In present paper, spatial Minkowski Pythagorean Hodograph (MPH) curves are characterized with Rational Rotation Minimizing Frames (RRMFs). We define Euler-Rodrigues Frame (ERF) for MPH curves and by means of this concept, we reach the definition of MPH curves of type (n, m). Expressing the conditions provided by these curves in the form of Minkowski-Hopf map that we define, it is aimed to establish a connection with the Lorentz force which occurs during the process of Computer Numerical Control (CNC) type sinker Electronic Discharge Machines (EDMs). This approach is reinforced by split quaternion polynomials. Finally, we give conditons satisfied by MPH curves of low degree to be type (n, m) and construct illustrative examples.

In this manuscript a new approach into analyzing the local stability of equilibrium points of non-linear Caputo fractional planar systems is shown. It is shown that the equilibrium points of such systems can be a stable focus or unstable focus. In addition, it is proposed that previous results regarding the stability of equilibrium points have been incorrect, the results here attempt to correct such results. Lastly, it is proposed that a Caputo fractional planar system cannot undergo a Hopf bifurcation, contrary to previous results prior. Though, it is shown that such systems can undergo a Hopf bifurcation (topologically).

Let $(A,\Delta)$ be a {\it weak multiplier Hopf algebra} as introduced in [VD-W3] (see also [VD-W2]). It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. If the algebra has an identity and the coproduct is unital, then we have a Hopf algebra. If the algebra has no identity, but if the coproduct is non-degenerate (which is the equivalent of being unital if the algebra has an identity), then the pair would be a multiplier Hopf algebra. If the algebra has an identity, but the coproduct is not unital, we have a weak Hopf algebra. In the general case, we neither assume $A$ to have an identity nor do we assume $\Delta$ to be non-degenerate and so we work with a {\it genuine} weak multiplier Hopf algebra. It is called {\it regular} if its antipode is a bijective map from $A$ to itself. \snl In this paper, we {\it continue the study of weak multiplier Hopf algebras}. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\ot C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a {\it separability idempotent} (as studied in [VD4.v2]). If the weak multiplier Hopf algebra is regular, then also $E$ is a {\it regular} separability idempotent. \snl We also consider {\it special cases and examples} in this paper. In particular, we see how for any weak multiplier Hopf algebra $(A,\Delta)$, it is possible to make $C\ot B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\ot C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

In this work, we discuss a newly proposed scalar-tensor theory in Minkowski spacetime. The basic frame in Minkowski spacetime is obtained. Consider a charged particle in the electromagnetic field, we make an attempt to connect this theory with conventional quantum theory. Two specific models are chosen to calculate values of the time delay.

In this paper, the concept of fuzzy locally convex spaces generated by a family of fuzzy seminorms is introduced. We prove that a Minkowski functional of zero neighborhoods generates a seminorm and finally deduce that the topology generated by the family of seminorms coincides with that of the Minkowski functional.

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica (or, Wolfram language) mathematical program package, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each has two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable that appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable.

In this paper, we define Ulisse Dini-type helicoidal hypersurface with spacelike axis in Minkowski 4-space ${\mathbb{E}}_1^4$ . We compute the Gaussian and the mean curvature of the hypersurface. Moreover, we obtain some special symmetry to the curvatures when they are flat and maximal.

The main objective of this study is to find out the influence of cooperation and intra-specific competition in escaping predation through refuge of the prey population and the effect of the two intra-specific interactions on the dynamics of prey-predator systems. For this purposes, two mathematical models with Holling type-II functional response function have been proposed and analyzed. The first model includes cooperation among prey populations whereas the second one incorporates intra-specific competition. Existence conditions and stability of different equilibrium points of both models have been carried out for the qualitative behaviour of the systems. Refuge through intra-specific competition has a stabilizing role, whereas cooperation has a destabilizing role on the system dynamics. Periodic oscillations are observed in both the systems through Hopf bifurcation. From the analytical and numerical findings, we conclude that intra-specific competition affects the prey population and continuously checks the refuge class under a critical value, and thus it never becomes too large to cause predator extinction due to food scarcity. Conversely, cooperation leads maximum individuals to escape predation through the refuge so that predators will suffer from low predation success.

We consider a model of predator-prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity as well as the boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon's theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurance of Hopf bifurcation around the interior point is also shown analytically. At the end, we implement the Predictor-Corrector scheme to perform some numerical simulations.

A novel geometric interpretation of the Minkowski metric is provided, which offers a different and more intuitive approach to phenomena in special relativity. First it is shown that a change of basis in Minkowski space is the equivalent of a change of basis in Euclidean space if a basis element is replaced by its dual element, constituting a mixed basis set. The methodology of the proof includes infinitesimal changes of basis using the Lie-algebras of the involved groups. As a consequence, a direct mapping between Euclidean and Minkowski space is defined. Second, a measuring device called a local, flat observer is defined in Euclidean space and it is shown, that this device uses a mixed basis when measuring distances. Combining these steps, it is concluded that a local, flat observer in a four-dimensional Euclidean spacetime measures a Minkowski spacetime.

