LaTeX vs MathPix image — 993787212-Burkhard-Heim-s-Unified-Field-Theory

c=\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}

x_{4}=\mathrm{i} c t

E=m c^{2} \quad \Longleftrightarrow \quad \text { Energy } \leftrightarrow \text { Mass (Inertia) }

C_{p} \phi_{k m}^{i}=\lambda_{p}(k, m) \phi_{k m}^{i}

(n-1)^{2}-1=p(p-1)(p-2)

(n-1)^{2}-1=4(3)(2)=24 \Longrightarrow(n-1)^{2}=25 \Longrightarrow n-1=5 \Longrightarrow \mathbf{n}=\mathbf{6}

\lim _{m \rightarrow 0}\left(r_{0} \cdot \lambda\right)=\tau \approx 6.15 \times 10^{-70} \mathrm{~m}^{2}

\sigma_{\text {Newton }}=\sigma_{(0) 0}=\frac{M_{(0)}}{V_{0}}

\operatorname{div} \vec{G}=\frac{\sigma}{\alpha}, \quad \text { where } \sigma=\sigma\left(M_{(0)}+\mu_{i}+\mu_{e}\right)

\left(=\mu_{e}+\mu_{i}+M_{(0)}=\mu+M_{(0)}\right)

g_{i k}\left(g_{i k}^{(1)}, g_{i k}^{(2)}, g_{i k}^{(3)}\right)=g_{k i}^{*}

M=M_{(0)}+\mu_{i}+\mu_{e}=\mathrm{const}

\frac{\mathrm{d} \sigma}{\mathrm{~d} t}=0

\frac{\mathrm{d} \sigma}{\mathrm{~d} t}=\dot{\sigma}+\sum_{k=1}^{3} \frac{\partial \sigma}{\partial x_{k}} \dot{x}_{k}=\dot{\sigma}+\sum_{k=1}^{3} \frac{\partial \sigma}{\partial x_{k}} \frac{\mathrm{~d} x_{k}}{\mathrm{~d} t}

\dot{\sigma}+\vec{v} \cdot \operatorname{grad} \sigma=0

0=\dot{\sigma}+\operatorname{div}(\sigma \vec{v}) \Longrightarrow \dot{\sigma}=-\operatorname{div}(\sigma \vec{v})

\alpha \operatorname{div} \dot{\vec{G}}=\dot{\sigma}

\alpha \operatorname{div} \dot{\vec{G}}=-\operatorname{div}(\sigma \vec{v})

0=\operatorname{div}(\alpha \dot{\vec{G}}+\sigma \vec{v})

b \operatorname{rot} \vec{\mu}=\alpha \dot{\vec{G}}+\sigma \vec{v}

b \operatorname{rot} \operatorname{rot} \vec{\mu}=\alpha \operatorname{rot} \dot{\vec{G}}+\operatorname{rot}(\sigma \vec{v})

b(\operatorname{grad} \operatorname{div} \vec{\mu}-\operatorname{div} \operatorname{grad} \vec{\mu})=\alpha \operatorname{rot} \dot{\vec{G}}+\operatorname{rot}(\sigma \vec{v})

\vec{w}=\operatorname{grad} \operatorname{div} \vec{\mu}-\frac{\operatorname{rot}(\sigma \vec{v})}{b}=\operatorname{div} \operatorname{grad} \vec{\mu}+\frac{\alpha}{b} \operatorname{rot} \dot{\vec{G}}

a^{2} \ddot{\vec{\mu}}=\operatorname{div} \operatorname{grad} \vec{\mu}, \quad a^{2} \neq 0

\vec{w}=a^{2} \ddot{\vec{\mu}}+\frac{\alpha}{b} \operatorname{rot} \dot{\vec{G}}

\vec{w}=-\alpha \beta \ddot{\vec{\mu}}+\frac{\alpha}{b} \operatorname{rot} \dot{\vec{G}} \quad\left[\frac{\mathrm{~kg}}{\mathrm{~m}^{3} \mathrm{~s}}\right]

\frac{\alpha}{b} \operatorname{rot} \dot{\vec{G}}=\alpha \beta \ddot{\vec{\mu}}+\dot{\sigma}_{\mu} \vec{f}(x)

\frac{\alpha}{b} \operatorname{rot} \vec{G}=\alpha \beta \dot{\vec{\mu}}+\sigma_{\mu} \vec{f}(x)

\frac{\alpha}{b} \operatorname{rot} \vec{G}=\alpha \beta \dot{\vec{\mu}}+\left(\sigma-\sigma_{(0)}\right) \vec{f}(x)

0=\alpha \operatorname{div} \beta \dot{\vec{\mu}}+\operatorname{div}\left(\left(\sigma-\sigma_{(0)}\right) \vec{f}(x)\right)

\begin{aligned} \operatorname{rot} \vec{G} & \sim \beta \dot{\vec{\mu}}+\frac{\sigma-\sigma_{(0)}}{\alpha} \vec{f}(x) \\ \alpha \beta \operatorname{div} \dot{\vec{\mu}} & =-\left(\sigma-\sigma_{(0)}\right) \operatorname{div} \vec{f} \end{aligned}

\vec{F}_{G}=m(\vec{G}+\vec{v} \times \vec{\mu})

\operatorname{rot} \vec{\mu}=\alpha \dot{\vec{G}} \quad \text { and } \quad \operatorname{rot} \vec{G}=\beta \dot{\vec{\mu}}

\operatorname{divgrad} \vec{p}+\frac{1}{\omega^{2}} \frac{\partial^{2} \vec{p}}{\partial t^{2}}=\overrightarrow{0}, \quad \text { where } \omega^{2}=\frac{1}{\alpha|\beta|}, \quad 0<\omega<\infty

\hat{\mathbf{B}}=\hat{\mathbf{A}}_{+} \hat{\mathbf{A}}_{-}

\Gamma_{i j}^{k}-\Gamma_{j i}^{k}=2 S_{i j}^{k}

G_{i k}=\kappa T_{i k}

\vec{G}=-\nabla \Phi

\left[\hat{A}_{+}, \hat{A}_{-}\right]=0

\nabla \cdot\left(\alpha \frac{\partial \vec{G}}{\partial t}+\sigma \vec{v}\right)=0

b \nabla \times \vec{\mu}=\alpha \frac{\partial \vec{G}}{\partial t}+\sigma \vec{v}

\sigma=\sigma\left(M_{(0)}+\mu_{i}+\mu_{e}\right)

\operatorname{div} \vec{G}=\frac{\sigma}{\alpha} \quad(\alpha=\text { scaling factor })

\operatorname{div}(\alpha \dot{\vec{G}}+\sigma \vec{v})=0

\begin{aligned} \operatorname{div} \vec{G} & =\frac{\sigma}{\alpha} \\ b \operatorname{rot} \vec{\mu} & =\alpha \dot{\vec{G}}+\sigma \vec{v}, \quad(b \neq 0) \end{aligned}

\operatorname{div} \operatorname{grad} \vec{p}+\alpha \beta \ddot{\vec{p}}=\overrightarrow{0}, \quad \omega^{2}=\frac{1}{\alpha|\beta|}

\left[\hat{\mathbf{A}}_{+}, \hat{\mathbf{A}}_{-}\right]=0

\vec{\xi}_{p}^{(j)}=\vec{\xi}_{1}^{(j)} \ldots \vec{\xi}_{4}^{(j)}=f\left(x_{1} \ldots x_{4}\right) \quad \text { for } j=1 \ldots n \text { interactions }

d s^{2}=\left(g_{i k}^{(1)}+g_{i k}^{(2)}+g_{i k}^{(3)}\right) d x^{i} d x^{k}

g_{i k}=g_{i k}^{(S)}+i g_{i k}^{(A)}

T_{i k}=\sum_{m=1}^{4} M_{i m} M_{m k}

\vec{\xi}_{p}^{(j)}=\vec{\xi}_{1}^{(j)} \ldots \vec{\xi}_{4}^{(j)}=f\left(x_{1} \ldots x_{4}\right) \quad(\text { for } j=1 \ldots n \text { interactions and } p=1 \ldots 4 \text { coordinates })

d s^{2}=\vec{z}_{, i}^{+} \vec{z}_{, k}^{+*} d x^{i} d x^{k}+\left(\vec{z}_{, i}^{-} \vec{z}_{, k}^{+*}+\vec{z}_{, i}^{+} \vec{z}_{, k}^{-*}\right) d x^{i} d x^{k}+\vec{z}_{, i}^{-} \vec{z}_{, k}^{-*} d x^{i} d x^{k}

