Nash Equilibrium of a Fixed-Sum Two-Player Game

It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method

For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error.(i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett.113 210501(2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O(1/λ01/2)rather than O(1/λ1/2), where λ0 is a known lower bound of λ.(ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.

Does the Total Soil N Determined by Kjeldahl Method Include Fixed NH4^+?

Forty soils from top layer (0-20 cm) were sampled in different regions of China and Kjeldahl, HF-Kjeldahl and double treatment methods were used to determine total N, total N plus fixed ammonium, and total N and the residual fixed ammonium left in soil after determination of total N, respectively, to evaluate if Kjeldahl’s method could include the fixed N by soil minerals. The fixed N by soil minerals was measured by Silva-Bremner procedure to make comparison. Results showed that total N determined by Kjeldahl’s method averaged 1.622 g kg-1, while that by HF- Kjeldahl’s method 1.633 g kg-1, and that by double procedure 1.666 g kg-1. Obviously results obtained by the last two methods, particularly the double treatment method, were higher than Kjeldahl’s, showing that Kjeldahl’s method could not or not fully release N fixed by 2:1 minerals in soil, and therefore the determined results would not be the true “total N” for soils that contained large amount of the fixed N. The mineral fixedN averaged 166 mg kg-1, accounting for 10.1% of the total N while the residual fixed N amounted to 30.4 mg kg-1, equivalent to 1.9% of the total N or 18.3% of the total fixed N. The residual fixed N was correlated neither to organic matter nor to total N, but closely related to the total fixed N with a correlation coefficient of 0.598 (n=40), showing that the fixed N was the sole source of the residues. Soils having high residues of the fixed N were just those containing high fixed N, and soils containing high fixed N were just those containing high amount of 2:1 minerals. As a result, Kjeldahl’s method could not give a true value of the total N for such soils. However, for those containing small or little amount of 2:1 minerals, there was no significant difference in results measured by these methods.

VISCOSITY ITERATIVE METHODS FOR COMMON FIXED POINTS OF TWO NONEXPANSIVE MAPPINGS WITHOUT COMMUTATIVITY ASSUMPTION IN HILBERT SPACES

In this article, we introduce a new viscosity iterative method for two nonexpansive mappings in Hilbert spaces. We also prove, without commutativity assumption, that the iterates converge to a common fixed point of the mappings which solves some variational inequality. The results presented extend the corresponding results of Shimizu and Takahashi IT. Shimizu, W. Takahashi, Strong convergence to common fixed point of families of nonexpansive mappings, J. Math. Anal. Appl. 211 （1997）, 71-83], and Yao and Chen [Y. Yao, R. Chert, Convergence to common fixed points of average mappings without commutativity assumption in Hilbert spaces, Nonlinear Analysis 67（2007）, 1758-1763].

New Modification of Fixed Point Iterative Method for Solving Nonlinear Equations

In this paper, we have modified fixed point method and have established two new iterative methods of order two and three. We have discussed their convergence analysis and comparison with some other existing iterative methods for solving nonlinear equations.

The Influence of Feeding Method on Fluidization Behavior of Fixed Fluidized Bed

An experimental installation of cold model simulation was set up to study the bed pressure drop in different regions of fixed fluidized bed reactor during top feeding and bottom feeding, respectively, at various gas velocities with the fluidization image of solid particles monitored at the same time. By comparing the changes in bed density and operating gas velocity in different regions of fixed fluidized bed reactor, the influence of top feeding and bottom feeding patterns on fluidization behavior could be investigated. The results showed that the bed density in top feeding reactor responded more stably to the change in gas velocity along with the advantage of working in a wider range of operating gas velocities. Based on this study, it is concluded that existing bottom feeding reactor configurations cannot meet the fluidization requirements; and optimization of bottom feeding reactor will be needed.

A Continuation Method for Solving Fixed Point Problems in Unbounded Convex Sets

In this paper, an unbounded condition is presented, under which we are able to utilize the interior point homotopy method to solve the Brouwer fixed point problem on unbounded sets. Two numerical examples in R3 are presented to illustrate the results in this paper.

Modified Subgradient Extragradient Method for Variational Inequality Problems and Fixed Point Problems

Many approaches inquiring into variational inequality problems have been put forward,among which subgradient extragradient method is of great significance.A novel algorithm is presented in this article for resolving quasi-nonexpansive fixed point problem and pseudomonotone variational inequality problem in a real Hilbert interspace.In order to decrease the execution time and quicken the velocity of convergence,the proposed algorithm adopts an inertial technology.Moreover,the algorithm is by virtue of a non-monotonic step size rule to acquire strong convergence theorem without estimating the value of Lipschitz constant.Finally,numerical results on some problems authenticate that the algorithm has preferable efficiency than other algorithms.

