Weighted <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-estimates of solutions for damped wave equations with variable coefficients

The authors establish weighted L²-estimates of solutions for the damped wave equations with variable coefficients u _tt ? divA(x)?u+au _t = 0 in ? ⁿ under the assumption a(x) ≥ a₀[1+ρ(x)]^?l, where a₀ > 0, l < 1, ρ(x) is the distance function of the metric g = A^?1(x) on ? ⁿ . The authors show that these weighted L²-estimates are closely related to the geometrical properties of the metric g = A^?1(x).     

Partially function linear error-in-response models with validation data

This paper considers partial function linear models of the form Y =∫X(t)β(t)dt + g(T)with Y measured with error. The authors propose an estimation procedure when the basis functions are data driven, such as with functional principal components. Estimators of β(t) and g(t) with the primary data and validation data are presented and some asymptotic results are given. Finite sample properties are investigated through some simulation study and a real data application.     

A unified approach to construct generalized B-splines for isogeometric applications

Generalized B-splines have been employed as geometric modeling and numerical simulation tools for isogeometric analysis(IGA for short). However, the previous models used in IGA,such as trigonometric generalized B-splines or hyperbolic generalized B-splines, are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper,a unified approach to construct the generalized non-uniform B-splines over the space spanned by{α(t), β(t), ξ(t), η(t), 1, t, ···, t~(n-4)} is proposed, and the corresponding isogeometric analysis framework for PDE solving is also studied. Compared with the NURBS-IGA method, the proposed frameworks have several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form. Furthermore, with the proposed spline models, isogeometric analysis can be performed on the computational domain bounded by transcendental curves/surfaces, such as the involute of circle, the helix/helicoid, the catenary/catenoid and the cycloid. Several numerical examples for isogeometric heat conduction problems are presented to show the effectiveness of the proposed methods.     

Construction of one-Lee weight and two-Lee weight codes over F<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> + <Emphasis Type="Italic">v</Emphasis>F<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is devoted to the construction of one-Lee weight codes and two-Lee weight codes over F _p + vF _p (v ² = v) with type \({p^{2{k_1}}}{p^{{k_2}}}{p^{{k_3}}}\) based on two different distance-preserving Gray maps from ((F _p + vF _p ) ⁿ , Lee weight) to (F _p ²ⁿ , Hamming weight), where p is a prime. Moreover, the authors prove that the obtained two-Lee weight codes are projective only when p = 2.     

A class of constacyclic codes over ring <Emphasis Type="Italic">R</Emphasis> + <Emphasis Type="Italic">vR</Emphasis>

By constructing a Gray map, a class of constacyclic codes over ring R = R+vR is studied. Using cyclic codes and negacyclic codes of length p ^s over ring R, the structure of (1?2v)-constacyclic codes and dual codes of length p ^s over ring R are given, the Gray images of (1 ? 2v)-constacyclic codes in a particular case are also studied. It is shown that linear codes of length p ^s over ring R are (1?2v)-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length 2p ^s over ring R.     

An FE-Inexact Heterogeneous ADMM for Elliptic Optimal Control Problems with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Control Cost

Elliptic PDE-constrained optimal control problems with L¹-control cost (L¹-EOCP) are considered. To solve L¹-EOCP, the primal-dual active set (PDAS) method, which is a special semismooth Newton (SSN) method, used to be a priority. However, in general solving Newton equations is expensive. Motivated by the success of alternating direction method of multipliers (ADMM), we consider extending the ADMM to L¹-EOCP. To discretize L¹-EOCP, the piecewise linear finite element (FE) is considered. However, different from the finite dimensional l1-norm, the discretized L¹-norm does not have a decoupled form. To overcome this difficulty, an effective approach is utilizing nodal quadrature formulas to approximately discretize the L¹-norm and L²-norm. It is proved that these approximation steps will not change the order of error estimates. To solve the discretized problem, an inexact heterogeneous ADMM (ihADMM) is proposed. Different from the classical ADMM, the ihADMM adopts two different weighted inner products to define the augmented Lagrangian function in two subproblems, respectively. Benefiting from such different weighted techniques, two subproblems of ihADMM can be efficiently implemented. Furthermore, theoretical results on the global convergence as well as the iteration complexity results o(1/k) for ihADMM are given. In order to obtain more accurate solution, a two-phase strategy is also presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the ihADMM. Numerical results not only confirm error estimates, but also show that the ihADMM and the two-phase strategy are highly efficient.     

Control and Stabilization of the Rosenau Equation Posed on a Periodic Domain

This paper considers the Rosenau equation with a moving control?t u + ?_t?_x~4 u + ?_xu + u?x u = a(x + ct)h(x, t), c = 0, x ∈ T = R/(2πZ), t 0.The authors prove that the Rosenau equation with a moving control is locally exact controllable in Hs(T) with s ≥ 0 and globally exponential stable in H~s(T) with s ≥ 2. The two results nontrivially extend the work of(Rosier L and Zhang B Y, 2013) from the BBM equation to the Rosenau equation.     

