Pure Thresholds Research Articles

Values of the F‐pure threshold for homogeneous polynomials

AbstractWe find a formula, in terms of , and , for the value of the ‐pure threshold for the generic homogeneous polynomial of degree in variables over an algebraically closed field of characteristic . We also show that in every characteristic and for all not divisible by , there always exist reduced polynomials of degree in whose ‐pure threshold is a truncation of the base expansion of at some place; in particular, there always exist reduced polynomials whose ‐pure threshold is strictly less than . We provide an example to resolve, negatively, a question proposed by Hernandez, Núñez‐Betancourt, Witt, and Zhang, as to whether a list of necessary restrictions they prove on the ‐pure threshold of reduced forms are “minimal” for . On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base expansion of that are always ‐pure thresholds for reduced forms of degree , and computations that show their conditions suffice (in every characteristic) for degrees up to eight and several other situations.

Optimal Secure Channel Access in Distributed Cooperative Networks With Untrusted Relay

This paper investigates optimal channel access strategies to improve secure communications over a distributed cooperative wireless network with multiple user pairs and an untrusted amplify-and-forward (AF) relay. The user pairs communicate directly or via the relay. To enable secure communication, we formulate a channel access problem to maximize the average secrecy system throughput with destination-based jamming. Using optimal sequential planned decision theory, we propose an optimal channel access strategy where the winning user pair uses direct transmission, cooperative AF relaying, or cooperative jamming in an adaptive manner. We rigorously prove the optimality of the solution and provide a distributed channel access algorithm based on the pure threshold structure, which is easy to implement. Numerical results demonstrate effectiveness and efficiency of the proposed strategy compared to alternatives.

Strategic behaviours analyses for the customers at both sides of a double-ended queueing system with asymmetrical information

Motivated by the applications of e-commerce platform services, a double-ended queueing system is considered in this paper. Clearly, the customers at both ends are strategic and their behaviours can affect the system performances. However, literatures almost only investigate one-side customers' strategic behaviours and ignore customers' behaviours on the other side. In this paper, we focus on the strategic behaviours of customers at both sides with asymmetrical information levels and study the interactions between them. Type-T customers are supposed to be insightful, who can observe system states and join the system following a pure threshold strategy. Type-P customers have no knowledge about the system states and join the system with equilibrium joining probability. An algorithm is provided to calculate customers' strategy behaviours. An application for the double-ended queueing system to a kitting process in the assembly system are investigated by numerical examples.

Allocation strategy for time‐sensitive tasks in mobile edge computing: An observable perspective

SummaryWith the development of computing technology and the popularization of application in 5G network, mobile‐edge computing (MEC), which can effectively reduce time delay and save energy consumption, has attracted extensive attention from the academic community. How to make a suitable allocation decision for tasks becomes one of the critical issues in MEC systems. In order to yield the most benefit to a newly arriving task, by placing a decision making module (DMM) in a mobile device, a MEC system architecture is presented. Based on the net benefit of a newly arriving task, the DMM makes a decision of dropping the task, allocating the task to the local execution system, or offloading the task to the MEC server via the transmission system. From the observable perspective of task individuals, a pure threshold strategy is proposed to show that a Nash equilibrium is always allowed. By constructing a two‐dimensional continuous‐time Markov chain, a social optimal threshold strategy is proposed. Numerical results show that threshold under the pure threshold strategy is always greater than that under the social optimal threshold strategy. For this, a charging policy is presented to coincide the pure threshold strategy with the social optimal threshold strategy.

Legendre polynomials roots and the F-pure threshold of bivariate forms

We provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomial in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific M\"{o}bius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the $F$-pure and the log canonical threshold, and roots of Legendre polynomials over $\mathbb{F}_p$.

Asymmetry in stochastic volatility models with threshold and time-dependent correlation

Abstract In this work we study the effects by including threshold, constant and time-dependent correlation in stochastic volatility (SV) models to capture the asymmetry relationship between stock returns and volatility. We develop SV models which include only time-dependent correlated innovations and both threshold and time-dependent correlation, respectively. It has been shown in literature that the SV model with only constant correlation does a better job of capturing asymmetry than threshold stochastic volatility (TSV) model. We show here that the SV model with time-dependent correlation performs better than the model with constant correlation on capturing asymmetry, and the comprehensive model with both threshold and time-dependently correlated innovations dominates models with pure threshold, constant and time-dependent correlation, and both threshold and constant correlation as well. In our comprehensive model, volatility and returns are time-dependently correlated, where the time-varying correlation is negative, and the volatility is more persistent, less volatile and higher following negative returns as expected. An empirical study is provided to illustrate our findings.

