Application of coding theory to the construction of the Singer difference sets†

In this paper we give a new and simple construction for the cyclic [(q ^m ? 1)/(q ? 1), q ^m?1, q ^m?2(q? 1)]—difference sets (q = p ^γ is a prime power) using the methods of coding theory. The construction is such that, in the case q = 2, the 2-ranks of both the incidence matrix and its complementary matrix are easily determined.     

Identification of certain noisy MA models: new results

In this paper, we address the identification problem of p-inputs q-outputs MA models, corrupted by a white noise with unknown covariance matrix, in the case where p<q. Under certain additional conditions, we show that the generating function of the MA model is identifiable (up to a p×p constant orthogonal matrix) from the autocovariance function of the observation. Our results extend those already obtained in Desbouvries et al. [5] and Desbouvries and Loubaton [6].     

The group of permutation automorphisms of a <Emphasis Type="Italic">q</Emphasis>-ary hamming code

We prove that the group of permutation automorphism of a q-ary Hamming code of length n = (q ^m − 1)/(q − 1) is isomorphic to the unitriangular group UT _m (q) if the code has a parity-check matrix composed of all columns of the form (0 ...0 1 * ... *)^T. We also show that the group of permutation automorphisms of a cyclic Hamming code cannot be isomorphic to UT _m (q). We thus show that equivalent codes can have different permutation automorphism groups.     

Identification of rank one rational spectral densities from noisy observations: A stochastic realization approach

Let S be a q × q spectral density matrix which is the sum of a rational rank one spectral density S₁ and of a constant positive definite matrix Q. The identification problem of S₁ and Q from S is addressed. The conditions under which S₁ and Q are identifiable are first derived. Then, an identification method is proposed. It is based on a parametrization of the external stochastic realizations of S whose innovation sequence has a prescribed dimension.     

Sensitivity minimization in an H ∞ norm: parametrization of all suboptimal solutions

A number of problems in the control of linear feedback systems can be reduced to the following: we are given three stable rational matrix functions K, ?, ψ of sizes p ₁ x q ₁, p ₁ x q ₂ and p ₂ x q ₁ respectively, and seek a stable rational q ₂ x p ₂ matrix function S so as to minimize ¦K + ?Sψ¦_∞. We assume that p ₁ ≥ q ₂, p ₂≤q ₁ and that ? and ψ have maximal rank (q ₂ and p ₂ respectively) on the jω-axis. Given a tolerance level μ sufficiently large, we obtain a linear fractional map G → F = [0 ₁₁ G + 0 ₁₂, 0 ₁₃][0 ₂₁ G + 0 ₂₂, 0 ₂₃]^?1 such that F = K + ?Sψ with S stable and ¦F _∞ ≤ μ if and only if G is a stable q ₂ x p ₂ matrix function with ¦G¦_∞ ≤ 1. The computation of 0 = [0 _ij ] (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) reduces to solving a pair of symmetric Wiener–Hopf factorization problems. For the special case where ? = [I _q2, 0]^T, ψ = [I _p2, 0] (and K not necessarily stable) to which the general case can be reduced, we provide explicit state-space formulae for 0 in terms of a state-space realization of K and the solutions of some related Riccati equations. The approach is a natural extension of that of Ball–Helton and Ball–Ran for the case p ₁ = q ₂ and p ₂ = q ₁.     

A novel multiple-attribute group decision-making method based on q-rung orthopair fuzzy generalized power weighted aggregation operators

In this paper, a novel approach is developed to deal with multiple-attribute group decision-making (MAGDM) problem under q-rung orthopair fuzzy environment. Firstly, some operators have been proposed to aggregate q-rung orthopair fuzzy information, such as the q-rung orthopair fuzzy generalized power averaging (q-ROFGPA) operator, the q-rung orthopair fuzzy generalized power weighted averaging (q-ROFGPWA) operator, the q-rung orthopair fuzzy generalized power geometric (q-ROFGPG) operator, and the q-rung orthopair fuzzy generalized power weighted geometric (q-ROFGPWG) operator. In addition, some desirable properties and special cases of these operators are discussed. Second, a novel approach is developed to solve MAGDM problem under the q-rung orthopair fuzzy environment based on the proposed q-ROFGPWA and q-ROFGPWG operators. Finally, a practical example is given to illustrate the application of the proposed method, and further the sensitivity analysis and comparative analysis are carried out.     

Quantum Algorithms for Weighing Matrices and Quadratic Residues

Abstract. In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to devise new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein and Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol χ (which indicates if an element of a finite field F _q is a quadratic residue or not). It is shown how for a shifted Legendre function f _s (i)=χ(i+s) , the unknown s ∈ F _q can be obtained exactly with only two quantum calls to f _s . This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log q + log ((1-ɛ )/2) queries to solve the same problem.     

Quantum codes based on fast pauli block transforms in the finite field

Motivated by the fast Pauli block transforms (or matrices) over the finite field GF(q) for an arbitrary number q, we suggest how to construct the simplified quantum code on the basis of quadratic residues. The present quantum code, which is the stabilizer quantum code, can be fast generated from an Abelian group with commutative quantum operators being selected from a suitable Pauli block matrix. This construction does not require the dual-containing or self-orthogonal constraint for the standard quantum error-correction code, thus allowing us to construct a quantum code with much efficiency.     

