In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications emerged and became widely accepted, which generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAEs systems with large dimension, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on its Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameter has been presented.It leads to a block Pantelides’ algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples. And numerical experiments show that the time complexity of BPA can be reduced by at least O(ℓ) compared to the MPA, which is mainly consistent with the results of our analysis.

The main ideas of F-transform came from representing expert rules. It would be therefore re reasonable to expect that the more accurately the membership functions describe human reasoning, the more efficient will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions -- however, somewhat surprisingly, most efficient applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon which are based on local behavior of the signal. In this paper, we supplement this local explanation by a global one: namely, we prove that triangular membership functions are the only one that provide the accurate description of appropriate global characteristics of the signal.

Intuitionistic fuzzy set, which can be represented using the triangular intuitionistic fuzzy number (TIFN), is a more generalized platform for expressing imprecise, incomplete and inconsistent information when solving multi-criteria decision-making problems, as well as for reflecting the evaluation information exactly in different dimensions. In this paper, the TIFN has been applied for solving some multi-criteria decision-making problems by developing a new triangular intuitionistic fuzzy geometric aggregation operator, that is the generalized triangular intuitionistic fuzzy ordered weighted geometric averaging (GTIFOWGA) operator, and defining some triangular intuitionistic fuzzy geometric aggregation operators including the triangular intuitionistic fuzzy weighted geometric averaging (TIFWGA) operator, the ordered weighted geometric averaging (TIFOWGA) operator and the hybrid geometric averaging (TIFHWGA) operator. Based on these operators, a new approach for solving multicriteria decision-making problems when the weight information is fixed has been proposed. Finally, the proposed method has been compared with some similar existing computational approaches by virtue of a numerical example to verify its feasibility and rationality.

This paper proposed a triangular inequality-based rewiring method for the Rapidly exploring Random Tree (RRT)-Connect robot path-planning algorithm that guarantees the planning time compared to the RRT algorithm, to bring it closer to the optimum. To check the proposed algorithm’s performance, this paper compared the RRT and RRT-Connect algorithms in various environments through simulation. From these experimental results, the proposed algorithm shows both quicker planning time and shorter path length than the RRT algorithm and shorter path length than the RRT-Connect algorithm with a similar number of samples and planning time.

To solve the problem that sampling-based Rapidly-exploring Random Tree (RRT) method is difficult to guarantee optimality. This paper proposed the Post Triangular Processing of Midpoint Interpolation method minimized the planning time and shorter path length of the sampling-based algorithm. The proposed Post Triangular Processing of Midpoint Interpolation method makes a closer to the optimal path and somewhat solves the sharp path problem through the interpolation process. The experiments were conducted to verify the performance of the proposed method. Applying the method proposed in this paper to the RRT algorithm increases the efficiency of optimization compared to the planning time.

This paper studied the natural convection heat transfer from two equal-sized cylinders of triangular cross-section confined in triangular enclosure. The inner cylinders have hot surfaces and the outer enclosure has a cold surfaces. The annular space between inner cylinders and the outer one was filled with incompressible Newtonian fluid. The work examined the behavior of fluid flow in the process of transferring heat energy. The work was achieved numerically using the commercial code ANSYS-CFX. The adopted initial conditions were selected for these ranges: Prantl number (Pr = 1 to 10³), Rayleigh number (Ra = 10³ to 10⁵). It was found that the increment of Ra number increases the rate of heat transfer. However, the effect of Pr number on heat transfer is almost negligible.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

Background: Post-operative pain and swelling are frequently observed after sinus lift procedures. The aim of the present study was a clinical evaluation of swelling and pain of two sinus lift different flaps technique measured through visual analogue scale (VAS), verbal rating scale (VRS) and infrared thermal imaging (i.e., thermography). Methods: A randomized controlled trial was conducted in fifteen patients treated for a total of 30 sinus randomly allotted into two groups. In patients of Group I, a triangular flap was performed, while in Group II modified triangular flap without anterior release was performed. Postoperative pain was scored by VAS and inflammation was recorded by VRS at 2, 4, 6 and 14 days. The facial temperature was recorded before and after sinus augmentation, at 2, 4, 6 day intervals for the first week and at 14 day, to check the course of healing. Results: In group I the pain intensity was recorded at 2 days after the surgery with a score of 38.67 ± 6.4 mm. Swelling was higher at 2 and 4 day, and it was absent at day 6. Facial temperature difference before and after the procedure was 3 (4.737 °C ± 0.37). In Group II the pain was lower than in Group I (p < 0,05). Swelling scored 2 on the first and second day, and it was reduced at day 4. After the second day the difference of temperature reduced significantly compared to day of surgery (0.77 °C), at 2 and 4 days no difference was registered. Conclusion: In conclusion based on the outcome of the present study, modified triangular flap can be used with success in sinus augmentation procedures and is to be preferred to trapezoidal flap.

This work presents a linear smoothing scheme over high-order triangular elements in the framework of a cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme for strain-smoothed finite element method (S-FEM) is no subdivision of finite element cells to sub-cells while the classical S-FEM needs sub-cells. Since the linear smoothing function is employed, S-FEM is able to use quadratic triangular or quadrilateral elements. The modified smoothed matrix obtained node-wise is evaluated. In the same manner with the computation of the strain-displacement matrix, the smoothed stiffness matrix and deformation graident are obtained over smoothing domains. A series of benchmark tests are investigated to demonstrate validity and stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using 2nd-order triangular element and the exact solutions.

Management practices must be developed to improve yam production sustainability. Image-based phenotyping techniques could help developing such practices based on non-destructive analyses of important plant traits. Our objective was to determine the potential of image-based phenotyping methods to assess traits relevant for tuber yield formation in yam grown in glasshouse and field. We took plant and leaf pictures with consumer cameras. We used the numbers of image pixels to derive the shoot biomass and the total leaf surface and calculated the ‘triangular greenness index’ (TGI) which is an indicator of the plant nitrogen (N) nutritional status. Under glasshouse conditions, the number of pixels obtained from nadir view (image taken top down) was positively correlated to the shoot biomass, and the total leaf surface, while the TGI was negatively correlated to the N content of diagnostic leaves. Under field conditions, pictures taken from the nadir view showed an increase in soil surface cover and a decrease in TGI with time. TGI was negatively correlated to SPAD measured on specific leaves but was not correlated to the N content of these leaves. In conclusion, these phenotyping techniques deliver relevant results but need to be further developed and validated for application in yam.

A triangular lattice model for pattern formation by core-shell particles at fluid interfaces is introduced and studied for the particle to core diameter ratio equal to 3. Repulsion for overlapping shells and attraction at larger distances due to capillary forces are assumed. Ground states and thermodynamic properties are determined analytically and by Monte Carlo simulations for soft outer- and stiffer inner shells, with different decay rates of the interparticle repulsion. We find that thermodynamic properties are qualitatively the same for slow and for fast decay of the repulsive potential, but the ordered phases are stable for temperature range depending strongly on the shape of the repulsive potential. More importantly, there are two types of patterns formed for fixed chemical potential - one for a slow and another one for a fast decay of the repulsion at small distances. In the first case two different patterns - for example clusters or stripes - occur with the same probability for some range of the chemical potential. For fixed concentration, an interface is formed between two ordered phases with the closest concentration, and the surface tension takes the same value for all stable interfaces. In the case of degeneracy, a stable interface cannot be formed for one out of four combinations of the coexisting phases, because of a larger surface tension. Our results show that by tuning the architecture of a thick polymeric shell, many different patterns can be obtained for sufficiently low temperature.

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.