-Related counter strings, that are here to be directly corresponding to the general phenomenology of Noether-Related superstrings of discrete energy permittivity -- are to tend to work to display an asymmetric distributional delineation, of their correlative partition-based discrepancies, when this is here to be  analogous, to their inter-relationship with their corollary-affiliated tense, of the stated superstrings (of discrete energy permittivity), to where these are here to bear an Inverted tense of mirrored symmetry (or in other words, the multiplicity of the said "counter strings," in a way, are here to bear a tense of "mirrored asymmetry." In the meanwhile; Each time that the inferred mass-bearing "team" of strings, that are here to be comprised of, by a cohesive set of such just mentioned superstrings and counter strings, is thereby to change in either its velocity and/or in its direction, in relation to the motion of electromagnetic energy, the net solved attribute, of such an eluded-to spatially translated wave of cohesive discrete energy, will thereby have a tendency, of changing in either its covariant rate of spatial translation and/or in its geometric association, when this is here to be considered, in its relation to the directly eminent motion and existence, of the electrodynamic Fourier-related conditions, of the phenomenology of electromagnetic energy. To Be Continued! SAMUEL ROACH.

Scalar Amplitude Of Kahler-Based Quotients -- Correlative Intensity Of Wave-Tug

 Let us initially consider two different Noether-Based mass-bearing cohesive sets of discrete energy quanta, that are here to be of basically the same general nature. Both of these two different inferred individually taken "teams" of discrete energy, are here to be exhibiting the mappable-tracing, of a "congruent" tense of a Lagrangian-Based path, as this is here to be taken into consideration, over the durational course of a relatively brief covariant Fourier-Transformation, that is here to work to bear the span of the same sequential series of group-related instantons. (As the inferred covariant duration that is to be considered here, in which these two different inferred "teams" of energy, are here to have had worked to map-out their correlative Lagrangian-Based paths, is to be indicated by a monomial tense, of an evenly-gauged relativistic Hamiltonian eigenmetric.) Next; The kinematic motion of one of these two said cohesive sets of discrete energy quanta, is here to work to mathematically bear a greater scalar amplitude of a Kahler-Based quotient, than the other indicated cohesive set of discrete energy quanta, as this is here to be taken, over the covariant span of time, in which these two earlier inferred sets of discrete energy quanta, have worked to map-out their "congruent" individually taken Lagrangian-Based traces. As a result; that cohesive set of discrete energy quanta of the two, that is here to have worked to mathematically bear a greater scalar amplitude of a Kahler-Based quotient, -- in the process of working to map-out its correlative Lagrangian-based path, -- will consequently tend to work to bear a higher relativistic intensity of a wave-tug, upon its immediately external environment, than that other cohesive set of discrete energy quanta of the two, that was instead, to mathematically bear a lower scalar amplitude of a Kahler-Based quotient. I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH.

As To Lagrangian-Based Tendency

 When any given arbitrary Noether-Based mass-bearing cohesive set of discrete energy quanta, is here to be working to exhibit a (relatively) smooth Ricci Flow, when in regards to its directly corresponding Lagrangian-Related tendency, the inferred set of energy of such a given case, will consequently, in the process of doing so, tend to result in displaying a homeomorphic spatial translation, of its correlative metric-gauge eigenstates, over the durational span, of a directly associated Fourier Transformation. Furthermore; when any given arbitrary Noether-Based mass-bearing cohesive set of discrete energy quanta, is here to be working to exhibit, a (relatively) spurious Ricci Flow, when in regards to its directly corresponding Lagrangian-Related tendency, the inferred set of energy of such a given case, will consequently, in the process of doing so, tend to result in displaying a heteromorphic spatial translation, of its correlative metric-gauge eigenstates, over the duration span, of a directly associated Fourier Transformation. I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH. 

Development From A Reminder -- As To Homotopic Residue

 All of the mass-bearing superstrings of discrete energy permittivity, -- that are here to work to comprise a given arbitrary Noether-based mass-bearing cohesive set of discrete energy quanta, -- per individually taken iteration of group-related instanton -- are to work to bear the same number of respective partition-based discrepancies. Yet; the differential geometric arrangement of such partition-based discrepancies, may vary, as this is here to depend upon their topological delineation-based distribution, as this is here to be taken, along the external shell, of the herein implied topological manifold, that is here to be of the earlier stated mass-bearing cohesive set of discrete energy quanta, of any one particular pertinent given arbitrary respective case scenarios. I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Residual Dispensation -- Partition-Based Discrepancies

 Therefore; the residual dispensation, of the indices of the multiplicity of those changes, that are here to happen, to a mass-bearing cohesive set of discrete energy quanta, -- as it is here to be altering in its dimensional-related pulsation, -- is to be compensated for, by a change or a set of changes, that is here to be exhibited, by either a change in the differential geometric arrangement of the respective correlative partition-based discrepancies, and/or, in a change of the correlative proximal local number (quantity), of  such respective partition-based discrepancies, as this is here to be happening, to all of the composite individually taken superstrings of discrete energy permittivity, that are here to work to comprise such a stated mass-bearing cohesive set of discrete energy quantity, that is here to be of such a general type of a case. I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Homomorphic Sets Of Discrete Energy Quanta -- Greater Probability -- Bosonic "Teams" Of Discrete Energy

Individually taken homomorphic mass-bearing cohesive sets of discrete energy quanta, of a Noether-based geometry, that are here to work to bear a homeomorphic covariant field of motion, with other of such inferred sets of discrete energy quanta, will often have the tendency, of working to bear a greater probability, of exhibiting the general characteristic, of displaying the behavior of a bosonic-related nature, than of a fermionic-related nature. To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

As To Externalized Dispensation Of Metric-Related Indices

This prior mentioned general case, of the externalized dispensation of metric-related indices, that I discussed yesterday, is physically applicable, to the general correlative alteration, in either the directoral geometric arrangement of the correlative "partition-based discrepancies," and/or of the correlative henceforth exhibited quantity of such said "partition-based discrepancies." And angular/directoral change, that is of the said set of energy, when in relation to the motion of electromagnetic energy, will consequently tend to alter the relativistic geometric arrangement, of the inferred partition-based discrepancies; whereas, -- a change in the rate of motion of the inferred respective mass-bearing set of discrete energy, as again, when this is here to be taken in a relationship with the motion of electromagnetic energy,  -- will thereby, in such a given type of a case, tend to work to alter the quantity of those partition-based discrepancies, that are here to be adjutant to the proximal local topological surface, of the composite superstrings of discrete energy permittivity, (that are here to come together, in so as to work to comprise the inferred cohesive set of discrete energy quanta, that is here to act as the inferred Hamiltonian Operator), of which are here to be directly appertaining, to such a given arbitrary general type of a case scenario. Sincerely, SAMUEL ROACH. (1989).

The Next Refresher-Related Development

 When the velocity of a Noether-based mass-bearing cohesive set of discrete energy quanta, is to alter, when in relation to the motion of electromagnetic energy, the directly corresponding dimensional-related pulsation, of such an inferred mass-bearing operator, will consequently tend to change. This tends to work to directly correspond, to the occurrence of both an eminence of the i*PI(del) Action, as well as to the directly corresponding occurrence, of a perturbation of those correlative Chern-Simons Invariants, that are here to be directly corresponding, to the respective given arbitrary flow, of the Lagrangian-based spatial translation, that is of the inferred respective given arbitrary mass-bearing cohesive set, of discrete energy quanta, that is here to be of this general type of a given case scenario. As this general type of a net gauged-action is here to occur, the metric-related change, that is here to be directly associated with the altering inferred set of energy, of such a respective given case, is then to work to bear an Externalized Dispensation, of its residual superfluous indices. I will continue with the suspense later! SAM ROACH.

