D with quotient field K, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one  factorization into irreducibles. In the case of the ring  Let associated in R if there exists a unit r = us. We denote this by Two factorizations into irreducibles of the same element,essentially the same if n = m and, after a suitable re-indexing, essentially different.are called tions in  are callDefinition 1.1.R be a commutative ring with identity. An irreducible element absolutely irreducible , if for all natural numbers n, every factorization of nc is essentially the same as Let R that is called “not absolutely irreducible” might not be irreducible at all, whereas a “non-absolutely irreducible” element is assumed to be irreducible, but not absolutely irreducible.Note the following fine distinction: an element of We now concentrate on integer-valued polynomials over a principal ideal domain.D is a principal ideal domain, is called primitive if the greatest common divisor of its coefficients is 1.Recall that a polynomial in Definition 1.2.D be a principal ideal domain with quotient field K, and f asI a finite (possibly empty) set, and each ig primitive and irreducible in standard form of f.Let b as the denominator, to a as the constant factor, and to numerator of f, keeping in mind that each of them is well-defined and unique only up to multiplication by units of D.We refer to Definition 1.3.fixed divisor of f, denoted D generated by f(D).For image-primitive.An integer-valued polynomial D is a principal ideal domain, we may, by abuse of notation, write the generator for the ideal, as in When Remark 1.4.D be a principal ideal domain with quotient field K, and f is in b divides Let Remark 1.5.D be a principal ideal domain with quotient field K. Then any non-constant irreducible element of f.Let a = 1 and Furthermore, Definition 1.6.D be a principal ideal domain. For p a prime element in D, we letLet Remark 1.7.D up to multiplication by units.By the above definition,By the nature of the minimum function, the fixed divisor is not multiplicative:2.We refer to, for instance,  for the Definition 2.1.D be a principal ideal domain, ig is essential for p among the jg with w is then called a witness for ig being essential for p.We say that ig is quintessential for p among the jg with w is called a witness for ig being quintessential for p.We say that Let jg with We will omit saying “among the Remark 2.2.ig with ig being essential  is an edge if and only if there exists a prime element p in D such that both ig and jg are essential for p among the kg with The quintessential graph of the indexed set of polynomials I, and in which  is an edge if and only if there exists a prime element p in D such that both ig and jg are quintessential for p among the kg with The Let Example 2.4.Let and setg is 15.w = 1, 2, 0, respectively, as witnesses, we see that g1 is not essential for 5 because a, also Taking w = 1, 0, 2 respectively, as witnesses, we see that g4 is quintessential for 3. The polynomial g3 is not essential for 3.Taking A quick check shows that the fixed divisor of Lemma 2.5.Let D be a principal ideal domain anda non-constant image-primitive integer-valued polynomial, written in standard form according to where T is a finite set of pairwise non-associated primes of D, and letEveryn can be written asdividing fwithforand uniquefor. Moreover, every such representation of h satisfies:Ifandj is quintessential for q among thesuch that g, thenj and gk are both quintessential for the same primeIn particular, whenever g, thenProof.nf is image-primitive, and, therefore, all polynomials in nf are image-primitive. Let h is written in standard form as in k. This is so because h and k are image-primitive; see We know jg is quintessential for q. Note that, by jg in the numerator of any factor of nf is unique.Now let Writing D.(i) follows from unique factorization in (ii) follows from unique factorization in w for jg being quintessential for q. Since f is image-primitive, To see (iii), consider a witness k instead of h).–(iii) together imply Theorem 1.Let D be a principal ideal domain with quotient field K. Letbe a non-constant image-primitive integer-valued polynomial, written in standard form aswith, andi is primitive and irreducible in, where each gIf the essential graph ofis connected, then f is irreducible inProof.f is irreducible in If f can be expressed as a product of m non-units if is a constant, and that Now assume kh primitive in I into non-empty pairwise disjoint subsets Write i to j in the essential graph of p0 in D dividing b, p0. As ig is essential for p0, p0 cannot divide any kb with p0 divides b1. For any kg essential for p0 it follows that kp for which both kI, and, finally, that Select m = 1. □As Theorem 2.Let D be a principal ideal domain andbe non-constant and image-primitive, written in standard form aswhereis a finite set and foris primitive and irreducible inIf the quintessential graph G ofis connected, then f is absolutely irreducible.Proof.i, p, Supposet with tf is a power of f by showing that each ig with tf with the same exponent.Fix I, h, a prime element hp in T such that hp. By tf with the same exponent. Eventually, ig and jg occur in the numerator of tf with the same exponent, for arbitrary tf is a power of f. Since tf is irreducible, tf = f. □Let or as in  and, henExample 2.6.The binomial polynomialThe converse of f is square-free, as we now proceed to show, cf. There is, however, a converse to 3.D be a principal ideal domain with quotient field K. When we talk of the denominator of a polynomial in Let Remark 3.1.D be a principal ideal domain. Suppose the denominator of Let f is irreducible in ig is essential for some ig. This further implies Then, if Theorem 3.Let D be a principal ideal domain andbe non-constant and image-primitive, with square-free denominator, written in standard form aswhereis a finite set and foris primitive and irreducible inLet G be the quintessential graph of Then f is absolutely irreducible if and only if G is connected.Proof.G is connected. Now assume G is not connected. We show that f is not absolutely irreducible. If f is not even irreducible, we are done. So suppose f is irreducible. This implies G is not connected, I is a disjoint union of J1 and J2, both non-empty, such that there is no edge  with In view of Theorem 2, we only need to show necessity. If T as a disjoint union of T1 and T2 by assigning every T1, every T2, and assigning each ig is quintessential to T1 or T2 arbitrarily. , each oRemark 3.2.D be a principal ideal domain. The proof of Theorem 3 shows that any non-absolutely irreducible element nf already for n = 3.Let D is a principal ideal domain, p a prime of D and each f is an irreducible element of ig is essential for p, that is, for each each If An analogous statement relates absolutely irreducible integer-valued polynomials with prime denominator to quintessential irreducible factors of the numerator:Corollary 3.3.Let D be a principal ideal domain,a prime, anda finite set. For, letbe primitive and irreducible in. LetThen f is an absolutely irreducible element ofandi is quintessential for p among the gi withg, that is, for eachthere existssuch thatandfor alleach Proof.f is in f is, in particular, irreducible and therefore ig is quintessential for p. □If We conclude by an example of how to apply Example 3.4.The following polynomial This is so because the essential graph of


Nature Communications 10.1038/s41467-020-18279-x, published online 7 September 2020.Correction to: The original version of this Article contained errors in Fig. 1. The label ‘a’ was omitted for the left panel and the label ‘b’ was omitted for the right panel. In Fig. 1b, the right parenthesis was incorrectly positioned indicating a 4-carbon linker, rather than the correct 6-carbon linker. These errors have been corrected in both the PDF and HTML versions of the Article.


