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1.
Cureus ; 14(2), 2022.
Article in English | ProQuest Central | ID: covidwho-1835727

##### ABSTRACT

IntroductionThe COVID-19 pandemic resulted in a transition to a virtual format for all medical residency and fellowship application processes. Previous studies have discussed the successful implementation of virtual interviews, but a deep analysis of how the application process has changed for orthopedic surgery fellowship programs during the pandemic is lacking. The purpose of this study was to assess how COVID-19 impacted the orthopedic spine fellowship application and selection process.MethodsA web-based survey was administered to the program directors of all 75 U.S. orthopedic surgery spine fellowship programs, which often can accept both orthopedic surgery and neurosurgery trained graduates. Questions focused on the changes from the 2019-2020 application cycle to the 2020-2021 cycle. We collected data on connecting with potential applicants, the general application process, and interviews offered by programs. Univariate analyses were used to compare data from the 2020-2021 cycle with the prior 2019-2020 cycle.ResultsTwenty-five of the 75 contacted program directors responded to our survey (33% response rate). The percentage of programs that offered virtual open houses/meet-and-greets increased from 20% in 2019-2020 to 52% in 2020-2021 (p=0.018). Social media use was unchanged (0.0% vs. 4.0%, p>0.05). Compared to the prior year, the number of interviews offered by programs increased by 1.5 (32.7 vs. 21.9 interviews, p=0.024). There were no significant differences in the numbers of applications received by programs, interview dates available, or separate interviews each candidate completed during an interview day (p>0.05 for all). The in-person interview was the most important factor in 2019-2020 for selecting applicants, whereas the virtual interview, letters of recommendation (LOR), and research were equally ranked as the most important factors in 2020-2021. Regarding interviews, 50% of respondents would “likely” consider virtual interviews as an option in addition to in-person interviews in the future, but most (55%) answered that it was “unlikely” that virtual interviews would entirely replace in-person interviews.ConclusionSpine fellowship programs were more likely to use virtual social events to recruit potential applicants, send out more interview invitations, and equally consider LOR and research with interview performance during an entirely virtual application cycle. Half of the program directors would consider offering virtual interviews as an option for future application cycles, which may help reduce costs associated with the process.

2.
Infectious Diseases of Poverty ; 11(32), 2022.
Article in English | CAB Abstracts | ID: covidwho-1833357

##### ABSTRACT

Background: Nucleic acid test (NAT) could effectively control the spread of COVID-19 caused by large-scale sports competitions. However, quantitative analysis on the appropriate frequency of NAT is scarce, and the cost-effectiveness and necessity of high-frequency NAT remain to be fully explored and validated. This study aims to optimize the COVID-19 surveillance strategies through cost-effectiveness analysis for the Tokyo 2020 Olympic Games and the upcoming Beijing 2022 Olympic Winter Games.

3.
Security and Communication Networks ; 2022, 2022.
Article in English | ProQuest Central | ID: covidwho-1832698

##### ABSTRACT

Robot Operating System (ROS) has received widespread utilization with the development of robotics, self-driving, etc., recently. Meanwhile, the other technology blockchain is frequently applied to various fields with its trustworthy characteristics and immutability in data storage. However, ROS has no ability to interact with the blockchain, which hinders research in related fields. Therefore, we wonder if we can develop a convenient tool to bridge ROS and blockchain. Inspired by this, we propose ROS-Ethereum. It bridges ROS and Ethereum, a widely used blockchain platform. ROS-Ethereum is based on the User Datagram Protocol (UDP) communication mechanism and the SM algorithm family along with Ethereum technology. Simply put, ROS-Ethereum allows users to invoke the contract when interacting with the blockchain, which makes this process easier and safer. We conduct experiments in real robots to verify the effectiveness of ROS-Ethereum and evaluate it from the following metrics: (1) the encryption efficiency and stability of the algorithm and (2) ROS-Ethereum transaction response time and packet loss rate.

4.
BMC Psychiatry ; 22(1):275-275, 2022.
Article in English | BioMed Central | ID: covidwho-1799113
5.
Cureus ; 14(2): e22630, 2022 Feb.
Article in English | MEDLINE | ID: covidwho-1761154

##### ABSTRACT

Introduction The COVID-19 pandemic resulted in a transition to a virtual format for all medical residency and fellowship application processes. Previous studies have discussed the successful implementation of virtual interviews, but a deep analysis of how the application process has changed for orthopedic surgery fellowship programs during the pandemic is lacking. The purpose of this study was to assess how COVID-19 impacted the orthopedic spine fellowship application and selection process. Methods A web-based survey was administered to the program directors of all 75 U.S. orthopedic surgery spine fellowship programs, which often can accept both orthopedic surgery and neurosurgery trained graduates. Questions focused on the changes from the 2019-2020 application cycle to the 2020-2021 cycle. We collected data on connecting with potential applicants, the general application process, and interviews offered by programs. Univariate analyses were used to compare data from the 2020-2021 cycle with the prior 2019-2020 cycle. Results Twenty-five of the 75 contacted program directors responded to our survey (33% response rate). The percentage of programs that offered virtual open houses/meet-and-greets increased from 20% in 2019-2020 to 52% in 2020-2021 (p=0.018). Social media use was unchanged (0.0% vs. 4.0%, p>0.05). Compared to the prior year, the number of interviews offered by programs increased by 1.5 (32.7 vs. 21.9 interviews, p=0.024). There were no significant differences in the numbers of applications received by programs, interview dates available, or separate interviews each candidate completed during an interview day (p>0.05 for all). The in-person interview was the most important factor in 2019-2020 for selecting applicants, whereas the virtual interview, letters of recommendation (LOR), and research were equally ranked as the most important factors in 2020-2021. Regarding interviews, 50% of respondents would "likely" consider virtual interviews as an option in addition to in-person interviews in the future, but most (55%) answered that it was "unlikely" that virtual interviews would entirely replace in-person interviews. Conclusion Spine fellowship programs were more likely to use virtual social events to recruit potential applicants, send out more interview invitations, and equally consider LOR and research with interview performance during an entirely virtual application cycle. Half of the program directors would consider offering virtual interviews as an option for future application cycles, which may help reduce costs associated with the process.

6.
Signal Transduct Target Ther ; 7(1): 62, 2022 02 25.
Article in English | MEDLINE | ID: covidwho-1758179
7.
Lancet Infect Dis ; 22(4): 447-448, 2022 Apr.
Article in English | MEDLINE | ID: covidwho-1757994
8.
Infect Dis Poverty ; 11(1): 32, 2022 Mar 18.
Article in English | MEDLINE | ID: covidwho-1745422

##### ABSTRACT

BACKGROUND: Nucleic acid test (NAT) could effectively control the spread of COVID-19 caused by large-scale sports competitions. However, quantitative analysis on the appropriate frequency of NAT is scarce, and the cost-effectiveness and necessity of high-frequency NAT remain to be fully explored and validated. This study aims to optimize the COVID-19 surveillance strategies through cost-effectiveness analysis for the Tokyo 2020 Olympic Games and the upcoming Beijing 2022 Olympic Winter Games. METHODS: A total of 18 scenarios were designed regarding the NAT frequency, symptom monitoring, and strengthening close-contact control. An agent-based stochastic dynamic model was used to compare the cost-effectiveness of different NAT scenarios and optimize the surveillance strategies. The dynamics of the proposed model included the arrival and departure of agents, transmission of the disease according to Poisson processes, and quarantine of agents based on regular NATs and symptom onset. Accumulative infections, cost, and incremental cost-effectiveness ratio (ICER) were simulated in the frame of the model. ICER was used to compare the cost-effectiveness of different scenarios. Univariate sensitivity analysis was performed to test the robustness of the results. RESULTS: In Scenario 16, where the competition-related personnel (CRP) received NAT daily and national sports delegation (NSD) with quarantined infections accepted an additional NAT daily, accumulative infection was 320.90 (90 initial infections), the total cost was (United States Dollar) USD 8 920 000, and the cost of detecting out each infection was USD 27 800. Scenario 16 would reduce the total cost by USD 22 570 000 (avoid 569.61 infections), USD 1 420 000 (avoid 47.2 infections) compared with Scenario 10 (weekly NAT, strengthened close contact control) and Scenario 7 (daily NAT, no strengthened close contact control), respectively. Sensitivity analysis showed that the result was most sensitive to the change in basic reproductive number. CONCLUSIONS: High-frequency NATs such as bidaily, daily, and twice a day were cost-effective. NAT daily for CRP with strengthening close-contact control could be prioritized in defense against COVID-19 at large-scale sports competitions. This study could assist policymakers by assessing the cost-effectiveness of NAT scenarios and provide the host country with an optimal COVID-19 surveillance strategy.

##### Subject(s)
COVID-19 , Beijing , COVID-19/prevention & control , Cost-Benefit Analysis , Humans
9.
Curr Med Chem ; 2022 Mar 14.
Article in English | MEDLINE | ID: covidwho-1742087

##### ABSTRACT

Pentamidine, an FDA-approved human drug for many protozoal infections, was initially synthesized in the late 1930s and first reported to be curative for parasitosis in the 1940s. After ninety years of sometimes quiet growth, pentamidine and its derivatives have gone far beyond antibacterial agents, including but not limited to the ligands of DNA minor groove, modulators of PPIs (protein-protein interactions) of the transmembrane domain 5 of lateral membrane protein 1, and the blockers of the SARS-CoV-2 3a channel. This mini review highlights the development and applications of pentamidine and its analogs, aiming to provide insights for further developing pentamidine derivatives in the following decades.

10.
EuropePMC; 2021.
Preprint in English | EuropePMC | ID: ppcovidwho-325041

##### ABSTRACT

Purpose: The epidemiological and clinical features, pathogenesis, and complications of patients with COVID-19 in the acute phase have been well described, but the long-term prognosis and rehabilitation of the patients remain largely unknown. Methods: 39 COVID-19 patients were followed up at 3, 6, and 12 months after discharge. The mental health, pulmonary function, exercise capacity, and quality of life were assessed by Self-Depression Scale and the Self Anxiety Scale, pulmonary function test, 6MWT, 36-Item Short-Form, respectively. Results: Total 33 survivors completed the assessment, 40.8 ± 0.8 years, body mass index= 22.7 ±1.3 kg/m 2 . The length of hospital stays was 19.6 ± 6.6 d. One year after discharge, the mean scores of SDS and SAS showed decreasing trends from 3-months to 12-months post-discharge. 6 patients (18.2%) had FVC <80% of the predicted value, 12 patients (36.4%) had FEV1 <80% of the predicted value. And 9 (27.3%), 3 (9.1%), and 2 (6.1%) of the patients showed reduced FEF25, FEF50, and FEF75 (<70% expected values), respectively. The mean 6MWD values increased significantly from 397±25.4 m at 3-months to 514±40.8 m at 12-months. Conclusions: the impaired pulmonary function in mild COVID-19 survivors was noted after 12 months discharging from hospital. The exercise capacity, mental status, and health status were lower than those of the normal population.

