Triangle Groups Research Articles

Imprecise earned duration model for time evaluation of construction projects with risk considerations

Monitoring and estimating the time performance of projects are two crucial factors which lead the companies to be prosperous. To address the issue, the most recent time-based method is earned duration management (EDM) which is used as an efficient technique. This research presents a new triangular intuitionistic fuzzy-EDM model to improve the applicability of time performance indices and to forecast under uncertain conditions. Since traditional fuzzy sets using membership degrees cannot deal with imprecise quantities in real cases, triangular intuitionistic fuzzy numbers (TIFNs) solve this issue by non-membership and hesitation degrees. In this sake, a TIFN as a special type of intuitionistic fuzzy sets (IFSs), defined on the set of real numbers, shows high capability of modeling ill-known and imprecise information in continuous domains. Besides, the importance of risks in time estimation of EDM is less considered in the literature. In this research, a new time-based risk performance indicator (TBRPI) is developed based on novel risk performance metrics (RPMs) to improve the project time performance estimating of EDM. In this respect, the significance of each RPM could increase the accuracy of the proposed approach, in which a new triangular intuitionistic fuzzy group decision method is presented for obtaining the RPMs and constituent factors (CFs)’ weights. Finally, a real case study about seawater intake basin construction project is provided to represent the applicability and feasibility of the proposed triangular intuitionistic fuzzy risk-based EDM model.

Syntheses, crystal structures, vibrational spectra, and luminescent properties of M3Eu2(BO3)4 (M = Ba, Sr)

Two alkali-earth and rare-earth metal borates, M3Eu2(BO3)4 (M ​= ​Ba, Sr), have been synthesized and their crystal structures determined by single-crystal XRD methods with the following crystal data: Ba3Eu2(BO3)4 (1), Pnma, a ​= ​7.6970(15) Å, b ​= ​16.554(3) Å, c ​= ​8.9300(18) Å, V ​= ​1137.8(4) Å3, Z ​= ​4; Sr3Eu2(BO3)4 (2), Pnma, a ​= ​22.2553(4) Å, b ​= ​15.9122(3) Å, c ​= ​8.7568(2) Å, V ​= ​3101.05(11) Å3, Z ​= ​12. The compound 1 is isotypic with Ba3RE2(BO3)4 (RE ​= ​Y, La, Pr, Nd) and its crystal structure features a 3D framework constructed by BO3 triangles and 8-fold coordinated positions M1, M2 and M3 with mixed Eu/Ba occupation. 2 adopts a threefold superstructure of the structure of 1. IR and Raman spectra further confirm that both 1 and 2 contain the planar triangular BO3 groups. The photoluminescence spectra of 1 and 2 consist of five groups of emission bands assigned to the 5D0→7Fj (j ​= ​0, 1, 2, 3, 4) transitions within Eu3+ ions and the decay curves show the single-exponential behavior with the lifetime being 1.358 and 0.887 ​ms for 1 and 2, respectively. 1 possess a very high luminescent efficiency (QE ​= ​90.09%) which benefited to being applied in warm white LEDs. Moreover, X-ray photoelectron spectroscopy (XPS) is used to analyze the chemical states of the elements and the band electronic structure on 1 has also been investigated.

Billiards on pythagorean triples and their Minkowski functions

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincaré disk model. Our tables have $ m\ge3 $ ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group $ {\rm{PSU}}^\pm_{1,1} \mathbb{Z}[i] $. The resulting billiard map $ \widetilde B $ acts on the de Sitter space $ x_1^2+x_2^2-x_3^2 = 1 $, and has a natural factor $ B $ on the unit circle, the pythagorean triples appearing as the $ B $-preimages of fixed points. We compute the invariant densities of these maps, and prove the Lagrange and Galois theorems: A complex number of unit modulus has a preperiodic (purely periodic) $ B $-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over $ \mathbb{Q}(i) $.Each $ B $ as above is a $ (m-1) $-to-$ 1 $ orientation-reversing covering map of the circle, a property shared by the group character $ T(z) = z^{-(m-1)} $. We prove that there exists a homeomorphism $ \Phi $, unique up to postcomposition with elements in a dihedral group, that conjugates $ B $ with $ T $; in particular $ \Phi $ -whose prototype is the classical Minkowski question mark function- establishes a bijection between the set of points of degree $ \le2 $ over $ \mathbb{Q}(i) $ and the torsion subgroup of the circle. We provide an explicit formula for $ \Phi $, and prove that $ \Phi $ is singular and Hölder continuous with exponent $ \log(m-1) $ divided by the maximal periodic mean free path in the associated billiard table.

