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Egyptian Journal of Medical Human Genetics
Prevalence of enterotoxin genes (SEA to SEE) and antibacterial resistant pattern of Staphylococcus aureus isolated from clinical specimens in Assiut city of Egypt
Ahmed A. Baz ORCID: orcid.org/0000-0002-8002-76621,
Elsayed K. Bakhiet1,
Usama Abdul-Raouf1 &
Ahmed Abdelkhalek2
Egyptian Journal of Medical Human Genetics volume 22, Article number: 84 (2021) Cite this article
Infections in communities and hospitals are mostly caused by Staphylococcus aureus strains. This study aimed to determine the prevalence of five genes (SEA, SEB, SEC, SED and SEE) encoding staphylococcal enterotoxins in S. aureus isolates from various clinical specimens, as well as to assess the relationship of these isolates with antibiotic susceptibility. Traditional PCR was used to detect enterotoxin genes, and the ability of isolates expressing these genes was determined using Q.RT-PCR.
Overall; 61.3% (n = 46) of the samples were positive for S. aureus out of 75 clinical specimens, including urine, abscess, wounds, and nasal swabs. The prevalence of antibiotic resistance showed S. aureus isolates were resistant to Nalidixic acid, Ampicillin and Amoxicillin (100%), Cefuroxime (94%), Ceftriaxone (89%), Ciprofloxacin (87%), Erythromycin and Ceftaxime (85%), Cephalexin and Clarithromycin (83%), Cefaclor (81%), Gentamicin (74%), Ofloxacin (72%), Chloramphenicol(59%), Amoxicillin/Clavulanic acid (54%), while all isolates sensitive to Imipinem (100%). By employing specific PCR, about 39.1% of isolates were harbored enterotoxin genes, enterotoxin A was the most predominant toxin in 32.6% of isolates, enterotoxin B with 4.3% of isolates and enterotoxin A and B were detected jointly in 2.1% of isolates, while enterotoxin C, D and E weren't detected in any isolate.
This study revealed a high prevalence of S. aureus among clinical specimens. The isolates were also multidrug resistant to several tested antibiotics. Enterotoxin A was the most prevalent gene among isolates. The presence of antibiotic resistance and enterotoxin genes may facilitate the spread of S. aureus strains and pose a potential threat to public health.
Staphylococcus aureus is a commensal and opportunistic human pathogen that can be found all over the world [1]. It causes a variety of clinical infections, including impetigo, furunculosis, and abscesses on the skin and soft tissues, as well as systemic infections, including pneumonia, bacteremia, endocarditis, and toxin-mediated diseases [2]. It is one of the most common pathogens linked to nosocomial infections in hospitals [3]. S. aureus cause a large amount of morbidity and mortality in developing countries as opposed to other infectious diseases like malaria, tuberculosis, and HIV infections [4]. The fast expansion of drug resistance as well as prominent virulence factors, surface proteins, metabolites and enzymes have all contributed to S. aureus clinical significance [5]. Large numbers of toxins, including hemolysin (α, β, γ, δ) and leukocidin (PVL, Luk E/D) are the most commonly associated virulence factors with these microorganisms [6]. Other virulence factors that cause enterocolitis, scalded skin syndrome, and toxic shock include heat-stable staphylococcal enterotoxins (SEs), exfoliative toxins (ETA and ETB), the toxin of toxic shock syndrome-1(TSST-1) [7, 8]. S. aureus is a concern not only because of its widespread distribution and pathogenicity but also because of its ability to resist antimicrobials [9]. In addition to drug resistance, monitoring of S. aureus strains and determination of susceptibility patterns are critical [10]. Staphylococcal enterotoxins have been identified as etiologic agents of human food poisoning and as active immunologic superantigens that induce non-specific T cell proliferation [11]. Enterotoxins are emetic toxins linked to a broad family of pyrogenic exotoxins produced by staphylococci and streptococci, they are active in concentrations ranging from nanograms to micrograms and are resistant to heat and low pH, also have proteolytic enzymes allowing them to remain active in the digestive tract after ingestion [7]. There are 23 staphylococcal superantigens that have been described, enterotoxins A to E, G to J, R to T, and staphylococcal enterotoxin-like toxins (SE1) K, Q, U, and X (SElK-SElQ, SE1U-SElX) [12,13,14,15,16,17]. Polymerase chain reaction (PCR), DNA probes, and reverse-transcription RT-PCR were used to detect enterotoxins and their activity in S. aureus strains [18]. So the aim of this research is to use PCR to detect enterotoxins genes in S. aureus, assess their distribution through clinical sources, and quantify their prevalence. Furthermore, we looked for any possible connection between enterotoxin prevalence and expression on the one side and antimicrobial susceptibility on the other.
Collection of samples
The present descriptive study was carried out over a six-month period, between January and July 2015, the research was approved by the ethics committee of the Botany and Microbiology department- the College of Science (Al-Azhar University, Assiut). A total of 75 clinical specimens were collected under aseptic conditions from patients admitted to Assiut University hospital—Faculty of Medicine, Political hospital, Al-Eman hospital, and El-Shmla hospital in Assiut governorate, Egypt. All clinical specimens collected from diverse infections were received from both genders and all age groups. Urine from the infected urinary tract (n = 29; 38.6%), Pus swabs from an abscess (n = 20; 26.6%), Swabs from septic wounds (n = 15; 20%), and Nasal swabs from cases with respiratory symptoms (n = 11; 14.6%). The samples obtained from patients were transported directly to the laboratory in an icebox within 4 h of collection and were immediately processed according to standard microbiological procedures. The sampling swabs were inoculated into a 5 ml Staphylococcus enrichment broth medium [19], blended briefly as necessary, and incubated at 37 °C for 24 h.
Prevalence and characterization of S. aureus isolates
The bacterial culture was streaked on mannitol salt agar (MSA) (Oxoid, Basingstoke, Hampshire, England), with a loopful of enrichment broth and incubated aerobically at 37 °C for 24 to 48 h. Suspected colonies were picked up, sub-cultured once on MSA and twice on 5% defibrinated sheep blood agar medium (Oxoid, Basingstoke, Hampshire, England). Bacterial colonies showing typical characteristics of S. aureus (i.e., beta-hemolytic on blood agar and colonies with golden yellow pigmentation on MSA) were subjected to subculture on to basic media, Gram stain, and biochemical tests catalase and coagulase. Catalase positive and Gram-positive bacteria appearing in the grape-like cluster was spot inoculated to DNase agar (Oxoid, Basingstoke, Hampshire, England). Inoculated DNase agar plates were incubated at 37 °C overnight and flooded with 1 N HCl (Merk, Darmstadt, Germany). Isolates that hydrolyzed DNA in DNase agar were considered S. aureus. Purified colonies were selected, propagated on nutrient agar slope, and preserved at 4 °C.
Antibiotics susceptibility testing
The disc diffusion method was used to determine the antibiotic susceptibility of S. aureus isolates on Mueller Hinton Agar (Oxoid, Basingstoke, Hampshire, England) [20]. Sixteen antibiotics were tested, Imipinem (IPM:10 μg), Nalidixic acid (NA:30 μg), Cephalexin (CN:30 μg), Chloramphenicol (CM:30 μg), Ofloxacin (OFX:5 μg), Amoxicillin (AMX:25 μg), Amoxicillin/Clavulanic acid(AMC:30 μg), Ampicillin (Amp:10 μg), Cefaclor (CEC:30 μg), Gentamycin (GM:30 μg), Erythromycin (E:15 μg), Ciprofloxacin (CIP:5 μg), Ceftriaxone (CRO:30 μg), Clarithromycin (CLR: 15 μg), Ceftaxime (CTX: 30 μg), Cefuroxime (CXM: 30 μg).
DNA extraction and detection of staphylococcal enterotoxin genes
DNA was extracted using a genomic DNA isolation kit (QIAGEN, Germany) according to the manufacturer's instructions. Specific PCR using specific primers was used to detect genes encoding enterotoxin A (SEA), enterotoxin B (SEB), enterotoxin C (SEC), enterotoxin D (SED), and enterotoxin E (SEE) [21,22,23,24,25] (Table 1). PCR reaction was conducted in the final volume 25 µl using 2.5 µl Taq polymerase buffer 10X (Promega, Madison, USA) containing a final concentration of 1 mM MgCl2, 2 µl of 2.5 mM dNTPs, 1 µl DNA (50 ng), 1 µl of 10 pmol/µl each specific primer and 0.2 µl Taq polymerase (5 U/µl). The PCR program started with an initial denaturation of 94 °C for 5 min followed by 35 cycles of denaturation at 94 °C for 30 s, annealing at (50 or 52 °C) for 30 s and extension at 72 °C for 1 min. Additional extension at 72 °C for 10 min was done. Aliquots of amplified products were loaded in 1.5% agarose gel, visualized, and photographed using a gel documentation system (Syngene, USA).
Table 1 Oligonucleotides primers used in the study
RNA extraction and expression of staphylococcal enterotoxin genes
Total RNA was extracted using the RNeasy Mini Kit according to the manufacturer's instructions (QIAGEN, Germany). The first strand of cDNA synthesis was performed in a total reaction volume of 25 µl. The reaction mixture contained 2.5 µl of 5X Reverse Transcriptase buffer with 1 mM MgCl2 (Fermentas, USA), 2.5 µl of 2.5 mM dNTPs, 0.5 µl of oligo dT primer 10 pmol/µl, 1 µl RNA (50 ng), 0.2 µl of 200 U/µl Reverse Transcriptase Enzyme in a final reaction volume up to 25 µl by RNase free water. The reverse transcriptase reaction was performed in a thermal cycler (Eppendorf, Germany) run at 42 °C for 1 h and 72 °C for 10 min.
Quantitative PCR was performed using SYBR Green PCR Master Mix (Fermentas, USA). Each reaction was performed in a 25 μl mixture, which contained 1 μl of 10 pmol/μl of each primer, 1 μl of cDNA (50 ng), 12.5 μl of 2X SYBR Green PCR Master and 9.5 μl of nuclease-free water. Samples were spin before loading in the rotor wells and each sample was run in triplicate. The amplification program proceeded at 95 °C for 10 min, followed by 40 cycles of denaturation at 95 °C for 15 s; annealing at 60 °C for 30 s and extension at 72 °C for 30 s, then followed by a final extension at 72 °C for 10 min. The reaction was performed using a Rotor-Gene 6000 (QIAGEN, ABI System, USA).
Real-Time Q-PCR data analysis
The relative expression ratio was quantified and calculated accurately. Accordingly, for each biological sample, the difference (Δ) in quantification cycle value (CT) between the target (CT (target) averaged from three technical repeats) and the reference 16S rRNA was used as the reference gene (CT (reference), a fixed CT value was used for all samples). The CT (threshold of the cycle) value of each detected gene was determined by automated threshold analysis on ABI System [26]. The CT value of each target gene was normalized to CT (reference) to obtain ΔCT (target) where
$$\begin{aligned} \Delta C_{{{\text{T }}({\text{target}})}} & = \left( {C_{{{\text{T }}({\text{target}})}} {-} \, C_{{{\text{T }}({\text{reference}})}} } \right), \\ \Delta C_{{{\text{T }}({\text{control}})}} & = \left( {C_{{{\text{T }}({\text{control}})}} {-}C_{{{\text{T }}({\text{reference}})}} } \right) \\ \end{aligned}$$
The relative expression quantity of the target gene was indicated with
$$\Delta \Delta {\text{CT}} = (\Delta C_{{{\text{T}}({\text{target}})}} {-} \, \Delta C_{{{\text{T}}({\text{control}})}} ){\text{according}}\;{\text{to}}\;{2} - \Delta \Delta {\text{Ct}}\;{\text{algorithm}}$$
Sequencing, sequence analysis, and phylogenetic tree construction of highly expression isolates
Amplified products of (16S rRNA gene and specific enterotoxin genes) for high expression isolates were purified according to (Maxim biotech INC, USA) and subjected to DNA sequencing using a forward primer in the sequencing reaction [27]. Sequencing was performed using BigDye® Terminator v3.1 Cycle Sequencing kit (Applied Biosystems, Foster City, CA, USA) and model 3130xl Genetic Analyzer (Applied Biosystems, Foster City, CA, USA). The obtained DNA nucleotide sequences were analyzed using NCBI-BLAST (http://blast.ncbi.nlm.nih.gov/Blast.cgi) for confirming the identity of the obtained sequences. Multiple sequence alignment of our sequences and the other published ones were performed using CLUSTLAW (1.83) [28] and a phylogenetic tree was analyzed and generated using MEGA 4 [29].
Correlations between data of antibiotic resistance and prevalence of enterotoxins were statistically analyzed using the Graphpad Instat 3. Fisher's exact test was used to evaluate these correlations, where P value less than 0.05 was considered statistically significant.
Isolation and identification of S. aureus isolates
In total, 46 S. aureus isolates were isolated from 75 patient samples (29 urine samples, 20 abscess samples, 15 wound samples, and 11 nasal swabs samples). Suspected colonies turned MSA medium yellow due to mannitol fermentation and were given a clear zone around colonies on blood agar because of β- hemolytic activity of isolates. S. aureus isolates were found most often in abscess swabs (15 isolates from 20 samples; 75%) and then wound swabs (9 isolates from 15 samples; 60%), urine samples (16 isolates from 29 samples; 55.1%), and lastly nasal swabs (6 isolates from 11 samples; 54.5%) (Table 2). Data are presented in Table 3 concluded that S. aureus isolates were positive for catalase, coagulase and, DNase, growth on crystal violet agar were purple, white and yellow colonies, also growth on Baird Parker agar medium were black and shiny with narrow white margins surrounded by clear zone, it also positive for Nitrate reduction, Methyl red test, Voges Prausker test and their ability for hydrolysis of Gelatin and Casein while negative for Indole production and Citrate utilization.
Table 2 Frequency of Staphylococcus aureus isolates among clinical specimens
Table 3 Biochemical characterization of S. aureus isolates
Susceptibility of S. aureus isolates to antibiotics
The disc diffusion method was used to screen all of the bacterial isolates on Muller–Hinton agar for sixteen antibiotics. The results revealed that all isolates were sensitive to imipinem (100%), 46% of isolates sensitive to amoxicillin/clavulanic acid, 41% sensitive to chloramphenicol, 28% sensitive to ofloxacin, 26% sensitive to gentamicin, 19% sensitive to Cefaclor, 17% sensitive to cephalexin and clarithromycin, 15% sensitive to erythromycin and ceftaxime, 13% sensitive to ciprofloxacin, 11% sensitive to ceftriaxone, 6% sensitive to cefuroxime, while current results demonstrated all isolates resistant to nalidixic acid, ampicillin, and amoxicillin (Fig. 1).
Antibiotic susceptibility pattern of S. aureus isolate from clinical specimens
PCR amplification of enterotoxin genes
The prevalence of different five enterotoxin genes in S. aureus isolates was investigated using PCR, it was reported that 15 isolates (32.6%) were positive for the SEA gene, 2 isolates (4.3%) were positive for the SEB gene, and only one isolate (2.1%) was positive for both the SEA and SEB genes (Fig. 2), SEC, SED and SEE genes were not found in any of the tested isolates.
The amplified products for SEA gene (A) and SEB gene (B) on 1.5% agarose gel electrophoresis which given positive with 16 isolates for SEA with 102 bp and positive with 3 isolates for SEB gene with 478 bp. A Lane M: DNA Ladder 100 bp (Thermo Scientific Fisher. USA), Lane 2 to Lane17: amplified products of 16 isolates for SEA gene. B Lane M: DNA Ladder 100 bp (Thermo Scientific Fisher, USA) lane 2 to Lane 4: amplified products of 3 isolates for SEB gene
Real-time PCR data analysis of S. aureus enterotoxins
The relative gene expression was examined of the tested harbored genes by isolation of RNA from the selected S. aureus isolates and gene expression was verified using a Reverse Transcription Real-time PCR, which enabled comparison of the target and reference genes, where the 16S rRNA was used as the reference gene and the enterotoxin genes as the target genes. The expression of the SEA and SEB genes were expressed to variable degrees in all isolates, it was revealed that isolate no: 10 had the highest SEA gene expression, followed by isolate no: 42 and isolate no: 37, while isolate no: 11 had the highest SEB gene expression, followed by isolate no: 8 and isolate no: 5, (Table 4, Fig. 3). The current report found that MDR of the tested isolates expressed SEA gene with (P value = 0.01) and SEB gene was expressed with (P value = 0.03).
Table 4 Distribution of enterotoxin genes pattern and relative gene expression among isolation sources
Histogram for estimation the quantitative expression level of specific enterotoxin A gene (1) and for estimation the quantitative expression level of specific enterotoxin B gene (2)
Identification of highly expression isolates and phylogenetic analysis
Completely identification of S. aureus isolates exhibited high level of enterotoxin A and enterotoxin B expression; 16S rRNA gene and enterotoxin genes were amplified and sequenced. The annotated sequences were deposited in GenBank under accession number MF563554 Azhar1 strain for enterotoxin A producing strain and MF563555Azhar2 strain for enterotoxin B producing strain. Sequence alignment and phylogenetic tree analysis revealed that MF563554 given the similarity of about 96% with LT677428 and LT677437 human strains isolated from the head and neck tissue in the USA, while MF563555 had similar of about 90% with HM452073 isolated from bovine mastitis in India (Fig. 4). High expression isolates for SEA and SEB genes using specific primers were amplified and sequenced. The annotated sequence was deposited in GenBank under accession number LC315607 ELBAZ1 for enterotoxin A strain and MF621929 ELBAZ2 for the enterotoxin B strain. Sequence analysis revealed that LC315607 ELBAZ1 strain given 98% similarity with KX777250 which was a local isolate in Egypt while MF621929 ELBAZ2 strain given similar 99% with AB860415 strain isolated from Tokyo food poisoning outbreak and AB716349 isolated from a human nasal swab in Japan. The relationship between ELBAZ1 and ELBAZ2 among other standard strains available in GeneBank were assessed by constructing a neighboring-Joining tree (Figs. 5, 6).
Phylogenetic tree of 1500 bp of 16S rRNA gene of S. aureus MF563554, MF563555, compared with other strains available in GeneBank using neighbor-joining distance method
The minimum evolution distance analysis phylogenetic tree of SEA gene encoding S. aureus MF563554 compared with other strains of SEA encoding gene available in GeneBank
The minimum evolution distance analysis phylogenetic tree of SEB gene encoding S. aureus MF563555 compared with other strains of SEB encoding gene available in GeneBank
S. aureus is a widespread human pathogen that can be found in both hospitals and the public, it's an opportunistic pathogen that can cause several diseases in humans, both self-limiting and life-threatening [30, 31]. In the current investigation revealed high frequency of S. aureus in (61.3%) of isolates. This is higher isolation rate than seen in earlier research (28.1% and 24.5%, respectively) [31, 32]. The higher isolation of S. aureus was observed especially in children and neonates and this finding consistent with report [31], it is believed that their immunity is not properly developed at this phase to cope with bacterial illness hence they are vulnerable and easily infected especially when hospitalized. The older children have also been observed to be more active than adults during their interaction with their playmates and while playing for hours, come in contact with various objects. In this process, they become a target to ubiquitous bacteria such as S. aureus. On MSA and blood agar media, purified S. aureus colonies were smooth, circular, convex, entire, and given different pigments: yellow (47.8%), white (47.8%), and lemony yellow (4.3%), the present data supported through other investigations [32,33,34]. S. aureus isolates were found to have the highest occurrence rate in abscess specimens, which is consistent with previous results [32, 33, 35]. In contrast, the highest incidence rate of S. aureus was showed in urine specimens [36], wounds infections [37, 38] and nasal swabs specimens [39, 40]. The high occurrence of S. aureus was found in abscess specimens which could be attributed to poor personal hygiene and abscess exposure, making it more susceptible to contamination and infection. In addition, some people in the study region treat their abscesses with self-medication or by employment unqualified or poorly trained quacks before seeking proper medical attention, which might account for the level of settlement by S. aureus. Regarding antibiotic resistance, the current results were explained, all S. aureus isolates were resistant to ampicillin, amoxicillin, and nalidixic acid, these findings are consistent with those published in other studies [41, 42]; this suggests that these antibiotics are no longer successful against infections caused by S. aureus. The other studied antibiotics showed a wide range of resistance like Cefuroxime (94%), Ceftriaxone (89%), Ciprofloxacin (87%), Ceftaxime and Erythromycin (85%), Cephalexin and Clarithromycin (83%), Cefaclor (81%), Gentamicin (74%), Ofloxacin (72%), Chloramphenicol (59%). and Amoxicillin/Clavulanic acid (54%), these data were compatible with some reports [32, 43, 44]. The present data were revealed significantly higher resistance to the tested antibiotics than other studies [33, 34, 45]. As a result of the high prevalence of antibiotic resistance in strains, antibiotics commonly used to treat S. aureus infections may not be sufficient. So, physicians must take into account the care guidelines for MRSA infections. All isolates, on the other hand, were found to be susceptible to Imipinem; this finding was consistent with prior investigations [46, 47] that found 87% and 98% of isolates to be susceptible to Imipinem, respectively. So this medication is still successful and can be used as an additional treatment choice for S. aureus infections in the study area. It's possible that the great prevalence of resistance to the antibiotics mentioned is attributable to their widespread usage in the treatment of human diseases. This suggests that these antibiotics are no longer effective as an empirical treatment for S. aureus infections in the research field. The low activity of these antibiotics could be related to earlier exposure to these drugs that would have accelerated the development of resistance. The rise in antibiotic abuse in our region, which stems from self-medication, failure to react to care, and antibiotic-sale actions, can bolster this assertion.
A long history of effective SE determination in epidemiology is cited in both clinical and environmental settings [48]. Enterotoxin genes were discovered in 39.1% of the isolates. Some investigations [49,50,51] found lower levels of toxigenicity (36%; 43%, and 23%, respectively). However, several studies revealed greater levels of enterotoxigenicity (88%, 76.4%, and 93.5%, respectively) [46, 52, 53]. Only two distinct enterotoxin genes were revealed among five enterotoxin genes, which was consistent with the findings of a previous investigation [50]. The enterotoxin A gene was found in 32.6% of isolates, which was similar to other studies [51, 54,55,56] that showed roughly similar levels of enterotoxin A (41%, 42.9%, 44%, and 40%, respectively). Previous studies [46, 57, 58] reported a greater frequency of the SEA gene (66%, 65.2%, and 60.6%, respectively). In contrast, other reported data [59,60,61] found lower prevalence rates of the SEA gene (15%, 18.8%, and 17%). Enterotoxin B was discovered in 4.3% of testing isolates, which was corroborated by investigations [47, 60,61,62], which indicated that (2%, 5%, 5.1%, and 5%, respectively) of isolates carried the SEB gene. According to several investigations, the detection rate of the SEB gene was 44.3%, 38%, and 19.6%, respectively [6, 46, 51]. The SEB gene was found in 21.6% of bovine mastitis cases [63] and 24% of cutaneous infections [64]. SEA and SEB genes were discovered together in only one isolate (2.1%), which is lower than studies [46, 65], which detected SEA and SEB genes combined in 11% and 22% of isolates, respectively. SEC, SED, and SEE genes were not found in any of the examined isolates; nevertheless, additional findings revealed the existence of the SEC, SED, and SEE genes in isolates [46, 51, 56, 66].
It is well recognized that the presence of toxin genes does not imply the potential to produce toxin [46, 67]. As a result, we used the real-time PCR technique to demonstrate the ability of chosen isolates to express the examined enterotoxin genes. With 100% and 80.66%, respectively, two isolates (10 and 42) showed extraordinarily high levels of expression. Furthermore, the SEB gene expression levels were demonstrated in varied degrees in isolates (5, 8, and 11) with a percentage ranging from 66 to 100%. These findings revealed considerable heterogeneity in the expression of enterotoxins among isolates, which was consistent with prior findings [46]. The discrepancies in the prevalence of enterotoxin encoding genes between studies can be due to a variety of factors, including the origin of the isolates, research locations, hygiene restrictions in different countries, and assay methods. The correlation between enterotoxin and resistance to antibiotics is unknown, but the current study demonstrated a strong connection between the determined toxin pattern distribution and antibiotic resistance and that was statically significant with (P value = 0.01 and > 0.05) for SEA and SEB genes respectively, and this was consistent with prior investigations where more association between level of enterotoxins and resistance to antibiotics [46, 68, 69], although some studies suggested a negative correlation between antibiotic resistance and enterotoxin investigation [70, 71]. Also, one of the most important findings in the current investigation was the pigmentation of isolates; although, the relationship between pigment creation on MSA and enterotoxigencity remains unclear; however, there was a putative link between isolate pigmentation and enterotoxin generation. These findings corroborated those investigations that noticed that SEs genes were more correlated with antibiotic resistance and pigment development [69, 70].
The current research looks at the prevalence of enterotoxin genes in clinical samples, especially the SEA gene, which is followed by the SEB gene. The source of isolation was not related to the enterotoxin genes. On the other hand, the SEA gene was discovered to be closely linked to clinical isolates. The existence of toxin genes does not necessarily mean that the toxin can be produced. Our results suggest that assessing S. aureus potential to cause serious disease requires evaluating the degree of expression of certain toxins at mRNA level. Despite the fact that the genetic relationship between resistance and enterotoxins is poorly understood, evidence from our study indicates that enterotoxin and antibiotic resistance are linked in a significant way.
S. aureus :
SEs:
Staphylococcal enterotoxins
SEA:
Staphylococcal enterotoxin A
SEB:
Staphylococcal enterotoxin B
Staphylococcal enterotoxin C
Staphylococcal enterotoxin D
Staphylococcal enterotoxin E
MSA:
Mannitol salt agar
Threshold of a cycle
Q.RT-PCR:
Quantitative Real Time-PCR
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All appreciation for workers in Plant Protection and Biomolecular Diagnosis Department, City of Scientific Research and Biotechnological Applications, Alexandria, Egypt, for their assistance during the study.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Botany and Microbiology Department, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt
Ahmed A. Baz, Elsayed K. Bakhiet & Usama Abdul-Raouf
Plant Protection and Biomolecular Diagnosis Department, ALCRI, City of Scientific Research and Technological Applications, Alexandria, Egypt
Ahmed Abdelkhalek
Ahmed A. Baz
Elsayed K. Bakhiet
Usama Abdul-Raouf
AB, EB, UA and AA study concept and design, development of the study, data interpretation and manuscript revision and drafting; AB, EB and UA contributed to sample collection, phenotypic studies; AB and AA contributed reagents/materials/analysis tools and molecular studies. All the authors have read and approved of the final version of the manuscript for publication.
Correspondence to Ahmed A. Baz.
The study protocol was approved by the ethics committee, Botany and Microbiology department- the College of Science (Al-Azhar University, Assiut) obtained the approval of the Ethics Committees and the research was conducted following the principles of the Declaration of Helsinki but Ethical approval number: not available. All specimens were collected aseptically and transported to the microbiology laboratory, where they were immediately processed according to the standard microbiological procedures. We would like to confirm that this material is the authors' own original work, which has not been previously published elsewhere. The paper is not currently being considered for publication elsewhere. The paper reflects the authors' own research and analysis in a truthful and complete manner. The paper properly credits the meaningful contributions of co-authors and co researchers. The results are appropriately placed in the context of prior and existing research. All authors have been personally and actively involved in substantive work leading to the manuscript and will hold themselves jointly and individually responsible for its content.
All data generated or analyzed during this study are included in this published article.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Baz, A.A., Bakhiet, E.K., Abdul-Raouf, U. et al. Prevalence of enterotoxin genes (SEA to SEE) and antibacterial resistant pattern of Staphylococcus aureus isolated from clinical specimens in Assiut city of Egypt. Egypt J Med Hum Genet 22, 84 (2021). https://doi.org/10.1186/s43042-021-00199-0
Accepted: 25 September 2021
Enterotoxin | CommonCrawl |
Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations
Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping
Mingqi Xiang 1, , Binlin Zhang 2,, and Die Hu 1,
College of Science, Civil Aviation University of China, Tianjin 300300, China
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
* Corresponding author: Binlin Zhang
Received December 2019 Revised March 2020 Published April 2020
The aim of this paper is to investigate the existence of weak solutions for a Kirchhoff-type differential inclusion wave problem involving a discontinuous set-valued term, the fractional $ p $-Laplacian and linear strong damping term. The existence of weak solutions is obtained by using a regularization method combined with the Galerkin method.
Keywords: Fractional $ p $-Laplacian, Kirchhoff-type problem, differential inclusion, Galerkin approximation.
Mathematics Subject Classification: 35L86, 35L70, 35L20.
Citation: Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034
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Mingqi Xiang Binlin Zhang Die Hu | CommonCrawl |
Treatment of chronic hepatitis B in sub-Saharan Africa: 1-year results of a pilot program in Ethiopia
Hailemichael Desalegn1,
Hanna Aberra1,
Nega Berhe2,3,
Bitsatab Mekasha1,
Kathrine Stene-Johansen4,
Henrik Krarup5,
Andre Puntervold Pereira6,
Svein Gunnar Gundersen7,8 &
Asgeir Johannessen ORCID: orcid.org/0000-0001-5966-71663,9
The World Health Organization has set an ambitious goal of eliminating viral hepatitis as a major public health threat by 2030. However, in sub-Saharan Africa, antiviral treatment of chronic hepatitis B (CHB) is virtually unavailable. Herein, we present the 1-year results of a pilot CHB treatment program in Ethiopia.
At a public hospital in Addis Ababa, CHB patients were treated with tenofovir disoproxil fumarate based on simplified eligibility criteria. Baseline assessment included liver function tests, viral markers, and transient elastography (Fibroscan). Changes in laboratory markers were analyzed using Wilcoxon signed-rank tests. Adherence to therapy was measured by pharmacy refill data.
Out of 1303 patients, 328 (25.2%) fulfilled the treatment criteria and 254 (19.5%) had started tenofovir disoproxil fumarate therapy prior to September 1, 2016. Of the patients who started therapy, 30 (11.8%) died within the first year of follow-up (28 of whom had decompensated cirrhosis), 9 (3.5%) self-stopped treatment, 7 (2.8%) were lost to follow-up, and 4 (1.6%) were transferred out. In patients who completed 12 months of treatment, the median Fibroscan value declined from 12.8 to 10.4 kPa (p < 0.001), 172 of 202 (85.1%) patients with available pharmacy refill data had taken ≥ 95% of their tablets, and 161 of 189 (85.2%) patients with viral load results had suppressed viremia. Virologic failure (≥ 69 IU/mL) at 12 months was associated with high baseline HBV viral load (> 1,000,000 IU/mL; adjusted OR 2.41; 95% CI 1.04–5.55) and suboptimal adherence (< 95%; adjusted OR 3.43, 95% CI 1.33–8.88).
This pilot program demonstrated that antiviral therapy of CHB can be realized in Ethiopia with good clinical and virologic response. Early mortality was high in patients with decompensated cirrhosis, underscoring the need for earlier detection of hepatitis B virus infection and timely initiation of treatment, prior to the development of irreversible complications, in sub-Saharan Africa.
NCT02344498 (ClinicalTrials.gov identifier). Registered 16 January 2015.
Chronic infection with hepatitis B virus (HBV) is a leading cause of cirrhosis, hepatic decompensation, and hepatocellular carcinoma (HCC). Approximately 257 million people worldwide are living with chronic hepatitis B (CHB), and an estimated 887,000 deaths per year are attributable to HBV infection [1]. Despite an effective vaccine and potent antiviral drugs, the number of HBV-related deaths has increased by 33% between 1990 and 2013 [2]. Indeed, in 2015, viral hepatitis claimed more lives than human immunodeficiency virus (HIV) [3].
Studies from high-income countries have shown that antiviral treatment of CHB reduces the risk of disease progression [4]. Long-term antiviral treatment has been shown to stop and even reverse liver fibrosis and prevent development of HCC [5, 6]. Hepatitis B surface antigen (HBsAg) loss is the optimal treatment endpoint, but is rarely achieved; thus, most patients need long-term treatment to suppress viral replication and prevent hepatic complications [7].
In sub-Saharan Africa, treatment for viral hepatitis is rarely available in the public sector. Patients with CHB are left untreated and physicians are left to merely follow the natural course of the disease and provide palliative care. Paradoxically, the recommended first-line drug for CHB, tenofovir disoproxil fumarate (TDF), is registered in most African countries for the treatment of HIV, but not for hepatitis. Therefore, patients mono-infected with HBV cannot access life-saving treatment, whereas those with HIV/HBV co-infection receive free treatment through donor agencies.
In 2016, the World Health Organization (WHO) published the Global Health Sector Strategy of Viral Hepatitis [3] and set an ambitious goal of eliminating viral hepatitis as a public health threat by 2030. To meet the WHO goals, countries and regions should reduce new infection by 90% and deaths by 65% by 2030 [3, 8]. However, to date, few African countries have developed national action plans for viral hepatitis, and only one published study – the PROLIFICA study in The Gambia [9] – has reported results of HBV treatment on the continent. Consequently, there is a lack of local data to direct guidelines and promote implementation.
Herein, we present results from one of the first and largest public treatment programs for hepatitis B in sub-Saharan Africa. Many of the barriers to treatment in Ethiopia, such as lack of diagnostic facilities, absence of public funding, and restrictions on antiviral drugs, are shared by most low-income countries; therefore, we believe that our findings can be relevant in the global scaling-up of antiviral treatment of CHB.
Study setting and participants
Ethiopia is a low-income country in East Africa with an estimated prevalence of hepatitis B at 7.4% [10]. A pilot program for the treatment of CHB was established in February 2015 at St. Paul's Hospital Millennium Medical College, which is a referral hospital in the capital Addis Ababa. A simplified approach to treatment and care of individuals with CHB was employed, wherein transient elastography rather than liver biopsy was used to assess liver fibrosis, nurses rather than physicians were responsible for most of the patient consultations, and treatment eligibility criteria were straight-forward and easy to use.
As this was the first public healthcare facility offering CHB treatment in the country, patients were referred from various hospitals, health centers, blood banks, and antenatal clinics for evaluation and treatment. Patients aged 18 years or above with CHB, defined as a persistently positive HBsAg test for more than 6 months, were enrolled in the program. Since previous HBsAg testing was usually performed in other hospitals or private clinics, and often years back, we accepted the patients' recollection of a previous positive result without demanding a written laboratory report. Patients who were HIV positive at presentation were not included, but rather transferred to the nearest HIV care and treatment center. Likewise, individuals with a known terminal disease such as HCC were referred for further management elsewhere.
Patient assessment and laboratory tests
At enrollment, all patients were interviewed by a trained nurse in their own language. A full diagnostic work-up was performed, including laboratory tests and transient elastography. The following laboratory tests were performed at the first visit:
Point-of-care rapid diagnostic tests: HBsAg, HIV (further tests were not performed in patients found to be HBsAg negative or HIV positive)
Routine chemistry: complete blood count, bilirubin, alanine aminotransferase (ALT), aspartate aminotransferase (AST), creatinine
Serology: HBsAg, hepatitis C virus (HCV) antibody, hepatitis D virus (HDV) antibody
HBV DNA viral load
Patients with signs or symptoms of decompensated liver disease, such as ascites or jaundice, were scheduled to see a physician within 1–2 weeks. These patients would usually receive symptom-directed therapy such as diuretics for ascites/edema; however, within our simplified setup, we did not systematically perform endoscopic treatment of esophageal varices or other more advanced procedures.
Patients without signs or symptoms of advanced liver disease were appointed to a physician after 3 months when the viral load result would usually be available. The decision to start therapy was made by a physician using the predefined criteria given below; otherwise, the follow-up was nurse led.
Those who started antiviral therapy were followed-up after 2 and 4 weeks, and thereafter 3-monthly. The main focus at each visit was adherence counseling (including pill count) and monitoring for side effects. Untreated individuals were followed-up at 3-month intervals.
The following laboratory tests were performed during follow-up (tests in parenthesis were only performed in patients on treatment):
3-monthly: complete blood count, ALT, AST, (creatinine, HIV rapid test)
6-monthly: HBsAg, HBV viral load
Blood tests were performed using commercially available kits and assays. HBsAg was detected on-site using a WHO-approved rapid diagnostic test (Determine, Alere Inc., USA). HIV testing was done in accordance with the National algorithm, i.e., using a WHO-approved rapid test kit (HIV 1+2 Antibody Colloidal Gold [KHB], Shanghai Kehua Bio-engineering co., China) for screening, and another rapid test kit (HIV 1/2 STAT-PAK, Chembio Diagnostics, USA) for confirmation. Other routine laboratory investigations for hematology (HumaCount 30, Human, Germany), biochemistry (Humalyzer 3000, Human, Germany), and serology (Elisys Uno, Human, Germany) were performed locally.
Aspartate aminotransferase to platelet ratio index (APRI) and FIB-4 were derived from standard blood test results using the following formulas:
$$ \mathrm{APRI}:\left(\mathrm{AST}\ \left[\mathrm{U}/\mathrm{L}\right]/\mathrm{upper}\ \mathrm{limit}\ \mathrm{of}\ \mathrm{normal}\ \mathrm{for}\ \mathrm{AST}\right)/\mathrm{platelet}\ \mathrm{count}\ \left({10}^9/\mathrm{L}\right)\times 100 $$
$$ \mathrm{FIB}-4:\left(\mathrm{age}\ \left[\mathrm{years}\right]\times \mathrm{AST}\ \left[\mathrm{U}/\mathrm{L}\right]\right)/\left(\mathrm{platelet}\ \mathrm{count}\ \left[{10}^9/\mathrm{L}\right]\times {\left(\mathrm{ALT}\ \left[\mathrm{U}/\mathrm{L}\right]\right)}^{1/2}\right) $$
HBV viral load testing was unavailable in Ethiopia at the time the program was set up; thus, baseline viral load testing was performed after shipment of samples to the Norwegian Public Health Institute (Oslo, Norway). The Abbott RealTime HBV assay (Abbott Molecular, Des Moines, USA) was used, following the manufacturer's instructions. From 2016, HBV viral load testing was established at a private laboratory in Addis Ababa using the Abbott RealTime HBV assay and therefore all follow-up samples for HBV viral load monitoring were tested locally.
HDV antibodies were detected using an enzyme-linked immunosorbent assay (ELISA) method (ETI-AB-DELTAK-2, Diasorin, Italy) from EDTA plasma samples. A second anti-HDV ELISA assay (Dia.Pro Diagnostic Bioprobes Srl, Milan, Italy) was used to confirm indeterminate or weak positive results obtained with the Diasorin assay, as suggested by the manufacturer when plasma is used instead of serum. These analyses were performed at the Centre national de référence des hépatites B, C et Delta, Hôpitaux universitaires de Paris-Seine-Saint-Denis, France.
Liver fibrosis assessment
Liver fibrosis was assessed using transient elastography (Fibroscan 402, Echosense, France). Patients were instructed to fast for at least 2 h prior to the examination, and the procedure was performed by an experienced operator as per the manufacturer's instructions. The median of 10 readings was employed, and the result was discarded if the interquartile range (IQR) divided by the median exceeded 30%.
Based on a previous meta-analysis and a study from West Africa [11, 12], we used a Fibroscan threshold of 7.9 kPa to define significant fibrosis (corresponding to Metavir score ≥ F2) and 9.9 kPa to define cirrhosis (corresponding to Metavir score F4). Ultrasound of the liver was performed at baseline, and thereafter annually, in all patients who started treatment, mainly to detect HCC.
Treatment eligibility
Since this program opened prior to the launch of the WHO Hepatitis B Guidelines in 2015 [13], treatment eligibility criteria were based on the European Association for the Study of the Liver (EASL) Guidelines from 2012 [7], with some modifications. Specifically, since liver biopsy was unrealistic in this setting, the two EASL criteria pertaining to liver inflammation (Metavir ≥ A2 with viral load > 2000 IU/mL and ALT > 80 U/L with viral load > 20,000 IU/mL) were merged into one, namely ALT > 80 U/L with viral load > 2000 IU/mL. Furthermore, since African patients with CHB are at particular risk of HCC, we created a new criterion for patients with HCC in their close family.
Thus, patients who fulfilled the following criteria were considered eligible for treatment:
Decompensated cirrhosis
Compensated cirrhosis (confirmed with ultrasound and/or Fibroscan)
Significant fibrosis (Fibroscan > 7.9 kPa) and viral load > 2000 IU/mL
Moderate/severe liver inflammation (ALT > 80 U/L) and viral load > 2000 IU/mL
HCC among first-degree relatives and viral load > 2000 IU/mL
Patients who met the treatment criteria were given adherence counseling and educated about the disease and the need for life-long follow-up. Preventive measures were recommended, including HBV vaccination to the patients' partner and children. Based on its potency, safety profile, and high barrier to resistance, TDF (Viread, Gilead Sciences, lnc., Foster City, CA, USA) 300 mg once daily was used in this program. Treatment for HCV and HDV co-infections were not available through this program.
Assessment of adherence to therapy
In patients who started antiviral treatment, TDF was initially dispensed for 1 month's duration, and thereafter at 3-monthly intervals. At each visit to the clinic, the patients were told to bring their old pill boxes so that the remaining pills could be counted. Adherence was calculated by dividing the total amount of tablets dispensed by the total number of days since initiating therapy, expressed as percentage. This method ('pharmacy refill' or 'pill count') has previously been proven accurate in HIV programs in resource-limited settings [14].
Baseline characteristics were summarized using descriptive statistics. Groups were compared using χ2 tests for categorical and Mann–Whitney U-tests for continuous variables. Changes over time in levels of ALT, viral load, and transient elastography were compared using Wilcoxon signed-rank tests. Intra-individual changes in Fibroscan measurements of more than 20% were considered significant [15].
Logistic regression models were used to study associations between baseline variables and clinically relevant outcomes (adherence, virologic failure, death). Variables with a p value below 0.2 in univariable analyses were included in multivariable logistic regression models, using a forward stepwise method. HBV viral load < 69 IU/mL was considered as viral suppression to allow comparison with previous CHB studies [6, 16], and > 1000 IU/mL was considered as major virologic failure.
SPSS version 23.0 software (SPSS Inc., Chicago, IL, USA) was used to analyze the data. The level of significance was set at p < 0.05. Results were reported in accordance with the Strengthening the Reporting of Observational studies in Epidemiology (STROBE) statement guidelines [17].
The study was approved by the National Research Ethics Review Committee (Ref. No.: 3.10/829/07) in Ethiopia and by the Regional Committees for Medical and Health Research Ethics (Ref. No.: 2014/1146) in Norway. The study was conducted in accordance with the Declaration of Helsinki. Written informed consent was obtained from all study subjects.
Patient characteristics
Between February 9 and December 14, 2015, a total of 1303 adults with CHB were enrolled in the program. Of these, 328 (25.2%) fulfilled the treatment criteria and 254 (19.5%) started treatment prior to September 1, 2016 (Fig. 1). Compared to those who were ineligible for treatment, individuals who met the treatment eligibility criteria were more likely to be men, older, anti-HDV positive, and to have elevated ALT, high viral load and increased liver stiffness (Table 1). Patients eligible for treatment but who did not start it were more likely to have a normal liver stiffness (< 8.0 kPa, 28.6 vs. 10.5%, p < 0.001) and a lower APRI score (median 0.31 vs. 0.54, p = 0.013) compared to those who started treatment; other distinguishing features could not be identified.
Profile of the hepatitis B treatment program, Addis Ababa, Ethiopia. CHB chronic hepatitis B
Table 1 Baseline characteristics of patients enrolled in a pilot treatment program for chronic hepatitis B, Addis Ababa, Ethiopia
Among those who started treatment, 197 (77.6%) were men and the median age was 35 years (IQR 27–42). The majority (n = 137, 53.9%) were from the capital city, Addis Ababa. The median ALT at baseline was 36 U/L (IQR 24–50), the median viral load was 26,700 IU/mL (IQR 568–9,480,000), and the median Fibroscan value was 15.5 kPa (IQR 9.0–28.9).
Antiviral treatment
Most patients in this cohort started treatment based on a diagnosis of cirrhosis. A total of 105 (41.3%) patients had clinical ascites or a history of ascites and were classified as decompensated cirrhosis, whereas 81 (31.9%) patients without ascites had a Fibroscan value above 9.9 kPa and were classified as compensated cirrhosis. The remaining started treatment based on significant liver fibrosis (n = 30, 11.8%), moderate/severe liver inflammation (n = 14, 5.5%), HCC in a first-degree relative (n = 3, 1.2%), or other criteria (n = 21, 8.3%). The latter groups mainly comprised patients who had initiated treatment through the private sector or 'black market', and who could not be assessed using the standard criteria, since most of had suppressed viral load at enrollment.
Among patients who initiated treatment, 111 (43.7%) started immediately (i.e., within 4 weeks of enrollment), mainly due to decompensated cirrhosis, whereas 116 (45.7%) started treatment between 1 and 6 months of enrollment. Only 27 (10.6%) started later than 6 months after enrollment.
Overall, 30 (11.8 %) patients died within the first 12 months after starting antiviral treatment, 28 of whom had decompensated cirrhosis at baseline. Among the decompensated patients, neither sex (men vs. women; odds ratio (OR) 1.54, 95% confidence interval (CI) 0.40–5.91, p = 0.531), age (per 1-year increment; OR 1.01, 95% CI 0.97–1.06, p = 0.549), nor co-infections (HCV and/or HDV; OR 1.11, 95% CI 0.20–6.06, p = 0.906) predicted subsequent death.
Another 20 (7.9%) patients failed to complete 12 months of TDF therapy; 7 (2.8%) were lost to follow-up, 9 (3.5%) refused to continue treatment for various reasons, and 4 (1.6 %) were transferred out. In the latter group, 3 were diagnosed with HCC and transferred to palliative care, and 1 was diagnosed with HIV on her 3-month follow-up visit and transferred to HIV care. Baseline characteristics of the 3 patients who developed HCC are summarized in Table 2.
Table 2 Characteristics of patients who developed hepatocellular carcinoma during the initial 12 months of antiviral treatment
Among the remaining 204 patients who completed 1 year of antiviral treatment, the median ALT was 36 U/L (IQR 24–47) at baseline and 32 U/L (IQR 24–39) at 12 months (p = 0.062), and the median Fibroscan value was 12.8 kPa (IQR 8.8–23.6) at baseline and 10.4 kPa (IQR 6.8–17.4) at 12 months (p < 0.001). A total of 140 patients had paired Fibroscan results at baseline and 12 months for comparison; the median intra-individual improvement in liver stiffness at 12 months was 3.4 kPa (IQR 0.4–7.0) (Fig. 2). After 12 months of treatment, 83 (59.3%) patients had a significant improvement in liver stiffness, 41 (29.3%) were unchanged (< 20% change from baseline), and 16 (11.4%) deteriorated.
Change in liver stiffness among patients who completed 12 months of hepatitis B treatment
Five individuals (2.5%) experienced HBsAg loss, i.e., they had two consecutively negative HBsAg results; 4 of these developed anti-HBs antibodies. In retrospect, however, 2 cases might have been acute hepatitis B since HBsAg positivity 6 months prior to inclusion could not be documented.
Out of 189 patients who had a viral load test performed at 12 months, 161 (85.2%) had suppressed viremia (i.e., <69 IU/mL) and only 6 (3.2%) had major virologic failure (i.e., > 1000 IU/mL). Genotypic resistance testing was performed in these 6 samples; 3 had wild type virus and 3 failed amplification due to low viral loads. The proportion of patients with suppressed viremia at 6 and 12 months is given in Fig. 3. Virologic failure (i.e., ≥ 69 IU/mL) at 12 months was associated with high baseline HBV viral load (> 1,000,000 IU/mL) and suboptimal adherence (< 95%) (Table 3).
Virologic response to therapy during the first 12 months of antiviral treatment
Table 3 Predictors of virologic failure (HBV viral load ≥ 69 IU/mL) after 12 months of antiviral treatment, Addis Ababa, Ethiopia
TDF therapy was generally well tolerated and only one individual with underlying comorbidities (diabetes mellitus, hypertension, gout, cardiomyopathy, alcohol abuse) discontinued due to progressive worsening of renal function. This patient was under treatment with several drugs, including diuretics and an angiotensin-converting enzyme inhibitor, and the role of TDF in the development of renal failure remains unclear.
Pharmacy refill data was available for 202 out of 204 patients who completed 12 months of antiviral treatment. Out of these, 172 (85.1%) individuals had excellent adherence, i.e., they took more than 95% of their tablets; 24 (11.9%) had medium adherence, i.e., they took between 80 and 95% of their tablets; and 6 had poor adherence, i.e., they took less than 80% of their tablets. Neither sex (men vs. women; OR 1.67, 95% CI 0.60–4.62, p = 0.326), age (per 1-year increment; OR 0.98, 95% CI 0.94–1.02, p = 0.221), nor decompensated liver disease (OR 1.11, 95% CI 0.20–6.06, p = 0.906) were significantly associated with suboptimal adherence (i.e., < 95%).
This report summarizes 1-year treatment results from the largest published cohort of CHB patients in sub-Saharan Africa. The findings indicate that antiviral treatment can be delivered safely and effectively in a low-income setting like Ethiopia using a simplified approach, similar to the early experiences from HIV treatment programs on the continent. In our own setup, the day-to-day management of patients, including the initial patient interview, liver stiffness measurement, blood testing and patient education, was performed by trained nurses, whereas physicians were involved only in the actual decision to start therapy and in the management of complications. Such task shifting, from physicians to trained nurses, has been successful in HIV programs throughout Africa [18], and may also prove useful in the management of other chronic infections like CHB.
The estimated 1-year mortality among patients who started treatment was 11.8%. The high early mortality likely reflects the absence of treatment options for CHB in the country to date; thus, patients with advanced cirrhosis, who are desperate for treatment, might have been overrepresented in our cohort. Indeed, 28 of 30 patients who died had decompensated cirrhosis at enrollment. On the other hand, approximately three-quarters of patients with decompensated cirrhosis in our study were still alive after 12 months of TDF treatment. These results are in line with a study from Korea [19], where 1-year transplantation-free survival was of 87.1% among 70 patients with decompensated cirrhosis treated with entecavir, suggesting that treatment of this group is safe and beneficial. Clearly, however, a major challenge in the Ethiopian setting is the identification of HBV-infected individuals earlier in the course of their illness, prior to the development of severe complications. This would require better access to HBV testing and screening, and improved access to antiviral therapy throughout the country.
In our study, 85.2% of patients achieved viral suppression 12 months after treatment initiation. Only 6 patients had major virologic failure and no genotypic resistance was discovered. This is comparable to studies from high-income settings. Indeed, in the phase 3 study of TDF supported by Gilead Sciences, viral suppression below 69 IU/mL was achieved in 76% and 93% of HBeAg-positive and HBeAg-negative patients, respectively [16]. Moreover, in a real-life multicenter study of 302 CHB patients from 19 countries in Europe, 68% of HBeAg-positive patients and 90% HBeAg-negative patients treated with TDF had a suppressed viral load after 12 months [20]. Studies from low- and middle-income countries are scarce, but in a retrospective study involving 220 cirrhotic patients from India [21], 91.8% had suppressed viremia after 12 months treatment with TDF. Furthermore, in a large real-life study from 50 sites in China [22], viral suppression after 12 months treatment with entecavir was achieved in 64% of patients with compensated liver disease and 68% with decompensated liver disease. Of note, detectable viremia at 12 months does not necessarily imply treatment failure, since it might take longer than 12 months to achieve full viral suppression in individuals with very high baseline viral loads [20].
Interestingly, liver stiffness (measured by Fibroscan) improved significantly during the initial 12 months of therapy. Reversal of liver fibrosis during antiviral therapy has previously been described by Marcellin et al. [6], who reported a significant improvement in fibrosis scores in repeated liver biopsies after 1 and 5 years of TDF treatment. Whether histological improvement translates to reduced risk of HCC and death has yet to be proven in Africa, but experiences from other settings indicate that antiviral treatment significantly reduces the risk of these complications [5].
The program loss in the present study was relatively low; overall, 6.3% were lost to follow-up or self-stopped treatment. Although the follow-up time in our study was shorter, this drop-out rate was lower than results from a recent multicenter study in Germany involving 33 sites across the country [23], where 75% remained in the study after 36 months of TDF treatment. In the only previous publication of CHB treatment in sub-Saharan Africa [9], there was no program loss after 12 months; however, the numbers were small.
Adherence to therapy was high in this cohort; 85.1% of patients took at least 95% of their medication; this is in line with studies from high-income countries. In a study from USA [24], non-adherence was reported in 10–12% of patients after 4 years of HBV therapy, whereas a study from France found non-adherence among 7% of patients after at least 3 months of CHB treatment [25]. High adherence is a prerequisite to achieve the clinical benefits of therapy and to avoid resistance; however, contrary to first-generation anti-HBV drugs, resistance does not seem to be a significant problem with TDF [26]. Previous studies from HIV programs have found that adherence rates in Africa are at least as good as in North America or Europe [27]; our study suggests that the same appears to be true for CHB treatment.
In the present program, treatment decisions were based on viral load testing and liver stiffness measurements. However, in most countries in sub-Saharan Africa, these tests are practically unavailable. Although the WHO recommends the initiation of treatment based on clinical and laboratory markers such as the APRI, experiences from our cohort shows that this marker fails to detect 90% of those in need of treatment [28]. Transient elastography, on the contrary, has shown excellent agreement with liver biopsy [29], and is non-invasive and easy to use, making it an appealing tool in resource-limited settings. Nevertheless, the retail price of a Fibroscan machine is currently beyond the budget of most low-income settings, for which generic competition or other financial mechanisms to improve access to this technology is an urgent priority.
With regard to viral load measurements, we have previously shown that dried blood spots can be used to reliably quantify HBV DNA [30], which means that samples can be sent to a central facility from smaller centers throughout the country. Additionally, the recent development of point-of-care molecular assays for various pathogens represent a new era of near-patient testing, and GeneXpert (Cepheid, Sunnyvale, CA, USA) is expected to launch their HBV DNA viral load kit later in 2018 (personal communication, Emiliano Leone, Cepheid). Improved access to low-cost, robust, point-of-care diagnostics will be a prerequisite to achieve the WHO goal of treating 80% of eligible persons with CHB within 2030 [3]. However, reaching this goal will also require concerted action from international stakeholders and local ministries of health to establish adequate funding mechanisms, develop local treatment guidelines, and remove legislative barriers to generic antiviral drugs.
Our study had certain limitations. First, although the treatment program was set up at a local public hospital, external financial support was provided to purchase a Fibroscan device and run viral load testing. Nevertheless, the clinic was run exclusively with local staff, most of whom had no previous experience with hepatitis B management, and we believe the setup can be duplicated in other countries in sub-Saharan Africa. Second, since this was the first treatment center for CHB in the country, patients with symptomatic (i.e., decompensated) liver disease were overrepresented. The high early mortality in the program reflects this, and as treatment becomes more accessible mortality can be expected to decline.
The current pilot study showed that treatment for CHB can be successful in terms of adherence to therapy and retention in care. Furthermore, there was significant improvement in liver stiffness during the initial 12 months of treatment, with the majority of patients achieving viral suppression. However, initial mortality was high, given that many patients presented with advanced cirrhosis. Therefore, earlier detection of HBV infection and timely initiation of treatment – prior to the development of irreversible complications – is needed to reach the WHO goal of eliminating viral hepatitis as a public health threat by 2030. This pilot program can provide valuable information for other African countries aiming to expand access to antiviral treatment of CHB.
ALT:
alanine aminotransferase
APRI:
aspartate aminotransferase to platelet ratio index
AST:
aspartate aminotransferase
CHB:
chronic hepatitis B
EASL:
European Association for the Study of the Liver
HBsAg:
hepatitis B surface antigen
HBV:
HCC:
HCV:
HDV:
hepatitis D virus
HIV:
IQR:
interquartile range
TDF:
tenofovir disoproxil fumarate
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We are indebted to the patients who participated in the study. We are thankful to the staff at the hepatitis clinic at St Paul's Hospital Millennium Medical College, as well as the laboratory staff at Aklilu Lemma Institute of Pathobiology and the Norwegian Institute of Public Health for their dedication and hard work.
This study was funded by The Norwegian Research Council, grant number 220622/H10, and the South-Eastern Norway Regional Health Authority, grant number 2011068. Antiviral drugs were donated by Gilead Sciences, Inc., Foster City, CA, USA. The sponsors had no role in the design, data collection, data analysis, data interpretation or writing of this paper, nor in the decision to submit the paper for publication.
Medical Department, St. Paul's Hospital Millennium Medical College, Addis Ababa, Ethiopia
Hailemichael Desalegn, Hanna Aberra & Bitsatab Mekasha
Aklilu Lemma Institute of Pathobiology, Addis Ababa University, Addis Ababa, Ethiopia
Nega Berhe
Centre for Imported and Tropical Diseases, Oslo University Hospital, Ullevål, PO Box 4956 Nydalen, 0424, Oslo, Norway
Nega Berhe & Asgeir Johannessen
Department of Molecular Biology, Norwegian Institute of Public Health, Oslo, Norway
Kathrine Stene-Johansen
Section of Molecular Diagnostics, Aalborg University Hospital, Aalborg, Denmark
Henrik Krarup
Faculty of Medicine, University of Oslo, Oslo, Norway
Andre Puntervold Pereira
Research Unit, Sørlandet Hospital HF, Kristiansand, Norway
Svein Gunnar Gundersen
Department of Global Development and Planning, University of Agder, Kristiansand, Norway
Department of Infectious Diseases, Vestfold Hospital Trust, Tønsberg, Norway
Asgeir Johannessen
Hailemichael Desalegn
Hanna Aberra
Bitsatab Mekasha
AJ conceived the study and wrote the protocol with significant contributions from NB and SGG. HD, HA, and BM were responsible for patient enrollment and data acquisition. KSJ and HK were responsible for the laboratory work and interpretation of results. APP collected adherence data and participated in data analysis. AJ and HD performed statistical analyses. AJ and HD drafted the manuscript, and all authors critically revised it and approved the final version.
Correspondence to Asgeir Johannessen.
Ethics, consent and permissions
The study was approved by the Regional Committee for Medical and Health Research Ethics in Norway and the National Research Ethics Review Committee in Ethiopia, as well as the pertinent institutional ethical review boards. Written informed consent was obtained from all study participants.
Desalegn, H., Aberra, H., Berhe, N. et al. Treatment of chronic hepatitis B in sub-Saharan Africa: 1-year results of a pilot program in Ethiopia. BMC Med 16, 234 (2018). https://doi.org/10.1186/s12916-018-1229-x
Accepted: 30 November 2018
DOI: https://doi.org/10.1186/s12916-018-1229-x
Antiviral therapy | CommonCrawl |
Some results on best proximity point on star-shaped sets in probabilistic Banach (Menger) spaces
Hamid Shayanpour1,
Maryam Shams1 &
Asiyeh Nematizadeh1
Fixed Point Theory and Applicationsvolume 2016, Article number: 13 (2016) | Download Citation
We first present the concepts of proximal contraction and proximal nonexpansive mappings on star-shaped sets in probabilistic Banach (Menger) spaces. We derive some results about the best proximity points for these mappings in probabilistic Banach (Menger) spaces. Next, we bring some examples that defend our main results.
Introduction and preliminaries
The equation $Tx = x$ for a mapping $T: A \rightarrow B $ may have no solution whenever $A\cap B =\emptyset$, where A, B are two nonempty subsets in a metric space $(X, d)$. Under this condition, it is beneficial to determine a point $a_{0} \in A $ such that $d(a_{0} , Ta_{0}) $ is minimal. If $d(a_{0} , Ta_{0})$ is the global minimum value of $\operatorname{dist}(A,B)$, i.e., $d(a_{0} , Ta_{0}) = \operatorname{dist}(A,B) = \min\lbrace d(a, b) : a \in A, b\in B\rbrace$, then $a_{0}$ is called best proximity point of T.
In 1969, Fan [1] proved one of the most classical theorems in best approximation theory. He showed that if $(V, \rho)$ is a topological vector space with seminorm p, $W\subseteq V$, and $T : W \rightarrow V$ is a mapping, then under certain conditions, there exists an element $w_{0}\in W$ such that
$$\rho(w_{0}-Tw_{0})=d(Tw_{0}, W). $$
Thereafter, this theorem has been generalized for continuous multivalued mappings by Reich [2, 3] and Sehgal and Singh [4].
Eldred et al. [5] showed that every relatively nonexpansive mapping has a proximal point under certain conditions. For further existence results of a best proximity point for several types of contractions, we refer to [6–25].
In 1942, a probabilistic metric (PM) space was introduced by Menger [26]. Schweizer and Sklar [27, 28] were two pioneers in the study of PM spaces.
PM spaces are very useful in probabilistic functional analysis, quantum particle physics, $\epsilon^{\infty} $ theory, nonlinear analysis, and applications; see [29–33].
Indeed, the study of fixed point results in PM spaces is one of the most active research areas in fixed point theory. Sehgal and Bharucha-Reid [34] were two pioneers in this study. For further existence results of a fixed point and common fixed point in PM spaces, we refer, for example, to [35–37]. In 2014, Su and Zhang [38], proved some best proximity point theorems in PM spaces.
Let $\Delta^{+}$ be the set of all distribution functions F (i.e., a nondecreasing and left-continuous function $F : \mathbb{R} \rightarrow [0, 1]$ such that $\inf_{t\in\mathbb{R}} F(t) = 0$ and $\sup_{t\in \mathbb{R}} F(t) = 1$) such that $F(0) = 0$. Let X be a nonempty set, $\epsilon_{0}=\chi_{(0,\infty)}\in\Delta ^{+}$, and $F: X \times X\rightarrow\Delta^{+}$ ($F(p,q)=F_{p,q}$) be a mapping such that
(PM1)
$F_{p,q}=\epsilon_{0}$ iff $p=q$,
$F_{ p,q}=F_{q,p}$, and
if $F_{p,q}(t) = 1$ and $F_{q,r}(s) = 1$, then $F_{p,r}(t+s) = 1$
for all $p, q, r\in X$ and $t,s\geq0$. Then $(X, F)$ is called a probabilistic metric space.
For well-known definitions (such as t-norm, t-norm of H-type, probabilistic Menger space, complete probabilistic Menger space, probabilistic normed (PN) space, etc.) and known results, we refer to [27, 39].
First, we state some notation, definitions, and known results; afterward, we introduce concepts of proximal contraction, proximal nonexpansive, P-property, weak P-property, and semisharp proximinal pair in PM spaces. Throughout this paper, the minimum t-norm will be denoted by $\Delta _{m}(a,b)=\min\{a,b\}$.
Let $( x_{n}) $ be a sequence in a probabilistic Menger space $(X, F,\Delta) $ such that Δ is a t-norm of H-type. If
$$F_{x_{n},x_{n+1}}(kt)\geq F_{x_{n-1},x_{n}}(t) \quad (n\geq1, t>0) $$
for some $k \in(0,1) $, then $( x_{n}) $ is a Cauchy sequence.
Suppose that A is a nonempty subset of a probabilistic Menger space $(X, F,\Delta)$. Then the probabilistic diameter of A is the mapping $D_{A}$ defined on $[0,\infty]$ by $D_{A}(\infty)=1$ and $D_{A}(x)=\lim_{t\rightarrow x^{-}}\varphi_{A}(t)$, where $\varphi_{A} (t)=\inf\{F_{a,b}(t): a,b\in A\}$.
A nonempty set A in a probabilistic Menger space is bounded if $\lim_{x\rightarrow\infty}D_{A}(x)=1$. It is easy to see that $F_{a,b}(t)\geq D_{A}(t)$ for all $a,b\in A$ and $t\geq0$.
Let $(X, F,\Delta)$ be a probabilistic Menger space, $A\subseteq X $, and $T:A\rightarrow A$ be a mapping. The mapping T is said to be an isometry if
$$F_{Tx,Ty}(t)=F_{x,y}(t) \quad \forall x,y\in X, \forall t\geq0. $$
Let $(X, F,\Delta)$ be a probabilistic Menger space, and $A, B\subseteq X $. A mapping $T:A\rightarrow B$ is said to be continuous at $x\in A$ if for every sequence $(x_{n})$ in A that converges to x, the sequence $(Tx_{n})$ in B converges to Tx.
If T is an isometry mapping on subset A of a probabilistic Menger space $(X,F, \Delta) $, then T is a continuous mapping because
$$F_{Tx_{n}, Tx}(t)=F_{x_{n},x}(t)\rightarrow1 \quad \forall t>0. $$
Also, it is easy to see that T is an injective mapping.
An immediate consequence of the definition of a PN space ([27], Section 15.1) is the following lemma.
Let $(X,\nu, \Delta) $ be a PN space, and $F^{\nu}$ be the function from $X\times X$ into $\Delta^{+} $ defined by
$$F^{\nu}(p,q)=\nu_{p-q}. $$
Then $(X,F^{\nu}, \Delta) $ is a probabilistic Menger space.
We call this probabilistic metric $F^{\nu}$ on X the probabilistic metric induced by the probabilistic norm ν.
A PN space $(X,\nu, \Delta) $ is said to be a probabilistic Banach space if $(X,F^{\nu}, \Delta) $ is a complete probabilistic Menger space.
Let A, B, C be a nonempty subsets of a PN space $(X,\nu, \Delta) $ such that Δ is continuous t-norm and $x\in A$. If two mappings $T:A\rightarrow B$ and $S:A\rightarrow C$ are continuous at x, then $T+S$ is continuous at x because
$$\nu_{(T+S)(x)-(T+S)(x_{n})}(t)\geq\Delta\biggl(\nu_{T(x)-T(x_{n})}\biggl(\frac {t}{2} \biggr),\nu_{S(x)-S(x_{n})}\biggl(\frac{t}{2}\biggr)\biggr)\rightarrow1 \quad \forall t>0. $$
Let A be a nonempty subset of a PM space $(X,F)$. A mapping $T:A\rightarrow X $ is called a contraction (nonexpansive) if $F_{Tx,Ty}(t)\geq F_{x,y} (\frac{t}{\alpha} )$ ($F_{Tx,Ty}(t)\geq F_{x,y} (t)$) for some $0<\alpha<1 $ and for all $x,y\in A $ and $t>0 $.
Definition 1.10
Suppose that A and B are nonempty subsets of a PM space $(X,F)$. Then the probabilistic distance of A, B is the mapping $F_{A,B}$ defined on $[0,\infty]$ by
$$F_{A,B}(t)=\sup_{x\in A, y\in B}F_{x,y}(t) \quad \forall t\geq0 . $$
Also, if A and B are nonempty subsets of a PN space $(X,\nu, \Delta ) $, then ${F^{\nu}}_{A,B}(t)=\nu_{A-B}(t)=\sup_{x\in A, y\in B}\nu _{x-y}(t)$, where $F^{\nu}$ is the probabilistic metric induced by the probabilistic norm ν.
Let $(X, F)$ be a PM space. For subsets A and B of X, define:
$$\begin{aligned}& A_{0}=\bigl\lbrace x\in A :\exists y\in B \text{ s.t. } \forall t \geq0, F_{x,y}(t)=F_{A,B}(t)\bigr\rbrace , \\& B_{0}=\bigl\lbrace y\in B :\exists x\in A \text{ s.t. } \forall t \geq0, F_{x,y}(t)=F_{A,B}(t)\bigr\rbrace . \end{aligned}$$
Clearly, if $A_{0}$ (or $B_{0}$) is a nonempty subset, then A and B are nonempty subsets.
Let $(X, F) $ be a PM space, and $(A,B) $ be a pair of nonempty subsets of X. A mapping $T : A\rightarrow B $ is called the proximal contraction (proximal nonexpansive) if there exists a real number $0<\alpha<1 $ such that
$$\begin{aligned}& F_{u,Tx}(t)=F_{A,B}(t)=F_{v,Ty}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{x,y}\biggl( \frac{t}{\alpha}\biggr) \\& \bigl(F_{u,Tx}(t)=F_{A,B}(t)=F_{v,Ty}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{x,y}(t) \bigr) \end{aligned}$$
for all $u ,v,x,y\in A$ and $t>0$.
Example 1.13
Let $X=[0,2] $, and $T:X\rightarrow X$ be the mapping defined by $Tx=\frac{1}{8}x $. If $F_{x,y}(t)=\frac{t}{t+|x-y|} $, then it is easy to check that $F_{X,X}(t)=1 $. If $F_{u,Tx}(t)=1=F_{v,Ty}(t)$, then for $\alpha=\frac{1}{8} $, we have $F_{u,v}(t)= F_{x,y}(\frac{t}{\alpha})$, where $u,v,x,y \in X$. Therefore, T is a proximal contraction.
Let X be a vector space, and A be a nonempty subset of X. Then the subset A is called a p-star-shaped set if there exists a point $p \in A $ such that $\alpha p + (1-\alpha)x \in A$ for all $x\in A$, $\alpha\in [0,1] $, and p is called the center of A.
Clearly, each convex set C is a p-star-shaped set for each $p \in C $. Let $(X,\nu,\Delta_{m} )$ be a PN space, A be a p-star-shaped set, B be a q-star-shaped set, and $\nu_{p - q}=\nu_{A-B}$. If $x\in A_{0}$, then there exists a point $y\in B$ such that $\nu _{x-y}(t)=\nu_{A-B}(t)$ for all $t>0$. So we have
$$\begin{aligned} \nu_{A-B}(t) \geq& \nu_{(\alpha p+(1-\alpha)x)-(\alpha q +(1-\alpha )y)}(t) \\ \geq& \Delta_{m}\bigl(\nu_{\alpha(p-q)}(\alpha t), \nu_{(1-\alpha )(x-y)}\bigl((1-\alpha)t\bigr)\bigr) \\ =& \Delta_{m}\bigl(\nu_{p-q}( t),\nu_{ x-y }(t) \bigr) \\ =& \Delta_{m}\bigl(\nu_{A-B}( t),\nu_{A-B}(t) \bigr) \\ =&\nu_{A-B}( t) \end{aligned}$$
for all $t>0$. Therefore, $\nu_{(\alpha p+(1-\alpha)x)-(\alpha q +(1-\alpha)y)}(t)=\nu_{A-B}( t)$, which means that $A_{0} $ is a p-star-shaped set and, similarly, that $B_{0} $ is a q-star-shaped set.
Let $(X, F) $ be a PM space. A pair $(A,B) $ of nonempty subsets of X is said to have the P-property (weak P-property) if $A_{0}\neq\emptyset $ and
$$\begin{aligned}& F_{u,x}(t)=F_{A,B}(t)=F_{v,y}(t) \quad \Longrightarrow\quad F_{u,v}(t)= F_{x,y}(t) \\& \bigl(F_{u,x}(t)=F_{A,B}(t)=F_{v,y}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{x,y}(t) \bigr) \end{aligned}$$
for all $u,v\in A_{0}$, $x,y\in B_{0}$, and $t>0$.
Let $X=\mathbb{R}^{2}$ and define
$$F_{(x,y),(u,v)}(t)=\frac{t}{t+\sqrt{(x-u)^{2}+(y-v)^{2}}}. $$
Clearly, $(X,F,\Delta_{m})$ is a complete probabilistic Menger space. Let
$$\begin{aligned}& A=\biggl\{ \biggl(0,\frac{1}{n}\biggr) : n\in\mathbb{N}\biggr\} \cup\bigl\{ (0,0)\bigr\} , \\& B=\biggl\{ \biggl(1,\frac{1}{n}\biggr) : n\in \mathbb{N}\biggr\} \cup\bigl\{ (1,0)\bigr\} . \end{aligned}$$
Then it is easy to check that $A_{0}=A $, $B_{0}=B $, and $F_{A,B}(t)=\frac{t}{t+1} $. If
$$F_{(0,x),(1,y)}(t)= F_{A,B}(t)=\frac{t}{t+1}=F_{(0,u),(1,v)}(t), $$
then $x=y $ and $u=v $, so that
$$F_{(0,x),(0,u)}(t)=\frac{t}{t+|x-u|}=\frac{t}{t+|y-v|}=F_{(1,y),(1,v)}(t). $$
Therefore, the pair $(A, B)$ has the P-property.
Let $A=\lbrace(0,0)\rbrace$ and $B=\lbrace(x,y)\in X : y=1+\sqrt {1-x^{2}} \rbrace$. Clearly, $A_{0}=\{(0,0)\} $ and $B_{0}=\{ (-1,1),(1,1)\} $. If
$$F_{(0,0),(x,y)}(t)=F_{A,B}(t)=\frac{t}{t+\sqrt{2}}=F_{(0,0),(u,v)}(t), $$
$$1=F_{(0,0),(0,0)}(t)\geq F_{(x,y),(u,v)}(t), $$
where $(x,y),(u,v)\in B_{0} $. Therefore, the pair $(A,B) $ has the weak P-property.
Let $(X, F) $ be a PM space. A pair $(A,B) $ of nonempty subsets of X is called a semisharp proximinal pair if there exists at most one $(x_{0}, y_{0}) \in A \times B$ such that $F_{x, y_{0}}(t) = F_{A,B}(t)=F_{x_{0}, y}(t) $ for all $(x, y) \in A \times B $.
It is easy to check that if a pair $(A,B) $ has the P-property, then the pair $(A,B) $ is a semisharp proximinal pair. Clearly, a semisharp proximinal pair $(A,B) $ does not necessarily have the P-property.
Suppose that $X =\mathbb{R} $, $A=\{-10,10\} $, $B=\{-2,2\} $, and $F_{x,y}(t)=\frac{t}{t+|x-y|} $. It is easy to verify that $F_{A,B}(t)=\frac{t}{t+8} $, $A_{0}= A$, $B_{0}= B$, and $( A,B) $ is a semisharp proximinal pair but does not have the P-property.
Remark 1.20
It is easy to check that the P-property is stronger than the weak P-property. If a pair $(A,B) $ has the weak P-property and $T : A\rightarrow B $ is a nonexpansive mapping, then for all $u,v, x,y\in A$, we have
$$ F_{u,Tx}(t)=F_{A,B}(t)=F_{v,Ty}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{Tx,Ty}(t)\geq F_{x,y}(t). $$
That is, T is a proximal nonexpansive mapping. Similarly, if a pair $(A,B) $ has the weak P-property and $T : A\rightarrow B $ is a contraction mapping, then T is a proximal contraction mapping. Also, a pair $(A,B) $ has the P-property if and only if both pairs $(A,B) $ and $(B,A) $ have the weak P-property.
Let X and Y be vector spaces. A mapping $T:X\rightarrow Y$ is affine if
$$T\Biggl(\sum_{i=1}^{n}\lambda_{i} x_{i}\Biggr)=\sum_{i=1}^{n} \lambda_{i} T(x_{i}) $$
for all $n\in\mathbb{N}$, $x_{1},\ldots,x_{n}\in X$, and $\lambda_{1},\ldots , \lambda_{n} \in\mathbb{R}$ such that $\sum_{i=1}^{n}\lambda_{i}=1$.
In Section 2, we show some results on the best proximity points in probabilistic Banach (Menger) spaces. For example, if $(A,B) $ is a semisharp proximinal pair of a probabilistic Banach space $(X, \nu,\Delta_{m}) $ such that A is a p-star-shaped set, $A_{0}$ is a nonempty compact set, B is a q-star-shaped set and $\nu_{p - q}(t)=\nu_{A-B}(t)$ for all $t>0$, then every proximal nonexpansive mapping $T : A\rightarrow B $ with $T(A_{0} )\subseteq B_{0}$ has a best proximity point. We also prove that if A is a nonempty, compact, and convex subset of a probabilistic Banach space $(X, \nu,\Delta_{m}) $ and $T:A\rightarrow A $ is a nonexpansive mapping, then T has a fixed point. Finally, we give some examples which defend our main results.
Proximity point for proximal contraction and proximal nonexpansive mappings
We first give the following lemma and then we state the main results of this paper. We recall that if $A_{0}$ (or $B_{0}$) is a nonempty subset, then A and B are nonempty subsets.
Let $(X, F,\Delta) $ be a complete probabilistic Menger space such that Δ is a t-norm of H-type, and $A,B \subseteq X $ be such that $A_{0} $ is a nonempty closed set. If $T : A\rightarrow B $ is a proximal contraction mapping such that $T(A_{0} )\subseteq B_{0}$, then there exists a unique $x\in A_{0} $ such that $F_{x,Tx}(t)=F_{A,B}(t) $ for all $t>0$.
Since $A_{0} $ is nonempty and $T(A_{0} )\subseteq B_{0}$, there exist $x_{1},x_{0}\in A_{0}$ such that $F_{x_{1},Tx_{0}}(t)=F_{A,B}(t) $. Since $Tx_{1}\in B_{0} $, there exists $x_{2}\in A_{0}$ such that $F_{x_{2},Tx_{1}}(t)=F_{A,B}(t) $. Continuing this process, we obtain a sequence $( x_{n})\subseteq A_{0} $ such that $F_{x_{n+1},Tx_{n}}(t)=F_{A,B}(t) $ for all $n\in\mathbb {N} $ and $t>0$. Since for all $n\in\mathbb{N} $,
$$F_{x_{n},Tx_{n-1}}(t)=F_{A,B}(t)= F_{x_{n+1},Tx_{n}}(t) \quad (t>0) $$
and T is a proximal contraction, we have
$$F_{x_{n+1},x_{n}}(t)\geq F_{x_{n},x_{n-1}}\biggl(\frac{t}{\alpha}\biggr)\quad (0< \alpha< 1, t>0). $$
Therefore, by Lemma 1.1, $( x_{n}) $ is a Cauchy sequence and so converges to some $x\in A_{0} $. Again by the assumption $T(A_{0} )\subseteq B_{0}$, $Tx \in B_{0}$. Then there exists an element $u\in A_{0} $ such that $F_{u,Tx}(t)=F_{A,B}(t)$ for all $t>0$. Since for all $n\in\mathbb{N} $,
$$F_{u,Tx}(t)=F_{A,B}(t)=F_{x_{n+1},Tx_{n}}(t)\quad (t>0), $$
by the hypothesis we have
$$F_{u,x_{n+1}}(t)\geq F_{x,x_{n}}\biggl(\frac{t}{\alpha}\biggr)\geq F_{x,x_{n}}(t) \quad (t>0). $$
Letting $n\rightarrow\infty$ shows that $x_{n}\rightarrow u $ and thus $x =u$, so $F_{x,Tx}(t)=F_{A,B}(t) $. If there exists another element y such that $F_{y,Ty}(t)=F_{A,B}(t) $, then by the hypothesis we have $F_{x,y}(t)\geq F_{x,y}(\frac{t}{\alpha}) $, which means that $x=y$. □
Let $(X, F,\Delta) $ be a probabilistic Menger space, and $A,B \subseteq X $ be such that $A_{0} $ is a nonempty set. Suppose that $T : A\rightarrow B $ is a proximal contraction mapping such that $T(A_{0} )\subseteq B_{0}$ and $g : A\rightarrow A $ is an isometry mapping such that $A_{0}\subseteq g(A_{0}) $. Denote $G = g(A)$ and
$$G_{0} =\bigl\lbrace z\in G : \exists y\in B \textit{ s.t. } \forall t \geq0, F_{z,y}(t)=F_{G,B}(t)\bigr\rbrace . $$
Then $Tg^{-1}$ is a proximal contraction, and $G_{0}=A_{0} $.
Since $G\subseteq A $, $F_{G,B}(t)\leq F_{A,B} (t)$ for all $t>0 $. Assume that $x\in A_{0} \subseteq g(A_{0})$. Then $x=g(x^{\prime}) $ for some $x^{\prime} \in A_{0}$, and so there exists $y\in B $ such that $F_{A,B}(t)=F_{g(x^{\prime}),y}(t)\leq F_{G,B}(t) $ for all $t>0 $. Thus, $F_{A,B}(t)=F_{G,B}(t) $ for all $t>0 $. Now we show that $Tg^{-1} $ is a proximal contraction. To this end, suppose that $u,v,x,y\in G $ are such that
$$F_{u,Tg^{-1}x}(t)=F_{G,B}(t)=F_{A,B}(t)=F_{v,Tg^{-1}y}(t) \quad (t>0). $$
$$F_{u,v}(t)\geq F_{g^{-1}x,g^{-1}y}\biggl(\frac{t}{\alpha } \biggr)=F_{gg^{-1}x,gg^{-1}y}\biggl(\frac{t}{\alpha}\biggr)=F_{x,y}\biggl( \frac{t}{\alpha }\biggr) \quad (t>0) $$
for some $\alpha\in(0,1) $. Therefore, $Tg^{-1}$ is a proximal contraction. If $x\in G_{0} $, then $x\in G\subseteq A$, and there exists $y\in B $ such that $F_{x,y}(t)=F_{G,B}(t)=F_{A,B}(t)$ for all $t>0 $, so that $x\in A_{0} $. If $x\in A_{0} \subseteq A$, then there exists $y\in B $ such that $F_{x,y}(t)=F_{A,B}(t)=F_{G,B}(t)$ for all $t>0 $. On the other hand, by the hypothesis $x\in G $, and therefore $G_{0}=A_{0} $. □
Corollary 2.3
Let the hypotheses of Lemma 2.1 be satisfied. Suppose that $T : A\rightarrow B $ is a proximal contraction mapping such that $T(A_{0} )\subseteq B_{0}$ and $g: A\rightarrow A$ is an isometry mapping such that $A_{0}\subseteq g(A_{0}) $. Then there exists a unique $x\in A_{0} $ such that $F_{gx,Tx}(t)=F_{A,B}(t) $.
By Proposition 2.2, $Tg^{-1}:G=g(A)\rightarrow B$ is proximal contraction, and $Tg^{-1}(G_{0})=Tg^{-1}(A_{0})\subseteq T(A_{0})\subseteq B_{0}$. Now by Lemma 2.1 there exists a unique $x'\in A_{0} $ such that $F_{x',Tg^{-1}x'}(t)=F_{A,B}(t) $. Since $A_{0}\subseteq g(A_{0})$, there exists $x\in A_{0} $ such that $x'=g(x)$, so that $F_{g(x),Tx}(t)=F_{A,B}(t) $. Note that g is an injective mapping, therefore, by Lemma 2.1, x is unique, and hence the result follows. □
Let $(X, \nu,\Delta_{m}) $ be a probabilistic Banach space, $A,B \subseteq X $ be such that A is a convex set, $A_{0}$ be a nonempty compact set, and B be a bounded convex set. Suppose that $T : A\rightarrow B $ is a continuous affine and proximal nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$ and $g : A\rightarrow A $ is an isometry mapping such that $A_{0}\subseteq g(A_{0}) $. Then there exists an element $x\in A_{0} $ such that $\nu _{gx-Tx}(t)=\nu_{A-B}(t) $ for all $t>0$.
Fix $z\in A_{0} $ and $i \in(0,1)$. We define the mapping $T_{i}:A\rightarrow B $ by
$$T_{i}x=(1-i)Tz+iTx. $$
We show that $T_{i} $ is a proximal contraction. Let $u,v,x,y\in A $ be such that
$$\nu_{u-T_{i}x}(t)=\nu_{A-B}(t)=\nu_{v-T_{i}y}(t)\quad (t>0). $$
Since T is an affine mapping, we have
$$\nu_{u-T((1-i)z+ix)}(t)=\nu_{A-B}(t)=\nu_{v-T((1-i)z+iy)}(t)\quad (t>0). $$
So by the hypothesis we have
$$\begin{aligned} \nu_{u-v}(t) & \geq\nu_{(1-i)z+ix-(1-i)z-iy}(t) \\ &= \nu_{i(x-y)}(t)=\nu_{x-y}\biggl(\frac{t}{i}\biggr)\quad (t>0). \end{aligned}$$
Hence, $T_{i} $ is a proximal contraction. Let $x\in A_{0} $, so that $Tx\in B_{0} $ and $Tz\in B_{0} $. Therefore, there exist $u,v\in A_{0} $ such that
$$\nu_{u-Tx}(t)=\nu_{A-B}(t)=\nu_{v-Tz}(t)\quad (t>0). $$
Put $y=iu+(1-i)v\in A $. Then
$$\begin{aligned} \nu_{y-T_{i}x}(t)&= \nu_{iu+(1-i)v-(1-i)Tz-iTx}(t) \\ &= \nu_{i(u-Tx)+(1-i)(v-Tz)}(t) \\ &\geq \Delta_{m}\bigl(\nu_{i(u-Tx)} (it ),\nu_{(1-i)(v-Tz)} \bigl((1-i)t \bigr)\bigr) \\ &= \Delta_{m}\bigl( \nu_{u-Tx}(t),\nu_{v-Tz}(t)\bigr) \\ &= \Delta_{m}\bigl(\nu_{A-B}(t),\nu_{A-B}(t)\bigr)= \nu_{A-B}(t) \quad (t>0), \end{aligned}$$
and thus $T_{i}(A_{0})\subseteq B_{0} $. By Corollary 2.3 there exists a unique $x_{i}\in A_{0} $ such that $\nu_{gx_{i}-T_{i}x_{i}}(t)=\nu _{A-B}(t) $ for all $t>0 $. Fix $j\in(0,1) $. Then
$$\begin{aligned} \nu_{gx_{i}-Tx_{i}}(t) & \geq\Delta_{m}\bigl(\nu_{gx_{i}-T_{i}x_{i}}(jt), \nu _{T_{i}x_{i}-Tx_{i}}\bigl((1-j)t\bigr)\bigr) \\ & =\Delta_{m}\bigl(\nu_{A-B}(jt),\nu_{(1-i)(Tz-Tx_{i})} \bigl((1-j)t\bigr)\bigr) \\ &= \Delta_{m} \biggl(\nu_{A-B}(jt),\nu_{Tz-Tx_{i}} \biggl( \frac {(1-j)t}{1-i} \biggr) \biggr) \\ &\geq\Delta_{m} \biggl(\nu_{A-B}(jt),D_{B} \biggl( \frac{(1-j)t}{1-i} \biggr) \biggr)\quad (t>0). \end{aligned}$$
Now letting $i\rightarrow1 $, we obtain
$$\lim_{i\rightarrow1}\nu_{gx_{i}-Tx_{i}}(t)\geq\Delta_{m} \bigl(\nu_{A-B}(jt),1\bigr)=\nu _{A-B}(jt)\quad \bigl(\forall j \in(0,1), t>0\bigr). $$
Then letting $j\rightarrow1 $, we have
$$\lim_{i\rightarrow1}\nu_{gx_{i}-Tx_{i}}(t)=\nu_{A-B}(t) \quad (t>0). $$
So we can create a sequence $(x_{n})$ in $A_{0}$ such that
$$\nu_{gx_{n}-Tx_{n}}(t)\rightarrow\nu_{A-B}(t) \quad (t>0). $$
Since $A_{0}$ is compact, the sequence $(x_{n})$ has a subsequence $(x_{n_{k}})$ such that $x_{n_{k}}\rightarrow x\in A_{0}$. By Remark 1.5, g is continuous mapping, and so $g-T$ is a continuous mapping by Remark 1.8. Indeed, since $\Delta_{m}$ is a continuous t-norm, $p\rightarrow \nu_{P}$ is continuous ([27], Chapter 12), and we get
$$\nu_{gx -Tx }(t)=\lim_{k\rightarrow\infty}\nu _{gx_{n_{k}}-Tx_{n_{k}}}(t)= \nu_{A-B}(t), $$
as required. □
Let $(X, F,\Delta) $ be a complete probabilistic Menger space such that Δ is a t-norm of H-type, and $(A,B) $ be a pair of subsets of X with the weak P-property such that $A_{0}$ is a nonempty closed set. If $T:A\rightarrow B $ is a contraction mapping such that $T(A_{0} )\subseteq B_{0}$, then there exists a unique x in A such that $F_{x,Tx}(t)=F_{A,B}(t) $ for all $t>0 $.
It is a direct consequence of Remark 1.20 and Lemma 2.1. □
Clearly, the pair $(A, A) $ has the P-property, so we have the following result.
Let $(X, F,\Delta) $ be a complete probabilistic Menger space such that Δ is a t-norm of H-type. Then every contraction self-mapping from each nonempty closed subset of X has a unique fixed point.
Let $(X, \nu,\Delta_{m}) $ be a probabilistic Banach space, and $(A,B) $ be a semisharp proximinal pair of X such that A is a p-star-shaped set, $A_{0}$ be a nonempty compact set, B be a q-star-shaped set, and let $\nu_{p - q}(t)=\nu_{A-B}(t)$ for all $t>0$. If $T : A\rightarrow B $ is a proximal nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$, then there exists an element $x\in A_{0} $ such that $\nu_{x - Tx}(t)=\nu_{A-B}(t)$ for all $t>0$.
For each integer $i\geq1 $, define $T_{i}:A_{0}\rightarrow B_{0} $ by
$$T_{i}(x)= \biggl(1-\frac{1}{i} \biggr) Tx +\frac{1}{i}q \quad ( x\in A_{0}). $$
Then by the hypothesis we have $T_{i}(A_{0} )\subseteq B_{0}$. Next, we show that for each i, $T_{i} $ is a proximal contraction with $\alpha=1-\frac{1}{i}< 1 $. To do this, suppose that $x,y,u,v,s,r\in A_{0} $ and $t>0$ are such that
$$\nu_{u-T_{i}x}(t)=\nu_{v-T_{i}y}(t)=\nu_{A_{0}-B_{0}}(t)= \nu_{A-B}(t)=\nu _{s-Tx}(t)=\nu_{r-Ty}(t). $$
Now we define
$$u'= \biggl(1-\frac{1}{i} \biggr) s +\frac{1}{i}p\in A_{0},\qquad v'= \biggl(1-\frac{1}{i} \biggr) r + \frac{1}{i}p\in A_{0}, $$
$$\begin{aligned} \nu_{A-B}(t)&\geq\nu_{u'-T_{i}x}(t) =\nu_{ (1-\frac{1}{i} ) s +\frac{1}{i}p- (1-\frac{1}{i} ) Tx -\frac{1}{i}q}(t) \\ & =\nu_{ (1-\frac{1}{i} )(s-Tx)+\frac{1}{i}(p-q)}(t) \\ & \geq\Delta_{m} \biggl(\nu_{ (1-\frac{1}{i} )(s-Tx)} \biggl( t \biggl(1- \frac{1}{i} \biggr) \biggr) ,\nu_{\frac{1}{i}(p-q)} \biggl( t \biggl( \frac {1}{i} \biggr) \biggr) \biggr) \\ & =\Delta_{m}\bigl( \nu_{s-Tx}(t),\nu_{p-q}(t) \bigr) \\ & =\Delta_{m}\bigl(\nu_{A-B}(t),\nu_{A-B}(t)\bigr)= \nu_{A-B}(t). \end{aligned}$$
Hence, $\nu_{u'-T_{i}x}(t)= \nu_{A-B}(t)$. Since $\nu _{u-T_{i}x}(t)= \nu_{A-B}(t)$ and $(A,B) $ is a semisharp proximinal pair, we have $u'=u$. By the same method we also have $v'=v $. Since T is a proximal nonexpansive mapping, we have
$$\begin{aligned} \nu_{u-v}(t)&=\nu_{u'-v'}(t) =\nu_{ (1-\frac{1}{i} )(s-r)}(t) \\ &= \nu_{s-r} \biggl(\frac{t}{1-\frac{1}{i}} \biggr)\geq\nu_{x-y} \biggl(\frac {t}{1-\frac{1}{i}} \biggr). \end{aligned}$$
Therefore, $T_{i} $ is a proximal contraction with $\alpha=1-\frac {1}{i}< 1 $. By Lemma 2.1, for each $i\geq1 $, there exists a unique $u_{i}\in A_{0} $ such that $\nu_{u_{i}-T_{i}u_{i}}(t)=\nu _{A_{0}-B_{0}}(t)= \nu_{A-B}(t)$. Since $A_{0} $ is compact and $( u_{i})\subseteq A_{0} $, without loss of generality, we can assume that $u_{i} $ is a convergent sequence and $u_{i}\rightarrow x \in A_{0} $.
For each $i\geq1$, since $T(u_{i})\in T(A_{0})\subseteq B_{0}$, there exists $v_{i}\in A_{0} $ such that $\nu_{v_{i}-Tu_{i}}(t)= \nu _{A-B}(t)$. So we have
$$\begin{aligned} \nu_{A-B}(t)&\geq \nu_{ (1-\frac{1}{i} )v_{i} +\frac {1}{i}p-T_{i}u_{i}}(t) \\ &= \nu_{ (1-\frac{1}{i} ) v_{i} +\frac{1}{i}p- (1-\frac {1}{i} ) Tu_{i} -\frac{1}{i}q}(t) \\ &\geq\Delta_{m} \biggl(\nu_{(1-\frac{1}{i})( v_{i}- Tu_{i})}\biggl( t\biggl(1- \frac {1}{i}\biggr)\biggr) ,\nu_{\frac{1}{i}(p-q)}\biggl( t\biggl( \frac{1}{i}\biggr) \biggr) \biggr) \\ &=\Delta_{m}\bigl( \nu_{v_{i}- Tu_{i}}(t),\nu_{p-q}(t)\bigr) \\ &= \Delta_{m}\bigl(\nu_{A-B}(t), \nu_{A-B}(t) \bigr)=\nu_{A-B}(t). \end{aligned}$$
Thus, $\nu_{A-B}(t)= \nu_{ (1-\frac{1}{i} )v_{i} +\frac {1}{i}p-T_{i}u_{i}}(t)$. Since $(A,B) $ is a semisharp proximinal pair and $\nu_{A-B}(t)=\nu _{u_{i}-T_{i}u_{i}}(t) $, we have $u_{i}=(1-\frac{1}{i}) v_{i} +\frac{1}{i}p$, and so
$$\nu_{u_{i}-v_{i}}(t)=\nu_{\frac{1}{i}(v_{i}-p)}(t)=\nu_{v_{i}-p}(it) . $$
Since $A_{0} $ is compact and $( v_{i})\subseteq A_{0} $, without loss of generality, we can assume that $v_{i} $ is a convergent sequence and $v_{i}\rightarrow z \in A_{0} $. For every $j\leq i$, we have
$$\nu_{u_{i}-v_{i}}(t)=\nu_{v_{i}-p}(it)\geq\nu_{v_{i}-p}(jt)\geq \Delta _{m} \biggl(\nu_{v_{i}-z}\biggl(\frac{j}{2}t \biggr),\nu_{z-p}\biggl(\frac{j}{2}t\biggr)\biggr). $$
Letting $i\rightarrow\infty$, we have
$$\lim_{i\rightarrow\infty}\nu_{u_{i}-v_{i}}(t)\geq\nu_{z-p} \biggl(\frac {j}{2}t\biggr) \quad (\forall j\geq1). $$
Now letting $j \rightarrow\infty$, we have
$$\lim_{i\rightarrow\infty}\nu_{u_{i}-v_{i}}(t)\geq\lim_{j \rightarrow\infty} \nu_{z-p}\biggl(\frac{j}{2}t\biggr)=1. $$
Therefore, $\nu_{u_{i}-v_{i}}(t)\rightarrow1 $, so that $z=\lim_{i\rightarrow\infty}v_{i}= \lim_{i\rightarrow\infty}u_{i}=x$. Since $Tx\in B_{0} $, there must exist $u\in A_{0} $ such that $\nu _{A-B}(t)=\nu_{u-Tx}(t)$. Since we know that $\nu_{A-B}(t)=\nu _{v_{i}-Tu_{i}}(t)$ and T is a proximal nonexpansive mapping, it follows that $\nu _{v_{i}-u}(t)\geq\nu_{u_{i}-x}(t) \rightarrow1$. This implies that $u= \lim_{i\rightarrow\infty}v_{i}=x$ and then $\nu_{A-B}(t)=\nu _{x-Tx}(t) $, as required. □
Let $(X,\nu,\Delta_{m}) $ be a probabilistic Banach space, $(A,B) $ be a semisharp proximinal pair of X with the weak P-property such that A is a p-star-shaped set, $A_{0} $ be a nonempty compact set, B be a q-star-shaped set, and let $\nu_{p - q}(t)=\nu_{A-B}(t)$ for all $t>0$. If $T:A\rightarrow B $ is a nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$, then T has a best proximity point in $A_{0} $.
It is a direct consequence of Remark 1.20 and Theorem 2.7. □
Let $(X, F,\Delta) $ be a probabilistic Menger space, and $A,B \subseteq X $ be such that $A_{0} $ is a nonempty set. Suppose that $T : A\rightarrow B $ is a proximal nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$ and $g : A\rightarrow A $ is an isometry mapping such that $A_{0}\subseteq g(A_{0}) $. Denote $G = g(A)$ and
Then $Tg^{-1}$ is a proximal nonexpansive, and $G_{0}=A_{0} $.
The result follows by using a similar argument as in the proof of Proposition 2.2. □
The following theorem is an immediate consequence of Theorem 2.7 and Proposition 2.9.
Theorem 2.10
Let $(X, \nu,\Delta_{m}) $ be a probabilistic Banach space, $(A,B) $ be a semisharp proximinal pair of X such that A is a p-star-shaped set, $A_{0}$ be a nonempty compact set, B be a q-star-shaped set, and let $\nu_{p - q}(t)=\nu_{A-B}(t)$ for all $t>0$. If $T : A\rightarrow B $ is a proximal nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$ and $g : A\rightarrow A $ is an isometry mapping such that $A_{0}\subseteq g(A_{0}) $, then there exists an element $x\in A_{0} $ such that $\nu_{gx - Tx}(t)=\nu_{A-B}(t)$ for all $t>0$.
Corollary 2.11
Let $(X, \nu,\Delta_{m}) $ be a probabilistic Banach space, and let $(A,B) $ be a pair of convex subsets of X with the P-property such that $A_{0}$ is a nonempty compact set. If $T : A\rightarrow B $ is a nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$ and $g : A\rightarrow A $ is an isometry mapping such that $A_{0}\subseteq g(A_{0}) $, then there exists an element $x\in A_{0} $ such that $\nu _{gx-Tx}(t)=\nu_{A-B}(t) $ for all $t>0$.
In Corollary 2.11, if $g(x)=x$, then we have the following corollary.
With the hypotheses of the previous corollary, if $T:A\rightarrow B $ is a nonexpansive mapping such that $T(A_{0} )\subseteq B_{0}$, then T has a best proximity point.
In Corollary 2.12, if $A=B$, then we have the following corollary.
If A is a nonempty, compact, and convex subset of a probabilistic Banach space $(X, \nu,\Delta_{m}) $ and $T:A\rightarrow A $ is a nonexpansive mapping, then T has a fixed point.
In the following, we give some examples that defend our main results.
Let $X=\mathbb{R}^{2} $, $A=\lbrace(0,y) : y\in\mathbb{R}\rbrace$ and $B=\lbrace(1,y): y\in\mathbb{R}\rbrace$. Suppose that $T :A\rightarrow B$ is defined by $T(0,y)= (1,\frac {y}{4} ) $, $g :A\rightarrow A$ is defined by $g(0,y)=(0,-y) $, and $F_{(x,x'),(y,y')}(t)=\frac{t}{t+|x-y|+|x'-y'|} $. It is easy to see that $(X,F,\Delta_{m})$ is a complete probabilistic Menger space, $F_{A,B}(t)=\frac{t}{t+1} $, $A_{0} =A$, $B_{0}=B $, $T(A_{0})\subseteq B_{0} $, and
$$F_{g(0,x),g(0,y)}(t)=F_{(0,-x),(0,-y)}(t)=\frac{t}{t+|x-y|}=F_{(0,x),(0,y)}(t). $$
If $(0,u),(0,x), (0,v),(0,y)\in A $ are such that
$$\frac{t}{t+1+|u-\frac{x}{4}|} = F_{(0,u),T(0,x)}(t)=F_{A,B}(t)= F_{(0,v),T(0,y)}(t)=\frac{t}{t+1+|v-\frac{y}{4}|}, $$
then $u= \frac{x}{4}$ and $v= \frac{y}{4}$, so that
$$F_{(0,u),(0,v)}(t)=F_{(0,\frac{x}{4}),(0, \frac{y}{4})}(t)=\frac {t}{t+\frac{1}{4}|x-y|}=F_{(0,x),(0,y)} \biggl(\frac{t}{\frac{1}{4}} \biggr). $$
Therefore, all the hypothesis of Corollary 2.3 are satisfied, and we also have
$$F_{(0,0),T(0,0)}(t)=F_{(0,0),(1,0)}(t)=\frac{t}{t+1}=F_{A,B}(t). $$
Let $X=\mathbb{R} $, $A=[0,2] $ and $B=[3,5] $. For every $x\in X $, define $\nu_{x} (t)=\frac{t}{t+|x|}$. It is easy to see that $(X, \nu,\Delta_{m}) $ is a probabilistic Banach space, $\nu_{A-B}(t)=\frac{t}{t+1} $, $A_{0}=\lbrace2\rbrace$, and $B_{0}=\lbrace3\rbrace$. For every $x\in A $, define $T:A\rightarrow B $ by $Tx=5-x $ and let g be the identity mapping. Clearly, T is a continuous affine and proximal nonexpansive mapping, and $T(A_{0})=\{T(2)\}=\{3\}=B_{0} $. Therefore, all the hypotheses of Theorem 2.4 are satisfied, and also we have
$$\nu_{2-T2}(t)=\nu_{2-3}(t)=\frac{t}{t+1}= \nu_{A-B}(t). $$
The following example shows that the weak P-property of the pair $(A, B) $ cannot be removed from Theorem 2.5.
Let $X =\mathbb{R} $, $A =\lbrace-10,10\rbrace$, $B = \lbrace-2, 2\rbrace$, and $F_{p,q}(t)=\frac{t}{t+|p-q|} $. Clearly, $(X, F,\Delta_{m}) $ is a complete probabilistic Menger space. Then $A_{0} = A $, $B_{0} = B $, and $F_{A,B}(t)=\frac{t}{t+8} $. Let $T : A \rightarrow B$ be a mapping given by $T (-10) = 2 $ and $T (10) = -2 $. It is easy to see that for $\alpha=\frac{1}{5} $, T is a contraction mapping with $T (A_{0})\subseteq B_{0}$. The mapping T does not have any best proximity point because $F_{x,Tx}(t)=\frac{t}{t+12} < \frac{t}{t+8}= F_{A,B}(t)$ for all $x \in A $. It should be noted that the pair $(A, B) $ does not have the weak P-property.
