WO2017155593A2 - Variable hydraulic press force f is derived from no fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol h gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid - Google Patents

Abstract

Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic- enthalpy a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction carbon dioxide C0₂, nitrogen, hydrogen, methane, and phosphane pipeline standard a heater fluid conducting operational flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity vector₁ q = ||u|| and is also a second scalar field at program temperature fluid periodic of the system fourier space shift distance modulation q Euclidean algorithm integers p and q that p ≥ q fluid dynamics, the continuous phase of dynamic fluid mechanics turbine-tubine powerplants.

Description

Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation ¾toms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

Specification

Background of the Invention

0001 Global warming and growing concerns of Greenhouse gas emissions from our dependency on fossil fuels are creating real demand and awareness for an powerplant and powerelectric alternative to fossil fuel engines and expensive replacement parts.

Brief Summary of the Invention

0001 Scientific utilization of preform technology the variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H ΔΕ = v = h Bo-Be (with γ gyromagnetic ratio) degenerate, and the gasiform species are derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter fluid or the quantumstatei particulate matter conducting operational flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity vectori q = ||u|| and is also a second scalar field at program temperature fluid variable hydraulic press periodic of the system fourier space shift distance modulation q Euclidean algorithm integers p and q that p > q fluid dynamics, and cryoplate refrigeration particles in the continuous phase of dynamic fluid mechanics, turbine-tubine powerplant.

Detailed Description of the Invention

0001 The variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) degenerate, and the gasiform species are derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter fluid or the quantumstatei particulate matter conducting operational flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity vectori q = ||u|| and is also a second scalar field at program temperature fluid Variable hydraulic press Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class¹ system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017. ^"■■

periodic of the system fourier space shift distance modulation q Euclidean algorithm integers p and q that p > q fluid dynamics, and cryoplate refrigeration particles in the continuous phase of dynamic fluid mechanics, turbine- tubine powerplant. Refrigeration vacuum pressure pumps rated alternating current AC or direct current DC drawing refrigerant through insulated super alloy feeder tubes in a parallel circuit to separate numeric super alloy curved or linear mathematic expression numbers, symbols and dynamics operators comprising a scientific aperture, control valve, or control valve venturi a vector calculus fundamental theorem limits of functions continuity, mean value theorem, by way of real-valued function / is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and /(a) = /(b), then there exists a c in the open interval (a, b) such that f (c) = 0, continuum mechanics differential, integral, series, vector, multivariable, and discrete entity, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point, condensed phases super- fluid as it is cooled, molecular diffusion coefficient' the relevant vector field is the velocity of motion fluid at a point, the fluid cools and contracts, the divergence has a negative value, as the region is a sink, measured in angulation and a scientific aperture, control valve, or control valve venturi molecular diffusive fluid refrigerant chemistry continuum vacuum mechanics pressure pumps negative or positive pressurization stream is the accelerator ± fluid mechanic the flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity-vector i q = ||u|| and is a scalar field, positive pressurization supercooling refrigeration deceleration temperature less than or equal to < -109°F degrees Fahrenheit or -78.5°C degrees Celsius supercooled fluid refrigerant passes over super alloy and through cooling fan. Numeric isolated super alloy cylinders in series and parallel refrigerant-dynamic conductive circuits are connected to the portable or stationary refrigeration unit from super alloy connecting rods supercooling the outside walls of the cylinders. The outer cylinder cools activating an electronic air compressor rated alternating current AC or direct current DC filling the inner variable cylinders with fluid compression providing diffusive flux compensation expanding or contracting the inner cylinders as fluctuating compensator forcing the carbon dioxide C0₂ filled outer cylinders through a continuous circuit of super alloy curved or linear mathematic expression numbers, symbols and dynamics operators comprising a scientific aperture, control valve, or control valve venturi a vector calculus fundamental theorem limits of functions continuity, mean value theorem, by way of real-valued function / is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and /(a) = /(b), then there exists a c in the open interval (a, b) such that f (c) = 0, continuum mechanics differential, Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

integral, series, vector, multivariable, and discrete entity, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point, condensed phases super-fluid as it is cooled, molecular diffusion coefficient the relevant vector field is the velocity of motion fluid at a point, the fluid cools and contracts, the divergence has a negative value, as the region is a sink, measured in angulation and a scientific aperture, control valve, or control valve venturi molecular diffusive fluid refrigerant chemistry continuum vacuum mechanics pressure pumps negative or positive pressurization stream is the accelerator ± fluid mechanic the flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity vectori q = ||u|| and is a scalar field, positive pressurization supercooling refrigeration deceleration temperature less than or equal to < -109°F degrees Fahrenheit or -78.5°C degrees Celsius carbon dioxide C0₂ forming dry ice in the outer superconducting cryocircuit numeric and hypernumeric cylinders vacuum component hypernumeric series and parallel superconducting cryocircuit processor molecules oscillations superconducting cryospheres,

superconducting cryoplates, superconducting cryotubes of organic dispersion computational integral analytical organic solid-state chemistry program processor flux compensators, superconducting cryocoils and substrate spectral parallel lines superconducting across transitive property of equality if a = b and b = c, then a = c having one of the equivalence property of equality and inequality cryolines. Supplemental component normalization of program temperature modulation q Euclidean algorithm integers p and q that p > q fluid dynamics, spectrometric computational integral analytical organic solid-state chemistry program processor, cryoplate refrigeration particles in the continuous phase of dynamic fluid mechanics, in transmittance U ratio of incident radiation to transmitted radiation function hypernumeric comparator processor hyper-transition superconducting cryotanks,

superconducting cryotubes of organic dispersion computational integral analytical organic solid-state chemistry program processor flux compensators, superconducting cryocoils and superconducting cryocircuit coupled superconducting cryogenic substrate transition refrigerant fluid filled or unfilled refrigeration natural gas, organic waste or biomass to synthesis gas chloride CI^", nitrogen, hydrogen, methane, phosphane, and argon glazing > - 45°C, comparative liquid and solid chloride CI^" superconducting cryotank, coupled substrate superconducting cryocircuit the transition refrigerant fluid filled or unfilled superconducting cryotanks and are vacuum component hypernumeric series and parallel superconducting cryocircuit processor molecules oscillations superconducting cryospheres, superconducting cryoplates, superconducting cryotubes of organic dispersion computational integral Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

analytical organic solid-state chemistry program processor flux compensators, superconducting cryocoils and substrate spectral parallel lines superconducting across transitive property of equality if a = b and b^'= c, then a = c having one of the equivalence property of equality and inequality cryolines neutral nitryl radical, symetrical corners chlorine CI^" molecule configuration joined at central carbon atom, tetrahedral arrangement neutral radical pairs oxygen 0₂ and helium He crystallographic phase network superconducting cryotube, flux compensator and superconducting cryocoil helium He gas, isolated antiprotonic helium three-body atoms He superconducting are the supercooling cryotube, flux compensator and is the superconducting cryocoil circuit in transmittance U ratio of incident radiation to transmitted radiation.

0001 Nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 , 1,1 , 1 , 1 , 1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1,1 ,1 , 1 ,1 and number base point k- face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- ^' poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

0001 Banded on equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring, nl , n2, n3 circumscript laterally through the interior shape area the intersection of three body in vector space with a plane, and function of maximum entropy rotation the base of the plane continue natural logarithm, resulting entropy by mathematical formalism, a constraint of a probability distribution in the case of a mean value constraint, the insertion of an exponential factor εχρ[-λί(η)] with an adjustable Lagrange multiplier λ maximum entropy

distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy distribution with specified mean μ and standard deviation σ, and the analog a in higher dimensional space symetry resection Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

position base of the plane in alignment with movement along equipotential line output the external consecutive magnetic fieldsi₂3 B₀ Bi, B₂, B₃, open dengenerate p(n), loop degenerate, or closed degenerate three quantum transitions between spin states Βι₂₃ ΔΕ = v = h Bo-Be (with γ gyromagnetic ratio) Bi + ΔΕ = v = h Bo-Be (with γ gyromagnetic ratio) B₂ + ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) B₃ + ΔΕ are the energy difference between spin states, space integral sub-plasma thermal cycle No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) degenerate, and the gasiform species heater derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation, atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid.

0001 The frequency v wavelength λ and speed of light c, are related by λν = c, amplitude of electric field E = F/q and magnetic field B, H, or Tesla nT. Triple-photon homodyne Fock reconstruction of the single-photon

consecutive three dimensional 3D optical drawing first and second common system with longinitude and exponential form shielding orders of optical cables projecting windows nT frames four rectangles A/B matter inner band red lines and ultra violet lines mass turbulent energy near and far begin visible violet, magenta, cyan, blue, yellow, and green wavenumbers wavelengths sine 12 wavenumbers wavelengths, triangle wavenumbers

wavelengths, square wavenumbers wavelengths, and sawtooth wavenumbers wavelengths continue by prismatic fourth band AU quantum transition ABCD circumscribed sphere, and rectilinear lines reconstruct 150-200 nm nanometer circuit band gap energy equilateral sides skew quantum energy wavenumbers inner and outer

descriptions bands balanced homodyne tomography reliable technique, high matter wave speed cl , high matter wave speed c2, and high matter wave speed c3 by way of exact relativistic linear mathematic expression numbers, symbols and dynamics operators for momentum and kinetic energy, equation polar, non-polar ionic Φ (ionic potential)* = Z/r; where Z = ionic charge (+1); and r = ionic radius (A) have energy the transfer of electrons in ionic bonds and covalent chemical bonds sharing a pair of valence electrons by two atoms, having a cumulative energy- gap are all related ΔΕ = hv.

0001 Tesla symbol nT magnetic flux density functional theory commonly B magnetic field symbol Ω, identified in the Planck constant denoted h is a physical constant that is the quantum of action in quantum mechanics, the Planck constant describe the proportionality constant between the energy E of a photon and the frequency v of its associated electromagnetic wave, this relation between the energy and frequency is called the Planck relation E = hv the frequency v wavelength λ and speed of light c, are related by λν = c, amplitude of electric field E = F/q and Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

magnetic field B, H, or Tesla degenerate nT. The spectroscopy surface analysis technique known as electron energy loss spectroscopy (EELS) quantum electrodynamic QED triple-photon homeodyne Fock reconstruction of the single-photon consecutive three dimensional 3D optical windows frames nanosecond balanced homodyne tomography reliable technique 150-200 nra nanometer circuit, the Planck relation for a photon is expressed as E = hc/λ, spin state, and applied magnetic field, the energy difference ΔΕ between spin states is proportional to the strength of B₀, applied magnetic field degenerate nT, the frequency c = λί of Bi₂₃ in frequency for matter, three quantum transitions between spin states are Βι₂₃, ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) degenerate, ΔΕ is the energy difference between spin states, h is Planck's constant, v is the frequency of Bi₂₃ field Be is the magnetic field generated by the circulation of electrons of specification matter molecules, B₀ is the strength of the magnetic field, γ gyromagnetic ratio is a constant which is a property of the particular nucleus, relationship between field B₀ and frequency v, ΔΕ = h x frequency nl + ΔΕ = h x frequency n2 + ΔΕ = h x frequency n3 + ΔΕ, is the importance of B_e.

