Irreducible Sequences Research Articles

Universality of Weyl unitaries

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p×p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v, satisfy up=vp=1p (the p×p identity matrix) and the commutation relation uv=ζvu, where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d×d unitary matrices such that up=vp=1d and uv=ζvu, then there exists a unital completely positive linear map ϕ:Mp(C)→Md(C) such that ϕ(u)=u and ϕ(v)=v. We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails.There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63]. This standard construction is generalised herein to the case p≥3, producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

Light and heavy particles on a fluctuating surface: Bunchwise balance, irreducible sequences, and local density-height correlations

We study the early time and coarsening dynamics in the Light-Heavy model, a system consisting of two species of particles ($light$ and $heavy$) coupled to a fluctuating surface (described by tilt fields). The dynamics of particles and tilts are coupled through local update rules, and are known to lead to different ordered and disordered steady state phases depending on the microscopic rates. We introduce a generalized balance mechanism in non-equilibrium systems, namely $bunchwise~balance$, in which incoming and outgoing transition currents are balanced between groups of configurations. This allows us to exactly determine the steady state in a subspace of the phase diagram of this model. We introduce the concept of $irreducible~sequences$ of interfaces and bends in this model. These sequences are non-local, and we show that they provide a coarsening length scale in the ordered phases at late times. Finally, we propose a $local$ correlation function ($\mathcal{S}$) that has a direct relation to the number of irreducible sequences, and is able to distinguish between several phases of this system through its coarsening properties. Starting from a totally disordered initial configuration, $\mathcal{S}$ displays an initial linear rise and a broad maximum. As the system evolves towards the ordered steady states, $\mathcal{S}$ further exhibits power law decays at late times that encode coarsening properties of the approach to the ordered phases. Focusing on early time dynamics, we posit coupled mean-field evolution equations governing the particles and tilts, which at short times are well approximated by a set of linearized equations, which we solve analytically. Beyond a timescale set by a lattice cutoff and preceding the onset of coarsening, our linearized theory predicts the existence of an intermediate power-law stretch, which we also find in simulations of the ordered regime of the system.

Additively irreducible sequences in commutative semigroups

Let S be a commutative semigroup, and let T be a sequence of terms from the semigroup S. We call T an (additively) irreducible sequence provided that no sum of some of its terms vanishes. Given any element a of S, let Da(S) be the largest length of an irreducible sequence such that the sum of all terms from the sequence is equal to a. In the case that any ascending chain of principal ideals starting from the ideal (a) terminates in S, we find necessary and sufficient conditions for Da(S) to be finite, and in particular, we give sharp lower and upper bounds for Da(S) in case Da(S) is finite. We also apply the result to commutative unitary rings. As a special case, the value of Da(S) is determined when S is the multiplicative semigroup of any finite commutative principal ideal unitary ring.

Improving ELM-based microarray data classification by diversified sequence features selection

In this paper, we focus on the problem of extreme learning machine (ELM)-based microarray data classification. Different from the traditional classification problem, the goal in this case is not just to predict the class labels for the unseen samples, but to make clear what lead to the results, i.e., the genes involving with a specific disease. This is especially significant for biologists, since they need to decipher the causes of disease. As a black-box method, ELM could not measure up to the task by itself. In this work, we propose a diversified sequence feature selection-based framework to address the problem. In this framework, (1) a sequence model, EWave, is introduced to ensure the structural ordering information among genes exploitable; (2) a concept of irreducible sequence is proposed, where the genes work as an orderly whole to keep high confidence with a specific class and any reduction in the genes decreases the confidence much. An efficient sequence mining algorithm together with some effective pruning rules is developed to mine such sequences; and (3) we study how to extract a set of diversified sequence features as the representative of all mined results. The problem is proved to be NP-hard. A greedy algorithm is presented to approximate the optimal solution. Experimental results show that the proposed approach significantly improves the efficiency and the effectiveness of ELM w.r.t some widely used feature selection techniques.