The authors wish to make the following correction to this paper :p > 0.001) compared...”. It should instead read “...LC-HS extract was statistically less (p < 0.001) compared... ”. Furthermore, the last sentence of the same paragraph reads “Interestingly, the differences observed in the reduction of fat mass showed the effectiveness of the HS-LC extract in reducing fat mass”. It should instead read “Interestingly, the differences observed in the reduction of fat mass showed the effectiveness of the LC-HS extract in reducing fat mass”. In this respect, there are several parts of the manuscript which reads HS-LC, it should be read as LC-HS instead. This includes: page 2, paragraph 2; Figure 1; page 4, last paragraph.There are text errors in Section 4.3.2 Densitometry, third paragraph. The second sentence states that “...LC-HS extract was statistically less (p < 0.003)”.There is also a missing explanation in the same section, regarding the intergroup analysis of the results. At the end of the third paragraph, it should include the following sentence: “Intergroup analysis revealed significant differences between both groups, where LC-HS significantly decreased body fat mass compared to placebo (p = 0.014)”, should read “Inter-group analysis revealed that the LC-HS extract significantly decreased the body weight compared to placebo (p = 0.014)”. In the same section, sentence “Therefore, the diminution of BMI after the consumption of the placebo was shown to be less than that observed for the LC-HS extract (p = 0.013)”, should read “Therefore, intergroup analysis revealed that the BMI of the LC-HS group significantly decreased with respect to the placebo group (p = 0.013)”.In Section 4.2 Weight and BMI, the sentence “Therefore, it can be determined that the LC-HS extract significantly decreased the body weight of the subjects (Information is missing from It should be as follows:Information is missing on It should be as follows instead:These changes have no material impact on the conclusions of our paper. The authors would like to apologize for any inconvenience caused to the readers by these changes.


N = 24 BN; N = 24 BED) in the analyses. Participants allocated to the food-specific, compared to the general intervention group, showed significantly greater reductions in reaction time to low energy-dense foods, compared to non-foods, by post-intervention. Commission errors significantly increased from baseline to post-intervention, regardless of stimulus type (food vs. non-food) and intervention group . There were no significant changes in omission errors. P3 amplitudes to “no-go” cues marginally, but non-significantly, decreased by post-intervention, but there was no significant interaction with stimulus type (high energy-dense food vs. non-food) or intervention group . There were no significant changes in N2 amplitudes to “no-go” cues, N2 amplitudes to “go” cues, or P3 amplitudes to “go” cues from baseline to post-intervention. Training effects were only marginally captured by these event-related potentials. We discuss limitations to the task paradigm, including its two-choice nature, ease of completion, and validity, and give recommendations for future research exploring ERPs using inhibitory control paradigms.In a feasibility trial comparing two forms of combined inhibitory control training and goal planning  among patients with bulimia nervosa (BN) and binge eating disorder (BED), we found evidence of symptomatic benefit, with stronger effects among participants receiving a food-specific intervention. The aim of the present study was to examine changes in behavioral outcomes and event-related potentials  from baseline to post-intervention that might suggest the mechanisms underpinning these effects. Fifty-five participants completed go/no-go tasks during two electroencephalography (EEG) sessions, at baseline and post-intervention. The go/no-go task included “go” cues to low energy-dense foods and non-foods, and “no-go” cues to high energy-dense foods and non-foods. Datasets with poor signal quality and/or outliers were excluded, leaving 48 participants ( The number of individuals receiving an eating disorder diagnosis has been increasing since the 1980s . This isImpulsivity is considered to be a risk factor for binge eating . ReviewsAs a multidimensional construct, impulsivity is thought to consist of two main components: decreased inhibitory control and increased reward sensitivity . AccordiEvent-related potentials (ERPs), derived from EEG recordings, offer the possibility of exploring cognitive processes within neural circuits . The N2,P3a among participants with healthy, overweight, and obese BMIs, amplitudes were enhanced when the task involved inhibiting to high energy-dense, as opposed to low-energy dense foods  and changes in N2 and P3 amplitudes to high energy-dense foods. Since the timing and onset of N2 and P3 can vary as a function of processing speed and training, we explore their respective latencies too.Recent evidence from a feasibility trial in our laboratory has found that an intervention combining go/no-go training and implementation intentions is associated with reductions in binge eating frequency among individuals with bulimia nervosa and BED . AccordiWe hypothesized that reaction time and omission errors to “go” cues  and commission errors to “no-go” cues  will decrease from baseline to post-intervention. These effects will be more pronounced for food cues, and among individuals receiving a food-specific intervention.In response to “no-go” cues, we hypothesized that mean “no-go” N2 amplitudes will increase from baseline to post-intervention, indicative of improved inhibitory control, and that mean “no-go” P3 amplitudes will decrease from baseline to post-intervention, indicative of reduced motivated attention. The opposite pattern is expected for “go” cues. Again, these effects will be more pronounced to food cues, and among individuals receiving a food-specific intervention. Finally, we predict that the training effects on binge eating will correlate with “no-go” N2 and P3 amplitudes to high energy-dense foods.N = 40) and BED (N = 38) were recruited through eating disorder charity websites, social media, flyers, and participant identification centers that supported the study. They were then randomly allocated to a food-specific or general intervention, which included both go/no-go training and goal planning . Within this manuscript, only participants who attended and completed both baseline and post-intervention EEG sessions were included (N = 55).Participants