11.
EuropePMC; 2020.
Preprint in English | EuropePMC | ID: ppcovidwho-325005

##### ABSTRACT

Background: At present, the epidemic of the novel coronavirus disease 2019 (COVID-19) has quickly engulfed the world. Inflammatory cytokines are associated with the severity and outcomes of patients with COVID-19. However, the effects of pro-inflammatory factors in cancer patients with COVID-19 are unknown. Methods: A multi-center, retrospective, cross-sectional study, based on 5 designated tertiary hospitals for the treatment of COVID-19 in Hubei Province, China. 112 cancer patients with COVID-19, and 105 COVID-19 patients without cancer were enrolled in the study between January 1 st , 2020 and April 30 th , 2020. The risk assessment of pro-inflammatory factors for disease severity and clinical adverse outcomes was identified by univariable and multivariable logistic regression models. Results: Of the 112 cancer patients with COVID-19, 40 (35.7%) patients were in critical condition and 18 (16.1%) patients died unfortunately. Univariate and multivariate analysis demonstrated that hemoglobin count and pro-inflammatory neutrophil and C-reactive protein, can be used as independent factors affecting the severity of COVID-19;Meanwhile, pro-inflammatory neutrophils and C-reactive protein can be used as an independent influencing factor for adverse clinical outcome. Moreover, the dynamic changes of neutrophils and C-reactive protein were also presented, and compared with COVID-19 patients without cancer, cancer patients with COVID-19 showed higher neutrophil counts and C-reactive protein levels. Conclusions: In cancer patients with COVID-19, the significant increase in pro-inflammatory neutrophil and C-reactive protein indicated a more critical illness and adverse clinical outcome, and pro-inflammatory neutrophils and C-reactive protein played a more adverse effect compare with COVID-19 patients without cancer, which may be the cause of critical illness and adverse clinical outcomes of cancer patients with COVID-19.

12.
EuropePMC; 2020.
Preprint in English | EuropePMC | ID: ppcovidwho-325004

##### ABSTRACT

The novel coronavirus disease (COVID-19) is a highly contagious respiratory disease that was first detected in Wuhan, China in December 2019, and has since spread around the globe, claiming more than 69,000 lives by the time this protocol is written. It has been widely acknowledged that the most effective public policy to mitigate the pandemic is \emph{social and physical distancing}: keeping at least six feet away from people, working from home, closing non-essential businesses, etc. There have been a lot of anecdotal evidences suggesting that social distancing has a causal effect on disease mitigation;however, few studies have investigated the effect of social distancing on disease mitigation in a transparent and statistically-sound manner. We propose to perform an optimal non-bipartite matching to pair counties with similar observed covariates but vastly different average social distancing scores during the first week (March 16th through Match 22nd) of President's \emph{15 Days to Slow the Spread} campaign. We have produced a total of $302$ pairs of two U.S. counties with good covariate balance on a total of $16$ important variables. Our primary outcome will be the average observed illness collected by Kinsa Inc. two weeks after the intervention period. Although the observed illness does not directly measure COVID-19, it reflects a real-time aspect of the pandemic, and unlike confirmed cases, it is much less confounded by counties' testing capabilities. We also consider observed illness three weeks after the intervention period as a secondary outcome. We will test a proportional treatment effect using a randomization-based test with covariance adjustment and conduct a sensitivity analysis.

13.
EuropePMC; 2020.
Preprint in English | EuropePMC | ID: ppcovidwho-325003

##### ABSTRACT

This paper evaluates the real-time forecast performance of alternative Bayesian Vector Autoregressive (VAR) models for the Australian macroeconomy. To this end, we construct an updated vintage database and estimate a set of model specifications with different covariance structures. The results suggest that a large VAR model with 20 variables tends to outperform a small VAR model when forecasting GDP growth, CPI inflation and unemployment rate. We find consistent evidence that the models with more flexible error covariance structures forecast GDP growth and inflation better than the standard VAR, while the standard VAR does better than its counterparts for unemployment rate. The results are robust under alternative priors and when the data includes the early stage of the COVID-19 crisis.

14.
EuropePMC; 2021.
Preprint in English | EuropePMC | ID: ppcovidwho-325002

##### ABSTRACT

Social distancing is widely acknowledged as an effective public health policy combating the novel coronavirus. But extreme social distancing has costs and it is not clear how much social distancing is needed to achieve public health effects. In this article, we develop a design-based framework to make inference about the dose-response relationship between social distancing and COVID-19 related death toll and case numbers. We first discuss how to embed observational data with a time-independent, continuous treatment dose into an approximate randomized experiment, and develop a randomization-based procedure that tests if a structured dose-response relationship fits the data. We then generalize the design and testing procedure to accommodate a time-dependent, treatment dose trajectory, and generalize a dose-response relationship to a longitudinal setting. Finally, we apply the proposed design and testing procedures to investigate the effect of social distancing during the phased reopening in the United States on public health outcomes using data compiled from sources including Unacast, the United States Census Bureau, and the County Health Rankings and Roadmaps Program. We rejected a primary analysis null hypothesis that stated the social distancing from April 27, 2020, to June 28, 2020, had no effect on the COVID-19-related death toll from June 29, 2020, to August 2, 2020 (p-value < 0.001), and found that it took more reduction in mobility to prevent exponential growth in case numbers for non-rural counties compared to rural counties.

15.
EuropePMC; 2020.
Preprint in English | EuropePMC | ID: ppcovidwho-323885

##### ABSTRACT

Objective: To conduct a meta-analysis of current studies that examined sex differences in severity and mortality in patients with COVID-19, and identify potential mechanisms underpinning these differences. Methods: We performed a systematic review to collate data from observational studies examining associations of sex differences with clinical outcomes of COVID-19. PubMed, Web of Science and four preprint servers were searched for relevant studies. Data were extracted and analyzed using meta-analysis where possible, with summary data presented otherwise. Publicly available bulk RNA sequencing (RNA-seq), single-cell RNA sequencing (scRNA-seq), and chromatin immunoprecipitation sequencing (ChIP-seq) data were analyzed to explore the potential mechanisms underlying the observed association. Results: 39 studies met inclusion criteria, representing 77932 patients, of which 41510 (53.3%) were males. Men were at a markedly increased risk of developing severe cases compared with women. Furthermore, the pooled odds ratio (OR) of mortality for male group compared with the female group indicated significant higher mortality rate for male. Data from scRNA-seq suggest that men have a higher amount of ACE2-expressing pulmonary alveolar type II cells than women. Sex-based immunological differences exist. The expression of androgen receptor (AR) is positively correlated with ACE2, and there is evidence that AR may directly regulate the expression of ACE2. Conclusions: This meta-analysis detected an increased severity and mortality rate in the male populations with COVID-19, which might be attributable to the sex-based differences in cellular compositions and immunological microenvironments of the lung. The host cell receptor ACE2 is likely regulated by AR signaling pathway, which is identified as a potential target for prevention and treatment of SARS-Cov-2 infections in men.

16.
EuropePMC; 2021.
Preprint in English | EuropePMC | ID: ppcovidwho-321371

##### ABSTRACT

Background The long-term consequences of human umbilical cord-derived mesenchymal stem cell (UC-MSC) treatment for COVID-19 patients are yet to be reported. This study assessed the 1-year outcomes in patients with severe COVID-19, who were recruited in our previous UC-MSC clinical trial.Methods: In this prospective, longitudinal, cohort study, 100 patients enrolled in our phase 2 trial were prospectively followed up at 3-month intervals for 1 year to evaluate the long-term safety and effectiveness of UC-MSC treatment. The primary endpoint was an altered proportion of whole-lung lesion volumes measured by high-resolution CT. Other imaging outcomes, 6-minute walking distance (6-MWD), lung function, plasma biomarkers, and adverse events were also recorded and analyzed. This trial was registered with ClinicalTrials.gov (NCT04288102).Findings: Within 3 months, MSC administration exerted numerical improvement in whole-lung lesion volume compared with the placebo, leading to a significant difference of −10.82% (95% CI: −20.69%, −1.46%, P=0.030) on day 10. MSC also reduced the proportion of solid component lesion volume compared with the placebo at each follow-up point, with a significant difference of − 9.02% (95%CI: − 17.44%, − 0.10%, P=0.045) at month 9. More interestingly, 17.86% (10/56) of patients in the MSC group had normal CT images at month 12 ( P= 0.013), but none in the placebo group. The incidence of symptoms was lower in the MSC group than in the placebo group at each follow-up time, particularly sleep difficulties at month 3 (OR 0.19, 95% CI 0.07,0.50;P=0.001), and usual activity at month 12 (OR 0.15, 95% CI 0.03,0.79;P=0.018). Neutralizing antibodies were all positive, with a similar median inhibition rate (61.6% vs. 67.55%) in both groups at month 12. No difference in adverse events at the 1-year follow-up and tumor markers at month 12 were observed between the two groups.Interpretation: UC-MSC administration achieves a long-term benefit in the recovery of lung lesions and symptoms in COVID-19 patients.Trial Registration: This trial was registered with ClinicalTrials.gov (NCT04288102).Funding The National Key R&D Program of China, the Innovation Groups of the National Natural Science Foundation of China, and the National Science and Technology Major Project.Declaration of Interest: None to declare. Ethical Approval: This study was approved by the Ethics Committee of the Fifth Medical Center, Chinese PLA General Hospital (2020-013-D).

17.
EuropePMC; 2020.
Preprint in English | EuropePMC | ID: ppcovidwho-321366

##### ABSTRACT

Background: Treatment of severe Corona Virus Disease 2019 (COVID-19) is challenging. We performed a phase 2 trial to assess the efficacy and safety of human umbilical cord-mesenchymal stem cells (UC‑MSCs) to treat patients with severe COVID-19 with lung damage, based on our phase 1 data.Methods: In this randomised, double-blind, and placebo-controlled trial, we recruited 101 eligible patients with severe COVID-19 with lung damage aged between 18–74 years from two hospitals. Enrolled patients were randomly assigned at a 2:1 ratio to receive either UC-MSCs (4 × 107 cells per infusion) or placebo on day 0, 3, and 6. We excluded patients with malignant tumours, shock, or other organ failure. The primary endpoint was an altered proportion of whole lung lesion areas from baseline to day 28, measured by chest computed tomography. Other imaging outcomes, 6-minute walk test, maximum vital capacity, diffusing capacity, plasma biomarkers, and adverse events were recorded and analysed. Primary analysis was done in the modified intention-to-treat (mITT) population and safety analysis was done in all patients who started their assigned treatment. Findings: From March 5, 2020, to March 28, 2020, 100 patients were finally enrolled and received either UC-MSCs (n = 65) or placebo (n = 35). During follow-up, the patients receiving UC-MSCs exhibited a trend of numerical improvement in whole lung lesions from baseline to day 28 compared with the placebo cases. UC-MSCs administration significantly reduced the proportions of consolidation lesions from baseline to day 28 in the treated patients compared with the placebo subjects. The 6-minute walk test showed an increased distance in patients treated with UC-MSCs. Notably, UC-MSCs delivery was well tolerated, with no serious adverse events.Interpretation: UC-MSCs treatment is a safe and potentially effective therapeutic approach for patients with severe COVID‑19. The trial suggests that UC-MSCs administration might benefit patients with COVID-19 with lung damage at the convalescent stage as well as the progression stage.Trial Registration: This trial is registered with ClinicalTrials.gov, number NCT04288102.Funding Statement: This trial was supported by The National Key R&D Program of China (2020YFC0841900, 2020YFC0844000, 2020YFC08860900);The Innovation Groups of the National Natural Science Foundation of China (81721002);The National Science and Technology Major Project (2017YFA0105703).Declaration of Interests: All authors declare no competing interests.Ethics Approval Statement: Ethical approval was obtained from the institutional review boards of each participating hospital. Written informed consent was obtained from all the enrolled patients or their legal representatives if they were unable to provide consent.

