One-dimensional Set Research Articles

Nonholonomic double-bracket equations and the Gauss thermostat

In this Rapid Communication we consider certain equations that arise from imposing a constant kinetic-energy constraint on a one-dimensional set of oscillators. This is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with Gauss's law of least constraint. Dynamics of this sort are of interest in nonequilibrium molecular dynamics. We show that under certain choices of external potential these equations give rise to a generalization of the so-called double-bracket equations which are of interest in studying gradient flows and integrable systems such as the Toda lattice. In the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained.

Independence-friendly cylindric set algebras

Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic. We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every one-dimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlying monadic Kleene algebras does not generate the variety of all monadic Kleene algebras. Finally, we offer a conjecture about which subvariety of monadic Kleene algebras the class of such monadic Kleene algebras does generate.

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

A family of fractal sets with Hausdorff dimension 0.618

In this paper, we introduce a class of fractal sets, which can be recursively constructed by two sequences { n k } k⩾1 and { c k } k⩾1 . We obtain the exact Hausdorff dimensions of these types of fractal sets using the continued fraction map. Connection of continued fraction with El Naschie’s fractal spacetime is also illustrated. Furthermore, we restrict one sequence { c k } k⩾1 to make every irrational number α ∈ (0, 1) correspond to only one of the above fractal sets called α-Cantor sets, and we found that almost all α-Cantor sets possess a common Hausdorff dimension of 0.618, which is also the Hausdorff dimension of the one-dimensional random recursive Cantor set and it is the foundation stone of E-infinity fractal spacetime theory.

COLLAGE-BASED INTERPOLATION OF REAL-DATA SETS

This paper presents a recurrent fractal interpolation method (approach) for one-dimensional sets of real-data. The method explores both the local collage idea, developed originally for image compression purposes, and the basic platform for generating of non-recurrent fractal interpolation functions – attractors of iterated function systems (IFS). The characteristic feature of the developed approach – the recurrent fractal interpolation functions are obtained by applying specialized correction procedures to the approximants of the real-data sets, i.e. to the attractors of local IFS, generated using self-similarities detected within the data under processing.

Design of interpretable fuzzy rule-based classifiers using spectral analysis with structure and parameters optimization

This paper presents a design method for fuzzy rule-based systems that performs data modeling consistently according to the symbolic relations expressed by the rules. The focus of the model is the interpretability of the rules and the model's accuracy, such that it can be used as tool for data understanding. The number of rules is defined by the eigenstructure analysis of the similarity matrix, which is computed from data. The rule induction algorithm runs a clustering algorithm on the dataset and associates one rule to each cluster. Each rule is selected among all possible combinations of one-dimensional fuzzy sets, as the one nearest to a cluster's center. The rules are weighted in order to improve the classifier performance and the weights are computed by a bounded quadratic optimization problem. The model complexity is minimized in a structure selection search, performed by a genetic algorithm that selects simultaneously the most representative subset of variables and also the number of fuzzy sets in the fuzzy partition of the selected variables. The resulting model is evaluated on a set of benchmark datasets for classification problems. The results show that the proposed approach produces accurate and yet compact fuzzy classifiers. The resulting model is also evaluated from an interpretability point of view, showing how the rule weights provide additional information to help data understanding and model exploitation.

Genome‐scale model for Clostridium acetobutylicum: Part II. Development of specific proton flux states and numerically determined sub‐systems

A regulated genome-scale model for Clostridium acetobutylicum ATCC 824 was developed based on its metabolic network reconstruction. To aid model convergence and limit the number of flux-vector possible solutions (the size of the phenotypic solution space), modeling strategies were developed to impose a new type of constraint at the endo-exo-metabolome interface. This constraint is termed the specific proton flux state, and its use enabled accurate prediction of the extracellular medium pH during vegetative growth of batch cultures. The specific proton flux refers to the influx or efflux of free protons (per unit biomass) across the cell membrane. A specific proton flux state encompasses a defined range of specific proton fluxes and includes all metabolic flux distributions resulting in a specific proton flux within this range. Effective simulation of time-course batch fermentation required the use of independent flux balance solutions from an optimum set of specific proton flux states. Using a real-coded genetic algorithm to optimize temporal bounds of specific proton flux states, we show that six separate specific proton flux states are required to model vegetative-growth metabolism and accurately predict the extracellular medium pH. Further, we define the apparent proton flux stoichiometry per weak acids efflux and show that this value decreases from approximately 3.5 mol of protons secreted per mole of weak acids at the start of the culture to approximately 0 at the end of vegetative growth. Calculations revealed that when specific weak acids production is maximized in vegetative growth, the net proton exchange between the cell and environment occurs primarily through weak acids efflux (apparent proton flux stoichiometry is 1). However, proton efflux through cation channels during the early stages of acidogenesis was found to be significant. We have also developed the concept of numerically determined sub-systems of genome-scale metabolic networks here as a sub-network with a one-dimensional null space basis set. A numerically determined sub-system was constructed in the genome-scale metabolic network to study the flux magnitudes and directions of acetylornithine transaminase, alanine racemase, and D-alanine transaminase. These results were then used to establish additional constraints for the genome-scale model.