R_{i k}^{(1)}-\frac{1}{2} g_{i k}^{(1)} R^{(1)} \sim V_{i k}

R_{i k}-\frac{1}{2} g_{i k} R \sim T_{i k}

R \sim-T

R_{i k} \sim T_{i k}-\frac{1}{2} g_{i k} T=W_{i k}

\text { Energy Density }=\frac{\text { Energy }}{\text { Volume }} \times \frac{\text { Time }}{\text { Time }}=\frac{\operatorname{Action}(\omega)}{\text { Space-Time }(\Omega)}

d \Omega=i c w d x^{1} d x^{2} d x^{3} d t

\omega_{i k}=h N_{i k} \quad\left(\text { where } N_{i k} \text { is a complex integer }\right)

\Delta \omega_{i k}=h \Delta N_{i k}

W_{i k}=\frac{\Delta \omega_{i k}}{\Delta \Omega} i c w=i c w h \frac{\Delta N_{i k}}{\Delta \Omega}

R_{i k} \sim w \cdot \eta_{i k} \quad \text { (2) }

\alpha W_{k m}=\sum_{j=1}^{4} G_{(j) k m}

G_{(p) k m} \Longrightarrow \lambda_{(p)}(k, m) \phi_{k m}^{(p)}

C_{(p)} \phi_{k m}^{(p)}=\lambda_{(p)}(k, m) \phi_{k m}^{(p)}

\begin{aligned} \alpha \nabla \cdot \vec{g} & =-\sigma \\ \beta \nabla \cdot \vec{\mu} & =-\sigma \nabla \cdot \vec{f} \\ \nabla \times \vec{g} & =-\beta \dot{\vec{\mu}}+\frac{\sigma}{\alpha} \vec{f} \\ \nabla \times \vec{\mu} & =\alpha \dot{\vec{g}}-\sigma \vec{v} \end{aligned}

\begin{aligned} \nabla \cdot \vec{g} & =-4 \pi G \sigma \\ \nabla \cdot \vec{B}_{g} & =0 \\ \nabla \times \vec{g} & =-\dot{\vec{B}}_{g} \\ \nabla \times \vec{B}_{g} & =\frac{1}{\omega^{2}}(\dot{\vec{g}}-4 \pi \sigma \vec{v}) \end{aligned}

T_{k}^{i}=f_{k n} f^{i n}

T_{i k}^{(E)}=W_{i k}+\Phi_{i k}, \quad W_{i k}=W_{k i} \approx V_{i k}

\Phi_{i k}=-\Phi_{k i} ; \quad \Phi_{12}=\varphi_{3} ; \quad \Phi_{13}=-\varphi_{2} ; \quad \Phi_{23}=\varphi_{1} ; \quad \Phi_{j, 4}=-\varphi_{j}

\vec{\varphi} \sim \vec{g} \times\left(\vec{E}+\vec{H} \sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\right)

V_{i}^{k}=f^{k l} f_{i l}-\frac{1}{4} \delta_{i}^{k} f_{m n} f^{m n}

\left\{\begin{array}{c} i \\ k l \end{array}\right\}=\left\{\begin{array}{c} i \\ l k \end{array}\right\}=\frac{1}{2} g^{i m}\left(\frac{\partial g_{k m}}{\partial x^{l}}+\frac{\partial g_{m l}}{\partial x^{k}}-\frac{\partial g_{k l}}{\partial x^{m}}\right)

R_{i k}-\frac{1}{2} g_{i k} R \sim V_{i k}

\widehat{\left.\left\{\begin{array}{c} i \\ k l \end{array}\right\} \neq \widehat{\left\{\begin{array}{c} i \\ l k \end{array}\right\}}, \quad R_{i k} \neq R_{k i}\right\}=0.0}

\hat{R}_{i k}-\frac{1}{2} \hat{g}_{i k} \hat{R} \sim \hat{T}_{i k}

T_{i k}=\sum_{m=1}^{4} M_{i m} M_{m k}=T_{i k}^{+}+T_{i k}^{-}

d s^{2}=\left(g_{i k}^{(1)}+g_{i k}^{(2)}+g_{i k}^{(3)}\right) d x^{i} d x^{k}

R_{i k}-\frac{1}{2} g_{i k} R \sim T_{i k}

\left.\begin{array}{r} k=1 \ldots 4 \\ m=1 \ldots 4 \\ p=1 \ldots 4 \end{array}\right\} \quad 4 \times 4 \times 4=64 \text { nonlinear eigenvalue equations }

R_{k m p}^{k}=\Gamma_{k p, m}^{k}-\Gamma_{k m, p}^{k}+\Gamma_{m s}^{k} \Gamma_{k p}^{s}-\Gamma_{p s}^{k} \Gamma_{k m}^{s}=A_{m p}

C_{m} \phi_{k m}^{k}=\lambda_{m}(k, m) \phi_{k m}^{k}=0

16+16-4=28 \text { empty spectra }

\lambda_{(m)}(m, p) \phi_{m p}^{i}=-\lambda_{(p)}(m, m) \phi_{m m}^{i}

\phi_{m p}^{i}=-\frac{\lambda_{(p)}(m, m)}{\lambda_{(m)}(m, p)} \phi_{m m}^{i} \xrightarrow{4 \mathrm{D} \text { limit }} \frac{0}{0} \phi_{m m}^{i}

g_{i k}^{(6)}=\left(\begin{array}{ccc|c|cc} g_{11} & g_{12} & g_{13} & g_{14} & 0 & 0 \\ g_{21} & g_{22} & g_{23} & g_{24} & 0 & 0 \\ g_{31} & g_{32} & g_{33} & g_{34} & 0 & 0 \\ \hline g_{41} & g_{42} & g_{43} & g_{44} & g_{45} & g_{46} \\ \hline 0 & 0 & 0 & g_{54} & g_{55} & g_{56} \\ 0 & 0 & 0 & g_{64} & g_{65} & g_{66} \end{array}\right)

r q e^{q}=A\left(1-\frac{\gamma m^{3} r}{\hbar^{2}}\right)^{2}

\lim _{m \rightarrow 0}\left(r_{0}^{*} \cdot \lambda\right)=\pi \gamma \hbar=\tau

\tau \approx 6.15 \times 10^{-70} \mathrm{~m}^{2}

\text { Total Spin }=\sigma \hbar, \quad \text { where } \sigma=\mathrm{i}\left(s+J(-1)^{P}\right)

B=k-1

P-P(k+1)+5 k=2\left(k^{2}+1\right) \quad \Longrightarrow \quad P=2-k \text { or } P=2 k-1

S_{k}=(-1)^{x} \Sigma_{s}\left(\text { sum of } q_{i} \text { of all possible multiplets for } k\right)

m(N, k, P, Q, \kappa)=m_{e} \cdot\left[1+\sum_{j} f_{j}(k, P, Q, \kappa) \cdot \mathrm{e}^{-N \lambda_{j}}\right]

m_{L} c s_{0}=4 \sqrt[4]{\pi} \sqrt[3]{3 \pi s_{0} \gamma \hbar} \sqrt{\frac{c \hbar}{3 \gamma}}

(2 \pi)^{5} \alpha \sqrt{1-\alpha^{2}}=9 \vartheta\left(1-A_{1} A_{2} Y_{3}\right), \quad \text { where } \alpha>0

\pi^{2} \varepsilon_{ \pm}= \pm 3 \sqrt{\hbar / R_{-}}

m_{\max }=\sqrt{\frac{c h}{\gamma}} \sqrt[4]{2} \eta_{q}

H \approx \sqrt{\pi e \gamma \sigma}

f\left(\frac{D f^{3}}{4 \sqrt{2 \tau}} \sqrt{3}-1\right)^{2} \sqrt{3 \tau}=D \sqrt{2} \quad \text { where } \quad f=\frac{\sqrt[4]{C}}{\sqrt{C-1}}, \quad C=\frac{e D \sqrt{\tau}}{\pi E}>1

T_{0}\left(D_{f m p}\right)=0 \leq t \leq \vartheta\left(D_{p m f}\right)=2 T_{A}<\infty

\Phi_{H}(r)=-\frac{G M}{r}\left(1-\mathrm{e}^{-r / R_{H}}\right)

R_{H}=\sqrt{\frac{1}{\pi e \gamma \rho_{a l l}}}

D=2 \cdot R_{0}\left(m_{\min }\right)

f\left(\frac{D f^{3}}{4 \sqrt{2 \tau}} \sqrt{3}-1\right)^{2} \sqrt{3 \tau}=D \sqrt{2} \quad \text { where } \quad f=\frac{\sqrt[4]{C}}{\sqrt{C-1}}, \quad C=\frac{e D \sqrt{\tau}}{\pi E}>1

n=1 \quad \Longrightarrow \quad \tau_{0}=\pi D_{0}^{2}

\eta^{7}-\eta= \pm a, \quad \text { where } \quad 2 \eta^{2}=f_{(0)} \sqrt[6]{6 / \pi}, \quad a \sqrt{\pi}=\sqrt[6]{\pi / 6}

t=n \cdot \vartheta \quad \text { (where } n \text { is an integer) }

T_{0}\left(D_{f m p}\right)=0 \leq t \leq \vartheta\left(D_{p m f}\right)=2 T_{A}<\infty

t=n \cdot \vartheta \quad \text { (where } n \text { is an integer) }

\int_{x_{0}}^{x_{n}} f(x) d x=n \tau \quad\left(\text { where } n \in \mathbb{Z}^{+}\right)

x_{n}=x(n), \quad y_{n}=f\left(x_{n}\right)=f(n)