18.
EuropePMC; 2020.
Preprint in English | EuropePMC | ID: ppcovidwho-318163

##### ABSTRACT

Outbreaks of coronavirus disease 2019 (COVID-19) have been recorded in different countries across the globe. The virus is highly contagious, hence early detection, isolation, and quarantine of infected patients will play an important role in containing the viral spread. Diagnosis in a mobile lab can aid to find infected patients in time. Here, we develop a field-deployable diagnostic workflow that can reliably detect COVID-19. Instruments used in this workflow could easily fit in a mobile cabin hospital and also be installed in the community. Different steps from sample inactivation to detection were optimized to find the fastest steps and portable instruments in detection of COVID-19. Each step was compared to that of the normal laboratory diagnosis set-up. From the results, our proposed workflow (80 min) was two times faster compared to that of the normal laboratory workflow (183 min) and a maximum of 32 samples could be detected at each run. Additionally, we showed that using 1% Rewocid WK-30 could inactivate the novel coronavirus directly without affecting the overall detection results. Comparison of our workflow using an in-house assay to that of a commercially acquired assay produced highly reliable results. From the 250 hospital samples tested, there was a high concordance 247/250 (98.8%) between the two assays. The in-house assay sensitivity and specificity were 116/116 (100%) and 131/134 (97.8%) compared to that of the commercial assay. Based on these results, we believe that our workflow is fast, reliable, adaptable and most importantly, field deployable.

19.
Frontiers in cardiovascular medicine ; 8, 2021.
Article in English | EuropePMC | ID: covidwho-1679286

##### ABSTRACT

Coronary artery disease (CAD) is a major contributor to morbidity and mortality worldwide. Myocardial ischemia may occur in patients with normal or non-obstructive CAD on invasive coronary angiography (ICA). The comprehensive evaluation of coronary CT angiography (CCTA) integrated with fractional flow reserve derived from CCTA (CT-FFR) to CAD may be essential to improve the outcomes of patients with non-obstructive CAD. China CT-FFR Study-2 (ChiCTR2000031410) is a large-scale prospective, observational study in 29 medical centers in China. The primary purpose is to uncover the relationship between the CCTA findings (including CT-FFR) and the outcome of patients with non-obstructive CAD. At least 10,000 patients with non-obstructive CAD but without previous revascularization will be enrolled. A 5-year follow-up will be performed. The primary endpoint is the occurrence of major adverse cardiovascular events (MACE), including all-cause mortality, non-fatal myocardial infarct, unplanned revascularization, and hospitalization for unstable angina. Clinical characteristics, laboratory and imaging examination results will be collected to analyze their prognostic value.

20.
Signal Transduct Target Ther ; 7(1): 43, 2022 02 09.
Article in English | MEDLINE | ID: covidwho-1683983


text:
# Math Help - Question on Trig Functions and Angles

1. ## Question on Trig Functions and Angles

I'm having trouble figuring out what to do on these two questions.

1. A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 21 degrees and 23 degrees.
How high (in feet) is the ballon?

I don't under stand how to figure this one out without the distances of the base.

2.For find the values of the trigonometric functions based on the given one.
Note: The answer must be given as a fraction. NO DECIMALS.

If then
=

I'm having trouble figuring out what to do on these two questions.

1. A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 21 degrees and 23 degrees.
How high (in feet) is the ballon?

I don't under stand how to figure this one out without the distances of the base.

2.For find the values of the trigonometric functions based on the given one.
Note: The answer must be given as a fraction. NO DECIMALS.

If then
=

Uhm...well I am not sure but you might have copied this down wrong but $arcse\bigg(\frac{7}{5}\bigg)=arcos\bigg(\frac{5}{7 }\bigg)$

Which does not have an exact value answer...so there would be decimals unless you are saying like $csc\bigg(arcsec\bigg(\frac{7}{5}\bigg)\bigg)=\frac {7\sqrt{6}}{12}$?

I'm having trouble figuring out what to do on these two questions.

1. A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 21 degrees and 23 degrees.
How high (in feet) is the ballon?

I don't under stand how to figure this one out without the distances of the base.

2.For find the values of the trigonometric functions based on the given one.
Note: The answer must be given as a fraction. NO DECIMALS.

If then
=

For the first question, use $tan(\alpha) = \frac{h}{d}$ and $tan(\beta) = \frac{h}{d+5280}$. Then you have:
$d tan(\alpha) = (d+5280) tan(\beta)$ and you can solve for d and then h.

4. I don't know this tan a and b thing you are showing me for the first question.

For the second question I copied it down right. I figured out that cos = 5/7. The rest i don't see how you find

format: plain


text:
# What is the largest possible mass for a stable planet made of iron?

Consider a large spherical planet made of pure iron. Think of something similar to Mercury or Earth without its mantle, only much bigger, though those planets have elements other than iron mixed in their core.

What is the highest possible mass such a planet could have while still being stable? Can you tell the approximate radius it would have (if this is meaningful), and the pressure at the center?

If you tried to make a planet with mass larger than that, what would stop you?

I can think of two possible obstacles that would stop a heavier iron planet to form.

• The gravity of the planet cannot keep all its mass, so some of the iron will leak away to open space.
• The planet (or at least its center) will collapse under its own pressure, and undergo some sort of phase change so it is no longer made of liquid iron, but of something else: plasma or some other exotic phase of matter made of mostly iron nucleuses, matter made of fused or split atoms other than iron, neutron degenerate matter, other degenerate matter not containing nucleuses, black hole.

I am specifically not asking how heavy a planet could form in nature, but rather, how heavy a planet would be stable if it somehow got formed.

I did not specify a temperature for the planet. Imagine the planet to orbit around a star or as a free planet, whichever is convenient for you, and choose a realistic equilibrium temperature it could attain from radiation in that case. The iron ball shall not be a core inside a star or gas planet.

The planet shall rotate slowly enough that the rotation cannot significantly reduce the gravitational acceleration on its surface.

• How could such a planet form? – HDE 226868 Jul 28 '15 at 22:01
• I don't understand the process you imagine by which iron would "leak away". The larger the planet, the stronger the gravitational field at the surface. – Floris Jul 28 '15 at 22:25
• en.wikipedia.org/wiki/Chandrasekhar_limit – Count Iblis Jul 28 '15 at 22:30
• Gravity increases with mass, so I do not think that such an event would occur. – Jimmy360 Jul 28 '15 at 22:36
• @WetSavanna: no, I'm not asking for the most stable form, but instead about the phase that's only one step more stable than an iron planet. There are likely multiple steps between an iron planet and a black hole. – b_jonas Jul 28 '15 at 22:37

You can create a massive sphere of cold iron up to about 1.1 solar masses that could be supported by electron degeneracy pressure.

The exact point of instability is likely to be caused by inverse beta decay reducing the electron number density (it could also be modified upwards slightly by extremely rapid rotation). [NB: This is lower than the "Chandrasekhar mass" commonly quoted for white dwarfs because (i) there are more mass units per electron in iron than for the usual carbon/oxygen assumed to be in most white dwarfs, and this reduces the degeneracy pressure for a given density; and (ii) more importantly, inverse beta decay sets in at much lower densities for iron than in carbon.]

The structure of such objects is studied in detail by Rotondo et al. (2011). The instability occurs at a finite density and hence radius, which is about 2200 km (the exact value is uncertain). The central density would be about $1.1\times 10^{12}$ kg m$^{-3}$ (this is where the electron Fermi energy equals the threshold energy for inverse beta decay in iron). The central pressure is essentially ideal relativistic electron degeneracy pressure - which I calculate to be about $5\times 10^{25}$ Pa. The temperature of the object would not really matter - the electrons would be completely degenerate unless it substantially exceeds $10^{8}$ K (e.g. see this link, setting the log density to 12 and the mass units per electron $\mu_e$ to $\sim 2.2$.)

Thus what you ask about is not in the planetary regime at all.

If you did try to add more iron to this maximal mass iron white dwarf, it would collapse. The outcome of that would likely be a low-mass neutron star.

Such spheres of iron (or iron peak elements) are created in the cores of massive stars, but are always thought to grow past the instability point and collapse, initiating a supernova explosion.

• I learnt a new expression today - "reverse beta decay". How is this different than "electron capture"? Either way - I understand the result to be "fewer electrons, more neutrons" and thus a collapse (into a neutron star, but with too much energy to be stable and hence the supernova?) – Floris Jul 30 '15 at 12:40
• @Floris Inverse beta decay. e + p -> n + nu The semantic difference between that and electron capture is that I think electron capture would normally be due to an electron that was bound to the nucleus in the first place. But they are essentially the same thing. Also called neutronisation. – Rob Jeffries Jul 30 '15 at 14:23
• Thanks for the clarification (and the correction - reminder to self: must not PSE on phone before coffee, without reading glasses...) – Floris Jul 30 '15 at 14:30


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Curiosity: Wouldn't the definition of the derivative always be 1 if it exists?

I'm pretty sure I have the wrong intuition here but I have a slight confusion about the way we could calculate the derivative at a certain point using (one of) the definition(s) of the derivative. See example bellow:

$$\frac{df(x)}{dx}= \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

let's see the case of $f(x) = \sqrt{5x+10}$

$$\frac{df(x)}{dx}=\lim_{h\to0}\frac{\sqrt{5h+5x+10}-\sqrt{5x+10}}{h}$$

If we want to calculate $f'(5)$

$$\left.\frac{df(x)}{dx}\right\rvert_5=\lim_{h\to0}\frac{\sqrt{5h+35}-\sqrt{35}}{h}$$

if we try to find the limits when $h\to0^+$:

• The numerator would be only slightly superior to 0

• The denominator would be only slightly superior to 0

$$\frac{\text{very small number above zero}} {\text{very small number above zero}}\approx 1$$

It should be the same for $h\to 0^-$

Hence: $f'(5)= 1$?

N.B: I know this result is wrong, I just want to know how the logic I used is faulty.

• Not all very small numbers are approximately equal, much like not all numbers in general are approximately equal. – Ivan Neretin Dec 13 '16 at 8:31
• $0.0000002$ is a $very\_small\_number\_above\_zero$, and so is $0.0000001$, yet $\frac{0.0000002}{0.0000001}\neq 1$. – Wojowu Dec 13 '16 at 8:32
• Dividing by small numbers makes big numbers. While both are a small number above $o.$ The denominator is more than twice as big as the numerator. – Doug M Dec 13 '16 at 8:35
• Multiply numerator an denominator by the $\sqrt {5h + 35} + \sqrt {35}$ This gets the $h$ out from under the radical. You will be able to cancel $h$ top and bottom. Then evaluate as $h$ approaches $0$ – Doug M Dec 13 '16 at 8:38
• Your calculation is a very rough approximation. You have shown that every derivative is very roughly equal to 1 -in the sense that every number is :) – Francesco Dondi Dec 14 '16 at 0:19

The intuitive approach to solving a limit, along the lines of "slightly more than zero" or "slightly less than zero" is just that - an intuitive approach. That is to say, it's a good rule of thumb that often gets you close to the right answer, but it's not actually correct. The issue is that when you have multiple expressions in play, how they synchronize is important.

To take an extremely simple example, consider $\lim_{x \to 0}\frac{2x}{x}$. $2x$ and $x$ are both "very small numbers" when $x$ is very small - but $x$ gets small twice as fast as $2x$ does. At any given instant, $\frac{2x}{x}$ will in fact always be $2$, so the limit is just $2$.

The key idea here is that the definition of the limit is what drives it. The definition of a limit states that $\lim_{x \to a}f(x) = L$ if and only if for every $\epsilon > 0$ there is a $\delta > 0$ so that whenever $|x - a| < \delta$, $|f(x) - L| < \epsilon$. What your example demonstrates is that the intuitive idea of replacing pieces of $f$ with "very small positive numbers" is not an accurate reflection of this definition.

