Bernoulli Series Research Articles

Estimating waiting time for compatible blood: A Negative Binomial approach.

Screening blood units for compatibility constitutes a Bernoulli series. Estimating the number of units needed to be screened represents a classic waiting time problem that may be resolved using the Negative Binomial Distribution. The currently recommended method for estimating the number of units screened, n, to find a required number of compatible units, r, with a given probability, p, is n = r/p. This coincides with the mean of the Negative Binomial Distribution so that the actual number of units screened will often be underestimated by the current method. The cumulative distribution function of the Negative Binomial Distribution provides the probability of success (compatibility), F(n;r,p), as a function of the number of trials performed (attempted crossmatches), n, the probability of success on each trial, p, and the number of successes (compatible units) required, r. Choosing a threshold cumulative probability sufficiently high, such as F ~ 0.9, for example, will provide confidence that the projected number of units screened will be underestimated less often (~10% of the time). With F ≥ 0.9, the estimated number of attempted crossmatches ranges from 1.3 to 2.3 times as many as the number calculated by the current method. As a rule of thumb approximately 1.6 times the current estimated number provides a similar estimate (n ~ 1.6∙r/p). Waiting time underestimation will be reduced significantly by using the Negative Binomial Distribution solution and should be accompanied by improved customer satisfaction.

Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.

The decomposition formula for Verlinde Sums

We prove a decomposition formula for Verlinde sums (rational trigonometric sums), as a discrete counterpart to the Boysal-Vergne decomposition formula for Bernoulli series. Motivated by applications to fixed point formulas in Hamiltonian geometry, we develop differential form valued version of Bernoulli series and Verlinde sums, and extend the decomposition formula to this wider context.

Numerical Solutions of System of First Order Normalized Linear Differential Equations by Using Bernoulli Matrix Method

Systems of first order differential equations have been arisen in science and engineering. Specially, the systems of normalized linear differential equations appear in differential geometry and kinematics problems. Solution of them is quite difficult analytically; therefore, numerical methods have need for the approximate solution. In this study, by means of a matrix method related to the truncated Bernoulli series we find the approximate solutions of the Frenet-Like system with variable coefficients upon the initial conditions. This method transforms the mentioned problem into a system of algebraic equations by using the matrix relations and collocation points; so, the required results along with the solutions are obtained and the usability of the method is discussed.

Eigenvalues and eigenfunctions of fourth-order sturm-liouville problems using Bernoulli series with Chebychev collocation points

Eigenvalues and eigenfunctions of fourth-order sturm-liouville problems using Bernoulli series with Chebychev collocation points

On Bernoulli series approximation for the matrix cosine

This paper presents a new series expansion based on Bernoulli matrix polynomials to approximate the matrix cosine function. An approximation based on this series is not a straightforward exercise since there exist different options to implement such a solution. We dive into these options and include a thorough comparative of performance and accuracy in the experimental results section that shows benefits and downsides of each one. Also, a comparison with the Padé approximation is included. The algorithms have been implemented in MATLAB and in CUDA for NVIDIA GPUs.

Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.

Convergence analysis of Bernoulli matrix approach for one-dimensional matrix hyperbolic equations of the first order

In this paper, an approximate approach based on Bernoulli operational matrices has been presented to obtain the numerical solution of first-order matrix hyperbolic partial differential equations under the given initial conditions. After using the operational matrices, we use the completeness of Bernoulli basis in which the main problem reduces to a system of algebraic equations. The solutions of this algebraic system are the coefficients of the truncated double Bernoulli series which are defined in the interval [0, 1]. Also, convergence analysis associated to the presented idea is provided under several mild conditions. Several numerical examples are considered to show the efficiency of the technique.

A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations

In this study, an approximate method based on Bernoulli polynomials and collocation points has been presented to obtain the solution of higher order linear Fredholm integro-differential-difference equations with the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Bernoulli polynomials and their derivatives by means of collocations. The solutions are obtained as the truncated Bernoulli series which are defined in the interval [a,b]. To illustrate the method, it is applied to the initial and boundary values. Also error analysis and numerical examples are included to demonstrate the validity and applicability of the technique.

Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

We study multiple Bernoulli series associated to a sequence of vectors generating a lattice in a vector space. The associated multiple Bernoulli series is a periodic and locally polynomial function, and we give an explicit formula (called wall crossing formula) comparing the polynomial densities in two adjacent domains of polynomiality separated by a hyperplane. We also present a formula in the spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula for the Bernoulli series describing it as a superposition of convolution products of lower dimensional Bernoulli series and multisplines. The study of these series is motivated by the work of E. Witten, computing the symplectic volume of the moduli space of flat G-connections on a Riemann surface with one boundary component.

