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In this paper, given an arbitrary set Ω, we study the main order and algebraic properties of some maps and set structures that are strictly related to dependence set relations on Ω, which are binary relations between subsets of Ω naturally arising when Ω is a topological space or an attribute set in rough set theory and granular computing based on information systems. The previous maps, that we call granular maps, have the families of the set systems, set operators, binary set relations or also of information systems on the ground set Ω as their domain and codomain. We make use of various algebraic methodologies on granular maps to determine the main order-theoretic and combinatorial properties of specific sub-collections of set systems, binary set relations and set operators naturally arising in the investigation of dependence set relations and of rough set theory. We introduce, in more detail, the notion of granular sub-bijection to formalize in all these situations the undefined notion of cryptomorphism, and through which we exhibit new equivalences between specific families of set systems, binary set relations and set operators strictly related to dependence set relations. By means of suitable granular maps we determine three granular sub-bijections between the family of all the closure operators, that of all the Moore set systems and that of all dependence set relations on the same ground set Ω. Next, through a property of adjunctivity, we see that in order to generate a dependence set relation it suffices to consider pointed relations on Ω, namely collections of pairs in Ω× ℘(Ω). Because of that, we study order-theoretical properties of some relevant subclasses of pointed relations and analyze the granular maps on Ω which determine two nontrivial granular sub-bijections between two subclasses of set operators and two corresponding subclasses of pointed relations. Next, we show that any dependence set relation has the form Dep픓 that is a dependence set relation induced by an information system 픓 on Ω and generalizes the Pawlak dependence set relation frequently used in rough set theory. With regard to this representation result, we characterize some set systems of minimal subsets with respect to the Pawlak indiscernibility relation on information systems. Finally, given an arbitrary binary set relation 풟 on Ω, we consider the smallest dependence set relation 풟+ on Ω containing 풟 and call it dependence closure of 풟. Then, when Ω is a finite set, we show how to generate 풟+ in four different and recursive ways by starting from 풟. Moreover, again in the finite case, given an information system 픓 on Ω, we also determine a binary set relation ℒ픓 on Ω for which ℒ픓+ agrees with Dep픓 and whose cardinality is minimum with respect to that of all binary set relations whose dependence closure agrees with Dep픓. [ABSTRACT FROM AUTHOR]

In this paper, the ordinal sum of quantales is provided. Based on this, the ordinal sums of many familiar ordered algebraic structures can be obtained, such as complete residuated lattice, spatial quantale, etc. Then we investigate the relationship between the quantale completion of partially ordered semigroups and the ordinal sum, and prove that for any family of two-side posemigroups, the ordinal sum of their quantale completions coincides with the quantale completion of their ordinal sum. [ABSTRACT FROM AUTHOR]

In order to deal with uncertainty, Molodtsov propoed soft set theory as a mathematical tool in 1999. Since that time, numerous forms of soft set operations have been defined and employed. By establishing the soft binary piecewise operations' distributions over complementary soft binary piecewise intersection and union operations, this study aims to contribute to the literature on soft set theory by giving inspiration to researchers as regards examining and obtaining some algebraic structures using these new soft set operations. [ABSTRACT FROM AUTHOR]

We investigate the thermodynamic stability of the (2+1)-dimensional β² = 0 ACGL Cern-Simons black hole solution of the topologically massive gravitoelectrodynamics. We show that the system is globally unstable against thermal fluctuations, but still able to retain some local regions of stability with respect to certain processes. As a consequence we are able to define an optimal Penrose process and study its properties via the concept of thermodynamic length. We estimate the thermodynamic time and speed of the process with respect to several Hessian metrics on the space of macro states. [ABSTRACT FROM AUTHOR]

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles. [ABSTRACT FROM AUTHOR]

In [4] stable structures had been described by SL-blocks. These blocks define a mathematical language with an extraordinary advantage of self-interlocking for some of the variations. In this manuscript, we give an extension of SL blocks to different fundamental surfaces (cylinder, torus, Möbius strip) which are obtained by deformed SL blocks (based on Differential Geometry). We believe that this extension can help make this idea applicable to architects, puzzle makers and more. [ABSTRACT FROM AUTHOR]