• 'a good rule of thumb that often gets you close to the right answer' - In this case, it provides an estimate with error less than (or equal to) infinity. Quick and dirty, but I'll take it! – ColinK Dec 13 '16 at 17:19
• This rule of thumb is useful to compare a number "slightly more than zero" with a number "clearly far away from zero". – Pere Dec 14 '16 at 9:13

$$\frac{\sqrt{5h+35}-\sqrt{35}}{h}$$

and make $h$ a very small number, say $0.0000001$, so we have

$$\frac{\sqrt{35.0000005}-\sqrt{35}}{0.0000001}$$

$$\frac{0.0000000422577}{0.0000001}$$ i.e. about $0.422577$ rather than $1$, even though it is a very small number divided divided by a very small number. It is in fact close to $\dfrac{\sqrt{35}}{14}$, which is what the calculus would suggest is the exact derivative $f'(x)=\frac{5}{2} (5x+10)^{-1/2}$ at $x=5$

The formal error is easy to explain, but maybe the question is about misunderstanding what is being done in this kind of problem more generally, so I give a more "meta" answer.

It is the very idea of analysis: we do not know exactly the quantities which we are dealing with, so we afford to change a little the problem in order to be able to solve it, hoping it will enlighten the original problem. We are able to handle only very few problems in analysis, with specific objects and strong hypothesis, but understanding those simpler cases helps sometimes to reach some better understanding of the original objects (see mainly the problems coming from physics).

Yet, that does not mean doing whatever you thing being reasonable an intuitively defendable. Indeed, Dieudonné said that analysis was only about "bounding by below, bounding by above, approximate", and we cannot forget the main idea behind this: we have to control the remainder, simplify but knowing what we do, not forgetting what we changed and assumed, otherwise we just change the problem and never will be able to come back. This idea, I think, must remain one of the greatest lighthouse for understanding and doing analysis.

You propose that those two quantities have quotient tending to one, but you have to really compare them (it is the meaning of doing a quotient), and saying "small positive number" is just blinding yourself, as if for treating basic number theory with small numbers, you just have a hike to a new spot like $10^{80}$ and say "well, there is not any problem, all those numbers seems to be $1$, hence all equations are trivial".

A simple example of why it is roughly wrong: when $x$ is a very small positive number, then so is $2x$, however you will agree that you do not want to conclude:

$$2 = \frac{2x}{x} \to_{x\to 0} 1$$

It is the very point you learn doing basic calculus like equivalents, developments and "indeterminate forms" (I do not know the word for it in english, but I mean evaluating limits of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$).

Both intuition and formalism are powerful in mathematics, and you have to make a perpetual dialogue between them: the intuition guide you sometimes where formalism would not have, and formalism is the method of walking correctly for finishing your hike, avoiding errors and getting lost, controlling validity. Sometimes it is the formalism that allows you to advance, even with a blind intuition, and intuition being the warden of validity. But if you remove one of those two pillars, as you have just done relying only on your intuition, everything quickly collapses.

• Interesting insight – Aetos Dec 13 '16 at 8:49

$\lim_{h\to0}$ doesn't really make any statement like “$h$ is a particular very small number”. Rather it considers its argument as a function of a general $h$, decides how small $h$ must be so that function becomes practically constant, and then yields that constant. In principle that doesn't require evaluating for any very small values at all, for example with $$f(x) = \begin{cases}1&\text{for }x<1\\x & \text{else}\end{cases}$$ we have $$\lim_{x\to0}f(h) = f(0.0001) = f(0.5) = f(0) = 1$$ because this function is really constant if only $x$ is smaller than 1. In practice, limits aren't usually over such functions that are constant in a whole region, but they are necessarily over continuous functions, which guarantee you that you can make the error-from-constness arbitrarily small by going to sufficiently small values; but the arguments are still always proper real numbers – no “this doesn't behave like numbers otherwise do”!

And for an ordinary “somewhat small” $h$, say $h=0.5$, you'd certainly agree that $\frac{\sqrt{5\cdot h + 35} - \sqrt{35}}{h}\neq 1$. In fact any pocket calculator will tell you that it is $\approx0.415$. If you then make $h$ yet smaller, the following will happen (on a computer with IEEE754 double precision arithmetic): \begin{align} \\ 10^{-1} &&& 0.4210786080687612 \\ 10^{-2} &&& 0.4224263146657137 \\ 10^{-3} &&& 0.4225620364017857 \\ 10^{-4} &&& 0.422575618168608 \\ 10^{-5} &&& 0.42257697643321984 \\ 10^{-6} &&& 0.42257711196924674 \\ 10^{-7} &&& 0.4225771199628525 \\ 10^{-8} &&& 0.42257708443571573 \\ 10^{-9} &&& 0.4225766403465059 \\ 10^{-10} &&& 0.4225775285249256 \\ 10^{-11} &&& 0.4225952920933196 \\ 10^{-12} &&& 0.4227729277772596 \\ 10^{-13} &&& 0.41744385725905886 \\ 10^{-14} &&& 0.4440892098500626 \\ 10^{-15} &&& 0.8881784197001251 \\ 10^{-16} &&& 0.0 \\ 10^{-17} &&& 0.0 \\ 10^{-18} &&& 0.0 \\ 10^{-19} &&& 0.0 \\ 10^{-20} &&& 0.0 \end{align}

Notice how the “moderately small” arguments give a very consistent value of 0.4225something – this corresponds to the actual “exact derivative”. But extremely small arguments suddenly give complete bogus. This is similar to your question: with extremely small numbers, the computer can't really calculate anymore (it basically runs out of digits to store the deviations in), so you then have a $0 \stackrel{?}= \frac00 \stackrel{?}= 1$ kind of situation.

Well – one could say that this is only an artifact of the floating-point processor. But IMO it gets to the very heart of how analysis works: it exploits the fact that certain functions behave in a certain regime (often close to a singularity of utter indeterminacy!) very predictable, so they can be well approximated by something simpler. That can then be used for further calculations, which would otherwise have been unfeasible. While it can be mathematically useful to describe the deviations as infinitesimally small, especially for physical applications it's actually more appropriate to say just small enough so we don't have to worry about higher-order effects, but not so small that the signal is lost in the noise floor.


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# Slope

The slope of a line in the plane is defined as the rise over the run, m = Δyx.

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line. The line may be practical - as set by a road surveyor : or in a diagram that models a road or a roof either as a description or as a plan.

The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2y1) = Δy. For relatively short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope m of the line is

$m=\frac{y_2-y_1}{x_2-x_1}.$

The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the grade m of a road is related to its angle of incline θ by

$m = \tan \theta\!$ (from the definition of tangent).

As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.

This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

## Definition

Slope illustrated for y = (3/2)x − 1. Click on to enlarge

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:

$m = \frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}}.$

(The Greek letter delta, Δ, is commonly used in mathematics to mean "difference" or "change".)

Given two points (x1,y1) and (x2,y2), the change in x from one to the other is x2x1 (run), while the change in y is y2y1 (rise). Substituting both quantities into the above equation generates the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}.$

The formula fails for a vertical line, parallel to the y axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined.

### Examples

Suppose a line runs through two points: P = (1, 2) and Q = (13, 8). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}.$

As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is

$m = \frac{ 21 - 15}{3 - 4} = \frac{6}{-1} = -6.$

## Geometry

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of −1. A vertical line's slope is undefined.

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:

$m = \tan\,\theta$

and

$\theta = \arctan\,m$

(see trigonometry).

Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes. Two lines are perpendicular if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). Also, another way to determine a perpendicular line is to find the slope of one line and then to get its reciprocal and then reversing its positive or negative sign (e.g. a line perpendicular to a line of slope  −2 is +1/2).

## Slope of a road or railway

There are two common ways to describe how steep a road or railroad is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway and rack railway. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are:

$\text{angle} = \arctan \frac{\text{slope}}{100} ,$

and

$\mbox{slope} = 100 \tan( \mbox{angle}),\,$

where angle is in degrees and the trigonometric functions operate in degrees. For example, a 100% or 1000 slope is 45°.

A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (etc.).

## Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form

$y = mx + b \,$

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.

If the slope m of a line and a point (x1,y1) on the line are both known, then the equation of the line can be found using the point-slope formula:

$y - y_1 = m(x - x_1).\!$

For example, consider a line running through the points (2,8) and (3,20). This line has a slope, m, of

$\frac {(20 - 8)}{(3 - 2)} \; = 12. \,$

One can then write the line's equation, in point-slope form:

$y - 8 = 12(x - 2) = 12x - 24 \,$

or:

$y = 12x - 16. \,$

The slope of the line defined by the linear equation

$ax + by +c = 0 \,$

is: $-\frac {a}{b} \;$.

## Calculus

At each point, the derivative is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition,

$m = \frac{\Delta y}{\Delta x}$,

is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

For example, the slope of the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at x = 32 is also 3—a consequence of the mean value theorem.)

By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that Δyx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero. Therefore, the slope of the tangent is the limit of Δyx as Δx approaches zero, or dy/dx. We call this limit the derivative.

$\frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}$


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Smooth function satisfying $f^{-1}(x)=f(x+1/2)-1/2$

Let $$f$$ be a smooth (infinitely differentiable) function satisfying $$f^{-1}(x)=f(x+1/2)-1/2$$ for all real $$x$$ (where $$f^{-1}$$ is an inverse function).

I have strong feeling that it also must satisfy $$f(x+1)=f(x)+1$$ for all real $$x$$, but cannot prove or disprove it.

I know that $$f$$ is monotone because it has inverse everywhere. Also if $$f$$ has fixed point $$p$$ (i.e. $$f^{-1}(p)=f(p)$$), then $$f(p+k/2)=p+k/2$$ where $$k$$ is any integer. But I don't know if $$f$$ even always has fixed point. In some cases $$f$$ can be calculated for all positive $$x$$ if values for $$0\le x <1/2$$ are given. I don't know how to invert that equation to get negative values.

Some examples which satisfy both relations:

• $$f(x)=x$$
• Define $$g(x)=x-\frac{\sin{2\pi x}}{4\pi}$$, then $$f(x)=2g^{-1}(x)-x$$, (basically a rotated and scaled sine wave)

Questions:

1. Does $$f$$ always satisfy $$f(x+1)=f(x)+1$$ for all real $$x$$?
2. Will $$f$$ be always increasing?
3. What else can we say about $$f$$?

Let $$g(x) = x+1/2$$. The equation can be written as $$f^{-1} = g^{-1} f g$$ Now operate with $$f$$, $$g$$ like elements of a group. We get from the above taking inverses $$g^{-1} f^{-1} g = f$$ and so $$g^{-2} f g^{2} = f$$ that is $$f(x+1) = f(x)+1$$
$$\bf{Added:}$$ If $$f$$ is continuous, then $$f$$ is monotone, and from the last equality we conclude $$f$$ strictly increasing.
$$\bf{Added:}$$ It's useful to look at graphs of the function $$f_1(x)\colon = f^{-1}(x)$$ and $$f_2(x) \colon = f(x+1/2)-1/2$$. We have $$\Gamma_{f_1} = S(\Gamma_{f})$$, and $$\Gamma_{f_2} = T(\Gamma_{f})$$, where $$S$$ is the symmetry w.r. the first bisector, and $$T$$ is the translation by the vector $$(-1/2, -1/2)$$. In other, words, the graph of $$f$$ is invariant under the transformation $$S T^{-1} = T^{-1} S$$, which is a glide reflection. Now we see there are many such functions $$f$$. $$f$$ will necessary have fixed points, and the set of fixed points is invariant under the translation $$x\mapsto x+1/2$$.
$$\bf{Added:}$$ Using the idea with the graph of functions, we see that the graph of $$f$$ can be parametrized by $$t\mapsto (t- h(t), t+ h(t))$$, where $$h$$ is a function satisfying $$h(t+1/2) = -h(t)$$. In general such functions $$h$$ are of the form $$h(t) = \sum_{k \ge 0} (a_k \cos 2(2k+1) \pi t + b_k \sin 2 (2k+1) \pi t)$$ that is a Fourier expansion with only terms of odd indexes. We want to make sure that $$t-h(t)$$ is strictly increasing, so require $$h'(t) <1$$ for all $$t$$.
• @Somnium: Yes indeed, works for all bijective functions $f$. Now it would be interesting to characterize such functions, perhaps satisfying other conditions (smooth, etc). Interesting question :-) Aug 6 at 0:17


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# zbMATH — the first resource for mathematics

The operator amenability of $$A(G)$$. (English) Zbl 0842.43004
Let $$G$$ be a locally compact group and $$A(G)$$ its Fourier algebra. The main results in this paper are: (1) $$A(G)$$ is operator amenable for every compact group $$G$$; (2) the group $$G$$ is amenable if and only if $$A(G)$$ is operator amenable; (3) the $$C^*$$-algebra $$A$$ is Banach algebra amenable if and only if $$A$$ is operator amenable.