Subgradient Extragradient Methods for Equilibrium Problems and Fixed Point Problems in Hilbert Space

Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to enhance the speed of the convergence and reduce computational cost,the algorithms used a new step size and a cutting hyperplane.The first algorithm was proved to be weak convergence,while the second algorithm used a modified version of Halpern iteration to obtain strong convergence.Finally,numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.

A Modified Fixed Point Method for the Perona Malik Equation

In this work we present a new method to solve the Perona Malik equation for the image denoising. The method is based on a modified fixed point algorithm which is fast and stable. We discretize the equation using a finite volume method by integrating the equation using a fuzzy measure on the control volume. To make our algorithm move faster in time, we have used an optimized domain decomposition which generalize the wave relaxation method. Several test of noised images illustrate this approach and show the efficiency of the proposed new method.

Solving a Class of Brouwer Fixed-point Problems via a Modified Aggregate Constraint Homotopy Method

In this paper, we provide an aggregate function homotopy interior point method to solve a class of Brouwer fixed-point problems. Compared with the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65）, the main adavantages of this method are as foUows： on the one hand, it can solve the Brouwer fixed-point problems in a broader class of nonconvex subsets Ω in R^n （in this paper, we let Ω={x∈ R^n ： gi（x） ≤0, i= 1,... , m}）; on the other hand, it can also deal with the subsets Ω with larger amount of constraints more effectively.

Modified Homotopy Method for a Class of Brouwer Fixed-point Problems

In this paper, we modify the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65） and hence make the modified method be able to solve Brouwer fixed-point problems in a broader class of nonconvex subsets in Rn. In addition, a simple example is given to show the effectiveness of the modified method.

一种改进的Fixed-base Comb安全快速算法

本文从分析椭圆曲线上Fixed-baseComb算法出发,根据其特点,利用牺牲乘法操作以降低求逆操作的方法,分别用2kP、2P+Q的快速算法对Fixed-baseComb算法的预计算阶段和赋值阶段进行改进,极大地提高了计算效率:在素数域上预计算阶段提高70%～80%,而赋值阶段提高38%～43%,同时,改进算法通过对k的预处理,使得算法能够抵抗边际信道攻击.

A FIXED POINT APPROACH TO THE STABILITY OF A GENERALIZED APOLLONIUS TYPE QUADRATIC FUNCTIONAL EQUATION

Using the fixed point method, this article proves the Hyers-Ulam-Rassias stability of a generalized Apollonius type quadratic functional equation in Banach spaces. The conditions of these results are demonstrated by the quadratic functional equation of Apollonius type.

FIXED POINTS AND STABILITY FOR QUARTIC MAPPINGS IN β-NORMED LEFT BANACH MODULES ON BANACH ALGEBRAS

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equationin β-Banach modules on Banach algebras. The concept of Ulam-Hyers-Rassias stability （briefly, HUR-stability） originated from Th. M. Rassias stability theorem that appeared in his paper： On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 （1978）, 297-300.

Performance of fixed geosynthetic technique of GRPS embankment

Geosynthetic-reinforced and pile-supported （GRPS） embankment has been increasingly constructed in a large number of regions and for a wide range of projects in the past decades. However, many disadvantages are exposed through a lot of applications on conventional technique of GRPS embankment （called CT embankment）, i.e., intolerable settlement and lateral displacement, low geosynthetic efficiency, etc. In view of these disadvantages, the fixed geosynthetic technique of GRPS embankment （called FGT embankment） is developed in this work. In this system, the geosynthetic is fixed on the pile head by the steel bar fulcrum and concrete fixed top. The principles and construction techniques involved in the FGT embankment are described firstly. Then, the numerical analysis method and two-stage analysis method are used to study the performance of FGT embankment, respectively. It is shown that the FGT embankment can provide a better improvement technique to construct a high embankment over soft ground.