0001 Optical depending on the value of n, there is an angular momentum quantum number i and the following series n = 1, L = λ, series ultraviolet, n = 2, L = λ, Balmer series visible or violet, n = 3, L = λ, Ritz-Paschen series near infrared, n = 4, L = λ, Brackett series short- wavelength infrared, n = 5, L = λ, Pfund series mid- wavelength infrared. Higher-order wave equations and matter waves the most common "wave equation" in one space and one time dimension is ^Ψ/^χ² = 1 /V² ^Ψ/δί² (1) where V is a constant equal to the phase velocity of a monochromatic wave. This has the general solution f(x+Vt) + g(x-Vt) for arbitrary functions f and g. Among the particular solutions of this equation are the familiar monochromatic solutions of the form Ψ(χ, t) = A cos(kx-cot) (2) where A is an arbitrary constant. The second partial derivatives of this function with respect to x and t are -Ak cos(kx-cot) and -Aco²cos(kx-oc>t) respectively, which implies V = co/k, showing that this particular Ψ is strictly a function of x- Vt, as a retarded wave. The amplitude of this function is purely real (assuming the constants and variables are real- valued), with a value that oscillates up and down corresponding to the projection onto the real axis of a point moving around in a circle in the complex plane. We could just as well include the imaginary component complex number i² = -1 , since the complex function Ψ(χ, t) = A [cos(kx-cot) + i sin (kx-oot)] = Ae'^(kx"wt) (3) is also a solution of (1). In fact, this simple function is a solution of a wide range of wave equations, some of which turn out to have important physical consequences. In general, for any positive integers r, s, consider the partial differential equation <9^χΨ/δχ^Γ = a5^S F/dt^s (4) where a is an arbitrary constant. Substituting the trial solution Ae^l(kx"wt) into this equation, evaluating the derivatives, and dividing through by ε'^(Ισ£"ω1), we get (ik)^r = a (- ico)^s This shows that (3) is a solution Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

of every one of the higher-order wave equations l/(ik)^r cf^/dx = l/(-ico)^s 3^S F/dt^s (5) The constant in equation (4) is a = (ik)7(-ico)^s The phase velocity of the wave is still V = ω/k, so if r = s we have a = (- 1/V)^r, and hence V = - l/a^1/r. (Noting that if m is even we can select the negative root to give a positive phase speed.) The energy E and momentum p (in one dimension) of a photon can be expressed in terms of the angular frequency ω and wave number k respectively as E = ηω p = hk where h is Planck's (reduced) constant. Substituting into the relativistic relation between energy, momentum, and rest mass m gives c⁴m² = E² - c²p² = h(oo² - c²k²) where c is the speed of light in vacuum. Consequently, if the phase speed of the photon equals c, meaning that V = ω /k = c, the right hand side vanishes, so the rest mass m is zero. On the other hand, if the phase speed V of the wave is^'not equal to e, the rest mass will be non-zero. This suggests that we might regard a particle of mass m and velocity V as a "matter wave" with the frequency ω and wave number k proportional to the energy and momentum respectively, in accord with the above equations for E and p. Returning to the generalized wave equation, we can express the constant k and ω in terms of the energy and momentum of the matter wave, and write the generalized wave equation in the form δ^ΓΨ/δχ^Γ = (-l )^s(i/h)^r"s p^r/E^s (6) For a photon we have E/p = c, so in order to eliminate E and p from the wave equation, arrive at a constraint that is independent of the wave, we need r = s, in which case (6) reduces to the form 3^S P/dx^s = (- l)7c^s d^S F/dt^s (7) With s = 0 this reduces to the trivial identity Ψ = Ψ, whereas with s = 2 it is the standard wave equation for electromagnetic waves. By construction, the simple wave function (3) satisfies this equation for every value of s, and so too (fortuitously) does the purely real wave function (2). In addition, each of these equations is consistent with all three of the conditions E = hco, p = hk, and E = pc, where c is the phase speed of the wave. Now, for a particle of rest mass m moving at the speed v, the momentum and energy are given by p = mv/V 1 -(v/c)² E = mc²/V 1 -(v/c)² (8) From the relations E = hoo and p = hk this implies that the phase velocity V is V = ω/k = E/p = c²/v (9) Of course, if v = c this gives the phase velocity V = c, which applies to the case of a photon. However, for a massive particle with v less than c, this formula may seem surprising at first, because it implies that the phase velocity of a matter wave exceeds c, and in fact goes to infinity as the speed v of the particle goes to zero. This might seem to suggest that the wave velocity has no significance, and that we cannot associate the wave with the particle. Fortunately, this apparent problem disappears when we remember that the actual transmittal of energy in a wave is propagated not at the phase velocity necessarily, but at the group velocity v_g,. For various velocities associated with another wave, we can determine phase, group, and signal velocity. The group velocity is given by v_g = dco/dk = dE/dp = v so in fact the matter wave does propagate at the same speed as the corresponding particle. Eliminating v from the energy and momentum equations (8) leads immediately to the relation c⁴m² = E² - c²p² =

r Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

h(co² - c²k²) In the case of a photon we were able to go from equation (6) to equation (7), eliminating any explicit reference to the momentum and energy, and expressing the wave equation in terms of the constant c, because of the relation p/E = 1/c for massless particles. Thus if the indices r and s are equal, we can eliminate the ratio p/E. Not true for a particle with non-zero mass, so continuing an analogous method of eliminating the explicit appearances of E and p in equation (6), as one method of replacing them with some function of the invariant rest mass m of the particle. First, we should note that when we carry over the equation E = co from photons to massive particles there is an ambiguity of interpretation of the quantity of energy represented by "E". Since a photon has no rest mass, its total energy and kinetic energy (total minus rest mass energy) are the same, so this is unambiguously the quantity represented by E. However, for a massive particle the total and kinetic energies are different, then determining which of these quantities is applicable for a matter wave. Continuing to determine if the wave that corresponds to the total energy of the particle, or in the wave that corresponds to just the kinetic energy associated with the motion of the particle. Since these two energies differ only by a constant (the rest mass of the particle), they propagate with the same group velocities, but they have very different phase velocities. We saw in equation (9) that the phase velocity V of the total energy of a particle moving at the speed v is c²/v, but the phase velocity of the particle's kinetic energy is given by V = cokinetic/k = E - mc²/p = c²/v (1 - Vl-(v/c)²) = v/2(l + ¼(v/c)² + ...) assuming the momentum is taken to be the same in both cases. Now, it might be argued that we are mixing incompatible quantities here, because the wave number k is still taken to be p/h, meaning that it is proportional to the momentum of the total mass, not just of the kinetic mass. We should attempt to interpret p in the relation p = hk as just the "kinetic momentum", the velocity times the difference between the total relativistic mass and the rest mass, and then we would (again) have the phase velocity V = c²/v. However, if we did this, the group velocity would no longer equal v, because the difference between the total momentum and the "kinetic momentum" is a function of v. Moreover, by analogy with the energy, we note that each particle can be associated with a certain "rest momentum" just as it has a certain rest mass-energy, but the rest momentum of a particle is identically zero, so there is no distinction between its total momentum and its kinetic momentum. Consequently, we conclude that it's reasonable to treat the wave of a particle of rest mass m and speed v as having the wave number k = p/h and the frequency ω = Ekinetic/ . And so the equation (6) governing this wave (free of external constraints) is expected to be of the form d ¥/dx^r = (-l)^s(i/h)^r"s p^r/Ekinetic^S d^S F/3t^s (10) If v is small compared with c, then the momentum is essentially p » mv and the kinetic energy is Emetic ~ (l/2)mv . Eliminating v from these two equations, we have the relation p /Eki„etic « 2m (1 1) Therefore, in order to create a general constraint out of (10) by eliminating the ratio p^r/Eki_netic^S, we must Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class, system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

have r = 2s, in which case the wave equation becomes d^ /dx²⁵⁵ = (-2m/ hi)^s 9^S F/<¾^s (12) Again, with s = 0 this reduces to the trivial identity Ψ = Ψ, whereas with s = 1 as is the Schrodinger equation for the complex wave function of a free particle (with no potential field). By its construction, this wave equation for any non-negative integer s is satisfied by the complex-valued wave function (3). However, it is satisfied by the real-valued wave function (2) only for even values of s. Equation (12) is based on the non-relativistic equation (1 1). If instead we use the exact relativistic expressions for momentum and kinetic energy, equation (1 1) would become p /Ekihetic ⁼ [1 + l/Vl-(v/c)²]m (13) which confirms that this ratio is essentially equal to 2m for non-relativistic velocities. In general it equals the rest mass plus the relativistic mass of the particle. Focusing on the traditional non-relativistic

Schrodinger equation (12), it's interesting to consider the consequences of setting the index s to a value other than 1. The complex-valued wave function (3) is a solution of every one of these equations, but the equations are nevertheless distinct, as illustrated by the fact that although every solution with s = 1 is also a solution with s = 0, the converse is not true. Also, with s = 2 the equation is satisfied by the real-valued wave function (2), even though this is not a solution with s = 1. Of course, the choice of s = 1 makes the left hand side equal to the Laplacian, which we expect on physical grounds, but it would be interesting to know what (if any) phenomena could be modeled by the higher-order wave equations, both in the case of massless (r = s) and massive (r = 2s) particles. At the very least, the higher-order wave equation (10) is satisfied by any function (or linear combination of functions) of the form Ψ(χ, t) = Ae ^,(kx"cot+e) + f(_Xj t) where « is a constant phase offset and f is a polynomial of degree r-1 in x and degree s-1 in t. Needless to say, the above does not constitute an a priori derivation of the Schrodinger equation of a free particle.

0001 A more natural and well-motivated approach can be given in terms of the Hamiltonian operator whose eignevalues are the possible energy levels of the particle. Nevertheless, it's interesting to see how the general outline of Schrodinger's equation in relation between the energy and momentum of a particle as represented by a wave with the same group velocity as the particle. This approach also clarifies that a matter wave actually has two distinct phase velocities, one (c²/v) corresponding to the total energy of the particle, and the other (~v/2)

corresponding to just the kinetic energy of the particle. For an atomic orbital an atomic electron's angular

2 2

momentum, L, is related to its quantum number I by the following equation L Ψ = h l(i + 1) Ψ where h is the reduced Planck's constant, L² is the orbital angular momentum operator and Ψ is the wavefunction of the electron. 0001 Catalysis of the wave equation circuit second coherent coordinates Wigner function common analogues optical windows frames between energy of amperes A = C/s and λ of refraction index henries v = Ldi/dt, Px(P), or Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

W(X, P), or Px(X) Minimum zero base define on a domain W(X, P) has a global : absolute extremum prime and bias value function takes at a point relative extremum are calculated on the function domain absolute extremum the maximum and minimum element of a set maxima and minima for ϋθ5(3πχ) χ, 0.1< x <1.1 rotation elementary particle with a half-integral spin that obeys the Pauli exclusion principle.

0001 Electrons, protons, and neutrons are types of the fermion thermal renormalization group for higher

dimensional space rotating on an axis-line quantumstatei₂₃ reconstruction of the single-photon Fock state relative 1 : 3 triple-photon Fock states i₂₃ spherically symmetrical s orbital, the B₀ field the magnetic field causes the electrons in the cloud surrounding the nucleus to circulate through the orbital, this circulation caused Be an induced magnetic field opposed to the B₀ field and the value B₀-B_e represents the actual magnetic field B experienced at the nucleus, the B_e field produced by electrons in p and d orbitals are more complex formulated by proton chemical shift, and the analog a in higher dimensional space symetry resection position base of the plane in alignment with movement along equipotential line output energy of the photon must exactly match the energy-gap ΔΕ between the these states are now reconstructed in atomic triple statei₂₃ A spin triplet is a set of three quantum numbers of a system, each with total spin S = 1 (in units of h), nucleus of the atom nl , n2, and n3 catalysis of the wave equation coherent coordinates Wigner function common analogues spectroscopy optical windows frames computing nanosecond processing optical control [0, 1] following spectroscopic notation, the magnetic quantum number, and the spin quantum number. It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as i quanta specific discrete codimension cAt, wavelength points Δχ, Ay and along Az the integral phase of wave time and space coherence coordinate j and mj, together with the parity of the quantumstatei, replace the three quantum numbers Z, mi and ms the projection of the spin along the specified axis. The former quantum numbers can be related to the latter, and the eigenvectors of j, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of I, ml and ms waveguide.