A model of electoral competition with abstaining voters

This work studies a two-candidate single-dimensional Downsian voting model with variable voter turnout. I consider a one-shot game between two candidates, where the payoffs of the candidates are their vote shares, and the strategy set of each candidate is a one-dimensional set of policy positions. The utility that a voter attributes to a candidate is linear in both the absolute distance between the candidate’s policy position and the voter’s preferred policy, and in the electorate’s bias toward the candidate (known as the candidate’s valence). I assume that the voter decides to abstain if either they are indifferent between the two candidates, or significantly dissatisfied with both candidates; otherwise, the voter votes for the candidate they prefer. I show that under any of the two assumptions about voting behavior, a pure-strategy Nash equilibrium exists under a broad range of conditions, even if the valence of one of the candidates is nonzero. In either case, the candidates do not always choose identical policy positions. I then study how the positions of the candidates, their relative shares of vote, and turnout depend on the distribution of voter preferences and candidate valence.

Neo-Liberal Constitutionalism: Ideology, Government and the Rule of Law

This article explores the centrality of constitutionalism and the rule of law in neo-liberal ideology. It argues that neo-liberalism is not simply a one-dimensional set of economic ideas directed at promoting the free market, but is an ideology with broader political dimensions. At the core of neo-liberalism is a serious doctrine about politics and the proper role of government. Neo-liberals like F.A. Hayek, Milton Friedman and James Buchanan recognised that in order to have a functioning market order, a corresponding political order is a vital corollary. However, the article points out that a number of contradictions and tensions sit at the heart of the neo-liberal conception of politics: those that exist between freedom and the state, liberty and democracy, and law and legislation. The article suggests that one of the most daunting tasks facing neo-liberal politicians and theorists in the twenty-first century will be to overcome the constitutional ‘ignorance’ of Western democracies and institute a framework of rules, conventions or procedures through which the powers of government can be adequately constrained.

On Stable Feasible Sets in Generalized Semi-infinite Programming

We consider the feasible set of a generalized semi-infinite programming problem with a one-dimensional index set of inequality constraints depending on the state variable. The latter dependence on the state variable gives rise to a complicated structure of the feasible set. Under appropriate transversality conditions we partially present the local description of feasible sets in new coordinates by means of the finitely many basic functions.

Asymtotically stable one-dimensional compact minimal sets

It is proved that an asymptotically stable, $1$-dimensional, compact minimal set $A$ of a continuous flow on a locally compact, metric space $X$ is a periodic orbit, if $X$ is locally connected at every point of $A$. So, if the intrinsic topology of the region of attraction of an isolated, $1$-dimensional, compact minimal set $A$ of a continuous flow on a locally compact, metric space is locally connected at every point of $A$, then $A$ is a periodic orbit.

Crack Initiation in Brittle Materials

In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → t g(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time t i > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional \(\epsilon(\nu, \Gamma)\,:=\int_{\Omega} f(x,\nabla v)\,{\rm d}x+k{\mathcal{H}}^{1} (\Gamma),\) where \(\Omega \subseteq {\mathbb{R}}^{2}\) , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with \({\mathcal{H}}^{1} (\Gamma)\) small enough are such that \(\epsilon(u,\emptyset) < \epsilon(u_{\Gamma},\Gamma)\) .

Assessment of integration of off-axis Fresnel lenses into a free-space interconnect

The integration of multiple optical functions into a single optical element is often seen as advantageous to improve practicality and/or performance. In free-space devices, a common way to achieve this is to use off-axis Fresnel lenses to simultaneously collimate and redirect light. We study the impact of this type of integration on assembly tolerances and performance of a free-space interconnect designed to format a one-dimensional set of inputs into a two-dimensional array. Two systems are built and compared: one where light collimation and redirection are performed by different elements and one where these functions were combined into off-axis Fresnel lenses. The integrated system has lower losses but exhibits higher polarization sensitivity.

Thin-film polarizer based on a one-dimensional–three-dimensional–one-dimensional photonic crystal heterostructure

The recent experimental demonstration of polarization stop bands in three-dimensional dielectric circular-spiral photonic crystals is extended in two ways. First, the combination with a one-dimensional set of lamellae on one side allows for “poor-man’s optical isolators” or for thin-film polarizers—depending on from which side light impinges onto the device. Second, a chiral three-dimensional photonic crystal sandwiched between two one-dimensional sets of lamellae acts as a thin-film polarizer from both sides. Corresponding polymeric heterostructures are fabricated by means of direct laser writing. Their performance is compared with theory.