Dimensional-Related Pulsation — Metric-Based Chern-Simons Singularity, — Refresher

 Whenever the dimensional-related pulsation, that is here to be of a correlative given arbitrary respective mass-bearing cohesive set of discrete energy quanta, is to go into a state of alteration (perturbation), this general type of a kinematic activity, will always tend to be directly associated, with the presence of a metric-based Chern-Simons singularity. Furthermore; whenever a mass-bearing cohesive set of discrete energy quanta, is to undergo the general process, of an alteration (perturbation), in either its direction and/or in its rate of spatial translation, over a correlative respective duration of discrete time, — as this is here to be taken, in its (the inferred set of discrete energy quanta) covariant relation to the motion of electromagnetic energy, such an inferred discrete set of energy, will also, over the durational course of this eluded-to self-same process, tend to inevitably be going through an alteration (perturbation) in its directly corresponding dimensional-related pulsation. Therefore, it follows, then; that whenever a mass-bearing cohesive set of discrete energy quanta, is to undergo the general process, of an alteration (perturbation), in either its direction and/or in its rate of spatial translation, over a correlative respective duration of discrete time, — as this is here to be taken, in its (the inferred set of discrete energy quanta) covariant relation to the motion of electromagnetic energy, such an inferred discrete set of energy, will also, over the durational course of this eluded-to self-same process, tend to inevitably be directly associated, with the presence of a metric-based Chern-Simons singularity. P.S.: I am starting with something that most of my readers already know, — so that I will tend to not “lose” anyone, with what I am going to describe later! (Simple is Best!) I will continue with the suspense later! SAM. (1989).

As To The i*PI(del) Action -- Dimensional-Related Pulsation -- Homotopic Residue

 Whenever the "i*PI(del) Action" is to occur (which is whenever the Chern-Simons Invariants, are to be undergoing perturbation/alteration), the directly associated mass-bearing cohesive set(s) of discrete energy quanta, are here to consequently tend to change, when this is in terms of the dimensional-related pulsation, of such said mass-bearing set(s) of discrete energy quanta. The residual dispensed Kahler-Metric-Based indices, will have then become externally tangential, -- in so as to then be working to be tugged-out of the correlative regional proximal locus, that is here to be of any one pertinent given arbitrary respective individually taken case scenarios of such, to where the consequentially resultant eminent implied covariant Ward-Cauchy-related kinematic field density, is to alter accordingly, in its kinematic interplay of gauged-action, when in relation to the said respective mass-bearing cohesive set of discrete energy quanta. S.D.R.. 

Chern-Simons Invariance -- Gauge-Invariance

 A given arbitrary discrete quantum of energy, is to tend to work to bear the general proximal local  characteristic, of what may here to be considered as being the Ward-Cauchy-related condition of Chern-Simon Invariance, -- when such a said discrete quantum of energy, is here to also work to bear the general proximal local characteristic, of what may here to be considered, as being the Ward-Cauchy-related condition of gauge-invariance. The general physical characteristic of Chern-Simons Invariance, is only of a heuristic manner of exemplification, when the directly associated discrete quantum of energy, is to be in the process of displaying both a constant rate of spatial translation,And, a constant direction of projected trajectory, as this is here to be taken, When in its covariant relationship with the motion of electromagnetic energy. Likewise; the general physical characteristic of Gauge-Invariance, is only of a heuristic manner of exemplification, when the directly associated discrete quantum of energy, is to be in the process of displaying both a constant rate of spatial translation, And, a constant direction of projected trajectory, as this is here to be taken, When in its covariant relationship with the motion of electromagnetic energy. So; whenever a Noether-based discrete quantum of energy, is to alter in either its speed and/or in its direction, in its relationship with electromagnetic energy, -- such a said quantum of energy, is then to alter out of the general Ward-Cauchy condition of gauge-invariance, -- to where its initial condition of Chern-Simons Invariance, is thereby to become variant in the process. To conclude: Both the physical condition of Gauge-Invariance, And, the physical condition of Chern-Simons Invariance, are only heuristically as such, as a given arbitrary directly corresponding cohesive set of discrete energy quantum, that is here to be in the process of being taken into consideration, is to be maintaining both the same rate of speed, And, a constant direction of its projected trajectory, when this is here to be considered, in its relation to the motion of electromagnetic energy. I will continued with the suspense later! To Be Continued! Sincerely, Samuel David Roach. (1989).

The Next Part; As To A Derivation Of, "Homotopic Residue"

 Initially, as a refresher; when a mass-bearing cohesive set of discrete energy quanta, is to change in its direction and/or in its rate of motion, when in relation to the motion or electromagnetic energy, the correlative Specifics, as to what the Chern-Simons Invariants are actually to be working to consequently express, will thereby tend to go into a state of alteration (perturbation). When Chern-Simons Invariants, are to go into a state of recursive perturbation, (as this is here to be appertaining, to the implied general physical conditions, of "excited" mass-bearing cohesive sets of discrete energy quanta), -- this will often tend, at least for some sort of a "Poincare level," to work to form a certain tense or manner, of what I term of as being the general physical attribute, of "cohomology-related dispensation." This said "cohomology-related dispensation," is consequently to tend to result, in having the general characteristic, of working to reverse-fractal, into what to us, is to tend to be the existence of electrodynamic charge. To Be Continued! Sincerely, SAMEUL ROACH. 

More As To A Derivation Of The Concept Of, “Homotopic Residue”

 Since most, if not all, of those mass-bearing superstrings of discrete energy permittivity, that are here to work to help in comprising their directly corresponding mass-bearing cohesive set of discrete energy quanta, will thereby tend to Not be able to purely move into their “innately desired” Holomorphic Direction, (due to the earlier inferred/stated info), — in which, that it is consequently to be the net overall mass-bearing cohesive set of discrete energy quanta, that is here to get “first-dibs,” as to what direction that the implied Hamiltonian Operator is to move into — such implied strings, will then, often tend to invariably, work to bear the general physical attribute, of something called, “Chern-Simons Invariants.” As I have indicated in some of may previous posts, Chern-Simons Invariants, are only maintained as actually being “Invariant” or unchanging (as being of the same specific nature), When the correlative mass-bearing cohesive set of discrete energy quanta, that is here to be of any one individually taken case scenario, is here to bear, both the same speed and the same direction, when this is here to be taken, in a covariant relationship, to the motion of electromagnetic energy. I will continue with The suspense later! To Be Continued! Sincerely, SAM ROACH. (1989).

