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In the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ₁(x)*γ₂(y), where γ₁ and γ₂ are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.

In this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔr_i = λ_ir_i. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.

Classification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.

The subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

Attempts to prove, in this second chapter of the author’s monograph, that with a new research programme, it is possible to build a methodological bridge between economics and all other natural sciences and the scientists should address this challenge. Reviews basic principles that govern nature, including Einstein’s findings along with such luminaries as Copernicus, Newton, Galileo and Jeans. Concludes that the future is safe, as a new generation of scientists is now emerging in the East and the West, and that the new methodology should provide enough space for new roads, ideas and interpretations, which may occur in the future. Closes by saying a new spirit should be initiated in economics and transplanted into natural sciences.

There is a double crisis in modern science and in particular in physics and mechanics. Among others Einstein and Stephane Lupasco, in the 1930s, warned about this crisis. The Quantum Theory cannot be reconciled with the Relativity Theory. Specifically there is a gap (cleavage) between micro – and macro‐physics and mechanics. Parallel or beneath there is also a second crisis derived from a discontinuity (again a cleavage) between classical and modern science, that is between two previous revolutions. A new research programme of a simultaneous equilibrium versus disequilibrium approach, initially applied in economics has now been extended to include natural sciences. It is the question of a new, more comprehensive methodology which is actually a sui generis synthesis between classical and modern heritage. The rigorous application of the new research programme leads to the organisation of an Orientation Table, that is, a methodological map of all possible combinations (systems). The Table shows, without any exaggeration, a few revolutionary results. For instance, with the help of the Table, modern science or the second revolution (Einstein, Bohr, Heisenberg) does not appear contradictory but rather complementary to classical science or the first revolution (Newton, Lavoisier). The Kuhnian thesis to the contrary is disproved and the second crisis is solved. With the help of the Universal Hypothesis of Duality (the basis of the Orientation Table), matter and energy, at the micro – and macro‐level, appear in a double form (the Principle of Duality): stable (equilibrium) particles and unstable (disequilibrium) waves. The strong interactions from modern physics are associated with the law of gravitation (attraction) or stable equilibrium which governs stable matter and energy. The weak interactions are associated with the law of disgravitation (dispersion or repulsion) including entropy or unstable equilibrium which governs unstable matter and energy. In this way the first crisis is also solved.

Presents the scientific methodology from the enlarged cybernetical perspective that recognizes the anisotropy of time, the probabilistic character of natural laws, and the entry that the incomplete determinism in Nature opens to the occurrence of innovation, growth, organization, teleology communication, control, contest and freedom. The new tier to the methodological edifice that cybernetics provides stands on the earlier tiers, which go back to the Ionians (c. 500 BC). However, the new insights reveal flaws in the earlier tiers, and their removal strengthens the entire edifice. The new concepts of teleological activity and contest allow the clear demarcation of the military sciences as those whose subject matter is teleological activity involving contest. The paramount question “what ought to be done”, outside the empirical realm, is embraced by the scientific methodology. It also embraces the cognitive sciences that ask how the human mind is able to discover, and how the sequence of discoveries might converge to a true description of reality.

Light, when viewed as a particle, reacts in a determinable manner with reference to the gravitational potential existing within the reference frame viewed. The elementary quanta of light, expressed under the terms of Planck, and as derived via the expressions of Einstein as a particle, may not reach a speed exactly equating to the speed (electromagnetic) of light of c. Here c is viewed as an electromagnetic constancy in any gravitational frame of reference. The theory is that a relative particle of mass may not achieve the speed of light, for the energy of that particle would then equate to infinity or in that the force required allowing the relative particle to reach c would then be infinite. The theory is then totally reliant upon the tenants of what has become to be known as the Special Theory of Relativity. As per the General Theory, light would be “bent”, more or less, from one gravitational reference frame as compared to another gravitational reference frame. The theory then evolves that light, when viewed as a particle, forms a curvilinear light path through the gravitational reference frame viewed. However, until now, the light path has been solely described on a linear basis. It is the result of the theory that the light path may be described on a curvilinear basis, under the method of Lagrange. This method, or model, allows a particle of light (viewed as a projectile of mass under a constant velocity, therefore under a constant acceleration) to achieve Newton's description of the path of a projectile. Note that the following paper is applicable to a previous paper, which proposes a displacement of light within the gravitational field.

