Triangular Numbers Research Articles

Power distribution network design considering the distributed generations and differential and dynamic pricing

In this study, a power distribution network design model is developed considering voltage control, as well as differential and dynamic pricing schemes. The objective is to maximize power distribution network profits under uncertain demands and parameters, such as power generation capacity, investment and generation cost of renewable distributed generation units, and the penalty budget for unsatisfied demands. The model determines the percentage of the power supply to the customer groups, the amount of power supplied to each customer group, the technology and capacity of renewable distributed generation units, and the prices offered. The proposed pricing scheme is differential for different types of customers with allowable voltage limits. The price is dynamic according to the planning time, including low, medium, and peak times. A fuzzy stochastic programming approach is developed to solve this problem. By simultaneously using triangular crisp numbers and probabilistic variables, the fuzzy stochastic programming integrates the fuzzy and stochastic approaches to deal with the uncertainty of demand and parameters. The numerical analysis shows that, as the confidence level decreases, the capacity of renewable distributed generation units and the profit increase. The results also show that when the decision maker responds to the uncertainty in demand with a high confidence level, the optimum profit can be obtained; approximately 55.90% of the power is supplied to low-demand customers.

A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching process. The limiting behaviour of the network is studied by mathematical methods. We prove that the number of triangles and edges have the same magnitude on the event of non-extinction, and it is eαt, where α is the Malthusian parameter. The probability of the extinction and the degree process of a fixed vertex are also studied. The results are illustrated by simulations.

Optimization of Travelling Salesman Problem on Single Valued Triangular Neutrosophic Number using Dhouib-Matrix-TSP1 Heuristic

The Travelling Salesman Problem is one of the fundamental operational research problems where the objective is to generate the cheapest route for a salesman starting from a given city, visiting all the other cities only once and finally returning to the starting city. In this paper, we study the Travelling Salesman Problem in uncertain environment. Particularly, the single valued triangular neutrosophic environment is considered viewing that it is more realistic and general in real-world industrial problems. Each element in the distance matrix of the Travelling Salesman Problem is presented as a single valued triangular neutrosophic number. To solve this problem, we enhance our novel column-row heuristic Dhouib-Matrix-TSP1 by the means of the center of gravity ranking function and the standard deviation metric. In fact, the center of gravity ranking function is applied for defuzzification in order to convert the single valued triangular neutrosophic number to crisp number.A stepwise application of several numerical Travelling Salesman Problems on the single valued triangular neutrosophic environment shows that the optimal or a near optimal solution can be easily reached thanks to the Dhouib-Matrix-TSP1 heuristic enriched with the center of gravity ranking function and the standard deviation metric.

Nuclear structure of Xe126

Background: The $(N<82)\mathrm{Xe}$ isotopes with only four valence protons are known to be \ensuremath{\gamma} soft and to have the spectrum similar to the O(6) limiting symmetry.Purpose: Study the validity of the $\mathrm{O}(6)\ensuremath{\supset}\mathrm{O}(5)$ symmetry in the various vibrational bands of the $N=72 ^{126}\mathrm{Xe}$ isotope.Methods: Comparison of the $^{126}\mathrm{Xe}$ spectrum with O(6) symmetry. Apply the microscopic theory of dynamic pairing plus quadrupole (DPPQ) model and the interacting boson model (IBM-1). The energy spectrum, absolute $B(\mathrm{E}2)$ values and E2 transition ratios are evaluated. The odd-even spin staggering in the \ensuremath{\gamma} band is displayed, and the potential-energy plot is used to determine the shape of the nucleus.Results: The predictions of the eigenvalues and the $B(\mathrm{E}2)$ values and the interband $B(\mathrm{E}2)$ ratios extended over the five excited bands up to $\ensuremath{\tau}=5$ in the O(6) multiplet view. The O(5) symmetry is well preserved, and the O(6) symmetry is slightly broken.Conclusions: The predictions in the DPPQ model provide an alternative framework to the fluctuations in the O(6) quantum numbers in the IBM framework. The DPPQ model predictions are in fair agreement with experiment. The IBM-1 predictions provide a complementary view.