Single versus multiple probe blocks of P300-based concealed information tests for self-referring versus incidentally obtained information

The aim of the present study was to compare two protocols and two information types on detection of concealed knowledge. Four independent groups of subjects were run. Two were tested on probe stimuli of a self-referring (AUTO) nature and two were tested on incidentally acquired details of a mock crime (MOCK). Each pair of groups was run either in a one-probe (1PB) or multiple probe (6PB) block. In the single probe block, which was repeated three times with a different probe on each block, one probe item was randomly repeated multiple times in a Bernoulli series with frequently presented, meaningless or irrelevant items. There was also a rare target item designed to force attention to the stimulus screen. All stimulus types were of the same category within each block. In contrast, in the multiple probe (and category) block, rare probes, rare targets and frequent irrelevant items were repeatedly presented in a Bernoulli series within one block. Major results: There was a difference in task demand as measured by reaction time between the two protocols (the multiple probe protocol was more demanding), and a difference trend in P300 detection sensitivity between protocols for both information types combined in favor of the 1PB ( p < .07). With both protocols combined, there was a trend ( p < .07) favoring detection of familiar versus incidentally learned information. In terms of P300 amplitudes, both protocols showed the usual result that P300 to probes was greater than that to irrelevants. Also, as with detection rates, self-referring information was better detected in terms of Probe-Irrelevant P300 amplitude differences than mock crime information, regardless of protocol. There was no effect of time passage across the repeated blocks of the one-probe protocol. Methodological problems with the multiple probe protocol as utilized in most recent publications are discussed.

Awareness of P300‐Related Cognitive Processes: A Signal Detection Approach

Although the P300 component of the human event-related potential is purported to indicate consciousness of an event, it is unclear which of the P300-related processes, if any, are accessible to awareness. This study explored awareness of such processes in Bernoulli series of auditory stimuli using signal detection theory and selective averaging of ERPs according to the subjects' classifications. Following targets eliciting appropriate P300 amplitudes, the subject was requested to classify his brain response as small, medium, or large, whereupon visual feedback was given about the correctness of the response. Averaging the ERPs according to the subjects' classifications showed P300 amplitudes to increase with the classification category, whereas the prestimulus and post-P300 event-related potentials were unrelated to the category. Signal detection analysis revealed non-zero sensitivity measures for the P300 amplitude categories. Although no conclusive suggestion can be made at present as to the basis of the P300 discrimination ability found here, it was possible to rule out electro-ocular and electromyographic activity, the preceding auditory stimulus, and the prior feedback given, as cues for discrimination.

Convergence of Bernoulli Series and the Set of Sums of a Conditionally Convergent Series of Functions

Convergence of Bernoulli Series and the Set of Sums of a Conditionally Convergent Series of Functions

Sequential expectancies and decision making in a changing environment: an electrophysiological approach.

ABSTRACTThis experiment addressed two questions. First, we sought to determine how sequential expectancies, as indexed by the amplitude of P300, are formulated when subjects are presented with a series in which the event probabilities are continuously changing. Second, we wanted to determine the effect on P300 amplitude of seeking information prior to a decision when the decisions are required at random intervals. Eight subjects were presented with a Bernoulli series of high and low tones in which the event probabilities reversed at random intervals. Thus, while the event probabilities were either .33 or .67 over short segments of the series, they were .50 over each trial block. The subject's task was to count the number of low‐pitched tones. In two separate conditions (“Unknown” and “Known”) the subjects received different information about the series. In the Unknown condition, the subjects were not informed of the reversals in stimulus probability and were only told that the events were equally probable. In the Known condition, the subjects were informed of the probability reversals and, in addition to the counting task, were asked to detect the probability reversals.The trials were divided into two categories: “transition” and “stable.” The transition trials were those that occurred between the actual probability reversal and the subject's reported detections. The stable trials were all those not included in the transition category. The data from the stable trials revealed that, in most cases, subjects generated their expectancies, as indexed by the amplitude of P300, in an identical manner during both conditions. Accurate knowledge of the stimulus probabilities did, however, influence the amplitude of P300 for events that were preceded by sequences of stimuli that occur only rarely at each level of probability. These data suggest that the mechanism responsible for generating sequential expectancies is largely automatic. Certain aspects of this mechanism are, however, modulated by the subject's knowledge of the parameters of the series. The data from the transition trials revealed that, in the Unknown condition, the subjects’ assignments of subjective probabilities, as indexed by the amplitude of P300, adapted rapidly to meet the new sequence‐generating rules following the probability transitions. In contrast, during the Known condition, the amplitude of the P300 and Slow Wave components steadily increased, as the decision point approached, to levels well above those in the Unknown condition. By the first trial following the decision, this enhancement in P300 and Slow Wave activity was no longer evident. These results are discussed in terms of a descriptive model of the subjects’ decision making behavior.

