Discrete Weight Research Articles

Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results

We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.

A Tauberian Theorem for Ergodic Averages, Spectral Decomposability, and the Dominated Ergodic Estimate for Positive Invertible Operators

Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:Lp(μ → L p(μ) be a bounded invertible linear operator such that T and T−1 are positive. Denote by \({\mathfrak{E}}\)n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {Tj+T−j}j =1∞. Martin-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of En(T)n=1∞; (b) a uniform App estimate in terms of discrete weights generated by the pointwise action on Ω of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {Tj+T−j}j=1∞ already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform Ap weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages En(T)n = 1∞.

Mutual learning in a tree parity machine and its application to cryptography

Mutual learning of a pair of tree parity machines with continuous and discrete weight vectors is studied analytically. The analysis is based on a mapping procedure that maps the mutual learning in tree parity machines onto mutual learning in noisy perceptrons. The stationary solution of the mutual learning in the case of continuous tree parity machines depends on the learning rate where a phase transition from partial to full synchronization is observed. In the discrete case the learning process is based on a finite increment and a full synchronized state is achieved in a finite number of steps. The synchronization of discrete parity machines is introduced in order to construct an ephemeral key-exchange protocol. The dynamic learning of a third tree parity machine (an attacker) that tries to imitate one of the two machines while the two still update their weight vectors is also analyzed. In particular, the synchronization times of the naive attacker and the flipping attacker recently introduced in Ref. 9 are analyzed. All analytical results are found to be in good agreement with simulation results.

Cryptography based on neural networks analytical results

Mutual learning process between two parity feed-forward networks with discrete and continuous weights is studied analytically, and we find that the number of steps required to achieve full synchronization between the two networks in the case of discrete weights is finite. The synchronization process is shown to be non-self-averaging and the analytical solution is based on random auxiliary variables. The learning time of an attacker that is trying to imitate one of the networks is examined analytically and is found to be much longer than the synchronization time. Analytical results are found to be in agreement with simulations.

ESTIMATING THE ASYMPTOTIC VARIANCE OF GENERALIZED L-STATISTICS

ABSTRACT Strong convergence of the estimated asymptotic variance of generalized L-statistics is demonstrated for smooth and discrete weighting functions. For smooth weighting functions that are trimmed, such as the 25%-trimmed mean, and for discrete functions, such as the median, minimal conditions are required for strong convergence of In a simulation study, for the trimmed mean, but not the median, appeared to converge to the sample variance of the statistic for samples from three distributions (normal, contaminated normal and Cauchy). For the smallest sample in the study tended to underestimate the sample variance of the GL-statistics.

Optimization of steel structures using distributed simulated annealing algorithm on a cluster of personal computers

This paper presents a distributed simulated annealing (SA) algorithm for optimal structural design of steel structures under stress, maximum displacement, and inter-story drift constraints. To effectively harness the distributed computing capabilities of a cluster of PCs, a two-phased SA algorithm consisting of simulated quenching (SQ) and SA is developed and used for development of the distributed algorithm. The distributed algorithm is based on two different levels of parallelism, design variable level for a distributed SQ algorithm and candidate design level for a distributed SA algorithm. The distributed SA algorithm is applied to the discrete minimum weight design of a verifying example and two steel braced frame structures. The results show that the distributed SA algorithm implemented on a network of PCs can reduce the computational requirement significantly, and yield more stable convergence histories.

Human cell DNA replication is mediated by a discrete multiprotein complex.

A discrete high molecular weight multiprotein complex containing DNA polymerase alpha has been identified by a native Western blotting technique. An enrichment of this complex was seen at each step in its purification. Further purification of this complex by ion-exchange chromatography indicates that the peak of DNA polymerase alpha activity co-purifies with the peak of in vitro SV40 DNA replication activity eluting from the column. The complex has a sedimentation coefficient of 18S in sucrose density gradients. We have designated this complex as the DNA synthesome. We further purified the DNA synthesome by electroeluting this complex from a native polyacrylamide gel. The eluted complex retains in vitro DNA synthetic activity, and by Western blot analysis, contains DNA polymerase delta, proliferating cell nuclear antigen, and replication protein A. Enzymatic analysis of the electroeluted DNA synthesome indicates that the synthesome contains topoisomerase I and II activities, and SDS-PAGE analysis of the electroeluted DNA synthesome revealed the presence of at least 25 major polypeptides with molecular weights ranging from 20 to 240 kDa. Taken together, our evidence suggests that the DNA synthesome may represent the minimal DNA replication unit of the human cell.