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams' Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional φ → ψ to the extent that (i) the conditional probability p (ψ | φ) is high, provided that (ii) φ is relevant for ψ . How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. [ABSTRACT FROM AUTHOR]

The importance of the theory of neutrosophy is due to its connections with several fields of sciences, engineering, and technology. So, the aim of this paper is to relate neutrosophy with some algebraic structures mainly the ordered semigroups. In this regard, we define single valued neutrosophic sets in ordered semigroups. More precisely, we study single valued neutrosophic ideals of ordered semigroups and single valued neutrosophic bi-ideals of ordered semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the periodicity problems in the finitely generated algebraic structures and exhibit their natural sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the imposing edifice of group theory. One of the deepest and important results of the related theory of finite groups is a complete classification of all periodic groups, that is, the finite groups with periodic properties. For every integer k ≥ 2 and a k-generated non-associative algebraic structure S =< A >, where A = fa1; a2;: ::; akg, the sequence...is called the k-nacci sequence of S with respect to the generating set A, denoted kA(S). If kA(S) is periodic, we call the length of the period of the sequence the periodicity length of S with respect to the generating set A, written LENA(S) and the minimum of the positive integers of LENA(S) will be mentioned as periodicity invariant of S, denoted by k(S). However, this invariant has been studied for groups and semi-groups during the years as well as the associative property of S where above sequence was reduced to xi = xi-kxi-k+1: :: xi-3xi-2xi-1, for every i ≥ k+1. Thus, we attempt to give explicit upper bounds for the periodicity invariant of two infinite classes of finite non-associative 3-generated algebraic structures. Moreover, two classes of non-isomorphic Moufang loops of the same periodicity length were obtained in the study. [ABSTRACT FROM AUTHOR]

The article discusses real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. It mentions Deduction Theorem is an another result from mathematical logic; and setting of sentential logic to larger classes of algebraic structures. It mentions about compound formulas built up recursively from atoms with propositional connectives.

We present a theoretical and experimental study of electrodynamics of a planar spiral superconducting resonator of a finite length. The resonator is made in the form of a monofilar Archimedean spiral. By making use of a general model of inhomogeneous alternating current flowing along the resonator and specific boundary conditions on the surface of the strip, we obtain analytically the frequencies fn of resonances which can be excited in such system. We also calculate corresponding inhomogeneous RF current distributions ψn(r) , where r is the coordinate across a spiral. We show that the resonant frequencies and current distributions are well described by simple relationships fn = f1n and ψn(r) ≃ ?sin[πn(r/Re)2] , where n = 1, 2... and Re is the external radius of the spiral. Our analysis of electrodynamic properties of spiral resonators' is in good agreement with direct numerical simulations and measurements made using specifically designed magnetic probe and laser scanning microscope. [ABSTRACT FROM AUTHOR]

Although a prototypical Su–Schrieffer–Heeger (SSH) soliton exhibits various important topological concepts including particle-antiparticle (PA) symmetry and fractional fermion charges, there have been only few advances in exploring such properties of topological solitons beyond the SSH model. Here, by considering a chirally extended double-Peierls-chain model, we demonstrate novel PA duality and fractional charge e/2 of topological chiral solitons even under the chiral symmetry breaking. This provides a counterexample to the belief that chiral symmetry is necessary for such PA relation and fractionalization of topological solitons in a time-reversal invariant topological system. Furthermore, we discover that topological chiral solitons are re-fractionalized into two subsolitons which also satisfy the PA duality. As a result, such dualities and fractionalizations support the topological Z 4 algebraic structures. Our findings will inspire researches seeking feasible and promising topological systems, which may lead to new practical applications such as solitronics. [ABSTRACT FROM AUTHOR]

In many previous papers, the authors used an algebra endomorphism to twist the original algebraic structures in order to produce the corresponding Hom-algebraic structures. In this work, we use a bijective linear map, an element of centroid, an averaging operator, a Rota-Baxter operator or a multiplier to produce a Hom-Poisson color algebra from a given one. [ABSTRACT FROM AUTHOR]

