Uniform Formula Research Articles

The score-goldilocks design for phase 3 clinical trials

ABSTRACT In this paper, we propose a new Bayesian adaptive design, score-goldilocks design, which has the same algorithmic idea as goldilocks design. The score-goldilocks design leads to a uniform formula for calculating the probability of trial success for different endpoint trials by using the normal approximation. The simulation results show that the score-goldilocks design is not only very similar to the goldilocks design in terms of operating characteristics such as type 1 error, power, average sample size, probability of stop for futility, and probability of early stop for success, but also greatly saves the calculation time and improves the operation efficiency.

Uniform Asymptotic Formulas of Ranks and Cranks for Cubic Partitions

Uniform Asymptotic Formulas of Ranks and Cranks for Cubic Partitions

Numerical simulation methods and analysis for the dynamics of the time-fractional KdV equation

In this paper, two classes of efficient difference schemes for the simulation of the time-fractional Korteweg–de Vries equation are carried out. The temporal derivative is approximated with the help of the uniform/nonuniform L1 formula and the uniform L2-1σ formula, respectively. The spatial derivative is done with the uniform centered difference scheme. Unique solvability, boundedness and convergence of the corresponding difference schemes are rigorously proved at length. As far as we know, the proposed schemes achieve the highest convergence order in space among all the difference methods for the time-fractional KdV equation. Several numerical examples confirm the theoretical results and demonstrate the dynamic behavior.

Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections RedW(g) of reduced reflection factorizations of g and RGS(W,g) of relative generating sets of g. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size #RedW(g) with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-0 Hurwitz numbers.

Elastic buckling formulas of multi-stiffened corrugated steel plate shear walls

Multi-stiffened corrugated steel plate shear walls (MCSPSWs) have desirable seismic behavior involving shear-resistant, ductile, and energy-dissipating capacities. The elastic buckling load of the MCSPSW is a major influencing factor on its seismic behavior and it lays a foundation for the further study and design of the elastoplastic shear bearing capacity of the MCSPSW. Current research on the elastic buckling load of the MCSPSW is limited. For the MCSPSW with corrugations laid horizontally, the number of pairs of stiffeners in engineering practice is usually larger than that of existing research. This study focused on the elastic buckling formulas of the MCSPSW with more than one pair of vertical stiffeners. Firstly, based on the orthotropic plate theory, the theoretical model of elastic shear buckling was established for the energy method derivation of the elastic buckling load of the MCSPSW. Then, for the stiffeners with transition rigidities, the effect of the stiffener layout on the elastic buckling load was investigated through theoretical analyses and numerical solutions. Accordingly, the optimal layout corresponding to the maximum elastic buckling load was determined. Considering the optimal layout of stiffeners, the elastic buckling load of the MCSPSW was solved using MATLAB. The uniform formulas for the elastic buckling coefficient of the MCSPSW involving more than one pair of vertical stiffeners were proposed by fitting the MATLAB results. Finally, eigenvalue buckling analyses based on finite element models were conducted to verify the accuracy of the optimal layout theory of stiffeners and proposed formulas.

Load-Side Inertia Estimation Method Based on Comprehensive Statistics

The development of modern power systems is accompanied by the problem of inertia deficit, and there is an urgent need to evaluate the system inertia level. Especially, the accurate estimation of load-side inertia level is extremely important. Based on comprehensive statistics, a load-side inertia estimation method is proposed. Starting from the inertia analysis of load-side base elements, the expressions for load-side inertia estimation under various operation modes are given based on the inertia modeling of basic load units; the uniform correction formula for load-side inertia estimation is given considering the hidden effect of behind-the-meter power supply on inertia estimation. The simulation analysis is carried out based on the IEEE9 bus standard system, and the results show that the proposed load-side inertia estimation method does not exceed 5%.

An Extension of the Order-Preserving Mapping to the WENO-Z-Type Schemes

In our latest studies, by introducing the novel order-preserving (OP) criterion, we have successfully addressed the widely concerned issue of the previously published mapped weighted essentially non-oscillatory (WENO) schemes that it is rather difficult to achieve high resolutions on the premise of removing spurious oscillations for long-run simulations of the hyperbolic systems. In the present study, we extend the OP criterion to the WENO-Z-type schemes as the forementioned issue has also been extensively observed numerically for these schemes. Firstly, we innovatively present the concept of the generalized mapped WENO schemes by rewriting the Z-type weights in a uniform formula from the perspective of the mapping relation. Then, we naturally introduce the OP criterion to improve the WENO-Z-type schemes, and the resultant schemes are denoted as MOP-GMWENO-X. Finally, extensive numerical experiments have been conducted to demonstrate the benefits of these new schemes. We draw the conclusion that, the convergence propoties of the proposed schemes are equivalent to the corresponding WENO-X schemes. The major benefit of the new schemes is that they have the capacity to achieve high resolutions and simultaneously remove spurious oscillations for long simulations. The new schemes have the additional benefit that they can greatly decrease the post-shock oscillations on solving 2D Euler problems with strong shock waves.

