Triangular Numbers Research Articles

Construction of Fuzzy Numbers via Cumulative Distribution Function

The first person to introduce possibility theory was Lotfi A. Zadeh, in 1977. It was, of course, of no coincidence that he directly combined it with the theory of fuzzy sets. Later, several researchers dealt with the mathematical foundations of the theory of possibilities. They introduced possibility distribution as a concept, and they directly combined it with fuzzy numbers. A fuzzy number corresponds to a possibility distribution and vice versa. This correspondence gave a key advantage to possibility theory over probability theory. This advantage is the facility of operations. However, there is also a basic: problem how is a possibility distribution generated? In this paper, we introduce a method of constructing a possibility distribution via a cumulative probability function. The advantage of this method is the simplicity of construction, which is nothing more than the construction of a fuzzy triangular or trapezoidal number via a cumulative probability function. This construction introduces a way to determine a fuzzy number without relying on the experience or intuition of the researcher. We should, of course, emphasize that this specific construction is within the framework of a theoretical model. We do not apply it to specific data. We also considered that the theoretical construction model should be presented through the theory of possibilities, thus avoiding the theory of probabilities.

Geometric properties of evolutionary graph states and their detection on a quantum computer

Geometric properties of evolutionary graph states of spin systems generated by the operator of evolution with Ising Hamiltonian are examined, using their relationship with fluctuations of energy. We find that the geometric characteristics of the graph states depend on properties of the corresponding graphs. Namely, it is obtained that the fluctuations of energy in graph states and therefore the velocity of quantum evolution, the curvature, and the torsion of the states are related to the total number of edges, triangles, and squares in the corresponding graphs. The obtained results give a possibility to quantify the number of edges, triangles, and squares in a graph on a quantum devise and achieve quantum supremacy in solving this problem with the development of a multi-qubit quantum computer. Geometric characteristics of graph states corresponding to a chain, a triangle, and a square are detected on the basis of calculations on IBM's quantum computer ibmq_manila.

Locally common graphs.

Goodman proved that the sum of the number of triangles in a graph on nodes and its complement is at least ; in other words, this sum is minimized, asymptotically, by a random graph with edge density 1/2. Erdős conjectured that a similar inequality will hold for in place of , but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called common graphs. Characterization of common graphs seems, however, out of reach. Franek and Rödl proved that is common in a weaker, local sense. Using the language of graph limits, we study two versions of locally common graphs. We sharpen a result of Jagger, Štovíček and Thomason by showing that no graph containing can be locally common, but prove that all such graphs are weakly locally common. We also show that not all connected graphs are weakly locally common.

Edge-disjoint rainbow triangles in edge-colored graphs

Let G be an edge-colored graph. A triangle of G is called rainbow if any two edges of the triangle have distinct colors. We use m ( G ) and c ( G ) to denote the number of edges of G and the number of colors appearing on the edges of G , respectively. Li et al. in 2014 proved that every edge-colored graph of order n with m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 contains a rainbow triangle and this result is best possible. In 2019, Fujita et al. characterized all graphs G satisfying m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 − 1 but containing no rainbow triangle. In this paper, we conjecture that every edge-colored graph of order n with m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 + 3 ( k − 1 ) contains k edge-disjoint rainbow triangles. We show that the conjecture holds for k = 2 and 3 and these results are best possible. Furthermore, we characterize all graphs G satisfying m ( G ) + c ( G ) ≥ n ( n + 1 ) / 2 + 2 but not containing two edge-disjoint rainbow triangles. At the end, we propose a conjecture on the number of vertex-disjoint rainbow triangles in an edge-colored graph.

Double nested Hilbert schemes and the local stable pairs theory of curves

We propose a variation of the classical Hilbert scheme of points, thedouble nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss$K$-theoretic refinements à la Nekrasov–Okounkov.

An Identity Relating Triangular and Pentagonal Numbers

Summary We give a visual proof for an identity relating triangular and pentagonal numbers.