The determination on weight hierarchies of <Emphasis Type="Italic">q</Emphasis>-ary linear codes of dimension 5 in class IV

The weight hierarchy of a linear [n; k; q] code C over GF(q) is the sequence (d ₁, d ₂, ···, d _k ) where d _r is the smallest support of any r-dimensional subcode of C. “Determining all possible weight hierarchies of general linear codes” is a basic theoretical issue and has important scientific significance in communication system. However, it is impossible for q-ary linear codes of dimension k when q and k are slightly larger, then a reasonable formulation of the problem is modified as: “Determine almost all weight hierarchies of general q-ary linear codes of dimension k”. In this paper, based on the finite projective geometry method, the authors study q-ary linear codes of dimension 5 in class IV, and find new necessary conditions of their weight hierarchies, and classify their weight hierarchies into 6 subclasses. The authors also develop and improve the method of the subspace set, thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV. It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes (classes III and VI), and break through the difficulties. Furthermore, the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough (not most tight nor best), so, it is important to excogitate further new necessary conditions for attacking and solving the k-dimensional problem.     

Construction of one-gray weight codes and two-Gray weight codes over ℤ<Subscript>4</Subscript> + <Emphasis Type="Italic">u</Emphasis>ℤ<Subscript>4</Subscript>

This paper firstly gives some necessary conditions on one-Gray weight linear codes. And then we use these results to construct several classes of one-Gray weight linear codes over ?₄+u?₄(u ² = u) with type \({16^{{k_1}}}{8^{{k_2}}}{8^{{k_3}}}{4^{{k_4}}}{4^{{k_5}}}{4^{{k_6}}}{2^{{k_7}}}{2^{{k_8}}}\) based on a distance-preserving Gray map from (?₄ + u?₄) ⁿ to ? ₄ ²ⁿ . Secondly, the authors use the similar approach to do works on two-Gray (projective) weight linear codes. Finally, some examples are given to illustrate the construction methods.     

A New Algorithm for Computing the Extended Hensel Construction of Multivariate Polynomials

This paper presents a new algorithm for computing the extended Hensel construction(EHC) of multivariate polynomials in main variable x and sub-variables u_1, u_2, ···, u_m over a number field K. This algorithm first constructs a set by using the resultant of two initial coprime factors w.r.t. x, and then obtains the Hensel factors by comparing the coefficients of x~i on both sides of an equation. Since the Hensel factors are polynomials of the main variable with coefficients in fraction field K(u_1, u_2, ···, u_m), the computation cost of handling rational functions can be high. Therefore,the authors use a method which multiplies resultant and removes the denominators of the rational functions. Unlike previously-developed algorithms that use interpolation functions or Gr?bner basis, the algorithm relies little on polynomial division, and avoids multiplying by different factors when removing the denominators of Hensel factors. All algorithms are implemented using Magma, a computational algebra system and experiments indicate that our algorithm is more efficient.     

On algebraic immunity of trace inverse functions on finite fields of characteristic two

The trace inverse functions Tr(λx~(-1)) over the finite field F_(2~n) are a class of very important Boolean functions and are used in many stream ciphers such as SFINKS,RAKAPOSHI,the simple counter stream cipher(SCSC) presented by Si W and Ding C(2012),etc.In order to evaluate the security of those ciphers in resistance to(fast) algebraic attacks,the authors need to characterize algebraic properties of Tr(λx~(-1)).However,currently only some bounds on algebraic immunity of Tr(λx~(-1)) are given in the public literature,for example,the NGG upper bound and the Bayev lower bound,etc.This paper gives the exact value of the algebraic immunity of Tr(λx~(-1)) over F_(2~n),that is,AI(Tr(λx~(-1))) =[2n~(1/2)]- 2,where n ≥ 2,A ∈ F_(2~n) and λ≠ 0,which shows that Dalai's conjecture on the algebraic immunity of Tr(λx~(-1)) is correct.What is more,the authors demonstrate some weak properties of Tr(λx~(-1)) against fast algebraic attacks.     

Polynomial dynamic programming algorithms for lot sizing models with bounded inventory and stockout and/or backlogging

This paper addresses a dynamic lot sizing problem with bounded inventory and stockout where both no backlogging and backlogging allowed cases are considered. The stockout option means that there is outsourcing in a period only when the inventory level at that period is non-positive. The production capacity is unlimited and production cost functions are linear but with fixed charges. The problem is that of satisfying all demands in the planning horizon at minimal total cost. We show that the no backlogging case can be solved in ) O(T ²) time with general concave inventory holding and outsourcing cost functions where T is the length of the planning horizon. The complexity can be reduced to O(T) when the inventory holding cost functions are also linear and have some realistic properties, even if the outsourcing cost functions remain general concave functions. When the inventory holding and outsourcing cost functions are linear, the backlogging case can be solved in O(T ³logT) time whether the outsourcing level at each period is bounded by the sum of the demand of that period and backlogging level from previous periods, or only by the demand of that period.     

Queue size distribution of Geo/G/1 queue under the Min(<Emphasis Type="Italic">N,D</Emphasis>)-policy

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first (Min(N,D)-policy). By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n ⁺. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution. Furthermore, the important relations between the steady state queue length distributions at different time epochs n ^-, n and n ⁺ are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.     

