Resolvability and the upper dimension of graphs

For an ordered set W = {w₁, w₂,…, w_k} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v | W) = (d(v, w₁), d(v, w₂),…, d(v, w_k)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations. A new sharp lower bound for the dimension of a graph G in terms of its maximum degree is presented.

A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). A resolving set S of G is a minimal resolving set if no proper subset of S is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim⁺(G). The resolving number res(G) of a connected graph G is the minimum k such that every k-set W of vertices of G is also a resolving set of G. Then 1 ≤ dim(G) ≤ dim⁺(G) ≤ res(G) ≤ n − 1 for every nontrivial connected graph G of order n. It is shown that dim⁺(G) = res(G) = n − 1 if and only if G = K_n, while dim⁺(G) = res(G) = 2 if and only if G is a path of order at least 4 or an odd cycle.

The resolving numbers and upper dimensions of some well-known graphs are determined. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dim(G) = dim⁺(G) = a and res(G) = b. Also, for every positive integer N, there exists a connected graph G with res(G) − dim⁺(G) ≥ N and dim⁺(G) − dim(G) ≥ N.     

Bivariate hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in R³ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points {(x_i, y_i)}_i=1^N from which none of the pairs (x_i,y_i) and (x_j,y_j) with i≠j coincide, it is proved that the resulting surface (function) is C¹. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is C² if the data are not ‘too irregular’. Finally the approximation properties of the methods are investigated.     

Analytic-numerical solutions with a priori error bounds for time-dependent mixed partial differential problems

The aim of this paper is double. First, we point out that the hypothesis D(t₁)D(t₂) = D(t₂)D(t₁) imposed in [1] can be removed. Second, a constructive method for obtaining analytic-numerical solutions with a prefixed accuracy in a bounded domain Ω(t₀,t₁) = [0,p] × [t₀,t₁], for mixed problems of the type u_t(x,t) − D(t)u_xx(x,t) = 0, 0 < x < p, t> 0, subject to u(0,t) = u(p,t) = 0 and u(x,0) = F(x) is proposed. Here, u(x,t) and F(x) are r-component vectors, D(t) is a C^{r × r} valued analytic function and there exists a positive number δ such that every eigenvalue z of (1/2) (D(t) + D(t)^H) is bigger than δ. An illustrative example is included.     

Nonlinear robust stabilization of multi-parameter families of systems

We show that given any family of asymptotically stabilizable LTI systems depending continuously on a parameter that lies in some subset [a₁,b₁]××[a_p,b_p] of , there exists a C⁰ time-varying state feedback law v(t,x) (resp. a C⁰ time-invariant feedback law v(x)) which robustly globally exponentially stabilizes (resp. which robustly stabilizes, not asymptotically) the family. Further, if these systems are obtained by linearizing some nonlinear systems, then v(t,x) locally exponentially stabilizes these nonlinear systems. Finally, v(t,x) globally exponentially stabilizes any time-varying system which switches “slowly enough” between the given LTI systems.     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

Theories of contact specified by connection matrices

We begin by characterizing notions of geometric continuity represented by connection matrices. Next we present a set of geometric properties that must be satisfied by all reasonable notions of geometric continuity. These geometric requirements are then reinterpreted as an equivalent collection of algebraic constraints on corresponding sets of connection matrices. We provide a general technique for constructing sets of connection matrices satisfying these criteria and apply this technique to generate many examples of novel notions of geometric continuity. Using these constraints and construction techniques, we show that there is no notion of geometric continuity between reparametrization continuity of order 3, (G³), and Frenet frame continuity of order 3, (F³); that there are several notions of geometric continuity between G⁴ and F⁴; and that the number of different notions of geometric continuity between Gⁿ and Fⁿ grows at least exponentially with n.     

Blossoms are polar forms

Consider the functions H(t):=t² and h(u,v):=uv. The identity H(t)=h(t,t) shows that h is the restriction of h to the diagonal u=v in the uv-plane. Yet, in many ways, a bilinear function like h is simpler than a homogeneous quadratic function like H. More generally, if F(t) is some n-ic polynomial function, it is often helpful to study the polar form of F, which is the unique symmetric, multiaffine function ƒ(u_1,…u_n) satisfying the identity F(t)=f(t,…,t). The mathematical theory underlying splines is one area where polar forms can be particularly helpful, because two pieces F and G of an n-ic spline meet at a point r with C^k parametric continuity if and only if their polar forms ƒ and g agree on all sequences of n arguments that contain at least n-k copies of r.