Let $X=\mathbb{R} $, $A=[0,1] $, and $B=[2,3] $. For every $x\in X $, define $\nu_{x}(t)=\frac{t}{t+|x|} $. It is easy to see that $(X, \nu,\Delta_{m}) $ is a probabilistic Banach space, A is 1-star-shaped set, B is 2-star-shaped set,
$$\nu_{A-B}(t)=\sup_{x\in A, y\in B}\nu_{x-y}(t)= \frac{t}{t+1},\qquad A_{0}=\lbrace1\rbrace,\qquad B_{0}= \lbrace2\rbrace, $$
$$\nu_{1-2}(t)=\frac{t}{t+|1-2|}=\frac{t}{t+1}= \nu_{A-B}(t). $$
Also, $(A,B) $ is a semisharp proximinal pair. Now for each $x\in A $, define $T:A\rightarrow B $ by $Tx=3-x $. If $u,v,x,y\in A $, then
$$\frac{t}{t+|u-3+x|} =\nu_{u-Tx}(t)=\nu_{A-B}(t)= \nu_{v-Ty}(t)=\frac {t}{t+|v-3+y|}, $$
so that $u= x=1 $ and $v= y=1 $. Thus,
$$\nu_{u-v}(t)=1=\nu_{x-y}(t). $$
So T is a proximal nonexpansive, and $T(A_{0})=B_{0}$. Therefore, all the hypotheses of Theorem 2.7 are satisfied, and we also have
Let $X=\mathbb{R}^{2} $, $A=\lbrace(x,0) : 0\leq x \leq1\rbrace$, $B_{1}=\lbrace(x,y): x+y=1, -1\leq x \leq0\rbrace$, $B_{2}=\lbrace(x,1): 0\leq x \leq1\rbrace$, $B=B_{1}\cup B_{2}$, and $\nu_{(x,x')}(t)=\frac{t}{t+ |x|+|x'|} $. It is easy to see that $(X, \nu,\Delta_{m}) $ is a probabilistic Banach space, $\nu_{A-B}(t)= \frac{t}{t+1} $, B is not convex but is a $(0,1) $-star-shaped set, and A is $(0,0)$-star-shaped set. Clearly, $A_{0}=A$ and $B_{0}=B_{2} $. So
$$\nu_{(0,0)-(0,1)}(t)=\frac{t}{t+ |0|+|1|}=\frac{t}{t+1}= \nu_{A-B}(t), $$
and $(A,B) $ is a semisharp proximinal pair. Suppose that $T:A\rightarrow B $ is defined by
$$ T(x,0)= \left\{ \textstyle\begin{array}{l@{\quad}l} (0,1), & x= 0, \\ (\sin x,1), & x\neq0, \end{array}\displaystyle \right. $$
and $(u,0),(v,0),(x,0),(y,0)\in A $ are such that
$$\nu_{(u,0)-T(x,0)}(t)=\nu_{A-B}(t)=\frac{t}{t+1}= \nu_{(v,0)-T(y,0)}(t). $$
If $x=y=0 $, then $u=v=0 $, and therefore
$$\nu_{(u,0)-(v,0)}(t)=\nu_{(0,0)-(0,0)}(t)=1=\nu_{(x,0)-(y,0)}(t). $$
If $x,y\neq0$, then $u=\sin x$, $v=\sin y $, and therefore
$$\begin{aligned} \nu_{(u,0)-(v,0)}(t)&=\nu_{(\sin x,0)-( \sin y,0)}(t)=\frac{t}{t+|\sin x-\sin y|} \\ &\geq\frac{t}{t+| x- y|} \\ &=\nu_{(x,0)-(y,0)}(t). \end{aligned}$$
If $x=0$ and $y\neq0$, then $u=0 $ and $v=\sin y $, and therefore
$$\nu_{(u,0)-(v,0)}(t)=\nu_{(0,0)-(\sin y,0)}(t)=\frac{t}{t+|\sin y|}\geq \frac{t}{t+| y|}\geq\nu_{(0,0)-(y,0)}(t). $$
If $x\neq0$ and $y= 0$, then $u=\sin x $ and $v=0 $, and therefore
$$\nu_{(u,0)-(v,0)}(t)=\nu_{(\sin x,0)-(0,0)}(t)=\frac{t}{t+|\sin x|}\geq \frac{t}{t+| x|}\geq\nu_{(x,0)-(0,0)}(t). $$
Hence, T is proximal nonexpansive, and $T(A_{0})\subseteq B_{2}=B_{0}$, so all the hypotheses of Theorem 2.7 are satisfied, and we also have
$$\nu_{(0,0)-T(0,0)}(t)=\nu_{(0,0)-(0,1)}(t)=\frac{t}{t+1}= \nu_{A-B}(t). $$
Let $X=\mathbb{R} $, $A=[0,1] $, $B=[\frac{15}{8},2] $, and $\nu _{x}(t)=\frac{t}{t+|x|} $. Clearly, $(X,\nu,\Delta_{m})$ is a probabilistic Banach space, $\nu_{A-B}(t)=\frac{t}{t+\frac{7}{8}} $, the pair $(A,B)$ has the P-property, $A_{0}=\lbrace1\rbrace$, and $B_{0}=\lbrace\frac{15}{8}\rbrace$. If $Tx=-\frac{1}{8}x+2 $, then $T(A_{0})=\{T(1)\}=\{\frac{15}{8}\}=B_{0} $. Let $x,y\in A$. Then we have
$$\nu_{Tx-Ty}(t)=\nu_{-\frac{1}{8}(x-y)}(t)=\nu_{x-y}(8t)\geq \nu_{x-y}(t). $$
Therefore, all the hypotheses of Corollary 2.12 are satisfied, and hence T has a best proximity point, and we also have
$$\nu_{1-T1}(t)=\nu_{1-\frac{15}{8}}(t)=\frac{t}{t+\frac{7}{8}}= \nu_{A-B}(t). $$
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Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851-862 (2003)
Mongkolkeha, C, Kumam, P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. (2012). doi:10.1007/s10957-012-9991-y
Mongkolkeha, C, Kumam, P: Some common best proximity points for proximity commuting mappings. Optim. Lett. (2012). doi:10.1007/s11590-012-0525-1
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Sintunavarat, W, Kumam, P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012, 93 (2012). doi:10.1186/1687-1812-2012-93
Srinivasan, PS: Best proximity pair theorems. Acta Sci. Math. (Szeged) 67, 421-429 (2001)
Wlodarczyk, K, Plebaniak, R, Banach, A: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 70(9), 3332-3341 (2009)
Wlodarczyk, K, Plebaniak, R, Banach, A: Erratum to: 'Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces'. Nonlinear Anal. 71, 3583-3586 (2009)
Wlodarczyk, K, Plebaniak, R, Obczynski, C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 72, 794-805 (2010)
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Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983)
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Chang, SS, Cho, YJ, Kang, SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers, New York (2001)
Chang, SS, Lee, BS, Cho, YJ, Chen, YQ, Kang, SM, Jung, JM: Generalized contraction mapping principle and differential equations in probabilistic metric spaces. Proc. Am. Math. Soc. 124, 2367-2376 (1996)
El Naschie, MS: Fuzzy dodecahedron topology and E-infinity spacetimes as a model for quantum physics. Chaos Solitons Fractals 30(5), 1025-1033 (2006)
El Naschie, MS: On gauge invariance, dissipative quantum mechanics and self-adjoint sets. Chaos Solitons Fractals 32(2), 271-273 (2007)
El Naschie, MS: P-Adic analysis and the transfinite E8 exceptional Lie symmetry group unification. Chaos Solitons Fractals 38(3), 612-614 (2008)
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Su, Y, Zhang, J: Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces. Fixed Point Theory Appl. 2014, 170 (2014)
Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001)
The authors would like to express their sincere appreciation to the Shahrekord University and the Center of Excellence for Mathematics for financial support.
Department of Pure Mathematics, University of Shahrekord, Shahrekord, 88186-34141, Iran
Hamid Shayanpour
, Maryam Shams
& Asiyeh Nematizadeh
Search for Hamid Shayanpour in:
Search for Maryam Shams in:
Search for Asiyeh Nematizadeh in:
Correspondence to Hamid Shayanpour.
All authors have contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.
https://doi.org/10.1186/s13663-015-0487-y
46S50
proximal contraction
proximal nonexpansive mappings
best proximity point
P-property
star-shaped | CommonCrawl |
Is There a Philosophical Problem of Reference?
Many philosophers worry about reference, and think we need to come up with a "theory of reference". This would explain how strings get "connected" to things: the string-thing relation. So, maybe this should be analysed in terms of causation. Causation glues cats to the word "cat". And so on. Or maybe one gets worried about how, e.g., "aleph_0" and $\aleph_0$ got glued together, when causation couldn't have done that---for $\aleph_0$ is a causally inert abstractum. In fact, the string "aleph_0" is also an abstractum. Worried by the unsurprising inability to reduce reference to causation, one might go deflationary and say that there is just a reference predicate, the binary predicate "$x$ refers to $y$", governed by disquotational axioms (e.g., ""red" refers to the set of red things") and one has this in one's language (which gives the illusion of genuine reference). Or perhaps one gives up, becomes a semantic nihilist and says outright: there is no reference relation at all.
Well, maybe there is no philosophical problem of reference at all: any relation of strings to things is a legitimate reference relation. So, the theory of reference is this:
$\mathcal{I}(\sigma) = x$
where $\mathcal{I}$ is some function: any function you like. Because $\mathcal{I}$ is a parameter here, reference is really a ternary relation,
$\sigma$ refers to $x$ relative to $\mathcal{I}$.
Many authors believe, however, that reference is a binary relation that strings bear to referents. That is,
$\sigma$ refers to $x$.
With no language or interpretation relativity! It seems to me that this violates the single most basic principle of language---that anything can mean anything in some language. The language relativity is essential. And yet the view that reference is a binary relation without language relativity seems fairly standard.
Chris Gauker gave a talk on a related topic at MCMP several months back. And Chris's view is an example of the conventional view of what is required from a theory of reference:
What is reference? Offhand, it appears to be a relation, just as being heavier than is a relation. Moreover, it is a relation that holds between words and other things. It holds between "chair" and the chair I'm sitting in, between "chair" and the chair next door, and between "chair" and the chairs of the past and future. It holds between "Socrates" and Socrates, between "meson" and mesons, and between "beautiful" and beautiful things. Of course, the reason it holds between these things is not just that it holds between every word and everything. For instance, it doesn't hold between "basketball" and daffodils. The fact that the relation being heavier than holds between an atom of oxygen and an atom of hydrogen, as well as between Pavarotti and Diana Ross, but not between everything and everything, ought to make us wonder what being heavier than amounts to, if we don't already know. Likewise, the combination of diversity and specificity exhibited by the reference relation ought to make us wonder what reference is. We mustn't just call it an "unanalyzable primitive".
Accounts of reference that might satisfy us fall into two classes. On the one hand, we might seek what I'll call an analysis of reference. An analysis takes the form:
$t$ refers to $a$ if and only if $\dots$.
(Gauker 1990, "Semantics without reference", Notre Dame Journal of Formal Logic, 438.)
This seems to me to be a bit like a grammatical mistake, because it omits the crucial language parameter,
$t$ refers to $a$ in $(\mathcal{L}, \mathcal{I})$ if and only if $\dots$.
And then the answer is:
$t$ refers to $a$ in $(\mathcal{L}, \mathcal{I})$ if and only if $\mathcal{I}(t) = a$.
There are no constraints at all on what the interpretation function $\mathcal{I}$ might be. It can be any old string-thing function. On this proposal, it isn't that the famous semantic indeterminacy puzzles disappear (e.g., Quinian inscrutability of reference, Kripkenstein, Putnam's "model-theoretic argument", ec.). Rather, the puzzles turn out, on reflection, to be puzzles about what language one speaks. The question isn't how languages get referentially glued to things---they already are, in every mathematically possible way (generating uncountably many different languages). The question is how the mind gets glued to some particular language: a problem of language cognition.
Published by Jeffrey Ketland at 3:49 am
Sam Roberts 3 August 2012 at 12:14
Thanks for the interesting post!
I want to get a bit clearer on what's been gained by the shift from reference to language cognition. Consider a really simple language L that just contains singular terms referring to objects. Two questions one might ask about L are:
(1) Why does t refer to x in L?
(2) Why do I (partially) speak L rather than L' (where t does not refer to x)?
(1) is a sharpening of the traditional metasemantic question which, as you point out, seems to have a trivial answer when L is understood as a function from terms to objects -- in particular, t refers to x in L because L(t) = x.
Of course, the traditional metasemanticist wouldn't be happy with that way of spelling out their question. Instead, I would imagine that they would claim that (2) is closer to the question they were asking, given the current framework.
Is there an easy way to see that (2) is a genuinely different question from the traditional one, and thus that this isn't just a difference in terminology?
Jeffrey Ketland 3 August 2012 at 13:41
Thanks - yes, as you say, for the first question (1), that is just what L does, on this Lewisian picture: t refers to x in L because L(t) = x. So, the question is a bit like "why is the value of sine on 30° equal to 0.5?", and the answer is "Because sin(30°) = 0.5". So, (1) gets a trivial answer: nothing explains why L(t) = x. It's a necessity.
I do think the view, even if it's only a reformulation, has benefits. It removes objections to the causal theory of reference; it is a competitor to deflationary and nihilist views about reference. It respects the Saussurean principle that anything can mean anything.
Then the second question, (2), I call the "cognizing problem", is a separate one; I think it is:
(a) very hard - the hard problem of representation.
(b) the problem meta-semanticists have normally been interested in.
But I don't think it's merely a difference of terminology, because, e.g., on this view, reference (and semantic notions generally) is language relative and reference facts are necessities (so, there's no hope of, or even need for, a causal reduction of "x refers to y").
But why do I speak L rather than L*? (I have a bit on this in the "There's Glory or You" post a couple of months ago.)
Here's my rough attempt at an explication of the contingent relation, "A speaks L":
(*) A cognizes/speaks L iff, for any concept C, string s: A assigns C to string s iff L(s) = C.
This assumes an intentional primitive, "A assigns C to s" (call this A's noetic function). So: A speaks L when A's noetic function matches the meaning function for L.
So what seems like referential or semantic indeterminacy arises when the speaker's noetic function is indeterminate. E.g., for the Kripkenstein case, It might be indeterminate whether the mind of Tim Gowers assigns + to "plus" or +* to "plus" (where +* is some odd function, whose restriction coincides with +).
In this sort of case, I don't think it is indeterminate, actually, but I think that is where the problem is: it concerns language cognition. Of course, I don't know how to work out the answer to that ... perhaps the mind selects certain concepts because the concept is simpler. So, some of the approaches advocated by (realist) meta-semanticists may well be similar on both ways of looking at these problems.
Cheers, Jeff
Thanks for the helpful response!
I'm glad we agree where the real meta-semantic action is on your framework -- namely with (2). Just to be clear, I do think the framework is neat; but I suppose I'm still not entirely clear what it says about existing debates about reference. For instance, take the causal theory of reference. In your framework, such a theory looks misplaced (as you say "there's no hope of, or even need for, a casual reduction"). But I don't think that the casual theorist will be happy equating "reference" in their sense with "reference in L" in your sense. Rather, I would imagine they would want their talk of "reference" to be interpreted in your framework by talk of "assigns".
Or is there some simple reason why that interpretation of the old reference language into your assigns language fail?
Thanks! With current debates I want to say there's a conflation of a semantic notion (the meaning relation for L, mapping strings to meanings) and a cognitive notion (the cognizing relation, between agent and language L). And the ternary nature of meaning/reference (language relativity) is important, as it yields that meaning facts like,
(i) "Schnee" denotes in German snow
(ii) "Schnee ist weiss" expressed in German that snow is white
are necessities. (This is a problem for naturalistic theories of reference, and tends to push people towards deflationism. It's not a problem for me because all the contingency resides in the "A speaks L" relation.)
But then to look at your question, suppose the meaning naturalist agrees, and then says, "Ok, fair enough. Let's grant you your "anything-can-mean-anything" account of reference/meaning, and conclude that there isn't a problem there, and discuss instead the intentional assigning relation - the agent assigns meaning m to string. Isn't that a causal relation?".
Then, I'd say that it's while it's temporally and modally contingent what concepts an agent is assigning to strings, I don't think it can be a causal relation, as the relata aren't always (or even sometimes) causally related: e.g., assigning say the concept of membership to the string "is a member of".
So, one has to buy into the idea that the mind can grasp concepts or rules or referents, understood as abstracta. (E.g., suppose the mind just can realize the addition function, somehow. It can concanenate mental strings, maybe. Then it's not so implausible that the mind can assign the genuine, abstract, addition function + to the string "plus")
I say all this because I don't want to do anything sneaky. I'm trying to be clear up front that the intentional primitive, "A assigns C to string s" is a rather metaphysically weighty notion. It's a bit like Fregean grasping or Putnam's noetic rays (he mocks it a bit, so I copied the word).
A slightly different approach, more in line with assertibility semantics, might be to take "A believes p" and "s is assertible for A" as primitive, and then define,
A speaks/cognizes L iff for any string s, and proposition p (if s means-in-L p then s is assertible for A iff A believes p).
But I'm unhappy with this approach. It's not fine-grained enough.
Thanks; that's really helpful!
One last thing: for singular terms t, do speakers and languages in your sense assign objects to t, or concepts? For instance, does my idiolect of English map "Sam Roberts" to me, or to some concept?
I have quite a long story to tell on this topic, but on the standard view in phil language, minds assign referents directly to words. So we have Davidsonian referential semantics, and I think that's a widely held account. (In parts of formal semantics the situation is different.) But I think that a reference-first view faces some independent problems. E.g., how does the mind assign the class S of spacetime points to "spacetime point" or assign the sets to "set". If p is a single spacetime point, then why does my word "spacetime point" refer to S, and not S / {p}? So, I think it must be intensional (or concepts a la Frege) rather than extensional, and reference is indirect.
And, in this case, it's your individual concept that I assign to "Sam Roberts". The referent is determined indirectly, since Sam is (by necessity) the unique satisfier of this intension. For the theory of meaning & reference for a language, this seems ok: what string refer to are the extensions of their intensions.
But for making sense of cognizing, maybe this is not ok. To pin down "A speaks L" (which is really what I want to do), one has to assume certain intentional primitives, though which ones, I'm not sure. E.g., "A uses string s to refer to x", "A assigns referent x to string s", "A assigns concept C to string s", etc.,
Even making sense of cognizing syntax is rather hard.
Basically, the idea of a physical system computing or realizing a program/function. Still, it's easier to make sense of cognizing a program for computing recursive function than it is to make sense of cognizing the referent of "spacetime point". (Which is why deflationism is so attractive.)
So there are two options; either we assign objects to singular terms or we assign their haecceities. The former option you think faces problems -- namely, on this option one needs to give an account of how agents can assign objects to a singular terms. And the latter is supposed to get around this problem.
It looks to me like a casual theory would be apposite in both cases. For instance, I can assign x to t because of some causal connection I have to x; and similarly I can assign H(x) to t because of some causal connection I have to x -- since having a relevant causal connection to x seems sufficient for having the relevant (perhaps non-casual) connection with H(x).
In general, I don't yet see why it is easier for an agent to assign a haecceity to t rather than an object. What is it about concepts that makes this easier?
Thanks for all the help so far!
There are two orthogonal bits here. First, resolving the philosophical demand for a theory of reference; in particular, a causal/naturalistic theory of reference, like "x refers to y iff Ryx", with R some causal relation. I want to resolve that by just saying reference isn't causal to begin with. It's a three-place relation of a string s, a language L and a referent x, and facts of that kind are necessities. I'm quite confident on that. Reference is ok; semantics, more generally, is ok. (Including all sorts of non-classical semantics as well.)
But then ... there's the cognizing problem(s). The usual debates become reformulated as questions about what language one speaks/cognizes, and these resolve into how the mind assigns meanings to strings. I think they become clearer problems in the way I set things up. But I am way less confident about how to make progress on any of these particular problems!
I have some preference for an internalist view: the mind grasps intensions/concepts more easily than it might grasp individual objects or sets. I nothing knock-down here. Maybe because there are logical operations on concepts. If I can grasp $C$ and negation, I can grasp $\neg C$. That's the sort of consideration.
For singular terms, and names in particular, maybe you're right. If x is a causally active concretum, then the choice between assigning x, and assigning H(x), to a string t might be about the same. But then what if x is a future event? Or x is an abstractum?
I tend to think the only way to solve these kinds of problems is to accept some form of grasping of abstract concepts, because trying to build things up from, e.g., causal contact with local concreta (or sense data) is not workable.
My not-so-hidden agenda here is an epistemology for logic, mathematics and modality!
Yes, I agree the problems become clearer in this framework! I also think that the problems you raise for causal accounts are right (if we're on board with abstract objects). But I think they are just as forcefully raised on the old framework for the reference relation. I wanted to see if your framework threw some more light on the casual theory, once we interpret it to be about the assigns relation.
Thanks so much - all helps me think out this stuff more carefully.
If I take the intentional relation "A assigns C to string s" as primitive, then I can define "A speaks L", and can shed light on the usual kinds of Quine-Kripke-Putnam style problem - so long as it's determinate. (Spoken languages are idiolects, and can fluctuate and change rapidly. They are very finely individuated - sameness of phonology, syntax, semantics and pragmatics.) But, on my set-up, a metasemantical sceptic (e.g., defending Quine on "gavagai") will just say "A assigns C to string s" is indeterminate, or even that it is just a mystical, unscientific grasping notion.
I can't yet say anything interesting about it. On the other hand, it's rather like the subsentential correlate of believing a proposition, so if that's an ok intentional primitive, then assigning should be too. I guess that causation, sense input, internal or innate properties of the mind, all certainly should play a role in making it determinate what concept/referent/extension etc. an agent assigns to a string.
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Would home heat pumps provide large benefits against climate change?
Some facts I have learnt
The Global Warming Potential (GWP) was developed to allow comparisons of the global warming impacts of different gases. Specifically, it is a measure of how much energy the emissions of 1 ton of a gas will absorb over a given period of time, relative to the emissions of 1 [US] ton of carbon dioxide (CO2).
Nitrous Oxide (N2O) has a GWP 265–298 times that of CO2 for a 100-year timescale.
— United States Environment Protection Agency
According to the UK Government's Department for Environment, Food & Rural Affairs (DEFRA, 2021):
With the average UK annual household gas usage being 12,000 kWh (UK Power, n.d.)
A heat pump that has a Coefficient of Performance (CoP) of three can create three kWh of heat from every one kWh of electricity. Therefore, compared to average gas usage, a heat pump with a CoP of three would use 4,000 kWh of electricity annually (Viessmann, n.d.)
The UK Government is looking for everyone to install Heat Pumps for home heating and will be banning the installation of gas combi boilers.
But, from my calculations, heat pumps could provide little benefit to the environment in comparison to gas central heating. Where is the major benefit I am missing?
DEFRA (2021) UK Government GHG Conversion Factors for Company Reporting https://www.gov.uk/government/publications/greenhouse-gas-reporting-conversion-factors-2021
UK Power (n.d.) Average gas and electric usage for UK households https://www.ukpower.co.uk/home_energy/average-household-gas-and-electricity-usage
Veissmann (n.d.) Do heat pumps use a lot of electricity? https://www.viessmann.co.uk/heating-advice/Do-heat-pumps-use-a-lot-of-electricity
climate-change air-pollution pollution carbon
Chris RogersChris Rogers
$\begingroup$ If the Government is promoting heat pumps as a form of heating for the future, it can be regarded as an admission that global warming is real. Heat pumps simply move heat from one side of a divide to the other side. Basically they transfer heat from outside a building to inside. This only works if there is heat to transfer and it is why heat pumps will not be used in cold environments such as Antarctica. If heat pumps are being recommended for the UK then it is an admission winters in the UK are getting warm enough for heat pumps to provide heat, instead of burning gas, coal or wood. $\endgroup$
– Fred
$\begingroup$ Interesting thought @Fred $\endgroup$
– Chris Rogers
$\begingroup$ @Fred heat pumps are also getting better: nrel.gov/news/features/2021/… $\endgroup$
– LShaver
$\begingroup$ @LShaver: Interesting! Thanks $\endgroup$
$\begingroup$ While heat pumps with current energy generation mixes (e.g. mostly burning fossil fuels, with some nuclear and some green) already have an advantage, this will increase when energy generation is fossil fuel free. The UK government has pledged to phase out fossil fuel electricity generation on the same timescales as the introduction of heat pumps. $\endgroup$
– Andy M
The values on the DEFRA table with the column heading $ kg\ \ce{N2O} $ already include the global warming potential (GWP) factor of 298, as indicated by a note on the column heading (looking at electricity):
This is pretty confusing and took me a while to track down, but I figured it out when I noticed that the value in the first column is simply the sum of the values in the next three columns, which only makes sense once you understand that those three values are already converted to equivalent $\ce{CO2}$ emissions (or $\ce{CO2e}$).
There are a couple other minor issues with your calculation:
GWP is a factor per unit mass, so there's no need to convert to US tons.
The 12,000 kWh figure is for gas usage, or what's recorded at the meter. Since no furnace is 100% efficient, this means that the heating load (how much heat is actually added to the home) will be less. For years, natural gas furnaces had an efficiency of 80%, but newer models can reach closer to 98%. Let's assume it's 90% on average. This means the actual heating load is $12\,000 \times 0.9 = 10\,800\ \text{kWh}$, which makes the electricity needed to provide the same amount of heat $10\,800 \div 3 = 3\,600\ \text{kWh}$.
Here's how the final table should look:
Heating fuel (kWh)
$ kg\ \ce{CO2e}$ per kWh
Total $ kg\ \ce{CO2e}$
Natural gas furnace (90%) $12\,000$ $0.18316$ $1\,978$
Heat pump (COP 3) $3\,600$ $0.21233$ $764$
Finally, while the equivalent emissions factor for electricity is currently higher than that for natural gas, it's trending down, as indicated by this note in the spreadsheet:
In the 2019 GHG Conversion Factors, there was a 10% decrease in the UK Electricity CO2e factor compared to the previous year. In the 2020 update, the CO2e factor decreased (compared with 2019) again by 9%. In the 2021 update, the CO2e factor has again decreased by 9% (in comparison to the 2020 update). The above decreases are all due to a decrease in coal use in electricity generation and an increase in renewable generation.
LShaverLShaver
$\begingroup$ "The 12,000 kWh figure is for gas usage, so you need to convert to the actual heating load before calculating the electric usage." - So Viessmann have given bad information then? $\endgroup$
$\begingroup$ @ChrisRogers it's an estimate and I didn't look into the accuracy, but they described it correctly as "gas usage." But 12,000 kWh of gas used (or gas recorded at the meter) doesn't translate to 12,000 kWh of heat energy, because no furnace is 100% efficient. $\endgroup$
$\begingroup$ As the proportion of renewable energy grows the emissions will go down when using a heat pump. With zero emissions electricity any electric heating can go all the way to zero with it but less clean energy needed with heat pump. With gas burning heating must always make emissions. $\endgroup$
– Ken Fabian
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Chapter 1: Fundamental Concepts
1.1 Preparatory Concepts
1.1.1 Scalar vs. Vector
1.1.2 Newton's Laws
1.1.3 Units
1.1.4 Measurement Conversions
1.1.5 Weight vs. Mass
1.1.6 Pythagorean Theorem
1.1.7 Sine/Cosine Law's
1.2 XYZ Coordinate Frame
1.2.1 Cartesian Coordinate Frame in 2D
1.2.2. Cartesian Coordinate Frame in 3D
1.3 Vectors
1.3.1 Vector Components
1.3.2 Componentizing a Vector
1.3.3 Position Vector
1.3.4 Vector Math
1.4 Dot Product
1.5 Cross Products
1.6 Torque/Moment
1.6.1 Moments
1.6.2 Scalar Method in 2 Dimensions
1.6.3 Vector Method in 3 Dimensions
1.7 Problem Solving Process
1.8 Examples
Example 1.8.1: Vectors, Submitted by Tyson Ashton-Losee
Example 1.8.2: Vectors, Submitted by Brian MacDonald
Example 1.8.3: Dot product and cross product, submitted by Anonymous ENGN 1230 Student
Example 1.8.4: Torque, Submitted by Luke McCarvill
Example 1.8.5: Torque, submitted by Hamza Ben Driouech
Example 1.8.6: Bonus Vector Material, Submitted by Liam Murdock
Chapter 2: Particles
2.1 Particle & Rigid Body
2.2 Free Body Diagrams for Particles
2.3 Equilibrium Equations for Particles
2.4. Examples
Chapter 3: Rigid Body Basics
3.1 Right Hand Rule
3.1.1 The Whole-Hand Method
3.1.2 Right Hand Rule and Torque
3.1.3 Three-Finger Configuration
3.2 Couples
3.3 Distributed Loads
3.3.1 Intensity
3.3.2 Equivalent Point Load & Location
3.3.3 Composite Distributed Loads
3.4 Reactions & Supports
3.5 Indeterminate Loads
Example 3.6.1: Reaction Forces, Submitted by Andrew Williamson
Example 3.6.2: Couples, Submitted by Kirsty MacLellan
Example 3.6.3: Distributed Load, Submitted by Luciana Davila
Chapter 4: Rigid Bodies
4.1 External Forces
4.2 Rigid Body Free Body Diagrams
4.2.1 Part FBD
4.2.2 System FBD
4.2.3 Examples
4.3 Rigid Body Equilibrium Equations
4.4 Friction and Impending Motion
Example 4.5.1: External Forces, submitted by Elliott Fraser
Example 4.5.2: Free-Body Diagrams, submitted by Victoria Keefe
Example 4.5.3: Friction, submitted by Deanna Malone
Example 4.5.4: Friction, submitted by Dhruvil Kanani
Example 4.5.5: Friction, submitted by Emma Christensen
Chapter 5: Trusses
5.1 Trusses Introduction
5.1.1 Two Force Members
5.1.2 Trusses
5.1.3 Parts of a Truss
5.1.4 Tension & Compression
5.2 Method of Joints
5.3 Method of Sections
5.4 Zero-Force Members
Example 5.5.1: Method of Sections – Submitted by Riley Fitzpatrick
Example 5.5.2: Zero-Force Members, submitted by Michael Oppong-Ampomah
Chapter 6: Internal Forces
6.1 Types of Internal Forces
6.1.1 Types of Internal Forces
6.1.2 Sign Convention
6.1.3 Calculating the Internal Forces
6.2 Shear/Moment Diagrams
6.2.1 What are Shear/Moment Diagrams?
6.2.2 Distributed Loads & Shear/Moment Diagrams
6.2.3 Producing a Shear/Moment Diagram
6.2.4 Tips & Plot Shapes
Example 6.3.1: Internal Forces – Submitted by Emma Christensen
Example 6.3.2: Shear/Moment Diagrams – Submitted by Deanna Malone
Chapter 7: Inertia
7.1 Center of Mass: Single Objects
7.1.1 Center of Mass of Two Particles
7.1.2 Center of Mass in 2D & 3D
7.1.3 The Center of Mass of a Thin Uniform Rod (Calculus Method)
7.1.4 The Center of Mass of a Non-Uniform Rod
7.2 Center of Mass: Composite Shapes
7.2.1 Centroid Tables
7.2.2 Composite Shapes
7.3 Types of Inertia
7.4 Mass Moment of Inertia
7.4.1 Intro to Mass Moment of Inertia
7.4.2 Inertia Table of Common Shapes
7.4.3 Radius of Gyration
7.5 Inertia Intro: Parallel Axis Theorem
Example 7.6.1: All of Ch 7 – Submitted by William Craine
Example 7.6.2 Inertia – Submitted by Luke McCarvill
Appendix A: Included Open Textbooks
Engineering Mechanics: Statics
To start, let's calculate the center of mass! This is a weighted function, similar to when we found the location of the resultant force from multiple distributed loads and forces.
[latex]\bar{x}=\frac{m_1*x_1}{m_1+m_2}+\frac{m_2*x_2}{m_1+m_2}[/latex]
When the density is the same throughout a shape, the center of mass is also the centroid (geometric center).
Consider two particles, having one and the same mass m, each of which is at a different position on the x axis of a Cartesian coordinate system.
Common sense tells you that the average position of the material making up the two particles is midway between the two particles. Common sense is right. We give the name "center of mass" to the average position of the material making up a distribution, and the center of mass of a pair of same-mass particles is indeed midway between the two particles. How about if one of the particles is more massive than the other? One would expect the center of mass to be closer to the more massive particle, and again, one would be right. To determine the position of the center of mass of the distribution of matter in such a case, we compute a weighted sum of the positions of the particles in the distribution, where the weighting factor for a given particle is that fraction, of the total mass, that the particle's own mass is. Thus, for two particles on the x axis, one of mass m1, at x1, and the other of mass m2, at x2,
the position x of the center of mass is given by equation 8-1:
Note that each weighting factor is a proper fraction and that the sum of the weighting factors is always 1. Also note that if, for instance, m1 is greater than m2, then the position x1 of particle 1 will count more in the sum, thus ensuring that the center of mass is found to be closer to the more massive particle (as we know it must be). Further note that if m1 = m2, each weighting factor is 1/2, as is evident when we substitute m for both m1 and m2 in equation 8-1:
$$\bar{x}=\frac{m}{m+m}x_1+\frac{m}{m+m}x_2\\\bar{x}=\frac{1}{2}x_1+\frac{1}{2}x_2\\\bar{x}=\frac{x_1+x_2}{2}$$
The center of mass is found to be midway between the two particles, right where common sense tells us it has to be.
Source: Calculus-Based Physics 1, Jeffery W. Schnick. p142, https://openlibrary.ecampusontario.ca/catalogue/item/?id=ce74a181-ccde-491c-848d-05489ed182e7
Below is a more visual representation of where the COM would be for two different weighing particles.
Source (image): Two_body_jacobi.svg: CWitte, from JPG by Brews oharederivative work: WillowW via Wikimedia Commons https://zh.wikipedia.org/wiki/File:Jacobi_coordinates.svg
A second explanation:
The most common real-life example of a system like this is a playground seesaw, or teeter-totter, with children of different weights sitting at different distances from the center. On a seesaw, if one child sits at each end, the heavier child sinks down and the lighter child is lifted into the air. If the heavier child slides in toward the center, though, the seesaw balances. Applying this concept to the masses on the rod, we note that the masses balance each other if and only if m1d1 = m2d2.
This idea is not limited just to two point masses. In general, if 𝑛 masses, 𝑚1, 𝑚2,…,𝑚𝑛, are placed on a number line at points 𝑥1,𝑥2,…,𝑥𝑛, respectively, then the center of mass of the system is given by:
$$ \bar x=\frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^nm_i}$$
Suppose four point masses are placed on a number line as follows:
𝑚1=30𝑘𝑔, placed at 𝑥1=−2𝑚
𝑚2=5𝑘𝑔, placed at 𝑥2=3𝑚
𝑚3=10𝑘𝑔,placed at 𝑥3=6𝑚
𝑚4=15𝑘𝑔,placed at 𝑥4=−3𝑚.
Find the moment of the system with respect to the origin and find the center of mass of the system.
First, we need to calculate the moment of the system (the top part of the fraction):
[latex]M =\sum_{i=1}^4 m_i *x_i \\\qquad \quad = (30kg)*(-2m) + (5kg)*(3m)+(10kg)*(6m)+(15kg)*(-3m) \\\qquad\quad = (-60+15+60-45)kg*m \\\qquad\quad = -30 kg*m[/latex]
Now, to find the center of mass, we need the total mass of the system:
$$ m = \sum_{i=1}^4 m_i = (30+5+10+15) kg = 60kg $$
Then we have [latex]\bar{x} = \frac{M}{m} = \frac{-30 kg*m}{60kg} = -0.5 m[/latex]
The center of mass is located 1/2 m to the left of the origin.
Source: "Moments and Centers of Mass" by LibreTexts, https://eng.libretexts.org/@go/page/67237
When we are looking at multiple objects in 2D or 3D, we perform the center of mass equation multiple times in the x, y, and z directions.
$$ \bar x=\frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^nm_i} \qquad \bar y=\frac{\sum_{i=1}^n m_i y_i}{\sum_{i=1}^nm_i} \qquad \bar z=\frac{\sum_{i=1}^n m_i z_i}{\sum_{i=1}^nm_i}$$
In some sense, one can think about the center of mass of a single object as its "average position." Let's consider the simplest case of an "object" consisting of two tiny particles separated along the x-axis, as seen below:
If the two particles have equal mass, then it's pretty clear that the "average position" of the two-particle system is halfway between them. If the masses of the two particles are different, would the "average position" still be halfway between them? Perhaps in some sense this is true, but we are not looking for a geometric center, we are looking for the average placement of mass. If m1 has twice the mass of m2, then when it comes to the average placement of mass, m1 gets "two votes." With more of the mass concentrated at the position x1 than at x2, the center of mass should be closer to x1 than x2. We achieve the perfect balance by "weighting" the positions by the fraction of the total mass that is located there. Accordingly, we define as the center of mass:
$$\bar x_{cm}=(\frac{m_1}{m_1+m_2})x_1+(\frac{m_2}{m_1+m_2})x_2=\frac{m_1x_1+m_2x_2}{M_{system}}$$
If there are more than two particles, we simply add all of them into the sum in the numerator. To extend this definition of center of mass into three dimensions, we simply need to do the same things in the y and z directions. A position vector for the center of mass of a system of many particles would then be:
$$\vec{r}_{cm}=\bar x_{cm}\underline{\hat{i}}+\bar y_{cm}\underline{\hat{j}}+ \bar z_{cm}\underline{\hat{k}}\\=\frac{[m_1 x_1+m_2 x_2+…]}{M}\underline{\hat{i}}+\frac{[m_1y_1+m_2y_2+…]}{M}\underline{\hat{j}}+\frac{[m_1 z_1+m_2 z_2+…]}{M}\underline{\hat{k}}\\=\frac{m_1[x_1\underline{\hat{i}}+y_1\underline{\hat{j}}+z_1\underline{\hat{k}}]+m_2[x_2\underline{\hat{i}}+y_2\underline{\hat{j}}+z_2\underline{\hat{k}}]+…}{M}\\=\frac{m_1\vec r_1+m_2\vec r_2+…}{M}$$
Source: " Center of Mass" by Tom Weideman, https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9A__Classical_Mechanics/4%3A_Linear_Momentum/4.2%3A_Center_of_Mass
Suppose three point masses are placed in the x-y plane as follows (assume coordinates are given in meters):
m1 = 2 kg placed at (-1, 3)m,
m2 = 6 kg placed at (1, 1)m, and
m3 = 4 kg placed at (2, -2)m.
Find the center of mass of the system.
First we calculate the total mass of the system:
$$ m = \sum_{i=1}^3 m_i = (2 + 6 + 4) kg = 12 kg $$
Next we find the moments with respect to the x- and y- axes:
[latex]M_x =\sum_{i=1}^3 m_i *x_i \\\qquad \quad = (2kg)*(-1m) + (6kg)*(1m)+(4kg)*(2m) \\\qquad\quad = (-2+6+8)kg*m \\\qquad\quad = 12 kg*m[/latex]
[latex]M_y =\sum_{i=1}^3 m_i *y_i \\\qquad \quad = (2kg)*(3m) + (6kg)*(1m)+(4kg)*(-2m) \\\qquad\quad = (6+6-8)kg*m \\\qquad\quad = 4 kg*m[/latex]
Then we have
[latex]\bar{x} = \frac{M_x}{m} = \frac{12 kgm}{12m} = 1 m[/latex]
[latex]\bar{y} = \frac{M_y}{m} = \frac{4 kgm}{12m} = 0.333 m[/latex]
The center of mass of the system is: (1, 0.333)m.
Quite often, when the finding of the position of the center of mass of a distribution of particles is called for, the distribution of particles is the set of particles making up a rigid body. The easiest rigid body for which to calculate the center of mass is the thin rod because it extends in only one dimension. (Here, we discuss an ideal thin rod. A physical thin rod must have some nonzero diameter. The ideal thin rod, however, is a good approximation to the physical thin rod as long as the diameter of the rod is small compared to its length.)
In the simplest case, the calculation of the position of the center of mass is trivial. The simplest case involves a uniform thin rod. A uniform thin rod is one for which the linear mass density µ, the mass-per-length of the rod, has one and the same value at all points on the rod. The center of mass of a uniform rod is at the center of the rod. So, for instance, the center of mass of a uniform rod that extends along the x axis from x = 0 to x = L is at (L/2, 0).
The linear mass density µ, typically called linear density when the context is clear, is a measure of how closely packed the elementary particles making up the rod are. Where the linear density is high, the particles are close together.
To picture what is meant by a non-uniform rod, a rod whose linear density is a function of position, imagine a thin rod made of an alloy consisting of lead and aluminum. Further imagine that the percentage of lead in the rod varies smoothly from 0% at one end of the rod to 100% at the other. The linear density of such a rod would be a function of the position along the length of the rod. A one-millimeter segment of the rod at one position would have a different mass than that of a one-millimeter segment of the rod at a different position.
People with some exposure to calculus have an easier time understanding what linear density is than calculus-deprived individuals do because linear density is just the ratio of the amount of mass in a rod segment to the length of the segment, in the limit as the length of the segment goes to zero. Consider a rod that extends from 0 to L along the x axis. Now suppose that ms(x) is the mass of that segment of the rod extending from 0 to x where x ≥ 0 but x < L. Then, the linear density of the rod at any point x along the rod, is just dm8/dx evaluated at the value of x in question.
Now that you have a good idea of what we mean by linear mass density, we are going to illustrate how one determines the position of the center of mass of a non-uniform thin rod by means of an example.
Find the position of the center of mass of a thin rod that extends from 0 to 0.890 m along the x axis of a Cartesian coordinate system and has a linear density given by µ = 0.650 kg/m3
In order to be able to determine the position of the center of mass of a rod with a given length and a given linear density as a function of position, you first need to be able to find the mass of such a rod. To do that, one might be tempted to use a method that works only for the special case of a uniform rod, namely, to try using m = µL with L being the length of the rod. The problem with this is, that µ varies along the entire length of the rod. What value would one use for µ ? One might be tempted to evaluate the given µ at x = L and use that, but that would be acting as if the linear density were constant at µ = µ(L). It is not. In fact, in the case at hand, µ(L) is the maximum linear density of the rod, it only has that value at one point on the rod.
Instead, using integration, we find the equation:
[latex]m=\frac{bL^3}{3}[/latex]
That can now be used to calculate the mass of a non-linear rod. The value of L is given as 0.890 m and we defined b to be the constant 0.650 kg/m3, therefore
$$m=\frac{0.650\frac{kg}{m^3}(0.890m)^3}{3}\\m=0.1527kg$$
That's a value that will come in handy when we calculate the position of the center of mass.
Now, when we calculated the center of mass of a set of discrete particles (where a discrete particle is one that is by itself, as opposed, for instance, to being part of a rigid body) we just carried out a weighted sum in which each term was the position of a particle times its weighting factor and the weighting factor was that fraction, of the total mass, represented by the mass of the particle. We carry out a similar procedure for a continuous distribution of mass such as that which makes up the rod in question.
Once again, using integration, we find the equation:
[latex]\bar{x}=\frac{bL^4}{4m}[/latex]
Now we substitute variables with values; the mass m of the rod that we found earlier, the constant b that we defined to simplify the appearance of the linear density function, and the given length L of the rod:
$$m= \frac{\left( 0.650\frac{kg}{m^3} \right) (0.890m)^4}{4(0.1527kg)}\\\bar{x}=0.668m$$
This is our final answer for the position of the center of mass. Note that it is closer to the denser end of the rod, as we would expect.
Basically: When there are multiple objects, the center of mass is the location in the x, y, and z directions between the objects.
Application: To calculate the acceleration or use F = ma, m is the total mass at the center of mass.
Looking Ahead: The next section will look at how to calculate the center of mass for a complex object.
Previous: Chapter 7: Inertia
Next: 7.2 Center of Mass: Composite Shapes
Engineering Mechanics: Statics by Libby (Elizabeth) Osgood; Gayla Cameron; and Emma Christensen is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. | CommonCrawl |
Software | Open | Published: 13 April 2018
Genexpi: a toolset for identifying regulons and validating gene regulatory networks using time-course expression data
Martin Modrák ORCID: orcid.org/0000-0002-8886-77971 &
Jiří Vohradský1
BMC Bioinformaticsvolume 19, Article number: 137 (2018) | Download Citation
Identifying regulons of sigma factors is a vital subtask of gene network inference. Integrating multiple sources of data is essential for correct identification of regulons and complete gene regulatory networks. Time series of expression data measured with microarrays or RNA-seq combined with static binding experiments (e.g., ChIP-seq) or literature mining may be used for inference of sigma factor regulatory networks.
We introduce Genexpi: a tool to identify sigma factors by combining candidates obtained from ChIP experiments or literature mining with time-course gene expression data. While Genexpi can be used to infer other types of regulatory interactions, it was designed and validated on real biological data from bacterial regulons. In this paper, we put primary focus on CyGenexpi: a plugin integrating Genexpi with the Cytoscape software for ease of use. As a part of this effort, a plugin for handling time series data in Cytoscape called CyDataseries has been developed and made available. Genexpi is also available as a standalone command line tool and an R package.
Genexpi is a useful part of gene network inference toolbox. It provides meaningful information about the composition of regulons and delivers biologically interpretable results.
Uncovering the nature of gene regulatory networks is one of the core tasks of systems biology. Identifying direct regulons of sigma factors/transcription factors can be considered the basic element of this task. In fact a large portion of software for network inference is limited to such direct interactions (e.g., [1,2,3]). It has however been shown that using only one source of data for network inference (e.g., only CHIP-seq experiment) can be misleading and combining multiple sources is necessary [4].
Primary focus of this paper is on CyGenexpi – a plugin for the Cytoscape platform [5] that uses time-course gene expression data to discover regulons among candidate genes obtained from other sources (literature, database mining, or ChIP experiments). CyGenexpi can be also used for de-novo network inference, although this is less reliable. CyGenexpi is built on top of the Genexpi software package that provides the core functionality also as a command-line tool and an interface to the R language.
Genexpi is based on an ordinary differential equation model of gene expression introduced in [6]. In the model, the synthesis of new mRNA for a gene is determined by a non-linear (sigmoidal) transformation of the expression of its regulators. The model also includes a per-gene decay rate of the mRNA, which is assumed to be constant.
While there are multiple tools for gene network inference from the command line or programming languages (see [7] for a recent review), there are currently, only two Cytoscape plugins for gene network inference: ARACNE [8] and Network BMA [9]. ARACNE is intended for steady-state expression data, while Network BMA handles time series, but assumes a simple linear model of regulation without regard to mRNA decay. CyGenexpi thus provides an alternative to Network BMA in that it builds on a non-linear model including decay.
A preliminary version of the method presented in this paper has been applied in our previous work [10]. The additional contribution of this paper is a) a polished and documented publicly available implementation of the method with well-defined API, b) improved workflow and software support for the workflow c) interfacing the method with Cytoscape and R and d) evaluation of the method on additional datasets. As Cytoscape does not natively support working with time series data, we also developed CyDataseries - a plugin for importing and handling time series and other forms of repeated measurements data in Cytoscape.
Both Genexpi and CyDataseries are imlemented in Java and are platform independent. Binaries, source code and documentation are available at http://github.com/cas-bioinf/genexpi/wiki/. The software is open source and licensed under LGPL version 3.
The core of Genexpi – the algorithm for fitting model parameters – is implemented in OpenCL, with a Java wrapper. Thanks to high portability of both Java and OpenCL, Genexpi can be executed on both GPUs and CPUs in any major operating system and has very good performance. There are currently three interfaces to Genexpi core: CyGenexpi (a Cytoscape plugin), a command-line interface and an R interface. In this section we describe the model and fitting method of Genexpi – the implementation of the interfaces is straightforward. Initial part of this section is taken from [10] and its supplementary material where we describe first use of Genexpi in practice. In addition we provide details of regularization and parameter fitting as well as further developments made to make the method usable by non-expert users, especially the semi-automatic evaluation of good fits and the "no change" and "constant synthesis" models.
Genexpi is based on an ordinary differential equation (ODE) model for gene regulation, inspired by the neural network formalism [6]. In this model the synthesis of new mRNA for a gene z controlled by set of m regulators y 1 ,..,y m (genes or any other regulatory influence) is determined by activation function f(ρ(t)) of the regulatory input \( \rho (t)={\sum}_{j=1..m}{w}_j{y}_j(t)+b \). Here w j is the relative weight of regulator y j and b is bias (inversely related to the regulatory influence that saturates the synthesis of the mRNA). In our case, f is the logistic soft-threshold function f(x) = 1/(1 + e-x). The transcript level of z is then governed by the ODE:
$$ \frac{dz}{dt}={k}_1f\left(\rho \right)-{k}_2z $$
where k 1 is related to the maximal level of mRNA synthesis and k 2 represents the decay rate of the mRNA. Both k 1 and k 2 must be positive. The complete set of parameters for this model is thus β = {k 1 , k 2 , b, w 1 ,…, w m }. Given N samples from a time series of gene expression taken at time points t 1 , …, t N the inference task can be formalized as finding β that minimizes squared error with regularization:
$$ \widehat{\beta}=\underset{\beta }{\mathrm{argmin}}\left[\sum \limits_{i=1}^N{\left({\widehat{z}}_{\beta}\left({t}_i\right)-z\left({t}_i\right)\right)}^2+r\left(\beta \right)\right] $$
Here z is the observed expression profile, \( {\widehat{z}}_{\beta } \) the solution to (1) given the parameter values β and the observed expression of y 1 ,..,y m , and r(β) is the regularization term. The regularization term represents a prior probability distribution over β that gives preference to biologically interpretable values for β and is discussed in more detail below. Assuming Gaussian noise in the expression data, (2) is the maximum a posteriori estimate of β.
Our model is similar to that used by the Inferelator algorithm [1], although there are important differences: the Inferelator does not model decay (k 2 ) – it assumes decay is always one. Further, Inferelator minimizes the error of the predicted derivative of the expression profile, while we minimize the prediction error for the actual integrated expression profile and introduce the regularization term.
Smoothing the expression profiles
Since the expression data is noisy, Genexpi encourages smoothing the data prior to computation. We have had good results with linear regression of B-spline basis with degrees of freedom equal to approximately half the number of measurement points. By smoothing we get more robust results with respect to low frequency phenomena, but sacrifice our ability to discover high-frequency changes and regulations (oscillations with frequency comparable to the measurement interval are mostly suppressed). Further our experiments with fitting raw data or tight interpolations of the data (e.g. cubic spline with knots at all measurement points) have had little success in fitting even the profiles that were highly correlated, due to the amplified noise in the data. Smoothing of time series profiles has been used previously for network inference [11].
Further advantage of smoothing is that it lets us subsample the fitted curve at arbitrary resolution. The subsampling then allows us to integrate (1) accurately with the computationally cheap Euler method, making evaluation of the error function fast and easy to implement in OpenCL.
Parameter fitting and regularization
Genexpi minimizes eq. 2 by simulated annealing. For each gene and candidate regulator set we execute 128 annealing runs with different initial parameter values. Using 128 runs was enough to achieve high replicability of the results. Annealing runs for the same target and regulator are executed on the same OpenCL compute unit, letting us to move all necessary data to local memory and thus increase efficiency. We use the XorShift1024* random generator [12] as a fast and high quality parallel source of randomness.
Note that in some cases, multiple vastly different combinations of parameters may yield almost identical regulatory profiles. For example, if the interval of attained regulatory input \( \left(\underset{i=1..N}{\min}\rho \left({t}_i\right);\underset{i=1..N}{\max}\rho \left({t}_i\right)\right) \) lies completely on one of the tails of f, the activation function becomes approximately linear over the whole interval, so increasing the weights and decreasing bias while decreasing k 1 yields a very similar \( {\widehat{z}}_{\beta } \). To discriminate between those models and to force the parameters into biologically interpretable ranges, we introduce the regularization term r(β). In particular, we expect k 1 smaller than the maximal expression level of the target gene (i.e., that maximal transcript level cannot be achieved in less than a unit time starting from zero), we put a bound on maximal steepness of the regulatory response: \( \underset{t}{\max}\mid {w}_j{y}_j(t)\mid <10 \) for all regulators j and we expect the regulatory input to come close to zero (the steepest point of the sigmoid function) for at least one time point: \( \underset{t}{\min}\mid \rho (t)\mid <0.5 \). For a suitable penalty function γ(x, ω) the regularization term becomes:
$$ r\left(\beta \right)=c\left(\gamma \left({k}_1,\underset{i=1..N}{\max }z\left({t}_i\right)\right)+\sum \limits_{j=1}^m\gamma \Big(\underset{i=1..N}{\max}\left|{w}_j{y}_j\left({t}_i\right)\right|,10\left)+\gamma \right(\underset{i=1..N}{\min}\left|\rho \left({t}_i\right)\right|,0.5\right) $$
where c is a constant governing the amount of regularization. In our work, the penalty for value x > 0 and bound ω is:
$$ \gamma \left(x,\omega \right)=\left\{\begin{array}{c}0,\kern0.5em x\le \omega \\ {}{\left(\frac{x}{\omega }-1\right)}^2,\kern0.5em x>\omega \end{array}\right. $$
Minimizing γ(x, ω) is then the same as maximizing log-likelihood, assuming that x is distributed uniformly over (0; ωx) with some probability p and as ωx + α|e| with probability (1 – p) where e ∼ N(0, 1). In this interpretation, the probability p is uniquely determined by c in the regularization term and by choosing α such that the resulting density function is continuous.
We have empirically determined the best value of c to be approximately one tenth of the number of time points after smoothing. While without regularization, many of the inferred models contained implausible parameter values, regularization forced almost all of those parameters into given bounds - r(β) was zero for most models. At the same time the mean residual error of the models inferred with regularization differed by less than one part in hundred from models inferred without regularization.
Evaluating good fits
To evaluate whether a fit is good, we have chosen a simple, but easily interpretable approach. The primary reason is that we intend to keep the human in the loop throughout the inference process and thus the human has to be able to understand the criteria intuitively. Since most published time series expression data is reported only as averages without any quantification of uncertainty, we let the user set the expected error margin based on their knowledge of the data. The error margin is determined by three parameters: absolute, relative and minimal error. These combine in a straightforward way to get an error margin for each time point, depending on the expression level z(t):
$$ error(t)=\max \left\{{e}_{minimal},{e}_{absolute}+z(t){e}_{relative}\right\} $$
Fit quality is then the proportion of time points where the fitted profile is within the error margin of the measured profile. A fit is considered good if fit quality is above a given threshold (the default value is 0.8).
No change and constant synthesis model
Prior to analyzing a gene as being regulated, we need to test for two baseline cases that would make any prediction useless. The obvious first case are genes that do not change significantly over the whole time range. Genes that do not change are excluded from further analysis as both regulators and targets as the Genexpi model contains no information in that case.
A slightly more complicated case is the constant synthesis model where we expect the mRNA synthesis to be constant over the whole time range:
$$ \frac{dz}{dt}={k}_1-{k}_2z $$
Note that this is the same as assuming there are 0 regulators. Since genes with constant synthesis could be fitted by any regulator by simply putting w = 0, and large b, those genes are excluded as targets. However, regulators that could be explained by constant synthesis are still analyzed, as there is meaningful information. Fitting the constant synthesis model is also done via simulated annealing in OpenCL.
For the putative regulations excluded this way, the correct interpretation is that the underlying dataset provides no evidence for or against such regulations. If there are biological justifications that the regulations should be visible in the data (e.g. that the regulatory effect should be larger than the measurement noise), it is possible to cautiously consider this as evidence against the regulations taking place.
In this section we describe the intended workflow for analysis with Genexpi and its user interface and then we discuss results of evaluation on real biological data.
The primary user interface for Genexpi is the CyGenexpi plugin for the Cytoscape software, but Genexpi can also be run directly from R and via a command line interface. For CyGenexpi, an important improvement over the Aracne or NetworkBMA Cytoscape plugins is the direct involvement of user in the process.
Genexpi workflow
The workflow for analysis with Genexpi is as follows:
Start with a network of putative regulations either obtained from database mining or experiments.
Import the time-course expression data and smooth them to provide a continuous curve.
Remove genes whose expression does not change significantly throughout the whole time-course.
Remove genes that could be modelled by the constant synthesis model.
Optional: Human inspection of the results of steps 3&4, possibly overriding the algorithm's decisions.
Finding best parameters of the Genexpi model for each gene-regulator pair. The fitted models are then classified into good and bad fits. Good fits indicate that the regulation is plausible, while bad fits show that the regulation either does not take place or involves additional regulators.
Optional: Human inspection of the fits, possibly overriding the algorithm's classification (shown in Fig. 1).
Human inspection of the model fits in CyGenexpi. The user is shown the profile of the regulator (blue) and target (red) as well as the best profile found by Genexpi (green). The red ribbon is the error margin of the measured profile. The algorithm classified the first profile as a good fit, while the second was considered implausible to be regulated. The user may however modify the classification based on their knowledge of the data and organism
This workflow is completely covered by CyGenexpi with the help of CyDataseries in a simple wizard-style interface. Alternatively, the same workflow, but without human intervention can be run by a single function call in R. All interfaces also provide the user with the ability to run individual steps separately.
While Genexpi can include multiple regulators for a gene, we found this not very useful in practice, as even for relatively long expression time series (13 time points), an arbitrary pair of regulators is able to model the expression of a large fraction of all genes, increasing the false positive rate. CyGenexpi therefore currently does not expose GUI for using more than one regulator in the model. Using more regulators is however available for more advanced users via the command-line or R interfaces.
For CyGenexpi, the time series data is imported with CyDataseries from either a delimited text file or the SOFT format used in Gene Expression Omnibus.
While Genexpi can be used for de-novo regulon identification from time-series expression data only, high rate of false positives should be expected. The main reason is that in real biological data, multiple sigma factors may have similar expression profiles and Genexpi thus considers all genes regulated by one of the sigma factors as possibly regulated by all of the similar sigma factors. The evaluation in this paper therefore focuses on identifying the regulated genes among a set of plausible candidates. Nevertheless, the workflow for de-novo inference is almost the same as described above, only the initial network should contain a link from each investigated regulator to all other genes.
We evaluated Genexpi in three ways: 1) direct biological testing of the suggested regulatory relationships, 2) comparing the ability of Genexpi and other tools to reconstruct two literature-derived regulons and 3) measuring computing time required to process the data. The first part of the evaluation is taken from our previous work [10], while the latter two are new contributions.
In-vitro biological evaluation
This section recapitulates the relevant results obtained with Genexpi, originally reported as a part of [10]. We performed a basic analysis of the predictive performance of Genexpi with the SigA regulon of Bacillus subtilis combined with the expression time series from GSE6865 [13]. We followed the Genexpi workflow outlined in the previous section, including evaluation of fits by human. Genexpi predicted 215 genes that were not known to be regulated by SigA as potential SigA targets. We selected 10 of those genes for in-vitro transcription assays.Footnote 1 We found that 5 of them were SigA-dependent (for the remaining five, the regulation could not be excluded). More details of the SigA analysis can be found in the aforementioned paper. We have however excluded the SigA regulon from purely computational evaluation as the method was developed and tweaked for the SigA data and any comparison would thus be likely biased.
Reconstructing bacterial regulons
To extend the biological evaluation from [10] and to better determine Genexpi's performance in identifying regulons, we took two bacterial regulons from the literature: a) the SigB regulon of B. subtilis from Subtiwiki [14] as of January 2017 combined with the GSE6865 expression time series [13] and b) two versions of the SigR regulon of Streptomyces coelicolor: one derived with ChIP-chip [15] and the one determined via knockouts [16]. Both versions of the SigR regulon were combined with the GSE44415 expression time series [17].
For each of the literature regulons we first exclude targets that were constant or had constant synthesis (steps 3&4 of the workflow) and determined how many of the remaining members were considered by Genexpi to be regulated by the respective sigma factor – these correspond to true positives. Then we generated a set of random expression profiles with similar magnitude and rate of change as the sigma factor. Inspired by [18] we draw random profiles from a Gaussian process with a squared exponential kernel with zero mean function, transformed to have positive values. See Fig. 2 for an example of the random profiles. We then tested how many targets were predicted to be regulated by this nonsensical profile – these correspond to false positives.
A sample of the random profiles tested against the SigB regulon. The dots represent the measured (not smoothed) profile of SigB
We consider testing a random regulator profile as a more reliable assessment than testing the complement of the literature-based regulon for two reasons. First, it is a better match for the intended Genexpi workflow, which starts with a set of candidate genes. Here, using a random profile for the regulator models the situation where the candidate list is wrong and we expect Genexpi to reject that there is regulatory influence on most genes. Second, the complement is usually composed of less characterized genes and there is little guarantee that the complement contains genes that are not regulated by the sigma factor. The complement may include genes that are regulated with the sigma factor, but were not annotated yet, and also genes that have expression profile similar to the profiles of the regulon of the analyzed sigma factor due to chance or non-regulatory interactions. Such profiles would be classified as false positives, while they in fact have nothing to do with the analyzed regulon and its sigma factor. Comparing the performance on regulon complement actually depends more on the uniqueness of the sigma factor profile than on the inference algorithm.
For this evaluation we ran Genexpi with default settings and without any human input. Complete code to reproduce all of the results for this and the following section is attached as an R notebook in Additional file 1.
For comparison, we performed the same analysis with TD-Aracne [19] – an extension of the frequently used Aracne algorithm designed for time series data. TD-Aracne was run both on the whole dataset at once and on each regulator-target pair separately. Running regulator-target pairs however had much worse performance than using the whole dataset, so those results are omitted here, but can be inspected in Additional file 1. We also compared the results for the whole regulon and for the subset of the regulon that was predicted by Genexpi, i.e. without the genes removed in steps 3&4 of the workflow.
For all analyses, we smoothed the raw data by linear regression over B-spline basis of order 3 with 3–10 degrees of freedom. TD-Aracne was tested with the raw data as well as the smoothed data subsampled to give lower number of equal-spaced time points as expected by TD-Aracne. For TD-Aracne we tested three methods of recovering the regulon from the inferred network over the full gene set: a) take only the genes that were marked as directly regulated by the sigma factor, b) take all genes connected by a directed path from the regulator and c) take all genes connected to the regulator. Variant a) had very low performance overall, among b) and c) we report the result more favorable to TD-Aracne. For the SigR regulon of Kim et al., the results were very similar when only the targets marked as having "strong" evidence were used. All results not shown here can be found in Additional file 1. See Table 1 for the main results.
Table 1 Main Evaluation Results
In the SigB regulon, the Genexpi performs slightly better than TD-Aracne. While TD-Aracne (in multiple settings) confirms almost all of the literature regulon while rejecting over half of the regulations by a random profile, Genexpi using spline with 4 degrees of freedom rejects two thirds of random regulations while also recovering 90% of the literature regulon. Moreover, Genexpi has the advantage of allowing for a sensitivity/specificity tradeoff by choosing the degree of freedom for the spline – with high degrees of freedom, almost all random regulations are rejected while still recovering majority of the literature regulon. The performance of TD-Aracne varied unexpectedly with the chosen degree of freedom. We also see, that running TD-Aracne with smoothed data and removing no change and constant synthesis genes as in Genexpi workflow, allows for only slight improvements for the performance of TD-Aracne over running directly with the raw data (as TD-Aracne is designed to work).
For both variants of the SigR regulon, TD-Aracne mostly found little difference between the literature based and random regulons. The few cases of better performance by TD-Aracne occurred unpredictably with certain smoothing of the data. At the same time, Genexpi was rarely misled by the random regulations and recovered large fractions of the literature regulon while behaving consistently: the proportion of both true and random regulations grows with more aggressive smoothing (less degrees of freedom).
Computing time required
For analysis of computing time, Genexpi was run on a mid-tier GPU (Asus Radeon RX 550) and TD-Aracne on an upper-level CPU (Intel i7 6700 K). Both algorithms were run on a Windows 10 workstation with only basic precautions to prevent other process from perturbing the system load. The numbers reported should therefore not be considered benchmarks but rather an informative estimate of the computing time during a normal analysis workflow. The results are shown in Table 2 and indicate that Genexpi was fast enough to be run repeatedly on commodity hardware with TD-Aracne being slower, but still fast enough for most practical use cases.
Table 2 Computing time [s] required for a single inference run on the given regulon
Reconstructing eukaryotic regulons
While Genexpi was designed for bacterial regulons, we also tested its performance on eukaryotic data, in particular the time series of gene expression throughout the cell cycle of Saccharomyces cerevisiae [20], deposited as GDS38. We chose the same 8 transcription factors regulating the cell cycle as in our previous work [21] and downloaded their regulons from the YEASTRACT database (as of 2018–02-09) [22]. We used spline with 6 degrees of freedom to smooth the data and interpolate missing values. After excluding constant and constant synthesis targets (steps 3&4 of the workflow), we selected 30 targets for each gene at random to reduce computational burden. We then proceeded as in the bacterial regulons evaluation by generating random profiles and comparing recovered regulations by both Genexpi and TD-Aracne across the measured regulator profiles and 20 random profiles. The results are shown in Table 3.
Table 3 Evaluation results for S. cerevisiae
In this case, the signal is weaker than in the prokaryotes, which is not unexpected given the increased complexity of eukaryotic regulation. Genexpi gives the worst (undistinguishable from random) results for MBP1, SWI4 and SWI6, which are known to regulate in complexes and thus break the model expected by Genexpi. Interestingly, TD-Aracne is able to determine some of those regulations. For the other genes, Genexpi provides consistent, but weak information while TD-Aracne provides strong signal for some genes, while performing very poorly on the others.
The full code to reproduce the analysis can be found in Additional file 1.
The Genexpi workflow was kept deliberately simple, but this involves some inaccuracies. Most notably, Genexpi masks uncertainty in the data and uses multiple hard thresholds. Following [18] that use a similar model of gene regulation in a fully Bayesian setting, we want to extend Genexpi to handle uncertainty explicitly and provide full posterior probability distributions for the quantities of interest.
Our evaluation has shown that Genexpi is a useful part of a bioinformatician's toolbox for uncovering and/or validating regulons in biological systems. Genexpi was designed for bacterial regulons, but can be – with caution – employed also for eukaryotic data. It also provides transparent results and – unlike other similar programs - lets the human to stay in the loop and apply expert knowledge when necessary. The parameters of the fitted models are biologically interpretable and thus can guide design of future experiments. Time-series expression data cannot in principle provide complete information about the regulatory interactions taking place and Genexpi is therefore best used as one of multiple sources of insight about a biological system.
Genexpi is equipped with both simple point&click interface for the Cytoscape application and with R and command-line interfaces for advanced users.
List of mathematical notation
k 1 synthesis rate of a gene at full activation
k 2 decay rate of a gene
w i weight of regulatory influence of putative regulator I on the gene
b bias of the activation function
ρ(t) regulatory response (weighed sum of regulator profiles) as a function of time
f activation function (logistic sigmoid in our case)
β vector of all model parameters
z(t) measured/smoothed mRNA levels of gene as a function of time
\( {\widehat{\boldsymbol{z}}}_{\boldsymbol{\beta}} \) mRNA levels estimated by a model with parameter vector β
N number of time points
y i (t) mRNA level of i-th regulator as a function of time
m number of regulators
Transcription of the gene within a solution with SigA present was compared to transcription without SigA (negative control) and transcription from a known strong SigA-dependent promoter (positive control).
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This work was supported by C4Sys research infrastructure project (MEYS project No: LM20150055).
The latest version of the Genexpi software is freely available (LGPL v3) at http://github.com/cas-bioinf/genexpi/wiki/, including source code (Java + OpenCL, optionally R; platform independent). The CyDataseries plugin is freely available (LGPL v3) at https://github.com/cas-bioinf/cy-dataseries, including source code (Java, platform independent). The CyGenexpi and Cydataseries plugins are also available via the Cytoscape App Store. The B. subtilis data is available in Gene Expression Omnibus as GSE6865 https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE6865, the S. coelicolor data is available in Gene Expression Omnibus as GSE44415 https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE44415. Full code to reproduce the evaluation is attached as an R notebook in the Additional file 1.
Institute of Microbiology of the Czech Academy of Sciences, Vídeňská, 1083, Prague, Czech Republic
Martin Modrák
& Jiří Vohradský
Search for Martin Modrák in:
Search for Jiří Vohradský in:
MM developed the Genexpi software, performed the validation and wrote most of the manuscript. JV conceived the work, provided the gene regulation model and relevant expertise, managed the research project and provided critical feedback on the manuscript. Both authors read and approved the final manuscript.
Correspondence to Martin Modrák.
Additional file 1:
evaluation.zip - an archive containing: • evaluation.Rmd – R Markdown notebook (best used with RStudio, https://www.rstudio.com/) to reproduce the evaluation on bacterial regulons in this paper. evaluation.nb.html – Compiled version of evaluation.Rmd for easy reading, including stored results produced by running all the code. • evaluation_sacharomyces.Rmd – R Markdown notebook to reproduce the evaluation on Sacharomyces data. • evaluation_sacharomyces.nb.html – Compiled version of evaluation_sacharomyces.Rmd, including stored results produced by running all the code.
https://doi.org/10.1186/s12859-018-2138-x
Gene network inference
Transcription regulation
Cytoscape | CommonCrawl |
To what fractional Sobolev spaces does the step function belong? (Sobolev-Slobodeckij norm of step function)
I'm new to fractional Sobolev spaces and I'm curious about the regularity of some simple functions like e.$\,$g. step functions in order to understand these spaces better.
In more detail, for $\Omega = [-1,1]^n \subseteq \mathbb{R^n}$ and $A = [-\frac{1}{2},\frac{1}{2}]^n\subseteq \Omega$ consider the function $$ \begin{align} f \colon \ \Omega & \longrightarrow \mathbb{R} \\ x & \longmapsto \begin{cases} 1 & \text{ for } x \in A \\ 0 & \text{ for } x \notin A\text{.}\end{cases} \end{align} $$ For which $s \in [0,1]$ does $f$ have a finite Sobolev-Slobodeckij norm? The norm that is meant here is defined by $$ \Vert f\Vert_{s}^2 := \int_\Omega \int_\Omega \frac{\vert f(x) - f(y)\vert^2}{\Vert x-y\Vert^{2s+n}} \, \mathrm{d}x \, \mathrm{d}y\text{.} $$
Is there a way to determine the value of the integral analytically in dependence of $n$ and $s$? Or can one at least easily determine those $s$ for which this integral would be finite? Can it at least be done for $n=2$?
So far I tried the simple case $n=1$ for which I get that $s\in [0,\frac{1}{2})$ has to be fulfilled. I expect that to be the case for any $n$ but at the moment I'm not quite sure since I did not prove it. For $n=2$ I would try to integrate by hand but with my approach it's about to become a rather long calculation. Is there maybe an elegant way to do it? I don't mind if $A$ is replaced by another set like for example a scaled $n$-Sphere or some simplex.
integration sobolev-spaces
MurpMurp
First I would make a quick guess based on the chart of function space. It groups together Sobolev spaces $W^{s,p}$ with the same value of $\frac n p -s$, because these are related by the embedding theorem. While the inclusion provided by this theorem is strict, the sharpness of the theorem still makes "the spaces with equal $\frac n p -s$ are similar" a useful heuristic.
Your function $f$ fails to be in $W^{1,1}$, because its gradient is not an $L^1$ function but rather a vector-valued measure supported on the boundary of $A$. On the other hand, this measure has finite mass (meaning $f\in BV$), which is pretty close to $W^{1,1}$. So, it seems that $s=1$, $p=1$ is at the edge of spaces to which $f$ belongs. From $$\frac{n}{1}-1 = \frac{n}{2} - s$$ we conclude that for $p=2$ (your question), the relevant $s$ is $s=1-\frac n2$.
This agrees with the situation in 1D, and suggests we won't find anything good when $n\ge 2$. Of course, this is just a guess; it may well be wrong.
Recall the layercake principle: the integral of a nonnegative function $g$ is equal to $\int_0^\infty |\{g>t\}|\,dt$ where $|\cdot|$ stands for the measure of the set. So, let's consider the inequality $$ \frac{|f(x)-f(y)|}{\|x-y\|^{2s+n}}>t \tag{1} $$ for large values of $t$ (only they are of interest when the measure space has finite measure).
For (1) to hold, exactly one of $x,y$ must be in $A$; also, both must be within distance $\delta \approx t^{-1/( 2s+n)}$ of the boundary of $A$. This constrains $x$ to a set of measure $\approx \delta $. Also, $y$ must lie in a ball of radius $\delta$ around $x$. So,
$$ \left|\left\{ (x,y) : \frac{|f(x)-f(y)}{\|x-y\|^{2s+n}}>t\right\}\right| \approx \delta^2 \approx t^{-(n+1)/(2s+n)} $$ The integral over $t\ge 1$ converges iff $$\frac{ n+1}{2s+n} >1$$ which is equivalent to $s<1/2$.
(In particular, the "guess" wasn't correct after all.)
$\begingroup$ Thank you a lot! I like that approach especially since I had not seen that strategy before. But shouldn't it be $\left|\left\{ (x,y) : \frac{|f(x)-f(y)}{\|x-y\|^{2s+n}}>t\right\}\right| \approx t^{-(n+1)/(2s+n)} =: c^{n+1}$? Because, roughly spoken, each $x \in A$ gets a ball of feasible $y$ that has area $c^n$ and the strip of feasible $x\in A$ has an area of order $c$ (because the area of the strip is basically $\vert\partial A\vert \cdot c$). Then we would get $s \in [0,\frac{1}{2})$ for any dimension $n$ and $p = 2$. $\endgroup$ – Murp Aug 4 '15 at 9:08
$\begingroup$ @Murp Good catch; I edited. Left the incorrect guess there since in other situations it actually works. [Now that I think of it more, it works for functions where the singular set is a point.] $\endgroup$ – user147263 Aug 4 '15 at 15:08
The function belongs to $W^{s,1}$ and BV space as in paper "The fractional Cheeger problem".
Not the answer you're looking for? Browse other questions tagged integration sobolev-spaces or ask your own question.
Compactly embedded in $L^p(0,1)$ but is not a subspace of $C^0[0,1]$
Proof that the characteristic function of a bounded open set is in $H^{\alpha}$ iff $\alpha < \frac{1}{2}$
What is the natural norm on these spaces?
Sobolev norm in the definition of Sobolev spaces
Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer
Relations between Fractional Sobolev spaces $H^s$ and $H^1$
Fractional Sobolev spaces are Banach spaces
Does multiplication by a test function stay in a Sobolev space? | CommonCrawl |
Mat. Sb.:
Mat. Sb., 2003, Volume 194, Number 8, Pages 139–160 (Mi msb764)
This article is cited in 9 scientific papers (total in 9 papers)
Surgery on triples of manifolds
Yu. V. Muranova, D. Repovšb, F. Spaggiaric
a Vitebsk Institute of Modern Knowledge
b University of Ljubljana
c University of Modena and Reggio Emilia
Abstract: The surgery obstruction groups for a manifold pair were introduced by Wall for the study of the surgery problem on a manifold with a submanifold. These groups are closely related to the problem of splitting a homotopy equivalence along a submanifold and have been used in many geometric and topological applications.
In the present paper the concept of surgery on a triple of manifolds is introduced and algebraic and geometric properties of the corresponding obstruction groups are described. It is then shown that these groups are closely related to the normal invariants and the classical splitting and surgery obstruction groups, respectively, of the manifold in question. In the particular case of one-sided submanifolds relations between the newly introduced groups and the surgery spectral sequence constructed by Hambleton and Kharshiladze are obtained.
DOI: https://doi.org/10.4213/sm764
References: PDF file HTML file
Sbornik: Mathematics, 2003, 194:8, 1251–1271
UDC: 513.8+515.1
MSC: 57R67, 57Q10, 19J25, 19G24, 18F25
Received: 11.07.2002
Citation: Yu. V. Muranov, D. Repovš, F. Spaggiari, "Surgery on triples of manifolds", Mat. Sb., 194:8 (2003), 139–160; Sb. Math., 194:8 (2003), 1251–1271
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http://mi.mathnet.ru/eng/msb764
https://doi.org/10.4213/sm764
http://mi.mathnet.ru/eng/msb/v194/i8/p139
This publication is cited in the following articles:
Yu. V. Muranov, R. Jimenez, "Structure sets of triples of manifolds", J. Math. Sci., 144:5 (2007), 4468–4483
Yu. V. Muranov, R. Himenez, "Transfer maps for triples of manifolds", Math. Notes, 79:3 (2006), 387–398
Cavicchioli A., Muranov Yu.V., Spaggiari F., "Mixed structures on a manifold with boundary", Glasg. Math. J., 48:1 (2006), 125–143
Yu. V. Muranov, D. Repovš, M. Cencelj, "The $\pi$-$\pi$-Theorem for Manifold Pairs", Math. Notes, 81:3 (2007), 356–364
Jimenez R., Muranov Yu.V., Repovš D., "Splitting along a submanifold pair", J. K-Theory, 2:2, Special issue in memory of Yurii Petrovich Solovyev, Part 1 (2008), 385–404
A. Bak, Yu. V. Muranov, "Splitting a simple homotopy equivalence along a submanifold with filtration", Sb. Math., 199:6 (2008), 787–809
A. Cavicchioli, Yu. V. Muranov, F. Spaggiari, F. Hegenbarth, "On Iterated Browder–Livesay Invariants", Math. Notes, 86:2 (2009), 196–215
Cavicchioli A., Muranov Yu.V., Spaggiari F., "Assembly maps and realization of splitting obstructions", Monatsh. Math., 158:4 (2009), 367–391
Cavicchioli A., Muranov Yu.V., Spaggiari F., "Surgery on pairs of closed manifolds", Czechoslovak Math. J., 59:2 (2009), 551–571
References: 13
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A Novel Indoor Positioning Algorithm for Wireless Sensor Network Based on Received Signal Strength Indicator Filtering and Improved Taylor Series Expansion
Jin Ren* | Songyang Huang| Wei Song| Jun Han
School of Information Science and Technology, North China University of Technology, Beijing 100144, China
Beijing Key Laboratory on Integration and Analysis of Large-scale Stream Data, Beijing 100144, China
School of Information Engineering, Minzu University of China, Beijing 100081, China
Corresponding Author Email:
[email protected]
https://doi.org/10.18280/ts.360113
| Citation
36.01_13.pdf
Considering the high accuracy needed for indoor positioning, this paper develops a novel indoor positioning algorithm for the wireless sensor network (WSN) in the following steps. First, the RSSIs of the network nodes were sampled and analyzed, and the excess errors were filtered to enhance positioning accuracy. Next, the initial position was iteratively obtained by the weighted centroid algorithm, and a correction matrix was developed to improve the Taylor series expansion (TSE), and the final position was determined through improved TSE iteration. The proposed positioning method was verified through simulation.
wireless sensor network (WSN), received signal strength indicator (RSSI), indoor positioning, Taylor series expansion (TSE), positioning accuracy
With the rapid development of mobile communication and terminals, the global positioning system (GPS), despite its excellent outdoor positioning effect, cannot fully support small-scale indoor location-aware services, owing to the complex channel environment and fast signal attenuation [1-3].
One of the solutions to the above problem is to develop a wireless local area network (LAN) with low cost and small power based on the wireless sensor network (WSN) technology. The WSN is an emerging technique integrating sensing, communication and distributed processing. Through the cooperation between its nodes, the network can capture and process the real-time information of multiple objects in the monitoring area, and transmit the processed information to interested users.
Node positioning technologies like the WSN takes positioning algorithm as the premise. The main positioning algorithms are based on field orientation, time different of arrival (TDoA), angle of Arrival (AoA), or received signal strength indicator (RSSI), or the coupling of these bases [4-6]. Thanks to its low hardware cost and simple implementation, the RSSI-based positioning algorithm becomes a research hotspot. In actual application, the proper position algorithm should be selected according to the specific environment and requirement.
In this paper, a novel indoor positioning algorithm is developed for the WSN in the following steps. First, the RSSIs of the network nodes were sampled and analyzed, and the excess errors were filtered to enhance positioning accuracy. Next, the initial position was iteratively obtained by the weighted centroid algorithm, and the correction matrix was improved by Taylor series expansion (TSE), and the final position was determined through iteration. The proposed positioning method was verified through simulation.
2. RSSI-Based Indoor Positioning Model
In the indoor environment, electromagnetic wave signals are affected by the propagation path, which is different from the propagation in free space. The distance between two nodes in the WSN can be estimated by the signal strength of the sending node and the signal strength at the receiving node, i.e. the RSSI. The RSSIs in wireless transmission can be explained by the shadowing model [7, 8]:
${{\left[ p\left( d \right) \right]}_{dBm}}={{\left[ p\left( {{d}_{0}} \right) \right]}_{dBm}}-10n\lg \left( \frac{d}{{{d}_{0}}} \right)+{{X}_{dBm}}$ (1)
where, p(d) is the RSSI of the receiving node separated from the sending node by a distance of d; p(d0) is the RSSI of the receiving node separated from the sending node by the reference distance d0; n is the path loss index; X is the cover factor. The path loss index depends on the external environment, and should be determined through actual measurement. The cover factor obeys the Gaussian random distribution and averages at zero.
In practice, the shadowing model is often simplified. For convenience, the reference distance can be set to 1m. Then, the $[p(d)]_{dBm}=A-10n lg(d)$ can be calculated by:
$RSSI=A-10n\lg \left( d \right)$ (2)
The above equation can be rewritten as:
$d\text{=1}{{\text{0}}^{\frac{A-RSSI}{10n}}}$ (3)
where, A is the attenuation factor of signal intensity when the sender-receiver distance is 1m. The value of A increases with its distance to the sending node.
Equation (2) is a popular model for RSSI positioning. With this equation, the sender-receiver distance can be calculated from the RSSI of the receiving node. This positioning method is suitable for the WSNs, which feature small-scale and low cost.
2.1 Gaussian filtering
In this paper, the measured RSSIs are processed by Gaussian filtering, to reduce probability error and improve the probability in the detection range. As a common filtering method for theoretical analysis, the Gaussian filter can eliminate excess errors and smooth measured RSSIs [9-12].
For the measured RSSIs obeying the distribution ($μ,σ^2$), the probability density function can be expressed as:
${{f}_{RSSI}}=\frac{1}{\sigma \sqrt{2\pi }}{{e}^{\frac{{{(RSSI-\mu )}^{2}}}{2{{\sigma }^{2}}}}}$ (4)
where, $μ=1/n ∑_{k=1}^nRSSI_k; σ=√(1/(n-1) ∑_{k=1}^n(RSSI_k-μ)^2 )$; $RSSI_k$ is the RSSI measured in the k-th round; n is the number of measurement rounds. The greater the σ value, the smoother the Gaussian filter. Through repeated experiments, it is confirmed that the σ value should be greater than 0.6. Thus, the RSSI scope is μ+0.15σ<x<μ+3.09σ. The measured RSSIs in the scope should be retained and taken average, and those outside the scope, i.e. excess errors, should be eliminated.
Taking 100 receiving nodes of the same sending node as samples, the filtering effect can be judged by the standard deviation σs of the samples:
${{\sigma }_{s}}=\sqrt{\frac{1}{N-1}{{\sum\limits_{i=1}^{N}{\left( RSS{{I}_{i}}-RSSI \right)}}^{2}}}$
where, RSSI is the mean value of the samples; $RSSI_i$ is the filtered value at each sampling point. The RSSIs processed by the Gaussian filter are presented in Figure 1. Obviously, the Gaussian filter eliminated the excess errors induced by environmental interference and short time interference.
Figure 1. The filtered RSSIs of the Gaussian filter
2.2 Kalman filtering
Next, the measured RSSIs were further optimized by the Kalman filter, which can effectively remove mutation points and noises and realize accurate and smooth outputs.
In Kalman filtering, the state prediction equations are:
$\begin{align} & X(k|k-1)=AX(k-1|k-1)+BU(k) \\ & P(k|k-1)=AP(k-1|k-1){{A}^{T}}+Q. \\\end{align}$ (5)
The state update equations are:
$\begin{align} & X(k|k)=X(k|k-1)+Kg(k)(Z(k)-HX(k|k-1)) \\ & Kg(k)=P(k|k-1){{H}^{T}}/(HP(k|k-1){{H}^{T}}+R) \\ & P(k|k)=(I-Kg(k)H)P(k|k-1). \\\end{align}$ (6)
where, X(k|k-1) and X(k-1|k-1) are the predicted values of the current and previous states, respectively; A and B are system parameters;U(k) is current state of the control volume (default value: 0); P(k|k-1) and P(k-1|k-1) are the covariances of X(k|k-1) and X(k-1|k-1), respectively; Q is system noise. Z(k) is the measured moment for k; H is the parameters of the measurement system; Kg(k) is the Kalman filter gain; R is the measurement noise; P(k|k) is the update value of the current state; I is the unit matrix (the value is 1 for single model and measurement).
The error distribution of the RSSIs after Kalman filtering is displayed in Figure 2 below. It can be seen that the Kalman filter smoothed the outputted sample values to some extent
Figure 2. The error distribution of the RSSIs after Kalman filtering
3. Initial Positioning Based on Weighted Centroid Algorithm
The TSE is applicable to all kinds of channel environments. The method enjoys a low complexity and converges rapidly under proper initial values. Here, the initial positions are estimated by the weighted centroid algorithm as the initial values for the TSE. Focusing on node connectivity, the algorithm achieves a low complexity and a light computing load. In this algorithm, the unknown node receives information from all anchor nodes within its communication scope, and all these anchor nodes are considered as weighted geometric centroids. The formula of the weighted centroid algorithm can be expressed as:
$\left( {{x}_{0}},{{y}_{0}} \right)=\left( {\left( \sum{{{{x}_{i}}}/{{{R}_{i}}}\;} \right)}/{\left( \sum{{1}/{{{R}_{i}}}\;} \right),\,\left( {\left( \sum{{{{y}_{i}}}/{{{R}_{i}}}\;} \right)}/{\sum{{1}/{{{R}_{i}}}\;}}\; \right)}\; \right)\,$ (7)
The accuracy of centroid positioning relies heavily on the density and distribution of anchor nodes [13-15]. The positioning is very accurate when the nodes are dense and uniformly distributed. Otherwise, the accuracy will decrease rapidly.
4. Proposed Indoor Positioning Algorithm
During transmission, electromagnetic wave signals may be subjected to multipath interference or obstruction. In this case, the signal strength received at the same position varies from time to time [16-18]. The reliability of the measured RSSIs should be enhanced before indoor positioning. In this paper, an indoor positioning algorithm is designed with the following steps. First, the RSSI threshold is determined according to the simulation of the test environment. Second, the measured values are compared against the threshold to remove the excess errors and retain the average of the proper values. Last, the TSE is introduced to obtain the distance by equation (3).
The TSE [4], as an iterative recursive algorithm, requires accurate initial estimation of node positions to achieve fast convergence and good real-time performance. The initial positions were estimated as follows:
To begin with, the RSSIs $r_i=√((x_i-x)^2+(y_i-y)^2 )$ of an unknown node to each anchor node were measured. Then, the unknown node-anchor node distances $R_i$,i=1,2,…,N were computed as $R_i$=$r_i$+$ε_i$, where $ε_i$ is error (mean value: 0), i.e. a normal random variable of mean square error $δ^2$. In this way, the function $f_i (x,y)=√((x_i-x)^2+(y_i-y)^2 )$ is close to the initial estimate position, laying a good basis ($x_0$, $y_0$) for the TSE. After ignoring the second-order partial derivatives, the matrix representation can be obtained as:
$h\text{=}G\delta \text{+}\varepsilon $ (8)
$h=\left[ \begin{matrix} -{{R}_{1}}+\sqrt{{{({{x}_{1}}+{{x}_{0}})}^{2}}+{{({{y}_{1}}-{{y}_{0}})}^{2}}} \\ -{{R}_{2}}+\sqrt{{{({{x}_{2}}+{{x}_{0}})}^{2}}+{{({{y}_{2}}-{{y}_{0}})}^{2}}} \\ \vdots \\ -{{R}_{N}}+\sqrt{{{({{x}_{N}}+{{x}_{0}})}^{2}}+{{({{y}_{N}}-{{y}_{0}})}^{2}}} \\\end{matrix} \right]$,
$A=\overline{RSSI}-\overline{n\rho }$,
$\delta \text{=}\left[ \begin{matrix} \Delta x \\ \Delta y \\\end{matrix} \right]$,
$\left( {{x}_{k}},{{y}_{k}} \right)k=1,2,\ldots ,50.$.
Using the weighted least squares (WLS) algorithm, the least squares estimation solutions can be derived from equation (8) as $δ={(G^T G)}^{-1} G^T h$. Then, the iterative process was initiated to determine whether |Δx|+|Δy| is less than the threshold (0.01). The estimated position was corrected iteratively until it satisfies the threshold. The iterated frequency distribution is shown in Figure 3. It can be seen that the estimated positions were stable and convergent and eventually satisfied the threshold. The last updated position was taken as the final position and compared with the actual position to obtain the positioning accuracy. The flow of the TSE algorithm is explained in Figure 4.
Figure 3. Iterated frequency distribution of the TSE
Figure 4. The flow of the TSE algorithm
The above TSE algorithm can carry out TSE at the given initial values, and correct the estimated position of the unknown node by the mean distance between the unknown node and each anchor node. This design does not consider the impact of the measured distance between the unknown node and the anchor node on Δx and Δy. In this case, the positioning error will increase significantly with the growth in the measured distance, resulting in a sharp decline in the positioning accuracy.
To solve the problem, the said measured distance was taken into account to locate the unknown node. As shown in equation (8), the weighted matrix h consists of measuring errors. According to the propagation law of radio signals, the positioning error is positively correlated with the measured distance of RSSI positioning. This correlation can be employed to reduce the impacts of excessively large or small measured distance on positioning accuracy. Specifically, the Taylor series can be improved as:
$Qh=G\delta $ (9)
where, Q is a positive definite diagonal matrix, in which the non-zero element in each row correspond to the element in the weighted matrix h. The matrix offsets the position change effect with the various elements in h, and calculates the amount of position correction of Δx and Δy according to the measured distances to the anchor nodes. Here, the Q is designed according to the calculated distances and RSSI positioning features:
$Q=\frac{1}{\left( N-1 \right)T}\left( \begin{matrix} T-{{d}_{1}}^{2} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & T-{{d}_{n}}^{2} \\\end{matrix} \right)$ (10)
where, $T=∑_{i=1}^Nd_i^2$; N is the total number of anchor node; $d_i$ is the measured distance from the unknown node to the i-th anchor node. Finally, the initial positions of the unknown node were substituted into the improved Taylor series for expansion.
5. Simulation and Results Analysis
The proposed method was verified through a simulation in a fixed area. The sending node was placed at a fixed position. Then, measuring points (i =1,2..., 100) were set up from the receiving node 20m away from the sending node to the fixed position at an interval of 0.2m. A total of 50 RSSIs were measured at these points, and averaged. Finally, a set of measured values were obtained from each measuring point. The measured RSSIs at each point correspond to the measured values at that distance. The 100 sets of data were plotted into a 2D coordinate system (Figure 5). Obviously, the RSSIs measured within 14m decreased gradually with the increase of the sender-receiver distance. Thus, 14m was taken as the detection range.
Figure 5. The 2D coordinates of the 100 sets of data
The radio signals propagate differently in different environments. Before simulation, the parameters A and n of the RSSI positioning model should be optimized to suit the specific environment. Using the first 70 sets of measured data $(RSSI_i,d_i )$, i=1, 2,..., 70, the A and n were calculated by linear regression analysis, and substituted into the model for further use. The optimized curve of the model demonstrates the good fitting effect of the linear regression. The linear regression can be expressed as:
$ρ_i=-10lgd_i$,i=1,2,…,70,
$n=∑_{i=1}^{70}(ρ_i -¯ρ)RSSI_i/∑_{i=1}^{70}(ρ_i -¯ρ)^2, A=¯RSSI-¯nρ$,
where, $¯RSSI=1/70 ∑_{i=1}^{70}RSSI_i$ and $¯ρ=1/70 ∑_{i=1}^{70}ρ_i$
To provide a reference for WSN positioning, several anchor nodes were randomly deployed in an area of 100*100 (m2). In other words, the positions of these nodes were randomly generated and known. The positioning accuracy was not evaluated by the deviation of the predicted position of an unknown node to the measured position. Instead, 50 unknown nodes were selected, and the positioning error of each of them was determined. Then, the mean positioning error was calculated on this basis. The actual position of each unknown node, estimated position of each unknown node and mean positioning error are respectively denoted as $(x_k,y_k)$,k=1,2,…,50, $(x ̄_k,y ̄_k )$, k=1,2,…,50 and $RMSE=1/50 ∑_{k=1}^{50}√((x_k-x ̄_k )^2-(y_k-y ̄_k )^2 )$.
First, the RSSI positioning was carried out with different filters and the TSE in the simulation environment. The results are shown in Figure 6. It can be seen that the Gaussian filter eliminated the excess errors induced by the environmental interference and removed many of the interference resulted from short-time RSSI measurement in the volatile situation. Meanwhile, the Kalman filter smoothed the sampling value. The indoor positioning algorithm outperformed the original algorithm, after coupling Gaussian filtering, and achieved the best performance, after coupling the Kalman filtering. As shown in the figure, the proposed method achieved a no-greater-than 0.5m positioning error, whichever the anchor node density.
Figure 6. RMSEs of RSSI positioning with different filters and the TSE
Figure 7. RMSEs of RSSI positioning with different filters and the improved TSE
Second, the RSSI positioning was carried out with different filters and the improved TSE in the simulation environment. The results in Figure 7 show that the improved TSE helped improve the positioning accuracy to various degrees under different filters.
This paper mainly explores the filter optimization of RSSI positioning and proposes a correction matrix of the TSE algorithm. First, the RSSI measurement was improved by Gaussian filtering and Kalman filtering. Then, the initial position was iteratively obtained by the weighted centroid algorithm, whose weight was computed by the optimized RSSI measuring distance, and a correction matrix was developed to improve the Taylor series expansion (TSE). The simulation results show the two filtering methods, especially the Kalman filter, can effectively reduce the RSSI measurement errors and improve positioning accuracy of unknown nodes, and the correction matrix of the TSE algorithm can further improve the positioning accuracy. In addition, the positioning accuracy of our method increases with the density of anchor nodes.
This work was supported by 2017 annual "Outstanding Young Teacher Training Program" project of North China University of Technology (No.: XN019009). Scientific Research Project of Beijing Educational Committee (No.: KM201710009004). 2018 Science and technology activities project for college students of North China University of Technology (No.: 110051360007). Research project on teaching reform and curriculum construction of North China University of Technology (No.: 18XN009-011). 2019 Beijing university student scientific research and entrepreneurship action plan project (No.: 218051360019XN004). 2019 Education and teaching reform general project of North China University of Technology.
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Solutions to a fluid-structure interaction free boundary problem
Box dimension and bifurcations of one-dimensional discrete dynamical systems
April 2012, 32(4): 1309-1353. doi: 10.3934/dcds.2012.32.1309
Computation of whiskered invariant tori and their associated manifolds: New fast algorithms
Gemma Huguet 1, , Rafael de la Llave 2, and Yannick Sire 3,
Center for Neural Science, New York University, New York, NY 10003, United States
Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-1082, United States
Université Paul Cézanne, Laboratoire LATP UMR 6632, Marseille, France
Received September 2010 Revised August 2011 Published October 2011
We present efficient (low storage requirement and low operation count) algorithms for the computation of several invariant objects for Hamiltonian dynamics, namely KAM tori (i.e diffeomorphic copies of tori such that the motion on them is conjugated to a rigid rotation) both Lagrangian tori(of maximal dimension) and whiskered tori (i.e. tori with hyperbolic directions which, together with the tangents to the torus and the symplectic conjugates span the whole tangent space). We also present algorithms to compute the invariant splitting and the invariant manifolds of whiskered tori. We present the algorithms for both discrete-time dynamical systems and differential equations.
The algorithms do not require that the system is presented in action-angle variables nor that it is close to integrable and are backed up by rigorous a-posteriori bounds. We will report on the implementation results elsewhere.
Keywords: Quasi-periodic solutions, whiskered KAM tori, numerical computation., invariant manifolds.
Mathematics Subject Classification: Primary: 70K43; Secondary: 37J4.
Citation: Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309
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Gemma Huguet Rafael de la Llave Yannick Sire | CommonCrawl |
Rational limit cycles of Abel equations
Pengyan Ding 1, and Zhijian Yang 2,,
College of Science, Henan University of Technology, Zhengzhou, 450001, China
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou, 450001, China
Received May 2020 Revised November 2020 Published March 2021 Early access January 2021
Fund Project: The research is supported by National Natural Science Foundation of China (Grant No. 11671367) and the Doctor Foundation of Henan University of Technology, China (No. 2019BS041). The first author is supported by the Doctor Foundation of Henan University of Technology, China (Grant No. 2019BS041). The second author is supported by NSFC (Grant No. 11671367)
The paper investigates the well-posedness and the existence of global attractor for a strongly damped wave equation on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u_{t}-\Delta u+u_{t}+u+g(u) = f(x) $. It shows that when the nonlinearity $ g(u) $ is of supercritical growth $ p $, with $ \frac{N+2}{N-2}\equiv p^*< p< p^{**} \equiv\frac{N+4}{(N-4)^+} $, (i) the initial value problem of the equation is well-posed and its weak solution possesses additionally partial regularity as $ t>0 $; (ii) the related solution semigroup has a global attractor in natural energy space. By using a new double truncation method on frequency space $ \mathbb{R}^N $ rather than approximating physical space $ \mathbb{R}^N $ by a sequence of balls $ \Omega_R $ as usual, we break through the longstanding existed restriction on this topic for $ p: 1\leqslant p\leqslant p^* $.
Keywords: Strongly damped wave equation, unbounded domain, well-posedness, regularity, global attractor.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35B33, 35B65.
Citation: Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006
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Pengyan Ding Zhijian Yang | CommonCrawl |
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
Let $a_1, a_2, \ldots , a_n, \ldots $ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq 0$ for $n = 1, 2, 3, \ldots$. Prove that \[0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots\]
Closing QOTD
Due to several emails requesting to close QOTD (people found them distracting to the main purpose of the blog), I will not be posting any more quotes. I have deleted the two quotes I posted as well. By the way, if you didn't know, you can contact me at [email protected] or at math4l#4750 on Discord for any questions or suggestions.
For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i \cdot 10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy.
1 million views!
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Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that\[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. | CommonCrawl |
Feature selection for high-dimensional temporal data
Michail Tsagris ORCID: orcid.org/0000-0002-2049-30631,
Vincenzo Lagani1 &
Ioannis Tsamardinos1
BMC Bioinformatics volume 19, Article number: 17 (2018) Cite this article
Feature selection is commonly employed for identifying collectively-predictive biomarkers and biosignatures; it facilitates the construction of small statistical models that are easier to verify, visualize, and comprehend while providing insight to the human expert. In this work we extend established constrained-based, feature-selection methods to high-dimensional "omics" temporal data, where the number of measurements is orders of magnitude larger than the sample size. The extension required the development of conditional independence tests for temporal and/or static variables conditioned on a set of temporal variables.
The algorithm is able to return multiple, equivalent solution subsets of variables, scale to tens of thousands of features, and outperform or be on par with existing methods depending on the analysis task specifics.
The use of this algorithm is suggested for variable selection with high-dimensional temporal data.
Temporal data measure a set of time-varying quantities over time on a population. They are often employed to understand the dynamics of evolution of a system, the effects of a perturbation (interventional studies), or the differences in dynamics between two groups (such as in case-control studies). Such data arise in many fields, namely bioinformatics, medicine, agriculture and econometrics, just to name a few.
Two broad categories of temporal data can be defined, depending on the sampling procedure: longitudinal data arise when the same samples are repeatedly measured at different times points, while time–course (a.k.a repeated cross-sectional) data are produced when distinct samples (from the same population) are measured at each time point (e.g., in case of destructive testing). In contrast, time-series data that often arise in econometrics, measure samples at regular time intervals and are often of a much larger temporal extent than temporal data in biology.
The correlation structure of temporal data, which includes auto-correlation of the same quantity over time or over the same sample requires special analysis techniques. For example, longitudinal data are often modeled with mixed models, which allow to properly account for within-subject correlations.
Feature selection (a.k.a. variable selection) in predictive modeling can be defined as the task of selecting one or more minimal-size and (collectively) optimally predictive feature subsets for a target outcome. Reducing the number of features results in smaller, easier-to-verify, understand, visualize, and apply predictive models; most importantly perhaps, it provides important insight to the data generating mechanism. This is no accident, as feature selection has been theoretically connected to causal discovery and the causal data generating model [1]. A typical example of a feature selection task is the identification of the genes whose expression allows the early diagnosis of a given disease. In the context of temporal data, each feature has a temporal extent and a time trajectory that can be employed for prediction.
To the best of our knowledge, most variable selection methods proposed so far for temporal data are devised for studies where the number of samples is larger than the number of predictors, i.e., p<n. This limits the applicability of these algorithms to "omics" types of data such as transcriptomics, epigenomics and genomics, where p is usually order of magnitudes larger than n.
Constraint-based, Markov-Blanket variable-selection methods form a class of algorithms that are inspired by the theory of (Causal) Bayesian Networks [2] and include HITON, MMMB, MMPC, SES and others [3–5]. The Markov Blanket of the target outcome T is defined as a minimal-size set that renders all other variables conditionally independent of T. Under certain broad conditions it has been shown to be the solution to the feature selection problem [1]. If the data distribution can be represented with a faithful Bayesian Network (BN) [6] then the Markov Blanket of T is unique and has an interesting graphical interpretation: it comprises of the neighbors of T (i.e., the parents and children of T) and the spouses (parents or common children) of T in any such (unknown) faithful BN graph.
The main contribution of this paper is adapting constraint-based, variable selection methods for temporal data. Constraint-based methods process the data exclusively through conditional independence tests, repetitively applying these tests for identifying variables that cannot be made independent of T conditioned on any other subset, and are thus needed for optimal prediction. As discussed in [7], employing a suitable conditional independence test is sufficient for extending constraint-based methods to new types of data. While such tests exist for various types of data, the idiosyncrasies of temporal data require the development of novel, specific conditional independence tests. We denote with Ind(X;T;Z) the test assessing the null hypothesis that X is independent of T given Z. For temporal data some of these variables (but not necessarily all of them) may have temporal extent and be better denoted as X t instead of X, with the index indicating the time-point. The independence test could be implemented as a log likelihood ratio test [8]. The latter fits two nested models, one modeling T on Z alone and the other on X∪Z. If the two models are equivalent, then the null hypothesis is not rejected. The modeling strategy used for creating the two nested models depends on the temporal characteristics of the variables involved in the test. However, for linear mixed models, likelihood ratio tests do not have the proper behaviour when the sample size is rather small and hence the use of F tests is suggested [9]. We depict four different scenarios with longitudinal and time course data, and for each scenario we define a suitable test of conditional independence.
The target variable is time-varying. In this scenario the task consists of identifying the predictors that are associated with the outcome of interest in the course of time. An example is modeling how a gene expression progresses over time on the basis of other gene expressions. Missing values can occur, or not all subjects may have measurements for all time points (unbalanced design). This case can be further subdivided in two sub-scenarios: the Temporal-longitudinal scenario, the same samples are being studied at all time points (longitudinal data), and the Temporal-distinct scenario, where different samples are being studied for each time points. The latter typically arises when it is impossible to repeat the measurements on the same sample: prototypical examples are animal studies where specimens are killed for collecting internal organs at different time points.
The target variable is a static (non-temporal) variable. In some studies the predictors are measured over time, however the dependent variable is static. An example is the study of gene-expressions differences between two mice groups (target). The task in this case is to identify the minimal set of genes whose trajectories, considered together, allow to best discriminate between the two groups. Also for this scenario we can identify two sub-cases, namely the Static-longitudinal scenario, where the same samples are measure over time, and the Static-distinct scenario, where different samples are considered at each time point.
Figure 1 graphically presents these four scenarios using data from some of the real datasets used in our experimentation. More information and example data for each scenario are presented in the Additional file 1.
Graphical representation of the four different scenarios. In all panel the x-axis reports the time dimension, while y-axis reports the log-transformed expression value of a randomly-chosen probeset from one of the datasets used in the experimentation. a Temporal-longitudinal scenario. All data, including the target variable, consists of longitudinal (repeated) measurements. Values from the same subject are linked with a dashed line. (data from the GDS3915 dataset). b Temporal distinct scenario. Each observed value refers to a different subject (data from the GDS964 dataset). c Static longitudinal scenario. There are two groups (red and black lines), and each group consists of trajectories of longitudinal measurements. Each trajectory refers to the same subject (data from the GDS4146 dataset). d Static distinct scenario. At every time point different subjects are measured. Green and red colors indicate the two populations from which the subjects are sampled from (data from the GDS2456 dataset)
These scenarios represent the most common designs for biological studies involving temporal data, and are widely applied in other fields as well. Other scenarios/study designs are of course possible (for example measurements might be repeatedly taken for each sample at each time point), however we consider them less relevant and out of the scope of the present paper.
In this paper we use the Statisticaly Equvialent Signatures (SES) algorithm [5, 10] as a prototype for the class of constraint-based algorithms. The predictors selected by SES (signature) are the neighbors of T in any BN (faithfully) representing the data distribution. This is a subset of the full Markov Blanket but it has been shown to be a good approximation for predictive purposes in extensive empirical studies [11]. Some algorithms (HITON, MMMB) do continue in trying to identify the full Markov Blanket which also includes the spouses of T at the expense of computational time. SES can successfully scale up to cases where p>>n, preserving excellent predictive capabilities [5]. We measure the time complexity of the algorithm in terms of the number of performed conditional independence tests. Each variable must be contrasted against each subset of the selected signature before being eliminated. This would require a number of tests in the order of O(p·2s), where p is the number of variables and s the number of selected variables. However, we only allow conditioning upon maximum k variables at the time, decreasing the complexity of the algorithm to O(p·sk). This means that the algorithm can still require an exponential number of tests with respect to the size of the selected signatures; however, in our experience the actual computational requirements of the algorithm are much lower, also due to the parsimonious signature often retrieved.
A desired feature of SES is the fact that it heuristically and efficiently attempts to identify statistically, equivalent solutions, i.e., minimal-sized feature subsets with the same optimal predictive performance. As mentioned before, when the distribution is faithful to a BN the solution is unique; however, in practice whether due to finite sample or deviations from assumptions there are multiple (empirically) equivalent solutions. Identifying all equivalent solutions is important when feature selection is employed for knowledge discovery and getting insight to the domain under study. Returning an arbitrarily-chosen single solution S may mislead the domain expert into thinking that all other variables are either redundant or irrelevant, when the situation can be reversed if selecting some other feature subset S′.
In our empirical study, we compare SES against the state-of-the-art feature selection algorithms for the above 4 scenarios on gene-expression data. SES successfully scales up to tens of thousands of gene trajectories. In terms of selection quality and predictive performance, SES outperforms other methods in the Temporal-longitudinal scenario, is on par or better in the Static-longitudinal and Static-distinct scenarios while selecting many fewer variables, while it is outperformed in the Temporal-distinct scenario.
