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.

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.

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).

Triangle分层护理管理对高血压患者血压水平和依从性的影响

目的 分析Triangle分层护理管理对高血压患者的影响。方法 抽取2019年6月至2022年12月共78例高血压住院患者作为研究对象,应用随机抽样法划分为两组,对照组中提供常规护理管理(n=39),研究组提供Triangle分层护理管理(n=39),对比分析两组患者血压、依从率、自我管理行为、并发症。结果 患者管理前收缩压和舒张压显示无差异(P>0.05),管理后研究组低于照组,差异具有统计学意义(P<0.05);自我管理行为评分对比,管理前比较两组无差异(P>0.05),管理后升高,经对比,研究组评分高于另一组,差异具有统计学意义(P<0.05);依从率对比差异具有统计学意义(P<0.05),对照组(79.49%)低于研究组(94.87%);并发症发生率经对比差异具有统计学意义(P<0.05),研究组(2.56%)更低。结论 Triangle分层护理管理对高血压患者具有重要影响,能积极有效地控制血压,提高其治疗依从性、自我管理行为,并能降低并发症。

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.

基于Triangle分层分级管理的LEARNS模式对妊娠期糖尿病孕期血糖控制水平及母婴结局

目的探讨基于Triangle分层分级管理的LEARNS模式对妊娠期糖尿病孕期血糖控制水平及母婴结局影响。方法选取山东省滕州市妇幼保健院2021年1月—2022年12月收治的116例妊娠期糖尿病孕妇,应用抽签法将孕妇分为观察组与对照组,每组58例。对照组实施常规护理,观察组实施基于Triangle分层分级管理的LEARNS模式干预。分别在干预前后应用一般自我效能量表(GSES)对两组孕妇的自我效能进行评价,应用健康促进生活方式量表Ⅱ(HPLPⅡ)评价两组孕妇生活健康水平,并对比两组产妇干预前后血糖水平及母婴结局。结果干预后,两组GSES、HPLPⅡ评分均升高,观察组GSES、HPLPⅡ评分高于对照组,差异有统计学意义(P<0.05);干预后,两组空腹血糖、餐后2 h血糖、糖化血红蛋白水平均降低,观察组空腹血糖、餐后2 h血糖、糖化血红蛋白水平低于对照组,差异有统计学意义(P<0.05);观察组妊娠不良结局、围产儿不良结局发生率明显低于对照组,差异有统计学意义(P<0.05)。结论基于Triangle分层分级管理的LEARNS模式可提升妊娠期糖尿病孕妇自我效能及健康生活水平,更有利于产妇孕期血糖控制,进而降低母婴不良结局发生率。

Effect of non-uniform swelling on coal multiphysics during gas injection: The triangle approach

In current dual porosity/permeability models,there exists a fundamental assumption that the adsorption-induced swelling is distributed uniformly within the representative elementary volume (REV),irrespective of its internal structures and transient processes.However,both internal structures and transient processes can lead to the non-uniform swelling.In this study,we hypothesize that the non-uniform swelling is responsible for why coal permeability in experimental measurements is not only controlled by the effective stress but also is affected by the adsorption-induced swelling.We propose a concept of the swelling triangle composed of swelling paths to characterize the evolution of the non-uniform swelling and serve as a core link in coupled multiphysics.A swelling path is determined by a dimensionless volumetric ratio and a dimensionless swelling ratio.Different swelling paths have the same start and end point,and each swelling path represents a unique swelling case.The swelling path as the diagonal of the triangle represents the case of the uniform swelling while that as the two perpendicular boundaries represents the case of the localized swelling.The paths of all intermediate cases populate inside the triangle.The corresponding relations between the swelling path and the response of coal multiphysics are established by a non-uniform swelling coefficient.We define this method as the triangle approach and corresponding models as swelling path-based ones.The proposed concept and models are verified against a long-term experimental measurement of permeability and strains under constant effective stress.Our results demonstrate that during gas injection,coal multiphysics responses have a close dependence on the swelling path,and that in both future experiments and field predictions,this dependence must be considered.

TRIANGLE operation,combined with adequate adjuvant chemotherapy,can improve the prognosis of pancreatic head cancer:A retrospective study