Homeomorphic/Heteromorphic Homotopic Vibrational Hamiltonian Field Operator -- De Rham/Dolbeault Cohomology

 A given arbitrary spatially transferred cohesive set of discrete energy quanta, that is to be respectively propagated along the general physical substrate of space-time-fabric, of which is here to work to display the general behavior, of acting as a homeomorphic homotopic vibrational Hamiltonian field operator, will consequently tend to have a greater tendency, of exhibiting a De Rham cohomology, than a Dolbeault cohomology. It follows, then; that a given arbitrary spatial transferred cohesive set of discrete energy quanta, that is to be respectively propagated along the general physical substrate of space-time-fabric, of which is here to work to display the general behavior, of acting as a heteromorphic homotopic vibrational Hamiltonian field operator, will consequently tend to have a greater tendency, of exhibiting a Dolbeault cohomology, than a De Rham cohomology. I will continue with the suspense later! SAM ROACH. (1989).

Homeomorphic Homotopic Wave -- When It Is Homomorphic

 When a given arbitrary homeomorphic homotopic wave, is here to bear the general Ward-Cauchy-related condition, of being of an isotropically stable nature, -- then, such a said respective homeomorphic homotopic wave,  may thenceforth be described of, as working to be exhibiting a homomorphic nature. I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Homeomorphic Homotopic Wave

 When the metric-based relationship of a given arbitrary homotopic wave, that is here to be directly involved with the Fourier-related spatial translation, that is here to be of the correlative dimensional-related pulsation, of a respective mass-bearing cohesive set of discrete energy quanta, to where these are thence to be of the Same Scalar Value, per each distinctively different individually taken directoral value, (I.E...In R3: One is to have the same type of a dimensional-related pulsation, as this is here to be taken, over each individually taken correlative axion, that is of all three directly pertinent directorals). -- then, one may consequently say, that such a herein stated general type of a respective homotopic wave, is here to tend to be spatially translated, in a relatively homeomorphic manner. To Be Continued! Sincerely, SAM.

More As To The Workings Of Homotopic Residue

 Superstrings of discrete energy permittivity Want to move in a certain given arbitrary innate direction, -- When given What the specific traits and other physical attributes, that are of such an individually taken  respective given arbitrary superstring, are here to be, in such an eluded-to type of a  theoretical case scenario. That directly corresponding respective mass-bearing cohesive set of discrete energy quanta, of which the directly corresponding composite individually taken superstrings of discrete energy permittivity, are to work to comprise, tends to Want to move Into its relatively Innate Holomorphic Direction. All of the  "strings" of discrete energy permittivity, that are thenceforth to work to comprise the earlier mentioned respective cohesive set of discrete energy quanta, are to be delineated during the general course of group-related instanton, At the external shell of the said correlative cohesive set of discrete energy quanta, that these are here to take part in working to be exhibited at, when such an inferred type of a set of interdependent energy, that is here to be operating to display one specific task, is here to be considered -- as a metaphorical "team" of reductional energy. The cohesive set of discrete energy quanta of such a respective given arbitrary case scenario, is to take precedence, -- as to the resultant accommodation of its "phenotypical" display, as to its tendency, to move Into its innately holomorphic direction, -- to where the inferred "team" of energy, will consequently tend to "win-out," as to the ensuing resultant spatial direction, at which the inferred net composite energy-related quantum, of such a general tense of a case scenario, is thereby to be transferred into, over the span of a directly corresponding duration of discrete time. SAM.

Homotopic Waves -- Homotopic Residue

 Homotopic waves may often help at working to describe metric(al)-related associations of dimensional-related pulsations.  So; when a given arbitrary physical entity, that is here to be exhibiting the general characteristic of acting as a topological manifold, in which it is here to be considered in lieu of its Ward-Cauchy-Based nature, is thence to change or alter in its respective physical attribute of dimensional-related pulsation, then, not only will the given arbitrary entity of topological manifold, of such a respective case, tend to work to form a metric-based Chern-Simons singularity -- at the covariant "point" in duration, at which such an inferred set of discrete energy quanta, is here to have phased into such a said change in its dimensional-related pulsation, yet, in the process of doing so, -- this general process that has just been eluded-to here, will also tend to work to involve, the eminent proximal adjutant flow, in the dispensation of its directly associated tense of energy-related indices, over a discrete increment of time. (That is, as this is taken over the course of a directly corresponding Fourier Transformation.) Such a general genus of an eminent adjutant flow, in the dispensation of a tense of energy-related indices, as taken over a Fourier Transformation, is directly associated with the general Ward-Cauchy-related concept, of what I term of as being the physical condition of "homotopic residue." To Be Continued! Sincerely, SAM ROACH. (1989).

"Phased-Out" Communicable Message (Signal) -- Dispersion

 When any given arbitrary electrodynamicly transmitted communicable signal-related message --  that is here to be delineated from one general proximal locus to another -- is here to have ended in its allocation, by the general process, in which such a said message is here to have been "phased-out" of being sent, to wherever it was initially distributed to; -- it follows then, that one may consequently say, that the respective given arbitrary cohesive set of vibrational oscillations, that were here to have initially been in the process of working, via a directly corresponding propagated electrodynamic-related spatial transmission in time and space, in order for such a said/inferred signal, to have been able to have been sent to such an inferred initially designated destination, in order to be able to transfer or carry the initially delineated given arbitrary respective electrodynamicly communicable message -- at the point in time in which such a said cohesive set of communicable vibrational oscillations, are to have been ended in their propagational spatial translation -- that the signal that is to here to have just been "disbanded," may thereby be described of, as having had gone through the general process, of electrodynamic "dispersion."

I will continue with the suspense later! To Be Continued! Sincerely, Samuel David Roach. (1989).

On The Coupling Of The Kinetic Motion Eigenstate, Of A Disturbance In Space-Time-Fabric

 When a Noether-based discrete kinetic eigenstate, that is of a perturbative disturbance in space-time-fabric (in units of "Joules"), is to couple with a discrete scalar quantum of time during instanton (in units of "seconds"), this general type of a kinematic display of behavior, will tend to work to form the general nature, of what may be termed of, as being the essence of "Planck Energy." (in units of "Joules*seconds"). The "thing" or "holonomic substrate" of such an inferred discrete quantum of "Planck Energy," is the general idea, behind what tends to be referred to, as being a superstring of discrete energy permittivity. Furthermore; the "action" or "motion" of such an inferred discrete quantum of "Planck Energy," is the general idea, behind what tends to be referred to, as being the "Lagrangian"of such an inferred discrete quantum of "Planck Energy." I will continue with the suspense later! Sincerely, Sam Roach. (1989).

