Design of Civil Engineering Frame Structures Using a Monomial/Newton Hybrid Method

This paper presents an application of a monomial approximation method for solving systems of nonlinear equations to the design of civil engineering frame structures. This is accomplished by solving a set of equations representing the state known as fully-stressed design, where each member of the structure is stressed to the maximum safe allowable level under at least one of the loading conditions acting on it. The monomial approximation method is based on the process of condensation, which has its origin in geometric programming theory. A monomial/Newton hybrid method is presented which permits some of the design variables to be free in sign, while others are strictly positive. This hybrid method is well suited to the structural design application since some variables are naturally positive and others are naturally free. The proposed method is compared to the most commonly used fully-stressed design method in practice. The hybrid method is shown to find solutions that the conventional method cannot find, while doing so with less computational effort. The impact of this approach on the activity of structural design is discussed.     

Summability of Product Jacobi Expansions

Orthogonal expansions in product Jacobi polynomials with respect to the weight function W_α, β(x)=∏^d_j=1 (1−x_j)^α_j (1+x_j)^β_j on [−1, 1]^d are studied. For α_j, β_j>−1 and α_j+β_j−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of L^p(W_{α, β}, [−1, 1]^d), 1p<∞, and C([−1, 1]^d) if

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Moreover, for α_j, β_j−1/2, the (C, δ) means define a positive linear operator if and only if δ∑^d_i=1 (α_i+β_i)+3d−1.     

Dedekind Complete Commutative BCK-algebras

We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubík known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic to direct products of basic BCK-chains, generalizing a result of Cignoli for MV-algebras.     

Stress State of Multilayer Structural Elements Subjected to Static Loading and Optimization of Laminated Beams and Bars

The results of FEM investigation of the triaxial stress state in multilayer structural elements subjected to axial and bending loads are presented. The distribution regularities of the stiffness and stresses or strains depending on the geometric and mechanical characteristics of layers and their position in the cross section of beams and bars are examined. The optimization of these elements is carried out using the dependences of the Bareisis—Paulauskas method and the Optim-98 computer program created by the present authors. As the optimization criteria, the strength, stiffness, mass, and cost of the structural elements are considered.     

Simulation optimization of buffer allocations in production lines with unreliable machines

We use a recent simulationbased optimization method, sample path optimization, to find optimal buffer allocations in tandem production lines where machines are subject to random breakdowns and repairs, and the product is fluidtype. We explore some of the functional properties of throughput of such systems and exploit these properties to prove the almost sure convergence of our optimization technique, under a regularity condition on the steady state. Utilizing a generalized semiMarkov process (GSMP) representation of the system, we derive recursive expressions to compute onesided directional derivatives of throughput, from a single simulation run. Finally, we give computational results for lines with up to 50 machines. We also compare results for smaller lines with the results from a more conventional method, stochastic approximation, whenever applicable. In these numerical studies, our method performed quite well on problems that are considered difficult by current computational standards.     

Mixed Partitions of Projective Geometries

Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qⁿ–1)/(q–1) or (qⁿ–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety of PG(8,q) into caps of size q²+q+1 which are Veronese surfaces.     

Quasi-Shuffle Products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, *, ); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (U, III, ) onto (U, *, ) the set L of Lyndon words on A and their images { exp(w) w L} freely generate the algebra (U, *). We also consider the graded dual of (U, *, ). We define a deformation *_q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.     

Global Optimization: On Pathlengths in Min-Max Graphs

Let f be a smooth nondegenerate real valued function on a finite dimensional, compact and connected Riemannian manifold. The bipartite min-max graph is defined as follows. Its nodes are formed by the set of local minima and the set of local maxima. Two nodes (a local minimum and a local maximum) are connected in by means of an edge if some trajectory of the corresponding gradient flow connects them. Given a natural number k, we construct a function f such that the length of the shortest path in between two specific local minima exceeds k. The latter construction is independent of the underlying Riemannian metric.     

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.