Negative linear compressibility of generic rotating rigid triangles

The compressibility properties of systems consisting of generic rotating rigid triangles are analyzed and discussed. It is shown that these systems which are usually associated with auxeticity can exhibit strongly anisotropic properties for certain conformations, which may give rise to the anomalous property of negative linear compressibility （NLC）, that is, the system with particular geometry will expand along one direction when loaded hydrostatically. It is also shown that through carefully choosing the geometric features （i.e. the dimensions and the alignment of the rotating triangles as well as the angles between them） and the direction along which the linear compressibility is measured, one may control the magnitude and range of the NLC. All this provides a novel but effective method of manufacturing the systems which can be tailored to achieve particular values of NLC to fit particular practical applications.

Feature detection on point clouds via Gabriel Triangles creation and l1 normal reconstruction

In this paper, we present a robust subneighborhoods selection technique for feature detection on point clouds scattered over a piecewise smooth surface. The proposed method first identifies all potential features using covariance analysis of the local- neighborhoods. To further extract the accurate features from potential features, Gabriel triangles are created in local neighborhoods of each potential feature vertex. These triangles tightly attach to underlying surface and effectively reflect the local geometry struc- ture. Applying a shared nearest neighbor clustering algorithm on ~ 1 reconstructed normals of created triangle set, we classify the lo- cal neighborhoods of the potential feature vertex into multiple subneighborhoods. Each subneighborhood indicates a piecewise smooth surface. The final feature vertex is identified by checking whether it is locating on the intersection of the multiple surfaces. An advantage of this framework is that it is not only robust to noise, but also insensitive to the size of selected neighborhoods. Ex- perimental results on a variety of models are used to illustrate the effectiveness and robustness of our method.

ON THE AREAS OF THE MINIMAL TRIANGLES IN VEECH SURFACES

Smillie and Weiss proved that the set of the areas of the minimal triangles of Veech surfaces with area 1 can be arranged as a strictly decreasing sequence{an}.And each an in the sequence corresponds to finitely many affine equivalent classes of Veech surfaces with area 1.In this article,we give an algorithm for calculating the area of the minimal triangles in a Veech surface and prove that the first element of{an}which corresponds to non arithmetic Veech surfaces is(5-√5)/20,which is uniquely realized by the area of the minimal triangles of the normalized golden L-shaped translation surface up to affine equivalence.

Torsion in Groups of Integral Triangles

Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0<x<1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.

SAS and SSA Conditions for Congruent Triangles

For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent angles are not between the corresponding congruent sides,then such triangles could be different.It turns out that it is possible to describe four cases in which triangles are congruent even though congruent angles.For two triangles to be congruent,SAS theorem requires two sides and the included angle of the first triangle to be congruent to the corresponding two sides and included angle of the second triangle.If the congruent angles are not between the corresponding congruent sides,then such triangles could be different.It turns out that it is possible to describe four cases in which triangles are congruent even though congruent angles are not between the corresponding congruent sides.Such a theorem could be named,for example,SSA theorem.Many texts state that two triangles cannot be shown to be congruent if the condition of SSA exists.However,the author describes cases in which such triangles could be proven congruent with the SSA theorem.An immediate consequence of this new understanding is the necessity of revising many problems and answers in high school and college-level texts related to congruent triangles.are not between the corresponding congruent sides.Such a theorem could be named,for example,SSA theorem.An immediate consequence of this new understanding is the necessity of revising many problems and answers in high school and college-level texts related to congruent triangles.

LES TRIANGLES DU COU ET SES APPLICATIONS CLINIQUES

Il existe des régions qui ont une forme triangulaire dans la région du cou.Ces triangles sont formés par des vaisseaux,des muscles ou des os.Ils sont importants en anatomie topographique et sont des parties essentielles dans le corps humain.Donc il est necessaire de bien apprendre les compositions de ces triangles et ses significations cliniques.En principe,les triangles du cou sont faits de 2 régions(divisés par le muscle sterno-cléido-mastodien).Tous ces triangles du cou sont très utiles en clinique.Quelques-uns sont les signes pour trouver les vaisseaux ou les nerfs en opération,soit sont les positions de paracentèse,soit sont pour observer l'évolution de maladie.Si on les matrise bien,on peut diminuer l'hémorragie en opration et découvrir des cancers ou maladies le plus tt possible ce qui est très important pour traitement.

Discrete Differential Geometry of Triangles and Escher-Style Trick Art

This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.

Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals

This paper provides several solutions to the degenerate scale for the shapes of triangles or quadrilaterals in an exterior boundary value problem （BVP） of the antiplane elasticity or the Laplace equation. The Schwarz-Christoffel mapping is used throughout. It is found that a complex potential with a simple form in the mapping plane satisfies the vanishing displacement condition （or w ---- 0） along the boundary of the unit circle when the dimension R reaches its critical value 1. This means that the degenerate size in the physical plane is also achieved. The degenerate scales can be evaluated from the partic- ular integrals depending on certain parameters in the mapping function. The numerical results of degenerate sizes for the shapes of triangles or quadrilaterals are provided.

On Quasiconformal Mappings Between Hyperbolic Triangles

Quasiconformal mappings between hyperbolic triangles are considered.We give an explicit estimate of the dilation of the quasiconformal mappings,which generalizes Bishop's results.