BACKGROUND The TRIANGLE operation involves the removal of all tissues within the triangle bounded by the portal vein-superior mesenteric vein,celiac axis-common hepatic artery,and superior mesenteric artery to improve patient prognosis.Although previously promising in patients with locally advanced pancreatic ductal adenocarcinoma(PDAC),data are limited regarding the long-term oncological outcomes of the TRIANGLE operation among resectable PDAC patients undergoing pancreaticoduodenectomy(PD).AIM To evaluate the safety of the TRIANGLE operation during PD and the prognosis in patients with resectable PDAC.METHODS This retrospective cohort study included patients who underwent PD for pancreatic head cancer between January 2017 and April 2023,with or without the TRIANGLE operation.Patients were divided into the PD_(TRIANGLE)and PD_(non-TRIANGLE)groups.Surgical and survival outcomes were compared between the two groups.Adequate adjuvant chemotherapy was defined as adjuvant chemotherapy≥6 months.RESULTS The PD_(TRIANGLE)and PD_(non-TRIANGLE) groups included 52 and 55 patients,respectively.There were no significant differences in the baseline characteristics or perioperative indexes between the two groups.Furthermore,the recurrence rate was lower in the PD_(TRIANGLE) group than in the PD_(non-TRIANGLE) group(48.1%vs 81.8%,P<0.001),and the local recurrence rate of PDAC decreased from 37.8%to 16.0%.Multivariate Cox regression analysis revealed that PD_(TRIANGLE)(HR=0.424;95%CI:0.256-0.702;P=0.001),adequate adjuvant chemotherapy≥6 months(HR=0.370;95%CI:0.222-0.618;P<0.001)and margin status(HR=2.255;95%CI:1.252-4.064;P=0.007)were found to be independent factors for the recurrence rate.CONCLUSION The TRIANGLE operation is safe for PDAC patients undergoing PD.Moreover,it reduces the local recurrence rate of PDAC and may improve survival in patients who receive adequate adjuvant chemotherapy.

Integrated Clustering and Routing Design and Triangle Path Optimization for UAV-Assisted Wireless Sensor Networks

With the development of the Internet of Things(IoT),it requires better performance from wireless sensor networks(WSNs),such as larger coverage,longer lifetime,and lower latency.However,a large amount of data generated from monitoring and long-distance transmission places a heavy burden on sensor nodes with the limited battery power.For this,we investigate an unmanned aerial vehicles assisted mobile wireless sensor network(UAV-assisted WSN)to prolong the network lifetime in this paper.Specifically,we use UAVs to assist the WSN in collecting data.In the current UAV-assisted WSN,the clustering and routing schemes are determined sequentially.However,such a separate consideration might not maximize the lifetime of the whole WSN due to the mutual coupling of clustering and routing.To efficiently prolong the lifetime of the WSN,we propose an integrated clustering and routing scheme that jointly optimizes the clustering and routing together.In the whole network space,it is intractable to efficiently obtain the optimal integrated clustering and routing scheme.Therefore,we propose the Monte-Las search strategy based on Monte Carlo and Las Vegas ideas,which can generate the chain matrix to guide the algorithm to find the solution faster.Unnecessary point-to-point collection leads to long collection paths,so a triangle optimization strategy is then proposed that finds a compromise path to shorten the collection path based on the geometric distribution and energy of sensor nodes.To avoid the coverage hole caused by the death of sensor nodes,the deployment of mobile sensor nodes and the preventive mechanism design are indispensable.An emergency data transmission mechanism is further proposed to reduce the latency of collecting the latency-sensitive data due to the absence of UAVs.Compared with the existing schemes,the proposed scheme can prolong the lifetime of the UAVassisted WSN at least by 360%,and shorten the collection path of UAVs by 56.24%.

Comparison of prognosis and postoperative morbidities between standard pancreaticoduodenectomy and the TRIANGLE technique for resectable pancreatic ductal adenocarcinoma

BACKGROUND Radical surgery combined with systemic chemotherapy offers the possibility of long-term survival or even cure for patients with pancreatic ductal adenocar-cinoma(PDAC),although tumor recurrence,especially locally,still inhibits the treatment efficacy.The TRIANGLE technique was introduced as an extended dissection procedure to improve the R0 resection rate of borderline resectable or locally advanced PDAC.However,there was a lack of studies concerning postoperative complications and long-term outcomes of this procedure on patients with resectable PDAC.PDAC.METHODS Patients with resectable PDAC eligible for PD from our hospital between June 2018 and December 2021 were enrolled in this retrospective cohort study.All the patients were divided into PDstandard and PDTRIANGLE groups according to the surgical procedure.Baseline characteristics,surgical data,and postoperative morbidities were recorded.All of the patients were followed up,and the date and location of tumor recurrence,and death were recorded.The Kaplan-Meier method and log-rank test were used for the survival analysis.RESULTS There were 93 patients included in the study and 37 underwent the TRIANGLE technique.Duration of operation was longer in the PDTRIANGLE group compared with the PDstandard group[440(410-480)min vs 320(265-427)min](P=0.001).Intraoperative blood loss[700(500-1200)mL vs 500(300-800)mL](P=0.009)and blood transfusion[975(0-1250)mL vs 400(0-800)mL](P=0.009)were higher in the PDTRIANGLE group.There was a higher incidence of surgical site infection(43.2%vs 12.5%)(P=0.001)and postoperative diarrhea(54.1%vs 12.5%)(P=0.001)in the PDTRIANGLE group.The rates of R0 resection and local recurrence,overall survival,and disease-free survival did not differ significantly between the two groups.CONCLUSION The TRIANGLE technique is safe,with acceptable postoperative morbidities compared with standardized PD,but it does not improve prognosis for patients with resectable PDAC.

