Optimal linear estimator of origin-destination flows with redundant data

Suppose given a network endowed with a multiflow. We want to estimate some quantities connected with this multiflow, for instance the value of an s–t flow for one of the sources–sinks pairs s–t, but only measures on some arcs are available, at least on one s–t cocycle (set of arcs having exactly one endpoint in a subset X of vertices with s∈X and t?X). These measures, supposed to be unbiased, are random variables whose variances are known. How can we combine them optimally in order to get the best estimator of the value of the s–t flow?This question arises in practical situations when the OD matrix of a transportation network must be estimated. We will give a complete answer for the case when we deal with linear combinations, not only for the value of an s–t flow but also for any quantity depending linearly from the multiflow. Interestingly, we will see that the Laplacian matrix of the network plays a central role.     

Locating Active Sensors on Traffic Networks

Sensors are used to monitor traffic in networks. For example, in transportation networks, they may be used to measure traffic volumes on given arcs and paths of the network. This paper refers to an active sensor when it reads identifications of vehicles, including their routes in the network, that the vehicles actively provide when they use the network. On the other hand, the conventional inductance loop detectors are passive sensors that mostly count vehicles at points in a network to obtain traffic volumes (e.g., vehicles per hour) on a lane or road of the network.This paper introduces a new set of network location problems that determine where to locate active sensors in order to monitor or manage particular classes of identified traffic streams. In particular, it focuses on the development of two generic locational decision models for active sensors, which seek to answer these questions: (1) “How many and where should such sensors be located to obtain sufficient information on flow volumes on specified paths?”, and (2) “Given that the traffic management planners have already located count detectors on some network arcs, how many and where should active sensors be located to get the maximum information on flow volumes on specified paths?”The problem is formulated and analyzed for three different scenarios depending on whether there are already count detectors on arcs and if so, whether all the arcs or a fraction of them have them. Location of an active sensor results in a set of linear equations in path flow variables, whose solution provide the path flows. The general problem, which is related to the set-covering problem, is shown to be NP-Hard, but special cases are devised, where an arc may carry only two routes, that are shown to be polynomially solvable. New graph theoretic models and theorems are obtained for the latter cases, including the introduction of the generalized edge-covering by nodes problem on the path intersection graph for these special cases. An exact algorithm for the special cases and an approximate one for the general case are presented.     

Capacity inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow \(x^0\), the inverse minimum cost flow problem is to modify the capacity vector or the cost vector as little as possible to make \(x^0\) form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the capacity inverse minimum cost flow problems under the weighted Hamming distance, where we use the weighted Hamming distance to measure the modification of the arc capacities. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in \(O(n\cdot m^2)\) time.     

An exact algorithm for finding a vector subset with the longest sum

We consider the problem: Given a set of n vectors in the d-dimensional Euclidean space, find a subsetmaximizing the length of the sum vector.We propose an algorithm that finds an optimal solution to this problem in time O(n^d?1(d + logn)). In particular, if the input vectors lie in a plane then the problem is solvable in almost linear time.     

Two-stage flexible flow shop scheduling subject to fixed job sequences

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that n _l jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m²Π_l=1 ^m n _l ²) dynamic programming algorithm coupled with numerical experiments is presented.     

An approximation scheme for a problem of search for a vector subset

We consider the following clustering problem: Given a vector set, find a subset of cardinality k and minimum square deviation from its mean. The distance between the vectors is defined by the Euclideanmetric. We present an approximation scheme (PTAS) that allows us to solve this problem with an arbitrary relative error ? in time O(n ^2/?+1(9/?)^3/? d), where n is the number of vectors of the input set and d denotes the dimension of the space.     