Discrete minimum weight design of steel structures using EC3 code

The paper presents a practical method of the discrete minimum weight design of steel structures. It is based on a concept of removal of the redundant material by successive diminishing of the size of the least stressed member. It is assumed that the member sizes are available from the European Steel Profiles Catalogues, and the design constraints are given by the rules of the code EC3 for nonsway buildings. Two numerical examples are presented: the classical benchmark problem of a ten-bar truss made of circular hollow sections (24 element catalogue, buckling taken into account) and a portal frame made of HEB sections with broad parallel flanges (11 element catalogue). The proposed method gives, after a reasonable calculation time, a very good approach to the exact optimum solution.

Design of Modal Transducers by Optimizing Spatial Distribution of Discrete Gain Weights

A general method is developed of designing distributed modal transducers, especially for use in the active vibration control of structure. For this purpose, a new two-dimensional modal transducer theory has been developed. This theory is based on the finite element model of the structure, which makes it possible to determine spatial gain distribution of the specific modal transducer without restrictions on the geometry and boundary conditions of the structure. Although the optimal gain distribution can he obtained theoretically, there is no practical means of implementing it. Therefore, two design methods of distributed modal transducer are developed, which optimize available parameters of piezoelectric film to approximate optimal gain distribution best. The first method uses multilayered polyvinylidene fluoride (PVDF) films as a single transducer. The electrode pattern, the lamination angle, and the relative poling direction of each PVDF layer are optimized to obtain the desired transducer. In the second method, the whole electrode area on a single PVDF film is divided into several segments, and the gain weight imposed on each segment by interface circuit is optimized. Sensor/actuator systems for the vibration control of cantilever composite plate are designed using the proposed methods. The performance of the designed transducers is verified experimentally. The real-time vibration control of integrated smart structure has been successfully achieved.

Training a perceptron in a discrete weight space.

Learning in a perceptron having a discrete weight space, where each weight can take 2L+1 different values, is examined analytically and numerically. The learning algorithm is based on the training of the continuous perceptron and prediction following the clipped weights. The learning is described by a new set of order parameters, composed of the overlaps between the teacher and the continuous/clipped students. Different scenarios are examined, among them on-line learning with discrete and continuous transfer functions. The generalization error of the clipped weights decays asymptotically as exp(-Kalpha(2)) in the case of on-line learning with binary activation functions and exp(-e(|lambda|alpha)) in the case of on-line learning with continuous one, where alpha is the number of examples divided by N, the size of the input vector and K is a positive constant. For finite N and L, perfect agreement between the discrete student and the teacher is obtained for alpha~Lsqrt[ln(NL)]. A crossover to the generalization error approximately 1/alpha, characterizing continuous weights with binary output, is obtained for synaptic depth L>O(sqrt[N]).

Convergence rates to discrete shocks for nonconvex conservation laws

This paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the Lax-Friedrichs scheme approximating non-convex scalar conservation laws. We assume that the discrete initial data tend to constant states as \(j\rightarrow \pm \infty\), respectively, and that the Riemann problem for the corresponding hyperbolic equation admits a stationary shock wave. If the summation of the initial perturbation over \((-\infty, j)\) is small and decays with an algebraic rate as \(|j|\rightarrow \infty\), then the perturbations to discrete shocks are shown to decay with the corresponding rate as \(n\rightarrow \infty\). The proof is given by applying weighted energy estimates. A discrete weight function, which depends on the space-time variables for the decay rate and the state of the discrete shocks in order to treat the non-convexity, plays a crucial role.