\frac{\partial \varphi}{\partial n}=\lim _{v \rightarrow+1} \frac{1}{v}(\varphi(n)-\varphi(n-v))=\varphi(n)-\varphi(n-1)

\partial \varphi(n)=\varphi(n)-\varphi(n-1) \quad(\text { for } 1 \leq n \leq N)

S_{n_{1}}^{n_{2}} \varphi \circlearrowright n=S_{n_{1}}^{n_{2}} \circlearrowright \phi=\sum_{n=n_{1}}^{n_{2}}(\phi(n)-\phi(n-1))=\phi\left(n_{2}\right)-\phi\left(n_{1}-1\right)

S_{n_{1}}^{n_{1}} \varphi \circlearrowright n=\phi\left(n_{1}\right)-\phi\left(n_{1}-1\right)=\varnothing \phi\left(n_{1}\right)=\varphi\left(n_{1}\right) \neq 0

C ; \varphi(n)=\Im \varphi(n)=\varphi(n)-\varphi(n-1)

\bar{Z}(i)=\bar{e}_{i}()_{i}, \quad\left(\bar{e}_{i}, \bar{e}_{k}\right)_{L}=\hat{A}\left(n_{i}\right)_{1}^{L}

{ }^{m} \bar{T}={ }^{m} \bar{C} ; n=\left(\prod_{k=1}^{m} C_{i_{k}}\right) ; n

\left(C_{i} \times C_{k}\right)_{ \pm} \neq 0 \Longrightarrow C_{i} C_{k}-C_{k} C_{i} \neq 0

{ }^{2} \bar{C}={ }^{2} \bar{C}_{+}+{ }^{2} \bar{C}_{-}

\partial^{2} \varphi-3 \breve{\partial} \varphi+\varphi=0 \quad \Longrightarrow \quad\left(\partial^{2}-3 \circlearrowright+E\right) ; \varphi=0

\hat{s}=\operatorname{ROT}_{N} \hat{\phi}=\left(\begin{array}{cc} 0 & { }^{2} \bar{s}_{12} \\ -{ }^{2} \bar{s}_{12} & 0 \end{array}\right), \quad \text { where }{ }^{2} \bar{s}_{\alpha \beta} ; n=S S \circlearrowright \bar{\xi}_{\alpha} \times \partial \bar{\xi}_{\beta}

\begin{aligned} \text { Gradient: } \bar{\delta} \varphi & =\mathrm{GRAD}_{\mathrm{L}} \varphi \\ \text { Divergence: } \quad \text { sp } \bar{\delta} ;{ }^{m} \bar{C} & ={\overline{\mathrm{DIV}_{\mathrm{L}}}}^{m} \bar{C} \\ \text { Rotation (Curl): } \quad \bar{\delta} ;{ }^{m} \bar{C}-\left(\bar{\delta} ;{ }^{m} \bar{C}\right)^{x} & =\operatorname{ROT}_{\mathrm{L}}{ }^{m} \bar{C} \end{aligned}

S_{\Omega(L)} \operatorname{DIV}_{\mathrm{L}} \bar{\phi} \breve{\partial} V=S_{\Omega(L-1)} \bar{\phi} \breve{\partial} \bar{V}

\underline{N}=S \bar{K} \breve{\partial} \bar{n}

{ }^{2} \bar{K}={ }^{2} \bar{\kappa} ; n

{ }^{2} \bar{\gamma}=\operatorname{sp}\left({ }^{2} \bar{\kappa} \times{ }^{2} \bar{\kappa}\right)

L=\binom{N}{p} \quad \text { and } \quad \frac{N}{p}=M \geq 1(\text { where } M \in \mathbb{Z})

\Gamma_{p k l}^{(a b)}(\tau)=[p k l(a b)] ; n, \quad{ }^{[3]}[p k l(a b)]=[\widehat{a b}]

\gamma \frac{i p}{(c d)}[p k l(a b)]=\left[\begin{array}{c} i \\ k l(c, d)-+(a, b) \end{array}\right]=\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]

\breve{\mathrm{O}}_{p}^{2} n^{\underline{i}}+\frac{\alpha_{k} \alpha_{l}}{\alpha_{i}} \breve{\mathrm{O}}_{p} n^{\underline{k}} \breve{\mathrm{O}}_{p} n^{\underline{l}}[k l(c, d)-+(a, b)] ; n=0

\widehat{[]}=\sum_{\alpha=1}^{\omega^{4}}\left(\left[\begin{array}{c} \widehat{(c d)} \\ -+(a b) \end{array}\right]+\operatorname{sp}^{2} \bar{Q}(\alpha) ;() \times\left[\begin{array}{c} \widehat{(c d)} \\ -+(a b) \end{array}\right]\right)

L ; \hat{[]}={ }^{4} \overline{0} \quad \text { where } L=K-\bar{\lambda} \times()

K_{m} ;\left[\begin{array}{l} i \\ k \\ l \end{array}\right]=\underline{\partial}_{l}\left[\begin{array}{l} i \\ k \\ m \end{array}\right]-\underline{\partial}_{m}\left[\begin{array}{l} i \\ k \\ l \end{array}\right]+\left[\begin{array}{l} i \\ l \end{array}\right] ;()\left[\begin{array}{l} { }_{k}^{s} \\ k \end{array}\right]-\left[\begin{array}{c} i \\ m \end{array}\right] ;()\left[\begin{array}{l} s \\ k \\ l \end{array}\right]=\lambda_{m}(k, l)\left[\begin{array}{l} i \\ k \\ l \end{array}\right]

a_{m l}=-\frac{\lambda_{l}(m, m)}{\lambda_{m}(m, l)} \Longrightarrow\left[\begin{array}{c} i \\ m l \end{array}\right]=a_{m l}\left[\begin{array}{c} i \\ m \end{array}\right]

\left((a(k, l)-1) \underline{\partial}_{l}-\sum_{l \neq m} \underline{\partial}_{m}\right) ; \varphi_{k l}+\varphi_{k l}^{2}=\lambda(k, l) \varphi_{k l}

\bar{a}_{k l} \mathrm{GRAD}_{q} \varphi_{k l}=\lambda(k, l) \varphi_{k l}-\varphi_{k l}^{2}

\frac{\partial u}{1-u^{2}}= \pm \frac{1}{2} \lambda(k, l) \partial N_{k l}= \pm \Lambda_{k l}

\left(E-\Psi_{k l}\right)^{\Lambda_{k l}+1} \cdot \Psi_{k l}^{\Lambda_{k l}-1}=2^{-2 \Lambda_{k l}} \cdot C_{k l} e^{-\lambda_{k l} \mu}

\Lambda_{k l}=\alpha_{l}(a(k, l)-1)^{-1}-\sum_{m \neq n} \alpha_{m}, \quad(a(k, l)-q) \cdot \lambda_{k l}=\lambda(k, l)

\bar{s}_{x}=\sum_{\mu, p, q} \mathbb{P}_{x}(\mu)^{q}= \pm \bar{s}_{0} \hbar m_{x} / 2

\gamma_{i k}^{(\mu v)}=\sum_{m=1}^{6} \kappa_{i m}^{(\mu)} \kappa_{m k}^{(v)}

g_{i k}^{(6)}=\left(\begin{array}{ccc|c|cc} g_{11} & g_{12} & g_{13} & g_{14} & 0 & 0 \\ g_{21} & g_{22} & g_{23} & g_{24} & 0 & 0 \\ g_{31} & g_{32} & g_{33} & g_{34} & 0 & 0 \\ \hline g_{41} & g_{42} & g_{43} & g_{44} & g_{45} & g_{46} \\ \hline 0 & 0 & 0 & g_{54} & g_{55} & g_{56} \\ 0 & 0 & 0 & g_{64} & g_{65} & g_{66} \end{array}\right)

(n-1)^{2}-1=p(p-1)(p-2) \Longrightarrow(n-1)^{2}-1=6(5)(4)=120 \Longrightarrow(n-1)^{2}=121 \Longrightarrow \mathbf{n}=\mathbf{1 2}

R_{12}=\underbrace{R_{3}(\text { Space })+T(\text { Time })+S_{2}(\text { Structure })}_{R_{6}(\text { Material World })}+\underbrace{I_{2}(\text { Information })+G_{4}(\text { Background })}_{V_{6}(\text { Non-Material Background })}

