Quantization Formula Research Articles

New gravitational model and quantization formula of the gravitational constant

The existing theories of gravitation cannot solve the problem of the gravitational quantization, and the exact value of the Newtonian constant of gravitation G has not been determined. Therefore, the essence of gravity needs to be deeply explored. Here, we show that a new gravitational model is proposed, which is of the viewpoint that the universe is a composition of elementary particles and gravitational lines, and the elementary particles are connected by gravitational lines, and the essence of gravity can be tension of the gravitational lines. Based on Newton's law of gravitation, quantum mechanics, and simple harmonic wave equation, the dynamic model of gravitational driven photons is constructed: <mml:math display="inline"> <mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> <mml:mi>/</mml:mi> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>Δ</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:mi>/</mml:mi> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> . The quantization formula of the gravitational constant is derived: <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:mi>/</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mi>g</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> </mml:mrow> </mml:mrow> </mml:math> . The calculated average is G = 6.678 804 09 × 10−11 m3 kg−1 s−2. Compared with the recommendations value of the CODATA in 2018, the deviation obtained is 0.067%. The quantum formula of gravity is derived: <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:mi>/</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:mrow> </mml:math> . The gravitational model proposed in this paper solves the problem of gravitation quantization, deduces the formula of the gravitational constant and gravitational quantum, and provides a new theoretical method for deeply revealing the essence of gravity.

Cervical Precancerous Lesion Image Enhancement Based on Retinex and Histogram Equalization

Cervical cancer is a prevalent chronic malignant tumor in gynecology, necessitating high-quality images of cervical precancerous lesions to enhance detection rates. Addressing the challenges of low contrast, uneven illumination, and indistinct lesion details in such images, this paper proposes an enhancement algorithm based on retinex and histogram equalization. First, the algorithm solves the color deviation problem by modifying the quantization formula of retinex theory. Then, the contrast-limited adaptive histogram equalization algorithm is selectively conducted on blue and green channels to avoid the problem of image visual quality reduction caused by drastic darkening of local dark areas. Next, a multi-scale detail enhancement algorithm is used to further sharpen the details. Finally, the problem of noise amplification and image distortion in the process of enhancement is alleviated by dynamic weighted fusion. The experimental results confirm the effectiveness of the proposed algorithm in optimizing brightness, enhancing contrast, sharpening details, and suppressing noise in cervical precancerous lesion images. The proposed algorithm has shown superior performance compared to other traditional methods based on objective indicators such as peak signal-to-noise ratio, detail-variance–background-variance, gray square mean deviation, contrast improvement index, and enhancement quality index.

A product design method for form and color matching based on aesthetic theory

Form and color matching design are vital links in product design. Recently, many scholars have focused on quantifying visual aesthetics to improve design efficiency in form or color matching design activities. However, few scholars have considered both form and color matching. Therefore, we propose a more comprehensive and practical design model based on the aesthetics theory of form and color matching. This model includes 3 phases: preparation (Phase I), form design (Phase II), and color matching design (Phase III). In Phase I, the target product's functional system and design objectives are clarified. In Phase II, the spatial layouts of the target product are analysed, and several reasonable layouts are designed in detail. Finally, the form with the highest aesthetic measurement is selected from alternatives using the modified equilibrium formula and questionnaire. In Phase III, a group of colors conforming to the target imagery is selected from a color palette. Meanwhile, a set of appropriate observation angles are chosen using an expert questionnaire. Subsequently, three-color combinations with higher color harmony are filled into images with different observation angles. Finally, the aesthetics of color matching schemes are obtained using the formula of color matching, and the scheme with a higher average value of the aesthetic measurement from all observation angles is considered the best scheme. This paper takes the fresh food vehicle and leafless fan as the cases. The best form and color matching scheme are obtained using the formula of aesthetic measurement and verified with an expert questionnaire. The result shows consistency between the results of the quantization formula and the expert questionnaire, which confirms the effectiveness of this design model. The proposed design model could improve the design efficiency and ensure the products' visual aesthetics.

A mechanical analog of Bohr's atom based on de Broglie's double-solution approach.

Motivated by recent developments of hydrodynamical quantum mechanical analogs [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)], we provide a relativistic model for a classical particle coupled to a scalar wave field through a holonomic constraint. In the presence of an external Coulomb field, we define a regime where the particle is guided by the wave in a way similar to the old de Broglie phase-wave proposal. Moreover, this dualistic mechanical analog of the quantum theory is reminiscent of the double-solution approach suggested by de Broglie in 1927 and is able to reproduce the Bohr-Sommerfeld semiclassical quantization formula for an electron moving in an atom.

