Spherical f-Tilings by (Equilateral and Isosceles) Triangles and Isosceles Trapezoids

Dihedral f-tilings by spherical parallelograms and spherical triangles were obtained in [3–5]. In this paper we extend these results presenting the study of all dihedral f-tilings of the sphere S ², whose prototiles are a spherical equilateral or isosceles triangle and a spherical isosceles trapezoid. The combinatorial structure, including the symmetry group of each tiling, is given in Table 1.     

The Number of Isosceles Right Triangles Determined by <Emphasis Type="Italic">n</Emphasis> Points in Convex Position in the Plane

Any set of n points in convex position in the plane induces at most 2n congruent copies of a fixed isosceles triangle. Furthermore, at most 2n–4 congruent isosceles right triangles can be induced by a set of n points in convex position, and in strictly convex position at most n congruent isosceles right triangles can be induced.     

Minimizing Neumann fundamental tones of triangles: An optimal Poincaré inequality

The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived.The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than π/3. Antisymmetry is proved for apertures greater than π/3.     

Error Bounds for the Large‐Argument Asymptotic Expansions of the Lommel and Allied Functions

In this paper, we reconsider the large‐z asymptotic expansion of the Lommel function and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re‐expansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger–Weber‐type functions, the Scorer functions, the Struve functions, and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.     

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle <90° admit locked chains, which is precisely the threshold beyond which the slender property no longer holds.     

Spherical f-Tilings by Scalene Triangles and Isosceles Trapezoids, Ⅱ

The study of the dihedral f-tilings of the sphere S ² whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.     

一类拟线性椭圆型方程的奇摄动

本文讨论了一类二阶拟线性椭圆型方程的奇摄动问题,给出了外部解和边界层项的N阶递推方程,并对余项进行了估计,从而导得了解的渐近展开式和摄动问题解的存在唯一性.     

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.     

Relative Newton numbers

We describe a method for estimating the number of congruent copies of a convex bodyC in the plane that can touch another given bodyK without having interior points in common. This applies also to the caseK=C thus giving bounds for the usual Newton numberN(K) ofK. Especially, we getN(T)=21 for the isosceles triangle having base angles of 30°.     

A new area-maximization proof for the circle

Conclusion  Finally, we compare the trianglesT _i with the isosceles triangles which would be obtained by slicing the circle by the same procedure. Since they all have the same short base length and the same angle opposite the base, the isosceles triangles composing the circle have more area than the trianglesT _i. Because the latter triangles cover all ofR ₁ (perhaps even with overlap and/or extension beyondR ₁), we see that the area of the quarter-circle is greater than that ofR ₁, and thus the circle's entire area is greater than that of our original regionR.     

Besicovitch triangles cover unit arcs

A long standing conjecture is that the Besicovitch triangle, i.e., an equilateral triangle with side is a worm-cover. We will show that indeed there exists a class of isosceles triangles, that includes the above equilateral triangle, where each triangle from the class is a worm-cover. These triangles are defined so that the shortest symmetric z-arc stretched from side to side and touching the base would have length one.      

伴有边界摄动的Volterra型积分微分方程组的奇摄动

讨论了伴有边界摄动的二阶非线性Volterra型积分微分方程组的奇摄动.在适当的条件下,利用对角化技巧证明了解的存在性,构造出解的渐近展开式并给出余项的一致有效估计.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

Some curious properties and loci problems associated with cubics and other polynomials

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of triangles associated with the family. Some fascinating connections were observed between the original family of the cubics and the loci of the centroids of these triangles. For example, we were able to prove that the locus of the centroid of certain triangles associated with the family of cubics is another cubic whose zeros are in arithmetic progression. Motivated by this, in the last section of the article, we considered families of cubic polynomials whose zeros are in arithmetic progression, along with the loci of the special points of certain triangles arising from such families. Special points include the centroid, circumcentre, orthocentre, and nine-point centre of the triangles. Throughout the article, we used the computer algebra system, Mathematica®, to form conjectures and facilitate calculations. Mathematica® was also used to create various animations to explore and illustrate many of the results.     

The operator formula for monotone triangles - simplified proof and three generalizations

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of −1s in the matrix when prescribing (1,2,…,n) as the bottom row of the monotone triangle.     

Numerical integrations over an arbitrary quadrilateral region

In this paper, double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method. The physical region is transformed into a standard quadrilateral finite element using the basis functions in local space. Then the standard quadrilateral is subdivided into two triangles, and each triangle is further discretized into 4 × n² right isosceles triangles, with area , and thus composite numerical integration is employed. In addition, the affine transformation over each discretized triangle and the use of linearity property of integrals are applied. Finally, each isosceles triangle is transformed into a 2-square finite element to compute new n² extended symmetric Gauss points and corresponding weight coefficients, where n is the lower order conventional Gauss Legendre quadratures. These new Gauss points and weights are used to compute the double integral. Examples are considered over an arbitrary domain, and rational and irrational integrals which can not be evaluated analytically.     

Error bounds for the uniform asymptotic expansion of the incomplete gamma function

We derive simple, explicit error bounds for the uniform asymptotic expansion of the incomplete gamma function Γ(a,z) valid for complex values of a and z as |a|→∞. Their evaluation depends on numerically pre-computed bounds for the coefficients c_k(η) in the expansion of Γ(a,z) taken along rays in the complex η plane, where η is a variable related to z/a. The bounds are compared with numerical computations of the remainder in the truncated expansion.     

Error bounds for asymptotic approximations of the partition function

We consider Markovian queueing models with a finite number of states and a product form solution for its steady state probability distribution. Starting from the integral representation for the partition function in complex space we construct error bounds for its asymptotic expansion obtained by the saddle point method. The derivation of error bounds is based on an idea by Olver applicable to integral transforms with an exponentially decaying kernel. The bounds are expressed in terms of the supremum of a certain function and are asymptotic to the absolute value of the first neglected term in the expansion as the large parameter approaches infinity. The application of these error bounds is illustrated for two classes of queueing models: loss systems and single chain closed queueing networks.     

Some bounds for incomplete gamma function

We establish some upper and lower bounds for the incomplete Gamma function Φ(a, x), with 0≤a<1 andx→+∞, by modifying the terms of its asymptotic expansion with a suitable parameter. Work sponsored by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica of Italy.     

Large Sets and Overlarge Sets of Triangle-Decomposition

There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even.