M(c, d)=T ; m=\mu_{+}\left[\sum_{j=1}^{4} \alpha_{j} G_{j}+\left(1-\frac{\alpha_{-}}{\alpha_{+}}\right) F_{S}+q \frac{\alpha_{-}}{\alpha_{+}}\right]

\bar{m}_{0}= \pm m_{0} \sqrt{ \pm i}

\tau \omega c^{2}=\pi \gamma \hbar

\Delta x \Delta p \geq \frac{\hbar}{2}=\frac{\omega c^{2}}{2 \pi \gamma} \tau

T_{i k} \sim \frac{d W_{i k}}{d \Omega}

W_{i k}=h\left(N_{i k}+i K_{i k}\right), \quad N, K \in \mathbb{Z}

\eta_{i k}=\frac{\Delta N_{i k}}{\Delta \Omega}

R_{i k} \sim w \cdot \eta_{i k}

\int_{x_{0}}^{x_{n}} f(x) d x=n \tau

x_{n}=x(n), \quad y_{n}=f\left(x_{n}\right)=f(n)

n=\binom{4}{2}=\frac{4 \times 3}{2 \times 1}=6

x_{i}(n)=\alpha_{i} \cdot n_{i} \cdot \sqrt{\tau}

L=\binom{N}{p}=\binom{4}{2}=6

\lim _{n \rightarrow \infty} \sum_{i} \Delta x_{i} \approx \int d x

\frac{d f(x)}{d x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

\frac{\Delta \varphi}{\Delta n}=\frac{1}{v}(\varphi(n)-\varphi(n-v))

\frac{\partial \varphi}{\partial n}=\lim _{v \rightarrow+1} \frac{1}{v}(\varphi(n)-\varphi(n-v))=\varphi(n)-\varphi(n-1)

\partial \varphi(n)=\varphi(n)-\varphi(n-1), \quad(1 \leq n \leq N)

\begin{gathered} S_{n_{1}}^{n_{2}} \varphi \circlearrowright n=S_{n_{1}}^{n_{2}} \partial \phi=\sum_{n=n_{1}}^{n_{2}}(\phi(n)-\phi(n-1))=\phi\left(n_{2}\right)-\phi\left(n_{1}-1\right) \\ J\left(n_{1}, n_{2}\right)=S_{n_{1}}^{n_{2}} \varphi(n) \circlearrowright n, \quad n_{1} \geq 1, n_{2}>n_{1} \end{gathered}

\begin{aligned} & \partial^{0} \varphi=\varphi \\ & \partial^{1} \varphi=\varphi(n)-\varphi(n-1) \\ & \partial^{2} \varphi=\varphi(n)-2 \varphi(n-1)+\varphi(n-2) \\ & \partial^{3} \varphi=\varphi(n)-3 \varphi(n-1)+3 \varphi(n-2)-\varphi(n-3) \end{aligned}

\delta^{k} \varphi=\sum_{v=0}^{k}(-1)^{v} a_{v}(k) \varphi(n-v), \quad a_{v}(k)=\binom{k}{v}, \quad 0 \leq k \leq N

\check{\partial} \sum_{j} u_{j}(n)=\sum_{j} \check{\partial} u_{j}

\partial C=0

\check{\partial}(C u)=C \check{u}

\begin{gathered} \partial(u v)=u v-u^{\prime} v^{\prime}=u v-(u-\Im u)(v-\Im v) \\ \partial(u v)=u \grave{v}+v \grave{v}-\grave{\partial} u \grave{v} \end{gathered}

\partial\left(\frac{u}{v}\right)=\frac{u}{v}-\frac{u^{\prime}}{v^{\prime}}=\frac{1}{v v^{\prime}}\left(u v^{\prime}-v u^{\prime}\right)=\frac{1}{v v^{\prime}}\left|\begin{array}{cc} u & u^{\prime} \\ v & v^{\prime} \end{array}\right|

\boldsymbol{\partial}\left(\frac{u}{v}\right)=\frac{1}{v v^{\prime}}(v \check{\partial} u-u \check{\partial} v)=\frac{1}{v v^{\prime}}\left|\begin{array}{cc} \Im u & \partial v \\ u & v \end{array}\right|

\Im\left(\frac{u}{v}\right)=\frac{1}{v}\left|\begin{array}{cc} \Im u & \partial v \\ u & v \end{array}\right| \cdot\left|\begin{array}{cc} v & \Im v \\ 1 & 1 \end{array}\right|^{-1}

C ; \varphi(n)=\check{\partial}(n)=\varphi(n)-\varphi(n-1)

\partial^{k} \varphi(n)=\sum_{v=0}^{k}(-1)^{v}\binom{k}{v} \varphi(n-v)

\begin{aligned} S_{n_{1}}^{n_{2}} \varphi \text { ð } n & =S_{n_{1}}^{v} \varphi \text { ð } n+S_{v+1}^{n_{2}} \varphi \text { ð } n \\ S_{n_{1}}^{v} \varphi \text { ð } n+S_{v+1}^{n_{2}} \varphi \text { ð } n & =\left(\phi(v)-\phi\left(n_{1}-1\right)\right)+\left(\phi\left(n_{2}\right)-\phi(v)\right) \\ & =\phi\left(n_{2}\right)-\phi\left(n_{1}-1\right)=S_{n_{1}}^{n_{2}} \varphi \text { ð } n \end{aligned}

\left|S_{n_{1}}^{n_{2}} \varphi \circlearrowright n\right| \neq\left|S_{n_{2}}^{n_{1}} \varphi \circlearrowright n\right|

\begin{aligned} S_{n_{1}}^{n_{2}} \varphi{ }^{\circ} n+S_{n_{2}}^{n_{1}} \varphi{ }^{\circ} n & =\phi\left(n_{2}\right)-\phi\left(n_{1}-1\right)+\phi\left(n_{1}\right)-\phi\left(n_{2}-1\right) \\ & =(\breve{\phi})_{n_{1}}+(\Im \phi)_{n_{2}}=\varphi\left(n_{1}\right)+\varphi\left(n_{2}\right) \end{aligned}

\begin{aligned} \Im S_{a}^{n} \varphi(v) ð v & =S_{a}^{n} \varphi \Im v-S_{a}^{n-1} \varphi \Im v \\ & =\phi(n)-\phi(a-1)-\phi(n-1)+\phi(a-1) \\ & =\phi(n)-\phi(n-1)=\Im \phi \end{aligned}

\phi(n)=S \varphi(n) \check{ } n+C, \quad \check{ }{ }

S \frac{\varphi}{\Psi} \breve{ } \partial=\frac{u}{v}, \quad \text { where } \frac{\varphi}{\Psi}=\frac{v \grave{ } u-u \grave{v}}{v(n) v(n-1)}

\partial_{\epsilon} \ln \varphi \approx \frac{\partial_{\epsilon} \varphi}{\varphi} \quad\left(0<\left|\partial_{\epsilon}\right| \ll 1\right)

\breve{\partial}_{\epsilon} e^{\varphi}=e^{\varphi}-\exp \left(\varphi-\breve{\partial}_{\epsilon} \varphi\right)=e^{\varphi}\left(1-\exp \left(-\breve{\partial}_{\epsilon} \varphi\right)\right)

\breve{\partial}_{\epsilon} e^{\varphi} \approx e^{\varphi} \breve{\partial}_{\epsilon} \varphi

f=f(\varphi), \quad \check{\partial}_{\varphi} f \cdot \check{\partial} \varphi=f(\varphi)-f(\varphi-\check{\partial})

S_{n}^{n} \varphi=\varphi(n)

S u ð v=u v-S v^{\prime} \partial u

1 \leqq i \leqq L<\infty, \quad 1 \leqq \kappa_{i} \leqq n_{i} \leqq N_{i}<\infty

\check{\partial}_{i} \varphi=\varphi-\varphi\left(\ldots, n_{i}-1, \ldots\right)

\left(\check{\partial}_{i} \times \check{\partial}_{k}\right)_{-}=0, \quad \text { where }(a \times b)_{ \pm}=a b \pm b a

\Im \varphi=\sum_{j=1}^{\lambda} \Im_{\Psi_{j}} \varphi \Im \Psi_{j}, \quad \partial_{\Psi_{j}} \varphi=\left(\varphi-\varphi\left(\ldots, \Psi_{j}-\Im \Psi_{j}, \ldots\right)\right)\left(\Im \Psi_{j}\right)^{-1}

\phi=S_{1+\kappa_{1}}^{N_{1}} \ldots S_{1+\kappa_{L}}^{N_{L}} \varphi\left(n_{i}\right)_{1}^{L} \prod_{k=1}^{L} \check{\partial} n_{k}, \quad \Im_{i} n_{k}=\delta_{i k}

\left|\varphi\left(n_{i}\right)_{1}^{L}-\varphi\left(n_{i}^{\prime}\right)_{1}^{L}\right|<\varepsilon, \quad\left|\varphi\left(n_{i}\right)_{1}^{L}-g\right|<\varepsilon