Time-Domain-Based Superposition Analysis for Triple Active Bridge and Its Application for ZVS and Current Stress Optimization

With the advantages of simple topology, voltage matching, electrical isolation, and bi-directional energy control, Triple Active Bridge (TAB) has become an active exploration and attempt for the flexible interconnection of DC ports. The high-frequency chain current (HFCC) calculation is the crucial step of zero voltage switching (ZVS) regions and current stresses, providing the basis for the subsequent performance optimization of TAB. Due to the number of switches increasing, TAB´s degrees of freedom (the phase shift angles and duty cycles) have multiplied compared with its two-port topology. The existing current calculation methods of TAB are challenging to balance accuracy and simplicity. Currently, the simple and accurate HFCC model considering all the TAB variables can hardly be found in the literature. This paper proposes a unified and novel solution for TAB HFCC calculation based on the superposition theorem, Thevenin theorem, and time domain method. It can get the uniform formula of the HFCC when all the control variables change. Based on the formula derived, this paper further presents the corresponding ZVS and current stress analysis of TAB. A multi-objective optimal control method is proposed, demonstrating efficiency improvement. Moreover, the trade-offs between the control simplicity and optimization effect are analyzed based on the proposed HFCC model. The modulation pattern with the least algorithmic complexity for efficiency maximization is found. The validity and accuracy of the proposed method have been verified experimentally.

Diagonal quartic surfaces with a Brauer–Manin obstruction

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

Uniform Asymptotic Estimate for the Ruin Probability in a Renewal Risk Model with Cox–Ingersoll–Ross Returns

Consider an insurance risk model with arbitrary dependence structures between the claim sizes. Suppose that the risky investment in the insurer can be established by the Cox–Ingersoll–Ross model. When the claim-size distribution is heavy-tailed, a uniform asymptotic formula for ruin probability is obtained.

Module intersection and uniform formula for iterative reduction of one-loop integrals

In this paper, we develop an iterative sector-level reduction strategy for Feynman integrals, which bases on module intersection in the Baikov representation and auxiliary vector for tensor structure. Using this strategy we have studied the reduction of general one-loop integrals, i.e., integrals having arbitrary tensor structures and arbitrary power for propagators. Inspired by these studies, a uniform and compact formula that iteratively reduces all one-loop integrals has been written down, where messy polynomials in integration-by-parts (IBP) relations have organized themselves to Gram determinants.

A semi-bijective algorithm for saturated extended 2-regular simple stacks

Combinatorics of biopolymer structures, especially enumeration of various RNA secondary structures and protein contact maps, is of significant interest for communities of both combinatorics and computational biology. However, most of the previous combinatorial enumeration results for these structures are presented in terms of generating functions, and few are explicit formulas. This paper is mainly concerned with finding explicit enumeration formulas for a particular class of biologically relevant structures, say, saturated 2-regular simple stacks, whose configuration is related to protein folds in the 2D honeycomb lattice. We establish a semi-bijective algorithm that converts saturated 2-regular simple stacks into forests of small trees, which produces a uniform formula for saturated extended 2-regular simple stacks with any of the six primary component types. Summarizing the six different primary component types, we obtain a bivariate explicit formula for saturated extended 2-regular simple stacks with n vertices and k arcs. As consequences, the uniform formula can be reduced to Clote's results on k-saturated 2-regular simple stacks and the optimal 2-regular simple stacks, and Guo et al.'s result on the optimal extended 2-regular simple stacks.

Equilibrium points of heterogeneous small body in finite element method

ABSTRACT This paper presents a finite element method to search for equilibrium points around a heterogeneous small body, which degenerates from the full two-body problem in finite element form. The gravitational potential, acceleration, gravitational gradient matrix, and the linearized perturbation equation, which are key formulas for solving the equilibrium points and discriminating their stabilities, are interpolated by the tetrahedral model nodes of the small body. The finite element method could capture the complex internal structures of small bodies and provide a uniform and simple formula for various configurations. The method is applied to the Kuiper Belt Object (486958) Arrokoth, which is a contact binary asteroid. Three types of heterogeneous structures are hypothesized to investigate the evolutions of equilibrium points, namely density disparity of two sub-lobes, hardcore structure, and cavity structure. Under the condition of constant Arrokoth mass, the density disparity of the two sub-lobes greatly influences the local gravitational field. It thus has a significant influence on the positions of equilibrium points. The hardcore and cavity structures have less impact on the equilibrium points. These three simulation groups verified that the equilibrium points of a heterogeneous small body are quite different from those of a homogeneous small body. Such investigation of the equilibrium points may give an in-depth understanding of the dynamical environment around the heterogeneous small body, which is significant for future deep-space missions.

Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

Abstract We show convergence of small eigenvalues for geometrically finite hyperbolic n-manifolds under strong limits. For a class of convergent convex sets in a strongly convergent sequence of Kleinian groups, we use the spectral gap of the limit manifold and the exponentially mixing property of the geodesic flow along the strongly convergent sequence to find asymptotically uniform counting formulas for the number of orthogeodesics between the convex sets. In particular, this provides asymptotically uniform counting formulas (with respect to length) for orthogeodesics between converging Margulis tubes, geodesic loops based at converging basepoints, and primitive closed geodesics.