Dark solitons impinging on interfaces in a superfluid Fermi gas

Ultracold quantum gases in the superfluid regime exhibit solitons, localized excitations that require nonlinearity of the underlying field equation in order to preserve their shape as they propagate. Here, we investigate the behavior of solitons at an inhomogeneity: an interface that separates two different interaction regimes of a superfluid Fermi gas. It is known that the soliton properties depend on the interaction regime, but what happens as a soliton impinges on such an interface is not clear. Using an effective field theory to describe the superfluid Fermi gas, we reveal the nontrivial dynamics of such a collision. Whether the original soliton makes it through the interface depends on the amplitude of the soliton. Regardless of whether the original soliton is transmitted or not, there will always be a shock wave with a phonon train created behind the interface and reflected secondary solitons. The details of this dynamics depends strongly on the equation of state corresponding to underlying microscopic theory describing the superfluid Fermi gas, and we argue that these collisions are realistic experimental probes to test microscopic theories of pairing in ultracold Fermi gases.

OBSTACLES TO SUSTAINABLE SECURITY IN THE KURDISTAN REGION USING THE FUZZY DELPHI

This study aimed to investigate the obstacles to achieving sustainable security in the Kurdistan region. The statistical analysis of data was performed through EXCEL software. Based on the studies' content analysis, nine obstacles to the Kurdistan region's sustainable security were finally identified to enter Delphi. The fuzzy Delphi method was used to screen and ensure the importance of the identified indicators and select the final indicators. Experts' opinions were used to assess the importance of the indicators. In this study, fuzzy triangular numbers were used to fuzzy experts' views. The first-round results showed that all indicators gained more than 7, and as a result, remained in the evaluation. Based on the second-round results, it was found that the difference of all measures was less than 0.5, and thus Delphi was ended. The DEMATEL results showed that the sum of elements (D) indicated the extent to which that factor influenced other factors in the system. Accordingly, C2 had the greatest impact. The sum of column elements (R) for each factor indicated how that factor was affected by other system factors. Thus, C7 was heavily influenced by other system factors. Based on the horizontal vector (D + R), C2 had the most interaction with other criteria.

Sums of triangular numbers and sums of squares

For non-negative integers a,b, and n, let N(a,b;n) be the number of representations of n as a sum of squares with coefficients 1 or 3 (a of ones and b of threes). Let N⁎(a,b;n) be the number of representations of n as a sum of odd squares with coefficients 1 or 3 (a of ones and b of threes). We have that N⁎(a,b;8n+a+3b) is the number of representations of n as a sum of triangular numbers with coefficients 1 or 3 (a of ones and b of threes). It is known that for a and b satisfying 1≤a+3b≤7, we haveN⁎(a,b;8n+a+3b)=22+(a4)+abN(a,b;8n+a+3b) and for a and b satisfying a+3b=8, we haveN⁎(a,b;8n+a+3b)=22+(a4)+ab(N(a,b;8n+a+3b)−N(a,b;(8n+a+3b)/4)). Such identities are not known for a+3b>8. In this paper, for general a and b with a+b even, we prove asymptotic equivalence of formulas similar to the above, as n→∞. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case b=0 and general a was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of N⁎(a,b;8n+a+3b) and N(a,b;8n+a+3b). The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients.

A parametric neutrosophic model for the solid transportation problem

PurposeThis research attempts to present a solid transportation problem (STP) mechanism in uncertain and indeterminate contexts, allowing decision makers to select their acceptance, indeterminacy and untruth levels.Design/methodology/approachDue to the lack of reliable information, changeable economic circumstances, uncontrolled factors and especially variable conditions of available resources to adapt to the real situations, the authors are faced with a kind of uncertainty and indeterminacy in constraints and the nature of the parameters of STP. Therefore, an approach based on neutrosophic logic is offered to make it more applicable to real-world circumstances. In this study, the triangular neutrosophic numbers (TNNs) have been utilized to represent demand, transportation capacity, accessibility and cost. Then, the neutrosophic STP was converted into an interval programming problem with the help of the variation degree concept. Then, two simple linear programming models were extracted to obtain the lower and upper bounds of the optimal solution.FindingsThe results reveal that the new model is not complicated but more flexible and more relevant to real-world issues. In addition, it is evident that the suggested algorithm is effective and allows decision makers to specify their acceptance, indeterminacy and falsehood thresholds.Originality/valueUnder the transportation literature, there are several solutions for TP and STP in crisp, fuzzy set (FS) and intuitionistic fuzzy set (IFS) conditions. However, the STP has never been explored in connection with neutrosophic sets to the best of the authors’ knowledge. So, this work tries to fill this gap by coming up with a new way to solve this model using NSs.