\lim _{n_{1} \rightarrow \infty} \varphi=\varphi_{1}\left(n_{i}\right)_{2}^{L} \ldots \lim _{n_{L} \rightarrow \infty} \varphi\left(n_{L}\right)=g

\varphi\left(t, n_{i}\right)_{1}^{L}=t^{h} \varphi\left(n_{i}\right)_{1}^{L}

\breve{\partial}_{n} \varphi=\sum_{i=1}^{L} n_{i} \breve{\partial}_{\eta_{i}} \varphi\left(\eta_{i}\right)_{1}^{L}

\partial_{n} \varphi=(-1)^{h+1} \sum_{v=0}^{h-1}(-1)^{v}\binom{h}{v} n^{v} \varphi\left(n_{i}\right)_{1}^{L}

\sum_{i=1}^{L} n_{i} \breve{\partial}_{\eta_{i}} \varphi\left(\eta_{i}\right)_{1}^{L}=(-1)^{h+1} \sum_{v=0}^{h-1}(-1)^{v}\binom{h}{v} n^{v} \varphi\left(n_{i}\right)_{1}^{L}

\pm \sum_{i=1}^{L} n_{i} \breve{\partial}_{i} \varphi\left(n_{i}\right)_{1}^{L}=(-1)^{h+1} \varphi \sum_{v=1}^{h-1}\binom{h}{v}(\mp 1)^{v}

\varphi=\sum_{j=1}^{\binom{L}{h}} S_{j}, \quad S_{j} \sim \prod_{a=1}^{h} n_{a}

\breve{\mathrm{O}}_{\varphi} F+\sum_{i=1}^{L} \breve{\partial}_{i} F=0

\sum_{i=1}^{L} n_{i} \Im_{i} \varphi=\lambda \varphi

\sum_{i=1}^{L} n_{i} \breve{\mathrm{O}}_{i} \varphi=\lambda \varphi

C ; \varphi=\lambda \varphi

Z(i)=()_{i}, \quad Z(i) ; n=n_{i}

Z(i)=()_{i}, \quad \varphi\left(n_{i}\right)_{1}^{L}=\phi ; n, \quad \phi=\phi\left(C_{k}, Z(i)\right)_{i, k=1}^{L, K}, \quad C_{k} ; n_{i}=f_{k}\left(n_{i}\right)

0 ; n=0

E ; n=1, \quad E ;() ; n=n

\left(C_{i} \times C_{k}\right)_{ \pm} \neq 0

C ; n=a=\operatorname{const}(n), \quad C=a \frac{()}{()}

\bar{Z}(i)=\bar{e}_{i}()_{i}, \quad\left|\bar{e}_{i}\right|=1, \quad\left(\bar{e}_{i}, \bar{e}_{k}\right)_{L}=\hat{A}\left(n_{i}\right)_{1}^{L}

\bar{\varphi}=\bar{C} ; n, \quad \bar{C}_{i}=\bar{e}_{i} C_{i}

T_{i_{1} \ldots i_{m}}=\prod_{k=1}^{m} \varphi_{i_{k}}=\left(\prod_{k=1}^{m} C_{i_{k}}\right) ; n

{ }^{m} \bar{C}={ }^{m}\left[\prod_{k=1}^{m} C_{i_{k}}\right]_{L}, \quad{ }^{m} \bar{T}={ }^{m} \bar{C} ; n, \quad 0 \leq m \leq L

\left(\bar{e}_{i}, \bar{e}_{k}\right)_{L}=\hat{A}\left(n_{i}\right)_{1}^{L}

g_{i k}={ }^{2} \bar{C} ; n=\left(C_{i} \cdot C_{k}\right) ; n

\prod_{k=1}^{l-2} C_{i_{k}} ; C_{l-1} ; C_{l} ; \prod_{k=l+1}^{m} C_{i_{k}}

C_{i_{1} \ldots i_{m}}^{T l-1, l}=\prod_{k=1}^{l-2} C_{i_{k}} ; C_{l} ; C_{l-1} ; \prod_{k=l+1}^{m} C_{i_{k}}

{ }^{m} \bar{C}_{ \pm(l-1, l)}=\frac{1}{2}\left({ }^{m} \bar{C} \pm{ }^{m} \bar{C}^{\times l-1, l}\right)

\begin{aligned} 2 C_{ \pm(l-1, l) i_{1} \ldots i_{m}} & =C_{i_{1} \ldots i_{m}} \pm C_{i_{1} \ldots i_{m}}^{T l-1, l} \\ & =\prod_{k=1}^{l-2} C_{i_{k}} ; C_{l-1} ; C_{l} ; \prod_{k=l+1}^{m} C_{i_{k}} \pm \prod_{k=1}^{l-2} C_{i_{k}} ; C_{l} ; C_{l-1} ; \prod_{k=l+1}^{m} C_{i_{k}} \\ & =\prod_{k=1}^{l-2} C_{i_{k}} ;\left(C_{l-1} ; C_{l} \pm C_{l} ; C_{l-1}\right) ; \prod_{k=l+1}^{m} C_{i_{k}} \\ & =\prod_{k=1}^{l-2} C_{i_{k}} ;\left(C_{l-1} \times C_{l}\right)_{ \pm} ; \prod_{k=l+1}^{m} C_{i_{k}} \end{aligned}

\left.{ }^{2} \bar{C}=\left[C_{i} ; C_{k}\right)_{ \pm}\right]_{L}, \quad{ }^{2} \bar{C}_{+}={ }^{2} \bar{C}_{+}^{x}, \quad{ }^{2} \bar{C}_{-}=-{ }^{2} \bar{C}_{-}^{x}

\mathrm{sp}_{j=l}{ }^{m} \overline{\mathrm{C}}=\left\{{ }^{[m-2]}\left[\sum_{l=1}^{L} \prod_{k=1}^{j-1} ; C_{i_{k}} ; C_{l} ; \prod_{k=j+1}^{l-1} ; C_{i_{k}} ; C_{l} ; \prod_{k=l+1}^{m} ; C_{i_{k}}\right]_{L}={ }^{m-2} \bar{C}\right.

\mathrm{sp}^{2} \bar{C}=\sum_{i=1}^{L} C_{i}^{2}, \quad \mathrm{sp}_{i=k}{ }^{m} \bar{C}_{+(i, k)}={ }^{m-2} \bar{C}, \quad \mathrm{sp}_{i=k}{ }^{m} \bar{C}_{-(i, k)}={ }^{m-2} \overline{0}

\begin{aligned} & \bar{D} ;{ }^{m} \bar{C}={ }^{m+1} \bar{W}, \quad \operatorname{sp} \bar{D} ;{ }^{m} \bar{C}={ }^{m-1} \bar{W}, \quad D ;{ }^{m} \bar{C}={ }^{m} \bar{W} \\ & \bar{\partial}=\sum_{i=1}^{L} \bar{e}_{i} \widetilde{\partial}_{i}, \quad \bar{\partial} \varphi=\operatorname{GRAD}_{\mathrm{L}} \varphi, \quad \bar{\partial} ;{ }^{m} \bar{C}=\widehat{\operatorname{DIV}}_{\mathrm{L}}{ }^{m} \bar{C} \\ & \operatorname{sp} \bar{\partial} ;{ }^{m} \bar{C}={\overline{\mathrm{DIV}_{\mathrm{L}}}}^{m} \bar{C}, \quad \bar{\partial} ;{ }^{m} \bar{C}-\left(\bar{\partial} ;{ }^{m} \bar{C}\right)^{x}=\operatorname{ROT}_{\mathrm{L}}{ }^{m} \bar{C} \end{aligned}