Scattering Of Communicable Signals -- Clarity Of Signal

When one is to have a given arbitrary cohesive set of vibrational oscillations -- that are here to be transferring a communicable message through time and space, in so as to relay a message to others, via the general venue of its Lagrangian-based spatial delineation, towards a given arbitrary respective technological device; It will consequently follow, that both the ensuing soon described general interdependent types of interactions, As Well As the ensuing soon described general interdependent types of scatterings, will thereby tend to influence the multiplicity of the general conveyance, of the earlier inferred communicable signal, in the following generic manner, that I am about to describe here: 

1) If there is to be a Wess-Zumino Interaction to be applied, in a  Gliosis-based manner,  upon the topological stratum, of the said cohesive set of vibrational oscillations, (of which is here, to act as that substrate of the implied communicable signal, of which is here to be relayed to a personal technological device), then, such a general type of a Wess-Zumino interaction, will consequently tend to work to form a Riemann scattering, upon the mentioned signal of communicable vibrational oscillation, to where the resultant of the spatial translation, that is of such a potentially obtained transferrable signal, that is here to be conveyed to another set of individuals, that are here to receive the inferred given arbitrary message, will thereby, in and of itself, make for a case scenario, to where there will consequently tend to be an increased probability, in which this will often tend to help at working to make for a crisper conveyance of a transferrable set of data. (In which there will consequently often tend to be the general physical condition, in which there will be relatively less of a chance, of the message being hindered, by the troublesome condition of "hissing.") ;

However:

2_ If there is to be a Civita Interaction to be applied, in a Gliosis-based manner, upon the topological stratum, of the said cohesive set of vibrational oscillations (of which is here, to act as that substrate of the implied communicable signal, of which is here to be relayed to a personal technological device), then, such a general type of a Civita interaction, will consequently tend to work to form a Rayleigh scattering, upon the mentioned signal of communicable vibrational oscillation, to where the resultant of the spatial translation, of such a potentially obtained transferable signal, that is here to be conveyed to another set of individuals, that are here to receive the inferred given arbitrary message, will thereby, in and of itself, make for a case scenario, to where there will consequently be an increased probability, in which this will often tend to work, at acting to decrease the general quality of the respective crispness of a transferred set of data. (In which there will consequently often tend to be the general physical condition, in which there will be a relatively heightened chance, of the message being hindered, by the troublesome conditions of "hissing.") Sam Roach.

Reason As To Tenses Of Co-Tangential Approach

 The reason as to why Noether-Based mass-bearing bosonic superstrings of discrete energy permittivity, are to work to bear a co-tangential approach, -- that is to tend to be of a hyperbolic manner -- is due to the general manner of the behavioral display of the Polyakov Action. Please let my explain: As the rate of the motion of a mass-bearing cohesive set of strings, that are here to work to perform a specific function (e.i. -- A Team Of String-Related Phenomenology), is to alter in either its rate and/or its direction of motion, when in relation to light, there is thence to be a perturbation of the directly corresponding individually taken light-cone-gauge eigenstates, that are here to be proximal local to the core-field-density, of each of the correlative respective superstrings of discrete energy permittivity, that are here to work to make-up/comprise this just inferred/mentioned team of superstring-related phenomenology (as being of a cohesive set of discrete energy quanta). As this is to occur, there is to be the proximal local physical attribute, of what may be termed of as being, the "Polyakov Action." (Which is the correlative inverse operation of the correlative Lorentz-Four-Contraction.) As this said "Polyakov Action" is thence to occur, -- the multiplicity of the correlative individually taken light-cone-gauge eigenstates, will quite often tend to work to bear a relatively peaking hyperbolic topological sway, in so as to work to accommodate the herein needed string-related flow, of the innate tendency of relativistic behavior, that is predominant in our world. Due to the general tendency of that wave-related arcing, that is here to be directly apperataining to the kinematic display of the exhibition of the Polyakov Action, the co-tangential approach of mini-string-related segmentation, as this is here to be in the general process of approaching the core-feild-density of the composite superstrings of discrete energy permititivty, that are here to work to comprse the multiplcity of the individually taken "teams" of such inferred quanta of energy, the co-tangential approach that is to be delineated towards such said strings, are thence to tend to be of a hyperbolic nature. Whereas; Since the earlier inferred "puncture," that is here to act as the individually taken "anuuli" of the core-field-density of such said strings, is inherently to not work to bear such an inextricable interdependence upon the general kinematic activiy of the Polyakov Action. This is, in general, a significant reason, as to Why the strings of such a case, in and of themselves, are to tend to be approached, in a hyperbolic manner; whereas, -- the multiplicity of the "puncture" of such inferred superstring-related individually taken fields (at the core-field-density), is to thereby tend to be approached, in a cartesian manenr. Sincerely, SAMUEL ROACH.

As To Hyperbolic/Cartesian Co-Tangential Approach

 Let us initally consider the core-field-density, of a given arbitrary Noether-based superstring of discrete mass-bearing energy permittivity. Let us next -- work to consider the general Ward-Cauchy-related condition, in which this inferred region of such a respective case scenario, is to act as a given arbitrary topological manifold, that is here to be converged upon, as a respective physical stratum, of what may here be termed of as being, the given arbitrary "co-tangential bundle," of such a given respective case scenario. Next; since the herein mentioned superstring of discrete energy permittivity of such a given case, is to be of a mass-bearing nature, -- the inferred mass-bearing region of discrete energy permittivity of this said case, is to be of the general nature, of being in the process of acting as a "bosonic" string. Let us next, consider the following logic, when this is here to be considered, in lieu of my particular model of string theory: 

Let us initially consider the actual holonomic topological structure, that is here to be of the actual proximal local physical entity, of the inferred closed-looped topological manifold of the said "bosonic" superstring itself, to act as the actual phenomenology of "holonomic substrate." 

Whereas; Let us initially consider the region of core-field-density, that is here to be Eminently Interior to the topological stratum of the said superstring itself, to act as the directly corresponding "puncture," of such a respective mentioned individually taken region of superstring-related core-field-density.

In a general description of the multiplicity, that is here to be of such a respective given case: As the general topological manifold of the actual string itself, is to act as being converged upon, by a tense of mini-string-related segmentation, it consequently follows, that this general genus of a co-tangential approach, is here to tend to be of a hyperbolic nature. (When of a general Noether-Related manner.)

Whereas; Also, as in a general description of the multiplicity, that is here to be of such a respective given case: As the general topological "manifold" of the earlier stated "puncture" itself, is to act as being converged upon, by a tense of mini-string-related segmentation, then it consequently follows, that this general genus of a co-tangential approach, is hereby to tend to be propagated towards the general inferred region, of that topological convergence, that is here to be approaching the inferred region of the "puncture"of the core-field-density, that is of such an inferred quantum of discrete energy permittivity, in a manner that is here to tend to be of a cartesian manner. 

I will continue with the suspense later! To Be Coninued! Sincerely, SAMUEL DAVID ROACH. (1989).

As To Hypothetically Piecing Together A Euclidean "Expansion" With A Clifford "Expansion"

 Let us initially say, that one were to be considering, the theoretical Laplacian-based piecing together, of two different cohomology-related projective trajectories. One of these two stated projective trajectories, is to work to be displaying the general nature, of exhibiting a euclidean-related "expansion"; whereas, -- the other of such stated projected physical trajectories, is here to work to be displaying the general nature, of exhibiting a Clifford-related "expansion." At the spot where these two just inferred cohomology-related patterns, are to be connectively braided, in a Laplacian-Based manner, there will often tend to be a greater probability, of having the proximal local presence, of a Lagrangian-Based Chern-Simons singularity, -- than if one were, instead, to be connectively braiding two different projected cohomology-related physical trajectories, that are here to be both of either a euclidean "expansion," or of a Clifford "expansion." SAM.