New algorithms and complexity status of the reducibility problem of sequences in open shop scheduling minimizing the makespan

We consider the problem of minimizing the makespan in open shop scheduling. The decision problem whether a given sequence in open shop scheduling is irreducible has already been considered, however, has not been solved yet. A sequence is an acyclic orientation of the Hamming graph K n ×K m . Irreducible sequences in open shop are the local optimal elements. We present two variants of algorithms based on the specific properties of the H-comparability graph. The first is polynomial whereas the second is exponential. The irreducibility is co-NP. The stated properties argue whether it belongs to P. The complexity status of the considered decision problem is updated.

Some Discrete Optimization Problems With Hamming and H-Comparability Graphs

Any H-comparability graph contains a Hamming graph as spanningsubgraph. An acyclic orientation of an H-comparability graph contains an acyclic orientation of the spanning Hamming graph, called sequence graph in the classical open-shop scheduling problem. We formulate different discrete optimization problems on the Hamming graphs and on H-comparability graphs and consider their complexity and relationship. Moreover, we explore the structures of these graphs in the class of irreducible sequences for the open shop problem in this paper.

Completely irreducible sequences of meromorphic functions

Given a sequence {fn} of meromorphic functions on a plane domain D, there exists a (possibly empty) open set U ⊂ D and a subsequence \( \{ f_{n_k } \} \) which converges uniformly (with respect to the spherical metric on Open image in new window ) on compact subsets of U, while no subsequence of \( \{ f_{n_k } \} \) converges uniformly on compact subsets of any larger open subset of D.

On the set of solutions of the open shop problem

In the classical open shop problem, n jobs have to be processedon m machines, where both job orders and machine orders can be chosen arbitrarily.A feasible (i.e., acyclic) combination of all job orders and machine orders iscalled a (multi‐) sequence. We investigatea set of sequences which are structurally optimal in the sense that there is at least oneoptimal sequence in this set for each instance of processing times. Such sequences arecalled irreducible. Investigations about irreducible sequences are believed to provide apowerful tool to improve exact and heuristic algorithms. Furthermore, structural propertiesof sequences are important for problems with uncertain processing times.We prove necessary and sufficient conditions for the irreducibility of a sequence. Forseveral values of n and m, we give the numbers of allsequences, of the sequences satisfying each of these conditions and of the irreduciblesequences, respectively. It turns out that only a very small fraction of all sequences isirreducible. Thus, algorithms which work only on the set of irreducible sequences insteadof the set of all sequences can potentially perform much better than conventional algorithms.

On the hardness of the classical job shop problem.

In a classical job shop problem, n jobs have to be processed onm machines, where the machine orders of the jobs are given. Computationalexperiments show that there are huge differences in the hardness of the job shop problem tominimize makespan depending on the given machine orders. We study a partial order“\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } \)” on the set of sequences, i.e., feasiblecombinations of job orders and machine orders, with the property thatB\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } \)B implies thatthe makespan of the semiactive schedule corresponding to sequence B isless than or equal to the makespan of any schedule corresponding to B.The minimal sequences according to this partial order are called irreducible.We present a polynomial algorithm to decide whether B\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } \)B holds andwe develop a new enumeration algorithm for irreducible sequences. To explain the differencesin the hardness of job shop problems, we study the relation between the hardness of a jobshop problem and the number of irreducible sequences corresponding to the given machine orders.

New steps in the amazing world of sequences and schedules

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timep i,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan. We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules. Furthermore, we investigate the problemO∥C max on an operation set with spanning tree structure.

Mathematical modelling of polymers, 3. Enumeration and generation of repeating irreducible sequences in linear bi‐, ter‐, quater‐ and quinquenary copolymers and in stereoregular homopolymers

AbstractIrreducible sequences in n‐ary copolymers are defined. An exhaustive and non‐redundant lexicographically ordered list of irreducible sequences of comonomers in binary (n = 2), ternary (n = 3), quaternary (n = 4) and quinquenary (n = 5) copolymers is presented. The numbers of irreducible sequences may be found either by an explicit formula in terms of the number n of comonomers and of the length m of the sequences, or by applying the Polya‐de Bruijn theorem, as shown in detail in the first two parts of this series. For stereoregular vinyl homopolymers, or for the geometrical (E/Z) isomerism in polyalkenes, irreducible sequences are the same as in binary copolymers. A nomenclature is proposed for m‐ads in stereoregular homopolymers with m > 2, i. e. beyond isotactic for m = 1, and syndiotactic for m = 2.