with bulimia nervosa  of at least 18.5, were between the ages of 18 and 60, did not have a visual impairment that could not be repaired with eyewear, a neurological impairment, an alcohol or drug dependence, or psychosis.Eligibility required that participants met criteria for bulimia nervosa or BED according to the Eligibility clearance. The Structured Clinical Interview for DSM-V SCID-5; , a semi-Over the last 28 days, how many times have you eaten what other people would regard as an unusually large amount of food?).Binge eating frequency. Item 13 of the eating disorder examination questionnaire EDE-Q;  was usedFood-specific go/no-go task. The present study used the food-specific go/no-go task, as implemented by EEG was recorded continuously throughout the experimental tasks using BrainVision Recorder, and amplified with two 32-channel BrainAmp DC amplifiers  An actiCAP 64Ch standard cap was equipped after the 10–20 system . FCz wasOffline, EEG data pre-processing was done using EEGLab  and compPrimarily driven by previous EEG studies using the go/no-go task e.g., , we had Similarly, based on previous literature  and our Individuals who expressed interest in learning about the study procedures were sent an information sheet detailing the procedure. Next, they were contacted for a 15-min eligibility phone interview. Those who met criteria were sent a consent form indicating their rights as participants. After informed consent, an appointment was booked for the first EEG session and participants were sent a battery of questionnaires to complete via an online platform . Before entering the laboratory for the first EEG session, participants were instructed to withhold from food, caloric drinks, and nicotine for 2 h, as well as caffeinated drinks and alcohol for 24 h. During the laboratory session, participants were briefed about what the session will involve. They were then asked to sign a hard copy of the consent form, and their weight and head circumference were measured. After the appropriate EEG cap size was selected, the researcher put the cap on and applied electro-gel into the electrodes.Before completing computerized tasks during EEG recording, participants were instructed to rest with their eyes open for 3 min and to rest with their eyes closed for 3 min. The researcher then explained the task rules and participants completed a practice trial of the go/no-go task, which included only 36 of the 216 trials of the full-length version. The average duration of the session was 2 h and 30 min. After 30 ± 2 days , the same procedure was followed for the second EEG session.All procedures were revised and approved by the London Westminster Research Ethics Committee and the Health Research Authority (IRAS Project ID: 209609).The study followed a mixed models design, with intervention  as the between subject variable and time (baseline vs. post-intervention) as the between subject variables.The inhibitory control training used was developed at the University of Exeter . ParticiImplementation intentions involved encouraging participants to identify an unhelpful habit, reflect on situations and motivations that are likely to precede the unhelpful behavior, and then design an alternative behavior that could replace the unhelpful behavior. Participants allocated to the food-specific intervention group were asked to select an unhelpful behavior that was related to food/eating, while those allocated to the general intervention group were asked to select an unhelpful behavior that was unrelated to food/eating . One example would be: “If I am home alone (situation) and feeling anxious (motivation), then I normally buy binge food (unhelpful eating-related habit),” would be replaced with “If I am home alone and feeling anxious, then I will meditate for 10 min .” Each participant was assigned a trained mentor who followed up with him/her weekly via email for 4 weeks.t-test was conducted to explore between-group differences in training task completion, and a chi-squared test was conduced to explore between group differences in implementation intention  engagement. Next, independent samples t-tests were conducted to ensure that the two interventional groups did not significantly differ on demographic and clinical characteristics.Statistical analyses were conducted using SPSS 24  for Mac.For behavioral data analysis, a repeated measures ANOVA was conducted to measure reaction time to “go” cues at two time points, across two types of stimuli, and between two intervention groups. The same ANOVA structure was used to analyze commission and omission errors, separately. ANOVAs followed the structure: 2 (time: pre- vs. post- intervention) × 2 (type of stimulus: low/high energy-dense food vs. non-food) × 2 . “No-go” analyses included high energy-dense foods and non-foods, while “go” analyses included low energy-dense foods and non-foods.t-tests were used to compare N2 and P3 amplitudes to “no-go” and “go” non-food cues.To ensure that ERP amplitudes reflected the expected task demands, two paired samples For the main analysis, two repeated measures ANOVAs (for P3 and N2 separately) were conducted to measure amplitudes to “no-go” cues at two time points, across two types of stimuli, and between two intervention groups. They followed the format: 2 (time: pre- vs. post- intervention) × 2 (type of stimulus: high energy-dense food vs. non-food) × 2 . This was repeated for “go” cues, with the following format: 2 (time: pre- vs. post- intervention) × 2 (type of stimulus: low energy-dense food vs. non-food) × 2 . The main analyses were repeated for ERP latencies.minus post-intervention no-go N2 amplitudes to high energy-dense food, (2) Baseline minus post-intervention no-go P3 amplitudes to high energy-dense food, and (3) Baseline minus post-intervention binge eating frequency. This analysis structure was repeated for ERP latencies.Finally, two-tailed Pearson’s correlations were used to assess the correlation between changes in binge eating frequency and changes in ERP amplitudes to high energy-dense foods across time. The variables were created using the following formulas: (1) Baseline p > 0.05; see SD = 6.79). There was no significant difference in the number of trainings completed between participants in the food-specific intervention group  and participants in the general intervention group . With regards to if-then planning, 50% of participants were minimally engaged at implementing their plan. There were no significant differences in engagement between participants in the food-specific vs. general