Let us consider that light, when viewed as a particle, forms a conic arc segment inscribed within the space viewed. The space (or frame) viewed is considered to exhibit a gravitational potential, and it is thus this potential that deforms the light path from a Euclidean/Newtonian derivation of a straight line to that of a relativistic curvilinear nature. Given a distance over this conic arc segment (assumed to form a parabolic arc segment) and a given time (considering the given distance involved), one derives a constancy of the speed of light of c, where c is considered as a constant regardless of the gravitational potential exhibited by the frame viewed. If we further consider that the Special Theory requires that light propagate on a linear measure as the velocity v (of necessity v being less than c on a comparable linear measure) between the axes concerned; then a displacement (in linear measure equal to c−v) occurs. The displacement evolved is then assumed to agree with the form of Maxwell. We assume that this linear displacement of c−v occurs upon the y‐axis of the frame viewed. Of necessity, a relative displacement must occur upon the x‐axis of the frame viewed. From the calculus, the dot products derived must vary in concept, in order to derive the totality of relative coordinate shifts occurring within any three‐dimensional space. One displacement is linear in nature, while the other is trigonometric in nature. We consider the displacement of Maxwell, Lorentz, Compton, and de Broglie to be linear in nature. Based on the principle of the Special Theory (and the other forms as mentioned), we consider the total displacement to be mechanically derivable. That derivation, once allowed, results the physics to agree with the observations complete to this moment in time. The paper concludes that the error in coordinate positioning shown by the global positioning satellite system (GPS satellite platform) is resolvable.

The purpose of this paper is to introduce new non‐classical implementations of neural networks (NNs). The developed implementations are performed in the quantum, nano, and optical domains to perform the required neural computing. The various implementations of the new NNs utilizing the introduced architectures are presented, and their extensions for the utilization in the non‐classical neural‐systolic networks are also introduced.

The introduced neural circuits utilize recent findings in the quantum, nano, and optical fields to implement the functionality of the basic NN. This includes the techniques of many‐valued quantum computing (MVQC), carbon nanotubes (CNT), and linear optics. The extensions of implementations to non‐classical neural‐systolic networks using the introduced neural‐systolic architectures are also presented.

Novel NN implementations are introduced in this paper. NN implementation using the general scheme of MVQC is presented. The proposed method uses the many‐valued quantum orthonormal computational basis states to implement such computations. Physical implementation of quantum computing (QC) is performed by controlling the potential to yield specific wavefunction as a result of solving the Schrödinger equation that governs the dynamics in the quantum domain. The CNT‐based implementation of logic NNs is also introduced. New implementations of logic NNs are also introduced that utilize new linear optical circuits which use coherent light beams to perform the functionality of the basic logic multiplexer by utilizing the properties of frequency, polarization, and incident angle. The implementations of non‐classical neural‐systolic networks using the introduced quantum, nano, and optical neural architectures are also presented.

The introduced NN implementations form new important directions in the NN realizations using the newly emerging technologies. Since the new quantum and optical implementations have the advantages of very high‐speed and low‐power consumption, and the nano implementation exists in very compact space where CNT‐based field effect transistor switches reliably using much less power than a silicon‐based device, the introduced implementations for non‐classical neural computation are new and interesting for the design in future technologies that require the optimal design specifications of super‐high speed, minimum power consumption, and minimum size, such as in low‐power control of autonomous robots, adiabatic low‐power very‐large‐scale integration circuit design for signal processing applications, QC, and nanotechnology.

1.1. Logical Necessity of the Three Dimensions as a Unit of Thought The mathematician does not look kindly on the simple question of why natural space should consist of precisely three dimensions. Instead of giving an answer he assumes a silent smile and shows us a version of space with an infinity of dimensions, as if space were some kind of toy for him to fiddle with to his heart's content.

A protocol showing the β ⁻ activity of high dilutions of nitric acid is described. It is given a physical mathematical frame founded on a an approach of quantum relativistic field based on the theory of Solovay (ethers theory, remanent wave).

The purpose of this paper is to state new formulation of the programme‐styled framework of pansystems research and related expansions.

Pansystems‐generalized extremum principle (0**: (dy/dx=0)**) is presented with recognitions to various logoi of philosophy, mathematics, technology, systems, cybernetics, informatics, relativity, biology, society, resource, communications and related topics: logic, history, humanities, aesthetics, journalism, IT, AI, TGBZ* <truth*goodness*beauty*Zen*>, etc. including recent rediscoveries of 50 or so pansystems logoi.

A keynote of the paper is to develop the deep logoi of the analytic mathematics, analytic mechanics, variational principles, Hilbert's sixth/23rd problems, pan‐axiomatization to encyclopedic principles and various applications. The 0**‐universal connections embody the transfield internet‐styled academic tendency of pansystems exploration.

The paper includes topics: history megawave, pansystems sublation‐modes, pan‐metaphysics, pansystems dialogs with logoi of 100 thinkers or so, and pansystems‐sublation for a series of logoi concerning the substructure of encyclopedic dialogs such as systems, derivative, extremum, quantification, variational principle, equation, symmetry, OR, optimization, approximation, yinyang, combination, normality‐abnormality, framework, modeling, simulation, relativity, recognition, practice, methodology, mathematics, operations and transformations, quotientization, product, clustering, Banach completeness theorem, Weierstrass approximation theorem, Jackson approximation theorem, Taylor theorem, approximation transformation theorems due to Walsh‐Sewell mathematical school, Hilbert problems, Cauchy theorem, theorems of equation stability, function theory, logic, paradox, axiomatization, cybernetics, dialectics, multistep decision, computer, synergy, vitality and the basic logoi for history, ethics, economics, society OR, aesthetics, journalism, institution, resource and traffics, AI, IT, etc.