High-temperature, high-pressure hydrothermal synthesis, crystal structure, and infrared and NMR spectroscopy of a barium lead borate with a 2D layer structure: [Ba3Pb(H2O)][B11O19(OH)3

With a high-temperature, high-pressure hydrothermal technique, a new barium lead borate, [Ba3Pb(H2O)][B11O19(OH)3] (1), has been synthesized and characterized by single-crystal X-ray diffraction, and infrared and solid-state NMR spectroscopy. The structure of 1 contains planar thick layers of borates with the Ba2+ cations at sites in the inter- and intralayer space. Each layer consists of three single sheets. The central sheet is very corrugated and is built up from the fundamental building block (FBB) 2Δ3□:Δ2□-Δ2□. On both sides of the central sheet there are borate single chains formed of the very rare FBB 2Δ4□:Δ2□-3□Δ via corner-sharing. This FBB was first observed in a high-pressure polymorph of CaB2O4. These chains are linked into a sheet by PbO5(H2O) polyhedra, which are further linked to the central sheet by sharing vertices between triangles and tetrahedra to form a thick layer. The IR spectrum shows the presence of hydroxyl groups of HBO4, water molecules, BO3 triangles, and BO4 tetrahedra. The presence of BO3 and BO4 polyhedra was also confirmed by 11B MAS NMR spectroscopy.

Minerals of the System Stichtite–Pyroaurite–Iowaite–Woodallite from Serpentinites of the Terekta Ridge (Gorny Altai, Russia)

Abstract —Hydrotalcite supergroup minerals stichtite, pyroaurite, iowaite, and woodallite form a complex solid-solution system at the Kyzyl-Uyuk locality (Terekta Ridge, Gorny Altai, Russia). The diversity of these minerals is due to: (1) subdivision by anionic interlayer composition into carbonate (stichtite and pyroaurite) and chloride (iowaite and woodallite) species and (2) isomorphism of M3+ cations, mainly between Cr- (stichtite and woodallite) and Fe3+-dominant species (pyroaurite and iowaite), with a quantitative predominance of stichtite and iowaite. Most of the studied samples correspond to stichtite and woodallite with high Fe3+ contents or pyroaurite and iowaite with high Cr3+ contents. According to vibrational (IR and Raman) spectroscopy data, the interlayer Cl– is partially substituted by OH– rather than CO32- groups. We suppose that the presence/absence of a band in the region 1400–1350 cm–1 in the Raman spectra of stichtite can be explained by the local distortion of triangular CO3 groups. Stichtite and iowaite occur here in polytypic modifications 3R and 2H that are the most widespread for the hydrotalcite supergroup iowaite and stichtite is 2:1. For both minerals, the polytype 3R strongly dominates over 2H. The lowest 3R/2H ratio determined for the Terekta iowaite and stichtite is 2:1. The Altai stichtite is close in 3R/2H to the stichtite from Tasmania (Australia) and differs significantly from that in Transvaal samples (South Africa).

Patient morbidity after impacted third molar extraction with different flap designs. A systematic review and meta-analysis.

Background To review the literature on the effect of different surgical flaps upon patient morbidity (pain perception, trismus, swelling and osteitis) after impacted third molar extraction. Material and Methods An electronic and complementary search of main databases and grey literature was performed up to January 2019 to retrieve randomized clinical trials. The Cochrane risk of bias assessment tool was used for methodological appraisal. A random-effects meta-analysis was conducted of pain perception and trismus. Results From the initially 1314 screened studies, only 11 were included in the qualitative synthesis, and 5 in the meta-analysis. There were no statistically significant differences in pain between the envelope and triangular flap designs over time, except on the sixth postoperative day, when the envelope flap proved more painful. Regarding trismus, statistically significant differences were observed on the seventh postoperative day, with greater mouth opening in the envelope flap group than in the triangular flap group. There were no clear differences in swelling and osteitis among the flap designs. Conclusions Despite its limitations, the present meta-analysis found no clear differences in patient morbidity between the different flap designs. Key words:Impacted third molar, triangular flap, envelope flap, surgical flap, pain perception, trismus.

