On polynomial solutions of the linear stationary control system

It is known that the controllable system x′ = Bx + Du, where the x is the n-dimensional vector, can be transferred from an arbitrary initial state x(0) = x ⁰ to an arbitrary finite state x(T) = x ^T by the control function u(t) in the form of the polynomial in degrees t. In this work, the minimum degree of the polynomial is revised: it is equal to 2p + 1, where the number (p ? 1) is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to n. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.     

Eigenvalue condition numbers and pseudospectra of Fiedler matrices

The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root \({\lambda }\) of a monic polynomial p(z) with the condition number of \({\lambda }\) as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of \({\lambda }\) as an eigenvalue of an arbitrary Fiedler matrix with the condition number of \({\lambda }\) as an eigenvalue of the classical Frobenius companion matrices, and (c) the pseudozero sets of p(z) and the pseudospectra of any Fiedler matrix of p(z). We prove that, if the coefficients of the polynomial p(z) are not too large and not all close to zero, then the conditioning of any root \({\lambda }\) of p(z) is similar to the conditioning of \({\lambda }\) as an eigenvalue of any Fiedler matrix of p(z). On the contrary, when p(z) has some large coefficients, or they are all close to zero, the conditioning of \({\lambda }\) as an eigenvalue of any Fiedler matrix can be arbitrarily much larger than its conditioning as a root of p(z) and, moreover, when p(z) has some large coefficients there can be two different Fiedler matrices such that the ratio between the condition numbers of \({\lambda }\) as an eigenvalue of these two matrices can be arbitrarily large. Finally, we relate asymptotically the pseudozero sets of p(z) with the pseudospectra of any given Fiedler matrix of p(z), and the pseudospectra of any two Fiedler matrices of p(z).     

Symmetric Masks for In-fill Pixel Interpolation on Discrete <Emphasis Type="Italic">p</Emphasis>:<Emphasis Type="Italic">q</Emphasis> Lattices

A 2D p:q lattice contains image intensity entries at pixels located at regular, staggered intervals that are spaced p rows and q columns apart. Zero values appear at all other intermediate grid locations. We consider here the construction, for any given p:q, of convolution masks to smoothly and uniformly interpolate values across all of the intermediate grid positions. The conventional pixel-filling approach is to allocate intensities proportional to the fractional area that each grid pixel occupies inside the boundaries formed by the p:q lines. However, these area-based masks have asymmetric boundaries, flat interior values and may be odd or even in size. Where edges, lines or points are in-filled, area-based p:q masks imprint intensity patterns that recall p:q because the shape of those masks is asymmetric and depends on p:q. We aim to remove these “memory” artefacts by building symmetric p:q masks. We show here that smoother, symmetric versions of such convolution masks exist. The coefficients of the masks constructed here have simple integer values whose distribution is derived purely from symmetry considerations. We have application for these symmetric interpolation masks as part of a precise image rotation algorithm which disguises the rotation angle, as well as to smooth back-projected values when performing discrete tomographic image reconstruction.     

On s-uniform property of compressing sequences derived from primitive sequences modulo odd prime powers

Let Z/(p^e) be the integer residue ring modulo p^e with p an odd prime and e ≥ 2. We consider the suniform property of compressing sequences derived from primitive sequences over Z/(p^e). We give necessary and sufficient conditions for two compressing sequences to be s-uniform with α provided that the compressing map is of the form ?(x₀, x₁,...,x_e?1) = g(x_e?1) + η(x₀, x₁,..., x_e?2), where g(x_e?1) is a permutation polynomial over Z/(p) and η is an (e ? 1)-variable polynomial over Z/(p).     

A real QZ algorithm for structured companion pencils

We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an \(N\times N\) structured matrix pencil using O(N) flops per iteration and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.     

Computing the periods of preimages in surjective cellular automata

A basic property of one-dimensional surjective cellular automata (CA) is that any preimage of a spatially periodic configuration (SPC) is spatially periodic as well. This paper investigates the relationship between the periods of SPC and the periods of their preimages for various classes of CA. When the CA is only surjective and y is a SPC of least period p, the least periods of all preimages of y are multiples of p. By leveraging on the De Bruijn graph representation of CA, we devise a general algorithm to compute the least periods appearing in the preimages of a SPC, along with their corresponding multiplicities (i.e. how many preimages have a particular least period). Next, we consider the case of linear and bipermutive cellular automata (LBCA) defined over a finite field as state alphabet. In particular, we show an equivalence between preimages of LBCA and concatenated linear recurring sequences (LRS) that allows us to give a complete characterization of their periods. Finally, we generalize these results to LBCA defined over a finite ring as alphabet.     

