Acute Triangles Research Articles

Intermediate spectral statistics of rational triangular quantum billiards.

Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle, which permits one to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion, and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short-range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

Plume active control in high-power fiber laser welding based on high-speed shielding gas microbeam

Plume is an inherent physical phenomenon in fiber laser keyhole welding, negatively affecting welding quality. This paper proposes a new type of high-speed shielding gas microbeam (HSGM) based on the limitation of conventional shielding gas regulating plumes. The gas flow characteristics of the HSGM on the molten pool surface were explored using numerical simulation. Then, the coaxial double-layer shielding nozzle was trial-manufactured to clarify the regulation law for the HSGM on the plume. The gas velocity and pressure produced by the HSGM on the molten pool surface were positively correlated with the shielding gas flow rate, and the gas flow pattern was an acute triangle shape. The suitable HSGM can significantly restrain plume formation, improve the weld surface forming quality, increase the weld penetration depth (about 15.4%), and reduce the mass loss of the weld (about 58.7%). The significant blowing pressure effect of HSGM on plumes and splashes was the key to regulating the welding process. Compared with conventional shielding gas, the use of HSGM was significantly reduced.

Parallel Covering an Obtuse Triangle with Squares

Suppose that 𝑇 (𝛼, 𝛽) is an obtuse triangle with base length 1 and with base angles 𝛼 and 𝛽 (where 𝛽 > 90◦). In this note a tight lower bound of the sum of the areas of squares that can parallel cover 𝑇 (𝛼, 𝛽) is given. This result complements the previous lower bound obtained for the triangles with the interior angles at the base of the measure not greater than 90◦.

Assisting autonomous precision medical rehabilitation through gesture recognition devices

With the aging of the population and the increase of various chronic diseases, there is a growing demand for effective and accessible medical rehab solutions. In order to overcome the above problems, this project innovatively adopts the gesture recognition technology to design and fabricate the realization of an autonomous accurate medical rehab gesture recognition assistance system. In terms of hardware, this paper uses Arduino as the main controller, Paj7620 as the gesture recognition device, and OLED 0.96 display as the system display. In terms of software, this paper utilizes Arduino IDE software to successfully write the code and realize all the functions. In terms of combining with actual medical rehab needs, the team chose elbow fracture rehab as a typical type of rehab, and designed a three-device obtuse triangle placement scheme for precise monitoring of rehab training. Initial simulation test results show that over 70% of participants reported an improvement in their rehabilitation outcomes, and about 90% found the system easy to use. The results of this project will directly benefit patients requiring long-term rehabilitation.

Adaptively Isotropic Remeshing Based on Curvature Smoothed Field.

With the development of 3D digital geometry technology, 3D triangular meshes are becoming more useful and valuable in industrial manufacturing and digital entertainment. A high quality triangular mesh can be used to represent a real world object with geometric and physical characteristics. While anisotropic meshes have advantages of representing shapes with sharp features (such as trimmed surfaces) more efficiently and accurately, isotropic meshes allow more numerically stable computations. When there is no anisotropic mesh requirement, isotropic triangles are always a good choice. In this paper, we propose a remeshing method to convert an input mesh into an adaptively isotropic one based on a curvature smoothed field (CSF). With the help of the CSF, adaptively isotropic remeshing can retain the curvature sensitivity, which enables more geometric features to be kept, and avoid the occurrence of obtuse triangles in the remeshed model as much as possible. The remeshed triangles with locally isotropic property benefit various geometric processes such as neighbor-based feature extraction and analysis. The experimental results show that our method achieves better balance between geometric feature preservation and mesh quality improvement compared to peers. We provide the implementation codes of our resampling method at github.com/vvvwo/Adaptively-Isotropic-Remeshing.

The Equality of the Values of the Area and Perimeter of a Number of Two– Dimensional Figures, Volume and Area - Three-Dimensional

Possible variants of the equality of the values of the area and perimeter of a number of two–dimensional figures (square, circle, rectangular, obtuse and equilateral triangles), volume and area - three-dimensional (Platonic bodies, cone, cylinder, pyramid and sphere) are considered.

Ultrafast Imaging of Jahn-Teller Distortion and the Correlated Proton Migration in Photoionized Cyclopropane.