Automorphism -- Homomorphism -- Isotropic Stability -- Group Action

 When the given arbitrary behaviors, of all of those given arbitrary superstrings of discrete energy permittivity, that are here to work to comprise the respective net gauged activity, that is here to be of the directly associated cohesive set of discrete energy quanta, that such inferred individually taken superstrings, are thence to act together in unison as a "team," in so as to kinematically "animate" the group action, that is of the respective inferred overall set of discrete energy quanta, -- are thence to display the exhibition of the same given arbitrary genus, of a respective relativistic distinct tense of automorphic qualities; then, -- it consequently will tend to follow, that the earlier stated cohesive group, of such inferred discrete energy eigenstates, will thereby consequently tend to smoothly act in unison, as a homomorphic manifold of topological substrate, when this is to be considered as that respective given arbitrary group action, that is here to be exhibited, via the general processes of the gauged-action, that is to tend to be exhibited by the Lagrangian-based motion, of the kinematic overall isotropically stable holonomic spatial translation, that is of the inferred Hamiltonian Operator, of this particular case here.  SAMUEL ROACH. (1989).

Isotropic Stability -- Homomorphic Interactive Relation -- Homeomorphic Field -- Smooth Metric Of Eigenstate Translation

 When a given arbitrary Noether-based field, that is here to be comprised of, by a set of one or more cohesive eigenstates of energy-related kinematic spatial translation, over a set proscribed given arbitrary period of discrete time, (as this is here to be taken, via the course of an evenly-gauged Hamiltonian eigenmetric), as such a stated field, is here to work to bear a relatively high scalar amplitude of group-related action, then, not only will such an inferred cohesive set of eigenstates, will thereby consequently tend to work to bear the general physical attribute, of displaying a relatively strong tense, of working to bear the general condition, of exhibiting a directly associated homomorphic tense of an interactive relationship, as this is here to be incurred interdependently, both amongst and between each of the individually taken composite eigenstates, that are here to work in helping to form the inferred cohesive set of such an inferred "team" of energy-related states, -- to where, in and of itself, (meaning here, in this given arbitrary case scenario, when basically all of the other physical attributes of a Ward-Cauchy-related nature, that one would otherwise fane to tend to be inclined to consider as being "experimental factors", were to theoretically be of the same general genus of physical behavior and characteristic), such a general inferred condition of a superstring-related field, is then to be in such a "position," to where it is here to work to bear such a tense of a relatively heightened isotropic inter-relation of homomorphically interactive field indices, then it is then to thereby consequently follow; -- that such a given arbitrary mentioned respective cohesive set of energy-related field eigenstate(s), -- will thereby often tend to have a higher probability, of consequently resulting in working to display, both a relatively smooth metric of spatial eigenstate translation  (when in regards to the Lagrangian-based flow, that is here to demonstrably be directly associated with such a set of one or more cohesive energy-related eigenstates, as such a set of inferred metric-gauge related field operators, are here to be tugged along the general "venue" of the Rarita Structure, over the course of such an inferred durational attribute, that is here to be directly corresponding to the fitting involvement, of a correlatively respective given arbitrary Fourier Transformation), as well as the general resulting tendency, in which the directly associated Lagrangian-based field, by which the inferred set of energy-related eigenstates, of such a respective case, are here to be transferred via a correlative Fourier Transformation, in so as to consequently tend to result, in working to also bear a heightened probability, of often tending to work to bear a relatively stronger homeomorphic field, than it otherwise would. Sincerely, SAM. (1989).

As To Altering The Initial Polarization-Related Effect, Of Two Different Physical Li-Based Groups

 Let us say, one were initially to have two different and distinct Li-Based groups, that were here, to initially be adjacent to one another, at a level that was here to be Poincare to the eminently inferred region, at which these two different and distinct inferred cohesive sets of discrete energy quanta, were here to be iterating at -- at a virtual metaphorical "snapshot" of duration, to where such a said "snapshot" in "time," was here to be taken, over the course of a respective given arbitrary iteration of BRST. Let us next stipulate, that these two different and distinct cohesive sets of discrete energy quanta, during such an iteration of BRST -- are to be delineated at such a general given arbitrary distribution-related manner -- to where such inferred sets of discrete energy quanta, are here to bear a covariant tense of a Nijenhuis Orthogonal nature (which is here to bear a relatively strong manner of conformal invariance, at a vantage-point, that is Poincare to that internal reference-frame, that is here to be of the eminent field-density, of the so-inferred region), at which the said sets of energy quanta, are to be both co-determinable and co-differentiable, as those composite superstrings of discrete energy permittivity, that are here to be working to make-up the external structure of such inferred "teams" of discrete energy quanta, are to be working to display the general gauge-metric-related exhibition, by which the general virtual standstill of the kinematic activity at BRST, is thence to occur. Let us next stipulate, that these two respective distinctly different "teams" of energy, are here to be Polarized -- the one from the other, -- due to the general Noether-related norm-based physical condition, that the covariant tense of the dual orthogonal projection, that is here to be existent between/amongst these two said "teams" of discrete energy, is here to be of a Nijenhuis dual state of norm-related projection  -- by which these two different Li-Based groups, that are here to be working to bear a tense of a Ward-Cauchy-related manner of norm-related projection -- are thence to be "encoded for," as belonging to two different universal settings. (Each of such individually taken Li-Based groups, is to be of a parallel universe from one another, when this is here to be taken, in correlation of the one to the other, and vice-versa), -- while all the more, this given arbitrary case scenario is to be considered, As this is here to be taken, over a basically "Laplacian" general format of stipulation. Now here's the "catcher." What If one were to alter the inferred covariant norm-conditions of each of such cohesive sets of discrete energy quanta -- the one towards the other, and vice-versa, to where the general initially considered Nijenhuis tense of orthogonal behavior, that we were just starting to discuss here, -- were now to bear a kinematic perturbation, into spontaneously and consequentially, ensuing to change into a set of two different and distinct geometric groups (groupings), that are thence to result, into working to form a case scenario, by which one is now to have two different and distinct cohesive sets of discrete energy quanta, that are now to work to bear a norm-related geometry, in which the inferred physical spaces discussed here, will now work to bear a covariant Riemann-Related geometry, that is here to be of a Real Riemann tense of norm-related projection. If and when this is here to happen in such a given arbitrary case scenario, these two different initially inferred unique "teams" of physical spaces, will then tend to end-up resulting, in working to display the exhibition, of two different and distinct sets of energy quanta, that are, at this point in duration, to be of the same universal setting. To Be Continued1 Sincerely, SAMUEL DAVID ROACH. (1989).

Civita Interaction Being Made Gliosis Upon Certain Phenomenology

 When a given arbitrary Civita interaction, is here to work to bear a Yukawa-Related Coupling, in a Gliosis-Based manner, as this is to be taken upon the respective given arbitrary physical phenomenology, that is here to behave, as working to bear the physical construct, of a given arbitrary initial mappable-tracing, that is here to act as the eminent entity of a given arbitrary topological manifold, that is here to be displayed as exhibiting a De Rham cohomology, as this is here to be taken over a directly corresponding Fourier-Transformation; -- that it then is to consequently follow, that this inferred general genus of a given  arbitrary respective gauged-action, will consequently often tend to work to form a perturbation (alteration), in the general cohomology-related pattern,  that was here to be of the earlier stated initially considered De Rham cohomology, -- to where the initially inferred given arbitrary hermitian cohomology (that is here to work to help at describing, the general behavior of the said De Rham cohomology), to where this will thereby tend to result, into spontaneously altering the general physical manner of its exhibited mappable-tracing (to where this eludes, to the initial covariant topological mapping of the said De Rham cohomology), to where such an initially inferred hermitian cohomology, will consequently tend to ensue, into changing in its general genus of topological construct, to where it will "now" tend to work to display, the exhibition of a  Dolbeault cohomology, instead. I will continue with the suspense later! Sincerely, SAM ROACH.