Design of modal transducers by optimizing spatial distribution of discrete gain weights

Design of modal transducers by optimizing spatial distribution of discrete gain weights

A generalized method to calculate diffusion rates in polydisperse systems. Application to miscible polymer pairs in the concentrated regime

A generalized method for calculating diffusion rates in polydisperse systems, valid in the concentrated regime, is outlined. In the formulation of the method the discrete variable that describes the molecular size, is replaced by a continuous variable in the same range. This replacement diminishes the number of degrees of freedom but keeping the essential physics of the original statement. The effects of monomeric friction coefficient, Flory-Huggins thermodynamic interaction parameter, individual species molecular weights, local molecular weights distribution and local T g are consistently included in the model. The method is used to calculate concentration distribution profiles generated by diffusion of polydisperse polymers blends, and experimentally tested. For this purpose polystyrene with discrete (bimodal and tetramodal) molecular weight distributions and polystyrene with wide and continuous molecular weight distributions were used to simulate polydisperse systems. They are allowed to diffuse in a blend of polyphenylene oxide and polystyrene. The simulated concentration profiles are compared with results obtained by using two experimental techniques based on independent physical properties, which give complementary information: Raman spectroscopy and DMA. The total PS concentration profiles calculated using the proposed method agree well with Raman spectroscopy results. Simulated DMA results—which are sensitive to the PS species molecular weight distribution—obtained using the concentration profiles calculated for each PS molecular weight species, agree well with the experimental DMA results. Calculations based on average molecular weights on the other hand, give incorrect results.

Predictions of linear viscoelastic properties for polydisperse entangled polymers

Different blending laws have been proposed in the literature to describe the polydispersity effect on the rheological behavior of polymer melts. In this paper predictions of linear viscoelastic properties of entangled polydisperse polymers have been derived from the double reptation mixing rule. The results in terms of the relaxation modulus, the zero shear-rate viscosity, η0, and the steady-state compliance, Je0, have been obtained using three different relaxation functions for the monodisperse fractions, namely the Tuminello step function, the single exponential function and the BSW function. Both discrete and continuous molecular weight distributions (MWDs) have been investigated. The Generalized Exponential Function (GEX) has been considered in the continuous case.

State space partition algorithms for stochastic systems with applications to minimum spanning trees

We investigated state space partition methods for computing probability measures related to the operation of stochastic systems and present new theoretical results concerning their efficiency. These methods iteratively partition the system state space, producing at each step progressively tighter bounds that can be used for constructing simple and efficient Monte Carlo routines for estimating the probabilities of interest. We apply our findings to the evaluation of measures related to minimum spanning trees in graphs whose edges have independent discrete random weights. Specifically, we seek to compute the distribution of the weight of a minimum spanning tree and the probability that a given edge belongs to a minimum spanning tree. Both of these unsolved problems are shown to be #P-hard. The algorithms for the minimum spanning tree problems are immediately applicable to other matroid problems with randomly weighted elements.

Weight Gain in FTT Population Post RD Intervention

Three months of outcome data on failure to thrive patients demonstrated the positive effects of intervention by an RD via improved weight status. All patients included in the study had to meet the criteria: 1) have a primary diagnosis of FTT, and 2) must be free from other diseases that have an impact on weight Within the data collection period 8/13/98-11/17/98,12 subjects met this criteria, to include 4 females and 8 males. Subjects ranged in age from 14 months to 9 years of age. The primary interventions provided to the subjects during the visit were nutritional counseling and medical intervention to rule out any organic causes for FTT. Weight information prior to the visit was obtained from the patient's medical record. Each patient's care giver was counseled for a 20-30minute period of time using an individualized education approach appropriate for the age of the patient The mam outcome, growth, was measured by weighing the child and converting to a percentage of expected weight gain based on Foman's expected weight gain for age data. Descriptive statistics to include percent of net change in weight were derived from this data. The z score was calculated to assess discrete changes in growth. Comparing baseline to post RD intervention data the percentage of children exhibiting > 100% of expected growth (catch up growth) improved from 16.7% (2/12) to 58.3% (7/12). The patients exhibiting weight gain of 50-100% of expected growth changed from 41.7% (5/12) to 16.7% (2/12). The patients exhibiting weight gain of 50% or less of expected growth changed from 33.3% (4/12) to 25% (3/12). The z score statistical analysis demonstrated that 75% (9/12) of patients experienced an improvement in weight status (z score increased post intervention.) The results illustrate a marked increase in the number of patients experiencing catch up growth (>100% of expected), as well as an increase in z score, a sensitive monitor of discrete weight changes. The integral role of RD intervention in the FTT population can be defined not only as improvement in patient care, but also as a significant cost savings per case. The ADA has derived that upwards of $13,758 can be saved when the complications of Pediatric FTT ( hospitalizations, and resulting mental and physical retardation) are treated.

