The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge $c=1$. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.

There exists an implicit potential limitation in every physical discoveries that has been implemented and understood. However, the limitations can be bounded within a safe limit to prevent any constructing theory to be free from errors. As, it’s the inert nature of the humans, to go far beyond the scope of experimental findings in order to pursue any studies with the sole help of logical reasoning and mathematics, the argument can be prevailed in the form of WEAK Clampdown Effect & STRONG Clampdown Effect. More, the theories are constructed out of physical nature, more the theory gets hypothetical without any finding evidence, but that does or doesn’t actually justify the phenomenon, that too with the more increment of KARDASHEV Scale, more moderate ways of experimentation got developed curbing down the limitations within the human limit of ‘ERRORS’, that does can be neglected by approximation. Relationship being cross-judgmental on the basis of the computational limits and calculation accuracy, leading to a soft singularity, as a warning, that if computer powers cannot be checked on the basis of error approximations, then this may lead to the hitting of a hard singularity, that in phase with the forbidden gap (or after the optimum limit that arises at the core constraints of nature) to prevent the computation being carried off with respect to super-intelligence machines that are cognitive capability oriented future computers responsible for self growth & reproduction with more improvement algorithm, restricting all forms of humanity & constraints the human growth by virtue of limiting capacities of the humans as compared to computers.

It is advisable to avoid and, even better, demystify such grandiose terms as "infinity" or "singularity" in the description of the cosmos. Its proliferation does not positively contribute to the understanding of key concepts that are essential for an updated account of its origin and evolutionary history. It will be here argued that, as a matter of fact, there are no infinities in physics, in the real world: all which appear, in any given formulation of nature by means of mathematical equations, actually arise from extrapolations, which are made beyond the bounds of validity of the equations themselves. Such crucial point is rather well-known, but too often forgotten, as discussed in this paper with several examples; namely, the famous Big Bang singularity and others, which appeared before in classical mechanics and electrodynamics, and notably in the quantization of field theories. A brief description of the Universe’s history and evolution follows. Special emphasis is put on what is presently known, from detailed observations of the cosmos and, complementarily, from advanced experiments of very-high-energy physics. To conclude, a future perspective on how this knowledge might soon improve is given.

The classical uncertainty principle inequalities were imposed as a mathematical constraint over the general relativity geodesic equation. In this way, the uncertainty principle was reformulated in terms of the proper space-time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirmed the necessity for a minimum length for the space-time line element in the geodesic, dependent on a geodesic-derived scalar which made the expression Lorentz-covariant. In agreement with quantum gravity theories, GeUP required the imposition of a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone was found around the singularity where uncertainty in space-time diverged to infinity.

When an edge is removed, a cycle graph Cn becomes a n-1 tree graph. This observation from extremal set theory leads us to the realm of set theory, in which a topological manifold of genus-1 turns out to be of genus-0. Starting from these premises, we prove a theorem suggesting that a manifold with disjoint points must be of genus-0, while a manifold of genus-1 cannot encompass disjoint points.

In this paper, we study singularities of the spherical indicatrix and evolute of spacelike ruled surface with spacelike ruling. Finally, we give an example to illustrate our results.