\left(\widehat{\mathrm{DIV}}_{\mathrm{L}} \mathrm{ROT}_{\mathrm{L}}\right)_{i k l}=\check{\partial}_{i}\left(\check{\partial}_{k}()_{l}-\check{\partial}_{l}()_{k}\right) \neq 0

\begin{aligned} \left(\overline{\mathrm{DIV}_{\mathrm{L}}} \mathrm{ROT}_{\mathrm{L}}\right)_{l} & =\sum_{i=1}^{L} \check{\partial}_{i}\left(\check{\partial}_{i}()_{l}-\check{\partial}_{l}()_{i}\right) \\ & =\sum_{i=1}^{L} \check{\partial}_{i}^{2}()_{l}-\sum_{i=1}^{L} \check{\partial}_{l} \check{\partial}_{i}()_{i} \\ & =\operatorname{DIV}_{\mathrm{L}} \operatorname{GRAD}_{\mathrm{L}}()_{l}-\left(\operatorname{GRAD}_{\mathrm{L}}\right)_{l} \operatorname{DIV}_{\mathrm{L}}() \end{aligned}

\begin{gathered} \operatorname{ImROT_{L}=2}={ }^{2} 0, \quad \operatorname{sp~ROT}_{L}=0 \\ \overline{\mathrm{DIV}_{L}} \mathrm{ROT}_{L}=\mathrm{DIV}_{L} \mathrm{GRAD}_{L}-\mathrm{GRAD}_{L} \mathrm{DIV}_{L} \\ \mathrm{DIV}_{L} \overline{\mathrm{DIV}}_{L} \mathrm{ROT}_{L}=0, \quad \mathrm{ROT}_{L} \mathrm{GRAD}_{L}={ }^{2} 0 \end{gathered}

\begin{gathered} S\left(\operatorname{GRAD}_{\mathrm{L}} \phi \breve{\partial} \bar{N}\right) ; n=\text { const }, \quad \mathrm{S}_{\mathbf{m}(\mathrm{L})} \operatorname{DIV}_{\mathrm{L}} \overline{\mathrm{G}} \partial \mathrm{~V}=\mathrm{S}_{\mathbf{m}(\mathrm{L}-1)} \overline{\mathrm{E}} \partial \overline{\mathrm{~V}} \\ S S R O \mathrm{~T}_{\mathrm{L}} \bar{\phi} \breve{\partial}^{2} \bar{F}=S \bar{\phi} \breve{\partial} \bar{N}, \quad \breve{\partial} \bar{N}=\sum_{i=1}^{L} \Im_{i} \overline{\mathrm{Z}}(i) \\ \partial V=\prod_{k=1}^{L} \Im_{k} Z(k), \quad \partial \bar{V}=\sum_{j=1}^{L} \bar{e}_{j} \partial V_{j} \\ \partial V_{j}=\prod_{k=1}^{j-1} \Im_{k} Z(k) \prod_{j+1}^{L} \Im_{k} Z(k), \quad\left(\bar{e}_{i} \bar{e}_{k}\right)_{L}=\hat{E} \\ \partial^{2} \bar{F}=\left[\Im_{i} Z(i) \Im_{k} Z(k)\right]_{L} \end{gathered}

\xi=\lim _{n \rightarrow \infty} \varphi(n) / \varphi(n-1)=1+1 / \xi

\partial^{2}-3 \partial+()=0, \quad 2 \xi=1+\sqrt{5}

C_{p} \phi_{k m}^{i}=\lambda_{p}(k, m) \phi_{k m}^{i}

H_{k m}^{(p)} \Psi=h_{k m}^{(p)} \Psi \quad \text { and } \quad L_{k m}^{(p)} \Psi=l_{k m}^{(p)} \Psi

H_{k m}^{(p)} \Psi=\lambda_{(p)}(k, m) L_{k m}^{(p)} \Psi

0=\int\left(\Psi^{*} H_{k m}^{(p)} \Psi-\Psi\left(H_{k m}^{(p)} \Psi\right)^{*}\right) d \Omega \Longrightarrow 0=\lambda_{(p)}(k, m)-\lambda_{(p)}(k, m)^{*}

m / p=M \geqq 1, \quad(M) \mathrm{MOD}(1)=0

\varphi\left(n_{i}\right)_{1}^{L}=\varphi\left(x_{k}\right)_{1}^{N}, \quad L=\binom{N}{p}

(n-1)^{2}-1=p(p-1)(p-2)

\begin{gathered} { }^{2} \bar{g}\left(x^{\underline{k}}\right)_{1}^{N}={ }^{2} \bar{\gamma}\left(Z^{\underline{k}}\right)_{1}^{N} ; n, \quad \kappa^{\underline{i}} \kappa^{\underline{k}} \gamma_{i k}-\alpha(p, \tau) \frac{()}{()}=0 \\ \alpha(p, \tau) \neq 1, \quad p \neq 2, \quad{ }^{2} \bar{\gamma}={ }^{2} \bar{\gamma}_{+}+{ }^{2} \bar{\gamma}_{-} \neq{ }^{2} \bar{\gamma}^{x}, \quad{ }^{2} \bar{\gamma}=\bar{\gamma} \times \bar{\gamma} \\ \left(\gamma_{i} \times \gamma_{k}\right)_{ \pm} \neq 0, \quad{ }^{2} \bar{\gamma}=\operatorname{sp}\left({ }^{2} \bar{\kappa} \times{ }^{2} \bar{\kappa}\right), \quad{ }^{2} \bar{\kappa} \neq{ }^{2} \bar{\kappa}^{x} \end{gathered}

\begin{gathered} \hat{\kappa}=\left(\kappa^{\underline{i}} \kappa^{\underline{k}}\right)_{N}, \quad|\hat{\kappa}|_{N} \neq 0, \quad 2 \bar{\gamma}=\bar{\gamma} \times \bar{\gamma}, \quad \gamma_{ \pm i k}=\frac{1}{2}\left(\gamma_{i} \times \gamma_{k}\right)_{ \pm} \\ \kappa^{\underline{i}} \kappa^{\underline{k}}\left(\gamma_{i} \times \gamma_{k}\right)_{+}-2 \alpha \frac{()}{()}=0 \end{gathered}

\gamma ; n=\left.\left.\right|^{2} \bar{\gamma}\right|_{N} ; n=w^{2}, \quad w=W ; n, \quad V=\kappa \tau^{M} S W ; n \breve{\partial}_{n}, \quad \kappa=\prod_{k=1}^{N} \kappa^{\underline{k}}

\Delta s^{2}=g_{i k} \Delta x^{i} \Delta x^{k}=\tau

{ }^{2} \bar{\gamma}=\operatorname{sp}\left({ }^{2} \bar{\kappa} \times{ }^{2} \bar{\kappa}\right)

w=\sqrt{-\left|g_{i k}\right|_{4}} \Longrightarrow \boldsymbol{\partial} V=\tau^{M} \cdot w

\begin{gathered} n_{i}\left(k_{l}^{(i)}\right)_{1}^{p}=c_{i} ; n, \quad P_{l}^{(i)} \leqq k_{l}^{(i)} \leqq Q_{l}^{(i)}, \quad c_{i}=c_{i}\left(\kappa_{l}^{(i)}\right)_{1}^{p}, \quad \kappa_{l}^{(i)} ; n=k_{l}^{(i)} \\ \varphi\left(n_{i}\right)_{1}^{L}=\varphi\left(c_{i} ; n\right)_{1}^{L}=\phi ; n, \quad \phi=\phi\left(K_{k}\right)_{1}^{G} \end{gathered}

{ }^{2} \bar{\gamma} ; n=\text { const }, \quad \xi^{\underline{k}}=X^{\underline{k}} ; n, \quad 1 \leqq k \leqq N

\begin{array}{cl} C_{k}=\kappa_{k} \sqrt[p]{\tau}()_{k}, & x_{k}=C_{k} ; n, \quad C_{k}=X_{k}, \quad{ }^{2} \underline{\gamma} ; n=\mathrm{const} \\ & C_{k} \neq X_{k}, \quad{ }^{2} \bar{\gamma} ; n={ }^{2} \bar{g} \end{array}

(n-1)^{2}-1=p(p-1)(p-2)

(n-1)^{2}-1=4(3)(2)=24 \Longrightarrow(n-1)^{2}=25 \Longrightarrow n-1=5 \Longrightarrow n=6

\hat{s}=\operatorname{ROT}_{\mathrm{N}} \hat{\mathrm{E}}, \quad(\hat{\mathrm{~s}} ; \mathrm{n})_{\mathrm{n}=1}=\hat{\varnothing}, \quad \hat{\mathrm{s}}=\left({ }^{2} \overline{\mathrm{~s}}_{\mathrm{fffi}}\right)_{\mathrm{p}}, \quad{ }^{2} \overline{\mathrm{~s}}_{\mathrm{fffi}} ; \mathrm{n}=\mathrm{SS}_{\mathrm{o}}^{-}{ }_{\mathrm{sff}} \times \breve{\partial}_{\mathrm{sfi}}^{-}

F_{i} ; n=S_{v=1}^{n_{i}} \int_{\tau} \prod_{l=1}^{p} d \xi_{(i)}^{l} \check{\partial v}, \quad \tau c_{i}\left(()_{(i)}^{l}\right)_{1}^{p}=F_{i}\left(C_{k}\right)_{1}^{N}

\left.\varphi\left(x^{\underline{k}}\right)_{1}^{N} \rightarrow \phi ; n, \quad \frac{\partial \varphi}{\partial x^{k}} \rightarrow\left(\frac{\breve{\partial}_{k} \phi}{\partial C^{\underline{k}}}\right) ; n, \quad d \varphi \rightarrow\left(\sum_{k=1}^{N} \partial_{(C}\right) \phi\right) ; n

\frac{\partial \varphi}{\partial x^{k}} \Longleftrightarrow \frac{\partial_{k} \phi}{\partial C^{k}}