Electron configuration quantum-mechanical nature of electrons an electron shell is the set of allowed states that share the same principal quantum number, n the number before the letter in the orbital label, which electrons may occupy. An atom's nth electron shell can accommodate 2n2 electrons, is the first shell can accommodate 2 electrons, the second shell 8 electrons, and the third shell 18 electrons. The factor of two arises because the allowed states are doubled due to electron spin-each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin + 1/2 noted by an up-arrow and one with a spin - 1/2 with a down-arrow. A subshell is the set of states defined by a common azimuthal quantum number, I, within a shell. The values 1 = 0, 1 , 2, 3 Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

correspond to the s, p, d, and f labels, respectively. The maximum number of electrons that can be placed in a subshell is given by 2(21 + 1). This gives two electrons in an s subshell, six electrons in a p subshell, ten electrons in a d subshell and fourteen electrons in an f subshell. The numbers of electrons that can occupy each shell and each subshell arises from the equations of quantum mechanics, in particular the Pauli exclusion principle, which states that no two electrons in the same atom can have the same values of the four quantum numbers. The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantumstatei₂₃ of an electron (the others being the principal quantum number N represents the relative overall energy of each orbital, and the energy of each orbital increases as the distance from the nucleus increases, the sets of orbitals with the same n value are often referred to as electron shells or energy levels, following spectroscopic notation, the magnetic quantum number, and the spin quantum number). It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as i. List of angular momentum quantum numbers derivation quantum mechanical orbital angular momentum. Connected with the energy states of the atom's electrons are four quantum numbers: n, i, mi, and ms. These specify the complete, unique quantumstatei₂₃ of a single electron in an atom, and make up its wavefunction or orbital.

0001 The wavefunction of the Schrodinger wave equation reduces to three equations that when solved, lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The azimuthal quantum number arose in the solution of the polar part of the wave equation and understanding the concept of the azimuth, spherical coordinate systems, and the cartesian coordinate system. Spherical coordinate system spherical models, the cylindrical system with cylinders, the cartesian with general volumes, the transverse Mercator projection is the transverse aspect of the standard or normal Mercator of the transverse cylinder to produce a secant projection with two standard transverse 2rp N-shift radius the midpoint of the segment (the transverse axis) connecting μΝ, and each of the different angular momentum states can take 2(2€ + 1) electrons. This is because the third quantum number mi (which can be thought of loosely as the quantized projection of the angular momentum vector on the z-axis) runs from - i to i in integer units, and so there are 2i + 1 possible states. Each distinct n, i, mi orbital can be occupied by two electrons with opposing spins (given by the quantum number ms), giving 2 (2i + 1) electrons overall. Orbitals with higher i than given in the table are perfectly permissible, but these values cover all atoms so far discovered. For a given value of the principal quantum number N, the possible Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

values ofi range from 0 to n - 1 ; therefore, the n = 1 shell only possesses an s subshell and can only take 2 electrons, the n = 2 shell possesses an s and a p subshell and can take 8 electrons overall, the n = 3 shell possesses s, p and d subshells and has a maximum of 18 electrons, or, the maximum number of electrons in the nth energy level is 2n2. The angular momentum quantum number, t, governs the number of planar nodes going through the nucleus.

0001 A planar node can be described in an electromagnetic wave as a median of a triangle is a line segment from a vertex of the triangle to the midpoint of the side opposite that vertex. Because there are three vertices, there are of course three possible medians i₂₃ exponents; all three always intersect at a single point. This point is called the centroid of the triangle. In quantum mechanics L by itself, L has meaning in its use as the angular momentum operator and simple use of the quantum number i. The principal quantum number is at the right of each row and the azimuthal quantum number is denoted by letter at top of each column. Atomic orbitals have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d describe the shape of the atomic orbital. Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials. The various orbitals relating to different values ofI are sometimes called sub-shells, and are referred to by letters, as follows i 0 letter s max electrons 2 shape sphere name sharp, i 1 letter p max electrons 6 shape three dumbbells name principle, i 2 letter d max electrons 10 shape four dumbbells or unique shape one name diffuse, i 3 letter f max electrons 14 shape eight dumbbells or unique shape two name fundamental, i 4 letter g max electrons 18, i 5 letter h max electrons 22, 16 letter i max electrons 26, and the letters after the f sub-shell follow f in alphabetical order except j and those already used.

0001 A maximum entropy probability distribution is the probability distribution whose entropy is at least as great as that of all other members of a specified class of distributions. Maximizing entropy minimizing the amount of prior information built into the distribution. Principle of maximum entropy the base of the logarithm is used consistently: change of base resulting rescaling of the entropy, utilization of base 2 or continue natural logarithm, result units for the entropy. In mathematical formalism, a constraint of some piecewise of information resulting to modify a probability distribution in the case of a mean value constraint, by inserting an exponential factor εχρ[-λί(η)] with an adjustable Lagrange multiplier λ.

0001 Maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ. The uniform distribution on the interval [a,b] is the maximum entropy distribution among all continuous distributions which are .

13

Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

supported in the interval [a, b] (which means that the probability density is 0 outside of the interval). More generally, if we're given a subdivision a=a0 < al < ... < ak = b of the interval [a,b] and probabilities pl,...,pk which add up to one, then we can consider the class of all continuous distributions such that pr(a,.i < x < aj) =p_j for j = \ ,...,k.

0001 The density of the maximum entropy distribution for this class is constant on each of the intervals [aj-l,aj). The uniform distribution on the finite set {xl,...,xn} (which assigns a probability of 1/n to each of these values) is the maximum entropy distribution among all discrete distributions supported on this set. Positive and given mean: the exponential distribution with mean l/λ is the maximum entropy distribution among all continuous distributions supported in [0,∞] that have a mean of l/λ. Discrete distributions with given mean supported on the set {χΐ,.,.,χη} with mean μ, the maximum entropy distribution has the following shape: pr(x = Xk) = cr^xk for k - 1 , ... ,n where the positive constants C and r can be determined by the requirements that the sum of all the probabilities must be 1 and the expected value must be μ. All the discrete distributions supported on the infinite set {xl,x2,...} with mean μ, the maximum entropy distribution has the shape: pr(x = Xk) = cr^xk for k =1 ,2,..., where again the constants C and r were determined by the requirements that the sum of all the probabilities must be 1 and the expected value must be μ. Such example, in the case that xk = k, this gives C = l/μ - 1, r = μ - 1/ μ, such that respective maximum entropy distribution is the geometric distribution.

0001 A polar plane is described in an electromagnetic field H equipotential wavefunction median of A triangle by A line segment from A vertex of the triangle to the midpoint of the side opposite that vertex having three vertices, continue three median number values 123 composite relating to or being A modification of order combining angular Ionic volutes with A circled whole number that can be divided evenly by numbers other than 1 or itself separate mathematic consecutive nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1 ,1,1,1 and number base point k- face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1,1,1,1,1 and number base point k-face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1 , 1,1 ,1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

0001 Banded on equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring, nl , n2, and n3 energy function, Constant are the first convex set definition, example, inner description, algebraic property, separation theorem, and characterization along with any pair of its points x, y, also the entire segment [x, y] linking the points in the multidimensional case the segment [x, y] linking the points x, y ε R" Convex set Let x, y be two points in Rⁿ the set [x, y] = {z = λχ + (1 - X)y | 0 < λ < 1 } segment with the endpoints x, y, Second, A subset M of Rⁿ is convex, if it contains, along with any pair of its points x, y, also the entire segment [x, y] then x, y & M, 0 < λ≤1 ^ λχ + (1 - X)y ε M the exact sense of the definition for the empty set, proof present the contradiction of counter example formulate that it is not convex simple nonempty convex sets singletons have { points { and the entire space R" is the solution set of an arbitrary possibly, infinite system < b_a, a £ A a^T _ax < b_a, a £ A of linear inequalities with n unknowns x { the set M = {x ε R" | <χ^τ _αχ < b_a, a £ A} is convex, In particular, the solution set of A finite system Ax < b of m inequalities with n variables A is m ^χ n matrix is convex, A set of this, latter type is three dimensions with flat faces, straight edges and sharp corners or vertices, Indeed, let x, y be two solutions to the system, does prove that any point z = λχ + λχ + (1 - Xy) with λ 6 |0, 11 also is A solution to the system, or

Continuous loop is having exponents the exponential function e^x, and the sum of the first n + 1 terms of its Taylor series at 0 or divided three number (n¹²³) mathematic placed on the right-hand side of the equals sign by factoring it's number (n')( n²)( n³) raised to the (n)^th power of itself, zero power (except 0); equals 1 ; or

0001 Diffeomorphism if it is A bijection and its inverse f-1 : N→ M having exponents b^m+n = b^m x bⁿ; (b^m)ⁿ = b^{m x n}; and (b x c)ⁿ = b" x c" mean value constraint, by inserting an exponential factor εχρ[-λί(η)] with an adjustable Lagrange multiplier λ, maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ, all three vertices continuously intersect at A single point named the centroid of the triangle continue mathematic consecutive nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1 ,1 ,1 ,1,1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1 ,1,1,1 ,1 and number base point k- face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1 ,1 ,1,1 , 1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

0001 Banded on equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring, nl , n2, and n3 energy function related to spin states^ composite relating to or being A modification of order combining angular Ionic volutes with A circled whole number that can be divided evenly by numbers other than 1 or itself of three body in vector space relativity, or closed at A second point symetry resection intrinsic position away nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1 ,1 ,1 ,1,1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1 ,1, 1,1,1 and number base point k- face element three quantum transitions two inner and outer description equilateral n-sides banded^' closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1 ,1 ,1,1 ,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring. Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

0001 Quadrupole magnetic field trapping function the starting point species dilute atoms derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid optically trapped and cooled to some 20 μΚ following species dilute atoms derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. The trapping potential is provided by A bisection measure of change method two pole magnetic field, bisection measure of change method in mathematics is A root finding method that repeatedly bisects an interval and then selects A subinterval in which A root must lie upon which A bias field position control of matter at the center of the trap having a cumulative energy-gap, two of the dimension of the product convex set of points of the unit sphere taking the maximum limit of any remaining electromagnetic field H equipotential wavefunction higher dimensions of the unit sphere, interior of the unit circle open unit disk, while the interior of the unit circle combined with the unit circle itself is the closed unit disk convex, two of the dimension of the product convex set of points in the n-dimensional space nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1,1, 1 ,1 ,1 and number base point k- face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1,1 and number base point k-face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1 ,1,1,1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

0001 Mangetic Bn, n-sides positioned symmetrical under electromagnetic field H equipotential wavefunction symetrical corners in A configuration joined regular sides ABCD by its circumscribed sphere unit cell axial circle continued in equilateral shape two lines of symmetry planes prism, or two lines of symmetry surface sides have divergence manifold is the sum of the dimensions of its factors manifolds M and N, A differentiable map f : M→ Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

N is called A diffeomorphism if it is A bijection and its inverse f-1 : N→ M is differentiable as well, if these functions are r times continuously differentiable, f is called A Cr-diffeomorphism, Two manifolds M and N are diffeomorphic symbol usually being if there is A diffeomorphism f from M to N, they are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times

continuously differentiable, equipotential wavefunction sine λ/2, cubic 1/4 and trigonl/3 qualifiers Lie Algebra solutions in harmonic simple fractions sine A unit circle S I , reduced from higher dimensions of the unit sphere, interior of the unit circle open unit disk, while the interior of the unit circle combined with the unit circle itself is the closed unit disk convex, The first convex set definition, example, inner description, algebraic property, separation theorem, and characterization along with any pair of its points x, y, also the entire segment [x, y] linking the points in the multidimensional case the segment [x, y] linking the points x, y S Rⁿ Convex set Let x, y be two points in R" the set [x, y] = {z = λχ + (1 - | 0 < λ < 1 } segment with the endpoints x, y, Second, A subset M of R" is convex, if it contains, along with any pair of its points x, y, also the entire segment [x, y] then x, y £ M, 0 < λ < 1 => λχ + (1 - X)y ε M the exact sense of the definition for the empty set, proof present the contradiction of counter example formulate that it is not convex simple nonempty convex sets singletons have { points { and the entire space R" is the solution set of an arbitrary possibly, infinite system < b_a, a £ A a^T _ax < b_a, a c A of linear inequalities with n unknowns x { the set M = {x £ Rⁿ | α^τ _αχ < b_a, a 6 A} is convex, In particular, the solution set of A finite system Ax < b of m inequalities with n variables A is m ^χ n matrix is convex, A set of this latter type is three dimensions with flat faces, straight edges and sharp corners or vertices, Indeed, let x, y be two solutions to the system, does prove that any point z = λχ + λχ + (1 - y) with λ £ |0, 11 also is A solution to the system, This is evident, since for every a 6 A is a _ax < b_a a _ay < b_a, when, multiplying the inequalities by nonnegative reals λ and 1 - λ and taking sum of the results, a _ax + (1 - λ) a _ay < b_a + (1 - λ) b_a = b_a, and what is in the left hand side is exactly α^τ _αζ, Note that any set given by example is not only convex, but also closed any plane in R" in particular any linear subspace is the figure whose sides are polygons sides that are line segments.