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.

Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

We consider the action of SL(2, R) on a vector bundle H preserving an ergodic probability measure nu on the base X. Under an irreducibility assumption on this action, we prove that if (nu) over cap is any lift of nu to a probability measure on the projectivized bunde P(H) that is invariant under the upper triangular subgroup, then (nu) over cap is supported in the projectivization P(E-1) of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [57]. Second, if P(V) is an irreducible, flat projective bundle over a compact hyperbolic surface Sigma, with hyperbolic foliation F tangent to the flat connection, then the foliated horocycle flow on (TF)-F-1 is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [13] to arbitrary dimension.

Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $\nu$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hat\nu$ is any lift of $\nu$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat \nu$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $\Sigma$, with hyperbolic foliation $\mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.

Clinico-pathological evaluation of cervical lymphadenopathy

Introduction: Cervical lymph nodes are peripheral lymphoid organs distributed in the neck. The term lymphadenopathy refers to nodes that are palpable and abnormal in size, consistency or numbers. Diagnosis of cervical lymphadenopathy can vary from neoplasm to various kinds of infection. Accurate diagnosis of the cause of lymphadenopathy can sometimes be challenging and can lead to delay in diagnosis causing delayed treatment and subsequent health issues.
 Methods: A prospective observational study was carried out on 78 patients of cervical lymphadenopathy who presented in the surgical outpatient department of KIST Medical College. Detailed history, physical examination and necessary investigations including fine needle aspiration cytology were done in all patients
 Results: Most patients were between 21 to 40 years of age. The commonest cause for cervical lymphadenopathy was reactive lymphadenitis (53.84%) followed by Tuberculosis (34.61%). Secondary metastasis was seen in 7 patients (8.97%). In tubercular lymphadenitis, the anterior triangle group was the most commonly involved group of cervical lymph nodes (74 %), followed by the posterior triangle and supraclavicular equally (11.11%). 
 Conclusions: Reactive Lymphadenopathy is the commonest cause of cervical lymphadenopathy. However, variable diagnosis can be possible from non-neoplastic to neoplastic condition.

FTIR, optical, and thermal studies of cadmium borate glass doped with Bi2O3 and effects of gamma irradiation

Bi2O3-doped glasses from CdO-B2O3 glasses were prepared by the melt-annealing method. Optical and FTIR spectroscopic properties of the prepared glasses were measured before and after gamma irradiation (dose 6 M rad). The undoped glass reveals distinct UV absorption within the spectrum of the undoped glass and extended to visible peaks upon the increase of the dopant Bi3+ ions. The extended identified UV to visible absorption is related to absorption of both trace Fe3+ ions and Bi3+ ions. Fourier transform IR spectra of the prepared glasses reveal distinct and composite vibrational bands within the wavenumber range 400–1650 cm−1 indicating the presence of both triangular and tetrahedral borate groups within their specific different wavenumbers and supporting the assumption of the role of CdO to convert some of the BO3 groups to BO4 groups. The introduction of Bi2O3 shows interfered vibrational bands due to additional structural BiO6 and BiO3 groups within their characteristic sites at about 450–480 cm−1 and 860–920 cm−1 and is assumed to be mixed or interfered with the two coordinated borate species. The effect of gamma irradiation on the optical spectra differs between the undoped and Bi2O3-doped glasses showing some shielding behavior in the latter Bi2O3-containing samples.

Abstract 266: The Effect of Different Kinds of Chest Compression Waveforms on the Heart Ischemia-Reperfusion Injury After Cardiac Arrest