Bio-inspired multi-agent coordination games with Poisson observations

Global games are a class of incomplete information games where the payoffs exhibit strategic complementarity leading to an incentive for the agents to coordinate their actions. Such games have been used to model scenarios in many socioeconomic phenomena, where the private signals available to the agents are typically assumed to be Gaussian. We study an instance of a global game where the agents observe Poisson random variables, which are inspired by applications in microbiology where information signals are disseminated via discrete molecular signals rather than continuous signals. Although this observation model violates the essential technical assumptions present in the Gaussian case, we present preliminary results on the existence of Nash equilibria in pure threshold policies in two variants of the underlying random state-of-the-world: an arbitrarily distributed discrete binary state and a continuous state with an (improper) uniform distribution.

The FrobeniusThresholds package for Macaulay2

This article describes the \emph{Macaulay2} package \emph{FrobeniusThresholds}, designed to estimate and calculate $F$-pure thresholds, more general $F$-thresholds, and related numerical invariants arising in the study of singularities in prime characteristic commutative algebra.

The effect factor and pure threshold average of the ear with normal hearing in patients with single sided deafness

Objective:To learn the hearing level and analyze the effect factors of hearing loss of the ear with normal hearing in patients with single sided deafness, and provide references for the treatment of single sided deafness. Methods:A retrospective analysis of pure threshold average of 89 patients with single sided deafness, it was divided into six groups,0-6 month group （14 cases）, >6-12 month group （17 cases）, >12-18 month group （15 cases）, >18-24 month group （26 cases）, >24-30 Month group （10 cases）, >30-36 months group （7 cases） in accordance with the different duration of deafness, and compare the pure threshold average of each group; In accordance with the different ages of onset of deafness, it was divided into four groups, 21-30 years old group（18 cases）, >30-40 years old group （24 cases）, >40-50 years old group （30 cases）, >50-60 years old group （17 cases）, and pure threshold average of matching age of normal hearing was compared. Spearman correlation was used to analyze whether gender, side, age of onset of deafness, etiology of deafness, duration of deafness, and pure threshold average hearing were correlated. Results:As the duration of deafness in the affected ear increased, the average pure threshold in patients with single sided deafness increased. As the age of onset of deafness increased, there was a statistically significant difference in age-matched normal hearing. Age of onset of deafness and duration of deafness were the main factors affecting the pure threshold average. Conclusion:In clinical work, the degree of attention to patients with single sided deafness should be increased, hearing of the better hearing ear patients should be protected, and the quality of life will be improved.

Dual Sensory Impairment and Cognitive Performance, MCI, and Dementia in Older Adults: The ARIC Neurocognitive Study

Dual sensory impairment (DSI) affects 11.3% of adults aged ≥80 years. Hearing and vision impairments are each associated with cognitive decline and dementia, but DSI’s impact is unknown. All-cause dementia and mild cognitive impairment (MCI) were adjudicated using longitudinal cognitive information. Ten neurocognitive tests were summarized using latent variable methods. Hearing was measured using pure tone better-ear thresholds (0.5-4 kHz) and vision with better-eye presenting distance visual acuity and/or contrast sensitivity. In 881 adults (79±4 years, 44% black, 64% female), DSI (vs. no hearing or vision impairment) was cross-sectionally associated with -0.17 standard deviations (SD) [95% confidence interval (CI): -0.32, -0.02] lower global cognitive score and an 87% increased odds (95% CI: 1.01, 3.45) of combined MCI/dementia, after full adjustment for demographic and clinical factors. Future longitudinal research should elucidate the mechanism underlying this association to determine if treatment can delay cognitive decline and MCI/dementia in older adults.

$F$-pure threshold and height of quasihomogeneous polynomials

The aim of this paper is to give a connection between the F-pure threshold of a polynomial and the height of the corresponding Artin–Mazur formal group. For this, we consider a quasihomogeneous polynomial f∈ℤ[x0,…,xN] of degree w equal to the degree of x0⋯xN and show that the F-pure threshold of the reduction fp∈𝔽p[x0,…,xN] is equal to the log-canonical threshold of f if and only if the height of the Artin–Mazur formal group associated to HN−1(X,𝔾m,X), where X is the hypersurface given by f, is equal to 1. We also prove that a similar result holds for Fermat hypersurfaces of degree greater than N+1. Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the F-pure threshold of a quasihomogeneous polynomial of degree w cannot be characterized by the height.