Metric-Related Spur In Pulsation

 When an initially hermitian given arbitrary cohomology, is to spontaneously undergo a metric-related spur, when in regards to the respective perturbation of its covariant pulsation, as the inferred cohomology-related entity, is here to work to bear a correlative spatial translation, through the general medium of space-time-fabric, as this is here to be taken through a discrete Lagrangian, over a proscribed given arbitrary discrete period of relativistic time, — this general process, will consequently tend to work to form a metric-related Chern-Simons singularity, at the covariant locus, (such a said locus, is to be super-conformally invariant, at the vantage-point of its internal reference-frame), to where at which such a mentioned “spur” was here to occur. To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Strings And "Touch"

 Superstrings of discrete energy permittivity that "directly interact," tend to interact -- via their immediately external field density; Superstrings tend to not directly strike each other verbatim. In my model of string theory, the general phenomenology, of which is to tend to directly come into contact with superstrings, in a heuristically Gliosis manner, are Associations of "first-order point particles," to where such inferred associations, tend to be of four different general varieties of norm-state-projections: zero-norm-state-projections; Campbell norm-state-projections; Hausendorf norm-state-projections; And -- Campbell-Hausendorf norm-state-projections. These just mentioned types of norm-state-projections, are the general phenomenology, of which are to directly "touch" the actual topological surface, of the multiplicity of superstrings of discrete energy permittivity. The multiplicity of the indistinguishably different interaction(s) of such "touch," works to form cohomology-related eigenstates. The general kinematic action, of the multiplicity of the herein inferred cohomology-related eigenstates, tends to reverse-fractal, into the general phenomenology of "charge." Sincerely, Samuel David Roach. To Be Continued! (1989).

Vibrational Oscillation -- Translated Through A Mass-Based Medium

 When a given arbitrary vibrational oscillation (of which is here, to simply be a vibrational disturbance in space-time-fabric), is to be harmonically translated through a mass-based medium, this general process, will often tend to work to form the mappable-tracing, of a De Rham cohomology. However; When a given arbitrary vibrational oscillation (of which is here, to simply be a vibrational disturbance in space-time-fabric), is to be anharmonic in its translation through a mass-based medium, this general process, will often tend to work to form the mappable-tracing, of a Dolbeault cohomology. SAM ROACH. (1989).

More As To Net Braided Cohomology-Related Mappable-Tracing

 When one is theoretically to consider having the mappable-tracing, of two different distinct Laplacian De Rham cohomology-related patterns, that are here to be linked, via a hermitian singularity -- the resultant net braided cohomology thus formed, will consequently tend to be of a De Rham cohomology-related pattern, as well. I will continue with the suspense later! To Be Continued! Sincerely, Samuel David Roach. (1989).

A Little As To Net Braided Cohomology-Related Mappable-Tracing

 When the Laplacian mappable-tracing of two different distinct De Rham cohomology-related patterns are linked, via a Lagrangian-Based Chern-Simons singularity, the resultant net braided cohomology thus formed, will consequently tend to be of a Dolbeault cohomology-related pattern. To Be Continued! Sincerely, SAM ROACH. (1989).

Tense Of Cohomology And Sound/Heat

 When a vibrational oscillation, (of which is simply here, in this given arbitrary case scenario, to be a general vibrational disturbance in space-time-fabric), is to be propagated through a mass-based medium, in a manner that is here to work to be exhibiting a De Rham cohomology -- this general process, will often tend to have a greater affinity, of working to form sound than heat. However; When a vibrational oscillation, (again -- of which is simply here, in this given arbitrary case scenario, to be a general vibrational disturbance in space-time-fabric), is to be propagated through a mass-based medium, in a manner that is here to work to be exhibiting a Dolbeault cohomology -- this general process, will often tend to have a greater affinity, of working to form heat than sound. Sincerely, SAM ROACH. (1989).

The Main Difference Between Voltage AND The Main Influence Formed By The Force Of Pressurized Vacuum

 The Main difference between Voltage AND the Main Influence that is created by the Force of Pressurized Vacuum (this Main Influence, of which is a general tense of a metric-gauge-related field), is:

Voltage is kinetic energy per charge; Whereas, the Main Influence that is created by the force of pressurized vacuum -- is equivalent to energy in static equilibrium, of which tends to be in a state of super-conformal invariance at an internal reference-frame, per charge. I will continue with the suspense later! To Be Continued! Sincerely, SAM. (1989).

Lack Of Holomorphic/Anti Holomorphic Flow Of Spatial Dimensions -- Betti Action Jacobian Stabilization (BJS)

 The Betti number, is an imaginary number ordering.  So, when I have accidentally implied that it is a Real number ordering, what I mean, is the equivocal Real Riemann value, that may be taken, in consideration of the respective Betti number  For instance; If the Betti number is “i,” the equivocal Real Riemann value for the Betti number would be “1.”  If the Betti number is “2i,” the equivocal Real Riemann value for the Betti number would be “2.” If the Betti number is “3i,” the equivocal Real Riemann value for the Betti number would be “3.”  This is the pattern that I have intended for this.  

When a superstring of discrete energy permittivity:  which is here to be both harmonically spatially translated, during the general course, of going into a correlative iteration of BRST, as well as that the general physical condition of such a said superstring, is here, to simultaneously, via the vantage-point of a central coni-axion, to thence Lack the general activity, of working to bear a holomorphic spin, in any number of spatial dimensions, during the respective general process of going Into the herein mentioned iteration of BRST; And,  -- When such a said superstring of discrete energy permittivity, which is both to be harmonically spatially translated, during the general course, of going into the correlative iteration of the directly corresponding Regge Action, as well as that the general physical condition of such a said superstring, is here to Lack the general activity, of working to bear an anti holomorphic spin, in any number of spatial dimensions, during the respective general process, of going into the mentioned iteration of the Regge Action; -- Then, -- one may consequently venture to say, -- that such a said superstring of discrete energy permittivity, that is here to be behaving in such a herein described manner, is thence to tend to not be in the process of altering, in the display of its resultant number of spatial dimensions, that it is here to be exhibiting, during the general course of such an inferred iteration of instanton. Thereby; such a said superstring of discrete energy permittivity, that is here to be of such an inferred case scenario, is consequently to result in tending to work at displaying, the general physical condition, of which I happen to name of as being, "(The) Betti Action Jacobian Stabilization." (BJS). I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

More As To Characteristics Of Lagrangian-Based Flow

 When a Noether-Based mass-bearing cohesive set of discrete energy quanta, is to work to bear a generative hermitian Lagrangian-Based flow, as taken Through a mass-bearing medium -- the directly corresponding motion of such a flow of an inferred set of mass-bearing energy quanta, will consequently tend to work to bear a stronger affinity for generating the general physical phenomenology of sound, than it will for generating the general physical phenomenology of heat. It follows then; that when a Noether-Based mass-bearing cohesive set of discrete energy quanta, is to work to bear a generative spurious Lagrangian-Based flow, as taken Through a mass-bearing medium -- the directly corresponding motion of such a flow of an inferred set of mass-bearing energy quanta, will consequently tend to work to bear a stronger affinity for generating the general physical phenomenology of heat, than it will for generating the general physical phenomenology of sound. To Be Continued! SAMUEL DAVID ROACH. (1989).

