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The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions W_Q on X₀(N) for N ≤ 50 are Weierstrass points or not.

The design is by using Lawittes's and Schoeneberg's theorems.

Finding all Weierstrass points on X₀(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.

The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X₀(N).

The purpose of this paper is to study the coupled fixed point problem and the coupled best proximity problem for single-valued and multi-valued contraction type operators defined on cyclic representations of the space. The approach is based on fixed point results for appropriate operators generated by the initial problems.

To model a bargaining environment where the tactic of qualitative commitment (staking a principle) can have a demonstrably strong effect.

Amend the two‐person alternating offers model due to Rubinstein to include simultaneously pre‐announced stakes, and a mechanism for altering agents' utilities by reference to substantial costs of “surrendering a principle” (capitulation costs). Identify all stationary, pure strategy equilibria. Isolate subgame perfect Nash equilibria (SPNE).

SPNE is unique and offers efficient re‐allocation. For high enough capitulation costs, gains at equilibrium (relative to the bench‐mark game without commitment) result to the first‐mover. The shift in the equilibrium allocation is linear in cost.

Symmetric fixed capitulation costs were assumed. Surrender of principle required introduction of discontinuities in the cost function (zero or fixed cost). No gradation of the cost was considered. Future directions could involve discovering what distortions are created by asymmetric costs, by gradations and from smoothing out of the discontinuity.

Practical implications at best point to the potential rewards available from identifying and introducing a high‐cost commitment tactics in a bargaining context.

The innovation is in the use of an explicit bargaining apparatus to examine the commitment tactic (previous research used an axiomatic treatment of bargaining), and in offering a model for the notion of a qualitative commitment. The result about the qualitative shift in the equilibrium is also new. The paper is thus a contribution to the understanding of how an agent may seek to influence the outcome of a game such as “divide the pie (the resource)” through the manufacture/introduction of a factor outside the original game.

This paper aims to prove some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results of this paper generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. Furthermore, the authors produce an example to demonstrate the applicability of the results.

The results of this paper are theoretical and analytical in nature.

The authors established some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. An example was constructed to demonstrate the applicability of the results.

Analytical and theoretical results.

The results of this paper can be applied in science and engineering.

The results of this paper is applicable in certain social sciences.

The results of this paper are new and will open up new areas of research in mathematical sciences.

The purpose of this paper is to propose a new trajectory planning algorithm for industrial robots, which can let the robots move through a desired spatial trajectory, avoid colliding with other objects and achieve accurate movements. Trajectory planning algorithms are the soul of motion control of industrial robots. A predefined space trajectory can let the robot move through the desired spatial coordinates, avoid colliding with other objects and achieve accurate movements.

The mathematical expressions of the proposed algorithm are deduced. The speed control, position control and orientation control strategies are realized and verified with simulations, and then implemented on a six degrees of freedom (6-DOF) industrial robot platform.

A fixed-distance trajectory planning algorithm based on Cartesian coordinates was presented. The linear trajectory, circular trajectory, helical trajectory and parabolic trajectory in Cartesian coordinates were implemented on the 6-DOF industrial robot.

A simple and efficient algorithm is proposed. Enrich the kind of trajectory which the industrial robot can realize. In addition, the industrial robot can move more concisely, smoothly and precisely.

In this paper, we use the notion of cyclic representation of a nonempty set with respect to a pair of mappings to obtain coincidence points and common fixed points for a pair of self-mappings satisfying some generalized contraction- type conditions involving a control function in partial metric spaces. Moreover, we provide some examples to analyze and illustrate our main results.

Theoretical method.

We establish some coincidence points and common fixed point results in partial metric spaces.

Results of this study are new and interesting.

In section II.1, “The Three Dimensions of Space”, it was argued that logical space comprised a “thought unit” of three dimensions, and it was then shown why natural space should consist of three dimensions as well. The attempt to provide a logically complete definition for the concept of time indicated that this too was to be viewed as a logical space of three dimensions. This raises the question of “how natural space and time are to be conceptually united in order to become intuitively perceptible from an operationist point of view”.

Discusses the notion of eigenform as explicated by Heinz von Foerster wherein an object is seen to be a token for those behaviors that lend the object its apparent stability in a changing world.

Describes von Foerster's model for eigenforms and recursions and puts this model in the context of mathematical recursions, fractals, set theory, logic, quantum mechanics, the lambda calculus of Church and Curry, and the categorical framework of fixed points of Lawvere.

Determines that iterating a transformation upon itself is seen to be a key to understanding the nature of objects and the relationship of an observer and the apparent world of the observer.

Contemplates the concept of recursion in the context of second‐order cybernetics.

The purpose of this paper is to obtain the single-tooth stiffness, single-tooth time-varying meshing stiffness and comprehensive meshing stiffness of the internal and external face gears and to analyze the influence of the modulus, pressure angle and tooth width of each face gear on the single-tooth stiffness of the gear in nutation face gear transmission.

From the point of view of material mechanics, the gear teeth of nutation face gear are simplified as spacial variable cross-section beams. The shear deformation of gear teeth, the bending deformation of tooth root and the additional elastic deformation caused by the base deformation are gotten by simplified trapezoidal section method, thus the stiffness of nutation face gear teeth can be obtained. The comparison with finite element method results verifies the rationality of simplified trapezoidal section method for calculating the tooth stiffness of nutation face gear.

The variation of stiffness of internal and external face gears along the meshing line and tooth height in nutation face gear transmission is studied, and the variation laws of single tooth stiffness, single-tooth-pair mesh stiffness and single tooth time-varying meshing stiffness of nutation face gear teeth are obtained.

Nutation face gear transmission is a new type of transmission. The stiffness of face gear teeth is analyzed, and the variation rules of single tooth stiffness, single-tooth-pair mesh stiffness and single tooth time-varying meshing stiffness of nutation face gear teeth are obtained, which not only enriches the research of nutation face gear transmission but also has important guiding significance for the application of nutation face gear in engineering practice.