\hat{s}=\left(\begin{array}{cc} 0 & { }^{2} \bar{s}_{12} \\ -{ }^{2} \bar{s}_{12} & 0 \end{array}\right)

M=\omega \frac{m}{p}=M \geqq 1, \quad(M) \mathrm{MOD}(1)=0

\alpha^{\underline{k}}=\alpha_{k}=\kappa_{k} \sqrt[p]{\tau}

\alpha_{i} \alpha_{k} \gamma_{i k}=\sum_{l, m=1}^{N} \Im_{i} \bar{\Psi}_{l} \Im_{k} \bar{\Psi}_{m}=\Im_{i} \bar{\Psi}_{k} \bar{\Psi}

\alpha_{k} \bar{\gamma}=\partial_{k} \bar{\Psi}

\partial_{k} n^{\underline{k}}=\sum_{l=1}^{N} \partial_{l} n^{\underline{k}}=\Im n^{\underline{k}}=1

\sum_{k=1}^{N} \bar{\gamma}_{k} ;() \breve{\partial} x^{\underline{k}}=\hat{\kappa} ;() \breve{\partial} x^{\underline{k}} \quad \text { where } \quad \hat{\kappa}={ }^{2} \bar{\kappa}

\bar{\Psi}=S^{2} \bar{\kappa} ;() \grave{\partial}, \quad{ }^{2} \bar{\gamma}=\operatorname{sp}\left({ }^{2} \bar{\kappa} \times{ }^{2} \bar{\kappa}\right), \quad{ }^{2} \bar{\gamma}_{+} \neq{ }^{2} \overline{0}, \quad \bar{\Psi} ; n=\bar{\xi}

2 \gamma_{-i k}=\gamma_{i k}-\gamma_{k i}^{*}=2 \sum_{\mu=1}^{N}\left(\kappa_{+i \mu} \kappa_{-\mu k}+\kappa_{-i \mu} \kappa_{+\mu k}\right)

S(\mu) ;{ }^{2} \bar{\kappa}_{(\mu)} ; n={ }^{2} \bar{E}

S\left(\mu_{j}\right)_{1}^{s} ; F\left(\mu_{j}\right)_{1}^{s}=F(E ;()), \quad\left(\mu_{j}\right)_{1}^{s} S ; F(E ;())=F\left(\mu_{j}\right)_{1}^{s}, \quad s \leqq \omega

\bar{\Psi}_{(v)}=S^{2} \bar{\kappa}_{(v)} ;() \breve{\partial} \bar{x}^{\prime}, \quad{ }^{2} \bar{\gamma}_{(v v)}=\operatorname{sp}\left({ }^{2} \bar{\kappa}_{(v)} \times{ }^{2} \bar{\kappa}_{(v)}\right),{ }^{2} \bar{\gamma}_{(v v)} ; n={ }^{2} \bar{g}_{(v)}\left(x^{l}\right)_{1}^{N}

{ }^{2} \bar{\gamma}_{(\mu v)}=\operatorname{sp}\left({ }^{2} \bar{\kappa}_{(\mu)} \times{ }^{2} \bar{\kappa}_{(v)}\right), \quad \hat{\gamma}=\left({ }^{2} \bar{\gamma}_{(\mu v)}\right)_{\omega}

d \vec{z}_{p}^{+}=\sum_{j=1}^{m} d \vec{\xi}_{p}^{(j)} \quad \text { and } \quad d \vec{z}_{p}^{-}=\sum_{j=m+1}^{n} d \vec{\xi}_{p}^{(j)}

J_{k m}^{i}=\int_{\Omega} \phi_{k m}^{i} \phi_{m k}^{i *} d \Omega<\infty \Longrightarrow J_{\ldots}^{i}=1

\Gamma_{a k j}=\frac{1}{2}\left(\frac{\partial g_{j a}}{\partial x^{k}}+\frac{\partial g_{k a}}{\partial x^{j}}-\frac{\partial g_{j k}}{\partial x^{a}}\right)=[j k, a]

\Gamma_{k j}^{i}=g^{i a}[j k, a]=\left\{\begin{array}{c} i \\ j k \end{array}\right\}

\underline{N}=S \bar{K}^{\partial} \bar{n}, \quad \bar{n}=\sum_{k=1}^{N} \bar{e}_{k} Z(k) ; n

{ }^{2} \bar{K}={ }^{2} \bar{\kappa} ; n

\partial A^{\underline{i}}=-\left(\Gamma_{k l}^{\underline{i}}\right)_{\tau} A^{\underline{k}} \alpha^{\underline{l}}

{ }^{2} \bar{\gamma}_{(a b)}=\operatorname{sp}\left({ }^{2} \bar{a} \times{ }^{2} \bar{b}\right), \quad \Gamma_{p k l}^{(a b)}(\tau)=[p k l(a b)] ; n, \quad{ }^{[3]}[p k l(a b)]=[\widehat{a} \widehat{b}]

{ }^{2} \bar{\gamma}_{(c d)}=\operatorname{sp}\left({ }^{2} \bar{c} \times{ }^{2} \bar{d}\right), \quad \gamma^{\frac{i p}{(c d)}}[p k l(a b)]=[k l(c, d)-+(a, b)], \quad[3][k l(c, d)-+(a, b)]=\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]

\Gamma_{k m}^{i}=\Gamma_{(+) k m}^{i}+\Gamma_{(-) k m}^{i}

\Gamma_{k m}^{i} \xrightarrow{\text { micro }} \phi_{k m}^{i} \quad\left(\phi_{k m}^{i} \neq \phi_{m k}^{i *}\right)

C_{(p)} \phi_{k m}^{(p)}=\lambda_{(p)}(k, m) \phi_{k m}^{(p)}

\Gamma_{k l}^{i} \rightarrow[k l(c, d)-+(a, b)] ; n

\check{\partial}_{p}^{2} n^{\underline{i}}+\frac{\alpha_{k} \alpha_{l}}{\alpha_{i}} \check{\partial}_{p} n^{\underline{k}} \check{\partial}_{p} n^{\underline{l}}[k l(c, d)-+(a, b)] ; n=0, \quad \mathrm{O}_{p}^{2} \xi^{\underline{i}}=0, \quad\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]_{(\xi)}=\hat{0}

{\underset{\partial}{p}}_{p}^{2} x^{\prime i}+{\underset{\partial}{p}}_{p} x^{\prime k}{\underset{\partial}{\partial}}_{p} x^{\prime l}[k l(c, d)-+(a, b)]^{\left(C^{\prime}\right)} ; n=0

{\underset{x^{\prime} \underline{m}}{x^{\prime} \underline{\mu}}}_{2}^{2} x^{\prime \prime \underline{i}}+[k l(c, d)-+(a, b)]^{\left(C^{\prime \prime}\right)} ; n \breve{\partial}_{x^{\prime \prime} \underline{m}} x^{\prime \prime k} \breve{\partial}_{x^{\prime} \underline{\mu}} x^{\prime \prime l}=[m \mu(c, d)-+(a, b)]^{\left(C^{\prime}\right)} ; n \breve{\partial}_{x^{\prime} \underline{p}} x^{\prime \prime \underline{i}}

\left.\partial_{l} \varphi=\alpha_{l}\left[k l \begin{array}{c} k \\ k \end{array}\right)(\kappa)\right] ; n, \quad \ln \sqrt{|g|}=\varphi ; n, \quad{ }^{2} \bar{g}={ }^{2} \bar{\gamma} ; n

\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]=\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]_{+}+\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]_{-}, \quad\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]_{ \pm}= \pm\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]_{ \pm}^{x}

\begin{gathered} \binom{\beta \pm}{\alpha}_{( \pm)}^{\left(s_{1}\right)\left(s_{2}\right)}=\sum_{k=1}^{N}\binom{\beta \pm}{\alpha}_{( \pm) k}^{\left(s_{1}\right)\left(s_{2}\right)} \\ \binom{\beta \pm}{\alpha}_{( \pm) k}^{\left(s_{1}\right)\left(s_{2}\right)}=\frac{1}{\alpha_{k}} \partial_{k}+\sum_{\lambda=\mu+1}^{m}()^{\underline{\sigma}}\left[\sigma k(\beta(\lambda))(\alpha(\lambda))( \pm)\left(\varepsilon_{\lambda}\left(s_{1}\right)\right)\right] ; n-\sum_{\lambda=1}^{\mu}()_{\sigma}\left[i_{\lambda} k(\beta(\lambda))(\alpha(\stackrel{\sigma}{\lambda}))( \pm)\left(\varepsilon_{\lambda}\left(s_{2}\right)\right)\right] ; n \end{gathered}