0001 A regular polygon has all its sides and angles equal. A polygon can be defined (as illustrated above) as A geometric object "consisting of A number of points (called vertices) and an equal number of line Specific polygons are named according to the number of sides, such as bisection two pole measure of change resection polygons μΝ continue named according to the number of sides and angles they have A plane shape with straight sides. Is it A Polygon are 2-dimensional shapes. They are made of straight lines, and the shape is" closed polyhedral set of all Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

solutions to some system of linear equations, Now, A system of linear equations is equivalent to A system of linear inequalities can equivalently represent A linear equality by A pair of opposite linear inequalities, It follows that A plane is A particular case of A three dimensions with flat faces, straight edges and sharp corners or vertices set and is therefore convex, and can obtain the conclusion directly convexity of A set means that it is closed with respect to taking certain restricted set of linear combinations of its members { namely, the pair combinations with nonnegative coefficients of unit sum, shown that any plane or an affine set is closed with respect to taking linear combinations not obligatory positive of its elements with unit sum, having the additional property that it is also A convex set of points in the n-dimensional space Rn, cylindrical integral, or equilateral integral joined symetrical corners attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1,1,1 ,1,1 and number base point k-face element in A tetrahedral configuration joined regular tetrahedron ABCD by its circumscribed sphere unit cell axial circle continued in equilateral shape two lines of symmetry planes prism, or two lines of symmetry surface sides geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).^tl][2]A paraxial ray is a ray which makes a small angle (Θ) to the optical axis of the system, and lies close to the axis throughout the system. ^tl] Generally, this allows three important approximations (for Θ in radians) for calculation of the ray's path:^tl] sin# « θ; ίαηθ « Θ and cosO « 1 The paraxial approximation is used in Gaussian optics and first-order ray tracing.^[1J Ray transfer matrix analysis is one method that uses the approximation. In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent do not change for the "second-order" paraxial approximation (the second term in their Taylor series expansion is zero), while for cosine the second order approximation is cosd « 1 - θ²/!. The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.^ For larger angles it is often necessary to distinguish between meridional rays, which lie in a plane containing the optical axis, and sagittal rays, which do not.

0001 The mathematical function that describes the Gaussian beam is a solution to the paraxial form of the

Ilelmholtz equation. The solution, in the form of a Gaussian function, represents the complex amplitude of the beam's electric field. The electric field and magnetic field together propagate as an electromagnetic wave. A description of just one of the two fields is sufficient to describe the properties of the beam. The behavior of the field of a Gaussian beam as it propagates is described by a few parameters such as the spot size, the radius of curvature, and the Gouy phase. ^ilJ Other solutions to the paraxial form of the Helmholtz equation exist. Solving the equation in Cartesian coordinates leads to a family of solutions known as the Hermite-Gaussian modes, while solving the Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class, system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

equation in cylindrical coordinates leads to the Laguerre-Gaussian modes. ^¹-³¹ For both families, the lowest-order solution describes a Gaussian beam, while higher-order solutions describe higher-order transverse modes in an optical resonator. Second higher order electromagnetic waves simultaneous or beam divergence wave in linear media, two-dimensional intensity by inverted-Gaussian having red edge reflectance Re or radius of curvature;

transmission Te simulation across the interface of two homogeneous media, paraxial light ray and red edge parallel to the axis of an optical system transmission Te of Gaussian beams measure small Θ angle with and position close to the axis of an optical paraxial light ray and red edge parallel to the axis of an optical system approximation to the full equations of optics that is valid in the limit of small Θ angles from the optical axis, having p primes represent projected Θ angles and distance of measure, and unprimed represent the object, with n the index of refraction to the left of the y-axis and the index of refraction to the right of the y-axis. The paraxial approximation is an

approximation to the full equations of optics that is valid in the limit of small angles from the optical axis. Primes representing here image angles and distances, and unprimed represent the object, with n the index of refraction to the left of the y-axis and n' the index of refraction to the right of the y-axis. The paraxial approximation then assumes that ξ « ξ (1); ϋθ3ξ « 1 (2); and ίΆηξ « ξ, (3) where ξ may be any of i, i', u, u', or Φ. This regime is known as first-order, paraxial light ray and red edge parallel to the axis of an optical system, or Gaussian optics.

0001 Spectrall reflectance measurements of single point, quantitative representation of the shape and position red edge reflectance in terms of parameters minimum reflectance Ro and where the amplitude A₀ is the maximum of the function, red edge inflection point wavelength λρ and reflectance minimum wavelength λθ, of the red edge by re- measurement of parameters are: Ro, λο and λρ ±2.0nm. High spectrall resolution reflectance, inverted-Gaussian spectrall position of the red edge direct spectrometry mathematic, a Taylor series represent of the function as an infinite sum of terms that are calculated from the quantum valuesi₂₃ of the function's derivatives at a single point. As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 1 1 and 13. The Taylor series of a real or complex- valued function f(x) that is infinitely differentiate at a real or complex number a is the power series /(a) + f(a)/l!(x - a) +f'(a)/2!(x - a)² +f"(a)/3!(x - a)³ + . Which can be written in the more compact sigma notation as

denotes the nth derivative of / evaluated at the point a. The derivative of order zero of/ is defined to be / itself and (x - )° and 0! are both defined to be 1.

When a = 0, the series is also called a Maclaurin series. The exponential function e^x, and the sum of the first n + 1 terms of its Taylor series at 0. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class, system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

series for (1 - x)^{_ l} is the geometric series 1 + x + x2 + x3 + ^" so the Taylor series for x^~l at a = 1 is 1 - (x - 1) + (x - I )² - ( x - l )³ + . By integrating the above Maclaurin series, we find the Maclaurin series for log(l - x), where log denotes the natural logarithm: -x - l/2x² - l/3x³ - l/4x⁴ - ... and the corresponding Taylor series for log(x) at = 1 is (x - 1) - l/2(x - l )² + l/3( x - l)³ - l/4( x - l)⁴ + ^{" '}, and more generally, the corresponding Taylor series for log(x) at some a = x₀ is: log(x₀) + l/x₀ (x - x₀) - l/x² ₀(x - x₀)²/2 + ^{" '}. The Taylor series for the exponential function e* at a = 0 is 1 + x'/l ! + x²/2! + x³/3 ! + x⁴/4! + x⁵/5 ! + = 1 + x + x²/2 + x³/6 x /24 + x⁵/120 + - =^∞∑_n=₀ x n! . The above expansion holds because the derivative of e* with respect to x is also e and e° equals 1. This leaves the terms (x - 0)" in the numerator and n\ in the denominator for each term in the infinite sum.

0001 Circular random variables for a continuous random variable 0j distributed about the unit circle, the Von Mises distribution maximizes the entropy when given the real and imaginary parts of the first circular moment or, equivalently, the circular mean and circular variance. When given the mean and variance of the angles 0j module 2π, the wrapped normal distribution maximizes the entropy. Non-existence of maximum for given mean, variance and skew exists an upper bound on the entropy of continuous random variables on R with a fixed mean, variance, and skew. A theorem by Boltzmann all the above examples are consequences of the following theorem by Ludwig Boltzmann. Continuous version suppose S is a closed subset of the real numbers R and we're given n measurable functions fl ,...,fn and n numbers al ,...,an. We consider the class S of all real- valued random variables which are supported on S (whose density function is zero outside of s) and which satisfy the n expected value conditions e(^(x)) = a,- for j = 1 , ...,n. If there is a member in C whose density function is positive everywhere in s, and if there exists a maximal entropy distribution for c, then its probability density p(x) has the following shape: p(x) = C exp ("∑=i fA ) for all x G S where the constants C and Xj have to be determined so that the integral of p(x) over S is 1 and the above conditions for the expected values are satisfied. Conversely, if constants C and Xj like this can be found, then p(x) is indeed the density of the (unique) maximum entropy distribution for our class C. This theorem is proved with the calculus of variations and Lagrange multipliers. Discrete version suppose S = {x l ,x2,...} is a (finite or infinite) discrete subset of the reals and we're given n functions fl ,...,fn and n numbers al ,...,an. We consider the class C of all discrete random variables X which are supported on S and which satisfy the n conditions e(#(x)) = a_j for j = Ι , . , . ,η. If there exists a member of C which assigns positive probability to all members of S and if there exists a maximum entropy distribution for C, then this distribution has the following shape: pr(x = Xk) = C exp

(^x ) for k = 1 ,2, .. .where the constants C and Xj have to be determined so that the sum of the probabilities is 1 and the above conditions for the expected values are satisfied. Conversely, if constants C and Xj like this can be Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

found, then the above distribution is indeed the maximum entropy distribution for our class C. This version of the theorem can be proved with the tools of ordinary calculus and Lagrange multipliers.

0001 Caveats note that not all classes of distributions contain a maximum entropy distribution. It is^'possible that a class contain distributions of arbitrarily large entropy (the class of all continuous distributions on R with mean 0 but arbitrary standard deviation), or that the entropies are bounded above but there is no distribution which attains the maximal entropy (the class of all continuous distributions X on R with E(X) = 0 and E(X2) = E(X3) = 1. It is also possible that the expected value restrictions for the class C force the probability distribution to be zero in certain subsets of S or by shrinking the set S.

0001 Turbine-turbine contain second common system bias Mersenne primes of form 2^Ap - 1 where p is a prime _. equivalently, primes of form 2^An - 1 for integers n values of p prime repunits in base second common system closed system hydraulic-force actuated, fermion A space function of normalization, mean value constraint, by inserting an exponential factor εχρ[-λί(η)] with an adjustable Lagrange multiplier λ, maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ, hydraulic-electric actuated hydraulic-force actuated, fermion A space function of normalization, mean value constraint, by inserting an exponential factor exp[- f(n)] with an adjustable Lagrange multiplier λ, maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ.

0001 Turboshaft, turbofan, turbofluid, or turbojet continue natural n composite number 2n - 1 fluid dynamics control enhance mathematic optimum series, and parallel integral to the current, optimal control dynamic system transition state second common component atmospheric, temperature, pressure correction, in the presence of constraints for the state or input [0, 1] optimal control n = c prime force validate cofactor of 3 coefficient process wavefunctions take the form of spherical harmonics, described by Legendre polynomials, mutual acoustic impedance, method of moments Let Ω be a bounded domain in RJV and let m be a ^-periodic function such that its restriction to Ω * (0, T) belongs to L\(0, T), Ζ^ν(Ω)) for some v > N/2 and s > 2v/2v - N, with v > 1 , number Nv > 0 some + positive or number Nv < 0 some - negative and s > 2. We give necessary and sufficient conditions on m for the existence, uniqueness, simplicity, and the normalization function of the principal eigenvalue for the Dirichlet periodic parabolic eigenvalue problem with weight m of load factor LF is n values = p prime repunits horizontal Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class- system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

weight, balance center of gravity CG, or vertical lift VL, weight L/W center of gravity CG, dynamic variable change match prime force 1 : 1 inertial frame of reference constant of quantum mechanics weight .- thrust ratio, or weight : power ratio coefficient of tractive force maximum to the normal force, F/N coefficient of friction = μ the power drivetrain system one or more wheels in contact with the roadway or railroad track having motion between the inertial frame of reference constant c is then a standard model of surface friction coefficient of kinetic friction = μk riding above the surface divided by the weight on the drive system equal to traction ratio = coefficient of static friction propulsion power-plant incorporate coherent whole of the state inertial space frame of reference that describe time and space composite modification of order matter and antimatter catylitic quantum number, homogeneous, isotropic, and time-independent manner state of constant rectilinear motion of each one another are zero acceleration momentum p intrinsic now it possesses the turbo-booster on occasion Arithmetic ordering sequence switch on : off engage super-fiuid dynamic function / the general derivative test for stationary points let / be a real-valued, differentiate function on the interval complex number i = - l , I c R, c G l and n > 1 an integer. If now holds f (c) = ^' = f ⁿ⁾ (c) = 0 and ί*^π+ι) (c)≠ 0 then, n is even and we have a (local) saddle point at c. More precisely: 1. f⁽ⁿ⁺^ (c) < 0→ c is a strictly decreasing point of inflection 2. f ⁿ⁺¹⁾ (c) > 0→ c is a strictly increasing point of inflection.