The Jahn-Teller (JT) distortion is one of the fundamental processes in molecules and condensed phase matters. For photoionized organic molecules with high symmetry, the JT effect leads to geometric instability in certain electron configurations and thus has a significant effect on the subsequent isomerization and proton migration processes. Utilizing the femtosecond pump-probe Coulomb explosion method, we probe the isomerization dynamics process of a monovalent cyclopropane cation (C3H6+) caused by proton migration and reveal the relationship between proton migration and JT distortion. We found that the C3H6+ cation evolves from the D3h symmetric equilateral triangle geometry either to the acute triangle via two elongated C-C bonds (JT1) or to the obtuse triangle via a single elongated C-C bond (JT2). The JT1 pathway does not involve proton migration, while the JT2 pathway drives proton migration and can be mapped into the indirect dissociation channel of Coulomb explosion. The time-resolved experiment indicates that the delay time between those two JT pathways can be as large as ∼600 fs. After the JT distortion, the cyclopropane cation undergoes a subsequent structural evolution, which brings a greater variety of dissociation channels.

Biomechanical Research of Three Parallel Cannulated Compression Screws in Oblique Triangle Configuration for Fixation of Femoral Neck Unstable Fractures.

Surgical treatment with internal fixation, specifically percutaneous fixation with three cannulated compression screws (CCSs), is the preferred choice for young and middle-aged patients. The mechanical advantage of the optimal spatial configuration with three screws provides maximum dispersion and cortical support. We suspect that the spatial proportion of the oblique triangle configuration (OTC) in the cross-section of the femoral neck isthmus (FNI) may significantly improve shear and fatigue resistance of the fixed structure, thereby stabilizing the internal fixation system in femoral neck fracture (FNF). This study aims to explore the mechanical features of OTC and provide a mechanical basis for its clinical application. Twenty Sawbone femurs were prepared as Pauwels type III FNF models and divided equally into two fixation groups: OTC and inverted equilateral triangle configuration (IETC). Three 7.3 mm diameter cannulated compression screws (CCSs) were used for fixation. The specimens of FNF after screw internal fixation were subjected to static loading and cyclic loading tests, respectively, with five specimens for each test. Axial stiffness, 5 mm failure load, ultimate load, shear displacement, and frontal rotational angle of two fragments were evaluated. In the cyclic loading test, the load sizes were 700 N, 1400 N, and 2100 N, respectively, and the fracture end displacement was recorded. Results were presented as means ± SD. Data with normal distributions were compared by the Student's t test. In the static loading test, the axial stiffness, ultimate load, shear displacement, and frontal rotational angle of two fragments were (738.64 vs. 620.74) N/mm, (2957.61 vs. 2643.06) N, (4.67 vs. 5.39) mm, and (4.01 vs. 5.52)° (p < 0.05), respectively. Comparison between the femoral head displacement after 10,000 cycles of 700N cyclic loading and total displacement after 20,000 cycles of 700-1400N cyclic loading showed the OTC group was less than the IETC group (p < 0.05). A comparison of femoral head displacement after 10,000 cycles of 1400N and 2100N cycles and total displacement after 30,000 cycles of 700-2100N cycles showed the OTC group was less than another group, but the difference was not significant (p > 0.05). When three CCSs are inserted in parallel to fix FNF, the OTC of three screws has obvious biomechanical advantages, especially in shear resistance and early postoperative weight-bearing, which provides a mechanical basis for clinical selection of ideal spatial configuration for unstable FNF.

Photonic generation of microwave waveform using delay feedback photonic reservoir computing system with SOA

This paper proposes and demonstrates a model for generating microwave waveforms using a delay feedback photonic reservoir computing system with SOA. Our scheme involves building a photonic reservoir computer with SOA, which provides rich nonlinearity functionalities and memory capability for the reservoir. The automatic optimization of output waveform is implemented through the incorporation of gradient descent regression and a reservoir output layer. In theory, the system is capable of generating arbitrary microwave waveforms towards the target automatically. In our work, the capacity of the model is tested through theoretical analysis and simulation. We also compared the optimization effects of several regression algorithms applied to the reservoir computing, and generated six typical microwave waveforms, including triangle, square, sawtooth, negative, oblique triangle 1, and oblique triangle 2.

Algebraic model studies of the structure and reactions of 12C and 16O

The role of algebraic models and geometric cluster models of alpha clusters in the structure of ground and first excited 0+ and second excited 2+ states in 12C are evaluated with the aim of establishing if the rotational bands are compatible with rigid structures or if they are instead overlaps of different configurations. A rigid equilateral triangle shape is assumed in Ref.[2, 3]: the energy spectrum, electromagnetic observables are calculated and the transition densities are compared with elastic and inelastic cross-sections for various processes. In a different approach [4, 5], we solve the three-body quantum problem for 3 alphas using the orthogonality condition and the stochastic variational method with correlated Gaussian basis functions. The two-body density distributions indicate that the main configurations of both the second 0+ and 2+ states are acute iscosceles triangle shapes with a 8Be(0+)+α configuration. We also find clues that the second 2+ state is probably not a member of the rigid rotational band built on the Hoyle state.

Ultrasonography as a Tool to Improve Preoperative Marking in Body Contour Surgery.