Articulated Estimator Random Field and Geometrical Approach Applied in System Identification

A new point of view of robust statistics based on a geometrical approach is tackled in this paper. Estimation procedures are carried out from a new robust cost function based on a chaining of elementary convex norms. This chain is randomly articulated in order to treat more efficiently natural outliers in data-set. Estimated parameters are considered as random fields and each of them, named articulated estimator random field (AERF) is a manifold or stratum of a stratified space with Riemannian geometry properties. From a high level excursion set, a probability distribution model M^sta is presented and a system model validation geometric criterion (SYMOVAGEC) for system model structures M^sys based on Riccian scalar curvatures is proposed. Numerical results are drawn in a context of system identification.     

An improved algorithm for solving all <Emphasis Type="Italic">d</Emphasis>-MPs in multi-state networks

Reliability is a desirable performance indicator of many real-world systems to measure the quality level. One general method for evaluating multi-state reliability is using d-minimal paths (d-MPs). However, being an NP-hard problem, searching for all d-MPs is a rather challenging task. This paper proposes an improved algorithm to solve the d-MP problem. To reduce the search space of d-MPs, a concept of lower capacity bound is introduced into the d-MP problem, and an effective technique is developed to find lower capacity bounds. Meanwhile, the fast enumeration method which is a recent improvement to the traditional enumeration method is employed to solve d-MPs. In addition, by introducing the operation of transforming undirected edges into directed edges, the proposed algorithm is applicable to solving both directed networks and undirected networks. Through numerical experiments, it is found that the proposed algorithm holds a distinct advantage over the existing methods in solving all d-MPs.     

On the axioms of revealed preference in fuzzy consumer theory

The revealed preference is a central subject in classical consumer theory.Authors likeSamuelson,Arrow,Richter,Sen,Uzawa and others have proposed an axiomatic setting of revealedpreference theory.Consequently revealed preference axioms WARP and SARP and congruence axioms WCA andSCA have been considered.An important theorem of Sen establishes the equivalence between these axioms provided thefamily of budgets includes all non-empty finite sets of bundles.Fuzzy consumer theory(＝fuzzy choice functions)is a topic that appears in a lot of papers.Particularly,Banerjee studies in fuzzy context axioms of revealed preference and congruenceextending some results of Arrow and Sen.In this paper we modify the Banerjee definition of a fuzzy choice function(＝fuzzy consumer)and we study some fuzzy versions of the axioms of revealed preference and congruence.Banerjeefuzzifies only the range of a consumer;we use a fuzzification of both the domain and the range of aconsumer.The axioms WAFRP,SAFRP,WFCA,SFCA generalize to fu     

The structure of departure process and optimal control strategy <Emphasis Type="Italic">N</Emphasis>* for <Emphasis Type="Italic">Geo/G</Emphasis>/1 discrete-time queue with multiple server vacations and Min(<Emphasis Type="Italic">N,V</Emphasis>)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n⁺ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0⁺, n⁺] from any initial state. Meanwhile, the relationship among departure process, server’s state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures, including the expected length of server busy period, server’s actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N* for minimizing the system cost under a given cost structure.     

An analogue of the ℤ<Subscript>4</Subscript>-Goethals code in non-primitive length

This paper constructs a cyclic ?₄-code with a parity-check matrix similar to that of Goethals code but in length 2^m + 1, for all m ≥ 4. This code is a subcode of the lifted Zetterberg code for m even. Its minimum Lee weight is shown to be at least 10, in general, and exactly 12 in lengths 33, 65. The authors give an algebraic decoding algorithm which corrects five errors in these lengths for m = 5, 6 and four errors for m > 6.     

Linear Bayes estimators applied to the inverse Gaussian lifetime model

For the two-parameter inverse Gaussian distribution denoted by I G(μ,λ),the authors employ a linear Bayes procedure to estimate the parameters μ and λ.The superiority of the proposed linear Bayes estimator(LBE) over both the classical UMVUE and the maximum likelihood estimator(MLE) is established in terms of the mean squared error matrix(MSEM) criterion.Compared with the usual Bayes estimator,which is obtained by an MCMC method,the proposed LBE is simple and easy to use.Some numerical results are presented to verify that the LBE performs well.     

Pricing and inventory control strategy for a periodic-review energy buy-back system

Along with the rapid development of economics and enhancement of industrialization, the power demand keeps rising and frequently creates mismatch between demand and supply in electricity. This provides miscellaneous energy buy-back programs with great opportunities. Such programs, when activated, offer certain amount of financial compensations to participants for reducing their energy consumption during peak time. They aim at encouraging participants to shift their electricity usage from peak to non-peak time, and thereby release the demand pressure during peak time. This paper considers a periodic-review joint pricing and inventory decision model under an energy buy-back program over finite planning horizons, in which the compensation levels, setup cost and additive random demand function are incorporated. The objective is to maximize a manufacturer’s expected total profit. By using Veinott’s conditions, it is shown that the manufacturer’s optimal decision is a state dependent (s, S, P) policy under a peak market condition, or partly an (s, S, A, P) policy under the normal market condition.