Quotients of Banach spaces of cotype 𝑞

Let Z Z be a Banach space and let X ⊂ Z X \subset Z be a B B -convex subspace (equivalently, assume that X X does not contain l 1 n l_1^n ’s uniformly). Then every Bernoulli series Σ n = 1 ∞ ε n z n \Sigma _{n = 1}^\infty {\varepsilon _n}{z_n} which converges almost surely in the quotient Z / X Z/X can be lifted to a Bernoulli series a.s. convergent in Z Z . As a corollary, if Z Z is of cotype q q , then Z / X Z/X is also of cotype q q . This extends a result of [4] concerning the particular case Z = L 1 Z = {L_1} .

Primary and Secondary Task Analysis of Step Tracking: An Event-Related Potentials Approach

The utility of the Event Related Brain Potential for the evaluation of task load was investigated. Subjects performed a discrete step tracking task with either first or second order control dynamics. In different conditions, the subject covertly counted auditory probes, visual probes, or tracking target steps presented in a Bernoulli series. In a fourth experimental condition subjects performed the primary tracking task without a secondary task. In the auditory condition, an increase in the difficulty of the primary task produced a decrease in the amplitude of the P300 elicited by the secondary count task. The introduction of the primary task in the visual condition resulted in an initial reduction in P300 amplitude but increasing task difficulty failed to attenuate the P300 further. A positive relationship between primary task difficulty and P300 amplitude was obtained in the step conditions. Furthermore, this effect did not require that the step changes be counted. The results are addressed in terms of the relative advantages of primary and secondary ERP workload assessment techniques.

The Dynamics of P300 during Dual-Task Performance

This chapter discusses the dynamics of p300 during dual-task performance. When a subject is presented with two stimuli in a Bernoulli sequence and is instructed to count one of the stimuli, a P300 component will be elicited. The amplitude of the P300 will be inversely proportional to the subjective probability of the stimulus. If, however, the same Bernoulli series is presented and the subject is instructed to ignore the stimuli while performing some other task (e.g., solving a crossword puzzle), no P300 is elicited by either of the stimuli, regardless of their probabilities. This observation is consistent with the generally accepted proposition that the amplitude of P300 depends, in part, on the task relevance of the eliciting stimuli. It appears, then, that the subject must in some way “attend” to the stimuli for P300 to be elicited. Attention, however, need not be invested entirely in one task, to the exclusion of all others. Several theories suggest that “attention” should be viewed as a resource of limited supply, various quantities of that can be allocated to different information-processing activities.

Application of Finite Orthogonal Polynomials to the Thermal Functions of Harmonic Oscillators. I. Reduced Partition Function of Isotopic Molecules

The utility as well as limitations of the Taylor series expansion of the quantum-mechanical partition function of a harmonic oscillator is discussed. Finite orthogonal polynomials are suggested as a basis for the approximation of the thermodynamic functions of an assumbly of harmonic oscillators. Shifted Chebyshev polynomials of the first kind are used to obtain finite expansions of arbitrary order of the reduced partition functions of isotopic molecules. The resulting series is similar, except for modulating coefficients, term by term to the Bernoulli series obtained from the Taylor expansion. The modulating coefficients of each term in the Chebyshev expansion approach unity as the order of the expansion increases. Thus, the Chebyshev series developed here approaches the Bernoulli series asymptotically. The Chebyshev expansion converges much faster than the Bernoulli expansion. The radius of convergence can be made arbitrarily large in contrast to the limit of 2π which exists for the Bernoulli expansion. These features of the Chebyshev expansion provide a basis for the extension of the principles of isotope effects, derived from quantum corrections of order (ℏ / kT)2n, to the complete spectrum of energy states. The mathematical behavior of the modulating Chebyshev coefficients is examined. Detailed application of the Chebyshev expansion to typical polyatomic molecules is given through numerical calculations.

On the Constants that occur in Certain Summations by Bernoulli's Series

On the Constants that occur in Certain Summations by Bernoulli's Series