糖尿病肾病开展Triangle分层分级管理+LEARNS模式的效果及心理状态的观察

目的探究在对糖尿病肾病患者护理时开展Triangle分层分级管理+LEARNS模式对患者血糖水平、肾功能水平以及心理状态等改善作用。方法在2022年2月至2023年1月云南省第三人民医院收治糖尿病肾病患者中随机抽选74例为对象,Excel表随机排序划分对照组(n=37,治疗期间接受常规护理)和观察组(n=37,接受Triangle分层分级管理+LEARNS模式护理)。针对患者血压水平、肾功能水平等改善情况进行评估。结果对比2组干预前后血糖水平,餐后2 h血糖、空腹血糖,干预后观察组低于对照组,差异有统计学意义(P<0.05)。对比患者生存质量,干预前无差异,干预后观察组QOLIE-31量表评分高于对照组,差异有统计学意义(P<0.05)。对比患者负面情绪评分,干预前BAI以及BDI评分无差异,干预后观察组评分低于对照组,差异有统计学意义(P<0.05)。对比患者肾功能水平,干预后观察组肾功能水平高于对照组,差异有统计学意义(P<0.05)。对比2组自我管理能力,干预后观察组高于对照组,差异有统计学意义(P<0.05)。结论在对糖尿病肾病患者护理时开展Triangle分层分级管理+LEARNS模式,可改善患者血糖、肾功能水平,缓解患者治疗期间负面情绪,有助于患者进行恢复。

Triangle分层分级护理在位置性眩晕患者复位成功后残余症状中的应用

目的:探讨Triangle分层分级护理在位置性眩晕患者复位成功后残余症状中的应用效果。方法:将2021年10月—2023年3月收治的60例眩晕复位成功后残余症状患者分为观察组和对照组,每组30例。对照组采取常规护理,观察组在对照组的基础上实施Triangle分层分级护理,两组均持续干预1周。比较两组眩晕程度、心理状态、跌倒发生情况以及患者满意度。结果:两组干预前眩晕残障程度评定量表(DHI)、医院焦虑抑郁量表(HADS)评分比较,差异无统计学意义(P>0.05);观察组干预后DHI,HADS各项评分均低于对照组,差异有统计学意义(P<0.05);观察组跌倒发生率低于对照组,差异有统计学意义(P<0.05);观察组护理满意度中质量及安全、人文关怀、健康教育、环境设施、服务可及性评分均高于对照组,差异有统计学意义(P<0.05)。结论:Triangle分层分级护理可改善位置性眩晕患者复位成功后残余症状,减轻患者消极情绪,减少跌倒发生,促进患者康复,从而获得更高的患者满意度。

2型糖尿病患者护理中Triangle分层分级管理联合LEARNS模式的应用效果

目的探究2型糖尿病(Type 2 Diabetes Mellitus,T2DM)患者护理中Triangle分层分级管理联合LEARNS模式的应用效果。方法选取武威市天祝藏族自治县人民医院于2021年2月—2023年2月收治的280例T2DM患者为研究对象,以随机信封法分为常规组(n=140,常规护理)、观察组(n=140,常规组基础上行Triangle分层分级管理联合LEARNS模式)。观察并对比护理前后组间、组内各指标之间的差异。结果干预前组间血糖水平、自我管理能力评分对比,差异无统计学意义(P均>0.05);组间干预6个月后血糖水平对比,观察组更低,差异有统计学意义(P<0.05);组间干预6个月后自我管理能力评分对比,观察组(85.41±3.12)分高于常规组(84.23±3.19)分,差异有统计学意义(t=3.129,P<0.05)。结论Triangle分层分级管理联合LEARNS模式护理T2DM,可改善血糖水平及提高患者自我管理能力。

基于Triangle模型的分层分级延续性护理在2型糖尿病病人中的应用研究

目的:分析基于Triangle模型的分层分级延续护理在2型糖尿病(T2DM)病人中的应用效果。方法:纳入2021年11月—2022年11月在徐州医科大学附属宿迁医院接受治疗的120例T2DM病人,按随机数字表法分为观察组与对照组,各60例。对照组接受延续性护理,观察组接受基于Triangle模型的分层分级延续性护理。于干预前、干预3个月时、干预6个月时评定病人血糖控制状况、自我管理能力、生活质量;并记录病人干预期间再住院次数。结果:干预3个月、6个月时,两组病人空腹血糖(FPG)、餐后2 h血糖(2 h PG)、糖化血红蛋白(HbA1c)水平明显低于干预前,糖尿病自我管理行为量表(SDSCA)得分高于干预前,中国糖尿病病人生存质量特异性量表(DSQL)得分低于干预前,且观察组病人各项指标改善优于对照组(P<0.05);但两组病人干预6个月时的各项指标与干预3个月时比较,差异无统计学意义(P>0.05);观察组病人再住院次数少于对照组(P<0.05)。结论:基于Triangle模型的分层分级延续性护理可提高T2DM病人的血糖控制水平、自我管理能力及生活质量,并减少再住院次数。

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.