Admissible Sequences Research Articles

Norm-attaining operators into Lorentz sequence spaces

We prove that the Lorentz sequence spaces do not have the property B of Lindenstrauss. In fact, for any admissible sequences w, v ∈ c0 \ l1, the set of norm-attaining operators from the Orlicz space hϕ(w) (ϕ is a certain Orlicz function) into d(v, 1) is not dense in the corresponding space of operators. We also characterize the spaces such that the subset of norm-attaining operators from the Marcinkiewicz sequence space into its dual is dense in the space of all bounded and linear operators between them.

Univoque numbers and an avatar of Thue–Morse

Univoque numbers are real numbers $\lambda > 1$ such that the number 1 admits a unique expansion in base $\lambda$, i.e., a unique expansion $1 = \sum_{j \geq 0} a_j \lambda^{-(j+1)}$, with $a_j \in \{0, 1, ..., \lceil \lambda \rceil -1\}$ for every $j \geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set $\{0, 1, >..., b\}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b, b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3 \to 31$, $2 \to 30$, $1 \to 03$, $0 \to 02$, occurs in a "universal" manner.

Monitoring and Active Diagnosis for Discrete-Event Systems

This article presents an original way to enrich the monitoring of discrete-event systems named active diagnosis. The objective of on-line active diagnosis is to find an admissible sequence of actions (or plan) that refines the diagnosis without radically changing the mission plan. This paper has two major contributions. First, active diagnosis is formally defined in the finite-state automata theory framework. This leads to the definition of a complete active diagnoser which on line monitors the system behavior. Secondly, from the complete active diagnoser is defined a planning problem. The goal is to find a conditional plan that defines an admissible sequence of actions. These actions are applied on the physical system and may conduct the active diagnoser into a diagnosable region.

Sturmian and spectral theory for discrete symplectic systems

We consider 2 n × 2 n 2n\times 2n symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n n . In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.

Optimisation of sampling windows design for population pharmacokinetic experiments

This paper describes an approach for optimising sampling windows for population pharmacokinetic experiments. Sampling windows designs are more practical in late phase drug development where patients are enrolled in many centres and in out-patient clinic settings. Collection of samples under the uncontrolled environment at these centres at fixed times may be problematic and can result in uninformative data. Population pharmacokinetic sampling windows design provides an opportunity to control when samples are collected by allowing some flexibility and yet provide satisfactory parameter estimation. This approach uses information obtained from previous experiments about the model and parameter estimates to optimise sampling windows for population pharmacokinetic experiments within a space of admissible sampling windows sequences. The optimisation is based on a continuous design and in addition to sampling windows the structure of the population design in terms of the proportion of subjects in elementary designs, number of elementary designs in the population design and number of sampling windows per elementary design is also optimised. The results obtained showed that optimal sampling windows designs obtained using this approach are very efficient for estimating population PK parameters and provide greater flexibility in terms of when samples are collected. The results obtained also showed that the generalized equivalence theorem holds for this approach.

Maximizing service availability for secure satellite broadcasting

AbstractCryptography is the most suitable technique to protect the access to subscribed value‐added services for mobile applications provided through satellite broadcasting (e.g. localization and mobile TV). However, if a temporary loss of signal is experienced by receivers, not only the data streaming is compromised, but also the key management‐related messages. Hence, when the signal is received again, it is impossible for the receivers to decrypt it. One way to overcome this issue is the re‐transmission of keys to guarantee that the largest set of legitimate receivers has the updated (set of) keys.In this paper we analyze the problem of key update in a mobile user context in order to maximize the service availability. In particular, we provide the following contributions: we propose a mathematical model that captures all the relevant features of mobile users. Then, we formally prove a few novel results on the structure of the admissible rekeying sequence. These results are used to design a scheduling algorithm for key broadcasting that, according to our model, minimizes the number of mobile users that are prevented from updating their crypto keys, hence continuing accessing the service. Finally, simulation results support our theoretical findings. Copyright © 2008 John Wiley & Sons, Ltd.

Examples of Generic Lattice Ideals of Codimension 3

In Peeva and Sturmfels (1998), the authors introduce a new notion of genericity for lattice ideals; however, the deterministic construction of this kind of “generic lattice ideals” with prescribed properties (such as Betti numbers) is still an open problem that appears to be difficult (cf. Miller and Sturmfels, 2005, Section 9.4). In this note, it is shown that any admissible sequence of Betti numbers for generic lattice ideals of codimension three in a polynomial ring in four variables can occur, by explicitly giving a family of generic lattice ideals, which includes ideals defining toric varieties, in each case.