0001 This analytical test classifies any extremum are calculated on the function domain absolute extremum the maximum and minimum element of a set maxima and minima for ϋο≤(3πχ)/χ, 0.1< x <1.1 rotating on an numeric-n cyclones vortex axisymmetric rotating on a turboshaft, turbofan, turbo-fluid, or turbojet acceleration, finding maxima, minima, tangents to curves, computation of areas, finding center of mass, the least action principle, approximate equality, correlated problems maxima of functions, and then finding tangent lines to curves valid identity between two constants. Transcendental curves as the cycloid mathematics bias numeric value built from position and momentum complex numbers from three discrete points, begin our particle has the whole x- axisymmetric to play on and in lieu of the wave function as three complex numbers, we continue to add a different complex number for every point on the x-axisymmetric, so then Y is now going to be a complex function of x, position probabilities with a continuous wave function as the standard part function product the magnitude of it's real number absolute extremum prime value |x| turning, bending, or curving a turn, bend, or curve an apparent magnitude volume of change inflection c prime force, changing meaning the force of load 0, 1 optimal control n = c prime force validate cofactor of 3 coefficient process wave functions take the form of spherical harmonics, described by Legendre polynomials, mutual acoustic impedance, method of moments Let Ω be a bounded domain Title of Invention: Variable hydraulic press force F is derived from No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H gasiform species dilute a heater fluid dynamic mechanic class system transfer the integral sub-plasma thermal cycle state of matter dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Application- number. Filing Date: 02/15/2017. Name of Inventor: Robert R. Matthews Tel. 202-676-2017.

in RN and let m be a -periodic function such that its restriction to Ω x (0, T) belongs to L\(0, 7), L^V(Q)) for some v > N/2 and s > 2v/2v - N, with v > 1 , stationary point of / The Potential function x has all of its derivatives at 0 equal and normalized Y so that the total probability sum is one equal to terms e, equal to terms π, or 0 equal to terms numeric-n, except for the 8th derivative, which is positive.

0001 Minimum at 0 defined on a domain X has a global : absolute extremum prime and bias numeric value built from calculated the expectation value of position and momentum complex function of this form Y=_e ^ipx/h ₅ where p is some number takes at a point relative number Nv > 0 some + positive or number Nv < 0 some - negative and s > 2. X=p/2 is some little point, one of infinitely many such points in the range 0<x<2p. The probability of being at exactly that point, or any other specific position, is zero. So Y is not a point but rather a probability density,.and now the golden rule is: the integral of\ Y\² between any two points, gives the probability of the particle appearing between those two points. The instance, ^J ro¹|Y| 2 dx gives the probability of finding the particle somewhere between x=0 and x=l . Now then we give necessary and sufficient conditions on m for the existence, uniqueness, simplicity, and the normalization function of the principal eigenvalue for the Dirichlet periodic parabolic eigenvalue problem with weight m of load factor LF magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, z*, where for any complex number z = a + bi, its complex conjugate is z* = a - bi.

Claims

Independent claim 1. Nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1 ,1 ,1 ,1 ,1 and number base point k- face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1,1 and number base point k-face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 , 1,1,1, 1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

Dependent claim 1. Banded on equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring, nl , n2, n3 circumscript laterally through the interior shape area the intersection of three body in vector space with a plane, and function of maximum entropy rotation the base of the plane continue natural logarithm, resulting entropy by mathematical formalism, a constraint of a probability distribution in the case of a mean value constraint, the insertion of an exponential factor εχρ[-λί(η)] with an adjustable Lagrange multiplier λ maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy distribution with specified mean μ and standard deviation σ, and the analog a in higher dimensional space symetry resection position base of the plane in alignment with movement along equipotential line output the external consecutive magnetic fieldsi₂3 B₀ Bi, B₂, B₃, open dengenerate p(n), loop degenerate, or closed degenerate three quantum transitions between spin states Βι₂₃ ΔΕ = v = h Bo-Be (with γ gyromagnetic ratio) Bi + ΔΕ = hv = h Bo- Be (with γ gyromagnetic ratio) B₂ + ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) B₃ + ΔΕ are the energy difference between spin states, space integral sub-plasma thermal cycle No fission product half-life range from 91 y to 210 ky magnetic-enthalpy symbol H ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) degenerate, and the gasiform species heater derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid. Dependent claim 2. The frequency v wavelength λ and speed of light c, are related by λν = c, amplitude of electric field E = F/q and magnetic field B, H, or Tesla nT. Triple-photon homodyne Fock reconstruction of the single- photon consecutive three dimensional 3D optical drawing first and second common system with longinitude and exponential form shielding orders of optical cables projecting windows nT frames four rectangles A/B matter inner band red lines and ultra violet lines mass turbulent energy near and far begin visible violet, magenta, cyan, blue, yellow, and green wavenumbers wavelengths sine 12 wavenumbers wavelengths, triangle wavenumbers

wavelengths, square wavenumbers wavelengths, and sawtooth wavenumbers wavelengths continue by prismatic fourth band AU quantum transition ABCD circumscribed sphere, and rectilinear lines reconstruct 150-200 nm nanometer circuit band gap energy equilateral sides skew quantum energy wavenumbers inner and outer descriptions bands balanced homodyne tomography reliable technique, high matter wave speed cl, high matter wave speed c2, and high matter wave speed c3 by way of exact relativistic linear mathematic expression numbers, symbols and dynamics operators for momentum and kinetic energy, equation polar, non-polar ionic Φ (ionic potential)* = Z/r; where Z = ionic charge (+1); and r = ionic radius (A) have energy the transfer of electrons in ionic bonds and covalent chemical bonds sharing a pair of valence electrons by two atoms, having a cumulative energy- gap are all related ΔΕ = hv. Tesla symbol nT magnetic flux density functional theory commonly B magnetic field symbol Ω, identified in the Planck constant denoted h is a physical constant that is the quantum of action in quantum mechanics, the Planck constant describe the proportionality constant between the energy E of a photon and the frequency v of its associated electromagnetic wave, this relation between the energy and frequency is called the Planck relation E = hv the frequency v wavelength λ and speed of light c, are related by λν = c, amplitude of electric field E = F/q and magnetic field B, H, or Tesla degenerate nT. The spectroscopy surface analysis technique known as electron energy loss spectroscopy (EELS) quantum electrodynamic QED triple-photon homeodyne Fock reconstruction of the single-photon consecutive three dimensional 3D optical windows frames nanosecond balanced homodyne tomography reliable technique 150-200 nm nanometer circuit, the Planck relation for a photon is expressed as E = hc/λ, spin state, and applied magnetic field, the energy difference ΔΕ between spin states is proportional to the strength of B₀, applied magnetic field degenerate nT, the frequency c = λί of Bi₂₃ in frequency for matter, three quantum transitions between spin states are B₁₂₃, ΔΕ = hv = h Bo-Be (with γ gyromagnetic ratio) degenerate, ΔΕ is the energy difference between spin states, h is Planck's constant, v is the frequency of Bi₂₃ field Be is the magnetic field generated by the circulation of electrons of specification matter molecules, B₀ is the strength of the magnetic field, γ gyromagnetic ratio is a constant which is a property of the particular nucleus, relationship between field B₀ and frequency v, ΔΕ = h x frequency nl + ΔΕ = h x frequency n2 + ΔΕ = h x frequency n3 + ΔΕ, is the importance of B_e. Optical depending on the value of n, there is an angular momentum quantum number i and the following series n = 1, L = λ, series ultraviolet, n = 2, L = λ, Balmer series visible or violet, n = 3, L = λ, Ritz-Paschen series near infrared, n = 4, L = λ, Brackett series short- wavelength infrared, n = 5, L = λ, Pfund series mid-wavelength infrared. Higher-order wave equations and matter waves the most common "wave equation" in one space and one time dimension is

(1) where V is a constant equal to the phase velocity of a monochromatic wave. This has the general solution f(x+Vt) + g(x-Vt) for arbitrary functions f and g. Among the particular solutions of this equation are the familiar monochromatic solutions of the form Ψ(^χ, t) = A cos(kx-G)t) (2) where A is an arbitrary constant. The second partial derivatives of this function with respect to x and t are -Ak²cos(kx-cot) and -Aoo²cos(kx-cot) respectively, which implies V = ω/k, showing that this

particular Ψ is strictly a function of x-Vt, as a retarded wave. The amplitude of this function is purely real

(assuming the constants and variables are real-valued), with a value that oscillates up and down corresponding to the projection onto the real axis of a point moving around in a circle in the complex plane. We could just as well include the imaginary component complex number i² = -1, since the complex function Ψ(χ, t) = A [cos(kx-cot) + i sin (kx-cot)] = Αε^{ι(ΙίΧ"ω1)} (3) is also a solution of (1). In fact, this simple function is a solution of a wide range of wave equations, some of which turn out to have important physical consequences. In general, for any positive integers r, s, consider the partial differential equation θ" Ψ/5χ^Γ = a<9^S P/dt^s (4) where a is an arbitrary constant. Substituting the trial solution Αε^{ι(Ιθ "ω1)} into this equation, evaluating the derivatives, and dividing through by e^l(kx"fflt), we get (ik)^r = a (- ioo)^s This shows that (3) is a solution of every one of the higher-order wave equations l/(ik)^r δ^ΓΨ/θχ^Γ = l/(-ico)^s 5^S iV5t^s (5) The constant a in equation (4) is a = (ik)^r/(-ia))^s The phase velocity of the wave is still V = co/k, so if r = s we have a = (-1/V)^r, and hence V = -l/a^1/r. (Noting that if m is even we can select the negative root to give a positive phase speed.) The energy E and momentum p (in one dimension) of a photon can be expressed in terms of the angular frequency ω and wave number k respectively as E = hoo p = k where h is Planck's (reduced) constant.

4 2 2 2 2 2

Substituting into the relativistic relation between energy, momentum, and rest mass m gives c m - E - c p = h(co - c²k²) where c is the speed of light in vacuum. Consequently, if the phase speed of the photon equals c, meaning that V = ω /k = c, the right hand side vanishes, so the rest mass m is zero. On the other hand, if the phase speed V of the wave is not equal to c, the rest mass will be non-zero. This suggests that we might regard a particle of mass m and velocity V as a "matter wave" with the frequency ω and wave number k proportional to the energy and momentum respectively, in accord with the above equations for E and p. Returning to the generalized wave equation, we can express the constant k and ω in terms of the energy and momentum of the matter wave, and write the generalized wave equation in the form 3^ΓΨ/5χ^Γ = (-l)^s(i/h)^r"s p7E^s cf^/d (6) For a photon we have E/p = c, so in order to eliminate E and p from the wave equation, arrive at a constraint that is independent of the wave, we need r = s, in which case (6) reduces to the form 5^S F/dx^s = (-l)^s/c^s d^S P/dt^s (7) With s = 0 this reduces to the trivial identity Ψ = Ψ, whereas with s = 2 it is the standard wave equation for electromagnetic waves. By construction, the simple wave function (3) satisfies this equation for every value of s, and so too (fortuitously) does the purely real wave function (2). In addition, each of these equations is consistent with all three of the conditions E = hco, p = hk, and E = pc, where c is the phase speed of the wave. Now, for a particle of rest mass m moving at the speed v, the momentum and energy are given by p = mv/Vl-(v/c)² E = mc²/ l-(v/c)² (8) From the relations E = hco and p = hk this implies that the phase velocity V is V = co/k = E/p = c N (9) Of course, if v = c this gives the phase velocity V = c, which applies to the case of a photon. However, for a massive particle with v less than c, this formula may seem surprising at first, because it implies that the phase velocity of a matter wave exceeds c, and in fact goes to infinity as the speed v of the particle goes to zero. This might seem to suggest that the wave velocity has no significance, and that we cannot associate the wave with the particle. Fortunately, this apparent problem disappears when we remember that the actual transmittal of energy in a wave is propagated not at the phase velocity necessarily, but at the group velocity v_g,. For various velocities associated with another wave, we can determine phase, group, and signal velocity. The group velocity is given by v_g = dco/dk = dE/dp = v so in fact the matter wave does propagate at the same speed as the corresponding particle. Eliminating v from the energy and momentum

