Form Of Formula Research Articles

Superconvergence analysis of the nonconforming FEM for the Allen–Cahn equation with time Caputo–Hadamard derivative

This paper first introduces a high-order approximate formula for the Caputo–Hadamard fractional derivative using interpolation theory, and analyzes its convergence and accuracy. Although the formula bears resemblance to the logarithmic L2-1σ formula in form, it differs in the selection of mesh points, making it suitable for more complex models in numerical computations. Subsequently, a numerical method for the Allen–Cahn equation with time Caputo–Hadamard derivative is proposed. The temporal direction is discretized using the newly derived difference formula, while the spatial direction is approximated by the anisotropic nonconforming quasi-Wilson finite element method (FEM). It is proven that the derived scheme has optimal convergence accuracy in the L2-norm and global superconvergence property in the H1-norm. Numerical examples are provided to corroborate the theoretical claims.

The measurement of China's provincial physical capital stock based on the improved method and indicators.

Physical capital stock is the basic indicator of macroeconomic empirical studies and also an important manifestation of macroeconomic activity capacity. When measuring China's provincial physical capital stock by the perpetual inventory method, the treatment for the depreciation rate and the initial physical capital stock in the existing literature has some defects in reflecting the economic performance. The purpose of this paper is to build the most reliable and reasonable statistical series of China's provincial physical capital stock. We establish a measurement formula in form of the variable depreciation rate and improve the treatment for the depreciation rate and the initial physical capital stock. For the depreciation rate, we measure the benchmark depreciation rate of physical capital and determine the variable depreciation rate according to the economic growth rate, which makes the variable depreciation rate reflect the economic growth. For the initial physical capital stock, we measure the initial physical capital stock according to the capital output ratio, which makes the initial physical capital stock reflect the economic aggregate. Based on these improvements, we measure China's provincial physical capital stock from 1952 to 2022 by the perpetual inventory method. Through analyzing the measurement results, we have a more distinct insight into the temporal and spatial characteristics of China's provincial physical capital stock: The growth rate of physical capital stock at the provincial level has slowed in spite of remaining high; The eastern region is much higher than the central and western regions. According to these two characteristics, it is suggested that policymakers need to improve market mechanisms and adjust investment policies to promote a benign profile of economic growth and a regional optimization of capital formation.

Neveu’s Exchange Formula for Analysis of Wireless Networks With Hotspot Clusters

Theory of point processes, in particular Palm calculus within the stationary framework, plays a fundamental role in the analysis of spatial stochastic models of wireless communication networks. Neveu’s exchange formula, which connects the respective Palm distributions for two jointly stationary point processes, is known as one of the most important results in the Palm calculus. However, its use in the analysis of wireless networks seems to be limited so far and one reason for this may be that the formula in a well-known form is based upon the Voronoi tessellation. In this paper, we present an alternative form of Neveu’s exchange formula, which does not rely on the Voronoi tessellation but includes the one as a special case. We then demonstrate that our new form of the exchange formula is useful for the analysis of wireless networks with hotspot clusters modeled using cluster point processes.

KILLING FORM ALJABAR LIE FROBENIUS BERDIMENSI ≤4 UNTUK MENENTUKAN KESEMISEDERHANAANNYA

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

Scattering amplitudes and simple canonical forms for simple polytopes

We provide an efficient recursive formula to compute the canonical forms of arbitrary d-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on d facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint ϕ3 theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in ϕ4 theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Subdivision schemes play a vital role in curve modeling. The curves produced by the class of 2 n + 2 -point ternary scheme (Deslauriers and Dubuc (1989)) interpolate the given data while the curves produced by a class of 2 n + 2 -point ternary B-spline schemes approximate the given data. In this research, we merge these two classes to introduce a consolidated and unified class of combined subdivision schemes with two shape control parameters in order to grow versatility for overseeing valuable necessities. However, the proposed class of subdivision schemes gives optimal smoothness in the final shapes, yet we can increase its smoothness by using a proposed general formula in form of its Laurent polynomial. The theoretical analysis of the class of subdivision schemes is done by using various mathematical tools and using their coding in the Maple environment. The graphical analysis of the class of schemes is done in the Maple environment by writing the codes based on the recursive mathematical expressions of the class of subdivision schemes.