-NONUNIFORM MULTIRESOLUTION ANALYSIS

AbstractGabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

TIGHT UNIVERSAL TRIANGULAR FORMS

AbstractFor a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$ . For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.

Ground State Properties and Shape Transition of Even-Even 76Os, 78Pt, 80Hg and 82Pb Isotopes in the Framework of Skyrme Hartree-Fock-Bogoliubov Theory

The Hartree-Fock-Bogolyubov (HFB) theory is a good theoretical approximation for describing the nuclear structure of neutron-rich deformed nuclei, it is generalized the HF theory with BCS pairing theory. In this work, ground-state properties of even-even 76Os, 78Pt, 80Hg and 82Pb isotopes from proton drip-line to neutron drip-line have been studied in the framework of the axially deformed Hartree-Fock-Bogoliubov (HFB) method with two Skyrme nucleon-nucleon force HFB9 and UNE1. The numerically calculated results of binding energy, two neutron separation energy, two neutron shell gap, quadrupole deformation, neutron pairing gap and charge radius are discussed and compared with the available values of the experiment, Finite Range Droplet Model (FRDM), Relativistic Mean-Field (RMF) and Hartree-Fock-Bogoliubov based on D1S Gognay force. The energy surface curves are plotted for some isotones of the investigated nuclei, the transition in the shapes was clear with the change in the number of neutrinos. Our calculations show good agreements in comparison with other data and results.

Faster counting empty convex polygons in a planar point set

In this paper, we study the problem of counting the number of convex empty polygons in a given set of n points in general position in the plane. We present two algorithms: one counts the number of all convex empty polygons in O(T) time and the other counts the number of convex empty m-gons for all 3≤m≤k in O(kT) time, where T denotes the number of empty triangles in S. Note that T varies from Ω(n2) to O(n3), while its expected value is known to be Θ(n2) when n input points are chosen uniformly and independently at random from a convex and bounded body.

On a generalized Erdős–Rademacher problem

AbstractThe triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers and an ‐vertex graph with edges and triangle covering number , we determine (for large ) sharp bounds on the minimum number of triangles in and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erdős, and Lovász–Simonovits whose results apply only to . Our results also address two conjectures of Xiao and Katona. We prove one of them and give a counterexample and prove a modified version of the other conjecture.

AN ANALYSIS OF ADJACENCY PAIRS ON CONVERSATION IN BLACK PANTHER MOVIE

This research investigated about adjacency pairs in conversation of the <em>Black Panther </em>movie from five main characters in the movie. This research used qualitative research by applied content analysis as a research design. The data obtained from transcript of <em>Black Panther </em>movie which has 134 minutes duration. The researchers transcribed the data in written form and analyzed by using theory of adjacency pairs. The researchers found 71 data of the conversation five main characters in the movie. Based on the research, there are 71 data of conversation that consist of thirteen types of adjacency pairs. From 71 data of conversation, there are 2 pairs of greeting-greeting sequence, 2 pairs of summons-answer sequence, 2 pairs of apology-minimization sequence, 24 pairs of question-answer sequence, 2 pairs of request-acceptance/refusal sequence, 4 pairs of offer-acceptance/refusal sequence, 2 pairs of blame-admission/denial sequence, 3 pairs of invitation-acceptance/refusal sequence, 8 pairs of assessment-agreement/disagreement sequence, 11 pairs of commands-compliance/incompliance sequence, 2 pairs of suggestion-acceptance/refusal sequence, 3 pairs of assertion-agreement/disagreement sequence and 6 pairs of announcement-acknowledgement sequence. From the research finding it can be concluded that question-answer was mostly-appeared type of adjacency pairs in the conversation by five main characters in the movie