Using the Sum of Triangular Numbers as the most Fundamental Number of Combinations Found using Heuristic Methods

The same mathematical meaning can be expressed in many different ways. Students should deduce this and thus perceive that the different expressions have the same meaning. In this work, a pedagogical approach to the determination of various equations related to the sum of triangular numbers is presented as an example to show how students can deduce these different expressions and their meanings. The sum of triangular numbers is closely related to the sums of natural numbers, square numbers, and two consecutive numbers. These relationships can be found using a heuristic method in which the equations representing the sum of natural numbers are simply listed. Guiding students to discover the formula by themselves is important, rather than giving them the formula from the beginning and asking them to prove it. Triangular numbers are the fundamental numbers of combinations. The sum of $n$ consecutive triangular numbers is equal to $_{n+2}C_{3}$, as shown by combinatoric tree diagrams. The sum of the triangular numbers can be regarded as an extended version of the sum of the natural numbers. The sum of square numbers is also related to the sum of triangular numbers, which is also easy to understand in terms of combinatorics. The combinatoric expression for the sum of triangular numbers can be extended to $\displaystyle \sum_{\ell=1}^{k-1}\ _{n-\ell}C_{k-1} =\ _{n}C_{k}$, which can be expressed as the sum of triangular numbers. This study provides an example of teaching material that broadens students’ view of a given equation by crossing different learning units such as combinatorics and algebra. The sum of triangular numbers discussed in this study is simply an example of developing the habit of considering various interpretations of the same formula, but even well known formulas can be utilized for this type of subject matter.

Asymptotics of the Determinant of Discrete Laplacians on Triangulated and Quadrangulated Surfaces

Consider a surface Omega with a boundary obtained by gluing together a finite number of equilateral triangles, or squares, along their boundaries, equipped with a vector bundle with a flat unitary connection. Let Omega ^{delta } be a discretization of this surface, in which each triangle or square is discretized by a bi-periodic lattice of mesh size delta , possessing enough symmetries so that these discretizations can be glued together seamlessly. We show that the logarithm of the product of non-zero eigenvalues of the discrete Laplacian acting on the sections of the bundle is asymptotic to A|Ωδ|+B|∂Ωδ|+Clogδ+D+o(1).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} A|\\Omega ^{\\delta }|+B|\\partial \\Omega ^{\\delta }|+C\\log \\delta +D+o(1). \\end{aligned}$$\\end{document}Here A and B are constants that depend only on the lattice, C is an explicit constant depending on the bundle, the angles at conical singularities and at corners of the boundary, and D is a sum of lattice-dependent contributions from singularities and a universal term that can be interpreted as a zeta-regularization of the determinant of the continuum Laplacian acting on the sections of the bundle. We allow for Dirichlet or Neumann boundary conditions, or mixtures thereof. Our proof is based on an integral formula for the determinant in terms of theta function, and the functional Central limit theorem.

Enumeration of Triangles and Hamiltonian Property of The Zero-Divisor Cayley Graph of The Ring G(Zₙ,⊕,⊙)

In this paper an enumeration method to find the number of triangles in the zero-divisor Cayley graph G(Zₙ,D₀ ) associated with the ring (Zₙ,⨁,⨀),n≥1 of integers modulo n, an integer and its subset D0 of zero-divisors is presented. Further it is shown that this graph is Hamiltonian, not bipartite and Eulerian graph when n is odd.