##### MSC:
 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 22D15 Group algebras of locally compact groups
Full Text:


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# Compute expectation value of an operator in openfermion

I am trying to compute the 2-electron reduced-density matrix (2-RDM) for a given quantum state with openfermion. The code is as follow

two_rdm = np.zeros((n_qubits,) * 4, dtype=complex)
psi =  = openfermion.haar_random_vector(2 ** n_qubits, random_seed).astype(numpy.complex64)
for i, j, k, l in itertools.product(range(n_qubits), repeat=4):
transformed_operator = jordan_wigner(
FermionOperator(((i, 1), (j, 1), (k, 0), (l, 0))))
two_rdm[i, j, k, l] = openfermion.linalg.expectation(transformed_operator, psi)

The problem is with the last line, which gives the following exception

TypeError: can only concatenate str (not "ABCMeta") to str

After examning the source code, I figured out that the problem is with the [operator * state][1]. Can anybody help to fix this?

• I think one would need more information in order to answer this question. Which line is throwing that exception? What sort of object is converter? You mention the package openfermion, but it looks like you're actually using Qiskit Nature? Oct 12 at 21:01
• If I had to guess, I'd say the normalization error you see is related to init_state. But it's hard to diagnose without a minimal working example: stackoverflow.com/help/minimal-reproducible-example Oct 13 at 14:31

From the openfermion documentation (see here), it says the operator must be a scipy.sparse matrix. So, replacing transformed_operator with get_sparse_matrix(transformed_operator, n_qubits=n_qubits) in the final line will give you a 2**n_qubits x 2**n_qubits dimensional sparse matrix, which should remove the error.
• After the modification, the program output the exception dimension mismatch for the state to multiply the operator. Oct 16 at 16:19
• I've edited my answer to something that should fix that problem. In get_sparse_matrix there is an optional argument n_qubits to specify the number of qubits you are working with. Oct 16 at 17:52
• Thanks for the answers. That really helps to fix the problem! I have one question about the quantum state of openfermion. The state is a $2^n$-length vector. I want to know what is the ordering of the vector. The first one, state[0] should be '0000' for a 4-qubit state. Then, is '1000' or '0001' the state[1]? Oct 17 at 3:13


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# Circular motion of a car and tires

## Homework Statement

A car is safely negotiating an unbanked circular turn at a speed of 21 m/s. The maximum static frictional force acts on the tires. Suddenly a wet patch in the road reduces the maximum static frictional force by a factor of three. If the car is to continue safely around the curve, to what speed must the driver slow the car?

## The Attempt at a Solution

Do I convert V to angular velocity and figure out the radius? Then ?_?
Do not know how to start, can someone help. Thanks

Doc Al
Mentor
Hint: How does the required centripetal force relate to the car's speed?

Hint: How does the required centripetal force relate to the car's speed?

Centripetal force = mv^2 / r ?
But I don't have an r or m. and what do I do without a given coefficient of friction?

(Sorry i just came back to class after a sick break)

(edit)
Am i starting it right?
Fc = friction = mv^2/r

u(mg) = mv^2/r?

Doc Al
Mentor
Centripetal force = mv^2 / r ?
Right.
But I don't have an r or m. and what do I do without a given coefficient of friction?
You won't need any of those. Hint: Think ratios.
Am i starting it right?
Fc = friction = mv^2/r
Yes.

Right.

You won't need any of those. Hint: Think ratios.

Yes.

Could you look at this:

[u(mg)]/3 = mv^2 / r

I divide by 3 because it was reduced by the factor of 3 ?
I canceled out the masses and:

3.266u = 441/r

do I solve for the radius? Then consider a case 2 and plug it in?

Doc Al
Mentor
As I said earlier, you don't need to know (or try to solve for) m, r, or mu, since they don't change. The only things that change are the force and the speed. Think ratios.

Try this:
$$F_1 = m v_1^2/r$$

$$F_2 = m v_2^2/r$$

You are given $$F_1/F_2 = 3$$; your job is to figure out $$v_1/v_2$$.


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Cite as:

### Purpose and Contribution

Multilevel Regression and Poststratification (MRP) is an increasingly popular method for analyzing surveys, and can be implemented on public datasets such as the CCES and ACS. Several helpful tutorials give introductions with sample R code (Kastellec, Lax, and Phillips, 2019; Hanretty, 2019),

But despite its increasingly popularity, doing your own MRP entails considerable upfront costs specific to the data: downloading the appropriate survey and contextual, recoding values and generating post-stratification frames so the survey and census datasets can be joined. While there already exist some attempts to make a “MRP package” (e.g. gelman/mrp, stan-dev/rstanarm, kuriwaki/sparseregMRP), often these packages define general functions with specific input requirements and not enough on data cleaning and data loading in order to use those functions.

This package instead provides data loading, processing, and formatting functions for a particular task: using CCES data for MRP. Limiting its usage to a fixed (but fairly widespread) set of survey data has several benefits. Its key contributions are functions that are calibrated to a consistent syntax, pre-built lookup tables and value-key pairs of data that are based upon a careful reading of data sources, and data loading functions that use APIs (IQSS/dataverse-client-r and walkerke/tidycensus) to reduce the dependency on downloading large files. Model fitting and visualization of MRP itself is handled elsewhere (See kuriwaki/ccesMRPrun). This package is focused on the preparation to get there.

## Installation

# remotes::install_github("kuriwaki/ccesMRPprep")
library(ccesMRPprep)

## Vignettes

See the vignettes for more long-form workflow overviews (vignette("overview")) and documentation and explanation of the nature of the data (vignette("acs")). Otherwise, each function and built-in data provides documentation as well.

## Workflow

See the overview vignette (vignette("overview")) from a illustrative workflow.

## Data Sources

Function-specific pages will detail the documentation used in each function. Here is a manual compilaiton:

Information Source Citation and URL (if public)
CCES Covariates Cumulative CCES Shiro Kuriwaki, “Cumulative CCES Common Content”. https://doi.org/10.7910/DVN/II2DB6
CCES Outcomes Each Year’s CCES Stephen Ansolabehere, Sam Luks, and Brian Schaffner. “CCES Common Content” (varies by year). https://cces.gov.harvard.edu/
Poststratification Census Bureau ACS American Community Survey. Extracted via tidycensus package. See ACS vignette
District-level Contestedness and Incumbency Collected mainly by Jim Snyder
CD-level Presidential Voteshare Daily Kos Daily Kos, The ultimate Daily Kos Elections guide to all of our data sets
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Research

Fourth order elliptic system with dirichlet boundary condition

Tacksun Jung1* and Q-Heung Choi2

Author Affiliations

1 Department of Mathematics, Kunsan National University, Kunsan 573-701, Korea

2 Department of Mathematics Education, Inha University, Incheon 402-751, Korea

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Journal of Inequalities and Applications 2011, 2011:60  doi:10.1186/1029-242X-2011-60

 Received: 13 February 2011 Accepted: 17 September 2011 Published: 17 September 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the multiplicity of the solutions of the fourth order elliptic system with Dirichlet boundary condition. We get two theorems. One theorem is that the fourth order elliptic system has at least two nontrivial solutions when λk < c < λk+1 and λk+n(λk+n - c) < a + b < λk+n+1(λk+n+1 - c). We prove this result by the critical point theory and the variation of linking method. The other theorem is that the system has a unique nontrivial solution when λk < c < λk+1 and λk(λk - c) < 0, a+b < λk+1(λk+1 - c). We prove this result by the contraction mapping principle on the Banach space.

AMS Mathematics Subject Classification: 35J30, 35J48, 35J50

Keywords:
Fourth order elliptic system; fourth order elliptic equation; variation linking theorem; contraction mapping principle

1. Introduction

Let Ω be a smooth bounded region in Rn with smooth boundary ∂Ω. Let λ1 < λ2 ≤ ... ≤ λk ≤ ... be the eigenvalues of -Δ with Dirichlet boundary condition in Ω. In this paper we investigate the multiplicity of the solutions of the following fourth order elliptic system with Dirichlet boundary condition

Δ 2 u + c Δ u = a ( ( u + v + 1 ) + 1 ) in  Ω , Δ 2 v + c Δ v = b ( ( u + v + 1 ) + 1 ) in  Ω , u = 0, v = 0, Δ u = 0, Δ v = 0 on  Ω , (1.1)

where c R, u+ = max{u, 0} and a, b R are constant. The single fourth order elliptic equations with nonlinearities of this type arises in the study of travelling waves in a suspension bridge ([6]) or the study of the static deflection of an elastic plate in a fluid and have been studied in the context of the second order elliptic operators. In particular, Lazer and McKenna [6] studied the single fourth order elliptic equation with Dirichlet boundary condition

Δ 2 u + c Δ u = b ( ( u + 1 ) + 1 ) , in  Ω , u = 0 , Δ u = 0 on  Ω . (1.2)

Tarantello [10] also studied problem (1.2) when c < λ1 and b λ1(λ1 - c). She show that (1.2) has at least two solutions, one of which is a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [8] proved that if c < λ1 and b λ2(λ2 - c), then (1.2) has at least four solutions by the Leray-Schauder degree theory. Micheletti, Pistoia and Sacon [9] also proved that if c < λ1 and b λ2(λ2 - c), then (1.2) has at least three solutions by variational methods. Choi and Jung [2] also considered the single fourth order elliptic problem

Δ 2 u + c Δ u = b u + + s in  Ω , u = 0 , Δ u = 0 on  Ω . (1.3)

They show that (1.3) has at least two nontrivial solutions when c < λ1, λ1(λ1 - c) < b < λ2(λ2 - c) and s < 0 or when λ1 < c < λ2, b < λ1(λ1 - c) and s > 0. They also obtained these results by using the variational reduction method. They [3] also proved that when c < λ1, λ1(λ1 - c) < b < λ2(λ2 - c) and s < 0, (1.3) has at least three solutions by using degree theory. In [7-9] the authors investigate the existence of solutions of jumping problems with Dirichlet boundary condition.

In this paper we improve the multiplicity results of the single fourth order elliptic problem to that of the fourth order elliptic system. Our main results are as follows:

THEOREM 1.1. Suppose that ab ≠ 0 and d e t 1 1 b - a 0 . Let λk < c < λk+1 and λk+n(λk+n - c) < a + b < λk+n+1(λk+n+1 - c). Then system (1.1) has at least two nontrivial solutions.