Constraint Solving for Synthesis and Verification of Threshold Logic Circuits

Threshold logic (TL) circuits gain increasing attention due to their feasible realization with emerging technologies and strong bind to neural network applications. In this work, we devise techniques for automatic synthesis and verification of TL circuits based on constraint solving. For synthesis, we formulate a fundamental operation to collapse TL functions, and derive a necessary and sufficient condition of collapsibility for linear combination of two TL functions. An approach based on solving the subset sum problem is proposed for fast circuit transformation. For verification, we propose a procedure to convert a TL function to a multiplexer (MUX) tree and to pseudo-Boolean (PB) constraints for formal Boolean and PB reasoning, respectively. Experiments on synthesis show that the collapse operation further reduces gate counts of synthesized TL circuits by an average of 18%. Experiments on verification demonstrate good scalability of the MUX-based method for equivalence checking of synthesized TL circuits, and efficiency of PB constraint conversion in cases where the conjunctive normal form (CNF) formula conversion and MUX tree conversion suffer from memory explosion.

Novel hybrid robust method for uncertain reliability analysis using finite conjugate map

Due to the limitation of measured data, the random and epistemic uncertainties must be simultaneously handled for the reliability analysis of curvilinear aircraft stiffened panels with multiple cutouts. A hybrid iterative conjugate first-order reliability method (CFORM) and adaptive dynamical harmony search (ADHS) optimization are developed for fuzzy reliability analysis (FRA) of stiffened panels. The proposed global level performs fuzzy analysis using ADHS, while the local level is utilized for reliability evaluation using efficient FORM. In ADHS, two adjusting procedures are implemented to update the new positions of reliability index bounds with dynamical parameters Due to the computational burden of fuzzy reliability analysis with robust iterative formula, a conjugate FORM is applied using finite-step size, which is formulated by applying the Armijo rule. A limited conjugate scalar factor is utilized to determine the conjugate nonlinear describe map. The proposed conjugate formula is examined by various FORM formulas. The fuzzy reliability index is obtained using alpha level set-based ADHS in terms of extracted results by conjugate FORM for stiffened panels. Furthermore, by the sensitivity measure of the fuzzy reliability index, boundaries of uncertainty interval for stiffener heights and thicknesses of the panel were proposed. Moreover, the influences of different stiffener height, design load, stiffener thickness, and skin thickness on buckling mode were discussed. Survey results demonstrated that the proposed FRA is more accurate to handle epistemic uncertainties of curvilinear aircraft stiffened panels than the classic reliability method-based FORM. The conjugate reliability method provides robust and efficient results compared to the modified iterative reliability methods.

A set-valued Markov chain approach to credit default

We propose a novel credit default model that takes into account the impact of macroeconomic factors and intergroup contagion on the defaults of obligors. We use a set-valued Markov chain to model the default process, which includes all defaulted obligors in the group. We obtain analytic characterizations for the default process and derive pricing formulas in explicit forms for synthetic collateralized debt obligations (CDOs). Furthermore, we use market data to calibrate the model and conduct numerical studies on the tranche spreads of CDOs. We find evidence to support that systematic default risk coupled with default contagion could have the leading component of the total default risk.

Generalized Optimal and Explicit PI/PID Tuning Formulas for Underdamped Second-order Systems

This study aims to propose generalized formulas in an explicit form to optimally tune the gains of the proportional-integral (PI) and proportional-integral-derivative (PID) controllers of an underdamped second-order system. Unlike other methods used in literature, the controller gains are tuned explicitly based on the known system parameters of the second-order system, namely: the damping ratio and the natural frequency. The proposed accurate and one-step formulas do not need any further effort to effectively control any given second-order system while keeping the desired performance indices within near-optimal specifications. The proposed work relies on a restricted deadbeat response criterion and cascaded surface curve fitting procedure to end by near-optimal controller gains tuning formulas. A particle swarm optimization (PSO) method is employed to optimize the primary tuned gains for wide-range of system frequency. The proposed formulas are tested and simulated to demonstrate their effectiveness with comparisons with the closest common related gains tuning methods. Based on the comprehensive analysis, the proposed work ensures 2% maximum percentage overshoot when the proposed formulas are adopted to tune the gains of the PID controllers. Different comparisons are carried out to show the effectiveness of the proposed work.