Parallel spinors on globally hyperbolic Lorentzian four-manifolds

We investigate the differential geometry and topology of globally hyperbolic four-manifolds (M, g) admitting a parallel real spinor $$\varepsilon $$ . Using the theory of parabolic pairs recently introduced in [22], we first formulate the parallelicity condition of $$\varepsilon $$ on M as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on an appropriate Cauchy surface $$\Sigma \hookrightarrow M$$ . The existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on $$\Sigma $$ , which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding $$\Sigma \hookrightarrow M$$ . Solutions of this differential system are precisely the allowed initial data for the evolution problem of a parallel spinor and define the notion of parallel Cauchy pair $$({\mathfrak {e}},\Theta )$$ , where $${\mathfrak {e}}$$ is a coframe and $$\Theta $$ is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of Leistner and Lischewski. Furthermore, we classify all compact three-manifolds admitting parallel Cauchy pairs, proving that they are canonically equipped with a locally free action of $${\mathbb {R}}^2$$ and are isomorphic to certain torus bundles over $$S^1$$ , whose Riemannian structure we characterize in detail. Moreover, we classify all left-invariant parallel Cauchy pairs on simply connected Lie groups, specifying when they are allowed initial data for the Ricci flat equations and when the shape operator is Codazzi. Finally, we give a novel geometric interpretation of a class of parallel spinor flows and solve it in several examples, obtaining explicit families of four-dimensional Lorentzian manifolds carrying parallel spinors.

On the Representation of Integers as Sums of a Class of Triangular Numbers

On the Representation of Integers as Sums of a Class of Triangular Numbers

Diagnosis of Melanoma Lesion Using Neutrosophic and Deep Learning

Melanoma is a kind of skin cancer which occurs due to too much exposure of melanocyte cells to the dangerous UV radiations, that gets damaged and multiplies uncontrollably. This is popularly known as malignant melanoma and is comparatively less heard of than certain other types of skin cancers; however it can be more detrimental as it swiftly spreads if not detected and attended at a primary stage. The differentiation between benign and melanocytic lesions sometimes may be confusing, but the symptoms of the disease can reasonably be discriminated by a profound investigation of its histopathological and clinical characteristics. In the recent past, Deep Convolutional Neural Networks (DCNNs) have advanced in accomplishing far better results. The necessity of the present day is to have faster and computationally efficient mechanisms for diagnosis of the deadly disease. This paper makes an effort to showcase a deep learning-based ‘Keras’ algorithm, which is established on the implementation of DCNNs to investigate melanoma from dermoscopic and digital pictures and provide swifter and more accurate result as contrasted to standard CNNs. The main highlight of this paper, basically stands in its incorporation of certain ambitious notions like the segmentation performed by a culmination of a moving straight line with a sequence of points and the application of the concept of triangular neutrosophic number based on uncertain parameters. The experiment was done on a total of 40,676 images obtained from four commonly available datasets— International Symposium on Biomedical Imaging (ISBI) 2017, International Skin Imaging Collaboration (ISIC) 2018, ISIC 2019 and ISIC 2020 and the end result received was indeed motivating. It attained a Jac score of 86.81% on ISIC 2020 dataset and 95.98%, 95.66% and 94.42% on ISBI 2017, ISIC 2018 and ISIC 2019 datasets, respectively. The present research yielded phenomenal output in most instances in comparison to the pre-defined parameters with the similar types of works in this field.

Consolidating Electromagnetic waves might embed more traceable Energy than the sum of the traceable Energies embedded in the waves before consolidation

The common notion is that Electromagnetic (EM) waves cannot consolidate. Contrary to that, an article [2] published by the author of this article, presented a scenario which argues that EM waves can and do consolidate. Another article [1] published by the author of this article presented one such scenario, in which consolidating EM waves generate an EM Null wave without any electric and magnetic fields, which seems to violate the Energy Conservation Principle. That article [1] resolves this paradox by introducing the novel Energy Pairs Theory, which implies that this Null wave conserves the energy as untraceable energy. This article presents another paradox manifested by another scenario of consolidating EM waves, in which the resultant EM wave contains more traceable energy than the sum of the traceable energies embedded in the consolidating waves. This scenario also seems to violate the Energy Conservation Principle, because energy seems to be created out of nothing. The Energy Pairs Theory resolves this paradox, which also significantly enhances the hypothesis presented in article [1], that the energies embedded in photons and EM waves are composed of traceable and untraceable energies.