Mechanical Parameters of Deep‐Buried Coal Goaf Rock Mass Based on Optimized GSI Quantitative Analysis

In order to obtain the accurate mechanical parameters of deep‐buried coal goaf rock mass, the limitation of geological strength index (GSI) in concealed rock mass is analyzed. Based on the test result and analysis of the current normative standards, the classification indexes of rock mass structural are optimized based on discontinuity distance d and rock mass integrity index Kv. The ratio of rock mass saturated strength to dry strength, η, is introduced, quantization formula of structural surface conditions is proposed, and the influence of groundwater and rock types is included in structural surface condition classification. The GSI system is improved to better suit all types of deep‐buried and water‐rich rock masses. Furthermore, the rock mass disturbance factor D’s quantitative formula is listed according to the Hoek–Brown (HB) criterion. Taking the goaf roof under railway as an example, the parameters of deep‐buried rock mass are obtained based on the improved quantitative GSI system and HB criterion. This research provides a scientific reference for achieving geological parameters and engineering designing in goaf areas.

A Matrix Aggregation Intelligent Decision Based on H-Convex Combination and Expert Preference

In the field of multi-dimensional intelligent decision-making, the evaluation results rely on the subjective experience of the evaluation experts. However, the evaluation information generated by personal expert preferences can has certain impact on the decision-making results.For this problem, we present an intelligent decision model based on H-convex combination and expert preference. The H-convex combinatorial matrix algorithm is used to assemble the individual judgment matrices, which reduces the inconsistency and non-reciprocity of the group decision matrix. The expert preference quantization formula is used to obtain the preference values of the corresponding experts, which are taken into the H-convex combination as power exponents. This solves the impact of expert preference differences on decision outcomes. The feasibility and effectiveness of the proposed method are verified by some evaluation cases.

The Hilb/Sym correspondence for $${\mathbb {C}}^2$$: descendents and Fourier–Mukai

We study here the crepant resolution correspondence for the $$\mathsf {T}$$ -equivariant descendent Gromov–Witten theories of $$\mathsf {Hilb}^n({\mathbb {C}}^2)$$ and $$\mathsf {Sym}^n({\mathbb {C}}^2)$$ . The descendent correspondence is obtained from our previous matching of the associated CohFTs by applying Givental’s quantization formula to a specific symplectic transformation $$\mathsf {K}$$ . The first result of the paper is an explicit computation of $$\mathsf {K}$$ . Our main result then establishes a fundamental relationship between the Fourier–Mukai equivalence of the associated derived categories (by Bridgeland, King, and Reid) and the symplectic transformation $$\mathsf {K}$$ via Iritani’s integral structure. The results use Haiman’s Fourier–Mukai calculations and are exactly aligned with Iritani’s point of view on crepant resolution.

Design optimization of vehicle EHPS system based on multi-objective genetic algorithm

Electric hydraulic power steering (EHPS) system has been widely used in large and medium cars, which plays an important role in determining the energy loss, driving safety and driving comfort of vehicles. This work mainly discusses the parameter design of the EHPS system based on the multi-objective optimization method. Since there is no explicit standard for the performance indexes of EHPS system, this work takes the energy consumption, steering road feel, steering sensibility and steering stability as the main performance indexes of EHPS system. The quantization formula of each performance index is explored and deduced. Based on these, a multi-objective optimization model is established. Then, the non-dominated sorting genetic algorithm-III based on fitness function (NSGA-III-FF) is proposed, which has better convergence than the original non-dominated sorting genetic algorithm-III (NSGA-III). Besides, the technique for order preference by similarity to an ideal solution (TOPSIS) is applied to select the ideal optimization solution. Simulation results show that the NSGA-III-FF enhances the comprehensive performance of the EHPS system, which can successfully achieve the goal of multi-objective optimization for steering flexibility, steering road feel, and steering energy loss while ensuring the steering stability.

Spectral Portraits in the Semi-Classical Approximation of the Sturm-Liouville Problem with a Complex Potential

We study the problem on the limit behavior of the discrete spectrum of the Sturm-Liouville problem − ε y ″ ( x ) + P ( x λ ) y ( x ) = 0 y ( a ) = y ( b ) = 0 , provided that the physical parameter ϵ tends to zero. It is assumed that λ is the spectral parameter, the function P is polynomial on x with analytic coefficients on λ varying on a domain G in the complex λ-plane ℂ. The case P(x, λ) = p(x) - λ with complex valued function p corresponds to the usual linear spectral problem. Boundary conditions are formed by arbitrary complex numbers a, b or ±∞. We shall show that in this case the eigenvalues are concentrated along the so-called limit spectral graph as ϵ → 0. We define three type of curves, forming this graph and find the asymptotic formulae for the eigenvalue distribution along the curves of various types. For the case P(x, λ) = p(x) - λ with real polynomial p(x) these formulae coincide with well known Bohr-Sommerfeld quantization formulae. Non-self-adjoint Sturm-Liouville problems are often found in mathematical physics. We demonstrate this considering the well-known in hydromechanics Orr-Sommerfeld spectral problem.