Hermitian Flow Of Set Of Energy -- Spurious Flow Of Set Of Energy

 The more of a generative hermitian Lagrangian-Based flow, through which a given arbitrary Noether-Based cohesive set of discrete energy quanta, is to undergo -- the more harmonic that the flow, via which its directly corresponding cohomology-related eigenstates are to be generated, will thereby tend to be. It follows, then; that the more of a generative spurious Lagrangian-Based flow, by which a given arbitrary Noether-Based cohesive set of discrete energy quanta, is to undergo -- the more anharmonic that the flow, via which its directly corresponding cohomology-related eigenstates are to be generated, will thereby tend to be. To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

A Little Bit, As To What Is Meant By Saying -- The "Vector Bundle"

What is meant by the term -- The "vector bundle," -- is a general means, of mathematically working to help  at describing, the topological construct, of a discrete quanta/quantum of energy. It consequently follows; that the general multiplicity of the Hamiltonian-Related "vector bundle," is a means of mathematically working to help at describing, the topological construct of a "Thing" or "Holonomic Entity" of a discrete quanta/quantum of energy. Whereas; the general multiplicity of the Lagrangian-Related "vector bundle," is a means of mathematically working to help at describing, the topological "construct,' of the Motion, of such an earlier mentioned Holonomic Entity of discrete energy quanta/quantum. To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989). 

Homeomorphic Field During The Betti Action

 The multiplicity of the general Noether-based field, that is here to exist, during any given arbitrary iteration of BRST -- when both in-between and amongst a superstring of discrete energy permittivity And its directly corresponding counter string (of discrete energy permittivity) -- is to tend to work to bear a homeomorphic nature, -- on account of the general Ward-Cauchy-related physical condition, that is here to tend to exist, during any of such individually taken implied iterations of BRST --  that, a Noether-based superstring, during the Betti Action, is to tend to work to bear a covariant asymmetric symmetry, with its directly corresponding counter string (of discrete energy permittivity). This general tendency, for individually taken Noether-related quanta of discrete energy permittivity, is to generally tend to be the norm -- in spite of any potentially proximal local rotational or spin-related gauged-actions, by which such strings and their correlative counter strings are to display, due to the general physical condition, that is here to occur during the process of the Betti Action, in which the multiplicity of the gauged-action of superstrings, when in relation to the gauged-action of their directly corresponding counter strings, is to tend to bear an inversely mirrored covariant delineation of asymmetry, to where the correlative phenomenology, that is here to exist at the general relatively internal region, that is here to be proximal local to the pressurized vacuum that is to exist in-between the said superstring and its counter string, is to work to bear a tense of a covariant indistinguishable difference, that is here to act as a fractal of a tense of an isotropic-related field mode. To Be Continued! (U of M). Sincerely, SAMUEL DAVID ROACH. (1989).

More About Possible Equations For Vector Bundles

 Let us now consider, in this particular case scenario, -- a Ward-Cauchy-related situation, in which one is to define the Motion of a given arbitrary cohesive set of discrete energy quanta -- to Act as being the "Lagrangian-Based Operator." Here and once again; this is to where one is to be considering, the actual phenomenology or "thing" of the said cohesive set of discrete energy quanta, to act as the prior referred to respective Hamiltonian Operator. This would consequently be working, to define the actual Motion of the mentioned Hamiltonian Operator (a.k.a the actual Motion of the mentioned entity, that is here to be of the phenomenology of the said cohesive set of discrete energy quanta, to be the Lagrangian of the herein considered set of energy quanta), to behave of as displaying the general physical characteristic, of exhibiting the nature of the said attribute of a Lagrangian-Based Operator. Think of the consequential directional-related metric-based phenomenology, that is here to be directly associated, with such a directly corroborative stated Lagrangian-Based Operator (a.k.a.; Think of the consequential directional-related metric-based phenomenology, that is here to be of such a directly corroborative stated Motion of such a mentioned Hamiltonian Operator), to act, as displaying the physical characteristic, of being exhibited, as a "vector bundle." In such a given arbitrary respective case; one is here to minimize the focus, upon any invariably latent Nijenhuis tensors. Consequently; one may be able to work to determine, certain equations, -- that may be derived, in so as to help to mathematically describe the general Nature, of both the non abelian tense of such an implied Lagrangian-Based Vector Bundle, as well as deriving the mathematical description of the general Nature, of the abelian tense of such an implied Lagrangian-Based Vector Bundle. The following two equations, is what I happened to come up with, in regards to this.:

The General Equation, For The Lagrangian-Based Reductional Non Abelian Vector Bundle:

[[[The Real Riemann-Related Component Of The Force Of The Lagrangian-Based Operator] - 

[The Imaginary Component Of The Force Of The Lagrangian-Based Operator] - 

[(i*The Directly Corresponding Lagrangian-Based Operator)/(2*PI*D)]](as taken, via a correlative set of directorals)].

&; 

The General Equation, Of The Lagrangian-Based Reductional Abelian Vector Bundle:

[[[The Real Riemann-Related Component Of The Force Of The Lagrangian-Based Operator] -

[The Imaginary Component Of The Force Of The Lagrangian-Based Operator] - 

[(The Directly Corresponding Lagrangian-Based Operator)/(2*PI*D)]](as taken, via a correlative set of directorals)].

Please Let Me Know If These Two Basic Equations Are Here To Make Adequate Sense To You!

To Be Continued! Sincerely, SAMUEL DAVID ROACH.

Possible Equations For Vector Bundles

 Here is something that I mathematically derived, as to what the Potential mathematical expressions for the Hamiltonian-Related Reductional Non Abelian/Abelian Vector Bundles are to be, for cotangent bundle R(N).  This is when one is to consider, that the energy of a cohesive set of discrete energy quanta, is in this particular case, to act as the directly associated Hamiltonian Operator (as the overall energy of the system); and, if one is to consider "D," to work to represent the number of overall proximal local spatial dimensions, that are here to be present for the inferred Hamiltonian Operator, -- to where, "D" = "N."

Hamiltonian-Related Reductional Non Abelian Vector Bundle:

[[[The Real Riemann-Related component of the Force of the directly associated Hamiltonian Operator] - 

[The Imaginary component of the Force of the directly associated Hamiltonian Operator]] + 

[(The directly corresponding Hamiltonian Operator)/(2*PI*D)]] -- (as taken, via a correlative set of directorals). 

Hamiltonian-Related Reductional Abelian Vector Bundle:

[[[The Real Riemann-Related component of the Force of the directly associated Hamiltonian Operator] -

[The Imaginary component of the Force of the directly associated Hamiltonian Operator] -

[(i*(The directly corresponding Hamiltonian Operator))/(2*PI*D)]] -- (as taken, via a correlative set of directorals).

(P.S.: -- To those who are not much familiar with math, "i" = (The square root of (-1).))

