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Consider a region that requires to be protected from unauthorized penetrations. The border of the region, modeled as a unit line segment, consists of high priority segments that require the highest level of protection separated by low priority segments that require to be visited infinitely often. We study the problem of patrolling the border with a set of k robots. The goal is to obtain a strategy that minimizes the maximum idle time (the time that a point is left unattended) of the high priority points while visiting the low priority points infinitely often. We use the concept of single lid cover (segments of fixed length) where each high priority point is covered with at least one lid, and then we extend it to strong double-lid cover where each high priority point is covered with at least two lids, and the unit line segment is fully covered. Let λk-1 be the minimum lid length that accepts a single λk-1-lid cover with k - 1 lids and Λ2k be the minimum lid length that accepts a strong double Λ2k-lid cover with 2k lids. We show that 2min(Λ2k, λk-1) is the lower bound of the idle time when the max speed of the robots is one. To compute Λ2k and λk-1, we present an algorithm with time complexity O(max(k, n) logn) where n is the number of high priority sections. Our algorithm improves by a factor of min(n, k) the previous O(knlogn) running time algorithm. For the upper bound, first we present a strategy with idle time λk-1 where one robot covers the unit line, and the remaining robots cover the lids of a single λk-1-lid cover with k - 1 lids. Then, we present a simple strategy with idle time 3Λ2k that splits the unit line into not-disjoint k segments of equal length that robots synchronously cover, i.e., reaching the leftmost and rightmost point simultaneously. Then, we present a complex strategy that split the unit line into k non-disjoint segments that robots asynchronously cover. We show that combining strategies one and two attain an approximation of 1.5 the optimal idle time and combining strategy one and third attain optimal idle time.

In this paper, a two-dimensional direction-of-arrival (DOA) estimation algorithm based on nested acoustic vector-sensor (AVS) array is introduced. The minimum unit interval of the used nested AVS array is integral multiple of half-wavelength of signal. Firstly, four selection matrices are used to recombine four received vectors, by which four special cross-covariance matrices are obtained to form an extended block covariance matrix. Then, a set of parameters on elevation angles are estimated by traditional estimation of signal parameter via rotational invariance technique (ESPRIT) algorithm. Using these parameters, unambiguous elevation angles can be obtained without spectral peak search and added eigenvalue decomposition. At last, using the estimated elevation angles, the azimuth angles can be estimated by array manifold matching algorithm. The proposed algorithm can distinguish more signals than some existing ESPRIT algorithms. Many simulation results can prove the performance of the proposed algorithm in improving the accuracy of DOA estimation. Simulation results also show that the unit interval of optimal nested AVS array should be larger than half-wavelength of received signal.

It is proved that, for any map of the unit interval onto the unit square, there exist two points in the interval such that the squared Euclidean distance between their images exceeds the distance between them on the interval at least by a factor of $$3.625$$ . The additional condition that the images of the interval endpoints belong to opposite sides of the square increases this factor to more than $$4$$ .

A new class of unimodal asymmetric distributions is introduced to the unit interval and these distributions are useful for modeling data of percentages, proportions and fractions. Therefore, we pro...

We consider a family of skew tent maps f a on the unit interval, determined by the parameter a, with 0 a 1 . We give a decision procedure, that on input a and a point x 0 in the unit interval, determines whether or not the sequence x 0 , f a ( x 0 ) , f a 2 ( x 0 ) , . . . of iterates of f a on x 0 reaches one of the two fixed points of f a after a finite number of iterations.

The story of the question “must commuting maps of the unit interval have a common fixed point” is used to illustrate strategies that advance mathematical research.

A 128-Gb 1-bit/cell 3-D flash memory chip has been developed with 96-word-line-layer technology. A novel chip floorplan architecture with less time constants of wordline and bitline realizes fast read access time. A newly introduced program sequence achieves for higher reliability with less read-retry even after write/erase cycles. External VPP supply (12 V), current mode reference distribution, and auto temperature code refresh are also adopted to boost the performance of the chip. A new duty cycle corrector succeeds in obtaining wider unit interval of the DQS. Consequently, the proposed chip has a read access latency of $4~\mu \text{s}$ and a program time of 75 $\mu \text{s}$ , which is 12–13 and 4–5 times faster than conventional 3-D flash memory with the same technology [Maejima et al. , (2018)]. The random read latency (tRRL) is estimated to be less than 50 $\mu \text{s}$ , which enables in the reduction of total read access time of solid state drive (SSD) system.

