Triangular Numbers Research Articles

HOW MANY TRIANGLES CAN A GRAPH HAVE?

Abstract We investigate the set $T_{n}$ of the possible number of triangles in a graph on n vertices. The first main result says that every natural number less than $\binom {n}{3} - (\sqrt {2}+o(1) )n^{3/2} $ belongs to $T_{n}$ . On the other hand, we show that there is a number $m = \binom {n}{3}-( \sqrt {2} +o(1))n^{3/2}$ that is not a member of $T_{n}$ . In addition, we prove that there are two interlacing sequences $\binom {n}{3} - ( \sqrt {2} + o(1) )n^{3/2} = c_{1} \le d_{1} \le c_{2} \le d_{2} \le \cdots \le c_{s} \le d_{s} = \binom {n}{3}$ with $| c_{t} - d_{t}| = n-2-\binom {s - t +1}{2}$ such that $( c_{t}, d_{t} ) \cap T_{n} = \emptyset $ for all t. Moreover, we obtain a generalisation of these results for the set of the possible number of copies of a connected graph H in a graph on n vertices.

Explicit description of viral capsid subunit shapes by unfolding dihedrons

AbstractViral capsid assembly and the design of capsid-based nanocontainers critically depend on understanding the shapes and interfaces of constituent protein subunits. However, a comprehensive framework for characterizing these features is still lacking. Here, we introduce a novel approach based on spherical tiling theory that explicitly describes the 2D shapes and interfaces of subunits in icosahedral capsids. Our method unfolds spherical dihedrons defined by icosahedral symmetry axes, enabling systematic characterization of all possible subunit geometries. Applying this framework to real T = 1 capsid structures reveals distinct interface groups within this single classification, with variations in interaction patterns around 3-fold and 5-fold symmetry axes. We validate our classification through molecular docking simulations, demonstrating its consistency with physical subunit interactions. This analysis suggests different assembly pathways for capsid nucleation. Our general framework is applicable to other triangular numbers, paving the way for broader studies in structural virology and nanomaterial design.

AI‐Based Segmentation for Corrosion Risk Prediction in Carline Designs

ABSTRACTWith shortening development cycles, digital development rises in importance. Therefore, we present an approach to segment digital carlines based on their corrosion risk using digital data sets. A typical carline is described via STL‐Files that accumulate to approximately 25 million triangles. While physics‐based simulation models exist capable of predicting the onset of corrosion in local geometry settings, an application to a complex surface mesh would be prohibitively expensive to compute. This calls for data reduction techniques that reduce the number of triangles by identifying areas prone to corrosion, as well as those that are protected by measures such as adhesives. Therefore, the implementation of corrosion‐prone areas of a body‐in‐white part, as presented in Waibel et al., is extended by implementing a design measure segmentation for adhesive pipes. This work introduces a method to predict corrosion‐protected areas for flanges with adhesives by implementing a feature extraction and a transfer learning‐based workflow for data‐efficient training. The results of the implementation are the validated prediction on body‐in‐white parts with high precision.

108.37 A parent figure for a family of triangle number identities

108.37 A parent figure for a family of triangle number identities

Wolfram alpha and maple as educational technology for future teachers of mathematics

The paper’s purpose is to present technology-enhanced activities with triangular, square, and other polygonal numbers arranged in basic geometric shapes – equilateral and isosceles triangles and squares. The author’s review of recently published mathematics education papers found no use of computing technology in teaching polygonal numbers either at the college or the pre-college levels worldwide. This omission served as a motivation to share and discuss computational algorithms designed by the author for the summation of such numbers within each geometric structure. A method through which those algorithms were used with teacher candidates is based on the technology-immune/technology-enabled problem-solving pedagogy. This method can be recommended for the modern-day teaching of elementary number theory and other mathematical topics across multiple grade levels and educational programs. The activities, supported by Wolfram Alpha and Maple, can be used by instructors of technology-motivated mathematics teacher education courses. The paper argues that the power of digital tools allows future teachers of mathematics, in the context of elementary number theory, to appreciate the use of simple algorithms in achieving sophisticated computational outcomes. Reflective comments of teacher candidates (the author’s students) are shared to indicate the importance of the history of mathematics and the focus on mathematical visualization as a pedagogical approach aimed at conceptual understanding of the subject matter.