Please tell me if these equations make an adequate amount of sense, since I have just recently derived this set of mathematical expressions! (P.S.: Dialogue works to indicate that NO ONE is perfect, -- and we can ALL learn from each other!) To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Non Abelian/Abelian Vector Bundle

 Let us, at first, consider the holonomic substrate of a given arbitrary cohesive set of discrete energy quanta, -- to act as being the overall energy, of a given arbitrary respective Ward-Cauchy-related system. This implied overall energy of a system, is often referred to as being called, the Hamiltonian Operator. The general physical attribute, of the group action-related behavior of a holonomic entity, may be thought of as being the general characteristic of isotropism. In this particular case scenario; let us minimize the focus upon the potentially proximal local presence, of any invariably latent Nijenhuis tensors. Let us next think of the general case scenario, in which the just mentioned Hamiltonian Operator, (in which the earlier stated cohesive set of discrete energy quanta, is here to act as the overall energy of the respective system), is to potentially directly interact, at the Poincare level, with the topological surface, of an ulterior Ward-Cauchy-related manifold, -- to where one may consequently say, that the said respective cohesive set of discrete energy quanta, that is here to be of the initially inferred general case at stake, is then to work to bear a potentially Gliosis contact, with the earlier inferred topological  manifold, that is here to act as the general type of a phenomenology, that is here to be potentially struck in a covariant manner, by the eminent impact of the motion of the said respective cohesive set of discrete energy quanta -- that I had initially brought-up in this discussion. Let us next call the direction-related tense of the holonomic entity, that is of the initially inferred set of energy quanta, to act as the Hamiltonian manner, of a kinematically delineated vector bundle. If the potentially impending direct contact of the Hamiltonian-related vector bundle, as interacting with the topological surface of that manifold, which is here to be potentially eminently struck, were to more than likely spontaneously work to bear the relative proximal local presence, of a tense of isotropic "slippage," then, one may consequetnly say, that the general characteristic of the reductional tense, of such an implied genus of a holonomic vector bundle, is then to be of the respective reductional nature, of acting as a Hamiltonian-related non abelian vector bundle. However; if the potentially impending direct contact of the Hamiltonian-related vector bundle, as with the topological surface of that manifold, which is here to be potentially eminently struck, were to more than likely spontaneously work to bear a relative lack of isotropic "slippage," then, one may consequently say, that the general characteristic of the reductional tense, of such an implied genus of a holonomic vector bundle, is then to be of the respective reductional nature, of acting as a Hamiltonian-related abelian vector bundle.  To Be Continued, As "Advertised"  -- "At A Computer Near You!" SAM.

Diffeomorphic Field Translated Through -- More Homeomorphic Transference

 The more diffeomorphic of a field of Hamiltonian Operand, that a given arbitrary mass-bearing cohesive set of discrete energy quanta, is to be spatially translated through, over time -- the more homeomorphic of a transference, that such a said cohesive set of discrete energy quanta, will consequently often tend to undergo. To Be Continued! Sincerely, SAM.

A Perception As To A View Of "Departure" Versus A Perception As To A View Of "Approach" (An Editorial)

 I perceive, that when the Zeta Function is set to "-i*(e^2)," that the directly corresponding Bessel Function, thus formed, will consequently be equal to "0."

Therefore; it follows then, that -- 

If one were to work to metaphorically define the "Departure," of what "moves" in the "direction" or "manner," of working to form the permittivity, of pressurizing the general eluded-to pure vacuum, -- that was here to exist, directly prior to the Action of the "Big-Bang," to tend to be:

[The eigenfunction "value" of {The Natural Log of {The Natural Log of (the hyperbolic sine of (-i*(e^2)));] which may be termed of as:

[The eigenfunction "value" of {ln(ln(sin(h)((Zeta^(-1)(0)))}]; ----

That; the directly corresponding metaphorical "Approach," of what "moves" in the "direction" or "manner," of working to form the impedance, of pressurizing the general eluded-to pure vacuum, -- that was here to exist, directly prior to the Action of the "Big-Bang," would thereby consequently tend to be:

[The eigenfunction "value" of {The Negative of the Natural Log of {The Natural Log of (the hyperbolic cosine of (-i(e^2)))]; which may be termed of as:

[The eigenfunction "value" of {-ln(ln(cos(h)((Zeta^(-1)(0))))}]. -- Please let me know what you think!

I will continue with the suspense later! To Be Continued! Sincerely, SAMUEL DAVID ROACH. (1989).

Smoothly Distributed Viscosity Of Space-Time-Fabric Traveled Through -- Less Spurious Of A Tense Of The i*PI(Del) Action

 The more smoothly distributed that the viscosity of space-time-fabric, of which a mass-bearing cohesive set of discrete energy quanta, is to be traveling through, over a given arbitrary discrete duration of time, the  greater of a correlative probability that there will consequently tend to be, as to there being a resultant relatively smooth proximal local Ricci Flow, of which the cohomology, that is here to be generated by the said mass-bearing cohesive set of discrete energy quanta, is thereby to consequently tend to exhibit, over the earlier inferred discrete period of time. The smoother of a Ricci Flow, that a given arbitrary cohomology-related tense, of a spatially transferred mass-bearing set of discrete energy quanta, is to work to display, the less spurious that the directly corresponding i*PI(Del) Action, of such an inferred set of mass-bearing energy, will consequently tend to result in displaying, over time. If follow then; that the more smoothly distributed that the viscosity of space-time-fabric, of which a mass-bearing cohesive set of discrete energy quanta, is to be traveling through, over a discrete period of time, the less spurious that the directly corresponding i*PI(Del) Action, of such an inferred set of mass-bearing energy, will consequently tend to result in displaying. Sincerely, SAMUEL DAVID ROACH. (1989). 

Initial Calabi-Yau Manifold And Ricci Flow

 Let us initially consider the general physical condition, that a heuristic Calabi-Yau Manifold, is to tend to work to exhibit a flat Ricci Curvature. (When it is to be exhibiting the general Ward-Cauchy condition, of a Noether-Based flow.) If one is to have a mass-bearing cohesive set of discrete energy quanta, that is to initially act as exhibiting the general characteristic, of working to bear the heuristic tense of a Calabi-Yau Manifold, -- to where such an inferred cohesive set of discrete energy quanta, is to subsequently begin to spin at a relatively high rate, due to the induced effect of having an anti gravitational field applied upon it -- this will consequently tend to result in  the general Ward-Cauchy-related condition, to where such a said respective mass-bearing discrete energy quanta, that is here to be brought into such an ensuing physical state, of such an inferred tense of spinning, will thereby tend to work to bear more of a recursively smooth Ricci Flow, -- than if such an inferred set of energy, were instead, to NOT initially work to bear such a stated heuristic tense, of acting as a Calabi-Yau Manifold. Sincerely, SAMUEL DAVID ROACH. (1989).

Electrodynamic Permeability -- Heightened Probability Of Smooth Ricci Flow

 The greater the electrodynamic permeability is to be, the more likely that the directly effected inferred respective mass-bearing cohesive set of discrete energy quanta, -- will consequently tend to potentially work to bear a greater probability, of exhibiting a relatively smooth Ricci Flow. Sincerely, SAM ROACH.

Electrodynamic Permeability -- Wave-Tug

 The greater that the electrodynamic permeability is to be, for a given arbitrary correlative mass-bearing cohesive set of discrete energy quanta, -- the more facilitated that the respective correlative wave-tug is to tend to be, for the corroborating angular momentum eigenstates, that are here to be most directly associated with the physical entity, that is here to be of the directional drive of the said respective mass-bearing cohesive set of discrete energy quanta. I will continue with the suspense later! Sincerely, SAMUEL DAVID ROACH. (1989).