In a multiple testing setting, we consider testing n null hypotheses, denoted by H 1 , … , H n , with the corresponding p-values P i for each H i . We propose a new method which ‘scans’ all intervals in the unit interval [ 0 , 1 ] , selects the longest interval with estimated false discovery rate not exceeding the target level, and then rejects a hypothesis if its p -value is contained in this interval. In contrast, the Benjamini–Hochberg method, which is a threshold-type procedure, does the same but over intervals with an endpoint at the origin. Following Storey et al. (2004), we show that our procedure controls the false discovery rate in an asymptotic sense. We also investigate its asymptotic false non-discovery rate, deriving conditions under which it outperforms the Benjamini–Hochberg procedure. This happens, for example, in power-law location models.

In this article, a maximum-eye-tracking clock and data recovery (MET-CDR) circuits for minimum bit error rate (BER) are presented. The proposed CDR does not require a BER counter or an eye-opening monitor with any iterative procedure to find the optimal sampling phase. The biased data level obtained from the weighted sum of error sampler outputs provides the actual eye height (EH) information in the presence of precursor intersymbol interference (ISI). Two samplers operating on two slightly different timings detect the current EH and the polarity of the eye slope so that the CDR can track the maximum EH where the slope becomes zero. Measured results show that the sampling phase of the maximum EH and that of the minimum BER match well. A prototype receiver fabricated in 28-nm CMOS process operates at 26 Gb/s with an eye-opening of 0.25 unit interval (UI) and consumes 87 mW while equalizing 23.5 dB of loss at 13 GHz.

The unit interval [0,1] is broken at random into n spacings with the breaking points given by a random sample of size n−1 from the uniform distribution on [0,1]. A decision-maker observes the lengt...

Studies of risk perceived using continuous scales of [0,100] were recently introduced in psychometrics, which can be transformed to the unit interval, but the presence of zeros or ones are commonly observed. Motivated by this, we introduce a full inferential set of tools that allows for augmented and limited data modeling. We considered parameter estimation, residual analysis, influence diagnostic and model selection for zero-and/or-one augmented beta rectangular (ZOABR) regression models and their particular nested models, which is based on a new parameterization of the beta rectangular distribution. Different from other alternatives, we performed maximum-likelihood estimation using a combination of the EM algorithm (for the continuous part) and Fisher scoring algorithm (for the discrete part). Also, we perform an additional step, by considering other link functions, besides the usual logistic link, for modeling the response mean. By considering randomized quantile residuals, (local) influence diagnostics and model selection tools, we identified that the ZOABR regression model is the best one. We also conducted extensive simulations studies, which indicate that all developed tools work properly. Finally, we discuss the use of this type of models to treat psychometric data. It is worthwhile to mention that applications of the developed methods go beyond to Psychometric data. Indeed, they can be useful when the response variable in bounded, including or not the respective limits.

In this paper, we introduce a new two-parameter distribution, called log-weighted exponential (log-WE) distribution, defined on the unit interval. Some of the statistical properties of the ...

When we take a decimal form of a number in the unit interval [0,1], we can form a new number by selecting certain digits and discarding the others. We analyze the properties of this process, partic...

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs---a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs. Clique-width is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tractable time when an efficient representation is given. Unfortunately, the exact computation of the clique-width representation is \NP-hard. Therefore, good upper-bounds on clique-width are highly appreciated, in particular, when such a bound is algorithmic., Accepted to Mathematical Foundations of Computer Science (MFCS 2020), 25 pages, 4 figures

Assume that \(m \ge 1\) autonomous mobile agents and \(0 \le b \le m\) single-agent transportation devices (called bikes) are initially placed at the left endpoint 0 of the unit interval [0, 1]. The agents are identical in capability and can move at speed one. The bikes cannot move on their own, but any agent riding bike i can move at speed \(v_i > 1\). An agent may ride at most one bike at a time. The agents can cooperate by sharing the bikes; an agent can ride a bike for a time, then drop it to be used by another agent, and possibly switch to a different bike.

This paper presents the design of the 8.4Gb/s transmitter with a two-tap feed-forward equalizer (FFE) and a 40:1 serializer. The transmitter includes an all-digital phased-locked-loop (ADPLL), a pre-driver and a driver. The simple architecture of the 5:1 serializer achieves low-power consumption by eliminating delay line buffers used to secure timing margin and the selection generator in the conventional 5:1 serializers. The prototype is fabricated in a 40-nm CMOS technology. It offers 72.5-ps eye width, which is 61% of the unit interval and exhibits energy efficiency of 1.66 pJ/bit at 8.4Gb/s.