The rest of the paper is organized as follows. The "Methods" section introduces conditional independence testing for temporal data, as well as the SES algorithm. A comparative evaluation of the proposed approaches against LASSO-inspired algorithms is then performed on real, high dimensional omics data. Discussion and conclusions end the paper.
In general, variable selection algorithms can be classified into two main categories, filter based and wrappers [12]. Methods of the first class select a subset of relevant features independently of the modeling algorithm that will be subsequently applied. On the other hand, wrapper methods try to select the set of features that optimize the performance of a specific classifier. A large bulk of literature has been published on the subject, with methods using several different approaches [13–23].
Finally, embedded methods are modeling algorithms whose operation automatically lead to the selection of the most relevant features (e.g., classification and regression trees [24]).
Many variable selection methods for classification of high dimensional biological data (particularly gene expression) have been proposed in the last decades [25]. For a recent review and open problems with regard to variable selection in high dimensional data the reader is addressed to Bolón-Canedo et al. [26].
In this work, we have carefully reviewed the current literature for identifying the most related and recent variable selection methods suitable for the four scenarios depicted above. Particularly, we have sought methods both applicable on temporal data and scalable to high-dimensional problems (i.e, thousands of candidate predictors).
In brief, the glmmLasso algorithm seems to be the most well-performing method for studies that belong to the Temporal-longitudinal scenario, according to the comparison performed in [27]. This algorithm combines mixed-models representation of complex variance structures with the sparsity of LASSO solutions; as a drawback, the resulting model is non-convex and difficult to optimize. In the Temporal-distinct and static-distinct scenarios there is no within-sample variance, and these two cases can be addressed with variable selection algorithms designed for non-temporal data. The Static-longitudinal scenario corresponds to discriminant analysis in longitudinal data, and not much research has been performed in the context of variable selection, see for example [28–30].
Available approaches for the Temporal-longitudinal scenario
Several approaches for variables selection were proposed in the last 15 years for studies where both the outcome and the predictors are measured over time on the same samples. Most of these approaches use either Generalized Linear Mixed Models (GLMM) or Generalized Estimating Equations (GEE).
On GLMM, Ni et al. [31] proposed a double-penalized likelihood approach in semi-parametric mixed models. Bondell and co-authors [32] proposed an algorithm that performs simultaneous selection of the fixed and random factors using a modified Cholesky decomposition and maximum penalized likelihood estimation, along with the smoothly clipped absolute deviation (SCAD). A similar approach, using adaptive LASSO penalty functions instead of SCAD, was presented as well [33]. Zhao et al. [34] suggested using a basis function approximations and a partial group SCAD penalty for semi-parametric varying coefficient partially linear mixed models, while Tang et al. [35] focused on quantile varying coefficient models via penalizing the L γ norm. Schelldorfer et al. [36] proposed an L1-penalty term for linear mixed models, and this work was later extended to include Poisson and binary logistic regression [37]. A method quite similar to the one of [37] was proposed in [27]; however the latter uses a gradient ascent algorithm whereas the former uses a coordinate gradient descent method based on a quadratic approximation of the penalized log-likelihood. Finally, a comparison of model selection methods for linear mixed models based on four major approaches is presented in [38]: information criteria such as AIC or BIC, shrinkage methods based on penalized loss functions such as LASSO, fence (ad-hoc procedures) and Bayesian techniques.
The literature is less extensive when it comes to GEE. The use of a modified AIC, termed quasi-likelihood information criterion (QIC), was proposed in [39]. Cantoni and co-authors [40] first used a generalised Mallow's criterion, and subsequently [41] used a Markov chain Monte Carlo (MCMC) procedure for variable selection without visiting all possible candidate models. The case of missing-at-random data was addressed in [42] by using a missing longitudinal information criterion selecting the optimal model and the correlation structure. Finally, a penalized GEE method that is consistent even when the working correlation structure is misspecified was presented in [43].
Some Bayesian techniques include [44–46] among others. The first used a Cholesky decomposition of the random effects covariance matrix and introduced a further decomposition of the Cholesky decomposed lower triangular matrix. The elements of the resulting diagonal matrix are assigned zero-inflated truncated-Gaussian priors and MCMC methods are applied. However, these types of approaches are discouraged [47], as they are computationally heavy and are prior dependent. Han and co-authors [45] compared a number of methods for comparing two linear mixed models using Bayes factors. They also mentioned that these kinds of methods require substantial human intervention and high computational power.
A common drawback of all the procedures presented so far is that they are applicable only on a small number of candidate predictors. The only exceptions are presented in [35–37], [43], that were tested on 100, 200, 500 and 1000 candidate predictors in their respective simulation studies. To note, these studies do not report information about the computational time required by the algorithms. Moreover, authors do not usually provide implementations of the methods they propose. The only methodologies available as R packages are the one presented by [36], under the name GLMMLasso, and the glmmLasso package by [27], which offers linear, Poisson and binary logistic mixed models.
Available approaches for the static-longitudinal scenario
This scenario refers to the task of discriminant analysis in longitudinal data. According to the concise review presented in [48], variable selection is somewhat not heavily researched in this context. More recently, L1 type constrains such as LASSO and SCAD allowing for grouped variables [28] were suggested. Matsui et al. [49] extended previous work to include multinomial logistic regression where the variables are selected in a grouped way. Finally, approaches based on functional regression also exist in the literature, see for example [50].
Available approaches for the Temporal-distinct and Static-distinct scenarios
Both the Temporal-distinct and Static-distinct scenarios are defined over time-course data measured at each time point on different samples. Thus, the within-sample variance cannot be modeled for these scenarios. This allows variable selection methods devised for non-temporal data, as the widely used LASSO [51], to be applied in this context.
The LASSO algorithm started gaining popularity after the work in [52] who suggested the least angle regression as a better and faster way to solve its underlying optimization problem. A coordinate descent algorithm, which allows using the LASSO penalty in the context of generalized linear models was then suggested [53]. This latter approach is implemented in the R package glmnet [54].
Grouped Lasso (gLASSO, [55]) was developed to handle categorical predictors, which are often encoded in linear modeling as groups of binary variables (dummy variables). For the sake of consistency, the dummy variables corresponding to a single categorical predictor should be either included or excluded altogether ("as a group") in the final LASSO solution. More recently, a quite efficient gLASSO implementation was proposed by [56], with their code made available in the R package gglasso [57].
In this section we discuss in detail how to adapt constraint-based method for temporal data analysis. First, we will briefly present Generalized Linear Mixed Models (GLMM) and Generalized Estimating Equations (GEE). Both techniques are suitable for devising conditional independence tests for temporal data with (un)balanced study designs. For a thorough comparison between GLMM and GEE see [58, 59].
Generalised linear mixed models
Let T i denote the n i -dimensional vector of observed values for the target (response) variable T in the i-th subject at the different d time-points. We model the link of T i with p covariates via the following equation:
$$ g\left(\mathbf{T}_{i}\right) = \mathbf{X}_{i}\pmb{\beta} + \mathbf{W}_{i}\mathbf{b}_{i} + \mathbf{e}_{i}, \ \ \ i= 1, \ldots, K. $$
The vector β is the (p+1)-dimensional vector of coefficients for the n i ×(p+1) fixed effects design matrix X i , which contains the predictor variables. The vector b i ∼N q (0,Σ) is the q-dimensional vector of coefficients for the n i ×q random effects matrix W i , while Σ is the random-effects covariance matrix. The vector \(\mathbf {e}_{i} \sim N_{n_{i}}\left (\mathbf {0}, \sigma ^{2}\mathbf {I}_{n_{i}}\right)\) is the n i dimensional within-group error vector which follows a spherical normal distribution with zero mean vector and fixed variance σ2.
We used the exchangeable or compound symmetry (CS) structure on the covariance matrix Σ. We decided not to use a first order autoregressive covariance (AR(1)) structure as a hyper-parameter of the GLMM method, since this type of structure did not improve the performance of generalised estimating equations (presented below) and would add a high computational burden to the fitting of GLMM.
K stands for the number of subjects and the total sample size (number of measurements) is equal to \(N=\sum _{i=1}^{K}n_{i}\). The link function g connects the linear predictors on the right hand side of (1) with the distribution of the target variable. Common link functions are the identity, for normally distributed target variables, and the logit function for binomial responses.
The possibility of specifying random effects allows mixed models to adequately represent between and within-subject variability, and to model the deviates of each subject from the average behavior of the whole population. These characteristics make GLMMs particularly suitable for temporal and longitudinal data [9].
Generalised estimating equations
Generalised Estimating Equations (GEE), developed by [60, 61], are an alternative to mixed models for modeling data with complex correlation structures. In contrast to GLMM which are subject specific, GEE contain only fixed effects and thus are population specific.
Using the notation defined in the previous section, in GEE the p covariates are related to the outcome as
$$ g\left(\mathbf{T}_{i}\right) = \mathbf{X}_{i} \pmb{\beta} + \mathbf{e}_{i}, \ \ \ i = 1, \ldots, K. $$
with the variance of the response variable T being modeled as Var(T ij )=ϕ·α ij , j=1…n i , where ϕ is a common scale parameter and α ij =α(T ij ) is a known variance function. We will focus on two different correlation structures for estimating α, the CS and the first order autoregressive AR(1):
$$ \begin{aligned} \text{CS:} \quad\text{Cor}\left(\mathbf{T}_{ij}, \mathbf{T}_{ij'}\right) & =\,\, \alpha \\ \text{AR(1):} \quad\text{Cor}\left(\mathbf{T}_{ij}, \mathbf{T}_{ij'}\right) & =\,\, a^{\left|j-j' \right|}. \end{aligned} $$
CS assumes that correlations of measurements for the same subject at different time-points are always the same, regardless of the temporal distance between them. Depending on the specific application, this might be not very realistic. In contrast, the AR(1) structure assumes that the correlation between measurements at different time points for the same subject decreases exponentially as the temporal gap between them increases.
A precise numerical estimation of α is critical in GEE modeling; we use the jackknife variance estimator suggested by [62], which is quite suitable for cases when the number of subjects is small (K≤30), as in many biological studies. The simulation studies conducted by [63] and [64] showed that the approximate jackknife estimates are in many cases in good agreement with the fully iterated ones.
Conditional independence tests for the Temporal-longitudinal scenario
We devise two independence tests based on GLMMs (Eq. 1) and GEEs (Eq. 2) respectively. This scenario assumes the predictors and the target variable are measured at a fixed set of time-points τ={τ1,…,τ m } in the same set of subjects. For balanced designs, all subjects are measured at all time-points, i.e, n i =n,∀i. The target variable is often a gene-expression trajectory and thus, in the rest of the paper and for this scenario we assume a continuous target.
Recall that the null hypothesis Ind(X;T|Z) implies that X is not necessary for predicting T when Z is given, and thus the conditional independence tests can be thought of as a testing the significance of the coefficient of x. The null and full models are written as
$$ \begin{aligned} H_{0}: \mathbf{T}_{i} =\,\, \mathbf{1} a \,\,+ \mathbf{1} b_{i} \,\,+ \,\, \gamma \pmb{\tau} \,\,+ \,\, \pmb{\delta} \mathbf{Z}_{i} & \\ H_{1}: \mathbf{T}_{i} =\,\, \mathbf{1} a \,\,+ \mathbf{1} b_{i} \,\,+ \,\, \gamma \pmb{\tau} \,\, + \,\, \pmb{\delta} \mathbf{Z}_{i} &\quad + & \beta \mathbf{X}_{i} \end{aligned} $$
where 1 is a vector of 1s, a is the global intercept, b i stands for the random intercept of the i-th subject, γ, δ and β are the coefficient of the predictors, and the generic link function g(.) (Eq. 1) has been substituted with the identity one.
This formulation stems from two specific modeling choices: (a) we use the vector of actual time points τ as a covariate, in order to model the baseline effect of the time on the trajectory of the target variable. Time becomes a linear predictor of the target. Other choices are possible, but would require more time-points that are typically not available in gene-expression data. (b) We include random intercepts, meaning we allow a different starting point for the estimated trajectory of each subject. This choice leads to W i =1 n i ,∀i, where 1 n is a vector of ones of size n. However, we do not allow random slopes, thus assuming all subjects have the same dynamics. This choice was dictated by the need of avoiding model over-specification, especially considering the small sample size of the datasets used in the experimentation.
Pinheiro and Bates [9] suggests the use of the F-test for comparing the two models, where only the model, the full, under the alternative is fitted and the significance of the coefficient β is tested. Another possible choice would be the log-likelihood ratio test, however the F-test is preferable for small samples, since the type I error is better controlled with the F distribution.
A second test is based on the GEE model. The null and alternative models now lose the random terms:
$$ \begin{aligned} H_{0}: \mathbf{T}_{i} = \,\, \mathbf{1} a &\quad + & \gamma \pmb{\tau} &\quad + & \mathbf{Z}_{i} \pmb{\delta} & & \\ H_{1}: \mathbf{T}_{i} = \,\, \mathbf{1} a &\quad + & \gamma \pmb{\tau} &\quad + & \mathbf{Z}_{i} \pmb{\delta} &\quad + & \beta \mathbf{X}_{i} \end{aligned} $$
GEE fitting does not compute a likelihood [59] and thus, no log-likelihood ratio test can be computed. A Wald test is used instead here again and the significance of the coefficient β is tested. Because of the lack of likelihood computation, its effectiveness in assessing conditional independence is questionable [65]. Despite these theoretical considerations, the experimental results proved the test to be quite effective in our context.
Conditional independence tests for the Static-longitudinal scenario
The Static-longitudinal scenario assumes longitudinal data with continuous predictors and a static target variable T that is either binary or multi-category. The goal is to discriminate between two or more groups on the basis of time-depending covariates. As in the Temporal-longitudinal scenario, the presence of longitudinal data requires to take into account the within-subject correlations.
We have devised a two-stage approach, partially inspired by the work of [66] and [67], for testing conditional independence in this scenario. In our approach a separate regression model is first fitted for each subject and predictor, using the time-points vector τ as unique covariate:
$$ \mathbf{G}_{i} = \gamma_{i0} + \gamma_{i1}\pmb{\tau}, \ \ \ i = 1, \ldots, n. $$
Here, G i is the vector of measurements for subject i and the generic predictor variable G. At the end of this step we end up with a matrix Γ with dimensions K×(2·p), containing all coefficients derived with the K models specified in (6). The two nested models needed for testing conditional independence can then be specified as:
$$ \begin{aligned} H_{0}: g\left(\mathbf{T}_{i}\right) = \,\, \mathbf{1} a &\quad + & \pmb{\delta} \pmb{\Gamma}_{\mathbf{Z}} \\ H_{0}: g\left(\mathbf{T}_{i}\right) = \,\, \mathbf{1} a &\quad + & \pmb{\delta} \pmb{\Gamma}_{\mathbf{Z}} &\quad + & \pmb{\beta} \pmb{\Gamma}_{X} \\ \end{aligned} $$
where Γ Z are the coefficients corresponding to the set of conditioning variables Z and Γ X are the coefficients corresponding to the variable X. A logit function g(.) is used for linking the linear predictors to the binomial (or multinomial) outcome. The log-likelihood ratio test (calibrated with a χ2 distribution) is used to decide which of the two models is to be preferred.
Conditional independence tests for the Temporal-distinct and Static-distinct scenarios
In these two scenarios different subjects are sampled at each time point (time-course data), and subject-specific correlation structures cannot be modeled. For the Temporal-distinct scenario, where the target variable is continuous, it is thus possible to use models (5) for assessing conditional independence. In absence of subject-specific correlation structures the GEE models reduce to standard linear models that can be compared with the standard F-test. A similar approach can be used for the Static-distinct scenario, where the outcome is binary or multinomial, by using a logit link function instead of the identity.
The SES algorithm
First introduced in [10], the SES algorithm attempts to identify the set(s) of predictors (signatures) that are minimal in size and provide optimal predictive performances for a target variable T. The basic idea is that if ∃Z, s.t., Ind(X;T|Z), then X is superfluous for predicting T. Thus, SES repetitively applies a test of conditional independence until it identifies the predictors that are associated with T regardless of the conditioning set used. Under certain conditions, these variables are the neighbors of T in a Bayesian Network representing the data at hand [2]. An interesting characteristic of SES is that it can return multiple, statistically indistinguishable predictive signatures. As discussed in [68], limited sample size, high collinearity or intrinsic characteristics of the data may produce several signatures with the same size and predictive power. From a biological perspective, multiple equivalent signatures may arise from redundant mechanisms, for example genes performing identical tasks within the cell machinery. The SES algorithm is further explained in the Additional file 1 and in [5].
Equipping constraint-based methods with conditional independence test for temporal data
SES belongs to the class of constraint-based feature-selection methods [4]. This type of algorithm processes the data exclusively through tests of conditional independence that assess Ind(X;T|Z). This means that in order to extend any constraint-based methods to temporal data it is sufficient to equip an appropriate test, such as the ones defined in Eqs.(4)-(7).
Experimentation on real data
The experimental evaluation aims at assessing the capabilities of the proposed conditional independence tests in real setting. For each scenario we identified several gene-expression datasets over which we applied the SES algorithm equipped with the conditional independence test most suitable for the data at hand. The feature subsets identified by SES were then fed to modeling methods for obtaining testable predictions.
Furthermore, in each scenario we contrasted SES against a feature selection algorithm belonging to the family of LASSO methods. This class of algorithms has proven to be well-performing in several applications, including variable selection in temporal data (see the Section regarding the literature review). Particularly, we compare against glmmLasso [27] for the Temporal-longitudinal scenario, with standard LASSO regression [51] for the Temporal-distinct scenario, and the grouped LASSO (GLASSO) for classification [54, 56] in the Static-longitudinal and Static-distinct scenarios.
We excluded from this comparative analysis approaches that a) do not scale-up to thousands of variables (e.g., Bayesian procedures), b) require a number of time points much larger than the applications taken into consideration in this work (as for functional regression, [69]), and c) in general do not have available implementations.
The configuration settings of all algorithms involved in the experimentation were optimized by following an experimentation protocol specifically devised for estimating and removing any bias in performance estimation due to over-fitting.
We thoroughly searched the Gene Expression Omnibus database (GEO, http://www.ncbi.nlm.nih.gov/) for datasets with temporal measurements. Keywords "longitudinal", "time course", "time series" and "temporal" returned nearly 1000 datasets. We only kept datasets having at least 15 measurement and at least three time points, and complete information about the design of the study generating the data. This resulted in at least 6 datasets for each scenario, except for the Static-longitudinal scenario, where we identified 4 datasets with at least 8 measurements. Detailed information on the selected datasets are available in the (Additional file 1: Tables S5 and S6).
Modeling approaches
For the Temporal-longitudinal scenario SES was coupled with either GLMM or GEE regression, so as to mirror the conditional independence test equipped to the algorithm. The glmmLasso algorithm is used for comparison, using a model similar to (4) defined over the whole predictors matrix X
$$ \begin{aligned} \mathbf{T}_{i} = &\quad \mathbf{1}a + b_{i} & + \quad& \gamma \pmb{\tau} & + \quad& \mathbf{X}_{i} \pmb{\beta}\\ \end{aligned} $$
For the Static-longitudinal scenario, logistic or multinomial regression was applied on the columns of the matrix Γ selected by SES, depending on the outcome at hand. The grouped Lasso (GLASSO, [56]) algorithm was used for comparison. GLASSO allows to specify groups of variables that can enter the final model only altogether. Particularly, the GLASSO was applied on the whole matrix Γ, forcing the algorithm to either select or discard predictors in pairs, following the way columns in Γ correspond to the original predictors.
For Temporal-distinct and Static-distinct scenarios SES was always coupled with standard linear, logistic or multinomial regression (depending on the specific outcome), while the standard LASSO algorithm (binary outcome) and GLASSO (multinomial outcome) were used for comparison (see Additional file 1 for further details).
In all analyses SES' hyper-parameters maximum conditioning variables size k and significance level a varied between {3,4,5} and {0.05,0.1}, respectively. The λ penalty values generated by the Least Angle Square (LARS) algorithm [52] were used for the LASSO models of all scenarios, apart from the temporal-longitudinal. LARS cannot be adapted to this latter scenario, and thus the range of values was separately determined for each dataset, by using all integer values between λ min , the smallest value guarantying the invertibility of the Hessian matrix in each fold, and λ max , the highest value after which no variable was selected.
Experimentation protocol
We used the m-fold cross-validation procedure with the Tibshirani-Tibshirani (TT) bias correction [70] for model selection and performance evaluation. In the standard cross-validation protocol the available samples are partitioned in m folds, with approximately an equal number of samples each. Each fold is in turn held-out for testing, while the remaining data form the training set. The current modeling approach is applied several times on the training set, once for each predetermined configuration setting, and the predictive performances of the corresponding models are evaluated on the hold-out fold. The configuration with the best average performance is then used for training a final model on the whole dataset. In all experimentation m was set to either 4 or 5, so that to have at least two measurements in each fold. Particularly, folds correspond to one or more subjects in the Static-longitudinal scenario, and to one or more time points in the other scenarios.
The performance of the best configuration is known to be optimistically biased, and thus a correction is needed for a fair evaluation. The TT method is a general methodology for estimating and removing the optimistic cross-validation bias. If the performance's metric is defined in terms of prediction error (the lower the error the better the performance), the bias estimation according to the TT method is the following:
$$ \hat{\text{bias}}=\frac{1}{m}\sum_{i=1}^{m}\left[e_{i}\left(\hat{\pmb{\theta}} \right) - e_{i}\left(\hat{\pmb{\theta}}_{i} \right) \right], $$
where e i is the performance on fold i, while \(\hat {\pmb {\theta }}\) and \(\hat {\pmb {\theta }}_{i}\) are the configurations corresponding to the best average performance and to the best performance of the i-th fold, respectively. Signs in (9) should be interchanged if the performance metric assigns higher scores to better models.
The statistical significance of the difference between average performances is computed through permutation-based t-tests, where single performances are randomly permuted for approximating the null distribution.
All of the simulations, computations and time measurements were performed on a desktop with Intel Core i5-3470 CPU @ 3.2 processor, 4 GB RAM memory using a 64-bit R version 3.2.2.
Coupling SES with GLMM and GEE
First, we contrasted the performances of GLMM and GEE-based conditional independence tests in the context of the Temporal-longitudinal scenario. Table 1 reports the results of the comparison.
Table 1 Temporal-longitudinal scenario: comparison between SES equipped with GLMM (SESglmm) and SES equipped with GEE
For each dataset the cross-validated, TT-corrected Mean Squared Prediction Error (MSPE) is reported (standard deviation in parenthesis), along with the respective computational time in Table 1. Average performances are reported at the bottom line. Methods are indicated as SESglmm, SESgee(CS)) and SESgee(AR(1)), corresponding to SES coupled with GLMM and GEE, the latter using either the CS or AR(1) covariance structure. All methods obtain statistically equivalent results in terms of MSPE (all paired permutation-based t-test p-values are above 0.37). The average computational time largely varies, with SESgee(AR(1)) being the fastest of the three methods (all paired permutation-based t-test p-values are below 0.002). For all methods, computational times strongly depend upon the number of variables of each dataset, in a log-linear way (see Additional file 1).
Since the three versions produced equally predictive results, in the remaining of the analysis we use only SESglmm, in order to ensure a comparison as fair as possible with the GLMM based method glmmLasso.
glmmLasso scalability in high-dimensional data
Preliminary analyses pointed out glmmLasso's limited ability of efficiently (in computational terms) scaling up to a few thousands of predictors (glmmLasso's implementation is limited to 17,000 variables). We characterized glmmLasso scalability by running the algorithm on increasingly larger numbers of randomly selected variables. Figure 2a reports the results obtained from dataset GSD5088. Different lines report time performances of glmmLasso, and SES equipped with different conditional independence tests. glmmLasso requirements in terms of computational time increase in a super-linear way with the number of predictors (see Additional file 1: Figures S1 and S2 for time comparisons with all datasets). An interesting feature of the SES implementation that is worthy to mention is the fact that information about the univariate associations (test statistics and associated p-values) is stored. Hence, when the hyper-parameters change values, the algorithm begins from the second step. For the 6 pairs of configurations (pairs of a and k ) used in our experimental analysis this results in a significant amount of computational savings.
a Temporal-longitudinal scenario: Time in seconds required by glmmLasso and SES equipped with different conditional independence tests on the GSD5088 dataset. The number of randomly selected predictors is reported on the x-axis, while y-axis reports the required computational time: glmmLasso rapidly becomes computationally more expensive than any SES variant. b Gene expression over time for the target gene CSHL1 in dataset GDS5088 (one line for each subject). c Average relative change for the target gene and predictors reported in model 10. The expression of the genes was averaged over subjects for each time point, and the logarithm of the change with respect to the first time point was then computed. The target gene appears as bold line, whereas the 5 predictor genes are reported as dashed lines. d Differences in performance between SESglmm and glmmLasso for the 20 replications on each dataset. Negative values indicate SESglmm outperforming glmmLasso; SESglmm is always comparable or better than glmmLasso, especially in dataset GDS5088 (excluded for sake of clarity). e Static-longitudinal scenario: Expressions over time of gene TSIX, selected by SES for dataset GDS4146. The plot show one line for each subject: there is a clear separation between the two classes included in the dataset (dashed and solid lines, respectively). f Static-distinct scenario: Expressions over time of gene Ppp1r42, selected by SES for dataset GDS2882. The dotted and dashed lines correspond to the average trend of the gene in two different classes; differences in intercept and trend are easily noticeable
The same analysis was repeated on all datasets selected for the Temporal-longitudinal scenario, consistently achieving similar results (Additional file 1). Consequently, for each dataset related to the Temporal-longitudinal scenario only 2000 randomly selected predictors were retained in all subsequent analyses, so that the experimentation could be performed in a reasonable time and to allow a fair comparison between SESglmm and glmmLasso (see Additional file 1: Table S9 for the values of the penalty parameter used in glmmLasso).
Results on the four scenarios
Table 2 reports the main results of the experimentation. For each dataset, cross-validated, TT-corrected performances are reported as average (st.d.). Zero standard deviations are caused by numerical rounding. For the Temporal-longitudinal and Temporal-distinct scenarios the MSPE metric is used, with lower values indicating better performances, while the Percentage of Corrected Classification (PCC) metric is used for the other scenarios, with higher values indicating better performances. Average differences (SES - LASSO) over all datasets are reported for each scenario and statistically significant differences at 0.01 and 0.05 significance level are indicated with ∗∗ and ∗, respectively.
Table 2 Cross-validated, TT-corrected performances of SES and LASSO-type methods on the four scenarios
On average, SES equipped with conditional independence tests for temporal data outperforms the corresponding LASSO algorithms, in terms of predictive performance, in all scenarios, except for the Temporal-distinct scenario. We also note that LASSO methods did not select any variable in at least one fold of cross validation for several datasets, as indicated by an average number of selected variables < 1 (baseline predictive models are produced in these cases). When LASSO methods select at least one variable in each fold, their variability in number of selected variables is considerably higher than the one of SES. Particularly, for the Temporal-longitudinal scenario SESglmm largely outperforms, in terms of predictive performance, glmmLasso in all datasets except one (GDS3181), where glmmLasso is only marginally superior (See Additional file 1: Table S10). For the Temporal-distinct scenario the results are quite turned around, with LASSO having better predictive performances than SES, although at the cost of identifying larger and unstable sets of variables. Finally, SES generally outperforms LASSO in the Static-longitudinal and Static-distinct scenarios, both in terms of average PCC and number of selected features. No variables were selected for dataset GDS3944 by neither method, and thus we excluded this dataset from the results.
Since the results for the Temporal-longitudinal scenario could depend on the specific randomly selected gene used as target variable, we repeated the whole comparison for this scenario 20 more times, each time with a different target gene. Table 3 contains the respective results: for 4 out of 8 datasets SESglmm had statistically significantly better performance (on average), whereas for the other 4, the average performances did not differ in a statistically significant way. By aggregating the results we see that 91 out 160 times SESglmm had better performance than glmmLasso (i.e., 56.88% of the times, significantly larger than 50%, p-value=0.0395, according to the one-sided asymptotic z-test). Figure 2d shows the difference between SESglmm and and glmmLasso performances over the 20 repetitions as boxplots. GDS5088 is not shown for the sake of clarity: SESglmm largely outperforms glmmLasso for this dataset and the difference is so out-of-scale that would overshadow the differences in the other datasets (see Additional file 1: Figure S3).
Table 3 Temporal-longitudinal scenario: comparison between SESglmm and glmmLasso based on 20 replications with different target variable (gene) and independently randomly selected 2000 genes as predictor variables
We give an example of how to interpret the models selected with SESglmm for Temporal-longitudinal datasets. Figure 2b reports the expression over time of the target gene CSHL1 for each subject in dataset GDS5088, while Fig. 2c shows the logarithm of the average relative change over time for the genes selected by SES as the best predictive signature for CSHL1. The fixed part of the corresponding mixed-effect model is
$$\begin{array}{@{}rcl@{}} {}\begin{array}{ll} CSHL1_{ij} = & 14.688 -0.092 \cdot \tau_{i} + 0.297 \cdot OR4G4P_{ij} \\ & -\, 0.628 \cdot RRNAD1_{ij} - 0.613 \cdot NDUFS2_{ij} \\ & +\, 0.212 \cdot {NC}_{ij} + 0.314 \cdot {ICMT}_{ij} \end{array} \end{array} $$
where i=1,2,3,4 represents the 4 time points, j=1,⋯,11 the 11 subjects, and NC stands for the genomic region chr1:232358622−232358886. The variance of the random intercepts is equal to 0.0018, corresponding to a 7.8% of the total variability. All coefficients are significant (maximum Wald test p-value 0.02), with positive coefficients indicating predictors whose trajectories over time agree with the one of the target genes, while the opposite holds for negative coefficients. The coefficient of the time effect has a negative sign; speculations on Fig. 2c suggest that a quadratic effect would perhaps perform better, but adding more parameters would easily lead to over-fitting, due to the limited number of time points and subjects. Figure 2e and f show the temporal trajectories of gene TSIX and Ppp1r42, respectively, which were included in the optimal predictive signatures of dataset GDS4146 (Static-longitudinal) and GDS2882 (Static-distinct). In both cases the trajectories of the two genes markedly differ between the two classes.
In this work we described how constraint-based, feature selection methods can be extended for the analysis of (high dimensional) temporal data, by equipping them with suitable conditional independence tests. The main contribution of this work is thus indicating how a whole class of state-of-the-art, provably well-performing feature selection methods [4] can be easily extended to data characterized by (a) measurements taken over time and (b) high dimensionality, settings frequently encountered in biological studies as well as in other fields. Furthermore, conditional independence tests are the cornerstone of any constrained-based method for (causal) network reconstruction [71]; under this respect, this work also paves the path for extending this type of algorithms to temporal data.
We assessed the performances of the proposed approach by evaluating a prototypical constraint-based method, the SES algorithm, on several real-world gene expression datasets. Each dataset belongs to one out of four different scenarios, which represent common study designs for temporal data. The Temporal-longitudinal and Static-longitudinal scenarios represent longitudinal studies with time-dependent or static target variable, respectively, while the Static-longitudinal and Static-distinct scenarios refer to the case of different samples measured at each time point. The Temporal-longitudinal and Temporal-distinct scenarios required devising conditional independence tests able to take into account the idiosyncrasies of their respective data. The conditional independence test devised for the Temporal-longitudinal scenario addresses the within sample variation by employing the GLMM and GEE modeling techniques; the tests devised for the Static-longitudinal scenario uses a two-step regression strategy for addressing the problem of discriminant analysis in longitudinal data.
In the context of our experimentation, SES outperformed state-of-the-art methods belonging to the class of LASSO algorithms in three out of four scenarios. Particularly, in the Temporal-longitudinal scenario, SESglmm clearly superseded the glmmLasso algorithm [27]. Moreover, SESglmm easily scales to tens of thousands of variables, while glmmLasso computation requirements become rapidly prohibitive.
A key feature of the SES algorithm is its ability to produce multiple solutions, signatures, i.e. more than one set of predictor variables, which are statistically equivalent, as demonstrated in a recent publication [5]. Many times in biological studies, and not only, the outcome of the study, the final model, or the selected variables, is not what expected by the expert in the field. This could be justified by the fact that the chosen model is statistically equivalent to the expected model. And hence, a degree of miss-information has been delivered. Multicollinearity among the predictor variables should not be treated as a disease, but rather as a means of extracting extra information about the data.
We assessed the equivalence of the signatures produced by SES also in the context of this experimentation. For every signature computed by SES we fitted a predictive model and calculated the corresponding MSPE. The distribution of these MSPE values confirmed that SES produces signatures whose predictive value is close to each other (see Additional file 1: Tables S7, and S8 and Figures S7, S8 and S9 for details).
The results showed in "Results on the four scenarios" section and Table 2 purposely do not report information regarding the computational time. In the Temporal-longitudinal scenario SES' computational requirements can be one or two orders of magnitude smaller than the ones of the glmmLasso. The opposite though is true for the other scenarios. A prototypical SES run on the GDS3859 dataset (45,100 variables and 23 samples, Temporal-distinct scenario) requires on average 278 s, while LASSO returns an answer in less than half a second. This difference is mainly due to implementation issues: the code of SES is written in R, while LASSO is based on a fast FORTRAN implementation. In addition, the SES algorithm needs to perform some additional computation for identifying multiple signatures. Other constraint-based methods, e.g., MMPC, that return only one signature are expected to be faster, in case the computational cost is an important parameter to consider.
Finally, we performed over-representation analysis on the pathways provided by the Kyoto Encyclopedia of Genes and Genomes (KEGG [72]) using the hyper geometric test. To ensure an adequate statistical power, we performed this analysis only for the 12 datasets where the probesets selected by SES correspond to five or more genes. For each dataset we used the pathways of the proper species. We found that on average 10 pathways are significant at FDR level 0.1 for each signature (the lists of enriched pathways are in the Additional file 1), meaning that the selected genes are significantly over-represented in known biological mechanisms. This indicates that performing feature selection with constraint-based methods coupled with conditional independence tests for temporal data can also provide biological insights, along with well-performing predictive models.
The main limitation of the present study is the relatively low sample size of several datasets, that makes difficult to precisely estimate performances. However, we note that the work presented by [73, 74] showed that the TT protocol is able to provide precise estimation even with relatively small sample size, computationally more efficiently than the more complex nested-cross validation protocol.
Future work will focus on several directions. Computationally, we have addressed the computational cost of the linear mixed models by using our implementations. We plan to add the other scenarios in this direction. Parallel computations for the first step of the algorithm will be made available for the other methods as well. More intriguing, the application of the tests introduced in this work on constraint-based, causal-discovery methods is currently under investigation.
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The first author would like to express his acknowledgments to Dimitris Rizopoulos and Janice Scealy for answering some of his questions.
The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 617393.
No funding body played any role in the design or conclusion of the present study.
All data are publicly available from the Gene Expression Omnibus database (GEO, http://www.ncbi.nlm.nih.gov/). The R codes for the Longitudinal-Temporal Scenario are available from the R package MXM https://cran.r-project.org/web/packages/MXM/index.html.
Department of Computer Science, University of Crete, Voutes Campus, Heraklion, 70013, Greece
Michail Tsagris, Vincenzo Lagani & Ioannis Tsamardinos
Michail Tsagris
Vincenzo Lagani
Ioannis Tsamardinos
MT has participated in the design of the study, wrote the code, performed the experiments and drafted the manuscript. VL has contributed to the manuscript and to the design and supervision of the study. IT contributed the initial idea, and to the design and supervision of the study. All authors have read and approved the final manuscript.
Correspondence to Michail Tsagris.
This file contains additional information regarding the datasets and the algorithms used in the paper as well as results on SES signatures equivalence and SES / Lasso computational requirements. (PDF 198 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Tsagris, M., Lagani, V. & Tsamardinos, I. Feature selection for high-dimensional temporal data. BMC Bioinformatics 19, 17 (2018). https://doi.org/10.1186/s12859-018-2023-7
Time course data
Longitudinal data | CommonCrawl |
SiPM-based gamma detector with a central GRIN lens for a visible/NIRF/gamma multi-modal laparoscope
Young Been Han, Seong Hyun Song, Han Gyu Kang, Ho-Young Lee, and Seong Jong Hong
Young Been Han,1 Seong Hyun Song,1 Han Gyu Kang,2 Ho-Young Lee,3 and Seong Jong Hong1,4,*
1Department of Senior Healthcare, Graduate School, Eulji University, Daejeon 34824, Republic of Korea
2Department of Nuclear Science, National Institute of Radiological Sciences (NIRS), National Institutes for Quantum and Radiological Science and Technology (QST), Chiba 263-8555, Japan
3Department of Nuclear Medicine, College of Medicine Seoul National University, Seoul 13620, Republic of Korea
4Department of Radiological Science, Eulji University, 553 Sanseongdae-ro, Seongnam-Si, Gyeonggi-Do, Republic of Korea
*Corresponding author: [email protected]
Han Gyu Kang https://orcid.org/0000-0001-5338-4819
Seong Jong Hong https://orcid.org/0000-0003-1252-5421
Y Han
S Song
H Kang
H Lee
S Hong
Young Been Han, Seong Hyun Song, Han Gyu Kang, Ho-Young Lee, and Seong Jong Hong, "SiPM-based gamma detector with a central GRIN lens for a visible/NIRF/gamma multi-modal laparoscope," Opt. Express 29, 2364-2377 (2021)
Proof-of-concept of a multimodal laparoscope for simultaneous NIR/gamma/visible imaging using wavelength division multiplexing (OE)
Linear array-based real-time photoacoustic imaging system with a compact coaxial excitation handheld probe for noninvasive sentinel lymph node mapping (BOE)
Electrically tunable fluidic lens imaging system for laparoscopic fluorescence-guided surgery (BOE)
Fiber optic sources
Gradient index optics
Manuscript Accepted: December 29, 2020
Intraoperative imaging has been studied using conventional devices such as near infrared (NIR) optical probes and gamma probes. However, these devices have limited depth penetration and spatial resolution. In a previous study, we realized a multi-modal endoscopic system. However, charge-coupled device (CCD)-based gamma imaging required long acquisition times and lacked gamma energy information. A silicon photomultiplier (SiPM)-based gamma detector is implemented in a multi-modal laparoscope herein. A gradient index (GRIN) lens and CCD are used to transfer and readout visible and NIR photons. The feasibility of in-vivo sentinel lymph node (SLN) mapping was successfully performed with the proposed system.
Image-guided surgery has been playing an important role in cancer surgery to improve the patient outcome significantly by detecting the tumor and sentinel lymph node (SLN) [1,2]. In particular, near-infrared fluorescence (NIRF) image-guided surgery [3] has been adopted rapidly into clinical practice in last two decades with the development of fluorescent agents such as methylene blue and indocyanine green (ICG) [4,5]. The NIRF image-guided surgery can visualize the tumor margin with high-resolution (∼1 mm) in real-time so as to help surgeon to remove the tumor tissues with precision while minimizing the unnecessary resection of normal tissue surrounding the tumor tissues. In the early 2000s, the NIRF image-guided surgery was first translated into clinical procedure for breast cancer surgery in which the cancer surgery can be performed with open surgery which does not demand miniaturized medical imaging systems [6]. Hence, the early stage of the NIRF imaging systems employed bulky charge-coupled device (CCD) cameras and illumination systems [1].
Recently, minimally invasive surgery which uses the limited size and number of incisions on the patient body, has been widely used for cancer surgery to reduce the patient recovery time significantly as compared to that of traditional open surgery which requires a long recovery time in the hospital [7,8]. The minimal invasive surgery includes an endoscopic surgery for gastric cancers [9], laparoscopic surgery for prostate cancers [10], and robot-assisted laparoscopic cancer surgery using the Da Vinci system [11]. Hence, NIRF imaging systems have been evolved toward minimally invasive surgery. As a result, various endoscopic and laparoscopic NIRF imaging systems have been developed with compact size to fit inside the laparoscopic port diameter of 12 mm [12–14].
NIRF image-guided surgery has transformed the cancer surgery procedure in clinical practice by providing real-time images and good spatial resolution of tumor margins [15,16]. However, tissue depth penetration using NIRF is limited to less than 10 mm because of significant optical scattering and absorption inside the tissues [3,17]. Therefore, tumors or SLNs located at deep tissue region are barely identified thereby increasing the false-positive rate on the detection of tumor tissues which may lead to a recurrence of cancer.
In order to solve the depth penetration issue of NIRF, a novel hybrid imaging system combining ultrasound, photoacoustic, and fluorescence device was proposed [18]. The handheld hybrid imaging system can provide real-time images with good spatial resolution for the detection of SLN. However, it is difficult to realize superficially located target images, and ultrasound probes need to make contact with the tissue for observation. Moreover, the size of the hybrid imaging system is bulky which is not applicable for minimal invasive surgery.
The depth penetration issue can also be solved by using gamma probes [19] or gamma cameras [20,21] which detect high-energy gamma photons emitted from radiopharmaceuticals such as 99mTc-nanocolloid [19]. Therefore, radionuclide surgery has been widely used for intraoperative cancer detection in clinical practice since the early 2000s [22]. Recently, a hybrid intraoperative surgery technique was proposed in which the standalone gamma probe and NIRF imaging system were used separately during robot-assisted laparoscopic cancer surgery [23–26]. The hybrid intraoperative cancer surgery technique could enhance the detection accuracy of SLN during prostate cancer surgery as compared to that of using only NIRF image guided surgery [27,28]. However, the gamma probe and NIRF imaging systems were physically separated thereby resulting in the potential mismatch on the information obtained from the two independent modalities. In order to address this issue, a novel hybrid optical/gamma probe was developed for SLN detection during laparosopic surgery [29]. The NIRF and gamma signals were detected simultaneously using an integrated probe system and the beep signals were generated in proportional to the number of detected NIRF and gamma signals. However, the hybrid optical/gamma probe cannot provide the images of SLN which are crucial for image-guided cancer surgery.
Toward the hybrid intraoperative imaging system combining gamma camera and NIRF devices for SLN mapping, our group has proposed several prototype hybrid laparoscopic imaging systems. A novel multi-modal laparoscopic system, realizing simultaneous annihilation-gamma (511 keV), NIRF, and visible images, has been developed [30]. However, the target radiopharmaceutical is [18F]-FDG; hence, the system cannot be used for SLN mapping using ICG-99mTc-nanocolloid, which emits relatively low-energy gamma photons (140 keV). In previous studies, we developed multi-modal laparoscopic systems to realize gamma, NIRF, and visible images [30–32]. The gamma images were obtained using a combination of a scintillation crystal, optical fiber bundles, and a charge-coupled device (CCD). However, CCD-based gamma imaging required long acquisition times, owing to light loss during transmission through the optical fiber bundle. Image acquisition times of more than 30 s were required to yield sufficiently high contrast-to-noise ratios (CNRs). Moreover, CCD-based gamma imaging is affected by scattering, which deteriorates the CNR. The long gamma acquisition time can be reduced substantially by coupling the scintillation crystal array to the photo-sensor. Furthermore, the gamma energy information can be recorded using a silicon photomultiplier (SiPM), thereby enabling the rejection of scattering events. In this study, we propose a novel intraoperative multi-modal imaging system comprised a SiPM-based gamma detector and NIRF/visible optical systems for image-guided laparoscopic surgery using ICG-99mTc-nanocolloid. The performance of the proposed multi-modal imaging system was evaluated using phantom studies, and the feasibility of in-vivo SLN imaging is discussed.
2.1 Overview of the multi-modal laparoscope system
The proposed visible/NIRF/gamma multi-modal system consists of a laparoscope tube, a beam splitter module, illumination components, and imaging devices, as shown in Fig. 1. When ICG-99mTc-nanocolloid hybrid tracer accumulates in tumors, the tracer emits NIRF and gamma (140 keV) signals, which are collected at the front of the laparoscope tube, along with visible photons from the surgical region. The beam splitter module was used to separate visible and NIRF photons using a dichroic long-pass filter. The illumination components illuminate the imaging object with white light and NIR excitation light simultaneously. The visible/NIRF/gamma multi-modal system acquires individual visible, NIRF, and gamma images to produce the comprehensive image of the lesion. The NIRF CCD band pass filter and illumination light source in Fig. 1 can be easily changed to match the optical properties of different fluorophores if different fluorophores are used.
Fig. 1. Schematic of the visible/NIRF/gamma multi-modal laparoscope imaging system using SiPM-based gamma detector.
2.2 Optical readout devices of the multi-modal laparoscope system
Visible and NIRF photons were collected using a gradient index (GRIN) objective lens (GT-IFRL-100-020-50-NC, GRINTECH, Germany) located at the center of the laparoscopic tube. The GRIN lens has a diameter of 1 mm and length of 2 mm. The collected optical photons were transferred from the GRIN lens to the beam splitter module using a GRIN relay lens (GT-IFRL-100-150-10-NC, GRINTECH, Germany) with a diameter of 1 mm and length of 67 mm. In the beam splitter module, transmitted visible and NIRF photons were collected separately by the dichroic long-pass filter, as shown in Fig. 1. The dichroic long-pass filter (750 nm, Edmund Optics, USA) reflects light of wavelengths 565-715 nm, such that visible light was reflected by a dichroic long-pass filter and NIRF light was transmitted to an optical fiber (OD = 1.00 mm, length = 840.8 mm, Schott, Germany). The transmitted NIRF light was filtered using a band-pass filter (832/37 nm, Edmund Optics, USA). The visible and NIRF photons were focused by an aspherized achromatic lens (f = 25 mm, Edmund Optics, USA). Subsequently, the images were acquired by a Basler CCD (acA1300-30uc, Basler, USA) for the visible photons and an Andor CCD (iKon-M 934, Andor Technology Ltd, Northern Ireland) for NIRF image photons.
The illumination light sources were a fiber optic illuminator LED (G180P-6500 K, ScopeLED, USA) for white light and an NIR light diode (M780LP1, Thorlabs, USA) for NIR excitation. Each light source was focused on the illumination fiber bundle in the multi-modal system.
2.3 Gamma detection of the multi-modal laparoscope system
2.3.1 SiPM-based gamma detector module
The SiPM-based gamma detector is shown in Fig. 1. A multi-modal gamma detector of 13 mm diameter was built for insertion into the laparoscope tip. To obtain the maximum effective area within a small diameter, SiPMs of areas 2 ${\times} $ 2 mm2 (S13360-2050VE, Hamamatsu, Japan) and 1 ${\times} $ 1 mm2 (S13615-1050, Hamamatsu Photonics, Japan) were used to cover the scintillation crystals. A cerium-doped gadolinium aluminum gallium garnet (Gd3Al2Ga3O12: Ce, GAGG: Ce; Furukawa Scintitech Corporation, Japan) of size of 1.4 ${\times} $ 1.4 ${\times} $ 4 mm3 was used as the scintillation crystal. All GAGG crystals were polished chemically, and barium sulfate (BaSO4) reflectors of thickness 100 µm were used to optically isolate each GAGG pixel. This resulted in a crystal pitch of 1.5 mm. The scintillation crystal array was coupled with the SiPM using a polyvinyl chloride (PVC) light guide as compensation. The PVC was used to compensate for the height difference between the two SiPM detectors, and spread the scintillation light effectively. The SiPM detector and crystal array have a hole in the center for inserting the GRIN lens, and the pinhole collimator of the gamma detector was designed by the Monte-Carlo simulation to obtain a gamma image that was the same as the optical focal plane. The field of views (FOVs) of the gamma, NIRF, and visible images are aligned in the identical optical axis. Consequently, image registration between the visible, NIRF, and gamma images can be performed, while minimizing any potential image registration errors. The bias voltage of the gamma detector was selected as an overvoltage of 5.6 V to acquire gamma events, and the threshold of the discriminator cut was 0.9 V.
2.3.2 Signal processing circuit and data analysis of the multi-modal gamma detector
Four position signals (A, B, C, and D) of the gamma detector were obtained using a discretized positioning circuit for each SiPM output signal, amplified by an amplifier board (BASP-10005, Brightonics Imaging, South Korea), and read out to a DRS4 waveform digitizer (DT5742B, Caen, Italy). The sampling rate of DT5742B is 5 GSa/s, and the sampling resolution is 12-bit for each channel. The position signals were used to calculate the gamma photon interaction position within the GAGG crystal array using the following equation [33]:
(1)$$X = \; \frac{{({A + B} )- ({C + D} )}}{{A + B + C + D}}$$
(2)$$Y = \; \frac{{ - ({A + D} )+ ({B + C} )}}{{A + B + C + D}}$$
where X and Y are the x-axis and y-axis positions at the gamma detector, respectively. To reject gamma scattering events, an energy window amounting to 20% of the 140 keV photo-peak was used.
To evaluate the performance of the gamma detector, the Voronoi diagram [34] was used to segment the crystal map so as to extract the energy information of individual crystal pixel. The photo-peak value in ADC unit for each crystal pixel was obtained and then normalized to compensate the non-uniformity of detection efficiency as well as light output variation over the crystal array. After the photo-peak normalization and detection efficiency compensation, a 2D gamma event map which represents the number of events per crystal pixel was generated to visualize the radiopharmaceutical inside the imaging object. In the gamma event map, a GRIN lens was placed at the center of the crystal array to acquire the visible and NIRF images. The gamma events at the central crystal were calculated from neighboring crystal events in the gamma event map, as shown in Fig. 2, using the following equation:
(3)$${N_4}(P )= ({x \pm 1,\; y} )or\; ({x,\; y \pm 1} )$$
(4)$${N_D}(P )= ({x \pm 1,\; y \pm 1} )or\; ({x \pm 1,y \pm 1} )$$
(5)$$P = {f_D}\frac{{\sum {N_D}(P )}}{4} + {f_4}\left( {\frac{{\sum {N_4}(P )}}{{\sum {N_D}(P )}}} \right)\left( {\frac{{\sum {N_4}(P )}}{4}} \right)$$
where ${N_4}$, ${N_D}$, ${f_D}$, ${f_4}$, and P, are 4 and diagonal neighboring events and weighting factors, and the number of central events in the gamma event map, respectively. In this study, we used 0.4 for ${f_D}$ and 0.6 for ${f_4}$ for best results.
Fig. 2. Gamma detector acquisition flow and gamma image inpainting method of the visible/NIRF/gamma multi-modal laparoscope system using SiPM-based gamma detector. (a) Gamma detector image acquisition diagram, (b) gamma image center hole inpainting method.
2.4 Image matching of the multi-modal laparoscope imaging system
The multi-modal laparoscope was designed for easy matching using the same FOV in the visible and NIRF imaging lenses and the gamma detector. The image fusion flow in the multi-modal imaging system is shown in Fig. 3. The gamma image was acquired based on the analog gamma signal at a 1 s interval. The 7 ${\times} $ 7 gamma event map was reshaped into an image of resolution 480 ${\times} $ 480 pixels using bicubic interpolation to match the image size of the visible CCD image. The 120 ${\times} $ 120 NIRF images were also reshaped to images of resolution 480 ${\times} $ 480 pixels image. The resized gamma and NIRF images were used to obtain fusion images with the visible image.
Fig. 3. Image acquisition flow diagram of the multi-modal laparoscope imaging system.
In the fusion images, the gamma and NIRF signals were pseudo-colored in the hot color map and green hot, respectively. The NIRF signal was filtered using a kernel with 11 ${\times} $ 11 median filters in the fusion visible/NIRF/gamma image. For the visible image, flat-field correction was employed to compensate for the non-uniformity in intensity across the FOV.
2.5 Phantom test of the proposed system
Figure 4 shows the experimental setup for the quantitative evaluation of gamma images. To mimic a patient's abdominal cavity during laparoscopic surgery, the multi-modal system and the solution of mixed 99mTc isotope and ICG NIR dye phantom source were placed in a dark box. A tungsten pinhole collimator of diameter 1.5 mm was used to acquire the gamma image. The distances from the phantom source to the pinhole collimator and that from the pinhole collimator to the GAGG crystals were 50 and 10 mm on the z-axis, respectively. The FOV of the visible, NIRF, and gamma images was 50 ${\times} $ 50 mm2. The 99mTc gamma source activity was 1.1 MBq, and the ICG dye volume was 0.1 mL with a concentration of 10 µM (Cardiogreen, Sigma Aldrich, USA). The gamma images were acquired with an acquisition time of 1 s, and then, the signal intensities in the gamma images were accumulated. The visible and NIRF images were obtained with acquisition times of 50 and 200 ms, respectively, and updated at the start of each acquisition. The CNR of the gamma image was calculated using the following equation [35]:
(6)$$CN{R_{lesion}} = \; \frac{{|{{C_{lesion}}} |}}{{{C_{noise}}}}$$
(7)$${C_{lesion}} = \; \frac{{{I_{sig}} - {I_{bkg}}}}{{{I_{bkg}}}}$$
(8)$${C_{noise}} = \; \frac{{\sqrt {{I_{bkg}}} }}{{{I_{bkg}}}}$$
where ${I_{sig}}$ and ${I_{bkg}}$ are the average pixel intensities of the region of interest (ROI) of the signal and background, respectively.
Fig. 4. Phantom test for gamma image acquisition time and tissue-equivalent phantom depth penetration test. (a) ICG-99mTc mixture phantom, (b) gelatin tissue-equivalent phantoms with different thicknesses, (c) phantom test setup.
To characterize the NIRF image for human tissue, tissue-equivalent phantoms were fabricated with 89% water, 10% gelatin from porcine skin (Sigma Aldrich, USA), and 20% weight of intralipid 1% (JW Pharmaceutical, Korea). The tissue-equivalent phantoms were molded into custom-made plastic frames of thicknesses 2, 4, 6, 8, and 10 mm. All images were acquired with the same FOV, and the signal-to-noise ratio (SNR) and CNR were calculated using the following equations [35]:
(9)$$SNR = \; \frac{{Mean}}{{{\sigma _{Background}}}}$$
(10)$$CNR = \; \frac{{Mean - Mea{n_{Background}}}}{{{\sigma _{Background}}}}$$
2.6 Testing with a rat model
Figure 5 shows photographs of the visible/NIRF/gamma multi-modal imaging system and a Sprague Dawley rat model, which was used to identify the SLN. To prevent motion artifacts in the rat model during in-vivo imaging, inhalation anesthetizing was maintained with Isoflurane 2%. The ICG-99mTc-nanocolloid (human serum albumin) source was used as the hybrid tracer. The injection radiation dose was 81.77 MBq of 99mTc, and ICG labeling was performed with a commercial kit-NH2 (Dojindo Laboratories, Japan). After the multi-modal images were acquired, the rat model was sacrificed to dissect the SLN. All rat model studies were approved by the Institutional Animal Care and Use Committee at Seoul National University Bundang Hospital. Imaging was performed 3 h after tracer injection for the axillary point image, and 4 h after tracer injection for the resected lymph node multi-modal images.
Fig. 5. Rat model test. (a) Setup, (b) injection position, (c) image acquisition from the axillary position.
3.1 Performance of the gamma detector
Figure 6 shows the performance of the proposed gamma detector for the multi-modal laparoscope system. Individual crystals were clearly resolved in the segmentation map (Fig. 6(a)), and the average energy resolution of the normalized gamma event map was 18.7 ± 1.4%, as shown in Fig. 6(c). The distance between the laparoscopic tip and source was 50 mm. As shown in Fig. 6(d), the signal was calculated from the ROI (white line), and the background (white dotted line) was calculated for the opposite side of the gamma image. Figure 6(e) shows the signal and noise intensity and CNR obtained for the gamma images of the 99mTc radioactive source (4.3 MBq/300 µL) for different acquisition times. The gamma signal increased linearly with the acquisition time, and the CNR increased with the acquisition time of the gamma detector. A gamma detector sensitivity of 0.035 cps/kBq was acquired for the 99mTc source.
Fig. 6. Intrinsic performance of the SiPM-based gamma detector. (a) Flood map of the detector, (b) raw energy spectra, (c) normalized energy spectra (W/ Gaussian fitting), (d) acquired gamma detector event map, (e) net count and CNR of the detector.
3.2 Evaluation of the multi-modal laparoscope system with a phantom test
Figure 7 shows the multi-modal visible, NIRF, and gamma images with different image acquisition times. The visible, NIRF, and gamma images shown in Fig. 7(a), (b), and (c) were acquired with acquisition times of 50 ms, 200 ms, and 30 s, respectively. The spatial resolution of the NIRF image was acquired at 5.4 mm in Fig. 7(d), and the gamma image spatial resolution was 8.2 mm, as shown in Fig. 7(e). The fusion images of visible, NIRF, and gamma images shown in Fig. 7(f) were obtained by combining Fig. 7(a) and (b) with gamma images obtained for 1, 5, 10, and 20 s.
Fig. 7. Phantom test images. (a) Visible image, (b) NIRF image, (c) gamma image, (d) NIRF image phantom spatial resolution plot (W/ Gaussian fitting), (e) gamma image phantom spatial resolution plot (W/ Gaussian fitting), (f) multi-modal fusion image different of gamma acquisition times.
Figure 8 shows the multi-modal laparoscope images of the 99mTc isotope and ICG dye mixture at 0, 4, 6, and 10 mm thicknesses of the tissue-equivalent gelatin phantom. All gamma images required an acquisition time of 10 s. The images in Fig. 8(a) were obtained by adjusting the contrast, while the images in Fig. 8(b) were obtained using the median filter. The scattered NIRF signals are shown in Fig. 8(b).
Fig. 8. Tissue-equivalent phantom depth penetration test images. (a) Multi-modal laparoscope fusion image of the tissue-equivalent phantom image in different depths (W/O histogram stretch), (b) tissue-equivalent phantom image (W/ histogram stretch and median filter).
Figure 9 shows the NIRF and gamma signal intensities at different depths of tissue-equivalent phantoms. As the depth of the phantom increases, the gamma and NIRF signals deteriorate by 13% and 80% at 10 mm, respectively.
Fig. 9. Relative and net pixel intensities by tissue-equivalent phantom depth: NIRF and gamma.
3.3 Evaluation of the multi-modal laparoscope system with the small animal test
Figure 10 shows the visible images of the ICG-99mTc-nanocolloid tracer injection points (paw and palm), axillary area, and resected lymph node. The images clearly show the characteristics of NIRF and gamma imaging, such as the limited depth penetration of NIR and poor spatial resolution of gamma. Immediately after injection, the gamma image was obtained with an acquisition time of 2 s, because the injection point radiation dose was high. The axillary gamma image acquired 3 h after injection was obtained with a 90 s acquisition time, because the distance from the multi-modal laparoscope to the rat model was increased from 50 mm to 100 mm. The resected lymph node gamma image acquired after 4 h of injection took 30 s. The NIRF image of the resected lymph node took 5 s, since the NIRF signal of the lymph node was very low.
Fig. 10. In-vivo imaging results of a rat with the proposed multi-modal laparoscope imaging system small animal image: (a) visible/NIRF image at paw (left), palm (middle), and axillary area (right), (b) visible/gamma image at paw (left), palm (middle), and axillary area (right), (c) visible/NIRF/gamma image at paw (left), palm (middle), and axillary area (right), (d) resected lymph node image of visible/NIRF, visible/gamma, and visible/NIRF/gamma.
Figure 10(a) shows the visible/NIRF images, and it can be seen that NIRF realizes good spatial resolution, but cannot penetrate the rat model's hand, as shown in the palm image. The yellow line in the palm image was caused by the NIRF concentration at the paw side. As shown in Fig. 10(b) of the visible and gamma image, the gamma signal can penetrate the rat model; however, this results in poor spatial resolution. Figure 10(c) shows the visible/NIRF/gamma fusion image, overcoming the weaknesses of each imaging modality. Figure 10(d) shows the lymph node image shown in Fig. 10(c), where the axillary area gamma hot spot region shows the detection ability of the multi-modal laparoscope image system.
Recently, intraoperative imaging methods have been studied using NIRF, ultrasound, and gamma probes to provide images and signals for surgeons. However, each imaging method has problems such as the penetration depth problem of NIRF, poor resolution of superficial regions in ultrasound, image acquisition time, and poor spatial resolution of the gamma probe. In addition, the existing imaging method can result in a high false-negative rate for lesions. The SLN biopsy using both 99mTc-nanocolloid and vital dye resulted in the greater detection ability than using separate markers [36]. Currently, the dual technique of using NIRF and gamma signal is the gold standard for SLN biopsy.
For the image guided surgery, the hybrid tracer study has been also increasing as the multi-modal system research. Even though there are some challenges generating the hybrid tracers, but the hybrid tracers showed better results than monomodal tracers [23,24,27,28].
To overcome the limitations of using a single imaging modality, the hand-held tri-modal probe was proposed using ultrasound, photoacoustic, and fluorescence. The tri-modal imaging method can show the obstructed SLNs and metastatic involvement [18]. In this study, we investigated a multi-modal laparoscope imaging system acquiring simultaneous visible, NIRF, and gamma images. The phantom study with the 99mTc isotope and ICG dye mixture showed that the fusion image was not affected by the depth penetration problem of the NIRF signal and the poor spatial resolution of the gamma signal. In addition, the small animal test demonstrated the detection ability of the proposed system in lymph node resection. The proposed system was focused on imaging the ICG-99mTc-nanocolloid hybrid tracer with a single system which is able to obtain visible, NIRF, gamma and hybrid images. The ICG fluorophore does not accumulate in the lymph node, while the ICG-99mTc-nanocolloid hybrid tracer is accumulated so that the lymph node could be located with the smaller false-positive rate using the multi-modal imaging than the ICG or 99mTc-nanocolloid alone imaging [20,23,24,27,28]. The single system which we proposed occupies less space in the laparoscopic surgery cavity than two imaging devices, one for ICG and the other for 99mTc-nanocolloid, something which could be quite convenient for surgeons considering that many devices compete for the laparoscopic surgery cavity. Moreover, it can be easily applied to other fluorophores by changing the NIR CCD bandpass filter and illumination NIR light diode.
The limitations of the multi-modal laparoscope imaging system are the central dead region of the gamma detector and the low sensitivity of NIRF and gamma signals. In the multi-modal system, the gamma detector was designed to acquire the optical and gamma images in the same FOV, owing to which the gamma detector has a central dead region for the optical GRIN lens system. To address this issue, inpainting of the center hole gamma event was implemented. In the phantom study, all the images matched well in the fusion image, and the gamma event center hole effect was not significant. However, the small animal test exposed the weakness of the large image defect at the central region, resulting in difficult interpretation of the gamma image. In addition, the central dead region caused worse gamma event defects as the FOV increased.
Gamma images were acquired at 1 s intervals (i.e., frame rate of 1 fps), and accumulated to produce the final image at the desired time. Visible and NIRF images were acquired at each acquisition time of the CCD camera single image and flushed. The gamma image was acquired for 30 s at a distance of 50 mm, and 90 s at a distance of 100 mm in the small animal test axillary image. As resection of the lymph nodes was performed 4 h after injecting the ICG-99mTc-nanocolloid, the concentration of lymph nodes was low, resulting in poor NIRF signal intensity in the resected lymph node image. The lymph node NIRF image also required an additional acquisition time of 5 s. The proposed multi-modal laparoscope system can provide visible, NIRF, and gamma image simultaneously. In surgical environment where the SLN is located under the tissue or fat, it can provide the rough SLN location with the gamma image which, with surgical tools, can be exposed to acquire the visible/NIRF image for the precise SLN location.
In future studies, we plan to investigate a multi-modal imaging system without a central dead region, assuming gamma detectors improve the source detection ability and reduces the gamma image acquisition time. The usefulness of the scattered NIRF signals at a thickness of 10 mm in tissue-equivalent phantoms will be also studied in future works.
We developed a novel prototype multi-modal laparoscope system for simultaneous NIRF/gamma/visible imaging, containing a gamma detector with a central dead region to accommodate the GRIN lens for visible and NIRF images. The feasibility of in-vivo SLN mapping was successfully performed with the proposed system, using the ICG-99mTc-nanocolloid. In the future, the laparoscopic system will be further optimized for a pilot clinical study.
National Research Foundation of Korea (NRF-2017M2A2A4A01071175, NRF-2020R1F1A1054317); Korea Medical Device Development Fund (202011D24).
National Research Foundation (NRF) of Korea of the Ministry of Science, ICT & Future Planning, Nuclear R&D Program (NRF-2017M2A2A4A01071175); National Research Foundation of Korea (NRF) grant funded by the Korea government Ministry of Science and ICT (MSIT) (NRF-2020R1F1A1054317); Korea Medical Device Development Fund grant funded by the Korea government R&D Program (202011D24).
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Alqahtani, M. S.
Aoki, T.
Ashitate, Y.
Aurenhammer, F.
Azar, F.
Bajaj, A.
Balm, A. J. M.
Bauwens, K. P.
Behnke, N. K.
Bhatia, B. S.
Blanco, J. I.
Boni, G.
Bonjer, H. J.
Bouvet, M.
Brouwer, O. R.
Buckle, T.
Bugby, S. L.
Bunschoten, A.
Carbajo, M. A.
Carroll, W. R.
Chambron, J.
Chang, J. H.
Cherry, S. R.
Choi, H. S.
Cotero, V. E.
Cuesta, M. A.
de Boer, E.
De Grand, A. M.
de Jong, M.
de la Cuesta, C.
Eiber, M.
Engelen, T.
Frangioni, J. V.
Gangadharan, S. P.
Garau, L. M.
Garcia-Allende, P. B.
Gibbs-Strauss, S. L.
Gioux, S.
Glatz, J.
Graefen, M.
Gray, D. C.
Greten, F. R.
Haglind, E.
Han, Y. B.
Hartmans, E.
Hawn, M. T.
Hehir, C. A. T.
Heller, S.
Hong, G. C.
Hong, S. J.
Hop, W. C. J.
Horenblas, S.
Hruby, S.
Hutteman, M.
Inglada, L.
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Jani, A. H.
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Kato, T.
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Khamene, A.
Kianzad, V.
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Kim, H.
Kim, K. M.
Kim, S. M.
KleinJan, G. H.
Klode, J.
Klop, W. M. C.
Koch, M.
Kuhry, E.
Kusano, M.
Lacy, A. M.
Lee, H. J.
Lee, H.-Y.
Lees, J. E.
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ACM Comput. Surv. (1)
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Ann. Surg. (2)
Biomed. Opt. Express (2)
Clin. Transl. Imaging (1)
Eur. J. Nucl. Med. Mol. Imaging (3)
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Expert Rev. Med. Devices (1)
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(1) X = ( A + B ) − ( C + D ) A + B + C + D
(2) Y = − ( A + D ) + ( B + C ) A + B + C + D
(3) N 4 ( P ) = ( x ± 1 , y ) o r ( x , y ± 1 )
(4) N D ( P ) = ( x ± 1 , y ± 1 ) o r ( x ± 1 , y ± 1 )
(5) P = f D ∑ N D ( P ) 4 + f 4 ( ∑ N 4 ( P ) ∑ N D ( P ) ) ( ∑ N 4 ( P ) 4 )
(6) C N R l e s i o n = | C l e s i o n | C n o i s e
(7) C l e s i o n = I s i g − I b k g I b k g
(8) C n o i s e = I b k g I b k g
(9) S N R = M e a n σ B a c k g r o u n d
(10) C N R = M e a n − M e a n B a c k g r o u n d σ B a c k g r o u n d | CommonCrawl |
•https://doi.org/10.1364/OME.444859
Plasmonic hot-carriers and their applications: opinion
Harsha Reddy and Vladimir M. Shalaev
Harsha Reddy1 and Vladimir M. Shalaev2,*
1Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA
2School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
*Corresponding author: [email protected]
H Reddy
V Shalaev
Harsha Reddy and Vladimir M. Shalaev, "Plasmonic hot-carriers and their applications: opinion," Opt. Mater. Express 11, 3827-3832 (2021)
Circular polarization
Light matter interactions
Quantum efficiency
Surface plasmons
Revised Manuscript: October 8, 2021
Manuscript Accepted: October 8, 2021
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The generation of nonequilibrium hot-carriers from the decay of surface plasmons has been attracting intense research attention in the last decade due to both the fundamental aspects of extreme light-matter interactions and potential practical applications. Here, we overview the physics associated with plasmon-assisted hot-carrier generation and outline the key applications of hot-carrier processes for photodetection, photovoltaics and photocatalysis. We also discuss the recent developments in employing molecular tunnel junctions as barriers for extracting hot-carriers and provide an outlook on the potential of this emerging field for sustainable energy.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Surface plasmons (SPs), namely the collective oscillations of free carriers in metal nanostructures that arise in response to incident electromagnetic fields, enable confinement and control over light at length scales well below the diffraction limit [1–4]. The localized SP energy, however, doesn't remain confined in the metal nanostructure indefinitely. Both geometry/boundary induced and material specific radiative and nonradiative decay channels eventually dissipate the confined SP energy, either as free-space radiation (radiative loss) or as absorption. Besides the conventional Ohmic losses, absorption also includes the generation of electron-hole pairs in the material (nonradiative loss). The radiative loss of SPs is an attractive feature for several important applications that leverage the radiative decay channel for engineering the spectral and directional emission properties of quantum emitters [5–7]. On the other hand, nonradiative losses have long been thought to be deleterious to the performance of plasmonic (metallic) structures and different approaches have been explored to reduce the nonradiative losses [8–10]. This notion has gradually changed in the last couple of decades, and there has been a concentrated effort to develop useful applications by leveraging nonradiative losses. One example is photothermal therapy [11–13], where the temperature increase of the plasmonic structure from the absorption of plasmon energy is utilized in medical treatments.
In addition to the thermalized carriers that are under thermal equilibrium with the lattice, the nonradiative decay of SPs also produces highly energetic carriers (both electrons and holes) that are not in thermal equilibrium with the lattice. Such nonequilibrium "hot-carriers", with distributions that substantially deviate from the equilibrium Fermi-Dirac distribution, are expected to enable efficient energy-harvesting applications, ranging from photodetection and photovoltaic schemes circumventing bandgap limitations [14–18] to catalyzing chemical reactions [19–24]. Below we provide a brief discussion of the recent developments in this field.
Among early suggested applications of plasmon assisted hot-carriers are photodetectors operating below the bandgap [15,16]. In this approach, hot-carriers are generated by the resonant SPs at telecommunications wavelengths in lithographically created gold plasmonic structures on silicon substrates. The generated energetic hot-carriers are distributed over a range of energies from $\{{{E_F} - \hbar \omega :\; {E_F} + \hbar \omega } \}$, where ${E_F}$ is the Fermi energy of gold and $\hbar \omega $ is the energy of the incident photon (those carriers above ${E_F}$ are the hot-electrons, while those below ${E_F}$ are the hot-holes). When $\hbar \omega $ is larger than the Schottky barrier height (${\mathrm{\Phi }_B}$), which for the Au-Si interface is ∼ 0.6 eV, some of the hot-carriers generated in the plasmonic structure have energies exceeding the barrier. Thus, they can flow to the semiconductor, provided the carriers have a positive momentum component that is directed from the plasmonic structure to the semiconductor (Fig. 1(A)). This flow of charge leads to the current, which is subsequently measured using suitable electronics, thereby detecting the incident photon flux. These generated photocurrents closely follow the SP absorption spectra, which along with the observed polarization dependence, strongly suggesting that the arising photocurrents are due to the generated nonequilibrium hot-carriers. A key advantage of this approach, as opposed to the conventional photodetection approach that employs pn semiconductor junctions, is that the generated photocurrent is no longer limited by the bandgap of the semiconductor or the interband transition energy in the metal. Instead, the Schottky barrier height is the limiting factor, which is usually much smaller than the constituent material bandgaps, thereby enabling the detection of the below-bandgap photons.
Fig. 1. A, (Left panel) Band diagram schematic of metal-silicon interface indicating the process of hot-carrier injection and photodetection. Schematic (Top right panel) and a scanning electron micrograph (Bottom right panel) of an Au nanoantenna device on Silicon [16]. B, Schematic showing the transfer of hot-carriers from a metal nanoparticle to nearby reactants surrounding the nanoparticle [19]. C, Schematic of a molecular junction-based light to electrical energy harvesting device, featuring a plasmonic structure (periodic metallic stripes) and a transparent electrode. (Figures (A) and (B) are adapted from Refs. [16] and [19], respectively)
Subsequently, several groups explored the "below bandgap" photodetection with hot-carriers. For instance, Chalabi et al. [14], instead of the Schottky barrier, explored the possibility of using a tunnel barrier for photodetection and investigated the role of the external voltage bias on the photodetection quantum efficiencies. Additionally, the strong polarization dependence of the underlying plasmonic structures can be used to design polarization selective photodetectors. Li et al. [18] in turn used chiral plasmonic structures, which exhibit strong plasmonic response for one helicity of incident circular polarization and negligible sensitivity for the other helicity. This allowed to detect and distinguish the right and left circular polarizations following the same principles as those outlined above. While all these works conclusively showed photon detection below the bandgap, the demonstrated quantum efficiencies were rather small, about 0.1% or even less. Such small quantum efficiencies can be attributed to the mismatch between the energies of hot-carriers under steady-state conditions and the associated Schottky/tunnel barrier heights, which for most metal-semiconductor/metal-insulator-metal junctions are about 0.5 eV or larger. As per Fowler theory [25,26], the quantum efficiencies of the above outlined photodetectors can be approximated as $c\frac{{{{({\hbar \omega - {\phi_B}} )}^2}}}{{\hbar \omega }}$, where c is a device specific emission coefficient. Although substantial number of carriers with energies exceeding the barrier height are generated soon after the plasmons nonradiatively decay, they rapidly relax to lower energies (due to electron-electron and electron-phonon collisions) before they can cross the barrier. As a result, under continuous wave illumination and steady-state conditions, only a tiny fraction of the generated hot-carries have energies exceeding ${\mathrm{\Phi }_B}$ and contribute to the photocurrent thereby limiting the quantum efficiencies.
A closely related application to photodetection is photovoltaics [27,28]. Hot-carrier based photovoltaics offers an enticing prospect of harvesting the energy of below bandgap photons. However, so far, there have been no demonstrations of hot-carrier based below bandgap photovoltaics with meaningful efficiencies, which again can be attributed to the mismatch between the steady-state hot-carrier energies and the barrier heights. Improving the quantum yields by engineering the barrier heights with either conventional metal-semiconductor junctions or via molecular junctions with versatile transmission characteristics (more on this below) could be the crucial next step to improve the efficiencies of hot-carrier based light-to-electrical energy harvesting devices to the point where they are of commercial value.
Catalysis is yet another avenue that has been extensively explored in the context of plasmon assisted hot-carrier generation [20–24]. It has been suggested that the hot-carriers generated via surface plasmon excitation can transfer some of their energy to the reactants/adsorbates surrounding the nanoparticle that contribute to the reaction (Fig. 1(B)). This process will push the reactants up in the energy landscape, effectively reducing the reaction activation energy barrier and catalyzing the reaction. Among prominent applications of plasmon assisted photocatalysis are dissociation of hydrogen [20,29], ethylene oxidation [22], ammonia decomposition [21], water splitting [24,30,31] and methane dry forming [32], to name a few.
While the observed plasmon assisted enhancements in the reaction rates is widely accepted, the governing physical processes contributing to the enhancements in the reaction rates are still under discussion [33]. The nonradiative decay of plasmons not only generates hot-carriers but also increases the temperature. Consequently, both hot-carrier effects and thermal effects contribute to plasmon driven chemical reactions and quantifying their relative contributions in plasmon driven chemical reactions has been a topic of intense debate [34]. We direct the readers to the recent articles [33,35,36] that discuss these issues in great detail. While the role of non-thermal hot-carriers in plasmon-induced chemical reactions is still under debate, there is a long-standing precedence describing the role of non-thermal effects in surface chemistry, both experimentally and theoretically [37–39].
A key aspect that strongly influences the performance efficiencies of all these applications is the energy distribution of the generated hot-carriers. Different theoretical frameworks involving either ab-initio first-principle calculations or semiclassical approaches have been developed to gain insights into these distributions [33,40–43]. These frameworks however involve different assumptions on the material properties and carrier relaxation processes, which have led to starkly different predictions on the energy distributions of the hot-carriers. To resolve these discrepancies and to gain insights into the physical processes governing hot-carrier generation, researchers have developed special experimental methods to quantify the hot-carrier distributions. Heilpern et al. [44] used an optical pump-probe technique in conjunction with a double-inversion approach to infer the transient hot-carrier distributions and their relaxation dynamics, while Pensa et al. [45] used the information from the energy barrier reduction for a plasmon assisted polymerization reaction to gain partial information on the steady-state hot-carrier energies.
Until recently, experimental elucidation of plasmonic hot-carrier energy distributions under steady-state conditions was lacking. Progress towards this goal was made in recent experiments where transport measurments through single molecule junctions (SMJs) were combined with plasmonics [46,47] and the steady-state energy distributions of hot carriers in plasmonic thin films were obtained. Specifically, SMJs featuring sharp peaks in the transmission characteristics were trapped between an ultra-thin Au film with an integrated grating coupler, and the Au tip of a scanning tunneling microscope probe [46]. Such SMJs act as spectral filters, wherein those carriers in a narrow energy window near the transmission function peak predominantly contribute to the current. Additionally, the spectral position of the molecular filter was tuned by varying the voltage bias between the STM probe and the Au film underneath. By measuring the differences in the bias voltage dependent currents flowing through such tunable molecular filters with and without plasmonic excitation (excited with an 830 nm continuous wave laser), the steady-state energy distributions of the generated hot-carriers were obtained. The measured distributions revealed that most of the hot-carriers are localized in an energy window from {-0.3 eV: 0.3 eV} relative to the Fermi level and substantially fewer hot-carriers were found at higher energies.
While past works combining plasmonics and molecular junctions have explored several interesting regimes of light-matter interactions, including the room-temperature strong coupling [48] and quantum plasmon resonances [49], the recent developments in probing the transport properties of molecular [46,47,50,51] and vacuum tunnel [52] junctions under the plasmonic excitation offer new avenues for energy harvesting applications. A distinct advantage with molecular junctions, as opposed to regular tunnel junctions and Schottky barriers, is that one could tailor the transmission characteristics by modifying the chemical structure of the molecule creating the junction. Additionally, advances in nanofabrication have enabled the production of large area molecular junctions [53,54]. Recent experiments have already established that plasmon induced photocurrents through large area alkanedithiols [51] to be comparable to metal-semiconductor Schottky barrier based photodetectors. One can thus envision having large area molecular junctions with transmission characteristics akin to a Heaviside function and sandwiched between plasmonic structures on one side and a contact electrode on the other side (Fig. 1(C)). Such junctions will allow the transfer of nearly all carriers of one kind (either electrons or holes) to flow from the plasmonic structure to the second contact, which would potentially enable the next generation of below bandgap photodetectors and photovoltaics with efficiencies exceeding the state-of-the-art hot-carrier devices.
The progress made in the field of plasmon-assisted hot-carrier generation has already contributed to crucial fundamental insights into the role of nonradiative loss in light-matter interactions, as well as to practical advances in energy conversion applications. We anticipate the future endeavors in the field to be aimed towards understanding and distinguishing the role of non-thermal and thermal carriers in plasmon driven processes, as well as exploring designer junctions for separating the energetic carriers. Insights from such studies would be crucial for rational design and development of future light energy harvesting technologies.
National Science Foundation (2029553-ECCS).
V. M. S. acknowledges support from NSF (Grant No. 2029553-ECCS).
The authors declare that there are no conflicts of interest related to this article.
No data was presented in this article.
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Andrea Alù, Editor-in-Chief | CommonCrawl |
Special issue: Carbonaceous Aerosols and Radiative Effects Study (CARES)
Research article 22 Aug 2012
Research article | 22 Aug 2012
Overview of the 2010 Carbonaceous Aerosols and Radiative Effects Study (CARES)
R. A. Zaveri et al.
Subject: Aerosols | Research Activity: Field Measurements | Altitude Range: Troposphere | Science Focus: Physics (physical properties and processes)
Effects of continental emissions on cloud condensation nuclei (CCN) activity in the northern South China Sea during summertime 2018
Mingfu Cai, Baoling Liang, Qibin Sun, Shengzhen Zhou, Xiaoyang Chen, Bin Yuan, Min Shao, Haobo Tan, and Jun Zhao
Cloud condensation nuclei activity in marine atmosphere affects cloud formation and the solar radiation balance over ocean. We employed advanced instruments to measure aerosol hygroscopicity and chemical composition in the northern South China Sea. Our results show that marine aerosols can be affected by local emissions or pollutants from long-range transport. Our study highlights dynamical variations in particle properties and the impact of long-range transport on this region during summertime.
Particle number concentrations and size distribution in a polluted megacity: the Delhi Aerosol Supersite study
Shahzad Gani, Sahil Bhandari, Kanan Patel, Sarah Seraj, Prashant Soni, Zainab Arub, Gazala Habib, Lea Hildebrandt Ruiz, and Joshua S. Apte
Delhi, India, has had the highest fine particle mass (PM2.5; diameter < 2.5 µm) concentrations of any megacity on the planet in recent years. Here, we undertook a year of detailed measurements of particle size distributions. We observed that the number count of ultrafine particles (diameter < 100 nm) – unlike PM2.5 – is not dramatically elevated in Delhi. Using observations and a simple model, we illustrate how the high amount of PM2.5 in Delhi may suppress ultrafine particle concentrations.
Airborne in situ measurements of aerosol size distributions and black carbon across the Indo-Gangetic Plain during SWAAMI–RAWEX
Mukunda Madhab Gogoi, Venugopalan Nair Jayachandran, Aditya Vaishya, Surendran Nair Suresh Babu, Sreedharan Krishnakumari Satheesh, and Krishnaswamy Krishna Moorthy
Extensive airborne measurements of aerosol number–size distribution and black carbon (BC) profiles are carried out for the first time across the IGP prior to the onset of the Indian summer monsoon. These measurements, combined with spaceborne sensors and model results, provided an east–west transect of the role of mineral dust (local and transported) in the aerosol loading across the IGP, with an increase in coarse mode concentration and coarse mode mass fraction with altitude.
Predictions of the glass transition temperature and viscosity of organic aerosols from volatility distributions
Ying Li, Douglas A. Day, Harald Stark, Jose L. Jimenez, and Manabu Shiraiwa
Viscosity is an important property of organic aerosols, but viscosity measurements of ambient organic aerosols are scarce. We developed a method to predict glass transition temperatures using volatility and the atomic oxygen-to-carbon ratio. The method was applied to field observations of volatility distributions to predict viscosity of ambient organic aerosols, yielding consistent results with ambient particle phase-state measurements and global simulations.
Marine productivity and synoptic meteorology drive summer-time variability in Southern Ocean aerosols
Joel Alroe, Luke T. Cravigan, Branka Miljevic, Graham R. Johnson, Paul Selleck, Ruhi S. Humphries, Melita D. Keywood, Scott D. Chambers, Alastair G. Williams, and Zoran D. Ristovski
We present findings from an austral summer voyage across the full latitudinal width of the Southern Ocean, south of Australia. Aerosol properties were strongly influenced by marine biological activity, synoptic-scale weather systems, and long-range transport of continental-influenced air masses. The meteorological history of the sampled air masses is shown to have a vital limiting influence on cloud condensation nuclei and the accuracy of modelled sea spray aerosol concentrations.
The vertical variability of black carbon observed in the atmospheric boundary layer during DACCIWA
Barbara Altstädter, Konrad Deetz, Bernhard Vogel, Karmen Babić, Cheikh Dione, Federica Pacifico, Corinne Jambert, Friederike Ebus, Konrad Bärfuss, Falk Pätzold, Astrid Lampert, Bianca Adler, Norbert Kalthoff, and Fabienne Lohou
We present the high vertical variability of the black carbon (BC) mass concentration measured with the unmanned aerial system ALADINA during the field experiment of DACCIWA. The COSMO-ART model output was applied for the campaign period and is compared with the observational BC data during a case study on 14–15 July 2016. Enhanced BC concentrations were related to transport processes to the measurement site by maritime inflow and not to local emissions as initially expected.
Effects of atmospheric circulations on the interannual variation in PM2.5 concentrations over the Beijing–Tianjin–Hebei region in 2013–2018
Xiaoyan Wang and Renhe Zhang
This study investigated the effects of meteorological elements on the interannual variation in air quality. It shows consistent variations of PM2.5 concentrations and the occurrence of unfavorable circulation types. The favorable meteorological conditions in 2017 contribute to 59.9 % of the interannual decrease in PM2.5 concentrations, which highlights the importance of atmospheric conditions when evaluating the interannual variations in local air quality.
On the relationship between cloud water composition and cloud droplet number concentration
Alexander B. MacDonald, Ali Hossein Mardi, Hossein Dadashazar, Mojtaba Azadi Aghdam, Ewan Crosbie, Haflidi H. Jonsson, Richard C. Flagan, John H. Seinfeld, and Armin Sorooshian
Understanding how humans affect Earth's climate requires understanding of how particles in the air affect the number concentration of droplets in a cloud (Nd). We use the air-equivalent mass concentration of different chemical species contained in cloud water to predict Nd. In this study we found that the prediction of Nd is (1) best described by total sulfate; (2) improved when considering up to five species; and (3) dependent on factors like turbulence, smoke presence, and in-cloud height.
Identifying a regional aerosol baseline in the eastern North Atlantic using collocated measurements and a mathematical algorithm to mask high-submicron-number-concentration aerosol events
Francesca Gallo, Janek Uin, Stephen Springston, Jian Wang, Guangjie Zheng, Chongai Kuang, Robert Wood, Eduardo B. Azevedo, Allison McComiskey, Fan Mei, Adam Theisen, Jenni Kyrouac, and Allison C. Aiken
Continuous high-time-resolution ambient data can include periods when aerosol properties do not represent regional aerosol processes due to high-concentration local events. We develop a novel aerosol mask at the U.S. Department of Energy's Atmospheric Radiation Measurement (ARM) facility in the eastern North Atlantic (ENA). We use two ground sites to validate the mask, include a comparison with aircraft overflights, and provide guidance to increase data quality at ENA and other locations.
Cloud condensation nuclei characteristics during the Indian summer monsoon over a rain-shadow region
Venugopalan Nair Jayachandran, Mercy Varghese, Palani Murugavel, Kiran S. Todekar, Shivdas P. Bankar, Neelam Malap, Gurnule Dinesh, Pramod D. Safai, Jaya Rao, Mahen Konwar, Shivsai Dixit, and Thara V. Prabha
Continuous aerosol and cloud condensation nuclei (CCN) measurements carried out at the ground observational facility of a CAIPEEX campaign, situated in the rain-shadow region of the Indian subcontinent, are illustrated. The variations in CCN characteristics within the monsoon period are investigated along with the aerosol physical and optical properties. The change in the dependency of CCN activity on aerosol size and composition due to the variations in air mass and meteorology is brought out.
Heavy air pollution with a unique "non-stagnant" atmospheric boundary layer in the Yangtze River middle basin aggravated by regional transport of PM2.5 over China
Chao Yu, Tianliang Zhao, Yongqing Bai, Lei Zhang, Shaofei Kong, Xingna Yu, Jinhai He, Chunguang Cui, Jie Yang, Yinchang You, Guoxu Ma, Ming Wu, and Jiacheng Chang
This study investigated the ambient PM2.5 variations over Wuhan, a typical urban Yangtze River middle basin (YRMB) region in central eastern China in January 2016. Through an analysis of observational data of the environment and meteorology, as well as via a FLEXPART-WRF simulation, it heavy air pollution is revealed with the unique "non-stagnant" atmospheric boundary layer in the YRMB region aggravated by regional transport of PM2.5 over central and eastern China.
Decreasing trends of particle number and black carbon mass concentrations at 16 observational sites in Germany from 2009 to 2018
Jia Sun, Wolfram Birmili, Markus Hermann, Thomas Tuch, Kay Weinhold, Maik Merkel, Fabian Rasch, Thomas Müller, Alexander Schladitz, Susanne Bastian, Gunter Löschau, Josef Cyrys, Jianwei Gu, Harald Flentje, Björn Briel, Christoph Asbach, Heinz Kaminski, Ludwig Ries, Ralf Sohmer, Holger Gerwig, Klaus Wirtz, Frank Meinhardt, Andreas Schwerin, Olaf Bath, Nan Ma, and Alfred Wiedensohler
To evaluate the effectiveness of emission mitigation policies, we evaluated the trends of the size-resolved particle number concentrations and equivalent black carbon mass concentration at 16 observational sites for various environments in Germany (2009–2018). Overall, significant decrease trends are found for most of the parameters and sites. This study suggests that a combination of emission mitigation policies can effectively improve the air quality on large spatial scales such as in Germany.
Condensation/immersion mode ice-nucleating particles in a boreal environment
Mikhail Paramonov, Saskia Drossaart van Dusseldorp, Ellen Gute, Jonathan P. D. Abbatt, Paavo Heikkilä, Jorma Keskinen, Xuemeng Chen, Krista Luoma, Liine Heikkinen, Liqing Hao, Tuukka Petäjä, and Zamin A. Kanji
Ice-nucleating particle (INP) measurements were performed in the boreal environment of southern Finland in the winter–spring of 2018. It was found that no single parameter could be used to predict the INP number concentration at the measurement location during the examined time period. It was also not possible to identify physical and chemical properties of ambient INPs despite the complexity of the instrumental set-up. Therefore, this paper addresses the necessity for future INP measurements.
Rapid reduction in black carbon emissions from China: evidence from 2009–2019 observations on Fukue Island, Japan
Yugo Kanaya, Kazuyo Yamaji, Takuma Miyakawa, Fumikazu Taketani, Chunmao Zhu, Yongjoo Choi, Yuichi Komazaki, Kohei Ikeda, Yutaka Kondo, and Zbigniew Klimont
Fundamental disagreements among bottom-up emission inventories exist about the sign of the black carbon emissions trend from China over the past decade. Our decadal observations on Fukue Island clearly indicate its rapid reduction, after correcting for interannual meteorological variability, which supports inventories reflecting governmental clean air actions after 2010. The reduction pace surpasses those of SSP1 scenarios for 2015–2030, suggesting highly successful emission control policies.
Variability in lidar-derived particle properties over West Africa due to changes in absorption: towards an understanding
Igor Veselovskii, Qiaoyun Hu, Philippe Goloub, Thierry Podvin, Mikhail Korenskiy, Yevgeny Derimian, Michel Legrand, and Patricia Castellanos
Atmospheric dust has a significant impact on the Earth's climate system, and this impact remains highly uncertain. The desert dust is always a mixture of various minerals, and the imaginary part of the complex refractive index often exhibits an increase in UV for dust containing iron oxides. Our results demonstrate that multiwavelength Raman lidar measurements allow for the characterization of the spectral dependence of the imaginary part of dust.
Hygroscopic properties and cloud condensation nuclei activity of atmospheric aerosols under the influences of Asian continental outflow and new particle formation at a coastal site in eastern Asia
Hing Cho Cheung, Charles Chung-Kuang Chou, Celine Siu Lan Lee, Wei-Chen Kuo, and Shuenn-Chin Chang
Air pollution can result in changes to the amount and properties of aerosols, which in turn could influence climate by acting as cloud condensation nuclei (CCN). This study investigated the composition and hygroscopicity of aerosols (PM2.5) at a station under influences of eastern Asian (EA) outflows and local air mass during respective seasons. The results show that the EA aerosols were more hygroscopic, whereas new particle formation and coagulation could also influence local CCN activity.
Mixing characteristics of refractory black carbon aerosols at an urban site in Beijing
Hang Liu, Xiaole Pan, Dantong Liu, Xiaoyong Liu, Xueshun Chen, Yu Tian, Yele Sun, Pingqing Fu, and Zifa Wang
The bare black carbon (BC) was in a fractal structure. With coating thickness increasing, BC changed from a fractal structure to a core–shell structure. In the ambient atmosphere, plenty of BC particles were not in a perfect core–shell structure. This study brought attention to the combined effects of morphology and coating thickness on the absorption enhancement of BC-containing particles, which is helpful for determining the climatic effects of BC.
Shipborne observations reveal contrasting Arctic marine, Arctic terrestrial and Pacific marine aerosol properties
Jiyeon Park, Manuel Dall'Osto, Kihong Park, Yeontae Gim, Hyo Jin Kang, Eunho Jang, Ki-Tae Park, Minsu Park, Seong Soo Yum, Jinyoung Jung, Bang Yong Lee, and Young Jun Yoon
The physical properties of aerosol particles throughout the Arctic Ocean and Pacific Ocean were measured aboard the Korean icebreaker R/V Araon during the summer of 2017. A number of new particle formation (NPF) events and growth were frequently observed in both Arctic terrestrial and Arctic marine air masses. This suggests that terrestrial ecosystems – including river outflows and tundra – strongly affect aerosol emissions in the Arctic coastal areas, affecting radiative forcing.
New particle formation and sub-10 nm size distribution measurements during the A-LIFE field experiment in Paphos, Cyprus
Sophia Brilke, Nikolaus Fölker, Thomas Müller, Konrad Kandler, Xianda Gong, Jeff Peischl, Bernadett Weinzierl, and Paul M. Winkler
Atmospheric particle size distributions with the focus on freshly nucleated particles were measured during the A-LIFE field experiment in Cyprus. A DMA-train was set up for the first time in an atmospheric environment and captures the sub-10 nm particle dynamics. Several new particle formation (NPF) events are studied in detail, of which some did not show particle growth beyond 10 nm indicating that NPF may occur more frequently than estimated when the sub-10 nm size range is not covered.
Large-scale ion generation for precipitation of atmospheric aerosols
Shaoxiang Ma, He Cheng, Jiacheng Li, Maoyuan Xu, Dawei Liu, and Kostya Ostrikov
Atmos. Chem. Phys. Discuss., https://doi.org/10.5194/acp-2020-23,https://doi.org/10.5194/acp-2020-23, 2020
Our work suggests that the large corona discharge system is an efficient and possibly economically sustainable way to increase the ion density in the open air and control precipitation of atmospheric aerosols. Once the system is installed on the mountain top, it will generate lots of charged nuclei, which may trigger water precipitation or fog elimination within a certain region in the downwind directions.
Overview of aerosol optical properties over southern West Africa from DACCIWA aircraft measurements
Cyrielle Denjean, Thierry Bourrianne, Frederic Burnet, Marc Mallet, Nicolas Maury, Aurélie Colomb, Pamela Dominutti, Joel Brito, Régis Dupuy, Karine Sellegri, Alfons Schwarzenboeck, Cyrille Flamant, and Peter Knippertz
This paper presents aircraft measurements of aerosol optical properties over southern West Africa. We show that aerosol optical properties in the boundary layer were dominated by a persistent biomass burning loading from the Southern Hemisphere. Biomass burning aerosols were more light absorbing that those previously measured in other areas (Amazonia, North America). Our study suggests that lens-coated black carbon particles were the dominant absorber for these biomass burning aerosols.
Haze pollution under a high atmospheric oxidization capacity in summer in Beijing: insights into formation mechanism of atmospheric physicochemical processes
Dandan Zhao, Guangjing Liu, Jinyuan Xin, Jiannong Quan, Yuesi Wang, Xin Wang, Lindong Dai, Wenkang Gao, Guiqian Tang, Bo Hu, Yongxiang Ma, Xiaoyan Wu, Lili Wang, Zirui Liu, and Fangkun Wu
Under strong atmospheric oxidization capacity, haze pollution in the summer in Beijing was the result of the synergistic effect of the physicochemical process in the atmospheric boundary layer (ABL). With the premise of an extremely stable ABL structure, the formation of secondary aerosols dominated by nitrate was quite intense, driving the outbreak of haze pollution.
On the annual variability of Antarctic aerosol size distributions at Halley Research Station
Thomas Lachlan-Cope, David C. S. Beddows, Neil Brough, Anna E. Jones, Roy M. Harrison, Angelo Lupi, Young Jun Yoon, Aki Virkkula, and Manuel Dall'Osto
We present a statistical cluster analysis of the physical characteristics of particle size distributions collected at Halley (Antarctica) for the year 2015. Complex interactions between multiple ecosystems, coupled with different atmospheric circulation, result in very different aerosol size distributions populating the Southern Hemisphere.
Absorption closure in highly aged biomass burning smoke
Jonathan W. Taylor, Huihui Wu, Kate Szpek, Keith Bower, Ian Crawford, Michael J. Flynn, Paul I. Williams, James Dorsey, Justin M. Langridge, Michael I. Cotterell, Cathryn Fox, Nicholas W. Davies, Jim M. Haywood, and Hugh Coe
Every year, huge plumes of smoke hundreds of miles wide travel over the south Atlantic Ocean from fires in central and southern Africa. These plumes absorb the sun's energy and warm the climate. We measured the properties of the smoke using instruments on board an aircraft, to work out how absorbing the smoke is, and which components of the smoke warm the air the most. We also tested different ways of simulating these properties that could be used in a climate model.
Seasonal contrast in size distributions and mixing state of black carbon and its association with PM1.0 chemical composition from the eastern coast of India
Sobhan Kumar Kompalli, Surendran Nair Suresh Babu, Sreedharan Krishnakumari Satheesh, Krishnaswamy Krishna Moorthy, Trupti Das, Ramasamy Boopathy, Dantong Liu, Eoghan Darbyshire, James D. Allan, James Brooks, Michael J. Flynn, and Hugh Coe
Vertical profiles of submicron aerosol single scattering albedo over the Indian region immediately before monsoon onset and during its development: research from the SWAAMI field campaign
Mohanan R. Manoj, Sreedharan K. Satheesh, Krishnaswamy K. Moorthy, and Hugh Coe
The study reports the observation of highly absorbing aerosol layers at high altitudes (1–2.5 km) prior to monsoon and during its development over the Indian region and quantifies its climate impacts. The absorption of solar radiation in these layers perturbs the onset of monsoon through the impact on the atmospheric stability. When height-resolved values of single scattering albedo (SSA) are used in a radiative transfer model, a maximum heating ~1 K d (~twice that using SSA) is obtained.
Vertical characteristics of aerosol hygroscopicity and impacts on optical properties over the North China Plain during winter
Quan Liu, Dantong Liu, Qian Gao, Ping Tian, Fei Wang, Delong Zhao, Kai Bi, Yangzhou Wu, Shuo Ding, Kang Hu, Jiale Zhang, Deping Ding, and Chunsheng Zhao
We present a series of aircraft-based in situ measurements of aerosol chemical components and size distributions over the North China Plain, and the hygroscopicity is derived from aerosol chemical composition. These results reveal the vertical characteristics of aerosol hygroscopicity, and we investigated their impacts on optical properties and activation under different moisture and pollution conditions over this polluted region.
The significant impact of aerosol vertical structure on lower atmosphere stability and its critical role in aerosol–planetary boundary layer (PBL) interactions
Tianning Su, Zhanqing Li, Chengcai Li, Jing Li, Wenchao Han, Chuanyang Shen, Wangshu Tan, Jing Wei, and Jianping Guo
We study the role of aerosol vertical distribution in thermodynamic stability and PBL development. Under different aerosol vertical structures, the diurnal cycles of PBLH and PM2.5 show distinct characteristics. Large differences in the heating rate affect atmospheric buoyancy and stability differently under different aerosol structures. As a result, the aerosol–PBL interaction can be strengthened by the inverse aerosol structure and potentially neutralized by the decreasing structure.
Characterising mass-resolved mixing state of black carbon in Beijing using a morphology-independent measurement method
Chenjie Yu, Dantong Liu, Kurtis Broda, Rutambhara Joshi, Jason Olfert, Yele Sun, Pingqing Fu, Hugh Coe, and James D. Allan
This study presents the first atmospheric application of a new morphology-independent measurement for the quantification of the mixing state of rBC-containing particles in urban Beijing as part of the UK–China APHH campaign. An inversion method has been applied for better quantification of rBC mixing state. The mass-resolved rBC mixing state information presented here has implications for detailed models of BC, its optical properties and its atmospheric life cycle.
Contrasting impacts of two types of El Niño events on winter haze days in China's Jing-Jin-Ji region
Xiaochao Yu, Zhili Wang, Hua Zhang, Jianjun He, and Ying Li
There are statistically significant positive and negative correlations, respectively, between winter haze days (WHD) in China's Jing-Jin-Ji region and Eastern Pacific and Central Pacific El Niño events. These opposite changes in WHD are attributable to the anomalies of both large-scale circulation and local synoptic condition corresponding to two types of El Niño. Our study highlights the importance of distinguishing the impacts of two types of El Niño on winter haze pollution in this region.
Cloud condensation nuclei properties of South Asian outflow over the northern Indian Ocean during winter
Vijayakumar S. Nair, Venugopalan Nair Jayachandran, Sobhan Kumar Kompalli, Mukunda M. Gogoi, and S. Suresh Babu
Extensive measurements of the aerosol and cloud condensation nuclei (CCN) properties in South Asian outflow to the northern Indian Ocean were carried out as a part of the ICARB-2018 experiment during winter. At high supersaturations, most of the aerosols in the South Asian outflow become activated as CCN, whereas the aerosol system over the equatorial Indian Ocean is less CCN efficient even at higher supersaturations.
Apparent dust size discrepancy in aerosol reanalysis in north African dust after long-range transport
Samantha J. Kramer, Claudia Alvarez, Anne Barkley, Peter R. Colarco, Lillian Custals, Rodrigo Delgadillo, Cassandra Gaston, Ravi Govindaraju, and Paquita Zuidema
Atmos. Chem. Phys. Discuss., https://doi.org/10.5194/acp-2020-1,https://doi.org/10.5194/acp-2020-1, 2020
Comparisons of sea salt and size-resolved dust mass concentration measurements over southeast Florida to those from the MERRA2/GEOS-5 FP aerosol reanalysis indicate the reanalysis depicts excessive sea salt and puts too much dust in larger intermediate sizes, than do the measurements. The dust vertical profile is approximately correct. The incorrect reanalysis aerosol speciation and dust sizes has implications for the modeling of their transport, deposition, and radiative impact.
In situ vertical characteristics of optical properties and heating rates of aerosol over Beijing
Ping Tian, Dantong Liu, Delong Zhao, Chenjie Yu, Quan Liu, Mengyu Huang, Zhaoze Deng, Liang Ran, Yunfei Wu, Shuo Ding, Kang Hu, Gang Zhao, Chunsheng Zhao, and Deping Ding
This study paints a full picture of the evolution of vertical characteristics of aerosol optical properties and shortwave heating impacts of carbonaceous aerosols during different stages of pollution events over the Beijing region and highlights the increased contribution of brown carbon absorption, especially at higher levels, during pollution.
Amplification of black carbon light absorption induced by atmospheric aging: temporal variation at seasonal and diel scales in urban Guangzhou
Jia Yin Sun, Cheng Wu, Dui Wu, Chunlei Cheng, Mei Li, Lei Li, Tao Deng, Jian Zhen Yu, Yong Jie Li, Qianni Zhou, Yue Liang, Tianlin Sun, Lang Song, Peng Cheng, Wenda Yang, Chenglei Pei, Yanning Chen, Yanxiang Cen, Huiqing Nian, and Zhen Zhou
Atmospheric aging processes (AAPs) can lead to black carbon (BC) light absorption enhancement (Eabs), which remained poorly characterized at a long timescale. By applying a newly developed approach, the minimum R squared method (MRS), this study investigated the temporal variations of BC Eabs at both seasonal and diel scales in an urban environment. Factors affecting the temporal variability of BC Eabs were also analyzed, including variability in emission sources and various types of AAPs.
Ultra-clean and smoky marine boundary layers frequently occur in the same season over the southeast Atlantic
Sam Pennypacker, Michael Diamond, and Robert Wood
Using observations from instruments deployed to a small island in the southeast Atlantic, we study days when the atmospheric concentrations of particles near the surface are exceptionally low. Interestingly, these ultra-clean boundary layers occur in the same months as the smokiest boundary layers associated with biomass burning in Africa. We find evidence that enhancements in drizzle scavenging, on top of a seasonal maximum in cloudiness and precipitation, likely drive these conditions.
Aerosol immission maps and trends over Germany with hourly data at four rural background stations from 2009 to 2018
Jost Heintzenberg, Wolfram Birmili, Bryan Hellack, Maik Merkel, Gerald Spindler, Thomas Tuch, Kay Weinhold, and Alfred Wiedensohler
A transition of atmospheric emissions of particles and gases from on-road heavy-duty trucks
Liyuan Zhou, Åsa M. Hallquist, Mattias Hallquist, Christian M. Salvador, Samuel M. Gaita, Åke Sjödin, Martin Jerksjö, Håkan Salberg, Ingvar Wängberg, Johan Mellqvist, Qianyun Liu, Berto P. Lee, and Chak K. Chan
The study reports the transition in the atmospheric emission of particles and gases from on-road heavy-duty trucks (HDTs) caused by the modernisation of the fleet. We measured particle number (PN), particle mass (PM), black carbon (BC), nitrogen oxides (NOx), carbon monoxide (CO), hydrocarbon (HC), particle size distributions, and volatility in the plumes of 556 individual HDTs. Significant but different changes in emissions were evident for various pollutants with respect to emission standards.