Body contouring surgery and surgical drawings are usually performed based on the surgeon's experience without considering the nature of the unique anatomical structures of each patient. Thus, we propose a more reliable surgical design approach that considers the anatomical structures of each patient. Ultrasonography is increasingly used in plastic surgery and helps plastic surgeons to highlight anatomical features representing results in their interventions by providing a better understanding of the patients' unique structures. This study presents a series of cases involving 100 recruited patients (36 men and 64 women) between 18 and 60 years of age. Five surgeons examined the patients and created a presurgical design based on palpation, which was validated later by a physician skilled in evaluating the anterior wall of the abdomen using ultrasonography. The concordance between the findings of the palpation and ultrasonography was assessed for each patient. The concordance rate for each structure in both evaluations was midline (49%) (P > 0.92), diastasis recti (15%), semilunar line (23%), upper edge of rectus abdominis muscles (12%), lower edge of pectoral muscles (16%), border of oblique muscle (13%), number of tendinous intersections (12%), shape of tendinous intersections (11%), serratus anterior muscle (15%), subcostal triangle (15%), and oblique triangle (26%) (P < 0.0001). All the structures evaluated by palpation in comparison with ultrasound show discordance, except the midline where agreement is evident, with a very good level of statistical significance.

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.

Structural evolution, electronic and magnetic properties investigation of V3Sin− (n = 14–18) clusters based on photoelectron spectroscopy and density functional theory calculations

Silicon clusters doped with a transition metal (TM) atom have been intensively studied due to their novel properties and superior stability for possible usage as building blocks of future nanoelectronics. Besides the selection of the doped element, introducing the number of dopant atoms as another dimension adds versatility to tuning the cluster properties. However, there are few studies of silicon clusters doped with multiple TM atoms. Here, we present a study of V3Sin− (n = 14–18) by combining high-resolution photoelectron spectroscopy (PES) measurements with a search of their global minimum energy structures based on a homemade genetic algorithm coupled with density functional theory (DFT) for energy calculations. The simulated photoelectron spectra of the putative global minimum structures are in fair agreement with the experimental ones, which gives evidence for their authenticity as the ground-state structures. In these clusters the three V atoms always bond with each other and form an acute triangle, which can be seen as a nanoscale analogue of phase segregation in the bulk. For the size range n = 14–17, each of the ground-state structures has a silicon basket-like structure with the vanadium triangle inside, while V3Si18− has a baseball-mitt-like structure almost completely encapsulating the V3 triangle. The average binding energy of TMSin− (TM = V1, V2, V3) indicates that the more V atoms are doped the more stable the clusters are. Among these clusters, only V3Si14− have a total magnetic moment of 2 μB, making it a potential structural unit for magnetic storage devices. Mulliken population analysis and the electron spin density show that V atoms at different positions have different contributions to the total magnetic moments, and d electrons in V atoms contribute the most.

Two inequalities involving circumradius, inradius and medians of an acute triangle

Two inequalities involving circumradius, inradius and medians of an acute triangle

Radiation Performance of Different Triangular Microstrip Patch Antenna Configuration Shapes Operating at 28 GHz

The radiation performance of different triangular microstrip patch antenna (TMPA) shapes such as right triangle (RTMPA), isosceles triangle (ITMPA), obtuse triangle (OTMPA), and equilateral triangle (ETMPA) operating at (28 GHz) are computed and compared using inset-fed techniques and Rogers-RT5880 substrate material of permittivity and (h=0.15 mm) height. The directivity, gain, efficiency, bandwidth, VSWR, S11 and 2D-radiation pattern for each mentioned triangular patch shapes are computed utilizing CST and HFSS method. The computed results reveal generally that the ETMPA provide better radiation performance whereas the OTMPA displays lower antenna radiation parameter values compared to the other considered ones. In addition, the antenna parameters of ETMPA with the use of coaxial probe fed are also simulated and the results are compared to those previously achieved experimentally and theoretically by other researchers. Generally, a good agreement between mentioned antenna parameter results is displayed and reliability of those achieved by CST with inset fed techniques is clearly observed. Moreover, the overall antenna parameter obtained, respectively, with CST and HFSS techniques for inset fed ETMPA are S11 (-28.68, -20.64), VSWR (1.076,1.20), gain (5.82, 6.29) dB, directivity (6.85, 7.09) dB, bandwidth (0.452, 0.369) GHz, efficiency (78.9%, 83.2%) and with a small antenna size of about (3.88 mm3) which is most reliable for 5G technology application systems.

