Simultaneous stabilization: Construction of a universal stabilizer for linear plants with delay with the use of spectral reducibility

We consider the problem of the construction of a common stabilizing controller for a family of k linear stationary nth-order plants with commensurable delays in the controls and the state variables. Our approach to stabilization is based on the construction of a common two-loop stabilizer. The controller of the first (internal) loop solves the problem of reducing each plant of the family to a finite spectrum, and the controller of the second (external) contour solves the stabilization problem.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

A combinatorial bijection between linear extensions of equivalent orders

Let P and Q be (partially) ordered sets with the same comparability graph. A bijection is constructed between the sets of linear extensions of P and Q such that the number of setups is preserved. This yields a common generalization of the comparability invariance of order dimension, setup number and number of linear extensions.     

Recognizing Intersection Graphs of Linear Uniform Hypergraphs

A hypergraph is linear if any two distinct hyperedges have at most one common vertex. The existence of a polynomial algorithm is shown for deciding whether a graph of minimum degree δ ≥ 19 is the intersection graph of a linear 3-uniform hypergraph. This result improves a corollary of the finite forbidden subgraph characterization proved for δ ≥ 69 by Naik et al. in [8]. Essentially the same methods yield a polynomial recognition algorithm for the intersection graph of a linear r-uniform hypergraph, r ≥ 3, provided the minimum edge-degree of the graphs is at least 2r ² ? 3r + 1. This improves on the cubic bound that follows from the corresponding finite characterization result in [8].     

A branch-and-cut algorithm for the plant-cycle location problem

The Plant-Cycle Location Problem (PCLP) is defined on a graph G=(I∪J, E), where I is the set of customers and J is the set of plants. Each customer must be served by one plant, and the plant must be opened to serve customers. The number of customers that a plant can serve is limited. There is a cost of opening a plant, and of serving a customer from an open plant. All customers served by a plant are in a cycle containing the plant, and there is a routing cost associated to each edge of the cycle. The PCLP consists in determining which plants to open, the assignment of customers to plants, and the cycles containing each open plant and its customers, minimizing the total cost. It is an NP-hard optimization problem arising in routing and telecommunications. In this article, the PCLP is formulated as an integer linear program, a branch-and-cut algorithm is developed, and computational results on real-world data and randomly generated instances are presented. The proposed approach is able to find optimal solutions of random instances with up to 100 customers and 100 potential plants, and of instances on real-world data with up to 120 customers and 16 potential plants.     

A new relation between molar sound velocity and molar volume in carboxy acid solutions

A new relationship between molar sound velocity ‘R ’ and molar volume (M/ρ) in carboxy acid solutions has been established. Using the composite ultrasonic interferometer, both monocarboxylic and dicarboxylic acid solutions in common solvents like methyl alcohol, carbontetrachloride have been studied and it has been established that a linear relationshipR =a ₀ +a ₁ · M/ρ holds good in all solutions of homologous series of acids in a common solvent at a particular temperature. The constantsa ₀ anda ₁ for a number of carboxy acid solutions have been evaluated.     

Pairwise commuting derivations of polynomial rings

We prove that n pairwise commuting derivations of the polynomial ring (or the power series ring) in n variables over a field k of characteristic 0 form a commutative basis of derivations if and only if they are k-linearly independent and have no common Darboux polynomials. This result generalizes a recent result due to Petravchuk and is an analogue of a well-known fact that a set of pairwise commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector.     

On stationary zeros of solutions of linear elliptic equations

For a nontrivial solution of a linear homogeneous elliptic equation, we study the dimension of the set of zeros whose multiplicity is not less than the order of the equation. In the case of a linear homogeneous differential operator P = P(D) with constant coefficients and three variables, we show that if, for a solution of the equation Pu = 0, a point x ⁰ is a zero of multiplicity not less than the order of the equation, then the intersection of a sufficiently small neighborhood of the point x ⁰ with the set of all other zeros of this kind is a finite set of segments with common endpoint x ⁰.     

Representing the GCD as linear combination in non-PID rings

We prove the following fact: If finitely many elements p ₁,p ₂,…,p _n of a unique factorization domain are given such that the greatest common divisor of each pair (p _i ,p _j ) can be expressed as a linear combination of p _i and p _j , then the greatest common divisor of all the p _i ’s can also be expressed as a linear combination of p ₁,…,p _n . We prove an analogous statement in general commutative rings.     

A remark on the simultaneous <Emphasis Type="Italic">α</Emphasis>-stabilization of linear plants

In the present paper, we consider the problem of simultaneous α-stabilization (i.e., stabilization with degree of stability exceeding a given number α > 0) for linear scalar stationary plants of arbitrary order, for which we obtain numerically verifiable necessary conditions for simultaneous α-stabilization and a sufficient condition together with a constructive algorithm for the stabilizing controller design. To solve these problems, we use an approach based on a one-to-one transformation of the α-stability domains in the spaces of coefficients of polynomials.     