In this paper, a one-to-one correspondence between Bessel sequences and bounded linear operators is provided. This leads to an algebra structure on the set of all Bessel sequences in a separable Hilbert space. Some kinds of frames as special classes of operators are considered. Also, normal Bessel sequences and positive frames are presented. Finally, power means of positive frames are introduced. These allow researches to construct a large number of new frames from existing frames. [ABSTRACT FROM AUTHOR]

In 1988, given a complete Brouwerian lattice L : = (L ; ∧ , ∨ ; 0 , 1) and a ring A : = (A ; + , · ; - ; 0) with unity 1, Swamy and Swamy (J Math Anal Appl 134:94–103, 1988) built a lattice structure, on the set of L-fuzzy ideals of A , and investigated some of its arithmetic properties. Since the residuation theory is richer than the lattice theory [see, Ciungu (Non-commutative multiple-valued logic algebras, Springer monographs in mathematics, Springer, Berlin, 2014), Galatos et al. (An algebraic glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics, Elsevier, Amsterdam, 2007), Jipsen and Tsinakis (in: Martinez (ed) Ordered algebraic structures, Kluwer Academic Publisher, Dordrecht, 2002), Piciu (Algebras of fuzzy logic, Editura Universitaria Craiova, Craiova, 2007)], in this paper, we consider the notion of fuzzy ideals rather under a complete Brouwerian residuated lattice L : = (L ; ∧ , ∨ , ⊖ , ↠ , ⊸ ; 0 , 1) . A residuated lattice F i d (A , L) : = (F i d (A , L) ; ∧ , + , ⊗ , ↪ , ↬ ; χ 0 , 1 ̲) is built on the set F i d (A , L) of L-fuzzy ideals of A and it is shown that the latter is both an extension of L and the residuated lattice I d (A) : = (I d (A) ; ∩ , + , ⊙ , → , ⇝ ; { 0 } , A) on the set I d (A) of ideals of A . [ABSTRACT FROM AUTHOR]

The notions of subalgebra-type hyper value and ideal-type hyper value are introduced, and related properties are investigated. The relation between subalgebra-type hyper value and ideal-type hyper value is considered. Conditions for a pair (α; β) in [0, 1] × [0, 1] to be subalgebra-type hyper value and ideal-type hyper value are discussed. For a hyperfuzzy structure, conditions for its level sets to be S-energetic, I-energetic, right vanished and right stable are founded. [ABSTRACT FROM AUTHOR]

This paper informs about the publication of the second edition of the Integrative Levels Classification (ILC2), a freely-faceted knowledge organization system (KOS), and reviews the main changes that have been introduced as compared to its first edition (ILC1). The most relevant changes are illustrated, with special reference to those of interest to general classification theory, by means of examples of notation for individual classes and combinations of them. Changes introduced in ILC2 include: the names and order of some main classes; the development of subclasses for various phenomena, especially quantities and algebraic structures; the order of facet categories and the new category of Disorder; notation for special facets; distinction of the semantical function of facets (attributes) from their syntactic function. The system can be freely accessed online through a PHP browser as well as in SKOS format. Only a selection of changed classes is discussed for space reasons. ILC1 has been previously applied to the BARTOC directory of KOSs. Update of BARTOC data to ILC2 and application of ILC2 to further information systems are envisaged. Possible methods for reclassifying BARTOC with ILC2 are discussed. ILC is a newly developed classification system, based on phenomena instead of traditional disciplines and featuring various innovative devices. This paper is an original account of its most recent evolution. [ABSTRACT FROM AUTHOR]

We give a brief overview how to couple general relativity to the Standard Model of elementary particles, within the higher gauge theory framework, suitable for the spinfoam quantization procedure. We begin by providing a short review of all relevant mathematical concepts, most notably the idea of a categorical ladder, 3-groups and generalized parallel transport. Then, we give an explicit construction of the algebraic structure which describes the full Standard Model coupled to Einstein-Cartan gravity, along with the classical action, written in the form suitable for the spinfoam quantization procedure. We emphasize the usefulness of the 3-group concept as a superior tool to describe gauge symmetry, compared to an ordinary Lie group, as well as the possibility to employ this new structure to classify matter fields and study their spectrum, including the origin of fermion families. [ABSTRACT FROM AUTHOR]