THEOREM 1.2. Suppose that ab ≠ 0 and d e t 1 1 b - a 0 . Let λk < c < λk+1 and λk(λk - c) < 0, a + b < λk+1(λk+1 - c). Then system (1.1) has a unique nontrivial solution.

In section 2 we define a Banach space H spanned by eigenfunctions of Δ2 + cΔ with Dirichlet boundary condition and investigate some properties of system (1.1). In section 3, we prove Theorem 1.1 by using the critical point theory and variation of linking method. In section 4, we prove Theorem 1.2 by using the contraction mapping principle.

2. Fourth order elliptic system

The eigenvalue problem Δ2u + cΔu = μu in Ω with u = 0, Δu = 0 on ∂Ω has also infinitely many eigenvalues μk = λk(λk - c), k ≥ 1 and corresponding eigenfunctions ϕk, k ≥ 1. We note that λ1(λ1 - c) < λ2(λ2 - c) ≤ λ3(λ3 - c) < ⋯.

The system

Δ 2 u + c Δ u = a ( ( u + v + 1 ) + 1 ) Δ 2 v + c Δ v = b ( ( u + v + 1 ) + 1 ) u = 0, v = 0, Δ u = 0, Δ v = 0 in  Ω , in  Ω , on  Ω

can be transformed to the equation

Δ 2 ( u + v ) + c Δ ( u + v ) = ( a + b ) ( ( u + v + 1 ) + 1 ) in  Ω , u = 0, v = 0, Δ u = 0, Δ v = 0 on  Ω . (2.1)

We also have

Δ 2 ( b u - a v ) + c Δ ( b u - a v ) = 0 in  Ω , u = 0 , v = 0 , Δ u = 0 , Δ v = 0 on  Ω .

It follows from the above equation that bu - av = 0. If u + v = w is a solution of (2.1), then the system

u + v = w , b u - a v = 0

has a unique solution of (1.1) since det 1 1 b - a 0 . Hence the number of the solutions w = u + v of (1.1) is equal to that of (2.1). To investigate the multiplicity of (1.1) it is enough to find the multiplicity of (2.1). Let us set w = u + v. Then (2.1) is equivalent to the equation

Δ 2 w + c Δ w = ( a + b ) ( ( w + 1 ) + 1 ) in  Ω , w = 0, Δ w = 0 , on Ω . (2.2)

Any element u L2(Ω) can be expressed by

u = h k ϕ k  with h k 2 < .

Let H be a subspace of L2(Ω) defined by

H = { u L 2 ( Ω ) | | λ k ( λ k - c ) | h k 2 < } .

Then this is a complete normed space with a norm

u = [ | λ k ( λ k - c ) | h k 2 ] 1 2 .

Since λk(λk - c) → + ∞ and c is fixed, we have

(i) Δ2u + cΔu H implies u H.

(ii) u C u L 2 ( Ω ) , for some C > 0.

(iii) u L 2 ( Ω ) = 0 if and only if || u || = 0.

For the proof of the above results we refer [1].

LEMMA 2.1. Assume that c is not an eigenvalue of , a + b λk(λk - c) and bounded. Then all solutions in L2(Ω) of

Δ 2 w + c Δ w = ( a + b ) ( ( w + 1 ) + 1 ) i n L 2 ( Ω )

belong to H.

Proof. Let us write (a + b)((w + 1)+ - 1) = ∑hkϕk L2(Ω).

( Δ 2 + c Δ ) 1 ( a + b ) ( ( w + 1 ) + 1 ) = 1 λ k ( λ k c ) h k ϕ k L 2 ( Ω ) . ( Δ 2 + c Δ ) 1 ( a + b ) ( ( w + 1 ) + 1 ) = | λ k ( λ k c ) | 1 ( λ k ( λ k c ) ) 2 h k 2 C h k 2 = C w L 2 ( ω ) 2 <

for some C > 0. Thus (Δ2 + cΔ)-1((a + b)((w + 1)+ -1)) ∈ H.   ■

With the aid of Lemma 2.1 it is enough that we investigate the existence of the solutions of (1.1) in the subspace H of L2(Ω).

Let us define the functional

F ( w ) = Ω 1 2 | Δ w | 2 - c 2 | w | 2 - a + b 2 | w + 1 | + - ( a + b ) w . (2.3)

If we assume that λk < c < λk+1 and a + b is bounded, F (u) is well defined. By the following lemma, F(w) ∈ C1. Thus the critical points of the functional F(w) coincide with the weak solutions of (2.2).

LEMMA 2.2. Assume that λk < c < λk+1 and a + b is bounded. Then the functional F(w) is continuous and Frechét differentiable in H and

D F ( w ) ( h ) = Ω [ Δ w Δ h c w h ( a + b ) ( w + 1 ) + h ( a + b ) h ] d x (2.4)

for h H.

Proof. First we shall prove that F(w) is continuous at w. Let w, z H.

F ( w + z ) F ( w ) = Ω [ 1 2 | Δ ( w + z ) | 2 c 2 | ( w + z ) | 2 a + b 2 | ( w + z + 1 ) + | 2 ( a + b ) ( w + z ) ] d x Ω [ 1 2 | Δ w | 2 c 2 | w | 2 a + b 2 | ( w + 1 ) + | 2 ( a + b ) w ] d x = Ω [ w ( Δ 2 z + c Δ z ) + 1 2 z ( Δ 2 z + c Δ z ) ( a + b 2 | ( w + z + 1 ) + | 2 a + b 2 | ( w + 1 ) + | 2 ( a + b ) z ) ] d x .

Let w = ∑hkϕk, z = h ̃ k ϕ k . Then we have

| Ω w ( Δ 2 z + c Δ z ) d x | = | Ω λ k ( λ k - c ) h k h ̃ k | w z , | Ω z ( Δ 2 z + c Δ z ) d x | = | λ k ( λ k - c ) h ̃ k 2 | z 2 .

On the other hand, by Mean Value Theorem, we have

a + b 2 | ( w + z + 1 ) + | 2 a + b 2 | ( w + 1 ) + | 2 ( a + b ) z .

Thus we have

a + b 2 | ( w + z + 1 ) + | 2 - a + b 2 | ( w + 1 ) + | 2 - ( a + b ) z 2 ( a + b ) z = O ( z ) .

Thus F(w) is continuous at w. Next we shall prove that F(w) is Fréchet differentiable at w H. We consider

| F ( w + z ) - F ( w ) - D F ( w ) z | = | Ω 1 2 z ( Δ 2 z + c Δ z ) - ( a + b 2 | ( w + z + 1 ) + | 2 - a + b 2 | ( w + 1 ) + | 2 + ( a + b ) ( w + 1 ) + z ) | 1 2 z 2 + ( a + b ) z + ( a + b ) ( w + 1 ) z = z ( 1 2 z + ( a + b ) + ( a + b ) ( w + 1 ) ) = O ( z ) .

Thus F(w) is Fréchet differentiable at w H.   ■

3. Proof of Theorem 1.1

Throughout this section we assume that λk < c < λk+1 and λk+n(λk+n - c) < a + b < λk+n+1(λk+n+1 - c). We shall prove Theorem 1.1 by applying the variation of linking method (cf. Theorem 4.2 of [8]). Now, we recall a variation of linking theorem of the existence of critical levels for a functional.

Let X be an Hilbert space, Y X, ρ > 0 and e X\Y , e ≠ 0. Set:

B ρ ( Y ) = { x Y : x X ρ } , S ρ ( Y ) = { x Y : x X = ρ } , Δ ρ ( e , Y ) = { σ e + v : σ 0 , v Y , σ e + v X ρ } , Σ ρ ( e , Y ) = { σ e + v : σ 0 , v Y , σ e + v X = ρ } { v : v Y , v X ρ } .

THEOREM 3.1. ("A Variation of Linking") Let × be an Hilbert space, which is topological direct sum of the subspaces X1 and X2. Let F C1(X, R). Moreover assume:

(a) dim X1 < +∞;

(b) there exist ρ > 0, R > 0 and e X1, e ≠ 0 such that ρ < R and

sup S ρ ( X 1 ) F < inf Σ R ( e , X 2 ) F ;

(c) - < a = in f Δ R ( e , X 2 ) F ;

(d) (P.S.)c holds for any c ∈ [a, b], where b = sup B ρ ( X 1 ) F .

Then there exist at least two critical levels c1 and c2 for the functional F such that :

inf Δ R ( e , X 2 ) F c 1 sup S ρ ( X 1 ) F < inf Σ R ( e , X 2 ) F c 2 sup B ρ ( X 1 ) F .

Let H+ be the subspace of H spanned by the eigenfunctions corresponding to the eigenvalues λk(λk - c) > 0 and H- the subspace of H spanned by the eigenfunctions corresponding to the eigenvalues λk(λk - c) < 0. Then H = H+ H-. Let Hk be the subspace of H spanned by ϕ1, ⋯, ϕk whose eigenvalues are λ1(λ1 - c), ⋯ , λk(λk - c). Let H k be the orthogonal complement of Hk in H. Then

H = H k H k .

Let e H+ Hk+n, e ≠ 0 and ρ > 0. Let us set

B ρ ( H k + n ) = { w H k + n | w ρ } , S ρ ( H k + n ) = { w H k + n | w = ρ } , Δ ρ ( e , H k + n ) = { σ e + w | σ 0 , w H k + n , σ e + w ρ } , Σ ρ ( e , H k + n ) = { σ e + w | σ 0 , w H k + n , σ e + w = ρ } { w | w H k + n , w ρ } .

Let L : H H be the linear continuous operator such that

( L w ) z = Ω ( Δ 2 w + c Δ w ) z d x - ( a + b ) Ω w z d x . (3.1)

Then L is an isomorphism and Hk+n, H k + n are the negative space and the positive space of L. Thus we have

( L w ) w - ( ( a + b ) - λ k + n ( λ k + n - c ) ) w 2 , w H k + n , (3.2)

( L w ) w ( λ k + n + 1 ( λ k + n + 1 - c ) - ( a + b ) ) w 2 , w H k + n . (3.3)

We can write

F ( w ) = 1 2 ( L w ) w - ψ ( w ) ,

where

ψ ( w ) = Ω a + b 2 | ( w + 1 ) - | 2 d x .

Since H is compactly embedded in L2, the map Dψ : H H is compact.

LEMMA 3.1. Let λk < c < λk+1 and λk+n(λk+n - c) < a + b < λk+n+1(λk+n+1 - c). Then F(w) satisfies the (P.S.)γ condition for any γ R.

Proof. Let (wn) be a sequence in H with DF(wn) → 0 and F(wn) → γ. Since L is an isomorphism and is compact, it is sufficent to show that (wn) is bounded in H. We argue by contradiction. we suppose that ||wn|| → +∞. Let z n = w n w n . Up to a subsequence, we have zn z in H. Since DF (wn) → 0, we get

D F ( w n ) w n w n 2 = Ω ( Δ 2 + c Δ ) z n 2 - Ω [ ( a + b ) ( z n + 1 w n ) + z n - ( a + b ) z n w n ] 0 . (3.4)

Let P + : H H k + n and P- : H Hk+n denote the orthogonal projections. Since P+ zn -P- zn is bounded in H, we have

Ω ( Δ 2 + c Δ ) ( P + z n + P - z n ) ( P + z n - P - z n ) - Ω [ ( a + b ) ( P + z n + P - z n + 1 w n ) + ( P + z n - P - z n ) ] 0 . (3.5)

Since P+ zn - P- zn P+ z - P- z in H, from (3.2) and (3.3) we get

0 Ω [ ( ( a + b ) z + ) ( P + z - P - z ) ] d x .

