Transformation Formula Research Articles

Elastoplastic solutions for deep-buried twin tunnels with arbitrary shapes and various arrangements under biaxial in-situ stress field based on Mohr-Coulomb and generalized Hoek-Brown criteria

Theoretical solutions for elastoplastic analysis of deep-buried twin tunnels with arbitrary shapes and various arrangements are developed under biaxial in-situ stress field based on Mohr-coulomb (M–C) and generalized Hoek-Brown (G-H-B) criteria. The boundary stress transformation formulas adapted for arbitrary stress conditions on non-circular tunnel wall are provided in terms of conformal transformations. Regarding the plastic analysis, analytical expressions for stress characteristic relationships along slip lines under generalized Hoek-Brown criterion are derived. Subsequently, the semi-analytical solutions for plastic stresses of rock surrounding arbitrary-shaped tunnels are constructed based on Mohr-Coulomb and generalized Hoek-Brown criteria applying conformal mapping technique, slip-line method and finite difference-numerical iterative approach. On the basis of the discrete plastic stresses, interpolation functions are generated according to three-dimensional linear interpolation technique. And then, solutions for elastic stress function and plastic radius of twin tunnels with various arrangements are developed with improved mapping functions and newly defined fitness functions through differential evolution method. The accuracy of the semi-analytical solutions is verified by the numerical simulation. Meanwhile, the proposed solutions have the advantages of meshless characteristic, better convergence and higher computing efficiency comparing with the numerical simulation. From the point of engineering, the developed elastoplastic solutions can provide theoretical supports for the stress analysis, plastic radius estimation, safety clearances optimizations of deep-buried twin tunnels.

Proof of conjectures of Sun on double basic hypergeometric sums

In this paper, we study some q-congruences on double basic hypergeometric sums. With the help of ‘creative microscoping’ method and several summation and transformation formulas for basic hypergeometric series, we confirmed several conjectures posed by Zhi-Wei Sun.

The Situation Assessment of UAVs Based on an Improved Whale Optimization Bayesian Network Parameter-Learning Algorithm

To realize unmanned aerial vehicle (UAV) situation assessment, a Bayesian network (BN) for situation assessment is established. Aimed at the problem that the parameters of the BN are difficult to obtain, an improved whale optimization algorithm based on prior parameter intervals (IWOA-PPI) for parameter learning is proposed. Firstly, according to the dependencies between the situation and its related factors, the structure of the BN is established. Secondly, in order to fully mine the prior knowledge of parameters, the parameter constraints are transformed into parameter prior intervals using Monte Carlo sampling and interval transformation formulas. Thirdly, a variable encircling factor and a nonlinear convergence factor are proposed. The former and the latter enhance the local and global search capabilities of the whale optimization algorithm (WOA), respectively. Finally, a simulated annealing strategy incorporating Levy flight is introduced to enable the WOA to jump out of the local optimum. In the experiment for the standard BNs, five parameter-learning algorithms are applied, and the results prove that the IWOA-PPI is not only effective but also the most accurate. In the experiment for the situation BN, the situations of the assumed mission scenario are evaluated, and the results show that the situation assessment method proposed in this article is correct and feasible.

On the two variables κ-Appell hypergeometric matrix functions

This work aims to introduce -Appell hypergeometric matrix functions and , where . New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with -Appell hypergeometric matrix functions are derived.

A note on odd zeta values over any number field and extended Eisenstein series

In this article, we study transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. Our transformation formula for the Dedekind zeta function at odd integers leads to a new number field extension of Eisenstein series, which satisfies the transformation z↦−1/z like an integral weight modular form over SL2(Z). The results provide a number of important applications, which are important in studying the behavior of odd zeta values as well as Lambert series in an arbitrary number field.

A group consensus model with prospect theory under probabilistic linguistic term sets

The challenges encountered in the realm of multi-attribute group decision-making (MAGDM) involving probabilistic linguistic term sets (PLTSs) have garnered substantial attention. Within the PLTS context, this study introduces a consensus reaching process (CRP) that iteratively refines the weights assigned to decision-makers (DMs) by leveraging the principles of prospect theory (PT). The primary goal of this iterative weight adjustment process is to enhance the overall decision-making procedure when dealing with PLTSs. To circumvent any data loss during transformation and compute the prospect values of PLTSs directly, a novel transformation formula is developed. Acknowledging the distinct cognitive levels among different DMs, the integration of multiple weights into the consensus process is characterized by its dynamic and iterative nature. Concerning the measurement of consensus, this study employs a method based on the gap between prospect values, which enhances objectivity while overcoming the limitations associated with the distance formula of PLTSs. Furthermore, the feedback mechanism incorporated into the modification process incorporates dynamic adjustment parameters that are tailored to different evaluation values, thereby preventing excessive adjustments that are either too low or too high. By utilizing the newly proposed prospect value function, this research aggregates the group evaluation value and identifies the optimal alternative. In conclusion, this paper concludes with a comparative analysis involving various counterparts, shedding light on the feasibility and validity of the proposed model.