Harris hawks optimization based on global cross-variation and tent mapping.

Harris hawks optimization (HHO) is a new meta-heuristic algorithm that builds a model by imitating the predation process of Harris hawks. In order to solve the problems of poor convergence speed caused by uniform choice position update formula in the exploration stage of basic HHO and falling into local optimization caused by insufficient population richness in the later stage of the algorithm, a Harris hawks optimization based on global cross-variation and tent mapping (CRTHHO) is proposed in this paper. Firstly, the tent mapping is introduced in the exploration stage to optimize random parameter q to speed up the convergence in the early stage. Secondly, the crossover mutation operator is introduced to cross and mutate the global optimal position in each iteration process. The greedy strategy is used to select, which prevents the algorithm from falling into local optimal because of skipping the optimal solution and improves the convergence accuracy of the algorithm. In order to investigate the performance of CRTHHO, experiments are carried out on ten benchmark functions and the CEC2017 test set. Experimental results show that the CRTHHO algorithm performs better than the HHO algorithm and is competitive with five advanced meta-heuristic algorithms.

Analysis of the Relationship of Video Text and Urban Image Communication Based on the Calculation Method of Wedge Diffraction in Geometrical Optics

City image reflects a city’s comprehensive competitiveness and is also an important indicator of a city’s spiritual civilization and urbanization process. A good city image is an intangible asset of a city, which can contribute to the political, economic, cultural, and social construction of a city and create more value for the city. This paper mainly discusses the research status and research methods of urban image at home and abroad. Based on the calculation method of wedge diffraction in geometrical optics, various heuristic uniform diffraction formulas of lossy wedge are compared and analyzed, and a better heuristic formula of uniform diffraction of lossy wedge is given. Finally, the selection of important channel parameters in the propagation channel is discussed, and a method for predicting the statistical parameters of the propagation channel of urban images based on the results of ray tracing is proposed. Then, the channel parameters are analyzed by using statistical parameters, and the channel parameters of the city image propagation model are analyzed.

Uniform asymptotics for ruin probabilities of a delayed renewal risk model with one-sided linear dependence and stochastic returns

In this article, we consider a renewal risk model with by-claims, where the price process of the investment portfolio follows an exponential Lévy process. We further assume that the main claim is a one-sided linear process and there exists a certain dependence structure between the innovations and by-claims. In the presence of heavy tails, we obtain a series of uniform formulas in finite and infinite intervals. In order to better describe the obtained results, we carry on the numerical simulations.

Design of Arbitrary Time Convergence Controller for n -th Order System

A class of arbitrary time convergence controllers based on a special time-varying scaling function provide designers with a good choice to realize prescribed time stable systems. However, these controllers have the problem of conservative control parameter range and lack a uniform formula for systems of different order. Herein, the arbitrary time convergence controller is improved, the unified formula for n -th order system is given, and a more accurate control parameter range is obtained. By constructing an ingenious auxiliary function and a novel Lyapunov function, the arbitrary time convergence of the n -th order controller is proved, and the reasonable parameter selection range is obtained using the integrator backstepping and the mathematical induction method. The effectiveness and advantages of the proposed arbitrary time convergence controllers under saturated input constraints are illustrated through numerical simulations by comparative studies in unmanned aerial vehicle (UAV) formation control.

Power Harmonic Analysis Based on Continuously Adjustable Asymmetric Window

The study of harmonics signal analysis in electric power system is a classic and important research subject. The signal by using traditional Fourier transform methods are nearly truncated by constant coefficient symmetry window or asymmetric window. This paper propose the improved continuously adjustable asymmetric window for precise harmonic parameters measurement and the uniform formulas for calculating the parameters of harmonics and interharmonic is obtained by the asymmetric window-based phase difference correction method. The major advantages of this method is easy to implement and independent of window spectrum. The simulation analysis results proves that there is obvious high precision, effectiveness and universal applicability of this method.

Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves

We study the combinatorial geometry of a random closed multicurve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components $$\gamma _1, \dots ,\gamma _k$$ of a random multicurve $$m_1\gamma _1+\dots +m_k\gamma _k$$ represent linearly independent homology cycles with asymptotic probability 1 and that all its weights $$m_i$$ are equal to 1 with asymptotic probability $$\sqrt{2}/2$$ . We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus g are both very well approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of $$3g-3$$ elements. In particular, we prove that the expected value of these quantities has asymptotics $$(\log (6g-6) + \gamma )/2 + \log 2$$ as $$g\rightarrow \infty $$ , where $$\gamma $$ is the Euler–Mascheroni constant. These results are based on our formula for the Masur–Veech volume $${\text {Vol}}{\mathcal {Q}}_g$$ of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform asymptotic formula for intersection numbers of $$\psi $$ -classes on $${\overline{{\mathcal {M}}}}_{g,n}$$ for large g proved by A. Aggarwal in 2020.