On the range maximum-sum segment query problem

The range minimum query problem, RMQ for short, is to preprocess a sequence of real numbers A[1…n] for subsequent queries of the form: “Given indices i, j, what is the index of the minimum value of A[i…j]?” This problem has been shown to be linearly equivalent to the LCA problem in which a tree is preprocessed for answering the lowest common ancestor of two nodes. It has also been shown that both the RMQ and LCA problems can be solved in linear preprocessing time and constant query time under the unit-cost RAM model. This paper studies a new query problem arising from the analysis of biological sequences. Specifically, we wish to answer queries of the form: “Given indices i and j, what is the maximum-sum segment of A[i…j]?” We establish the linear equivalence relation between RMQ and this new problem. As a consequence, we can solve the new query problem in linear preprocessing time and constant query time under the unit-cost RAM model. We then present alternative linear-time solutions for two other biological sequence analysis problems to demonstrate the utilities of the techniques developed in this paper.     

The Sylvester-Gallai theorem, colourings and algebra

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem:

LetSbe a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct pointsA,B∈Ssharing a colour there is a third pointC∈S, of the other colour, collinear withAandB. Then all the points inSare collinear.

We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A,B∈S sharing a colour there is a third point C∈S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly’s complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a three-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem.     

Parallelism in a restricted linear space

In this paper two lines are called parallel if they are identical or if they have no point in common. It is then shown that in a restricted linear space of square order n parallelism is an equivalence relation on the n-lines.     

On the global stability of SIS, SIR and SIRS epidemic models with standard incidence

In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic models with constant recruitment, disease-induced death and standard incidence rate. We will make ingenious linear combination of known functions, common quadratic and Volterra-type, and of a new class of functions, we call composite-Volterra function, for obtain a suitable Lyapunov functions. In particular, for SIRS model we prove the global stability of the endemic equilibrium under a condition of parameters.     

Modelling and computing (Rn, Sn) policies for inventory systems with non-stationary stochastic demand

This paper addresses the single-item, non-stationary stochastic demand inventory control problem under the non-stationary (R, S) policy. In non-stationary (R, S) policies two sets of control parameters—the review intervals, which are not necessarily equal, and the order-up-to-levels for replenishment periods—are fixed at the beginning of the planning horizon to minimize the expected total cost. It is assumed that the total cost is comprised of fixed ordering costs and proportional direct item, inventory holding and shortage costs. With the common assumption that the actual demand per period is a normally distributed random variable about some forecast value, a certainty equivalent mixed integer linear programming model is developed for computing policy parameters. The model is obtained by means of a piecewise linear approximation to the non-linear terms in the cost function. Numerical examples are provided.     

The fractional quantum derivative and its integral representations

The quantum fractional derivative is defined using formulations analogue to the common Grünwald–Letnikov derivatives. While these use a linear variable scale, the quantum derivative uses an exponential scale and is defined in R⁺ or R⁻. Two integral formulations similar to the usual Liouville derivatives are deduced with the help of the Mellin transform.     

The categorical approach to social choice theory

A number of different problems in social choice theory are given a common treatment in the framework of category theory. A social choice function is a map μ from a set Hom(A, PX) of profiles - in applications functions from a set A of agents to the set PX of linear orders on a set X of alternatives - to a set GPX the nature of which depends on the application. The main theorem established is applied to social welfare functions and strategy-proof voting schemes.     

The total linear discrepancy of an ordered set

In this paper we introduce the notion of the total linear discrepancy of a poset as a way of measuring the fairness of linear extensions. If L is a linear extension of a poset P, and x,y is an incomparable pair in P, the height difference between x and y in L is |L(x)−L(y)|. The total linear discrepancy of P in L is the sum over all incomparable pairs of these height differences. The total linear discrepancy of P is the minimum of this sum taken over all linear extensions L of P. While the problem of computing the (ordinary) linear discrepancy of a poset is NP-complete, the total linear discrepancy can be computed in polynomial time. Indeed, in this paper, we characterize those linear extensions that are optimal for total linear discrepancy. The characterization provides an easy way to count the number of optimal linear extensions.     

Revisiting queueing output processes: a point process viewpoint

After some historical notes concerning queueing output processes N _dep??, the paper discusses methods for establishing asymptotic linear relations for var??N _dep??(0,t], whether in the crude form B ₁ t or the more detailed form B ₁ t+B ₀+o(1) for t→∞. The crude form holds whenever the process N _adm of customers admitted to service has a linear asymptote, and then (var??N _dep??(0,t])/t and (var??N _adm(0,t])/t share a common limit (that may be infinite) in stationary G/G/k/K systems. A standard integral formula for the variance of a stationary orderly point process shows that, if N _dep?? is a renewal process whose generic lifetime X has finite second moment, then B ₁=(var??X)/([E(X)]²), and the more detailed linear asymptote holds when E(X ³) is finite. Geometric ergodicity of the queue size process Q(?) in stationary M/M/k/K systems establishes that the more detailed linear asymptote is true for them. It is conjectured that var??N(0,t]～B ₁ t for any stationary point process N possessing an embedded regenerative structure.