The current work aims to develop an approximation of the slice of Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space $\mathbb{R}^3$ that, to the best of the author's knowledge, has not been addressed yet. This approximation of the actual slice is in a closed form of an explicit parametric equation in the case that the slice is not passing through those zones of the Minkowski surface with high curvatures, namely the "corners". At corners with high curvatures an alternative computational algorithm is introduced in which a family of ellipsoidal inner and outer bounds of the Minkowski's sum are used to construct a "narrow strip" for the actual slice of Minkowski sum. This strip can narrow persistently for a few more number of constructing bounds to precisely coincide on the actual slice of Minkowski sum. This algorithm is also applicable to the cases of slice of Minkowski sum of ellipsoids with high aspect ratio. In line with the main goal, some ellipsoidal inner and outer bounds of the Minkowski sum are reviewed, including the so called "Kurzhanski's" bounds. Also some ellipsoidal approximations are suggested for the Minkowski sum, which they can be used, as well as the inner and outer bounds, in calculation of the suggested parametric approximation of the slice of Minkowski sum.

We give an exact mathematical construction of a spacelike order $<$, which is dual to the standard chronological order $\ll$ in the $n$-dimensional Minkowski space $M^n$ and we discuss its order-theoretic, geometrical as well as its topological implications, conjecturing a possible extension to curved spacetimes.

We give a series of numerical examples of competitive evolution in the predation system, showing in some cases how the choice is made to increase the efficiency of the predation mechanism (or other significant parameters) to the detriment of populations (both of prey and predators). We then develop the mathematical theory that enables us to understand the causality involved, and identify a trend towards the emergence of the functional predation mechanism as such (and not of populations of the species involved). The realization of this trend only takes place when the conditions for it are offered by the hazards proposed to successive competitive choices. The logical structure of this trend is similar to that of the « tendency of rate of profit to fall » in certain economic models.

We here investigate the dynamical behavior of a fractional-order predator-prey system with anti-predator behavior and Holling IV type functional response. First we study the non-negativity, existence, uniqueness, and boundedness of solutions to the system from a mathematical analysis perspective. Then, we analyze the stability of its equilibrium points and the possibility of bifurcations using stability analysis methods and bifurcation theory, and prove the existence of a supercritical Hopf bifurcation in the system. After providing numerical simulations to illustrate the conclusions theoretically derived, by summarizing those various analytical results obtained, we finally present three interesting conclusions that can contribute to better understanding and preservation of ecological systems.

The complete-lattice approach to optimization problems with a vector- or even set-valued objective already produced a variety of new concepts and results and was successfully applied in finance, statistics and game theory. For example, the duality issue for multi-criteria and vector optimization problems could be solved using the complete-lattice approach, compare [11]. So far, it has been applied to set-valued dynamic risk measures (in the stochastic case), as discussed in Feinstein, Rudloff etc. (see [11], for example), but it has not been applied to deterministic calculus of variations and optimal control problems. In this paper, the following problem of set-valued optimization is considered: minimize the functional $$ \overline J_t[y]=\int_0^t \overline L(s,y(s),\dot y(s))\ ds + U_0(y(0)) $$ over all admissible arcs $y$, where $\overline L$ is the associated multifunction to a vector-valued Lagrangian $L$, the integral is in the Aumann sense and $U_0$ is the initial cost. A new concept of \emph{value function}, for which a Bellman's optimality principle holds, is introduced. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time $t$ and a directional derivative with respect to $x$ of the value function are defined, based on ideas close to the concepts in [12]. The value function is proved to be solution of a suitable Hamilton-Jacobi equation.