To solve a series of major problems, such as the singularity before the Big Bang, the inflation of the early universe and the accelerating expansion of the present universe, this series of studies start with super gravity, dark energy and dark matter, go beyond the surface of a lot of observation data, and systematically reveal the arche-internal mechanism of all things, to establish the arche-unity of physics, and give a natural and reasonable explanation. At present, there are at least three basic pathways that are leading our research field to repulsion against gravitation, quantum repulsion against quantum gravitation and high-dimensional superrepulsion against high-dimensional supergravity, that is, the research on the accelerating expansion mechanism of the current universe, the research on the dynamic mechanism of the early cosmic inflation, and the research on the rebound mechanism near the singularity before the Big Bang. In this series of studies, the arche problem of everything is always placed under the extreme environmental conditions of the initial universe, the starting point of scientific logic is closely combined with the starting point of the universe history, the pseudo vacuum energy considered by the inflationary universe model, the Higgs field predicted by the standard physical model and the dark energy speculated by the observational cosmology are reduced to the positive energy system with strong negative pressure, and they are reduced to the repulsion (all the resistance to gravity) that forms a conjugate relation with gravity. In our opinion, dark energy and dark matter are essentially at the level of quantum gravity and quantum repulsion. Therefore, we not only consider the conventional quantum effect and non commutative quantum effect, but also further consider the arche-conjugate effect and the cluster resonance-at-cofrenquence effect. For noncommutative space-time, we put forth the ultimate supersymmetry, the arche-conjugation and the cluster resonance-at-cofrenquence. A relatively complete natural mechanism forms such an important inference: the closer they get to Planck scale, the stronger the conventional quantum effect, non commutative quantum effect, arche-conjugate effect and co-frequency resonance effect are. In the high-dimensional space-time (the 11 dimensional hyperspace compactly reduced to M 4 × N r ), corresponding to the introduction of supergravity with the supersymmetry between the graviton and the gravitino, we put forth the superrepulsion with the supersymmetry between the repulsion and the repulsitino, and establish the arche-conjugate relationship between the high-dimensional supergravity and the high-dimensional superrepulsion. Therefrom, we propose a basic inference that can be promoted to the principle of hyperholography of high-dimensional universe (or multiuniverse): in a parallel universe formed by mutual coupling and holographic correspondence between the P universe dominated by negative pressure (high-dimensional superrepulsion and quantum repulsion of accumulating positive energy, mainly composed of dark energy) and the N universe dominated by positive pressure (high-dimensional supergravity and quantum gravity of accumulating negative energy, mainly composed of dark matter), there is a correspondence between the small-scale physics (ultraviolet cutoff) of the P universe and the large-scale physics (infrared cutoff) of the N universe, while there is a correspondence between the large-scale physics (ultraviolet cutoff) of the P universe and the small-scale physics (infrared cutoff) of the N universe. In the new research paradigms set in this series, the arche-conjugation between the high-dimensional supergravity and the high-dimensional super repulsion, as well as between the quantum gravity and the quantum repulsion, is higher than the supersymmetry, so it becomes the core concept of the new theory. The arhce-unity of physics is a super unity that transcends the grand unification of four basic interaction, which is formed between high-dimensional supergravity (related to dark matter) and high-dimensional superrepulsion (related to dark energy), between quantum gravity (related to dark matter) and quantum repulsion (related to dark energy). From the basic component point of view, before or after the big bang, the gravitons and repulsons are coupled together, the negative energy contraction dominated by the positive pressure and the positive energy expansion dominated by the negative pressure work alternately, forming the arche-conjugate pulsaton of neither the point particle nor the linear superstring. Under the big unified framework of quantum gravitational field and quantum repulsive field, this series of work comprehensively expands the general relativity and loop quantum ring gravitation theory, sets up the quantum repulsive field equation corresponding to the quantum gravitational field equation, and then establishes the quantum hedge-unified field equation describing the interaction of quantum gravitational field and quantum repulsive field; On this basis, the quantum repulsive universe equation corresponding to the quantum gravitational universe equation is established, thus a set of new equations describing the expansive universe with quantum hedge-unified field are established. On the one hand, scalar disturbance is considered as the exhibition of repulsive effect, on the other hand, tensor disturbance is considered as the exhibition of gravitational effect. In the initial stage of the big ripping, due to the mutual restraint of quantum repulsion and quantum gravity, the scalar and tensor perturbations are not obvious. However, at the end of the big ripping, the scalar and tensor perturbations are very significant, and there is a large power spectral exponential run. Between particle physics and cosmology, we improve and expand the path integral of quantum gravity, create the path integral of quantum hedge-unified field, and set up the basic equation of quantum cosmic-synergetic dynamics, and then establish the basic equation of quantum-cosmic paradigm dynamics. Thus we put up the operator distribution function and its dynamic equation of the quantum arche- pulsaton, and finally use the Wigner-Ville distribution as a non-linear time-frequency distribution to establish the Wigner joint-distribution of quantum-conjuagted pulsation function, so as to analyze the arche of all things with three evolutionary forms: quantum chaotic-pulsaton, quantum quasi-spherical pulsaton and quantum spherical-pulsaton. In this series of work, the mechanism to avoid the singularity of the universe is put forward fundamentally, and the dark energy and dark matter are explained through new mechanisms and effects at the level of quantum gravity. Between particle physics and cosmology, we improve and expand the path integral of quantum gravity, establish the path integral of quantum hedge-unified field, thus establish the basic equation of quantum cosmic-synergy dynamics, and then establish the basic equation of quantum cosmic-paradigm dynamics. Thus the covariant gravitational mass and the covariant inertial mass formed by the ultra-synergy between particles and the universe are presented, and the equivalence hypothesis on the covariant non-inertial system which can be elevated to the basic principle, is presented. Based on this hypothesis, a systematic calculation and analysis of binary stars composed of neutron stars and black holes are made. The master equation, variance and Fokker-Planck equation for the birth-death and migration of a large number of particles are established between binary stars composed of neutron stars and black holes.