Simple Pulse-Step Model Predictive Controller

This paper describes a novel simple model based predictive controller with manipulated value constraints. This controller is suitable for substitution of the classical PID controller used in industrial practice. It is assumed that the controlled system is stable, linear and t-invariant FIR system. The discrete step response sequence is used as the process model. Alternatively it is possible to use three-parameter model. To make the open-loop optimization easier the set of admissible control sequences is restricted to stepwise pulse-step sequences. The optimization procedure is then executable in reasonable time. A single tuning parameter is available for manual fine-tuning of the controller - the control moves penalty coefficient.

Sequences of reflection functors and the preprojective component of a valued quiver

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander–Reiten quiver. The results apply to the study of reduced words in the Weyl group associated with an indecomposable symmetrizable generalized Cartan matrix.

First Observations on Prefab Posets’ Whitney Numbers

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers’ triangular array — are then calculated and the explicit formula for them is provided. Next — in the second construction — we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F-sequence independent while the second partial order is F-sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F-determined cobweb poset’s Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section.

Discrete Systems and their Characteristic Spectral Functions

We define first-order discrete systems in the matrix-valued case. They are characterized by sequences of pair of matrices, called admissible sequences. We present two important examples of such sequences, called Szego and Nehari sequences. We introduce the characteristic spectral functions associated to a first-order system. We define in particular the scattering function, the Weyl function and the reflection coefficient function and we study the relationships between these functions.

Chaos in digital filters: identification of all periodic symbolic sequences admissible adjacent to zero

We study the symbolic sequences associated with a two-dimensional nonlinear map describing a digital filter with 2's complement arithmetic, and the role of the filter parameter a = 2cos θ in determining admissibility of these sequences. We solve an appropriately specialized form of Chua and Lin's inequalities governing admissibility, and obtain a complete identification of the infinite class of periodic symbolic sequences admissible in an open interval having parameter value θ = 0 as left end-point. A key step in the argument, easily established by parity for sequences having odd period, requires a more penetrating analysis in the even case. Computer searches for sequences with periods up to 200 have revealed that this class accounts for the greater number of admissible periodic sequences associated with the map. Our solution provides an efficient construction for the periodic symbolic sequences admissible adjacent to zero, parametrized by the period and the number of +− digit pairs present. We also show that the precise interval of admissibility is (0, π/N), where N is the least period, which permits us to relate the sequences to the points of state space that give rise to them. We thereby obtain a bound on the measure of the subset of state space consisting of the points associated with periodic symbolic sequences, improving previously published bounds. A similar identification of the periodic sequences admissible at the far end of the parameter range, adjacent to θ = π, is deduced.

Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

The notions of a perfect element and an admissible element of the free modular lattice D r generated by r ≥ 1 elements are introduced by Gelfand and Ponomarev in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901–1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3–58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1–56]. We recall that an element a ∈ D of a modular lattice L is perfect, if for each finite-dimension indecomposable K -linear representation ρ X : L → L ( X ) over any field K , the image ρ X ( a ) ⊆ X of a is either zero, or ρ X ( a ) = X , where L ( X ) is the lattice of all vector K -subspaces of X . A complete classification of such elements in the lattice D 4 , associated to the extended Dynkin diagram D ˜ 4 (and also in D r , where r > 4 ) is given in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901–1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3–58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1–56; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, I, Uspehi Mat. Nauk 31 (5(191)) (1976) 71–88 (Russian); English translation: Russian Math. Surv. 31 (5) (1976) 67–85; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, II. Uspehi Mat. Nauk 32 (1(193)) (1977) 85–106 (Russian); English translation: Russian Math. Surv. 32 (1) (1977) 91–114]. The main aim of the present paper is to classify all the admissible elements and all the perfect elements in the Dedekind lattice D 2 , 2 , 2 generated by six elements that are associated to the extended Dynkin diagram E ˜ 6 . We recall that in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901–1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3–58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1–56], Gelfand and Ponomarev construct admissible elements of the lattice D r recurrently. We suggest a direct method for creating admissible elements. Using this method we also construct admissible elements for D 4 and show that these elements coincide modulo linear equivalence with admissible elements constructed by Gelfand and Ponomarev. Admissible sequences and admissible elements for D 2 , 2 , 2 (resp. D 4 ) form 14 classes (resp. 8 classes) and possess some periodicity. Our classification of perfect elements for D 2 , 2 , 2 is based on the description of admissible elements. The constructed set H + of perfect elements is the union of 64 -element distributive lattices H + ( n ) , and H + is the distributive lattice itself. The lattice of perfect elements B + obtained by Gelfand and Ponomarev for D 4 can be imbedded into the lattice of perfect elements H + , associated with D 2 , 2 , 2 . Herrmann in [C. Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbänden. (German) [Frames and generating quadruples in modular lattices], Algebra Universalis 14 (3) (1982) 357–387] constructed perfect elements s n , t n , p i , n in D 4 by means of some endomorphisms γ i j and showed that these perfect elements coincide with the Gelfand–Ponomarev perfect elements modulo linear equivalence. We show that the admissible elements in D 4 are also obtained by means of Herrmann’s endomorphisms γ i j . Herrmann’s endomorphism γ i j and the elementary map of Gelfand–Ponomarev φ i act, in a sense, in opposite directions, namely the endomorphism γ i j adds the index to the beginning of the admissible sequence, and the elementary map φ i adds the index to the end of the admissible sequence.