Localization of Eigenstates in One-Dimensional Infinite Disordered Systems with Off-Diagonal Randomness

Problem of localization of eigenstates is examined for one-dimensional infinite disordered systems with off-diagonal randomness. For this purpose Matsuda and Ishii's theory, based on Furstenberg's convergent theorem on products of random matrices, is generalized by intro­ ducing irreducible sequences S and irreducible transfer Q* as useful mathematical tools. A Furstenberg-type theorem is established for the product of matrices associated with a Markov-chain. This theorem leads to some conclusions about the localization of eigenstates, which are very similar, except for some minor differences, to those obtained. by Matsuda and Ishii for systems with diagonal-randomness only.

Layer-by-layer low-energy electron diffraction and two-dimensional space group symmetry

The wave field of LEED in a 2D structure is symmetrised with respect to the structure's 2D space group (2D structure meaning diperiodic structure composed of one or several 2D periodic overlayers lying in register on the 2D periodic face of a crystal). Attention is focused on the layer-KKR (Korringa-Kohn-Rostoker) method, the wave representations of which are beams outside layers and partial waves inside layers. It is shown how the field of beams and the field of partial waves are made globally symmetric representations of the 2D space group, how the 2D translations of an individual layer (giving rise to local 2D von Laue diffraction conditions) divide the beams into independent bundles and, further, how local point-group symmetry arranges the bundles in 'irreducible sequences' and combines the partial waves into 'irreducible showers'. The theory is applied to three examples of overlayer structures on surfaces of face-centered cubic crystals, and the coding of the scheme of symmetrisation is discussed.

On the Status of Irreducibility Problems of Open Shop Sequencing Problems

In an open shop scheduling problem (OSSP) one of the major tasks is to minimize the makespan over the sequences. It has practical values, for instance, in manufacturing, service industries, communication, and facility assignments. The concept of irreducibility has proven one of the most important and fertile research topics as the set of all irreducible sequences attains optimality independent of processing times. This work presents the recent complexity status of irreducibility of OS sequences. The recent works give encouraging results. The results obtained in past several years are viewed with useful hints for the developments of exact or heuristic algorithms and generalize the idea to general regular objectives. Key words: scheduling/sequencing theory; open shop; reducibility; potential optimality; complexity DOI: 10.3126/njst.v11i0.4147Nepal Journal of Science and Technology 11 (2010) 205-214

Investigation of lattice dynamics by means of vibronic spectra

Electronic transitions at a foreign probe ion in a lattice exhibit vibrational sidebands. They may be used as a new tool to investigate lattice dynamics. To make this too effective the theory of vibronic spectra has been refined. The principal refinement is that of a partial Born-Oppenheimer approximation which establishes a rigid coupling of the electronic wavefunctions of the foreign ion to its own nucleus. The structural form of the one-phonon sideband is given as a projection of the classical Greens' function (imaginary part) onto single multipole-fields in lattice space. Employing a variety of electronic transitions and by careful group-theoretical analysis a complete irreducible sequence of projections can be extracted from the experiments. These functional forms depend sensibly on the eigenvectors of the lattice modes, whence they contain information, which cannot be received from neutron scattering. Moreover, the spectra related to the higher multipoles give insight into the disturbed (local) dynamics, whereas the lower type spectra concern the undisturbed (ideal) one. Systems, where the foreign probe ion has high symmetry, are most suitable for the new method. For systems of low local symmetry approximations are suggested by the low-frequency behaviour of the measured one-phonon band. Because of the high resolution of optical measurements, the information from vibronic spectra is comparable or even superior in detailed richness to neutron scattering data.