Efficient Verified (UN)SAT Certificate Checking

SAT solvers decide the satisfiability of Boolean formulas in conjunctive normal form. They are commonly used for software and hardware verification. Modern SAT solvers are highly complex and optimized programs. As a single bug in the solver may invalidate the verification of many systems, SAT solvers output certificates for their answer, which are then checked independently. However, even certificate checking requires highly optimized non-trivial programs. This paper presents the first SAT solver certificate checker that is formally verified down to the integer sequence representing the formula. Our tool supports the full DRAT standard, and is even faster than the unverified state-of-the-art tool drat-trim, on a realistic set of benchmarks drawn from the 2016 and 2017 SAT competitions. An optional multi-threaded mode further reduces the runtime, in particular for big certificates.

Evaluation of LFP Battery SOC Estimation Using Auxiliary Particle Filter

State of charge (SOC) estimation of lithium batteries is one of the most important unresolved problems in the field of electric vehicles. Due to the changeable working environment and numerous interference sources on vehicles, it is more difficult to estimate the SOC of batteries. Particle filter is not restricted by the Gaussian distribution of process noise and observation noise, so it is more suitable for the application of SOC estimation. Three main works are completed in this paper by taken LFP (lithium iron phosphate) battery as the research object. Firstly, the first-order equivalent circuit model is adapted in order to reduce the computational complexity of the algorithm. The accuracy of the model is improved by identifying the parameters of the models under different SOC and minimum quadratic fitting of the identification results. The simulation on MATLAB/Simulink shows that the average voltage error between the model simulation and test data was less than 24.3 mV. Secondly, the standard particle filter algorithm based on SIR (sequential importance resampling) is combined with the battery model on the MATLAB platform, and the estimating formula in recursive form is deduced. The test data show that the error of the standard particle filter algorithm is less than 4% and RMSE (root mean square error) is 0.0254. Thirdly, in order to improve estimation accuracy, the auxiliary particle filter algorithm is developed by redesigning the importance density function. The comparative experimental results of the same condition show that the maximum error can be reduced to less than 3.5% and RMSE is decreased to 0.0163, which shows that the auxiliary particle filter algorithm has higher estimation accuracy.

Triangulation of 2-loop MHV amplituhedron from sign flips

In this paper, we consider the triangulation of the 2-loop MHV amplituhedron from “sign flip” definition. Using the isomorphism between the m = 2, k = 2 tree amplituhedron and the 1-loop MHV physical amplituhedron, we found the direct triangulation of the 2-loop MHV amplituhedron from sign flips. This triangulation is different from the BCFW triangulation because of the structure of the cells. And we also found a formula of the canonical form of the n-point 2-loop MHV amplituhedron. This formula looks like a 2-loop version of the Kermit representation of the 1-loop MHV amplitude. We checked that the sum of these cells is consistent with the BCFW up to at least 22-pt numerically.

К вопросу об ошибках в терминологии по теории поверхностей и геометрического моделирования

At present, a great amount of scientific papers, monographs, and reference books dealing with analytical and differential geometry of surfaces have been published. They contain materials for following geometric investigations, for implementation of received earlier geometrical results into architecture, building, and machinery manufacturing. In the paper it has been shown on the specific examples that sometimes the results of geometric investigations for shells’ middle surfaces taken in published references for the following application without check could lead to serious errors because of ones in the surfaces equations or inexactitudes in a surfaces class definition. At present, 38 classes of surfaces, uniting more than 600 ones that have their own names and are described in scientific publications, are known. The author has worked up a great number of researches and found errors, inaccuracies, and alternative versions in monographs and scientific papers, related to questions on geometry of developable surfaces (conic and torse surfaces), surfaces of rev olution (paraboloid and ellipsoid of revolution, nodoid), minimal surfaces (catenoids), conoids, and cyclic surfaces including the canal ones. In actual practice there are much more geometric errors, but in this paper are discussed only well-known geometricians and architects’ works, as well as in this paper there is no information on surfaces that are presented at specialized sites in Internet. Here are encountered misreckoned coefficients for surfaces’ fundamental quadratic forms, there are errors in the formulae for the quadratic forms’ coefficients determination, as well as in the formulae for the calculation a surface element’s area, surface’s principle curvatures, and so on. All of encountered errors have been divided into four groups. The fourth group’s errors named as “typographical errors and authors’ slips of the pen” have been considered fragmentarily because they are encountered the most frequently, and can be corrected by the authors themselves in the following papers.