A 0 0 0 0 0 0

equations (8) leads immediately to the relation c m = E - c p = h(co - c k ) In the case of a photon we were able to go from equation (6) to equation (7), eliminating any explicit reference to the momentum and energy, and expressing the wave equation in terms of the constant c, because of the relation p/E = 1/c for massless particles. Thus if the indices r and s are equal, we can eliminate the ratio p/E. Not true for a particle with non-zero mass, so continuing an analogous method of eliminating the explicit appearances of E and p in equation (6), as one method of replacing them with some function of the invariant rest mass m of the particle. First, we should note that when we carry over the equation E = hco from photons to massive particles there is an ambiguity of interpretation of the quantity of energy represented by "E". Since a photon has no rest mass, its total energy and kinetic energy (total minus rest mass energy) are the same, so this is unambiguously the quantity represented by E. However, for a massive particle the total and kinetic energies are different, then determining which of these quantities is applicable for a matter wave. Continuing to determine if the wave that corresponds to the total energy of the particle, or in the wave that corresponds to just the kinetic energy associated with the motion of the particle. Since these two energies differ only by a constant (the rest mass of the particle), they propagate with the same group velocities, but they have very different phase velocities. We saw in equation (9) that the phase velocity V of the total energy of a particle moving at the speed v is c²/v, but the phase velocity of the particle's kinetic energy is given by V = C0kinetic/ = E - mc /p = c²/v (1 - Vl-(v/c)²) = v/2(l + ¼(v/c)² + ...) assuming the momentum is taken to be the same in both cases. Now, it might be argued that we are mixing incompatible quantities here, because the wave number k is still taken to be p/h, meaning that it is proportional to the momentum of the total mass, not just of the kinetic mass. We should attempt to interpret p in the relation p = hk as just the "kinetic momentum", the velocity times the difference between the total relativistic mass and the rest mass, and then we would (again) have the phase velocity V = c /v. However, if we did this, the group velocity would no longer equal v, because the difference between the total momentum and the "kinetic momentum" is a function of v. Moreover, by analogy with the energy, we note that each particle can be associated with a certain "rest momentum" just as it has a certain rest mass-energy, but the rest momentum of a particle is identically zero, so there is no distinction between its total momentum and its kinetic momentum. Consequently, we conclude that it's reasonable to treat the wave of a particle of rest mass m and speed v as having the wave number k = p/h and the frequency ω = E_kj_netic/h■ And so the equation (6) governing this wave (free of external constraints) is expected to be of the form δ^χΨ/δχ^Γ = (-l)^s(i/h)^r"s p7E_kinetic^S ^Ψ/θί⁵ (10) If v is small compared with c, then the momentum is essentially p « mv and the kinetic energy is E_kj_netic ^¾ (l/2)mv². Eliminating v from these two equations, we have the relation p²/Ekj_netic ~ 2m (1 1) Therefore, in order to create a general constraint out of (10) by eliminating the ratio p7Ekinetic^S, we must have r = 2s, in which case the wave equation becomes ^Ψ/δχ²⁵ = (-2m/ hi)^s <9^S P/dt^s (12) Again, with s = 0 this reduces to the trivial identity Ψ = Ψ, whereas with s = 1 as is the Schrodinger equation for the complex wave function of a free particle (with no potential field). By its construction, this wave equation for any non-negative integer s is satisfied by the complex-valued wave function (3). However, it is satisfied by the real-valued wave function (2) only for even values of s. Equation (12) is based on the non-relativistic equation (1 1). If instead we use the exact relativistic expressions for momentum and kinetic energy, equation (1 1) would become p²/Eki_netic ⁼ [1 + l/Vl-(v/c)²]m (13) which confirms that this ratio is essentially equal to 2m for non-relativistic velocities. In general it equals the rest mass plus the relativistic mass of the particle. Focusing on the traditional non-relativistic Schrodinger equation (12), it's interesting to consider the consequences of setting the index s to a value other than 1. The complex-valued wave function (3) is a solution of every one of these equations, but the equations are nevertheless distinct, as illustrated by the fact that although every solution with s = 1 is also a solution with s = 0, the converse is not true. Also, with s = 2 the equation is satisfied by the real-valued wave function (2), even though this is not a solution with s = 1. Of course, the choice of s = 1 makes the left hand side equal to the Laplacian, which we expect on physical grounds, but it would be interesting to know what (if any) phenomena could be modeled by the higher-order wave equations, both in the case of massless (r = s) and massive (r = 2s) particles. At the very least, the higher-order wave equation (10) is satisfied by any function (or linear combination of functions) of the form Ψ(χ, t) = Ae ^{ι(Ιο<"ω1+θ)} + f(_X) ) where « is a constant phase offset and f is a polynomial of degree r-1 in x and degree s-1 in t. Needless to say, the above does not constitute an a priori derivation of the Schrodinger equation of a free particle. A more natural and well- motivated approach can be given in terms of the Hamiltonian operator whose eignevalues are the possible energy levels of the particle. Nevertheless, it's interesting to see how the general outline of Schrodinger's equation in relation between the energy and momentum of a particle as represented by a wave with the same group velocity as the particle. This approach also clarifies that a matter wave actually has two distinct phase velocities, one (c /v) corresponding to the total energy of the particle, and the other (~v/2) corresponding to just the kinetic energy of the particle. For an atomic orbital an atomic electron's angular momentum, L, is related to its quantum number i by the following equation L F = ²Z(i + 1) Ψ where h is the reduced Planck's constant, L² is the orbital angular momentum operator and Ψ is the wavefunction of the electron. Catalysis of the wave equation circuit second coherent coordinates Wigner function common analogues optical windows frames between energy of amperes A = C/s and λ of refraction index henries v = Ldi/dt, Px(P), or W(X, P), or Px(X) Minimum zero base define on a domain W(X, P) has a global : absolute extremum prime and bias value function takes at a point relative extremum are calculated on the function domain absolute extremum the maximum and minimum element of a set maxima and minima for COS(3TCX)/X, 0.1< X <1.1 rotation elementary particle with a half-integral spin that obeys the Pauli exclusion principle. Electrons, protons, and neutrons are types of the fermion thermal renormalization group for higher dimensional space rotating on an axis-line quantumstatei₂₃ reconstruction of the single-photon Fock state relative 1 : 3 triple-photon Fock statesi₂₃ spherically symmetrical s orbital, the B₀ field the magnetic field causes the electrons in the cloud surrounding the nucleus to circulate through the orbital, this circulation caused Be an induced magnetic field opposed to the B₀ field and the value B₀-B_e represents the actual magnetic field B experienced at the nucleus, the B_e field produced by electrons in p and d orbitals are more complex formulated by proton chemical shift, and the analog a in higher dimensional space symetry resection position base of the plane in alignment with movement along equipotential line output energy of the photon must exactly match the energy-gap ΔΕ between the these states are now reconstructed in atomic triple state^ A spin triplet is a set of three quantum numbers of a system, each with total spin S = 1 (in units of h), nucleus of the atom nl, n2, and n3 catalysis of the wave equation coherent coordinates Wigner function common analogues spectroscopy optical windows frames computing nanosecond processing optical control [0, 1] following spectroscopic notation, the magnetic quantum number, and the spin quantum number. It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as i quanta specific discrete codimension cAt, wavelength points Δχ, Ay and along Δζ the integral phase of wave time and space coherence coordinate j and mj, together with the parity of the quantumstatei, replace the three quantum numbers i, mi and ms the projection of the spin along the specified axis. The former quantum numbers can be related to the latter, and the eigenvectors of j, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of I, mi and ms waveguide. Electron configuration quantum-mechanical nature of electrons an electron shell is the set of allowed states that share the same principal quantum number, n the number before the letter in the orbital label, which electrons may occupy. An atom's nth electron shell can accommodate 2n2 electrons, is the first shell can

accommodate 2 electrons, the second shell 8 electrons, and the third shell 18 electrons. The factor of two arises because the allowed states are doubled due to electron spin-each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin + 1/2 noted by an up-arrow and one with a spin - 1/2 with a down- arrow. A subshell is the set of states defined by a common azimuthal quantum number, i, within a shell. The values i = 0, 1 , 2, 3 correspond to the s, p, d, and f labels, respectively. The maximum number of electrons that can be placed in a subshell is given by 2(2i + 1). This gives two electrons in an s subshell, six electrons in a p subshell, ten electrons in a d subshell and fourteen electrons in an f subshell. The numbers of electrons that can occupy each shell and each subshell arises from the equations of quantum mechanics, in particular the Pauli exclusion principle, which states that no two electrons in the same atom can have the same values of the four quantum numbers. The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantumstatei₂₃ of an electron (the others being the principal quantum number N represents the relative overall energy of each orbital, and the energy of each orbital increases as the distance from the nucleus increases, the sets of orbitals with the same n value are often referred to as electron shells or energy levels, following spectroscopic notation, the magnetic quantum number, and the spin quantum number). It is also known as the orbital angular momentum quantum number, orbital quantum number or second quantum number, and is symbolized as i. List of angular momentum quantum numbers derivation quantum mechanical orbital angular momentum. Connected with the energy states of the atom's electrons are four quantum numbers: n, i, mi, and ms. These specify the complete, unique quantumstatei₂₃ of a single electron in an atom, and make up its wavefunction or orbital.

Dependent claim 3. The wavefunction of the Schrodinger wave equation reduces to three equations that when solved, lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The azimuthal quantum number arose in the solution of the polar part of the wave equation and understanding the concept of the azimuth, spherical coordinate systems, and the cartesian coordinate system.

Spherical coordinate system spherical models, the cylindrical system with cylinders, the cartesian with general volumes, the transverse Mercator projection is the transverse aspect of the standard or normal Mercator of the transverse cylinder to produce a secant projection with two standard transverse 2rp N-shift radius the midpoint of the segment (the transverse axis) connecting μΝ, and each of the different angular momentum states can take 2(2 £ + 1) electrons. This is because the third quantum number mi (which can be thought of loosely as the quantized projection of the angular momentum vector on the z-axis) runs from - i to i in integer units, and so there are 2i + 1 possible states. Each distinct n, i, mi orbital can be occupied by two electrons with opposing spins (given by the quantum number ms), giving 2 (2£ + 1) electrons overall. Orbitals with higher i than given in the table are perfectly permissible, but these values cover all atoms so far discovered. For a given value of the principal quantum number N, the possible values of i range from 0 to n - 1 ; therefore, the n = 1 shell only possesses an s subshell and can only take 2 electrons, the n = 2 shell possesses an s and a p subshell and can take 8 electrons overall, the n = 3 shell possesses s, p and d subshells and has a maximum of 18 electrons, or, the maximum number of electrons in the nth energy level is 2n2. The angular momentum quantum number, i, governs the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as a median of a triangle is a line segment from a vertex of the triangle to the midpoint of the side opposite that vertex. Because there are three vertices, there are of course three possible mediansi₂₃ exponents; all three always intersect at a single point. This point is called the centroid of the triangle. In quantum mechanics L by itself, L has meaning in its use as the angular momentum operator and simple use of the quantum number Z. The principal quantum number is at the right of each row and the azimuthal quantum number is denoted by letter at top of each column. A polar plane is described in an

electromagnetic field H equipotential wavefunction median of A triangle by A line segment from A vertex of the triangle to the midpoint of the side opposite that vertex having three vertices, continue three median number values i₂₃ composite relating to or being A modification of order combining angular Ionic volutes with A circled whole number that can be divided evenly by numbers other than 1 or itself separate mathematic consecutive nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1 , 1,1 , 1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1 ,1 and number base point k- face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1 ,1,1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