Recently, unit-Lindley distribution and its associated regression models have been developed as an alternative to Beta regression model for which continuous outcome in the unit interval [Image: see text] . Proportion data usually occur in clinical trials, economics and social studies with hierarchical structures. In this study, unit-Lindley mixed-effect model is proposed and the appropriate likelihood analysis methods for parameter estimation are investigated. In the case of clustered or longitudinal proportion data in mixed-effect models, the full-likelihood function does not have a closed form. Parameter estimations of unit-Lindley mixed-effect model are obtained with Laplace and adaptive Gaussian quadrature approximation methods in this study. We analyzed a dataset on the proportion of households with insufficient water supply and sewage with some sociodemographic variables in the cities of Brazil by using unit-Lindley mixed-effect model including a random intercept as federative states of Brazil. Analysis results indicate that the proposed unit-Lindley mixed-effect model provides better fit than unit-Lindley regression model and beta mixed model. Also, in the simulation study the accuracy of the estimates of approximation methods are evaluated and compared via Monte Carlo simulation study in terms of bias and mean square error.

In this paper, two new general families of distributions supported on the unit interval are introduced. The proposed families include several known models as special cases and define at least twenty (each one) new special models. Since the list of well-being indicators may include several double bounded random variables, the applicability for modeling those is the major practical motivation for introducing the distributions on those families. We propose a parametrization of the new families in terms of the median and develop a shiny application to provide interactive density shape illustrations for some special cases. Various properties of the introduced families are studied. Some special models in the new families are discussed. In particular, the complementary unit Weibull distribution is studied in some detail. The method of maximum likelihood for estimating the model parameters is discussed. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Applications to the literacy rate in Brazilian and Colombian municipalities illustrate the usefulness of the two new families for modeling well-being indicators.

A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with ot...

In this article, considering the two-parameter Johnson SB distribution, bounded on the unit interval, we derived, for the first time, the analytical expressions for bias-reduction of maximum likeli...

A digital-intensive on-chip serial link achieving a 10 Gb/s data rate over a 10-mm interconnect was demonstrated in a 65-nm GP process. A three-tap half-rate feed-forward equalizer was implemented for signal pre-emphasis in the transmitted block. On the receiver (RX) side, a two-tap half-rate time-based decision feedback equalizer was employed to cancel out inter-symbol interference noise. A 215 – 1 pseudorandom binary sequence generator and an in situ bit error rate (BER) monitor were designed for bit stream generation and convenient eye-diagram measurements. The measured energy efficiency of the transmitter and RX was 31.9 and 45.3 fJ/b/mm, respectively, for a data rate of 10 Gb/s. A BER less than 10−12 was verified for an eye width of 0.43 unit interval.

Assume that n mobile sensors with identical sensing range r = 1 2 n are distributed independently at random on the unit interval with the uniform distribution. We study the maximum sensor’s displacements to the power a > 0 for full coverage of the unit interval in the sense that every point in the unit interval is within the range of a sensor. We derive the asymptotic result about the maximum of moments absolute deviation around the mean for order statistics from uniform distribution.

The Beta distribution is the standard model for quantifying the influence of covariates on the mean of a response variable on the unit interval. However, this well-known distribution is no longer useful when we are interested in quantifying the influence of such covariates on the quantiles of the response variable. Unlike Beta, the Kumaraswamy distribution has a closed-form expression for its quantile and can be useful for the modeling of quantiles in the absence/presence of covariates. As an alternative to the Kumaraswamy distribution for the modeling of quantiles, in this paper the unit-Weibull distribution was considered. This distribution was obtained by the transformation of a random variable with Weibull distribution. The same transformation applied to a random variable with Exponentiated Exponential distribution generates the Kumaraswamy distribution. The suitability of our proposal was demonstrated to model quantiles, conditional on covariates, with two simulated examples and three real applications with datasets from health, accounting and social science. For such data sets, the obtained fits of the proposed regression model were compared with those provided by the Beta and Kumaraswamy regression models.