Functional relationships between the conventional steady-state error characteristics and the weighting matrices in the quadratic performance index†

The requirements for achieving a desired value of steady-state error for an optimal control system utilizing the quadratic performance index with a linear plant arc derived for step, ramp and parabolic inputs. These requirements result in a number of the elements of the Riccati matrix being specified in terms of known parameters. It is shown that for an all-pole plant the control system cannot follow a parabolic input. For the stop input the q ₁₁ element of the Q matrix is specified as one for an all-pole plant or in terms of the fixed plant parameters for a pole-zero plant. Depending on the number of requirements and the order of the system it may be necessary to assume values for the remaining elements of the Q matrix in order to solve the reduced Riccati equation.     

A Fast Output-Sensitive Algorithm for Boolean Matrix Multiplication

We use randomness to exploit the potential sparsity of the Boolean matrix product in order to speed up the computation of the product. Our new fast output-sensitive algorithm for Boolean matrix product and its witnesses is randomized and provides the Boolean product and its witnesses almost certainly. Its worst-case time performance is expressed in terms of the input size and the number of non-zero entries of the product matrix. It runs in time [(O)\tilde](n²s^w/2-1)\widetilde{O}(n^{2}s^{\omega/2-1}), where the input matrices have size n×n, the number of non-zero entries in the product matrix is at most s, ω is the exponent of the fast matrix multiplication and [(O)\tilde](f(n))\widetilde{O}(f(n)) denotes O(f(n)log  ^d n) for some constant d. By the currently best bound on ω, its running time can be also expressed as [(O)\tilde](n²s^0.188)\widetilde{O}(n^{2}s^{0.188}). Our algorithm is substantially faster than the output-sensitive column-row method for Boolean matrix product for s larger than n ^1.232 and it is never slower than the fast [(O)\tilde](n^w)\widetilde{O}(n^{\omega})-time algorithm for this problem. By applying the fast rectangular matrix multiplication, we can refine our upper bound further to the form [(O)\tilde](n^{w(\frac12log_ns,1,1)})\widetilde{O}(n^{\omega(\frac{1}{2}\log_{n}s,1,1)}), where ω(p,q,t) is the exponent of the fast multiplication of an n ^p ×n ^q matrix by an n ^q ×n ^t matrix.     

Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems

In this paper, we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the preconditioned conjugate gradient solution of the linearized Newton system to solve Au=q(u) u, q(u) being the Rayleigh quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi–Davidson method on all the test problems.     

q-Rung orthopair fuzzy sets-based decision-theoretic rough sets for three-way decisions under group decision making

As an extension of Pythagorean fuzzy sets, the q-rung orthopair fuzzy sets (q-ROFSs) can easily solve uncertain information in a broader perspective. Considering the fine property of q-ROFSs, we introduce q-ROFSs into decision-theoretic rough sets (DTRSs) and use it to portray the loss function. According to the Bayesian decision procedure, we further construct a basic model of q-rung orthopair fuzzy decision-theoretic rough sets (q-ROFDTRSs) under the q-rung orthopair fuzzy environment. At the same time, we design the corresponding method for the deduction of three-way decisions by utilizing projection-based distance measures and TOPSIS. Then, we extend q-ROFDTRSs to adapt the group decision-making (GDM) scenario. To fuse different experts’ evaluation results, we propose some new aggregation operators of q-ROFSs by utilizing power average (PA) and power geometric (PG) operators, that is, q-rung orthopair fuzzy power average, q-rung orthopair fuzzy power weighted average (q-ROFPWA), q-rung orthopair fuzzy power geometric, and q-rung orthopair fuzzy power weighted geometric (q-ROFPWG). In addition, with the aid of q-ROFPWA and q-ROFPWG, we investigate three-way decisions with q-ROFDTRSs under the GDM situation. Finally, we give the example of a rural e-commence GDM problem to illustrate the application of our proposed method and verify our results by conducting two comparative experiments.     

A joint economic-lot-size model for purchaser and vendor in fuzzy sense

This paper investigates a group of computing schemas for joint economic lot size as fuzzy values of the economic lot size model for purchaser and vendor. We express the fuzzy order quantity/production lot size for the purchaser/vendor as the normal triangular fuzzy number (q₁, q₀, q₂) and then we solve the aforementioned optimization problem under the condition 0 < q₁ < q₀ < q₂. We find that, after defuzzification, the joint total relevant cost is slightly higher than in the crisp model.     

Entanglement and Yangian in a {V^{\otimes 3}} Yang-Baxter system

In this paper, a 8 × 8 unitary Yang-Baxter matrix \breveR₁₂₃(q₁,q₂,f){\breve{R}_{123}(\theta_{1},\theta_{2},\phi)} acting on the triple tensor product space, which is a solution of the Yang-Baxter Equation for three qubits, is presented. Then quantum entanglement and the Berry phase of the Yang-Baxter system are studied. The Yangian generators, which can be viewed as the shift operators, are investigated in detail. And it is worth mentioning that the Yangian operators we constructed are independent of choice of basis.     