Above-cloud aerosol optical depth from airborne observations in the southeast Atlantic
Samuel E. LeBlanc, Jens Redemann, Connor Flynn, Kristina Pistone, Meloë Kacenelenbogen, Michal Segal-Rosenheimer, Yohei Shinozuka, Stephen Dunagan, Robert P. Dahlgren, Kerry Meyer, James Podolske, Steven G. Howell, Steffen Freitag, Jennifer Small-Griswold, Brent Holben, Michael Diamond, Robert Wood, Paola Formenti, Stuart Piketh, Gillian Maggs-Kölling, Monja Gerber, and Andreas Namwoonde
The southeast Atlantic during August–October experiences layers of smoke from biomass burning over marine stratocumulus clouds. Here we present the light attenuation of the smoke and its dependence in the spatial, vertical, and spectral domain through direct measurements from an airborne platform during September 2016. From our observations of this climatically important smoke, we found an average aerosol optical depth of 0.32 at 500 nm, slightly lower than comparative satellite measurements.
Characterization of aerosol particles at Cabo Verde close to sea level and at the cloud level – Part 1: Particle number size distribution, cloud condensation nuclei and their origins
Xianda Gong, Heike Wex, Jens Voigtländer, Khanneh Wadinga Fomba, Kay Weinhold, Manuela van Pinxteren, Silvia Henning, Thomas Müller, Hartmut Herrmann, and Frank Stratmann
We characterized the aerosol particles in Cabo Verde at sea and cloud levels. We found four well-separable types of PNSDs, with the strongest differences between air masses coming from the ocean compared to from the African continent. During the strongest observed dust periods, CCN concentrations were 2.5 higher than during clean marine periods. The hygroscopicity of the particles did not vary much between different periods. Aerosol at sea level and on the mountaintop was well in agreement.
Variation of size-segregated particle number concentrations in wintertime Beijing
Ying Zhou, Lubna Dada, Yiliang Liu, Yueyun Fu, Juha Kangasluoma, Tommy Chan, Chao Yan, Biwu Chu, Kaspar R. Daellenbach, Federico Bianchi, Tom V. Kokkonen, Yongchun Liu, Joni Kujansuu, Veli-Matti Kerminen, Tuukka Petäjä, Lin Wang, Jingkun Jiang, and Markku Kulmala
In this study, we focus on explaining the concentration variations in the observed particle modes, by relating them to the potential aerosol sources and sinks, and on understanding the connections between these modes. Interestingly, even in the atmospheric cocktail in urban Beijing, secondary new particle formation (NPF) drives the particle number concentration, especially in the sub-3 nm range. We found that the total number concentration is ~ 4 times higher on NPF days than on haze days.
Significant contribution of organics to aerosol liquid water content in winter in Beijing, China
Xiaoai Jin, Yuying Wang, Zhanqing Li, Fang Zhang, Weiqi Xu, Yele Sun, Xinxin Fan, Guangyu Chen, Hao Wu, Jingye Ren, Qiuyan Wang, and Maureen Cribb
In this study the aerosol liquid water content (ALWC) is determined from aerosol hygroscopic growth factor (GF) measurement (ALWCHTDMA) and also simulated by the ISORROPIA II thermodynamic model (ALWCISO). We found that ALWC contributed by organics (ALWCOrg) accounts for 30 % ± 22 % of the total ALWC in winter in Beijing. A case study reveals that ALWCOrg plays an important role in the formation of secondary aerosols through multiphase reactions at the initial stage of a heavy-haze episode.
Distinct diurnal variation in organic aerosol hygroscopicity and its relationship with oxygenated organic aerosol
Ye Kuang, Yao He, Wanyun Xu, Pusheng Zhao, Yafang Cheng, Gang Zhao, Jiangchuan Tao, Nan Ma, Hang Su, Yanyan Zhang, Jiayin Sun, Peng Cheng, Wenda Yang, Shaobin Zhang, Cheng Wu, Yele Sun, and Chunsheng Zhao
A new method was developed to calculate hygroscopicity parameter κ of organic aerosols (κOA) based on aerosol light-scattering measurements and bulk aerosol chemical-composition measurements. Derived high-time-resolution κOA varied in a wide range (near 0 to 0.25), and the organic aerosol oxidation degree significantly impacts variations in κOA. Distinct diurnal variation in κOA is found, and its relationship with oxygenated organic aerosol is discussed.
Evidence for impacts on surface-level air quality in the northeastern US from long-distance transport of smoke from North American fires during the Long Island Sound Tropospheric Ozone Study (LISTOS) 2018
Haley M. Rogers, Jenna C. Ditto, and Drew R. Gentner
This study combines surface-level air quality measurements with satellite imagery and back-trajectory modeling to investigate the long-distance transport of these emissions to the New York City metropolitan area and the northeastern US. Two events in August 2018 were traced to biomass burning on the western coast of Canada and from the southeastern US, highlighting the importance of understanding long-distance transport of fire emissions in air quality planning.
Altitude profiles of cloud condensation nuclei characteristics across the Indo-Gangetic Plain prior to the onset of the Indian summer monsoon
Venugopalan Nair Jayachandran, Surendran Nair Suresh Babu, Aditya Vaishya, Mukunda M. Gogoi, Vijayakumar S. Nair, Sreedharan Krishnakumari Satheesh, and Krishnaswamy Krishna Moorthy
Concurrent measurements of the altitude profiles of the concentration of cloud condensation nuclei (CCNs), as a function of supersaturation (ranging from 0.2 % to 1.0 %), and aerosol optical properties were carried out aboard an instrumented aircraft across the Indo-Gangetic Plain (IGP) just prior to the onset of the 2016 Indian summer monsoon (ISM). A high CCN concentration is observed up to 2.5 km across the IGP, indicating the significant possibility of aerosol indirect effects.
Daytime aerosol optical depth above low-level clouds is similar to that in adjacent clear skies at the same heights: airborne observation above the southeast Atlantic
Yohei Shinozuka, Meloë S. Kacenelenbogen, Sharon P. Burton, Steven G. Howell, Paquita Zuidema, Richard A. Ferrare, Samuel E. LeBlanc, Kristina Pistone, Stephen Broccardo, Jens Redemann, K. Sebastian Schmidt, Sabrina P. Cochrane, Marta Fenn, Steffen Freitag, Amie Dobracki, Michal Segal-Rosenheimer, and Connor J. Flynn
Rapid formation of intense haze episodes via aerosol–boundary layer feedback in Beijing
Yonghong Wang, Miao Yu, Yuesi Wang, Guiqian Tang, Tao Song, Putian Zhou, Zirui Liu, Bo Hu, Dongsheng Ji, Lili Wang, Xiaowan Zhu, Chao Yan, Mikael Ehn, Wenkang Gao, Yuepeng Pan, Jinyuan Xin, Yang Sun, Veli-Matti Kerminen, Markku Kulmala, and Tuukka Petäjä
Atmos. Chem. Phys., 20, 45–53, https://doi.org/10.5194/acp-20-45-2020,https://doi.org/10.5194/acp-20-45-2020, 2020
We found a positive particle matter-mixing layer height feedback at three observation platforms at the 325 m Beijing meteorology tower, which is characterized by a shallower mixing layer height and a higher particle matter concentration. Measurements of solar radiation, aerosol chemical composition, meteorology parameters, trace gases and turbulent kinetic energy (TKE) could explain the feedback mechanism to some extent.
Traffic-originated nanocluster emission exceeds H2SO4-driven photochemical new particle formation in an urban area
Miska Olin, Heino Kuuluvainen, Minna Aurela, Joni Kalliokoski, Niina Kuittinen, Mia Isotalo, Hilkka J. Timonen, Jarkko V. Niemi, Topi Rönkkö, and Miikka Dal Maso
Atmos. Chem. Phys., 20, 1–13, https://doi.org/10.5194/acp-20-1-2020,https://doi.org/10.5194/acp-20-1-2020, 2020
Photochemically formed sulfuric acid is generally considered the main source for new particle formation in the atmosphere. Contrary to current understanding, our measurements of nanoclusters and gaseous sulfuric acid performed in an urban area imply that traffic contributes to sulfuric acid concentration and that even for the smallest particles, the traffic-emitted fraction mostly exceeds the photochemistry-driven regional new particle formation.
Multi-method determination of the below-cloud wet scavenging coefficients of aerosols in Beijing, China
Danhui Xu, Baozhu Ge, Xueshun Chen, Yele Sun, Nianliang Cheng, Mei Li, Xiaole Pan, Zhiqiang Ma, Yuepeng Pan, and Zifa Wang
Wet deposition is one of the most important and efficient removal mechanisms in the evolution of the air pollution. Due to the lack of a localized parameterization scheme and some mechanisms being neglected in theoretical estimations and modeling calculations, below-cloud wet scavenging coefficients (BWSC) have large uncertainties. We compare the BWSCs under the same conditions to perform a multi-method evaluation in order to describe their characteristics.
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June 2016, 36(6): 3077-3106. doi: 10.3934/dcds.2016.36.3077
Global existence and optimal decay rates of solutions for compressible Hall-MHD equations
Jincheng Gao 1, and Zheng-An Yao 1,
School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China, China
Received April 2015 Revised October 2015 Published December 2015
In this paper, we are concerned with global existence and optimal decay rates of solutions for the compressible Hall-MHD equations in dimension three. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for higher order spatial derivatives of classical solutions in $H^3$-framework, which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].
Keywords: global solutions, optimal decay rates, Compressible Hall-MHD equations, Fourier splitting method..
Mathematics Subject Classification: Primary: 76W05; Secondary: 35A01, 35D35, 74H4.
Citation: Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077
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What is the difference between dynamic and static electronic correlation
I am confused. In the static correlation, we use combination of Slater determinants to account for electronic correlation (known as CI or configuration interaction) and the wave function is represented as a sum of various SD where the HF SD is the base function and higher excited SD contributes some parts. Similarly, in dynamic electronic correlation, we use basis set expansion to account for higher excitation orbitals. What is the fundamental difference between these two correlations? Both are essentially expanding the electronic space to allow more relaxation. What am I missing here? Thanks
theoretical-chemistry
Osman MamunOsman Mamun
$\begingroup$ Related: chemistry.stackexchange.com/questions/50804/…. $\endgroup$
– Wildcat
$\newcommand{\el}{_\mathrm{e}}$In quantum chemistry, when a nomenclature in which one distinguishes between "static" and "dynamic" correlation is used, "correlation" referrers to all the deficiencies of the Hartree-Fock (HF) single-determinantal approach. For instance, the the correlation energy is defined as the difference between the exact (non-relativistic) energy and the HF energy (calculated with a complete basis), $$ E_{\mathrm{corr}} = E_{\mathrm{exact}} - E_{\mathrm{HF}} \, . $$
Now, what are the deficiencies of a Hartree-Fock approach?
First, electrons in this model do not instantaneously interact with each other, as they do in reality, but rather each and every electron interacts with the average, or mean, field created by all other electrons. Classically speaking, each electron moves in a way so that it avoids locations in a close proximity to the instantaneous positions of all other electrons. And the failure of the HF model to correctly reproduce such motion of electrons is the first source of $E_{\mathrm{corr}}$. This type of correlation is called dynamic correlation since it is directly related to electron dynamics.
Secondly, the wave function in the HF model is a single Slater determinant, which might be a rather poor representation of a many-electron system's state: in certain cases an electronic state can be well described only by a linear combination of more than one (nearly-)degenerate Slater determinants. This is the second reason why $E_{\mathrm{HF}}$ may differ from $E_{\mathrm{exact}}$ and the corresponding type of correlation is called static, or nondynamic, since it ist not related to electron dynamics.
Interestingly, both static and dynamic correlation effects can be taken into account by "mixing in" more Slater determinants $\Phi_i$ to the Hartree-Fock one $\Phi_0$, $$ \Psi\el(\vec{r}\el) = c_0 \Phi_0 + \sum_i c_i \Phi_i \, , $$ Here, if $c_0$ is assumed to be close to $1$ and a large number of excited determinants $\Phi_i$ are added each of which is assumed to give only a small contribution, then the method primarily treats dynamic correlation. And if, on the other hand, it is assumed that there are just a few excited determinants $\Phi_i$ with weights close to that of the reference determinant $\Phi_0$, then the method primarily treats static correlation.
An example of a method that recovers primarily dynamic correlation is Møller–Plesset perturbation theory (MP$n$), while multi-configurational self-consistent field (MCSCF) method primarily takes account of static correlation. Note the word "primarily" here and above. It is almost impossible in principle to keep dynamic and static correlation effects separated since they both arise from the very same physical interaction. Thus, methods that typically cover dynamical correlation effects include at high-order also some of the non-dynamical correlation effects and vice versa.
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A longitudinal genome-wide association study of anti-tumor necrosis factor response among Japanese patients with rheumatoid arthritis
Kyoko Honne1,
Ingileif Hallgrímsdóttir2,
Chunsen Wu3,4,
Ronnie Sebro5,
Nicholas P. Jewell6,
Takeo Sakurai7,
Masahiro Iwamoto1,
Seiji Minota1 &
Damini Jawaheer8
Studies of Caucasian patients with rheumatoid arthritis (RA) to identify genetic biomarkers of anti-tumor necrosis factor (TNF) response have used response at a single time point as the phenotype with which single nucleotide polymorphism (SNP) associations have been tested. The findings have been inconsistent across studies. Among Japanese patients, only a few SNPs have been investigated. We report here the first genome-wide association study (GWAS) to identify genetic biomarkers of anti-TNF response among Japanese RA patients, using response at 2 time-points for a more reliable clinical phenotype over time.
Disease Activity Scores based on 28 joint counts (DAS28) were assessed at baseline (before initial therapy), and after 3 and 6 months in 487 Japanese RA patients starting anti-TNF therapy for the first time or switching to a new anti-TNF agent. A genome-wide panel of SNPs was genotyped and additional SNPs were imputed. Using change in DAS28 scores from baseline at both 3 (ΔDAS-3) and 6 months (ΔDAS-6) as the response phenotype, a longitudinal genome-wide association analysis was conducted using generalized estimating equations (GEE) models, adjusting for baseline DAS28, treatment duration, type of anti-TNF agent and concomitant methotrexate. Cross-sectional analyses were performed using multivariate linear regression models, with response from a single time point (ΔDAS-3 or ΔDAS-6) as phenotype; all other variables were the same as in the GEE models.
In the GEE models, borderline significant association was observed at 3 chromosomal regions (6q15: rs284515, p = 6.6x10−7; 6q27: rs75908454, p = 6.3x10−7 and 10q25.3: rs1679568, p = 8.1x10−7), extending to numerous SNPs in linkage disequilibrium (LD) across each region. Potential candidate genes in these regions include MAP3K7, BACH2 (6q15), GFRA1 (10q25.3), and WDR27 (6q27). The association at GFRA1 replicates a previous finding from a Caucasian dataset. In the cross-sectional analyses, ΔDAS-6 was significantly associated with the 6q15 locus (rs284511, p = 2.5x10−8). No other significant or borderline significant associations were identified.
Three genomic regions demonstrated significant or borderline significant associations with anti-TNF response in our dataset of Japanese RA patients, including a locus previously associated among Caucasians. Using repeated measures of response as phenotype enhanced the power to detect these associations.
Anti-tumor necrosis factor (TNF) agents have been particularly effective in controlling disease activity and joint erosions in rheumatoid arthritis (RA) [1–3]. Nonetheless, a considerable proportion of patients (30–40 %) demonstrate only partial response or no response to these therapies. This complex phenotype of anti-TNF responsiveness has a genetic component, as demonstrated by heritability estimates of 0.59 or more for different clinical measures of response [4]. Several studies conducted mainly in Caucasian populations [5–23] have attempted to identify genetic biomarkers that can reliably predict response to anti-TNF agents in patients with RA. However, there have been no reports of genome-wide association studies (GWAS) for biomarkers of anti-TNF response among Japanese or other East Asian RA patients. Only single nucleotide polymorphisms (SNPs) in the TRAF1 [24] and CD84 genes [18] have been examined for association in two small samples of 101 and 151 Japanese patients, respectively. It has been demonstrated for several RA risk loci that there are both similarities and ethnic differences in disease associations between Caucasian and Japanese populations [25–29]. Similarly, although efficacy of anti-TNF therapies appears to be similar in these populations [30, 31], there may be similarities and differences in genetic predictors of anti-TNF response.
Previous studies investigating genetic predictors of anti-TNF response focused on a limited number of candidate genes [5–15, 20–22, 32], and few GWAS have been performed [9, 16–19, 23]. The findings from these different studies have been largely inconsistent. Only a couple of loci (PTPRC and PDE3A-SLCO1C1) have been associated in more than one study [5, 11, 33], but these associations were not consistently replicated in other studies [8, 14, 20]. Findings at other loci have not been independently replicated [12, 22]. All of these studies used a single assessment of anti-TNF response.
In the present study, we have performed a GWAS of anti-TNF response in a population of Japanese RA patients. In contrast to previous studies, we have used repeated measures of response at 3 and 6 months after initiation of anti-TNF therapy in order to have a more reliable assessment of clinical phenotype over time, and have used longitudinal statistical models to accommodate the repeated measures of the outcome.
A total of 487 RA patients were recruited from the Jichi Medical University and from Inoue Hospital within the Gunma prefecture in Japan, and prospectively followed for 6 months. All included patients: (1) were of Japanese descent; (2) satisfied the 1987 American College of Rheumatology (ACR) criteria for RA [34]; (3) were not in remission (Disease activity score in 28 joints (DAS28) <2.6) at baseline; and (4) were starting treatment with the anti-TNF agents etanercept (ETN), infliximab (INF) or adalimumab (ADA) for the first time or were switching to a new anti-TNF drug due to inefficacy or adverse events.
Data and sample collection
Study baseline
Detailed clinical and demographic data, including age, sex, disease duration, smoking history, and concurrent treatment with methotrexate and prednisolone, were collected on each patient prior to initiation of anti-TNF therapy (study baseline). A serum sample was collected, and used to measure baseline titers of anti-citrullinated protein antibodies (ACPA), rheumatoid factor (RF) using the RA particle agglutination test (RAPA), and C-reactive protein (CRP) levels. Patients were considered to be seropositive for ACPA and/or RF auto-antibodies if ACPA and RAPA titers were ≥4.5 U/ml and ≥40 units, respectively. Whole blood samples (8.5 ml) were collected from each patient and stored at −80 °C for subsequent DNA extraction.
All patients were evaluated at 3 and 6 months after initiation of anti-TNF therapy. Tender joint counts (TJC28) and swollen joint counts (SJC28) for 28 joints, patient global scores and CRP levels were assessed at each of these time points. The study protocol was approved by the ethics committee of the Jichi Medical University and by the Institutional Review Board of the Children's Hospital Oakland Research Institute. Signed informed consent was obtained from each study participant.
Assessment of response to anti-TNF therapy
RA disease activity was assessed at baseline, 3 months and 6 months by the DAS28 calculated on three variables, including 28-joint counts and CRP, i.e., DAS28CRP3 [35, 36], as follows:
$$ \mathrm{D}\mathrm{A}\mathrm{S}28\mathrm{C}\mathrm{R}\mathrm{P}3=\left(0.56\ast \sqrt{TJC28}\right)+\left(0.28\ast \sqrt{SJC28}\right)+\left(0.36\ast ln\left(CRP+1\right)\ast 1.10\right)+1.15 $$
We will refer to the DAS28CRP3 as DAS28. The change in DAS28 scores from baseline at 3 months and 6 months were calculated as ΔDAS-3 and ΔDAS-6, respectively.
Genotyping and data cleaning
DNA was extracted from frozen whole blood samples using a standard protocol and a Gentra Autopure system (Gentra Systems, Minneapolis, MN, USA). All samples were genotyped for a total of 1,133,484 SNPs on an Illumina BeadLab1000 platform using the HumanOmni1-Quad BeadChip and the Infinium HD assay (Illumina, San Diego, CA, USA). Genotypes were called using a score threshold of 0.15 in the Illumina BeadStudio software. Data cleaning was performed using the PLINK software [37] (http://pngu.mgh.harvard.edu/~purcell/plink/) SNPs with genotyping rates ≤98 %, minor allele frequencies (MAF) ≤1 %, or not in Hardy Weinberg equilibrium (HWE) (p ≤0.0001) were excluded from further analyses due to possible genotyping error. To test for population stratification, principal component analysis (PCA) was performed using the EIGENSTRAT software [38, 39].
Imputation of genotypes
The cleaned genotypes were phased using the ShapeIT (v2) software [40] and imputation of genotypes was performed using the Impute2 software [41]. All available multi-population haplotypes from the 1000 Genomes haplotypes Phase I integrated variant set (June 2014 release) were used as reference panels both for phasing and imputation, as recommended [42]. The probability distribution of three possible genotypes generated by Impute2 at each imputed SNP was converted to genotypes using the GTOOL software (v 0.7.5) (http://www.well.ox.ac.uk/~cfreeman/software/gwas/gtool.html) and a stringent probability threshold of 0.9 was applied. Imputed SNPs with genotyping rates ≤98 % or MAF <5 % were excluded from subsequent analyses.
Variables influencing response to anti-TNF therapy
To identify potential confounder variables that influence patient response to anti-TNF therapy over time, a longitudinal analysis was performed using GEE models to accommodate response at two time-points for each patient and to adjust for within-patient correlation [43]. Repeated measures of the change in DAS28 at 3 and 6 months (i.e., ΔDAS-3 and ΔDAS-6) were used as the outcome variable in the models, and the explanatory variables included baseline DAS28, duration of anti-TNF therapy, age at baseline, RA duration, sex, concurrent methotrexate use (yes/no), concurrent prednisolone use (yes/no), type of anti-TNF agent (ETN, INF or ADA), RAPA (yes/no), smoking status (never/ever) and ACPA seropositivity (yes/no). Each variable was tested for association with repeated measures of the change in DAS28 in univariate and multivariate models.
Longitudinal genome-wide association analyses
Longitudinal GEE models were used to investigate associations between each SNP on a genome-wide scale and patient response to anti-TNF therapy over time (response at two time points for each patient). For each SNP, repeated measures of the change in DAS28 at 3 and 6 months (i.e., ΔDAS-3 and ΔDAS-6) were used as the outcome variable in the model, with the SNP being the explanatory variable. Covariates included in the model were baseline DAS28, concurrent methotrexate use (yes/no), type of anti-TNF agent (ETN, INF or ADA) and duration of anti-TNF therapy. These covariates were selected on the basis of their association with change in DAS28 from the GEE models described earlier. Principal components were not adjusted for in the main model because the genomic control inflation factor (λGC) was estimated at 1.001. These analyses were repeated after excluding patients who had mild disease activity at baseline.
All analyses were performed using the STATA package (version 13). P values below a non-stringent threshold of 1×10−6 were taken as evidence of borderline significant association.
Cross-sectional genome-wide association analyses at each time point
Cross-sectional genome-wide association (GWA) analyses were performed using response data from a single time point, at 3 months or at 6 months, as the phenotype in two separate multivariate linear regression models. These models (model 1 and model 2) differed from the GEE models only in terms of the response phenotype: ΔDAS-3 was the outcome variable in model 1, and ΔDAS-6 in model 2. All other explanatory variables and covariates were the same as those in the GEE models. The GWA analyses for models 1 and 2 were performed using the PLINK software.
Identifying regions of association
Independent chromosomal regions of association were identified using the LD clumping option within PLINK, based on p values and patterns of LD in the data.
A total of 444 patients had genotype data and data on ΔDAS-3, ΔDAS-6, and covariates available for analysis. Among these, ΔDAS-3 was missing for 3 patients and ΔDAS-6 was missing for 22 patients. The clinical, demographic and treatment characteristics of the patients at baseline are summarized in Table 1. The majority of patients were female (84 %), with RA duration of 8.1±8.5 years, had moderate to severe disease activity at baseline (93 %), were naïve to anti-TNF drugs (94 %), were concurrently being treated with methotrexate (80 %) and were positive for ACPA antibodies (89 %). Overall, response to anti-TNF therapy was significantly better at 6 months than at 3 months (ΔDAS-3 (mean ± SD): 1.50±1.01, ΔDAS-6: 1.72±1.12; p = 0.002).
Table 1 Patient characteristics at baseline
In univariate and multivariate GEE models, only baseline DAS28 (p <0.0005), concurrent methotrexate use (p <0.0005) and duration of anti-TNF therapy (p <0.0005) were associated with change in DAS28 (Table 2).
Table 2 Results from multivariate GEE model to identify predictors of response
Data quality and population stratification
A total of 4,253,138 autosomal SNPs were available for statistical analyses after imputation and after quality control thresholds were applied. Of these, 738,576 had been genotyped, and the rest had been imputed with high accuracy (imputation accuracy score >0.96).
SNP associations
GEE models using repeated measures of change in DAS28 as the phenotype
Three independent genomic regions showed borderline significant (p <1×10−6) association with repeated measures of change in DAS28 (i.e., both ΔDAS-3 and ΔDAS-6) (Fig. 1); the results for the index SNPs within each region are shown in Table 3. Two of these regions mapped to chromosome 6: one at 6q15 (rs284515: p = 6.6×10−7) approximately 15 Kb downstream from the Mitogen-activated protein kinase kinase kinase 7 (MAP3K7) gene and 572 Kb upstream from the Basic leucine zipper transcription factor 2 (BACH2) gene, and the other at 6q27 (rs75908454: p = 6.3×10−7) within a locus containing the WD repeat-containing protein 27 (WDR27) gene. The third locus (rs1679568: p = 8.1×10−7) mapped to the 3' untranslated region (UTR) of the Glial cell line-derived neurotrophic factor family receptor alpha 1 (GFRA1) gene at 10q25.3. As shown in Fig. 1, at each locus, numerous SNPs within the region of LD containing the index SNP also showed evidence of association with change in DAS28. Similar results were obtained when the analysis was restricted to patients with moderate or severe disease activity at baseline (n = 413) (see Additional file 1). In addition, the results were again similar when the analysis was restricted only to the subset of patients who were anti-TNF naïve at baseline (n = 419). In both cases though, the significance levels were lower than in the full dataset of 444 patients. Chromosomal regions showing moderate evidence of association (p <1×10−5) are shown in Additional file 2, and the index SNPs within those regions are listed in Additional file 3.
Regions demonstrating significant or borderline significant association with anti-TNF response. The regional plots show three regions with borderline significant association (p <1×10−6) at 6q15 (top row), 6q27 (middle row) and 10q25.3 (bottom row), based on results from all patients (n = 444) and generalized estimating equations (GEE) models using anti-TNF response at two time points, i.e., 3 and 6 months (left panels). Association results at the same genomic regions are shown for models using anti-TNF response at a single time point only: 6 months (middle panels) and 3 months (right panels). Note: at 6 months, another single nucleotide polymorphism (SNP) in linkage disequilibrium with rs284515 the 6q15 locus (i.e., rs284511) was significantly associated with response at 6 months (change in Disease Activity Score at 6 months, ΔDAS-6); this SNP was not associated at 3 months, and was borderline significant in the GEE model. Chr chromosome
Table 3 Index SNPs from independent chromosomal regions showing borderline significant association with repeated measures of response at 3 and 6 months
The CD84, PTPRC, PDE3A-SLCO1C1 and MED15 loci reported to be significantly associated with anti-TNF response in RA in previous GWAS showed no evidence of association in our dataset.
Cross-sectional analyses at 3 months and at 6 months
When response at only the 3-month or 6-month follow up (ΔDAS-3 or ΔDAS-6) was used as the phenotype instead of ΔDAS at both time points, SNP rs284511 mapping close to and in LD with rs284515 near the MAP3K7 locus was significantly associated with ΔDAS-6 at the genome-wide significance level (p = 2.5×10−8). No other SNPs showed significant or borderline significant associations with either ΔDAS-6 or ΔDAS-3.
Of interest, as shown in Fig. 1 and Table 4, compared to the cross-sectional analyses which used response phenotype from a single time point, the GEE models provided increased power to detect associations at loci that demonstrated modest association at both time points, i.e., rs284515, rs75908454 and rs1679568, but not rs284511.
Table 4 SNPs showing significant or borderline significant association in the cross-sectional analyses or in the GEE model
This is the first GWAS investigating genetic biomarkers of response to anti-TNF therapy in Japanese patients with RA, utilizing a longitudinal approach to examine associations between genome-wide SNPs and repeated measures of anti-TNF response at 3 and 6 months. We found borderline significant association (p <1×10−6) at three non-correlated regions within our study population, with the associated SNPs mapping to or close to the following genes: MAP3K7, BACH2 (6q15), WDR27 (6q27) and GFRA1 (10q25.3). Each of these regions harbored numerous SNPs demonstrating evidence of association with change in DAS28. Furthermore, the 6q15 locus was significantly associated with response at 6 months (p = 2.5×10−8), and the association at the GFRA1 locus represents a replication of a previously reported association among Caucasian patients. We therefore considered these regions worthy of being reported so that they may be investigated further in larger datasets.
The MAP3K7 gene encodes transforming growth factor beta-activated kinase 1 (TAK1) which is a key regulator in multiple inflammatory signaling pathways [44, 45], including the p38 MAPK and nuclear factor kappa B signaling pathways. TAK1 deficiency leads to reduced pro-inflammatory cytokine production in cultured RA synoviocytes [46]. It is thus an excellent candidate that may influence the effect of anti-TNF agents, as proposed [47], and is already a candidate therapeutic target to block pro-inflammatory pathways in RA [48, 49]. Transcription factor BACH2, on the other hand, appears to be a key negative regulator of effector T cell differentiation, promoting immune homeostasis [50]. In the mouse, it appears to be a super-enhancer repressing a network of genes critical for T cell function [51]. Of interest, variants at the BACH2 locus have been associated with multiple autoimmune diseases, including RA [52–57]. The GFRA1 protein is a member of the Glial cell line-derived neurotrophic factor (GDNF) receptor family and mediates activation of the RET tyrosine kinase receptor. GDNF is produced by astrocytes in response to pro-inflammatory cytokines including TNFα [58] and appears to suppress interleukin-17 (IL-17)-mediated inflammation via the NF-kappa B pathway [59]. The function of the WDR27 protein has not been established.
The association with the GFRA1 gene was previously identified by Plant et al. [16]. SNP rs7070180 mapping to an intron of the GFRA1 gene, was associated with anti-TNF response in a cohort of 566 Caucasian RA patients in the UK (p = 2.24×10−4) and in a meta-analysis including a cohort of 379 additional patients (p = 6.42×10−5). However, this locus was not reported among the major findings of that study as SNP rs7070180 failed to genotype in one of the cohorts. While this SNP was not genotyped or imputed in our data, several other SNPs within the 3' untranslated region (UTR) of the GFRA1 gene were associated with anti-TNF response among our patients. There are no reports of MAP3K7 being associated with anti-TNF response in RA, although it has been proposed as a good candidate for pathway pharmacogenetics relating to TNF inhibitors [47]. Among the genes of the p38 MAPK network that have been investigated [7, 9, 13], suggestive evidence of an association with MAP2K6 was reported [9], though not replicated [13]. Other associations for p38 MAPK candidate genes were reported in a sample of 1,102 patients using a generously non-stringent significance threshold of p <0.1 [7].
A major strength of the present study is the use of repeated measures of anti-TNF response at 3 months and 6 months after treatment was started. Previous studies included clinical response from a single time point in standard linear or logistic regression models [5–9, 11, 12, 16–19]. However, assessment of response at a single point in time may not adequately reflect a patient's response to therapy, as response may fluctuate over time. Hence, using response data from at least two time points is more reliable and clinically relevant. The longitudinal approach enables the use of repeated measures of response from different time-points for each patient, thus increasing the power to detect an association as we have demonstrated, while taking into account within-patient correlation. Patients with missing data at one time point were still included in the analyses, as the GEE uses all available data. However, while the association with the MAP3K7 locus achieved genome-wide significance for SNP rs284511 when using anti-TNF response at only 6 months, the lack of association with this SNP at 3 months led to the GEE model only detecting a borderline significant association.
We did not identify any other overlap between our results and previous findings, possibly due to differences in ethnicity, response variable, i.e., two time points vs a single time point, or duration of anti-TNF treatment (3–12 months in previous studies). Further, the lack of consistent findings between previous studies may also have been the result of differences in a number of factors including study design, clinical outcomes examined (DAS28ESR vs DAS28CRP), specific anti-TNF medications used, concomitant disease modifying anti-rheumatic drugs (DMARDs), sex ratios, and small sample sizes in some cases [9, 12, 19]. Another important difference between studies is the heterogeneity in phenotypes introduced by differences in time from baseline to assessment of response ranging from 3 to 12 months in different studies. As seen in our data, treatment duration is significantly associated with response, and should be given due consideration when using multiple datasets for combined analyses to minimize phenotypic heterogeneity or when comparing results between studies. Further, the low significance thresholds used to identify previously reported associations may have led to false positive associations being included. We also cannot exclude the possibility that our findings may include false positives until they can be replicated in independent datasets or that our sample size was not adequately powered to detect some of the previously reported findings.
The present study has a number of limitations. First, the sample size of 444 patients is modest compared to previous GWAS, which combined data from different populations of European ancestry to achieve large sample sizes [16, 18]. Nonetheless, it represents the largest reported sample size for studies of anti-TNF response among Japanese or any East Asian RA population [18, 24]. Given the lack of pharmacogenomics studies examining associations with anti-TNF response among Japanese RA patients, these results represent an important contribution to the field. Although power may be limited due to the modest sample size, this may in part be compensated for by the ethnic homogeneity of the patient population and the use of response data from two time points as described earlier. Second, the response phenotype in our study included response to three different anti-TNF agents, which may have introduced some bias in responder status because a patient who was a non-responder to one drug, might have responded well to another drug. For example, of the 18 patients who had switched to a new anti-TNF drug at baseline due to inefficacy of the previous drug, 11 had a good response to the second line of anti-TNF agent. However, all patients starting a new anti-TNF agent at study baseline - i.e., those who switched and those who were anti-TNF naïve - were followed for 6 months to assess response as is routinely done in clinical practice. Hence, none of them were switched to a new agent for the duration of the study. To mitigate misclassification bias in the response phenotype that might have arisen from inclusion of three different anti-TNF agents in the analysis, we adjusted for the type of anti-TNF agent used. Third, as there is no gold standard measure to evaluate treatment response in RA, we used the DAS28CRP3 as a surrogate to assess disease activity although it may not be a perfect measure of response. The possibility that such a complex phenotype may be associated only with modest genetic effects has been raised [4, 16]. A comparison of the component variables, i.e., tender and swollen joint counts and CRP levels, showed similar trajectories from baseline to 6 months to the composite DAS28 score. This suggests that no major differences in associations would be expected by focusing on components of the DAS28 as the outcome in our dataset. In order to more closely capture variations in patient response over time, we chose to use repeated measures of the change in DAS28 from baseline. We did not categorize the outcome into European League Against Rheumatism (EULAR) responses as this would have led to a reduction in power. Last, the association between a genetic predictor and clinical response could be confounded by factors that influence response. As far as possible, we adjusted for likely confounders associated with change in DAS28 in our dataset. We did not, however, adjust for variations in drug dosage. We had previously reported that response to anti-TNF therapy is influenced by sex in the long term [60], but that these sex differences were not observed during the first 6 months of treatment. Thus, the lack of an association with sex in the present dataset, which was followed for only 6 months, is in agreement with our previous findings.
In summary, we have identified three chromosomal regions demonstrating significant or borderline significant association with response to anti-TNF therapy among Japanese RA patients. The replication in our Japanese dataset of a previous association with the GFRA1 locus among Caucasians provides evidence for a trans-ethnic association of this locus with anti-TNF response. We have also demonstrated the importance of including response at more than one time point in order to enhance power to detect associations in such pharmacogenomics studies of RA.
ΔDAS-3 and ΔDAS-6:
change in Disease Activity Scores based on 28 joint counts (from baseline) at 3 months and 6 months
anti-citrullinated protein antibodies
ACR:
American College of Rheumatology
BACH2:
basic leucine zipper transcription factor 2
base pairs
DAS28:
Disease Activity Scores based on 28 joint counts
DMARD:
disease modifying anti-rheumatic drug
ETN:
GDNF:
glial cell line-derived neurotrophic factor
GEE:
generalized estimating equations
GFRA1:
glial cell line-derived neurotrophic factor family receptor alpha 1
GWAS:
genome-wide association study
HWE:
Hardy Weinberg equilibrium
IL-17:
INF:
LD:
linkage disequilibrium
MAF:
minor allele frequency
MAP3K7:
PCA:
principal component analysis
RAPA:
rheumatoid arthritis particle agglutination
SJC28:
swollen joint counts for 28 joints
SNP:
TAK1:
transforming growth factor beta-activated kinase 1
TJC28:
tender joint counts for 28 joints
untranslated region
WDR27:
WD repeat-containing protein 27
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Jawaheer D, Olsen J, Hetland ML. Sex differences in response to anti-tumor necrosis factor therapy in early and established rheumatoid arthritis – results from the DANBIO registry. J Rheumatol. 2012;39(1):46–53.
We thank the patients for their participation, which made this study possible. This work was supported in part by funds from Chugai Pharmaceutical, Mitsubishi-Tanabe Pharma, Eisai, and Takeda Pharmaceutical (Japan). These funders did not have any role in conducting this study or in interpretation and reporting of results. Dr. Honne was supported by the Jichi Medical University Young Investigator Award.
Division of Rheumatology and Clinical Immunology, Jichi Medical University, Shimotsuke, Tochigi, Japan
Kyoko Honne, Masahiro Iwamoto & Seiji Minota
23andme, Mountain View, CA, USA
Ingileif Hallgrímsdóttir
Research Unit of Gynecology and Obstetrics, Institute of Clinical Research, University of Southern Denmark, Odense, Denmark
Chunsen Wu
Department of Obstertics & Gynecology, Odense University Hospital, Odense, Denmark
Department of Radiology, University of Pennsylvania School of Medicine, Philadelphia, PA, 19104, USA
Ronnie Sebro
School of Public Health, University of California, Berkeley, CA, 94720, USA
Nicholas P. Jewell
Inoue Hospital, Takasaki, Gunma, Japan
Takeo Sakurai
Center for Genetics, Children's Hospital Oakland Research Institute, Oakland, CA, 94609, USA
Damini Jawaheer
Kyoko Honne
Masahiro Iwamoto
Seiji Minota
Correspondence to Damini Jawaheer.
MI is a member of the speakers' bureau for Astellas Pharma and Takeda; SM has been awarded research grants from Astellas, Pfizer Inc, Essai, AbbVie, Mitsubishi-Tanabe, Takeda and Chugai. KH, IH, CW, RS, NPJ, TS and DJ have no competing interests.
KH participated in the conception and design of the study, collected samples and clinical data from patients, and contributed to drafting the manuscript. IH participated in the analysis and interpretation of data and critically revised the manuscript. CW participated in the analysis and interpretation of data and critically revised the manuscript. RS participated in the analysis and interpretation of data and critically revised the manuscript. NPJ participated in the analysis and interpretation of data and critically revised the manuscript. TS collected samples and clinical data from patients. MI collected samples and clinical data from patients and critically revised the manuscript. SM conceived the study, participated in its design, and in data collection, and critically revised the manuscript. DJ was involved in the study conception and design, in the analysis and interpretation of data, and in drafting the manuscript. All authors have given final approval of the version to be published, and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.
Seiji Minota and Damini Jawaheer Joint senior authors.
Takeo Sakurai deceased.
Regional plots showing association results from GEE models at 6q15, 6q27 and 10q25.3, when the analyses were restricted to patients with moderate or severe disease activity at baseline (n = 413). (PDF 350 kb)
Regions showing moderate evidence of association (p < 1x10−5) with anti-TNF response (GEE models). (PDF 324 kb)
SNPs showing moderate evidence of association (p < 1x10−5) with anti-TNF response (GEE models). (PDF 183 kb)
Honne, K., Hallgrímsdóttir, I., Wu, C. et al. A longitudinal genome-wide association study of anti-tumor necrosis factor response among Japanese patients with rheumatoid arthritis. Arthritis Res Ther 18, 12 (2016). https://doi.org/10.1186/s13075-016-0920-6
Anti-TNF
BACH2
GFRA1 | CommonCrawl |
(14)/(15)+(13)/(20) - adding of fractions
(14)/(15)+(13)/(20) - step by step solution for the given fractions. Adding of fractions, full explanation.
If it's not what You are looking for just enter simple or very complicated fractions into the fields and get free step by step solution. Remember to put brackets in correct places to get proper solution.
+ - * /
fill out with example data
Solve the problem
Solution for the given fractions
$ \frac{14}{15 }+\frac{ 13}{20 }=? $
The common denominator of the two fractions is: 60
$ \frac{14}{15 }= \frac{(4*14)}{(4*15)} =\frac{ 56}{60} $
Fractions adjusted to a common denominator
$ \frac{14}{15 }+\frac{ 13}{20 }=\frac{ 56}{60 }+\frac{ 39}{60} $
$ \frac{56}{60 }+\frac{ 39}{60 }= \frac{(39+56)}{60} $
$ \frac{(39+56)}{60 }=\frac{ 95}{60} $
$ \frac{95}{60 }=\frac{ 19}{12} $
see mathematical notation
| (6)/(11)+(1)/(22) - adding of fractions | | (14)/(9)+(7)/(6) - addition of fractions | | (1)/(x)*(1)/(x+1) - multiply fractions | | (13)/(35)+(5)/(21) - add fractions | | (7)/(10)+(8)/(25) - add fractions | | (-3)/(4)/((2x-3))/(20) - divide fractions | | (19)/(14)+(6)/(21) - adding of fractions | | (5)/(8)+(1)/(26) - addition of fractions | | (13)/(20)+(1)/(6) - addition of fractions | | (25)/(24)-(19)/(16) - subtract fractions | | (4)/(5)+(7)/(20) - addition of fractions | | (7)/(3)*(3)/(14) - multiplication of fractions | | (4)/(5)+(16)/(20) - add fractions | | (4)/(5)+(16)/(20) - adding of fractions | | (-15)/(2)*(1)/(100) - multiplication of fractions | | (13)/(18)-(5)/(9) - subtraction of fractions | | (10)/(8)-(5)/(4) - subtract fractions | | (11)/(15)-(18)/(25) - subtract fractions | | (7)/(15)-(3)/(25) - subtract fractions | | (1)/(9)-(1)/(36) - subtraction of fractions | | (2)/(3)-(4)/(6) - subtract fractions | | (23)/(12)-(5)/(6) - subtraction of fractions | | (2)/(7)-(5)/(8) - subtraction of fractions | | (2)/(7)+(5)/(8) - addition of fractions | | (44)/(9)+(16)/(3) - addition of fractions | | (67)/(10)+(393)/(100) - add fractions | | (6)/(10)+(93)/(100) - addition of fractions | | (61)/(10)+(393)/(100) - addition of fractions | | (5(2m-7)+8m)/(2)+(17)/(2) - addition of fractions | | (5(2m-7)+8m)/(5)+(17)/(2) - adding of fractions | | (3)/(4)+(1)/(24) - add fractions | | (7y)/(8)/(-9)/(5y) - divide fractions | | (-11a)/(4)+(11)/(8a) - adding of fractions | | (7)/(2)/(8)/(7) - dividing of fractions | | (-8)/(-5)/(7)/(5) - divide fractions | | (11)/(12)/(11)/(4) - dividing of fractions | | (a)/(7)-(5)/(7) - subtract fractions | | (13)/(56)+(5)/(7) - adding of fractions | | (5)/(16)-(1)/(20) - subtraction of fractions | | (1)/(5)+(3)/(20) - adding of fractions | | (5-2)/(3)+(2-2)/(4) - adding of fractions | | (x-2)/(3)+(x-2)/(4) - adding of fractions | | (7)/(1/4)*(1)/(18) - multiplication of fractions | | (13)/(56)+(5)/(7) - add fractions | | (2)/(15)-(1)/(20) - subtraction of fractions | | (35)/(36)*(6)/(7) - multiplying of fractions | | (150)/(45)+(180)/(45) - adding of fractions | | (150)/(45)*(180)/(45) - multiplication of fractions | | (2)/(5)+(1)/(9) - adding of fractions | | (10)/(81)+(25)/(72) - adding of fractions | | (2a)/(15)-(1)/(3) - subtraction of fractions | | (7)/(9)-(14)/(45) - subtraction of fractions | | (12)/(15)/(6)/(5) - dividing of fractions | | (2)/(7)/(10)/(3) - dividing of fractions | | (9)/(1)*(1)/(5) - multiplying of fractions | | (12)/(7)*(9)/(8) - multiply fractions | | (1)/(4)*(2)/(1) - multiplication of fractions | | (4)/(1)*(1)/(16) - multiplication of fractions | | (1)/(16)*(1)/(16) - multiplication of fractions | | (1)/(3)+(7a)/(8) - adding of fractions | | (16)/(1)*(25)/(36) - multiplication of fractions | | (12)/(1)*(4)/(3) - multiplication of fractions | | (12)/(1)+(4)/(3) - addition of fractions | | (12)/(1)*(25)/(36) - multiplication of fractions | | (15(2-x))/(1)+(13(3-x))/(1) - adding of fractions | | CommonCrawl |
How to solve this puzzle?
There are $N$ consecutive doors. Two players 'B' and 'J' plays a game. Both take turns alternately, and in each turn a player can open any one door. They define a block of 3 consecutive open doors as "hole". The first one to create a "hole" wins. Player 'J' plays first.
Given 'N', how could we determine the winner assuming both plays optimally?
Example: Let us denote the open door by 'o' and closed door by '-'.
If N = 3: Initially all cells are closed (---), if 'J' opens first door configuration is (o--). 'B' can either open second cell leading to configuration (oo-) or third cell (o-o). In both cases 'J' wins.
If N = 5:
'J' wins by opening the 3rd cell i.e. configuration (--o--).
recreational-mathematics puzzle game-theory
superAnnoyingUser
$\begingroup$ Your question seems weird...and it looks like you wrote this question. First, you mentioned that a block of 3 consecutive open doors is a 'hole'. but in your N=5 example, there are no 3 consecutive open doors ie "ooo" so there is no hole but 'J' wins. Second, you use the words "cells" and "doors"...can they be assumed to refer to the same thing? Third, are the doors only lined up in 1 direction (left - right) etc? Fourth, what happens if enough doors are opened (say $N-1$) and then no doors are left? Does any player win? Finally, what do you mean by a player playing optimally? $\endgroup$ – bryanblackbee Feb 17 '13 at 11:30
$\begingroup$ J wins there because whatever door B opens in the next step, J will make a "hole" $\endgroup$ – ciceksiz kakarot Feb 17 '13 at 11:32
$\begingroup$ @bryansis2010 J wins, whatever B opens. $\endgroup$ – superAnnoyingUser Feb 17 '13 at 11:35
$\begingroup$ It seems that there are n choose 3 = n!/((n-3)!*3!) possible combinations of consecutive open doors. My intuition tells me that there's some kind of finite subcovering optimization that can be used here but it's too late/early for me to say anything more concrete than that :-) Cool puzzle! $\endgroup$ – user39898 Feb 17 '13 at 11:36
In the course of a the game, the players will want to avoid forming configurations o-o or oo, because that means the other player will win in the next move. Thus they will have to open doors in order to split consecutive sequences of closed doors into two smaller consecutive sequences of closed doors.
If a player creates a sequence of length $0$ or $1$ bounded by opened doors (o-o or oo) by opening a door, then the next player wins, so until the last move, the game will only consist of sequences of length $\ge 2$ (except maybe at the boundaries)
Add three imaginary doors on each side of the original sequence, and open the two extreme doors. Then you have an equivalent game, since the players will never try to open the $4$ closed imaginary doors (this would create a sequence of length $<2$ allowing the next player to win). Therefore we only need to think about games with sequences of closed doors bounded by two open doors.
Since players play into only one sequence at a time, we can consider that each sequence is its own game, and we only need to compute their respective Grundy numbers. Letting $\otimes$ be the bitwise xor operations on natural numbers on $\oplus$ be the superposition of games, we have $G(x \oplus y) = G(x) \otimes G(y)$, and $G(x) = \min \Bbb N - \{G(y); x \to y\}$ (where $x \to y$ means there is a move to play on $x$ to get $y$)
If we note $n$ the game with a sequence of $n$ closed doors bounded by two open doors, and looking at the allowed moves, we get :
$G(n) = \min \Bbb N - \{G(i)\otimes G(j); 2 \le i,j ; i+j+1=n\}$
This allows you to compute the Grundy number of $(n)$ in $O(n^2)$ steps. For example, $G(2) = G(3) = G(4) = 0$ because you can't make any move in there (you can't play in there without creating a $(0)$ or $(1)$ sequence, which means the opponent wins globally on the next turn).
If $G(N+4) = 0$ then the first player loses, and if $G(N+4)>0$, then there is a winning strategy for the first player.
The first few $N$ where the first player loses are $6,12,22,30,32,44,54,64,76,86,98,\ldots$
Andreas Caranti
merciomercio
$\begingroup$ +1 Enlightening post on a subject about which I do not know much, thanks! I have taken the liberty to add a reference for the Grundy numbers. $\endgroup$ – Andreas Caranti Feb 17 '13 at 15:27
$\begingroup$ I should have a look at Winning Ways to see whether this game appears in that book. $\endgroup$ – Gerry Myerson Feb 17 '13 at 22:30
$\begingroup$ It is! Editing my answer.... $\endgroup$ – Gerry Myerson Feb 18 '13 at 6:12
EDITED in response to comment by Gugg:
If $N$ is odd, it's a 1st player win. Let $N=2n+1$ and label the doors $-n,\dots,n$. 1st player opens door zero and for the rest of the game plays a game-ending move if possible, otherwise replies to 2nd player opening door $k$ by opening door $-k$. Opening door $-k$ can never permit a game-ending move by 2nd player since if it did then by symmetry 2nd player allowed a game-ending move by opening door $k$, and 1st player would have made it.
MORE EDIT: Berlekamp, Conway, and Guy, Winning Ways, Volume 1, page 93: "Treblecross is a Tic-Tac-Toe game played on a $1\times n$ strip in which both players use the same symbol (X). The first person to complete a line of three consecutive crosses wins." They find the "nim-value" for $n\le12$, but so far as I can see don't work out any general results.
EVEN MORE EDIT: There are explorations of Treblecross available on the web.
http://lbv-pc.blogspot.com.au/2012/07/treblecross.html discusses it.
http://www.calstatela.edu/faculty/sheubac/presentations/EllieSJhandout.pdf is a discussion by Heubach, Chinn, Dufour, and Stevens, from May 2009. They say there is no complete analysis; that Grundy values have been computed up to $n=2^{21}=2097152$, finding 37 "P" positions: $0, 1, 2, 8, 14, 24, 32, 34, 46, 56, 66, 78, \dots, 16170$
http://www.mathstat.dal.ca/~rjn/papers/UnsolvedCGT.pdf is Guy and Nowakowski, Unsolved problems in combinatorial games, Feb 2008. Treblecross comes in bottom of page 3, top of page 4. Grundy number computation pushed to $2^{25}$. They call this game "Perhaps the most notorious and deserving of attention" of the "finite octal games".
Flammenkamp keeps an up-to-date website on these games, http://wwwhomes.uni-bielefeld.de/achim/octal.html He has pushed the Grundy number computation to $2^{28}$.
Gerry MyersonGerry Myerson
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The gear ratio of a gear train, also known as its speed ratio, is the ratio of the angular velocity of the input gear to the angular velocity of the output gear.[1] The gear ratio can be calculated directly from the numbers of teeth on the gears in the gear train. The torque ratio of the gear train, also known as its mechanical advantage, is determined by the gear ratio.[2] The speed ratio and mechanical advantage are defined so they yield the same number in an ideal linkage.
File:NonSynchronousGearBoxSF.jpg
Gear trains with two gears Edit
In a gear train, the input gear, or drive gear, transmits power to the output gear, also known as the driven gear. The input gear, which is usually connected to a power source, such as a motor or engine, transmits power through any other gears that may be in the gear train to the output gear.
Two meshing gears transmit rotational motion.
Formula Edit
The simplest gear train is a pair of meshing gears in which the input gear drives the output gear. The gear teeth are designed so that the pitch circles of the two gears roll on each other without slipping. The velocity, represented by v, of the points of contact of the two pitch circles are the same. If the input gear GA has the radius rA and angular velocity $ \omega_A \! $, and meshes with output gear GB of radius rB and angular velocity $ \omega_B \! $, then:
$ v = r_A \omega_A = r_B \omega_B, \! $
The number of teeth on a gear is proportional to the radius of its pitch circle, which means that the ratios of the gears' angular velocities, radii, and number of teeth are equal. Where NA is the number of teeth on the input gear and NB is the number of teeth on the output gear, the following equation is formed:
$ \frac{\omega_A}{\omega_B} = \frac{r_B}{r_A} = \frac{N_B}{N_A}. $
This shows that a simple gear train with two gears has the gear ratio R given by
$ R = \frac{\omega_A}{\omega_B} = \frac{N_B}{N_A}. $
This equation shows that if the number of teeth on the output gear GB is larger than the number of teeth on the input gear GA, then the input gear GA must rotate faster than the output gear GB.
Speed ratio Edit
Gear teeth are distributed along the circumference of the pitch circle so that the thickness t of each tooth and the space between neighboring teeth are the same. The pitch p of a gear, which is the distance between equivalent points on neighboring teeth along the pitch circle, is equal to twice the thickness of a tooth,
$ p=2t.\! $
The pitch of a gear GA can be computed from the number of teeth NA and the radius rA of its pitch circle
$ p = \frac{2\pi r_A}{N_A}. $
In order to mesh smoothly two gears GA and GB must have the same sized teeth and therefore they must have the same pitch p, which means
$ p = \frac{2\pi r_A}{N_A} = \frac{2\pi r_B}{N_B}. $
This equation shows that the ratio of the circumference, the diameters and the radii of two meshing gears is equal to the ratio of their number of teeth,
$ \frac{r_B}{r_A} = \frac{N_B}{N_A}. $
The speed ratio of two gears rolling without slipping on their pitch circles is given by,
$ R =\frac{\omega_A}{\omega_B}=\frac{r_B}{r_A}, $
In other words, the gear ratio, or speed ratio, is inversely proportional to the radius of the pitch circle and the number of teeth of the input gear.
Torque ratio Edit
A gear train can be analyzed using the principle of virtual work to show that its torque ratio, which is the ratio of its output torque to its input torque, is equal to the gear ratio, or speed ratio, of the gear train.
This means that the input torque TA applied to the input gear GA and the output torque TB on the output gear GB are related by the ratio
$ R = \frac{T_B}{T_A}, $
where R is the gear ratio of the gear train.
The torque ratio of a gear train is also known as its mechanical advantage, thus
$ MA = \frac{T_B}{T_A} = \frac{N_B}{N_A}. $
Idler gears Edit
In a sequence of gears chained together, the ratio depends only on the number of teeth on the first and last gear. The intermediate gears, regardless of their size, do not alter the overall gear ratio of the chain. However, the addition of each intermediate gear reverses the direction of rotation of the final gear.
An intermediate gear which does not drive a shaft to perform any work is called an idler gear. Sometimes, a single idler gear is used to reverse the direction, in which case it may be referred to as a reverse idler. For instance, the typical automobile manual transmission engages reverse gear by means of inserting a reverse idler between two gears.
Idler gears can also transmit rotation among distant shafts in situations where it would be impractical to simply make the distant gears larger to bring them together. Not only do larger gears occupy more space, the mass and rotational inertia (moment of inertia) of a gear is proportional to the square of its radius. Instead of idler gears, a toothed belt or chain can be used to transmit torque over distance.
Formulae Edit
If a simple gear train has three gears, such that the input gear GA meshes with an intermediate gear GI which in turn meshes with the output gear GB, then the pitch circle of the intermediate gear rolls without slipping on both the pitch circles of the input and output gears. This yields the two relations
$ \frac{\omega_A}{\omega_I} = \frac{N_I}{N_A}, \quad \frac{\omega_I}{\omega_B} = \frac{N_B}{N_I}. $
The speed ratio of this gear train is obtained by multiplying these two equations to obtain
Notice that this gear ratio is exactly the same as for the case when the gears GA and GB engaged directly. The intermediate gear provides spacing but does not affect the gear ratio. For this reason it is called an idler gear. The same gear ratio is obtained for a sequence of idler gears and hence an idler gear is used to provide the same direction to rotate the driver and driven gear, if the driver gear moves in clockwise direction, then the driven gear also moves in the clockwise direction with the help of the idler gear.
Example Edit
2 gears and an idler gear on a piece of farm equipment, with a ratio of 42/13 = 3.23:1
Assuming that in the photo the smallest gear is connected to the motor, it is the driver gear. The somewhat larger gear on the upper left is called an idler gear. It is not connected directly to either the motor or the output shaft and only transmits power between the input and output gears. There is a third gear in the upper-right corner of the photo. Assuming that that gear is connected to the machine's output shaft, it is the output or driven gear.
The input gear in this gear train has 13 teeth and the idler gear has 21 teeth. Considering only these gears, the gear ratio between the idler and the input gear can be calculated as if the idler gear was the output gear. Therefore, the gear ratio is driven/driver = 21/13 = ~1.62 or 1.62:1.
This ratio means that the driver gear must make 1.62 revolutions to turn the driven gear once. It also means that for every one revolution of the driver, the driven gear has made 1/1.62, or 0.62, revolutions. Essentially, the larger gear turns more slowly.
The third gear in the picture has 42 teeth. The gear ratio between the idler and third gear is thus 42/21, or 2:1, and hence the final gear ratio is 1.62x2=~3.23. For every 3.23 revolutions of the smallest gear, the largest gear turns one revolution, or for every one revolution of the smallest gear, the largest gear turns 0.31 (1/3.23) revolution, a total reduction of about 1:3.23 (Gear Reduction Ratio (GRR) = 1/Gear Ratio (GR)).
Since the idler gear contacts directly both the smaller and the larger gear, it can be removed from the calculation, also giving a ratio of 42/13 = ~3.23.
Belt drives Edit
Belts can have teeth in them also and be coupled to gear-like pulleys. Special gears called sprockets can be coupled together with chains, as on bicycles and some motorcycles. Again, exact accounting of teeth and revolutions can be applied with these machines.
Valve timing gears on a Ford Taunus V4 engine — the small gear is on the crankshaft, the larger gear is on the camshaft. The crankshaft gear has 34 teeth, the camshaft gear has 68 teeth and runs at half the crankshaft RPM.
(The small gear in the lower left is on the balance shaft.)
For example, a belt with teeth, called the timing belt, is used in some internal combustion engines to synchronize the movement of the camshaft with that of the crankshaft, so that the valves open and close at the top of each cylinder at exactly the right time relative to the movement of each piston. A chain, called a timing chain, is used on some automobiles for this purpose, while in others, the camshaft and crankshaft are coupled directly together through meshed gears. Regardless of which form of drive is employed, the crankshaft to camshaft gear ratio is always 2:1 on four-stroke engines, which means that for every two revolutions of the crankshaft the camshaft will rotate once.
Automotive applications Edit
Automobile drivetrains generally have two or more major areas where gearing is used. Gearing is employed in the transmission, which contains a number of different sets of gears that can be changed to allow a wide range of vehicle speeds, and also in the differential, which contains the final drive to provide further speed reduction at the wheels. In addition, the differential contains further gearing that splits torque equally between the two wheels while permitting them to have different speeds when travelling in a curved path. The transmission and final drive might be separate and connected by a driveshaft, or they might be combined into one unit called a transaxle. The gear ratios in transmission and final drive are important because different gear ratios will change the characteristics of a vehicle's performance.[citation needed]
A 2004 Chevrolet Corvette C5 Z06 with a six-speed manual transmission has the following gear ratios in the transmission:
1st gear 2.97:1
2nd gear 2.07:1
3rd gear 1.43:1
4th gear 1.00:1
reverse 3.38:1
In 1st gear, the engine makes 2.97 revolutions for every revolution of the transmission's output. In 4th gear, the gear ratio of 1:1 means that the engine and the transmission's output rotate at the same speed. 5th and 6th gears are known as overdrive gears, in which the output of the transmission is revolving faster than the engine's output.
The Corvette above has an axle ratio of 3.42:1, meaning that for every 3.42 revolutions of the transmission's output, the wheels make one revolution. The differential ratio multiplies with the transmission ratio, so in 1st gear, the engine makes 10.16 revolutions for every revolution of the wheels.
The car's tires can almost be thought of as a third type of gearing. This car is equipped with 295/35-18 tires, which have a circumference of 82.1 inches. This means that for every complete revolution of the wheel, the car travels 82.1 inches (209 cm). If the Corvette had larger tires, it would travel farther with each revolution of the wheel, which would be like a higher gear. If the car had smaller tires, it would be like a lower gear.
With the gear ratios of the transmission and differential, and the size of the tires, it becomes possible to calculate the speed of the car for a particular gear at a particular engine RPM.
For example, it is possible to determine the distance the car will travel for one revolution of the engine by dividing the circumference of the tire by the combined gear ratio of the transmission and differential.
$ d = \frac{c_t}{gr_t \times gr_d} $
It is also possible to determine a car's speed from the engine speed by multiplying the circumference of the tire by the engine speed and dividing by the combined gear ratio.
$ v_c = \frac{c_t \times v_e}{gr_t \times gr_d} $
Distance per engine revolution
Speed per 1000 RPM
1st gear 8.1 in (210 mm) 7.7 mph (12.4 km/h)
2nd gear 11.6 in (290 mm) 11.0 mph (17.7 km/h)
3rd gear 16.8 in (430 mm) 15.9 mph (25.6 km/h)
4th gear 24.0 in (610 mm) 22.7 mph (36.5 km/h)
6th gear 42.9 in (1,090 mm) 40.6 mph (65.3 km/h)
Wide-ratio vs. close-ratio transmission Edit
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Main article: Close-ratio transmission
A close-ratio transmission is a transmission in which there is a relatively little difference between the gear ratios of the gears. For example, a transmission with an engine shaft to drive shaft ratio of 4:1 in first gear and 2:1 in second gear would be considered wide-ratio when compared to another transmission with a ratio of 4:1 in first and 3:1 in second. This is because the close-ratio transmission has less of a progression between gears. For the wide-ratio transmission, the first gear ratio is 4:1 or 4, and in second gear it is 2:1 or 2, so the progression is equal to 4/2 = 2 (or 200%). For the close-ratio transmission, first gear has a 4:1 ratio or 4, and second gear has a ratio of 3:1 or 3, so the progression between gears is 4/3, or 133%. Since 133% is less than 200%, the transmission with the smaller progression between gears is considered close-ratio. However, the difference between a close-ratio and wide-ratio transmission is subjective and relative.[3]
Close-ratio transmissions are generally offered in sports cars, sport bikes, and especially in race vehicles, where the engine is tuned for maximum power in a narrow range of operating speeds, and the driver or rider can be expected to shift often to keep the engine in its power band.
Factory 4-speed or 5-speed transmission ratios generally have a greater difference between gear ratios and tend to be effective for ordinary driving and moderate performance use. Wider gaps between ratios allow a higher 1st gear ratio for better manners in traffic, but cause engine speed to decrease more when shifting. Narrowing the gaps will increase acceleration at speed, and potentially improve top speed under certain conditions, but acceleration from a stopped position and operation in daily driving will suffer.
Range is the torque multiplication difference between 1st and 4th gears; wider-ratio gear-sets have more, typically between 2.8 and 3.2. This is the single most important determinant of low-speed acceleration from stopped.
Progression is the reduction or decay in the percentage drop in engine speed in the next gear, for example after shifting from 1st to 2nd gear. Most transmissions have some degree of progression in that the RPM drop on the 1-2 shift is larger than the RPM drop on the 2-3 shift, which is in turn larger than the RPM drop on the 3-4 shift. The progression may not be linear (continuously reduced) or done in proportionate stages for various reasons, including a special need for a gear to reach a specific speed or RPM for passing, racing and so on, or simply economic necessity that the parts were available.
Range and progression are not mutually exclusive, but each limits the number of options for the other. A wide range, which gives a strong torque multiplication in 1st gear for excellent manners in low-speed traffic, especially with a smaller motor, heavy vehicle, or numerically low axle ratio such as 2.50, means that the progression percentages must be high. The amount of engine speed, and therefore power, lost on each up-shift is greater than would be the case in a transmission with less range, but less power in 1st gear. A numerically low 1st gear, such as 2:1, reduces available torque in 1st gear, but allows more choices of progression.
There is no optimal choice of transmission gear ratios or a final drive ratio for best performance at all speeds, as gear ratios are compromises, and not necessarily better than the original ratios for certain purposes.
Outline of machines
Gear train
Wheel train (horology)
Epicyclic gearing - related to turboprop reduction gear boxes
Mechanism (engineering)
Virtual work
↑ J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.
↑ B. Paul, 1979, Kinematics and Dynamics of Planar Machinery, Prentice Hall.
↑ Cangialosi, Paul (2001). "TechZone Article: Wide and Close Gear Ratios". 5speeds.com. Medatronics. Retrieved on 28 October 2012.
Gear ratio at How Stuff Works
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Does it matter if immigrants work in jobs related to their education?
Jason Dean ORCID: orcid.org/0000-0001-7015-28761
IZA Journal of Development and Migration volume 8, Article number: 7 (2018) Cite this article
A common finding throughout the Canadian immigration literature is that, despite having high levels of education, recent immigrants endure substantial earnings disadvantages upon arrival that persist throughout their working career. This paper investigates the role of "qualitative" education-job matches in explaining these poor labor market outcomes. Using a self-reported match measure, available in the Survey of Labour and Income Dynamics (SLID), the incidence and wage penalties associated with being mismatched are found to be higher among immigrants relative to Canadian-born workers. As a consequence, mismatches on the part of immigrants are a mechanism behind the immigrant wage disadvantages reported throughout the literature. Successful matching is also found to significantly improve the return to pre-migration education and work experience.
There exists a sizable literature on the economic assimilation of Canadian immigrants. The evidence clearly demonstrates that over the last four decades, immigrants endure substantial wage disadvantages upon arrival which persist over their entire working career (Aydemir and Skuterud 2005; Baker and Benjamin 1994; Bloom et al. 1994; Campolieti et al. 2013; Grant 1999; Hum and Simpson 2004; Picot and Sweetman 2005; Skuterud and Su 2012). These disadvantages are observed in tandem with high levels of education reflecting a culmination of over four decades of a point system targeting highly skilled immigrants. These unsettling observations have raised concerns about the effectiveness of the immigrant selection system and the labor market suitability and recognition of immigrants' human capital. In fact, the evidence reveals that foreign human capital, most notably foreign work experience, is discounted in the Canadian labor market (Aydemir and Skuterud 2005; Ferrer and Riddell 2008; Schaafsma and Sweetman 2001; Skuterud and Su 2012). This article examines the degree to which education-job mismatches are responsible for the wage and human capital return disparities between Canadian-born and immigrant workers.
The analysis employs an alternative self-reported match measure that captures the "qualitative" aspects of matching that are associated with skills acquired through formal education and those required in employment. In contrast, the mainstay of education-job matching research focuses on estimating the effects of "quantitative" matching which compares a worker's observed level of schooling to the level required in a particular occupationFootnote 1. Using quantitative matching, Sharaf (2013) shows there is a high incidence of over-education among Canadian immigrants and these mismatches are associated with significant wage penalties. In the USA, Chiswick and Miller (2008; 2009a, b) show that two thirds of the lower returns to immigrants' schooling is due to different payoffs to under- and over-education.
A shortcoming of quantitative matching is that it fails to capture whether education-related skills match those required in employment. A worker could have a similar level of schooling as his/her colleagues but may have acquired very different, possibly irrelevant, occupational skills. Alternatively, qualitative matching considers the amount of education-related skills being utilized by an employer. Through formal education, students acquire a bundle of general and specific skills. General skills are transferable across occupations, while specific skills are important only in particular occupations. Students, who acquire credentials in fields such as business administration, acquire relatively more general skills compared to students who acquire certificates in engineering or in the skilled trades. Similarly, the skill needs of employers can vary greatly; while some employers require more general skills such as those hiring a salesperson, others require more occupation-specific skills such as those of an electrician. A close match is where an employee is utilizing many of their education-related skills in employment, whether they are general or specific.
Qualitative mismatches can arise due to the extensive heterogeneity in workers and jobs, mentioned above, coupled with imperfect information and labor market frictions (Sattinger 2012). In a perfectly competitive labor market with complete information on the worker's skills and employer requirements, optimal sorting would immediately occur and no mismatching would persist. In reality, workers and firms need to engage in costly search efforts to learn about optimal jobs and employees, respectively. A close match is desirable as productivity and wages are higher when workers can effectively implement their education-related skills. As a consequence of the time and cost involved in searching, mismatches are an unavoidable characteristic of the labor market. Mismatches can also occur as a result of changes in the incentives behind acquiring formal education, such as changes in government subsidies, as this can change the distribution of the supply of skills. Furthermore, structural and cyclical changes in economic conditions can also affect the distribution of demand for skills by employers.
There are reasons to suspect that mismatches could be more prevalent among immigrant, relative to Canadian-born, workers. In response to many new economic challenges, the Liberal government made dramatic changes to immigration policy in 1995 (Green and Green 1999). Their new objective was to select immigrants based on their potential to contribute to the economy in the long run as opposed to the previous short-term tap-on/tap-off (absorptive capacity) approach that regulated the inflow based on specific occupational needs of the labor market. The immediate skill needs of the labor market were not a significant criterion used to evaluate an immigrant's potential contribution. As such, the point system assessed a candidate's education solely on their accumulated years of schooling, treating all credential fields equally. As a result, many immigrants could have been admitted with education-related skills not in high demand, further reducing their already limited employment opportunities upon arrivalFootnote 2.
A notable exception to this long-term focus was the increased emphasis in the late 1990s on admitting IT professionals and engineers, but the IT bust in 2001 left many of these workers mismatched (Picot and Hou 2009). Similarly, sizable increases in provincial nominees and temporary foreign workers in the early 2000s affected the flow of immigrants in favor of those that have specific occupational skills (Lu and Hou 2017). In recent years, Canada's selection system has indeed become more favorable to immigrants with skills in high demand by employersFootnote 3. Nonetheless, past immigration policies, to the extent that an economic penalty exists for unmatched workers, may have contributed in part to the poor immigrant outcomes observed in the literature.
The quality and relevancy of foreign-acquired education can vary across source country and can further contribute to immigrant mismatches (Friedberg 2000). Sharaf (2013) finds that the quality of foreign education is linked to the incidence of over-education and can explain a significant portion of immigrant-native wage gaps in Canada. Not only is education quality important, but also a portion of education-related skills could be irrelevant to Canadian employers (e.g., legal professions). Despite the in-equivalency of these credentials, immigrants dedicated many years in acquiring them and would have likely got the impression they were desirable as education is heavily weighted in the point system and a key reason for their admittance. Thus, if immigrants find themselves unable to utilize the very skills deemed important enough for admission, they would certainly consider themselves mismatched. In general, the more similar are the educational institutions, economic development, and labor market characteristics between a source and host country, the higher the likelihood that immigrants will find close matches.
Nevertheless, even if foreign education is not devalued by the above noted aspects, immigrant mismatches can arise because of imperfect information on the part of employers. Immigrants face an additional burden of proving the merits of foreign-acquired credentials in contrast to Canadian graduates. Moreover, immigrant professionals such as doctors, lawyers, and nurses cannot legally work in their fields unless their credentials are officially assessed and deemed creditable. A myriad of government agencies are available to assist immigrants in this regard, although their effectiveness is uncertainFootnote 4. These services tend to focus their resources on the disproportionate share of immigrants who intend to work in regulated occupationsFootnote 5. Regardless of whether they are officially recognized or not, the decision to hire is ultimately made by employers who must themselves be convinced of its equivalency. Employer discrimination is also an obstacle as highlighted by Oreopoulos (2009) who found considerable discrimination against applicants with ethnic names and that when employers choose among candidates to interview, foreign education and experience is substantially devalued.
A close match is thought to be desirable as productivity and wages are higher when workers can effectively implement their education-related skills (Sattinger 2012). The wage of a worker is jointly influenced by the characteristics of their skills and those of their current job. The larger the discrepancy (i.e., the degree of mismatch), the greater is the potential for the worker to increase their wage by finding a match. Furthermore, collaboration among similarly educated colleagues may lead to a more stimulating work environment and lead to peer effects that can further enhance productivity. These notions are corroborated by the substantial wage effects found in Lemieux (2014) and Yuen (2010) for Canada and Robst (2007) for the USA using a similar self-reported measure as in this article. Furthermore, similar to that found in Lemieux (2014), there is no evidence that these matching premiums are an artifact of unobserved ability bias. No study has examined the role of qualitative matching in the context of earnings differentials between native-born and immigrant workers.
The remainder of the paper is structured as follows. The subsequent section discusses the data and estimation sample. The third section explores the validity of using a self-reported job match measure. The fourth section outlines the empirical methodology. Section 5 contains the estimation results and begins by presenting some descriptive statistics on the wage effects, incidence, and transition of job matches. This section also reports results on the importance of mismatches in the context of economic assimilation and whether matching has an effect on the returns to foreign-acquired human capital. The final section concludes with a general discussion.
The data are from the Survey of Labour and Income Dynamics (SLID) collected by Statistics Canada for the years 2001–2011. The SLID is a rich source of information on income, human capital, and labor market activity. It consists of 6-year panels that interview roughly 15,000 households. Since a new panel is introduced every 3 years, there are two concurrent samples of roughly 30,000 households available to researchers in each year. Respondents are interviewed twice a year in each of the panel's 6 years—once in January to collect labor information and once in May (near income tax time) to collect income information. In the event of a household split, all individuals are followed.
The estimation sample starts in 2001, the initial year of data collection on education-job match, through to the final year of data collection in 2011Footnote 6. The sample is restricted to full-time, full-year, non-self-employed, male, and female workers aged 18–64 who had at least one post-secondary degree or diploma. Respondents without any post-secondary education were not included as secondary education is usually not associated with a field of study. A small number of respondents who acquired further schooling over the panel were also excluded. Child immigrants who migrated before 10 years of age are excluded given that they are typically well acculturated upon labor force entry and achieve comparable outcomes as natives (Schaafsma and Sweetman 2001). Immigrants with missing information on years since migration are also excluded.
The primary outcome variable is the composite hourly wage measure provided in the SLID. The wage measure includes tips, bonuses, and commissions and is imputed using earnings and information on hours, months, or weeks worked. In cases where the respondents directly reported their salary as an hourly amount, no imputation was made. For workers with multiple jobs, the measure is a weighted average based on the hours worked in each job. Wages are adjusted for inflation using the provincial all-items CPI for base year 2002.
The SLID self-reported match measure
The SLID collects information on how related a worker's current job is to their education. Immigrant and Canadian-born respondents, after providing information about their job, are asked, "How closely was this job related to your education?" The available responses encompass a range of matches—"not at all related," "somewhat related," or "closely related" to their education. Similar to Lemieux (2014) and Yuen (2010), I assume that respondents consider the qualitative aspects of their job when making an assessment of their degree of match. Specifically, respondents consider their acquired skills and contemplate their degree of utilization at their current job, rather than make quantitative assessments based on, for example, discrepancies in their acquired years of schooling relative to the occupation norm.
The SLID contains detailed records on field of study and occupation which can be used to provide some external validity to my assertion that the self-reported measure captures qualitative assessments. I first select a sample of respondents with highly specialized degree and diploma fields (see Appendix Table 16) and then create a closely matched indicator variable for cases when there is an exact match between the observed field and occupation namesFootnote 7.
The first panel of Appendix Table 17 shows the distribution of SLID's self-reported match measure when considering the subset of workers that are closely matched according to their field and occupation names. Across all fields of study categories, the vast majority of Canadian-born workers (95.6%) reported being closely matched (row 2, column 5). The final column further examines a subset of immigrant workersFootnote 8. To the extent that immigrants have different perceptions than the Canadian-born on what constitutes a job match, the degree of job match could be systematically misreported rendering the subsequent analysis misleading. The immigrant figures are almost identical to those of the Canadian-born revealing that when both groups are truly matched (according to field of study and occupation), roughly 95% self-report being closely matched (row 2, columns 5 and 6). Thus, it does not appear that immigrants tend to overstate or understate their degree of match relative to the Canadian-born.
The second panel shows that roughly 61% (row 2, columns 5 and 6) of those not exactly matched according to their field-occupation names, regardless of immigrant status, self-report a close match. A close examination of the field-occupation pairs within this subset shows that many are in fact closely matched. For example, a large portion (roughly 25%) of the sample had studied to be a "Registered Nurse" or "Nurse Practitioner" and work in occupations that are clearly a close matchFootnote 9. These observations highlight the challenges of identifying matches based solely on occupation-field titles and were behind Statistics Canada's decision to use a self-reported measure (Lathe 1996; Yuen 2010)Footnote 10. According to Statistics Canada, posing the question directly to respondents is conceptually superior as they themselves are best qualified to make an assessment of job match. It is respondents who fully understand the skill requirements of their job and what their program of study was designed to do (Yuen 2010). This notion is corroborated in Lemieux (2014) who assesses the self-reported match measure in Statistics Canada's National Graduates Survey (NGS) and asserts its effectiveness at capturing the degree of mismatch between worker skills and job.
To further corroborate that SLID's self-reported measure is based on qualitative, rather than quantitative, assessments, I constructed a quantitative-based match measure using the realized matches (RM) procedure (Chiswick and Miller 2008). This procedure compares workers' acquired years of schooling to a summary measure of the typical level within an occupation. If workers' years of schooling equals the mode of their occupation, they are deemed adequately matchedFootnote 11. Column 1 in Appendix Table 18 reports the distribution of quantitative matches (i.e., adequately matched, over-educated, and under-educated) for those that self-report being closely matched and also have exact occupation-field name. If the self-reported measure reflects quantitative matching, then the vast majority of these truly matched workers should be categorized as adequately matched, but only 27% are categorized as such. The remaining columns are a cross-tabulation of the self-reported and quantitative measures of the entire estimation sample. The lack of association shown between these measures provide further evidence that the self-reported measure is based on qualitative assessments.
Empirical methodology
The analysis begins with a preliminary examination of the potential role of mismatches in explaining the observed immigrant-wage disadvantages. Descriptive statistics on mean wages by degree of job match are estimated to explore if mismatched workers suffer an economic penalty. The distribution of workers by job match is then examined with all estimates stratified by Canadian-born and immigrant workers. To the extent that immigrants are more mismatched and as a result have lower wages, qualitative matching may be an endogenous mechanism behind the poor labor market performance of immigrants and justifies a more complex multivariate analysis. It is reasonable to suspect a dynamic element to matching. New arrivals are probably not familiar with the intricacies associated with the Canadian labor market, such as resume writing and successful interview techniques, which could hinder their ability to secure related jobs. Further, their language skills may not yet allow them to effectively promote the relevancy of their skills and understand the skill requirements of employers. However, over time, their employment opportunities may improve as better job searching skills and social and professional networks are developed.
To assess the degree of mismatch over time while holding various determinants of matching constant, a series of ordered logit models are estimated. The respective specifications for the Canadian-born and immigrant samples are:
$$ y_{it}^{*}=\beta_{1}\text{exp}_{it}+\beta_{2}\text{exp}_{it}^{2}+\beta_{3}s_{it}+X_{it}\lambda+\epsilon_{it}, \text{and} $$
$$ y_{it}^{*}=\beta_{1}\text{exp}_{it}+\beta_{2}\text{exp}_{it}^{2}+\beta_{3}s_{it}+\beta_{4}\text{ysm}_{it} +\beta_{5}\text{ysm}_{it}^{2}+X_{it}\lambda+\epsilon_{it}, $$
where \(y_{it}^{*}\) represents a continuous (unobserved latent) variable capturing qualitative matches in time period t for individual i. The term exp is potential years of work experience (age−years of schooling−6), s denotes years of schooling, ysm is years since migration, and X contains controls for field of study, socio-demographic variables—including indicators for marital status, non-English (Ontario and Québec) or non-French (Québec) mother tongue, region (British Columbia, Prairies, Ontario, Québec, Maritimes), and major cities (Toronto, Montréal, Vancouver)—and the panel and year observedFootnote 12. Conditional probabilities of the SLID match categories are obtained by assuming ε is logistically distributed and estimated by maximum likelihood. Standard errors are adjusted for clustering at the individual level. A series of predicted probabilities are then obtained by varying Canadian experience (ysm for immigrants) from 1 to 25 years. Other variables are held constant at their mean values.
The standard empirical strategy used in estimating the relative economic performance of immigrants is based on a variant of the standard human-capital-corrected earnings function (Mincer 1974). The typical specification models earnings as a function of the standard human capital variables, schooling, and experience but additionally includes a quadratic in years since migration (Borjas 1999; Chiswick 1978). The estimation sample combines natives and immigrant workers, and since time in the host country is conditioned on, the coefficient estimate on an immigrant indicator variable captures any wage disparities upon migration and is commonly referred to as an "entry effect."Footnote 13 Estimates of the quadratic terms in years since migration reveals the return to a year in the host country, commonly referred to as the "assimilation effect."
The standard immigrant assimilation model is first estimated to obtain the entry and assimilation effects experienced by Canadian immigrants. Subsequently, controls for the categories of job match are included and the proportion of the entry effect explained by immigrant mismatches is estimated. In other words, the aim here is to identify a possible endogenous mechanism that is responsible for immigrant-wage disadvantages and then subsequently estimate the degree to which an exogenous intervention can improve immigrant outcomes. The full empirical specification is:
$$\begin{array}{*{20}l} ln\left(w_{it}\right)&=\beta_{0}+\beta_{1}\text{exp}_{it}+\beta_{2}\text{exp}_{it}^{2}+\beta_{3}s_{it} +\beta_{4}\text{some}_{it}+\beta_{5}\text{close}_{it} \\ &\quad+I_{i}\left(\alpha_{0}+\alpha_{1}\text{ysm}_{it}+\alpha_{2}\text{ysm}_{it}^{2}+\alpha_{3}\text{some}_{it} +\alpha_{4}\text{close}_{it}\right)+X_{it}\lambda+\epsilon_{it}, \\ \end{array} $$
where w it is an individual's composite wage in year t and exp and s are as defined aboveFootnote 14. The letter I denotes an immigrant indicator variable. Qualitative matching is incorporated into the model with indicator variables, i.e., some is an indicator variable for respondents in a "somewhat related" job, while close is an indicator variable for respondents in a "closely related" job. The omitted category refers to those in a job "not at all related" to their education.
There is a significant degree of heterogeneity in the education fields acquired by immigrant and Canadian-born workersFootnote 15. Consequently, the estimated impact of job matching will be misleading if the incidence of matching varies across fields and is not held constant in the regression analysis. The vector X additionally contains indicator variables for major field of study groups in order to mitigate any potential biasesFootnote 16. Due to the limited size of the immigrant sample, more detailed field of study controls cannot be included. However, the potential bias, due to variation in job match within the aggregated fields, has been shown by Lemieux (2014) to be inconsequential.
The estimated returns to matching could also be contaminated by unobserved heterogeneity bias. Consistent and unbiased estimates of the return to matching are guaranteed if the idiosyncratic error in (3) is not correlated with job match. Specifically, workers with higher innate ability and motivation may be more likely to obtain better matches which could impart an upward bias on the returns to matching. Standard fixed effects estimation offers a solution by removing unobserved time-invariant heterogeneity; however, this methodology does not permit identification of the immigrant time-invariant variables. Furthermore, it is entirely plausible that respondents make random errors in judgment when reporting their degree of match. It is well known that fixed effects estimation substantially exaggerates the attenuation bias when estimated on variables suffering from classical measurement error (Angrist and Krueger 1999). Parental level of education (available in the SLID) is a candidate for use as an instrumental variable; however, its correlation with matching is insignificant—a necessary condition of a valid instrumentFootnote 17.
Nonetheless, potential biases can be mitigated if a proxy variable, a measure highly corrected with ability, is available (Wooldridge 2010). Lamo and Messina (2010) who examined wage returns of a self-reported quantitative match measure found little evidence their results were tainted by unobserved ability bias. Nordin et al. (2010) using cognitive test scores and Lemieux (2014) using a worker's self-reported rank in their graduating class and enrollment in a co-op program as proxies for ability found they had little impact on the effect of matching when included in their wage regression. Although these proxies are not available in the SLID, data on whether a paid worker's job involved supervising the work of other employees is available. A profit maximizing firm certainly has an incentive to only give such responsibility to workers with high ability and/or motivation. Since it is reasonable to assume a high degree of correlation between supervisor responsibility and innate ability and motivation, the vector X additionally contains this proxy variable.
Estimates of Eq. (3) are obtained by pooling the individual data over time, and using OLS and the reported standard errors are adjusted for clustering at the individual level. The estimates of Eq. (3) are first used to predict the entry effects experienced by immigrants upon arrival. The predictions are calculated using the immigrant-native match disparities reported in Tables 3 and 4 (row 6 for Canadian-born; row 9 for immigrants, i.e., at ysm=0). Then, the entry effect is predicted assuming immigrants had the same distribution of matches as the Canadian-born at arrival. The difference in these predicted entry effects reflect the extent to which the entry effects are due to the lower incidence of matching among immigrants when first entering Canada's labor market. These entry effects and their percentage change is reported. In addition, the estimates obtained from (3) are used to explore the consequences of job matching on the economic assimilation experience. Specifically, the years to earnings parity with natives is estimated for each category of job matchFootnote 18.
As mentioned previously, Canadian studies have shown that an important contributor to the poor entry effects observed for immigrants is their lower returns to foreign human capital, most notably foreign work experience. It is plausible that foreign human capital has a complementary effect with better matches that can boost productivity. Employers who require many education-related skills may put more effort into researching the merits of an applicant's relevant pre-migration schooling and work experience. Following Friedberg (2000), Eq. (3) is reformulated with human capital variables decomposed into their foreign and Canadian components:
$$\begin{array}{*{20}l} ln(w_{it})&=\beta_{0}+\beta_{1}\text{canexp}_{it}+\beta_{2}\text{cans}_{it}+\beta_{3}\text{some}_{it}+\beta_{4}\text{close}_{it} \\ &\quad\,+\,I_{i}(\alpha_{0}+\alpha_{1}\text{canexp}_{it}\,+\,\alpha_{2}\text{forexp}_{i}\,+\,\alpha_{3}\text{cans}+\alpha_{4}\text{for}s_{i}+\alpha_{5}\text{some}_{it}+\alpha_{6}\text{close}_{it} \\ &\quad+\alpha_{7}\text{for}s_{i}\times\text{some}_{it}+\alpha_{8}\text{for}s_{i}\times \text{close}_{it}+\alpha_{9}\text{forexp}_{i}\times\text{some}_{it}\\&\quad+\alpha_{10}\text{forexp}_{i}\times\text{close}_{it}) \\ &\quad+X_{it}\lambda+\epsilon_{it}, \\ \end{array} $$
where the foreign and Canadian components are identified using age at migration as commonly done in the literature (Aydemir and Skuterud 2005; Betts and Lofstrom 2000; Friedberg 2000). More specifically, it is assumed that there are no breaks in the acquisition of schooling which permits an estimate of the age when an immigrant started to work full-time (i.e., 6+ years of schooling). This age is used in conjunction with the age at migration to create foreign and Canadian measures of schooling and potential experience. It has been shown that potential experience can understate (overstate) the returns to experience (education) when used in wage regressions (Regan and Oaxaca 2009). More precise foreign and Canadian human capital measures are created using unique information provided in the SLID on actual experience, an actual full-time work-starting age, and the age respondents received their credentialsFootnote 19. Estimates using the potential and actual experience measures are reported and compared. All foreign human capital variables are interacted with the degree of matching to ascertain if matching plays a role in boosting their returns in the labor market. Similar to Eq. (3), estimates are obtained from OLS on a sample of individual data pooled over time with standard errors adjusted for clustering.
The immigrant sample used in the analysis is stratified by traditional and non-traditional source regions. The former include those born in either the UK, USA, Europe, Australia, or New Zealand. The non-traditional group are those born in the Middle East, Caribbean, Mexico and Central America, Southern America, Asia, Southeast Asia, or Africa. Immigrants from the latter regions, who comprise a substantial portion of arrivals over the last couple of decades, can have vast cultural, institutional, and linguistic differences compared to those from traditional source regions and have been shown to experience greater difficulties integrating into western labor markets (Aydemir and Skuterud 2005; Ferrer and Riddell 2008; Friedberg 2000; Picot 2004; Skuterud and Su 2012).
Qualitative matching: wages, prevalence, and transition
Table 1 shows male and female mean hourly wages by immigrant status and degree of job match. There is clear pattern of increasing wages as the quality of job match improves for all groups (rows 1–3). The wage penalties (relative to a close match) for Canadian-born males are 4.8 and 27.2% for somewhat and closely related jobs, respectively. The comparable figures for females are higher, especially when partially matched, at 13.2 and 29.5% (column 3; rows 4, 5). All immigrant groups are found to endure significantly higher penalties than the Canadian-born with the largest penalties at 41.0 and 36.2% (column 2; row 3), respectively, for non-traditional males and females. These unadjusted figures imply that foreign-born workers may have more difficulties adapting to the labor market when unable to secure a close match.
Table 1 Mean wages by immigrant status and job match
Table 2 shows the unadjusted frequency distribution of qualitative matching by immigrant status. Among Canadian-born males, 61.3, 17.3, and 21.4% (column 3; rows 1–3) are in closely, somewhat, and unrelated jobs, respectively. The comparable frequencies for females suggest a slightly higher incidence of matching at 65.2, 16.2, and 18.7%. This pattern closely resembles the 2006 SLID estimates reported in Yuen (2010) and also the US tabulations reported by Robst (2007)Footnote 20. In contrast to popular perceptions, on the whole, immigrants do not appear to be substantially mismatched relative to native-born workers. The largest disparities are among females from non-traditional source regions who are about 8.6 (columns 2–3; row 3) percentage points less likely to be closely matched. However, immigrants differ in terms of their geographic region, field of study, time in the host country, and human capital acquisition. The subsequent multivariate analysis reveals that immigrants do experience a sizable degree of mismatch relative to natives upon arrival in Canada.
Table 3 reports the multivariate estimates of Eqs. (1) and (2) for males. The predicted distribution of job match for the Canadian-born and for immigrants at various stages of assimilation are calculated at the means of the respective samples. Upon arrival, traditional and non-traditional male immigrants are more mismatched than natives with a disparity of 9.1 (61.8–52.7) and 14.9 (61.8–46.9) percentage points, respectively. The former group, however, are faster at securing close matches and catching up to comparably skilled natives. The slower transition for non-traditionals may reflect the higher burden and lengthy time required to prove the merits of their foreign education through formal recognition or acquiring complementary Canadian credentials. The degree of mismatch is much higher among female immigrants in Table 4 with the respective figures for traditionals and non-traditionals at 15.9 (66.6–50.7) and 22.3 (66.6–44.3). Furthermore, neither group catches up to natives in a reasonable time frame. Intermittent labor force participation as a result of child rearing and cultural differences in the role of females in the household could explain their lower incidence of, and transition into, closely matched jobs.
The statistics presented thus far imply that mismatches on the part of immigrants may be an important mechanism behind their poor labor market outcomes. Not only are immigrants more mismatched than natives, but they also face steeper wage penalties when mismatched. Thus, the immigrant assimilation experience is likely quite different depending on the degree of job match.
The role of matching on immigrant assimilation
Tables 5 and 6 contain estimates of the immigrant assimilation model (i.e., Eq. (3)) for males and females, respectively. The immigrant sample is stratified by traditional and non-traditional source regions. Consistent with related studies, the assimilation experience is worse among non-traditionals who earn much less upon arrival (columns 1; rows 1). The estimated assimilation profiles are similar to those reported by Hum and Simpson (2004) whose sample contained males of all education levels and estimated on SLID's first panel. A major distinction gleaned from the current analysis is that only slight differences exist in the assimilation profiles between genders when considering post-secondary graduates. Their finding of improved entry effects and worsening assimilation effects when actual, rather than potential, experience was corroborated using the more current SLID data. The estimated profiles, based on actual experience, are not reported in order to remain consistent with the majority of the literature and also because, when actual experience is used, any improvements in the entry effect are mostly offset by a fall in the assimilation effect. Thus, no matter which measure is used, the number of years it takes to reach parity with natives is similar.
Table 6 Effect of qualitative matching on wages (females)
Across all groups, qualitative matching has a significant effect on wages. A partial match yields roughly a 16% premium over an unrelated job (the excluded category), while a close match yields a premium of roughly 25 and 30% for males and females, respectively (Tables 5 and 6; rows 7 and 8; columns 2 for each group)Footnote 21. These wage effects are consistent with the notion that productivity is higher when workers are able to use more of their education-related skills on the job. The immigrant interaction terms for partial and close matches reveal that there is no statistically significant difference in the returns between traditional and Canadian-born males (rows 9 and 10; columns 3). In contrast, non-traditional males receive higher returns than natives by 11 and 23 percentage points for partial and close matches, respectively. Thus, matching boosts their productivity more than both natives and traditionals but conversely face greater difficulties adapting in the labor market when not matched. Conceivably, since traditionals likely face less obstacles such as language difficulties and discrimination, their disadvantage when not matched is mitigated. The results reported in column 4 show that when supervisory responsibility (a proxy for innate ability and motivation) is conditioned on, the returns fall only slightly across all stratification groups (column 4; rows 7–11). Similar to that found in Nordin et al. (2010) and Lemieux (2014), there is little evidence that the estimated returns to job match are an artifact of higher ability workers (with higher earning potential) being more likely to secure a job match. The pattern of results for females is broadly similar except that both immigrant groups receive a premium for better matching compared to the Canadian-born (Table 6).
The amount of effort put into searching for more closely matched positions may differ systematically between graduates with general and specialized credentials. It is not clear which graduates would put more or less effort into the search process. If general-education graduates dedicate less effort relative to graduates with specialized credentials, then the matching estimates above may simply reflect employers' willingness to offer a premium for specialized skills. However, general-education graduates possess many important labor market skills such as problem-solving, communication, and on-the-job learning abilities which can provide them greater flexibility to move across industries and occupations (Giles and Drewes 2001; Heijke et al. 2003). Given the productive potential of their dynamic skill sets, those with general fields may put more effort into securing a match. Lin et al. (2003) show there is little difference in the perceived possession of relevant labor market skills between those with general and specialized education. Thus, the search effort among graduates with general skills may in fact not be very different from those with specialized skills.
The ongoing analysis is further stratified into general and specialized fields and by college and university credentialsFootnote 22. The results are presented in Table 7 and focus on non-traditional immigrants as they comprise the majority of new arrivals in past decades. Regardless of having a general or specialized credential field, a close match is associated with substantial premiums for males and females (rows 7–10; columns 1 and 2 for each group). The returns to using university-acquired skills on the job are much higher, especially for non-traditional males, despite being less specific than those acquired in vocational institutions (rows 7–10; columns 3 and 4 for each group). If it were true that the returns to job match simply captured a premium for specialized skills, then one would expect little or no returns for those with general fields of study.
Table 7 The effect of qualitative matching on wages—field of study and level of education (non-traditional immigrants)
Tables 8, 9, and 10 highlight the role of mismatches in explaining the poor entry effects experienced by immigrants. The first column under each group reports an entry effect that is predicted at the relative degree of match faced by immigrants upon arrival using the match distributions reported in Tables 3 and 4.Footnote 23 In the second column, the predicted entry effect assumes that immigrants had the same degree of match upon arrival as their Canadian-born peers. To the extent that immigrants are more mismatched, the percentage change in these two figures, reported in column 3, reveals how much the initial wage disadvantages experienced by immigrants are due to being more mismatched than the Canadian-born. Table 8 reveals that mismatching is a mechanism behind the initial disadvantages experienced by all groups of immigrants. Among traditional males, mismatches explain about 9% of their negative entry effect, while the comparable figure for non-traditionals is 11%. Among females, mismatches explain a slightly higher percentage at 14 for traditionals and 12% for non-traditionals. The following Tables (9, 10, 11, 12) focus on non-traditional immigrants—a group that makes up the vast majority of arrivals over the last three decades. When further considering the importance of matching across education type and field category (Tables 9 and 10), between 10 and 13% of the entry effects are due to mismatches upon arrival.
Although mismatches explain a sizable portion of immigrants' initial wage disparity, the results do not point to systemic problems of highly educated immigrants working in low paying unrelated jobs. Indeed, as shown in Tables 3 and 4, the degree of relative mismatch upon arrival is not drastically lower for immigrants. This lends some support to the previous long-term goals of immigration policy that gave a low priority to matching the skill sets of newcomers with the immediate needs of the labor market. Nevertheless, recent changes to our point system favoring those with greater matching potential can go a long way at improving the assimilation experience because substantial consequences exist for unmatched immigrants as the next set of tables will highlight.
Tables 11 and 12 highlight important differences in the predicted assimilation profiles across the varying degrees of job match. The calculations are also based on the results reported in Tables 5 and 6 (column 4).Footnote 24 The entry effect and years to earnings equality with natives is reported for each group by degree of match. A clear pattern emerges in Table 11 across gender and immigrant group indicating that if an immigrant is unable to obtain at least a partial job match, economic assimilation is unattainable (row 1). The exception is for unmatched traditional males, but their estimated years to equality is almost 22 years. Traditional males who begin their Canadian career as partially matched have entry earnings similar to their Canadian-born peers, while the estimated years to assimilation is 13 for traditional females (row 2). Male and female traditional immigrants who can secure a closely related job upon arrival are not economically disadvantaged in the labor market (row 3).
The pattern of assimilation among non-traditional males and females improves slightly for those partially matched, but full assimilation requires at least 20 years. On a positive note, those able to secure a close match over their career can fully assimilate in 6 and 11 years for males and females, respectively. The predicated results for non-traditionals in Table 12 reveal a similar pattern of reduced entry effects with improved matches. Among those who have acquired a specialized field or a university degree, assimilation is unattainable in unrelated jobs but is expected in under 10 years if in a closely related job (rows 1 vs. 3). The impact of matching is less helpful at improving assimilation among college graduates and females with general degrees. Nevertheless, their entry effects are reduced by more than half when closely matched. The ability of immigrants to obtain higher returns to their foreign-acquired human capital when matched could potentially be behind the strong wage effects of matching reported here.
The role of matching and the returns to foreign human capital
Tables 13 and 14 contain estimates of Eq. (4), stratified by traditional and non-traditional immigrants, for males and females, respectively. Separate estimates, based on potential and actual measures of experience, are also reported. Consistent with the Canadian literature, foreign experience is virtually worthless regardless of an immigrant's source country or gender (columns 1, row 6)Footnote 25. The return to a year of foreign schooling is also discounted but to a much lesser extent for males than females (columns 1, rows 4, 7). These findings are consistent to Census estimates from Pendakur and Pendakur (1998), who show there is a sizable earnings disadvantage for female immigrants with foreign education from Asia or Africa, but not for male immigrants from these same regions. Similar returns to foreign education based on SLID data are also reported by Skuterud and Su (2012). Similarly, Alboim et al. (2005) using the Survey of Literacy Skills find high returns to foreign education, and Buzdugan and Halli (2009) using the Ethnic Diversity Survey reveal that for recent arrivals, discounted foreign experience is the driving force behind immigrant-wage disadvantages regardless of the origin of education. Taken as a whole, the main culprit behind immigrant-native wage gaps is the complete lack of return to foreign experience.
The second column under each stratification group in Tables 13 and 14 interacts with the foreign human capital variables (based on potential experience) with the degree of qualitative match (i.e., somewhat and closely related). The estimated interactions suggest that, even if immigrants are successful at finding a better match, their accumulated foreign work experience seems to be irrelevant to employers (columns 2, rows 6, 8, 9). Alternatively, there is a clear pattern across immigrant group and gender of increasing returns to foreign education with better matching. Additionally, only those closely matched receive the sizable returns to foreign education observed in columns 1. The pattern of results here echo those of Goldmann et al. (2009) that successfully matched immigrants, in terms of their pre- and post-migration occupations, increase the return to foreign schooling, but not to potential foreign experience.
The results from estimating Eq. (4), using more accurate measures of foreign human capital and based on actual years of work experience, are reported in Tables 13 and 14 (columns 3 and 4) and also in Table 15 which is stratified by college, university, and general and specialized fields. Across all genders, immigrant groups, and education types, those with better matches earn much higher returns to their foreign education (Tables 13, 14, 15; rows 10, 11). The effect tends to be moderated among college graduates which is consistent with the more modest effects of matching on assimilation presented in Table 12. However, some immigrant groups are now found to earn positive and significant returns to their foreign work experience. Traditional females are found to earn yearly returns of 1.1% regardless of job match (Table 14, row 6, column 4). Non-traditional males receive an 1.1 percentage point boast in their returns when closely matched, but those not closely matched earn negative, albeit small, returns to foreign experience. In contrast to Goldmann et al. (2009) and Warman (2007) who also find negative returns, this disadvantage is only experienced by unmatched immigrants whose unobservable motivation or willingness to adapt could be lower compared to older immigrants who have acquired more foreign experienceFootnote 26. If true, this negative correlation could be responsible for the negative returns as it would impart a downward bias on the estimates of foreign experience. When considering non-traditional males with specialized fields or university degrees, matching is also found to boost their returns to foreign experience (Table 15, row 9)
It is plausible that a portion of education-related skills have a complementary effect with work experience such that it boasts productivity. As well, employers who require many education-related skills may put more effort into exploring the merits of an applicant's pre-migration work experience. Some skilled immigrants may also cluster in certain field-occupation pairs where the productivity enhancing attributes of their foreign experience become widely known among employers. If clustering is less prevalent among traditional immigrants, this could explain why matching has little effect on the returns to their foreign experience. Further research based on a larger immigrant sample than that collected in the SLID could help resolve this issue and provide a more comprehensive picture.
Canadian immigration studies have repeatedly found that immigrant earnings are substantially below that of similarly skilled native-born workers. This article examined the role of qualitative education-job matching in explaining these poor labor market outcomes. Upon arrival, immigrants, especially females, are more likely than natives to be mismatched. Further, when immigrants are mismatched, they generally experience larger wage penalties compared to natives suggesting they have a greater difficulty adapting when not able to work in their field. As such, mismatches are an important mechanism behind the negative entry effects experienced by Canadian immigrants reported throughout the literature. On the whole, mismatches can explain in the range of 9 to 14% of the entry wage disadvantage experience by immigrants upon arrival. Among immigrants from non-traditional source regions, only those who are closely matched upon arrival can expect to reach parity with comparably skilled natives over their working career in Canada.
An overarching theme in the immigration literature is that immigrants receive little to no returns to foreign experience. Consequently, this discounting is the main culprit behind immigrant-native wage disparities. The analysis here examined the effects of matching on the wage returns to foreign experience and education. Matching is associated with higher wage returns to foreign education and, in some cases, to foreign work experience with the estimated effect increasing with the degree of job match. Clearly, immigration policy aimed at increasing the share of immigrants who are able to work in their field of study can significantly reduce immigrant earnings disadvantages.
Table 16 Specialized occupation and field of study matches
Table 17 SLID job match vs. occupational/field matches
Table 18 Distribution of quantitative match by self-reported SLID match variable (males and females)
Table 19 Means of key variables (Continued)
Hartog (2000) provides a literature review on quantitative matching.
Chiswick and Miller (2009b) suggest that point systems used to ration visas such as those in Canada and Australia could benefit more if additional points are added for pre-arranged employment or for having particular skills highly demanded in the labor market—criteria that could reduce the level of education-job mismatches.
On July 1, 2012, as a result of changes to the Immigration and Refugee Protection Act, immigrants entering under the federal skilled worker program are now assessed based on similarities between their host-country occupation and occupations in Canada that are deemed to have a labor shortage.
The most recent agency to be established is the Foreign Credential Recognition Program administered by Human Resources and Skills Development Canada. This is the first national agency created that works with provincial agencies to develop comprehensive policies and procedures to help immigrants gain recognition for their foreign credentials.
According to data obtained from Citizenship and Immigration Canada between 1995 and 2006, the average percent of immigrants entering under the skilled worker class (the largest component of the economic class) who are principal applicants that self-identified themselves as being in regulated occupations is 31% at the national level.
Access to the unrestricted micro-data file was provided through the Toronto Research Data Center (RDC).
Occupations were classified according to the National Occupational Classification for Statistics 2006. Field of study was classified according to the Classification of Instructional Programs code (CIP Canada 2000).
