Applying a procedure similar to that of E.S. Bring, by using a 4^th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a De Moivre form (DMQ), so that it could be solved by radicals. The general solution by radicals of the De Moivre equations of any degree is presented. By the same procedure the BJSx (normal sextic) equation was taken to another one without the 2^nd, 4^th and 6^th terms which was transformed into a cubic (solvable) equation. By applying a 6^th degree Tschirnhaus transformation to the BJSp (normal septic) equation its Bring-Jerrard binormal (without the 2^nd, 3^rd, 4^th and 5^th terms) form was obtained.

The paper briefly reviews the Clifford algebras of space Cl(3,0) and anti-space Cl(0,3) with a particular focus on the paravector representation, emphasizing the fact that both algebras have an isomorphic center represented just by two coordinates. Since the paravector representation allows assigning the scalar element of grade 0 to the time coordinate, we consider the relativity in such two-dimensional spacetime for a uniformly accelerated frame with the constant acceleration 3Hc. Using the Rindler coordinate transformations in two-dimensional spacetime and then applying it to Minkowski coordinates, we obtain the FLRW metric, which in the case of the Clifford algebra of space Cl(3,0) corresponds to the anti-de Sitter (AdS) flat (k=0) case, the negative cosmological term and an oscillating model of the universe. The approach with anti-Euclidean Clifford algebra Cl(0,3) leads to the de Sitter model with the positive cosmological term and the exact form of the scale factor used in modern cosmology.

This research answers the knowledge gap regarding the explanation of the quantum jump of the electron. This scientific paper aims to complete Einstein’s research regarding general relativity and attempt to link general relativity to quantum laws.

This research answers the knowledge gap regarding the explanation of the quantum jump of the electron. This scientific paper aims to complete Einstein’s research regarding general relativity and attempt to link general relativity to quantum laws.

This research answers the knowledge gap regarding the explanation of the quantum jump of the electron. This scientific paper aims to complete Einstein’s research regarding general relativity and attempt to link general relativity to quantum laws.

This paper contains a comparative review of existing Zitterbewegung models. It attempts to synthesize them by offering a toy model containing an extra hypothesis of electromagnetic self-oscillation at the Fermi scale (10−15m). Inspired by Bell violation experiments, Zitterbewegung models and hydrodynamic walker studies, we revisit standing waves in quantum mechanics, echoing de Broglie-Bohm pilot wave theories. Starting with the electromagnetic self-oscillation hypothesis at the Fermi scale we propose an emergent and deductive hierarchical electron structure across multiple scales. The model spans Fermi, Compton (10−13m), Bohr (10−11m) and finally de Broglie (>10−10m). At the Fermi scale, the hypothesis is one of "source-free chiral light", known as the "zilch", emerging as a self-oscillating electromagnetic field at the Schwinger limit. This self-interaction of retarded potentials due to real relativistic delays is a generic self-oscillation of Liénard-Wiechert potentials. This primary Fermi oscillation accounts for charge and mass at the Fermi scale. It generates secondary stable "auto-orbits" dynamics at the Compton scale, manifesting as observable spin and other quantum properties of the usual Dirac zitterbewegung models. These zitter models account for known electron characteristics, including mass, charge, spin and g/2 anomalous factors up to 9 digits precision. We explain the de Broglie wavelength through Doppler-shifted Compton-scale standing waves, in the tradition of de Broglie. The framework extends to electron-positron pair production at the Schwinger limit, described as a result of natural light focusing in non-homogeneous media, utilizing René Thom’s catastrophe theory. This offers a new perspective on baryogenesis and matter emergence from electromagnetic fields. We dub this toy model the "sub-standard" model of the electron.

In this paper, we extend the KdVSKR equations in (1+1)-dimension and (2+1)-dimension to fractional KdVSKR equations with modified Riemann-Liouville derivative. The (2+1)-dimensional KdVSKR equation, which is a recent extension of (1+1)-dimensional KdVSKR equation, can model the resonances of solitons in shallow water. By means of the Hirota bilinear method, finite symmetry group method and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton-cnoidal interaction solution of the (1+1)-dimensional fractional KdVSKR equation has been derived for the first time. For the (2+1)-dimensional frac-tional KdVSKR equation, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. Impact of the fractional order on the shapes of these solutions has been illustrated by figures. Three-wave interaction solutions of fractional systems have not been reported in the existing references. The research idea in this paper can be applied to other fractional differential equations.

Cities are complex systems requiring urban design models that balance order and disorder. Collective creativity initiatives engage citizens in these processes, empowering bottom-up approaches that prioritize people and social well-being within urban development. This paper investigates an 'Urban Laboratory' as a case study, examining the potential of collective creativity to address urban complexity. The successful and ongoing project 'El Campo de Cebada' in Madrid, Spain, demonstrates how a community transformed a vacant lot into a vibrant social hub. The phases of this study include case selection, data collection, data analysis, and presentation of the results. This study identifies key enabling factors, including agents, management, social dynamics, infrastructure, and actions. These insights offer a methodological framework for designing future collaborative, resilient, and inclusive urban spaces, addressing the complex needs of communities within our cities.

This paper introduces an innovative nanotechnology-based approach that provides a pathway to enhance the energy release efficiency of nanoenergetic materials (nEMs) by harnessing self-synchronized collective atomic vibrations and phonon wave resonance within the transition domains between nanocomponents, without altering the material composition. The key innovation involves incorporating finely-tuned 2D-ordered linear-chain carbon-based multilayer nano-enhanced interfaces as programmable nanodevices into the transition domains using advanced multistage processing and assembly techniques. These programmable nanodevices enable precise control over self-synchronized collective atomic vibrations and phonon wave propagation, leading to synergistic effects. To activate and optimize these effects, a combination of various methods is employed, including energy-driven initiation of allotropic phase transformations, surface acoustic wave-assisted micro/nano-manipulation, heteroatom doping, directed self-assembly using high-frequency electromagnetic fields, and data-driven inverse design approaches. By leveraging a data-driven inverse design strategy and uncovering hidden structure-property relationships, we maximize energy release efficiency using the carbon nanomaterials genome approach derived from multifactorial neural network-based predictive models. This approach not only unlocks new functionalities in nEMs but also improves environmental performance and safety levels. By pioneering transformative pathways for nEMs through harnessing phonon wave resonance in low-dimensional nanocarbon transition interfaces, this research brings significant advancements in the field.

The real birth of Brazil can be traced to the remote Jesuit village of São Paulo de Piratininga, founded in 1554, after arduous efforts to ascend the Serra do Mar. This isolated settlement, inhabited by a mixture of Indigenous peoples and Portuguese colonizers, became the pivotal point for the expansion of Portuguese territory in South America. The people of São Paulo, or paulistas, shaped by their harsh environment and violent history, forged a unique identity. Through their relentless expeditions, the bandeirantes ventured deep into the interior, hunting Indigenous peoples for slavery and expanding the Portuguese domain beyond the boundaries defined by European treaties. This essay explores how the humble origins of São Paulo de Piratininga grew into a force that shaped the modern Brazilian state and its borders.