Bounding Monochromatic Triangles Using Squares

Summary Goodman gave the lower bound on the number of monochromatic triangles of the complete graph on n vertices when the edges are colored with two colors. We reprove his result by giving an algebraic reformulation and utilizing that squares of real expressions are nonnegative.

TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

AbstractTwisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

Triangle of Triangular Numbers

Among several interesting number triangles that exist in mathematics, Pascal’s triangle is one of the best triangle possessing rich mathematical properties. In this paper, I will introduce a number triangle containing triangular numbers arranged in particular fashion. Using this number triangle, I had proved five interesting theorems which help us to generate Pythagorean triples as well as establish bijection between whole numbers and set of all integers.

Economic Upper Bound Estimation in Hausdorff Distance Computation for Triangle Meshes

AbstractThe Hausdorff distance is one of the most fundamental metrics for comparing 3D shapes. To compute the Hausdorff distance efficiently from a triangular mesh to another triangular mesh , one needs to cull the unnecessary triangles on quickly. These triangles have no chance to improve the Hausdorff distance estimation, that is the parts with local upper bound smaller than the global lower bound. The local upper bound estimation should be tight, use fast distance computation, and involve a small number of triangles in during the reduction phase for efficiency. In this paper, we propose to use point‐triangle distance, and only involve at most four triangles in in the reduction phase. Comparing with the state‐of‐the‐art proposed by Tang et al. in 2009, which uses more costly triangle‐triangle distance and may involve a large number of triangles in reduction phase, our local upper bound estimation is faster, and with only a small impact on the tightness of the bound on error estimation. Such a more economic strategy boosts the overall performance significantly. Experiments on the Thingi10K dataset show that our method can achieve several (even over 20) times speedup on average. On a few models with different placements and resolutions, we show that close placement and large difference in resolution bring big challenges to Hausdorff distance computation, and explain why our method can achieve more significant speedup on challenging cases.

Multibranch Rao–Wilton–Glisson Basis Functions for Electromagnetic Scattering Problems

A multibranch Rao–Wilton–Glisson (MB-RWG) basis function is proposed in this article, which is composed of one positive triangle with a larger edge length and several negative triangles with smaller edge lengths. All negative triangles are aggregated to replace the negative triangle in the traditional RWG basis function, with exactly the same function expression defined on the triangles. The number of negative triangles can be changed in different mesh structures. Similar to traditional RWG basis functions, the proposed MB-RWG basis functions guarantee normal current continuity across the common edges. No line charges exist and the total charge on one MB-RWG is zero. MB-RWG basis functions can be used very conveniently to connect one surface with a coarse mesh scheme to another with a fine mesh scheme and can be flexibly applied in the domain decomposition method (DDM). Numerical results validate the accuracy and demonstrate the versatility of the proposed basis function in modeling multiscale perfect electrically conducting (PEC) objects.

Fuzzy-Based EOQ Model With Credit Financing and Backorders Under Human Learning

In this paper, an inventory model has been developed with trade credit financing and back orders under human learning. In this model, it is considered that the seller provides a credit period to his buyer to settle the account and the buyer accepts the credit period policy with certain terms and conditions. The impact of learning and credit financing on the size of the lot and the corresponding cost has been presented. For the development of the model, demand and lead times have been taken as the fuzzy triangular numbers are fuzzified, and then learning has been done in the fuzzy numbers. First of all, the consideration of constant fuzziness is relaxed, and then the concept of learning in fuzzy under credit financing is joined with the representation, assuming that the degree of fuzziness reduces over the planning horizon. Finally, the expected total fuzzy cost function is minimized with respect to order quantity and number of shipments under credit financing and learning effect. Lastly, sensitive analysis has been presented as a consequence of some numerical examples.