Due to confidentiality concerns raised by Statistics Canada, the "somewhat related" and "not at all related" categories of the SLID match variable are collapsed into one category in Appendix Table 18. For similar reasons, only the results for the entire sample are reported for immigrants.
A frequency distribution of the occupations and fields of study for the subsets in panel 2 (rows 2 and 3) of Appendix Table 17 are not included in the manuscript due to their length and confidentiality concerns raised by Statistics Canada. However, the list of these occupation/field of study pairs without their associated frequency is available upon request from the author.
Statistics Canada attempted to derive a match measure by linking the categories of the 1980 Standard Occupational Classification to the 1991 Census Major Field of Study classification (Lathe 1996).
Two alternative statistics have been used to estimate the typical level of schooling within an occupation. Chiswick and Miller (2008), Cohn and Khan (1995), and Kiker et al. (1997) prefer using modal years of schooling within an occupation while Verdugo and Verdugo (1989) deem workers fully matched if their years of schooling fall within one standard deviation above or below the occupation mean. Generally, empirical results are not overly sensitive to the measurement used (Hartog 2000)
Appendix Table 19 contains summary statistics of the key variables employed in the multivariate analysis.
It is common in the literature to include indicators for immigrant arrival cohorts. However, unlike the Census, the SLID only covers 2001–2007. When the immigrant indicator is replaced with indicators for arrival cohorts, I find a decline in cohort quality not consistent with Census-based studies. The reason for this discrepancy is that the sample does not have immigrant wages upon arrival for earlier cohorts. Thus, any estimates are based on an out-of-sample projection based on the assimilation experience of recent immigrants. As such, the sample is stratified by traditional and non-traditional immigrants with this latter group comprising the majority of the intake in recent decades.
Although a measure of actual experience is available in SLID, the reported estimates are based on potential experience in order to report estimates consistent with the literature. Similar to Hum and Simpson (2004) when actual experience is used instead of potential experience, I find that the entry effects improve but the assimilation effect deteriorates. On the whole, despite which measure is employed, qualitative matches are found to be a significant mechanism behind the poor economic performance of immigrants.
See Appendix Table 19.
Field of study categories are aggregated according to the first two digits of the Classification of Instructional Programs code (CIP Canada 2000).
See Card (1999) for a survey of studies who have used parental education as an instrumental variable in estimating the returns to education.
Years to earnings parity is obtained by solving for the quadratic term in ysm in \(\alpha _{0} +\alpha _{1} \text {ysm}_{\text {it}} +\alpha _{2} \text {ysm}_{\text {it}}^{2} = 0\) from Eq. (3).
The actual foreign and Canadian experience variables are based on the proportions of potential experience acquired in the source country and Canada, respectively. For immigrants who received a credential after their migration age, its duration is correctly attributed to Canadian schooling rather than foreign schooling as is done in other studies that need to assume continuously acquired education. Likewise, this time in school is not incorrectly attributed to Canadian experience.
Neither Yuen (2010) nor Robst (2007) include immigrants in their analysis.
Table 2 Frequency distribution of job match by immigrant status
Table 3 Proportion in categories of job match by years since migration (male ordered logit results)
Table 4 Proportion in categories of job match by years since migration (female ordered logit results)
Table 5 The effect of qualitative matching on wages (males)
The reported figures are transformed using (eβ−1) as suggested by Halvorsen and Palmquist (1980) when interpreting indicator variables in a semi-log specification.
General fields of study refer to those with the highest credential in Humanities, Social Sciences, Psychology, or Liberal Arts. Specialized fields of study refer to those with the highest credential in Sciences, Technology, Engineering, or Math.
All of the predicted figures in Tables 8, 9, 10, 11, and 12 make use of the transformation (eβ−1) suggested by Halvorsen and Palmquist (1980) when interpreting indicator variables in a semi-log specification.
Table 8 Percent of entry effect due to mismatch: traditional vs. non-traditional immigrants
Table 9 Percent of entry effect due to mismatch: general vs. specialized fields (non-traditional immigrants)
Table 10 Percent of entry effect due to mismatch: college vs. university (non-traditional immigrants)
Table 11 Entry earnings and assimilation by job match—traditional vs. non-traditional immigrants
Table 12 Entry earnings and assimilation by job match—general vs. specialized and college vs. university (non-traditionals)
Separate specifications were run (not shown) with interaction terms of ysm with the categories of match and were found to be statistically insignificant.
See Green and Worswick (2010); Aydemir and Skuterud (2005); Schaafsma and Sweetman (2001); Skuterud and Su (2012); Goldmann et al. (2009); Warman (2007).
Table 13 The effect of job match on the returns to foreign human capital (males)
Table 14 The effect of job match on the returns to foreign human capital (females)
Introducing age at migration into the model to capture these unobservable effects would introduce perfect collinearity with foreign experience and foreign years of schooling.
Table 15 The effect of job match on the returns to foreign human capital (non-traditional males and females)
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I am indebted to Jennifer Hunt, Mary MacKinnon (1959–2010), Daniel Parent, Katerina Koka, Bok Hoong Young Hoon, Samer Atallah, Charbel Rizk, Jonathon Gravel, to seminar participants at McGill University for valuable comments and suggestions, and to the Social Sciences and Humanities Research Council (SSHRC) for their financial support. I would also like to thank the anonymous referee and the editor for useful comments.
Department of Economics, Wilfrid Laurier University, Waterloo, Canada
Jason Dean
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Correspondence to Jason Dean.
The IZA Journal of Development and Migration is committed to the IZA Guiding Principles of Research Integrity. The author declares that he has observed these principles.
Dean, J. Does it matter if immigrants work in jobs related to their education?. IZA J Develop Migration 8, 7 (2018) doi:10.1186/s40176-017-0114-2
Job match | CommonCrawl |
Machine learning analysis plans for randomised controlled trials: detecting treatment effect heterogeneity with strict control of type I error
James A. Watson ORCID: orcid.org/0000-0001-5524-03251,2 &
Chris C. Holmes ORCID: orcid.org/0000-0002-6667-49432,3
Trials volume 21, Article number: 156 (2020) Cite this article
Retrospective exploratory analyses of randomised controlled trials (RCTs) seeking to identify treatment effect heterogeneity (TEH) are prone to bias and false positives. Yet the desire to learn all we can from exhaustive data measurements on trial participants motivates the inclusion of such analyses within RCTs. Moreover, widespread advances in machine learning (ML) methods hold potential to utilise such data to identify subjects exhibiting heterogeneous treatment response.
We present a novel analysis strategy for detecting TEH in randomised data using ML methods, whilst ensuring proper control of the false positive discovery rate. Our approach uses random data partitioning with statistical or ML-based prediction on held-out data. This method can test for both crossover TEH (switch in optimal treatment) and non-crossover TEH (systematic variation in benefit across patients). The former is done via a two-sample hypothesis test measuring overall predictive performance. The latter is done via 'stacking' the ML predictors alongside a classical statistical model to formally test the added benefit of the ML algorithm. An adaptation of recent statistical theory allows for the construction of a valid aggregate p value. This testing strategy is independent of the choice of ML method.
We demonstrate our approach with a re-analysis of the SEAQUAMAT trial, which compared quinine to artesunate for the treatment of severe malaria in Asian adults. We find no evidence for any subgroup who would benefit from a change in treatment from the current standard of care, artesunate, but strong evidence for significant TEH within the artesunate treatment group. In particular, we find that artesunate provides a differential benefit to patients with high numbers of circulating ring stage parasites.
ML analysis plans using computational notebooks (documents linked to a programming language that capture the model parameter settings, data processing choices, and evaluation criteria) along with version control can improve the robustness and transparency of RCT exploratory analyses. A data-partitioning algorithm allows researchers to apply the latest ML techniques safe in the knowledge that any declared associations are statistically significant at a user-defined level.
In the medical sciences, randomised controlled trials (RCTs) provide the gold standard for evidence evaluation of novel treatments and health interventions. The growing accessibility and recording of data modalities, arising from genetics, medical imaging, mobile devices, genomics, and electronic health records captured on trial participants, alongside breakthroughs in machine learning (ML) provide new opportunities for scientific discovery of patient strata exhibiting systematic variation in treatment effect. This can improve patient outcomes and optimise treatment recommendations. However, exploratory analyses of RCTs and correct interpretations of these analyses are difficult [1, 2] and controversial [3]. Data analytic tools such as ML algorithms [4] are particularly attractive for identifying treatment effect modifiers in RCTs due to their hypothesis-free nature and ability to learn by example. Although there have been numerous recent papers on technical developments and novel methods for subgroup analysis and treatment effect heterogeneity (TEH) [5–14], we know of none to date that have considered ML paradigms purely from a testing perspective that provides strict control of the false positive rate (type I error) for the quantities we consider here, namely the evidence of crossover TEH and the evidence of predictive improvement of an ML model over a conventional statistical model. Some recent papers, e.g. [15], have derived test statistics for detecting global heterogeneity using ML, yet they lack the simplicity of our approach and the broad applied nature of our work. Moreover, we focus on detecting actionable (crossover) interactions as well as quantifying the evidence for the added predictive benefit of ML over simpler statistical models. A key component of this work is to provide concrete recommendations for how subgroup statistical analysis plans (subgroup-SAPs) can incorporate ML methods (summarised in Panel 1).
Medical statisticians know how to assess the evidence when the subgroups or interactions are predefined and the models are explicit, by counting the 'degrees of freedom', or number of free parameters, in the model and using formal tests of hypotheses [16–18]. But for ML algorithms the models are designed to adapt their complexity and dependency structures to the underlying problem during the training phase, and hence notions of counting parameters become meaningless. The question then remains: How to assess the true evidence of an effect following ML discovery?
We show that it is possible to train such methods, alongside conventional statistical models, to analyse RCT data and provide a global hypothesis test for the presence of TEH. The methodology explicitly uses the underlying treatment randomisation to test for TEH. We show that it is possible to formally test for the presence of patient subgroups (crossover TEH) and also formally test the added predictive benefit of the ML algorithm by 'stacking' the ML predictions alongside predictions from a baseline 'vanilla' statistical model. ML algorithms should only be used if their predictive benefit can be proven superior to that of simpler and more interpretable methods. This framework has important implications for how existing data can be used in a principled manner for trusted hypothesis generation. We hope that it will motivate careful a priori construction and monitoring of statistical analysis plans utilising the latest ML techniques. This is necessary to ensure optimal evidence evaluation and learning through retrospective discovery of TEH.
Our formal approach is illustrated step by step via an open source R Markdown computational notebook [19] which uses random forests (RF) [20] to retrospectively analyse a randomised treatment trial in severe malaria [21]; see the Methods section for further details on RF. Throughout this paper we refer to subgroup analysis and TEH interchangeably. Clinically relevant subgroups are a consequence of TEH. We take the convention that a subgroup is said to occur when the optimal treatment allocation changes, whereas heterogeneity more broadly suggests a systematic differential benefit of any one treatment. It is important to distinguish between such crossover and non-crossover TEH (see Methods), the former directly resulting in a treatment allocation that is dependent on patient characteristics [22]. Non-crossover TEH can result in patient-dependent optimal treatment allocation, but only when additional factors (e.g. cost or side effects) are brought into account to calculate the overall utility of each treatment.
We reiterate the principle that subgroups of clinical importance identified through a retrospective data analysis, from a trial not explicitly designed to identify these subgroups, ultimately need to be validated in a focussed, independent, follow-up RCT [1]. Subgroup analysis typically exploits data from trials that were designed to answer a different primary question not involving subgroups, and hence the analysis cannot by itself provide a complete picture of the evidence. In this respect, any ML subgroup analyses should seek to establish the strength of evidence that heterogeneous treatment effects are real (true positives). Establishing and controlling the false positive rate of the discovery procedure mitigates the risk of following false leads in subsequent confirmatory trials targeting the putative subgroup, and aids in the communication of evidence from the analysis. The following sections outline a formal methodology for exploratory analysis with strict control of the type I error.
Predefining an ML subgroup statistical analysis plan (ML subgroup-SAP)
Modern statistical and ML methods are able to automate the discovery of subgroups in high-dimensional data, and statistical scripting and programming packages such as R or Python allow the analyst to construct routines that take trial data as input and apply statistical or ML models to the data to identify potential heterogeneity. Here we consider both crossover TEH, whereby the subgroup is characterised by the set of patients predicted to benefit from a change in treatment compared to the current standard of care, and non-crossover TEH, whereby the standard of care is everywhere optimal but the benefits vary systematically across patient strata. The standard of care should be defined prospectively (before looking at the data), even if the analysis is retrospective.
In order to maintain the transparency of the evidence, an ML subgroup-SAP should be prespecified before any exploration of the primary RCT data has taken place. Failure to do so runs the risk of biasing the results [23]. When formulating the analysis plan, covering either the ML or statistical method (model) used for discovery, and the set of potential stratifying measurements used by the method, researchers should be cautioned against throwing in every possible variable and every flexible method. There is a principle here of 'no free lunch', or rather 'no free power'. The choice of discovery method and the potential variables to include is an important step. Methods that trawl through measurements looking for interactions are not panaceas or substitutes for careful thought, and the more judicious the a priori data selection and choice of discovery model, the higher the expected power and ability of the analysis to detect true effects [24].
The analysis plan should also include the specification of a test statistic that can compare overall patient benefit between any two groups and that can be used to quantify the type I error when declaring beneficial subgroups. The form of this test statistic is study-specific and should relate to the clinical outcome of interest, such as survival time, cure rate, or a quantitative measurement of treatment benefit. This will typically match that used in the original study protocol of the primary trial.
False positive control of crossover interactions: subgroup detection
Subgroup detection refers to the discovery of crossover TEH whereby the optimal treatment allocation changes. We propose using a held-out data approach to construct a test for a global null hypothesis of 'no true crossover TEH (no subgroups)'. Figure 1 illustrates this procedure using the example of a primary two-arm RCT where the original trial failed to detect an overall benefit of the experimental treatment. The approach is as follows. The trial data are repeatedly randomly divided into two subsets, with the ML method fitted independently and separately to each subset. Each ML algorithm (or statistical model), trained on one half of the data, is used to predict the individual treatment effects and thus the optimal treatments for subjects in the corresponding other half of the 'held-out' data, and vice versa. Combining the resulting subjects whose held-out predicted optimal treatment assignment differs from the standard of care forms a held-out subgroup of size ns from the original trial of sample size n. The actual treatment administered to these subjects in the primary RCT is random, such that in a balanced two-arm trial we would expect half of the subjects, \(\frac {1}{2} n_{s}\), to have received the standard of care and the other half the experimental treatment. This then facilitates a two-sample hypothesis test, using the test statistic defined in the analysis plan, with a null hypothesis of 'no improved subject benefit identified through the subgroup analysis plan'. The hypothesis test compares the outcomes of the patients who were predicted to benefit from the experimental treatment and who received the experimental treatment, to those predicted to benefit from the experimental treatment but who received the standard of care. A one-sided test would be appropriate if the test statistic measures patient benefit. If there is no true benefit arising from the alternative treatment in the subgroup identified by the ML model, then the distribution of outcomes should be the same in both groups, and thus the resulting p value is uniformly distributed over [0,1]. If K iterations of this procedure are run, randomising the 50-50 data-split at each iteration, then we obtain corresponding K distinct p values {p1,..,pK}. We note that each of these is conservative in that the discovery model on each subset has half the sample size to identify the subgroups. Finally it is possible to form a conservative aggregated p value, summarising {p1,..,pK}, to compute a global significance test for the presence of a benefitting subgroup. This aggregation can be done by adapting a method developed for p-value aggregation in high-dimensional regression [25]. In brief, if α is the level of control of the type I error (this is usually set to 0.05), then the set of p values can be merged into one test using the following formula (adapted from [25]):
$$\begin{array}{@{}rcl@{}} p_{\text{aggregate}} = \min_{\gamma \in [\alpha,1]} \left[1, (1-\log\alpha) Q_{\gamma}\left(\{ p_{i}\}_{i=1}^{K}\right) \right], \end{array} $$
Illustrative example of hypothesis testing in exploratory subgroup discovery using 1000 iterations of twofold cross-prediction. The example considers a primary RCT with two arms where a null hypothesis of 'no improvement from the experimental treatment' is not rejected; i.e. there is no significant evidence of the experimental treatment providing improvement over the standard of care. Each random division results in a corresponding p value against the null hypothesis of no benefitting subgroup. The p values are then aggregated for the overall test (Eq. 1)
where \(Q_{\gamma }\left (\{ p_{i}\}_{i=1}^{K}\right) = \min \left [ 1, Quantile_{\gamma }\left (\left \{\frac {p_{i}}{\gamma }\right \}_{i=1}^{K} \right)\right ] \). Quantileγ(·) computes the γ quantile of the set of p values which have been scaled by \(\frac {1}{\gamma }\). This procedure sweeps over γ∈[α,1] to find the minimum value in Qγ. The term 1− logα corrects for any inflation from searching over multiple values of γ. Alternately the analyst could fix γ in the analysis plan, such as γ=0.5 to select the median p value, and then compute:
$$ \begin{aligned} p_{\text{aggregate}}^{(\text{median})} &= Q_{0.5}\left(\left\{\frac{p_i}{0.5}\right\}_{i=1}^{K}\right)\\ &=\min\left(1, \text{Median}[ 2p_{1},2p_{2},.., 2p_{K}]\right). \end{aligned} $$
A proof of correctness for this aggregation procedure, for any value of γ∈(0,1), is provided in the supplementary Appendix, adapted from [25].
Note that if a true subgroup exists in the population from which the RCT trial participants are drawn, then \(\frac {n_{s}}{n} \times 100\%\) estimates the subgroup prevalence in that population. The more refined the subgroup, the smaller ns will tend to be, and hence the resulting test will have lower power to detect a true effect. That is, rarer subgroups are harder to detect. Intuitively this highlights how the original trial has reduced power to support more intricate subgroup discovery.
Optimality of this procedure is obtained when the random partitioning splits the data into two equal-size subsets. The standard error across the predictions will be proportional to \(1/(\sqrt {n_{1}} + \sqrt {n_{2}})\), where n1+n2=n is the total trial sample size. This is minimised for n1=n2=n/2. We illustrate this optimality using RF applied to simulated data; see supplementary material (Additional file 1). The number of random partitions, K, should be chosen large enough such that the aggregate p value stabilises, rendering the analysis reproducible under different initial random seeds. Stability with respect to K can be visualised by the traceplot of the aggregated p value for values k<Kmax. The exact number of random splits required will depend on the context. In our simulation studies, K=1000 is more than sufficient, with results stabilising around K=200. However, an appropriate choice of K is context-dependent.
False positive control of the added predictive benefit of the ML analysis
The primary outcome in a standard RCT will often be strongly associated with particular baseline covariates and prognostic factors which are predictive of the event rate, e.g. severity of disease or co-morbidities. Adjusting for these differences in baseline risk greatly enhances the power to detect subgroups of interest [26, 27]. Generalised linear models (GLMs) provide one of the most interpretable statistical model types for relating clinical outcome to a multivariate combination of prognostic factors and the randomised treatment. Using more complex and therefore less interpretable ML methods needs to be justified with respect to the added benefit over such a baseline model. In this context, the utility of ML methods is in their ability to detect non-linear interactions between prognostic factors and the randomised intervention. Using exactly the same data-splitting approach as for the discovery of statistically significant crossover subgroups, we can objectively evaluate the added benefit of the ML method. We illustrate the approach using a binary clinical outcome, yi∈{0,1} for the ith subject, and a logistic regression GLM, where
$$Pr(Y_{i} = 1) = \frac{\exp(Z_{i})}{1 + \exp(Z_{i})} $$
with the linear predictor Zi=Xiβ+Tiα, for prognostic variables X and randomised treatment indicator T. The procedure is summarised as follows.
For K iterations:
Split the data into two equally sized subsets with a balanced number of treated and untreated individuals in each subset.
Fit a GLM to each subset separately and record for each individual their out-of-sample linear predictor \(\widehat {Z}_{i}^{GLM} = X_{i} \widehat {\beta } + T_{i} \widehat {\alpha }\), where \(\left (\widehat {\beta }, \widehat {\alpha }\right)\) are obtained from the in-sample data fit.
Fit the ML method to each subset separately and predict the out-of-sample outcome probabilities, \( Pr(Y_{i} = 1) = \widehat {P}_{i} = \widehat {f}_{\text {ML}}(X_{i}, T_{i}), \) to obtain the corresponding log-odds out-of-sample prediction \(\widehat {Z}_{i}^{ML} = \log \left (\frac {\widehat {P}_{i}}{1 - \widehat {P}_{i}} \right) \) for each individual i.
Fit a 'stacked' GLM model to the full dataset using the n×2 matrix of prediction values \(\left (\widehat {Z}^{GLM}, \widehat {Z}^{ML}\right)\) as two independent covariate variables,
$$Pr(Y_{i} = 1) = \frac{\exp\left(\widehat{Z}_{i}^{GLM} \theta_{GLM} + \widehat{Z}_{i}^{ML} \theta_{ML}\right)}{1 + \exp\left(\widehat{Z}_{i}^{GLM} \theta_{GLM} + \widehat{Z}_{i}^{ML} \theta_{ML}\right)} $$
to obtain \(\left (\widehat {\theta }_{GLM}, \widehat {\theta }_{ML}\right)\). Record the p value, pk, assigned to an analysis of variance (ANOVA) test for the model with θML≠0 versus a model with θML=0.
Construct the aggregate p value from the set p1,..,pK using the adjustment method from Eq. 1.
This method is analogous to 'stacking', a popular ML technique whereby multiple competing models are aggregated to form a more powerful ensemble model [28]. We propose 'stacking' the standard accepted 'vanilla' statistical model (a GLM) alongside the predictions from an ML model. The aggregate p value formally tests the added benefit of the ML-based predictions.
Exploratory analysis
These ML-driven procedures for both testing the presence of crossover subgroups and for testing the added benefit of ML-predicted TEH provide valid p values. Under the null hypothesis, the probability of falling below the significance level α is upper bounded by α. However, this approach is by definition non-constructive: the output does not report an estimate of the discovered subgroup or an estimate of the treatment effect heterogeneity. A useful analogy is a conventional ANOVA F test of significance for levels of a factor. The ANOVA F test is an example of an 'omnibus test', which reports the significance (p value) that the outcome varies across the factors, rather than an estimate of the individual factor effects themselves. In a similar manner, our procedure simply reports a p value, subsequent to which further exploratory data analysis may be warranted. If the aggregated p value falls below a prespecified significance level, the ML model can be fit to the full dataset to estimate the individual treatment effects. This could use methods developed specifically for the determination of the structure of the TEH, e.g. [12, 14, 29], which use RF, answering questions such as: Which individuals are contained within the subgroup? Which covariates are predictive of the treatment effect heterogeneity? Is the subgroup clinically relevant? For example, this could be done via scatter plots of important covariates against the individual treatment effects. It is often possible to characterise a method detecting a true signal in the data by a few simple rules, for example using a decision tree (e.g. Fig. 2, panel D). By proceeding in this order, first evaluating the p value for the null hypothesis, then undertaking the exploratory analysis using the full data, formal control of the type I error is obtained.
Graphical visualisation and validation of treatment heterogeneity defined by non-crossover interactions in the SEAQUAMAT trial. Panels a and b show the univariate relationships to the individual predicted treatment effect for total parasite biomass and base deficit, respectively. The thick blue lines show spline fits to the data. Panel c shows the cumulative distribution of the p values for the added benefit of the ML model obtained by repeated data-splitting and stacking of the standard model alongside the ML model. Significance (at the 5% level) is obtained if the black line crosses above the red boundary. Panel d summarises the overall non-crossover interaction found by the random forest model with a pruned regression tree model fitted to the individual treatment effects. The leaves of the tree in panel d show the mean treatment effect (difference in mortality between artesunate and quinine)
Transparency and reproducibility
It is essential that all the findings and analysis paths taken are transparent and auditable to an external researcher. This can be achieved through the use of statistical notebooks, akin to the laboratory notebook in experimental science. Mainstream programming environments for data analysis (such as R and Python) provide open source notebooks such as R Markdown or Jupyter which seamlessly combine the analysis and the reporting. This allows all the exploratory analysis paths to be curated. Research recorded in a computational notebook is transparent, reproducible, and auditable. Auditability can be further improved without becoming burdensome through the use of version control repositories such as github (https://github.com) which record, timestamp, and preserve all versions and modifications of the analysis notebooks. In this way all of the steps, time lines, and historical evolution of the subgroup analysis are included, and the work flow is open to external criticism and interrogation. Any published results can be audited back to the original RCT. Any p values or statistical estimates that point toward subgroup effects that are reported subsequent to the heterogeneity tests need to be clearly labelled as such and treated with caution, due to the potential for evidence inflation and post selective inference that arises from using the data twice. We prefer to label such measures that follow after data interrogation as qualitative, or q values, as the formal statistical sampling uncertainty is often unknown [30].
Statistical and ML algorithms for subgroup detection
The optimal choice of statistical or ML algorithm will depend on the context of the data and on the primary endpoint of interest. When the number of candidate predictors is large but where the effects are likely to be linear, then penalised regression models such as the least absolute shrinkage and selection operator (lasso) or ridge are generally recommended [31]. An alternative, particularly if non-linear effects are expected, is random forests (RF). RF are one of the most popular and general ML methods in use today, in part as they consistently exhibit good empirical performance with little to no tuning of parameters. RF work by repeatedly building deep decision treesFootnote 1 on bootstrapped subsamples of the data, and then aggregating predictions made by the individual trees. RF can be applied to both classification and regression. Chapter 15 of reference [31] provides a detailed overview.
In brief, the standard RF algorithm for binary classification problems proceeds as follows (for example, as implemented in the R package randomForest). A user-determined number of binary decision trees are constructed, where each tree is constructed independently of one another. Usually 500 trees are sufficient to obtain approximate convergence, and this is the default setting in the R package. Each tree is built on a random bootstrapped version of the training data (using sampling with replacement). At each node in the decision tree, a user-determined number of predictive variables are sampled without replacement (for classification problems the default setting is the square root of the number of available predictors). The node is then defined as the optimal data partition over all splits amongst the sampled variables with respect to a user-defined objective function (as default the Gini impurity is used for classification). The decision tree is grown until the number of training cases in each leaf reaches a lower bound (the default is 1 for classification). Note that as the training data are split at each internal node in the tree, the sample size on each branch decreases monotonically down the tree. Prediction on a new test case is done by aggregating the individual tree predictions, thus giving a classification probability. RF are also applicable to data with continuous endpoints, with extensions to survival data [32], and further extensions to the general detection of treatment effect heterogeneity [14]. Some of the well-known advantages of RF are that they are generally insensitive to the tuning parameters used in the model (e.g. the number of trees, the parameters governing the depth of trees), and they can implicitly handle missing values. In our illustrative application, we use RF with the default parameter settings from the R package randomForest. This analysis can be exactly replicated using the compute capsule available on Code Ocean [19], and readers are encouraged to play with the default parameter settings should they wish to explore further.
ML-driven exploratory RCT subgroup analysis
Panel 1 summarises how ML methods can be used for exploratory analyses testing for the presence of significant crossover TEH which results in statistically significant subgroups. The framework we propose is novel, and it relies on recent results in the statistics literature for aggregating correlated p values into a single, reproducible p value for the null hypothesis 'no crossover TEH'. The core of the framework relies on random data-splitting and cross-prediction, leading to unbiased optimal treatment predictions. To increase transparency, we recommend using computational notebooks to document the process, ideally prespecified via an ML subgroup-SAP. In the following we illustrate how this framework is applied to a large randomised treatment trial in severe malaria, the analysis of which provides an open source computational template for ML exploratory subgroup analysis [19].
Antimalarial pharmacodynamics of artemisinin in severe malaria
Severe Plasmodium falciparum malaria is a medical emergency with case fatality rates ranging from 10 to 40% [33]. A recent major advance in the treatment of severe malaria has been the introduction of parenteral artesunate. In Asia, this has been shown to reduce mortality by a third [21], and in Africa by a fifth [34]. To illustrate the methodology advocated in this work, we use data from the definitive study of artesunate for severe malaria in Asia (SEAQUAMAT: South East Asian Quinine Artesunate Malaria Trial). This was a large multi-country randomised trial comparing intravenous quinine to intravenous artesunate [21].
The superiority of parenteral artesunate for severe malaria is now well established [35]. Thus, in this retrospective analysis the artesunate arm is considered 'standard of care'. The complete statistical analysis is published as an open source Code Ocean capsule and is entirely reproducible [19]. This analysis provides an easily adjusted template for new exploratory subgroup analyses of different datasets.
We chose to use RF to fit the data, one of the most popular and important ML methods in use today [20]. The RF method deals well with multiple correlated covariates, as is the case in these data. We first evaluate whether there is evidence for a subgroup of patients who would benefit from quinine treatment as opposed to artesunate. The subgroup analysis does not reject the null hypothesis of 'homogeneous optimal treatment allocation' (p=1), showing that there is no evidence in the data of any subgroup benefitting from quinine.
This analysis was followed by examining the added benefit of the predictive RF ML model relating patient survival to the baseline measurements and treatment. An aggregation of the p values obtained by repeated data-splitting and 'stacking' of the out-of-sample ML model predictions alongside the validated best linear predictor (the linear predictor on the logistic scale comprising Glasgow coma scale, base deficit, and treatment [36]) showed a strongly significant added benefit of the RF ML model (p = 10−6, Fig 2, panel C). The statistical significance of the repeated data-splitting and cross-prediction procedure can be assessed visually by comparing the cumulative distribution of the resulting p values against the boundary curve as given by Eq. 1.
Further exploratory analysis attempted to characterise possible interactions explaining this variation in predicted individual treatment effect. This analysis showed that significant TEH can be partially explained by the total non-sequestered parasite biomass (panel A) and the base deficit (panel B). This treatment heterogeneity can be summarised using a pruned classification and regression tree (CART) model decision tree (panel D). This suggests that the greatest benefit of parenteral artesunate (estimated as 20 percentage points difference in mortality) is seen in patients with large numbers of circulating young ring stage parasites (an interaction between total parasitaemia and % of young rings). This is not highlighting a clinically relevant subgroup, but it helps elucidate the mechanism of action of artemisinin, a useful exercise in light of emerging drug resistance [37]. Moreover, these results are concordant with the current proposed mechanism of action of the artemisinin derivatives and the importance of the artemisinin-specific mode of action in the treatment of severe malaria. Artemisinin derivatives kill a broader range of parasite stages compared to quinine, notably the younger circulating ring forms, thereby reducing further sequestration and subsequent death in patients with a high parasite biomass [38].
This work demonstrates how modern machine learning algorithms can be trained safely to discover treatment effect heterogeneity in a way that rigorously controls for type I error. The validity of our data-splitting and cross-prediction procedure holds irrespective of the method used, provided that samples are independently recruited from the study population—the same assumption necessary for the validity of cross-validation methods. If this is not the case, for example if patients are recruited in pairs, or are related in some manner, then adjustments need to be made to ensure that the p value reports the correct out-of-sample evidence. The choice of discovery algorithm should depend on the measurement variables collected (how many, and of which type) and the primary or secondary outcomes of the study for which subgroup analysis is to be applied, e.g. survival time, binary outcome, continuous risk score. The specification of the stratifying measurements used by the method needs careful thought under a principle of 'no free power' in that feeding in irrelevant predictor variables will reduce the ability to detect true signals [24].
The approach we advocate here is generic. Exploring the benefit of one predictive model over another, either traditional or machine learning, can be done within a common statistical machine learning analysis plan, where the null hypothesis is that Model B provides no additional benefit in prediction over that of Model A. In our corresponding compute capsule available on Code Ocean [19], we implemented a test for the added benefit of random forests over a generalised linear model, and the reader can easily adapt this code to compare other models, traditional or otherwise, as long as each model can provide a prediction of the outcome following treatment.
It is important that the analysis is transparent, that the methods, data transformations, and analytic procedures are laid out and documented in an auditable plan, and that any code base used is properly documented and available for scrutiny. We recommend the use of open source repositories such as github or cloud computing services such as Code Ocean for fully reproducible data analyses. By following some simple guidelines, we hope to improve upon the reliability and stability of subgroup analysis reported in the literature. Recent advances in statistical machine learning algorithms along with recent advances in measurement technologies have the potential to impact heavily and positively in the advancement of medical science. However, alongside these advances great care must be taken to ensure that the integrity of the statistical analysis and the validity of the evidence base are upheld at all times.
All data are available via Code Ocean at the following url: https://codeocean.com/capsule/2760408/tree/v2.
Figure 2, panel D gives an example of a shallow decision tree. In contrast, RF build deep decision trees from subsamples of the data where the branches (questions) descend until only a small number of samples lie within each leaf of each tree. Predictions on new data are then averaged across all trees.
GLM:
Generalised linear model
RCT:
Randomised controlled trial
Random forest(s)
ML subgroup-SAP:
Machine learning subgroup statistical analysis plan
TEH:
Treatment effect heterogeneity
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We thank the Mahidol Oxford Research Unit for providing us with the data from the SEAQUAMAT trial. We are grateful to Nicholas White and Stije Leopold for valuable input concerning the interpretation of the SEAQUAMAT analysis. We thank Aimee Taylor for help with the graphical representation of the method. We thank Dr Rajen Shah for pointing us to the work by Meinshausen et al. on p-value aggregation.
The authors received no specific funding for this work.
Mahidol Oxford Tropical Medicine Research Unit, Faculty of Tropical Medicine, Mahidol University, Rajvithi Road, Bangkok, 10400, Thailand
James A. Watson
Nuffield Department of Medicine, University of Oxford, Oxford, OX3 7LF, UK
James A. Watson & Chris C. Holmes
Department of Statistics, University of Oxford, 29 Saint Giles', Oxford, OX1 3LB, UK
Chris C. Holmes
CCH and JAW contributed equally to all parts of this work. Both authors read and approved the final manuscript.
Correspondence to James A. Watson.
All patients in the SEAQUAMAT study gave full informed consent for study participation. The study was approved by the Oxford University ethics review board and local ethics committees.
Additional file 1
Proof of correctness for p value adjustment.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Watson, J.A., Holmes, C.C. Machine learning analysis plans for randomised controlled trials: detecting treatment effect heterogeneity with strict control of type I error. Trials 21, 156 (2020). https://doi.org/10.1186/s13063-020-4076-y
Heterogeneous treatment effects
Subgroup statistical analysis plan | CommonCrawl |
Accepted Manuscript: Finite chemical potential equation of state for QCD from an alternative expansion scheme
Title: Finite chemical potential equation of state for QCD from an alternative expansion scheme
The Taylor expansion approach to the equation of state of QCD at finite chemical potential struggles to reach large chemical potential μ B . This is primarily due to the intrinsic diffculty in precisely determining higher order Taylor coefficients, as well as the structure of the temperature dependence of such observables. In these proceedings, we illustrate a novel scheme [1] that allows us to extrapolate the equation of state of QCD without suffering from the poor convergence typical of the Taylor expansion approach. We continuum extrapolate the coefficients of our new expansion scheme and show the thermodynamic observables up to μ B / T ≤ 3.5.
Parotto, Paolo; Borsányi, Szabolcs; Fodor, Zoltan; Guenther, Jana N.; Kara, Ruben; Katz, Sandor D.; Pásztor, Attila; Ratti, Claudia; Szabó, Kalman K.
David, G.; Garg, P.; Kalweit, A.; Mukherjee, S.; Ullrich, T.; Xu, Z.; Yoo, I.-K.
Diffusions interacting through a random matrix: universality via stochastic Taylor expansion
Dembo, Amir ; Gheissari, Reza ( August 2021 , Probability Theory and Related Fields)
Abstract Consider $$(X_{i}(t))$$ ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ J = ( J ij ) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ ( X i ( t ) ) , initialized from some $$\mu $$ μ independent of $${\mathbf {J}}$$ J , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.
Power expansion for heavy quarkonium production at next-to-leading order in e+e− annihilation
Lee, Kyle ; Sterman, George ( September 2020 , Journal of High Energy Physics)
A bstract We study heavy quarkonium production associated with gluons in e + e − annihilation as an illustration of the perturbative QCD (pQCD) factorization approach, which incorporates the first nonleading power in the energy of the produced heavy quark pair. We show how the renormalization of the four-quark operators that define the heavy quark pair fragmentation functions using dimensional regularization induces "evanescent" operators that are absent in four dimensions. We derive closed forms for short-distance coefficients for quark pair production to next-to-leading order ( $$ {\alpha}_s^2 $$ α s 2 ) in the relevant color singlet and octet channels. Using non-relativistic QCD (NRQCD) to calculate the heavy quark pair fragmentation functions up to v 4 in the velocity expansion, we derive analytical results for the differential energy fraction distribution of the heavy quarkonium. Calculations for $$ {}^3{S}_1^{\left[1\right]} $$ 3 S 1 1 and $$ {}^1{S}_0^{\left[8\right]} $$ 1 S 0 8 channels agree with analogous NRQCD analytical results available in the literature, while several color-octet calculations of energy fraction distributions are new. We show that the remaining corrections due to the heavy quark mass fall off rapidly in the energy of the produced state. To explore the importance of evolutionmore »at energies much larger than the mass of the heavy quark, we solve the renormalization group equation perturbatively to two-loop order for the $$ {}^1{S}_0^{\left[8\right]} $$ 1 S 0 8 case.« less
Phase transitions and resilience of the magnetic dual chiral density wave phase at finite temperature and density
https://doi.org/https://doi.org/10.1103/PhysRevD.106.016011
William Gyory and Vivian de la Incera ( July 2022 , Physical review and Physical review letters index)
We study the phase transitions at finite temperature and density of the magnetic dual chiral density wave (MDCDW) phase. This spatially inhomogeneous phase emerges in cold, dense QCD in the presence of a strong magnetic field. Starting from the generalized Ginzburg-Landau (GL) expansion of the free energy, we derive several analytical formulas that enable fast numerical computation of the expansion coefficients to arbitrary order, allowing high levels of precision in the determination of the physical dynamical parameters, as well as in the transition curves in the temperature vs chemical potential plane at different magnetic fields. At magnetic fields and temperatures compatible with neutron star (NS) conditions, the MDCDW remains favored over the symmetric ground state at all densities. The phase's "resilience" manifests in (1) a region of small but nonzero remnant mass and significant modulation at intermediate densities, originating in part from the nontrivial topology of the lowest Landau level, and (2) a region of increasing condensate parameters at high densities. Our analysis suggests the MDCDW condensate remains energetically favored at densities and temperatures much higher than previously considered, opening the possibility for this phase to be a viable candidate for the matter structure of even young neutron stars producedmore »by binary neutron star (BNS) mergers.« less
A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system
https://doi.org/10.1090/mcom/3642
Liu, Chun ; Wang, Cheng ; Wise, Steven ; Yue, Xingye ; Zhou, Shenggao ( September 2021 , Mathematics of Computation)
null (Ed.)
In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough errormore »estimate (to establish the ℓ ∞ \ell ^\infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.« less
Semi-implicit hybrid discrete (H$^T_N$) approximation of thermal radiative transfer
https://doi.org/https://link.springer.com/article/10.1007/s10915-021-01686-7
R.G. McClarren, J.A. Rossmanith ( November 2021 , Journal of scientific computing)
The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P$_N$ (spherical harmonics) or the S$_N$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$^T_N$) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P$_N$ and S$_N$, and indeed reduces to each of these approximations in various limits: H$^1_N$ $\equiv$ P$_N$ and H$^T_0$ $\equiv$ S$_T$. We prove that H$^T_N$ results in a system of hyperbolic partial differential equations for all $T\ge 1$ and $N\ge 0$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensivemore »due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop a numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge-Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H$^T_N$ ansatz and the numerical discretizations.« less
Parotto, Paolo, Borsányi, Szabolcs, Fodor, Zoltan, Guenther, Jana N., Kara, Ruben, Katz, Sandor D., Pásztor, Attila, Ratti, Claudia, and Szabó, Kalman K.. Finite chemical potential equation of state for QCD from an alternative expansion scheme. Retrieved from https://par.nsf.gov/biblio/10336539. EPJ Web of Conferences 259. Web. doi:10.1051/epjconf/202225910015.
Parotto, Paolo, Borsányi, Szabolcs, Fodor, Zoltan, Guenther, Jana N., Kara, Ruben, Katz, Sandor D., Pásztor, Attila, Ratti, Claudia, & Szabó, Kalman K.. Finite chemical potential equation of state for QCD from an alternative expansion scheme. EPJ Web of Conferences, 259 (). Retrieved from https://par.nsf.gov/biblio/10336539. https://doi.org/10.1051/epjconf/202225910015
Parotto, Paolo, Borsányi, Szabolcs, Fodor, Zoltan, Guenther, Jana N., Kara, Ruben, Katz, Sandor D., Pásztor, Attila, Ratti, Claudia, and Szabó, Kalman K.. "Finite chemical potential equation of state for QCD from an alternative expansion scheme". EPJ Web of Conferences 259 (). Country unknown/Code not available. https://doi.org/10.1051/epjconf/202225910015. https://par.nsf.gov/biblio/10336539.
place = {Country unknown/Code not available}, title = {Finite chemical potential equation of state for QCD from an alternative expansion scheme}, url = {https://par.nsf.gov/biblio/10336539}, DOI = {10.1051/epjconf/202225910015}, abstractNote = {The Taylor expansion approach to the equation of state of QCD at finite chemical potential struggles to reach large chemical potential μ B . This is primarily due to the intrinsic diffculty in precisely determining higher order Taylor coefficients, as well as the structure of the temperature dependence of such observables. In these proceedings, we illustrate a novel scheme [1] that allows us to extrapolate the equation of state of QCD without suffering from the poor convergence typical of the Taylor expansion approach. We continuum extrapolate the coefficients of our new expansion scheme and show the thermodynamic observables up to μ B / T ≤ 3.5.}, journal = {EPJ Web of Conferences}, volume = {259}, author = {Parotto, Paolo and Borsányi, Szabolcs and Fodor, Zoltan and Guenther, Jana N. and Kara, Ruben and Katz, Sandor D. and Pásztor, Attila and Ratti, Claudia and Szabó, Kalman K.}, editor = {David, G. and Garg, P. and Kalweit, A. and Mukherjee, S. and Ullrich, T. and Xu, Z. and Yoo, I.-K.} } | CommonCrawl |
Machine Vision and Applications
June 2019 , Volume 30, Issue 4, pp 629–644 | Cite as
Gait-based age progression/regression: a baseline and performance evaluation by age group classification and cross-age gait identification
Chi Xu
Yasushi Makihara
Yasushi Yagi
Jianfeng Lu
Gait is believed to be an advanced behavioral biometric that can be perceived at a large distance from a camera without subject cooperation and hence is favorable for many applications in surveillance and forensics. However, appearance differences caused by human aging may significantly reduce the performance of gait recognition. Modeling the aging process on gait features is one of the possible solutions to this problem, and it may inspire more potential applications, such as finding lost children and examining health status. To the best of our knowledge, this topic has not been studied in the literature. Motivated by the fact that aging effects are mainly reflected in the shape and appearance deformations of the gait feature, we propose a baseline algorithm for gait-based age progression and regression using a generic geometric transformation between different age groups, in conjunction with the gait energy image, which is an appearance-based gait feature frequently used in the gait analysis community, to render gait aging and reverse aging effects simultaneously. Various evaluations were conducted through gait-based age group classification and cross-age gait identification to validate the performance of the proposed method, in addition to providing several insights for future research on the subject.
Gait aging modeling Age progression/regression Performance evaluation Age group classification Cross-age gait identification
The online version of this article ( https://doi.org/10.1007/s00138-019-01015-x) contains supplementary material, which is available to authorized users.
Gait recognition is a useful technique to authenticate a person from his/her walking style and has increasingly attracted attention in the computer vision community. Compared with other physiological biometric modalities, such as DNA, fingerprints, irises, and faces, gait has promising advantages because it is difficult to obscure and imitate and is available at a large distance from a camera without subject cooperation (e.g., closed-circuit television (CCTV) footage with low image resolution). Gait recognition has therefore been in increasing demand in many applications for surveillance and forensics [4, 14, 25].
GEI examples of different subjects at various ages
However, gait recognition often suffers from the influence of various covariates, such as view [28, 44], walking speed [10, 45], clothing [12, 20], and elapsed time [31]. Among these covariates, a long elapsed time period, that is, aging, is one of the challenging factors that often occurs in real application situations (e.g., looking for long-lost children and the recognition of fugitives that escaped many years ago). Although the study [31], which is the only work in which researchers have investigated the effect of elapsed time on gait recognition performance, demonstrated that gait does not drastically change within 1 year, studies on gait-based age group classification [1, 6, 7, 30] and age estimation [27] have reported that gait has obvious differences among children, adults, and the elderly in terms of, for example, stride length, body length, and head-to-body ratio, which can be easily understood from Fig. 1. In Fig. 1, the gait of different subjects at various ages are shown with the gait energy image (GEI) [11], which is a simple yet effective gait feature frequently used in the gait recognition community. We can observe an obvious change in the body shape such as head-to-body ratio during the growth of children (e.g., less than 20 years old), which indicates body height growth in the size-normalized gait silhouette images. Additionally, the relative stride to body also tends to be larger for the children and teenagers. When people get older (e.g., after 50 years old), the stride tends to be smaller due to the decrease of physical strength, whereas the middle-age spread and stoop appear for most middle-aged and elderly subjects simultaneously. Consequently, these types of common changes in gait features may therefore significantly degrade the performance of gait recognition under age variation.
To reduce the appearance difference caused by the age gap between the gallery and probe, modeling the gait aging process is one of the possible solutions. More specifically, gait aging modeling includes a simulation with natural aging and reverse aging effects, that is, age progression (i.e., prediction of the future gait) and age regression (i.e., estimation of the previous gait), which is similar to the definition of aging and reverse aging effects considered in existing studies on face analysis [18, 38, 39, 40, 41, 43, 47]. However, in real scenarios, such as surveillance, facial images may not work well because of low image resolution or even occlusion by a face mask. By contrast, human gait can be well perceived because of its unique ascendancy, and hence, gait-based age progression/regression has high application potential. In addition to enhancing the surveillance capabilities for locating lost children and escaped criminals, it could also help with some latent applications, such as entertainment and health examination (e.g., a young person with large middle-age spread in age progression may need to be conscious of staying in shape). However, to the best of our knowledge, there is no prior work that considers age progression/regression using gait features.
A few works [9, 18, 21] have tackled face-based age regression; however, most existing facial aging modeling works have focused on age progression, which is mainly classified into two categories [38, 47]: (i) physical model-based, which models the change in biological facial parameters for texture/shape with age, and (ii) prototype-based, which transforms aging patterns between the prototypes (e.g., averaged faces) of two predefined age groups. Although these methods have achieved reasonable performance, it is not feasible to extend most of them to gait-based age progression/regression for the following two reasons.
Most existing face-based age progression/regression methods require paired samples for different ages of the same individual, or even real aging sequences over a long age span to better model the complex biological pattern and physical mechanisms of aging [47], particularly for physical model-based approaches. It is still difficult to collect multiple facial images over a long age span; however, short-term (e.g., 10 years) aging sequences are available in some face databases (e.g., MORPH dataset [34]) or can be obtained from social network sites (e.g., photographs of celebrities on Facebook and Twitter [38]). However, there is no existing gait database that contains such aging sequences over a 1-year span because it is quite difficult or almost impossible to collect gait sequences of the same person for his/her different ages in real-world surveillance scenarios. Therefore, most face-based age progression/regression methods cannot be applied in the case of gait.
Existing face-based age progression/regression studies focus more on handling color and texture changes (e.g., wrinkles, muscles, and skin) rather than geometric variations to better reflect the aging patterns of facial images. However, these methods are unsuitable for modeling gait aging patterns because texture and color changes are not apparent in gait features during the human aging process [22], as shown in Fig. 1. By contrast, geometric transformations are more important for gait because the variations caused by aging mainly occur in shape and appearance deformations.
Motivated by these problems, we propose a baseline algorithm for gait-based age progression/regression using generic warping between two age groups, and evaluate the performance of simulations using both human perception and automatic algorithms. Note that, in this paper, we do not aim at technical novelty for the age progression/regression method, but at providing a baseline algorithm and suitable set of experimental evaluation results that contribute to the human gait analysis research community because there is no prior work on image-based gait age progression/regression to the best of our knowledge. The main contributions of this work are twofold.
1. The first attempt at gait-based age progression and regression.
This is the first work on the topic of gait-based age progression/regression to the best of our knowledge. Because of the lack of multiple gait aging samples of the same individual, we adopt a similar notion to prototype-based methods [5, 15, 35] in face-based age progression/regression as a baseline algorithm. More specifically, the GEIs are first averaged over subjects that belong to each age group (referred to as the mean-GEI), and then, the transformation between the mean-GEIs of two neighboring age groups is generated using free-form deformation (FFD) [37] to render the general aging process of humans and preserve some personalized gait characteristics (e.g., walking style and clothing appearance) for a specific subject, simultaneously.
2. Subjective and objective quantitative evaluation through age group classification and cross-age gait identification.
We evaluated the simulation results of the proposed method on the OU-ISIR Gait Database, Large Population Dataset with Age (OULP-Age) [46], which is the world's largest gait database with age information. Similar to face-based age progression/regression works [40, 41], we conducted both subjective (human perception) experiments and objective (machine perception) experiments as quantitative measurements by implementing age group classification and cross-age gait identification to validate both the quality of aging patterns and preservation of identity for the simulation results.
The outline of the paper is as follows. In Sect. 2, we review existing face-based age progression/regression methods and studies on gait analysis relevant to age. We then introduce the proposed gait-based age progression/regression method in Sect. 3 and present various performance evaluations of the proposed method in Sect. 4. In Sect. 5, we analyze several failure cases in the experiments and finally conclude this paper in Sect. 6 and discuss future work on this topic.
2.1 Face-based age progression/regression
Currently, extensive studies on face-based age progression have been conducted, with approaches mainly classified into two groups: physical model-based and prototype-based. Physical model-based methods correlate biological facial aging patterns with human age using complex models, such as the statistical face model [18] and-or graph [40, 41], concatenational graph evolution aging model [39], and craniofacial growth model [33]. By contrast, a limited number of works on facial age regression [9, 18, 21] have been based on physical models by simply removing textures from facial surfaces. However, to model the intricate aging mechanisms, sufficient training samples with aging sequences over a long age span for each individual are required, which are almost unlikely to be collected for gait videos.
In an early study on face-based age progression, Burt and Perrett [5] divided the training faces into several age groups and created an averaged face as the prototype of each group, with the transformation between prototypes serving as the aging model. Based on this, other prototype-based approaches [13, 15, 36] were proposed subsequently to involve more individual characteristics. Shu et al. [38] proposed an age group-specific dictionaries-based age progression method that synthesizes the aging pattern formed by the learned dictionary bases and an extra personalized facial pattern (e.g., a mole). Although some of these approaches have achieved relatively better performance, they still require short-term paired samples of the same subject. Moreover, to present bidirectional simulations, that is, both age progression and age regression, prototype-based methods may need to inversely retrain the model [47].
To date, deep learning-based approaches [43, 47] have achieved state-of-the-art performance. In particular, Zhang et al. [47] proposed a conditional adversarial autoencoder (CAAE) that first achieved age progression and regression simultaneously without using the paired samples of each person. Although it presented reasonably good simulation results, it is still unsuitable to be simply applied in the case of gait, considering the differences between face and gait aging patterns (e.g., texture changes, such as wrinkles on a face, do not exist in a gait).
Overview of the proposed method
2.2 Age-related gait analysis
Effect of elapsed time on gait recognition In [31], Matovski et al. studied the effect of elapsed time on gait recognition performance, which is the only work that considers this problem. To investigate the pure influence of elapsed time, the other covariates, such as the clothing worn by the subjects and the environment, were controlled during image capture. The experimental results illustrated that gait is relatively invariant over a short time period (i.e., 9 months), and hence, a short-elapsed time period between the gallery and probe does not significantly affect gait recognition performance. However, no study has examined the effects of a long-elapsed time period (e.g., 10 years) on gait recognition because of the difficulties of data collection.
Gait-based age group classification and age estimation By contrast, there is a rich body of literature on gait-based age group classification and age estimation [1, 6, 7, 22, 24, 27, 30], which are based on the analysis of various gait features. Davis [7] classified children (3–5 years old) and adults (30–52 years old) by analyzing the gait differences in terms of leg length, stride width, and stride frequency; Begg et al. [1] classified young people (28.4 years old average and 6.4 years standard deviation) and the elderly (69.2 years old average and 5.1 years standard deviation) using minimum foot clearance data. In [30], the frequency-domain feature [28], which is another famous appearance-based gait feature, was used to classify four age groups: children (under 15 years old), adult males, adult females, and the elderly (over 65 years old). Chuen et al. [6] investigated the correlation of gait features (e.g., stride length, body length, and head-to-body ratio) among children and adults.
Researchers have also made great progress on gait-based age estimation, with the approaches typically using the gait feature (e.g., GEI) combined with an age estimation model, including Gaussian process regression [26, 27], multilabel-guided subspace [22], and ordinary preserving manifold analysis [23, 24].
The aforementioned studies on gait-based age group classification and age estimation provide evidence that the human gait or more specifically, the gait feature, contains discriminative aging patterns, which display the feature differences among different ages, and provides the possibility of gait-based age progression and regression.
3 Gait-based age progression/regression using FFD
An overview of the proposed method is shown in Fig. 2. Using the training set composed of multiple GEI samples with various age values, we first compute the mean-GEIs by averaging all GEI samples that belong to each predefined age group, which represent a general gait of each age group that mitigates the gait variations among different individuals. We then generate the transformation fields between mean-GEIs from two adjacent age groups because the gait changes between neighboring age groups are relatively small, and this results in less distortion in the deformation. In the testing phase, given an input GEI sample and its age, we first morph it into its nearest neighboring age groups with the corresponding generic transformation fields and then simulate the subsequent age groups by taking the morphed GEI as the input, which renders the entire aging process from the current age step by step.
The details of the proposed method are given in the following sections.
3.2 GEI representation
Compared with the other gait feature representations [16, 28, 42], GEI is the most popular and widely used gait feature in gait analysis studies because it is simple yet effective. Hence, we also use GEI as our gait representation.
As a preprocess, gait silhouettes are first extracted from the captured gait video sequences through background subtraction-based graph-cut segmentation [29]. The size-normalized and registered silhouette sequences are then obtained by normalizing the height and registering the region center [28]. After detecting the gait period from the size-normalized and registered silhouette sequences [28], GEI G(x, y) is extracted by averaging the silhouettes over one gait period T [frames] as
$$\begin{aligned} G(x,y) = \frac{1}{T} \sum _{t = 1}^{T} I(x,y,t), \end{aligned}$$
where I(x, y, t) is the binary silhouette value at position (x, y) from the tth frame in the size-normalized silhouette sequence, with 0 and 255 indicating the background and foreground, respectively. Examples of GEI can be found in Fig. 1. We can see that GEI not only successfully reflects the static body parts (i.e., where the intensity value is 255), but also represents well the dynamic gait feature (i.e., where the intensity is between 0 and 255), such as arms and legs.
3.3 Geometric transformation using FFD
To better render the change of gait characteristics during the human aging process, we first divide the ages into several age groups. The mean-GEI is then computed by averaging over all GEI templates for each age group considering its insensitiveness to the variations among individual subjects (e.g., different clothes and various walking poses), and hence, the gait diversity, in addition to the effects of gait fluctuations within the gait period (e.g., look down temporarily), is mitigated by the mean-GEI. Consequently, the mean-GEI is able to represent a general gait of each age group (i.e., a prototype), and the differences among mean-GEIs can well reflect the inter-variations among different age groups for a general gait aging process by mitigating intra-variations for each age group. In addition, the mean-GEI also contains the gait individuality with different intensities (e.g., an intensity value close to 255 in the torso indicates a slim subject, whereas an intensity value between 0 and 255 indicates an overweight subject) simultaneously.
Because there are large variations among the GEIs of different subjects even within the same individual age, the finer age groups (e.g., individual ages) retain more detailed aging patterns but result in more variations in the mean-GEI of each age group due to the reduced number of training samples (especially for elderly subjects), whereas the coarser age groups lead to less detailed descriptions of the change in gait characteristics but create a more stable mean-GEI representation for each age group. Thus, we consider the appropriate age groups realizing the trade-off between the stability in representation of mean-GEI and the sufficient descriptions of the whole gait aging process.
We next utilize a warping field between two neighboring mean-GEIs to reflect the general gait changes between adjacent age groups. To achieve this, we use the notion of FFD with piecewise linear interpolation because FFD is suitable for representing the transformation of non-rigid objects (e.g., the human body) due to its high flexibility [37]. Moreover, it can preserve the derivative continuity of adjacent regions and never corrupt appearance-based gait features [8], which helps to maintain the personalized gait characteristics of specific subjects. Given a mean-GEI pair from neighboring age groups, that is, a source and target, a traditional transformation typically transforms the source into the target so as to minimize the difference between the target and transformed source. However, to achieve a bidirectional simulation, that is, age progression and regression, an inverse transformation from the target to the source needs to be retrained, which causes asymmetry between the source and target and increases the computational cost. We therefore adopt an intermediate transformation that transforms both the source and target into the intermediate state, which has been applied in cross-view gait recognition [8] and thus could freely render age progression and regression simultaneously because of the symmetric and equalized deformation degree between two directions (i.e., deformation from the source to the intermediate state equals the reverse of that from the target to the intermediate state).
Illustration of the transformation. a Original source mean-GEI from age group 5–9. b Intermediate warping field from the source to the intermediate state. c Transformed source mean-GEI using the warping field in b. d Original target mean-GEI from age group 10–14. e Intermediate warping field from the target to the intermediate state. f Transformed target mean-GEI using the warping field in e. Comparing c and f, it is obvious that the source and target mean-GEI become very similar after the transformation, which demonstrates the effectiveness of the FFD-based intermediate transformation
Let a pair of source and target mean-GEIs from neighboring age groups i and \(i+1\) be \(\bar{G}_{i}, \bar{G}_{i+1}\in \mathbb {R}^{H \times W}\)\((i = 1, \ldots , N-1)\), respectively, where W and H are the width and height of the GEI, respectively, and N is the number of defined age groups. We first allocate a set of control points on both \(\bar{G}_{i}\) and \(\bar{G}_{i+1}\) and then define the transformation from source \(\bar{G}_{i}\) to the intermediate state using a set of two-dimensional displacement vectors \(\vec {u}_{i}\) on the control points. An entire warping field \(F(\vec {u}_{i})\) from the source to the intermediate state is subsequently obtained using piecewise linear interpolation, and similarly, the warping field from target \(\bar{G}_{i+1}\) to the intermediate state is represented by \(F(-\vec {u}_{i})\). Consequently, we optimize the displacement vectors \(\vec {u}_{i}\) by minimizing the difference between the transformed source \(\bar{G}_{i} \circ F(\vec {u}_{i})\) and transformed target \(\bar{G}_{i+1} \circ F(-\vec {u}_{i})\) in the intermediate domain as
$$\begin{aligned} \vec {u}^{*}_{i} = \arg \min _{\vec {u}_{i}} \Vert \bar{G}_{i} \circ F(\vec {u}_{i}) - \bar{G}_{i+1} \circ F(-\vec {u}_{i}) \Vert _{F}^{2} + \lambda R(\vec {u}_{i}), \end{aligned}$$
where \(\circ \) indicates a transformation operator and \(R(\vec {u}_{i})\) is a linear elastic smoothness term on the displacements between adjacent control points [19], with a hyperparameter \(\lambda \) controlling the smoothness. The optimization of Eq. (2) is solved using gradient descent, and the visualization of the intermediate transformation is illustrated in Fig. 3.
Example of year-by-year age progression. a Input testing GEI with the age of 5 years old. b Intermediate warping field between the age groups 5–9 and 10–14. c Simulations from 6 to 10 years old, with the corresponding morphing ratio used for each simulation result. From the year-by-year simulations in c, we can clearly observe the gradual transition from 5 to 10 years old
3.4 Morphing-based age progression/regression
Once we obtain the warping fields between each pair of adjacent mean-GEIs, we simulate the future/previous gaits of a given subject by first morphing the input GEI into its nearest neighboring age group and then extending it to the other age groups sequentially. Additionally, based on the assumption about a uniform transition between adjacent age groups, we can present the simulation of any target age by adjusting the morphing ratio during the transformation, which helps to improve the effectiveness for cross-age gait recognition in surveillance and the attractiveness for applications, such as entertainment, with a complete year-by-year aging/reverse aging process. By contrast, many face-based age progression/regression approaches [38, 40, 41, 47] only generate a single simulated face for each age group.
Let the absolute age difference between two neighboring mean-GEIs \(\bar{G}_{i}\) and \(\bar{G}_{i+1}\) be \(|\bar{a}_{i} - \bar{a}_{i+1}|\). Because we consider an intermediate transformation between the source and target mean-GEI pairs, the absolute age difference between the intermediate state and original source/target approximates to \(|\bar{a}_{i} - \bar{a}_{i+1}|/2\) because we assume a uniform transition between adjacent age groups. Given input testing GEI \(G_{\mathrm{in}}\) with its age of \(a_{\mathrm{in}}\) that belongs to age group i, simulation \(G_{\mathrm{sim}}\) at the age \(a_{\mathrm{sim}}\) that belongs to age group \(i+1\) is obtained as
$$\begin{aligned} G_{\mathrm{sim}} = {G}_{\mathrm{in}} \circ F(\alpha \vec {u}^{*}_{i}), \end{aligned}$$
where morphing ratio \(\alpha \) is determined based on the ratio of the age difference between the simulation and input GEI, and the age difference between the intermediate state and original source as
$$\begin{aligned} \alpha = \frac{2|a_{\mathrm{sim}} - a_{\mathrm{in}}|}{|\bar{a}_{i} - \bar{a}_{i+1}|}. \end{aligned}$$
Thus, we render the simulation of any intermediate age inside the adjacent age groups, as shown in Fig. 4.
4.1 Datasets and parameter settings
We adopted the OULP-Age dataset [46],1 which is the world's largest gait database with age information, to evaluate the proposed method. It is composed of 63,846 subjects with ages ranging from 2 to 90 years old and has a good gender balance (the ratio of males to females is close to one). The detailed distribution of subjects' gender and age groups in 5-year intervals is shown in Fig. 5, which illustrates that the dataset provides a large number of samples for training reliable transformations for age progression/regression. Half the samples (i.e., 31,923 subjects) randomly chosen from the entire dataset were used to train the general transformation for each adjacent age group pair, and the other half were used for the qualitative and quantitative evaluation of the proposed method. One gait period is detected for constructing a single GEI image for each subject in this dataset.
We defined the age groups based on the physical growing process (e.g., body height) of a natural human. According to the biological constraints of human [2, 3], the children and teenagers (i.e., less than 20 years old) tend to have relatively rapid growing speed, whereas the growth of adults (i.e., more than 20 years old) becomes much slower because their bodies have grown into a mature physical state [2, 48]. We therefore divided the ages into 11 age groups, that is, 0–4, 5–9, 10–14, 15–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70–79, and over 80, where the age interval was set to be finer (i.e., 5 years) for children and teenagers while that was set to be coarser for adults (i.e., 10 years),2 which is also often done in face-based age progression/regression studies [38, 47].
4.2 Qualitative evaluation
4.2.1 Comparison with ground truth
To evaluate the performance of the proposed method, we first qualitatively compared the simulation results of the mean-GEI with the corresponding real images for each age group in Fig. 6, which were the only samples that had ground truth. We performed long age span progression that started from the input mean-GEI of the first age group (i.e., 0–4) to validate the quality of the simulated aging patterns through the entire aging process, from a child to an elderly person. Compared with the true mean-GEIs, we observe that the simulation results of the proposed method were similar to the ground truths, which presented obvious gait changes during the aging process, such as the head-to-body ratio reduction in the growth of children, in addition to the middle-age spread and stoop for the middle-aged and elderly.
Distribution of subjects' gender and age groups in the OULP-Age dataset
Comparison of the simulated mean-GEIs and true mean-GEIs. The leftmost image is the input of the age progression, whereas the other images in the first row show the simulation results of a typical age within each age group. The second row provides the true mean-GEIs that correspond to each simulation image above. The digit under each image indicates the age of the simulation or age group to which the true mean-GEI belongs
More age progressed/regressed examples using the proposed method. The first two samples are male subjects and the last two are female subjects
4.2.2 Simulation examples of specific subjects
We picked up two male subjects and two female subjects from different age groups to test the simulation results of the proposed method, as shown in Fig. 7. More simulation examples can be found in Fig. 11. It is obvious that the simulations present the natural aging patterns in a human's gait, in addition to maintaining the personalized gait characteristics (e.g., walking pose and body appearance) of different individuals. Moreover, although the proposed method used a generic transformation across the population, the examples in Figs. 7 and 11 still illustrate that subjects with different walking styles or body shapes could result in different gait changes in the proposed age progression/regression that are consistent with our intuition (e.g., an overweight young person may have a larger spread in his/her middle-age or when elderly than a slim person, and a young person that appears to stoop may become heavily stooped, whereas a young person that has an almost straight back may also remain relatively straight while aging) because the mean-GEIs are averaged across a large number of subjects with various appearances (e.g., walking style and body shape), which makes the warping fields contain some type of different transformations for different gait appearances (e.g., the displacement vectors for a slim body and an overweight body could be different). The examples of the year-by-year aging/reverse aging process using the proposed method are shown in a movie in the supplementary material.
4.3 Quantitative evaluation
Similar to face-based works [40, 41], we quantitatively evaluated the performance of the proposed method for two aspects: (i) the quality of aging patterns, that is, whether the age progressed/regressed image truly presented the age characteristics of the intended age, and (ii) the preservation of identity, that is, whether the simulations retained the identity information of the original subject. We conducted both subjective (human perception) experiments and objective (machine perception) experiments by implementing age group classification and cross-age gait identification to quantitatively measure the two criteria, respectively. Twenty-four participants were asked to participate in experiments for age group classification and cross-age gait identification.
4.3.1 Age group classification
We first investigated the human perception of classifying the full age range into fine age groups (i.e., 11 age groups as defined in the training stage) using real GEI samples. Twenty images were randomly selected for each age group from the entire testing set and then randomly reordered to present to the participants for classification. However, the overall correct classification rate averaged over all age groups and participants was only 28%, and, particularly for age groups over 20 years old, the classification accuracies were only close to the chance-level (i.e., 9%), which is unsuitable for quantitative evaluation. The low accuracy in this human test was mainly because of the slow growth over 20 years old, which resulted in small appearance differences across different age groups.
Correct classification rates [%] of simulations and real images for each age group in the subjective evaluation, objective evaluation using the same samples, and objective evaluation using more samples
Therefore, we reselected seven discontinuous age groups with sparse intervals, 0–4, 5–9, 10–19, 20–29, 30–39, 50–59, and over 70, to be involved in the final subjective evaluation of the performance of the proposed age progression/regression method. Specifically, we randomly chose 40 subjects (20 females and 20 males) aged 20–29 with diverse gait appearances (e.g., different walking poses and different clothes) from the testing set as the inputs, and for each subject, simulated the age progressed and regressed GEI images for the other six age groups; thus, we prepared 240 simulated images with 40 images for each age group. Similar to [40, 41], we also prepared a set of real GEI images with 40 images randomly selected for each age group to compare the performance of age group classification between simulation and real images, which validated the quality of the simulated aging patterns. Some simulated and real images are shown in Fig. 9.
Overall correct classification rates [%] of simulations and real images for all age groups in the subjective evaluation, objective evaluation using the same samples, and objective evaluation using more samples
Objective (increased)
Similarly, we executed objective age group classification on both sets of simulations and real images using directed acyclic graph support vector machine (DAGSVM) [32], which is more suitable for classifying groups with ordered information using a rooted binary directed acyclic graph that integrates multiple binary SVM classifiers. Additionally, we also increased the evaluated samples by eight times, that is, 320 simulations and 320 real images for each age group, to objectively evaluate the performance of the proposed method in a more statistically reliable manner.
The correct classification rates of simulations and real images for each age group for both subjective and objective evaluations using the same samples, in conjunction with the objective evaluation using more samples, are shown in Fig. 8. Additionally, the overall correct classification rates for each image set are reported in Table 1.
Clearly, the automatic algorithm yielded much higher accuracies than human perception because it is more difficult for a human to correctly perceive age-related variations from grayscale GEI images compared with more informative facial images that contain aging patterns in, for example, muscles, wrinkles, and skin. By contrast, for both subjective and objective evaluations of the same samples, the classification of simulations for the middle three age groups, that is, 5–9, 10–19, and 30–39, resulted in better performance or competitive accuracies compared with that for real images; for the other three age groups, there was relatively worse classification performance for simulations than real images. However, the differences in classification performance between simulations and real images for specific age groups has also been observed in the face-based age progression/regression literature [17, 41], and the differences can be reduced by increasing samples in the objective evaluation, which provides more statistically reliable evaluation results. Additionally, because we performed a longer age progression for the last two groups, that is, 50–59 and over 70, for which the performance for simulations was relatively worse, it is understandable that the performance would be improved if we started from a middle-aged input (e.g., 40 or 50 years old), which would lead to a shorter age progression for the elderly.
The overall correct classification rates in Table 1 also illustrate that the age group classification results of the simulations were almost consistent with those of the real images in all three evaluation experiments, which indicates that our age progressed and regressed images contained most of the reasonable age-related variations.
4.3.2 Cross-age gait identification
Rank-1 identification rate [%] for each age group, and the overall rank-1 identification rate [%] for both subjective and objective evaluations
Failures and successes from the subjective evaluation of age group classification. a Samples with the ground truth age group of 0–4. b Samples with the ground truth age group of over 70. The first row shows simulation examples, whereas the second row shows real image examples. For each sample, the digits in black represent the ground truth age group and the bracketed digits represent the predicted age group; blue indicates failure and red indicates success (color figure online)
To validate the preservation ability in terms of the personal identity information for the proposed method, we prepared one simulation set and one real image set as the probe and gallery, respectively, in the experiment for cross-age gait identification. However, the OULP-Age dataset contains only a single GEI per subject; hence, we used a sequence from another camera that was different to that used for the OULP-Age dataset, but from a similar viewing angle (i.e., \(90^{\circ }\) azimuth angle),3 to construct a gallery. Specifically, the 240 simulation images, in addition to their corresponding inputs of real images aged 20–29 (i.e., totally 280 images) used in the experiment for age group classification constituted the probe set. Because we found that it was difficult for a human to identify one probe subject from a large gallery set (e.g., over 40 gallery subjects) using GEI images, we further divided the probe set into two subsets of equal size (i.e., 20 subjects for each age group in each probe subset) to reduce the gallery size. For each probe subset, the gallery set was composed of 30 real images for age group 20–29, including the same 20 individuals in the probe subset and another 10 distracting samples. Thus, we could evaluate the gait identification (i.e., match one probe from 30 galleries) performance in both cases for the same age and cross-age.
We simply adopted direct matching, that is, we computed the Euclidean distance between the probe and each gallery, as the dissimilarity measure, and choose the nearest neighbor from all the galleries as the final identity of that probe for the objective evaluation of the performance of cross-age gait identification. The rank-1 identification rate for each age group, together with the overall rank-1 identification rate averaged over all probes for both subjective and objective evaluations, are shown in Table 2.
Generally, the experimental results demonstrate that the proposed method maintained most identity information of the input image during the age progression and regression. For both subjective and objective evaluations, the same-age case (i.e., 20–29) obtained the highest rank-1 identification rate, and the identification accuracy decreased with the increase of the age gap between the probe and gallery, which indicates the difficulty of gait identification under a large age difference, in addition to the increased challenge of age progression/regression for a longer age span. Again, the performance of human perception was still much lower than that of the machine, which was even true for the same-age case because the existence of slight differences between the probe and genuine caused by segmentation and/or gait fluctuation easily led to an incorrect matching for human eyes.
In this section, several failure cases in the subjective evaluation of age group classification and cross-age gait identification are discussed.
Failures from the subjective evaluation of cross-age gait identification. a Probes with the ground truth age group of 20–29. b Probes with the ground truth age group of 0–4. Each column shows a different failed example. The first row is the probe, the second row is the imposter (false match) that most participants mismatched, and the third row is the genuine (true match). The fourth and fifth row shows the difference images between the first and second rows and the differences between the first and third rows, respectively. The imposters and genuines (i.e., galleries) were both in the 20–29 age group
5.1 Failures in age group classification
We first focus on the failure modes from the evaluation of age group classification. The samples were selected from the ground truth age group of 0–4 and over 70, which were located at the end of the age progression/regression sequences and obtained relatively worse results than the other age groups. Both failed simulation and real image examples are shown in Fig. 9 (with blue digits). Additionally, some successful examples are also shown in Fig. 9 (with red digits) for comparison.
From our observation, the failures were mainly because of two reasons: (i) the inconsistency between the person's chronological age and physiological age and (ii) the unrealistic walking pose or fashion style for the person's age. The first type of failures existed for both simulations and real images, such as children with relatively small head-to-body ratios, and the middle-aged and elderly who looked very slim or had no apparent stoop in their gaits. These types of failures also occur in the field of gait-based age estimation [27, 46], which often leads to the large overestimation or underestimation of a person's age from the gait features.
The second failure more frequently occurred in the case of simulations. For example, children appeared to have relatively mature walking poses that were rarely observed in the real 0–4 group because they were transformed from mature adults aged 20–29 using the mean-GEI, which did not present an obvious immature walking pose because of the large diversity in the poses of young children; thus, this is a limitation of the proposed method, which uses a single generic subject-independent transformation. Similarly, this problem also appeared in the fashion style of the elderly (e.g., very few elderly people wear long dresses or have long hair in real scenes), which was particularly true for females, and it is difficult to solve this well using a generic transformation for all subjects.
5.2 Failures in cross-age gait identification
We next analyze some failures in the evaluation of cross-age gait identification. Considering the large performance differences between the subjective and objective evaluations, we chose the examples in which humans failed but the machine succeeded to investigate the difference in the gait recognition capability between the human and machine using the appearance-based gait feature, which has not been discussed to the best of our knowledge.
We provided the probe, the imposter that most participants mismatched, and the genuine for each example in Fig. 10. We first focus on the failures in Fig. 10a for both probes and galleries aged 20–29, that is, gait identification for the same age. Although the probe and genuine were captured almost simultaneously by two cameras with a similar view angle, because of the different segmentation results and gait fluctuation across gait periods, there still existed slight differences between the appearances of the probe and genuine, which easily caused the mismatch for human eyes, particularly when other imposters had some similarities (e.g., similar clothes and similar body shape) to the probe. By contrast, it was much easier for the machine to make a correct match by computing the dissimilarity scores, which can be understood from the difference images shown in Fig. 10, where smaller differences are obtained for the genuine pairs compared with the imposter pairs.
Next, we consider the examples in Fig. 10b, in which the age difference between the probe and gallery was relatively large (i.e., 0–4 vs. 20–29). The appearance differences between the probe and genuine greatly increased, which made it more difficult for a human to determine the true match by simply observing the 30 gallery candidates and thus led to the low accuracy for this age group in the subjective evaluation. The identification rate of the objective evaluation also degraded in this case; however, it still had superior accuracy to that of the human because the genuine had a smaller dissimilarity score than the other imposters for most subjects.
Although the performance of subjective evaluation is much worse than that of the objective evaluation, we noticed that there are large differences among the results originated from different annotators. Actually, two of the twenty-four participants in our experiments were experts in gait recognition, whereas the remaining twenty-two participants were general people that were not familiar with observing GEI images. The statistics of the rank-1 identification rates received from all participants for identification under both the cases of same age and cross-age are shown in Table 3. Based on the standard deviation, it is obvious that there exist large variations among the identification performance by different people. While the experts gained almost 100% identification rates which even outperform that of the machine, most of the participants obtained much lower accuracies because they were not skilled at finding dissimilarities from the GEI images compared with the raw gait videos, which resulted in the worse identification rates than the direct matching consequently.
5.3 Impact of gender
To analyze the impact of gender on the proposed method, we retrained two warping fields for female and male subjects, respectively (i.e., gender-dependent morphing) to compare the performance with the original proposed general warping for both genders (i.e., gender-independent morphing). Because there is no existing work that considers gait-based age progression/regression, we also compared the simulation results of our gender-dependent morphing with a state-of-the-art approach in face-based age progression/regression that considers gender in the training stage, that is, the CAAE [47], which is a unique study that does not use paired face samples from each individual. The same GEI training samples with the same age group definition used in our method were fed into the network for CAAE.
5.3.1 Qualitative evaluation
We first qualitatively compared the simulations of the proposed gender-independent and gender-dependent morphing, in addition to the results of CAAE from two male subjects and two female subjects in Fig. 11.
Statistics of the rank-1 identification rates [%] from all participants for both the cases of same age and cross-age identification
Same age
Cross-age
Std indicates the standard deviation
Comparison of CAAE, proposed gender-dependent and gender-independent morphing. The leftmost column shows the inputs together with their ages. For each subject, the first row on the right shows the results of the CAAE for each age group; the second and third row shows the results of the proposed gender-dependent and gender-independent morphing for a typical age within each age group corresponding to the CAAE results above, respectively. The first two samples are male subjects and the last two are female subjects
Although the gait aging patterns reflected in the results of the CAAE were consistent with our observations from Fig. 1, we observe that the CAAE lost the gait individuality of a specific subject because it generated the same age progression/regression results for different males/females, which demonstrates that the face-based age progression/regression method is unsuitable to be directly applied to the gait because of the differences between the face and gait in terms of aging mechanisms and data types (i.e., facial images contain rich texture information that is relevant to aging, such as wrinkles, whereas silhouette-based gait does not contain such information).
By contrast, both the proposed gender-independent and gender-dependent morphing keep the personalized gait characteristics while presenting the reasonable gait aging patterns. The results generated by gender-dependent warping have slight differences from that of gender-independent warping mainly in terms of the degree of stoop appeared in the middle-aged and elderly, and the simulations using these two morphing models are, however, still very similar to each other generally, which demonstrates that the females and males have similar gait changes during the aging/reverse aging process.
5.3.2 Quantitative evaluation
The comparison of the CAAE, the proposed gender-independent and gender-dependent morphing in terms of correct classification rate and rank-1 identification rate
Ours (gender-independent)
Ours (gender-dependent)
CAAE
(a) Correct classification rate (%)
(b) Rank-1 identification rate (%)
The results of females, males and both genders from each age group, in addition to the overall results, are shown for each method, respectively
We next quantitatively evaluated the performance of CAAE, the proposed gender-independent and gender-dependent morphing again using age group classification and cross-age gait identification. The same 40 test subjects aged 20–29 that used in Sect. 4.3 were chosen as the inputs to simulate for the other six age groups, i.e., 0–4, 5–9, 10–19, 30–39, 50–59, and over 70. Only objective evaluation was executed for the simulation images considering its reliability and higher accuracies compared with subjective evaluation.
(a) Age group classification The correct classification rates of simulations by each method are computed for females, males and both genders from each age group, respectively, as shown in Table 4(a).
Because the CAAE generated the same simulation results for different males/females (i.e., one female and one male simulation for each age group), we observe either 0% or 100% correct classification rate for each gender in each age group, which illustrates the results of CAAE somewhat render gait aging patterns although the gait individuality is lost. By contrast, both the proposed gender-independent and gender-dependent morphing obtain better accuracies than CAAE in terms of the overall correct classification rate.
Generally, the gender-dependent morphing yields slightly better results than the gender-independent one, which mainly results from the minor improvements for the elderly females. Therefore, the effects of gender on the proposed geometric warping are subtle considering the quality of simulated aging patterns, which is consistent with the observations from the simulation examples in Sect. 5.3.1.
(b) Cross-age gait identification Similar to Sect. 4.3.2, direct matching was applied to identify the probes including simulations of aforementioned six age groups from the galleries of real images aged 20–29. We compared the rank-1 identification rates of each gender and both genders from each age group, in addition to the overall rank-1 identification rate for the CAAE, the proposed gender-independent and gender-dependent morphing in Table 4(b).
The low identification accuracies of CAAE clearly show that the personal identity is corrupted in its simulated results, which quantitatively demonstrates it is unsuitable to directly apply face-based age progression/regression method to the gait scenario.
Similar to the results of age group classification, only the accuracies of females are slightly improved by considering gender in the training of the proposed geometric warping fields, and hence, similar preservation ability of the gait identity are yielded for the gender-independent and gender-dependent morphing, which is easily understood from the statistics of both genders.
In this paper, we presented a baseline algorithm for gait-based age progression and regression, which has not been addressed in the literature to the best of our knowledge. We first divided the entire age range into several age groups and then generated the FFD-based general geometric transformation between adjacent age groups to render age progression and regression for the full age range simultaneously. Qualitative evaluation in conjunction with the subjective and objective quantitative evaluations through age group classification and cross-age gait identification were executed to validate the proposed method for the quality of aging patterns and preservation of identity and to provide several insights for future research on this topic.
One important future direction is to extend age progression/regression to silhouette sequences to render the change in motion by simulating different gait phases, which is more beneficial for potential applications, such as entertainment and health examination. Moreover, rather than using a general transformation across the population, the performance improvement of the proposed baseline method by involving more individuality, such as incorporating clothing-dependent transformations, also needs to be investigated. Additionally, to collect paired samples of individuals with a short age span (e.g., 5 years or 10 years) remains challenging but meaningful future work, which will contribute to the development of a better gait-based age progression/regression approach and the research on cross-age gait recognition tasks.
Publicly available at http://www.am.sanken.osaka-u.ac.jp/BiometricDB/index.html.
For ages over 80 years old, we simply grouped them together considering the lack of such training samples compared with the other age groups.
These two cameras captured images almost simultaneously for each subject. The reader may refer to [26] for more details on the data collection system.
This work was supported by JSPS Grants-in-Aid for Scientific Research (B) (16H02848), the Jiangsu Provincial Science and Technology Support Program (No. BE2014714), 111 Project (No. B13022), and Priority Academic Program Development of Jiangsu Higher Education Institutions. We thank Xiang Li for helping to implement the algorithm in the objective evaluation and providing valuable suggestions for the subjective evaluation. We thank Maxine Garcia, Ph.D., from Edanz Group (www.edanzediting.com/ac) for editing a draft of this manuscript.
Supplementary material 1 (mp4 11777 KB)
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1.School of Computer Science and EngineeringNanjing University of Science and TechnologyNanjingChina
2.Department of Intelligent Media, The Institute of Scientific and Industrial ResearchOsaka UniversityOsakaJapan
Xu, C., Makihara, Y., Yagi, Y. et al. Machine Vision and Applications (2019) 30: 629. https://doi.org/10.1007/s00138-019-01015-x
Revised 04 January 2019
DOI https://doi.org/10.1007/s00138-019-01015-x | CommonCrawl |
Potential of entomopathogenic nematode isolates from Rwanda to control the tomato leaf miner, Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae)
Assinapol Ndereyimana ORCID: orcid.org/0000-0003-4239-70961,2,
Samuel Nyalala1,
Patrick Murerwa1 &
Svetlana Gaidashova2
Tomato leaf miner, Tuta absoluta (Meyrick), is a major threat to tomato production as it can cause up to 100% yield loss under both greenhouse and open-field conditions. Chemical control, which is associated with several undesirable effects, remains the only option readily available for this pest since its invasion of Rwanda in the year 2015. This study assessed the potential of using local isolates of entomopathogenic nematodes (EPNs) in management of T. absoluta in Rwanda. Six EPNs including four locally isolated strains: Steinernema sp. RW14-M-C2a-3, Steinernema sp. RW14-M-C2b-1, S. carpocapsae RW14-G-R3a-2 and Heterorhabditis bacteriophora RW14-N-C4a, and two exotic species: S. carpocapsae All and H. bacteriophora H06 were evaluated. Three bioassays were conducted in the laboratory, using a tomato leaflet with third instar T. absoluta larva in gallery and 9-cm Petri dishes as bioassay arenas in a completely randomized design with three replications. The EPNs were applied at a volume of 1 ml containing 500 infective juveniles per leaflet, while sterile tap water was used as negative control. Larval mortality was checked continuously for 96 h at 24 h interval. The results revealed that all the tested EPNs were able to find and kill T. absoluta larvae inside the leaf galleries; and their efficacy increased with exposure time. The pathogenicity effects were significantly different (p < 0.05) among EPNs. In the first 24 h after inoculation, the efficacy of local EPN isolates (53.3–96.7%) was significantly higher than the one of exotic species (0.0–26.7%). The efficacy of three Rwandan EPN isolates, Steinernema sp. RW14-M-C2a-3, Steinernema sp. RW14-M-C2b-1, and S. carpocapsae RW14-G-R3a-2 was not significantly different from 24 to 96 h after inoculation, except for S. carpocapsae RW14-G-R3a-2 during 24 h after inoculation in bioassay 3. There was insignificant difference among all the EPN isolates after 96 h of exposure. This is the first study carried out in Rwanda that investigated the potential of locally isolated EPNs against T. absoluta. Field experiments should be conducted to fully explore the possibilities of using local EPN isolates in integrated pest management of T. absoluta in Rwanda.
Tomato leaf miner, Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae), is an invasive pest originating from South America and was detected for the first time in Rwanda in the year 2015 (FAO, 2015). This pest is the major threat to tomato production as it can cause up to 100% yield loss under both greenhouse and open-field conditions (Desneux et al., 2010 and Biondi et al., 2018). It has also been reported in other Solanaceae crops, like eggplant (Solanum melongena L.), potato (Solanum tuberosum L.), pepper (Capsicum annuum L.), nightshade (Solanum nigrum L.), and tobacco (Nicotiana tabacum L.) (Ferracini et al. 2012).
Chemical control remains the only option readily available in areas newly infested by T. absoluta (Brévault et al. 2014). However, the short developmental period of this pest and its many generations per year lead to numerous insecticide sprays in one season (Biondi et al. 2018). This facilitates the development of resistant pest strains to frequently used insecticides (Haddi et al. 2017) and leads to the destruction of natural enemies (Macharia et al. 2009). The mine-feeding behavior of T. absoluta larva also limits the effectiveness of synthetic insecticides (Gebremariam, 2015). The limited effectiveness in addition to hazardous nature of chemical insecticides (Macharia et al. 2009), trigger the need for integrated pest management (IPM) and use of pest control actions that assure positive economic, ecological, and sociological effects (Blake et al. 2007).
Since T. absoluta is a major threat to tomato production in Rwanda, it is important to develop sustainable mechanisms to cope with it so as to sustain the role of tomato in diversifying the economy, alleviating poverty, and improving nutrition (Clay and Turatsinze, 2014). Biological control is one of the safe ways of managing agricultural pests as it has no harmful effects on environment and human health. Among biological control methods, entomopathogenic nematodes (EPNs) have the potential of being used effectively against T. absoluta (Van Damme et al. 2016 and Biondi et al. 2018). The potential of EPNs was also evidenced against a diversity of other economically important pests (Wraight et al. 2017). However, no research had been conducted in Rwanda to explore the possibilities of including these EPNs in IPM of tomato crop.
Entomopathogenic nematodes belong predominantly to Steinernematidae and Heterorhabditidae families; they are obligate parasites that kill insects with the help of mutualistic bacteria (Xenorhabdus spp. and Photorhabdus spp., respectively) which live in gut of infective juveniles (IJs) (Campos-Herrera, 2015). They can be effective against many soilborne pests and others which live in galleries due to the conducive environment (protection against desiccation and ultraviolet light) for their IJs (Garcia-del-Pino et al., 2013). The IJs, which are the only free-living stage (third stage) enters the host body via the natural openings or even through the soft body. Once inside the body, the bacteria cells released by IJs multiply quickly and kill the host in 24–72 h (Gözel and Kasap, 2015). Moreover, these bacterial cells digest host tissues and release antibiotics which protect the killed host against saprophytes and scavengers, thus permitting the nematodes to develop and reproduce (Griffin et al. 2005). There might be one to three nematode generations depending on the size of the host. When host nutrients are exhausted, the IJs sequester the bacteria in their intestines, leave the host and search for a new one; but when they miss a new host, the IJs are able to persist for months in moist soil (Stock, 2015).
It is recognized that the environment determines the success or failure of EPNs because of the possible differences in persistence, virulence, host range, and familiarity to habitats between local and non-local EPN isolates (Lacey and Georgis, 2012). The target host and the environment where EPNs will be applied should be considered when designing a control program using EPNs. Thus, screening several nematode isolates against a particular target host in a specific environment is prerequisite in development of any control program using EPNs (Biondi et al. 2018).
Four new EPN strains were isolated from semi-natural and small-holder farming habitats of Rwanda and maintained in Biological Control Laboratory at Rwanda Agriculture and Animal Resources Development Board (RAB) (Yan et al. 2016).
The objective of this study was to determine the potential of the EPNs against T. absoluta under laboratory conditions.
The present study was carried out in Biological Control Laboratory—EPN Production Facility of Rwanda Agriculture and Animal Resources Development Board (RAB) (Holmes et al. 2015). The average annual rainfall and temperature of the locality are 1039 mm and 19 oC, respectively (Ndabamenye et al. 2013).
Source and mass production of EPNs
Six EPNs including four local isolates and two exotic species, maintained in the Biological Control Laboratory—EPN Production Facility of RAB were used for the study. The local EPNs were Steinernema sp. RW14-M-C2a-3, Steinernema sp. RW14-M-C2b-1, S. carpocapsae RW14-G-R3a-2, and Heterorhabditis bacteriophora RW14-N-C4a, which were isolated from semi-natural and small-holder farming habitats of Rwanda in the year 2014 (Yan et al. 2016). The two exotic EPN species, S. carpocapsae All and H. bacteriophora H06, were obtained from Lvbenyan Biotech Ltd., Guangdong Institute of Applied Biological Resources (GIABR) in China (Kajuga et al. 2018). The exotic species were used as standard checks since they are among the most used for control of foliar and soil insect pests (Lacey and Georgis, 2012).
In vivo method of EPN mass production was followed in the aforesaid laboratory, using last instar larvae of Galleria mellonella L. (Lepidoptera: Pyralidae), which is mostly used for this purpose as it is easily infested by nematodes (Kaya and Stock, 1997). The G. mellonella larvae, killed by EPNs, were moved to a white trap (White, 1927) for the IJs to come out of the cadavers. IJs were harvested, rinsed in distilled water, and stored at 7 oC for less than 1 week. Since EPNs do not go through complete dormancy stage and continue to consume their limited energy during storage (Mahmoud, 2016), they were used in bioassays when still fresh within a period of less than 7 days following their harvesting from white trap. Before using the EPNs, they were allowed to acclimatize at room temperature for 1 h, and their viability was checked under a stereomicroscope with × 60 magnification, where live IJs were moving actively (Garcia-Del-Pino et al. 2013). These EPNs were used in bioassay once more than 90% of IJs were viable (Kajuga et al. 2018).
Source of T. absoluta larvae
To secure the source of laboratory specimens (tomato leaflets with T. absoluta larvae inside the galleries) a field of tomato, Solanum lycopersicum cv. Roma, was established in Bugesera District, Rweru Sector. This area was selected because it is the hot spot of T. absoluta infestation in Rwanda. Tomato crop was established in November, 2018 (2 months before starting bioassays) following the package recommended for field cultivation of the crop in Rwanda. However, insecticides were not applied from 2 weeks after planting to favor quick development of T. absoluta in the field. Tomato leaflets containing T. absoluta larvae inside the galleries were harvested from this naturally infested tomato field with caution to have only one larva per leaflet. These leaflets were transported in a cloth bag to the laboratory and kept for a maximum of 12 h before being used in bioassays.
Pathogenicity test
An hour before each bioassay, nematode concentrations were calculated as per Navon and Ascher (2000) and adjusted to the required concentration of 500 IJs/ml (Batalla-Carrera et al. 2010; Youssef, 2015 and Mutegi et al. 2017) using sterile tap water. The leaflets containing third instar T. absoluta larvae in galleries were carefully selected for use in bioassay. Each leaflet was positioned in a 9-cm diameter Petri dish having 3 moistened filter paper discs at its base and 10 Petri-dishes were used per treatment in each replication. Thereafter, 1 ml of water containing 500 IJs of EPNs was applied on both sides of each leaflet, using sterilized pipette for each EPN isolate (Batalla-Carrera et al. 2010) and sterile tap water was used as control. The Petri dishes were sealed by parafilm to protect them against dehydration and prevent escape of the larvae. The Petri dishes were then maintained at a temperature of 25±2 oC in dark. Fresh leaflets were added after 2 days to prevent starvation of larvae. Each bioassay was carried out in a completely randomized design with three replications and three bioassays were conducted from January to February, 2019.
Petri dishes were opened 24, 48, 72, and 96 h after inoculation to check the status of larvae whether dead or alive. Dead larvae were recognized by being unable to respond to stereoscopic light or to probing by camel's hair brush tip (Van Damme et al. 2016). Dead and alive larvae at the end of bioassay were dissected under the stereomicroscope to determine the presence or absence of nematodes in their body (Kajuga et al. 2018). The number of dead larvae at each observation period was used to compute the observed mortality (%), which was obtained by dividing the number of dead larvae per treatment by the total number of larvae per treatment and multiplying the result by hundred. When there was mortality in Petri dishes treated with water, this was considered as natural mortality and was used to correct the mortality observed in Petri dishes, where EPNs were applied. This correction was done using Schneider-Orelli's formula (Püntener, 1981) as follows:
$$ \mathrm{Corrected}\kern0.5em \mathrm{mortality}=\frac{\mathrm{Observed}\kern0.5em \mathrm{mortality}\kern0.5em \left(\%\right)\hbox{-} \mathrm{control}\kern0.5em \mathrm{mortality}}{100\hbox{-} \mathrm{control}\kern0.5em \mathrm{mortality}}\times 100. $$
Data were checked for normality before subjecting them to statistical analysis, using proc univariate procedures of SAS. Data on corrected mortality at 48 h and 72 h after inoculation were arcsine transformed as per Rangaswamy (2013). General linear model (GLM) was used to determine whether the effect of EPNs on T. absoluta larvae mortality was significantly different or not. Tukey's honestly significant difference (HSD) test was used to separate the means. These analyses were performed, using Statistical Analysis System package SAS software version 9.2 (SAS Institute, 2010), and the level of significance was fixed at 5%.
All EPN isolates evaluated were able to find, infect, and kill T. absoluta larvae inside the tomato leaf galleries, however, with different levels of pathogenicity (Table 1). The recorded mortality differed significantly among the tested EPNs. However, there was an insignificant difference among EPNs (P > 0.05) after 72 h of exposure for bioassay 2 and after 96 h of exposure for all bioassays. When data (arcsine transformed) were analyzed considering exposure time and EPNs × exposure time combination as treatments, it was observed that both affected significantly (P < 0.001) T. absoluta larval mortality (Table 1). This implies that efficacy of the tested EPNs depends on exposure time.
Table 1 GLM analysis for mortality of Tuta absoluta larvae caused by different EPNs used at 500 IJs/ml
In the first 24 h after inoculation, the efficacy of all local EPN isolates was significantly higher than the two exotic EPN species in the three bioassays (P < 0.001), except for H. bacteriophora RW14-N-C4a in bioassay 2. At 48 h after inoculation, the efficacy of H. bacteriophora H06 was insignificantly different from the one of local EPN isolates in bioassays 1 and 3; but this lasted 72 h in bioassay 2. In 72 h post-inoculation, all local EPN isolates had achieved 100% mortality in all bioassays, except H. bacteriophora RW14-N-C4a, which had 96.3% in bioassay 3. The maximum mortality was achieved on different times of exposure to EPNs, ranging from 48 to 96 h (Table 2).
Table 2 Daily mortality of T. absoluta larvae (mean ± SD) in leaf galleries treated with local and exotics EPNs using the concentration of 500 IJs/ml
Among the local (Rwandan) EPN isolates, three, Steinernema sp. RW14-M-C2a-3, Steinernema sp. RW14-M-C2b-1 and S. carpocapsae RW14-G-R3a-2, remained insignificantly different in efficacy from 24 to 96 h after inoculation, except that S. carpocapsae RW14-G-R3a-2 was significantly different from them only during 24 h after inoculation in bioassay 3. H. bacteriophora RW14-N-C4a joined their group 48 h after inoculation for bioassays 1 and 2 and 72 h for bioassay 3; which revealed that it was not as effective. Here, EPNs belonging to Steinernema genus were more effective than the ones belonging to Heterorhabditis (Table 2).
The ability of EPNs to reach and kill T. absoluta larvae in leaf galleries was also reported by Batalla-Carrera et al. (2010), Van Damme et al. (2016), and Kamali et al. (2018). The local EPN isolates were able to kill T. absoluta and had even been found to be effective against white grubs in Rwanda (Kajuga et al. 2018), while it was not easy to find EPNs which can kill them (Laznik and Trdan, 2015). Although EPNs live naturally in soil, different researchers found that they can be used on above-ground parts of the plant to control effectively the pests living in cryptic habitats like in leaf galleries (Batalla-Carrera et al. 2010 and Garcia-del-Pino et al. 2013); which concurs with the results of the present study under laboratory conditions.
In the present study, the third instar larvae were used; other studies revealed that EPNs were able to find and kill all the four larval instars inside or outside the leaf galleries (Batalla-Carrera et al. 2010 and Van Damme et al. 2016). Batalla-Carrera et al. (2010) found that T. absoluta larval stage was the most vulnerable to EPNs. They thus emphasized the necessity to apply EPNs on the above-ground part of tomato plant to ensure effective control of this pest using the most suitable isolates against a particular pest in a given environment.
Different pathogenicity levels displayed by the studied EPNs agree with other studies, using different EPN isolates (Gözel and Kasap, 2015 and Van Damme et al. 2016). This underlines the necessity of EPNs screening and selection as emphasized by Sharma et al. (2011) and Biondi et al. (2018) in a view to boost their efficacy of EPNs. The highest efficacy of local EPN isolates than the exotics could be explained by the fact that these EPNs were isolated in Rwanda (Yan et al. 2016), and they might be more adapted to the local conditions than the exotics, which were isolated in a completely different environment. These results agree with the earlier findings where locally isolated biological control agents, including EPNs, performed better than exotics (Lima et al. 2017).
The highest pathogenicity of EPN isolates belonging to Steinernema genus than Heterorhabditis had also been reported by Batalla-carrella et al. (2010) who obtained 76.3% mortality of T. absoluta larvae inside the leaf galleries using H. bacteriophora, while it was 88.6 and 92.0%, using S. carpocapsae and S. feltiae, respectively, at a dosage of 60 IJs/cm2. Furthermore, Steinernema genus was observed to be the most virulent among different EPN species by different other researchers (Van Damme et al. 2016; Mutegi et al. 2017, and Kamali et al. 2018) which is inconsistent with the obtained findings.
Higher efficacy of EPNs belonging to Steinernema genus could be due to the bacteria associated with their genus and ambusher strategy for host scavenging with standing and jumping behaviors, which helps them to attach on the host (Lacey and Georgis, 2012 and Campos-Herrera, 2015). Furthermore, it was reported that some species of Steinernema genus possess both cruiser and ambusher strategies; which make them more efficient in finding their host. The stand and jump behaviors as well as this intermediate foraging strategy have not been reported in Heterorhabditis genus (Lewis et al. 2006). However, because of their dorsal tooth EPNs belonging to Heterorhabditis genus would be expected to penetrate directly the insect body through the thin wall area between the segments (Griffin et al. 2005), and they would be more pathogenic than Steinernema genus. However, this thought was not evidenced by the results of the present study.
It is established that after EPNs have entered the host, the bacteria cells released by the IJs multiply quickly and kill the host in 24–72 h after infection (Gözel and Kasap, 2015 and Van Damme et al. 2016). This was verified in the present study where all local EPN isolates caused between 53.3 and 96.7% mortality just within the first 24 h after inoculation, while in 72 h, they all had caused between 96.3 and 100% mortality. This is not the case of other entomopathogens like entomopathogenic fungi, which require 3–5 days or even more time to kill their host (Reda and Hatem, 2012). The quick kill behavior of EPNs is beneficial for foliar applications where it can be guaranteed that before EPNs are killed by adverse environmental conditions such as desiccation and ultraviolet light, they would have searched, found and invaded their hosts in leaf galleries. Kim et al. (2006) reported that EPNs were able to survive 12 h after foliar spray on Chinese cabbage. This gives hope that application of EPNs on the aboveground part of the plant would yield good results because EPNs will have at least 12 hrs to find and invade the host no matter whether it is on the leaf surface or inside the leaf gallery where more convenient environment is guaranteed for survival.
During this study, one fixed dosage of 500 IJs/ml was used because the main purpose was to screen the local EPNs isolates and find out the most effective ones against T. absoluta for further investigations. Other researchers found that the higher the dosage, the higher the efficacy. For instance, Youssef (2015) observed that with 3 dosages of 250, 500, and 1000 IJs/ml for S. carpocapsae, mortality rates of T. absoluta larvae reached to 80, 100, and 100%, respectively. A similar trend was obtained by Batalla-Carrera et al. (2010), Mutegi et al. (2017), Yuksel et al. (2018), and Kajuga et al. (2018) on various pests. This could be due to the fact that high number of EPNs would result in high number of symbiotic bacteria released in the host's body and thus enhanced killing speed owing to increase digestion of host tissues by toxins and hydrolytic enzymes secreted by these bacteria (Van Damme et al. 2016).
The observed efficacy and rapid action of EPNs make them able to compete with conventional insecticides that are preferred owing to their quick action among others (Macharia et al. 2009 and Biondi et al. 2018). The safety, high virulence, ability to actively search for their hosts, mass production possibility, and compatibility with many pesticides are the other traits, which make EPNs a good option in IPM and potential substitutes to synthetic insecticides (Lima et al. 2017). Thus, further investigations should be carried out to determine their effectiveness under field conditions.
The results of this study revealed that local EPN isolates were able to find and kill T. absoluta larvae inside the leaf galleries under laboratory conditions and their efficacy increased with exposure time. The efficacy of local EPN isolates was significantly superior to that of the exotic species. This is the first study carried out in Rwanda on the potential of locally isolated EPNs against T. absoluta. The results of this study form the basis for further research. High EPN efficacy obtained under laboratory conditions cannot easily be extrapolated to field efficacy. Therefore, field experiments on tomato crop are justified to fully determine the potential of local EPN isolates against T. absoluta in Rwandan conditions.
EPN:
FAO:
Food Agriculture Organization
GLM:
General linear model
HSD:
Honestly significant difference
IPM:
P :
RAB:
Rwanda Agriculture and Animal Resources Development Board
SAS:
Statistical Analysis System package
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Authors would like to express gratitude to Rwanda Agriculture and Animal Resources Development Board (RAB) and Egerton University for their great contribution toward the achievement of this work. Special thanks go out to Mr Kagiraneza Boniface, Ms Bancy W. Waweru, Ms Kajuga Joelle and Dr Rukundo Placide for their inspirations and technical support throughout this study. Mr Bazagwira Didace, Ms Ingabire Geraldine and Ms Ishimwe Mukundwa Primitive are also acknowledged for their assistance in laboratory work.
This material is based upon work supported by United States Agency for International Development, as part of the Feed the Future initiative, under the CGIAR Fund, award number BFS-G-11-00002, and the predecessor fund the Food Security and Crisis Mitigation II grant, award number EEM-G-00-04-00013.
Department of Crops, Horticulture and Soils, Egerton University, Nakuru, Kenya
Assinapol Ndereyimana, Samuel Nyalala & Patrick Murerwa
Agriculture Research and Technology Transfer Department, Rwanda Agriculture and Animal Resources Development Board (RAB), Kigali, Rwanda
Assinapol Ndereyimana & Svetlana Gaidashova
Assinapol Ndereyimana
Samuel Nyalala
Patrick Murerwa
Svetlana Gaidashova
All authors jointly designed the experiment. AN conducted the laboratory bioassays, performed data analysis and drafted the manuscript with inputs from all authors. SN, PM and SG collaborated closely with AN in the whole process especially during data analysis. All authors read and approved the final manuscript.
Correspondence to Assinapol Ndereyimana.
Ndereyimana, A., Nyalala, S., Murerwa, P. et al. Potential of entomopathogenic nematode isolates from Rwanda to control the tomato leaf miner, Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae). Egypt J Biol Pest Control 29, 57 (2019). https://doi.org/10.1186/s41938-019-0163-3
Heterorhabditis
Steinernema
Local isolates | CommonCrawl |
Fixed Point Theory and Applications
Research Article | Open | Published: 18 July 2015
An iterative method for split hierarchical monotone variational inclusions
Qamrul Hasan Ansari1,2 &
Aisha Rehan1
Fixed Point Theory and Applicationsvolume 2015, Article number: 121 (2015) | Download Citation
In this paper, we introduce a split hierarchical monotone variational inclusion problem (SHMVIP) which includes split variational inequality problems, split monotone variational inclusion problems, split hierarchical variational inequality problems, etc., as special cases. An iterative algorithm is proposed to compute the approximate solutions of an SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. We present an example to illustrate our algorithm and convergence result.