The Truth of the Experiments of Invariable Speed of Light in Special Relativity

It is pointed out that the most experiments on the invariant speed of light in special relativity proves the round-trip invariant speed of light, not one-way invariant. This paper makes a distinction between them. The Michelson-Morley experiment and the experiment of high-energy particles emitting photons and so on are the round-trip experiment of light’s speed, showing that the average speed of light is a constant. But the Sagnac effect experiment, the Michelson-Gale's experiment that the earth's rotation effects the speed of light, and the satellite signal propagation between China and Japan are the one-way experiments of light’s speed, showing that the speed of light is variable, satisfying the Galilean velocity addition rule. The orbit shape changes of binary stars and the phenomenon of charm stars cannot be observed due to that the calculated observing directions are different from the practical observing direction for the observers on the Earth. The Fizeau water flow experiment and the Sagnac experiment are combined to prove that the rotation of optical fiber would affects the speed of light. The problem that the Sagnac effect is independent of the refractive index of optical fiber is explained well. It is proved that the Sagnac effect formula derived from special relativity is consistent with that derived from classical mechanics when the optical fiber’s refractive index . When , there is no the Sagnac effect according to special relativity. While according to classical mechanics, there is the Sagnac effect. The original experiment of Sagnac effect in 1913 was carried out in atmosphere with . Therefore, the original Sagnac experiment became a judgment experiment. It certainly jugated that the velocity of light satisfied the Galilean addition rule rather than the Lorentz transformation formula.

A reflection formula for the Gaussian hypergeometric function of matrix argument

We obtain a reflection formula for the Gaussian hypergeometric function of real symmetric matrix argument. We also show that this result extends to the Gaussian hypergeometric function defined over the symmetric cones, and even to generalizations of the Gaussian hypergeometric function defined in terms of series of Jack polynomials. Finally, we obtain a quadratic transformation formula for the Gaussian hypergeometric function of Hermitian 2×2 matrix argument.

Simultaneous uniqueness in determining the space‐dependent coefficient and source for a time‐fractional diffusion equation

This article concerns the uniqueness of an inverse problem of simultaneously identifying the space‐dependent coefficient and source in a one‐dimensional time‐fractional diffusion equation with derivative order and the zero Neumann boundary value. By additional boundary measurements, we first obtain the uniqueness of the coefficient from the Laplace transform and a transformation formula. Then, we further show the uniqueness of the source through the asymptotic behavior of solutions to the corresponding forward problem. The result shows that the uniqueness of the simultaneous identification can be obtained under the condition that the prior information only on one set of parameters in the model is given other than that of two sets.

Ramanujan-type series for , revisited

AbstractIn this note, we revisit Ramanujan-type series for $\frac {1}{\pi }$ and show how they arise from genus zero subgroups of $\mathrm {SL}_{2}(\mathbb {R})$ that are commensurable with $\mathrm {SL}_{2}(\mathbb {Z})$ . As illustrations, we reproduce a striking formula of Ramanujan for $\frac {1}{\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for $\frac {1}{\pi }$ . As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.

Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus

In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric 4F3 and 5F4 functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities of the same nature successively. These identities are used to derive transformation formulas between the Srivastava–Daoust double hypergeometric function (S–D function) and Kampé de Fériet’s double hypergeometric function (KDF function) with equal arguments. We also demonstrate reduction formulas from the S–D function or KDF function to the generalized hypergeometric function pFq. Additionally, we provide general summation formulas for the pFq and S–D function (or KDF function) with specific arguments. We further highlight the connections between the results presented here and existing identities.

Bergman kernels of monomial polyhedra

The Bergman kernels of monomial polyhedra are explicitly computed. Monomial polyhedra are a class of bounded pseudoconvex Reinhardt domains defined as sublevel sets of Laurent monomials. Their kernels are rational functions and are obtained by an application of Bell's transformation formula.

