Search Results (3)

An inverse-square probability mass function (PMF) is at the Newcomb–Benford law (NBL)’s root and ultimately at the origin of positional notation and conformality. $Pr\left(Z\right)={\left(2Z\right)}^{-2}$, where $Z\in {\mathbb{Z}}^{+}$. Under its tail, we find information as harmonic likelihood $\mathcal{L}\left(\left[s,t\right)\right)=H_{t-1}-H_{s-1}$, where $H_n$ is the nth harmonic number. The global $\mathbb{Q}$-NBL is $Pr\left(b,q\right)={\displaystyle \raisebox{1ex}{$\mathcal{L}\left(\left[q,q+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\mathcal{L}\left(\left[1,b\right)\right)$}\right.}={\left(q{H}_{b-1}\right)}^{-1}$, where b is the base and q is a quantum ($1\le q<b$). Under its tail, we find information as logarithmic likelihood $\ell \left(\left[i,j\right)\right)=ln{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The fiducial $\mathbb{R}$-NBL is $Pr\left(r,d\right)={\displaystyle \raisebox{1ex}{$\ell \left(\left[d,d+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\ell \left(\left[1,r\right)\right)$}\right.}={log}_r\left(1+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\right)$, where $r\ll b$ is the radix of a local complex system. The global Bayesian rule multiplies the correlation between two numbers, s and t, by a likelihood ratio that is the NBL probability of bucket $\left[s,t\right)$ relative to b’s support. To encode the odds of quantum j against i locally, we multiply the prior odds ${\displaystyle \raisebox{1ex}{$Pr\left(b,j\right)$}\!\left/ \!\raisebox{-1ex}{$Pr\left(b,i\right)$}\right.}$ by a likelihood ratio, which is the NBL probability of bin $\left[i,j\right)$ relative to r’s support; the local Bayesian coding rule is $\tilde{o}\left(j:i|r\right)={\displaystyle \raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$j$}\right.}{log}_r{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The Bayesian rule to recode local data is $\tilde{o}\left(j:i|r^{\prime }\right)=\tilde{o}\left(j:i|r\right){\displaystyle \raisebox{1ex}{$lnr$}\!\left/ \!\raisebox{-1ex}{$ln{r}^{\prime }$}\right.}$. Global and local Bayesian data are elements of the algebraic field of “gap ratios”, ${\displaystyle \frac{A-B}{C-D}}$. The cross-ratio, the central tool in conformal geometry, is a subclass of gap ratio. A one-dimensional coding source reflects the global Bayesian data of the harmonic external world, the annulus $\left\{x\in \mathbb{Q}|1\le \left|x\right|<b\right\}$, into the local Bayesian data of its logarithmic coding space, the ball $\left\{x\in \mathbb{Q}|\left|x\right|<1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}\right\}$. The source’s conformal encoding function is $y={log}_r\left(2x-1\right)$, where x is the observed Euclidean distance to an object’s position. The conformal decoding function is $x={\displaystyle \frac{1}{2}}\left(1+r^y\right)$. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality. Full article

It is demonstrated, that recently proposed by the author Q-ball mechanism of the pseudogap state and high-Tc superconductivity in cuprates may be detected in micro X-ray diffraction, since it imposes inverse correlations between the size and scattering intensities of the Q-ball charge-density-wave (CDW) fluctuations in these compounds. The Q-ball charge Q gives the number of condensed elementary bosonic excitations in a CDW fluctuation of finite amplitude. The attraction between these excitations inside Euclidean Q-balls is self-consistently triggered by the simultaneous condensation of Cooper/local pairs. Euclidean Q-ball solutions, analogous to the famous Q-balls of squarks in the supersymmetric standard model, arise due to the global invariance of the effective theory under the $U\left(1\right)$ phase rotation of the Fourier amplitudes of the short-range CDW fluctuations. A conserved ‘Noether charge’ Q along the Matsubara time axis equals $Q\propto T{M}^{2}V$, where the temperature T, Q-ball’s volume V, and fluctuation amplitude M enter. Several predictions are derived in an analytic form that follow from this picture. The conservation of the charge Q leads to an inverse proportionality between the volume V and X-ray scattering intensity ∼$M^2$ of the CDW puddles found in micro X-ray scattering experiments. The theoretical temperature dependences of the most probable Q value of superconducting Q-balls and their size and scattering amplitudes fit well the recent X-ray diffraction data in the pseudogap phase of high-Tc cuprates. Full article

The origin of the pseudogap and superconducting behaviors in high-$T_c$ superconductors is proposed, based on the picture of Euclidean Q-balls formation that carry Cooper/local-pair condensates inside their volumes. Euclidean Q-balls that describe bubbles of collective spin-/charge density fluctuations (SDW/CDW) oscillating in Matsubara time are found as a new self-consistent solution of the Eliashberg equations in the ‘nested’ repulsive Hubbard model of high-$T_c$ superconductors. The Q-balls arise due to global invariance of the effective theory under the phase rotation of the Fourier amplitudes of SDW/CDW fluctuations, leading to conservation of the ‘Noether charge’ Q in Matsubara time. Due to self-consistently arising local minimum of their potential energy at finite amplitude of the density fluctuations, the Q-balls provide greater binding energy of fermions into local/Cooper pairs relative to the usual Frohlich mechanism of exchange with infinitesimal lattice/charge/spin quasiparticles. We show that around some temperature $T^{*}$ the Q-balls arise with a finite density of superconducting condensate inside them. The Q-balls expand their sizes to infinity at superconducting transition temperature $T_c$. The fermionic spectral gap inside the Q-balls arises in the vicinity of the ‘nested’ regions of the bare Fermi surface. Solutions are found analytically from the Eliashberg equations with the ‘nesting’ wave vectors connecting ‘hot spots’ in the Brillouin zone. The experimental ‘Uemura plot’ of the linear dependence of $T_c$ on superconducting density $n_s$ in high-$T_c$ superconducting compounds follows naturally from the proposed theory. Full article