The notion of a double-framed soft (normal) subalgebra in a B-algebra is introduced and related properties are investigated. We consider characterizations of a double-framed soft (normal) subalgebra and establish a new double-framed soft subalgebra from old one. Also, we show that the int-uni double-framed soft of two double framed soft subalgebras is a double framed soft subalgebra. [ABSTRACT FROM AUTHOR]

Let G be a finite group. In this paper, we study the structure of finite groups having |G| - r cyclic subgroups for 3 < r < 5. [ABSTRACT FROM AUTHOR]

After the establishment of the finite embeddability property for integral and commutative residuated ordered monoids by Blok and Van Alten in 2002, the finite embeddability property for some other types of residuated ordered algebraic structures have been extensively studied. The main purpose of this paper is to construct a finite quantale X̂F from a finite partial subalgebra F of an increasing quantum B-algebra X so that F can be embedded into X̂F, that is, the class of increasing quantum B-algebras has the finite embeddability property. [ABSTRACT FROM AUTHOR]

In this paper we define the Smarandache hoop-algebras and Q-Smarandache filters, we obtain some related results. After that, by considering the notions of these filters we determine relationships between filters in hoop-algebras and Q-Smarandache filters in hoop- algebras. Finally, we introduce the concept of Smarandache 2-structure and Smarandache 2-filter on hoop-algebras. [ABSTRACT FROM AUTHOR]

Ordered algebraic structures are examined within the framework of Bishop-style constructive mathematics. In the construc- tive approach, the partial order is replaced by the classically equiva- lent, but constructively stronger, notion of co-order. While one could define an ordered algebraic structure by requiring certain properties of monotonicity of the algebraic operations, the constructive counterpart of strong monotonicity could be more appropriate for a constructive examination. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a non-commutative hypervaluation as a generalization of a non-commutative valuation is introduced and some results are proved in this respect. Also, we define the notion of a non-commutative hypervaluation over a Krasner hyperfield and some basic results and characterizations are obtained. Finally, we introduce the non-commutative discrete hypervaluation of a hyperfield and investigate the important properties. [ABSTRACT FROM AUTHOR]

In this note the simple conditions are provided to produce directed partial orders on a unital algebra A over a non-archimedean totally ordered field F to make A into a partially ordered algebra over F with a directed partial order. [ABSTRACT FROM AUTHOR]

We propose a unifying approach to the problem of measuring the inconsistency of judgments. More precisely, we define a general framework to allow several well-known inconsistency indices to be expressed as special cases of this new formulation. We consider inconsistency indices as aggregations of 'local', i.e. triple-based, inconsistencies. We show that few reasonable assumptions guarantee a set of good properties for the obtained general inconsistency index. Under this representation, we prove a property of Pareto efficiency and show that OWA functions and t-conorms are suitable aggregation functions of local inconsistencies. We argue that the flexibility of this proposal allows tuning of the index. For example, by using different types of OWA functions, the analyst can obtain the desired balance between an averaging behavior and a 'largest inconsistency-focused' behavior. [ABSTRACT FROM AUTHOR]

Given two cyclic A∞-algebras A and B, in this paper we prove that there exists a cyclic A∞-algebra structure on their tensor product A ⊗ B which is unique up to a cyclic A∞-quasi-isomorphism. Furthermore, the Kontsevich class of A ⊗ B is equal to the cup product of the Kontsevich classes of A and B on the moduli space of curves. [ABSTRACT FROM AUTHOR]

Abstract Working in the context of symmetric spectra, we consider algebraic structures that can be described as algebras over an operad O. Topological Quillen homology, or TQ -homology, is the spectral algebra analog of Quillen homology and stabilization. Working in a framework for homotopical descent, the associated TQ -homology completion map appears as the unit map of an adjunction comparing the homotopy categories of O -algebra spectra with coalgebra spectra over the associated comonad via TQ -homology. We prove that this adjunction between homotopy categories can be turned into an equivalence by replacing O -algebras with the full subcategory of 0-connected O -algebras. We also construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ -homology groups TQ ⁎ (X) of an O -algebra X , and prove that it converges strongly to π ⁎ (X) when X is 0-connected. [ABSTRACT FROM AUTHOR]