General wisdom thinks that the central of blackholes are singular and featured by mass, charge and spin exclusively; the microstate reflected in the Bekenstein-Hawking entropy can be accounted for only in some quantum gravity. We challenge this wisdom with two exact solution families to the sourceful Einstein equation which describe dust ball with oscillatory inner structure and variety of mass profiles. We attribute the matter content's oscillation across the central point of the system to the forward scattering domination in high energy collisions induced by inter-gravitating acceleration and provide their Penrose-Carter diagram representation and quantum description briefly. Our solution family provides a very simple semi-classic picture for the micorstate definition of black holes.

Toby Walsh in “The Singularity May Never Be Near” gives six arguments to support his point of view that technological singularity may happen but that it is unlikely. In this paper, we provide analysis of each one of his arguments and arrive at similar conclusions, but with more weight given to the “likely to happen” probability.

The Chebyshev collocation method (ChC) implemented as Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for toughest problems we have to loosen the default Chebfun tolerance in Newton's solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

The anisotropy in the CMBR detected by the COBE satellite has sprouted the research and theories of the void that got created, after being washed away by a parallel universe after its collision. This details of the roadmap to the parallel universe, that are previously been entangled, but now, separated, with the concept of 2-time dimensions as a time curve of containing the closed curves embedded on the surface area, thereby time running backward in orientation in the parallel universe, along with the perspective of eternal inflation and divided subspaces giving birth to twin universes are provided here. This time curve being responsible for creating an exponentiality of time frames, growing in layers, giving us the perception of the confabulations being an indication of the previously connected worlds.

Quantum gravity theories rely on a minimal measurable length for their formulations, which clashes with the classical formulation of the uncertainty principle and with Lorentz invariance from general relativity. These incompatibilities led to the development of the generalized uncertainty principle (GUP) from string theories and its various modifications. GUP and covariant formulations of the uncertainty principle are discussed, together with implications for space-time quantization.

The first part of this paper contains a brief description of the beginnings of modern cosmology, which, the author will argue, was most likely born in the Year 1912. Some of the pieces of evidence presented here have emerged from recent research in the history of science, and are not usually shared with the general audiences in popular science books. In special, the issue of the correct formulation of the original Big Bang concept, according to the precise words of Fred Hoyle, is discussed. Too often, this point is very deficiently explained (when not just misleadingly) in most of the available generalist literature. Other frequent uses of the same words, Big Bang, as to name the initial singularity of the cosmos, and also whole cosmological models, are then addressed, as evolutions of its original meaning. Quantum and inflationary additions to the celebrated singularity theorems by Penrose, Geroch, Hawking and others led to subsequent results by Borde, Guth and Vilenkin. And corresponding corrections to the Einstein field equations have originated, in particular, R², f(R), and scalar-tensor gravities, giving rise to a plethora of new singularities. For completeness, an updated table with a classification of the same is given.

The geodesics on a relativistic manifold are given by the well-known equation involving the Christoffel coefficients. This equation can be solved numerically step by step and transposed into a quantization. We study here the effect of this quantization on the Schwarzschild spacetime, more precisely in the Kruskal-Szekeres map.

In this paper we present a review of recent developments in AI towards the possibility of an artificial intelligence equals that of human intelligence. AI technology has always shown a stepwise increase in its capacity and complexity. The last step took place several years ago, with the increased progress in deep neural network technology. Each such step goes hand in hand with our understanding of ourselves and our understanding of human cognition. Indeed, AI was always about the question of understanding human nature. AI percolates into our lives, changing our environment. We believe that the next few steps in AI technology, and in our understanding of human behavior, will bring about much more powerful machines, flexible enough to resemble human behavior. In this context, there are two research fields: Artificial Social Intelligence (ASI) and General Artificial Intelligence (AGI). The authors also allude to one of the main challenges for AI, embodied cognition, and explain how it can viewed as an opportunity for further progress in AI research.

The present study aims to develop an efficient numerical method for computing the diffraction and radiation of water waves with horizontal long cylindrical structures, such as floating breakwaters. A higher-order scheme is used to discretize geometry of the structure as well as the relevant physical quantities. As the kernel of this method, Wehausen’s free-surface Green function is calculated by a newly-developed Gauss-Kronrod adaptive quadrature algorithm after elimination of its Cauchy-type singularities. To improve computational efficiency, a Chebyshev approximation approach is applied to a fast calculation of the Green function that needs evaluation thousands of times. In addition, OpenMP parallel technique is used to the formation of influence coefficient matrix, which significantly reduces CPU time. Finally, computations are performed on wave exciting forces and hydrodynamic coefficients for the long cylindrical structures, either floating or submerged. Comparison with other numerical and analytical methods demonstrates good performance of the present method.