intervention group .No significant differences in demographic and clinical characteristics were found between the two intervention groups  = 28.12, p < 0.001, η2 = 0.379] and a significant interaction between time x type of stimulus x intervention group , indicating that participants in the food-specific intervention, compared to the general intervention group, showed a significantly greater reduction (training effect) in reaction time to low energy-dense foods, compared to non-foods, by post-intervention (see p > 0.05). Although there was a main effect of time on commission errors , the direction, indicating that participants made more errors by post-intervention, compared to baseline, was unexpected. No other significant main effects of interaction effects were significant  see There was a significant main effect of time on reaction time . Neither was there a time × type of stimulus × intervention group interaction effect . There was no significant main effect of time  and no time x type of stimulus x intervention group interaction effect  on “no-go” N2 latency.There was no main effect of time on “no-go” N2 amplitudes , but no significant time x type of stimulus x intervention group interaction . There was a significant main effect of time on “no-go” P3 latency , indicating that “no-go” P3 latency decreased from baseline to post-intervention regardless of stimulus type. There was no time x type of stimulus x intervention group interaction effect  on “no-go” P3 latency.There was a marginal, but non-significant, main effect of time on “no-go” P3 amplitudes . Moreover, the expected 3-way interaction was not significant . There was no significant main effect of time  and no time × type of stimulus × intervention group interaction effect  on “go” N2 latency.There was no main effect of time on “go” N2 amplitudes , and the expected 3-way interaction was not significant . There was a significant main effect of time on “go” P3 latency , indicating that “no-go” P3 latency decreased from baseline to post-intervention regardless of stimulus type. There was no time x type of stimulus x intervention group interaction effect  on “go” P3 latency.There was no main effect of time on “go” P3 amplitudes [See SD = 10.28).Across all participants, the mean reduction in binge eating frequency was 3.95  or N2 latency to high energy-dense food . Moreover, there was no significant correlation between changes in binge eating frequency and changes in no-go P3 amplitude to high-energy dense food  or P3 latency to high energy-dense food .There was no significant correlation between changes in binge eating frequency and changes in no-go N2 amplitude to high energy-dense foods  and event-related potential  changes at baseline and at the end of an intervention designed to modify inhibitory control for bulimia nervosa and BED. The research aimed to explore whether changes would be present, whether they would be specific to food, and whether they would differ between the two intervention groups.In line with our hypothesis, individuals allocated to the food-specific intervention group, compared to the general intervention group, showed significantly greater reductions in reaction time to low-energy dense foods, compared to non-foods, from baseline to post-intervention. These indicate that successful stimulus-response learning to “go” cues had taken place, which may have induced a beneficial attentional bias toward these foods. Participants in both intervention groups showed significant reductions in P3 latency over time, indicating a speeding of task-related information processing . ContrarAt baseline, “no-go” cues elicited larger N2 amplitudes and smaller P3 amplitudes compared to “go” cues, suggesting that there was greater inhibition and less attention/salience toward “no-go” cues, lending evidence to the fundamental validity of the task and training.Despite this, no significant increases in “no-go” N2 amplitudes were found from baseline to post-intervention, and no significant differences were found between the two intervention groups. This was in line with findings from previous research, which reported no increase in N2 amplitude as a result of inhibitory control training . Such fiAlbeit marginally and non-significantly, P3 amplitudes to “no-go” cues descriptively decreased from baseline to post-intervention. Nonetheless, there was no significant interaction between time x type of stimulus x intervention group, and no significant correlation between changes in binge eating frequency and no-go P3 amplitudes to high energy-dense foods. These indicate that the marginal decrease in P3 to “no-go” cues may reflect a general depreciation that comes with repeated task completion. Given that the P3 amplitude is sensitive to the amount of attentional resources engaged , it is lOur hypotheses regarding changes in N2 to “go” cues were not supported and no significant 3-way interaction was found. In addition, while we expected less negativity over time (suggestive of reduced inhibition), there was a non-significant trend in the opposite direction. These findings might have been influenced by the inclusion of a binary “left-right” decision to the “go” instruction . This additional attentiontal control may have slowed down the “go” process and engaged inhibitory processes.In addition, no significant changes in P3 amplitudes to “go” cues were found from baseline to post-intervention, and no significant 3-way interaction was found. While previous research using similar methodology had found enhanced P3 amplitudes in frontal electrodes over the course of training , the go/In light of these findings, we were unable to identify changes in neural components that correlate with changes in eating behavior over time. Therefore, while changes in binge eating frequency were found on a behavioral level , it remains unclear what change processes had occurred on a neural level. A consideration of limitations is essential. For instance, the negligible error rates during task completion may suggest that the task was not challenging and hence, did not recruit inhibitory circuits. This is consistent with the absence of a P3 in fronto-central electrodes . To incrThe present research shows that, while participants in the intervention showed reductions in binge eating frequency , the neural processes supporting this clinical effect could not be entirely uncovered. It remains unclear whether the null findings reflect an absence of change in neural activity over time, or an inability of the measures to detect change. It is advisable for future research to explore different task parameters, by potentially differing the ratio of “go” to “no-go” trials and increasing the speed-accuracy trade-off.The datasets generated for this study are available on request to the corresponding author.The London Westminster Research Ethics Committee and the Health Research Authority reviewed and approved all the procedures involved in this study (IRAS Project ID: 209609). The patients/participants provided their written informed consent to participate in this study.RC has recruited participants and collected the data. GM has offered use of EEG equipment. RC and ML-M have pre-processed the data. KE has contributed to epoching the EEG data. JB has assisted in the creation of figures. RC has written the manuscript, with suggestions and corrections from JT, VC, ML-M, KE, GM, and JB.ML-M was employed by company CiberObn. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Reducing energy consumption has become a critical issue in today data centers. Reducing the number of required physical and Virtual Machines results in energy-efficiency. In this paper, to avoid the disadvantages of VM migration, a static VM placement algorithm is proposed which places VMs on hosts in a Worst-Fit-Decreasing (WFD) fashion. To reduce energy consumption further, the effect of job scheduling policy on the number of VMs needed for maintaining QoS requirements is studied. Each VM is modeled by an M/M/* queue in space-shared, time-shared, and hybrid job scheduling policies, and energy consumption of real-time as well as non-real-time applications is analyzed. Numerical results show that the hybrid policy outperforms space-shared and time-shared policies, in terms of energy consumption as well as Service Level Agreement (SLA) violations. Moreover, our non-migration method outperforms three different algorithms which use VM migration, in terms of reducing both energy consumption and SLA Violations. Ever-increasingly, companies move to the cloud to lower their budgets and reduce costs by benefiting from pooled hardware and software resources, delivered as IT services. These shared, on demand services are mainly offered in three forms: Infrastructure as a Service (IaaS), Platform as a Service (PaaS), and Software as a Service (SaaS). In the scope of IaaS clouds ), server virtualization is a key factor for elasticizing the data center by sharing computing resources between several users and simultaneously, isolating them from each other. Elasticity, the ability to expand and contract resources as workloads change, is an important aspect of IaaS clouds, which profits both cloud providers and users by removing the extra resources. Elasticity not only decreases green-house gasses and global warming by reducing the energy consumption of data centers, but also cuts the costs for cloud users, while providing sufficient resources to maintain QoS requirements. Server virtualization benefits IaaS datacenters, in fundamental different ways :Physical servers are usually too big for a single user. Virtual machines can divide them into smaller virtual servers, and assign each one to a different user.The Virtual Machine (VM) technology is capable of isolating different users and workloads from each other, while sharing hardware resources between them.Virtual servers are easier to manage because they are software-based and expose a uniform interface through standard abstractions.QoS requirements and obligations such as response time, reliability, availability, and security are summarized in Service Level Agreement (SLA), and a penalty is charged for any violation . In thisSome papers consider using VM migration to dynamically place as many VMs as possible on a server and reduce the number of active servers , 7. VM mIn this paper, we propose a method to place VMs on less number of servers statically in a Worst-Fit-Decreasing (WFD) fashion. WFD is a bin packing algorithm that according to our results performs better than other bin packing algorithms, namely, First-Fit-Decreasing (FFD) and Best-Fit-Decreasing (BFD) when placing VMs on servers. To further make our method more energy-efficient, we purpose a hybrid job scheduling algorithm using queuing theory that reduces the number of VMs by placing as many jobs as possible on a VM. By using less VMs, we utilize less servers and eventually reduce energy consumption.W + Ts)/Ts, where W is the job’s waiting time and Ts is the expected service time), for the next slice, respectively. Feedback scheduling is another pre-emptive scheduling algorithm which concentrates on the time spent in execution so far, rather than the remaining time to execute. The hybrid mode includes algorithms such as co-operative scheduling and scheduling with Message Passing Interface (MPI). Detailed discussion of job scheduling disciplines is beyond the scope of this paper.Generally, Virtual Machines (VM) use two main scheduling policies, namely, space- and time-shared to assign resources to the jobs appointed to them. Additionally, hybrid algorithms, combine the two aforementioned policies. While in space-shared, which is non-preemptive, the processing space is shared among different jobs, in time-shared, which is pre-emptive, time is divided between them , 13. In Inspired with static VM allocation superiority, in this paper, we propose a non-migration method which does not necessarily need to predict the workload, when placing VMs on hosts in a WFD fashion. As an extension to our research and to reduce the energy consumption further, a novel queue-based algorithm is proposed to schedule jobs on fewer number of VMs by finding the right number of VMs required for each cloud user to maintain SLA conditions. By removing the extra VMs, we aim to utilize fewer number of servers and reduce energy consumption. Since a VM with space-share policy shares the processor space between jobs non-preemptively, it executes the jobs on different cores of the processor. Therefore, it is modeled by an M/M/m queue. Normally, all the cores of a processor are the same and have equal capacity. Furthermore, we make sure jobs are assigned to the VMs which have enough capacity for executing them. In such a situation, FCFS with backfilling, which performs very well  and evenλmax of each VM is derived. By modeling SLA in terms of response times or latencies, both real-time and non-real-time applications are taken into consideration. Our findings show that there is a trade-off between server utilization and SLA violation: The busier a server gets, the more time is required for each job to be completed. Analyzing VM request queues, we balance those two metrics to design an energy-aware non-migration algorithm for placing VMs on servers in a WFD bin-packing fashion, as WFD is found in our work to have better performance in placing VMs on physical hosts compared with the other two algorithms, namely BFD and FFD bin-packing.In time-shared policy, since Processor Sharing (PS) discipline is a good approximation for RR algorithm, a VM is modeled by an M/M/1/PS queue. In hybrid mode, each core devotes a time slot to a different job in an RR manner. Therefore, it is modeled by an M/M/m/PS queue. In each scenario, the maximum request arrival rate To simulate our algorithm CloudSim  is interThe remainder of the paper is organized as follows. Related work is reviewed in Section 2. Section 3 and 4 describe our proposed model. Section 5 presents the experimental results. A conclusion is made in section 6.