Using the notion of generalized fuzzy sets, we introduce the notions of generalized fuzzy hyperideals, generalized fuzzy bi-hyperideals, and generalized fuzzy normal bi-hyperideals in an ordered nonassociative and non-commutative algebraic structure, namely an ordered LA-semihypergroup, and we characterize these hyperideals. We provide some results related to the images and preimages of generalized fuzzy hyperideals in ordered LA-semihypergroups. [ABSTRACT FROM AUTHOR]

Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$ -ary optimal locally repairable codes from subcodes of Reed-Solomon codes given by Tamo and Barg have length upper bounded by $q$. Recently, it was shown through extension of construction by Tamo and Barg that length of $q$ -ary optimal locally repairable codes can be $q+1$ by Jin et al.. Surprisingly it was shown by Barg et al. that, unlike classical MDS codes, $q$ -ary optimal locally repairable codes could have length bigger than $q+1$. Thus, it becomes an interesting and challenging problem to construct $q$ -ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we make use of rich algebraic structures of elliptic curves to construct a family of $q$ -ary optimal locally repairable codes of length up to $q+2\sqrt {q}$. It turns out that locality of our codes can be as big as 23 and distance can be linear in length. [ABSTRACT FROM AUTHOR]

In this paper we extend the notion of classical (pre-)semiadditive category to (pre-)semihyperadditive category. Algebraic hyperstructures are algebraic systems whose objects possessing the hyperoperations or multi-valued operation. We introduce categories in which for objects A and B, the class of all morphisms from A to B denoted by Mor(A,B), admits an algebraic hyperstructures, such as semihyper-group or hypergroup. In this regards we introduce the various types of pre-semihyperadditive categories. Also, we construct some (pre-)semi-hyperadditive categories by introducing a class of hypermodules named general Krasner hypermodules. Finally, we investigate some properties of these categories. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of ordered Krasner hyperrings. Also we state some definitions and basic facts and prove some results on ordered Krasner hyperring (R,+,.,≥). In particular, we introduce the concepts of prime hyperideals and semiprime hyperideals of an ordered Krasner hyperring and present several examples of them. [ABSTRACT FROM AUTHOR]

We construct a nontrivial definable type V field topology on any dp-minimal field K that is not strongly minimal, and prove that definable subsets of K n have small boundary. Using this topology and its properties, we show that in any dp-minimal field K , dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and | K × / (K ×) n | is finite for all n ≥ 0. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)]. [ABSTRACT FROM AUTHOR]

We give a generating set of the generalized Reidemeister moves for oriented singular links. We then introduce an algebraic structure arising from the axiomatization of Reidemeister moves on oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links. [ABSTRACT FROM AUTHOR]

In this paper, for any non-empty subset A of a BCI-algebra X, we introduce the concept of p-closure of A, denoted by Apc, and investigate some related properties. Applying this concept, we characterize the minimal elements of BCI-algebras. We also give a characterization of the p-closure of subalgebras of X by some branches of X. We state a necessary and sufficient condition for a BCI-algebra (1) to be p-semisimple; (2) to be a BCK-algebra. Moreover, we show that the p-closure can be used to define a closure operator. We investigate the relationship between f(Apc) and (f(A))pc for a BCI-homomorphism f. Finally, we prove that Apc is the least closed p-ideal containing A for any ideal A of X. [ABSTRACT FROM AUTHOR]

In Panaite [Iterated crossed products, J. Algebra Appl. 13(7), 14580036 (2014)], Panaite studied iterated crossed product construction from the point of algebraic structures. In this paper, we study iterated crossed product from the point of Combinatorial Group Theory and define a new version of the crossed product of groups. First, we give some conditions for this new product to be a group, then we obtain a presentation for iterated crossed product of cyclic groups. Additionally, using this presentation, we find a complete rewriting system and thus we obtain normal form structure of elements of this new group construction. This gives us the solvability of the word problem for this product. [ABSTRACT FROM AUTHOR]