\binom{\widehat{\beta \pm}}{\alpha}=\left(\binom{\beta \pm}{\alpha}_{( \pm)}^{\left(s_{1}\right)\left(s_{2}\right)}\right)_{P, Q}, \quad \widehat{()}=\left(\binom{\widehat{\beta \pm}}{\alpha}\right)_{V, W}

d \vec{s}_{ \pm}=d \vec{s}_{+}+d \vec{s}_{-}

\lambda_{(m)}(m, p) \phi_{m p}^{i}=-\lambda_{(p)}(m, m) \phi_{m m}^{i}

\phi_{m p}^{i}=-\frac{\lambda_{(p)}(m, m)}{\lambda_{(m)}(m, p)} \phi_{m m}^{i} \xrightarrow{\text { empty spectra }} \phi_{m p}^{i}=-\frac{0}{0} \phi_{m m}^{i}

\lim \frac{\lambda_{(p)}(m, m)}{\lambda_{(m)}(m, p)}=a_{m p}=\text { const } \neq 0

\begin{aligned} & { }^{2} \bar{\gamma}_{(\mu v)}={ }^{2} \bar{E}, \quad(\mu, v) \neq 1, \quad{ }^{2} \bar{\gamma}(11)={ }^{2} \bar{\gamma} \neq{ }^{2} \bar{E} \\ & { }^{2} \bar{\gamma}=\operatorname{sp}\left({ }^{2} \bar{\kappa} \times{ }^{2} \bar{\kappa}\right) \neq{ }^{2} \bar{\gamma}^{x}, \quad\left[\begin{array}{c} \widehat{c d} \\ -+a b \end{array}\right]=\widehat{[\kappa]} \\ & \binom{\beta \pm}{\alpha} \\ & ( \pm)=\left(s_{1}\right)\left(s_{2}\right) \\ & ( \pm) \end{aligned}

\lim _{{ }_{2} \rightarrow 2^{2} \bar{E}}(\kappa)_{( \pm)}^{\left(s_{1}\right)\left(s_{2}\right)}=\widehat{\mathrm{DIV}}_{(x)}

(\kappa) l=\frac{1}{\alpha_{l}} \Im_{l}-[l s(\kappa)(\kappa)+] ; n, \quad \lim _{2 \bar{\gamma} \rightarrow{ }^{2} \bar{E}}(\kappa)=\operatorname{GRAD}_{(\mathrm{x})}

\begin{aligned} & \frac{1}{\alpha_{m}} \mho_{m} \frac{1}{p}(\kappa)_{l} ; p-\frac{1}{\alpha_{l}} \mho_{l} \frac{1}{p}(\kappa)_{m} ; p \\ & =\frac{1}{\alpha_{l}} \mho_{l}[m s \stackrel{s}{(\kappa)}(\kappa)+] ; n-\frac{1}{\alpha_{m}} \mho_{m}[l s \stackrel{s}{(\kappa)}(\kappa)+] ; n, \quad[k l \stackrel{i}{(\kappa)}(\kappa)]=\left[{ }_{k}{ }^{i}{ }_{l}\right] \end{aligned}

\begin{aligned} & s p\left((\kappa)_{(+)}^{(1)}+(\kappa)_{(+)}^{(2)}\right) ; \underline{\bar{A}}=2 \widehat{\mathrm{DIV}}_{(x)} \underline{\bar{A}} \\ & s p\left((\kappa)_{(+)}^{(1)}-(\kappa)_{(+)}^{(2)}\right) ; \underline{\bar{A}}=2 \underline{A}^{\underline{k}}\left[s k(\kappa)_{(\kappa)-]}^{s}\right] ; n \\ & (\kappa)_{(+) k^{\prime}}^{(1,2)} ; \underline{\gamma}^{\underline{i k}}=\frac{1}{\alpha_{k}} \partial_{k} \gamma^{\underline{i k}}-[k s \stackrel{s}{(\kappa)}(\kappa)-] ; n \cdot \underline{\gamma} \underline{i k} \\ & \gamma_{i k}(\kappa)_{(+) l}^{(1,2)} ; \underline{\gamma}^{\underline{i k}}=(N-2)(\kappa)_{l} ; w \end{aligned}

\widehat{[]}=\sum_{\alpha=1}^{\omega^{4}}\left(\left[\begin{array}{c} \widehat{(c d)} \\ -+(a b) \end{array}\right]+s p^{2} \bar{Q}(\alpha) ;() \times\left[\begin{array}{c} \widehat{(c d)} \\ -+(a b) \end{array}\right]\right), \quad \alpha \widehat{=}\left(\begin{array}{ll} c & d \\ a & b \end{array}\right)

R_{v \lambda \kappa}^{\mu}=\partial_{\lambda} \Gamma_{v \kappa}^{\mu}-\partial_{\kappa} \Gamma_{v \lambda}^{\mu}+\Gamma_{\eta \lambda}^{\mu} \Gamma_{v \kappa}^{\eta}-\Gamma_{\eta \kappa}^{\mu} \Gamma_{v \lambda}^{\eta}

\zeta_{k l m}^{\underline{i}}=\underline{\partial}_{l}\left[\begin{array}{cc} i \\ k & m \end{array}\right]-\underline{\partial}_{m}\left[\begin{array}{c} i \\ k \end{array}\right]+\left[\begin{array}{l} i \\ l \end{array}\right] ;()\left[\begin{array}{cc} s & \\ k & m \end{array}\right]-\left[\begin{array}{c} i \\ m \end{array}\right] ;()\left[\begin{array}{c} s \\ k \\ l \end{array}\right]

L_{;} \widehat{[]}={ }^{4} \overline{0}

K_{m} ;\left[\begin{array}{cc} i & i \\ k & \end{array}\right]=\zeta_{k l m}^{i}=\lambda_{m}(k, l)\left[\begin{array}{cc} i \\ k & l \end{array}\right]

\underline{\partial}_{l}\left[{ }_{m}{ }^{i}{ }_{m}\right]-\underline{\partial}_{m}\left[{ }_{m}{ }^{i} l\right]+\left[{ }_{l}{ }^{i}{ }_{s}\right] ;\left[{ }_{m}{ }^{i}{ }_{s}\right] ;()\left[{ }_{m}{ }^{s} l\right]=\lambda_{m}(m, l)\left[{ }_{m}{ }^{i} l\right]

a_{m l}=-\frac{\lambda_{l}(m, m)}{\lambda_{m}(m, l)} \Longrightarrow\left[\begin{array}{c} i \\ m \end{array}\right]=a_{m l}\left[\begin{array}{c} i \\ m \end{array}\right] \Longrightarrow\left[\begin{array}{c} i \\ m \end{array}\right]=\frac{a_{l m}}{a_{m l}}\left[\begin{array}{l} i \\ l \end{array}\right]

\left((a(k, l)-1) \underline{\partial}_{l}-\sum_{l \neq m} \underline{\partial}_{m}\right) ; \varphi_{k l}+\varphi_{k l}^{2}=\lambda(k, l) \varphi_{k l}

\bar{a}_{k l}=\frac{\bar{e}_{l}}{\alpha_{l}}(a(k, l)-1)-\sum_{m \neq l} \frac{\bar{e}_{m}}{\alpha_{m}}

\bar{a}_{k l} \mathrm{GRAD}_{q} \varphi_{k l}=\lambda(k, l) \varphi_{k l}-\varphi_{k l}^{2}

\frac{\breve{\partial} u}{1-u^{2}}= \pm \frac{1}{2} \lambda(k, l) \circlearrowright N_{k l}= \pm \Lambda_{k l}

\left(E-\Psi_{k l}\right)^{\Lambda_{k l}+1} \cdot \Psi_{k l}^{\Lambda_{k l}-1}=2^{-2 \Lambda_{k l}} \cdot C_{k l} e^{-\lambda_{k l} \mu}

\begin{aligned} & \Lambda_{k l}=\alpha_{l}(a(k, l)-1)^{-1}-\sum_{m \neq n} \alpha_{m} \\ & (a(k, l)-q) \cdot \lambda_{k l}=\lambda(k, l) \end{aligned}

\begin{aligned} \sum_{m=1}^{q} & \frac{1}{\lambda_{m}(k, l)} \sum_{m=1}^{q}\left[\left(\frac{\lambda_{i}(l, l) \lambda_{m}(k, k) \lambda_{l}(l, k) \lambda_{k}(k, i)}{\lambda_{k}(l, l) \lambda_{i}(k, k) \lambda_{l}(l, i) \lambda_{k}(k, m)}\right)\right. \\ & \left.-\left(\frac{\lambda_{m}(i, i) \lambda_{m}(k, k) \lambda_{m}(m, k) \lambda_{k}(k, l)}{\lambda_{i}(i, m) \lambda_{k}(k, m) \lambda_{k}(m, m) \lambda_{l}(k, k)}\right)\right]\left[k_{k}{ }^{i}\right] \\ = & \left(E+C_{k l} e^{-\sum \lambda_{m}(k, l) \mu}\right)^{-1} \end{aligned}