This paper presents impact of PCB and Package trace lengths on signal integrity of LPDDR4X interface in a SoCDRAM automotive system. The DRAM communicates with SoC in an external configuration. LPDDR4X interface was running at 4266 Mbps (UI =234ps), 0.6V VDDQ (TT) and VOH=360mV (Writes). The system trace lengths of data bits consist of SoC package traces, PCB traces and DRAM traces. The signal propagation happens in a mismatched termination environment which leads to reflection-based ISI causing eye-aperture degradation. The analysis presented shows that the total system trace length is optimal (best-case) when signal propagation round-trip delay is 2*UI (Unit Interval) from TX to RX and is worst-case when the round-trip delay is 1*UI. Pulse responses and eye-aperture comparison results are presented for the best- and worst-case scenarios.

Consider n mobile sensors placed independently at random with the uniform distribution on a barrier represented as the unit line segment 0 , 1 . The sensors have identical sensing radius, say r. When a sensor is displaced on the line a distance equal to d it consumes energy (in movement) which is proportional to some (fixed) power a 0 of the distance d traveled. The energy consumption of a system of n sensors thus displaced is defined as the sum of the energy consumptions for the displacement of the individual sensors.We focus on the problem of energy efficient displacement of the sensors so that in their final placement the sensor system ensures coverage of the barrier and the energy consumed for the displacement of the sensors to these final positions is minimized in expectation. In particular, we analyze the problem of displacing the sensors from their initial positions so as to attain coverage of the unit interval and derive trade-offs for this displacement as a function of the sensor range. We obtain several tight bounds in this setting thus generalizing several of the results of 10 to any power a 0 . We prove that total movement to power a of the sensors with range r = 1 2 n is ( a 2 ) ! 2 a 2 ( 1 + a ) 1 n a 2 - 1 + O ( 1 n a 2 ) , when a is an even natural number.We give the Algorithm 1 that uses expected O ( 1 n a 2 - 1 ( ln ź n n ) a 2 ) , total movement to power a, where a 0 and r ź 6 2 n .

Consider a region that require to be protected from unauthorized penetrations. A team of robots can patrol (perpetually move along) the border of the region looking for intruders. The problem is known as the patrolling problem and has been extensively studied.

In this article, we have studied a 1D map, which is formed by combining the two well-known maps i.e. the tent and the logistic maps in the unit interval i.e. [0, 1]. The proposed map can behave as the piecewise smooth or non-smooth maps (depending on the behaviour of the map just before and after the border) and then the dynamics of the map has been studied using analytical tools and numerical simulations. Characterization has been done by primarily studying the Lyapunov spectra and the corresponding bifurcation diagrams. Some peculiar dynamics of this map have been shown numerically. Finally, a Simulink implementation of the proposed map has been demonstrated., Comment: 25 pages

Often, the dependent variable in regression models has a fractional nature, being bounded in the unit interval. Several variants of cross-sectional and panel data fractional regression models have recently been proposed. This chapter shows how to estimate most of those models by using three R packages: frm , frmhet , and frmpd .

In contrast to the previous chapters, in analyzing the term structure of interest rates, it is suggested to treat the time variable not as a duration of a short-term interest rate (where the time variable depends on the parameters of the models in question, which makes it difficult to compare yields for the same real maturities), but as a nonlinear transformation of the time term, not dependent on model parameters, that allows us to map the whole of the time semi axis to a unit interval. The use of this approach is illustrated by analyzing the properties of the yield curve and the forward curve for one-, two- and three-factor interest rate models.

Let the unit interval I represent a cake to be divided among n players estimating the measurable subsets of I by nonatomic probability n measures. We show a method of obtaining an exact fair division of the cake such that i-th each part has i-th measure equals 1/n for all players.

In this paper, we propose a dynamic comparator that improved the operation performance of receiver (RX) with the effort to reduce power consumption. It is implemented via double-tail StrongARM latch comparator with an active inductor and efforts are made to minimize power consumption for high-speed resulting in better energy efficiency at the targeted high frequency. In this regard, our comparator is suitable for memory application RX to satisfy both low-power and high-speed. It is applied to the single-ended RX designed with a continuous-time linear equalizer, a clock generator and a quarter-rate 2-tap decision-feedback equalizer which is appropriate for the high-frequency memory application. Compared to the conventional one, our design, fabricated in 55nm CMOS process, provides an improvement of 7% in unit interval (UI) margin under the same power consumption and receives up to 10Gb/s PRBS15 data at BER

Impact of non-target ODT (On-Die Termination) in dual-rank DRAM is investigated on SoC-DRAM SI (signal integrity). Analysis at data rate of 4266Mbps was performed. It shows that terminating non-target DRAM improves SI of the target DRAM by ~3-5% of unit interval due to mitigation of reflections. This added timing margin is significant at high data rates.