UNCERTAIN EQUIVALENCE METHOD FOR CONTINUOUS ONE-DIMENSIONAL UTILITY ELICITATION

The paper proposes a new risk-risk method for elicitation of utility over a continuous one-dimensional set of prizes, called uncertain equivalence method. The method adapts the probability equivalence – certainty equivalence transition over the lottery equivalence method, and on the other hand it adapts the probability equivalence-lottery equivalence transition over the certainty equivalence method. A modified midpoint chaining uncertain equivalence recurrent method is developed, which improves the elicitation procedure on the expense of accumulation of error. A generic theoretical representation of the elicitation techniques is presented, and seven methods are analyzed as its special cases. Three requirements for the elicitation procedures of one-dimensional utility are formulated. The proposed methods are compared with the existing techniques against these requirements.

Quantum Path Integral Algorithm and Its Application in Magnetotelluric Inversion

AbstractA quantum path integral algorithm (QPIA) is proposed in this paper by introducing quantum conceptions, such as transverse field and propagator in quantum mechanics. Different from general methods, QPIA, which fully applies quantum‐tunneling effect, can improve inversion efficiency greatly in some aspects of convergent rate and avoid to be trapped in local minimum. By composing the objective function and regarding Feynman propagator as the acceptable probability, we accomplish the algorithm. Inversion results of one‐dimensional synthetic data set and real data set show that QPIA is more precise, stable convergent and fast compared with traditional nonlinear inversion methods to approach the real models.

On Scheduling Fees to Prevent Merging, Splitting, and Transferring of Jobs

A deterministic server is shared by users with identical linear waiting costs, requesting jobs of arbitrary lengths. Shortest jobs are served first for efficiency. The server can monitor the length of a job but not the identity of the job’s user, thus merging, splitting, or partially transferring jobs offer cooperative strategic opportunities. Can we design cash transfers to neutralize such manipulations? We prove that mergeproofness and splitproofness are not compatible, and that it is similarly impossible to prevent all transfers of jobs involving three or more agents. On the other hand, robustness against pairwise transfers is feasible and essentially characterizes a one-dimensional set of scheduling methods. This line is borne by two outstanding methods: the merge-proof S+ and the split-proof S−. Splitproofness, unlike mergeproofness, is not compatible with several simple tests of equity. Thus, the two properties are far from equally demanding.

Supermodularity and Complementarity in Economics: An Elementary Survey

The literature on supermodular optimization and games is surveyed from the perspective of potential users in economics. This methodology provides a new approach for comparative statics based only on critical assumptions, and allows a general analysis of games with strategic complementarities. The results are presented in a simplified yet rigourous manner, without reference to lattice theory, for the special case of onedimensional parameter and actions sets, with the emphasis being on wide accessibility. Detailed applications are presented for well-known models of consumer behavior, monopoly pass-through, Bertrand and Cournot competition, strategic R&D, search and matching. Wherever appropriate, useful tricks for applications and comparative comments are inserted.

Asymptotic partial decomposition of domain for spectral problems in rod structures

We consider the vibrations of a bundle composed of two co-linear rods. The thickness of the rods is of order ε, where ε is a small parameter. Let ε / 2 be the width of one of the rods, and let ε be the width of the other one. Considering the associated spectral problem, we study the asymptotic behavior, as ε → 0 , of the eigenelements. In particular, we obtain asymptotics for these eigenelements providing correcting terms and precise bounds for convergence rates of the eigenelements. These eigenelements are approached up to the first order by the eigenelements of a one-dimensional Dirichlet problem set in ( − 1 , 1 ) . We also provide alternative approaches via the eigenelements of the spectral problem obtained by asymptotic partial decomposition of domain, which is known to be useful in numerical computations. Finally, we show that the technique developed in the paper can be applied to other spectral problems for thin structures.

Manufacture of pupil filters for 3D beam shaping

In a previous work we presented a new method for binarizing pupil filters designed to control the three-dimensional (3D) irradiance distribution in the focal volume of apodized systems. The method is based in the fact that the 3D amplitude point spread function of an axially-symmetrical system can be recovered entirely from a one-dimensional (1D) set of regularly spaced amplitude samples. Hence we proposed the use of 1D iterative Fourier transform algorithm to binarize a, properly mapped, version of the amplitude transmittance of the filter. The binary masks obtained consist of a set of opaque and transparent concentric annular zones. In this paper we have built two of these masks with opposing properties and we have experimentally verified their abilities in a focusing system. These experimental results strongly agree previous theoretical predictions.