Let $H_{1}$ and $H_{2}$ be real Hilbert spaces, $C \subseteq H_{1}$ and $Q \subseteq H_{2}$ be nonempty, closed, and convex sets, $A : H_{1}\rightarrow H_{2}$ be a bounded linear operator, and $f : H_{1} \rightarrow H_{1}$ and $g : H_{2}\rightarrow H_{2}$ be two given operators. Recently, Censor et al. [1] introduced the following split variational inequality problem (SVIP):
$$ \mbox{Find } x^{*}\in C \mbox{ such that } \bigl\langle f \bigl(x^{*}\bigr), x-x^{*}\bigr\rangle \geq0,\quad \mbox{for all } x \in C, $$
and such that
$$ y^{*} := Ax^{*} \in Q \mbox{ solves } \bigl\langle g\bigl(y^{*}\bigr),y-y^{*} \bigr\rangle \geq0, \quad \mbox{for all } y \in Q. $$
Let Λ denote the solution set of the SVIP, that is,
$$\Lambda= \bigl\{ x \mbox{ solves } (1.1) : Ax \mbox{ solves } (1.2)\bigr\} . $$
If f and g are convex and differentiable, then the SVIP is equivalent to the following split minimization problem:
$$ \min f(x), \quad\mbox{subject to } x \in C, $$
$$ y^{*} := Ax^{*} \in Q \mbox{ solves } \min g(y), \quad \mbox{subject to } y \in Q. $$
For further details on the equivalence between a variational inequality and an optimization problem, we refer to [2]. The SVIP also contains the split feasibility problem (SFP) [3] as a special case. For further details of the SFP, we refer to [4, 5] and the references therein.
If the sets C and Q are the set of fixed points of the operators $T : H_{1} \to H_{1}$ and $S : H_{2} \to H_{2}$, respectively, then the SVIP is called a split hierarchical variational inequality problem (SHVIP). It is introduced and studied by Ansari et al. [6]. Several special cases of a SHVIP, namely, the split convex minimization problem, the split variational inequality problem defined over the solution set of monotone variational inclusion problem, the split variational inequality problem defined over the solution set of equilibrium problem, are also considered in [6].
Let $B_{1} : H_{1} \rightrightarrows H_{1}$ and $B_{2} : H_{2} \rightrightarrows H_{2}$ be set-valued mappings with nonempty values, and let $f : H_{1} \to H_{1}$ and $g : H_{2} \to H_{2}$ be mappings. Then, inspired by the work in [1], Moudafi [7] introduced the following split monotone variational inclusion problem (SMVIP):
$$ \mbox{Find } x^{*}\in H_{1} \mbox{ such that } 0 \in f\bigl(x^{*}\bigr)+B_{1}\bigl(x^{*}\bigr), $$
$$ y^{*} := Ax^{*} \in H_{2} \mbox{ solves } 0 \in g\bigl(y^{*}\bigr)+B_{2}\bigl(y^{*} \bigr). $$
Let Γ denote the solution set of SMVIP, that is,
$$\Gamma= \bigl\{ x \mbox{ solves } (1.5) : Ax \mbox{ solves } (1.6)\bigr\} . $$
To solve the SMVIP, Moudafi [7] proposed the following iterative method: Let $\lambda> 0$ and $x_{0}$ be the initial guess. Compute
$$ x_{n+1} = U\bigl(x_{n}+\gamma A^{*}(T-I)Ax_{n}\bigr), \quad \mbox{for all } n \in \mathbb{N}, $$
where $\gamma\in(0, 1/L)$ with L being the spectral radius of the operator $A^{*}A$, $U = J_{\lambda}^{B_{1}}(I-\lambda f)$, $T = J_{\lambda}^{B_{2}}(I-\lambda g)$, and $J_{\lambda}^{B_{1}}$ and $J_{\lambda}^{B_{2}}$ are the resolvents of $B_{1}$ and $B_{2}$, respectively, with parameter λ (see [8]). He obtained the following weak convergence result for iterative method (1.7).
Theorem 1.1
([7], Theorem 2.1)
Given a bounded linear operator $A : H_{1} \rightarrow H_{2}$. Let $f : H_{1}\rightarrow H_{1}$ and $g : H_{2}\rightarrow H_{2}$ be $\alpha_{1}$ and $\alpha_{2}$ inverse strongly monotone operators on $H_{1}$ and $H_{2}$, respectively, and $B_{1}$, $B_{2}$ be two maximal monotone operators, and set $\alpha:= \min\{\alpha_{1},\alpha_{2}\}$. Consider the operator $U:= J_{\lambda}^{B_{1}}(I-\lambda f)$, $T:= J_{\lambda}^{B_{2}}(I-\lambda g)$ with $\lambda\in(0,2\alpha)$. Then the sequence $\{x_{n}\}$ generated by (1.7) converges weakly to an element $x^{*}\in\Gamma$, provided that $\Gamma\neq\emptyset$ and $\gamma \in(0, 1/L)$.
Let $T : H_{1}\rightarrow H_{1}$ and $S : H_{2}\rightarrow H_{2}$ be operators such that $\operatorname{Fix}(T)\neq\emptyset$ and $\operatorname{Fix}(S)\neq\emptyset$, where $\operatorname{Fix}(T)$ and $\operatorname{Fix}(S)$ denote the set of fixed points of T and S, respectively. Inspired by the work in [6] and [7], in this paper, we introduce the following split hierarchical monotone variational inclusion problem (SHMVIP):
$$ \mbox{Find } x^{*}\in\operatorname{Fix}(T) \mbox{ such that } 0 \in f\bigl(x^{*}\bigr)+B_{1}\bigl(x^{*}\bigr), $$
$$ y^{*} := Ax^{*} \in\operatorname{Fix}(S) \mbox{ solves } 0 \in g\bigl(y^{*}\bigr)+B_{2}\bigl(y^{*}\bigr). $$
We denote by Ψ the set of solutions of the SHMVIP, that is,
$$\Psi= \bigl\{ x \mbox{ solves } (1.8) : Ax \mbox{ solves } (1.9)\bigr\} . $$
We propose an iterative algorithm to compute the approximate solutions of the SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. An example is presented to illustrate the proposed algorithm and result.
Let H be a real Hilbert space whose inner product and norm are denoted by $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$, respectively. We denote by $x_{n} \rightarrow x$ (respectively, $x_{n}\rightharpoonup x$) the strong (respectively, weak) convergence of the sequence $\{x_{n}\}$ to x. Let $T : H \rightarrow H$ be an operator whose range is denoted by $R(T)$. The set of all fixed points of T is denoted by $\operatorname{Fix}(T)$, that is, $\operatorname{Fix}(T) = \{ x \in H : x = Tx \}$.
Definition 2.1
An operator $T : H \rightarrow H$ is said to be:
nonexpansive if $\|Tx-Ty\| \leq\|x-y\|$, for all $x, y \in H$;
strongly nonexpansive [6, 9, 10] if T is nonexpansive and
$$\lim_{n\rightarrow\infty}\bigl\| (x_{n}-y_{n})-(Tx_{n}-Ty_{n}) \bigr\| =0, $$
whenever $\{x_{n}\}$ and $\{y_{n}\}$ are bounded sequences in H and $\lim_{n\rightarrow\infty}(\|x_{n}-y_{n}\|-\|Tx_{n}-Ty_{n}\|)=0$;
averaged nonexpansive [11] if it can be written as
$$T = (1-\alpha)I +\alpha S, $$
where $\alpha\in(0,1)$, I is the identity operator on H, and $S:H\rightarrow H$ is a nonexpansive mapping;
firmly nonexpansive if $\|Tx-Ty\|^{2}\leq\langle x-y, Tx-Ty\rangle$, for all $x,y \in H$;
α-inverse strongly monotone (α-ism) if there exists a constant $\alpha> 0$ such that
$$\langle Tx-Ty, x-y\rangle\geq\alpha\|Tx-Ty\|^{2}, \quad \mbox{for all } x ,y \in H. $$
The following example shows that every nonexpansive operator is not necessarily strongly nonexpansive.
Example 2.1
Let $T : [-1,1] \to \mathbb {R}$ be defined by $Tx = -x$, for all $x \in[-1,1]$. Then T is nonexpansive but not strongly nonexpansive.
Indeed, let $x_{n} = 1$ and $y_{n}= 0$, for all n. Then $\{x_{n}\}$ and $\{ y_{n} \}$ are bounded sequences. Also,
$$\lim_{n\rightarrow\infty}\bigl|(x_{n}-y_{n})-(Tx_{n}-Ty_{n})\bigr| = \lim_{n\rightarrow\infty}|1+1| = 2 \neq0. $$
Thus, T is not strongly nonexpansive.
The following result will be used in the sequel.
Proposition 2.1
Let $T : H \rightarrow H$ be an operator.
If T is ν-ism, then for $\gamma> 0$, γT is $\frac{\nu}{\gamma}$-ism.
T is averaged if and only if the complement $I-T$ is ν-ism for some $\nu> \frac{1}{2}$. Indeed, for $\alpha\in(0,1)$, T is α-averaged if and only if $I-T$ is $\frac{1}{2\alpha}$-ism.
The composite of finitely many averaged mappings is averaged.
Let $\varphi: H \rightarrow H$ be a given single-valued α-inverse strongly monotone operator and $\lambda\in(0, 2\alpha)$. Then $(I-\lambda\varphi)$ is averaged. Indeed, since φ is α-inverse strongly monotone, λφ is $\frac{\alpha}{\lambda}$-ism. Thus, $I-\lambda\varphi$ is averaged as $\frac{\alpha}{\lambda} > \frac{1}{2}$.
Recall that a Banach space X is said to satisfy Opial's condition [12] if whenever $\{x_{n}\}$ is a sequence in X which converges weakly to x as $n\to\infty$, then
$$\limsup_{n\to\infty}\|x_{n}-x\|< \limsup _{n\to\infty}\|x_{n}-y\|, \quad\mbox{for all } y\in X, y\neq x. $$
It is well known that every Hilbert space satisfies Opial's condition.
Lemma 2.1
(Demiclosedness principle) [12], Lemma 2
Let C be a nonempty, closed, and convex subset of a real Hilbert space H and $T : C \rightarrow C $ be a nonexpansive operator with $\operatorname{Fix}(T) \neq\emptyset$. If the sequence $\{x_{n}\} \subseteq C$ converges weakly to x and the sequence $\{(I-T)x_{n}\}$ converges strongly to y, then $(I-T)x = y$. In particular, if $y = 0$, then $x \in\operatorname{Fix}(T)$.
Let $B : H \rightrightarrows H$ be a set-valued mapping. The domain, range, and inverse of B are denoted by
$$\begin{aligned}& D(B) = \bigl\{ x \in H : B(x) \neq\emptyset\bigr\} , \qquad R(B) = \bigcup _{x \in D(B)}B(x),\quad \mbox{and}\quad B^{-1}(y) = \bigl\{ x \in H : y \in B(x) \bigr\} , \end{aligned}$$
The set-valued mapping $B : H \rightrightarrows H$ with nonempty values is said to be:
monotone if
$$\langle u-v, x-y\rangle\geq0, \quad\mbox{for all } u \in Bx, v \in By; $$
maximal monotone if it is monotone and the graph
$$G(B) =\bigl\{ (x,u) \in H \times H : u \in Bx \bigr\} $$
of B is not properly contained in the graph of any other monotone operator.
Algorithm and convergence result
It is well known that when the set-valued mapping $B : H \rightrightarrows H$ is maximal monotone, then for each $x \in H$ and $\lambda> 0$, there is a unique $z \in H $ such that $x \in(I+\lambda B)z$ [13, 14]. In this case, the operator $J_{\lambda}^{B} := (I+\lambda B)^{-1}$ is called resolvent of B with parameter λ. It is well known that $J_{\lambda}^{B}$ is a single-valued and firmly nonexpansive mapping.
Indeed, for any given $u \in H$, let $x, y \in J_{\lambda}^{B}(u)$. Then $x, y \in(I+\lambda B)^{-1}(u)$, and thus $u-x \in\lambda Bx$ and $u-y \in\lambda B y$. The monotonicity of λB implies that
$$\bigl\langle u-x-(u-y), x-y \bigr\rangle \geq0. $$
This implies that $\|x-y\| \leq0$, and thus $x=y$. Hence, $J_{\lambda }^{B}$ is single-valued.
Next we show that $J_{\lambda}^{B}$ is firmly nonexpansive mapping. For any $x, y \in H $, let $J_{\lambda}^{B}(x)=(I+\lambda B)^{-1}(x)$ and $J_{\lambda}^{B}(y)=(I+\lambda B)^{-1}(y)$. This implies that $x \in(I+\lambda B)(J_{\lambda}^{B}(x))$ and $y \in(I+\lambda B)(J_{\lambda}^{B}(y))$. It follows that $\frac{1}{\lambda} ( x-(J_{\lambda}^{B}(x)) ) \in B(J_{\lambda}^{B}(x))$ and $\frac{1}{\lambda} ( y-(J_{\lambda}^{B}(y)) ) \in B(J_{\lambda}^{B}(y))$. The monotonicity of B implies that
$$\biggl\langle J_{\lambda}^{B}(x)-J_{\lambda}^{B}(y), \frac{1}{\lambda }\bigl(x-J_{\lambda}^{B}(x)\bigr) - \frac{1}{\lambda}\bigl(y-J_{\lambda}^{B}(y)\bigr) \biggr\rangle \geq0, $$
that is,
$$\bigl\langle J_{\lambda}^{B}(x)-J_{\lambda}^{B}(y), x-y \bigr\rangle \geq \bigl\Vert J_{\lambda}^{B}(x) - J_{\lambda}^{B}(y) \bigr\Vert ^{2}. $$
Thus, $J_{\lambda}^{B}$ is firmly nonexpansive.
Let $\phi: H \rightarrow H$ be a given single-valued operator. Then
$$ 0 \in\phi\bigl(x^{*}\bigr)+B\bigl(x^{*}\bigr) \quad\Leftrightarrow\quad x^{*} \in\operatorname{Fix} \bigl(J_{\lambda}^{B}(I-\lambda\phi) \bigl(x^{*}\bigr) \bigr). $$
Indeed, let $x^{*} \in\operatorname{Fix}(J_{\lambda}^{B}(I-\lambda\phi )(x^{*}))$. Then $x^{*} = J_{\lambda}^{B}(I-\lambda\phi)(x^{*})$. It follows that
$$x^{*} = (I+\lambda B)^{-1}(I-\lambda\phi) \bigl(x^{*}\bigr) \quad\Leftrightarrow\quad x^{*}-\lambda\phi \bigl(x^{*}\bigr) \in(I+\lambda B) \bigl(x^{*}\bigr) \quad \Leftrightarrow \quad 0 \in\phi\bigl(x^{*}\bigr) + B \bigl(x^{*}\bigr). $$
Since $J_{\lambda}^{B}$ is firmly nonexpansive, and therefore, averaged. It is well known that the composition of averaged mapping is averaged, thus $J_{\lambda}^{B}(I-\lambda\varphi)$ is averaged. Since every averaged mapping is strongly nonexpansive [10], it follows that $J_{\lambda }^{B}(I-\lambda\varphi)$ is also strongly nonexpansive.
Let $B_{1} : H_{1} \rightrightarrows H_{1}$ and $B_{2} : H_{2} \rightrightarrows H_{2}$ be set-valued mappings with nonempty values, and let $f : H_{1} \to H_{1}$ and $g : H_{2} \to H_{2}$ be mappings. Let $T : H_{1}\rightarrow H_{1}$ and $S : H_{2}\rightarrow H_{2}$ be operators such that $\operatorname{Fix}(T)\neq\emptyset$ and $\operatorname{Fix}(S)\neq\emptyset$. Let $U := J_{\lambda}^{B_{1}}(I-\lambda f)$ and $V := J_{\lambda}^{B_{2}}(I-\lambda g)$. With the help of (3.1), (1.8), and (1.9) can be rewritten as
$$ \mbox{find } x^{*}\in\operatorname{Fix}(T) \mbox{ such that } x^{*} \in \operatorname{Fix}\bigl(J_{\lambda}^{B_{1}}(I-\lambda f)\bigr), $$
$$ y^{*} := Ax^{*} \in\operatorname{Fix}(S) \mbox{ solves } y^{*}\in \operatorname{Fix}\bigl(J_{\lambda}^{B_{2}}(I- \lambda g)\bigr). $$
Now we propose the following algorithm to compute the approximate solutions of the SHMVIP.
Algorithm 3.1
Initialization: Let $\lambda> 0$ and take arbitrary $x_{0} \in H_{1}$.
Iterative step: For a given current $x_{n}\in H_{1}$, compute
$$ x_{n+1} = TU\bigl(x_{n}+\gamma A^{*}(SV-I)Ax_{n}\bigr), $$
where $\gamma\in (0, \frac{1}{\|A\|^{2}} )$.
Last step: Update $n := n+1$.
Next we prove the weak convergence of the sequence generated by Algorithm 3.1.
Let $A : H_{1}\rightarrow H_{2}$ be a bounded linear operator, $f : H_{1}\rightarrow H_{1}$ be an $\alpha_{1}$-inverse strongly monotone operator, $T : H_{1}\rightarrow H_{1}$ be a strongly nonexpansive operator such that $\operatorname{Fix}(T) \neq\emptyset$, $g : H_{2}\rightarrow H_{2}$ be an $\alpha_{2}$-inverse strongly monotone operator, $S : H_{2}\rightarrow H_{2}$ be a nonexpansive operator such that $\operatorname{Fix}(S)\neq\emptyset$, and $\alpha:= \min\{\alpha_{1},\alpha_{2}\}$. Consider the operator $U := J_{\lambda}^{B_{1}}(I-\lambda f)$ and $V := J_{\lambda}^{B_{2}}(I-\lambda g)$ with $\lambda\in(0,2\alpha)$, and $B_{1} : H_{1} \rightrightarrows H_{1}$ and $B_{2} : H_{2} \rightrightarrows H_{2}$ are two maximal monotone set-valued mappings with nonempty values. Then the sequence $\{x_{n}\}$ generated by Algorithm 3.1 converges weakly to an element $x^{*}\in\Psi$, provided $\Psi\neq\emptyset$.
Let $p\in\Psi$. Then $Tp=p$, $Up=p$, $SAp=Ap$, and $VAp=Ap$. Let $y_{n}:= x_{n}+\gamma A^{*}(SV-I)Ax_{n}$ and consider
$$\begin{aligned} \|y_{n}-p\|^{2}={}&\bigl\| x_{n}+\gamma A^{*}(SV-I)Ax_{n}-p\bigr\| ^{2} \\ ={}&\|x_{n}-p\|^{2}+\gamma^{2} \bigl\| A^{*}(SV-I)Ax_{n}\bigr\| ^{2} \\ &{}+2\gamma\bigl\langle x_{n}-p, A^{*}(SV-I)Ax_{n} \bigr\rangle \\ \leq{}&\|x_{n}-p\|^{2}+\gamma^{2}\|A \|^{2}\bigl\| (SV-I)Ax_{n}\bigr\| ^{2} \\ &{}+2\gamma\bigl\langle x_{n}-p, A^{*}(SV-I)Ax_{n} \bigr\rangle . \end{aligned}$$
Consider the third term of inequality (3.5), we have
$$\begin{aligned} &\bigl\langle x_{n}-p,A^{*}(SV-I)Ax_{n} \bigr\rangle \\ &\quad= \bigl\langle Ax_{n}-Ap, (SV-I)Ax_{n}\bigr\rangle \\ &\quad=\bigl\langle (SV-I)Ax_{n}-Ap+Ax_{n}-(SV-I)Ax_{n},(SV-I)Ax_{n} \bigr\rangle \\ &\quad=\langle SVAx_{n}-Ap, SVAx_{n}-Ax_{n} \rangle-\bigl\| (SV-I)Ax_{n}\bigr\| ^{2} \\ &\quad=\frac{1}{2}\| SVAx_{n}-Ap \|^{2}+ \frac{1}{2}\|SVAx_{n}-Ax_{n}\| ^{2}- \frac{1}{2}\|Ax_{n}-Ap\|^{2}-\bigl\| (SV-I)Ax_{n} \bigr\| ^{2} \\ &\quad=\frac{1}{2}\|SVAx_{n}-SVAp\|^{2}+ \frac{1}{2}\|SVAx_{n}-Ax_{n}\| ^{2}- \frac{1}{2}\|Ax_{n}-Ap\|^{2} - \bigl\| (SV-I)Ax_{n}\bigr\| ^{2} \\ &\quad\leq\frac{1}{2}\|Ax_{n}-Ap\|^{2}+ \frac{1}{2}\|SVAx_{n}-Ax_{n}\| ^{2}- \frac{1}{2}\|Ax_{n}-Ap\|^{2} - \bigl\| (SV-I)Ax_{n}\bigr\| ^{2} \\ &\quad=-\frac{1}{2}\bigl\| (SV-I)Ax_{n}\bigr\| ^{2}. \end{aligned}$$
Combining (3.5) and (3.6), we obtain
$$\begin{aligned} \|y_{n}-p\|^{2}&\leq\|x_{n}-p \|^{2}+\gamma^{2}\|A\|^{2}\bigl\| (I-SV)Ax_{n}\bigr\| ^{2}-\gamma\bigl\| (SV-I)Ax_{n}\bigr\| ^{2} \\ &=\|x_{n}-p\|^{2}-\gamma\bigl(1-\gamma\|A\|^{2} \bigr)\bigl\| (SV-I)Ax_{n}\bigr\| ^{2}. \end{aligned}$$
Since $\gamma\in(0,\frac{1}{\|A\|^{2}})$, we have $\gamma(1-\gamma\| A\|^{2}) >0$, and thus,
$$ \|y_{n}-p\|\leq\|x_{n}-p\|. $$
From the above inequality (3.7), we have
$$\begin{aligned} \|x_{n+1}-p\|^{2}&=\|TUy_{n}-TUp \|^{2} \\ &\leq\|Uy_{n}-Up\|^{2} \\ &\leq\|y_{n}-p\|^{2} \\ &\leq\|x_{n}-p\|^{2}-\gamma\bigl(1-\gamma\|A\|^{2} \bigr)\bigl\| (SV-I)Ax_{n}\bigr\| ^{2} \\ &\leq\|x_{n}-p\|^{2}. \end{aligned}$$
This shows that $\|x_{n+1}-p\|\leq\|x_{n}-p\|$ and this implies that $\{\|x_{n}-p\|\}_{n=1}^{\infty}$ is a monotonic decreasing sequence, also $\{x_{n}\}$ is a bounded sequence, see [15], Theorem 4.5.10 and, hence, $\lim_{n\rightarrow\infty}\|x_{n}-p\|$ exists. Taking the limit at both sides in (3.9), and noticing that $\gamma(1-\gamma\|A\|^{2}) >0$, we have
$$ \lim_{n\rightarrow\infty}\bigl\| (SV-I)Ax_{n}\bigr\| =0, $$
and since $y_{n} := x_{n}+\gamma A^{*}(SV-I)Ax_{n}$, we have $\| y_{n}-x_{n}\|= \gamma\|A\|\|(SV-I)Ax_{n}\|$ thus in view of (3.10) we have
$$ \lim_{n\rightarrow\infty}\|y_{n}-x_{n}\|=0. $$
Since $\{x_{n}\}$ is a bounded sequence and it has a weakly convergent subsequence say, $x_{n_{i}}\rightharpoonup x^{*}$, [10] or with the help of Opial's condition [12], we can see that $x_{n}\rightharpoonup x^{*}$. Thus, we have $Ax_{n}\rightharpoonup Ax^{*}$. Since SV is nonexpansive, from (3.10) and the closedness of $SV-I$ at 0 we obtain $SV Ax^{*}=Ax^{*}$. Next we show that $VAx^{*}=Ax^{*}$. We have
$$\bigl\vert \|SV Ax_{n}-Ap\|-\|Ax_{n}-Ap\| \bigr\vert \leq\|SV Ax_{n}-Ax_{n}\|. $$
Taking the limit at both sides and by using (3.10), we obtain
$$ \begin{aligned} &\lim_{n\rightarrow\infty}\bigl\vert \bigl(\|SV Ax_{n}-Ap\|-\|Ax_{n}-Ap\| \bigr)\bigr\vert =0, \\ &\Bigl\vert \lim_{n\rightarrow\infty} \bigl(\|SV Ax_{n}-Ap\|- \|Ax_{n}-Ap\| \bigr)\Bigr\vert =0, \\ &\lim_{n\rightarrow\infty} \bigl(\|SV Ax_{n}-Ap\|-\|Ax_{n}-Ap \| \bigr)=0. \end{aligned} $$
Since $S Ap = Ap$ and $V Ap = Ap$, by the nonexpansiveness of S and V, we have
$$\|SV Ax_{n}-Ap\| \leq\|V Ax_{n}-Ap\| \leq \|Ax_{n}-Ap\|, $$
and therefore
$$\|SV Ax_{n}-Ap\|-\|Ax_{n}-Ap\| \leq\|V Ax_{n}-Ap \|-\|Ax_{n}-Ap\|\leq0. $$
Thus, we have
$$ \lim_{n\rightarrow\infty} \bigl(\|SV Ax_{n}-Ap\|- \|Ax_{n}-Ap\| \bigr) \leq\lim_{n\rightarrow\infty} \bigl(\|V Ax_{n}-Ap\|-\|Ax_{n}-Ap\| \bigr)\leq0. $$
From (3.12) and (3.13), we obtain
$$ \lim_{n\rightarrow\infty} \bigl(\|V Ax_{n}-Ap\|- \|Ax_{n}-Ap\| \bigr)= 0. $$
Since V is averaged nonexpansive and every averaged nonexpansive map is strongly nonexpansive, we see that V is strongly nonexpansive. Since $\{Ap\}$ and $\{Ax_{n}\}$ are bounded sequences, by the definition of strong nonexpansiveness of V, we have
$$\lim_{n\rightarrow\infty}\|VAx_{n}-Ax_{n}\|=0. $$
Since V is nonexpansive, by the demiclosedness principle, we have
$$V Ax^{*}=Ax^{*}. $$
Now, we show that $Tx^{*}=x^{*}$ and $Ux^{*}=x^{*}$. By using the nonexpansiveness of T and U, in view of (3.4) and (3.8), we have
$$0 \leq\|Uy_{n}-p\|-\|TUy_{n}-p\| \leq\|y_{n}-p\|- \|TUy_{n}-p\| \leq\| x_{n}-p\|-\|x_{n+1}-p\|. $$
This implies that
$$ \lim_{n\rightarrow\infty}\bigl(\|Uy_{n}-p\|- \|TUy_{n}-p\|\bigr)=0. $$
From (3.8), we see that $\{y_{n}\}$ is a bounded sequence and thus $\{Uy_{n}\}$ is bounded and since $\{p\}$ is a constant sequence thus $\{p\}$ is also bounded and since T is strongly nonexpansive, we have
$$ \lim_{n\rightarrow\infty}\|Uy_{n}-TUy_{n} \|=0. $$
In view of (3.4) and (3.8), by using the nonexpansiveness of TU, we have
$$0 \leq\|y_{n}-p\|-\|TUy_{n}-p\| \leq\|x_{n}-p\|- \|x_{n+1}-p\|. $$
$$ \lim_{n\rightarrow\infty}\bigl(\|y_{n}-p\|- \|TUy_{n}-p\|\bigr)=0. $$
By using the nonexpansiveness of T and U, we have
$$\|TU y_{n}-p\|\leq\|Uy_{n}-p\|\leq\|y_{n}-p\|, $$
$$\|TUy_{n}-p\|-\|y_{n}-p\| \leq\|Uy_{n}-p\|- \|y_{n}-p\|\leq0. $$
Thus, from (3.17), we have
$$ \lim_{n\rightarrow\infty}\bigl(\|Uy_{n}-p\|- \|y_{n}-p\|\bigr)=0. $$
From (3.8) we see that $\{y_{n}\}$ is a bounded sequence and $\{p\}$ being a constant sequence, also bounded, by the strong nonexpansiveness of U, we have
$$ \lim_{n\rightarrow\infty}\|Uy_{n}-y_{n} \|=0. $$
Next consider, for all $f \in H$,
$$\begin{aligned} \bigl\| f(y_{n})-f\bigl(x^{*}\bigr)\bigr\| &=\bigl\| f(y_{n})-f(x_{n})+f(x_{n})-f \bigl(x^{*}\bigr)\bigr\| \\ &\leq\bigl\| f(y_{n})-f(x_{n})\bigr\| +\bigl\| f(x_{n})-f \bigl(x^{*}\bigr)\bigr\| \\ &\leq\|f\|\|y_{n}-x_{n}\|+\bigl\| f(x_{n})-f \bigl(x^{*}\bigr)\bigr\| . \end{aligned}$$
Since $x_{n}\rightharpoonup x^{*}$ and from (3.11), we have $\lim_{n\rightarrow\infty}\|f(y_{n})-f(x^{*})\|=0$, thus $y_{n}\rightharpoonup x^{*}$. Thus, in view of (3.19) and by applying the demiclosedness principle, we have $Ux^{*}=x^{*}$. Again, since $y_{n}\rightharpoonup x^{*}$, in view of (3.19), we have $U y_{n}\rightharpoonup x^{*}$. Thus, again in view of (3.16) and by applying the demiclosedness principle, we have $Tx^{*}=x^{*}$. □
Now, we illustrate Algorithm 3.1 and Theorem 3.1 by the following example.
Let $H_{1} = H_{2} = H = \mathbb{R}$ and $B : H \rightrightarrows H$ be defined by
$$ B(x) = \textstyle\begin{cases} \{1\}, &\mbox{if }x > 0, \\ {[}0,1], &\mbox{if }x =0, \\ \{0\}, &\mbox{if }x < 0. \end{cases} $$
Then, as shown in [16], B is a set-valued maximal monotone mapping. We define the mappings $A, f, h, T, S : H \rightarrow H$ by
$$\begin{aligned}& Ax = \frac{x}{2}, \quad \mbox{for all } x \in H, \\& fx = hx = \frac{2x}{3}, \quad\mbox{for all } x \in H, \\& Tx = \frac{x}{3}, \quad\mbox{for all } x \in H, \end{aligned}$$
$$Sx = \frac{4x}{5} , \quad\mbox{for all } x \in H, $$
respectively. It is easy to show that A is a bounded linear operator, f and h are $\frac{1}{3}$-ism, T is firmly nonexpansive, and thus T is strongly nonexpansive [10] and S is nonexpansive. Let $B_{1}(x)= B_{2}(x)=Bx$. Then $B_{1}$ and $B_{2}$ are maximal monotone set-valued mappings. Let $J_{\lambda}^{B_{1}}(x) = J_{\lambda}^{B_{2}}(x)= \frac{x}{2}$ be the resolvent operator. The values of $\{x_{n}\}$ with different values of n are reported in the Table 1. All codes were written in Matlab R2010.
Table 1 The values of $\pmb{\{x_{n}\}}$ with initial guess $\pmb{x_{1}=10}$ , $\pmb{x_{1}=15}$ , and $\pmb{x_{1}=20}$
Table 1 shows that the sequence $\{x_{n}\}$ converges to 0, which is the required solution.
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Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India
Qamrul Hasan Ansari
& Aisha Rehan
Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia
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Correspondence to Qamrul Hasan Ansari.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
split hierarchical monotone variational inclusions
fixed point problems
iterative method
maximal monotone set-valued mappings
resolvent operators
convergence analysis | CommonCrawl |
MIDI mike
Signal path
Digitizing the analog signal
Frequency and pitch detection
Note level segmentation and buffering
Johan Vonk
Describes frequency and pitch detection for pitch detection on Arduino (frequency.cpp, pitch.cpp). The core of the application: the frequency detection algorithm. Part of the project Arduino Pitch Detector.\(\)
Each musical instruments create an unique set of harmonics [demo]. In the commonly used Even Tempered Scale, the A4 key on the piano corresponds a fundamental frequency \(f_0=440\mathrm{\ Hz}\). Other frequencies follow as:
$$ f=2^{\frac{n}{12}}\times 440\,\rm{Hz} $$ where \(n\) is the number of half-steps from middle A (A4).
Designs considered
The main challenge of this project is to detect the fundamental frequency of the notes played using an embedded system. The fundamental frequency is defined as the lowest frequency produced by an oscillation.
The following three methods were considered
Using a time-domain feature such as zero crossings. This means that you find the distance between when the waveform goes from negative to positive the first time and when it does that a second time.
Using autocorrelation to determine the frequency of instrumental sounds as published by Judith C. Brown and Bin Zhang [Brown, Monti]. Autocorrelation is a math tool for finding repeating patterns. It estimates the degree to which a signal and a time lagged version of itself are correlated. A high correlation indicates a periodicity in the signal at the corresponding time lag.
An alternate method of calculating autocorrelation is by using a Fast Fourier Transform and approaching it similar to convolution. To get cross-correlation instead of convolution, I time-reverse one of the signals before doing the FFT, or take the complex conjugate of one of the signals after the FFT as shown in $$ R_{xx}(k) = \mathcal{F}^{-1}\left(\mathcal{F}(t)\times\mathcal{F}(t+k)^*\right) $$
A literature study revealed that using time-domain features (1) will not perform well for musical instruments, such as a clarinet, that produce harmonics that are stronger than the fundamental frequency.
Brown's method (2) is more promising. It calculates the autocorrelation \(R_{xx}\) at lag \(k\) by the equation [wiki, Lyon]
$$ \begin{align} R_{xx}(k) & =\frac{1}{\sigma^2} \sum_{t=1}^N(s_t-\bar{s})(s_{t+k}-\bar{s})\\ \rm{where}\quad \bar{s}&=\frac{1}{N}\sum_{t=1}^Ns_t,\quad \sigma=\sqrt{\frac{1}{N}\sum_{t=1}^N(s_t-\bar{s})^2}\nonumber \end{align} $$
The symbols:
\(s\) are audio samples
\(N\) is the total number of samples
\(k\) is the lag
\(\bar{s}\) is the mean signal value
\(\sigma^2\) is a normalization factor.
However, calculating the autocorrelation requires \(2N\) subtractions, \(N\) additions, \(2N\) multiplications, and \(N\) divisions. This is likely to exceed the design constraints.
The alternate method of calculating autocorrelation (3) reduces the processing requirement to \(N-\log(N)\), but the algorithm uses significantly more memory. This leaves less memory to store audio samples thereby reducing the window size and consequently limits the ability to recognize low frequencies.
Once it determines the frequency, the MIDI pitch \(m\) follows as
$$ m = 69+12\log_2\frac{f}{440} $$
Design to find the frequency
To detect the fundamental frequency, I simplified Brown's method by making two assumptions
The signal has no DC bias, \(\bar{s}=0\).
We're only interested in the lag for which the autocorrelation peaks, not the absolute value of the autocorrelation. Therefore, the normalization factor \(\sigma^2\) that is independent of the lag \(k\) can be ignored.
If the term \(t+k\) extends past the length of the series, the series is considered to be \(0\).
Based on these assumptions, the autocorrelation can be estimated as:
$$ R_{xx}(k) = \sum_{t=1}^Ns_t\,s_{t+k} $$
The figure below shows a visualization of the term . The original waveform is shown in blue, and the time lagged version in red. The black curve shows the multiplication of these signals.
The term s(t) s(t+k). for one value of k
The plot below shows an example of the estimated autocorrelation for \(R_{xx}(k)\) as a function of the lag \(k\). By definition the maximum autocorrelation \(R_{xx}(0)\) is at lag \(k=0\).
I ported my frequency detection code to GNU Octave to enhance my visual understanding of the algorithm. This was especially helpful in determining the empirical threshold for the peak finding algorithm.
A peak finding algorithm looks for the first peak that exceeds a threshold at \(\frac{2}{3}R_{xx}(0)\). The corresponding lag \(k_0\) is considered the period time \(T_0\). The fundamental frequency \(f_0\) follows as the inverse of \(T_0\).
Pitch Rxx
The listing below shows a code fragment from frequency.cpp that implements the autocorrelation function.
View the code on Gist.
Autocorrelation function
Design to find the peak
By definition the autocorrelation is maximum at lag \(k=0\). If we find the maximum value for \(R_{xx}(k)\), for \(0 \lt k \lt N\), then \(k\) is the period time. This requires calculating \(R_{xx}(k)\) for all values of \(k\).
To make it faster, I accept the first maximum that is above \(\frac{2}{3}R_{xx}(0)\). The first peak that exceeds this value is considered the period time \(T_0\). The algorithm is shown below.
Peak finding algorighm
A complementary simulation in GNU Octave visualizes the algorithm, making the process easier to understand and fine tune. The video below shows the calculation of \(R_{xx}(k)\), and the peak finding algorithm. To run the simulation yourself, load the file simulation/file2pitch.m. in GNU Octave or Matlab.
https://coertvonk.com/wp-content/uploads/pitch-rxx-peak.mp4
Analyzing accuracy
Analysis revealed that the sample rate and window size determine the maximum and minimum frequency that can be recognized. These variables can be configured in config.h.
If the sample rate is to low, high frequencies will only have a few audio samples per period, causing these frequencies not to be accurately recognized.
The window size is the number of audio samples that are processed at the time in the autocorrelation loop. If the windows size is too small, low frequencies cannot be recognized.
The delay is caused by the sampling of the input, calculating the frequency and the segmentation algorithm. The highest delay occurs at the lowest frequency, approximately 60 milliseconds. This was noticeable but acceptable. I observed that my simple synthesizer software introduced a noticeable additional delay. The delay was minimized by sampling audio while doing the calculations, and by stopping the autocorrelation as soon as the frequency could be determined.
The project's aim is to recognize the range of notes produced by a B♭ clarinet. This clarinet can produce notes from E♭3 to G6, corresponding to a fundamental frequencies \(f_L\) and \(f_H\) $$ \shaded{ \left\{ \begin{align} f_L &= 155.6 \, \rm{Hz} \nonumber \\ f_H &= 1.568 \, \rm{kHz} \nonumber \end{align} \right. } $$
For the 12-note-per-octave equal-tempered scale, each note or semi-tone is \(5.946309436\%\) "higher" in frequency than the previous note. Given that the frequency will be rounded to a note pitch, we can allow for an error rate \(\varepsilon\) $$ \varepsilon \approx 0.05946\% $$
The highest frequency \(f_H\), determines the sample rate. To stay within the error rate \(\varepsilon\), the sample rate \(f'_{s}\) follows as $$ f_s' = \frac{f_H}{\varepsilon} = \frac{1568}{0.05946} \approx 26.37\,\rm{kHz} $$
The Arduino can only sample a signal at \(2^a\,\frac{16\,\rm{MHz}}{128\times 13}\), where \(a\in \mathbb{N}\). As a consequence, the sampling rate has to be rounded up to \(f_{s}^{\prime\prime}\) $$ f_s^{\prime\prime} = 38.461\,\rm{kHz} $$
The lowest frequency \(f_L\) determines the window size, where the number of audio samples \(N^\prime\) should cover at least twice period time of the lowest frequency \(f_L\) $$ N^\prime = 2\,\frac{f_s^{\prime\prime}}{f_L} = 2\,\frac{38461}{155.6} \approx 495 $$
Each audio sample requires 1 byte of the Arduino's SDRAM. With only about \(200\) bytes left available to store audio samples \(N'\) will not fit.
Instead, we use the available \(200\) bytes to store the samples, so the window size \(N\) is $$ \shaded{ N = 200 } $$
In order to recognize the lowest frequency \(f_L\), the sample frequency \(f_s^{""}\) follows as $$ \begin{align} f_s^{"'} &\le f_L\,\frac{N}{2} \nonumber \\ &\le 155.6\,\frac{200}{2} \nonumber \\ &\le 15.560\,\rm{Hz} \end{align} $$
For the Arduino sample rate scaler, this needs to be rounded down to \(f_s\) $$ \shaded{ f_s = 9.615\,\rm{kHz} } $$
The resulting frequency range can be expressed as $$ \begin{align} \frac{f_s}{N/2} \lt &f \lt \Delta\varepsilon\,f_s \nonumber \\ \frac{9615}{200/2} \lt &f \lt 0.0595\times 9615 \nonumber \\ 96.2\,\rm{Hz} \lt &f \lt 572\,\rm{Hz} \end{align} $$
This implies that it will only reach D♭5. Let's see how we can improve the accuracy.
Low notes are meased accurately, but errors increase with frequency.
B♭ Clarinet pianissimo, N=200, S=9615, threshold=67%
With the base algorithm in place, time has come to focus on improvements.
Improving speed
The fundamental frequency requires calculating \(R_{xx}\) for all values of \(k\) . However, the possible values of \(k\) are limited by the window size and sample frequency.
The window size limits the lowest frequency, while
the sample frequency limits the highest frequency.
The range for the lag \(k\) is determined by the highest frequency \(f_H\) and the windows size \(N\) $$ \begin{align} \frac{f_s}{f_H} \leq &k \leq \frac{N}{2} \nonumber \\[0.5em] \implies \frac{9615}{1568} \leq &k \leq \frac{200}{2} \end{align} $$
Rounding down the minimum and rounding up the maximum values, the range for the lag \(k\) follows as $$ \shaded{ 6 \leq k \leq 62 } $$
Improving accuracy of high notes
We can improve the accuracy of especially the high notes, by using interpolation. Fitting a parabolic curve to the prior, current and next autocorrelation values \(k_1,k_2,k_3\). The value for \(k\) that corresponds to the top of the parabola, is the estimate lag \(k_m\).
The difference between the estimated autocorrelation value \(k_m\) and the measured factor \(k_2\) is the correction factor \(\delta\). [^1][^2] [^1]: [Polynomial Interpolation, Abdulkadir Hassen [^2]: Cross-Correlation, Douglas Lyon] $$ \delta = \frac{k_3-k_1}{2(2k_2-k_1-k_3)} $$
The corresponding sample window size \(N'\), is determined by the lowest frequency \(F_L\) $$ N' = 2\frac{f_{s}'}{f_L}=2\,\frac{9615}{155.6}\approx 125 $$
I rounded the window size \(N"\) up to 200 bytes. $$ \shaded{ N = 200 } $$
The accuracy of high note dramatically improves as shown below.
B♭ Clarinet pianissimo, N=200, S=9615, interpolation, threshold=67%
Improving accuracy of low notes
As the peak finding algoritm considers higher values of the lag \(k\), the autocorrelation values decrease because of the zeroes introduced in the shifted signal.
I tried normalizing the autocorrelation for these introduced zeroes, by multiplying with a normalization factor of \(\frac{N}{N-k}\). As a result the normalized autocorrelation can be expressed as $$ R_{xx}(k)=\frac{N}{N-k}\sum_{t=1}^Ns_t\,s_{t+k} $$
For the clarinet the nomalization makes it drop some high notes as shown in the figure below. The clarinet doesn't benefit from improving accuracy on low notes, because the lowest notes it can play is \(155.6\,\rm{Hz}\) compared to the Arduino that can detect up to \(96\,\rm{Hz}\).
B♭ Clarinet pianissimo, N=200, S=9615, interpolation, normalization, threshold=80%
The results for piano samples using interpolation are shown below for reference.
Piano mezzo-forte, N=200, S=9615, interpolation, threshold=80%
For piano it greatly benefits the low notes as shown below.
Piano mezzo-forte, N=200, S=9615, interpolation, normalization, threshold=80%
Continue reading on the next page to learn about note level segmentation.
Tagsarduino uno, atmega328p, auto-correlation
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March 2020, 2(1): 19-33. doi: 10.3934/fods.2020002
Semi-supervised classification on graphs using explicit diffusion dynamics
Robert L. Peach 1,2, , Alexis Arnaudon 2,†, and Mauricio Barahona 2,,
Department of Mathematics and Imperial College Business School, Imperial College London, London SW7 2AZ, UK
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
* Corresponding author: Mauricio Barahona
† Current address: Blue Brain Project, Éole polytechnique fédérale de Lausanne (EPFL), Campus Biotech, 1202 Geneva, Switzerland.
Fund Project: All authors acknowledge funding through EPSRC award EP/N014529/1 supporting the EPSRC Centre for Mathematics of Precision Healthcare at Imperial
Classification tasks based on feature vectors can be significantly improved by including within deep learning a graph that summarises pairwise relationships between the samples. Intuitively, the graph acts as a conduit to channel and bias the inference of class labels. Here, we study classification methods that consider the graph as the originator of an explicit graph diffusion. We show that appending graph diffusion to feature-based learning as an a posteriori refinement achieves state-of-the-art classification accuracy. This method, which we call Graph Diffusion Reclassification (GDR), uses overshooting events of a diffusive graph dynamics to reclassify individual nodes. The method uses intrinsic measures of node influence, which are distinct for each node, and allows the evaluation of the relationship and importance of features and graph for classification. We also present diff-GCN, a simple extension of Graph Convolutional Neural Network (GCN) architectures that leverages explicit diffusion dynamics, and allows the natural use of directed graphs. To showcase our methods, we use benchmark datasets of documents with associated citation data.
Keywords: Semi-supervised learning, graph convolutional neural networks, deep learning, Laplacian dynamics, graph diffusion.
Mathematics Subject Classification: Primary: 05C81, 05C85, 05C21, 68R10, 62M45; Secondary: 34B45, 60J60.
Citation: Robert L. Peach, Alexis Arnaudon, Mauricio Barahona. Semi-supervised classification on graphs using explicit diffusion dynamics. Foundations of Data Science, 2020, 2 (1) : 19-33. doi: 10.3934/fods.2020002
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Table 1. Statistics of datasets as reported in [35] and [30]
Datasets Nodes Edges Classes Features
Citeseer $ 3,327 $ $ 4,732 $ $ 6 $ $ 3,703 $
Cora $ 2,708 $ $ 5,429 $ $ 7 $ $ 1,433 $
Pubmed $ 19,717 $ $ 44,338 $ $ 3 $ $ 500 $
Wikipedia $ 20,525 $ $ 215,056 $ $ 12 $ $ 100 $
Table 2. Percentage classification accuracy before and after application of relabelling by GDR for various classifiers. We present the improvement of GDR on the uniform prediction (which ignores features). We also consider four supervised classifiers (which learn from features without the graph): projection, RF, SVM and MLP. For RF, we use a maximum depth of $ 20 $; for SVM, we set $ C = 50 $; for MLP, we implement the same architecture as GCN ($ d_1 = 16 $-unit hidden layer, $ 0.5 $ dropout, $ 200 $ epochs, $ 0.01 $ learning rate, $ L^2 $ loss function). Finally, we compare with two semi-supervised graph classifiers: GCN [20] and Planetoid [35]. The numbers in brackets record the change in accuracy accomplished by applying GDR on the corresponding prior classifier. Boldface indicates the method with highest accuracy for each dataset
Method Citeseer Cora Pubmed Wikipedia
Uniform 7.7 13.0 18.0 28.7
GDR (Uniform) 50.6 (+42.9) 71.8 (+58.8) 73.2 (+55.2) 31.4 (+2.7)
Projection 61.8 59.0 72.0 32.5
RF 60.3 58.9 68.8 50.8
SVM 61.1 58.0 49.9 31.0
MLP 57.0 56.0 70.7 43.0
GDR (Projection) 70.4 (+8.7) 79.7 (+20.7) 75.8 (+3.8) 36.9 (+4.4)
GDR (RF) 70.5 (+10.2) 78.7 (+19.8) 72.2 (+3.2) 50.8 (+0.0)
GDR (SVM) 70.3 (+9.2) 81.2 (+23.2) 52.4 (+2.5) 41.9 (+10.8)
GDR (MLP) 69.7(+12.7) 78.5 (+22.5) 75.5 (+4.8) 40.5 (-2.5)
Planetoid 64.7 75.7 72.2 -
GCN 70.3 81.1 79.0 39.2
GDR (GCN) 70.8 (+0.5) 82.2 (+1.1) 79.4 (+0.4) 39.5 (+0.3)
Table 3. Percentage classification accuracy of GCN and its extension diff-GCN, which has an explicit diffusion operator (16)
Model Citeseer Cora Pubmed Wikipedia
diff-GCN 71.9 82.3 79.3 45.9
Table 4. Accuracy of GDR using the undirected, directed, and reverse directed graphs of the Cora dataset
Undirected Directed (fw) Directed (bw)
Method $ A $ $ A_\text{dir} $ $ A_\text{dir}^T $
GDR (Projection) 79.7 62.1 64.6
GDR (RF) 78.7 58.0 57.6
GDR (SVM) 81.2 63.6 62.1
GDR (MLP) 78.5 57.3 56.4
Table 5. Accuracy of GCN and diff-GCN using the undirected, directed, reverse directed, and bidirectional (augmented) graphs of the Cora dataset. The highest accuracy is achieved by diff-GCN with the augmented graph (boldface)
Undirected Directed (fw) Directed (bw) Augmented (fw, bw)
Method $ A $ $ A_\text{dir} $ $ A_\text{dir}^T $ $ \begin{bmatrix} A_\text{dir} \, A_\text{dir}^T \end{bmatrix} $
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Robert L. Peach Alexis Arnaudon Mauricio Barahona | CommonCrawl |
Integrability, Supersymmetry and Coherent States
Integrability, Supersymmetry and Coherent States pp 231-253 | Cite as
Trace Formulas Applied to the Riemann ζ-Function
Mark S. Ashbaugh
Fritz Gesztesy
Lotfi Hermi
Klaus Kirsten
Lance Littlejohn
Hagop Tossounian
Part of the CRM Series in Mathematical Physics book series (CRM)
We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.
Dirichlet Laplacian Trace class operators Trace formulas Riemann zeta function Spectral theory
We are indebted to the anonymous referee for kindly bringing references [24] and [27] to our attention.
Klaus Kirsten was supported by the Baylor University Summer Sabbatical and Research Leave Program.
Appendix: Basic Formulas for the Riemann ζ-Function
We present a number of formulas for ζ(z) and special values of ζ(⋅). It goes without saying that no such collection can ever attempt at any degree of completeness, and certainly our compilation of formulas is no exception in this context.
$$\displaystyle \begin{aligned} \zeta(z) &= \sum_{k \in {\mathbb{N}}} k^{-z}, \quad z \in {\mathbb{C}}, \, \operatorname{Re}(z) > 1 {} \end{aligned} $$
(A.1)
Functional Equation
$$\displaystyle \begin{aligned} \zeta(z) = 2^z \pi^{z-1} \sin{}(\pi z/2) \varGamma(1 - z) \zeta(1 - z), \quad z \in {\mathbb{C}}, \, \operatorname{Re}(z) < 0. \end{aligned} $$
Alternative Formulas
$$\displaystyle \begin{aligned} \zeta(z) &= \varGamma(z)^{-1} \int_0^{\infty} dt \, \frac{t^{z - 1}}{e^t - 1}, \quad z \in {\mathbb{C}}, \, \operatorname{Re}(z) > 1 \end{aligned} $$
$$\displaystyle \begin{aligned} &= \varGamma(z)^{-1} [1 - 2^{1 - z}]^{-1} \int_0^{\infty} dt \, \frac{t^{z - 1}}{e^{t} + 1}, \quad z \in {\mathbb{C}}, \, \operatorname{Re}(z) > 0 \end{aligned} $$
$$\displaystyle \begin{aligned} \varGamma(z) = \int_0^{\infty} dt \, t^{z - 1} e^{- t}, \quad z \in {\mathbb{C}}, \, \operatorname{Re}(z) > 0. \end{aligned} $$
In addition,
(A.10)
$$\displaystyle \begin{aligned} &= \pi^{z/2} \varGamma(z/2)^{-1} \int_0^{\infty} dt \, t^{(z/2) - 1} \sum_{k \in {\mathbb{N}}} e^{- k^2 \pi t}, \end{aligned} $$
$$\displaystyle \begin{aligned} &= \frac{2^{z - 1}}{z - 1} - 2^z \int_0^{\infty} dt \, \frac{\sin{}(z \arctan(t))}{(1 + t^2)^{z/2} (e^{\pi t} + 1)}, \quad z \in {\mathbb{C}}\backslash\{1\}, \end{aligned} $$
$$\displaystyle \begin{aligned} &= \varGamma(z+1)^{-1} 4^{-1} (2a)^{z + 1} \big[1 - 2^{1 - z}\big]^{-1} \int_0^{\infty} dt \, \frac{t^{z}}{[\cosh(at)]^2}, \end{aligned} $$
Specific Values
$$\displaystyle \begin{aligned} \zeta (2n) = \frac{(-1)^{n + 1}(2 \pi)^{2n}B_{2n} }{2 (2n)!}, \quad n \in {\mathbb{N}}_0, {} \end{aligned} $$
where Bm are the Bernoulli numbers generated, for instance, by
$$\displaystyle \begin{aligned} \frac{w}{e^{w} - 1} = \sum_{m \in {\mathbb{N}}_0} B_m \frac{w^m}{m!}, \quad w \in {\mathbb{C}}, \, |w| < 2 \pi, \end{aligned} $$
in particular,
$$\displaystyle \begin{aligned} & B_0 = 1, B_1 = - 1/2, \; B_2 = 1/6, \; B_3 = 0, B_4 = - 1/30, B_5 = 0, B_6 = 1/42, \text{etc.,} \end{aligned} $$
$$\displaystyle \begin{aligned} & B_{2k+1} = 0, \; k \in {\mathbb{N}}. {} \end{aligned} $$
Moreover, one has the generating functions for ζ(2n),
$$\displaystyle \begin{aligned} - (\pi z/2) \cot{}(\pi z) &= \sum_{n \in {\mathbb{N}}_0} \zeta(2n) z^{2n}, \quad |z| < 1, \; \zeta(0) = -1/2, {} \end{aligned} $$
$$\displaystyle \begin{aligned} - (\pi z/2) \coth(\pi z) &= \sum_{n \in {\mathbb{N}}_0} (-1)^n \zeta(2n) z^{2n}, \quad |z| < 1, \; \zeta(0) = -1/2, {} \end{aligned} $$
and [32]
$$\displaystyle \begin{aligned} (n!/6)[\zeta(n - 2) - 3 \zeta(n - 1) + 2 \zeta(n)] = \int_0^{\infty} dt \, \frac{t^n e^t}{(e^t - 1)^4}, \quad n \in {\mathbb{N}}, \, n \geq 4. \end{aligned} $$
Choosing k = 2n, \(n \in {\mathbb {N}}\), even, employing (A.30) for ζ(2n), ζ(2n − 2), yields a formula for ζ(2n − 1). Moreover,
$$\displaystyle \begin{aligned} \zeta(2n+1) &= \frac{1}{(2n)!} \int_0^{\infty} dt \, \frac{t^{2n}}{e^t - 1}, \quad n \in {\mathbb{N}} \end{aligned} $$
where Bm(⋅) are the Bernoulli polynomials,
$$\displaystyle \begin{aligned} B_m(z) = \sum_{j = 0}^m \begin{pmatrix} m \\ j \end{pmatrix} B_j z^{m - j}, \quad t \in {\mathbb{C}}, \end{aligned} $$
generated, for instance, by
$$\displaystyle \begin{aligned} \frac{w e^{z w}}{e^{w} - 1} = \sum_{m \in {\mathbb{N}}_0} B_m(z) \frac{w^m}{m!}, \quad w \in {\mathbb{C}}, \, |w| < 2 \pi. \end{aligned} $$
Explicitly,
$$\displaystyle \begin{aligned}& \begin{aligned} & B_0(x) = 1, \; B_1(x) = x - (1/2), \; B_2(x) = x^2 - x + (1/6), \\ & B_3(x) = x^3 - (3/2)x^2 + (1/2)x, \; \text{etc.,} {} \end{aligned} \end{aligned} $$
$$\displaystyle \begin{aligned} & B_n(0) = B_n, \; n \in {\mathbb{N}}, \quad B_1(1) = - B_1 = 1/2, \; B_n(1) = B_n, \; n \in {\mathbb{N}}_0 \backslash \{1\}, {} \end{aligned} $$
$$\displaystyle \begin{aligned} & B_n^{\prime}(x) = n B_{n-1}(x), \quad n \in {\mathbb{N}}, \; x \in {\mathbb{R}}. {} \end{aligned} $$
In addition, for \(n \in {\mathbb {N}}\),
Just for curiosity,
$$\displaystyle \begin{aligned} \zeta(3) = 1.2020569032 \ldots.. \end{aligned} $$
Apery [1] proved in 1978 that ζ(3) is irrational (see also Beukers [2], van der Poorten [29], Zudilin [35], and [31], [33]).
$$\displaystyle \begin{aligned} \zeta(3) &= -\dfrac{2}{7} \pi^2 \operatorname{ln}(2) - \dfrac{16}{7} \, \int_0^1 dt \, \dfrac{\mathrm{arctanh}(t) \operatorname{ln} (t)}{t(1-t^2)} {} \end{aligned} $$
$$\displaystyle \begin{aligned} &= - \dfrac{4}{3} \, \int_0^1 dt \, \dfrac{\operatorname{ln}(t) \operatorname{ln}(1+t)}{t} \end{aligned} $$
$$\displaystyle \begin{aligned} & = - 8 \, \int_0^1 dt \, \dfrac{\operatorname{ln}(t) \operatorname{ln}(1+t)}{1+t} \end{aligned} $$
$$\displaystyle \begin{aligned} &= \int_0^1 dt \, \dfrac{\operatorname{ln}(t) \operatorname{ln}(1-t)}{1-t} = \int_0^1 dt \, \dfrac{\operatorname{ln}(t) \operatorname{ln}(1-t)}{t} \end{aligned} $$
$$\displaystyle \begin{aligned} &= \dfrac{1}{4} \pi^2 \operatorname{ln}(2) + \int_0^1 dt \, \dfrac{\operatorname{ln}(t) \operatorname{ln}(1+t)}{1-t} \end{aligned} $$
$$\displaystyle \begin{aligned} &= \dfrac{2}{13} \pi^2 \operatorname{ln}(2) + \dfrac{8}{13} \, \int_0^1 dt \, \dfrac{\operatorname{ln}(t) \operatorname{ln}(1-t)}{1+t} \end{aligned} $$
$$\displaystyle \begin{aligned} &= \dfrac{2}{7} \, \int_0^{\pi/2} dt \, \dfrac{t (\pi - t)}{\sin{}(t)}. {} \end{aligned} $$
Formulas (A.63)–(A.69) were provided by Glasser and Ruehr and can be found in [21, Problem 80-13]. Finally, we also recall,
$$\displaystyle \begin{aligned} &= \dfrac{6}{7} + \dfrac{2}{7} \,\int_0^\infty dt \, \dfrac{\sin{}(3 \arctan(2t))}{[(1/4) + t^2]^{3/2}} \, \dfrac{1}{e^{2\pi t} - 1} {} \end{aligned} $$
$$\displaystyle \begin{aligned} &= \dfrac{6}{7} + \dfrac{8}{7} \,\int_0^\infty dt \, \dfrac{\sin{}(3 \arctan(t))}{(1 + t^2)^{3/2}} \, \dfrac{1}{e^{\pi t} - 1} {} \end{aligned} $$
$$\displaystyle \begin{aligned} &= 2 - 8 \,\int_0^\infty dt \, \dfrac{\sin{}(3 \arctan(t))}{(1 + t^2)^{3/2}} \, \dfrac{1}{e^{\pi t} + 1} {} \end{aligned} $$
$$\displaystyle \begin{aligned} &= 1 + 2 \,\int_0^\infty dt \, \dfrac{\sin{}(3 \arctan(t))}{(1 + t^2)^{3/2}} \, \dfrac{1}{e^{2\pi t} - 1}. {} \end{aligned} $$
Formulas (A.71)–(A.73) are due to Jensen (1895) and are special cases of results to be found in [30, p. 279] (cf. (A.16), (A.29)); finally, (A.74) is a consequence of (A.72) and (A.73).
For more on ζ(3) see also [12, p. 42–45].
For a wealth of additional formulas, going beyond what is recorded in this appendix, we also refer to [24] and [27].
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1.Department of MathematicsUniversity of MissouriColumbiaUSA
2.Department of MathematicsBaylor UniversityWacoUSA
3.Center for Mathematical ModelingUniversidad de ChileSantiagoChile
4.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
5.GCAP-CASPER, Department of MathematicsBaylor UniversityWacoUSA
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The influence of arm composition on the self-assembly of low-functionality telechelic star polymers in dilute solutions
Invited Article
Esmaeel Moghimi1,2,
Iurii Chubak3,
Dimitra Founta1,2,
Konstantinos Ntetsikas4,
George Polymeropoulos4,
Nikos Hadjichristidis4,
Christos N. Likos3 &
Dimitris Vlassopoulos1,2
Colloid and Polymer Science volume 299, pages 497–507 (2021)Cite this article
We combine synthesis, physical experiments, and computer simulations to investigate self-assembly patterns of low-functionality telechelic star polymers (TSPs) in dilute solutions. In particular, in this work, we focus on the effect of the arm composition and length on the static and dynamic properties of TSPs, whose terminal blocks are subject to worsening solvent quality upon reducing the temperature. We find two populations, single stars and clusters, that emerge upon worsening the solvent quality of the outer block. For both types of populations, their spatial extent decreases with temperature, with the specific details (such as temperature at which the minimal size is reached) depending on the coupling between inter- and intra-molecular associations as well as their strength. The experimental results are in very good qualitative agreement with coarse-grained simulations, which offer insights into the mechanism of thermoresponsive behavior of this class of materials.
Self-organization of building blocks due to external stimuli is ubiquitous in most materials and all living organisms in nature. Inspired by this, a remarkable body of work has been performed to understand and emulate their response to temperature [1,2,3], pH [4,5,6] and light [7, 8], enabling the controlled design of their structure assembly. Recent advances in polymer chemistry have led to the synthesis of various building blocks with complex architectures and functionalized properties. Such responsive building blocks that can self-organize into higher-order structures may form soft patchy particles, which have directional interactions and varying softness. Furthermore, topological effects that arise in systems with complex architecture can alone lead to a range of interesting phenomena in and out of thermodynamic equilibrium for both low [9,10,11,12,13,14,15] and high system densities [16,17,18,19,20,21]. Functionalized biomolecules such as DNA-grafted colloidal particles represent a typical example where patchiness reflects the competition between inter- and intra-particle associations [22,23,24,25]. However, despite its significance, DNA-based research is very specialized and yields limited amounts of samples. An alternative design of patchy particles is based on the so-called Telechelic Star Polymers (TSPs), that is star polymers with functionalized end groups [26,27,28,29]. A TSP consists of f amphiphilic AB-block copolymer arms grafted on a common center. The solvophilic A-block is attached at the center of the star, whereas the solvophobic B-block is exposed to the exterior of the star. On changing the solvent quality through temperature variation, the outer blocks become attractive and form patches on the surface of the particle [30,31,32]. With such a TSP system, it is thus possible to cover the entire range of inter-particle interactions, from purely repulsive to attractive soft colloids simply by changing the solvent quality. The self-assembly of TSPs at the single molecule level and in concentrated solutions depends mainly on three parameters: (i) the functionality f of the stars, (ii) the outer block size ratio α = NB/(NA + NB) (NA and NB denote the length of the respective block), and (iii) the attraction strength between the outer solvophobic blocks which is enhanced upon worsening solvent quality.
TSPs with low functionality (f ≤ 5) collapse into a watermelon structure with one single patch on the surface of the particle [30]. However, more complex structures with a richer distribution of patches are formed in TSPs with higher functionality [31]. Such soft patchy particles can preserve their properties such as the size, number, and arrangement of patches upon increasing the TSP concentration [33]. The inherent flexibility of such soft-particles leads to formation of ordered structures in the case of high functionality [33]. On the other hand, low-functionally TSPs tend to form micellar aggregates [34,35,36,37], which at relatively high concentrations self-assemble into long worm-like micelles [35, 38]. In previous work, we have examined the effects of temperature (or attraction strength) on the self-assembly of these low-f TSPs in dilute solution [32]. In the present work, we extend these investigations by addressing the effects of the block size ratio and arm length on the self-organization of TSPs with f = 3 in dilute solutions. The new results provide insight into the responsive behavior of TSPs, paving the way for the design of functional materials with tunable properties.
Telechelic star polymers (TSPs) with three arms made of 1,4-polybutadiene (PB) as the inner A-block and polystyrene (PS) as the outer B-block were synthesized by anionic polymerization and chlorosilane chemistry using high-vacuum techniques. Detailed information on the synthesis procedure can be found in Ref. [32]. Three different TSP samples have been used in the present study. Two of them have a similar total molar mass of about 40000 g/mol, albeit a different PS weight fraction of fPS = 0.14 and 0.33. The third sample, which was used in our previous work [32], has a lower molar mass of MW = 26700 g/mol and fPS = 0.23. The molar mass distribution in all three TSP samples is rather narrow with the corresponding polydispersity being around Ð = 1.03. The detailed molecular characteristics of the samples are listed in Table 1.
Table 1 Molecular characteristics of investigated star diblock copolymers (PS-b-PB)3
We have used 1-phenyldodecane as the solvent. It has a cloud-point at 53 ∘C for PS [39] and 22 ∘C for PB (the corresponding 𝜃-temperatures are expected to be slightly higher). Solutions were prepared by mixing an appropriate amount of the TSP with the solvent to reach the desired concentration. The sample degradation was inhibited by adding 0.1 wt% of the TSP of the antioxidant BHT (2,6-Di-tert-butyl-4-methylphenol). In order to fully dissolve TSPs, methylene chloride was used as the cosolvent. Then, the cosolvent was evaporated under ambient conditions until a constant weight was achieved.
Dynamic Light Scattering (DLS) was used to investigate the dynamics and self-assembly of TSPs in dilute solutions. In DLS, the normalized autocorrelation function \(G(q, t) = \left \langle I(q, 0) I(q, t)\right \rangle / \left \langle I(q, 0) \right \rangle ^{2}\) of the total scattered light intensity I(q) at the wave vector \(q = \left (4 \pi n_{0} / \lambda \right ) {\sin \limits } \left (\theta / 2\right )\) (n0 is the refractive index, 𝜃 denotes the scattering angle, and λ is the wavelength of the incident laser beam) is related to the normalized time correlation function of the scattered electric field E(q,t) by the Siegert relation:
$$ G(q,t) = 1 + f^{*}\left|\tilde{\alpha} g(q,t)\right|^{2} = 1 + f^{*}\left|C(q,t)\right|^{2}, $$
where f∗ stands for the coherence instrumental factor, \(\tilde {\alpha }\) is the fraction of I(q) associated with fluctuations relaxing with times longer than 0.1 µs [40,41,42], and C(q,t) denotes the intermediate scattering function (ISF). The inverse Laplace transformation using the constrained regularization method was applied to compute the relaxation spectrum \(H(\ln \tau )\). This method assumes that C(q,t) can be expressed as the superposition of exponentials:
$$ C(q, t) = {\int}_{-\infty}^{+\infty} H(\ln \tau) \exp{\left( - t / \tau \right)} \mathrm{d}\left( \ln \tau \right) $$
The characteristic relaxation times correspond to the peak positions of \(H(\ln \tau )\), whereas the area under the peak defines the value of \(\tilde {\alpha }\) in Eq. 1 and hence the intensity \(\tilde {\alpha } I(q)\) associated with the particular dynamic process. The transformation was performed with the program CONTIN [43] that yielded the relaxation time and intensity of the partitioning modes.
DLS experiments were performed on an ALV-5000 goniometer/correlator setup (ALV-GmbH, Germany). The light source was a Nd:YAG dye-pumped, air-cooled laser (100 mW) with the wavelength λ = 532 nm. The refractive index of 1-phenyldodecane is n0 = 1.482. Before each DLS experiment, the samples were equilibrated at T = 60 ∘C, which is above the cloud point of the outer PS-block, for 10 min to erase thermal history. Then, the sample was quenched to the desired temperature and equilibrated. The equilibration process was probed by measuring the ISF until it reached steady values over time. The duration of equilibration depended on the temperature and ranged from 10 min for T = 60 ∘C to 10 h for T = 20 ∘C.
Simulation details
To model TSP dynamics under worsening solvent conditions for its outer block, we have employed a coarse-grained dissipative particle dynamics (DPD) model with explicit solvent. In what follows, the inner blocks of a TSP are labeled "A," the outer ones—"B," whereas solvent particles—"S." In DPD, the total force Fi acting on the i th particle is composed of the conservative FC, dissipative FD, and random FR contributions [44]:
$$ \mathbf{F}_{i} = \sum\limits_{j \neq i} \left( \mathbf{F}_{ij}^{\text{C}} + \mathbf{F}_{ij}^{\text{D}} + \mathbf{F}_{ij}^{\text{R}} \right). $$
In Eq. 3 above, \(\mathbf {F}_{ij}^{\text {C}}\) is the conservative force acting between the i th and j th particle separated by a distance rij (here and in what follows, rij = ri −rj, rij = |rij|, \(\hat {\mathbf {r}}_{ij} = \mathbf {r}_{ij} / r_{ij}\), and vij = vi −vj):
$$ \mathbf{F}_{ij}^{\text{C}} = A_{ij} w(r_{ij}), $$
where Aij is the maximal repulsion between the particles and w(rij) is given by
$$ w(r_{ij}) = (1 - r_{ij}/r_{\text{cut}}) \theta(r_{\text{cut}}-r), $$
with 𝜃(x) denoting the Heaviside step function and the cutoff distance rcut being chosen as the unit of length (rcut = 1). Furthermore, \(\mathbf {F}_{ij}^{\text {D}}\) is the pairwise dissipative force
$$ \mathbf{F}_{ij}^{\text{D}} = - \gamma w(r_{ij})^{2} \left( \hat{\mathbf{r}}_{ij} \cdot \mathbf{v}_{ij} \right), $$
and \(\mathbf {F}_{ij}^{\text {R}}\) is the pairwise random force
$$ \mathbf{F}_{ij}^{\text{R}} = -\sqrt{2 \gamma k_{\text{B}} T/ {\Delta} t} \cdot \eta_{ij} w(r_{ij}), $$
where ηij is a Gaussian random number with zero mean and unit variance. The unit of mass is set by the (same) mass of every particle m, whereas the unit of energy was chosen to be kBT (kBT = 1). The simulation were performed using the HOOMD-blue simulation package [45,46,47,48] using friction coefficient γ = 4.5mτ− 1 and the equations of motion were integrated using the Velocity-Verlet algorithm [49] with time step Δt = 0.04τ, where \(\tau = r_{\text {cut}} \sqrt {m/k_{\text {B}}T}\) is the DPD unit of time.
To obtain the total polymerization degree N of the star arms in the experimental samples considered, we first estimated the molar volumes of PB and PS, given by vPS/PB = MW/ρPS/PB, where ρPS/PB is the corresponding molar density (ρPS = 1.05 g/mL and ρPB = 0.892 g/mL). N and αPS/PB were then computed on the basis of the PS reference segment volume 99.2 mL/mol and are listed in Table 2. In general, we are interested in the behavior of experimental systems in a rather narrow temperature range 20 ∘C < T < 60 ∘C, where the Flory-Huggins incompatibility parameter χPS-PB = 18.78/T − 9.6⋅10− 4 [50] does not change substantially (experimental values of χPS-PBN for the three samples in such temperature range are \(\chi _{\text {PS-PB}}N \lesssim 10\)), implying that the self-assembly is mainly controlled by the solvent selectivity towards the outer block.
Table 2 Composition of investigated star diblock copolymers (PS-b-PB)3
Given the computational cost of simulations with explicit solvent particles, we focused on a star polymer model with f = 3 arms containing N = 64 monomers, and systematically varied the outer block ratio α by changing NA and NB. All our simulations were performed at total particle density \(\rho r_{\text {cut}}^{3} = 3\). Bonded interactions were given by \(V_{\text {bond}}(r) = \tfrac {K}{2} (r - r_{\text {cut}})^{2}\) with K = 50kBT. The central particle, to which all arms were connected to, was treated as a monomer of type A. In DPD, repulsion amplitudes Aij can be directly related to the Flory-Huggins χij parameters [51]:
$$ A_{ij} \approx A_{ii} + \kappa(\rho) \chi_{ij}, $$
where κ(ρ) depends on the DPD density such that κ(3) = 3.49. In all simulations, we fixed the inter-block incompatibility parameter χAB = 0.23, which corresponds to AAB = 25.8 at \(\rho r_{\text {cut}}^{3} = 3\). Such value of χAB was obtained from a conservative experimental value \((\chi _{\text {AB}}N)_{\exp } = 10\) by taking into account finite polymer chain length corrections: \(\chi _{\text {AB}} = (\chi _{\text {AB}}N)_{\exp } \cdot (1 + 3.9 N^{2/3-2\nu }) / N\) with N = 64 and ν = 0.588 [51]. The incompatibility parameter χAS for the outer block and effective solvent particles was always set to χAS = 0, which corresponds to AAS = 25. Moreover, χBS was systematically varied between 0 and 7 with step ≈ 0.72, corresponding to ABS in the range between 25 and 50. Finally, note that the main goal of our simulations using such a coarse-grained model is not to quantitatively reproduce the change of star properties with increasing χBS, for which atomistic simulations with realistic solvent interactions would be necessary, but to qualitatively assess the effect of self-associations on the change of TSPs static and dynamic behavior.
We first focus on the effect of the outer PS-block fraction on the dynamic relaxation of the TSP system at low densities. In Fig. 1, we show the experimental ISFs at a fixed wave vector and various temperatures for the two samples with comparable total molar masses but with distinct PS weight fractions. The ISFs show two distinct trends upon changing temperature. At high temperatures, the ISFs show a single exponential decay that demonstrates the existence of individual stars in solution. However, when temperature is reduced below the cloud temperature of the outer-block, the ISF features a two-step decay which indicates the coexistence of two distinct populations in the system. The first decay (fast process) in the ISF is similar to the one observed at high temperatures and hence represents the individual stars in solution. On the other hand, the second decay (slow process) taking place at longer times suggests the presence of larger aggregates (clusters of TSPs). The slow process becomes more pronounced as the temperature is decreased. Interestingly, the two-step decay in the ISF appears at a slightly higher temperature for the TSP with a larger PS-fraction.
Experimental ISFs at constant wave vector q = 0.02475 nm− 1 and different temperatures for TSPs with the outer PS-block fractions of fPS = 0.14 (a) and fPS = 0.33 (b). Note that the plateau values of the ISF at short times are well below one. This is due to the fact that a part of the scattered intensity originates from density fluctuations of solvent molecules
To extract hydrodynamic sizes associated with the two processes in the solution, that is individual TSPs and clusters, the relaxation spectrum is calculated from the inverse Laplace transformation of the ISF using the constraint regularized method [43] discussed in Section 4. Typical results of such analysis for the TSPs with two different PS fractions at T = 60 ∘C and 30 ∘C are shown in Fig. 2. The relaxation spectrum at 60 ∘C shows a single peak, which is rather sharp, reflecting single exponential decay of the ISF mode. The position of the peak shifts to a slightly longer time for the TSP with a smaller PS fraction, indicating a larger hydrodynamic radius. At the lower temperature of 30 ∘C, the relaxation spectrum exhibits two well-separated peaks, as seen in Fig. 2b. Similarly to high temperatures, the position of the first peak shifts to a slightly longer time for the TSP with a smaller PS fraction, whereas the position of the second peak in both TSP samples is located at a similar time. These two peaks represent the relaxation times associated with individual TSPs and clusters, respectively. Subsequently, the two relaxation times are used to calculate the diffusion coefficients associated with each component. The diffusion coefficient for the fast mode (where qR < 1) is q-independent, whereas D for the slow mode (where qR ≥ 1) shows some q-dependence. In the latter case, D extrapolated to q = 0 is used to calculate Rh. Then, the hydrodynamic sizes of TSPs and clusters are obtained using the Stokes-Einstein-Sutherland relation.
ISF (open symbols, left axis) and its corresponding relaxation times spectrum (closed symbols, right axis) deduced from the constrained regularization method for the TSPs with PS fractions of fPS = 0.14 (black squares) and 0.33 (red circles) at q = 0.02475nm− 1 for T = 60 ∘C (a) and T = 30 ∘C (b)
We first examine the effect of the block size ratio α on the single TSP size upon cooling. In experiments, the radius of gyration was too small to be probed by DLS. Instead, we focused on the hydrodynamic radius Rh of individual TSPs in dilute solution, as calculated from the fast process in the ISF using the Stokes-Einstein-Sutherland relation. The temperature dependence of Rh for the three studied TSP samples is shown in Fig. 3. The single star size exhibits a two-step shrinkage upon reducing temperature or equivalently worsening the solvent quality. The first decay in size takes place at temperatures well below the cloud-point of outer PS-blocks, whereas the second drop is seen when temperature is reduced further below the cloud-point of the inner PB-block. Hence, the first decrease in size is associated with the collapse of outer blocks, whereas the second decay corresponds to the case when inner blocks start to collapse. At high temperatures, for the TSP with fPS = 0.14, we find Rh ≈ 5 nm and for fPS = 0.33, Rh ≈ 4.6 nm (see Fig. 3a). Although both TSPs have almost the same molar mass of about 40000 g/mol, the difference in their size originates from the difference in the fraction of PS. The radius of gyration of a star homopolymer in good solvent conditions is given by Rg ≈ κ(f)bNν, where ν = 0.588, N is the number of Kuhn segments in a star arm, b is the size of a Kuhn segment, and κ(f), which depends on the number of star arms f, is a numeric constant that takes into account the star functionality [52,53,54]. Using the latter relation, it can be found that the size of a star made of purely PB is about 65% larger compared with that made of purely PS, \(R_{\text {g}}^{\text {PB}} \approx 1.65 R_{\text {g}}^{\text {PS}}\), given that their molar mass is the same [55]. Hence, it is expected that the increase in PS fraction reduces the size of a TSP. In order to compare the collapse process for stars of different size, we have normalized the TSP size by the plateau value of Rh at high temperatures.
a The hydrodynamic radius, Rh, of individual TSPs in the dilute solution calculated from the fast process in the experimentally determined ISFs. bRh normalized by its plateau value at high temperatures. The dashed lines serve as a guide to the eye. The black arrows indicate the cloud-points of inner PB and outer PS blocks
The TSP with a larger fraction of outer PS-blocks (fPS = 0.33) shows the first-step reduction in size at higher temperatures and the second-step drop at slightly lower temperatures compared with the TSP with a smaller PS fraction (fPS = 0.14). In both stars, the decrease in size is about 15%, as seen in Fig. 3b. In addition, in Fig. 3, we show the results for a TSP with a smaller molar mass (26700 g/mol) with the outer PS-block fraction fPS = 0.23, which is between the other two higher molar mass TSPs with fPS = 0.14 and 0.33. The main difference is that the TSP with the smaller molar mass exhibits the decay in size at a much lower temperature compared with the other two, which originates from a smaller value of the incompatibility parameter \(\sim \chi _{\text {PS-S}}N\). Moreover, the decrease in size is also rather weaker (about 10%), which can be attributed to a shorter length of its arms.
In Fig. 4, we present the temperature dependence of clusters' Rh for the systems of TSPs with the same molar mass but two different PS fractions (MW = 40000 g/mol, fPS = 0.14 and 0.33). In both TSPs, the cluster size shrinks on cooling. However, a slight but consistent increase in the cluster size is observed when temperature is further reduced below the cloud-point of the inner PB-block. Moreover, the hydrodynamic cluster size does not show change with the fraction of outer PS block. In Fig. 4a, we additionally show the results for the TSP with smaller molar mass (MW = 26700 g/mol) and fPS = 0.23. In this case, the temperature dependence of the cluster size is the same as for the other two TSPs (fPS = 0.14,0.33). However, the smaller molar mass TSP shows a cluster size that is nearly three times larger. This could be due to a higher concentration of these TSPs. To rule out the effect of concentration, we have normalized the cluster size by the number density of TSPs in solution (Fig. 4b). The number density takes into account for the number of stickers available in the solution. With such normalization, the differences in cluster size between different TSPs are reduced to a great extent. The minor differences could originate from the complex nature of self-organization of TSPs due to differences in their molecular characteristics.
a The hydrodynamic radius, Rh, of clusters extracted from the slow process in the experimental ISFs. b The ratio of Rh to the number density n (see Table 1) of TSPs in the solution. The black arrow indicates the cloud-point of inner PB blocks
Simulation results
We now focus on static and dynamic properties of single TSPs under worsening solvent conditions for the outer B block, that is under increasing χBS. To do so, we simulated single stars with f = 3 arms of length N = 64 using a coarse-grained DPD model with explicit solvent particles, as described in detail in Section 4. The outer block ratio was systematically varied from 0.1 to 0.5 with step 0.1. The exact number of A- and B-type monomers in an arm was NA = 58, 52, 45, 39, 32 and NB = 6, 12, 19, 25, 32, respectively (the corresponding α = 0.1, 0.2, 0.3, 0.4, 0.5). For each state point (α, χBS), we performed 10–12 independent simulation runs of length 105τ, followed after a shorter equilibration period of 104τ. Single TSPs were simulated in a box of size L = 30rcut at the total particle density \(\rho r_{\text {cut}}^{3} = 3\). To check if such box size is sufficient to accommodate a TSP, we initially simulated the same star in good solvent conditions (χBS = 0) in a larger box with L = 35rcut, and we did not observe any substantial changes in its properties. In selective solvents, the TSP size is even smaller due to the formation of patches, which justifies the use of the same box size L = 30rcut in this case.
To assess single star shape properties, we computed the eigenvalues λi (i = 1,2,3, λ1 ≥ λ2 ≥ λ3) of the star's gyration tensor
$$ G_{ij} = \frac{1}{fN+1} \sum\limits_{k=1}^{fN + 1} {\Delta} r_{i}^{(k)} {\Delta} r_{j}^{(k)}, $$
where \({\Delta } r_{i}^{(k)}\) is the i th component of the k th monomer's position in the star's center of mass frame. In Fig. 5, we report the TSP's mean radius of gyration \(R_{\text {g}} = {\langle R_{\text {g}}^{2} \rangle }^{1/2}\) (\(R_{\text {g}}^{2} = \lambda _{1} + \lambda _{2} + \lambda _{3}\)) as well as the mean asphericity parameter \(\langle \lambda _{1} - \tfrac {1}{2}(\lambda _{2} + \lambda _{3}) \rangle \), which is positive and can vanish only for a completely symmetric configuration, as a function of χBS for different block length ratios α. The angles 〈⋯ 〉 denote an ensemble and time average. We find that the behavior of a single TSP size is generally very similar to the experimental one (see Fig. 3): upon increasing χBS, we first observe a rather small decrease in Rg, followed by a major drop at higher χBS. Such behavior of Rg is associated with the formation of a single patch, where all three arms of a TSP clump together (see Fig. 6). We find that the transition point shifts towards a higher χBS, that is a lower temperature because \(\chi \sim 1/T\), with decreasing α, which is in full accordance with the experimental behavior of the two samples with fPS = 0.14 and 0.33 that have a very similar total molar mass (see Fig. 3b). Afterwards, only a small reduction of Rg is observed upon increasing χBS, as seen in Fig. 3a. We also note that simulations do not capture the second drop in size which is observed in experiments for temperatures below the cloud-point of inner-block. The reason for this discrepancy is that in simulation, for simplicity reasons, the inner-block is assumed to be always in a good solvency condition. Hence, it only captures single step shrinkage process due to collapse of outer-block monomers.
a The mean radius of gyration of a TSP Rg as a function of χBS for different fractions of the outer block α = NB/N. b The mean asphericity of a TSP, computed as \(\langle \lambda _{1} - \tfrac {1}{2}(\lambda _{2} + \lambda _{3}) \rangle \), scaled with its mean radius of gyration Rg as a function of χBS for different α
Characteristic TSP conformations with α = 0.1 (left) and α = 0.3 (right) at a high χBS ≈ 7. B-monomers are blue, A-monomers—red, and star centers are black. Solvent particles are not shown for clarity
We furthermore find that the final TSP size decreases with increasing α, also in accordance with the experimental findings for the samples with fPS = 0.14 and 0.33 (see Fig. 3b). This behavior is associated with generally more open configuration of collapsed TSPs with small α that permit solvent flow through the TSP's interior. On the other hand, in the case of larger α, the solvophobic B-blocks form a single large patch that expels the solvent from its interior, resulting in more compact and symmetric configurations (see Figs. 6 and 5b). Interestingly, as can be seen in Fig. 5b, upon slightly increasing χBS from 0, the stars first become more aspherical, which can be attributed to the formation of transient patches between two out of three star arms. This is confirmed in Fig. 7 that reports the average number of patches formed by the star, the average number of arms in a patch, and the fraction of free arms as a function of χBS for different arm compositions. In agreement with earlier results [30], we find that such TSP with f = 3 forms only one patch for all α with all three arms contributing to it at high enough χBS. In addition, as seen from Fig. 7a and 7b, the point when all arms start to form a single patch corresponds to the point when Rg drops significantly (see Fig. 5a). Finally, to assess the influence of arm length N on the transition point for the watermelon-like structure formation, for α = 0.3, we additionally simulated stars with arm length N = 32,48,80 for different values of χBS. The comparison between the behavior of the radius of gyration of TSPs with different N for α = 0.3 is shown in Fig. 8. We find that the TSP with shorter arms features the star collapse at higher values of χBS, which therefore corresponds to lower temperatures in the experiments, being in line with the trend observed for the experimental sample with fPS = 0.23 that has a lower molar mass, see Fig. 3.
a The total number of formed patches as a function of χBS for different α. Two arms are defined as being in a common patch if there is at least one pair of monomers from the two distinct arms lying at a distance r ≤ rcut. The average number of arms in a patch (b) and the average fraction of free arms (c) as a function of χBS for different α. a, b, and c share the same legend shown in (a)
The mean radius of gyration Rg of a TSP as a function of χBS for the same fraction of the outer block α = 0.3 but different arm lengths N
The presence of faithful hydrodynamic interactions in DPD allows us to assess the influence of patch formation on the dynamics of single stars in solution. We did this by considering the mean-square displacement of TSP's center of mass, computed as:
$$ {\text{MSD}}(t) = \frac{1}{T-t} {\int}_{0}^{T-t} \left\langle [ \mathbf{R}(t^{\prime} + t) - \mathbf{R}(t^{\prime})]^{2} \right\rangle \text{d}t^{\prime}, $$
where R(t) is the position of the star's center of mass at time t, T is the total simulation time, and 〈⋯〉 stands for the average over independent simulations runs. Typical behavior of the MSD for different α as well as the extracted diffusion coefficients D is shown in Fig. 9. We find that the tendency to form patches, causing more compact watermelon-like structures, increases the diffusivity of the TSP. In the experiments, this behavior corresponds to a reduction in the hydrodynamic radius Rh, which is in good agreement with results shown in Fig. 3. Furthermore, this effect is especially significant for the case of high α, where D at high χBS can become about two times bigger compared with athermal conditions with χBS = 0, as seen in Fig. 9a, again in agreement with the experiments, where a larger reduction in Rh is seen for the TSP with a higher PS fraction. Finally, more open conformations of collapsed TSPs with low α make the increase in diffusivity less pronounced (for example, about 30% increase for α = 0.10).
a The diffusion coefficient D of the TSP's center of mass as a function of χBS for different α. D was extracted from the long-time behavior of the MSD of the star's center of mass, MSD = 6Dt. Typical MSD as a function of χBS are shown for α = 0.5 (b) and α = 0.1 (c)
In addition, we have considered the dynamics of internal patch reorganizations at the single-star level. In Fig. 10, we show the times tfree for an arm spent in the free state, that is not forming an association with other arms, as a function of χBS for different α. We find that for all α, the mean value of tfree initially decreases exponentially fast with increasing χBS (Fig. 10a), up to a point where a single patch forms. At this point, 〈tfree〉 drops to 0, indicating that the single patch is stable over the course of the whole simulation. The value of χBS where it happens compares with the point where a significant reduction of Rg occurs (Fig. 5a). Furthermore, in Fig. 10b, we show the distribution for tfree for α = 0.1 at various χBS, featuring tails that decay exponentially fast with increasing tfree in all cases.
a The mean time of an arm not belonging to a patch 〈tfree〉 as a function of χBS for different α. b The normalized distribution of tfree for α = 0.1 and various χBS
Finally, we consider the formation of inter-star aggregates in the dilute solution. To asses such behavior, it is necessary to simulate a sufficiently large number of stars, which becomes computationally restrictive if using the model with N = 64 that was employed for single star behavior discussed previously. We therefore resort to an even coarser model with, similarly, f = 3 but with N = 10, and in what follows we focus on the case with α = 0.3. We simulated 2000 such stars in a box with L = 70rcut at particle density \(\rho r_{\text {cut}}^{3} = 3\). In this model, the corresponding star concentration is c ≈ 0.4c∗, which is higher than the one used in the experiments (to reach an equivalent experimental concentration of c ≈ 0.04c∗, it would be necessary to simulate about a ten times bigger system containing \(\sim 10^{7}\) particles). Nevertheless, even in this regime, we remain at concentrations considerably below c∗. Initially, TSPs were initialized uniformly in the box and subsequently equilibrated in athermal solvent conditions for both blocks with χAS = χBS = 0 (χAB was set to χAB = 28.6 to match the experimental value (χABN)eff = 10, as explained in Section 4). Afterward, the incompatibility parameter for the outer block was increased to 4.3 (ABS = 40) over 2⋅106 integration time steps and then further equilibrated for another 2⋅106 steps. During the latter stages, the TSPs initially began to form small micelles that subsequently merged into worm-like structures, which then again merged into a single giant cylindrical aggregate, shown in the left column of Fig. 11. Note that such cylindrical architecture is rather a consequence of periodic boundary conditions. It is likely that symmetric spherical aggregates would form in a more dilute system with a larger simulation box size, as, for instance, recently shown in Ref. [56]. Nonetheless, this illustrates the tendency of TSPs to form large aggregates even at dilute conditions, as previously shown in the experimental cluster sizes in Fig. 4. We also considered the effect of lowering temperature on the structure of such aggregate by quenching χBS to 10 (ABS = 60) and equilibrating the system for another 106 integration time steps. As shown in the right column of Fig. 11, as a result of the χBS increase, the aggregate shrinks in the two transverse directions. This occurs because the solvophobic TSP blocks that lie in the aggregate's interior become more ordered and thus push away the remaining solvent (see the bottom row of Fig. 11). This further agrees with the experimental trend of decreasing cluster size with decreasing temperature (Fig. 4).
Left column: self-assembly of three-arm TSPs with N = 10 and α = 0.3 into a giant cylindrical aggregate at c ≈ 0.4c∗. The bottom row shows only the solvophobic B-blocks. Right column: the aggregate shrinks in the transverse directions upon increasing the solvophobicity of the B-monomers that become more ordered (bottom image)
In summary, we have investigated the self-assembly of TSPs with a variable size of the outer block as well as the arm length, which are subject to worsening solvent conditions, in dilute solutions. We find that two distinct modes in the experimental ISF appear upon lowering the temperature below a critical value: the fast-relaxing mode that corresponds to free stars in solutions as well as a slow-relaxing mode that indicates the presence of large aggregates. We find that the size of both populations decreases upon cooling. For single TSPs, the decay is associated with the formation of a single patch, where all three arms come together. From both experiments and simulations, we find that the temperature that corresponds to such transition increases with growing fraction of solvophobic monomers. However, we find that the transition temperature increases with the polymerization degree of TSP arms, when keeping the fraction of solvophobic monomers constant. The formed aggregates in solution are much bigger than single stars (\(\sim 100\) nm versus \(\sim 5\) nm). In simulations, albeit at a higher concentration of TSPs, we have found the formation of micellar aggregates with complex internal structure. Upon worsening the solvent quality for the outer block, the solvent is becoming more strongly expelled from the aggregate's interior, which causes the reduction of its size, similarly to the experimental behavior. We therefore speculate that similar objects also form at concentrations similar to the experimental ones, in line with assembled structures that have been recently observed in large-scale DPD simulations of linear diblock co-polymers in dilute conditions [56].
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I. C. acknowledges useful discussions with B. Capone. We are also grateful for a generous computational time at the Vienna Scientific Cluster.
Open access funding provided by University of Vienna. We would like to acknowledge financial support from KAUST under grant OSR-2016-CRG5-3073-03. We also acknowledge the additional support provided by a STSM Grant from COST Action CA17139.
Institute of Electronic Structure and Laser, FORTH, 71110, Heraklion, Crete, Greece
Esmaeel Moghimi, Dimitra Founta & Dimitris Vlassopoulos
Department of Materials Science, Technology, University of Crete, 71003, Heraklion, Crete, Greece
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria
Iurii Chubak & Christos N. Likos
Physical Sciences and Engineering Division, KAUST Catalysis Center, Polymer Synthesis Laboratory, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955, Kingdom of Saudi Arabia
Konstantinos Ntetsikas, George Polymeropoulos & Nikos Hadjichristidis
Esmaeel Moghimi
Iurii Chubak
Dimitra Founta
Konstantinos Ntetsikas
George Polymeropoulos
Nikos Hadjichristidis
Christos N. Likos
Dimitris Vlassopoulos
Correspondence to Christos N. Likos or Dimitris Vlassopoulos.
The authors declare that they have no conflict of interest.
E. M. and I. C. contributed equally.
Moghimi, E., Chubak, I., Founta, D. et al. The influence of arm composition on the self-assembly of low-functionality telechelic star polymers in dilute solutions. Colloid Polym Sci 299, 497–507 (2021). https://doi.org/10.1007/s00396-020-04742-0
Revised: 15 August 2020
Accepted: 23 August 2020
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Chi-square Distribution
The chi-square distribution is a continuous probability distribution that describes the distribution of the sum of squares of independent standard normal random variables. It is commonly used in statistical hypothesis testing to evaluate the goodness of fit of a model to a set of data. The chi-square distribution is defined by a single parameter, k, which is the number of degrees of freedom.
Properties of Chi-Square Distribution:
The chi-square distribution is a continuous probability distribution that is defined by a single parameter called the degrees of freedom. It has several important properties, including:
Right-skewed shape: The chi-square distribution is right-skewed, meaning that it has a long tail on the right side of the distribution.
Non-negative: The chi-square distribution is always non-negative, since the squared values of random variables can never be negative.
Asymptotic behaviour: The chi-square distribution approaches the x-axis as the values of x become increasingly large. This means that the probability of a value being extremely large is always non-zero but becomes increasingly small as the value becomes more extreme.
Non-symmetrical: The chi-square distribution is not symmetrically distributed.
Use in statistical hypothesis testing: The chi-square distribution is often used in statistical hypothesis testing to determine the goodness of fit of a model to a set of data or to test the independence of two variables.
Chi-square Statistic
To use the chi-square distribution in statistical hypothesis testing, you need to calculate the chi-square statistic, which measures how well a model fits a data set. The chi-square statistic is calculated by summing the squared differences between the observed values and the expected values, divided by the expected values.
$$\chi^2 = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}$$
n is the number of data points, O is the observed value for each data point, E is the expected value for each data point
Probability Density Function (PDF):
The probability density function (PDF) of the chi-square distribution is a function that describes the probability of a given value occurring under the distribution. The PDF of the chi-square distribution is defined as:
$$f(x) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2-1} e^{-x/2}$$
\(k\) is the degrees of freedom
\(\Gamma(k/2)\) is the gamma function
Mean, Mode and Variance of Chi-square Distribution
Mean: The mean of the chi-square distribution is equal to the degrees of freedom (df).
$$\mu = df$$
Mode: The mode of the chi-square distribution is equal to the degrees of freedom minus 2 (if the degrees of freedom are greater than 2).
$$mode = (df - 2) \text { for df} \geq 2 $$
Variance: The variance of the chi-square distribution is equal to twice the degrees of freedom.
$$\sigma^2 = 2 \cdot df$$
Standard deviation: The standard deviation of the chi-square distribution is equal to the square root of twice the degrees of freedom.
$$\sigma = \sqrt{2 \cdot df}$$
These are the mean, mode, standard deviation, and variance of the chi-square distribution. They are all related to the degrees of freedom and can be used to describe the shape and spread of the distribution.
Using Microsoft Excel
CHISQ.DIST:
The CHISQ.DIST function in Microsoft Excel calculates the probability density function (PDF) or cumulative distribution function (CDF) of the chi-square distribution. The function has the following syntax:
CHISQ.DIST(x, degrees_freedom, cumulative)
x: is the value for which you want to calculate the probability density or cumulative probability.
degrees_freedom: is the number of degrees of freedom in the chi-square distribution.
cumulative: is a logical value that specifies whether you want to calculate the PDF (FALSE) or CDF (TRUE) of the chi-square distribution.
The function returns the probability density or cumulative probability of the given value under the chi-square distribution with the specified degrees of freedom.
CHISQ.INV:
The CHISQ.INV function in Microsoft Excel calculates the inverse of the chi-square distribution for a given probability and degrees of freedom. The function has the following syntax:
CHIINV(probability, degrees_freedom)
probability: is the probability for which you want to find the corresponding value under the chi-square distribution. This should be a value between 0 and 1.
The function returns the value of the chi-square statistic corresponding to the given probability and degrees of freedom.
CHISQ.DIST.RT:
The CHISQ.DIST.RT function in Microsoft Excel calculates the right-tailed probability of the chi-square distribution for a given value and degrees of freedom. The function has the following syntax:
CHISQ.DIST.RT(x, degrees_freedom)
x: is the value for which you want to calculate the right-tailed probability.
The function returns the right-tailed probability of the given value under the chi-square distribution with the specified degrees of freedom.
CHISQ.INV.RT:
The CHISQ.INV.RT function in Microsoft Excel calculates the inverse of the right-tailed probability of the chi-square distribution for a given probability and degrees of freedom. The function has the following syntax:
CHIINV.RT(probability, degrees_freedom)
probability: is the right-tailed probability for which you want to find the corresponding value under the chi-square distribution. This should be a value between 0 and 1.
The function returns the value of the chi-square statistic corresponding to the given right-tailed probability and degrees of freedom.
In conclusion, the chi-square distribution is a probability distribution used in statistical analysis to test the goodness of fit of a model to a set of data and to test the independence of two variables. It is defined by the degrees of freedom.
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•https://doi.org/10.1364/OE.27.029332
Interference-free and single exposure to generate continuous cycloidal alignment for large-area liquid crystal devices
Da Chen, Huijie Zhao, Kexin Yan, Dong Xu, Qi Guo, Linghao Sun, Fan Wu, Vladimir G. Chigrinov, and Hoi-Sing Kwok
Da Chen,1 Huijie Zhao,1 Kexin Yan,1 Dong Xu,1 Qi Guo,1,2,* Linghao Sun,1 Fan Wu,1 Vladimir G. Chigrinov,3,4 and Hoi-Sing Kwok3
1School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing, 100191, China
2National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
3State Key Laboratory on Advanced Displays and Optoelectronics Technologies, HKUST, Hong Kong
4School of Physics and Optoelectronic Engineering, Foshan University, Foshan, Guangdong, 528000, China
*Corresponding author: [email protected]
Da Chen https://orcid.org/0000-0002-2944-1163
Qi Guo https://orcid.org/0000-0003-1936-195X
D Chen
H Zhao
K Yan
D Xu
Q Guo
L Sun
F Wu
V Chigrinov
H Kwok
Da Chen, Huijie Zhao, Kexin Yan, Dong Xu, Qi Guo, Linghao Sun, Fan Wu, Vladimir G. Chigrinov, and Hoi-Sing Kwok, "Interference-free and single exposure to generate continuous cycloidal alignment for large-area liquid crystal devices," Opt. Express 27, 29332-29339 (2019)
Single-exposure fabrication of tunable Pancharatnam-Berry devices using a dye-doped liquid crystal
Yan Li, et al.
Holographic binary grating liquid crystal cells fabricated by one-step exposure of...
Kotaro Kawai, et al.
Dual-mode liquid crystal grating based on photo- and nanoparticle-induced alignment effects
Hui-Chi Lin, et al.
Opt. Express 27(8) 11264-11272 (2019)
Nanophotonics, Metamaterials, and Photonic Crystals
Circular polarization
Liquid crystal devices
Optical elements
Wave plates
Original Manuscript: July 26, 2019
Revised Manuscript: September 12, 2019
Manuscript Accepted: September 12, 2019
Principle and experiments
An approach for generating cycloidal pattern of liquid crystal (LC) molecules based on interference-free and single exposure is illustrated. The spatial manipulation of polarization state is achieved using birefringent prism and wave plates. And then, the spatially variant polarization of exposure beam is transferred to LC molecules by azo-dye photo-sensitive layer. Consequently, the LC samples fabricated shows periodically cycloidal texture and diffraction efficiency more than 99%. The measured period Λ and diffraction angle are in good consistency with theoretical results. Thus, this exposure method provides an effective and robust way for fabricating large-area LC elements, therefore paving the way for widespread applications of high-performance diffractive LC devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical elements incorporating geometric phase (GP) effect have attracted extensive research interest, due to their capability of generating light waves with complex amplitude, phase and polarization by embodying inhomogeneous anisotropy [1–4]. One of the great benefits is to replace massive optical system by flat and ultrathin GP elements to achieve highly efficient beam deflecting, focusing, filtering, etc. [5–8]. Among various implementation methods, liquid crystals (LCs) based GP elements shows high flexibility in developing highly functional optical devices due to their high birefringence, high sensitivity to external field and surfaces anchoring forces [9–12]. Recently, LC GP elements, such as lenses, gratings, vortex phase plate, or ones with arbitrary phase profile, were generated by patterning the directors of LCs [10,13–16]. Specifically, LC elements with a cycloidal molecular orientation pattern, known as polarization grating (PG), are extensively used in displays, optical communications, and imaging systems [17–21]. There are basically two methods for generating cycloidal patterns of LC directors utilizing photo-sensitive aligning. One is based on polarization holograms by exposing photo-aligning layer with two orthogonal circular polarization [22,23]. This method possesses the characteristics of high spatial resolution, but its complex optical setup and vulnerability to environmental disturbance hinders its industrial application [24]. The other is digitalized polarization holographic approach. For instance, DMD (digital mirror device) based microlithography or laser direct writing provides flexibility of arbitrary pattern, but multi-exposure process and high-precision mechanical rotation of polarizer limit the productivity [13,25]. Recently improved digital polarization holography employing phase-type spatial light modulator (SLM) for single-exposure process [24,26]. However, the alignment patterns of LCs are quasi-continuous, and in one exposure the spatial resolution and effective area are in conflict. Thus, the challenge still exists for fabricating continuous cycloidal LC plate with large-area and high-precision.
Here we propose a novel method to generate cycloidal pattern of LC directors that enables highly efficient and large area thin-film LC PGs for visible wavelengths. The polarization manipulation of exposure beam was achieved using birefringent prism and quarter wave plate, and then the polarization orientation pattern created was recorded on photo-sensitive aligning layer and passed on to LC molecules. To prove the concept, the fabricated LC PGs and binary gratings were tested and compared with theoretical results. This fabrication technique features low cost, continuous pattern, high stability, and single-step exposure. In addition, this non-interferometric and passive optical elements-based design provides a novel approach for fabrication GP elements. Thus, it could significantly facilitate the fabrication and practical usage of LC PG.
2. Principle and experiments
LC gratings with cycloidal pattern of liquid crystal alignment are efficient polarization-sensitive GP elements which separate monochrome plane wave to sub-waves with corresponding polarization state. The spatially continuous in-plane axis variance of liquid crystal director within one period is critical for theoretical 100% diffraction efficiency, since non-continuous distribution will lead to higher order diffraction and result in stray light and the loss of first order efficiency. Since the grating pattern generated based on digitalized polarization holography method is widely employed [24–26], it is necessary to evaluate the influence of domain number per period on diffractive efficiency for quasi-continuous gratings.
The simulated result shown in Fig. 1 was done using FDTD method for evenly segmented LC PG with domain number N when incident is linearly polarized. The cycloidal pattern degenerates to binary pattern with domain number of two. In this case, the polarization of output beam follows that of incidence to be linear polarized. With half-wave condition satisfied, the diffractive efficiency of fist orders, defined as intensity of + 1 and −1 order divided by that of incidence, is ${\eta _{ \pm 1}} = 81\%$. With N increases from 3, the intensity evolves into first order gradually, also the higher order vanish gradually, finally the efficiency η±1 increase to saturation. Thus, to achieve more than 95% of first order efficiency, at least eight domains are required when fabricated using digitalized polarization holography method. As for the polarization state of diffractive beams in the case of cycloidal pattern, the zeroth order follows incidence to be linearly polarization, and the + 1 and −1 order are left-hand circular (LHC) and right-hand circular (RHC) polarization respectively. The simulated results show that the first order beams are almost circular polarized when N is greater than 3, as illustrated in Fig. 1 by the ratio of horizontal and vertical electrical field E⊥/E∥. For the applications of LC PG with high requirement of polarization and certain tolerance of efficiency loss, small domain number is preferable considering the fabrication complexity. However, for large area devices, fabrication methods based on digitalized polarization holography confront the tradeoff between spatial resolution and effective area.