Parallel Packing Squares into an Obtuse Triangle

Parallel Packing Squares into an Obtuse Triangle

New accurate algorithms of circularity evaluation

The minimum circumscribed circle (MCC), maximum inscribed circle (MIC), and minimum zone circle (MZC) methods for circularity evaluation are difficult to execute due to the lack of specific rules in mathematics, especially the MZC. New accurate algorithms have been proposed to realize the MIC, MCC and MZC evaluation methods. First, the diameter criterion and acute triangle criterion of control points defining the MIC or MCC are presented. The definition of the crossing sector structure is introduced in the minimum zone criterion and transformed into an angular relationship of control points, making it easy to identify the MZC. Then the key points of the algorithms are presented in details including the initial circle or the initial region, centre moving direction, minimum step size and updating rules of control points. Flow charts are provided to make the algorithm producible. Finally, four cases are selected to demonstrate the advantages of the algorithms. The results show that the proposed MZC algorithm is more general than that in the published reference. Compared to the traditional ergodic searching algorithms, the efficiency of the new algorithm for the MIC, the MCC and the MZC evaluations of circularity can be improved by 100% when the point set increases to 100 and above.

Ideal Configuration Distribution of Femoral Neck Cannulated Screw in High Altitude Population: A Finite Element Study

For people in high altitude areas, the shape of the femoral neck is different due to the different living environment. To deduce the optimal insertion angle and position of the cannulated screw using three-dimensional finite element technology in patients at high altitude. The data of 100 volunteers were used to establish a finite element model. The stress and displacement of cannulated screws of equilateral and oblique triangle screw placement in femoral neck fracture model and complete femoral neck model were evaluated. On the narrowest plane, the average values of ∠A, B, and C were 47.63o, 75.49o, and 56.88o, respectively, and the shape was an oblique triangle. In complete femoral neck model, the maximum Von Mises stress of the three cannulated screws of equilateral triangle screw placement was slightly larger than oblique triangle screw placement, and this difference was more obvious in femoral neck fracture model. Under same loads, the overall maximum displacement of femur in oblique triangle screw placement was less than equilateral triangle screw placement in two models. For people in high-altitude areas, the three screws should be implanted in an oblique triangle configuration in cannulated screw treatment of femoral neck fractures.

Evaluation of earth observation datasets for LST trends over India and its implication in global warming

Land surface temperature (LST) is crucial in surface energy balance, urban climatology, intensifying global change, ecological and environmental concerns. The present study examined the LST trends and spatio-temporal variation over India from 2002 to 2022. This includes comparison of LST for the summer and winter seasons over two decades. Secondly, the present study examined the LULC category wise LST variability during the day and night-time using MODIS (The Terra Moderate Resolution Imaging Spectroradiometer) derived products. This study explored the feasibility of cloud computing for big data analysis for LST distribution in seven landuse categories over India, providing a conceptual response to global warming. Results showed the existence of spatial LST variation due to changes in land-use patterns and MODIS derived vegetation indices- NDVI. Daytime LST for the summer and winter seasons of 2002 was found to be 45.17 °C and 39.13 °C, respectively. Outcomes illustrate declining trends in range (LSTmin-LSTmax) for winter seasons, initially, it was observed as 56.29 °C for 2002 while later on it was observed to be 20.21 °C and 20.87 °C for 2021 and 2022 respectively. The LSTmin (summer) has shown an increasing trend towards upper LST values, from −3.01 °C to 12.21 °C from 2002 TO 2022. LSTmin_winter has shown a rising trend towards the upper LST values from −17.16 °C to 9.15 °C in 2022. The maximum LST for the DRs was observed to be 61.56 °C, followed by UR as 56.24 °C. The findings demonstrate that daytime LSTmin and LSTmax are found to be 19.50 °C to 56.24 °C for UR, 29.5 °C to 61.56 °C for DR, 19.24 °C to 54.08 °C for SAR, −21.1 °C to 0.05 °C for SCR and 15 °C to 32.18 °C for FHR. NDVI-LST (daytime, nighttime and diurnal temperature range) feature space generates an obtuse triangle and depicts a negative correlation of vegetation for a few LULC categories. The outcomes indicated that desert and snow regions have highest LSTmax followed by urban and semiarid regions during daytime. During the nighttime, desert and urban regions have the highest temperature followed by semi-arid and forest regions. The outcomes support the efficacy of earth observation datasets and help to facilitate a better understanding of LULC and its impact on regional climate.

A proof of the triangular Ashbaugh–Benguria–Payne–Pólya–Weinberger inequality

In this paper, we show that for all triangles in the plane, the equilateral triangle maximizes the ratio of the first two Dirichlet–Laplacian eigenvalues. This is an extension of work by Siudeja (2010), who proved the inequality in the case of acute triangles. The proof utilizes inequalities due to Siudeja and Freitas (2010), together with improved variational bounds.