Purpose: This study was aimed to compare the effect of different kinds of chest compression waveforms on the heart ischemia-reperfusion injury after cardiac arrest (CA) in a rat model. Materials and methods: After the CA model was successfully established with continuous electrical stimulation of ventricular fibrillation, rats were divided into four different chest compression groups randomly, including manual group, triangular wave group, sine wave group and trapezoidal wave group. The success rates of return of spontaneous circulation (ROSC) of those four groups were calculated, and all ROSC rats were executed 30 minutes after ROSC. Further comparisons were made among those four groups, including pathological changes, superoxide dismutase (SOD) activities, malondialdehyde (MDA) and superoxide (O 2 - ) content. Results: The success ROSC rates for those four groups (manual group, triangular wave group, sine wave group and trapezoidal wave group) were 30%, 40%, 40% and 90% respectively. There were inflammatory cell infiltration and edema in the heart tissue of those four groups. The activities of SOD in trapezoidal wave group were 85.91±1.91 U/mg, much higher than those of the other three groups (P <0.05). On the other hand, the MDA and O 2 - levels of trapezoidal wave group were much lower than those of the other three groups (p<0.05). Conclusion: Our study demonstrated that trapezoidal wave chest compression may be beneficial in improving the success rate of ROSC and alleviate heart injury. Its possible mechanism might lie in its anti-oxidative stress capability during the chest compression.

Haar’ s Measure using Triangular Fuzzy Finite Topological Group

In this paper, A new approach is used to apply Haar’s measure theory to triangular fuzzy number theory for comprehending and generalizing the uniqueness of invariant measure when there are uncertainty and risk. If T ~ is a triangular fuzzy finite Topological group and X ~ is its subgroup, X ~ also being a triangular fuzzy number, then ) ~ ( (~) T x 

Preparation and characterization of invert ZnO–B2O3 glasses and its shielding behavior towards gamma irradiation

Binary zinc borate glasses with invert compositions (50–75 mol % ZnO) were prepared by melting and annealing technique. The glasses were collectively characterized by measuring and identifying their structural building groups through FTIR spectroscopy and their optical transparency beside also their thermal expansion parameters. FTIR spectra reveal compact vibrational bands within the mid region 400-1700 cm−1 reflecting both triangular and tetrahedral borate groups beside the sharing of ZnO as either modifier oxide or as B–O–Zn linkages and possibly ZnO4 groups. The optical absorption measurements reveal that the studied undoped ZnO–B2O3 glasses show distinct UV absorption extending from 200 to 370 nm with three peaks and these UV absorption peaks are related to unavoidable trace iron (Fe3+) impurities contaminated within the chemicals used for the preparation of the zinc borate glasses. The optical absorption spectra after gamma irradiation with a dose of 5 Mrad reveal the maintenance of the UV spectra identified before irradiation with the appearance of a minor curvature at 510 nm in the spectra of two samples and disappeared at high ZnO contents. Such optical spectral result after irradiated refers to suggested shielding effect towards gamma irradiation with high ZnO content in the studied invert compositions. Both the dilatometric softening temperatures and density data show obvious increase with the first increase of ZnO content up to 70% and then reveal constancy. This behavior is correlated with the mode of housing Zn2+ ions in the network structure which is assumed to form some ZnO4 groups at high percent of ZnO giving compactness to the network of such glasses.

Composition of permutation representations of triangle groups

A triangle group is denoted by and has finite presentationWe examine a method for composition of permutation representations of a triangle group that was used in Everitt’s proof of Higman’s 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations. In particular situations, where the representations being composed are all equivalent copies of an alternating group in the same degree containing at least one handle, then has a quotient . We also prove that if G has a quotient containing at least two handles, then G has a quotient for many . This article contains the main results of the author’s PhD thesis.

Maps Corresponding to the Subgroups $$\varGamma _0(N)$$ of the Modular Group

The normalizer $$\varGamma _B(N)$$ of $$\varGamma _0(N)$$ in $$PSL(2,{\mathbb {R}})$$ is the triangle group $$(2,3,\infty )$$ for $$N=2^\alpha 3^\beta $$ where $$\alpha =0,2,4,6$$; $$\beta =0,2$$. In this paper we examine relationship between the normalizer and the regular maps for these values. We define a family of subgroups of the normalizer and then we study maps with triangular faces using these subgroups and calculating the associated arithmetic structure.

