Interaction between three moving Griffith cracks at the interface of two dissimilar elastic media

The paper deals with the interaction between three Griffith cracks propagating under antiplane shear stress at the interface of two dissimilar infinite elastic half-spaces. The Fourier transform technique is used to reduce the elastodynamic problem to the solution of a set of integral equations which has been solved by using the finite Hilbert transform technique and Cooke’s result. The analytical expressions for the stress intensity factors at the crack tips are obtained. Numerical values of the interaction effect have been computed for and results show that interaction effects are either shielding or amplification depending on the location of each crack with respect to other and crack tip spacing.     

Cosine of angle and center of mass of an operator

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.     

A generalization of the Leibnitz rule of successive differentiation

This note generalizes the well-known Leibnitz rule of successive differentiation for the product of two functions to a similar result for the product of three functions.     

A Sharp Upper Bound for the Number of Spanning Trees of a Graph

Let G = (V,E) be a simple graph with n vertices, e edges and d₁ be the highest degree. Further let λ_i, i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ₂ = λ₃ = ... =  λ_n-1 if and only if G is a complete graph or a star graph or a (d₁,d₁) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d₁ only:

On properties of <Emphasis Type="Italic">G</Emphasis>-expansive homeomorphisms

We show that LUB of the set of G-expansive constants for a G-expansive homeomorphism h on a compact metric G-space, G compact, is not a G-expansive constant for h. We obtain a result regarding projecting and lifting of G-expansive homeomorphisms having interesting applications. We also prove that the G-expansiveness is a dynamical property for homeomorphisms on compact metric G-spaces and study G-periodic points.     

Two-warehouse production model for deteriorating inventory items with stock-dependent demand under inflation over a random planning horizon

This paper develops a production-inventory model for a deteriorating item with stock-dependent demand under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. The effects of learning in set-up, production, selling and reduced selling is incorporated. Different inflation rates for various inventory costs and time value of money are also considered. A hybrid genetic algorithm is designed to solve the optimization problem which is hard to solve with existing algorithms due to the complexity of the decision variable. To illustrate the model and to show the effectiveness of the proposed approach a numerical example is provided. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.     

Oscillation of higher order delay differential equations

A sufficient condition was obtained for oscillation of all solutions of theodd-order delay differential equation $$x^{(n)} (t) + \sum\limits_{i = 1}^m {p_i (t)} x(t - \sigma _{_i } ) = 0,$$ wherep _i (t) are non-negative real valued continuous function in [T ∞] for someT≥0 and σ_i,∈(0, ∞)(i = 1,2,…,m). In particular, forp _i (t) =p _i ∈(0, ∞) andn > 1 the result reduces to $$\frac{1}{m}\left( {\sum\limits_{i = 1}^m {(p_i \sigma _i^m )^{1/2} } } \right)^2 > (n - 2)!\frac{{(n)^n }}{e},$$ implies that every solution of (*) oscillates. This result supplements forn > 1 to a similar result proved by Ladaset al [J. Diff. Equn.,42 (1982) 134–152] which was proved for the casen = 1.     

Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.     

Billingsley's packing dimension

For a stochastic process on a finite state space, we define the notion of a packing measure based on the naturally defined cylinder sets. For any two measures , , corresponding to the same stochastic process, if

then we prove that

     

WEB-spline based mesh-free finite element analysis for the approximation of the stationary Navier–Stokes problem

The objective in this paper is to discuss the existence and the uniqueness of a weighted extended B$B$-spline (WEB-spline) based discrete solution for the stationary incompressible Navier–Stokes equations. The WEB-spline discretization is newly developed methodology which satisfies the inf–sup condition or Ladyshenskaya–Babus?ka–Brezzi (LBB) condition. The main advantage of these new elements over standard finite elements is that they use regular grids instead of irregular partitions of the domain, thus eliminating the difficult and time-consuming pre-processing step. An error estimate for this WEB-spline based discrete solution is also obtained.