Dependent claim 4. Banded on equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring, nl , n2, and n3 energy function, Constant are the first convex set definition, example, inner description, algebraic property, separation theorem, and characterization along with any pair of its points x, y, also the entire segment [x, y] linking the points in the multidimensional case the segment [x, y] linking the points x, y€ Rⁿ Convex set Let x, y be two points in R" the set [x, y] = {z = λχ + (1 - X)y | 0 < λ < 1 } segment with the endpoints x, y, Second, A subset M of R" is convex, if it contains, along with any pair of its points x, y, also the entire segment [x, y] then x, y & M, 0 < λ < 1 =? λχ + (1 - 6^· M the exact sense of the definition for the empty set, proof present the contradiction of counter example formulate that it is not convex simple nonempty convex sets singletons have { points { and the entire space Rⁿ is the solution set of an arbitrary possibly, infinite system < b_a, a 6 A a _ax < b_a, a £ A of linear inequalities with n unknowns x { the set M = {x Ξ Rⁿ | α^τ _αχ < b_a, a 6 A} is convex, In particular, the solution set of A finite system Ax < b of m inequalities with n variables A is m ^χ n matrix is convex, A set of this latter type is three dimensions with flat faces, straight edges and sharp corners or vertices, Indeed, let x, y be two solutions to the system, does prove that any point z = λχ + λ^χ + (1 - λγ) with λ ε |0, 1| also is A solution to the system, or Continuous loop is having exponents the exponential function e^x, and the sum of the first n + 1 terms of its Taylor series at 0 or divided three number (n¹²³) mathematic placed on the right-hand side of the equals sign by factoring it's number (n')( n²)( n³) raised to the (n)^th power of itself, zero power (except 0); equals 1; or

diffeomorphism if it is A bijection and its inverse f-1 : N→ M having exponents b^m+n = b^m x b"; (b^m)ⁿ = b^{m x n}; and (b x c)ⁿ = b" x c" mean value constraint, by inserting an exponential factor exp[- f(n)] with an adjustable Lagrange multiplier λ, maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ, all three vertices continuously intersect at A single point named the centroid of the triangle continue mathematic consecutive nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1,1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1, 1, 1, 1 ,1 and number base point k- face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1, 1,1,1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

Dependent claim 5. Banded on equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring, nl, n2, and n3 energy function related to spin statesi₂₃ composite relating to or being A modification of order combining angular Ionic volutes with A circled whole number that can be divided evenly by numbers other than 1 or itself of three body in vector space relativity, or closed at A second point symetry resection intrinsic position away nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1, 1 ,1 ,1 ,1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1 ,1 ,1 ,1 , 1 ,1 and number base point k- face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n- poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

Dependent claim 6. Quadrupole magnetic field trapping function the starting point species dilute atoms derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid optically trapped and cooled to some 20 μΚ following species dilute atoms derived from organic fresh and salt water algae dilute synthesis extraction higher aggregation atoms and hybridization of natural gas, organic waste or biomass to synthesis gas pipeline standard a heater fluid.

Dependent claim 7. The trapping potential is provided by A bisection measure of change method two pole magnetic field, bisection measure of change method in mathematics is A root finding method that repeatedly bisects an interval and then selects A subinterval in which A root must lie upon which A bias field position control of matter at the center of the trap having a cumulative energy-gap, two of the dimension of the product convex set of points of the unit sphere taking the maximum limit of any remaining electromagnetic field H equipotential wavefunction higher dimensions of the unit sphere, interior of the unit circle open unit disk, while the interior of the unit circle combined with the unit circle itself is the closed unit disk convex, two of the dimension of the product convex set of points in the n-dimensional space nuclear-mangetic fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1 ,1,1,1 and number base point k-face element three quantum transitions single inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1 ,1,1 and number base point k-face element three quantum transitions two inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring; or fourth band AU quantum transition ABCD circumscribed sphere position optimal control symetrical corners quadrupoles extend four circular sides and four poles sides attractive and repulsive number μΝ area A spaces two coordinates position normal vector

1,1 ,1 ,1,1 ,1 and number base point k-face element three quantum transitions three inner and outer description equilateral n-sides banded closed-chain, n-poles banded closed-chain, or circular closed-ring.

Dependent claim 8. Mangetic Bn, n-sides positioned symmetrical under electromagnetic field H equipotential wavefunction symetrical corners in A configuration joined regular sides ABCD by its circumscribed sphere unit cell axial circle continued in equilateral shape two lines of symmetry planes prism, or two lines of symmetry surface sides have divergence manifold is the sum of the dimensions of its factors manifolds and N, A differentiate map f : M→ N is called A diffeomorphism if it is A bijection and its inverse f-1 : N→ M is differentiate as well, if these functions are r times continuously differentiable, f is called A Cr-diffeomorphism, Two manifolds M and N are diffeomorphic symbol usually being ^ if there is A diffeomorphism f from M to N, they are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable, equipotential wavefunction sine λ/2, cubicl/4 and trigonl/3 qualifiers Lie Algebra solutions in harmonic simple fractions sine A unit circle SI , reduced from higher dimensions of the unit sphere, interior of the unit circle open unit disk, while the interior of the unit circle combined with the unit circle itself is the closed unit disk convex, The first convex set definition, example, inner description, algebraic property, separation theorem, and characterization along with any pair of its points x, y, also the entire segment [x, y] linking the points in the multidimensional case the segment [x, y] linking the points x, y€ Rⁿ Convex set Let x, y be two points in Rⁿ the set [x, y] = {z = λχ + (1 - λ)γ | 0 < λ < 1 } segment with the endpoints x, y, Second, A subset M of R" is convex, if it contains, along with any pair of its points x, y, also the entire segment [x, y] then x, y€E^: M, 0 < λ < 1 =» λχ + (1 - € M the exact sense of the definition for the empty set, proof present the contradiction of counter example formulate that it is not convex simple nonempty convex sets singletons have { points { and the entire space Rⁿ is the solution set of an arbitrary possibly, infinite system < b_a, a e A a^T _ax < b_a, a e A of linear inequalities with n unknowns x { the set M = {x £ R" | α^τ _αχ < b_a, a ε A} is convex, In particular, the solution set of A finite system Ax < b of m inequalities with n variables A is m ^χ n matrix is convex, A set of this latter type is three dimensions with flat faces, straight edges and sharp corners or vertices, Indeed, let x, y be two solutions to the system, does prove that any point ζ = λχ + λχ + (1 - λν) with λ ε |0, 11 also is A solution to the system, This is evident, since for every a€ A is a^T _ax < b_a a^T _ay < b_a, when, multiplying the inequalities by nonnegative reals λ and 1 - λ and taking sum of the results, α^τ _αχ + (1 - λ) <x^T _ay < b_a + (1 - λ) ba^— b_a, and what is in the left hand side is exactly α^τ _αζ, Note that any set given by example is not only convex, but also closed any plane in R" in particular any linear subspace is the figure whose sides are polygons sides that are line segments.

Dependent claim 9. A regular polygon has all its sides and angles equal. A polygon can be defined (as illustrated above) as A geometric object "consisting of A number of points (called vertices) and an equal number of line Specific polygons are named according to the number of sides, such as bisection two pole measure of change resection polygons μΝ continue named according to the number of sides and angles they have A plane shape with straight sides. Is it A Polygon are 2-dimensional shapes. They are made of straight lines, and the shape is" closed polyhedral set of all solutions to some system of linear equations, Now, A system of linear equations is equivalent to A system of linear inequalities can equivalently represent A linear equality by A pair of opposite linear inequalities, It follows that A plane is A particular case of A three dimensions with flat faces, straight edges and sharp corners or vertices set and is therefore convex, and can obtain the conclusion directly convexity of A set means that it is closed with respect to taking certain restricted set of linear combinations of its members { namely, the pair combinations with nonnegative coefficients of unit sum, shown that any plane or an affine set is closed with respect to taking linear combinations not obligatory positive of its elements with unit sum, having the additional property that it is also A convex set of points in the n-dimensional space Rn, cylindrical integral, or equilateral integral joined symetrical corners attractive and repulsive number μΝ area A spaces two coordinates position normal vector 1,1,1,1,1 ,1 and number base point k-face element in A tetrahedral configuration joined regular tetrahedron ABCD by its circumscribed sphere unit cell axial circle continued in equilateral shape two lines of symmetry planes prism, or

Dependent claim 10. Two lines of symmetry surface sides geometric optics, the paraxial approximation is a small- angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as

a lens).^[l][2]A paraxial ray is a ray which makes a small angle {&) to the optical axis of the system, and lies close to the axis throughout the system. ^[1] Generally, this allows three important approximations (for Θ in radians) for calculation of the ray's path:^[1] sin# « θ; ΐαηθ « Θ and cosd « 1 The paraxial approximation is used in Gaussian optics and first-order ray tracing. Ray transfer matrix analysis is one method that uses the approximation. In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent do not change for the "second-order" paraxial approximation (the second term in their Taylor series expansion is zero), while for cosine the second order approximation is cos6 « / - Q /2. The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.^[3J For larger angles it is often necessary to distinguish between meridional rays, which lie in a plane containing the optical axis, and sagittal rays, which do not.

Dependent claim 1 1. The mathematical function that describes the Gaussian beam is a solution to the paraxial form of the Helmholtz equation. The solution, in the form of a Gaussian function, represents the complex amplitude of the beam's electric field. The electric field and magnetic field together propagate as an electromagnetic wave. A description of just one of the two fields is sufficient to describe the properties of the beam. The behavior of the field of a Gaussian beam as it propagates is described by a few parameters such as the spot size, the radius of curvature, and the Gouy phase. ^[l] Other solutions to the paraxial form of the Helmholtz equation exist. Solving the equation in Cartesian coordinates leads to a family of solutions known as the Hermite-Gaussian modes, while solving the equation in cylindrical coordinates leads to the Laguerre-Gaussian modes. ^[2]t3] For both families, the lowest-order solution describes a Gaussian beam, while higher-order solutions describe higher-order transverse modes in an optical resonator. Second higher order electromagnetic waves simultaneous or beam divergence wave in linear media, two-dimensional intensity by inverted-Gaussian having red edge reflectance Re or radius of curvature; transmission Te simulation across the interface of two homogeneous media, paraxial light ray and red edge parallel to the axis of an optical system transmission Te of Gaussian beams measure small Θ angle with and position close to the axis of an optical paraxial light ray and red edge parallel to the axis of an optical system approximation to the full equations of optics that is valid in the limit of small Θ angles from the optical axis, having p primes represent projected Θ angles and distance of measure, and unprimed represent the object, with n the index of refraction to the left of the y-axis and the index of refraction to the right of the y-axis.

Dependent claim 12. The paraxial approximation is an approximation to the full equations of optics that is valid in the limit of small angles from the optical axis. Primes representing here image angles and distances, and unprimed represent the object, with n the index of refraction to the left of the y-axis and n' the index of refraction to the right of the -axis. The paraxial approximation then assumes that 3Ϊηξ « ξ (1); cost, « 1 (2); and ta^ « ξ, (3) where ξ may be any of i, i', u, u', or Φ. This regime is known as first-order, paraxial light ray and red edge parallel to the axis of an optical system, or Gaussian optics. Spectrall reflectance measurements of single point, quantitative

representation of the shape and position red edge reflectance in terms of parameters minimum reflectance R₀ and where the amplitude A₀ is the maximum of the function, red edge inflection point wavelength λρ and reflectance minimum wavelength λθ, of the red edge by re-measurement of parameters are: R₀, λο and λρ ±2.0nm. High spectrall resolution reflectance, inverted-Gaussian spectrall position of the red edge direct spectrometry mathematic, a Taylor series represent of the function as an infinite sum of terms that are calculated from the quantum valuesi₂₃ of the function's derivatives at a single point. As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1 , 3, 5, 7, 9, 1 1 and 13. The Taylor series of a real or complex- valued function f(x) that is infinitely differentiable at a real or complex number is the power series /(a) +f(a)/l!(x - a) +f'(a)/2!(x - a)² +f"(a)/3!(x - a)³ + . Which can be written in the more compact sigma notation as °°∑_n=o ⁽ⁿ⁾(a)/n!(x - a)ⁿ where n denotes

the factorial of n and / ⁽ⁿ⁾(a) denotes the nth derivative of / evaluated at the point a. The derivative of order zero of/ is defined to be / itself and (x - of and 0! are both defined to be 1. When a = 0, the series is also called a Maclaurin series. The exponential function e^x, and the sum of the first n + 1 terms of its Taylor series at 0. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for (1 - x)^~l is the geometric series 1 + x + x2 + x3 + " so the Taylor series for x^~l at a - 1 is 1 - (x - 1) + (x - l)² - ( χ - I)³ + . By integrating the above Maclaurin series, we find the Maclaurin series for log(l - x), where log denotes the natural logarithm: -x - l/2x² - l/3x³ - l/4x⁴ - ... and the corresponding Taylor series for log(x) at = 1 is (x - 1) - l/2(x - l)² + l/3( x - I)³ - l/4( x - l)⁴ + ^', and more generally, the corresponding Taylor series for log( r) at some a = x₀ is: log(x₀) + 1/xo (x - xo) - l/x²o(x - x₀)²/2 + ^"'. The Taylor series for the exponential function e* at a = 0 is 1 + xVl ! + x²/2! + x³/3 ! + x⁴/4! + x⁵/5! + = 1 + x + x²/2 + x³/6 x⁴/24 + x⁵/120 + ^' = °°∑_n=o x n! . The above expansion holds because the derivative of e* with respect to x is also e and e° equals 1. This leaves the terms (x - 0)" in the numerator and n in the denominator for each term in the infinite sum.