This paper introduces two general models for computing centiles when the response variable Y can take values between 0 and 1, inclusive of 0 or 1. The models developed are more flexible alternatives to the beta inflated distribution. The first proposed model employs a flexible four parameter logit skew Student t (logitSST) distribution to model the response variable Y on the unit interval (0, 1), excluding 0 and 1. This model is then extended to the inflated logitSST distribution for Y on the unit interval, including 1. The second model developed in this paper is a generalised Tobit model for Y on the unit interval, including 1. Applying these two models to (1-Y) rather than Y enables modelling of Y on the unit interval including 0 rather than 1. An application of the new models to real data shows that they can provide superior fits.

A signal with a jitter in excess of one unit interval will result in a failure of functional testing because the signal and its expected data are misaligned. Additionally, jitter measurements performed without considering the correct jitter transfer function (JTF) will yield incorrect results. Here, two methods are presented: one for functional testing while countering jitter and the other that considers the JTF for measuring jitter. The new methods use a time measurement unit (TMU) in automatic test equipment (ATE) and signal processing.

[EN] In this paper, the full subcategory Hcomp of Top whose objects are Hausdorff compact spaces is identified as the orthogonal hull of the unit interval I = [0,1]. The family of continuous maps rendered invertible by the reflector β◦ρ is deduced., The authors thank the referee for his/her comments, corrections and suggestions improving both the presentation and the mathematical content of this paper. Lazaar would like to thank The laboratory of research LATAO (Faculty of sciences of Tunis, University Tunis El Manar,Tunisia) for its support (LR11ES12).

In this paper we introduce the class of beta seasonal autoregressive moving average ($\beta$SARMA) models for modeling and forecasting time series data that assume values in the standard unit interval. It generalizes the class of beta autoregressive moving average models [Rocha and Cribari-Neto, Test, 2009] by incorporating seasonal dynamics to the model dynamic structure. Besides introducing the new class of models, we develop parameter estimation, hypothesis testing inference, and diagnostic analysis tools. We also discuss out-of-sample forecasting. In particular, we provide closed-form expressions for the conditional score vector and for the conditional Fisher information matrix. We also evaluate the finite sample performances of conditional maximum likelihood estimators and white noise tests using Monte Carlo simulations. An empirical application is presented and discussed., Comment: 26 pages, 5 figures, 4 tables

Publisher Summary This chapter provides a number of techniques that use minimal prior assumptions about the statistics of data and briefly discusses context-based algorithm. The best-known context-based algorithm is the “ppm algorithm,” first proposed by Cleary and Witten. The idea of the ppm algorithm is elegantly simple. The basic algorithm initially attempts to use the largest context. The size of the largest context is predetermined. If the symbol to be encoded has not previously been encountered in this context, an escape symbol is encoded and the algorithm attempts to use the next smaller context. If the symbol has not occurred in this context either, the size of the context is further reduced. This process continues until either one obtains a context that has previously been encountered with this symbol or one arrives at the conclusion that the symbol has not been encountered previously in any context. The chapter proposes arithmetic coding and the basic idea behind arithmetic coding is the division of the unit interval into subintervals, each of which represents a particular letter. The smaller the subinterval, the more bits are required to distinguish it from other subintervals. If one can reduce the number of symbols to be represented, the number of subintervals goes down as well. This in turn means that the sizes of the subintervals increase, leading to a reduction in the number of bits required for encoding. The exclusion principle used in ppm provides this kind of reduction in rate.

Simulating/Measuring the jitter tolerance of clock and data recovery (CDR) circuits, and confirming if the associated jitter tolerance meets the required specification for a specified communication standard, is an important consideration for designing/testing high-speed serial link interface circuits. However, conducting such performance evaluations are costly and time-consuming. In this paper, a simple but effective testing method for evaluating the tracking capability of bang-bang CDR circuits is introduced. The tracking capability of the CDR loop is obtained by simply inverting the recovered clock to produce a 0.5 unit interval (UI) phase shift and capture the tracking time. The proposed technique is easily implemented, because of its fully-digital characteristic, and suitable for testing CDRs that is embedded in a complex interface transceiver. Then, a quick jitter tolerance estimation technique based on the obtained tracking capability is proposed to simplify the time-consuming process as well as avoid the costly test equipment required for designing and/or testing CDR circuits. Experimental results show that the proposed techniques could precisely evaluate the tracking capability and efficiently reduce test costs in acquiring complete jitter tolerance testing.