Near-wellbore fracture propagation physical simulation based on innovative interlaced fixed perforation

An innovative perforation method of interlaced fixed perforation was put forward based on the analysis of the characteristics of fractures in various periods of perforation and conventional perforation modes.By conducting a large-scale perforation shooting experiments,we investigated the morphology,propagation mechanism and propagation law of the near-wellbore fractures generated during perforating processes under different fixed angle and interlaced angle combinations,and discussed the control method of near-wellbore fractures in different types of unconventional oil and gas reservoirs.The experimental results show that:(1)The interlaced fixed perforation strengthens the connectivity between the perforation tunnels not only in the same fixed plane but also in adjacent fixed planes,making it likely to form near-wellbore connected fractures which propagate in order.(2)Three kinds of micro-fractures will come up around the perforation tunnel during perforation,namely typeⅠradial micro-fracture,typeⅡoblique micro-fracture and typeⅢdivergent micro-fracture at the perforation tip,which are interconnected into complex near-wellbore fracture system.(3)Different types of perforation bullets under different combinations of fixed angles and interlaced angles result in different shapes of near-wellbore fractures propagating in different patterns.(4)By using the interlaced perforation on fixed planes,arranging fixed planes according to the spiral mode or the continuous"zigzag"shape,the desired near-wellbore fractures can be obtained,which is conducive to the manual control of main fractures in the fracturing of unconventional or complex conventional reservoirs.

Nodulation and Fixed Atmospheric Nitrogen of Some Local Lima Bean (<i>Phaseolus lunatus</i>L.) Cultivars Grown in a Coastal Savannah Environment

Legumes, in symbiotic association with Rhizobia, are able to fix atmospheric N. Six local lima bean (Phaseolus lunatus) cultivars were grown under rainfed conditions in a coastal savannah environment. Objectives of the study were to evaluate the nodulation and fixed atmospheric N levels of the six local lima bean cultivars using both the 15N isotope dilution method and N difference method (NDM). The linear relationship between fixed atmospheric N estimated using the 15N isotope dilution method and NDM, was also assessed. The experiment was arranged in a randomized complete block design (RCBD) in three replicates with seven treatments, comprising six lima bean cultivars (B1, B2, B3, B4, B5 and B6) and the early maturing local maize variety, “Doke”, as the reference crop. Total, effective nodules (EN) and non-effective nodules (NEN) were determined on 42 and 56 days after planting (DAP). The 15N isotopic dilution method and NDM were used to quantify the fixed atmospheric N by the lima bean cultivars on 60 DAP. Effective root nodules per plant (EN) on 56 DAP ranged from 0.71 to 1.22, with the lima bean cultivar B4 having the highest value and cultivars B2 and B5 having the lowest value of EN, respectively. Similarly on 56 DAP, the lima bean cultivar B4 had the highest NEN value while cultivars B1, B2 and B5 had the lowest NEN value of 0.71 per plant. The mean fixed atmospheric N was 8.98 kg&middotha-1, based on the 15N isotope dilution method, which was lower than 10.13 kg&middotha-1 of fixed atmospheric N determined using NDM. The linear relationship between fixed atmospheric N estimated using the 15N isotope dilution method and that estimated using the NDM, was positive but of average quality as the R2 value was 0.56. Consequently, the linear model obtained from this relationship is moderate as 56% of the data used for the linear regression analysis were accounted for by the linear regression model developed. However, NDM could be used for fast screening to select lima bean cultivars for a more detailed study to identify cultivars with promising fixed atmospheric N capabilities. Generally, results of the study provide opportunities for designing breeding and other agronomic programmes for enhancing the productivity and N-fixing capacity of local lima beans in the coastal savannah environment.

Common Fixed Point Iterations of Generalized Asymptotically Quasi-Nonexpansive Mappings in Hyperbolic Spaces

We introduce a general iterative method for a finite family of generalized asymptotically quasi- nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multi-step iterative method of Khan et al. [1] as a special case. Our results are new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) spaces, simultaneously.

Fixed-node Quantum Monte Carlo: A Novel Approach

In this paper, a novel method for fixed-node quantum Monte Carlo is given. We have derived an expansion of the eigenvalue of the energy for a system and proved that the value of the energy calculated using the traditional fixed-node quantum Monte Carlo method is only the zero order approximation of the eigenvalue of the energy. But when using our novel method, in the case of only increasing less computing amounts (<1%), we can obtain conveniently the first order approximation, second order approximation, and so on. We have calculated the values of the zero, first and second approximation (0, 1 and 2) of the energies of 11A1 state of CH2, 1Ag (C4h, acet) state of C8 and the ground-state of H2O using this novel method. The results indicate that for 11A1 state of CH2, 1Ag (C4h, acet) state of C8 and the ground-state of H2O it needs only the second order approximation to obtain electronic correlation energy with over 97%. This demonstrates that this novel method is very excellent in both the computing accuracy and the amount of calculation required.