Hydrodynamic interaction between two polygonal cylinders in uniform potential flow

The analytical solution for uniform potential flow around two similar or dissimilar polygonal cylinders is obtained by using complex potential function for a doublet trapped in an annulus. The annulus of two circles is mapped conformally to the pair of polygonal cylinders by unique combination of bilinear and hypotrochoidal transformation and mathematical formulas to get velocity, pressure and hydrodynamic force distribution around two polygonal cylinders are presented. The effects of center distance between the cylinders, shape, orientation and corner radii of cylinders, and flow angle on the hydrodynamic interaction between the cylinders are also investigated and presented. The comparison of some of the numerical results obtained using present method is done with the results obtained from commercial CFD software package and results from literature.

The Truths of Space-time Contractions of Special Relativity

This paper points out that there is no any experimental evidence for the length contraction of a moving object in special relativity. It is just a theoretical prediction based on the Lorentz transformation formula. Einstein put forward the concept of simultaneous relativity in order to explain the length contraction. In this paper, an example called Ji Hao's bridge-breaking paradox is provided to prove that this kind of paradoxes cannot be explained by simultaneous relativity. A completely symmetric method is introduced to prove that the famous twin paradox is unsolvable. The time delay experiments of special relativity, such as the life time of μ meson and the atomic clocks moving around the Earth are discussed. It is proved that time slows down of a moving clock does not exist too. It is a misunderstanding to use the lifetime of μ mesons to prove the time delay of special relativity because μ mesons decay prematurely due to strong collisions with other nuclei in the Earth's surface atmosphere. What calculated in theory is the time difference between two atomic clocks flying east and west observed in the stationary reference frame of the Earth's mass center. But the measurement of time difference is on the surface of the Earth. Because of the symmetry of motion speed, there is no time difference caused by the motion speed between the two atomic clocks observed on the Earth’s surface, so the experiment of atomic clocks moving around the Earth is invalid. The experiment is also suspected of fabricating experimental data. The conclusion of this paper is that the space-time contraction of special relativity and its relativity cannot happen in real nature, time and space are absolute concepts, and the Lorentz transformation cannot be correct.

Equivalent criterion for the grand Riemann hypothesis associated with Maass cusp forms

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated with the primitive Maass cusp form over $\Gamma _0(N)$.

A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the so-called Itô-Lyons map in the energy spaces L∞(0,T;Hk)∩L2(0,T;Hk+1) for any k≥1. The proof proceeds in two steps. First, based on an energy estimate in the aforementioned space together with a compactness argument we prove existence of a unique solution, implying the continuous dependence in a weaker norm. This is then strengthened in the second step where the continuity in the optimal norm is established through an application of the rough Gronwall lemma. Our approach is direct and does not rely on any transformation formula, which permits to treat multidimensional noise. As an easy consequence we then deduce a Wong-Zakai type result, a large deviation principle for the solution and a support theorem.

Vector localized and periodic waves for the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation

Dynamical properties of vector localized and periodic waves hold significant importance in the study of physical systems. In this work, we investigate the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation. For the two interacting components, we construct the binary Darboux transformation formulas, vector localized, and periodic wave solutions. Via those solutions, different kinds of nonlinear waves can be achieved, including rogue waves, solitons, positons, and periodic waves. When the imaginary part of the spectral parameter is not zero, eye-shaped rogue waves appear in one component, and the twisted rogue wave pairs in the other component. As the spectral parameter is real, we derive distinct forms of vector localized and periodic waves on the non-zero background, such as the vector solitons, positons, periodic waves, breathers on the periodic wave background, and rational solitons. These results may be valuable in this investigation of nonlinear waves in physical systems.

Q-Difference Equation for the Operator E ̃(x,a;θ) and their Applications for the Polynomials h_n (a,b,x|q^(-1))

q-Difference Equation for the Operator E ̃(x,a;θ) and their Applications for the Polynomials h_n (a,b,x|q^(-1))

ON CERTAIN RESULTS ASSOCIATED WITH THE TRANSFORMATIONS OF GENERALIZED HYPERGEOMETRIC SERIES

In this paper, certain transformation formulas for ordinary hypergeometric series and also for q-hypergeometric series have been established.

Wall-crossing for vortex partition function and handsaw quiver variety

We investigate vortex partition functions defined from integrals over the handsaw quiver varieties of type A1 via wall-crossing phenomena. We consider vortex partition functions defined by two types of cohomology classes, and get functional equations for each of them. We also give explicit formula for these partition functions. This gives proofs to formula suggested by physicists. In particular, we obtain geometric interpretation of formulas for multiple hypergeometric functions including rational limit of the Kajihara transformation formula.