The polar approach to the theory of splines emerged in rather different guises in three independent research efforts: Paul de Faget Casteljau called it ‘shapes through poles’; Carl de Boor called it ‘B-splines without divided differences’; and Lyle Ramshaw called it ‘blossoming’. This paper reviews the work of de Casteljau, de Boor, and Ramshaw in an attempt to clarify the basic principles that underly the polar approach. It also proposes a consistent system of nomenclature as a possible standard.     

A study of neural network based inverse kinematics solution for a three-joint robot

A neural network based inverse kinematics solution of a robotic manipulator is presented in this paper. Inverse kinematics problem is generally more complex for robotic manipulators. Many traditional solutions such as geometric, iterative and algebraic are inadequate if the joint structure of the manipulator is more complex. In this study, a three-joint robotic manipulator simulation software, developed in our previous studies, is used. Firstly, we have generated many initial and final points in the work volume of the robotic manipulator by using cubic trajectory planning. Then, all of the angles according to the real-world coordinates (x, y, z) are recorded in a file named as training set of neural network. Lastly, we have used a designed neural network to solve the inverse kinematics problem. The designed neural network has given the correct angles according to the given (x, y, z) cartesian coordinates. The online working feature of neural network makes it very successful and popular in this solution.     

Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σ_j=1^N d_jφ(x−x_j), where φ(u) = |u|²log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ε, can be cut to O(1+|log ε|) operations. In particular, if A is the interpolation matrix, a_i,j = φ(x_i−x_j, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N²) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.     

Optimal and nearly optimal algorithms for approximating polynomial zeros

We substantially improve the known algorithms for approximating all the complex zeros of an n^th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of Schönhage [1], Pan [2], and Neff and Reif [3]. In parallel (NC) implementation, we dramatically decrease the number of processors, versus the parallel algorithm of Neff [4], which was the only NC algorithm known for this problem so far. Specifically, under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p(x) to the unit disc x : |x| ≤ 1, our algorithms (which promise to be practically effective) approximate all the zeros of p(x) within the absolute error bound 2^−b, by using order of n arithmetic operations and order of (b + n)n² Boolean (bitwise) operations (in both cases up to within polylogarithmic factors). The algorithms allow their optimal (work preserving) NC parallelization, so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or (b + n)n² Boolean processors. All the cited bounds on the computational complexity are within polylogarithmic factors from the optimum (in terms of n and b) under both arithmetic and Boolean models of computation (in the Boolean case, under the additional (realistic) assumption that n = O(b)).     

Generalization of rectangular element stiffness matrix and thermal load vector associated with a0 + a1x + a2y + a3xy type interpolation rule

Explicit expressions for the element stiffness matrix K and element load vector p for the rectangular plane-stress and plane-strain finite elements associated with Ψ(x, y) = a₀ + a₁x + a₂y + a₃xy type interpolation rule are given for the general anisotropic material in xy-planc subjected to non-uniform temperature increases. The expressions are optimized with respect to the numerical operations required for the computation of K and p, and they are valid for special cases of material properties and thermal loading.     

Efficient enumeration of all minimal separators in a graph

This paper presents an efficient algorithm for enumerating all minimal a-b separators separating given non-adjacent vertices a and b in an undirected connected simple graph G = (V, E), Our algorithm requires O(n³R_ab) time, which improves the known result of O(n⁴R_ab) time for solving this problem, where ¦V¦= n and R_ab is the number of minimal a-b separators. The algorithm can be generalized for enumerating all minimal A-B separators that separate non-adjacent vertex sets A, B < V, and it requires O(n²(n − n_A − n_b)R_AB) time in this case, where n_a = ¦A¦, n_B = ¦B¦ and r_AB is the number of all minimal A−B separators. Using the algorithm above as a routine, an efficient algorithm for enumerating all minimal separators of G separating G into at least two connected components is constructed. The algorithm runs in time O(n³R⁺_Σ + n⁴R_Σ), which improves the known result of O(n⁶R_Σ) time, where R_σ is the number of all minimal separators of G and R_ΣR⁺_Σ = ∑_1i, v_j) ER_vivj n − 1)/2 − m)R_Σ. Efficient parallelization of these algorithms is also discussed. It is shown that the first algorithm requires at most O((n/log n)R_ab) time and the second one runs in time O((n/log n)R⁺_Σ+n log nR_Σ) on a CREW PRAM with O(n³) processors.     

