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In this paper we show that the area of any quadrilateral can be estimated from the four lengths sides. With the Triangle Inequality Theorem and a novel provided diagonal's formula, the boundaries of quadrilateral diagonals are found. Finally, Bretschneider's formula can be applied to find a set of possible areas.

An early two-dimensional element iQ8 exhibits excellent performance, but its three-dimensional counterpart has not yet been reported. This study focuses on extending iQ8 to three dimensions. Initially, a geometric extension is attempted based on the concept of a virtual node, but all three cases fail, yielding no similar results to the two-dimensional scenario. Next, the underlying nonconforming modes are derived by reorganizing the nonconforming displacement of iQ8; these modes, combined with a special 9-point integration rule, successfully facilitate the extension, with the non-central integration points located outside the element domain. The numerical results indicate that the three-dimensional version performs well when the element aspect ratio is moderate. For plate and shell problems that experience self-locking due to large aspect ratios, a meaningful finding shows that moving the integration points closer to the element domain can quickly alleviate the locking issue and yield surprisingly favorable outcomes.

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

We calculate the modulus of curve families inside a hyperbolic quadrilateral and a hyperbolic annulus.

A full solution to the recently proposed problem of determining the probability that no \\(k\\)-gon can be built from \\(n\\) independently and uniformly chosen sticks in \\([0,1]\\) is proposed. This extends the known results for triangles and quadrilaterals to general \\(k\\)-gons and offers a clearer interpretation of the connection to products of \\(k\\)-bonacci numbers.

Three-term relations of the form AB+CD=EF arise in multiple mathematical contexts, including the Ptolemy equation for a cyclic quadrilateral, Casey's theorem on bitangents, Penner's relation for lambda lengths, and Pl\"ucker's identity for the maximal minors of a 2x4-matrix. In this note, we explain how these different occurrences of the 3-term relation can be directly obtained from each other.

Via continuous deformations based on natural flow evolutions, we prove several novel monotonicity results for Riesz-type nonlocal energies on triangles and quadrilaterals. Some of these results imply new and simpler proofs for known theorems without relying on any symmetrization arguments.

Given a symmetric social network, we are interested in testing whether it has only one community or multiple communities. The desired tests should (a) accommodate severe degree heterogeneity, (b) accommodate mixed memberships, (c) have a tractable null distribution and (d) adapt automatically to different levels of sparsity, and achieve the optimal phase diagram. How to find such a test is a challenging problem. We propose the Signed Polygon as a class of new tests. Fixing m ≥ 3, for each m-gon in the network, define a score using the centered adjacency matrix. The sum of such scores is then the mth order Signed Polygon statistic. The Signed Triangle (SgnT) and the Signed Quadrilateral (SgnQ) are special examples of the Signed Polygon. We show that both the SgnT and SgnQ tests satisfy (a)–(d), and especially, they work well for both very sparse and less sparse networks. Our proposed tests compare favorably with existing tests. For example, the EZ and GC tests behave unsatisfactorily in the less sparse case and do not achieve the optimal phase diagram. Also, many existing tests do not allow for severe heterogeneity or mixed memberships, and they behave unsatisfactorily in our settings. The analysis of the SgnT and SgnQ tests is delicate and extremely tedious, and the main reason is that we need a unified proof that covers a wide range of sparsity levels and a wide range of degree heterogeneity. For lower bound theory, we use a phase transition framework, which includes the standard minimax argument, but is more informative. The proof uses classical theorems on matrix scaling.