Fig. 1. Simulated results of the degeneration of efficiency and polarization of first order beams due to number of domains N of non-continuous LC PG.
A simple exposure scheme with high robustness is achieved in this work for fabricating continuous cycloidal pattern on a large area device by employing a birefringent prism aligned with quarter wave plate (QWP). Cycloidal distribution of linear polarization direction of exposure beam is created and then embedded on photosensitive SD1 layer with a single exposure, as illustrated in Fig. 2(a). By setting the optical axis of birefringent prism with an angle of 45° to the polarization direction of incidence, the oscillation pattern between circular polarization and linear polarization is generated due to continuously varied optical path length. In this case, the linear polarization is with ± 45°to the optical axis of birefringent prism.
Fig. 2. (a) Schematics of generation of continuous cycloidal pattern using birefringent prism aligned with QWP. (b) Fabrication of large area LC PG. (c) Top view and (d) Side view of LC PG's molecules directors
Since the incident polarization has an angle of 45° with respect to the optical axis of the birefringent prism, it can be considered as a waveplate with spatially varied phase retardation. As illustrated is Fig. 2(a), the phase retardation induced varies linearly along y-axis with a coefficient, which is determined by the wedge angle and birefringence of the prism:
(1)$$\delta = ky=\frac{{2\pi }}{\lambda }\Delta n\tan \alpha y$$
Thus, the polarization distribution of exposure beam analyzed using Jones calculus is as follows:
(2)$${J_{exposure}} = M_{QWP}^{\prime}{J_1} = M_{QWP}^{\prime}{M_{prism}}{J_{in}}=M_{QWP}^{\prime}\left[ {\begin{array}{{c}{c}} {\exp \left( {i\frac{\delta }{2}} \right)}&0\\ 0&{\exp \left( { - i\frac{\delta }{2}} \right)} \end{array}} \right]\left[ {\begin{array}{{c}} {\cos {{45}^ \circ }}\\ {\sin {{45}^ \circ }} \end{array}} \right]$$
The Jones vector of beam passed through the birefringent prism is:
(3)$${J_1} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{{c}} {\exp \left( {i\frac{\delta }{2}} \right)}\\ {\exp \left( { - i\frac{\delta }{2}} \right)} \end{array}} \right]$$
As we can see from Eq. (3), the linearly polarized light is transferred to spatial variant polarization between linear and circular due to the continuously varying phase retardation of birefringent prism. With the fast axis of QWP aligned 45° to the optical axis of birefringent prism, the slope of the spatially variant phase retardation is converted to the change of linearly polarized direction:
(4)$${J_{exposure}}={\textrm{R}^T}({45^\circ } )M_{QWP}^{}\textrm{R}({45^\circ } ){J_1} = \left[ {\begin{array}{{c}} {\cos (\frac{\delta }{2} + \frac{\pi }{4})}\\ {\sin (\frac{\delta }{2} + \frac{\pi }{4})} \end{array}} \right]$$
As we can see from Eq. (4), the emergent light is linearly polarized with polarization direction rotating along y-axis. Based on this configuration, the exposure setup is designed and realized as shown in Fig. 3. A semiconductor laser with wavelength of 450 nm and a crystalline quartz prism (Crystock, Inc) with optical aperture of 1*1 cm2 are arranged as the geometry shown in Fig. 2(a) with 3D-printed holders. This method is much simpler and more robust compared to the existing exposure techniques. Moreover, our approach expands the range of choice for exposure light source compared with method based on polarization holography, since the latter one requires laser source with sufficient coherent length.
Fig. 3. (a)Geometry of the exposure path. (b) Photograph of fabricated sample. (c) Experiment setup formed by cascaded 3D-printed holders.
The emergent light irradiates on glass substrate coated with photosensitive alignment layer. In this research, sulfonic azo-dye SD1 (DIC. Japan) is selected as photo aligning layer fabricated in following steps: dissolved in dimethylformamide (DMF) solvent at concentration of 0.4% wt./wt., then spin-coated and soft-baked to form a 10 nm-thick layer, and exposed on the designed stage with the dosage of 1.5 J/cm2. The polymerizable LC material UCL-P100 (DIC. Japan) is adopted as the birefringent layer aligned by SD1 and cured by UV to form a PG due to its easy adjustment of layer thickness, which is controlled during fabrication by spin-coating speed. Similarly, nematics or smectics can also be aligned by exposed SD1 layer. The photograph of fabricated sample with effective area of approximate 0.7*0.8 cm2 shown as the black square in Fig. 3(b). The effective area is determined by optical aperture of prims and shows no conflict with spatial resolution.
Based on our method, the textures of fabricated sample are shown in Fig. 4. Observed under polarized microscope with crossed polarizer and analyzer, bright and dark strips appear due to rotation of directors along y-direction. The polarization direction of linearly-polarized exposure beam is illustrated by blue arrows at the bottom of Fig. 4(d), and the photoaligned directors are represented by red bars since the preferable orientation of SD1 is perpendicular to the exposure polarization. The period of grating measured using microscope image is 89.5 µm. The corresponding diffraction pattern of LC PG is shown in Fig. 4(b) using optical setup in Fig. 4(a).
Fig. 4. (a) Optical setup for measuring fabricated grating samples. Diffraction pattern of (b) polarization grating and (c)binary grating. Textures of (d) polarization grating and (c)binary grating observed under polarized microscope with black arrows showing polarizer and analyzer direction. The blue arrows/ellipses represent the polarization state of exposure beam, and the red bars represents the aligned LC director.
To realize a binary phase grating with only two director orientations instead of continuously rotating ones, we simply remove the QWP in exposure path in Fig. 2(a). One unique property of photoalignment layer SD1 playing a crucial role here is that the easy axis is determined by the long axis of the ellipse and unsensitive to the ellipticity of exposure light [27]. The exposure beam passed through birefringent prism shows the oscillation of ellipticity with polarization direction fixed at ± 45° to y-axis, as described by J1 in Eq. (3) and shown by blue ellipses at the bottom of Fig. 4(e). Consequently, the binary phase grating with orthogonal alignment directions is obtained, as texture shown in Fig. 4(e) and corresponding alignment directions represented by red bars. This method for binary phase grating has the potential in achieving submicron period compared with commonly used chrome mask-based method [28,29] since no diffraction arises during exposure process.
As described by Eq. (3), the orientation of LC director changes periodically. The period of LC grating fabricated can be derived:
(5)$$\Lambda = \frac{\lambda }{{\Delta n\tan \alpha }}$$
The crystalline quartz prism with birefringence of $\Delta n = 0.0094$ and wedge angle of 30° provides grating period with the value of 82.9 µm.
In the real case, the exposure beam is bended from the incidence due to refraction. Considering the deflection of propagating direction, the modified calculation gives the period value of 90.6 µm in the case of laser arranged vertically and no inclination of the sample (inclined angle $\beta = {0^ \circ }$ in Fig. 3(a)). The grating period is adjusted by the inclined angle β of exposed sample. The minimum value of period occurs when the exposure beam is normal to the substrate surface, which is 86.6 µm with corresponding $\beta = {18^ \circ }$ by calculation, as shown in Fig. 5.
Fig. 5. The dependence of grating period on inclined angle of sample during exposure.
The diffraction efficiency of LC PG, as summarized in Table 1, is determined by the thickness and birefringence of LC layer and the incident polarization [20,30,31].
Table 1. Summarization of Polarization and Efficiency of Diffractive Orders
With half-wave condition satisfied $\Delta nd = \lambda /2$, the 0th order disappears, and efficiency of ± 1st orders achieve maximum value of 100%. The intensity ratio between + 1st and −1st orders depends on the RHC or LHC fraction of the incidence. Here the corresponding index is defined from the Stokes parameter of the output as ${\gamma _{RHC}}=(1+{\textrm{S}_3} + {S_0})/2$ and ${\gamma _{LHC}}=(1-{\textrm{S}_3} + {S_0})/2$ respectively.
The measured efficiency, calculated from the intensity of ± 1st orders divided by that of incidence, is shown in Fig. 6. The thickness of LC layer is controlled by the concentration of monomer and the spin-coating speed, and measured by a stepper (KOSAKA ET150). The maximum value of LC PG efficiency measured reaches 99.74% using 532 nm green laser, and over 95% for 450 nm blue laser and 633 nm red laser. The optimal thickness in terms of efficiency varies for different wavelengths based on the configuration. The achromatic LC PG by Oh expands the high-efficiency band by a double twist configuration [32].
Fig. 6. (a) The PG efficiency of + 1st and −1st orders vs. thickness of LC layer. (b) Diffraction pattern of LCPG using blue green and red lasers with corresponding thickness.
This research put forward a new approach to fabricate a cycloidal pattern of LC directors to form PGs based on photoalignment technology. The method is to employ birefringent elements to continuously modulate the polarization state of exposure beam instead of digitalized phase or amplitude masks. Based on this, a cascaded configuration including polarizer, birefringent prism and quarter wave plate with designed orientation is demonstrated for generating a cycloidal distribution of linearly polarized direction. And then the quality of the fabricated gratings has been verified by the efficiency and ellipticity measurement. The advantages of this method include high robustness, low cost, less optical element deployed and simple optical design. Hence, the single-step exposure process can greatly simplify the fabrication and increase the yield with large area and high quality maintained, and the method has potential in designing and developing various geometric phase LC devices.
National Natural Science Foundation of China (61405009, 61875004).
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16. A. Shishido, "Rewritable holograms based on azobenzene-containing liquid-crystalline polymers," Polym. J. 42(7), 525–533 (2010). [CrossRef]
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32. C. Oh and M. J. Escuti, "Achromatic diffraction from polarization gratings with high efficiency," Opt. Lett. 33(20), 2287–2289 (2008). [CrossRef]
J. Anandan, "The geometric phase," Nature 360(6402), 307–313 (1992).
S. Pancharatnam, "Generalized theory of interference and its applications," Proc. - Indian Acad. Sci., Sect. A 44(6), 398–417 (1956).
M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392(1802), 45–57 (1984).
D. M. Tong, E. Sjöqvist, L. C. Kwek, and C. H. Oh, "Kinematic approach to the mixed state geometric phase in nonunitary evolution," Phys. Rev. Lett. 93(8), 1–4 (2004).
M. W. Kudenov, M. N. Miskiewicz, M. J. Escuti, and E. L. Dereniak, "Spatial heterodyne interferometry with polarization gratings," Opt. Lett. 37(21), 4413–4415 (2012).
J. Kobashi, H. Yoshida, and M. Ozaki, "Planar optics with patterned chiral liquid crystals," Nat. Photonics 10(6), 389–392 (2016).
G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, "Metasurface holograms reaching 80% efficiency," Nat. Nanotechnol. 10(4), 308–312 (2015).
N. Yu and F. Capasso, "Flat optics with designer metasurfaces," Nat. Mater. 13(2), 139–150 (2014).
Y. Shi, Y. J. Liu, F. Song, V. G. Chigrinov, H.-S. Kwok, M. Hu, D. Luo, and X. W. Sun, "Photoalignment-induced two-dimensional liquid crystal polarization structure via multi-beam polarization interferometry," Opt. Express 26(6), 7683–7692 (2018).
A. M. W. Tam, F. Fan, T. Du, W. Hu, W. Zhang, C. Zhao, X. Wang, K. L. Ching, G. Li, H. Luo, V. G. Chigrinov, S. Wen, and H.-S. Kwok, "Bifocal Optical-Vortex Lens with Sorting of the Generated Nonseparable Spin-Orbital Angular-Momentum States," Phys. Rev. Appl. 7(3), 034010 (2017).
K. Hisano, M. Aizawa, M. Ishizu, Y. Kurata, W. Nakano, N. Akamatsu, C. J. Barrett, and A. Shishido, "Scanning wave photopolymerization enables dye-free alignment patterning of liquid crystals," Sci. Adv. 3(11), e1701610 (2017).
C. Provenzano, P. Pagliusi, and G. Cipparrone, "Electrically tunable two-dimensional liquid crystals gratings induced by polarization holography," Opt. Express 15(9), 5872–5878 (2007).
J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, "Fabrication of ideal geometric-phase holograms with arbitrary wavefronts," Optica 2(11), 958–964 (2015).
W. Duan, P. Chen, S.-J. Ge, X. Liang, and W. Hu, "A Fast-Response and Helicity-Dependent Lens Enabled by Micro-Patterned Dual-Frequency Liquid Crystals," Crystals 9(2), 111 (2019).
Y. Weng, D. Xu, Y. Zhang, X. Li, and S.-T. Wu, "Polarization volume grating with high efficiency and large diffraction angle," Opt. Express 24(16), 17746–17759 (2016).
A. Shishido, "Rewritable holograms based on azobenzene-containing liquid-crystalline polymers," Polym. J. 42(7), 525–533 (2010).
X. Xiang, J. Kim, and M. J. Escuti, "Bragg polarization gratings for wide angular bandwidth and high efficiency at steep deflection angles," Sci. Rep. 8(1), 7202 (2018).
M. W. Kudenov, M. J. Escuti, N. Hagen, E. L. Dereniak, and K. Oka, "Snapshot imaging Mueller matrix polarimeter using polarization gratings," Opt. Lett. 37(8), 1367–1369 (2012).
T. Zhan, Y.-H. Lee, G. Tan, J. Xiong, K. Yin, F. Gou, J. Zou, N. Zhang, D. Zhao, J. Yang, S. Liu, and S.-T. Wu, "Pancharatnam–Berry optical elements for head-up and near-eye displays," J. Opt. Soc. Am. B 36(5), D52–D65 (2019).
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(1) δ = k y = 2 π λ Δ n tan α y
(2) J e x p o s u r e = M Q W P ′ J 1 = M Q W P ′ M p r i s m J i n = M Q W P ′ [ exp ( i δ 2 ) 0 0 exp ( − i δ 2 ) ] [ cos 45 ∘ sin 45 ∘ ]
(3) J 1 = 2 2 [ exp ( i δ 2 ) exp ( − i δ 2 ) ]
(4) J e x p o s u r e = R T ( 45 ∘ ) M Q W P R ( 45 ∘ ) J 1 = [ cos ( δ 2 + π 4 ) sin ( δ 2 + π 4 ) ]
(5) Λ = λ Δ n tan α
Summarization of Polarization and Efficiency of Diffractive Orders
Diffraction order 0 +1 −1
Polarization Same as incidence Left-hand circular Right-hand circular
Efficiency cos 2 ( π Δ n d λ ) γ R H C sin 2 ( π Δ n d λ ) γ L H C sin 2 ( π Δ n d λ ) | CommonCrawl |
Research | Open | Published: 20 February 2017
Efficient network coding solutions for limiting the effect of packet loss
Samih Abdul-Nabi1,2,
Ayman Khalil1,
Philippe Mary2 &
Jean-François Hélard2
In traditional store and forward protocols, lost packets have no impact on the delivery of other transmitted packets. With network coding, the impact of a packet loss may affect the decoding of other transmitted packets thus affecting the entire process of communication between nodes. In this work, we propose a new network coding model that allows generating, coding, decoding and transmission activities on the packets. Based on this model, the impact of lost packets on buffering and the complexity at the receiving nodes is studied and two new mechanisms are proposed to allow the recovery of lost packets. Compared to traditional linear network coding protocol, our mechanisms provide a significant performance amelioration in terms of number of transmissions required to recover from packet loss.
Recent advancements in network coding (NC) implementations have allowed substantial throughput gain in wireless networks. The simple idea of combining several received or generated packets into one outgoing packet was the starting point for the NC paradigm [1, 2]. Since the publication of [1], researchers have concentrated on the use of NC for optimizing throughput in multicast networks [1, 3, 4] and for improving network reliability through random linear coding [5, 6]. However, only few published works have considered packet loss and packet loss recovery in the analysis of network coding performance [7, 8]. Figure 1 illustrates a NC example with one sender and three receivers over a network with a random packet erasure process. Three cases are considered, a reliable transmission in Fig. 1 a, one different packet loss on each link in Fig. 1 b, c. To recover the three packets P 1, P 2, and P 3, the scheme in Fig. 1 c using NC for the additional retransmission requires only one channel use instead of 3 channel uses with the scheme in Fig. 1 b, which does not use NC. Recently, Instantly Decodable NC (IDNC) [9] with its fast decoding potential becomes of major interest for real time applications. IDNC uses XOR coding operations and offers instantaneous packet decoding when enough information is available at the receiver.
Network coding example. a Channel uses, b six channel uses, and c four channel uses
Another popular NC scheme, known as linear network coding (LNC), has been applied to save packet transmission in wireless networks [3, 10, 11]. Using the example of Fig. 1, LNC consists of improving reliability by initially sending linearly independent combinations of the three packets. Suppose the network eraser rate is 20% for example and that the sender broadcasts four linear combinations of the three packets as follows:
$$ \underbrace{\left(\begin{array}{c} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \end{array}\right)}_{R} = \underbrace{\left(\begin{array}{ccc} \alpha_{1} & \beta_{1} & \gamma_{1} \\ \alpha_{2} & \beta_{2} & \gamma_{2} \\ \alpha_{3} & \beta_{3} & \gamma_{3} \\ \alpha_{4} & \beta_{4} & \gamma_{4} \end{array}\right)}_{A} \underbrace{\left(\begin{array}{c} P_{1} \\ P_{2} \\ P_{3} \\ \end{array}\right)}_{P} $$
where A is the coding matrix and P the packet vector. In this case, each receiving node can successfully decode all transmitted packets when receiving at least three of the linear combinations. However, the eraser rate is usually variable and unknown and hence the required number of transmissions is very difficult to predict. To guarantee the delivery of all packets, the number of transmitted linear combinations should be large enough to compensate for any loss, consequently, this can compromise the throughput benefit of NC. Thus, one of the main contributions in the solution detailed later is that it allows, with an optimized number of transmissions, to decode all received combined messages independently of the loss rate.
In the traditional store and forward protocols, when a packet is lost, reliability is achieved by requesting the retransmission of that packet. However, with NC, the decoding of new packets relies on previously received ones. Thus, losing few packets might have a domino effect and decoding further packets could become impossible. This tradeoff between throughput gain with NC and decoding complexity was studied in [12] and [13], for instance, from a decoding delay point of view.
Motivated by this tradeoff and by the decoding feasibility at end nodes, a complete study on the impact of packet loss is conducted in this paper in the case of linear networks [14] and solutions are proposed to deliver all packets in a near optimal number of transmissions.
Background and prerequisites
Network model: In this paper, the adopted network model is represented as an undirected graph G=(N,E) where the set of nodes N is divided into two exhaustive and mutually exclusive subsets, EN, the set of end nodes and IN, the set of intermediate (or middle) nodes. As illustrated in Fig. 2, A and B are the two end nodes and n i i={1,2,…,N−2} are the intermediate nodes. End nodes generate packets to be transmitted to each other via intermediate nodes responsible for coding and routing tasks. We denote by packet a newly generated message by an end node ∈E N and by coded message a combination of packets coded together using the XOR operation. Without the lost of generality, NC is assumed to be applied in GF(2). The network in Fig. 2 can be the support of a video conference, data exchange, or other type of information.
A network with four intermediate nodes
The coded message has its own header identifying the two correlated addresses, the number of coded packets, a header for each packet, and some adaptive memory to help guiding the coded message to its destination as illustrated in Table 1. The payload of the coded message consists of all packets coded together by the XOR operation. By inspecting the adaptive memory, a node in the network can determine whether the message should be broadcasted to all neighboring nodes or unicasted to next hop, i.e., next node. The header of each packet in the coded message identifies its source, its destination, a flag that determines if it exists at destination, and its age.
Table 1 Routing concept
[15] The age of a packet is defined as the number of times the packet contributes to a coded message and the age of a coded message is defined as the age of the oldest packet in the coded message.
[15] The maturity of a network is defined as the highest allowed age for all the packets in the network.
Packets reaching the maturity age are prohibited from further coding. The idea is to prevent packets from living forever in the network. Packets in a coded message are address-correlated and only those that are classified as address-correlated are allowed to be coded together [16, 17].
Routing rules: Figure 3 clarifies the setting of the "Exist at dest." flag. At iteration l, the routing table of node n i shows that the next hop for packet P A is n j , and the routing table of node n k shows that the next hop for packet P B is n j . When n i receives back P A at iteration l+1 within a coded message it knows that a copy of P A is in its way to destination thus the "Exist at dest." flag is set. Similarly, node n k sets the "Exist at dest." flag of packet P B .
"Exist at dest." flag
Table 1 shows the header of a newly coded message at node n i . The created message is broadcasted to neighboring nodes (unicast is unset) and both packets in the coded message are unknown to their destinations A and B, respectively, as shown in Table 1a. When the coded message reaches the node n i+1, the header is updated by the node server and the "Exist at dest." flag of the packet originated by node B is set as shown in Table 1b. Thus, the message continue its way to the destination using unicast. The reader is referred to [16] for more details about routing coded messages in NC. The "Exist at dest." flag, initially set to false, is updated by the routing nodes and set to true whenever a packet is received back within a coded message after being sent to its destination. For decodability purposes, the NC scheme guarantees that, in any coded message, at most one, packet is unknown by each destination.
Impact of losing a packet: Figure 4 shows a detailed trace of packets exchanged between end nodes A and B where node A generates packets ai i=1,2,… and node B generates packets bi i=1,2,…. A maturity value of 3 is set and each packet is upper-indexed by its age. In this figure, solid arrows indicate a unicasting activity to the next hop and dashed arrows indicate a broadcasting activity of newly coded messages. It is clearly observed that the multiplicity of a packet, given for example a1, is limited and controlled by the use of aging where the age is visible as superscript on each packet. Packet a1 generated at iteration 1 is removed from all nodes of the network at iteration 7.
Delivery of packets in a network with no loss
In this context, we are mainly interested in the impact of losing a coded message, for example (a1⊕b2) between nodes n 2 and B at iteration 3, on the decoding capability at node B. Failing to receive and decode a1, node B will successively receive undecodable packets.
In traditional communication protocols, reliability relies on retransmissions, and end to end reliability is assured by the transport layer of the OSI layered communication stack. Moreover, hop to hop reliability is assured by the PHY and MAC layers by retransmitting unacknowledged packets. Recently, hop by hop transport layer reliability protocols have been developed where intermediate nodes store and recover packets to deliver data more efficiently [18, 19]. With NC, reliability and throughput gain are achieved through linear NC [3, 10] where several packets are linearly combined together and broadcasted into the network. Many works in the literature have dealt with the benefits of NC from the reliability point of view [6, 20].
In [6], authors have presented analytical and numerical results for the performance of end-to-end and link-by-link reliability mechanisms based on automatic repeat request (ARQ), forward error correction (FEC) and NC in a tree topology. Using an access point topology with k receivers, they demonstrated that the reliability gain of NC compared to ARQ is of Θ(logk). They showed also that allowing intermediate nodes to recover lost packets remains in Θ(logk) while further minimizing the number of transmissions. However, it should be noted that buffering packets in intermediate nodes requires decoding coded messages which might be costly. In our proposed solution, decoding only occurs at end nodes while intermediate nodes are only allowed to perform coding activity to minimize the number of transmissions.
Due to lossy wireless channel conditions, recent works have focused on loss recovery taking into consideration the overhead on the network due to retransmissions. In [21], authors have proposed Codecast protocol suitable to low-loss, low-latency constraint applications such as video streaming. Their work is based on random NC [3] implementing localized loss recovery and path diversity with very low overhead. They demonstrated throughout simulations that codecast achieves a near-perfect packet delivery ratio while maintaining at the same time lower overhead than conventional multicast. Routing and route selection have a major impact on reliability and time delivery.
To reduce loss impact, routing selection mechanisms have been investigated. In [22], overlay routing networks have been proposed to monitor communication between nodes in order to improve routing of packets and provide loss reduction over traditional routing algorithms. Opportunistic routing [23–25] dynamically selects paths based on loss conditions. However, in [26], authors conducted a comparison of best-path routing and opportunistic routing and found that the benefit of opportunistic routing is much less than commonly believed. It should be well noted that these mechanisms introduce latency on the overall communication time and overhead on the routing nodes. The mechanisms we are proposing are network independent and require no probe from the network.
In the majority of cases, NC is considered as block coding where packets can only be decoded in batches [27]; a packet needs to wait for the arrival of other packets before being able to be decoded. While NC increases throughput and lowers congestion, block decoding may cause major delays and reduction in the quality of service (QoS). Recent works have focused on QoS and proposed methods to reduce the average waiting time in the buffer of the intermediate nodes [28, 29]. Our work differs from existing works in the sense that we use NC for data exchange activity and not for multicast purposes. With address-correlated NC protocol, no delay is engendered by the coding activity and no packet is delayed waiting for a match. Moreover, instantaneous decoding at end nodes is guaranteed when packets arrive in order. Decoding remains assured in real networks where messages arrive unordered.
Contributions and paper organization
This work aims to study the impact of packet loss on the decoding capability and resource management. NC schemes where decoding only occurs at destination nodes are investigated. Undelivered packets have impact on some major resources especially buffer size and processing time.
Our contributions are summarized as follows:
A new NC decision model is presented and implemented at each node of the network. The model is used to manage node queues in a way to avoid packet drop due to the limited queue capacity.
The impact of packet loss on the decoding capability at the receiver and the decoding perturbation caused by each loss are investigated and analyzed.
Two new recovery mechanisms initiated by end nodes are proposed to handle packet loss with NC. (i) The immediate retransmission request (IRR) mechanism that uses information extracted from the undecoded messages to request the retransmission of lost packets. (ii) The recovery from packet loss mechanism (BCSD) with near optimal number of retransmissions.
The paper is organized as follows; the NC decision model to generate, route, code and decode messages is presented in Sections 2 and 3. In Section 4 different scenarios for packet loss are investigated and the impact on decoding capabilities and buffering requirements are recorded. Based on the results from Section 4, solutions are proposed in Section 5.
Simulations and results are presented in Section 6 and we conclude with Section 7.
A new NC decision model
General description of the NC decision model
In the considered network, each node consists of a server managing two queues; the receiving queue that holds messages waiting to be coded and the outgoing queue that holds messages waiting to be transmitted (Fig. 5). The receiving queue collects received messages to investigate coding opportunities for address correlated packets which are only allowed to be coded together. On the other hand, the outgoing queue holds messages ready to be transmitted into the network in broadcast or unicast modes. The presence of two queues facilitates the scheduling of tasks at the node and reduces the time needed to search for matching messages to be coded.
Node model
The state diagram of an intermediate node is shown in Fig. 6. In this diagram, the coding phase is represented by a link from the receiving queue to the server where two or more messages are selected from the receiving queue to be coded together and sent to the outgoing queue. Note that with aging, some mature messages are transferred from receiving to outgoing queue without being coded. The transmission of a message to a neighboring node is done through the server.
Intermediate node state diagram
Similarly, the state diagram of an end node is shown in Fig. 7. In this diagram, the decoding phase is represented by a link to the server where one message is selected to be decoded. A message containing several packets can be decoded if a maximum of one of these packets does not exist at destination. When a packet is decoded, it is transferred to the upper layer. If the server is unable to decode a message, the message is returned to the receiving queue and further retries are performed later when more received packets are available. The generation phase is represented by a link from the server which generates packets according to a model that will be described later. Finally, the transmission of a message to the neighboring node is done through the server.
End node state diagram
The receiving queue is used to store all received messages and has a capacity denoted by $C_{r_{q}}$. We also denote by $x_{r_{q}}(t)$ the number of messages at time t waiting in the receiving queue. The outgoing queue holds messages ready to be transmitted to the network. We denote by $C_{o_{q}}$ and $x_{o_{q}}(t)$ the capacity and the number of coded messages at time t in the outgoing queue, respectively. The server mainly performs one of the following activities: (1) packet generation denoted by g,, coding activity denoted by c, (3) sending message denoted by s, and (4) decoding activity denoted by d. The possible activities at nodes are:
Sending node activity set = {g,s}
Intermediate node activity set = {c,s}
Receiving node activity set = {d}
Note that, when a node plays more than one role, its set of activity is the union of the activity set associated to each role. For example, the set of activities at an end node is {g,s,d}.
A decision model is needed at each node to select the best activity. Two decision models are adopted for our implementation, a deterministic decision model for intermediate nodes and a probabilistic model for end nodes.
Deterministic model for intermediate nodes
In an intermediate node where the decision fluctuates between coding or sending messages, the selection of the activity to be performed largely depends on the number of messages in the queues. The decision model needs to constantly balance the occupation of both receiving and outgoing queues. When balanced, the decision model preserves QoS and favors early received messages in both queues to be processed. For that purpose, we propose an integer programming problem where a binary variable y i is assigned to each possible activity with i∈{c,s}. y i is 0 if the activity is not performed and 1 elsewhere. The optimization problem handled at each intermediate node is described in (2).
$$\begin{array}{@{}rcl@{}} \underset{y_{c},y_{s}}{\max} & & y_{c}(t) \sum_{i=1}^{x_{r}(t)}{m_{r_{i}}(t)} + y_{s}(t) \sum_{i=1}^{x_{o}}{m_{o_{i}}(t)} \end{array} $$
$$\begin{array}{@{}rcl@{}} \text{subject to} && y_{c}(t)+y_{s}(t) = 1 \end{array} $$
$$\begin{array}{@{}rcl@{}} & &y_{c}(t) \geq \frac{(x_{r}(t)-x_{o}(t)) - d_{ro}}{C_{r}} \end{array} $$
$$\begin{array}{@{}rcl@{}} & &y_{s}(t) \geq \frac{(x_{o}(t)-x_{r}(t)) - d_{or}}{C_{o}} \end{array} $$
$$\begin{array}{@{}rcl@{}} & &y_{c},y_{s} \in \{0, 1\} \end{array} $$
In this problem, d ro (respectively, d or ) is the acceptable difference in number of messages between the receiving queue and the outgoing queue (respectively, between the outgoing queue and the receiving queue), and $m_{r_{i}}(t)$ is the amount of time message i has been in the receiving queue while $m_{o_{j}}$ is the amount of time message j has been in the outgoing queue. The objective of (4) is to force y c (t) to take the value 1 whenever the difference between the number of messages waiting to be further coded and the number of messages waiting to be transmitted reaches or exceeds d ro . Similar objective for (5) by forcing y s (t) to be 1 when the difference between the number of messages waiting to be transmitted and the number of messages waiting to be further coded reaches or exceeds the value d or . Constraint (3) guarantees that the server performs exactly one task at a time. It should be noted that (4) and (5) are built to guarantee that only one of the binary variables y c (t) and y s (t) is forced to take the value 1, however, both variables are allowed to take the value 0 at the same time which explains the need for (3). In that case, the choice of activity, i.e., (y c ,y s )=(1,0) or (y c ,y s )=(0,1) is the choice maximizing (2). The objective function forces the selection of the activity that helps releasing messages with the largest waiting time.
Probabilistic model for end nodes
A binary variable y i is assigned to each possible activity with i∈{d,g,s}. y i can take the values 0 if the activity is not performed and 1 elsewhere. As for the case of intermediate nodes, the server can perform exactly one task at a time. We denote by P(d), P(g) and P(s), respectively, the probability of performing a decoding (y d =1), a generation (y g =1), or a sending activity (y s =1).
The generation of packets is considered as an independent process from other activities and the decision of the activity to be undertaken at end nodes is performed in two phases. During the first phase, a decision is taken on whether or not the node should generate a packet. If no packets are to be generated, we move to the second phase to select, according to a stochastic decision model, the node activity to be performed.
The stochastic decision model is based on [30] where routing packets are given by some probability functions. For our model, we have the following equations:
$$\begin{array}{@{}rcl@{}} P(d) &=& \frac{e^{-\beta x_{oq}(t)}}{e^{-\beta x_{oq} (t)}+e^{-\beta x_{rq}(t)}} \end{array} $$
$$\begin{array}{@{}rcl@{}} P(s) &=& \frac{e^{-\beta x_{rq}(t)}}{e^{-\beta x_{oq}(t)}+e^{-\beta x_{rq}(t)}} \end{array} $$
where β is a control parameter and P(d)+P(s)=1. In (7), the probability of decoding decreases when the number of messages in the outgoing queue increases favoring in this case the sending activity. Furthermore, a similar argument can be given to (8). However, even when the number of coded messages in the outgoing queue (respectively, receiving queue) exceeds the number of coded messages in the receiving queue (respectively, outgoing queue), the activity selection process remains stochastic and there is still some probability of selecting the decoding (respectively, sending) activity.
As noted in [30], if we set β=0, both probabilities will be equal to 1/2 and decoding or sending are chosen randomly. Furthermore, if β grows infinitely, the limits of P(d) and P(s) will be, respectively, 0 and 1 if x oq (t)>x rq (t) thus creating a deterministic decision model where the activity is selected according to queues occupation.
Note that, a delayed message at an intermediate node has a domino effect on the delivery of other messages. This imposes a QoS requirement to avoid such effect which is assured by the deterministic model. In revenge, delayed messages at end nodes do not have such effect and the probabilistic model reveals to be flexible and requires less computational time.
Activity selection algorithms
Models presented in Section 2 are used to propose two-low cost algorithms to select the best activity to be performed at intermediate and end nodes, respectively. These algorithms are initiated each time the node is requested to select and complete a task. The activity selection at the intermediate node is shown in Algorithm 1.
In this algorithm, steps 1 and 3 represent constraints (4) and (5) in the optimization problem presented in (2) and select the activity to perform depending on the occupation of the receiving and outgoing queues. When queues are almost balanced, steps 6 and 17 attempt to maximize (2) helping late packets to be released from the queues and improving QoS.
The activity selection at end nodes is presented in Algorithm 2. In this algorithm, step 2 independently selects the generation activity with a certain predefined probability. If no new packets are generated, the code between lines 5 and 10 give higher probability to the activity that balances the receiving and outgoing queues.
Packet loss simulation
Algorithms presented in Section 3 are validated by running simulations on a linear network of 6 nodes, i.e., with four intermediate nodes as shown in Fig. 2. Statistics are recorded about the loss impact on decoding opportunity and buffering time at end nodes. For this purpose, the new NC decision model presented in Section 2 is used to built a specific application to simulate the network layer at each node of the network. As example, only the hop n 3→n 2 is considered with loss and the packet loss is modeled as an independent and identical Bernoulli distributed process. The contribution of this section is of twofold; first, to understand the correlation between the packet loss and the number of undecoded packets at the receiver, and second, to aid to establish a recovery process to minimize the number of retransmissions.
Correlation between lost and undecoded messages
In this simulation, the number of packets exchanged between A and B is set to be 10,000 packets. During the same simulations, the impact of aging is also observed [15]. The results of simulation for each specific percentage of packet loss are presented in Fig. 8. The percentage of undecoded packets corresponds to the percentage of different packets that have been identified in the header of the received messages without being decoded. The simulation is repeated for three different maturity values. Figure 8 shows the impact of the aging concept on the number of undecoded messages. As shown in the figure, the number of undecoded packets at the receiving node is proportional to the maturity. This is justified by the fact that when maturity increases, coded packets live longer in the network and are further coded with other packets leading to an increase in the number of undecoded packets at the receiver. Moreover, the number of undecoded packets at the receiver is finite and the system resumes receiving decodable packets after each loss.
Percentage of undecoded packets with different loss probabilities and different maturities
In Fig. 9, the buffering of received packets at the destination nodes is studied. Destination nodes need to buffer sent and received packets so that they will be used in the decoding process. Buffering is costly since it requires significant memory space. Buffering time of a packet is computed as the laps of time between the moment the packet was received and the last time the packet was used for decoding other packets. The figure gives the percentage of the buffering time compared to the total communication time versus the loss probability P, for three different maturity values. Furthermore, for each case, the highest values and the average values are represented. These results show that the buffering time decreases when the loss percentage increases. This is justified by the fact that less decoding occurs when more coded messages are lost. Moreover, the buffering also varies with maturity. When maturity is increased, more buffering time is required since packets live longer in the network.
Buffering versus loss percentage
Decoding opportunity
In this section, we study the multiplicity of the lost packets in the network and its impact on the decoding process. By multiplicity, we mean the number of copies of the packet that exist in the network. Recall that, when a packet is coded with others, the coded message is broadcasted into the network creating duplicates of these coded packets. When a packet is lost, other copies of that packet still exist in the network creating undecoding opportunities at the receiver.
To clarify how end nodes can receive undecodable messages, consider the example shown in Fig. 10. In this example, a coded message m is created and broadcasted at iteration l by node n j (shown in the figure as m 1 and m 2 which are identical). Consider now that message m 1 is lost and m 2 reaches node n i . All packets in m 2 originated from A are flagged as exist at destination ("Exist at dest." flag set to true). Node n i uses message m 2 in further coding activity and broadcasts at itration l+1 a new coded message $ m^{'} \oplus m_{2}\phantom {\dot {i}\!}$. Node B cannot decode $m^{'} \oplus m_{2}\phantom {\dot {i}\!}$ since it holds two packets originated by A and unknown to B.
Lost packet discovery
Let us denote by μ∈ℜ+ the maturity age of the messages. Theorem 4.1 sets a boundary on the dissemination of each packet in the network.
Theorem 4.1
[15] Given a network G=(N,E) and a maturity scalar μ in ℜ+. The number of coded messages in the network containing a specific packet is bounded by 2μ.
Note that, due to the linear nature of the network, each end node receives a maximum of $\frac {2^{\mu }}{2}$ messages containing the same packet. This upper bound on the multiplicity of a packet is used in the following theorem to estimate the number of undecoded messages when a packet is lost.
Given a network G=(N,E) and a maturity scalar μ in ℜ+. When a packet loss occurs in the direction of end node E, the estimated number of undecoded messages that can be received by E as a result of that loss is given by
$$ \frac{1}{\mu + 1} \sum_{i=2}^{\mu} 2^{\mu - i} $$
If a packet in the direction of E is lost at age 0, i.e., before being coded, no other copies of that packet exist in the network. Consequently, no undecodable messages are received by E. On the other hand, if a packet is lost in the direction of E at maturity age, the other copy of that packet is unicasted in the opposite direction. Consequently, no undecodable messages are received by E. Let P a be a packet with age a, and $\phantom {\dot {i}\!}N_{P^{a}}$ the number of messages received by E that are created starting from P a. According to Theorem 4.1, $N_{P^{^{a}}} = \frac {2^{\mu -a}}{2}$. Refering to the coding graph presented in [15], we assume that in a network with maturity μ, there is an equal probability that a lost packet has an age between 0 and μ. The total number of undecoded messages that can be received by E is:
$$\frac{1}{\mu+1}\sum_{i=1}^{\mu-1} N_{P^{^{i}}} = \frac{1}{\mu+1}\sum_{i=1}^{\mu-1} \frac{2^{\mu-i}}{2} = \frac{1}{\mu+1}\sum_{i=1}^{\mu-1} 2^{\mu-(i+1)} $$
By taking i+1=j, we can write
$$\frac{1}{\mu+1}\sum_{j=2}^{\mu} {2^{\mu-j}} $$
As an example, a packet loss might lead to receiving four undecodable messages when maturity is set to 6, after which the system resumes receiving decodable messages.
Another important factor that helps in reducing the number of received undecodable messages is related to the model presented in Section 2. Recall that the model allows packets to be transmitted without coding and never waits for a match. A simulation is performed to count the number of packets that are traveling through the network without being coded. Figure 11 shows the number of uncoded messages in both queues of each node of the network. End node generates uncoded messages while they receive the lowest percentage of uncoded messages. There is, however, a significant percentage of uncoded packets floating between the nodes of the network. This is in fact due to the following factors:
Percentage of uncoded messages at each node
Aging: preventing packets from being further coded.
Decision model: forcing packets to be transmitted without waiting for a match.
Analysis in this section shows an interesting characteristic of NC when it comes to lossy network. The intensive use of NC, represented by a large maturity value, keeps duplicates of a lost packet in the network and the number of undecodable packets degenerate. One of the advantages of aging is that it helps creating mature messages in the network thus preventing coupling with duplicates of lost packets.
Proposed mechanisms
Two new mechanisms are detailed in this section. To further evaluate the performance of these mechanisms, a revised version of the LNC protocol is also presented in this section.
Immediate retransmission request
The end node receiving an undecodable message is able to identify, by reading the header of the undecoded message, those packets that are flagged as "exist at dest." but are not in its buffered packets for decoding purpose. Without waiting for a retransmission event from the upper layer (i.e., transport layer) a request of retransmission is initiated by the NC protocol for these packets. Note that the retransmission request initiated by the network layer of the receiving node is answered by the network layer of the sending node since it buffers also transmitted packets for decoding purposes. These requesting and retransmission operations are transparent to the TCP protocol.
Due to delays and unordered arrival of messages at the receiver, one undecodable message is not a sign of packet loss. The receiving node suspects that a loss occurs when undecodable messages remain undecodable for a specific period of time, denoted by t time (threshold on time) or when receiving a certain number of consecutive undecodable messages, denoted by t cup (threshold on consecutive undecoded packets).
The IRR algorithm is designed to take into consideration the previously mentioned concerns. The IRR algorithm is shown in Algorithm 3 where UM denotes an undecoded message. The algorithm works as follows: initially, consecutive undecoded packets (cup) is set to 0, request retransmission (rr) is set to false and the list of requested packets (l rp ) is empty. The algorithm then iterates on each entry in the list of undecoded messages. Lines 4 to 11 are dedicated to identify consecutive undecoded messages and a request for retransmission is initiated if cup reaches a predefined threshold. Lines 12 to 14 investigate the buffering time of each undecoded message and request retransmission if a threshold on time is reached. The request for packet retransmissions are sent in the header of newly generated packets or immediately if no new packets are available for transmission. Since no buffering of packets is performed in the intermediate nodes, the request for retransmission is sent to the generator nodes. The algorithm is initiated each time the node is about to perform a decoding activity.
IRR differs from the ARQ of the end to end transport layer in that IRR is faster to identify lost packets and to initiate a retransmission request. However, it should be noted that IRR does not replace ARQ. As shown in the proof of Theorem 4.2, a packet might be lost without decoding consequences at the receiving nodes. Such a loss can only be detected and requested by the transport layer using ARQ.
The simulation results of IRR are shown in Fig. 12. The simulation ran with the three values of maturity and with the same conditions than in Fig. 8 of Section 4. Figure 12 shows that the number of requested packets increases with maturity for the same packet loss percentage. Furthermore, the number of retransmissions compared to the number of received undecodable messages is presented in Table 2 for two loss percentages, where the undecoded and the requested columns are extracted from the results of Figs. 8 and 12, respectively. Table 2 reveals a reduction in the percentage of requested packets when maturity increases. This is justified by the fact that, with larger maturity, a loss generates more undecodable messages and a retransmission helps also decoding more packets.
IRR: percentage of requested packets versus the loss probability
Table 2 Percentage of retransmission versus undecoded
Recovery from packet loss
With NC, recovery from packet loss differs from traditional retransmission protocols since packets are interconnected together by the mean of coded messages. We showed that losing one packet might lead the way to lose additional packets due to the nature of the coded messages and of the decoding process. For this reason, it is obvious that an efficient algorithm is needed in order to minimize the number of retransmissions.
Due to NC, end nodes receive multiple copies of the same packet as shown previously in Fig. 4. When a loss occurs on a link, a copy of a packet is lost but other copies might still be present in the network. End nodes receive messages that they are unable to decode but might contain such copies. In what follows, we propose algorithms that request the minimal number of retransmissions in order to retrieve lost packets. Note that retransmitted packets can be coded with newly generated packets leading to further reduction in the number of retransmissions.
Before continuing with the recovery process, we give some definitions and then propose a mechanism to identify the packets to be requested from the sender in order to recover all lost packets with minimal retransmissions.
Closure and covering sets
Let S be a set of packets and C be a set of coded messages.
When XOR operation is used to code packets together, the simple closure of S, denoted by $S^{+}_{C}$, is the set of packets that can be decoded from C using packets in S.
To better understand the simple closure definition, we illustrate the following example.
Let S={p 1,p 2,p 3} and C={p 1 p 4,p 1 p 3 p 5,p 4 p 5 p 7,p 5 p 6 p 7,p 6 p 8 p 9,p 7 p 8 p 9}.
To build the simple closure of S, we proceed as follows:
$$\begin{array}{lll} S^{+}_{C} = \{p_{1},p_{2},p_{3} \} & &\\ S^{+}_{C} = \{p_{1},p_{2},p_{3},p_{4} \} &\text{using} & p_{1}p_{4} \\ S^{+}_{C} = \{p_{1},p_{2},p_{3},p_{4},p_{5} \} &\text{using} & p_{1}p_{3}p_{5} \\ S^{+}_{C} = \{p_{1},p_{2},p_{3},p_{4},p_{5},p_{7} \} &\text{using} & p_{4}p_{5}p_{7} \\ S^{+}_{C} = \{p_{1},p_{2},p_{3},p_{4},p_{5},p_{7},p_{6} \}&\text{using} & p_{5}p_{6}p_{7} \\ \end{array} $$
When linear combination is used to code packets together, the complex closure of S, denoted by $S^{++}_{C}$, is the set of packets that can be decoded, by the mean of Gaussian Elimination, from C using packets in S.
Going back to the previous example, the complex closure of S, is denoted by: $S^{++}_{C} = S^{+}_{C} \cup \{p_{8},p_{9} \} $ where p 8 and p 9 can be obtained by solving the system:
$$\left\{ \begin{array}{l l l} \alpha_{1}p_{8} + \alpha_{2}p_{9} & = & R_{1} \\ \beta_{1}p_{8} + \beta_{2}p_{9} & = & R_{2} \\ \end{array}, \right. $$
where R 1 and R 2 are the received coded packets and (α 1,α 2)t and (β 1,β 2)t are the corresponding coding vectors normally included in the header of the received coded packets.
S is a covering set of C if the closure of S includes all packets that are part of the coded messages in C.
Going back again to the previous example, we see that $S^{++}_{C}$ is a covering set of C but $S^{+}_{C}$ is not.
A basic covering set of C is a covering set of C with minimal cardinality (having the least number of packets).
Recovery mechanism
The recovery process starts when the receiving end node receives a message that cannot be decoded, i.e., it has more than one unknown packet. At that time, the end node expects to receive additional undecodable packets. Each received undecodable message is XOR-ed with all known packets and the resulting chunk of undecoded packets XOR-ed together is added to an originally empty set C.
The gathering period of undecodable chunks of messages is controlled by a timer that starts immediately after the reception of the first undecodable coded message. As stated before, undecodability might be caused by unordered arrival of messages and the problem might be solved by itself when more messages arrive to the end node. The duration of the timer is an application-specific parameter based on the communication medium and the type of application. It can vary from milliseconds in the case of instant messaging where messages need to be delivered in order to the upper layer, to seconds in the case of file exchange where only the complete file needs to be ordered. When the timer times out, the recovery mechanism is lunched. The recovery mechanism consists of finding a basic covering set and packets in that covering set are requested from the sender. The Basic Covering Set Discovery (BCSD) algorithm is presented in Algorithm 4. The algorithm follows a greedy mechanism that tries, within a reasonable amount of time, to identify a basic covering set for all chunks of messages in C. The algorithm starts by building the incident matrix where columns identify the packets in C and rows represent undecoded chunks of messages sorted by receiving time. For the set C={p 1 p 4,p 1 p 3 p 5,p 4 p 5 p 7,p 5 p 6 p 7,p 6 p 8 p 9,p 7 p 8 p 9} the corresponding incident matrix M is
$$\begin{aligned} &\qquad \qquad \qquad \quad \!\!p_{1} \quad p_{3} \quad p_{4} \quad p_{5} \quad p_{6} \quad p_{7} \quad p_{8} \quad p_{9}\\ &M=\begin{array}{l} p_{1}p_{4}\\ p_{1}p_{3}p_{5}\\ p_{4}p_{5}p_{7}\\ p_{5}p_{6}p_{7}\\ p_{6}p_{8}p_{9}\\ p_{7}p_{8}p_{9} \end{array} \left(\begin{array}{cccccccc} 1{\phantom{00}}&{\phantom{00}}& 1{\phantom{00}}&{\phantom{00}} &{\phantom{00}} &{\phantom{00}} & {\phantom{00}}& \\ 1{\phantom{00}}& 1{\phantom{00}} &{\phantom{00}}& 1{\phantom{00}}&{\phantom{00}} &{\phantom{00}} &{\phantom{00}} & \\ {\phantom{00}}&{\phantom{00}} & 1{\phantom{00}}& 1{\phantom{00}}& & 1{\phantom{00}}&{\phantom{00}} & \\ {\phantom{00}}& {\phantom{00}}& {\phantom{00}}& 1{\phantom{00}}& 1{\phantom{00}}& 1{\phantom{00}}& & \\ {\phantom{00}}& {\phantom{00}}& {\phantom{00}}& {\phantom{00}}& 1{\phantom{00}}& {\phantom{00}}& 1{\phantom{00}}& 1\\ {\phantom{00}}& {\phantom{00}}& {\phantom{00}}& {\phantom{00}}& {\phantom{00}}& 1{\phantom{00}}& 1{\phantom{00}}& 1 \end{array}\right). \end{aligned} $$
The algorithm then iterates on the incident matrix until the covering set is built. At each iteration, the column with the highest number if 1's is selected (i.e., highest occurrence in the undecoded messages) and the corresponding packet is added to the covering set B. Rows with one entry are then removed together with the column of the entry since the corresponding packet can be decoded. The algorithm stops when the incident matrix becomes empty.
By applying BCSD to the incident matrix M, the basic covering set found is $S^{+}_{C} = \{p_{5},p_{7},p_{8} \}$.
Revised linear network coding
To make a fair comparison with both mechanisms presented earlier, a revised version of the LNC mechanism is introduced in this section. The idea of the revised LNC protocol is to use LNC for unicast and to send enough packets through a lossy network that uses the new NC decision models for intermediate and end nodes, in order to:
overcome the loss of packets in the links of the network,
overcome the undecodability of packets at the end nodes.
The revised LNC protocol uses block coding as follows: each sending node groups b packets together and sends b Δ linearly independent combinations of the packets in each group. The cardinality b of each block and the number of combinations b Δ are determined according to Theorem 4.2 and according to the loss percentage between the end nodes of the network given by
$$ P_{n} = 1- (1-P)^{N-1}, $$
where N is the total number of nodes and P is the probability of loss in a link.
Once P n is identified, the sending node needs to send $b_{\Delta } = \left \lceil \frac {b}{1-P_{n}} \right \rceil $ independent linear combinations of the packets of each block to guarantee the delivery of the packets in the block. In order to minimize the number of transmissions, the cardinality of the block should be selected to satisfy $\frac {b}{1-P_{n}} = \left \lceil \frac {b}{1-P_{n}} \right \rceil $.
In addition, each loss in the network leads, by Theorem 4.2, to have $U_{n} = \frac {1}{\mu + 1} \sum _{i=2}^{\mu } 2^{\mu - i}$ undecodable messages that might reach the destination. With P n as loss percentage in the network, U n ×P n undecodable messages can reach the end node. To bypass the undecodability problem, b Δ is augmented by P n ×b Δ ×U n ×(1−P n ).
Simulation and results
In this section, both IRR and BCSD algorithms are implemented and tested. Simulations are run on linear network of six nodes with different loss probabilities on any link. Results are compared to both traditional store and forward with ARQ protocols and to the revised LNC protocol presented in Section 5.3. Along the simulations, 10,000 packets are exchanged between both end nodes of the network. Algorithms are compared according to the total number of transmissions needed to deliver all the exchanged packets between the end nodes. In the store and forward protocol, the transport layer at the end nodes automatically claims unacknowledged packets. The total number of recorded transmissions to reliably deliver all the packets is recorded. In a reliable network of six nodes, five transmissions are needed to deliver each packet from one end node to another. For our algorithms, two series of runs are performed and compared to the other solutions. The first one with NC using IRR and the second one with NC using BCSD.
Results are presented in Fig. 13. Traditional store and forward algorithm starts with a total of 50,000 transmissions on a reliable network and ends with an increase of 34% when the loss probability reaches 20%.
Total number of transmissions required to deliver 10,000 packets
Also on a reliable network, revised LNC, IRR, and BCSD start with almost the same number of required transmissions to deliver all the packets. When the loss probability increases, the number of transmissions with the revised LNC increases about 56% when the loss probability reaches 20%. Both NC with IRR and NC with BCSD show an important saving in the number of transmissions compared to the revised LNC. With a loss probability of 20%, NC with BCSD represent a saving of 22% in the number of transmissions over the revised LNC. IRR lightly increases the number of transmissions compared to BCSD.
Figures 14 and 15 show, respectively, the total delivery time and the average delivery time for the three competing mechanisms. The total delivery time is the time elapsed between the transmission of the first packet and the reception of the last one while the delivery time of a packet is computed as the time between the first transmission of the packet and its delivery. In these simulations, each transmission from one node to an adjacent one requires 1 ms of time. As shown in Fig. 15, the revised LNC overcomes the proposed mechanisms when it comes to average delivery time. This is justified by the fact that no retransmission is requested. With the revised LNC, the slow increase in the average delivery time is due to the time needed at the receiver to gather enough messages of each block in order to decode the packets of the block. As for our proposed mechanisms, IRR bypasses BCSD when it comes to average delivery time, since IRR reacts immediately when a loss is detected while BCSD waits longer to gather undecoded messages and reduces the size of the covering set; thus, the number of retransmissions.
Total delivery time of the 10,000 packets exchanged
Average delivery time
This paper deals with the impact of packet loss in NC. First of all, a new decision model has been presented that prioritizes the tasks that should be performed by each node of the network. The model balances the receiving and the sending queues of the node and manages to guarantee QoS by releasing late packets to destination. The new model has been used to study packet loss effect on the decoding process at end nodes. Simulations have shown that with the proposed model and with the use of aging, any perturbation caused by losing packets is finite in time and the receiving nodes receive a limited number of undecoded packets. The number of undecoded packets at end nodes has been analytically and experimentally studied under the effect of aging and NC activities and two new mechanisms have been proposed to improve NC performance in lossy networks. The first mechanism, called IRR, immediately requests lost packets when a packet loss occurs and the second mechanism, called BCSD, requests missing packets in a reduced number of transmissions. Simulation results have shown that in typical cases, both algorithms help reduce by 22% the number of transmissions compared to the revised LNC protocol. Finally, it should be noted that IRR and the revised LNC are more suitable to some applications like video conferencing where the average delivery time is needed to be reduced while BCSD is more convenient to file exchange where the total delivery time needs to be optimized.
ARQ:
Automatic repeat request
BCSD:
Basic covering set discovery
FEC:
IRR:
LNC:
Linear network coding
NC:
Network coding
QoS:
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Lebanese International University, Beirut, Lebanon
Samih Abdul-Nabi
& Ayman Khalil
INSA de Rennes, IETR UMR CNRS, Rennes, 6164, France
, Philippe Mary
& Jean-François Hélard
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Correspondence to Samih Abdul-Nabi.
Decision model | CommonCrawl |
Vibrational Motion
You have a pendulum pulled to the side, a heavy beach ball pushed partly under water, and a medicine ball lifted some height above the ground. Each object is then released. (a) Draw a picture of the motion of each object and a motion diagram for each. (b) Indicate the equilibrium position. (c) Draw two or more force diagrams at key points in the motion. (d) Use (a) through (c) to reason whether the motions of these objects can be considered vibrational motions.
Dading C.
Consider the three objects described in Problem 1. (a) Choose a system for each and construct energy bar charts for each motion. What initial and final states did you choose? (b) What assumptions did you make?
Exercise stretch cord You want to determine the spring constant of an exercise stretch cord. You pull the cord with a force probe that exerts a 50-N force on the cord, causing it to stretch 20 cm. (a) What is the spring constant of the cord? Describe your reasoning and assumptions. (b) How can you test your answer?
Winnie W.
Two exercise cords In order to increase resistance, you put two exercise cords together as shown in Figure P19.4. What is the effective spring constant of the two bands compared to one? Use data from
Problem 3. Explain.
You have a ball bearing and a bowl. You let the ball roll down from the top of the bowl; it moves up and down the bowl's wall for some time. (a) Is this vibrational motion? Explain why or why not. (b) Draw a force diagram for the ball for four different positions and indicate what force or force component is responsible for the acceleration toward the equilibrium position. (c) Draw energy bar charts for these four ball-Earth system positions.
(a) Draw a sketch of an object attached to a vertical spring. Indicate the equilibrium position. (b) Show this object vibrating up and down and indicate the amplitude of vibrations on the sketch. (c) Describe an experiment you would perform to determine the period of vibration of the object. (d) Perform the experiment. Include experimental uncertainty in your result.
Draw a sketch of a pendulum. Indicate the equilibrium position. Show this object vibrating and indicate the amplitude of vibration on the sketch.
Concert $A$ and $O_{2}$ vibration (a) Musicians in an orchestra tune their instruments to what is called "concert $A, "$ a frequency of 440 $\mathrm{Hz}$ . Determine the period for one vibration. (b) The atoms in an oxygen molecule complete one vibration in a time interval of $2.11 \times 10^{-14}$ s. Determine the frequency of vibration of $\mathrm{O}_{2}$ .
Bl0 Hearing range A doctor is checking your hearing. (a) If the period of the lowest-frequency sound you can hear is 0.050 $\mathrm{s}$ , then what is its frequency? (b) If the highest-frequency sound you can hear is $20,000 \mathrm{Hz}$ , then what is its period?
Draw a graph showing the position-versus-time curve for a simple harmonic oscillator (a) with twice the frequency of that shown in Figure P19.10 and (b) with the same frequency but twice the amplitude as shown in the figure.
Suppose that at time zero the cart attached to the spring such as shown in Figure 19.8 is released from rest at position $x=+A$ and that its period of vibration is 8.0 $\mathrm{s}$ . Draw nine sketches showing the cart's approximate position each second starting at 0 s and ending at 8.0 s. Draw and label arrows indicating the relative velocity and acceleration of the cart at each position.
(a) Sketch a motion diagram and a position-versus-time graph for the motion of a cart attached to a spring during one period. It passes at high speed through the equilibrium position at time zero. (b) Sketch a velocity-versus-time graph for the cart. (c) Sketch an acceleration-versus-time graph for the
cart. (d) Draw another representation of the process; explain how this representation allows you to learn more about the process than the kinematics graphs do.
The motion of a cart is described as $x=$ $(0.17 \mathrm{m}) \sin \left(\pi s^{-1}\right) t .$ Say everything you can about this motion and represent it using graphs, motion diagrams, and energy bar charts.
Sketch an acceleration-versus-time graph for the simple harmonic motion of an object of your choice. Indicate the amplitude of the acceleration and the period of the vibration. Underneath the graph draw corresponding position-versus- time and velocity-versus-time graphs with the correct shape but not necessarily the correct amplitudes.
Devise a position-versus-time function that describes the simple harmonic motion of an object of your choice. Choose the amplitude and the period of vibration, and draw corresponding velocity-versus-time and acceleration-versus time graphs. Draw a motion diagram for one period of its motion.
* The position of a vibrating object changes as a function of time as $x=(0.2 \mathrm{m}) \cos \left(\pi s^{-1}\right) t$ . Say everything you can about this motion. Write an expression for the velocity and acceleration as functions of time. Draw graphs for each function.
The position of a vibrating object changes as a function of of time as $v=-(0.6 \mathrm{m} / \mathrm{s}) \cos (2 \pi) .$ (a) Say everything you can about the motion. Draw a sketch of the situation when the motion starts. (b) Represent the motion with a position- versus-time graph and a work-energy bar chart.
$*$ A cart at the end of a spring undergoes simple harmonic motion of amplitude 10 $\mathrm{cm}$ and frequency 5.0 $\mathrm{Hz}$ . (a) Determine the period of vibration. (b) Write an expression for the cart's position at different times, assuming it is at $x=-A$ when $t=0 .$ Determine its position $(\mathrm{c})$ at 0.050 $\mathrm{s}$ and (d) at 0.100 $\mathrm{s}$ .
You exert a $100-\mathrm{N}$ pull on the end of a spring. When you in- crease the force by 20$\%$ to $120 \mathrm{N},$ the spring's length increases 5.0 $\mathrm{cm}$ beyond its original stretched position. What is the spring constant of the spring and its original displacement?
Metronome You want to make a metronome for music practice. You use a $30-\mathrm{g}$ object attached to a spring to serve as the time standard. What is the desired spring constant of the spring if the object needs to make 1.00 vibrations each second? What are the assumptions that you made?
Determine the frequency of vibration of the cart shown in Figure P19.21.
$*$ A spring with a cart at its end vibrates at frequency 6.0 $\mathrm{Hz}$ . (a) Determine the period of vibration. (b) Determine the frequency if the cart's mass is doubled while the spring constant remains unchanged and (c) the frequency if the spring constant doubles while the cart's mass remains the same.
A cart with mass $m$ vibrating at the end of a spring has an extra block added to it when its displacement is $x=+A .$ What should the block's mass be in order to reduce the frequency to half its initial value?
$*$ A $2.0-\mathrm{kg}$ cart vibrates at the end of an $18-\mathrm{N} / \mathrm{m}$ spring. (a) Make a list of physical quantities you can determine about the vibrations and determine two of them. (b) If a second $18-\mathrm{N} / \mathrm{m}$ spring is attached beside the first one, what will be the period
of the vibration?
$*$ What were the main ideas that we used to derive the expression for the period of an object vibrating at the end of a spring? Explain the assumptions that we made. When is the expression for the period not valid for a spring of known $k$ ?
$*$ A spring with a spring constant of 1200 $\mathrm{N} / \mathrm{m}$ has a $55-\mathrm{g}$ ball
at its end. (a) If the energy of the system is $6.0 \mathrm{J},$ what is the amplitude of vibration? (b) What is the maximum speed of the ball? (c) What is the speed when the ball is at a position $x=+A / 2 ?$ What assumptions did you make to solve the problem? How do you know if your answer makes sense?
A person exerts a $15-\mathrm{N}$ force on a cart attached to a spring and holds the cart steady. The cart is displaced 0.060 $\mathrm{m}$ from its equilibrium position. When the person stops holding
the cart, the system cart + spring undergoes simple harmonic motion. (a) Determine the spring constant of the spring. (b) Determine the energy of the system. (c) Write an expression $x(t)$ for the motion of the cart. (d) Draw as many graphical representations of the motion as you can.
$*$ A spring with spring constant $2.5 \times 10^{4} \mathrm{N} / \mathrm{m}$ has a $1.4-\mathrm{kg}$ cart at its end. (a) If its amplitude of vibration is $0.030 \mathrm{m},$ what is the total energy of the cart $+$ spring system? (b) What is the maximum speed of the cart? (c) If the energy is tripled, what is the new amplitude? (d) What is the maximum speed of the cart? (e) What assumptions did you make to solve the problem? If the assumptions were not reasonable, how would the answers change?
$*$ Proportional reasoning By what factor must we increase the amplitude of vibration of an object at the end of a spring in order to double its maximum speed during a vibration? Explain.
Proportional reasoning By what factor must we increase the amplitude of vibration of an object at the end of a spring in order to double the total energy of the system? Explain. What will happen to the speed the object? What assumptions did you make?
Monkey trick at zoo A monkey has a cart with a horizontal spring attached to it that she uses for different tricks. In one trick, the monkey sits on the vibrating cart. When the cart reaches its maximum displacement from equilibrium, the monkey picks up a 0.30-kg cantaloupe from a trainer. The mass of the monkey and the cart together is 3.0 kg. The spring constant is 660 N/m. The amplitude of horizontal vibrations is 0.24 m. Determine the ratio of the maximum speed of the monkey before and after she picks up the cantaloupe.
A cart attached to a spring vibrates with amplitude $A$ (a) What fraction of the total energy of the cart spring system is elastic potential energy and what fraction is kinetic energy when the cart is at position $x=A / 2 ?$ (b) At what position is the cart when its kinetic energy equals its elastic potential energy?
A 2.0 -kg cart attached to a spring undergoes simple harmonic motion so that its displacement is described by $x=(0.20 \mathrm{m}) \sin [(2 \pi / 2.0 \mathrm{s}) t] .$ Construct energy bar charts for the cart-spring system at times $t=0, t=T / 4$ $t=T / 2, t=3 T / 4,$ and $t=T$
$* /$ Equation Jeopardy The following expression describes a situation involving vibrational motion. Sketch a process and devise a problem for which the expression might be an answer.
\begin{aligned} \frac{1}{2}(20,000 \mathrm{N} / \mathrm{m})(0.20 \mathrm{m})^{2}=& \frac{1}{2}(100 \mathrm{kg}) v^{2} \\ &+\frac{1}{2}(20,000 \mathrm{N} / \mathrm{m})(0.10 \mathrm{m})^{2} \end{aligned}
Pendulum clock Shawn wants to build a clock whose pendulum makes one swing back and forth each second. (a) What is the desired length of the rod (assumed to have negligible mass) holding the metal ball at its end? (b) Will the rod need to be shorter or longer if he includes the mass of the rod in the
calculations? Explain.
Show that the expression for the frequency of a pendulum as a function of its length is dimensionally correct.
A pendulum swings with amplitude 0.020 m and period of 2.0 s. What is its maximum speed?
$*$ Proportional reasoning You are designing a pendulum clock whose period can be adjusted by 10$\%$ by changing the length of the pendulum. By what percent must you be able to change the length to provide this flexibility in the period? Explain.
Building demolition $A 500$ -kg ball at the end of a $30-m$ cable suspended from a crane is used to demolish an old building. If the ball has an initial angular displacement of $15^{\circ}$ from the vertical, determine its speed at the bottom of the arc.
$*$ You have a pendulum with a long string whose length you can vary but cannot measure, a small ball, and a stopwatch. Describe two experiments that you can design to determine the height of a ladder. Indicate the assumptions that you use in each method.
" Variations in $g$ The frequency of a person's pendulum is 0.3204 $\mathrm{Hz}$ when at a location where $g$ is known to be exactly 9.800 $\mathrm{m} / \mathrm{s}^{2} .$ Where might the same pendulum be when its frequency is 0.3196 $\mathrm{Hz}$ ? What is $g$ at that location?
A graph of position versus time for an object undergoing simple harmonic motion is shown in Figure
P19.42. Estimate from the graph the amplitude and period of the motion and determine the object's frequency. If the object is a pendulum, what is its length?
Determine the period of a 1.3-m-long pendulum on the Moon.
$*$ Trampoline vibration When a $60-\mathrm{kg}$ boy sits at rest on a trampoline, it sags 0.10 $\mathrm{m}$ at the center. (a) What is the effective spring constant for the trampoline? (b) The trampoline is pulled downward an extra 0.050 $\mathrm{m}$ by a strap sewed under the center of the trampoline. When the strap is released, what are the energy and frequency of the trampoline? What assumptions did you make?
$*$ A 1.2 -kg block sliding at 6.0 $\mathrm{m} / \mathrm{s}$ on a frictionless surface runs into and sticks to a spring. The spring is compressed 0.10 $\mathrm{m}$ before stopping the block and starting its motion back in the opposite direction. What can you determine about the vibrations that start after the collision? Make a list of physical quantities and determine four of them.
$*$ Proportional reasoning If you double the amplitude of vi- bration of an object at the end of a spring, how does this affect the values of $k, T, U,$ and $v_{\max } ?$
$*$ EST Willis Tower vibration The mass of the Willis (formerly Sears) Tower in Chicago is about $2 \times 10^{8} \mathrm{kg}$ . The tower sways back and forth at a frequency of about 0.10 $\mathrm{Hz}$ . (a) Estimate the effective spring constant for this swaying motion. Explain why you included a particular number of significant figures. (b) A gust of wind hitting the building exerts a
force of about $4 \times 10^{6} \mathrm{N}$ . By approximately how much is the top of the building displaced by the wind? Explain why you included a particular number of significant figures. State the
assumptions that you made. Does your make sense?
* EST BIO Annoying sound Low-frequency vibrations (less than 5 $\mathrm{Hz}$ ) are annoying to humans if the product of the amplitude and the frequency squared $\left(A f^{2}\right)$ equals
$0.5 \times 10^{-2} \mathrm{m} \cdot \mathrm{s}^{-2}(\text { or more). Estimate the frequency and }$
amplitude of a buzzer that produces these annoying vibrations if the device vibrates a total mass of 0.12 $\mathrm{kg}$ and has a vibrational energy of 0.012 $\mathrm{J}$ .
$* *$ You shoot a 0.050 -kg arrow into a 0.50 -kg wooden cart that sits on a horizontal, frictionless surface at the end of a spring that is attached to the wall at the other end. The arrow hits the cart and sticks into it. The cart and arrow together compress the spring and start the system vibrating at a frequency of 2.0 Hz with a $0.20-$ -m amplitude. How fast was the arrow moving? State any assumptions you made to solve the problem.
$*$ Pendulum on Mars The frequency of a pendulum is 39$\%$ less when on Mars than when on Earth's surface. Use this fact to determine Mars's gravitational constant.
* This chapter stated that when damping is present, the period of vibration of a system is more than without damping. How can you test this assertion? Provide the details.
$*$ You have a pendulum whose length is 1.3 $\mathrm{m}$ and bob mass is 0.20 $\mathrm{kg}$ . The amplitude of vibration of the pendulum is 0.07 $\mathrm{m}$ . (a) Determine the maximum energy of the bob-Earth system. Explain why you included a particular number of significant figures. (b) How much mechanical energy is converted to internal energy before the pendulum stops?
You have a $0.10-\mathrm{kg}$ cart on a spring. The spring constant of the spring is 20 $\mathrm{N} / \mathrm{m}$ . The cart's initial vibration amplitude is 0.10 $\mathrm{m}$ . ( a Make a list of physical quantities you can determine using this information and determine three of them. (b) For approximately how long will the vibrations last if 10$\%$ of the mechanical energy during each cycle is converted into
internal energy during each cycle?
You have a spring that stretches 0.070 $\mathrm{m}$ when a 0.10 -kg block is attached to and hangs from it. Imagine that you slowly pull down with a spring scale so the block is now below the
equilibrium position where it was hanging at rest. The scale reading when you let go of the block is 3.0 $\mathrm{N}$ . (a) Where was the block when you let go? (b) Determine the work you did stretching the spring. (c) What was the energy of the spring- Earth system when you let go? (d) How far will the block rise after you release it? (e) The vibrations last for 50 cycles. Qualitatively represent the beginning of cycle $\# 1$ and beginning of cycle $\# 25$ with an energy bar chart.
Imagine that you have a cart on a spring that moves on a rough surface. (a) Represent the cart's motion with a motion diagram for one period. (b) Draw force diagrams for each quarter of a period. (c) Draw energy bar charts for each quarter of a period. (d) Draw position-, velocity-, and acceleration-versus-time graphs for each period. (e) On these graphs, use a dashed line to sketch graphs for the same cart
on the same spring, assuming no friction between the cart and the surface.
Twins on a swing How frequently do you need to push a swing with twin brothers on it compared to when pushing the swing with one on it? What assumptions did you make?
$*$ (a) Determine the maximum speed of a girl on a $3.0-\mathrm{m}$ -long swing when the amplitude of vibration is 1.2 $\mathrm{m} .$ (b) What assumptions did you make? Is the child's vibrational motion
damped? Explain. (c) Under what conditions is the motion a forced vibration? Explain.
You have a 0.20 -kg block on a $10-\mathrm{N} / \mathrm{m}$ spring that oscillates up and down. How often do you need to push the block upward when it passes the bottom of its motion to increase the
amplitude of its vibrations?
Sloshing water You carry a bucket with water that has a natural sloshing period of 1.7 s. At what walking speed will water splash if your step is 0.90 $\mathrm{m}$ long?
$*$ Feeling road vibrations in a car If the average distance between bumps on a road is about 10 $\mathrm{m}$ and the natural frequency of the suspension system in the car is about 0.90 $\mathrm{Hz}$ , at what speed will you feel the bumps the most?
$*$ EST H atom vibration A hydrogen atom of mass $1.67 \times 10^{-27} \mathrm{kg}$ is attached to a very large protein by a bond that behaves much like a spring. (a) If the vibrational frequency of the hydrogen is $1.0 \times 10^{14} \mathrm{Hz}$ , what is the "effective" spring constant of this spring-like bond? (b) If the total vibrational energy is $k T(k \text { is Boltzmann's constant and } T \text { is }$ the temperature in kelvins), approximately what is the classical amplitude of vibration at room temperature $(T=300 \mathrm{K})$ ? By comparison, the diameter of a hydrogen atom is about $10^{-10} \mathrm{m} .$ State any assumptions that you made.
Child's bouncy chair You are designing a bouncy chair for a neighbor's child. A 1.0 -kg chair hangs from a spring that is attached to the ceiling. When you put the $10.0-\mathrm{kg}$ child in the chair, the system vibrates with a period of 3.0 $\mathrm{s}$ . What would be the period if the child's mother (mass 50 $\mathrm{kg} )$ sits on the same chair? What assumptions did you make?
$* /$ You attach a block (mass $m )$ to a spring (spring constant $k )$ that oscillates in a vertical direction. Determine an expression for the period of its vibrations. Sketch a position-versus-time graph for the vibrations, assuming that the motion starts when the block is at its lowest maximum displacement from the equilibrium position. What assumptions did youmake?
$*$ You attach a $1.6-\mathrm{kg}$ object to a spring, pull it down 0.12 $\mathrm{m}$ from the equilibrium position, and let it vibrate. You find that it takes 5.0 $\mathrm{s}$ for the object to complete 10 vibrations. Make a list of physical quantities that you can determine about the motion of the object and determine five of them.
$*$ Traveling through Earth A hole is drilled through the center of Earth. The gravitational force exerted by Earth on an object of mass $m$ as it goes through the hole is $m g(r / R)$ , where $r$ is the distance of the object from Earth's center and $R$ is the radius of Earth $\left(6.4 \times 10^{6} \mathrm{m}\right) .$ Will an object dropped into the hole execute simple harmonic motion? If yes, find the period of the motion. How does one-half this time compare with the time needed to fly in an airplane halfway around Earth? What assumptions did you make to solve the problem?
EST Estimate the effective spring constant of the suspension system of a car. Describe your technique carefully. How can you test your answer to determine if it makes sense?
* Use dimensional analysis to show that the expressions for the periods of an object attached to a spring and for a simple pendulum are reasonable.
* BIO Vibration amplitude in ear The weakest sound a human ear can possibly hear makes the ear vibrate with the energy of about $10^{-19} \mathrm{J}$ . If the spring constant of the ear is $20 \mathrm{N} / \mathrm{m},$ what is the amplitude of vibration of the eardrum? What assumptions did you make? How does this compare to the size of an atom (about $0.5 \times 10^{-10} \mathrm{m} ) ?$
$* *$ A 5.0 -g bullet traveling horizontally at an unknown speed hits and embeds itself in a $0.195-\mathrm{kg}$ block resting on a frictionless table. The block slides into and compresses a $180-\mathrm{N} / \mathrm{m}$ spring a distance of 0.10 $\mathrm{m}$ before stopping the block and bullet. Determine the initial speed of the bullet. State any assumptions that you made to solve the problem.
$* *$ You have a pendulum of mass $m$ and length $L$ that is displaced an angle $\theta$ at the start of the swinging. (a) Determine an expression for the energy of the bob-Earth system. (b) Determine an expression for the maximum vertical height of the bob with respect to the equilibrium position.
(c) Determine an expression for the maximum speed of the bob. (d) What assumptions did you make? (e) Discuss how the answer to each of the questions changes if relevant assumptions are not valid.
A pendulum clock works for many days before the swinging stops. Describe several mechanisms that allow an actual pendulum clock to continue working without decreasing amplitude. Write a report explaining how it works.
$* *$ Foucault's pendulum In 1851 , the French physicist Jean Foucault hung a large iron ball on a wire about 67 $\mathrm{m}(220 \mathrm{ft})$ long to show that Earth rotates. The pendulum appears to
continuously change the plane in which it swings as time elapses. Determine the swinging frequency of this pendulum. Explain why the behavior of Foucault's pendulum provides supporting evidence for the hypothesis pendulum provides ertial reference frame and helps reject the model of a geocentric universe (Earth at the center).
You push down on a raft floating on a lake. The raft sinks and then vibrates up and down for a short time. How does the period of its vibrations depend on the size and mass of the raft?
Which expression below represents the mass $m$ of Earth inside a sphere of radius $r$ smaller than the radius $R$ of Earth? Note that $\rho$ is the density of Earth, assumed uniform.
\begin{array}{ll}{\text { (a) }(4 / 3) \pi r^{3} \rho} & {\text { (b) } G(4 / 3) \pi r^{3} \rho} \\ {\text { (c) } G M_{\text { Earth }} / r^{2}} & {\text { (d) } G(4 / 3) \pi r \rho} \\ {\text { (e) }m g} \end{array}
Which expression below represents the magnitude of the re- storing force that Earth exerts on an object of mass $m$ when a distance $r$ from the center of Earth? Note that $\rho$ is the density of Earth, assumed uniform.
\begin{array}{ll}{\text { (a) } m(4 / 3) \pi r^{3} \rho} & {\text { (b) } m G(4 / 3) \pi r^{3} \rho} \\ {\text { (c) } G M_{\mathrm{E}} m / r^{2}} & {\text { (d) } m G(4 / 3) \pi r \rho} \\ {\text { (e) } m g r}\end{array}
Which expression below represents the period $T$ for one oscillation from Earth's surface through the center, to the other side, and then back again?
$\begin{array}{ll}{\text { (a) } 2 \pi(3 / G 4 \pi \rho)^{1 / 2}} & {\text { (b) } 2 \pi(G 4 \pi \rho / 3)^{1 / 2}} \\ {\text { (c) } 2 \pi(3 m / G 4 \pi \rho)^{1 / 2}} & {\text { (d) } 2 \pi(m G 4 \pi \rho / 3)^{1 / 2}} \\ {\text { (e) } 2 \pi(m / k)^{1 / 2}} \end{array}$
During the trip, when will your acceleration be greatest?
(a) At the beginning and end of the trip
(b) At the end of the trip
(c) When passing through the center of Earth
(d) The same for entire the trip
During the trip, when will your speed be greatest?
(a) At the beginning and ends of the trip
If the radius of Earth is $6.4 \times 10^{\circ} \mathrm{m},$ its mass is $6.0 \times 10^{24} \mathrm{kg},$ and $G=6.67 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2},$ which answer is closest to the time interval for one trip through Earth?
$$\begin{array}{llll}{\text { (a) } 0.7 \mathrm{h}} & {\text { (b) } 1.4 \mathrm{h}} & {\text { (c) } 3.2 \mathrm{h}}\\{\text { (d) } 4.8 $\mathrm{h}}&{\text { (e) } 6.3 $\mathrm{h}}\end{array}$$
What causes vibrations in the fluid in the cochlea?
(a) The eardrum pushes against the cochlea.
(b) Sound waves push against the cochlea.
(c) A bone in the middle ear pushes against the cochlea.
(d) The fluid vibrates when something outside the ear vibrates.
(e) The fluid does not vibrate—the hair cells vibrate.
How does the basilar membrane distinguish different frequency vibrations?
(a) The hair cells have different lengths.
(b) The dimensions and stiffness of the cochlea vary along its length.
(c) The basilar membrane vibrates where the cochlea dimension and stiffness match the vibration frequency.
(d) The cochlea fluid resonates in only one part of the cochlea.
(e) All the hair cells bend back and forth at only the frequency of the vibration.
If you shake the board shown in Figure 19.20 at a frequency higher than the natural frequency of the rod on the right, then what happens?
(a) None of the rods vibrate.
(b) All of the rods vibrate.
(c) The shortest rod vibrates.
(d) The longest rod vibrates.
(e) The middle rods vibrate.
If you were to shake the special board (the one that has 15,000 rods of varying length) at one particular frequency, then what would happen?
(a) None of the rods would vibrate.
(b) All of the rods would vibrate.
(c) A small number of rods at one location would vibrate.
(d) A disturbance would travel back and forth along the board.
You hang four pendulum bobs from strings connected to a wooden dowel. The strings are different lengths. How can you get the second longest pendulum bob to vibrate while the other three do not—without touching the pendulums?
(a) Shake the dowel back and forth.
(b) Shake the dowel back and forth at the resonant frequency of that pendulum.
(c) Move the dowel sideways at any frequency.
(d) Blow air on that bob. | CommonCrawl |
Theorists vs multi-muons
There has not been much talking recently about the CDF multi-muon anomaly. Unlike the PAMELA/ATIC cosmic-ray anomaly, the CDF one did not trigger a lot of theoretical activity. There is more than one reason for this shroud of silence. On one hand, even though it is possible to write an ad-hoc particle models that describe various characteristics of the multi-muon signal, it seems hopeless to fit that in a bigger picture. On the other hand, multiple members of the CDF collaboration refer to the multi-muon publication as "that crap" (when being polite), while those who signed it admit the fact with certain embarrassment. Besides, the main author of the analysis is, hmm, a controversial figure, which does not help either (to understand the context, see Tommaso's account of the superjets saga).
Nevertheless, there is always a possibility that the multi-muon anomaly signals genuine new physics rather than mice in the detector, and few theorists try their luck. Today there was a paper on arXiv that sheds some light on the possible production mechanism of the mysterious ghost particles. As explained earlier, the multi-muon signal can be a result of a pair of "ghost" scalar particles with the mass around 15 GeV cascade-decaying into four tau leptons each. But the question how these ghosts particles are produced in the first place was not addressed in the original publications. It turns out that a vialable possibility is to couple the ghost field $\phi$ to the Standard Model quarks q via higher-dimensional operators. The non-renormalizable dimension-5 operator:
$\frac{1}{\Lambda} (\bar q q) \phi^2$
provides a pretty good fit to the invariant mass distribution of the ghost muons, see the plot.
Dimension-six operators involving the ghost fields coupled to quarks or gluons are disfavored.
One can think of this dimension-5 operator as an effective interaction left after integrating out a heavier particle with renormalizable interactions (in analogy to the Fermi theory of weak interactions after integrating out the W boson). For example, what would do here is a heavy doublet field $H_q$ (but not the Higgs!) interacting with the quarks via $Q u H_q$ and with the ghost pair via $H \phi^2$. But there is a tension here. The cross-section for the ghost pair production is required to be very large for the particle physics standard: 200 picobarns or so. To match that, the scale $\Lambda$ suppressing the dimension-5 operator has to be as low as 200 GeV. In consequence, the integrated-out particle cannot be too heavy and there is a danger that it violates some of the known experimental bounds. In particular, it could generate other higher-dimension effective operators, like the four-quark operator $(q q)^2/\Lambda^2$ that would affect dijet distributions at the Tevatron. Surprisingly, unlike four-lepton operators that were extremely well constrained by LEP, there is no strong bounds in the literature on four-quark effective operators (except for the bound on $(Q \gamma_\mu Q)^2/2\Lambda^2$ which is $\Lambda > 700$ GeV, but that's not directly applicable here). Improving the bounds on four-quark operators could clarify the situation and, in fact, would be extremely interesting for many other applications.
As a CDF'er I think the common opinion is that the effect is very real - which has been verified by countless other experimenters - but that the new physics interpretation is premature at best and downright silly at worst. Comments like "that crap" are either referring to the written paper - which I would agree with - or are an emotional reaction based on the person performing the analysis.
I do agree that no theorist should be spending time on this though. It's an interesting experimental problem which happened to be noticed by a very polarizing person.
Anonymous2 said...
hi Anonymous, do you have a SM interpretation in terms of some detector effect?
Given that theorists are spending time on "LHC olympics", working on the muon anomaly at worst is a "CDF olympics".
Underestimating hadronic punch-through.
Dr Jekyll and Mr Higgs
What's up at Susy's?
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Recent progress and prospects of topological quantum material-based photodetectors
ZHANG Xing-chao, PAN Rui, HAN Jia-yue, DONG Xiang, WANG Jun
Progress on two-dimensional quantum sheets and their optics
CHEN Zhe-xue, WANG Wei-biao, LIANG Cheng, ZHANG Yong
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A Novel Methane and Hydrogen sensor with Surface Plasmon Resonance-Based Photonic Quasi-crystal Fiber
LIU Qiang, ZHAO Jin, SUN Yudan, LIU Wei, WANG Jianxin, LIU Chao, LV Jingwei, WANG Shimiao, JIANG Yu, CHU Paul K
, Available online , doi: 10.37188/CO.2022-0025
Abstract(7) FullText HTML(1) PDF 4148KB(5)
A novel photonic quasi-crystal fiber (PQF) sensor based on surface plasmon resonance (SPR) is designed for simultaneous detection of methane and hydrogen. In the sensor, Pd-WO3 and cryptophane E doped polysiloxane films deposited on silver films are the hydrogen and methane sensing materials, respectively. The PQF-SPR sensor is analyzed numerically by the full-vector finite element method and excellent sensing performance is demonstrated. The maximum and average hydrogen sensitivities are 0.8 nm/% and 0.65 nm/% in the concentration range of 0% to 3.5% and the maximum and average methane sensitivities are 10 nm/% and 8.81 nm/% in the range between 0% and 3.5%. The sensor provides the capability of detecting multiple gases and has large potential in device miniaturization and remote monitoring.
High precision structural light scanning viewpoint planning for aircraft blade morphology
LI Mao-yue, CAI Dong-chen, ZHAO Wei-xiang, XIAO Gui-feng
The machining quality and detection accuracy of aero-engine blades have a very important influence on the service life of blades. Aiming at the problem of improving the accuracy of blade detection, a high-precision scanning viewpoint planning method based on structured light was proposed in this paper. Firstly, the coarse model data were obtained by coarse scanning under the overall size of the blade, and the field of view was determined according to the camera resolution and acquisition accuracy. Secondly, the improved Angle Criterion algorithm was used to extract the boundary, and the boundary segmentation points were determined according to the boundary coordinates and the range of visual field. The coarse model was sliced by the section line method of surface, and the internal segmentation points were determined according to the slice results, so as to complete the uniform segmentation of point clouds. Then, a directed bounding box was established for the segmented point cloud data to obtain the coordinates of the center point, and the normal vector was statistically analyzed to determine the orientation of the main normal, so as to generate the viewpoint coordinates of high-precision scanning. Finally, the surface morphology of the blade was tested and verified. The experimental results show that the average standard deviation of the proposed method is reduced by 0.0054mm and the collected viewpoint is reduced by 1/3 compared with the viewpoint acquisition result of the supervoxel segmentation, which has a good application prospect in machining inspection of thin-walled blades.
SSFM-global-error-local-energy method for improving computational efficiency of passively mode-locked fiber laser
YAN Run-bin, HE Xiao-ying, ZHANG Chuan, ZHANG Yin-dong, RAO Lan
, Available online , doi: 10.37188/CO.EN.2022-0016
Abstract(28) FullText HTML(16) PDF 4568KB(27)
We propose a method for improving the computational efficiency of passively mode-locked fiber laser, which is composed by symmetric split-step Fourier method (SSFM) and the global-error-local-energy (GELE) method for solving propagating equations. Our proposed method relies on the limitation of local energy increment related with global error within a certain value to control the selection of step size. This method has advantage of an automatic step adjustment mechanism. To achieve the same order of computation accuracy, the computational time of our method is 255 s, while SSFM with small constant step size method needs to calculate 3855 s. The computational time of our proposed method is one or two orders of magnitude less than that of the SSFM, which indicates our method can enhance the computational efficiency by a factor up to 10. It could be expanded with high-order algorithms, such as RK4IP, Adams, predictor–corrector, etc. for improving the accuracy.
Research on monocular camera edge spectrum based improved algorithm for ranging by defocused images
JIE Deng-fei, WANG Hao, LV Hui-fang, TIAN Bo-tao, ZHANG Zhan-xiang
In order to achieve accurate target ranging of weak or surface-free texture features based on a monocular camera, an improved difocus image ranging algorithm based on preserving edge spectral information is presented. By comparing two classical disfocal ranging theories with Fourier transform and Laplace transform as the calculation core, the corresponding definition evaluation function is constructed, select the method based on spectrum definition function with better sensitivity, and select the calculation range of frequency domain by retaining the information on the target edge. To verify the feasibility of the algorithm, 6 sets of different duck eggs samples were used to obtain scattered focus images of different aperture and different distances, and use the improved algorithm to solve the distance of duck eggs from the camera lens. The experimental results show that the improved algorithm based on the edge spectrum preservation has good ranging effect, with a correlation coefficient of 0.986 and Root mean square error (RMSE) of 11.39 mm, and it is found that the range ability can be effectively improved after the image rotation processing of the duck egg image taken at an oblique Angle, the root mean square error 11.39 mm to 8.76 mm, from 2.85% to 2.28% and the correlation coefficient to 0.99. It basically meets the requirements of stability and high accuracy of target ranging with weak or no surface texture features.
Output characteristics of all-fiber laser with 2 μm MOPA structure
WU Ling, LOU Yan, HOU Xin-yi, LI Bao-qun, LI Yong-liang, WANG Tian-shu, ZHAO Yi-wu
Abstract(19) FullText HTML(5) PDF 4243KB(24)
In order to improve the output performance of a high-power thulium-doped fiber laser (TDFL) and increase the optical-optical conversion efficiency of the system, a high-power TDFL with an all-fiber main oscillation power amplification structure (MOPA) was developed, which can operate in both continuous (CW) and quasi-continuous (QCW) modes. First, a laser oscillator was built to study the output characteristics of the seed source laser. Then, a thulium-doped fiber amplifier is built and connected to the laser oscillator to study the output characteristics of the MOPA structured fiber laser. Finally, the pulse characteristics of the MOPA structured fiber laser are analyzed under QCW modulation mode. The laser oscillator achieved a continuous and stable laser output with a central wavelength of 1940 nm, and the highest average output power was 18.56 W. The slope efficiency is 54.84%, and the spectrum was free of Raman components. Using this low-power continuous laser as the seed source through the homemade thulium-doped fiber amplifier, the average output power can reach 66.9 W, and the slope efficiency is 48.48%. When the system is operated in QCW mode, the frequency and duty cycle can be adjusted, and the peak power is calculated to be 80.3 W when the frequency is 75 Hz and the duty cycle is 10%. This research proposal is of reference significance for the development of higher power MOPA lasers in the 2 μm band.
Research progress on high-resolution imaging system for optical remote sensing in aerospace
SU Yun, GE Jing-jing, WANG Ye-chao, WANG Le-ran, WANG Yu, ZHENG Zi-xi, SHAO Xiao-peng
Abstract(214) FullText HTML(143) PDF 9744KB(222)
With the continuous development of optical imaging technology and the growing demand for remote sensing applications, cross-scale high-resolution optical technology has been widely used in the field of remote sensing. In order to obtain more detailed information on the target, domestic and foreign researchers have carried out relevant research in different technical directions. In this paper, through the technical classification of remote sensing imaging, we introduce a representative aerospace optical remote sensing high-resolution imaging system. It focuses on monomer structure, block expandable imaging, optical interference synthesis aperture imaging, diffraction main mirror imaging, optical synthetic aperture and other technologies. It provides a new idea for the development of high-resolution optical remote sensing loads on the ground.
Theoretical design and preparation of high performance mwir notch filter
SHANG Peng, CHEN Bei-xi, SUN Peng, LIU Hua-song, BAI Jin-lin, JI Yi-qin, CAO Bo, MA Yuan-fei, LIN Quan
, Available online , doi: 10.37188/CO.2022-0193_en
In order to effectively suppress the interference of CO2 radiation from 4.3 μm attachment on 3 μm−5 μm MWiR target signal, based on the Needle random intercalation optimization algorithm, an accurate inversion correction model for the growth error of multi-layer ultra-thick Ge/Al2O3 films under quartz crystal monitoring is established by the electron beam evaporation method, thus realizing the design, the accurate inversion and the accurate preparation of MWiR notch filter; in order to solve the problem that the surface profile of the MWiR notch filter changes greatly, the preset substrate surface method is used to realize the low surface profile regulation of MWiR notch filter. The results show that the high refractive index Ge film has good deposition stability with the increase of coating time, while the deposition scale factor of low refractive index Al2O3 thin film changes up to 11.9% in a regular gradual trend. For the prepared MWiR notch filter, the average cut-off transmittance is <0.3% at 4.2 μm−4.5 μm, and the average transmittances are >95% at 3.5 μm−4.05 μm and 4.7 μm−5.0 μm. The surface profile of the substrate after coating can be effectively controlled in a small range. The film has good adaptability to complex environment, and has successfully passed the environmental test of firmness, high temperature, low temperature and damp heat specified in GJB 2485-95.
Theoretical investigation on super-resolving refractive index measurement with parity detection
WANG Qiang, WANG Qian-qian, WANG Zhen, HAO Li-li
Abstract(164) FullText HTML(35) PDF 4112KB(100)
The refractive index measurements based on traditional wave optical methods are mainly depended on intensity and wavelength detection strategies. Interferometric spectrometers are widely used as the most ideal wavelength detecting devices. Interference spectrometers measure the signal intensity, analyze the change of fringe numbers and the corresponding optical path difference by means of optical power meter, and then calculate the wavelength of signal light. Therefore, its essence is still based on intensity detection. However, the resolution of interference signal with intensity detection is restricted by classical diffraction limit, thus its resolution is difficult to further improve. In order to solve this bottleneck parity detection which could break through the classical resolution limit and realize super-resolving refractive index measurement is proposed in this paper. According to the quantum detection and estimation theory, the expressions for signals and their corresponding sensitivities of refractive index measurement with parity and intensity detections were derived respectively and their numerical comparison analysis was carried out. In addition, the effects of loss on resolution and sensitivity of the output signal were investigated. Numerical results show that the resolution of parity detection is \begin{document}${\text{2{\text{π}} }}\sqrt {\text{N}} $\end{document} times that of intensity detection, achieving super-resolving refractive index measurement. Moreover, the optimal sensitivity reaches the refractive index measurement shot noise limit\begin{document}${\lambda \mathord{\left/ {\vphantom {\lambda {\left( {2{\text{π}} l\sqrt N } \right)}}} \right. } {\left( {2{\text{π}} l\sqrt N } \right)}}$\end{document}. The loss reduces the sensitivity and resolution of the signal. The resolution of the parity detection signal is consistently better than that of intensity detection except for the very large loss and very low photon number. Finally, the physical essence of the super-resolving refractive index measurement is analyzed from the detection means itself.
Recent progress of Non-Line-of-Sight imaging reconstruction algorithm in typical imaging modalities
ZHAO Lu-da, DONG Xiao, XU Shi-long, HU Yi-hua, ZHANG Xin-yuan, ZHONG Yi-cheng
Abstract(48) FullText HTML(41) PDF 10303KB(61)
Non-Line-of-Sight (NLoS) imaging is a promising technique developed in recent years, which can reconstruct hidden scene by analyzing the information in the intermediate surface, and achieve "see around the corner", and has strong application value in many fields. In this paper, we review the reconstruction algorithm for NLoS imaging tasks. Firstly, considering the crossover and non-independent phenomena existing in the NLoS imaging classification, we use the different features of physical imaging model and algorithm model to reclassify it. Secondly, according to the proposed classification criteria, we respectively review the traditional and deep learning-based NLoS imaging reconstruction algorithms, summarize the state-of-the-art algorithms, and push the technical methods. And compared the results of deep learning-based and traditional NLoS imaging reconstruction algorithms for reconstruction tasks. Finally, the current challenges and the future development of NLoS imaging are summarized. Different types of NLoS imaging reconstruction algorithms are comprehensively analyzed in this review, which provides important support for the further development of NLoS imaging reconstruction algorithm.
Effect of atmospheric turbulence on tracking accuracy of high-resolution remote sensing satellites
CAO Zong-xin, CAO Nan, YANG Yan-yan, DING Zhi-ya, MAO Hong-min, PENG Jian-tao, FAN Li-na, LU Huan-jun, SUN Hui-juan, HU Li-fa, CAO Zhao-liang
Atmospheric turbulence affects the tracking and positioning accuracy of high-resolution remote sensing satellites seriously. This paper focuses on the effects of camera aperture, atmospheric turbulence intensity and satellite orbit height on the positioning accuracy. Firstly, we establish the turbulence model and turbulence simulation method based on Kolmogorov turbulence theory for earth observation. Then, the influence of camera aperture, satellite orbit height and atmospheric coherence length on the positioning accuracy of the satellite is simulated and analyzed, and then, the universal formula is deduced to calculate the tilt aberration of turbulence wavefront. Finally, based on this universal formula, the theoretical calculation formula of jitter is derived for earth observation. The research work can provide a theoretical basis of the influence of atmospheric turbulence for design, analysis and evaluation of very high-resolution remote sensing satellites.
Design of vanadium dioxide-assisted switchable multifunctional metamaterial structure
CHEN Xin-yi, YAN De-xian, FENG Qin-yin, LU Zi-jun, ZHANG He, LI Xiang-jun, LI Ji-ning
In this paper, a multifunctional metamaterial device based on the phase transition properties of vanadium dioxide (VO2) is proposed. The metamaterial structure consists of a top layer with the combined VO2-filled SRR and metal cross, a polyimide (PI) dielectric layer, and a metal substrate. When VO2 is in the insulating state, the cross-polarization conversion function can be realized, and the polarization conversion rate (PCR) is greater than 90% in the range of 0.48−0.87 THz. When VO2 is in the metallic state, the device can realize dual-frequency absorption and high-sensitivity sensing functions. The absorption rates are higher than 88% at the frequencies of 1.64 THz and 2.15 THz. By changing the refractive index of the sample material, the sensing sensitivities at the two related frequencies are about 25.6 GHz/RIU and 159 GHz/RIU, and the Q-factors are 71.34 and 23.12, respectively. The proposed metamaterial multifunctional device exhibits the advantages of simple structure, switchable function, and high-efficiency polarization conversion, and provides potential application values in the future terahertz communication, imaging and other fields.
Microwave photonic RF frequency multiplying phase shifter with tunable multiplication factor and a full 360-deg tunable range
YAN Ying, MA Jian-xin
A filterless microwave photonic phase shifter (MPPS) with a tunable frequency multiplication factor (FMF) and a full 360-deg tunable range is theoretically analyzed and verified by simulation. In the scheme, two parallel Mach-Zehnder modulators (MZM), cascaded with two dual-parallel integrated Mach-Zehnder modulators (DPMZM) by a 2×2 optical coupler (OC), are used to generate the ±1st- to 4th-order sidebands adjustably, and a phase modulator (PM) is used to phase shift one of the two lightwaves. After photodetection, the 2nd- to 8th- order harmonics with a continuously tunable phase shift from 0 to 360-deg can be generated by adjusting the RF driving signal and the DC bias voltage of the DPMZM, and the DC voltage of the PM. Simulation results demonstrate that both 360-deg continuously tunable phase shift and frequency multiplication can be implemented. Large Optical Sideband Suppression Ratio (OSSR) and Electrical Spurious Suppression Ratio (ESSR) of around 20 dB can be obtained. The phase shifter wavelength insensitive performance has been also evaluated by simulation.
Design of reflector assembly and adhesive layer under airborne wide temperature condition
ZHANG Jia-qi, GUO Yi-bo, ZHANG You-jian, ZHANG Zhi-hua
Airborne ambient temperature varies widely and airborne vibration is strong. And the mirror coating temperature is higher, the traditional bonding process will lead to bonding failure. Because of the difference of thermal expansion coefficient between Invar inlay and mirror material, the surface precision of mirror can not meet the requirement of system. Therefore, this paper puts forward a method of bonding the mirror after processing and coating, and designs some important parameters of the adhesive layer. In the scheme, RTV is used as the main binder, the mirror and the inlay are bonded together, and the effect of RTV curing on the structure is alleviated by good elasticity. The thickness of RTV is 1.1 mm, the width of RTV is 7.2 mm and the thickness of epoxy adhesive is 0.022 mm. The simulation results show that the RMS of the mirror shape is 25.91 nm and the first-order frequency of the mirror group mode is 242 Hz under gravity and 60 °C temperature change. The final surface detection RMS is 15.8 nm and the resonance frequency is 213 Hz. The experimental results show that the design of structure and bonding layer can meet the requirements of wide temperature and vibration condition.
3D reconstruction method based on rotating 2D laser scanner and multi-sensor
ZHANG Xin-rong, WANG Xin, WANG Yao, XIANG Gao-feng
3D reconstruction technology is one of the most popular research directions in machine vision, and has been widely used in the fields of unmanned driving and digital processing and production. Traditional 3D reconstruction methods include depth cameras and multi-line laser scanners, but the point clouds obtained by depth cameras have incomplete and inaccurate information, and the high cost of multi-line laser scanners hinders the application of this technology and Research. To solve the above problems, a three-dimensional reconstruction method based on a rotating two-dimensional laser scanner was proposed. First, a stepper motor was used to drive a 2D laser scanner to rotate to obtain 3D point cloud data. Then, the position of the laser scanner was calibrated by the method of multi-sensor fusion, and the point cloud data matching was completed by the coordinate system transformation. Finally, the collected point cloud data were filtered and simplified. The experimental results show: Reconstruction method compared with depth camera/IMU data fusion, the average error is reduced by 0.93 mm, the average error is 4.24 mm, the accuracy has reached the millimeter level, and the error rate is also controlled within 2%. The cost of the whole set of equipment is also greatly reduced compared to the reconstruction method of the multi-line laser scanner. It basically meets the requirements of retaining the shape characteristics of the object, high precision and low cost.
带宽可调谐的太赫兹超构材料半波片器件
Abstract(32) PDF 1139KB(39)
基于二氧化钒(vanadium dioxide, VO2)的相变原理,提出了一种"树叶型"复合超构材料,能够实现带宽可调谐的半波片功能。VO2薄膜为绝缘态时,复合超构材料可以看作是空芯"树叶型"金属结构,能够实现双频带的半波片功能。在1.01~1.17 THz和1.47~1.95 THz 频带范围内能够将y偏振光转换成x偏振光,偏振转换率大于0.9且平均相对带宽为26%。VO2 薄膜为金属态时,实芯"树叶型"金属结构的超构材料在1.13~2.80 THz范围内能够实现反射型的宽频带半波片功能,相对带宽为85%。利用瞬时表面电流分布和电场理论详细地分析了带宽可调谐半波片器件的工作原理。本文所提出的"树叶型"复合超构材料半波片器件在太赫兹成像、传感和偏振探测等领域具有潜在的应用前景。
基于互补集合经验模态分解的光频扫描干涉信号相位提取方法
针对光频扫描非线性对光频扫描干涉(FSI)信号相位提取精度存在影响,进而降低扫频干涉测距精度这一问题,本文提出了一种基于互补集合经验模态分解结合希尔伯特变换(CEEMD-HT)算法的干涉信号相位提取方法。在CEEMD-HT算法进行理论推导和仿真分析的基础上,通过仿真验证了该算法对非平稳扫频干涉干涉信号相位求解的有效性。进一步采用FSI实验系统中的真实输出光频率作为仿真条件进行了仿真实验,仿真结果表明CEEMD-HT算法对干涉信号相位的求解精度以及FSI测距精度都有显著的改善。最后,通过FSI测距系统的测距实验对所提出的干涉信号相位提取方法进行验证,测量结果表明:在2米自由空间测量范围内,基于CEEMD-HT算法的重复测距精度为2.79 μm,相较于EMD-HT和直接测量法重复测量精度分别提高了5.19倍和8.28倍。
A method for aligning point cloud prism boundaries of cultural relics based on normal vector and faceted index features
YANG Peng-cheng, YANG Zhao, MENG Jie, XIAO Yuan, Cui Jia-bao
Three-dimensional reconstruction is a common method for cultural relics information conservation, mainly through point cloud alignment technology to reorganize the spatial point cloud information of cultural relics, and its alignment accuracy has an important impact on cultural relics recovery. To address the problems of low accuracy and poor robustness in the alignment of complex point cloud texture features on the surface of cultural relics, this paper proposes a local point cloud alignment method based on normal vector angle and faceted index features. Firstly, the normal vector angle and covariance matrix thresholds are set according to the point cloud planar characteristics, and the point cloud feature points satisfying both features are extracted; secondly, the K-nearest neighbor search extracts the point cloud local feature point set, and the two sets of point cloud center-of-mass positions are overlapped by rigid transformation for coarse alignment; finally, the nearest points are iterated based on ICP for fine alignment. By comparing with the traditional ICP, this method reduces the point cloud alignment error by 3% and reduces the matching time by 50%, which effectively improves the accuracy and efficiency of alignment and enhances the robustness of point cloud alignment.