The sixteen generators of Abelian integral I(h)=∮Γhg(x,y)dx-f(x,y)dy, which satisfy eight different Picard-Fuchs equations respectively, are obtained, where Γh is a family of closed orbits defined by H(x,y)=ax4+by4+cx8=h, h∈Σ, Σ is the open intervals on which Γh is defined, and f(x, y) and g(x, y) are real polynomials in x and y of degree n. Moreover, an upper bound of the number of zeros of I(h) is obtained for a special case f(x,y)=∑0≤i≤4k+1=naix4k+1-iyi,g(x,y)=∑0≤i≤4k+1=nbix4k+1-iyi. [ABSTRACT FROM AUTHOR]

We come up with a semantic method of forcing formulas by finite structures in an arbitrary fixed Fraïssé class . Both known and some new necessary and sufficient conditions are derived under which a given structure will be a forcing structure. A formula φ is forced on a¯ in an infinite structure ╟φa¯ if it is forced in by some finite substructure of . It is proved that every ∃∀∃-sentence true in a forcing structure is also true in any existentially closed companion of the structure. The new concept of a forcing type plays an important role in studying forcing models. It is proved that an arbitrary structure will be a forcing structure iff all existential types realized in the structure are forcing types. It turns out that an existentially closed structure which is simple over a tuple realizing a forcing type will itself be a forcing structure. Moreover, every forcing type is realized in an existentially closed structure that is a model of a complete theory of its forcing companion. [ABSTRACT FROM AUTHOR]

The concepts of a positive implicative BMBJ-neutrosophic ideal is introduced, and several properties are investigated. Conditions for an MBJ-neutrosophic set to be a (positive implicative) BMBJ-neutrosophic ideal are provided. Relations between BMBJ-neutrosophic ideal and positive implicative BMBJ-neutrosophic ideal are discussed. Characterizations of positive implicative BMBJ-neutrosophic ideal are displayed. [ABSTRACT FROM AUTHOR]

The concept of neutrosophy and indeterminacy I was introduced by Smarandache, to deal with neutralies. Since then the notions of neutrosophic rings, neutrosophic semigroups and other algebraic structures have been developed. Neutrosophic duplets and their properties were introduced by Florentin and other researchers have pursued this study.In this paper authors determine the neutrosophic duplets in neutrosophic rings of characteristic zero. The neutrosophic duplets of hZ , and the neutrosophic ring of integers, neutrosophic ring of rationals and neutrosophic ring of reals respectively have been analysed. It is proved the collection of neutrosophic duplets happens to be infinite in number in these neutrosophic rings. Further the collection enjoys a nice algebraic structure like a neutrosophic subring, in case of the duplets collection {a-aI|a∊Z} for which 1-I acts as the neutral. For the other type of neutrosophic duplet pairs {a-aI,1-dI} where a ∊R+ and d ∊ R, this collection under component wise multiplication forms a neutrosophic semigroup. Several other interesting algebraic properties enjoyed by them are obtained in this paper. [ABSTRACT FROM AUTHOR]

We propose a way to understand the three fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e. the emergence of three fermion generations and the mass relationships among those generations. We define the coordinate fiber space of the bundle of the manifold as the space in which the classical noncommutative geometry is expressed, then the tensor product extension explicitly shows the contribution of structures in the non-coordinate base space of the bundle to the action. The quaternion extension plays an essential role to reveal the physical effect of the structure in the non-coordinate base space. [ABSTRACT FROM AUTHOR]

A class of complex-valued convolutional codes, wavelet codes, that encode numbers into numbers, is constructed by assigning consecutively indexed discrete Fourier transform (DFT) vectors to the parity-checking requirements. For a rate $k/n$ code with constraint length $m$ , the parameters must satisfy (m + 1)(n-k)

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue's paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837]. [ABSTRACT FROM AUTHOR]

First, we extend the notion of stratified spaces to diffeology. Then we characterise the subspace of stratified differential forms, or zero-perverse forms in the sense of Goresky–MacPherson, which can be extended smoothly into differential forms on the whole space. For that we introduce an index which outlines the behaviour of the perverse forms on the neighbourhood of the singular strata. [ABSTRACT FROM AUTHOR]