v-Regular Ternary Menger Algebras and Left Translations of Ternary Menger Algebras

Let n be a fixed natural number. Ternary Menger algebras of rank n, which was established by the authors, can be regarded as a suitable generalization of ternary semigroups. In this article, we introduce the notion of v-regular ternary Menger algebras of rank n, which can be considered as a generalization of regular ternary semigroups. Moreover, we investigate some of its interesting properties. Based on the concept of n-place functions (n-ary operations), these lead us to construct ternary Menger algebras of rank n of all full n-place functions. Finally, we study a special class of full n-place functions, the so-called left translations. In particular, we investigate a relationship between the concept of full n-place functions and left translations

General Fractional Integrals and Derivatives of Arbitrary Order

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

Depth Contours and Coastline Generalization for Harbour and Approach Nautical Charts

Generalization of nautical charts and electronic nautical charts (ENCs) is a critical process which aims at the safety of navigation and clear cartographic presentation. This paper elaborates on the problem of depth contours and coastline generalization—natural and artificial—for medium-scale charts (harbour and approach) taking into account International Hydrographic Organization (IHO) standards, hydrographic offices’ (HOs) best practices and cartographic literature. Additional factors considered are scale, depth, and seafloor characteristics. The proposed method for depth contour generalization utilizes contours created from high-resolution digital elevation models (DEMs) or those already portrayed on nautical charts. Moreover, it ensures consistency with generalized soundings. Regarding natural coastline generalization, the focus was on managing the resolution, while maintaining the shape, and on the islands. For the provision of a suitable generalization solution for the artificial shoreline, it was preprocessed in order to automatically recognize the shape of each structure as perceived by humans (e.g., a pier that looks like a T). The proposed generalization methodology is implemented with custom-developed routines utilizing standard geo-processing functions available in a geographic information system (GIS) environment and thus can be adopted by hydrographic agencies to support their ENC and nautical chart production. The methodology has been tested in the New York Lower Bay area in the U.S.A. Results have successfully delineated depth contours and coastline at scales 1:10 K, 1:20 K, 1:40 K and 1:80 K.

Ternary Menger Algebras: A Generalization of Ternary Semigroups

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented.

Cyclic Mechanics: the Principle of Cyclicity

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows:1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist.2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and especially that of conservation of energy: This is the conservation of action or information.3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete.4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole.5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality.6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice.The list is not intended to be exhausted or disjunctive, but only to give an introductory idea.

FRIEZE PATTERNS WITH COEFFICIENTS

Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.

DECIDABILITY OF THE THEORY OF MODULES OVER PRÜFER DOMAINS WITH INFINITE RESIDUE FIELDS

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary.

Hot-electron relaxation in metals within the Götze–Wölfle memory function formalism

We consider nonequilibrium relaxation of electrons due to their coupling with phonons in a simple metal. In our model, electrons are living at a higher temperature than that of the phonon bath, mimicking a nonequilibrium steady-state situation. We study the relaxation of such hot electrons proposing a suitable generalization of the memory function formalism formulated by Götze and Wölfle (GW) [W. Götze and P. Wölfle, Phys. Rev. B 6, 1226 (1972)]. We derive analytical expressions for both the DC or zero frequency scattering rates and the optical scattering rates in various temperature and frequency regimes. Limiting cases are in accord with the previous studies. An interesting feature that the DC scattering rate at high temperatures and optical scattering rate at high frequencies are independent of the temperature difference between the electrons and the phonons is found in this study. The present formalism forms a basis which can also be extended to study hot-electron relaxation in variety of complex materials.