Cryptanalysis of an MOR cryptosystem based on a finite associative algebra基于有限结合代数的 MOR 公钥密码安全性分析

The Shor algorithm is effective for public-key cryptosystems based on an abelian group. At CRYPTO 2001, Paeng (2001) presented a MOR cryptosystem using a non-abelian group, which can be considered as a candidate scheme for post-quantum attack. This paper analyses the security of a MOR cryptosystem based on a finite associative algebra using a quantum algorithm. Specifically, let L be a finite associative algebra over a finite field F. Consider a homomorphism φ: Aut(L) → Aut(H)×Aut(I), where I is an ideal of L and H ? L/I. We compute dim Im(φ) and dim Ker(φ), and combine them by dim Aut(L) = dim Im(φ)+dim Ker(φ). We prove that Im(φ) = Stab_Comp(H,I)(μ + B²(H, I)) and Ker(φ) ? Z¹(H, I). Thus, we can obtain dim Im(φ), since the algorithm for the stabilizer is a standard algorithm among abelian hidden subgroup algorithms. In addition, Z¹(H, I) is equivalent to the solution space of the linear equation group over the Galois fields GF(p), and it is possible to obtain dim Ker(φ) by the enumeration theorem. Furthermore, we can obtain the dimension of the automorphism group Aut(L). When the map ? ∈ Aut(L), it is possible to effectively compute the cyclic group 〈?〉 and recover the private key a. Therefore, the MOR scheme is insecure when based on a finite associative algebra in quantum computation.     

An Improved Approximation Algorithm for the Most Points Covering Problem

In this paper we present a polynomial time approximation scheme for the most points covering problem. In the most points covering problem, n points in R ², r>0, and an integer m>0 are given and the goal is to cover the maximum number of points with m disks with radius r. The dual of the most points covering problem is the partial covering problem in which n points in R ² are given, and we try to cover at least p≤n points of these n points with the minimum number of disks. Both these problems are NP-hard. To solve the most points covering problem, we use the solution of the partial covering problem to obtain an upper bound for the problem and then we generate a valid solution for the most points covering problem by a careful modification of the partial covering solution. We first present an improved approximation algorithm for the partial covering problem which has a better running time than the previous algorithm for this problem. Using this algorithm, we attain a \((1 - \frac{{2\varepsilon }}{{1 +\varepsilon }})\)-approximation algorithm for the most points covering problem. The running time of our algorithm is \(O((1+\varepsilon )mn+\epsilon^{-1}n^{4\sqrt{2}\epsilon^{-1}+2}) \) which is polynomial with respect to both m and n, whereas the previously known algorithm for this problem runs in \(O(n \log n +n\epsilon^{-6m+6} \log (\frac{1}{\epsilon}))\) which is exponential regarding m.     

Efficient distance-based representative skyline computation in 2D space

Representative skyline computation is a fundamental issue in database area, which has attracted much attention in recent years. A notable definition of representative skyline is the distance-based representative skyline (DBRS). Given an integer k, a DBRS includes k representative skyline points that aims at minimizing the maximal distance between a non-representative skyline point and its nearest representative. In the 2D space, the state-of-the-art algorithm to compute the DBRS is based on dynamic programming (DP) which takes O(k m ²) time complexity, where m is the number of skyline points. Clearly, such a DP-based algorithm cannot be used for handling large scale datasets due to the quadratic time cost. To overcome this problem, in this paper, we propose a new approximate algorithm called ARS, and a new exact algorithm named PSRS, based on a carefully-designed parametric search technique. We show that the ARS algorithm can guarantee a solution that is at most ?? larger than the optimal solution. The proposed ARS and PSRS algorithms run in O(klog2mlog(T/??)) and O(k ² log3m) time respectively, where T is no more than the maximal distance between any two skyline points. We also propose an improved exact algorithm, called PSRS+, based on an effective lower and upper bounding technique. We conduct extensive experimental studies over both synthetic and real-world datasets, and the results demonstrate the efficiency and effectiveness of the proposed algorithms.     

Noise Level Estimation in High-Dimensional Linear Models

We consider the problem of estimating the noise level σ² in a Gaussian linear model Y = Xβ+σξ, where ξ ∈ ?ⁿ is a standard discrete white Gaussian noise and β ∈ ?^p an unknown nuisance vector. It is assumed that X is a known ill-conditioned n × p matrix with n ≥ p and with large dimension p. In this situation the vector β is estimated with the help of spectral regularization of the maximum likelihood estimate, and the noise level estimate is computed with the help of adaptive (i.e., data-driven) normalization of the quadratic prediction error. For this estimate, we compute its concentration rate around the pseudo-estimate ||Y ? Xβ||²/n.     

Algorithmizing design of a class of combinatorial block diagrams

An algorithm to design combinatorial symmetrical block diagrams of the class B(n, s = p + 1, σ = 1), where p is a simple number, was described and illustrated by an example for s = 7 + 1.     