Global existence of real roots and random Newton flow algorithm for nonlinear system of equations

To solve nonlinear system of equation, F(x) = 0, a continuous Newton flow x _t (t) = V (x) = ?(DF(x))^?1 F(x), x(0) = x ⁰ and its mathematical properties, such as the central field, global existence and uniqueness of real roots and the structure of the singular surface, are studied. We concisely introduce random Newton flow algorithm (NFA) for finding all roots, based on discrete Newton flow x ^j+1 = x ^j + hV (x ^j ) with random initial value x ⁰ and h ∈ (0, 1], and three computable quantities, g _j , d _j and K _j . The numerical experiments with dimension n = 300 are provided.     

A note on scheduling jobs with equal processing times and inclusive processing set restrictions

We consider the problem of scheduling n jobs on m parallel machines with inclusive processing set restrictions. Each job has a given release date, and all jobs have equal processing times. The objective is to minimize the makespan of the schedule. Li and Li (2015) have developed an O(n²+mn log?n) time algorithm for this problem. In this note, we present a modified algorithm with an improved time complexity of O(min{m, log?n} ? n log?n).     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Asymptotics of the Velocity Potential of an Ideal Fluid Flowing around a Thin Body

We consider the Neumann problem outside a small neighborhood of a planar disk in the three-dimensional space. The surface of this neighborhood is assumed to be smooth, and its thickness is characterized by a small parameter ε. A uniform asymptotic expansion of the solution of this problem with respect to ε is constructed by the matching method. Since the problem turned out to be bisingular, an additional inner asymptotic expansion in the so-called stretched variables is constructed near the edge of the disk. A physical interpretation of the solution of this boundary value problem is the velocity potential of a laminar flow of an ideal fluid around a thin body, which is the neighborhood of the disk. It is assumed that this flow has unit velocity at a large distance from the disk, which is equivalent to the following condition for the potential: u(x₁, x₂, x₃, ε) = x₃+O(r^?2) as r → ∞, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: ?u/?n = 0 at the boundary. After subtracting x₃ from the solution u(x₁, x₂, x₃, ε), we get a boundary value problem for the potential ?(x₁, x₂, x₃, ε) of the perturbed motion. Since the integral of the function ??/?n over the surface of the body is zero, we have ?(x₁, x₂, x₃, ε) = O(r^?2) as r → ∞. Hence, all the coefficients of the outer asymptotic expansion with respect to ε have the same behavior at infinity. However, these coefficients have growing singularities at the approach to the edge of the disk, which implies the bisingularity of the problem.     

On homeomorphisms of the plane which have no fixed points

Homeomorphisms of the planeR ² onto itself are studied, subject to the restriction that they should preserve the sense of orientation and have no fixed points. The results of this investigation are then applied to the problem of determining which homeomorphisms can be embedded in flows, i.e., in one-parameter subgroups of the full homeomorphism group of the plane. A “free mapping” ofR ² onto itself is defined to be a homeomorphismT, without fixed points, such thatC∩TC=0 impliesC∩T ⁿ C=0 for alln≠0 wheneverC is a compact connected subset ofR ². Free mappings turn out to be just those homeomorphisms ofR ² onto itself that preserve orientation and have no fixed points. A fundamental property of free mappingsis the fact that ifT is a free mapping andA is any compact subset ofR ² then\(\mathop U\limits_{ - \infty }^{ + \infty } T^n A\) does not meet some unbounded connected subsetB ofR ². The proof of this theorem is lengthy, and will appear elsewhere. The theorem can be weakened by adding the extra assumption thatT be embedded in a flow; the proof of this weakened version is much easier, and is included in the present article. It is found that for an arbitrary free mappingT there exists a natural partition of the plane into a collection of “fundamental regions”, with the property that ifT is embedded in a flow then each of the fundamental regions is invariant under the flow. An example is given of a free mapping whose fundamental regions are bad enough so that the mapping cannot be embedded in a flow. It is proved, on the other hand, that if a free mappingT has just one fundamental region thenT is equivalent to a translation, i.e., there is a homeomorphismU ofR ² onto itself such thatUTU ^?1 is just the translation(x, y)→(x+1, y). Indeed,T is equivalent to a translation if and only ifT has just one fundamental region.     