λ. In order to satisfy QoS requirements, they define a limit on maximum waiting times, corresponding to deadlines. By Little’s law, they transfer this limit to an upper bound on the queue’s length. Using this upper bound, they adjust the number of VMs to reduce energy consumption.To reduce the power consumption, the proposed methods in  and 7]  7] decreThe reported works in –27 try tThe authors of  and 28]28] use FNone of the works explained above consider the VM’s job scheduling policy, when rightsizing the allocated resources. Moreover, they do not bring real-time applications under consideration. EARH  proposesIn this paper, we propose an energy-aware non-migration VM placement algorithm which benefits real-time applications the most. We show that taking the job scheduling policy into consideration and adopting hybrid policy instead of space-shared or time-shared can remarkably reduce the energy consumption while maintaining SLA conditions. For this purpose, a resource-rightsizing algorithm for a space-shared policy with FCFS and a time-shared policy with RR discipline, and hybrid policy: the combination of space-shared and time-shared policies, are tested. We take both real-time and non-real-time applications into consideration. The results show that the hybrid policy excels space-shared and time-shared policies lower energy consumption and fewer SLA violations. In addition, the hybrid policy outperforms WFD, BFD, and FFD with VM migration, in terms of reducing both energy consumption and SLA violation. The next two sections explain our proposed model. The terminology used is summarized in According to what has been stated in the SLA, some minimum number of VMs are initially assigned to each cloud user. It has been shown in  and 31]31] that We aim to execute the jobs on minimum number of VMs possible, and therefore, reduce the number of active hosts. Since it takes considerable amount of time to turn on a server, a host whose all its VMs are idle will not be switched off, instead, it will be asleep. This is extremely useful for real-time applications. As depicted in λmax .In our method, when a job is executed and completed, the VM updates the maximum request arrival rate, λmax to the maximum number of jobs in the VM, Nmax (Nmax = 1 + λmax.waiting_time where λmax.waiting_time number of jobs are waiting for service and 1 job is being served). If the number of jobs that the VM is serving is smaller than Nmax, the VM reports to the job scheduler that it “can accept more jobs” otherwise, it reports that it “cannot accept more jobs”.The job scheduler unit keeps a record for each VM, stating whether the VM can or cannot accept more jobs. The job scheduler uses this record to decide whether or not it should give a newly arrived job to a VM. Each time a new job is assigned to a VM, or completed and removed, the VM sends a report to the job scheduler unit and updates its record. Algorithm 1 shows a VM sending a report to the job scheduler unit upon a new job arriving, or a job departing. In this algorithm, the VM uses Little’s law to convert the maximum request arrival rate Algorithm 1. VM Report Upon a New Job Arriving or a Job Departingλmax, the maximum request arrival rate the VM is allowed to have;Input: W, maximum waiting time.job_number: the number of jobs being served by this VM.Output: Report stating whether it can accept more jobs or not.Nmax = λmax.W + 1 1: jobnumber < Nmax then 2: if  3: report = can_accept_more_jobs 4: else 5: report = cannot_accept_more_jobs 6: send (report)When the service is entering busy hours, or the job request rate is increasing, the job scheduler unit might find that none of the active VMs are able to undertake the new job, therefore, a new VM should be activated or created. We designed the job scheduler unit to make a fair decision in the four following steps:If there are idle sleeping VMs on active hosts, activate the largest one and assign the new job to it. The host is already active and much of its energy is being consumed. By activating the largest VM, we make better use of the host and provide more processing capacity to the increasing number of arriving jobs.If there isn’t any idle VM on the active hosts, but there is enough empty space on them, select the largest space and create the largest VM which fits in that space. Aside from making better use of active hosts and presenting biggest processing capacity possible to the increasing number of arriving jobs, this decision is the closest to WFD policy.If no VM can be activated or created on the active hosts, but there are idle VMs on sleeping hosts, select the smallest sleeping host and activate the largest VM on it. Since we are activating only one VM of a host, a great portion of the hosts’ CPU capacity will be unused. With a larger host, which consumes more energy, more CPU capacity is unused and wasted. Additionally, the largest VM consumes a greater portion of hosts’ processing capacity and provides greater processing capacity to the increasing number of arriving jobs.If no VM can be activated or created on the active hosts, and there is no idle VM on the sleeping ones, select the smallest host and create the largest VM on it. Other than the reason explained in the previous part, this decision is the closest to WFD policy.