Some q‐Rung Orthopai Fuzzy Bonferroni Mean Operators and Their Application to Multi‐Attribute Group Decision Making

In the real multi‐attribute group decision making (MAGDM), there will be a mutual relationship between different attributes. As we all know, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between parameters. In addition, in describing uncertain information, the eminent characteristic of q‐rung orthopair fuzzy sets (q‐ROFs) is that the sum of the qth power of the membership degree and the qth power of the degrees of non‐membership is equal to or less than 1, so the space of uncertain information they can describe is broader. In this paper, we combine the BM operator with q‐rung orthopair fuzzy numbers (q‐ROFNs) to propose the q‐rung orthopair fuzzy BM (q‐ROFBM) operator, the q‐rung orthopair fuzzy weighted BM (q‐ROFWBM) operator, the q‐rung orthopair fuzzy geometric BM (q‐ROFGBM) operator, and the q‐rung orthopair fuzzy weighted geometric BM (q‐ROFWGBM) operator, then the MAGDM methods are developed based on these operators. Finally, we use an example to illustrate the MAGDM process of the proposed methods. The proposed methods based on q‐ROFWBM and q‐ROFWGBM operators are very useful to deal with MAGDM problems.     

半张量积压缩感知模型的l0-范数解

目的 半张量积压缩感知模型是一种可以有效降低压缩感知过程中随机观测矩阵所占存储空间的新方法,利用该模型可以成倍降低观测矩阵所需的存储空间。为寻求基于该模型新的重构方法,同时提升降维后观测矩阵的重构性能,提出一种采用光滑高斯函数拟合l₀-范数方法进行重构。方法 构建降维随机观测矩阵,对原始信号进行采样;构建可微且期望值为零的光滑高斯函数来拟合不连续的l₀-范数,采用最速下降法进行重构,最终得到稀疏信号的估计值。结果 实验分别采用1维稀疏信号和2维图像信号进行测试,并从重构概率、收敛速度、重构信号的峰值信噪比等角度进行了测试和比较。验证结果表明,本文所述算法的重构概率、收敛速度较该模型的l_q-范数（0 <q <1）方法有一定的提升,且当观测矩阵大小降低为通常的1/64,甚至1/256时,仍能保持较高的重构性能。结论 本文所述的重构算法,能在更大程度上降低观测矩阵的大小,同时基本保持重构的精度。     

Proving hypergeometric identities by numerical verifications

It is known that proper and q-proper hypergeometric identities can be certified by checking a finite number, say n₁, of initial values. By studying the degree and the height of the determinant of a polynomial matrix, we give a new method to estimate n₁. Examples show that the new estimates are considerably smaller than the previous results.     

The structure factor of dense two-dimensional polymer solutions

According to the generalized Porod law the intramolecular structure factor F(q) of compact objects with surface dimension ds scales as F(q)/N≈1/(R(N)q)^2d−ds in the intermediate range of the wave vector q with d being the dimension of the embedding space, N the mass of the objects and R(N)∼N1/d their typical size. By means of molecular-dynamics simulations of a bead-spring model with chain lengths up to N=2048 it is shown that dense self-avoiding polymers in strictly two dimensions (d=2) adopt compact configurations of surface dimension ds=5/4. In agreement with the generalized Porod law the Kratky representation of F(q) thus reveals a nonmonotonous behavior with q²F(q)∼1/(N1/2q)^3/4. Using a similar data analysis we argue briefly that melts of non-concatenated rings in three dimensions become marginally compact with ds=d=3, i.e. q²F(q)∼N⁰/q, for asymptotically long chains.     

RBF neural network based on q-Gaussian function in function approximation

To enhance the generalization performance of radial basis function (RBF) neural networks, an RBF neural network based on a q-Gaussian function is proposed. A q-Gaussian function is chosen as the radial basis function of the RBF neural network, and a particle swarm optimization algorithm is employed to select the parameters of the network. The non-extensive entropic index q is encoded in the particle and adjusted adaptively in the evolutionary process of population. Simulation results of the function approximation indicate that an RBF neural network based on q-Gaussian function achieves the best generalization performance.     

Use of the q-Gaussian mutation in evolutionary algorithms

This paper proposes the use of the q-Gaussian mutation with self-adaptation of the shape of the mutation distribution in evolutionary algorithms. The shape of the q-Gaussian mutation distribution is controlled by a real parameter q. In the proposed method, the real parameter q of the q-Gaussian mutation is encoded in the chromosome of individuals and hence is allowed to evolve during the evolutionary process. In order to test the new mutation operator, evolution strategy and evolutionary programming algorithms with self-adapted q-Gaussian mutation generated from anisotropic and isotropic distributions are presented. The theoretical analysis of the q-Gaussian mutation is also provided. In the experimental study, the q-Gaussian mutation is compared to Gaussian and Cauchy mutations in the optimization of a set of test functions. Experimental results show the efficiency of the proposed method of self-adapting the mutation distribution in evolutionary algorithms.