Hence z ≠ 0. On the other hand, from (3.5), we get

0 = Ω ( Δ 2 + c Δ ) ( P + z + P z ) ( P + z P z ) Ω [ ( a + b ) z + ( P + z P z ) ] Ω ( Δ 2 + c Δ ) ( P + z + P z ) ( P + z P z ) Ω [ ( a + b ) z ( P + z ) P z ) ] = Ω ( Δ 2 + c Δ ) ( P + z + P z ) ( P + z P z ) Ω ( a + b ) ( P + z ) + P z ) ( P + z ) P z ) = Ω ( Δ 2 + c Δ ( a + b ) ) ( P + z ) 2 d x Ω ( Δ 2 + c Δ ( a + b ) ) ( P z ) 2 ( λ k + n + 1 ( λ k + n + 1 c ) ( a + b ) ) P + z L Ω 2 ( λ k + n ( λ k + n c ) ( a + b ) ) P z L 2 ( Ω ) 2 . (3.6)

The last line of (3.6) is positive or equal to 0 since λk+n+1(λk+n+1 - c) - (a + b) > 0 and - (λk+n(λk+n - c) - (a + b)) > 0. Thus the only possibility to hold (3.6) is that P+ z = 0 and P- z = 0. Thus z = 0, which gives a contradiction.

LEMMA 3.2. Let λk < c < λk+1 and λk+n(λk+n - c) < b < λk+n+1(λk+n+1 - c).

Then

(i) there exists Rk+n > 0 such that the functional F(w) is bounded from below on H k + n ;

inf w H k + n | | w | | = R k + n F ( w ) > 0 a n d inf w H k + n | | w | | < R k + n F ( w ) > - . (3.7)

(ii) there exists ρk+n > 0 such that

sup w H k + n | | w | | = ρ k + n F ( w ) < 0 a n d sup w H k + n | | w | | ρ k + n F ( w ) < . (3.8)

Proof. (i) Let w H k + n . Then we have

lim w H k + n | | w | | + F ( w ) lim w H k + n | | w | | 1 2 ( 1 - r λ k + n + 1 ( λ k + n + 1 - c ) ) w 2 - lim w H k + n | | w | | + Ω [ a + b 2 | ( w + 1 ) + | 2 - ( a + b ) w - r 2 w 2 ] d x lim w H k + n | | w | | 1 2 ( 1 - r λ k + n + 1 ( λ k + n + 1 - c ) ) w 2 - lim w H k + n | | w | | + Ω [ a + b 2 ( w 2 + 1 ) - r 2 w 2 ] d x lim w H k + n | | w | | + 1 2 ( 1 - r λ k + n + 1 ( λ k + n + 1 - c ) ) w 2 - lim w H k + n | | w | | + 1 2 ( ( a + b ) - r ) Ω w 2 - a + b 2 | Ω | + ,

since a + b - r < λ k + n + 1 ( λ k + n + 1 - c ) - r = λ k + n + 1 ( λ k + n + 1 - c ) - λ k + n ( λ k + n - c ) 2 . Thus there exists Rk+n > 0 such that inf   w H k + n | | w | | = R k + n F ( w ) > 0 . Moreover if w H k + n with ||w|| < Rk+n, then we have

F ( w ) 1 2 ( λ k + n + 1 ( λ k + n + 1 - c ) ) | | w | | L 2 ( Ω ) 2 - Ω [ a + b 2 ( w + 1 ) 2 - - ( a + b ) w ] d x (1) > 1 2 { ( λ k + n + 1 ( λ k + n + 1 - c ) ) - ( a + b ) } w L 2 ( Ω ) 2 - Ω a + b 2 d x > - . (2) (3)

Thus we have inf   w H k + n | | w | | < R k + n F ( w ) > - - .

(ii) Let w Hk+n. Then

( L w ) w ( λ k + n ( λ k + n - c ) - r ) Ω w 2 d x λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 Ω w + 2 ,

Ω [ 1 2 ( a + b ) | ( w + 1 ) + | 2 - ( a + b ) w - r w 2 ] d x Ω [ 1 2 ( a + b ) | w + | 2 - ( a + b ) w - r w + 2 ] d x , ,

so that

F ( w ) 1 2 λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 Ω w + 2 (1)  - a + b - r 2 Ω w + 2 + Ω ( a + b ) w d x (2)  1 2 { λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 - ( a + b - r ) } w + L 2 ( Ω ) 2 (3)  + ( a + b ) w L 2 ( Ω ) . (4)  (5)

Since λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 - ( a + b - r ) < 0 , there exists ρk+n > 0 such that if w Hk+n and ||w|| = ρk+n, then sup F(w) < 0. Moreover, if w Hk+n and ||w|| ≤ ρk+n, then we have F ( w ) 1 2 { λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 - ( a + b - r ) } w + L 2 ( Ω ) 2 + ( a + b ) w L 2 ( Ω ) ( a + b ) w L 2 ( Ω ) < .   ■

LEMMA 3.3. Let λk < c < λk+1, λk+n (λk+n - c) < a + b < λk+n+1 (λk+n+1 - c) and let e1 H+ Hk+n with ||e1|| = 1. Then there exists R k + n - such that R k + n - > ρ k + n ,

inf w Σ R k + n - ( e 1 , H k + n ) F ( w ) 0 a n d inf w Δ R k + n - ( e 1 , H k + n ) F ( w ) - .

Proof. Let us chose w H k + n and σ ≥ 0 and e1 H+ Hk+n with ||e1|| = 1. Then we get

F ( w + σ e 1 ) 1 2 λ k + n + 1 ( λ k + n + 1 - c ) w L 2 ( Ω ) 2 + σ 2 2 e 1 2 (1) - Ω [ a + b 2 ( w + σ e 1 + 1 ) 2 - ( a + b ) ( w + σ e 1 ) ] d x (2) = 1 2 { λ k + n + 1 ( λ k + n + 1 - c ) - ( a + b ) } w L 2 ( Ω ) 2 + σ 2 2 ( Λ - ( a + b ) ) e 1 L 2 ( Ω ) 2 (3) - ( a + b ) σ 2 w L 2 ( Ω ) e 1 L 2 ( Ω ) - a + b 2 | Ω | , (4) (5)

where λk+1 (λk+1 - c) ≤ Λ ≤ λk+1 (λk+1 - c). Choose σ > 0 so mall that σ 2 e 1 2 is small. We can choose a number R k + n - > 0 such that R k + n - > σ , R k + n - > ρ k + n , and inf w H k + n , σ 0 | | w + σ e 1 | | = R k + n F ( w + σ e 1 ) 0 : Moreover if w H k + n , σ 0 w + σ e 1 R k + n - , then F ( w ) σ 2 2 ( Λ - b ) e 1 L 2 ( Ω ) 2 - ( a + b ) σ w L 2 ( Ω ) e 1 L 2 ( Ω ) - a + b 2 | Ω | - . Thus we prove the lemma.   ■

Proof of Theorem 1.1

By Lemma 2.2, F(w) is continuous and Frechét differentiable in H. By Lemma 3.1. F(w) satisfies the (P.S.)γ condition for any γ R. We note that the subspace S ρ k + n H k + n and the subspace Σ R k + n - ( e 1 , H k + n ) link at the subspace {e1}. By Lemma 3.2 and Lemma 3.3, we have

sup w S ρ k + n H k + n F ( w ) < inf w Σ R k + n - ( e 1 , H k + n ) F ( w ) .

By Lemma 3.3, we also have inf w Δ R k + n - ( e 1 , H k + n ) F ( w ) > - Thus by the variation of linking theorem, there exists at least two nontrivial solutions of (2.2). Thus we complete the Theorem 1.1.

4. Proof of Theorem 1.2

Proof of Theorem 1.2

Assume that λk < c < λk+1 and λk(λk - c) < 0, b < λk+1(λk+1 - c). Let r = 1 2 { λ k ( λ k - c ) + λ k + 1 ( λ k + 1 - c ) } . We can rewrite (2.2) as

( Δ 2 + c Δ - r ) w = ( a + b ) ( w + 1 ) + - r ( w + 1 ) + + r ( w + 1 ) + - r w - ( a + b ) in  L 2 ( Ω ) , w = 0 , Δ w = 0 on  Ω . (4.1)

or

w = ( Δ 2 + c Δ - r ) - 1 [ ( a + b ) ( w + 1 ) + - r ( w + 1 ) + + r ( w + 1 ) + - r w - ( a + b ) ] in  L 2 ( Ω ) , w = 0 , Δ w = 0 on  Ω . (4.2)

We note that the operator (Δ2 +cΔ - r)-1 is a compact, self-adjoint and linear map from L2(Ω) into L2(Ω) with norm 2 λ k + 1 ( λ k + 1 - c ) - λ k ( λ k - c ) , and

( ( a + b ) - r ) { ( w 2 + 1 ) + - ( w 1 + 1 ) + } + r { ( w 2 + 1 ) + - ( w 1 + 1 ) + } - r ( w 2 - w 1 ) max { ( a + b ) - r , r } | | w 2 - w 1 | | < 1 2 { λ k + 1 ( λ k + 1 - c ) - λ k ( λ k - c ) } | | w 2 - w 1 | | .

Thus the right hand side of (4.2) defines a Lipschitz mapping from L2(Ω) into L2(Ω) with Lipschitz constant < 1. By the contraction mapping principle, there exists a unique solution w L2(Ω) of (4.2). By Lemma 2.1, the solution u H. We complete the proof.   ■

Abbreviations

(FESDBC): fourth-order elliptic system with Dirichlet boundary condition.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

TJ carried out (FESDBC) studies, participated in the sequence alignment and drafted the manuscript. QC participated in the sequence alignment. All authors read and approved the final manuscript.

Acknowledgements

This work(Tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (KRF-2010-0023985).

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9. Micheletti, AM, Pistoia, A, Sacon, A: Three solutions of a 4th order elliptic problem via variational theorems of mixed type. Applicable analysis. 75, 43–59 (2000). Publisher Full Text

10. Tarantello: A note on a semilinear elliptic problem. Diff Integ Equat. 5(3), 561–565 (1992)


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chapter  11
6 Pages

## Energy expenditure measurement in badminton players during a training camp using doubly-labelled water

ByE. WATANABE, S. IGAWA, T. SATO, M. MIYAZAKI, S. HORIUCHI, K. SEKI

The quantity of physical activity and energy expenditure of badminton players have been studied in general using heart rate monitoring. There are many research reports concerning the physical activity of a game of badminton (Seki et al., 1982; Araragi et al., 1999; Cabello et al., 2003). In order to obtain good results in an international match, it is necessary to prepare for the target game. For that purpose, we need to understand the habitual energy consumption (in training camp, on tour, during competition games and so on) that a player expends.


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# help

0
59
4

Find the number of equilateral triangles in the grid.

Dec 18, 2019

#1
+2551
+1

We classify the types of equ-triangles starting with biggest to smallest.

The biggest is obviously the equ-triangle with a side length of 5, the smallest equ-triangle has a side length of 1.

Count the number of equ-triangles for each class. The first one has already been filled out for you.

 Side Length Number of $$\Delta$$ 5 1 4 ? 3 ? 2 ? 1 ?

Now once you complete the table, just find the total up the number of equ-triangles for the table.

Calcuser not logged in

Dec 18, 2019
#2
+1

and don't forget the upside-down triangles

Guest Dec 18, 2019
#3
+2551
+1

yes! remember that! very important.