Exact solution of the differential equation of transverse oscillations of the rod taking into account own weight

The free transverse oscillations of the rod of uniform cross section taking into account its own weight are considered in this work. The appropriate partial differential equation of transverse oscillations of a rod was reduced to two ordinary differential equations for the time function and the amplitude function of deflections. Concurrent with the differential equations for the amplitude state, the equivalent system of differential equation is considered. In total, the exact solution of the initial partial differential equation of transverse oscillations of a rod, expressed in nondimensional fundamental functions and initial parameters is attained. The method of power series was used for the construction of fundamental functions. Due to the exact solution, the formulas in an explicit form for dynamic variables of the state of a rode – deflections, angular displacement, bending moment and transverse force – were defined. The analytical form for equation of free oscillation frequency is defined. That has limited the finding of frequency to definition the unknown non-dimensional parameter through the frequency equation. As a result, the presence of derivative exact solutions provides the possibility to investigate the free oscillations of rod with various types of boundary conditions.

Model Checking Software with First Order Logic Specifications Using AIG Solvers

Static verification techniques leverage Boolean formula satisfiability solvers such as SAT and SMT solvers that operate on conjunctive normal form and first order logic formulae, respectively, to validate programs. They force bounds on variable ranges and execution time and translate the program and its specifications into a Boolean formula. They are limited to programs of relatively low complexity for the following reasons. (1) A small increase in the bounds can cause a large increase in the size of the translated formula. (2) Boolean satisfiability solvers are restricted to using optimizations that apply at the level of the formula. Finally, (3) the Boolean formulae often need to be regenerated with higher bounds to ensure the correctness of the translation. We present a method that uses And-Inverter-Graph (AIG) sequential circuits, and AIG synthesis and verification frameworks to validate programs. An AIG is a Boolean formula with memory elements, logically complete negated conjunction gates, and a hierarchical structure. Encoding the validation problem of a program as an AIG (1) typically provides a more succinct representation than a Boolean formulae encoding with no memory elements, (2) preserves the high-level structure of the program, and (3) enables the use of a number of powerful automated analysis techniques that have no counterparts for other Boolean formulae such as CNF. Our method takes an imperative program with a first order logic specification consisting of a precondition and a postcondition pair, and a bound on the program variable ranges, and produces an AIG with a designated output that is ${true}$ when the program violates the specification. Our method uses AIG synthesis reduction techniques to reduce the AIG, and then uses AIG verification techniques to check the satisfiability of the designated output. The results show that our method can validate designs that are not possible with other state of the art techniques, and with bounds that are an order of magnitude larger.

Fixed Point Theorems of Fuzzy Integer Value Mappings and Optimization Management to Balance Problem

In this paper, the concepts of supremum and infimum of bounded Fuzzy integer set are given, their existence is proved, and their expressions formula in level set form are set up. Then some fixed point theorems for Fuzzy integer value mapping are obtained, and a balance problem is proposed. And then, using the new theories established by us about Fuzzy integers and Fuzzy integer value mappings, we give two examples (one is about the protection of some animal species whose surviving numbers are declining year by year, and the other one is about sustainable production problems to some kind of renewable energy) to show the optimal management method for the proposed balance problem.

Voronovskaya–Bernstein Type Asymptotic Formulas in Strong Form

We establish Voronovskaya–Bernstein type asymptotic formulas in strong form for approximation methods of different type.

Temporal normal form for Linear Temporal Logic formulae1

In the present paper, the concepts of characters as well as the least characters for LTL (Linear Temporal Logic) formulae are introduced. It is pointed out that those LTL formulae with characters can always be checked within finite steps during model checking even in some cases when the underlying transition system contains infinite states. What is more, the class of LTL formulae with characters can be characterized by full LTLn, the bounded case for LTL. Meanwhile, two types of temporal normal form for LTL formulae are proposed. A necessary and sufficient condition is given in which an LTL formula has an equivalent formula in temporal normal form.

SYMMETRIES IN MODAL LOGICS

AbstractIn this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [5]. Hence, the results apply to a wide class of modal logics including, for example, hybrid logics. We present two graph constructions that enable the reduction of symmetry detection in modal formulas to the graph automorphism detection problem, and we evaluate the graph constructions on modal benchmarks.