Research on hot news discovery model based on user interest and topic discovery

In order to find a way to process network data and discover hot news based on user’s interest and topic, through research on hot news discovery algorithm, a double-layer text clustering model based on density clustering strategy and Single-pass strategy is proposed. In view of the huge network data characteristics, DBSCAN algorithm is firstly used to cluster the single-crawled network data into small-scale clusters. Then, the Single-pass strategy is used to perform incremental clustering on the micro-classes to create the topic classes. In the hot news part of the network, the media and the user’s attention to the topic is combined to design a model. The heat quantization formula is obtained. Based on the research of related technologies, a network hot topic detection model is designed and implemented by using web crawler, news discovery and hot news discovery technology. By comparing the two-layer model used in the model with the traditional Single-pass strategy, the feasibility of the two-layer model is verified, and the discovery of network hot news is realized.

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle’s energy is less than the potential at both negative and positive infinity. The particle’s energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions, which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues, angular momentum numbers, screening parameters, reduced mass, and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.

Multidisciplinary hybrid hierarchical collaborative optimization of electric wheel vehicle chassis integrated system based on driver’s feel

The optimization design of chassis integrated system mainly involves steering, suspension and brake subsystems, which is essentially a multidisciplinary design optimization. This paper mainly researches the multidisciplinary optimization of the chassis integrated system for the electric wheel vehicle, from the view of ensuring a favorable feel for the driver. The dynamic models of differential steering system, brake system, active suspension system and vehicle are established. Then, taking the coupling relationship of the chassis subsystems into account, this paper proposes an evaluating index of driver’s ride comfort (Drc), which is composed of the steering road feel, brake feel and suspension ride comfort. In order to determine the weight coefficient in the quantization formula of Drc, the technique for order preference by similarity to ideal solution (TOPSIS) method is used to overcome the subjectivity in the selection. Based on these, a multidisciplinary hybrid hierarchical collaborative optimization (HHCO) method is proposed on the basis of the collaborative optimization (CO), which consists of a system level coordinator and a coupling analyzer to solve the problem of poor convergence and the low efficiency of CO method. The optimization results show that the proposed HHCO method has excellent computational efficiency and better convergence compared with the CO method, which can further improve the steering road feel and the drive ride comfort, on the premise of ensuring the brake feel and suspension ride comfort.

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt counting formula (without any truncations) for the number [Formula: see text] of zeroes in the critical strip with imaginary part greater than [Formula: see text] and less than or equal to [Formula: see text].

Action Quantization, Energy Quantization, and Time Parametrization

The additional information within a Hamilton-Jacobi representation of quantum mechanics is extra, in general, to the Schr\"odinger representation. This additional information specifies the microstate of $\psi$ that is incorporated into the quantum reduced action, $W$. Non-physical solutions of the quantum stationary Hamilton-Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue $J$. Eigenvalues $J$ and $E$ mutually imply each other. Jacobi's theorem generates a microstate-dependent time parametrization $t-\tau=\partial_E W$ even where energy, $E$, and action variable, $J$, are quantized eigenvalues. Substantiating examples are examined in a Hamilton-Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of $W$ is shown to ease numerical and analytic computations. Second, a Hamilton-Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics..

Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.

Exact energy quantization condition for single Dirac particle in one-dimensional (scalar) potential well

We present an exact quantization condition for the time independent solutions (energy eigenstates) of the one-dimensional Dirac equation with a scalar potential well characterized by only two ‘effective’ turning points (defined by the roots of ) for a given energy E and satisfying . This result generalizes the previously known non-relativistic quantization formula and preserves many physically desirable symmetries, besides attaining the correct non-relativistic limit. Numerical calculations demonstrate the utility of the formula for computing accurate energy eigenvalues.

Unified and simple derivation of energy levels for the Pöschl–Teller-type and Scarf-type potentials

A general class of potentials covering the Pöschl–Teller-type and Scarf-type is considered. The analytical expression of the energy eigenvalues is established with the help of the exact quantization formula. Then, using appropriate coordinate transformations, the exact spectrums of five well-known potentials: Pöschl–Teller, Pöschl–Teller I and Pöschl–Teller II, Scarf I, and Scarf II, are deduced in a simple way.

The Necessity of Wheels in Universal Quantization Formulas

In the context of formal deformation quantization, we provide an elementary argument showing that any universal quantization formula necessarily involves graphs with wheels.

Riemann surface dynamics of periodic non-Hermitian Hamiltonians

A number of physical problems, including statistical mechanics of 1D multivalent Coulomb gases, may be formulated in terms of non-Hermitian quantum mechanics. We use this example to develop a non-perturbative method of instanton calculus for non-Hermitian Hamiltonians. This can be seen as an extension of semiclassical methods in conventional quantum mechanics. Treating momentum and coordinate as complex variables yields a Riemann surface of constant complex energy. The classical and instanton actions are given by periods of this surface; we show how to obtain these via methods from algebraic topology. We demonstrate the accuracy of this analytic procedure in comparison with numerical simulations for a class of periodic non-Hermitian Hamiltonians, as well as the validity of the Bohr–Sommerfeld quantization and Gamow's formula in these cases.

Geometric quantization for proper moment maps: the Vergne conjecture

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.