Higher-order polynomial approximation

A new approach to polynomial higher-order approximation (smoothing) based on the basic elements method (BEM) is proposed. A BEM polynomial of degree n is defined by four basic elements specified on a three-point grid: x ₀ + α < x ₀ < x ₀ + β, αβ <0. Formulas for the calculation of coefficients of the polynomial model of order 12 were derived. These formulas depend on the interval length, continuous parameters α and β, and the values of f ^(m)(x ₀+ν), ν = α, β, 0, m = 0,3. The application of higher-degree BEM polynomials in piecewise-polynomial approximation and smoothing improves the stability and accuracy of calculations when the grid step is increased and reduces the computational complexity of the algorithms.     

Computing the Ramsey number R(4,3,3) using abstraction and symmetry breaking

The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for almost 50 years. This paper presents a methodology based on abstraction and symmetry breaking that applies to solve hard graph edge-coloring problems. The utility of this methodology is demonstrated by using it to compute the value R(4, 3, 3) = 30. Along the way it is required to first compute the previously unknown set \(\mathcal {R}(3,3,3;13)\) consisting of 78,892 Ramsey colorings.     

Multigrid algorithms for $$\varvec{}$$-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied.     

A metric for total tardiness minimization

In this paper we consider the NP-hard 1|r _j |ΣT _j scheduling problem, suggesting a polynomial algorithm to find its approximate solution with the guaranteed absolute error. The algorithm employs a metric introduced in the parameter space. In addition, we study the possible application of such an approach to other scheduling problems.     

Non-binary entanglement-assisted quantum stabilizer codes

In this paper, we present the p^m-ary entanglement-assisted (EA) stabilizer formalism, where p is a prime and m is a positive integer. Given an arbitrary non-abelian “stabilizer”, the problem of code construction and encoding is settled perfectly in the case of m = 1. The optimal number of required maximally entangled pairs is discussed and an algorithm to determine the encoding and decoding circuits is proposed. We also generalize several bounds on p-ary EA stabilizer codes, such as the BCH bound, the G-V bound and the linear programming bound. However, the issue becomes tricky when it comes to m > 1, in which case, the former construction method applies only when the non-commuting “stabilizer” satisfies a sophisticated limitation.     

Counting the Number of Perfect Matchings in K5-Free Graphs

Counting the number of perfect matchings in graphs is a computationally hard problem. However, in the case of planar graphs, and even for K_3,3-free graphs, the number of perfect matchings can be computed efficiently. The technique to achieve this is to compute a Pfaffian orientation of a graph. In the case of K₅-free graphs, this technique will not work because some K₅-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K₅-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC². We remark that our results generalize to graphs without singly-crossing minor.     

Some lower bounds on the algebraic immunity of functions given by their trace forms

The algebraic immunity of a Boolean function is a parameter that characterizes the possibility to bound this function from above or below by a nonconstant Boolean function of a low algebraic degree. We obtain lower bounds on the algebraic immunity for a class of functions expressed through the inversion operation in the field GF(2 ⁿ ), as well as for larger classes of functions defined by their trace forms. In particular, for n ≥ 5, the algebraic immunity of the function Tr _n (x ^?1) has a lower bound ?2√n + 4? ? 4, which is close enough to the previously obtained upper bound ?√n? + ?n/?√n?? ? 2. We obtain a polynomial algorithm which, give a trace form of a Boolean function f, computes generating sets of functions of degree ≤ d for the following pair of spaces. Each function of the first (linear) space bounds f from below, and each function of the second (affine) space bounds f from above. Moreover, at the output of the algorithm, each function of a generating set is represented both as its trace form and as a polynomial of Boolean variables.     

On a new extension of BTP for binary CSPs

The study of broken-triangles is becoming increasingly ambitious, by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through either value merging or variable elimination. Considerable progress has been made in extending this important concept, such as dual broken-triangle and weakly broken-triangle, in order to maximize the number of captured tractable CSP instances and/or the number of merged values. Specifically, m-wBTP allows us to merge more values than BTP. DBTP, ??-BTP, k-BTP, WBTP and m-wBTP permit us to capture more tractable instances than BTP. However, except BTP, none of these extensions allows variable elimination while preserving satisfiability. Moreover, k-BTP and m-wBTP define bigger tractable classes around BTP but both of them generally need a high level of consistency. Here, we introduce a new weaker form of BTP, called m-fBTP for flexible broken-triangle property, which will represent a compromise between most of these previous tractable properties based on BTP. m-fBTP allows us on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define a new bigger tractable class for which arc consistency is a decision procedure. Likewise, m-fBTP permits to merge more values than BTP but fewer than m-wBTP. The binary CSP instances satisfying m-fBTP are solved by algorithms of the state-of-the-art like MAC and RFL in polynomial time. An open question is whether it is possible to compute, in polynomial time, the existence of some variable ordering for which a given instance satisfies m-fBTP.