λmax.The job scheduler unit checks the VM reports in the order of VM creation. In other words, the VMs which were created earlier are checked sooner. This way, when the busy hours are over, or the job request rate is decreasing, new jobs are assigned to older VMs and the younger ones would gradually become idle and asleep in turn. A host can go to sleep as soon as all of its VMs are idle. Once a VM is created, we do not delete it anymore, because it is very likely that it would be needed again soon when the job request rate increases. With this paradigm, we place the VMs on servers in a WFD fashion, and reduce the number of servers used and the amount of energy consumed without migrating any VM. We elasticize the data center by adding and removing VMs from the service. In other words, we divided the VMs into two parts: (1) the minimum number of VMs which are almost always running, because the load seldom gets lower than the sum of their capacities, and (2) additional VMs, which are added to the service as the load increases, and removed again when the load decreases. The minimum number of VMs are created initially and located on the servers according to WFD algorithm. The additional VMs are located on the hosts later on in our aforementioned four steps with a policy very close to WFD. Algorithm 2 provides the code of the job scheduler unit. In the next section, we explain how VMs calculate their maximum request arrival rate Algorithm 2. The job scheduler unitVM, a list of all VMs in the order of their age, stating whether they can accept more jobs or not,Input: ctive_hosts, a list of active hosts,   asleeping_hosts, a list of sleeping hosts.   Output: A suitable VM for the new job. 1: selected_vm = nullfor each vm inVMdo 2: if report(vm) = = can_accept_more_jobs then 3:   4:   selected_vm = vm 5:   break;end if 6:  end for 7: if selected_vm = = null then 8:  9:    largest_idle_vm = nullfor each host inactive_hostsdo 10:  for each idle_vm in host do 11:   if largest_idle_vm = = null or largest_idle_vm < idle_vm then 12:     13:     largest_idle_vm = idle_vmend if 14:    end for 15:   end for 16:   17:  selected_vm = largest_idle_vmend if 18: if selected_vm = = null then 19:  20:  largest_idle_vm = nullfor each host insleeping_hostsdo 21:  for each idle_vm in host do 22:   if largest_idle_vm = = null or largest_idle_vm < idle_vm then 23:     24:     largest_idle_vm = idle_vmend if 25:    end for 26:   end for 27:   28:  selected_vm = largest_idle_vmend if 29: if selected_vm = = null then 30:  31:  smallest_host = nullfor each host insleeping_hostsdo 32:  if smallest_host = = null or smallest_host > host then 33:    34:    smallest_host = hostend if 35:   end for 36:   37:  selected_vm = create_largest_vm end if 38: return selected_vm 39: We define a start time, an execution time, and a deadline for each job. Therefore, the waiting time is calculated by:W is as small as 100 (ms), but in non-real-time procedures, it can be greater [λ. In fact, it is very common to assume the arrival time and service time of the jobs have an exponential distribution [λ, increases, the VM gets busier and it takes more time for each job to complete. Our goal is to discover the maximum value of λ, which keeps the time jobs spend in the VM lower than W and no SLA violation occurs. Taking t as a random variable denoting the time a job spends in the VM (sojourn time), we can find a lower bound on the probability In real-time applications,  greater . Since rribution –24. As tequality Pr. Thus, the utilization factor is ρ = λ/(mμ) = λTs. According to t, which can be obtained for FCFS discipline by Eqs In space-shared policy the VM shares the CPU cores between jobs, non-preemptively . A VM wiπw is the probability of the number of requests waiting in the queue (n), being greater or equal to m. In other words, the probability that a request must wait for service, when it arrives.Where π0 is the probability of no requests waiting in the queue. Hence, πw is Erlang’s C formula. We want b, therefore by substituting Eqs And ρ with maximum order 2m + 2. For m = 2, we findb is to 1, the higher is the possibility of roots being lower than 1. Once the roots of ρ are found, the largest positive answer is chosen, and the maximum arrival rate λmax is obtained byWe solve this equation with Newton-Raphson’s method . The cloλmax. We calculated the mean execution time (Ts = 1/(mμ)) using the moving average method [That implies that in our algorithm, this VM is allowed to accept the coming jobs, as long as its arrival rate is not greater than e method .αϵ is the smoothing factor, and i is the index of the job submitted to the VM. By experiment we found that the best results are obtained when α = 0.2.Where λ and mean service time Ts = 1/μ. Therefore, the utilization factor is ρ = λ/μ = λTs. In order to find λmax, we find a lower limit on In time-shared policy, in each time slot, the VM devotes all of its processing space to the selected job. Thus, it is modeled by an M/M/1/PS queue with arrival rate PS queue , 43:μt=b for Again we consider a fixed lower limit ρ, which can be solved by Newton-Raphson’s method [λmax by Ts is acquired by s method , and thrλ and mean service time Ts = 1/(mμ), and thus, the utilization factor is ρ = λ/(mμ) = λTs. Again we need the mean and variance of the sojourn time to acquire a lower limit for M(1) and M(2) [m = 2 the variance of the sojourn time isIn the hybrid mode, in each time slot, the VM assigns each one of its cores to a different job and therefore serves more than one job in a time slot. This scenario, which is a combination of space-shared and time-shared policies, is modeled using an M/M/m/PS queue with arrival rate and M(2) . Accordin, the number of requests in the queue is zero. It has thoroughly been explained in [Where ained in  how M0 (see ) is solv(2)  + (1 − α).Ts 1:  2: ifroot = = ϕthen 3: λmax = 0 4:  else 5:  6:  max = 0for eachρ ϵ rootdo 7:  ifρ > maxthen 8:   ρ 9:    max = end if 10:   end for 11:  ifmax = = 0 then 12:  λmax = 0 13:   else 14:  λmax = ρ/Ts 15:   end if 16:  end if 17: Based on experimental analysis, we found that our proposed queuing theory methods for finding the right number of jobs in a VM with space-shared, time-shared, and hybrid job scheduling policies have a small transition time and converge to steady state fast enough, so that sufficient number of VMs are activated on time.b = 0.99, 0.9, and 0.8 for our simulation. b = 0.99, 0.9, and 0.8.To simulate and evaluate our proposed model, we interfaced CloudSim with Matlab. We used Google cluster data  that wasIt t can be seen from the diagrams that as maximum waiting time increases, energy consumption and SLA violations are both reduced. This indicates that managing real-time jobs is much harder and more complicated than non-real-time jobs. Moreover, the diagrams indicate that the hybrid policy outperforms both space-shared and time-shared policies and has lower energy consumption