Ternary Sum Research Articles

Ternary plots of public interest in cultural human-nature interactions: Google Trends-based culturomics of the mediterranean olive triptych

Abstract The Mediterranean olive triptych –i.e. the conceptual integration of an ecosystem (the olive-grove), a product (olive-oil) and a nutrition/diet scheme (Mediterranean olive-based diet)– is a discernable cultural model in Europe. In this paper, we attempt to explore the potential and limitations of statistical analysis of compositional data –especially ternary plotting techniques– vs. typical representations of Google Trends-based Conservation Culturomics. In that perspective, typical representations of GT-data in published Culturomics literature, e.g. reference term/country/language/time, are considered as descriptors of variation of public interest (2D trend lines, 3D graphics and triangular projections). Ternary plots, i.e. a typical representation of compositional data, conventionally differ significantly from the previous representations, since the primary GT-data are weighted so that ternary sum equals to 1 (or 100%). We present the technique for such a calculation method. The comparison of these representation techniques allows for strongly diverging interpretations. Lower level interpretations show significant differences between (Mediterranean) countries and/or groups-of-countries (Mediterranean vs. non-Mediterranean) where higher ternary compilations do suggest that cultural pattern across Europe actually diverge. The overall conclusion of this research is that GT-based Culturomics should overcome earliest simplistic approaches, such as linear regressions; and, it has to progressively construct a new epistemology where the significance of accumulation of empirical evidence should be strictly connected to scientific paradigm shift.

Троичные коды с суммированием и их модификации

Троичные коды с суммированием и их модификации

Fixed Points for a Pair of F-Dominated Contractive Mappings in Rectangular b-Metric Spaces with Graph

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results.

Ternary universal sums of generalized polygonal numbers

An integer of the form [Formula: see text], for some integer [Formula: see text], is called a generalized polygonal number of order [Formula: see text]. A ternary sum [Formula: see text] of generalized polygonal numbers, for some positive integers [Formula: see text] and some integers [Formula: see text], is said to be universal over [Formula: see text] if for any nonnegative integer [Formula: see text], the equation [Formula: see text] has an integer solution [Formula: see text]. In this paper, we prove the universalities of [Formula: see text] ternary sums of generalized polygonal numbers, which was conjectured by Sun.

Almost universal ternary sums of polygonal numbers

For a natural number m, generalized m-gonal numbers are those numbers of the form $$p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$$ with $$x\in \mathbb Z$$ . In this paper we establish conditions on m for which the ternary sum $$p_m(x)+p_m(y)+p_m(z)$$ is almost universal.

Sums of multiplicative characters with additive convolutions

We obtain new estimates for binary and ternary sums of multiplicative characters with additive convolutions of characteristic functions of sets with small additive doubling. In particular, we improve a result of Mei-Chu Chang. The proof uses the Croot–Sisask almost periodicity lemma.

Ternary cyclic redundancy check by a new hardware-friendly ternary operator

This paper presents a new ternary operator for cyclic redundancy check in ternary logic with high hardware efficiency. It shows the essential properties of a ternary operator for calculating CRC. Gate-level implementation of CRC-16 is exemplified and comprehensively explained in this paper. Transistor-level realizations of the new and the previous operators are also given in full detail. This paper demonstrates that the employment of the proposed operator reduces hardware components such as the elimination of one entire operator in both transmitter and receiver sides in comparison with ternary SUM. In addition, simulation results by HSPICE and 32nm CNTFET technology shows about 20.6% energy reduction for one of the implementation methods of the proposed operator with respect to the previously presented one (CCW CYCLE). Furthermore, five fewer transistors are needed to design the proposed operator in contrast with both operators SUM and CCW.

The partially alternating ternary sum in an associative dialgebra

The alternating ternary sum in an associative algebra, gives rise to the partially alternating ternary sum in an associative dialgebra with products ⊣ and ⊢ by making the argument a the center of each term: We use computer algebra to determine the polynomial identities in degree ⩽9 satisfied by this new trilinear operation. In degrees 3 and 5, we obtain these identities define a new variety of partially alternating ternary algebras. We show that there is a 49-dimensional space of multilinear identities in degree 7, and we find equivalent nonlinear identities. We use the representation theory of the symmetric group to show that there are no new identities in degree 9.

Polynomial identities for the ternary cyclic sum

We simplify the results of Bremner and Hentzel [J. Algebra 231 (2000) 387–405] on polynomial identities of degree 9 in two variables satisfied by the ternary cyclic sum [a, b, c] = abc + bca + cab in every totally associative ternary algebra. We also obtain new identities of degree 9 in three variables which do not follow from the identities in two variables. Our results depend on (i) the LLL algorithm for lattice basis reduction, and (ii) linearization operators in the group algebra of the symmetric group which permit efficient computation of the representation matrices for a non-linear identity. Our computational methods can be applied to polynomial identities for other algebraic structures.

Almost universal ternary sums of triangular numbers

For any integer x, let T x denote the triangular number x(x+1)/2. In this paper we give a complete characterization of all the triples of positive integers (α, β, γ) for which the ternary sums αT x + βT y +γT z represent all but finitely many positive integers, which resolves a conjecture of Kane and Sun.

Some Applications of a Theorem of M. Kneser.

A Theorem of M. Kneser relating densities of sumsets to densities of their summands (see [H. Halberstam and K. F. Roth, "Sequences," Chap. 1. Oxford Univ. Press, Oxford], 1966]) is used to set up a machine for settling questions regarding the generalized form of Waring′s problem in additive number theory. For example, if a strictly increasing sequence of integers represents all sufficiently large integers by ternary sums, and an element is removed from the sequence such that the terms remaining represent all sufficiently large integers by sums of some fixed number of terms, then that fixed number is at most seven and this result is best possible.