Admissible and Asymptotically Optimal Linear Congruential Generators

The paper is devoted to the theoretical study of the so-called spectral test for Linear Congruential Generators (briefly, LCGs) of pseudorandom numbers with moduli 2 p and full periods. Using the technique developed in V. Gerlovina and V. Nekrutkin, Monte Carlo Methods and Appl. 11:2, 2005 , we present several examples of admissible and asymptotical optimal sequences of multiplicators. These examples give rise to well-directed search methods for LCGs with good equidistribution properties.

The approach‐dependent, time‐dependent, label‐constrained shortest path problem

AbstractIn this article, we consider an approach‐dependent, time‐dependent, label‐constrained shortest path problem. This is a variant of the shortest path problem defined on a transportation network comprised of a set of nodes and a set of directed arcs such that each arc has an associated label designating a mode of transportation, and an associated travel‐time function that depends on the time of arrival at the tail node, as well as on the link via which this node was approached. A label constraint is specified that controls the admissible sequence of transportation modes, and the approach‐dependent travel‐time feature models turn‐penalties and prohibitions in transportation networks, whereby the time spent at an intersection before entering the next link depends on whether we travel straight through the intersection, or make a right turn at it, or make a left turn at it, as permissible. We propose two effective algorithms to solve this problem and present related theoretical complexity results. In addition, we also explore various heuristic delay estimation and network curtailment schemes that can be used to confine the search to only the more promising subsets of the network to reduce computational effort while preserving near‐optimality. Extensive empirical results using available data for the Portland, OR, and Blacksburg, VA, transportation networks are presented to demonstrate the efficacy of the developed procedures. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(2), 57–67 2006

A property ofm-tuplings morse sequence

Them-tuplings Morse sequence, as a fixed point of a constant length substitution, was discussed. An equivalent definition was given by the property of them-adic development of the integers. Using the combinatorial properties of them-tuplings Morse sequence, we mainly obtain this sequence is an admissible sequence and the other two sequences generated by it are also admissible sequences, which extends the conclusion that the Thue-Morse sequence is an admissible sequences.

The Number of Admissible Sequences for Indecomposable Serial Rings

We give a formula for the number of admissible sequences for indecomposable serial rings with $n$ indecomposable projective modules whose minimum composition length is less than or equal to $m$. In particular, if $n=m$ is prime, we show that the number of such admissible sequences is $${{2n-1}\choose n} +{1\over n} \bigg[ (n-1)^2 - {{2n-2}\choose {n}}\bigg] .$$

On the quasi-nodal map for the Sturm–Liouville problem

We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L1 (0, 1) × [0, π)2 such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L1 convergence of the reconstruction formula for q by the exactly nodal sequence.

Stabilizing long-period orbits via symbolic dynamics in simple limiter controllers

We present an efficient approach to determine the control parameter of simple limiter controllers by using symbolic dynamics of one-dimensional unimodal maps. By applying addition- and subtraction-symbol rules for generating an admissible periodic sequence, we deal with the smallest base problem of the digital tent map. The proposed solution is useful for minimizing the configuration of digital circuit designs for a given target sequence. With the use of the limiter controller, we show that one-dimensional unimodal maps may be robustly employed to generate the maximum-length shift-register sequences. For an arbitrary long Sarkovskii sequence, the control parameters are analytically given.

Crop succession requirements in agricultural production planning

A method is proposed to write crop succession requirements as linear constraints in an LP-based model for agricultural production planning. Crop succession information is given in the form of a set of inadmissible successions of crops. The decision variables represent the areas where a certain admissible sequence of crops is cultivated. The number of decision variables may be reduced by forming suitable combinations of crop sequences. For this purpose, an algorithm is presented. Also, multi-year linear programming models for farm production planning containing crop succession constraints are considered. It is shown that, under some regularity conditions, a stationary cropping plan is an optimal solution of such a model. Finally, it is discussed how to determine, given a collection of inadmissible sequences, crop sequences which are inadmissible but do not contain inadmissible subsequences. The length of the longest of these sequences determines the length of the crop sequences taken into account in the model.