Sobolev Mapping of the Bergman Projections on Generalized Hartogs Triangles

In this paper,we investigate Sobolev mapping properties of the Bergman projection.Th domain we focus on is defined byΩm/n:={(z,w)∈C^(1+N):|w1|<|z|m1/n1<1,…|wN|<|z|/m_(N)/n_(N)<1},where m=(m1,…mN)∈(Z^(+))^(N),n=(n1,…nN)∈(Z^(+))N,N∈Z^(+).Sobolev irregularity of the Bergman projections on 2 is shown.We also prove some Sobolev regularity results of the Bergman projections onΩm/n for m=(1,…1).

Crashworthiness Optimization Design of Thin-Walled Tube Filled with Re-entrant Triangles Honeycombs

A novel re-entrant triangles-filled tube(RTT)is proposed through decoupling structural stiffness and energy absorption.Inner re-entrant triangles are employed to satisfy energy absorption,and outer thin wall is used to acquire high stiffness.This paper starts from establishment of theoretical models between geometric parameters of re-entrant triangles and relative density,equivalent elastic modulus and energy absorption characteristics,which are validated by experiments.On this basis,the optimal geometric parameters of unit cell are sought to maximize unit volume energy absorption and minimize relative density by adopting NSGA-II method.Subsequently,the cross-section of tube with optimal stiffness is obtained with targets for maximizing axial stiffness and lateral stiffness by employing static topology optimization method.To verify the proposed optimization method,RTT is analyzed and compared with positive Poisson’s ratio foam-filled tube(PFT),non-filled traditionally optimized tube(NTT)and pre-optimized square tube(PST).The results show that the novel RTT can improve stiffness and energy absorption performance simultaneously.Compared with the positive Poisson’s ratio material,re-entrant triangles honeycomb shows superior advantages in energy absorption.In comparison with the PFT,energy absorption of the RTT increases by 17.23%,and the peak crush force reduces by 5.04%.Therefore,the proposed decoupling design method demonstrates superiority in satisfying various performance requirements simultaneously.

Notes on Proper Class of Triangles

Let T be a triangulated category and ζ a proper class of triangles. Some basics properties and diagram lemmas are proved directly from the definition of ζ.

A Note on Tropical Triangles in the Plane

We define transversal tropical triangles （affine and projective） and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.

Distance-Regular Graphs of Diameter 3Without Triangles with c_(2)=2

Earlier it was proved that some distance-regular graphs of diameter 3 with c_(2)=2 do not exist.Distance-regular graphΓwith intersection array{17,16,10;1,2,8}has strongly regular graphΓ_(3)(pseudo-geometric graph for the net pG_(9)(17,9)).By symmetrizing the arrays of triple intersection numbers,it is proved that the distanceregular graphs with intersection arrays{17,16,10;1,2,8}and{22,21,4;1,2,14}do not exist.

A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA

We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain.

The Classification of Proper Holomorphic Mappings Between Special Hartogs Triangles of Different Dimensions

The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.

Back-to-Front Ordering of Triangles in Digital Terrain Models over Regular Grids

Visiting triangles that conform a digital terrain model is a core operation in a number of fields like animationand video games or generating profiles, cross-sections, and contours in civil engineering. Perfornfing the visit in an efficientmanner is an issue specially when the output of the traversal depends in some way on additional parameters or informationchanging over time, for example, a moving point of view. In this work we report a set of rules that, given a digital terrainmodel defined over a regular grid and an arbitrary point of view outside the terrain, define a total back-to-front order in theset of digital terrain model triangles with respect to the point. The set of rules is minimal, complete and correct. To assesshow the rules perform, we have implemented a CPU-based algorithm for realistically rendering height fields defined overregular grids. The algorithm does not make use of the z-buffer or shaders featured by our graphics card. We show how ouralgorithm is implemented and show visual results obtained from synthetic and real data. We further discuss the algorithmperformance with respect to two algorithms： a naive algorithm that visits triangles according to grid indices and does notsolve the hidden line problem, and the z-buffer provided by the graphics card featured by our computer. Our algorithmallows real-time interaction when the point of view arbitrarily moves in 3D space and we show that its performance is asgood as that of the z-buffer graphics card.

Sierpiński triangles formed by molecules with linear backbones on Au(111)

Fractals play an important role in mathematics, aesthetic, science, and engineering. The representative Sierpinski-triangle fractals have been successfully constructed by V-shape molecules in experiments. The molecular Sierpinski triangles formed by molecules with linear backbones have been theoretically predicted but not experimentally discovered. To achieve this goal in the experiment, we used[1,1’;4’,1’’;4’’,1’’’]-quaterphenyl-3,40 0-dicarbonitrile molecules as building blocks and employed cobalt atoms as cements, then successfully obtained metal-organic Sierpinski triangles with an order up to 2 on the Au（111） surface. There are twenty-four types of three-fold coordination nodes formed between the metal atom and organic ligands via coordinate interactions. The coexistence of various nodes is responsible for that the highest order of Sierpinski triangles is limited to 2.

Proper Holomorphic Mappings Between n-generalized Hartogs Triangles

In this paper,we first introduce the notion of n-generalized Hartogs triangles.Then,we characterize proper holomorphic mappings between some of these domains,and describe their automorphism groups.

Areas of Triangles

We know that the area of any right triangle is half the area of the rectangle which has the same height and the same base with the triangle.The area of rectangle DFEG below is bh.So the area of triangle DFG is 1/2bh. Now we can give a proof of the conclusion that every triangle has half the area of a related rectangle.