杨辉三角矩阵(英文)

杨辉三角矩阵的性质吸引许多研究者．本文进一步研究杨辉矩阵,获得一些颇为有趣的性质     

Triangular and parametric approximations of fuzzy numbers—inadvertences and corrections

Recent papers were dedicated to approximate fuzzy numbers by triangular, trapezoidal or parametric fuzzy numbers, with or without additional conditions. Unfortunately, the result of approximation is not always a fuzzy number, sometimes it is not a fuzzy set. We point out the wrongs and inadvertences in some recent papers, then we correct the results.     

The lattice-theoretic structure of the sets of triangular norms and semi-copulas

The lattice-theoretic structure of the set of semi-copulas and some of its subsets is investigated. As a relevant case, we obtain that the set of commutative semi-copulas is a lattice completion of the set of triangular norms.     

On the Hochschild Cohomology of Triangular Algebras

Abstract

In order to study the Hochschild cohomology of an n-triangular algebra 𝒯 _n , we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with 𝒯 _n , and which converges to the Hochschild cohomology of 𝒯 _n . We describe explicitly its components and its differentials which are sums of cup products. In case n = 3 we study some properties of the differential at level 2. We give some examples of use of the spectral sequence and recover formulas for the dimension of the cohomology groups of particular cases of triangular algebras.     

On the adjacencies of triangular meshes based on skeleton-regular partitions

For any 2D triangulation τ, the 1-skeleton mesh of τ is the wireframe mesh defined by the edges of τ, while that for any 3D triangulation τ, the 1-skeleton and the 2-skeleton meshes, respectively, correspond to the wireframe mesh formed by the edges of τ and the “surface” mesh defined by the triangular faces of τ. A skeleton-regular partition of a triangle or a tetrahedra, is a partition that globally applied over each element of a conforming mesh (where the intersection of adjacent elements is a vertex or a common face, or a common edge) produce both a refined conforming mesh and refined and conforming skeleton meshes. Such a partition divides all the edges (and all the faces) of an individual element in the same number of edges (faces). We prove that sequences of meshes constructed by applying a skeleton-regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements, faces, edges and vertices of the nth and (n−1)th meshes. By using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes (number of triangles by node and number of tetrahedra by vertex) is constant when n goes to infinity. We relate these results with the non-degeneracy properties of longest-edge based partitions in 2D and include empirical results which support the conjecture that analogous results hold in 3D.     

Algebra structures on hom(C,L)

We consider the space of linear maps from a coassociative coalgebra C into a Lie algebra L. Unless C has a cocommutative coproduct, the usual symmetry properties of the induced bracket on Hom(C L) fail to hold. We define the concept of twisted domain (TD) algebras in order to recover the symmetries and also construct a modified Chevalley-Eilenbcrg complex in order to define the cohomology of such algebras.     

Finite Eulerian posets which are binomial, Sheffer or triangular

In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows:

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We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets.

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We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases.

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In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets.

We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.     

Effective algorithms for computing triangular operator in Schubert calculus

We develop two parallel algorithms progressively based on C++ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.     

模糊关系性质的模糊T-近似算子刻画

在定义一般模糊关系下的模糊T-近似的基础上，用模糊T-近似算子来刻画模糊关系的自反性、对称性、T-传递性等性质，并讨论了模糊T-粗糙集的一些性质。     

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge-Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.     

广义F积分的推广

减弱吴从[4]的广义三角模的条件,给出广义三角半模的概念,并以此定义广义半模F积分,使其是广义F积分的推广,指出关于广义F积分的主要性质对广义半模F积分仍然成立。     

On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the 2-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.     

Biderivations of triangular algebras

Let be a triangular algebra. A bilinear map is called a biderivation if it is a derivation with respect to both arguments. In this paper we define the concept of an extremal biderivation, and prove that under certain conditions a biderivation of a triangular algebra is a sum of an extremal and an inner biderivation. The main result is then applied to (block) upper triangular matrix algebras and nest algebras. We also consider the question when a derivation of a triangular algebra is an inner derivation.     

Possibility measures and possibility integrals defined on a complete lattice

We consider the definition of possibility measures on complete lattices rather than on complete Boolean algebras of sets. We give a necessary and sufficient condition for the extendability of any mapping to such a possibility measure. We also associate two types of integrals with these possibility measures, and discuss some of their more important properties, amongst which a monotone convergence theorem.     

Hamiltonian properties of triangular grid graphs

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly-convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearly-convex hull of the Star of David. We extend this result to a wider class of locally connected triangular grid graphs. Namely, we prove that all connected, locally connected triangular grid graphs (with the same exception of graph D) are hamiltonian. Moreover, we present a sufficient condition for a connected graph to be fully cycle extendable. We also show that the problem Hamiltonian Cycle is NP-complete for triangular grid graphs.     

Bit-size estimates for triangular sets in positive dimension

We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm.This extends the results by the first and the last author, which were confined to the case of dimension 0. Our strategy is to get back to dimension 0 by evaluation and interpolation techniques. Even though the main tool (height theory) remains the same, new difficulties arise to control the growth of the coefficients during the interpolation process.     

Adaptive cubature over a collection of triangles using the d-transformation

We describe an automatic routine to integrate a function over a collection of triangles where on each triangle the function is well behaved, or has singularities of certain types at one or more vertices or edges. The underlying algorithm is globally adaptive and incorporates the d-transformation for extrapolation. Results from performance profile testing indicate that the routine is superior to other published routines when the singularities are located along edges of the triangles.     

The restricted three body problem revisited

We present a new computation of the Birkhoff normal form for the Hamiltonian of the restricted three body problem near the Lagrangian libration points. This leads to a new proof of the Lyapunov stability of these points.     

Analysis of time-varying delay systems via triangular functions

This paper presents a method for finding the analysis of time-varying delay systems using triangular functions. We present the properties of the triangular functions. The operational matrices of integration, delay and product are utilized to reduce the solution of delay systems to the solution of algebraic equations. Illustrative examples is included to demonstrate the validity and applicability of the technique.