Modification and verification of the Infant\u2013Toddler Meaningful Auditory Integration Scale: a psychometric analysis combining item response theory with classical test theory

BackgroundEarly prelingual auditory development (EPLAD) is a fundamental and important process in the speech and language development of infants and toddlers. The Infant–Toddler Meaningful Auditory Integration Scale (ITMAIS) is a widely used measurement tool for EPLAD, however it has not yet undergone a comprehensive psychometric analysis. The aim of this research was to modify and verify the psychometric properties of ITMAIS using a combination of Item Response Theory (IRT) and Classical Test Theory (CTT).MethodsStage 1—1730 children were retrospectively recruited to enable the application of an IRT model, specifically the graded response model, to modify the ITMAIS. Stage 2—another 450 infants and toddlers with normal hearing or permanent hearing loss before auditory intervention were recruited to verify the psychometric properties of the modified ITMAIS (ITMAIS-m) using the CTT method.ResultsUsing the metric of the graded response model, by removing item 2 from the ITMAIS, ITMAIS-m demonstrated discrimination parameters ranging from 3.947 to 5.431, difficulty parameters from − 1.146 to 1.150, item information distributed between 4.798 and 9.259 and a test information score of 48.061. None of the items showed differential item functioning. ITMAIS-m was further verified in Stage 2, showing Cronbach’s α of 0.919 and item-total correlations ranging from 0.693 to 0.851. There was good convergent validity of ITMAIS-m with other auditory outcome measure (r = 0.932) and pure tone average thresholds (r ranging from − 0.670 to − 0.909), as well as a high ability to discriminate between different hearing grades (Cohen d ranging from 0.41 to 5.83).ConclusionsThe ITMAIS-m is a reliable and valid tool for evaluating EPLAD in infants and toddlers, which can be efficiently and precisely applied in clinical practice. The combined use of IRT and CTT provides a powerful means to modify psychometrically robust scales aimed at childhood auditory outcome measurements.

Equilibrium in a finite capacity M/M/1 queue with unknown service rates consisting of strategic and non-strategic customers

We consider an $$M/M/1/{\overline{N}}$$ observable non-customer-intensive service queueing system with unknown service rates consisting of strategic impatient customers who make balking decisions and non-strategic patient customers who do not make any decision. In the queueing game amongst the impatient customers, we show that there exists at least one pure threshold strategy equilibrium in the presence of patient customers. As multiple pure threshold strategy equilibria exist in certain cases, we consider the minimal pure threshold strategy equilibrium in our sensitivity analysis. We find that the likelihood ratio of a fast server to a slow server in an empty queue is monotonically decreasing in the proportion of impatient customers and monotonically increasing in the waiting area capacity. Further, we find that the minimal pure threshold strategy equilibrium is non-increasing in the proportion of impatient customers and non-decreasing in the waiting area capacity. We also show that at least one pure threshold strategy equilibrium exists when the waiting area capacity is infinite.

The characteristics of monosyllable recognition in Mandarin-speaking patients with auditory neuropathy

Background: A set of Chinese Mandarin monosyllable test lists has been widely used in clinical diagnosis, while the performance of the Mandarin-speaking patients with auditory neuropathy (AN) in the monosyllable tests was still unknown.Objectives: To analyze the characteristics of monosyllable recognition in Mandarin-speaking patients with AN.Materials and methods: Nineteen Mandarin-speaking patients diagnosed with AN were recruited to obtain the performance-intensity (P-I) functions. They were divided into two paired groups according to two conditions: (1) the rising-slope (RS) and the non-rising-slope pattern (NRS) group for the audiogram; or (2) the male and the female group for gender.Results: The performance at 20 and 30 dB SL showed negative correlation with the pure tone-averaged thresholds of 0.5, 1.0, 2.0 and 4.0 kHz (4FA). We found significant differences between the P-I functions of the RS and the NRS group, and surprisingly found that between the male and female as well although there was no difference in 4FA between the paired groups. A notable rollover phenomenon in P-I functions was detected in each group.Conclusions: The monosyllable recognition scores of Mandarin-speaking AN patients showed numerous particular characteristics reflected by P-I functions. Acquiring the complete P-I functions at multiple presentation levels and the rollover index may give us more information about the diagnosis and prognosis.

Ascending chain condition for -pure thresholds on a fixed strongly -regular germ

In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

Ascending Chain Condition for F-Pure Thresholds with Fixed Embedding Dimension

Abstract In this paper, we prove that the set of all $F$-pure thresholds of ideals with fixed embedding dimension satisfies the ascending chain condition. As a corollary, given an integer $d$, we verify the ascending chain condition for the set of all $F$-pure thresholds on all $d$-dimensional normal locally complete intersection varieties. In the process of proving these results, we also show the rationality of $F$-pure thresholds of ideals on non-strongly $F$-regular pairs.