We propose a description of the electromagnetic field in the form of a four-component complex spinor, from which a vector of electromagnetic potential with two degrees of freedom, calibrated by two conditions - zero length and zero component along the y-axis - is obtained by using Pauli matrices. A similar approach is applied to the field of a fermion, in particular, the electron. It is known that the quantum field of the electron and the electron itself is a four-component complex spinor, so, existing in the Minkowski vector space, we cannot observe it directly. But with the help of Pauli matrices a vector is formed from the electron spinor, which is known to us as an electric current vector, and this current vector describes exactly a single particle. As a vector, it is available to us for observation in our vector space. Similarly, the electromagnetic field and its photon particle is also a four-component spinor, from which the universal formula using Pauli matrices produces a vector, it is known to us as the electromagnetic potential vector, and it too describes even a single photon. All the differences in the properties of the current vector and the electromagnetic potential vector, and hence the electron and the electromagnetic field, are due only to a slight difference in the structures of their four-component spinors and inextricable linked to them momentum spinors and coordinate spinors. Expressions for the electric and magnetic fields of a photon during its interaction with an electron.Thus, a unified way to describe bosons and fermions in spinor space is proposed. Each spinor using the same formula corresponds to a vector, in the case of a fermion it is a current vector, in the case of a boson it is a vector, for example, of the electromagnetic potential. Each spinor of a field is matched with a spinor of coordinates and a spinor of momentum, which are transformed by the same Lorentz transformations and which have the same structure as their corresponding field spinor, that is, the momentum and coordinates of boson have a bosonic spinor structure, while momentum and coordinates of fermion are a spinor with a fermionic structure. Field, coordinates and momentum vectors of boson automatically have a zero length, while in the case of fermion they all have a nonzero length, so the fermion, in contrast to the boson, has a nonzero mass, nonzero charge and moves with a sub light speed.While quantum mechanics treats probability as a real number, quantum field theory deals with probability as a four-dimensional real vector. The place of the probability amplitude, which in quantum mechanics is a complex number, in quantum field theory is taken by a complex spinor.In the same way as in quantum mechanics the processes of propagation and interaction are described at the level of complex probability amplitudes, and the final result is translated into a real probability value, so in quantum field theory all processes should be described in terms of complex spinor space, and the result is translated into the form of a real vector in Minkowski space.The presented approach, at its proper development, makes it possible to carry out calculations of the interaction of particles in two-dimensional spinor space, and to interpret in terms of the Minkowski vector space only the final results.

Starting from Thomas Friedrich’s work “Weak Spin(9) structures on 16-dimensional Riemannian manifolds”, we review several interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry, the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.

By generalizing the notion of pointwise 1-type Gauss map, the generalized 1-type Gauss map has been recently introduced. Without any assumption, we classified all possible ruled surfaces with generalized 1-type Gauss map in a 3-dimensional Minkowski space. In particular, null scrolls do not have the proper generalized 1-type Gauss map. In fact, it is harmonic.

We obtain some generalised Minkowski-type integral formulas for compact Riemannian (resp. spacelike) hypersurfaces in Riemannian (resp. Lorentzian) manifolds admitting an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has constant mean curvature.

A gravitational lens is mass-curved spacetime. Light that propagates freely, locally always on the fastest way, follows curved paths in it. The same applies to all effects propagating with the speed of light, also to gravity itself. They all have to follow the curvature of spacetime. A gravitational lens not only enhances the light from background objects but concurrently also their gravitational effect ! The "Gravitationally Lensed Gravity" (GLG) is a still scarcely noticed peculiar feature of general relativity. In [1] I presented a thought experiment to make the existence of this effect plausible. Here these considerations are described in terms of future and past light cones or null cones commonly used in relativity theory to illustrate the causal structure of spacetime. In the framework of general relativity, the GLG effect can be understood as a consequence of the folding of the past null cone by a gravitational lens. Starting from the basic ideas of the theory, the terms are introduced and applied. The extensive description on textbook level should be understandable also for beginners. In order not to obstruct the view on the essential physical relations, I avoid a mathematical formalism and use instead diagrams and figures, based on realistic numerical calculations, to reinforce my logical argumentation. Experts will recognize many long-known facts, but may still be surprised by the resulting conclusions. First, I present some particular aspects of the historical evolution of the concept of gravitational lensing and give a brief overview of the work of other authors on the subject GLG. I then explain the limitations of the applicability of the "Newtonian Limit" and move on to the key role of the speed of gravity in understanding GLG. Finally, I use a simple example to illustrate the possible significance of the GLG effect. With this work I intend to elicit greater interest in "Gravitationally Lensed Gravity" and to initiate a broader discussion about it.

The effects of photon inertia on the determination of its trajectory were verified and the representation of a displacement mass characterized by the flow of the number of wavefronts and the decomposition of photon inertia into parts associated with translation and rotation motions was considered. It was found that with the relativistic increase of the photon's resistance to change its directional properties, it inhibits the relativistic trajectory of the second torque, of Minkowski, in an angular range of incidence. After synchronizations, in the OAM inversions, there are reductions of the inertia associated to the translational part that assumes classical predominance, where the relativistic trajectory is allowed while the photon offers less resistance to changes in its directional properties. The classical-relativistic variability of the photon inertia characterizes the classical or relativistic profile of the energy distribution in forms of motion, where adjustments of the rotational and translational parts can be performed as a function of the refractive index rate, temperature and angle of incidence. It was found that with increasing temperature of the refringent medium, the synchronizations displacement in the sense of the normal incidence. A specific vacuum temperature for the refringent medium was characterized, where the photon exhibits a classical-relativistic synchronization under all angles of incidence, characteristic of its immaterial state in vacuum.