A Polynomial Time Algorithm for the Resource Allocation Problem with a Convex Objective Function

Consider the resource allocation problem:minimize ∑ⁿ_i=1 f_i(x_i) subject to ∑ⁿ_i=1 x_i = N and x_i's being nonnegative integers, where each f_i is a convex function. The well-known algorithm based on the incremental method requires O(N log n + n) time to solve this problem. We propose here a new algorithm based on the Lagrange multiplier method, requiring O[n²(log N)²] time. The latter is faster if N is much larger than n. Such a situation occurs, for example, when the optimal sample size problem related to monitoring the urban air pollution is treated.     

On the asymptotic optimality of a solution of the euclidean problem of covering a graph by <Emphasis Type="Italic">m</Emphasis> nonadjacent cycles of maximum total weight

In the problem of covering an n-vertex graph by m cycles of maximum total weight, it is required to find a family of m vertex-nonadjacent cycles such that it covers all vertices of the graph and the total weight of edges in the cover is maximum. The paper presents an algorithm for approximately solving the problem of covering a graph in Euclidean d-space R^d by m nonadjacent cycles of maximum total weight. The algorithm has time complexity O(n³). An estimate of the accuracy of the algorithm depending on the parameters d, m, and n is substantiated; it is shown that if the dimension d of the space is fixed and the number of covering cycles is m = o(n), then the algorithm is asymptotically exact.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Let L _∞,s ¹ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that ?f/?x _i ∈ L _s (? ^{m) for each i = 1, ...,m} . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D ^α f∥_∞ of functions f ∈ L _∞,s ¹ (? ^m ). Stechkin’s problem on approximation of unbounded operators D ^α by bounded operators on the class of functions f ∈ L _∞,s ¹ (? ^m ) such that ∥?f∥ _s ≤ 1 and the problem of optimal recovery of the operator D ^α on elements from this class given with error δ are solved.     

Operations policy for a supply chain system with fixed-interval delivery and linear demand

This research addresses a production-supply problem for a supply-chain system with fixed-interval delivery. A strategy that determines the optimal batch sizes, cycle times, numbers of orders of raw materials, and production start times is prescribed to minimize the total costs for a given finite planning horizon. The external demands are time-dependent following a life-cycle pattern and the shipment quantities follow the demand pattern. The shipment quantities to buyers follow various phases of the demand pattern in the planning horizon where demand is represented by piecewise linear model. The problem is formulated as an integer, non-linear programming problem. The model also incorporates the constraint of inventory capacity. The problem is represented using the network model where an optimal characteristic has been analysed. To obtain an optimal solution with N shipments in a planning horizon, an algorithm is proposed that runs with the complexity of Θ(N²) for problems with a single-phase demand and O(N³) for problems with multi-phase demand.     

Approximation of circular arcs by parametric polynomial curves

In this paper the approximation of circular arcs by parametric polynomial curves is studied. If the angular length of the circular arc is h, a parametric polynomial curve of arbitrary degree \(n \in {\mathbb{N}}\) , which interpolates given arc at a particular point, can be constructed with radial distance bounded by h ²ⁿ . This is a generalization of the result obtained by Lyche and Mørken for odd n.     

Parabolic equation with a singular potential

We study the existence of a nonnegative generalized solution of an initial-boundary value problem for the heat equation with a singular potential in an arbitrary bounded domain Ω ? R ⁿ , n ≥ 3, containing the unit ball. We show that if the condition ∝ _Ω V ^n/2+s |x| ^s dx ≤ c _n is satisfied for some s ≥ 0 and c _n = c _n (n, s, Ω) > 0, then the problem in question has a nonnegative solution.     

Value sharing problems for differential and difference polynomials of meromorphic functions in a non-Archimedean field

In this paper we discuss the uniqueness problem for differential and difference polynomials of the form (f ^nm (z)f ^nd (qz + c))^(k) for meromorphic functions in a non-Archimedean field.