RATIONAL SHIFTS OF THE BASIC RACK’S PROFILE FOR WHEELS OF A CYLINDRICAL SPUR GEAR PAIR TO DECREASE TEETH WEAR

Purpose. It is to create a method for determination of such shift coefficients of the basic rack’s profile for driving and driven wheels of a cylindrical spur gear pair, that during gear pair operation time the wear rate of a surface layer on the most wearing sections of the teeth’ involute surfaces will be minimal. Research methods are based on integrated application of the involute gear pair theory and tribological laws. The golden section method is used to minimize the function. Results. We have obtained a dimensionless quantity, which is a function of shift coefficients of the basic rack’s profile for driving and driven wheels of a cylindrical spur gear pair. Values of this function are proportional to the greatest wear rate in the neighborhood of characteristic points on teeth’ involute profiles of these wheels. The method is created for determination of rational shift coefficients of the basic rack’s profile, which minimize the obtained dimensionless function, and which maximize the service life of the spur gear pair until the greatest worn-layer thickness on the teeth’ active surfaces reaches limiting permissible value. Scientific novelty. It is proved, that increasing of the worn-layer thickness during spur gear pair operation time will be occurring most rapidly in the neighborhood of certain characteristic points on teeth’ involute profiles of the wheels. Calculation model is elaborated for determination of the worn-layer thicknesses in the neighborhood of extreme active points on teeth’ profiles of the cylindrical spur gear pair and in the neighborhood of the lowest and the highest bounding points of single-pair contact. This model takes into account the influence of specific slides and numbers of wheels’ teeth as well as hardness values of teeth’ surface layer on the wear rates at characteristic points of teeth’ profiles. In addition, the model takes into account sharing of total force of the wheels interaction between two teeth pairs in double-pair contact and also a change of force transmitted by single pair of teeth when it comes into or go out of engagement. Practical value. Application of the created method for determination of shift coefficients of the basic rack’s profile is expedient in design of cylindrical spur gear pairs for machines and equipment, which operate in conditions when ingress of abrasive particles in the teeth engagement region is possible. An example of application of this method in design calculation of the cylindrical spur gear pair is given.

Tri-Hepta-Penta

Summary We offer a visual proof of a relationship between triangular, pentagonal, and heptagonal numbers.

The dual problem of space: Relative player positioning determines attacking success in elite men’s football

ABSTRACT The concept of space has been successfully modelled in football using spatiotemporal player data and Voronoi diagrams. Current approaches, however, are narrow in scope by focusing on an inter-team allocation of space to measure space control. The present work extends this widespread perspective with an intra-team application of the Voronoi diagram and its dual Delaunay triangulation to measure space management. Both models are leveraged to derive novel performance metrics, which assess how teams use triangular positioning and how players tie up defenders during attacks. The outcome of N = 128,187 attacking sequences from 306 elite men’s football matches is analysed using linear mixed-effects models to validate the proposed performance metrics. Results show that attacking success is characterized by player positioning which promotes forming of large triangles especially in ball proximity, whereas the overall number of triangles is of no relevance. Furthermore, players tie up more defenders and thus create free teammates more often during successful attacks. The results demonstrate that a new perspective on space is helpful to better quantify and understand the effect of space management and player positioning on attacking performance in football.

An Attempt on Investment Selection Problem under Neutrosophic Environment based on Weighted Value Method

The main aim of this study is to develop a ranking algorithm in neutrosophic environment and applying this algorithm to choose the best option for investment selection problems. Firstly, the concept of single valued triangular neutrosophic numbers (SVTrN-numbers) and it’s characteristics are presented briefly. Here we define the λ-weighted value of SVTrN-numbers and applied it to the best selection of investment selection problem (ISP). Finally, to determine the feasibility and validity of this algorithm, we compared the resultant ranking by the proposed method with other existing approaches.

Closeness Centrality of Asymmetric Trees and Triangular Numbers

The combinatorial problem in this paper is motivated by a variant of the famous traveling salesman problem where the salesman must return to the starting point after each delivery. The total length of a delivery route is related to a metric known as closeness centrality. The closeness centrality of a vertex v in a graph G was defined in 1950 by Bavelas to be CC(v)=|V(G)|−1SD(v), where SD(v) is the sum of the distances from v to each of the other vertices (which is one-half of the total distance in the delivery route). We provide a real-world example involving the Metro Atlanta Rapid Transit Authority rail network and identify stations whose SD values are nearly identical, meaning they have a similar proximity to other stations in the network. We then consider theoretical aspects involving asymmetric trees. For integer values of k, we considered the asymmetric tree with paths of lengths k,2k,…,nk that are incident to a center vertex. We investigated trees with different values of k, and for k=1 and k=2, we established necessary and sufficient conditions for the existence of two vertices with identical SD values, which has a surprising connection to the triangular numbers. Additionally, we investigated asymmetric trees with paths incident to two vertices and found a sufficient condition for vertices to have equal SD values. This leads to new combinatorial proofs of identities arising from Pascal’s triangle.

The maximum number of pentagons in a planar graph

AbstractIn 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an ‐vertex planar graph. They precisely determined the maximum number of triangles and four‐cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the ‐vertex, planar graphs with the maximum number of pentagons.