On the type of a polynomial f, when f(0)f−f(0)f is a monomial

A polynomial is said to be of type (p₁, p₂, p₃) relative to a directed line in the complex plane if, counting multiplicities, it has p₁ zeros to the left of, p₂ zeros on, and p₃ zeros to the right of the line. In this paper we determine explicitly the types of all polynomials belonging to a very restricted (but infinite) family of polynomials. A polynomial ƒ belongs to this family if and only if its coefficients are such that the polynomial ƒ^*(0)ƒ(z)−ƒ(0) ƒ^*(z) is a monomial; here ƒ^* denotes the reflection of ƒ in the directed line.

A special case of the present result appeared in an earlier publication.     

Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem

Consider the cubic sensor dx = dw, dy = x³dt + dv where w, v are two independent Brownian motions. Given a function φ(x) of the state x let φ_t(x) denote the conditional expectation given the observations y_s, 0 s t. This paper consists of a rather detailed discussion and outline of proof of the theorem that for nonconstant φ there cannot exist a recursive finite-dimensional filter for φ driven by the observations.     

“Probability transforms” of digital pictures

Let g be any local property (e.g., gray level or gradient magnitude) defined on a digital picture. Let p_g(z) be the relative frequency with which g has value z. At each point (x,y) of the picture we can display p_g[g(x,y)], appropriately scaled; the result is called the p_g transform of the picture. Alternatively, we can use joint or conditional frequencies of pairs of local properties to define transforms. This note gives examples of such transforms for various gs and discusses their possible uses and limitations.     

On Stieltjes functions and Hankel operators

Let G be a Stieltjes function which is analytic in the open right half plane. It is shown that G is in H^∞(RHP) if and only if the Hankel operator H_G on H²(RHP) with symbol G is nuclear. If G is in H^∞(RHP) it is shown that the non-tangential limit of G at s = 0 equals twice the nuclear norm of H_G.     

An element for static, vibration and buckling analysis of thick laminated plates

A conforming finite element formulation of the equations governing composite multilayered plates using Reddy's higher-order theory is presented. The element has eight degrees of freedom, u₀, v₀, w, ∂w/∂x, ∂w/∂y, ∂²w/∂x∂y, γ_x, γ_y, per node. The transverse displacement of the present element is described by a modified bicubic displacement function while the in-plane displacements and shear-rotations are interpolated quadraticly. The element is evaluated for its accuracy in the analysis of static, vibration, and buckling of anisotropic rectangular plates with different lamination schemes and boundary conditions. The conforming finite element described here for the higher-order theory gives fairly accurate results for displacements, stresses, buckling loads, and natural frequencies.     

Integral averages and oscillation of second-order nonlinear differential equations

We present some criteria for the oscillation of the second-order nonlinear differential equation

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where a ε C¹([t₀, ∞)) is a nonnegative function, q ε C ([t₀, ∞)) are allowed to change sign on [t₀, ∞), ψ, f ε C¹ , ψ(x) > 0, xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. These criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques and represent extension, as well as improvement of known oscillation criteria of Philos and Purnaras for the generalized Emden-Fowler equation.     

The filtering problem for continuous-time linear systems with Markovian switching coefficients

The stochastic model under consideration is a Markovian jump process θ, with finite state space, feeding the parameters of a linear diffusion process x. The processes y and z observe linearly and separately x and θ in independent white noises. Some properties of the finite optimal filter for the x and θ processes given the history of measurements z are investigated. Apart from their theoretical interest, these results have an interesting practical bearing on the general filtering problem, by providing a natural finite suboptimal solution. Preliminary experimental results show the effectiveness of our approach to estimate the state trajectory, even with a relatively low signal-to-noise ratio on the measurement processes.     

Hybrid stabilization of planar linear systems with one-dimensional outputs

We consider a linear control system x′=Ax+Bu with output y=Cx, where xR², u, yR, and give necessary and sufficient conditions in order that it can be stabilized by a hybrid, linear feedback, where the action of the “switch” just depends on the sign of y. We also show, on these conditions, that the use of two control functions is enough for getting the goal.