CalculatorUser  Dec 18, 2019
#4
+23809
+1

Find the number of equilateral triangles in the grid ($$n=5$$).

$$a(n) = \Big\lfloor \dfrac{n(n+2)(2n+1)}{8} \Big \rfloor$$

Source:  http://oeis.org/A002717

$$\begin{array}{|rcll|} \hline a(5) &=& \Big\lfloor \dfrac{5(5+2)(2*5+1)}{8} \Big\rfloor \\ &=& \Big\lfloor \dfrac{5*7*11}{8} \Big\rfloor \\ &=& \lfloor 48.125 \rfloor \\ &=& \mathbf{48} \\ \hline \end{array}$$

Dec 18, 2019


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2013
Impact Factor

# Sergei Tabachnikov

University Park, PA 16802, USA
Department of Mathematics, Pennsylvania State University

## Publications:

 Tabachnikov S. Two Variations on the Periscope Theorem 2020, vol. 25, no. 1, pp.  11-17 Abstract A (multidimensional) spherical periscope is a system of two ideal mirrors that reflect every ray of light emanating from some point back to this point. A spherical periscope defines a local diffeomorphism of the space of rays through this point, and we describe such diffeomorphisms. We also solve a similar problem for (multidimensional) reversed periscopes, the systems of two mirrors that reverse the direction of a parallel beam of light. Keywords: periscope, optical reflection, projectively gradient vector field Citation: Tabachnikov S.,  Two Variations on the Periscope Theorem, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 11-17 DOI:10.1134/S1560354720010037
 Khesin B. A., Tabachnikov S. Contact complete integrability 2010, vol. 15, no. 4-5, pp.  504-520 Abstract Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical $\mathbb{R} \times \mathbb{R}^{n−1}$ structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories. Keywords: complete integrability, contact structure, Legendrian foliation, pseudo-Euclidean geometry, billiard map Citation: Khesin B. A., Tabachnikov S.,  Contact complete integrability, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 504-520 DOI:10.1134/S1560354710040076
 Dogru F., Tabachnikov S. On Polygonal Dual Billiard in the Hyperbolic Plane 2003, vol. 8, no. 1, pp.  67-81 Abstract We study the polygonal dual billiard map in the hyperbolic plane. We show that for a class of convex polygons called large all orbits of the dual billiard map escape to infinity. We also analyse the dynamics of the dual billiard map when the dual billiard table is a regular polygon with all right angles. Citation: Dogru F., Tabachnikov S.,  On Polygonal Dual Billiard in the Hyperbolic Plane, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 67-81 DOI:10.1070/RD2003v008n01ABEH000226


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# Math Help - Advanced Calculus 1 question

1. ## Advanced Calculus 1 question

Here is the question from Multivariable Calculus by James Stewart 6th Edition:

"Find the angle between a diagonal of a cube and a diagonal of one of its faces." This is no differentiation or integration question. It is under the dot product section. I think you need to solve it through direction angles and direction cosines.

I drew a cube with one of its corner points on the origin of the x, y, and z-axis. The diagonal of the cube can be seen as the diameter of a sphere and therefore is 2r (r=radius). What can the length of the diagonal of one of the faces of the cube be? I don't think its half that of the diagonal of the whole cube. But if it is then it would seem one line is perpendicular to the other and the angle would be PI/2 in radians.

2. Originally Posted by Undefdisfigure
Here is the question from Multivariable Calculus by James Stewart 6th Edition:

"Find the angle between a diagonal of a cube and a diagonal of one of its faces." This is no differentiation or integration question. It is under the dot product section. I think you need to solve it through direction angles and direction cosines.

I drew a cube with one of its corner points on the origin of the x, y, and z-axis. The diagonal of the cube can be seen as the diameter of a sphere and therefore is 2r (r=radius). What can the length of the diagonal of one of the faces of the cube be? I don't think its half that of the diagonal of the whole cube. But if it is then it would seem one line is perpendicular to the other and the angle would be PI/2 in radians.
Take one corner as origin and the others as (0,0,1), (0,1,0), ...

The diagonal of the cube starting at the origin is then the vector d1=(1,1,1),
and a face diagonal starting from the same point is d2=(1,1,0).

Now the dot product of these is:

|d1| |d2| cos(theta) = (1,1,1).(1,1,0) = 2.

so:

theta= arccos(2/[|d1| |d2|])

RonL


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1. ## Confusing Limit problem

Hi, I'm confused on how to figure out the problem $\lim_{x \to 27} \frac{x-27}{x^\frac{1}{3}-3}$. I have the problem worked out, but I am having difficulty understanding how to get from step one to step 2:
$\lim_{x \to 27} \frac {(x^\frac{1}{3})^3-(3)^3}{x^\frac{1}{3}-3}$

to

$\lim_{x \to 27}\frac {(x^\frac{1}{3}-3)(x^\frac{2}{3}+3x^\frac{1}{3}+9)}{(x^\frac{1}{3}-3)}$

Thanks for the help in advance.

2. It was done using the difference of 2 cubes.

$a^3- b^3 = (a - b)(a^2 + ab + b^2)$

3. It is just factorisation.

For example,

If $f(x)=x^{3}-27$

Since $f(3)=0$, by factor theorem, $(x-3)$ is a factor.

So $f(x)=(x-3)(ax^{2}+bx+c)$

Do either long division or comparison to get the other quadratic factor, you will get $f(x)=(x-3)(x^{2}+3x+9)$.

In your limit problem, my $x$ is your $x^{\frac{1}{3}}$

4. $(x^\frac{1}{3})^3-(3)^3$

You can do synthetic division:

$(x^\frac{1}{3})^3+0*x^\frac{2}{3}+0*x^\frac{1}{3}-27$

$x^\frac{1}{3}-3=0$
$x^\frac{1}{3}=3$

So:

3 | 1 0 0 -27
_ |___3 9 27
___ 1 3 9 0

hmm don't know how to do this in latex, nor can i get it to line up . . . sorry.

So you lower the power thing and:

$1*x^\frac{2}{3}+3*x^\frac{1}{3}+9$

hence:
$(x^\frac{1}{3}-3)(x^\frac{2}{3}+3x^\frac{1}{3}+9)$

5. A slightly different way to do this problem would be to "rationalize" the denominator.

Since $a^3- b^3= (a- b)(a^2+ ab+ b^2)$, [tex](x^{1/3}- 3)(x^{2/3}+ 3x^{1/3}+ 9)= x- 27. Multiplying both numerator and denominator by $x^{2/3}+ 3x^{1/3}+ 9$ gives
$\frac{(x- 27)(x^{2/3}+ 3x^{1/3}+ 9)}{x- 27}= x^{2/3}+ 3x^{1/3}+ 9$. Now you can let x= 27 in that.

6. Originally Posted by cynlix

Hi, I'm confused on how to figure out the problem $\lim_{x \to 27} \frac{x-27}{x^\frac{1}{3}-3}$.
put $t=x^{\frac13}$ and the limit becomes $\underset{t\to 3}{\mathop{\lim }}\,\frac{t^{3}-27}{t-3},$ which you know how to solve.


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# Local Langlands correspondence: Weil-Deligne group

While reading the book 'Langlands correspondence for loop groups', I came across the definition of the Weil group $W_F$ and the Weil-Deligne group $W'_F = W_F \ltimes \mathbb{C}$ with action $$\sigma x\sigma^-1 = ||\sigma||x, \sigma \in W_F,x\in \mathbb{C}.$$ In it, they give the definition of an $n$-dimensional complex representation of $W'_F$:

"An $n$-dimensional complex representation of $W'_F$ is by definition a homomorphism $\rho': W'_F \rightarrow GL_n(\mathbb{C})$, which may be described as a pair $(\rho,u)$, where $\rho$ is an $n$-dimensional representation of $W_F$, $u \in \mathfrak{gl}_n (\mathbb{C})$, and we have $\rho(\sigma)u\rho(\sigma) = ||\sigma||u$."

Now, I understand what $\rho$ means and why you define that action of $\rho(\sigma)$ on $\rho'(\mathbb{C})$, but I don't understand the meaning of $u \in \mathfrak{gl}_n (\mathbb{C})$, shouldn't that be $u \in GL_n(\mathbb{C})$ ?

For a reference of the book: http://goo.gl/pq5XZ, page 4.

-

In this formulation (which is completely standard), the image of $1\in \mathbb{C}$ under $\rho'$ is not, as you might expect, $u$, but rather $\text{exp}(u)$, which is indeed in $GL_n(\mathbb{C})$. $u$ itself is nilpotent, and is usually called the monodromy operator attached to the Weil-Deligne representation $\rho'$. You should check that the claimed relation $\rho(\sigma)u\rho(\sigma)^{-1}=||\sigma||u$ then holds (you forgot the inverse).
Thank you very much, but does $u$ have to be nilpotent ? In the book I mentioned, they make a distinction, with nilpotent (and $\rho$ continous) only necessary for admissible representations. –  KevinDL Oct 17 '11 at 12:38
@KevinDL I guess, conventions might vary there slightly. I have only heard the word "admissible" being used on the reductive side, but it does make sense to use it here, since the Frobenius semi-simple Weil-Deligne representations with nilpotent $u$ conjecturally correspond to admissible irreducible representations of $GL_n$. Tate on the other hand just defines a Weil-Deligne representation to come with a nilpotent monodromy operator. –  Alex B. Oct 17 '11 at 13:19


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# Intuition behind Pontryagin duality

We have that for a locally compact abelian group,

$$A \mapsto \operatorname{Hom}(A, \mathbb{R}/\mathbb{Z})$$

is a biduality map when endowing $$\operatorname{Hom}$$ with the compact-open topology.

My question is why $$\mathbb{R}/\mathbb{Z}$$? This sounds like a rather counter-intuitive result. I can see one why one would give $$\operatorname{Hom}$$ the compact-open topology (there are many categorical and topological reasons to do this) but it looks rather ad-hoc that we use the circle group.

How did this come up? Are there other bidualities using different groups?

• mathoverflow.net/questions/294019/… Feb 15 at 22:02
• It is in fact explained in this post. Read the first comment by Kcd. His $U(1)$ is your $\Bbb R/\Bbb Z$. And he cited Hewitt and Ross vol. 1 p. 424 as well as Pontryagin's book Topological Groups Example 72. I didn't check any of the citations. Feb 15 at 22:27

The fundamental notion behind the theory of groups is the concept of symmetries and the best place to visualize symmetries is Euclidean space and, by extension, Hilbert's space.

By a symmetry on a Hilbert space $$H$$ one means any map $$U:H\to H$$ that preserves distance, and sends the origin to itself. In fact any such map is necessarily a unitary operator.

The group $$\mathscr U(H)$$, formed by all unitary operators on Hilbert's space is, according to this, the archetype of symmetry!

Given any group $$G$$, it therefore makes a lot of sense to try to model $$G$$ via $$\mathscr U(H)$$, and this is usually done by considering group homomorphisms $$u:G\to \mathscr U(H),$$ often called group representations.

If $$u_1$$ and $$u_2$$ are representations of $$G$$, one may define a direct sum representation $$u=u_1\oplus u_2 \tag 1$$ on $$H\oplus H$$, but everything there is to know about $$u$$ is already present in either $$u_1$$ or $$u_2$$.

On the other hand, should we be interested in studying a given representation $$u$$, we might ask whether or not it has an expression such as (1), in which case it would be better to study $$u_1$$ and $$u_2$$ separately.

This means that one should concentrate in the study of irreducible group representations, namely those which cannot be split as in (1).

For abelian groups, Schur's Lemma says that all irreducible unitary representations are necessarily one-dimensional, meaning that the underlying Hilbert space is $$\mathbb C$$, and since $$\mathscr U(\mathbb C) = \mathbb T,$$ we are inexorably led to studying homomorphisms from $$G$$ to $$\mathbb T$$!

• This is a very vivid, nice explanation. Jun 15 at 1:39