A channel-aware expected energy consumption minimization strategy in wireless networks

With the rapid development of wireless network technology, energy saving has become a very important topic to build a green network in wireless networks. Due to the time-varying characteristics of the channel, it is possible to obtain a higher utilization for energy by using the channel with good state in wireless communications. From the view of the data transmission energy consumption of the whole wireless network, this paper proposes an expected energy consumption minimization strategy (E2CMS) for data transmission based on the optimal stopping theory. Considering the maximum transmission delay and a given receiving power, E2CMS delays data to transmit until the best expected channel state is detected. In order to solve the problem, firstly, we construct an energy consumption minimization problem with quality of service constraints. Then, we prove that E2CMS is a pure threshold strategy by the optimal stopping theory and obtain the power threshold by solving a fixed-point equation using backward induction. Finally, simulations are performed in a typical small-scale fading channel model. E2CMS is compared with a variety of different transmission scheduling strategies. The results show that E2CMS has lower average energy consumption for per unit data and significantly improves the network performance.

Impact of Strong Sound of Starting Guns on the Hearing Loss of Track and Field Athletes Based on Brainstem Auditory Evoked Potential

To explore the impact of the sound of starting guns on the hearing of track referees (starters); to measure and study the noise characteristics of the muzzle and ear when track referees issue the order. The survey, ear routine examination, pure tone audiometry and brainstem auditory evoked potential were applied to conduct the audiometric test analysis and auditory function test on 20 sports college students (40 ears) and 10 college teachers (20 ears) engaged in the starting work in different time period before and after (before starting, 3h and 8d after starting) the starting (instantaneous strong noise). Under the noise intensity of 158~156 dB(Lp), the pure tone average threshold, high pure tone average threshold and I wave latency of BAEP of left and right ears 3h after the starting were significantly higher than those before the start (P<0.01). After 8 days, it gradually returns to the level before the starting. The auditory condition of teachers who regularly engage in the starting work is significantly different from that of ordinary people (P<0.01). For those who are exposed to the high-intensity noise of the starting gun for the first time, it may cause the reversible recoverable damage of hair cells of the ear and temporary threshold shift.

Using perceptual cues for brake response to a lead vehicle: Comparing threshold and accumulator models of visual looming

Previous studies have shown the effect of a lead vehicle’s speed, deceleration rate and headway distance on drivers’ brake response times. However, how drivers perceive this information and use it to determine when to apply braking is still not quite clear. To better understand the underlying mechanisms, a driving simulator experiment was performed where each participant experienced nine deceleration scenarios. Previously reported effects of the lead vehicle’s speed, deceleration rate and headway distance on brake response time were firstly verified in this paper, using a multilevel model. Then, as an alternative to measures of speed, deceleration rate and distance, two visual looming-based metrics (angular expansion rate θ˙ of the lead vehicle on the driver’s retina, and inverse tau τ-1, the ratio between θ˙ and the optical size θ), considered to be more in line with typical human psycho-perceptual responses, were adopted to quantify situation urgency. These metrics were used in two previously proposed mechanistic models predicting brake onset: either when looming surpasses a threshold, or when the accumulated evidence (looming and other cues) reaches a threshold. Results showed that the looming threshold model did not capture the distribution of brake response time. However, regardless of looming metric, the accumulator models fitted the distribution of brake response times better than the pure threshold models. Accumulator models, including brake lights, provided a better model fit than looming-only versions. For all versions of the mechanistic models, models using τ-1 as the measure of looming fitted better than those using θ˙, indicating that the visual cues drivers used during rear-end collision avoidance may be more close to τ-1.

Joint Listening, Probing, and Transmission Strategies for the Frame-Based Equipment in Unlicensed Spectrum

The recent efforts of extending cellular technologies to unlicensed spectrum have led to concerns about the coexistence between cellular and incumbent systems. To harmonize coexistence, regions including European Union and Japan require that a radio equipment operating in unlicensed spectrum shall perform listen-before-talk (LBT). In this paper, we study joint listening, probing, and transmission strategies for a frame-based equipment (FBE) in unlicensed spectrum. We first investigate throughput optimal strategy and find that the optimal rule for an FBE is to transmit whenever allowed by the regulation. We then turn to nominal throughput optimal transmission that takes into account transmission costs such as power consumption. We find that the nominal throughput optimal rule is a pure threshold policy: The FBE should stop listening and transmit once the channel quality exceeds an optimized threshold. The optimal threshold can be found by solving a fixed point equation, but the fixed point equation in general does not admit a closed-form solution. We then derive a lower bound and an upper bound on the optimal threshold. We further devise an iterative algorithm with convergence analysis to compute the optimal threshold. Our results shed further light on LBT strategies for radio equipment operating in unlicensed spectrum.