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.     

Some remarks on uniform asymptotic expansions for Bessel functions

The uniform asymptotic expansion of the modified Bessel function of the first kind I_ν(νz), except for an auxiliary multiplier, is in the form

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is a polynomial in u² of degree k and U_k(u) is determined by use of a recurrence equation which is in the form of a difference–differential–integral equation. As a consequence these coefficients are not easily evaluated and the calculation becomes more cumbersome as k increases. In this paper, we show that the above series can be rearranged as where c_k(ν) satisfies a difference equation of order three, but there are only three terms as the coefficient of one of the terms is always zero. We also show how K_ν(νz), the Hankel functions, etc. can be handled in a similar fashion.     

Input constrained funnel control with applications to chemical reactor models

Error feedback control (in the presence of input constraints) is considered for a class of exothermic chemical reactor models. The primary control objective is regulation of a setpoint temperature T^* with prescribed accuracy: given λ>0 (arbitrarily small), ensure that, for every admissible system and reference setpoint, the regulation error e=T−T^* is ultimately smaller than λ (that is, ||e(t)||<λ for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the regulation error should be contained in a prescribed performance funnel F around the setpoint temperature T^*. A simple error feedback control with input constraints of the form

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, u^* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)−T^* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T^*.     

Biinfinite words with maximal recurrent unbordered factors

A finite non-empty word z is said to be a border of a finite non-empty word w if w=uz=zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form ^ωuvu^ω, where u and v are finite words, in terms of its unbordered factors.

The main result of the paper states that the words of the form ^ωuvu^ω are precisely the biinfinite words w=a₋₂a₋₁a₀a₁a₂ for which there exists a pair (l₀,r₀) of integers with l₀<r₀ such that, for every integers ll₀ and rr₀, the factor a_la_l0a_r0a_r is a bordered word.

The words of the form ^ωuvu^ω are also characterized as being those biinfinite words w that admit a left recurrent unbordered factor (i.e., an unbordered factor of w that has an infinite number of occurrences “to the left” in w) of maximal length that is also a right recurrent unbordered factor of maximal length. This last result is a biinfinite analogue of a result known for infinite words.     

Oscillation criteria for second-order quasi-linear neutral difference equations

In this paper, we obtain some oscillation criteria for the second-order quasi-linear neutral delay difference equation

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, where > 0, τ ≥ 0, and σ ≥ 0 are constants, {a_n}, {p_n}, {q_n} are nonnegative sequences and f ε C(R,R).     

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.     

A limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

Hybrid method for a general optimal sensor scheduling problem in discrete time

In this paper, we consider a general class of optimal sensor scheduling problems in discrete time. There are N₁ sensors available for acquiring data so as to estimate the needed but unknown signal. Only N₂ out of the N₁ sensors can be turned on at any moment, while different weights can be assigned to different sensors. This problem is formulated as a discrete time deterministic optimal control problem involving both discrete and continuous valued controls. A computational method is developed for solving this discrete time deterministic optimal control problem based on a branch and bound method in conjunction with a gradient-based method. The branch and bound method is used to determine the optimal schedule of sensors, where a sequence of lower bound dynamic systems is introduced so as to provide effective lower bounds for the construction of the branching rules. Each of the branches is an optimal weight vector assignment problem and a gradient-based method is developed for solving this optimal control problem. For illustration, two numerical examples are solved.     

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.     

Finding the minimum vertex distance between two disjoint convex polygons in linear time

Let V = v₁, v₂, …, v_m and W = w₁, w₂, …, w_n be two linearly separable convex polygons whose vertices are specified by their cartesian coordinates in order. An algorithm with O(m + n) worst-case time complexity is described for finding the minimum euclidean distance between a vertex v₁ in V and a vertex w_j in W. It is also shown that the algorithm is optimal.