An all-digital 2-tap half-rate time-based decision feedback equalizer (TB-DFE) was demonstrated on a 10mm on-chip serial link. Implemented in a 65nm GP technology, the transmitter and receiver achieve an energy-efficiency of 31.9 and 45.3 fJ/b/mm, respectively, at a data rate of 10Gb/s. A Bit Error Rate (BER) less than 10−12 was verified for an eye width of 0.43 Unit Interval (UI) using an in-situ BER monitor.

In this paper, we use a compensation method of the measured short interval to improve the measurement accuracy by two groups of multiplex. By using two groups of multi-channel monostable pulse signal as a compensation signal, we respectively compensate time difference between the starting/ending measured short time interval and the reference signal. Pulse width of each compensation signal is uniform increasing and incremental sum is a reference signal cycle. We select the average value of two compensation signals pulse width as the beginning and the end of the time difference when the reference signal value jumps after compensation. Then according to the number of reference signal and two compensation time difference we can obtain measured short time interval. Error analysis and evaluation of uncertainty measurement show that the precision of this method is decided by pulse width increment of adjacent compensation signals. The experimental data shows that the method effectively reduces the measurement error of one reference signal cycle in the measurement of time interval.

A new method to determine the minimum measurement time interval needed to obtain the equivalent continuous A-weighted sound pressure level with a designated reliability is presented. This method considers variations in the noise emission from passing vehicles. To verify the validity and availability, simulation experiments based on our dynamic model are examined under various traffic conditions. A statistic quantity, the mean time interval between two maximum sound pressure levels consecutively observed during the reference measurement time interval, is introduced in the experimental analysis. Additionally, the theoretical analysis includes another statistic quantity corresponding to the mean time interval, the mean recurrence time of the maximum sound pressure level, when the transition probability, rate of heavy vehicles, and probability distribution of vehicles passing the observation point during the reference measurement time interval are known.

This paper describes the working principle of hall sensor and its application in DC stepper motor. It designs a speed detection system for DC stepper motor based on P89V51 Single Chip Microcontroller (SCM), i.e. hardware circuit diagram and software flow chart. It utilizes Hall elements to complete the conversion from speed to pulse and then process the pulse signal to get the quantity of pulse during unit interval. By means of comparison of test and theoretical data, it indicates that the speed measurement system is satisfactory for application in reality.

Assume that [Formula: see text] mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors [Formula: see text] from [Formula: see text] to [Formula: see text] where the contribution of the [Formula: see text] sensor is its displacement from the current location to the anchor equidistant point [Formula: see text] raised to the [Formula: see text] power, when [Formula: see text] is an odd natural number. As a consequence, we derive the following asymptotic identity. Fix [Formula: see text] positive integer. Let [Formula: see text] denote the [Formula: see text] order statistic from a random sample of size [Formula: see text] from the Uniform[Formula: see text] population. Then [Formula: see text] where [Formula: see text] is the Gamma function.

To reduce pattern-dependent jitter in four-level pulse-amplitude modulation (PAM-4) signalling, a transition-aware feed-forward equaliser (TFFE) is proposed. By allowing the transition time to vary according to the transition types of the PAM-4 symbols, switching jitter in TFFE-based PAM-4 transmitters can be greatly suppressed, down to the non-return-to-zero (NRZ) level. Simulation results confirm that a PAM-4 transmitter that employs a TFFE can reduce the jitter level by 0.18 unit interval peak-to-peak, and this proposed equaliser can achieve nearly identical eye height as the conventional PAM-4 feed-forward equalisers design without introducing any signal-to-noise ratio penalty.

This work provides rates of convergence in the Darling Kac law for infinite measure preserving Pomeau-Manneville (unit interval) maps. Along the way we obtain error rates for the stable law associated with the first return map and the first return time to some suitable set inside the unit interval.

In batch-oriented processes, the quality of the process tends to deteriorate over time. A bounded reset chart (BRC) using the integral controller is proposed to adjust such processes effectively. The effectiveness of the BRC procedure is evaluated in terms of the average cost per unit interval (ACU). Optimal values for the reset limit, considering various process and cost parameters, are studied to minimize the ACU. The BRC procedure is shown to be effective for adjusted processes with deterioration over time compared to the traditional adjustment procedure where resetting is not considered.