Independent claim 2. Turbine-turbine contain second common system bias Mersenne primes of form 2^Ap - 1 where p is a prime equivalently, primes of form 2^An - 1 for integers n values of p prime repunits in base second common system closed system hydraulic- force actuated, fermion A space function of normalization, mean value constraint, by inserting an exponential factor exp[- f(n)] with an adjustable Lagrange multiplier λ, maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ, hydraulic-electric actuated hydraulic-force actuated, fermion A space function of normalization, mean value constraint, by inserting an exponential factor exp[- f(n)] with an adjustable Lagrange multiplier λ, maximum entropy distribution given mean and standard deviation: the normal distribution Ν(μ,σ2) has maximum entropy among all real-valued distributions with specified mean μ and standard deviation σ.

Dependent claim 1. Turboshaft, turbofan, turbofluid, or turbojet continue natural n composite number 2n - 1 fluid dynamics control enhance mathematic optimum series, and parallel integral to the current, optimal control dynamic system transition state second common component atmospheric, temperature, pressure correction, in the presence of constraints for the state or input [0, 1] optimal control n = c prime force validate cofactor of 3 coefficient process wavefunctions take the form of spherical harmonics, described by Legendre polynomials, mutual acoustic impedance, method of moments Let Ω be a bounded domain in RN and let m be a -periodic function such that its restriction to Ω ^χ (0, T) belongs to L\(0, T), L^V(Q)) for some v > N/2 and s > 2v/2v - N, with v > 1, number Nv > 0 some + positive or number Nv < 0 some - negative and s > 2.

Dependent claim 2. We give necessary and sufficient conditions on m for the existence, uniqueness, simplicity, and the normalization function of the principal eigenvalue for the Dirichlet periodic parabolic eigenvalue problem with weight m of load factor LF is n values = p prime repunits horizontal weight, balance center of gravity CG, or vertical lift VL weight L/W center of gravity CG, dynamic variable change match prime force 1 : 1 inertial frame of reference constant of quantum mechanics weight : thrust ratio, or weight : power ratio coefficient of tractive force maximum to the normal force, F/N coefficient of friction = μ the power drivetrain system one or more wheels in contact with the roadway or railroad track having motion between the inertial frame of reference constant c is then a standard model of surface friction coefficient of kinetic friction = μk riding above the surface divided by the weight on the drive system equal to traction ratio μ≤ = coefficient of static friction propulsion power-plant incorporate coherent whole of the state inertial space frame of reference that describe time and space composite modification of order matter and antimatter catylitic quantum number, homogeneous, isotropic, and time-independent manner state of constant rectilinear motion of each one another are zero acceleration momentum p intrinsic now it possesses the turbo-booster on occasion arithmetic ordering sequence switch on : off engage super-fluid function / the general derivative test for stationary points let /be a real-valued, differentiable function on the interval complex number i² = - 1, 1 c R, e e l and n > 1 an integer. If now holds f (c) = = f ⁿ⁾ (c) = 0 and i ^n+l) (c)≠ 0 then, n is even and we have a (local) saddle point at c. More precisely: 1. ί ^η+1) (c) < 0→ c is a strictly decreasing point of inflection 2. f*^n+l) (c) > 0→ c is a strictly increasing point of inflection.

Dependent claim 3. This analytical test classifies any extremum are calculated on the function domain absolute extremum the maximum and minimum element of a set maxima and minima for cos(37tx)/x, 0.1< x <1.1 rotating on an numeric-n cyclones vortex axisymmetric rotating on a turboshaft, turbofan, turbo-fluid, or turbojet acceleration, finding maxima, minima, tangents to curves, computation of areas, finding center of mass, the least action principle, approximate equality, correlated problems maxima of functions, and then finding tangent lines to curves valid identity between two constants. Transcendental curves as the cycloid mathematics bias numeric value built from position and momentum complex numbers from three discrete points, begin our particle has the whole x- axisymmetric to play on and in lieu of the wave function as three complex numbers, we continue to add a different complex number for every point on the x-axisymmetric, so then Y is now going to be a complex function of x, position probabilities with a continuous wave function as the standard part function product the magnitude of it's real number absolute extremum prime value |x| turning, bending, or curving a turn, bend, or curve an apparent magnitude volume of change inflection c prime force, changing meaning the force of load 0, 1 optimal control n = c prime force validate cofactor of 3 coefficient process wave functions take the form of spherical harmonics, described by Legendre polynomials, mutual acoustic impedance, method of moments Let Ω be a bounded domain in RN and let m be a -periodic function such that its restriction to Ω χ (0, T) belongs to L^s((0, 7), Ι^ν(Ω)) for some v > N/2 and s > 2v/2v - N, with v > 1 , stationary point of f. The Potential function x has all of its derivatives at 0 equal and normalized Y so that the total probability sum is one equal to terms e, equal to terms π, or 0 equal to terms numeric-n, except for the 8th derivative, which is positive. Minimum at 0 defined on a domain X has a global : absolute extremum prime and bias numeric value built from calculated the expectation value of position and momentum complex function of this form Y=_e ^ipx/h _; where p is some number takes at a point relative number Nv > 0 some + positive or number Nv < 0 some - negative and s > 2. X=p/2 is some little point, one of infinitely many such points in the range 0<x<2p. The probability of being at exactly that point, or any other specific position, is zero. So Y is not a point but rather a probability density, and now the golden rule is: the integral of \ Y\ between any two

f¹ 2

points, gives the probability of the particle appearing between those two points. The instance, ^Jo|Y| dx gives the probability of finding the particle somewhere between x=0 and x=l . Now then we give necessary and sufficient conditions on m for the existence, uniqueness, simplicity, and the normalization function of the principal eigenvalue for the Dirichlet periodic parabolic eigenvalue problem with weight m of load factor LF magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, z*, where for any complex number z = a + bi, its complex conjugate is z* = a - bi.

Independent claim 3. Refrigeration vacuum pressure pumps rated alternating current AC or direct current DC drawing refrigerant through insulated super alloy feeder tubes in a parallel circuit to separate numeric super alloy curved or linear mathematic expression numbers, symbols and dynamics operators comprising a scientific aperture, control valve, or control valve venturi a vector calculus fundamental theorem limits of functions continuity, mean value theorem, by way of real- valued function / is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and /(a) = /(b), then there exists a c in the open interval (a, b) such that f (c) = 0, continuum mechanics differential, integral, series, vector, multivariable, and discrete entity, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point, condensed phases super-ii id as it is cooled, molecular diffusion coefficient the relevant vector field is the velocity of motion fluid at a point, the fluid cools and contracts, the divergence has a negative value, as the region is a sink, measured in angulation and a scientific aperture, control valve, or control valve venturi molecular diffusive fluid refrigerant chemistry continuum vacuum mechamcs pressure pumps negative or positive pressurization stream · is the accelerator. Fluid mechanic the flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity vectori q = ||u|| and is a scalar field, positive pressurization supercooling refrigeration deceleration temperature less than or equal to < -109°F degrees Fahrenheit or -78.5 °C degrees Celsius supercooled fluid refrigerant passes over super alloy and through cooling fan. Numeric isolated super alloy cylinders in series and parallel refrigerant-dynamic conductive circuits are connected to the portable or stationary refrigeration unit from super alloy connecting rods supercooling the outside walls of the cylinders. The outer cylinder cools activating an electronic air compressor rated alternating current AC or direct current DC filling the inner variable cylinders with fluid compression providing diffusive flux compensation expanding or contracting the inner cylinders as fluctuating compensator forcing the carbon dioxide C0₂ filled outer cylinders through a continuous circuit of super alloy curved or linear mathematic expression numbers, symbols and dynamics operators comprising a scientific aperture, control valve, or control valve venturi a vector calculus fundamental theorem limits of functions continuity, mean value theorem, by way of real-valued function / is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and /(a) = /(b), then there exists a c in the open interval (a, b) such that f (c) = 0, continuum mechanics differential, integral, series, vector, multivariable, and discrete entity, divergence is a vector operator that measures the

magnitude of a vector field's source or sink at a given point, in terms of a signed scalar, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point, condensed phases super-fiuid as it is cooled, molecular diffusion coefficient the relevant vector field is the velocity of motion fluid at a point, the fluid cools and contracts, the divergence has a negative value, as the region is a sink, measured in angulation and a scientific aperture, control valve, or control valve venturi molecular diffusive fluid refrigerant chemistry continuum vacuum mechanics pressure pumps negative or positive pressurization stream is the accelerator.

Dependent claim 1. Fluid mechanic the flow velocity u of a fluid is a vector field u = u x, t which gives the velocity of an element of a fluid at position x and time t, the flow speed modulate q is the length of the flow velocity vectori q = ||u|| and is a scalar field, positive pressurization supercooling refrigeration deceleration temperature less than or equal to < -109°F degrees Fahrenheit or -78.5°C degrees Celsius carbon dioxide C0₂ forming dry ice in the outer superconducting cryocircuit numeric and hypernumeric cylinders vacuum component hypernumeric series and parallel superconducting cryocircuit processor molecules oscillations superconducting cryospheres,

superconducting cryoplates, superconducting cryotubes of organic dispersion computational integral analytical organic solid-state chemistry program processor flux compensators, superconducting cryocoils and substrate spectral parallel lines superconducting across transitive property of equality if a = b and b = c, then a = c having one of the equivalence property of equality and inequality cryolines.

Dependent claim 2. Supplemental component normalization of program temperature modulation q Euclidean algorithm integers p and q that p > q fluid dynamics, spectrometric computational integral analytical -organic solid- state chemistry program processor, cryoplate refrigeration particles in the continuous phase of dynamic fluid mechanics, in transmittance U ratio of incident radiation to transmitted radiation function hypemumeric comparator processor hyper-transition superconducting cryotanks, superconducting cryotubes of organic dispersion

computational integral analytical organic solid-state chemistry program processor flux compensators,

superconducting cryocoils and superconducting cryocircuit coupled superconducting cryogenic substrate transition refrigerant fluid filled or unfilled refrigeration natural gas, organic waste or biomass to synthesis gas chloride CI^", nitrogen, hydrogen, methane, phosphane, and argon glazing > - 5°C, comparative liquid and solid chloride CI^" superconducting cryotank, coupled substrate superconducting cryocircuit the transition refrigerant fluid filled or unfilled superconducting cryotanks and are vacuum component hypemumeric series and parallel superconducting cryocircuit processor molecules oscillations superconducting cryospheres, superconducting cryoplates,

superconducting cryotubes of organic dispersion computational integral analytical organic solid-state chemistry program processor flux compensators, superconducting cryocoils and substrate spectral parallel lines

superconducting across transitive property of equality if a = b and b = c, then a = c having one of the equivalence property of equality and inequality cryolines neutral nitryl radical, symetrical corners chlorine CI^" molecule configuration joined at central carbon atom, tetrahedral arrangement neutral radical pairs oxygen 0₂ and helium He crystallographic phase network superconducting cryotube, flux compensator and superconducting cryocoil helium He gas, isolated antiprotonic helium three-body atoms He superconducting are the supercooling cryotube, flux compensator and is the superconducting cryocoil circuit in transmittance U ratio of incident radiation to transmitted radiation.