Unveiling novelty evolution in the field of library and information science in China

PurposeThis study aims to analyze the distribution of novelty among scholarly papers in the field of library and information science (LIS) in China. Specifically, this study explores the distribution of novelty of papers in various journals, research topics and different periods. It is possible to understand the characteristics of LIS research in China and what factors have influenced it.Design/methodology/approachThis paper collects articles published in Chinese library science journals indexed by the Chinese Social Sciences Citation Index from 2000 to 2022. The BERTopic model is used based on abstracts of the papers and to obtain the topic of each paper. Based on the combination innovation theory of reference pairs cited by focal papers, novelty scores of all papers are calculated. Next, this paper analyzes the novelty of papers under different topics. Finally, this paper analyzes the differences in author collaboration patterns across various topics, aiming to explain how these differences relate to the novelty of papers from a collaborative perspective.FindingsThis study shows that archival research topics have lower novelty than papers on journal evaluation and patent technology in Chinese LIS. Research papers in this field are gradually becoming more novel over time. Papers on different topics and with varying degrees of novelty exhibit distinct author collaboration patterns, with low-novelty topics more frequently featuring solo authorship, while high-novelty topics tend to involve a higher percentage of inter-institutional collaboration.Originality/valueThis study investigates the novelty characteristics of research papers on different topics in the field of LIS in China. The authors’ contribution includes visualizing research hotspots and trends in the field and analyzing authors’ collaboration patterns at the level of research topics, thereby providing new perspectives on the factors affecting the novelty of these papers.

Triangular Number Products

Summary We show visually that twice the product of a pair of consecutive triangular numbers is triangular, and discuss some consequences.

A Few Linear Combinations of Triangular Numbers that are the Difference of Two Squares

Summary We provide visual proofs for two general identities to show that the sums of certain triangular numbers result in the difference of two squares. Since these linear combinations represent some other sequences like the polygonal numbers, we show those sequences are the difference of two squares as well.

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that | E ( G )| ≥ α | V ( G )| 2 for some fixed α > 0. While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given ε > 0, returns an integer g such that G has an embedding in a surface of genus g , and this is ε-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G) \) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G) \) . The running time of the algorithm is O ( f (ε) n 2 ), where f (·) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time O ( f 1 (ε) n 2 ). Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [25]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [17] and to Fox, Lovász, and Zhao [14, 15]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ε-approximation ν ( G ) for the maximum number of edge-disjoint triangles in G . The value ν ( G ) can be computed by solving a linear program whose size is bounded by certain value f 2 (ε) depending only on ε. After solving the linear program, the genus can be approximated (see Corollary). The proof of this result is long and will be of independent interest in topological graph theory.

The Various Radii Circle Packing Problem in a Triangle

Malfatti’s problem is the problem of fitting three circles into a triangle such that they are tangent to each other and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to include Tn = 1 + 2 + … + n circles inside the triangle with special tangency properties among the circles and triangle sides; this problem is referred to as the extended Malfatti problem or the Tri(Tn) problem. In the extended Malfatti problem, the number of circles in the triangle is a triangle number because the tangency properties between the internal circles and the three sides of the triangle have a special type of structure; that is, the corner circle is tangent to two sides of the triangle and two other circles, the boundary circles are tangent to one side of the triangle and four other circles, and the inner circles are always tangent to six other circles. The circles we find in the extended Malfatti problem have the following property: the smallest and largest radii of the circles differ to a great extent. In the study presented herein, we propose algorithms to solve the problem that the tangency properties between the circles and the sides of the triangle are not fixed, so that the number of circles in the triangle is not necessarily a triangle number. The purpose of this change is to attempt to establish the radii of the circles in the triangle within a small range.

Kruskal–Katona-type problems via the entropy method

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with R red, G green, B blue edges, the number of rainbow triangles is at most 2RGB, which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.

Layer and orientation dependent optical anisotropic response of CVD grown MoS2 films under resonance/non-resonance excitation

The MoS2hasgained worldwide research interest owing to their strong light-matter interaction and their rich physics that offers unprecedented potential for next-generation nanotechnology applications. Theirelectrical and thermal properties are mainly governed by interaction mechanism of electrons with phonons and thus, studying the polarization-dependent anisotropic phonon response ofMoS2are of great interest. Here, we discuss the evolution of phonon modes with increasing layer number and their respective phonon DOS. Further, the optical anisotropic responseis experimentally investigated using ARPRSstudy for CVD-grown triangular MoS2with different layer numbers and different excitations, and has been discussedbased ontheory that includes deformation potential and Fröhlich interaction.Further, the ARPRS study of horizontally and vertically oriented MoS2films are studied. Thedifferent anisotropic response of vertically oriented film is observed due to symmetry breaking of the vertical nanosheets, thus providing wide-field applications in MoS2-based photonic and optoelectronic devices.

Higher Dimensional Triangular and Square Numbers

Higher Dimensional Triangular and Square Numbers

GEOMETRIC PATTERN OF POWERS OF THREE VIA SIERPINSKI TRIANGULAR NUMBERS

In this study, we describe a new number sequence based on integers arranged in a fractal like structure to visualize powers of three. Moreover, we associate these new numbers with triangular numbers