On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

Large Sets and Overlarge Sets of Triangle-Decomposition

There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even.     

Asymptotic behaviour of large solutions of quasilinear elliptic problems

The paper deals with the large solutions of the problems \triangle u=u^p\triangle u=u^p and \triangle u = e^u.\triangle u= e^u. They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P-function to establish the asymptotic behaviour of the gradients.     

Triply Existentially Complete Triangle‐Free Graphs

A triangle‐free graph G is called k‐existentially complete if for every induced k‐vertex subgraph H of G, every extension of H to a ‐vertex triangle‐free graph can be realized by adding another vertex of G to H. Cherlin  11 , 12 asked whether k‐existentially complete triangle‐free graphs exist for every k. Here, we present known and new constructions of 3‐existentially complete triangle‐free graphs.     

The Lucas p-triangle

In this note, we study the Lucas p-numbers and introduce the Lucas p-triangle, which generalize the Lucas triangle is defined by Feinberg. We derive an expansion for the Lucas p-numbers by using some properties of our triangle.     

Independence,odd girth,and average degree

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle‐free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle‐free graphs. For connected triangle‐free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n?m?1)/7. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:96‐111, 2011     

The construction of a special equilateral triangle

This article deals with the construction of an equilateral triangle that must satisfy the following special constraint conditions. If the equilateral triangle is denoted by ΔABC, then the radii of the inscribed circle, the three escribed circles of ΔABC, and the circumcircle of ΔABC all must have positive integral radii. The inscribed circle radius is required to be 1 unit. The three escribed circles that have equal radii must have 3 units each, and the circumcircle of the triangle must have 2 units. All these requirements may seem outlandish. The aim is to teach crucial Geometric principles that Geometric designs must take into account before the constructions are implemented. This article hopefully may be useful to students of College Geometry as well as teachers.     

Triangle centres: some questions in Euclidean geometry

The circumcentre E of a triangle ABC is defined, as in figure 1, by the two relations EA = EB EB = EC The other centres (such as the incentre, the centroid, etc.) can be defined by two similar relations. This note is an elaboration on the simple fact that if two centres of a triangle coincide then it is equilateral. We take a certain centre of a given triangle and investigate what can be deduced from the assumption that it satisfies one of the two defining relations of another centre. This is done for each pair of, what one may think of as, the seven most natural centres.     

Analagmatic cubics through quadratic involutive transformations (I)

We generalize the precedent study of two particular cubics, related to any given triangle, by the construction of analagmatic class six cubics under a common quadratic involutive transformation referred again to any given triangle. As resume, we point out some insights for future extensions to generalized theories about the cubics.     

On simplices in diameter graphs in ℝ4

Agraph G is a diameter graph in ?^d if its vertex set is a finite subset in ?^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

     

Optimization of nested polygons

The optimization problem under consideration requires to find the largest regular polygon withk sides to be fitted into a regular polygon withk – 1 sides. If the sequence of these maximal polygons is started with an equilateral triangle, then the final nested polygon, a circle, possesses a radiusr=0.3414r ₃, wherer ₃ is the radius of the inscribed circle of the equilateral triangle. Lower bounds for the ratior/r ₃ are also obtained.     

Left-inverses of fractional Laplacian and sparse stochastic processes

The fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ ^d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I _γ which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I _γ to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I _γ f is not p-integrable. We then introduce the new unique left-inverse I _{γ, p} of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} with the property that I _{γ, p} is dilation-invariant (but not translation-invariant) and that I _{γ, p} f is p-integrable for any Schwartz function f. We finally apply that linear operator I _{γ, p} with p = 1 to solve the stochastic partial differential equation (-\triangle)^g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w.     

Countable decompositions ofR 2 andR 3

If the continuum hypothesis holds,R ² is the union of countably many sets, none spanning a right triangle. Some partial results are obtained concerning the following conjecture of the first author:R ² is the union of countably many sets, none spanning an isosceles triangle. Finally, it is shown thatR ³ can be colored with countably many colors with no monochromatic rational distance.     

A decomposition of (<Emphasis Type="Italic">λK</Emphasis><Subscript><Emphasis Type="Italic">υ</Emphasis></Subscript>)<Superscript>+</Superscript> with extended triangles

By an extended triangle, we mean a loop, a loop with an edge attached (known as a lollipop), or a copy of K ₃ (known as a triangle). In this paper, we completely solve the problem of decomposing the graph (λK _υ )⁺ into extended triangles for all possible number of loops. Dedicated to Professor Alex Rosa on the occasion of his 70th birthday     

Exploring the Role of Individual and Socially Constructed Knowledge Mobilization Tasks in Revealing Preservice Elementary Teachers' Understandings of a Triangle Fraction Task

This study compares the effectiveness of two forms of a knowledge mobilization task on preservice elementary teachers' (n= 65) performance in solving a triangle fraction problem. The study then identifies the source of the successful solutions by linking solutions to earlier activities. One group worked with the triangle fraction task individually; a second worked with the triangle fraction task in a social constructivist setting; a control group had no knowledge mobilization pretask. Although there was no significant difference in the frequency of successful solutions among treatment groups, a chi‐square analysis found that the social‐constructivist pretask group applied fewer ideas from the manipulative lessons as solutions to the posttask than did the comparison groups. The social constructivist group was, however, most successful at generating novel solutions to the triangle problem. The potential benefits of individual and socially constructed knowledge mobilization tasks are discussed.     

Asymptotic behavior of large solutions to -Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|^p−2u)=b(x)f(u) in a bounded ΩR^N subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.     

Probabilistic metric spaces constructed from stationary Markov chains

Cumulative transition density functions of stationary Markov chains are shown to sausfy the Menger triangle inequality under thet-normT = Prod.     

The Smallest Convex Cover for Triangles of Perimeter Two

We find the unique smallest convex region in the plane that contains a congruent copy of every triangle of perimeter two. It is the triangle ABC with AB=2/3, B=60°, and BC1.00285.     

Parameterizations of the Chazy Equation

The Chazy equation y?= 2yy″? 3y′² is derived from the automorphic properties of Schwarz triangle functions S(α, β, γ; z) . It is shown that solutions y which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of α, β, γ . The solutions are then systematically constructed. These analytic solutions provide all known and one new parameterization of the Eisenstein series P, Q, R introduced by Ramanujan in his modular theories of signature 2, 3, 4, and 6.     

Cubic curves in the triangle plane

The subject of this paper are two pencils of cubic curves that are the result of certain geometrical constructions in the triangle plane. One of them turns out to be the probably most significant pencil of anallagmatic cubics that are associated with triangle geometry. Both contain virtually all important single cubics, and other well known curves appear closely connected with them.Dedicated to Professor T.G. Ostrom on the occasion of his 80th birthday