Hormonally regulated programmed cell death in barley aleurone cells

Cell death was studied in barley (cv Himalaya) aleurone cells treated with abscisic acid and gibberellin. Aleurone protoplasts incubated in abscisic acid remained viable in culture for at least 3 weeks, but exposure to gibberellin initiated a series of events that resulted in death. Between 4 and 8 days after incubation in gibberellin, >70% of all protoplasts died. Death, which occurred after cells became highly vacuolated, was manifest by an abrupt loss of plasma membrane integrity followed by rapid shrinkage of the cell corpse. Hydrolysis of DNA began before death and occurred as protoplasts ceased production of alpha-amylase. DNA degradation did not result in the accumulation of discrete low molecular weight fragments. DNA degradation and cell death were prevented by LY83583, an inhibitor of gibberellin signaling in barley aleurone. We conclude that cell death in aleurone cells is hormonally regulated and is the final step of a developmental program that promotes successful seedling establishment.

Mechanism of Transformation of Precursors into Nanoslabs in the Early Stages of MFI and MEL Zeolite Formation from TPAOH−TEOS−H2O and TBAOH−TEOS−H2O Mixtures

The formation of silicate particles upon gradual addition of TEOS to concentrated TPAOH and TBAOH solutions, and upon dilution with water and aging, was studied with in situ X-ray scattering (XRS) and gel permeation chromatography (GPC). The samples for the GPC analyses were obtained by extraction of the particles from solution via a sequence of acidification with HCl, salting out with NaCl, and phase transfer into tetrahydrofuran. XRS and GPC reveal the presence of populations of particles with discrete sizes and number molecular weights, respectively, growing through aggregation. The entities forming in TPAOH are identified on the basis of their size and number molecular weight relationships with species previously identified in these suspensions by 29Si NMR and other independent techniques. A mathematical expression for the X-ray scattering function is derived for particles with slab shape. When dimensions of 1.3 × 4.0 × 4.0 nm, corresponding to the nanoslab with MFI structure, or integer multiples of them are assumed, the derived function resembles the measured intensity pattern in shape and position. The observed scattering at 6.6° 2θ at very early stages is assigned to a trimer of tetracyclic undecamer. All larger particles including the nanoslab observed by XRS and GPC are multiples of this trimer. The evolution of the system in TBAOH also obeys a stacking sequence of particles of the same size, number and molecular weights, although the kinetics are totally different. Up to the formation of nanoblocks, TBAOH and TPAOH affect the reaction mixture in a very similar way, indicating that the first molecular steps in the early stages of formation of MFI and MEL zeolite structures in these systems are very similar. The nanoslabs representing a specific fragment of MFI or MEL structures have channel intersections which are at least to two sides open and contain TPA or TBA molecules, respectively. The surface of these Silicalite-1 and -2 fragments is decorated on the ac and bc planes with alkyl groups sticking out of the surface. In nanoslabs with TBA, unfavorable template−template interaction suppresses aggregation along a and b. The final product is a “double” nanoslab, probably connected along the later c direction of the MEL structure in which there is only a very small repulsion of the butyl chains. TPA favors particle aggregation to give larger particles measuring up to 15.6 × 8 × 8 nm, even at room temperature.

Two approximations to learning from examples

We investigate the learning of a rule from examples of the case of boolean perceptron. Previous studies of this problem have been made using the full quenched theory. We consider here two alternative approaches that can be applied easily. the two-replicas interactions approach considerably improves upon the well-known first-order approach. The mean field approach proved some results that have been obtained previously using the complex full quenched theory. Both approximations have been applied to both continuous weights and discrete weights perceptron.

Asymptotic stability of stationary discrete shocks of Lax-Friedrichs scheme for non-convex conservation laws

The nonlinear stability of stationary discrete shocks for the Lax-Friedrichs scheme approximating non-convex scalar conservation laws is proved without smallness restrictions on the initial perturbations provided that the summation of initial perturbation is zero. The long time behavior of the Lax-Friedrichs is also investigated for a rather broad class of initial data. The selection of the discrete weight function plays a crucial role. The detailed study of discrete error equations is a major technical involvement.