Folding polyiamonds into octahedra

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid – the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exists a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.

"A case study in set-mapping pair theory: the set-mapping pair bijections"

"In a recent paper (I. A. Rus. Sets with structure, mappings and fixed point property: fixed point structures. Fixed Point Theory 23 (2022), No. 2) the author introduced, amongst others, the following notions: set-mapping pair and set-mapping pair bijection. In this paper we study the set-mapping pair bijections in connection with the isomorphisms structure, and their impact on the fixed point theory in a set-mapping pair. Some open problems are also formulated"

Scalable probabilistic truss decomposition using central limit theorem and H-index.

Truss decomposition is a popular notion of hierarchical dense substructures in graphs. In a nutshell, k-truss is the largest subgraph in which every edge is contained in at least k triangles. Truss decomposition aims to compute k-trusses for each possible value of k. There are many works that study truss decomposition in deterministic graphs. However, in probabilistic graphs, truss decomposition is significantly more challenging and has received much less attention; state-of-the-art approaches do not scale well to large probabilistic graphs. Finding the tail probabilities of the number of triangles that contain each edge is a critical challenge of those approaches. This is achieved using dynamic programming which has quadratic run-time and thus not scalable to real large networks which, quite commonly, can have edges contained in many triangles (in the millions). To address this challenge, we employ a special version of the Central Limit Theorem (CLT) to obtain the tail probabilities efficiently. Based on our CLT approach we propose a peeling algorithm for truss decomposition that scales to large probabilistic graphs and offers significant improvement over state-of-the-art. We also design a second method which progressively tightens the estimate of the truss value of each edge and is based on h-index computation. In contrast to our CLT-based approach, our h-index algorithm (1) is progressive by allowing the user to see near-results along the way, (2) does not sacrifice the exactness of final result, and (3) achieves all these while processing only one edge and its immediate neighbors at a time, thus resulting in smaller memory footprint. We perform extensive experiments to show the scalability of both of our proposed algorithms.

The Frobenius Number for Sequences of Triangular Numbers Associated with Number of Solutions

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However, no explicit form has been known even in any special case of three variables. In this paper we are successful to show explicit forms of the generalized Frobenius numbers of the triples of triangular numbers. When $p=0$, their Frobenius number is given by Robles-P\'erez and Rosales in 2018.

Upper tails via high moments and entropic stability

Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph Gn,p. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in Gn,p. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in Gn,p for all p=p(n) satisfying n−1logn≪p≪1.

Regularity in Ramanujan Summation Function Results for the Triangular Number Series

The Triangular Number Series, defined as 1 + 2 + 3 + 4 + ..., when subjected to Ramanujan Summation, gives the known and somewhat controversial result of -1/12. The Ramanujan Summation Function is defined in such a way as to accept any such series and produce corresponding values for them, thus allowing for the aforementioned result to be obtained as well. In this paper, the Triangular Number Group Series Function is defined as a function that generates members of a sum, whereas the values of those members depend on a parameter g, representing the number of elements from the Triangular Number Series in each member of the new sum. This paper shows that the results of the Ramanujan Summation Function upon any such Triangular Number Group Series follow a regularity that is also dependent on the parameter g. Once such a regularity is obtained, the scope of g is extended to real and complex numbers as well.

Swarm Intelligence Algorithms for Optimal Scheduling for Cloud-Based Fuzzy Systems

A fuzzy cloud resource scheduling model with time-cost constraints is built using fuzzy triangular numbers to represent uncertain task execution time. Task scheduling reduces total time and cost spent on a project. It connects virtual machines and functions. Particle swarm optimization (HPO) is used to plan cloud resources (HSOA). The approach uses orthogonal particle swarm initialization to increase the quality of the initial particle exploration, rerandomization to regulate the particle search range, and real-time updating of inertia weights to control particle speed. The suggested problem model and optimization approach are evaluated using random simulation data provided by the CloudSim simulation platform. Less overall execution time and a lower cost are shown to have fast convergence and solution capabilities in experiments.