and fewer SLA violations compared to space-shared and time-shared policies.b = 0.99 and b = 0.9.Next, our algorithm is compared with three different energy-aware VM placement algorithms which use VM migration. Initially, VMs are randomly placed on hosts. Every five seconds, VM placement is checked and optimized using WFD, BFD, and FFD algorithms. The diagrams show that our algorithm outperforms WFD, BFD, and FFD algorithms with VM migration. The reason is that our algorithm places VMs on physical machines in a WFD fashion without migration. VM migration prolongs the execution process and raises the probability of SLA violation. Even in live migration the jobs have to stop and wait until their essential codes and data stacks are moved, and then continue their execution. Moreover, VM migration introduces a burst to the network and causes even more latency. Thus, especially for real-time applications, VM migration brings about SLA violation. In addition, some additional amount of energy is consumed by the network to migrate the VMs. Therefore, the energy consumption and SLA violation of the three algorithms with VM migration degrades and our energy-aware non-migration VM placement excels them. Corresponding to any SLA violation level, the energy consumption of our method is better than that of dynamic FFD, BFD, and WFD methods. So, our hybrid algorithm presents lower energy consumptions and fewer SLA violations and outperforms the three dynamic algorithms. We also found that the energy consumption of the three dynamic algorithms is highly affected by the number of migrations. Therefore, since the number of migrations of the three dynamic algorithms does not noticeably change as the waiting time rises, in λmax by studying the sojourn time of the jobs. λmax was used to determine whether a VM can accept more jobs or not. An energy-aware non-migration algorithm was designed to place the VMs on servers in a WFD fashion.In this paper, an energy-aware non-migration VM placement and hybrid job scheduling policy was proposed to make the cloud elastic. Generally, job scheduling policies can be classified into two main groups, namely, space-shared and time-shared. Moreover, some algorithms, which we call hybrid, combine the two aforementioned algorithms. Proposed methods which completely avoid VM migration, usually require the workload to be predictable. However, in our approach, the workload do not necessarily need to be predictable. In addition, our method is applicable for both real-time and non-real-time applications. We showed in our paper that hybrid policy excels space-shared and time-shared policies in terms of reducing both energy consumption and SLA violation. We adjusted the resources allocated to each cloud user according to their QoS by modeling the VMs of a data center with a queuing theory technic. We applied our idea to space-shared, time-shared, and hybrid policies, developed different equations for each one, and obtained the maximum acceptable arrival rates We interfaced CloudSim with Matlab to simulate and evaluated our algorithm. The results show that as waiting time increases, both energy consumption and SLA violation decrease. This clearly shows that although real-time jobs are not minded in previous research deservedly, managing real-time jobs are much more complicated than non-real-time jobs. The experimental results indicate that the hybrid policy is superior to space-shared and time-shared policies in reducing both energy consumption and SLA violation. Moreover, we compared hybrid algorithm with three different algorithms, WFD, BFD, FFD with migration. The results show that our energy-aware non-migration VM placement algorithm with hybrid job scheduling policy outperforms all three in terms of reducing both energy consumption and SLA violation.S1 Data(ZIP)Click here for additional data file.


To quantify coherence, two proper measures were introduced in the literature, the one is the relative entropy of coherence  Quantum computing utilizes the superposition and entanglement of quantum states to operate and process information. Its most significant advantage lies in the parallelism of operations ,2,3. To In 2014, Baumgratz et al.  proposedoherence , fidelitoherence , robustnoherence , geometroherence , coherenoherence , relativoherence , trace doherence ,23,24.In addition, some research related to quantum coherence have been developed, including quantum coherence and quantum correlations ,28,29,30In this paper, we discuss some inequalities on the measures of quantum coherence. The organization of this paper is as follows: In In this section, we give a review of some fundamental notions about quantification of coherence, such as incoherence states, incoherence operations, and measures of coherence.H be a d-dimensional Hilbert space, whose elements are denoted by the Dirac notations H. The adjoint operator of an operator T in H is denoted by I. We use H, whose elements are said to be the states of the quantum system S described by H. Fixed an orthonormal basis (ONB) H, a state S is said to be incoherent with respect to (w.r.t.) the basis e if coherent w.r.t. e. Let S that are incoherent w.r.t. e, that is,Let e if and only if e, i.e.,e if and only if it can not be written as a diagonal matrix under this basis.For every According to , a lineae-incoherent operation (IO) if it has Kraus operators A CPTP map e-incoherent Kraus operators of In this case, we call In order to measure coherence, Baumgratz et al.  presentee-incoherent operation e-incoherent operation e-incoherent Kraus operators It was proved in  that theNotably, for a bipartite quantum system Moreover, let quations  and 4),,ρA:=trB,,ρABC satded thatλS≥ows thatCr(ρABC)≤n-partite state As a generalization of inequalities  and 13)13), we cTheorem 1.For any state where Proof. To prove that the first inequality in Equation  holds, wN-partite state Next, let us prove that the second inequality in  holds byality in  holds foEquation , we knowEquation  is proveAs immediate application of Theorem 1, we have the following corollaries.Corollary 1.Let Corollary 2.Let It is remarkable that the equalities in Equation  may holdEquation  also becEquation  are tighIn this section, we discuss the relation between t state ≤Cℓ1 proved ≤Cℓ1Proposition 1.Let ρ be a state of S satisfying Equation  and let Proof. The convexity of d-dimensional system, it holds thatRana et al. proved in (, Proposiived in (, Equation-partite quantum states In this paper, we have established lower and upper bounds for relative entropy of coherence