Improving the Mechanism of Detecting and Measuring Holes in Ad hoc Wireless Sensor Network

Abstract Area detection and measuring is one of the most important problems in wireless sensor network because it mainly relates to the continuity and functionality of most routing protocols applied to the region of interest (ROI). Electronics failure, random deployment of nodes, software errors or some phenomena such as fire spreading or water flood could lead to wide death of sensor nodes. The damage on ROI can be controlled by detecting and calculating the area of the holes, resulting from the damaged sensor networks. In this paper, a new mathematical algorithm, wireless sensor hole detection algorithm (WHD), is developed to detect and calculate the holes area in ROI where the sensor nodes are spread randomly. WHD is developed for achieving quality of service in terms of power consumption and average hole detection time. The dynamic behavior of the proposed WHD depends on executing the following steps. Firstly, WHD algorithm divides down the ROI into many cells using the advantage of the grid construction to physically partition the ROI into many small individual cells. Secondly, WHD algorithm works on each cell individually by allocating the nearest three sensor nodes to each of the cell’s coordinates by comparing their positions, WHD connects each cell’s coordinate points with the selected sensor nodes by lines that construct a group of triangles, then WHD calculates the area of upcoming triangles. Repeating the previous step on all the cells, WHD can calculate and locate each hole in the ROI. The performance evaluation depends on the NS-2 simulator as a simulation technique to study and analyze the performance of WHD algorithm. Results show that WHD outperforms, in terms of average energy consumption and average hole discovery time, path density algorithm, novel coverage hole discovery algorithm and distriputed coverage hole Detection.

Order-disorder Twinning of Malachite

Abstract Malachite has a layered structure composed of triangular CO3 groups and CuO2(OH)2 coordination squares of two types. In 2D projection, each layer can be decomposed into two OD strips: the A strip, containing CO3 groups and square co-ordinations of Cu2, with idealized symmetry pgm2, and the B strip, without CO3 groups, with symmetry p2 (the b axis of malachite is the infinite strip direction). In the 3D structure, these strips form the ‘A' OD-layers with symmetry reduced by coordination requirements to approximately p2212, which alternate with the ‘B' OD-layers with layer symmetry . The B-layer can assume two positions, which are related by a 180° rotation axis running through the C1 and O1 atoms of the CO3 group in the A-layer. This axis is oriented close to [100]*. The fit of the A-layer and of the coplanar rotated B-layer is hindered by the presence of important partial gaps and overlaps at the boundary. Inclining these two structure portions towards one another, together with small positional adjustments, appears to alleviate the misfit problems; the two layer portions so related form an angle of about 124°, i.e., they form a twin relationship. This desymmetrization of the malachite structure away from the ideal model results in occasional twinning instead of the fully developed monoclinic polytype structure which is derived in this paper. In the structure of the Cu-Zn analogue, rosasite, the above-described match problem is solved in a different way, by shifting a Zn coordination octahedron (which replaces Cu2 in malachite) into the interlayer space. Malachite polytypes 1 and 2 and rosasite illustrate three different orientations of P21/a symmetry elements with respect to the structural layers, resulting in three different structure configurations.

Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

Fix an equilateral triangle group Ti=〈a,b;ai,bi,(ab)i〉 with i≥6 arbitrary. Our main result is: for every presentation P of every countable group Q there exists an HNN-extension TP of Ti such that Out(TP)≅Q. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of “malcharacteristic” subgroups.

Peroxo–tungstate(VI) complexes: syntheses, characterization, reactivity, and DFT studies

Three new oxodiperoxo–tungsten(VI) complexes containing benzene core carboxylic acids, viz., benzoic acid, 2-chlorobenzoic acid, and 3-aminobenzoic acid as co-ligands have been synthesized from reaction of Na2WO6H4, 30% H2O2 and the corresponding co-ligands in aqueous medium. The compounds have been comprehensively characterized by elemental analyses, FT-IR, 1H NMR, UV–Vis spectral studies as well as by mass spectrometric and TGA analyses. The infrared spectra suggest occurrence of terminally bonded W=O as well as triangular bidentate peroxo groups (C2v) and monodentate carboxylate group bound to the WO4+ center. The mass spectra of the compounds are in good agreement with proposed molecular formulations. Thermogravimetric analyses indicate the existence of both lattice and coordinated water molecules in the complexes. Density functional theory (DFT) calculations were used to compute the frequencies of relevant vibrational modes, electronic properties and also to investigate structure of the compounds. Compound potassium(aquo)(2-chlorobenzoato)oxodiperoxo–tungstate(VI)dihydrate acts as an oxidant for bromide ion in aqueous phase bromination of chosen organic substrates to their corresponding bromo-organics.