A study of the method for sea-sky-line detection based on polarization difference images
SU De-zhi, LIU Liang, WANG Kun, WU Shi-yong, LIU Ling-shun, MING Rui-long, GONG Jian
Aiming at the problem of sea-sky-line detection in low-contrast infrared images being difficult and easily affected by such interference factors as clouds, strip waves and sea clutter, this paper proposes a method of using polarization difference images for sea-sky-line detection. Firstly, polarization difference imaging (PDI) is used to enhance the local contrast of the sea surface area and the signal-to-noise ratio (SNR) of the sea-sky-line. A method of large-scale local contrast accumulation of the polarization difference images is then used to determine the sea-sky-line area. Finally, the accurate detection of small-scale sea-sky-line is completed via combining the methods of gradient significance and polynomial fitting in the sea-sky-line area. Overall, the methodology integrates multi-dimensional information such as the degree of linear polarization (DOLP) and the angle of polarization (AOP) for sea-sky-line detection, and combines large-scale and small-scale detection, which can effectively overcome the interference of factors such as clouds, strip waves and sea clutter. The experimental results show that the accuracy of this algorithm for sea-sky-line detection is 98.5%, and the average time consumed is 16 ms, and that fast and accurate sea-sky-line detection can be realized through this algorithm, which in turn has wide applicability to different scenes.
太赫兹人工表面等离子体共面激发与高Q传感
Abstract(29) PDF 396KB(55)
本文提出使用单层光栅超表面结构耦合的方式实现太赫兹人工表面等离子体激元(SSP)共面激发,克服了通过介质耦合器在实际应用时需要反射测量不便等缺点。在单层金属结构上同时构造周期性光栅和太赫兹SSP复合结构,当太赫兹波垂直入射时,实现光栅波矢和SSP波矢相匹配,激发SSP模式,在透射谱中可以产生高Q值谐振峰,其Q因子可以达到1923。分析了结构参数对光栅耦合超表面透射谱以及色散特性的影响。其次,基于该结构透射谱中的高Q谐振峰,进行传感研究,在谐振中心频率为0.22THz时,实现传感灵敏度为67GHz/RIU。本文所提出的光栅耦合超表面结构,仅仅使用单层超表面结构实现了太赫兹SSP模式的激发以及高Q传感,在诸多实际应用领域具有较大的研究潜力。
Multiple scattering transmission characteristic of polarized light in ellipsoidal fine particles
WANG Peng-cheng, ZHANG Su, SHEN Cheng-biao, ZHAN Jun-tong, DUAN Jin, LI Ying-chao, LIU Zhuang
Objetive
To investigate the multiple scattering transmission characteristic of polarized light in ellipsoidal fine particles, the simulation and experiment verification system of black carbon aerosol particles was established.
The polarization transmission characteristic after multiple scattering of the randomly oriented ellipsoidal fine particles are studied by combining T-matrix with Monte Carlo method. The half-material simulation environment testing was established to verify the simulation algorithm, and the ellipsoidal fine particles were prepared by extending ganoderma lucidum spores burning time. The size distribution and optical thickness of the ellipsoidal fine particles were measured by malvern spraytec and light power meter respectively. The simulation result can be proved by combining the experiment with simulation.
: The results show that, with the increasing of the concentration of black carbon ellipsoidal fine particles, the degree of polarization (DOP) of the horizontal, vertical, 45° linearly polarized light and the right circularly polarized light all decrease, and the polarization preservation ability of three kinds of linear polarizations are basically consistent. The polarization preservation ability of circularly polarized light is gradually superior to the linearly polarized light with the increasing of concentration. The gap between the linear and circular polarizations becomes more larger and the maximum value is reached at 3.12 optical thickness. When optical thickness is greater than 3.12, the DOP difference between the circularly and the linearly polarized lights tend to be stable. By calculation, the percent agreement between simulation and experiment is better than 70.84%.
These results can expand the environmental applicable range of polarization detection and provide theoretical support for studying the polarization detection of atmospheric non-spherical particles.
Study on Flux measurement method for High Flux Spot
WEI Xiu-dong, ZHAO Yu-hang, ZHANG Ya-nan, XU Ying-chao
A new method for measuring the flux distribution of high-magnification convergent radiation spot is proposed. The radiation flux sensor is used to measure the flux density at different positions of the spot, and the calibration curve of the grayscale and flux density at different positions of the spot is fitted by polynomial, and finally the radiation spot is obtained. In order to verify the accuracy and feasibility of the measurement method, a high-magnification convergent radiation spot flux distribution measurement experiment was carried out, and the direct measurement results of the radiant flux sensor were compared. The results show that the measurement results of the new method are consistent with the direct measurement results, and the average deviation is < 0.54%. Through analysis, the measurement uncertainty of this measurement method is 4.35%, and the measurement accuracy is higher than the traditional measurement. The method has been improved to meet the needs of practical applications.
Phase measurement with dual-frequency grating in nonlinear system
QIAO Nao-sheng, SHANG Xue
Abstract(76) FullText HTML(59) PDF 618KB(96)
To gain better phase measurement results in the nonlinear measurement system, a phase measurement method which uses dual-frequency grating after reducing nonlinear effect almost is proposed. Firstly, the nonlinear effect of the phase measurement system is discussed, the basic reason for the existence of high-order spectra components in the frequency domain is analyzed, and the basic method to reduce the nonlinear effect and separate the fundamental frequency information is given. Then, on the basis of reducing the nonlinear effect influence for the system, the basic principle of the phase measurement for the fringe image of the measured object using the dual-frequency grating method is analyzed. To verify the correctness and effectiveness for the proposed phase measurement method, computer simulation and practical experiments are implemented, and good results are obtained. In the simulation, the error value of this method is 27.97% of the method with nonlinear influence, and 52.51% of with almost no nonlinear influence. In the experiment, the effect of phase recovery is the best. It shows that the phase measurement by using this method mentioned in this paper has good effect and small error.
Design and fabrication of an optical film for an external cavity diode laser
YOU Dao-ming, TAN Man-qin, GUO Wen-tao, CAO Ying-chun, WANG Zi-jie, YANG Qiu-rui, WAN Li-li, WANG Xin, LIU Hen
Abstract(88) FullText HTML(81) PDF 6458KB(118)
The optical film is one of the most crucial components of the external cavity diode laser (ECL). However, the widely used optical film that employs the plane wave method (PWM) is not ideal in ECL. The finite-difference time-domain (FDTD) method is used to analyze this problem with the effect facet dimensions and structure taken into account. According to the simulation, PWM's film suffers from poor reflectivity and deviation of the reflection curve, which significantly affects performance. Therefore, the optical film design is optimized and verified by experiments. Magnetron sputtering is used to fabricate the optical film, which is then applied to ECL. The measurement results show that the reflectivity of Anti-Reflection (AR) film is reduced by 30% after optimization, while the reflectivity of High-Relection (HR) film increased by 7% to 96%. The prepared ECL has more than 650 mW of single-mode power with a 3 dB linewidth of less than 90 pm. In this paper, the optical film suitable for ECL is designed and fabricated, and provides a reference for optical films in ECLs and other semiconductor optoelectronic devices.
Research progress of grating projection on machine 3D topography inspection technology
LYU Hong-yu, LI Mao-yue, CAI Dong-chen, ZHAO Wei-xiang
Vision-based measurement has good application prospects and far-reaching development significance for advanced manufacturing fields such as aerospace, the military industry and electronic chips. Among them, on-machine 3D vision detection technology based on structured light is one of the hotspots and challenges in the field of precision machining. Based on the on-machine 3D measurement process of structured light, this paper discusses and summarizes the key technologies, including its technical requirements, methods and principles involved, related research status and existing problems in the measurement calibration, phase optimization solution, on-machine 3D point cloud processing and reconstruction of different feature surfaces. Finally, according to the actual needs of relevant technologies in the future, predictions are made with regard to processing field calibration, dynamic real-time 3D reconstruction, sub-micron and nano measurement, and measurement processing integrated data transmission technology, with the corresponding research ideas put forward.
White light interferometry micro measurement algorithm based on principal component analysis
CHEN Hao-bo, ZHANG Li-wei, SUN Wen-qing, CHEN Bao-hua, CAO Zhao-liang, WU Quan-ying
in order to save the problem of the phase solution in white light interferometry and realize the height measurement of micro morphology, white light interferometry micro measurement algorithm based on principal component analysis was proposed. White light microscopic interference system is used to collect multiple interferograms and reconstruct them into vector form. From a set of interferograms, the background illumination can be estimated by a temporal average, eliminating background light components. Then, the eigenvalues and eigenvectors representing the original data are obtained by matrix operation. Finally, the phase distribution is calculated by arctangent function. Experimental results indicate that the measurement result of standard step height of 956.05 nm by the proposed method is about 953.66 nm, the solution is approximately consistent with the iterative algorithm, in comparison to the iterative algorithm, the average time of the proposed method is 2 orders of magnitude faster. The interference fringes with surface roughness of 0.025 μm were analyzed, the mean of surface roughness calculated by the proposed method was 24.83 nm, and the sample standard deviation is 0.3831 nm. The proposed method improves the deficiency of monochromatic interferometry and has advantages of low computational requirements, fast and high accuracy.
Imaging comparison experiment of an underwater imaging system with a semiconductor white laser, a monochromatic laser and an led white light as the light source
JIANG Zi-qi, LIU Xiao-mei, CAI Fu-hong, ZHANG Dian, CAI Wei-yu, LIU Hua
To solve the problems of short illumination distance and narrow spectral range in the current underwater detection technology, an underwater semiconductor white laser imaging system was established. The quality of the images captured by the system under different light sources and different conditions was studied. A white laser with a power of 220 mw and a color temperature of 6469 K synthesized by an RGB semiconductor laser is used as the underwater lighting source, which is respectively compared with three RGB monochromatic lasers and an LED white light source under different conditions. For these images, different algorithms are used to process, analyze and evaluate their quality. The results indicate that when the white laser is used as the underwater light source, the collected image is not only better than that with the LED white light source with respect to information detail and structural integrity, but also better than the monochrome laser in color reproduction of the target and the integrity of the edge feature information. The semiconductor white laser has the advantages of concentrated energy, strong color rendering, and high illuminance, and its light source performance can meet the requirements of underwater low-illumination imaging. With the same imaging system and imaging distance, images with stronger authenticity, better texture and more target feature information can be obtained.
Research progress of gas detection based on thermoelastic spectroscopy
LOU Cun-guang, DAI Jia-liang, LI Rui-kai, LIU Xiu-ling, YAO Jian-quan
Laser-induced thermoelastic spectroscopy (LITES) is a new developed gas detection technology that based on the thermoelastic effect of quartz tuning fork (QTF). The QTF has the advantages of low cost, small volume, high sensitivity and wide spectral response range, and the LITES is becoming a vital method for trace gas detection. In this paper, the basic principle of measuring gas concentration based on LITES is firstly analyzed. Secondly, from the perspective of various technical methods, this paper introduces the methods for improving the sensitivity of the QTF detector, and reviews the research progress of LITES system in recent years. The performance of these systems is evaluated by the signal amplitude, signal-to-noise ratio (SNR), minimum detection limit, and normalized noise equivalent absorption coefficient (NNEA). Finally, the practical application of LITES in the field of gas detection technology is briefly reviewed, and the methods for further improving the sensitivity are summarized and prospected.
Study on measurement repeatability of high power laser measuring device based on light pressure
ZHAO Li-qiang, SUN Zhen-shan, YU Dong-yu, YANG Hong, ZHANG Yun-peng, SUN Qing
Measurement repeatability is the largest uncertainty component of a light pressure measurement device, which directly affects the accuracy of the measurement results. In order to improve the accuracy of the measurement power in the process of high-power laser measurement, a high-power laser measuring device based on light pressure is built. Quality measurement repeatability and laser power measurement repeatability experiments were carried out, and the results of the two experiments were compared and analyzed. The experimental results show that the measurement repeatability of the light pressure measuring device gradually decreased with the increase of the measured mass and the measured laser power, indicating that the light pressure method has more advantages in measuring high-power lasers. In the laser power measurement repeatability experiment, the influence of eccentric loads and airflow disturbance is avoided, so the laser power measurement repeatability is better than the measurement repeatability calculated according to the equivalent mass. The research results have guiding significance for further improving the measurement accuracy of the light pressure method in the future.
Spatial pulse position modulation multi-classification detector based on deep learning
WANG Hui-qin, HOU Wen-bin, HUANG Rui, CHEN Dan
In order to effectively avoid high computational complexity when using maximum likelihood (ML) detection, a deep learning-based spatial pulse position modulation (SPPM) multi-classification detector is proposed by combining a deep neural network (DNN) and step detection. In the detector, the DNN is used to establish a non-linear relationship between the received signal and the PPM symbols. Thereafter, the subsequent received PPM symbols are detected according to this relationship, so as to avoid the exhaustive search process of PPM symbol detection. The simulation results show that with the proposed detector, the SPPM system approximately achieves optimal bit error performance on the premise of greatly reducing detection complexity. Meanwhile, it overcomes the error platform effect caused by K-means clustering (KMC) step classification detection. When the PPM order is 64, the computational complexity of the proposal is about 95.45% and 33.54% lower than that of ML detectors and linear equalization DNN detectors, respectively.
Research progress of raman lidar temperature and humidity pressure detection technology
LIU Dong, YAO Qing-rui, ZHANG Si-nuo, GAO Jia-xin, WANG Nan-chao, WU Jiang, LIU Chong
Abstract(133) FullText HTML(75) PDF 5304KB(93)
Atmospheric temperature, humidity and pressure are deemed important atmospheric parameters. Quickly and accurately understanding the temperature, humidity and pressure information of the atmosphere and their changing trends is of great significance to research on meteorology, climatology, and artificial weather research. Raman lidar can obtain various atmospheric environment-related parameters by separating Raman scattering signal inversion, which can achieve high accuracy detection of atmospheric parameter profile information. Raman Lidar has unique advantages and potential in atmospheric temperature, humidity and pressure detection. With an introduction to the principle and inverse analysis algorithm of Raman Lidar for atmospheric temperature, humidity and pressure detection, this paper also highlights the advantages and disadvantages along with related advances of spectral devices such as filters, etalons and gratings commonly used in Raman Lidar. The detection techniques involved in Raman Lidar are also included. Finally, typical applications of meteorological parameter measurements by Raman Lidar are shown.
Study of short pulse laser drive technology in a distance-selective imaging system
WANG Chong, YANG Jia-hao, Zhu Bing-li, Han Jiang-hao, Dang Wen-bin
In a distance-selected imaging system based on single-photon detection, a short-pulse laser is emitted and between the transmitter and receiver for synchronization control, and the detector operates in photon counting mode and integrates in time to complete the imaging. In this paper, in order to obtain a short pulse laser that meets the system requirements while reducing the system's size and cost, we propose to apply these two types of narrow pulse generation circuits to single photon distance selective imaging systems. We introduce the principle and design method of both types and verify the system through simulation, physical fabrication and testing. The characteristics of the pulse generator and factors affecting its pulse width and amplitude are analyzed. The physical test results show that the transistor-based method can generate a narrow pulse with a rise time of 903.5 ps, a fall time of 946.1 ps, a pulse width of 824 ps, and an amplitude of 2.46 V; the SRD-based method can generate a narrow pulse with a rise time of 456.8 ps, a fall time of 458.3 ps, a pulse width of 1.5 ns, and an amplitude of 2.38 V; and the repetition frequency of both can reach 50 MHz. Both design methods can be used with external current-driven laser diodes to achieve excellent short pulse laser output.
Real-time detection of infrared ammonia leakage through lightweighted shuffling self-attention
ZHANG Yin-hui, ZHUANG Hong, HE Zi-fen, YANG Hong-kuan, HUANG Ying
Ammonia gas is an important basic industrial raw material, and realizing its non-contact detection is of great significance for the timely detection of ammonia gas leaks to avoid major safety incidents. Aiming at the shortcoming of conventional ammonia leak detection devices that can only respond when ammonia diffuses to a certain range and makes contact with a sensor, a shuffling self-attention network (SSANet) model is proposed to realize the infrared non-contact detection of ammonia leaks. Due to the high noise and low contrast of ammonia leakage images obtained by infrared cameras, an infrared detection dataset of ammonia leakage was established through non-local mean denoising and contrast-limited adaptive histogram equalization preprocessing. On the basis of YOLOv5s, the SSANet model uses the K-means algorithm to cluster and analyze the candidate frame suitable for the infrared detection of ammonia gas leakage to preset the model's parameters. Using the lightweight ShuffleNetv2 network, the depth of 3×3 in the Shuffle Block can be adjusted. The separate convolution kernel is replaced with a 5×5 depth, and the feature extraction network is reconstructed with an SK5 Block containing a new convolution module, which makes the model size, calculation and parameters non-intensive while improving the detection accuracy. The Transformer module is used instead of its original version. The C3 module in the network bottleneck module realizes the bottom-up fusion of multi-head attention in the leake area, and further improves the detection accuracy. The experimental results show that the size and parameter requirements of the SSANet model are reduced by 76.40% and 78.30%, respectively, to 3.40 M and 1.53 M compared with the basic model of YOLOv5s; the average detection speed of a single image is increased by 1.10% to 3.20 ms; and the average detection accuracy is increased by 3.50% , reaching 96.30%. This paper provides an effective detection algorithm for the development of a non-contact detection device for ammonia leakage to ensure the safe production and stable operation of ammonia-related enterprises.
Development and algorithm research of optical alignment system for a high precision flip chip bonder
HAN Bing, MA Hong-tao, XU Hong-gang, YAN Ying, JU De-han, ZHAO Chun-yu
Aiming at the urgent demand of high-precision optical alignment systems for a domestic infrared focal plane flip chip bonder, an optical alignment system was designed and verified, and the parallel adjustment, optical alignment and coordinate system error compensation algorithms applied to the system were researched.
Firstly, this paper analysed the optical alignment process of a flip chip bonder, then introduced the parallelism adjustment and optical alignment algorithm, and proposed a more reasonable error compensation algorithm according to the test process of the optical alignment system. finally, based on the above calculation algorithm, the optical alignment system was designed including three parts: a collimation system, a microscopic imaging system and a laser ranging system. The functions of parallel coarse adjustment, feature point recognition and parallel fine adjustment were realized.
The experimental results show that the collimation system has a good collimation effect, the microscopic imaging system has high resolution and good imaging quality, and the ranging accuracy of the laser ranging system is 0.084μm.
The designed high-precision optical alignment system solves the urgent need for a domestic infrared focal plane flip chip bonder for high-precision optical alignment systems. It has been applied in a certain types of flip chip bonders, and has very important social significance for improving the independent research and development and production capacity of domestic high-end large-scale integrated circuits.
Sub region curvature sensing method for survey telescope with larger aperture
AN Qi-chang, WU Xiao-xia, ZHANG Jing-xu, LI Hong-wen, ZHU Jia-kang
The large aperture sky survey telescope needs closed-loop error correction based on the feedback of its wavefront sensing system, so as to give it a better confirm to its limit detection ability. In this paper, firstly, the basic theoretical expression of sub region curvature sensing is derived. Then, a joint simulation model is established. The process of sub region curvature sensing is simulated and analyzed by using the combination of optical design software and numerical calculation software. Finally, by setting up a desktop experiment, the cross-comparison of single- and multi-target curvature sensors is carried out to verify the correctness of the algorithm. Compared to the traditional active optical technology, the method proposed in this paper can improve the detection signal-to-noise ratio and sampling speed by expanding the available guide stars. For the standard wavefront, compared with the single guide star curvature sensor, the error is 0.02 operating wavelengths (RMS), and the error is less than 10%, which can effectively improve the correction ability of the active optical system.
Spectral diagnosis of an arc jet actuator
YUAN Ye, TIAN Leichao, GUO Cheng, ZHAO Qing
At present, the simulation research of arc actuator is only limited to obtaining the working characteristics of the plasma generated by the actuator, such as potential, pressure, temperature and velocity, while the plasma state is only limited to diagnosing its electron temperature and electron density by spectrum. The two are separated. This paper attempts to unify the two. Therefore, the arc jet plasma actuator designed here adopts the finite element method to solve the nonlinear multi physical equations. The working characteristics of the arc jet plasma actuator are numerically simulated, and the potential, pressure, temperature and velocity distributions inside the actuator are obtained. On this basis, the electron density is calculated, The simulation calculation model of the plasma state (electron temperature and electron density) of the actuator is obtained from the working condition of the actuator. Then the spectral diagnosis of jet plasma is carried out by using the emission spectral diagnosis method, and the electron density of plasma is calculated by using the intensity ratio method of discrete spectral lines. The diagnostic experiment of arc plasma actuator shows that the maximum electron temperature is 10505.8 k and the maximum electron density is 5.75 e + 22 m−3. For the plasma electron temperature and plasma density under different working conditions, the experimental and simulation results increase with the increase of inlet gas flow and discharge current. It shows that our simulation model of plasma state is reasonable and applicable for our miniaturized arc jet actuator with high jet velocity. At the same time, it also shows that our unified consideration is basically successful. Of course, there are still areas worthy of further improvement.
Positioning algorithm for laser spot center based on BP neural network and genetic algorithm
ZHANG Jing-yuan, CHEN Bei-bei, YANG Yong-xing, ZHU Qing-sheng, LI Jin-peng, ZHAO Jin-biao
The traditional laser spot center positioning algorithm in a vibrating environment has problems such as long processing time and low accuracy. This paper proposed a laser spot center positioning method based on a genetic algorithm optimized BP neural network. This algorithm uses a BP neural network to predict the spot center position and a genetic algorithm to optimize the neural network. Based on the BP neural network, the spot center position derived by the gray weighted centroid method, centroid method, Gaussian fitting method, and the radius of laser spot obtained by the centroid method are used to predict the actual center position of the spot. Genetic algorithms are used to optimize the weights and thresholds of neural networks to improve prediction accuracy. An experimental platform is established to simulate the vibration environment by applying perturbations to the optical system and the data is collected for neural network training and algorithm verification. The experimental results show that the number of calibration test iterations before and after optimization is 55 and 29, and the average errors are 0.81 pixels and 0.45 pixels, respectively. Under the optimization of the genetic algorithm, the iteration speed and prediction accuracy of the neural network algorithm is improved.
Improving sensitivity by multi-coherence of magnetic surface plasmons
YANG Zongmeng, XING Qian, CHEN Yian, HOU Yumin
This paper studies the coherence of magnetic surface plasmons in one-dimensional metallic nano-slit arrays and proposes a double-dip sensing method to improve sensitivity. Different from the conventional way of scanning wavelength at a fixed incident angle, coherence of surface plasmons is investigated by changing the incident angle at a fixed wavelength. Due to the retardation effect, two coherence dips move in opposite directions as the refractive index of the surrounding medium changes. Compared with one dip used for sensing, two oppositely moving dips can efficiently improve the sensitivity. The sensitivity of two dips can reach 141.6°/RIU while the sensitivities of two single dips are 39.2°/RIU and 102.4°/RIU respectively. Besides, the inconsistency between the refractive index of slit medium and upper medium has few influences on the sensing performance, which can lead to wide practical applications.
Temperature control method of CO2 laser operating in airborne wide temperature range
ZHAO Zi-yun, CHEN Fei, ZHANG Kuo, HE Yang
Abstract(84) FullText HTML(60) PDF 750KB(132)
Airborne lidar is an important means to achieve long-range accurate atmospheric monitoring. The laser wavelength is consistent with the absorption spectrum of most atmospheric pollutants and chemical substances, which makes it an important laser source for airborne lidar. However, it is difficult to design a temperature control system for airborne CO2 laser to work in a wide temperature range of −40 °C−55 °C under the condition of controlling volume and weight. This paper proposes a temperature control method. In the method, the laser characteristic and environment temperature are used as input, thermo electric cooler and forced air cooling are combined to control the laser temperature. According to the structure and heat transfer characteristics of laser, thermo electric cooler and forced air cooling, the finite element model of temperature control method is established, and the temperature control performance of laser is optimized based on the model. In the high temperature environment of 55 °C, the temperature of the laser is controlled at 40 °C, after the temperature control system works for 25 min. In the low temperature environment of −40 °C, the laser temperature is controlled at 25 °C after the temperature control system works for 20 minutes, which meets the normal working requirements of the laser. According to the laser and the established temperature control method, the experimental research on the working ability of the laser in high and low temperature environment is carried out, the temperature data of the laser in the experimental process is collected, and the laser output power is measured under high and low temperature conditions. The experimental results show that the experimental measured temperature data is consistent with the finite element simulation results, the error between them is less than 10%. The laser with proposed temperature control method can work steadily, and the output power of the laser is consistent with that of the laser at room temperature.
Flexible fiber grating hydrophone array theoretical and experimental research
ZHU Miao, GU Hong-can, SONG Wen-zhang
In order to improve the suitability of the fiber hydrophone towing line array, a flexible fiber grating hydrophone array was proposed.The sound pressure sensitivity of three flexible fiber grating hydrophones was calculated according to the mechanical theoretical model,and the influence factors were compared and analyzed. The 2-element flexible fiber hydrophone sample arrays with diameters of 10 mm,12 mm and 16 mm were developed through finite element simulation for frequency response analysis. The sensitivity was measured by vibration liquid column experiment. The experimental results show that the response is flat within the frequency range of 200−800 Hz, and the average sound pressure sensitivities of hydrophone arrays with different structural parameters are −160.87 dB, −154.59 dB, and −156.73 dB, respectively. The theory and simulation analysis are verified. By further optimizing the material and structure parameters and using weak reflection fiber grating, the integrated flexible hydrophone array with hundreds elements can be constructed according to the design in this paper.
Double doughnut-shaped focal spots with controllable position in axial direction
TIAN Yu-yuan, ZHANG Jia-qi, JIANG Xiao-tong, SUN Mei-yu, SHI Qiang, ZHU Lin-wei
In order to generate double doughnut-shaped focal spots in adjustable position along axial direction. Based on a formula of annular radius derived from vector diffraction integral, a vortex phase zone plate was designed to produce the double doughnut-shaped focal spots in axial direction. The focusing properties of the modulated vortex phase zone plate was further investigated in tightly focused system. First, integral formulas of linearly and circularly polarized vortex beams were calculated under high NA focusing condition. Then the intensity distributions of linearly and circularly polarized vortex beams in high NA focusing system were simulated by the integral formulas with various axial shifting distances and topological charges. Finally, the corresponding experimental results of linearly and circularly polarized light were also given, utilizing a spatial light modulator loaded on double doughnut-shaped phase patterns. The double doughnut-shaped focal spots with the topological charge of 1 and axial distances of ±10 μm and ±15 μm were produced when the incident light was linear polarization. As well as the double doughnut-shaped focal spots with axial distances of ±20 μm and topological charge of 1−4 were also produced when the incident light was circular polarization. The simulated and experimental results demonstrated that two doughnut-shape focal spots with controllable axial shifting distance and dark spot size could be produced in the tight focusing region of a high NA objective when it modulated by the vortex phase zone plate. This kind of vortex phase zone plate could be applied in the field of optical micromanipulation, two-beam super-resolution nanolithography, and stimulated-emission-depletion fluorescence microscopy (STED).
Laser intensity distribution measurement method based on tomographic imaging
WANG Qian, CAI Wei-wei, TAO Bo
, Available online
Abstract(206) FullText HTML(180) PDF 9813KB(73)
In order to accurately measure the laser intensity distribution, this paper proposes a method based on tomographic imaging. Firstly, numerical studies were performed to validate the correctness of the imaging model and convergence of the reconstruction algorithm. Reconstruction errors were less than or equal to 7.02% with different laser intensity distribution phantoms employed and less than 8.5% with the addition of different random noise levels under 10%. Additionally, a demonstration experiment was performed with the employment of a customized fiber bundle to realize the measurement from seven views. Seven views are distributed along a semi-circle plane which is perpendicular to the propagation direction of the laser beam. The distance from the laser beam to each view is nearly 160 mm and the angle coverage range of the seven views is about 150°. Laser-induced fluorescence obtained after the laser passed through a rhodamine-ethanol solution was collected by the tomographic imaging system. Then, the laser intensity distribution was obtained through absorption-corrected three-dimensional (3D) reconstruction. The correlation of the projection and re-projection of the one view was used to quantitatively access the accuracy after the other six views were adopted in the reconstruction. The results show the feasibility of the method with a correlation coefficient of 0.9802. It can be predicted that the 3D laser intensity measurement scheme proposed in this work has a broad prospect in the field of laser applications.
Ground electronics verification of inter-satellites laser ranging in the taiji program
DENG Ru-jie, ZHANG Yi-bin, LIU He-shan, LUO Zi-ren
In the Taiji program, laser interferometry is utilized to detect the tiny displacement produced by the gravitational wave signals. Due to the large-scale unequal arm, the laser frequency noise is the largest noise budget in the space interferometer system. To reduce the influence of laser frequency noise, a technology called the Time Delay Interferometry (TDI) is utilized to deal with it. The TDI is a kind of data post-processing method, which forms the new data stream by the method of the time delay to initial data. But the premise of TDI needs to obtain accurate absolute arm length between satellites. Thus, for that requirement, we discuss the ranging system scheme and implement a ground electronics verification experiment. The ranging system is based on Direct Sequence Spread Spectrum (DS/SS) modulation, and it mainly includes three parts, which are the signal structure, a Delay Locked Loop (DLL), and a data processing algorithm. In DS/SS modulation, types of pseudo-random code can make a difference to the quality of correlation and the ranging accuracy. Therefore, to design the optimal pseudo-random code, we compare the correlation and flexibility in choosing lengths of the m sequence, gold sequence, and Weil code. Weil code that has a shift-cutoff combination with the best autocorrelation is chosen as the ranging code. The ground electronics verification experiment is set up for simulating the physical process of signal transmission and verifying system performance. The main device of the experiment is a FPGA card based on the K7 chip from Xilinx, which is used to simulate the function of communication and ranging between satellites. Meanwhile, we change the length of the Radio Frequency (RF) coaxial cable to correspond to different ranges. The experimental process can be summarized as follows. Firstly, 16-bit data at 24.4 kbps and 1024-bit Weil code at 1.5625 Mbps are modulated with Binary Phase Shift Keying (BPSK) in the 50 MHz sampling frequency. Then the signal is transmitted through RF coaxial cables of 10 to 60 m in length. In receiving end, the signal is consolidated by DLL and the ranging information is collected. To measure the range accurately, we use a centroid method to optimize the collected data. The results show that the ranging accuracy is better than 1.6 m within 60 m. In conclusion, this experiment proves the principle of the scheme and its feasibility, laying a technical foundation for optical system verification in the future.
Influence of radiation coupling effect on polarization characteristics of targets
SU De-zhi, LIU Liang, WU Shi-yong, ZHANG Ji-lei, WANG Kun, Ling-shun LIU
Infrared polarization imaging technology has the advantages of long detection range and high rate of target recognition. However, the polarization characteristics of targets are easily affected by background radiation in complex environments, which significantly reduces the detection capability of infrared polarization equipment. Based on the polarized Bidirectional Reflectance Distribution Function (pBRDF), this paper establishes a calculation model for the target's degree of linear polarization (DoLP), comprehensively considering the radiation coupling effect between the target and the background. The variation of the target's DoLP under two conditions - with and without a strong radiation backplate – is then comparatively studied. Additionally, in order to solve problems of land-based and airborne small-angle detection, simulation research is done to find out how the target's DoLP is influenced by parameters such as the temperatures and the included angle between the target and the backplate. Research results show that the radiation coupling effect significantly reduces the target's degree of polarization when the temperatures of the target and the backplate are the same, but it does not change the trend of the target's degree of polarization, which increases with an increase in temperature. When the temperature of the target and the backplate is 30 °C, 40 °C, and 50 °C, the maximum degree of polarization of the target is 63.7%, 44.9%, and 42.2% of those without a strong radiation backplate, respectively. It can be concluded then that the higher the temperature, the stronger the radiation coupling effect between the target and the backplate, and the greater the reduction of the target's degree of polarization; and that the strength of the radiation coupling effect is not only related to the temperature, but also to the included angle between the target and the backplate. With the increase of the included angle, the target's DoLP first increases and then decreases, and the maximum value is obtained when the included angle is about 105°. Therefore, the radiation coupling effect changes the target's DoLP to a certain extent, thereby affecting the detection ability of the infrared polarization equipment. Finally, through building a long-wave infrared polarization imaging system, the established calculation model of the target's degree of polarization is verified by experiments, whose results are basically consistent with those of the simulation analysis. Overall, the research results in this paper have certain guiding significance for improving the detection and identification capabilities of land-based and airborne infrared polarization equipment.
Lane detection based on dual attention mechanism
REN Feng-lei, ZHOU Hai-bo, YANG Lu, HE Xin
In order to improve the performance of lane detection algorithms under complex scenes like obstacles, we have proposed a multi-lane detection method based on dual attention mechanism in this paper. Firstly, we designed a lane segmentation network based on a spatial and channel attention mechanism. With this, we obtain a binary image which shows lane pixels and the background region. Then, we introduced HNet which can output a perspective transformation matrix and transform the image to a bird's eye view. Next, we did curve fitting and transformed the result back to the original image. Finally, we defined the region between the two-lane lines near the middle of the image as the ego lane. Our algorithm achieves a 96.63% accuracy with real-time performance of 134 FPS on the Tusimple dataset. In addition, it obtains 77.32% of precision on the CULane dataset. The experiments show that our proposed lane detection algorithm can detect multi-lane lines under different scenarios including obstacles. Our proposed algorithm shows more excellent performance compared with the other traditional lane line detection algorithms.
Synchronization transmission technology of semiconductor lasers with transverse effect
ZHAO Li-na, WEI Qing-tao
In order to account for the transverse effect of semiconductor lasers, the dynamic equation of a semiconductor laser with transverse effect is given by modifying the dynamic model describing it, and the influence of the transverse effect on its output characteristics is analyzed. On this basis, the synchronization transmission technology of a semiconductor laser's output signal with transverse effect is further studied. The results show that the output of the semiconductor laser presents a new spatiotemporal chaotic state after considering the transverse effect, and is very sensitive to the dependence of the initial value. At the same time, whether the synchronization transmission of single-channel or multi-channel signals is carried out by a semiconductor laser, its transmission performance is very stable. The synchronization technology is very simple and easy to apply in practice.
Trend of action on the display effect based on pepper's ghost images affected by illumination and color temperature from led light sources
LIN Jia-yuan, ZOU Nian-yu, LIANG Jing, HE Xiao-yang, MU Yu-fan, HUANG Xia, HE Qi-peng
With the continuous development of digital display technology, display methods have also changed. Pepper's ghost images that adopt modern display methods require the light environment of the exhibition space to ensure the effect and ensure better visual comfort. In order to explore the influence of the lighting environment on the display effect based on Pepper's ghost images, a virtual imaging display space is set up to analyze the factors and trends affecting the imaging effect. A virtual imaging display space is set up in which 12 sets of LED lighting conditions with different illuminances and color temperatures are generated. 25 observers were used to conduct a psychophysical experiment. Conclusion: color temperature has no significant effect on the evaluation of color authenticity, detail expressiveness and stereoscopic expressiveness for Pepper's ghost images; Illumination has no strong effect on the evaluation of the color authenticity of Pepper's ghost images, but has a significant effect on their detail expressiveness and stereoscopic expressiveness. Under the lighting environment where the color temperature is 3500 K and the illumination is 10 lx, the detailed expressiveness and stereoscopic expressiveness of the display effect are relatively high and the visual comfort of 2500 K and 10 lx is better.
Experimental study on CCD damage by Multi-wavelength raman lasers
LIU Jin-sheng, LIU Jin-bo, LI Xiao-wen
The damage threshold of an interline transfer CCD irradiated by different wavelength nanosecond Raman lasers was studied and an experiment with 496 nm, 532 nm, 632 nm Raman and multispectral Raman laser-irradiated CCD was carried out. The damage threshold interval of dot damage, line damage and total damage were observed and collected by adjusting the energy of each focused Raman laser. By careful fitting, the damage threshold interval and the damage possibility curve of the CCD at different laser energy densities with each Raman laser were estimated. Results showed that the multispectral Raman laser including a residual pump laser is most effective for damaging the CCD than the monochrome Raman laser, and the 630 nm Raman laser acts better than 574 nm and the 496 nm Raman laser. The microscopic images of the damaged CCD were reviewed, and the electronic characters of the damaged CCD were also tested to understand the damage and blindness mechanism of a Raman laser pulse-irradiated CCD.
A study on the measurement method of heliostat surface shape error based on photogrammetry
WEI Xiu-dong, ZHANG Fan, XU Ying-chao
In the tower solar thermal power plant, the heliostat mirror shape errors have an important impact on the optical efficiency of the heliostat field, so it is necessary to measure the heliostat surface shape error. The heliostat is generally made up of splicing multiple sub-mirrors, the tilt angle error of the sub-mirror is an important part of the heliostat mirror shape errors. This paper proposes a measurement method for the tilt angle errors of the heliostat sub-mirror based on the photogrammetry. That is, under the condition of known the shape size of the heliostat sub-mirror, the spatial position coordinates of the 4 corner points of the heliostat sub-mirror are calculated by using the principle of photographic imaging. Then the normal direction of the sub-mirror is found, and the tilt angle of the sub-mirror is calculated by using the normal line obtained. Finally, the purpose to measure the tilt angle error of the heliostat sub-mirror is achieved. The measurement principle of the method is elaborated, the calculation formula is derived, and relevant verification experiments were carried out using planar mirrors and cameras. By measuring the plane mirror with different tilt angles at different distances, the deviation between the measured tilt angle and the actual tilt angle of the plane mirror is about 0.1°−0.3°, and the experimental results show that the method can accurately measure the tilt angle error of the sub-mirror of heliostat, thus the correctness and feasibility of the method are verified.
Bound states in continuum in periodic optical systems
YAO Jian-quan, LI Ji-tao, ZHANG Ya-ting, LI Jie, YUE Zhen, XU Hang, YANG Fan
2023, 16(1): 1-23. doi: 10.37188/CO.2022-0189
Periodic optical systems, such as photonic crystals and optical metamaterials, can localize high-density electromagnetic field energy at subwavelength scales and obtain extremely small mode volumes, so they have great application potential in the field of light manipulation. In recent years, a strong interaction between light and matter in periodic optical systems has been discovered, which is called Bound States in Continuum (BIC). Optics BICs are special electromagnetic eigenstates whose frequencies lie in the radiation continuum but are completely localized, and have shown interesting physics and rich application scenarios. This paper systematically reviews the classification and theory of BICs in periodic optical systems, and summarizes their basic physical properties and the latest application development. BICs in periodic optical systems are injecting new impetus into the fields of integrated optics, information optics, bio-optics, topological optics, and nonlinear optics.
Recent progress on synthesis and optical characterization of two-dimensional Bi2O2Se
XIE Bing, AN Xu-hong, ZHAO Wei-wei, NI Zhen-hua
2023, 16(1): 24-43. doi: 10.37188/CO.2022-0071
Abstract(123) FullText HTML(96) PDF 12124KB(154)
Two-dimensional (2D) Bi2O2Se has attracted broad attention in the field of electronic and optoelectronic applications in the UV-Vis-NIR region due to its unique crystal structure, energy band, high carrier mobility, and excellent stability. In this paper, we review the recent research progress in the material synthesis and optical characterization of Bi2O2Se. Firstly, the synthetic method and growth mechanism of 2D Bi2O2Se are introduced, including Chemical Vapor Deposition (CVD), wet chemical process, Molecular Beam Epitaxy (MBE) and Pulsed Laser Deposition (PLD), etc. Via steady-state spectrum study, the properties change of 2D Bi2O2Se with thickness change can be studied, such as the band gap. The defect type, temperature coefficient and thermal conductivity of 2D Bi2O2Se material can be further studied by focusing on the crystal vibration mode. Transient spectrum techniques can benefit the study of relaxation process and carriers transport properties in 2D Bi2O2Se materials. Finally, we summarize the existing challenges and application prospects for the promising Bi2O2Se field.
Panoramic endoscopic imaging technology and it's applications
HUO Jia-yi, LI Mian-hao, WANG Zi-chuan, YUAN Bo, YANG Qing, WANG Li-qiang
Panoramic endoscopic imaging technology can effectively reduce the observation blind area of internal organs. It has many advantages, such as shortening the operation time, reducing the risk of intraoperative bleeding, improving the prognosis and shortening the postoperative recovery time. It has important application value in minimally invasive surgery and preoperative examination. It is a research hotspot in recent years. This paper combs the panoramic endoscopic imaging technology from two aspects: principle and product applications. Firstly, various panoramic endoscopic imaging technologies based on two-dimensional and three-dimensional imaging are reviewed, their implementation methods are described, and their key indexes and performances are analyzed. Secondly, the capsule endoscope, panoramic enteroscope and other different types of products derived from panoramic endoscopic imaging technology are compared and analyzed, and the development trend and application prospect of panoramic endoscopic imaging technology are prospected.
Research progress of monolithic integration master-oscillation power-amplifiers
TAN Man-qing, YOU Dao-ming, GUO Wen-tao, LIU Wei-hua
Besides its advantages in volume, power and beam quality, a monolithic integration Master-Oscillation Power-amplifier (MOPA) can also realize a narrower linewidth and dynamic single-mode by integrating Bragg grating. Its application value is high in the fields of frequency doubling, pumping, optical communication and sensing, which makes it a popular research topic in recent years. This paper firstly went over the mainstream structure and characteristics of monolithic integrated MOPA, including a tapered amplifier, ridge amplifier, Bragg grating and three-section MOPA. Based on their working principles and performance characteristics, we introduce the main research directions and the latest development trends in combination with their problems. Aiming at the problem of beam quality degradation at high power in monolithic integrated MOPA, the optimal design of epitaxial layer structure, facet optical film and electrode aspects are then summarized for monolithic integrated MOPAs. After that, we sort out the research progress of MOPAs with different performance characteristics for various application requirements including high power, narrow linewidth, high beam quality and high brightness. Finally, we prospect the development trend of monolithic integrated MOPA.
Design, preparation and application of orthogonal excitation-emission upconversion nanomaterials
JIA Heng, FENG Xiao-rui, LI Da-guang, QIN Wei-ping, YANG Long, HE Wei-yan, MA Hui-yan, TENG Ying-yue
Rare earth-doped upconversion luminescence nanomaterials have received considerable attention from researchers due to their great potential for applications in many fields such as information security, biomedicine, optical fiber communication, digital displays, and energy. The recently-developed upconversion luminescence nanoparticles with orthogonal excitation-emission properties have attracted especially strong research interest because their distinct luminescence outputs can be dynamically modulated by switching the excitation conditions. The orthogonal luminescence properties further endow such nanocrystals with a set of new features and functionalities, which largely expands their potential applications. This review summarizes the progress in the development of orthogonal upconversion luminescence of rare earth ions, and provides a systematic discussion on design principles and construction strategies of orthogonal excitation-emission systems based on core-shell structures, as well as introduces their recent advances in various fields of applications including data storage, security anti-counterfeiting, digital displays, sensing, bioimaging and therapy. Furthermore, the prospective opportunities and challenges in the future research of orthogonal luminescence systems are also provided.
Analysis of influence of diffraction effect of microlens array on Shack-Hartmann wavefront sensor
ZHU Qin-yu, CHEN Mei-rui, LU Huan-jun, FAN Li-na, PENG Jian-tao, SUN Hui-juan, XU Guo-ding, MAO Hong-min, CAO Zhao-liang
2023, 16(1): 94-102. doi: 10.37188/CO.2022-0176
The diffraction effect of microlens array will affect the detection accuracy of Shack-Hartmann wavefront sensor. Based on Huygens-Fresnel diffraction theory, a two-dimensional microlens array diffraction model is established to simulate and analyze the two-dimensional diffraction spot array generated in the focal plane when the ideal parallel light is incident on the microlens array. First, the maximum centroid calculation error is determined by calculating the centroid error in the process of diffraction spot shifting by one pixel. Then the wavefront is reconstructed by using the modal method to obtain the wavefront detection error. The simulation results show that the maximum wavefront error caused by diffraction is 0.125 λ at 0.21 and 0.79 pixels offset, that is, when the wavefront deflection is 0.03° and 0.13°. Finally, an experiment is performed to verify the effectiveness of the error calculation method. This work provides a theoretical basis for the design of shack-Hartmann wavefront detector.
Quantitative analysis of thorium in graphite using femtosecond laser-induced breakdown spectroscopy
LIU Xiao-liang, WANG Lan, PENG Ling-ling, LI Xiao-yan, LIU Yun-hai, ZOU Chun-yan
2023, 16(1): 103-112. doi: 10.37188/CO.2022-0082
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In order to promote the application of Laser-Induced Breakdown Spectroscopy (LIBS) in the nuclear industry, in this paper, a femtosecond LIBS(fs-LIBS) system was used to quantitatively analyze Thorium (Th) in a highly pure graphite matrix. According to the Th concentrations in the Thorium-based fuel, a total of 9 homemade Th2O3-graphite mixture samples with Th concentrations that varied from 0.35% to 35.15% were prepared by the standard addition method. The favorable experimental parameters such as the treatment methods for LIBS detection, laser pulse energy and delay times were studied before the quantitative analysis. The results show that the signal intensity of the fs-LIBS spectrum acquired by the scanning with moving method is significantly higher than that without the moving method. For the Th I 396.21 nm line, the Relative Standard Deviation (RSD) value of multiple measurements for the scanning method was just 5.7%, which was much lower than that of without the moving method (20.4%). The Th spectral lines show obvious saturation due to the self-absorption effect in the higher concentration region, and thus the basic calibration method was no longer applicable. Therefore, an exponential function was used to fit the spectral line intensity and concentration in the whole concentration region, and the concentration saturation threshold values corresponding to the analytical lines Th I 394.42, 396.21, and 766.53 nm were obtained. The basic calibration method has good detection performance when the calibration curves were constructed by using a lower concentration below the saturation threshold. For the peak area and peak intensity of each analytical line, using the internal standard method with the internal standard line (C I 247.85 nm), a good linear relationship can be found between them and the Th concentrations in the whole concentration region, especially for analytical line Th I 766.53 nm with a higher saturation threshold. The internal standard method had good prediction performance for unknown samples with higher concentrations. The above results show that fs-LIBS has the potential to monitor and analyze the thorium concentration in the thorium-based fuel cycle.
Viewpoint planning of surface structured light scanning for complex surface parts
REN Ming-yang, WANG Li-zhong, ZHAO Jian-bo, TANG Zheng-zong
In order to realize the efficient and automatic measurement of complex curved surface parts, we propose a viewpoint planning method of surface structured light scanning based on an improved grid method, and apply it to the automatic measurement of automobile parts with complex curved surface. Firstly, aiming at the problem of serious redundancy and poor scanning integrity of the manual teaching viewpoint, a scanning viewpoint planning algorithm for surface structured light based on an improved grid method is proposed. According to the effective measurement range of a surface structured light scanner, the grid size is determined, and the candidate viewpoint generation strategy is improved. The effective measurement range of candidate viewpoints is obtained by the measurement constraint condition of the scanner, and the optimal viewpoint is determined by the viewpoint quality evaluation function. Secondly, in view of the low efficiency of the algorithm and the low accuracy of feature reconstruction in the process of viewpoint planning, the voxel grid method is used to simplify the model. The complex surface model is segmented by the octree algorithm, and the voxel grid size is determined according to the normal vector consistency error. For the models with different geometric characteristics, the influence of the weight coefficient on the scanning quality is analyzed, and the optimal weight coefficient is given. Finally, the scanning viewpoint planning and measurement experiments of automobile sheet metal parts and reducer shell are carried out. The results show that the viewpoint planning of the automobile sheet metal parts takes 21.93 s, the scanning integrity is 99.124%, and the scanning accuracy is 0.025 mm. The viewpoint planning of automobile reducer shell takes 158.29 s, its scanning integrity is 93.231%, and its scanning accuracy is 0.032 mm. This method can quickly complete the viewpoint planning of complex curved surfaces, and the model obtained by planning viewpoint scanning has good integrity and high precision, which can meet the requirements of complex curved surface parts for automatic measurement.
Design of a radiometric calibration light source based on a freeform reflector
TIAN Jie-wen, YE Xin, FANG Wei
We propose a design method of a free-form reflector for collimating illumination of integrating spherical light sources to reduce the space occupation on a satellite. By using this method, a square irradiance distribution with large area can be achieved through a integrating sphere with small diameter. Firstly, the mathematical model of off-axis reflection lighting of free-form surface is established through the point light source model, then the free-form surface is discretized by Chebyshev points, and the free-form surface model that satisfies the point light source illumination is solved. Finally, the light source characteristics of the integrating sphere are analyzed. The transformation from the point light source illumination model to the integrating sphere illumination model is achieved by the optimization of the free-form surface energy distribution. The analysis results show that when the illumination area is set as 140 mm×140 mm, the irradiance non-uniformity of the target surface is less than 0.02. This method can meet the requirements of spaceborne calibration for light weight, short light path and simple structure.
Multi-channel optical switching based on scanning mirror instrumentation
XIAO Ze-hua, LI Ming-yu, SU Guo-shuai, He Ze-nan
An optical switch based on a scanning mirror was designed in this paper. The optical switch is programmable and controlled by an embedded Linux system that switches between the fiber array channels on the UI of the touch display. Meanwhile, the switching sequence and residence time of the optical switch can be preset. In addition, the optical switch can be self-calibrated to obtain the maximum output power of each channel. The principle of the optical switch is analyzed and the performance of the optical switch is tested experimentally. The experimental results show that the average insertion loss is less than 17 dB for the single mode fiber array, the average crosstalk between adjacent channels is more than 30 dB, and the switching time between the adjacent channel is less than 1.3 ms. The average insertion loss is less than 2.4 dB for the multi-mode fiber array. It has the advantages of low loss, low delay, high precision, good stability, high repeatability,low cross-talk between the adjacent channel, and good man-machine interaction for the application of theWavelength Division Multiplexing (WDM) and multi-channel optical waveguide sensors test device.
Design of solar concentrated uniform linear light source of composite ellipsoid cavity
LU Yu, YANG Yi, ZHANG Xiang-xiang, KONG Wei-jing
In order to implement a solar direct pumping slab high power laser, a linear uniform high-power density pump source is studied. In this paper, we propose a design method of a high-power density uniform linear light source by combining the first-order concentrating system with seven confocal ellipsoids to form a composite ellipsoid cavity. The equal radiation flux segmentation of the circular first focal spot is realized by each ellipsoid. The mirror imaging characteristics do not significantly decrease the peak power density. After decomposition, the mirror spot forms a uniform linear light source at the second point of focus. The mathematical model of equal radiation flux is given by coordinate transform, and the rotation and translation parameters of each ellipsoid are solved by the annealing algorithm. The first-order system is composed of a Fresnel lens with a radius of 30 mm, a focal length of 70 mm and a single ellipsoidal cavity with a of 3.4 mm, c of 3.15 mm, The second-order composite ellipsoidal cavity concentrating system is attached. The line source is realized with effective length of 12 mm, the peak power density of 1.09 × 106 W/m2, and the uniformity of 95.46 %. Compared with the contribution of each ellipsoid parameter to the uniformity, the uniformity effect is significantly improved when the rotation parameter θ of the middle ellipsoid is 1.4°. The change of the edge ellipsoid parameter Δ has a significant influence on the effective length of the linear light source, and its optimal value is 0.53 mm.
Research on highly sensitive detection of oxygen concentrations based on tunable diode laser absorption spectroscopy
YANG Shu-han, QIAO Shun-da, LIN Dian-yang, MA Yu-fei
Tunable Diode Laser Absorption Spectroscopy (TDLAS) is a recently developed laser spectral gas detection technology. Compared with common oxygen sensors such as electrochemical devices and ionic conductive ceramics, TDLAS has the advantages of high selectivity and sensitivity, fast response, on-line measurement and strong anti-background spectral interference ability. Oxygen (O2) is an important gas in habitable environments and is greatly significant to industrial production and human life, and the detection of O2 concentration is also widely used in these fields. Based on this, we adopt TDLAS technology to carry out high sensitivity measurements of O2 in air. Using a semiconductor laser with an output wavelength of 760 nm as the light source, the oxygen concentration in the environment is 20.56% by direct absorption spectroscopy, and the minimum detection limit is 5.53×10−3. In the wavelength modulation spectroscopy method, the laser wavelength modulation depth is optimized to obtain a complete second harmonic waveform, which can be used to calibrate the oxygen concentration. The SNR of the system is 380.74, and the minimum detection limit is about 540×10−6. The system realized in this paper has good oxygen detection ability and can be widely used in various fields of oxygen concentration detection.
Design of compound eye optical system with hexagonal band arrangement and common optical path
FAN Chen, LIU Jun, GAO Ming, LV Hong
Abstract(93) FullText HTML(41) PDF 11578KB(124)
In order to solve the challenges of low space utilization and small aperture size for the sub-eye in bionic compound eye systems, a design method for a large aperture compound eye system with a hexagonal band arrangement is proposed in this paper. Using the filling factor theory, taking the traditional curved surface circular arrangement as the control group, it is demonstrated that the hexagonal band arrangement model can effectively improve the space utilization of a large-aperture compound eye system. Aiming at the limited target information acquisition of a single-band compound eye system, an infrared dual-band common optical path imaging form was designed, supplemented by a two-color image sensor, which enhanced the multi-dimensional ability of the compound eye system to obtain information. At the same time, a mathematical model of the sub-aperture positioning of the hexagonal band arrangement is established. The bionic compound eye system is composed of 91 sub-apertures with an entrance pupil diameter of 16 mm, a focal length of 48 mm and a field of view of 9°. The combined total field of view of the sub-apertures is 96°×85°. The focal length of the relay system is 6.14 mm. In a temperature range of −40 °C~+60 °C, the sub-aperture and the relay systems basically have no influence from thermal differences. The cold reflection effect of the detector can be ignored. The simulation results show that the Root Mean Square (RMS) radius of each sub-channel is smaller than the airy spot and the optical distortion value of each sub-channel is less than 0.1%. The Modulation Transfer Function (MTF) of the edge sub-channel in the MWIR/LWIR band is above 0.5 at 17 lp/mm. The system has a compact structure and strong detection ability, and can be used for multi-target detection and recognition in complex environments.
LIU Qiang, ZHAO Jin, SUN Yu-dan, LIU Wei, WANG Jian-xin, LIU Chao, LV Jing-wei, WANG Shi-miao, JIANG Yu, PAUL K CHU
2023, 16(1): 174-183. doi: 10.37188/CO.EN.2022-0006
A novel Photonic Quasi-crystal Fiber (PQF) sensor based on Surface Plasmon Resonance (SPR) is designed for simultaneous detection of methane and hydrogen. In the sensor, Pd-WO3 and cryptophane E doped polysiloxane films deposited on silver films are the hydrogen and methane sensing materials, respectively. The PQF-SPR sensor is analyzed numerically by the full-vector finite element method and excellent sensing performance is demonstrated. The maximum and average hydrogen sensitivities are 0.8 nm/% and 0.65 nm/% in the concentration range of 0% to 3.5% and the maximum and average methane sensitivities are 10 nm/% and 8.81 nm/% in the same concentration range. The sensor has the capability of detecting multiple gases and has large potential in device miniaturization and remote monitoring.
Dynamic 3D measurement error compensation technology based on phase-shifting and fringe projection
CAO Zhi-rui
In the process of dynamic 3D measurement based on phase-shifting and fringe projection, the ideal correspondence between object points, image points and phases in different fringe images is destroyed. On this condition, the application of traditional phase formulas will cause significant measurement errors. In order to reduce the dynamic 3D measurement error, the basic principle of the error is firstly analyzed, and the errors are equivalent to the phase-shifting errors between different fringe images. Then, a dynamic 3D measurement error compensation method is proposed, and this method combines the advanced iterative algorithm based on least squares and the improved Fourier assisted phase-shifting method to realize the high-precision calculation of random step-size phase-shifting and phase. The actual measurement results of a precision ground aluminum plate show that the dynamic 3D measurement error compensation technology can reduce the mean square errors of dynamic 3D measurement by more than one order of magnitude, and the dynamic 3D measurement accuracy after compensation can be better than 0.15mm.
The poynting vector and angular momentum density of Cosh-Pearcey-Gaussian vortex beams in uniaxial crystals
LIANG Meng-ting, CHENG Ke, SHU Ling-yun, LIAO Sai, YANG Ceng-hao, HUANG Hong-wei
We investigate a family of Cosh-Pearcey-Gaussian Vortex (CPeGV) beams, obtain the general propagation expressions of a CPeGV beam, and study the longitudinal and transverse Poynting vector and Angular Momentum Density (AMD) when the CPeGV beams propagate in uniaxial crystals. The effects of the cosh modulation parameter, topological charge, and propagation distance on the propagation properties of CPeGV beams are discussed. A larger cosh modulation parameter can lead the energy transfer significantly along the transverse Poynting vector direction. Moreover, we also investigate how the cosh modulation parameter and topological charge influence the propagation properties in the far-field. A larger cosh modulation parameter can lead AMD to present four-lobe structures rather than their usual parabolic curve. Our investigation will provide a better understanding of the state of the CPeGV beams propagating in uniaxial crystals and be useful for applications in information transmission.
Visible light emission of ultraviolet polarization sensitive CsPbBr3 nano-films
JI Yu-jin, CHU Xue-ying, DONG Xu, LI Jin-hua
In order to detect polarized ultraviolet light by visible optical elements, CsPbBr3 nanocrystal/metal wire-grid composited films were prepared. The stability of its fluorescence was improved by depositing Al2O3 passivation layer. The green fluorescence of polarization-sensitive perovskite nanocrystals film was obtained under ultraviolet exciting light. The results show that the crystal structure of the CsPbBr3 nanocrystals obtained by hot-injection method have a cubic crystal system structure with a square shape and an average size of about 39 nm. An obvious green fluorescence at about 530 nm were observed under ultraviolet light excitation of the nanocrystal colloidal solution. The fluorescence intensity of the CsPbBr3 nanocrystal/metal wire-grid composited film obtained by self-assembly changed periodically with the polarization direction of the excited light. The luminous polarization ratio is about 0.54. The fluorescence intensity of this composite film was enhanced when Al2O3 was deposited on its surface by atomic layer deposition technology. The polarization ratio of the passivated film can still reach 0.36. The above results show that the fluorescence stability and polarization of perovskite nanocrystals film can be optimized by the surface passivation and the introduction of metal wire-grids, respectively. The obtained ultraviolet polarization sensitive CsPbBr3 nanocrystals composited film exhibits important application value in the fields of ultraviolet polarization detection and liquid crystal display.
Multiple Fano resonance properties of nanoring-heptamer metal-dielectric structures
LV Jing-wei, WANG De-bao, LIU Chao, LIU Qiang, WANG Jian-xin, YANG Lin, MU Hai-wei, PAUL K CHU
In order to achieve tunable multiple Fano resonance characteristics and design a refractive index sensor with high sensitivity, a nanoring-heptamer metal-dielectric composite nanoantenna structure is proposed, and the influencing factors and variation rules of its Fano resonance characteristics are studied by using the Finite Element Method (FEM). Researches show that Fano resonance characteristics of the hybrid nano-antenna is sensitive to the changes of the height, incident angle and internal gap. In addition, the electric intensity and the Purcell factor (PF) under the excitation of the electric dipole source can reach 134.74 V/m and 3214 respectively, which greatly enhances the electric intensity near the center of the nanoantenna. The hybrid nanoantenna has high Sensitivity (S) (1400 nm/RIU) and Figure of Merit (FOM) (17 RIU−1), respectively, which can be used as two significant performance indices for evaluating the refractive index sensor with high sensitivity. This paper provides a feasible way to realize the tunability of Fano resonance in the composite nanoantenna and a solid theoretical basis for practical applications such as surface-enhanced Raman scattering, quantum emitters, and refractive index sensors.
2023, 16(1): 228-228.
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Resolution, super-resolution and spatial bandwidth product expansion——some thoughts from the perspective of computational optical imaging
ZUO Chao, CHEN Qian
2022, 15(6): 1105-1166. doi: 10.37188/CO.2022-0105
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Conventional optical imaging is essentially a process of recording and reproducing the intensity signal of a scene in the spatial dimension with direct uniform sampling. In this process, the resolution and information content of imaging are inevitably constrained by several physical limitations such as optical diffraction limit, detector sampling, and spatial bandwidth product of the imaging system. How to break these physical limitations and obtain higher resolution and broader image field of view has been an eternal topic in this field. In this paper, we introduce the basic theories and technologies associated with the resolution, super-resolution, and spatial bandwidth product expansion, as well as some examples in the field of computational optical imaging. By placing these specific cases into the higher dimensional framework of "computational optical imaging", this paper reveals that most of them can be understood as a "spatial bandwidth regulation" scheme, i.e., a process of exploiting the available degrees of freedom of the imaging system to optimally encode, decode, and transmit information within the constraints of the limited spatial bandwidth of the imaging system, or figuratively speaking - "dancing with shackles". This is essentially a legal trade-off and choice between "gain" and "loss" under physical constraints. The conclusions of this paper are expected to provide valuable insights into the design and exploration of new imaging mechanisms and methods for various complex practical imaging applications.
In-vivo across-scales two-photon microscopic imaging technique
CHEN Shuai, REN Lin, ZHOU Zhen-qiao, LI Min, JIA Hong-bo
Two-photon microscopy's ability to maintain good spatial resolution in thick biological tissues has led to its application in in-vivo brain imaging studies soon after its conception. As neural networks have cross-scale multidimensional spatio-temporal properties, two-photon microscopy has developed rapidly and significantly in recent years to meet the demand for in-vivo cross-scale imaging of the brain. This paper firstly introduces the working principle of two-photon microscopy, then reviews the progress of two-photon microscopy from five perspectives: imaging field of view, imaging flux, imaging depth, resolution, miniaturization, and analyzes the difficulties and future challenges of cross-scale two-photon in-vivo microscopic imaging technology.
Multi-channel multiplexing digital holographic imaging for high throughput
HUANG Zheng-zhong, CAO Liang-cai
Optical imaging has become the dominant method for characterizing information in biological systems. The rapid, non-destructive and comprehensive characterization of biological samples in recent years has placed high demands on the resolvable volume of imaging systems. Digital holography records an entire complex wavefront including both the amplitude and phase of the light field by interference imaging. Due to fast, non-destructive, and 3D imaging abilities, digital holography has been used in numerous applications such as digital pathology, label-free observation and real-time monitoring of in vitro cells. First, this paper introduces the main ways to achieve high-throughput imaging, and analyzes the advantages of digital holography and the evolution of spatial bandwidth. Secondly, a theoretical framework for high-throughput multi-channel multiplexing digital holography based on the Hilbert transform is presented. Then, an extended field of view digital holographic microscope is introduced based on this theoretical framework. Experimental results indicate that the system achieves 8 times the space-bandwidth product higher than that of conventional off-axis holographic microscopes without sacrificing spatial and temporal resolution. This high-throughput digital holographic multiplexing technology can make full use of the redundant spatial bandwidth of single intensity image, which verifies the feasibility of high-throughput multi-channel multiplexing digital holography.
Large field-of-view optical microscopic imaging technology
WANG Yi-qiang, LIN Fang-rui, HU Rui, LIU Li-wei, QU Jun-le
With the characteristics of real-time, high-resolution and non-invasive, optical microscopy can scale from cells, tissues to whole living organisms, which has greatly expanded our understanding to the nature of life. However, due to the limited Space-Bandwidth Product (SBP), it is hard for a conventional optical microscope to achieve a large field of view with a high resolution. This makes it very difficult for microscopic imaging in large field of view biological imaging applications, such as imaging of neural circuits between the synapse of the brain neural networks. Recently, large field-of-view imaging technology has received increasing attention and experienced rapid development. The SBP has been improved ten times or even a hundred times as compared to a traditional optical microscope and the field-of-view has been expanded without sacrificing resolution, which, in turn, has resolved some major problems in biomedical research. This review introduces the progress, characteristics and corresponding biological applications of several typical trans-scale optical imaging techniques in recent years, and gives an outlook on their future development.
Recent progress on the reconstruction algorithms of structured illumination microscopy
ZHOU Bo, WANG Kun-hao, CHEN Liang-yi
2022, 15(6): 1211-1227. doi: 10.37188/CO.EN.2022-0011
As an early component of modern Super-Resolution (SR) imaging technology, Structured Illumination Microscopy (SIM) has been developed for nearly twenty years. With up to ~60 nm wavelengths and 564 Hz frame rates, it has recently achieved an optimal combination of spatiotemporal resolution in live cells. Despite these advantages, SIM also suffers disadvantages, some of which originated from the intrinsic reconstruction process. Here we review recent technical advances in SIM, including SR reconstruction, performance evaluation, and its integration with other technologies to provide a practical guide for biologists.
Advances in organic fluorescent probes for super-resolution imaging of cellular lipid droplets
ZHOU Ri, WANG Chen-guang, LU Ge-yu
Lipid droplets are a kind of spherical organelle in eukaryotic cells and are relevant to many cellular physiological processes. Fluorescence imaging techniques are one of the most powerful tools to visualize and study lipid droplets. However, conventional wide-field microscopy and confocal microscopy can only provide a resolution of about 250 nm due to the limitation of optical diffraction. This resolution is quite insufficient for visualizing the small lipid droplets, especially the nascent ones (size of about 30~60 nm). Emerging super-resolution microscopes that can break the diffraction limit (such as stimulated emission depletion microscopy, structured illumination microscopy and photoactivated localization microscopy) have gradually attracted much interest in recent years. To obtain high-resolution fluorescence images of lipid droplets, the advanced fluorescent probes which meet the special requirements of the corresponding super-resolution microscopes are highly essential. This review paper will briefly introduce the working principles of various super-resolution microscopes, discuss the special requirements on the photophysical properties of fluorescent probes, and systematically summarize the research progress of super-resolution imaging of lipid droplets by employing these fluorescent probes. Meanwhile, this review will compare the advantages and shortcomings of different super-resolution techniques for lipid droplets imaging, and prospect their future possible trends.
Advances in multi-dimensional single molecule imaging
LI Meng-fan, CHEN Jian-wei, SHI Wei, FU Shuang, LI Yun-ze, LUO Ting-dan, CHEN Jun-fan, LI Yi-ming
Single-molecule imaging is widely used for the reconstruction of three-dimensional subcellular structures. The point spread function is an important window to analyze the information of a single molecule. Besides 3D coordinates, it also contains abundant additional information. In this paper, we reviewed the recent progress of multi-dimensional single-molecule imaging, including spatial location, fluorescence wavelength, dipole orientation, interference phase, etc. We also briefly introduced the latest methods for molecule localization and proposed the further directions for its research.
Multi-target panoramic digital pathology: from principle to application
ZHANG Xin-hua, LI Cai-wei, ZHANG Yu, HUANG Sheng-nan, SHI Han, WU Jun-nan, REN Shi-jie, LIU Ke-han, GAO Tong-lu, SHI Bing
Digital pathology has brought new opportunities for remote pathological consultation and joint consultation owing to its convenient storage, management, browsing and transmission. However, because of the limited field of view of a microscope, panoramic imaging cannot be achieved while ensuring a high resolution. The proposal of panoramic digital pathology makes up for this defect and achieves panoramic imaging while ensuring high resolution. However, a single slice can only detect a single target, and disease diagnosis needs to observe the expression of multi-target at the same time. In recent years, multi-target panoramic digital pathology technology has developed rapidly. It has attracted much attention because of its great application potential in drug research and development, clinical research and basic research. Owing to its large field of view, wide range of colors and high flux, the system can detect the expression of various biomarkers on a whole tissue section in situ in a short time to identify the phenotype, abundance, state, and relationship of each cell. Firstly, this paper reviews the development process of digital pathology, panoramic digital pathology and multi-target panoramic digital pathology, as well as the update and iteration of technology in the development process, and illustrates the importance of developing multi-target panoramic digital pathology. Then, the multi-target panoramic digital pathology is described in detail from three perspectives: biological sample preparation, multi-color imaging system and image processing. Next, the applications of multi-target panoramic digital pathology in biomedical fields, such as tumor microenvironments and tumor molecular typing are described. Finally, the advantages, challenges and future development of multi-target panoramic digital pathology are summarized.
Recent development of cryo-correlated light and electron microscopy
LU Jing, LI Wei-xing, XU Xiao-jun, JI Wei
Cryo-electron tomography (cryo-ET) has become a cutting-edge technology in life sciences for the investigation of protein complexes directly in their natural state. In cryo-ET, the sample's thickness must be less than 300 nm and the target molecule must be within the lamella, which is prepared by cryo-Focus Iron Beam (FIB) milling. In order to precisely navigate molecules and to improve the efficiency of sample preparation, cryo-Correlative Light and Electron Microscopy (cryo-CLEM) has been introduced to perform in-situ imaging on the frozen samples. The cryo-CLEM combines the localization advantages of fluorescence imaging with the resolution advantages of electron microscopy. By registering images of light and electrons, frozen samples can be thinned by FIB milling, so the efficiency of cryo-ET sample preparation can be improved. In this paper, we review the latest progress and applications of cryo-CLEM technologies, with a particular focus on super-resolution cryo-CLEM imaging and integrated cryo-CLEM. The advantages and limitations of various methodologies, as well as their application scope, are discussed. A discussion on cryo-CLEM's limitations and potential directions for its future development are also presented.
Trans-scale optical endoscopy imaging technology
WANG Zi-chuan, ZHANG Wei, GUO Fei, JIA Zhi-qiang, WANG Li-qiang, DONG Wen-fei, YANG Qing
Due to the advantages of high resolution, multi-scale, multi-dimension, low radiation and easy to integrate, optical imaging technology plays an important role in biomedical field. In the field of endoscopy, how to obtain, process and visualize the endoscopic image information is the core of the problem what optical imaging technology need to solve. The obtaining of trans-scale endoscopic image of patients in the medical clinical is more advantageous to the surgeon for the diagnosis of patients and can improve in accuracy of the operation. The review starts with the application of trans-scale optical imaging technology in the field of endoscopy, focusing on the different optical systems to obtain trans-scale images in clinical endoscopy, including trans-scale optical zoom system, multi-channel imaging system, fiber-scanning imaging system, and expounds its progress and future trends.
Review of optical systems′ desensitization design methods
MENG Qing-yu, QIN Zi-chang, REN Cheng-ming, QI Yun-sheng
The effective realization of desired optical system performances depends not only on the design results of imaging quality, but also on the realizability of various tolerances such as optical manufacturing tolerances, assembly tolerances, and environmental tolerances. An optical system with low error sensitivity relaxes tolerance requirements, which can better resist image quality degradation disturbed by errors. While reducing manufacturing costs, it effectively improves the realizability of an optical system, thereby reducing error sensitivity. It is an important link that should be considered in optical system design. This paper analyzes and summarizes the research status of optical system error sensitivity, summarizes typical optical system desensitization methods, and summarizes the application of these methods in optical system design. Finally, potential future development directions for low error sensitivity design methods for optical systems are provided.
Research progress of single molecule biological detection methods and applications
ZHOU Wen-chao, LI Zheng-hao, WU Jie
Single molecule biological detection technology is an efficient technology to understand the dynamic characteristics of various biomolecules at the single molecule level and explore their structure and function. The advantage of this technology is that it can detect the heterogeneity of free energy on a single molecule, which is beyond the traditional methods. Therefore, researchers use it to solve long-standing problems in complex biological systems, heterogeneous catalysis, biomolecular interactions, enzyme systems and conformational changes. In terms of medical detection, detecting specific information about single molecules or their interactions with biological factors is not only crucial for the early diagnosis and treatment of various diseases such as cancer, but also has great potential for real-time detection and precision medicine. The advantages of high specificity and high precision of single-molecule bioassays are used to real-time detection of single biomolecules in molecular populations, and can be combined with multiple high-throughput analysis for the precise diagnosis of clinical samples. In this paper, the principle of single molecule detection and the application of biosensing are introduced, and the detection methods and related applications are summarized. Finally, the prospect and development direction of this research direction are discussed.
Recent advances in lateral mode control technology of diode lasers
WANG Li-jie, TONG Cun-zhu, WANG Yan-jing, LU Huan-yu, ZHANG Xin, TIAN Si-cong, WANG Li-jun
High power diode lasers are widely used for pumping solid-state lasers and fiber lasers, material processing, laser radars, free-space optical communication, security and defense. However, conventional diode lasers suffer from large far-field divergence angles, poor beam quality and low brightness, which restricts their direct applications. Broad-Area diode Lasers (BALs) can achieve high output power and efficiency. However, their lateral mode is usually influenced by many physical mechanisms, leading to a large number of guided lateral modes at high-power operation. It results in a rapid increase of the far-field width and strongly deteriorated beam quality, limiting the improvement of diode lasers′ brightness. Therefore, the lateral modes should be carefully controlled. In this paper, the factors influencing the diode lasers′ lateral modes are reviewed, and the lateral mode characteristics, optical field distribution and their relations with the device construction are analyzed. Then, the current lateral mode control technologies are described in detail. The beam quality and brightness of the output beam can be enhanced via the suppression of high-order lateral modes and the far-field blooming effect. As a result of advanced lateral mode control, novel high-brightness diode lasers can be developed at the chip level, which is beneficial for developing new diode lasers applications and reducing their system cost.
Review of ultraviolet photodetectors based on micro/nano-structured wide bandgap semiconductor oxide
CHEN Xing, ZHOU Chang, LIU Ke-wei, SHEN De-zhen
Ultraviolet photodetection technology is another dual-use detection technology after infrared detection and laser detection technology, which has broad application prospects. Vacuum photomultiplier tubes and Si-based photodiodes are common commercial UV detectors, but vacuum photomultiplier tubes are susceptible to high temperatures and electromagnetic radiation, and need to work under high pressure while Si-based photodiodes require expensive filters. Wide bandgap semiconductor ultraviolet photodetectors have overcome some of the problems faced by the above two devices, and are becoming the research hotspot. Among them, wide bandgap oxide materials have attracted extensive attention, due to the advantages of easy preparation for high response and high gain devices, and rich micro-structures and nano-structures. In this paper, ultraviolet photodetectors based on micro/nano-structured wide bandgap semiconductor oxide are combed, and some related researches in recent years are reviewed. The oxide materials involved include ZnO, Ga2O3, SnO2 and TiO2, etc. and the device structures involved include metal-semiconductor-metal devices, Schottky junction devices and heterojunction devices, etc.
Review of physical implementation architecture in compressive spectral imaging system
LI Yun-hui
Different from the traditional point-to-point mapping imaging method, computational optical imaging combines the physical regulation of the front-end optical signal with the processing of the back-end digital signal to make the image information acquisition more efficient. This new imaging mechanism is expected to alleviate the contradiction between low manufacturing cost and high performance indicators under the framework of traditional imaging technology, especially in the acquisition of high-dimensional image information. Since the system architecture supported by physical devices is the cornerstone of computational optical imaging, aiming at the sub-technical field of compressive spectral imaging, in this paper, we introduce the existing optical devices that can realize spatial or spectral modulation. Based on this, the architecture of multi-type compressive spectral imaging system is sorted out and summarized, which can be categorized as single-pixel spectral imaging, coded aperture spectral imaging, spatial-spectral dual-coded spectral imaging, microarray spectral imaging and scattering medium spectral imaging, based on the information modulation process. We focus on the information modulation and acquisition principles of various system architectures and their modulation effects on the spatial-spectral data cube, and then analyze and explore the common issues. Finally, the technical challenges faced are given, and the future development trend is discussed.
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Tagged: linear combination
by Yu · Published 03/19/2018
Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices
Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.
Read solution
Click here if solved 86
Add to solve later
Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
-1 \\ -2 \\ -1
2 \\ 6 \\ -2
\right\}$.
How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix
Let $A=\begin{bmatrix}
2 & 4 & 6 & 8 \\
1 &3 & 0 & 5 \\
1 & 1 & 6 & 3
\end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.
(b) Find a basis for the row space of $A$.
(c) Find a basis for the range of $A$ that consists of column vectors of $A$.
(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.
Click here if solved 104
Can We Reduce the Number of Vectors in a Spanning Set?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?
Does an Extra Vector Change the Span?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set
\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.
Find a Basis for Nullspace, Row Space, and Range of a Matrix
If $\mathbf{v}, \mathbf{w}$ are Linearly Independent Vectors and $A$ is Nonsingular, then $A\mathbf{v}, A\mathbf{w}$ are Linearly Independent
Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent.
Compute $A^5\mathbf{u}$ Using Linear Combination
\[A=\begin{bmatrix}
-4 & -6 & -12 \\
-2 &-1 &-4 \\
\end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix}
6 \\
\end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix}
-2 \\
\end{bmatrix}, \quad \text{ and } \mathbf{w}=\begin{bmatrix}
\end{bmatrix}.\]
(a) Express the vector $\mathbf{u}$ as a linear combination of $\mathbf{v}$ and $\mathbf{w}$.
(b) Compute $A^5\mathbf{v}$.
(c) Compute $A^5\mathbf{w}$.
(d) Compute $A^5\mathbf{u}$.
Spanning Sets for $\R^2$ or its Subspaces
In this problem, we use the following vectors in $\R^2$.
\[\mathbf{a}=\begin{bmatrix}
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
\end{bmatrix}, \mathbf{c}=\begin{bmatrix}
\end{bmatrix}, \mathbf{d}=\begin{bmatrix}
\end{bmatrix}, \mathbf{e}=\begin{bmatrix}
\end{bmatrix}, \mathbf{f}=\begin{bmatrix}
\end{bmatrix}.\] For each set $S$, determine whether $\Span(S)=\R^2$. If $\Span(S)\neq \R^2$, then give algebraic description for $\Span(S)$ and explain the geometric shape of $\Span(S)$.
(a) $S=\{\mathbf{a}, \mathbf{b}\}$
(b) $S=\{\mathbf{a}, \mathbf{c}\}$
(c) $S=\{\mathbf{c}, \mathbf{d}\}$
(d) $S=\{\mathbf{a}, \mathbf{f}\}$
(e) $S=\{\mathbf{e}, \mathbf{f}\}$
(f) $S=\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$
(g) $S=\{\mathbf{e}\}$
How to Obtain Information of a Vector if Information of Other Vectors are Given
Let $A$ be a $3\times 3$ matrix and let
\[\mathbf{v}=\begin{bmatrix}
\end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix}
\end{bmatrix}.\] Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$.
Then find the vector
\[A^5\begin{bmatrix}
by Yu · Published 01/15/2018 · Last modified 01/16/2018
The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$
For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,
\[T (f) (x) = x f(x).\]
Prove that $T$ is a linear transformation, and find its range and nullspace.
Prove that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $2$ or Less
Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less.
(a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$.
(b) Write the polynomial $f(x) = 2 + 3x – x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 \}$.
A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix
Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.
Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$.
Write a Vector as a Linear Combination of Three Vectors
Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors
\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]
Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent
Prove that any set of vectors which contains the zero vector is linearly dependent.
Orthogonal Nonzero Vectors Are Linearly Independent
Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$.
Suppose that $S$ is an orthogonal set.
(a) Show that $S$ is linearly independent.
(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.
Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less
Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x)\}$, where
\[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.\]
(a) Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for $P_2$.
(b) Find the coordinate vector of $p(x)=x^2+2x+3\in P_2$ with respect to the basis $S$.
The Subspace of Linear Combinations whose Sums of Coefficients are zero
Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } a_1+a_2+\cdots+a_k=0\}.\] So each element of $W$ is a linear combination of vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ such that the sum of the coefficients is zero.
Prove that $W$ is a subspace of $V$.
Determine Whether Each Set is a Basis for $\R^3$
Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix} \,\right\}$
(b) $S=\left\{\, \begin{bmatrix}
(c) $S=\left\{\, \begin{bmatrix}
(d) $S=\left\{\, \begin{bmatrix}
If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\] is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
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Pharmacist intervention in the prevention of heart failure for high-risk elderly patients in the community
Vivian W. Lee1,
Libby M. Choi1,
Winki J. Wong1,
Ho Wing Chung1,
Carman K. Ng1 &
Franco W. Cheng1
Heart failure has become one of the major causes of hospitalization worldwide. Hypertension, diabetes mellitus and hyperlipidemia are the major causes of heart failure. In order to effectively prevent heart failure, blood pressure, blood glucose and cholesterol levels shall be closely monitored and controlled as well as medication adherence. This study aimed to investigate the role of pharmacist intervention in prevention of heart failure in high risk elderly patients in the community of Hong Kong.
This study aimed to investigate the role of pharmacist intervention in prevention of heart failure in highrisk elderly patients in the community of Hong Kong.
This prospective uncontrolled study was conducted between July 2012 and April 2013 with two revisits every 3 to 4 months to review elderly patients' medication profiles as well as assess their blood pressure (BP), random capillary blood glucose (RCBG) level, cholesterol levels, signs and symptoms of heart failure and the compliance level. The results collected at the baseline data were analyzed and compared with those collected at the last follow-up visit.
A significant increase in number of subjects free of symptoms of heart failure (31.88%, p < 0.001) was found. For chronic disease management, significant reduction in LDL-cholesterol level (-0.86 ± 0.56mmol/L, p = 0.038) and triglyceride level (-1.15 ± 1.09mmol/L, p < 0.001) was observed in overall participants. Yet, no significant reduction in BP level or RCBG level was observed in overall subjects. Significant reduction in mean Morisky Medication Adherence Score (-0.54 ± 1.50, p = 0.005) indicated improvement in medication compliance in participants.
The Pharmacy Outreach service has a significant role in prevention of heart failure, by means of minimizing heart failure symptoms, improving medication compliance and enhancing chronic disease management, particularly cholesterol management in community elderly patients. This study provided a reference for further investigation and evaluation of the role of pharmacists in preventing heart failure in the high-risk community elderly patients.
Heart Failure (HF) is a serious condition worldwide, with around 5.7 million people in United States of America being the victims. Also, the disease was found to be a contributing cause in more than 280,000 deaths (1 in 9) in 2008 [1].The American Heart Association has also suggested about half of people who have heart failure die within 5 years of diagnosis [1] and hence the problem should not be neglected. It has been reviewed that the population suffering from heart failure is expanding in Asia, particularly in areas like Hong Kong, due to improved living standards [2]. In Hong Kong, cardiovascular diseases rank the second leading cause of death in age group 65 or above according to the Department of Health [3]. These findings imply primary interventions must be carried out promptly to deal with the problem, in accordance with the Heart Failure Society of America (HFSA) suggestion that early identification and treatment of risk factors are recognized of utmost importance in limiting the public health impact of heart failure [4].
It has been investigated that hypertension [4, 5, 7], diabetes mellitus [4, 5, 6, 7] and hyperlipidemia [4, 6] are the major causes of HF. To effectively prevent HF, blood pressure (BP), blood glucose and cholesterol level shall be closely monitored and controlled. Moreover, medication compliance plays an incumbent role to achieve desirable therapeutic outcomes [7]. It reduces incidents of hospitalization as well as the healthcare cost [8] in community elderly patients, their compliance to medical regimen shall be emphasized to prevent HF. A foreign study has shown that home-based intervention performed by pharmacists could significantly reduce mortality and recurrent hospitalization in HF patients [9]. Meanwhile, local studies suggest that a pharmacy outreach service (POS) for community elderly patients can effectively improve BP control [10] and drug related problems (DRPs) [11]. Besides, it has been illustrated that pharmacists have an established role in managing DRPs in elderly patients, including drug-drug interactions and therapeutic duplications [12]. This study aimed to investigate the role of pharmacist intervention in the early identification of HF symptoms in high-risk elderly patients in the community. We therefore conducted a series of POS visits to perform pharmacist interventions on managing the risk factors of HF in the high-risk community elderly patients, hypertension, diabetes, hyperlipidemia as well as reviewing their medication compliance, for the prevention of HF.
This was a prospective uncontrolled study of which subjects were recruited from seven elderly centers in Hong Kong during the Pharmacy Outreach Service in July and August 2012. Patients with recent symptoms of HF were included in this study. These symptoms were evaluated by a questionnaire designed to correspond to the New York Heart Association (NYHA) functional classification of HF. High-risk subjects would also be selected, defined as two out of three of the following criteria were met:
High blood pressure as defined in Joint National Committee (JNC) 7 [13]: ≥ 130/80mmHg for diabetic patients and ≥ 140/90mmHg for non-diabetic patients
High blood glucose as defined in Standard of Medical Care in Diabetes 2012 [14]: RCBG ≥ 11.1mmol/L
Dyslipidemia as defined in ATP III [15]: LDL level ≥ 2.59mmol/L or TG ≥ 2.3mmol/L
Patients who were younger than 65 years old or diagnosed with the following medical conditions, namely malignancies, dementia, psychiatric disorders, Alzheimer's diseases, Parkinson's diseases or epilepsy, were excluded in the study. Patients without the need of pharmacological management of their hypertension, diabetes, or dyslipidemia were also excluded. Other exclusion criteria include previous participations in the Chinese University of Hong Kong (CUHK) Outreach Service, current involvement in other clinical trials and the presence of communication barriers. The current project was approved by the Joint Chinese University of Hong Kong-New Territories East Cluster (CUHK-NTEC) Clinical Research Ethics Committee (Reference number: 2014.012). All recruited subjects had been consented prior to joining the current project.
Subjects were revisited twice every 3 to 4 months from October 2012 to November 2012 and from February 2013 to April 2013. At each visit, subjects' BP, RCBG level and cholesterol level were measured, and their demographic data and medication profiles were reviewed and recorded. The Omron HEM-7011 electronic blood pressure monitor (Omron Healthcare, Kyoto, Japan) was used for measuring the blood pressure. This electronic device had achieved an "A/A" performance classification under the British Hypertension Society criteria and passed the Association for the Advancement of medical instrumentation requirements [16]. The random blood glucose level was measured with Accu-Chek® Performa (Roche Diagnostics, Switzerland). The lipid panel of each subject was measured with a CardioChek® Analyzer (Polumer Technology Systems Inc., USA). By using the battery-operated analyzer, four categories of measurements were collected, namely total cholesterol (TC), high-density lipoprotein (HDL), low-density lipoprotein (LDL) and triglyceride (TG), from fingerstick capillary samples [17]. The test system is intended for in vitro diagnostic use to test whole blood [18].
The workflow of each visit is summarized in Fig. 1. Pharmacists counseled on the use of medications, disease knowledge and provided non-pharmacological measures for disease management. Interventions were made with regard to any DRPs identified. Additionally, evaluation on signs and symptoms of HF by a questionnaire (Additional file 1) corresponding to the NYHA functional classification and assessment on medication compliance by the Morisky 8-item Medication Adherence Scale were performed at the first and last visit.
Workflow of each visit
The baseline data were compared to those obtained in the last follow-up visit in order to analyze the impacts of POS on community elders' management of risk factors associated with HF. McNemar's test was used for the assessment of HF symptoms while paired t-test was utilized to detect changes in BP, RCBG, cholesterol levels and Morisky Medication Adherence Scale. According to previous POS studies [19–21], mean standard deviation of the change of systolic and diastolic blood pressures are 10.67 and 8.55mmHg respectively.
The number of subject needed was calculated using the equation below [22]:
$$ n=2\left[\frac{\sigma \left({z}_{\alpha }+{z}_{\beta}\right)}{\mu_1-{\mu}_2}\right] $$
At least 40 subjects would be required to identify a 10mmHg change of the systolic blood pressure and at least 26 subjects would be needed to identify a change in 10mmHg of diastolic blood pressure with a power of 80%, a significance level of 0.05 and with an assumption of 40% drop-out rate. Therefore, at least 66 subjects would be required.
All statistical analysis were performed using SPSS 16.0 and a p value of <0.05 was defined as statistically significant.
We screened 297 patients and a total of 103 patients were recruited. The demographic characteristics and medical history of participants is summarized in Table 1. The mean age of participants was 78.19 ± 6.87 years old, and they were taking an average of 5.08 ± 3.02 chronic medications. For the HF symptoms assessment, a significant increase in number of 31.88% subjects (p < 0.001) were found to be free of symptoms of HF, spanning the period between baseline and the last visit. A significant reduction of mean number of HF symptoms experienced by each subject was observed (p < 0.001). In addition, there were fifteen less elderly patients (-21.74%, p = 0.001, Table 2) experienced four or more symptoms at the end of the study. It was observed that more than half (52.17%) of the subjects reported to have limitation of physical activity at baseline and the number was significantly reduced at the end of the study (-26.09%, p = 0.001) based on the questionnaire. The other HF symptoms that were found to be sensitive to pharmacist intervention included, shortness of breath (-18.84%, p = 0.004), fatigue easily (-17.39%, p = 0.029), edema (-17.39, p = 0.008) and physical activity limitation when climbing stairs in particular (-18.84%, p = 0.015) (Table 2).
Table 1 Baseline characteristics of the study participants
Table 2 Summary of various heart failure symptoms experienced by subjects based on the heart Failure Symptoms Assessing Questionnaire at baseline and latest follow-up visit, N = 69
For the impact of POS on the risk factors management of heart failure, namely hypertension, diabetic mellitus and hyperlipidemia, no significant improvement was observed in BP and glycemic control in overall participants. However, significant reduction in SBP level (-14.45 ± 17.97mmHg, p = 0.024) was achieved in subjects with Stage II hypertension. The reduction of RCBG level (-3.70 ± 2.44mmol/L, p = 0.004) was also significant in not well-controlled subjects (Table 3). For lipid control, significant improvement was achieved in overall participants, with significant reduction in LDL-cholesterol level (-0.86 ± 0.56mmol/L, p = 0.038), and in triglyceride level (-1.15 ± 1.09mmol/L, p < 0.001).
Table 3 Comparison of blood pressure, random capillary blood glucose and cholesterol level between baseline and the latest follow-up visit
Non-compliance to medications is also a contributing factor of HF, 66.99% of the elderly patients were identified to be medication non-compliant. The Morisky score results showed a significant reduction in mean score by 0.54 ± 1.50 (p = 0.005) (Table 4). At the baseline, 27.54% of the subjects were considered as low compliance subjects and there was a slight but insignificant reduction in the number at the end of the study (-4.35%, p = 0.607). In addition, results showed a significant increase by 23.19% (p = 0.001) in the number of subjects with high compliance level. Forgetfulness (30.4%) was found to be the most common reason for non-compliance, followed by improper administration time (20.3%) and fear of adverse drug reaction (18.8%).
Table 4 Identified causes of non-compliance and the compliance score at Baseline and the latest Follow-up visit, N = 69
It has been suggested by HFSA that patients at risk of cardiovascular disease are also at high risk for developing HF, and early identification and treatment of risk factors are recognized of utmost importance in limiting the public health impact of HF [4]. Hypertension, hyperlipidemia, diabetes mellitus and unhealthy lifestyle are recognized as potential risk factors for the development of myocardial remodeling, cardiac dysfunction and hence HF. In the study, subjects were being assessed with the presence of HF symptoms using the HF Symptoms Assessment Questionnaire via self-reporting method. There were a total of 16 questions in the questionnaire that correspond to ten symptoms of HF. Significant reduction in number of HF symptoms experienced by each subject as well as number of subjects experienced four or more symptoms were observed, which demonstrated the changes before and after the pharmacists' interventions. Although the changes may not be solely due to the pharmacists' intervention due to the lack of control group in the current study. It demonstrated the potential essential role of pharmacist intervention upon better disease control and minimizing risk of HF in community elderly patients. These results correlate to a previous study suggesting that pharmacists have an incumbent role in the community in HF management [16].
In addition, the role of pharmacist in managing risk factors of HF was also investigated in the study. According to the US NHANES I Epidemiologic Follow-up Study, not well-controlled hypertension was found to be positively and significantly associated to increased risk of HF [17]. This study has shown no significant improvement in the BP management in overall participants, however in subgroup analysis, statistically significant reduction in SBP level was detected in participants with stage-II hypertension at baseline. This result is consistent with a previous study which has shown that hypertensive patients with baseline SBP ≥160mmHg would have a more significant decrease in BP than the others who have lower BP baseline after pharmacist intervention [25]. Therefore, hypertensive patients, who are of higher risk to developing HF, are more sensitive to pharmacist intervention in the prevention of HF.
It was clearly established in prior studies that diabetes is a risk factor for HF [26-29]. In the study, no significant improvement in glycemic control was observed in overall participants. The reason for the insignificant result might be due to the fact that RCBG level was highly subjected to the influence of various confounding factors, such as time elapsed after food intake. As a result, RCBG level might not able to truly reflect the impact of pharmacist intervention on glycemic control in overall participants. Improvement in study design, such as measuring HbA1c level instead of RCBG level, is suggested in future investigation.
HFSA has suggested that hyperlipidemia is a risk factor for the development of HF [4]. Meanwhile, since hyperlipidemia is one of the risk factors for the development of CHD [30], well control of cholesterol levels, particularly LDL-cholesterol level, can reduce risk of CHD and subsequently myocardial remodeling and heart failure. This study has showed that pharmacist intervention significantly reduce LDL-cholesterol level as well as triglyceride level in overall participants. In the sub-group analysis, subjects with not well-controlled cholesterol levels were found to be more sensitive to pharmacist intervention in managing hyperlipidemia. Precedent studies have found that regular pharmacist follow-up and intervention for patients on disease education, cholesterol measurement, medication compliance and referral to physician can improve cholesterol management in high-risk patients [31, 32]. Systemic review also showed that community pharmacy-based service contributed to the reduction in risk factors for CHD regarding lipid management [17]. Therefore, pharmacist intervention in POS may contribute in prevention HF in community elderly patients but further investigation shall be conducted to provide solid evidence on this issue.
Several studies have suggested that non-compliance can be a precipitating factor of HF exacerbation [18, 33] as well as impairing the chronic disease management [34]. Pharmacist therefore are in ideal position to evaluate patients' compliance and aid them to improve their medication compliance. Therefore, it is essential for pharmacist to ensure patients with chronic diseases are compliance with medications to prevent HF or its progression. In this study, the Morisky score results showed a significant improvement in medication compliance among subjects with pharmacist intervention, which provide medical education on individual's regimen. Over 66% of subjects were identified to have non-compliance issue, which is one of the well-known common DRPs in elderly patients [35]. Out of the various reasons behind the non-compliance issue, forgetfulness contributed the most (30.4%). Forgetfulness is classified as an unintentional cause of non-compliance. Previous study [36] illustrates the importance of pharmacist's intervention in providing memory aids, educating measures to take if a dose is missed and suggesting the use of medication compliance calendars. Therefore, additional compliance aids and education shall be provided in the future POS.
The current study has several limitations. There was selection bias during sampling since this was not a randomized trial, which was also reflected in the unequal distribution in gender in the sample. There were more female subjects involved in the POS probably due to their better health-seeking behavior but this was consistent with the previous POS [10]. A randomized controlled trial with cross-over design might be considered not only to reduce the selection bias but also to provide a more concrete evidence for the benefits of pharmacist interventions. Furthermore, only seven elderly centers participated in this study and this may not be truly reflecting the situation in Hong Kong. We did not assess the echocardiography results to confirm the impact of POS. The compliance assessment was based on self-reporting from the patients. The current HF questionnaire developed for this study was not validated and therefore may not be applicable for patients outside Hong Kong. Not all risk factors were examined in this study. The sample size of the current study was small and Hong Kong-based. As a result, the study may be underpowered and not be able to generalize to other countries with a larger population.
Pharmacy outreach service has significantly improved HF symptoms management and medication compliance in the high-risk community elderly patients. It has also improved cholesterol management in overall subjects. Further investigation with improved study design shall be conducted in order to evaluate and provide sound evidence on pharmacist role in preventing HF in community elderly patients.
Ethical approval
The study was approved by the Ethics Committee, CUHK-New Territories East Cluster, Hospital Authority, Hong Kong.
CUHK:
CUHK-NTEC:
Chinese University of Hong Kong-New Territories East Cluster
DRPs:
Drug-related problems
HF:
HFSA:
HDL:
High Density Lipoprotein
IFCC:
International Federation of Clinical Chemistry and Laboratory Medicine
JNC:
Joint National Committee
LDL:
Low Density Lipoprotein
NYHA:
New York Heart Association
Pharmacy Outreach Service
RCBG:
Random Capillary Blood Glucose
Total Cholesterol
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This study was supported by the School of Pharmacy, The Chinese University of Hong Kong and Give Back to Community Foundation.
School of Pharmacy, Faculty of Medicine, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Vivian W. Lee, Libby M. Choi, Winki J. Wong, Ho Wing Chung, Carman K. Ng & Franco W. Cheng
Vivian W. Lee
Libby M. Choi
Winki J. Wong
Ho Wing Chung
Carman K. Ng
Franco W. Cheng
Correspondence to Vivian W. Lee.
All authors declared that there was no conflict of interest during the study and the preparation of the manuscript.
LMC, WJW, CKN and HWC were the pharmacy students who collected and analyzed data and prepared report for this project. FC was responsible for the interpretation of data and preparation of manuscript. VL was responsible for study design, interpretation of data and logistics of this project. All authors read and approved the final manuscript.
Appendix 1 - Heart Failure Symptoms AssessingQuestionnaire. (DOCX 27 kb)
Lee, V.W., Choi, L.M., Wong, W.J. et al. Pharmacist intervention in the prevention of heart failure for high-risk elderly patients in the community. BMC Cardiovasc Disord 15, 178 (2015). https://doi.org/10.1186/s12872-015-0173-3
Pharmacist intervention | CommonCrawl |
random flight model for polymer
The random-flight model assumes there are no long-range interactions: The most obvious long-range interaction is that a chain must avoid itself: two chain segments cannot be in the same place at the same time. Polymer-RISM theory is applied to study athermal polymeric lattice fluids. A linear flexible polymer chain in the melt- or theta-state can be modeled as a random flight in three dimensions where all intermolecular interactions between polymer chains are neglected which is equivalent to the Gaussian chain model. Can Excluded-volume Inter- Actions Be Nonlocal, And Why? # The end-to-end vector is the position of the last. A continuum version of self-consistent field model for polymer adsorption at the solid−liquid interface has been formulated and solved to obtain configurational statistics of an adsorbed polymer chain. R max = N K l K = N l. As Kuhn and others have shown, the spacial properties of such a model chain are the same as those of a random flight in three-dimensions. Free Aircraft 3D models. Thanks for letting me know,Christian, # Number of random flight segments in the polymer, # pos olds the locations of each segment (Cartesian coordinates), # Pick a new random direction for the next segment uniformly on. Ideal Polymer Chains: Distribution of End-to-End Distance. \Delta x &= a\sin\theta\cos\phi \\ For sure FSCaptain (payware) will generate the flights for you. β m are the roots of a zero order Bessel function of the first kind and r i polymer molecules prefer strongly one conformational state to other states, which are stiff polymers have similar preferences for a number of different torsional states, which are flexible. In polymer physics two common modifications to the Random Walk approach are usually considered and these are called Short Range and Long Range Interactions. The theory of the second virial coefficient, A 2 (M 2,T), for dilute polymer solutions is investigated from the point of view, first developed by Zimm, of the Ursell‐Mayer‐like development in terms which successively represent increasing numbers of contacts between molecular subunits of flexible chain molecules. # flight polymers with the predicted probability distribution function. The inertia ellipsoid displayed in blue is that for the polymer at an instant in time. As a reference, we have indi-cated the 〈R g〉 of both the native and the denatured state. \end{align*}. This random polymer was grown in accordance with the stated scheme. Although the flexible cylinder ellipse polymer model, fitted to the data in SASView, fit the LAR12P scattering data well (Figure S4C), a more detailed model may be justified. This technique has actually become a "routine" analytic characterization method even for the non-experts. The partition coefficient is given by ∞ p RW = 4 -----1 -exp[–(β ) 2 ∑ m r i /R ] (2) β 2 m = 1 m where . The fraction of those configurations corresponding to a displacement length L which are acceptable is calculated through use of a model which considers the chain to be composed of a sequence of identical subchains. The polymer, segment positions are already given relative to the centre of. Kirill Titievsky : 15: Brownian Motion in Energy Landscapes. monochromated neutron beams or time-of-flight instruments at pulsed neutron sources. To illustrate the vast structural diversity of the random coil ensemble of conformations described by the random flight model, Figure 2 shows the distribution of R g values # segment, since we started at the origin. In the above code calculate the position of the polymer's centre of mass, $\mathbf{r}_\mathrm{CM}$, and then shift the origin of the polymer's segment coordinates so that they are measured relative to this point (that is, the segment coordinates have their origin at the polymer centre of mass). Polymer basics and definitions The ideal chain Freely jointed chain (FJC) Freely rotating chain (FRC) Kuhn length and persistence length End-to-end vector distribution function Gaussian chain Force extension relation Worm-like-chain Real chains Conclusion Polymers Overview. These materials have microstructures that exhibit heterogeneity over multiple length scales, which obscures the relationship between properties, processing conditions, and device performance. Along the way a number of key tools from probability theory are encountered and applied. Polymer chains exhibit in several cases a random chain conformation , i.e. Branching formation is known from the beginning of polymer chemistry and modern synthesis ... EJ, Dekmezian AH (2002) Long chain branch polymer chain dimensions: Application of topology to the Zimm–Stockmayer model. The freely-rotating chain improves the freely-jointed chain model by taking into account that polymer segments make a fixed bond angle to neighbouring units because of specific chemical bonding. dimensional random-walk calculation, also called a random-flight calculation. Ah yes, you're right: set_aspect('equal') never worked properly with Matplotlib and now it raises a NotImplementedError:https://github.com/matplotlib/matplotlib/issues/1077#issuecomment-14006279I've removed the offending line. 8 2.2 Conformation of an ideal chain See the answer. Exact Expression for the Bond Probability 11 5b. E 76, 051805 – Published 27 November 2007 # A polymer with 1000 segments of length 0.5, # Compare the observed distribution of end-to-end distances for Np random-. SANS ... SANS is a heavily model-dependent method. Polymer Chain Conformations: Random flight model for polymer, The Gaussian Chain, Chain Conformation under an External Field, Excluded Volume Effect and Theta Condition, Rotational Isomeric States, Scaling Concept. A random sequential adsorption (RSA) model, which focuses on the random distribution of polymer chains on the surface leading to polymer-free spots, was studied. You can pick your airplane and it generates one random flight for you to fly. A class representing a random-flight polymer in solution. Macromolecular chains with random conformation ( random coils or Gaussian chains ) are found in solutions of polymers in good solvents, in polymer melts and in glassy amorphous polymers. Generating random polymers Generalization to graphs Definition: Let G be a graph with edge-lengths e. # Start our polymer off at the origin, (0,0,0). $$, One way to pick the location of the next segment is to pick a random point on the surface of the unit sphere and use the corresponding pair of angles in the spherical polar coordinate system, $\theta$ and $\phi$ (where $0 \le \theta < \pi$ and $0 \le \phi < 2\pi$) to set the displacement from the previous segment's position as In the following code, a random flight polymer with $N$ segments, each of length $a$, is constructed and displayed. A continuum version of self-consistent field model for polymer adsorption at the solid−liquid interface has been formulated and solved to obtain configurational statistics of an adsorbed polymer chain. Such a hypothetical chain is assigned n freely jointed links of equal length, l. If one end of this hypothetical The solid surface is viewed as a singular phase (having zero thickness but finite adsorption capacity) in equilibrium with the solution. Exact Expression for the Bond Probability 11 5b. The random flight model [16] represents a polymer chain as a three dimensional random walk and accounts for the walks that are lost as a result of confinement. sity of the random coil ensemble of conformations described by the random flight model, Figure 2 shows the distribution of R g values calculated for the above-men-tioned freely jointed chain. We have conceived a computer-aided strategy to enable polymeric micelle-based delivery of poorly soluble drugs. The simplest off-lattice model of polymer chains is the random flight model , , , . We can test the Polymer class by importing it in the Python shell: Let's now compare the distribition of the end-to-end distances with the theoretically-predicted probability density function: To describe the conformation of polymer molecules, a model of a random-flight or freely jointed and volumeless chain is often used as the starting point. Computer simulations of polymer chains in restricted environments were carried out by the Monte Carlo technique. A simple model of a polymer in solution treats its monomer units as totally uncorrelated in position (each monomer unit adopts a random orientation with respect to all the others): this is the random flight model. Random loop model for long polymers Manfred Bohn, Dieter W. Heermann, and Roel van Driel Phys. (b) In General, The Loop Closure Probability For A Loop With K Bonds Is Approximately Proportional To 1/k". The total polymer volume fraction is equal to 0,5, while the volume fraction of … Calculates and returns the radius of gyration, Rg. random flight model for polymer conformation applied to flexible polymer chains in the bulk. Polymers For range limited interactions this will always be the case θ θ 1 cos 1 cos − + C∞= C∞is called Flory's characteristic ratio, and can be seen as a measure of the stiffness of the polymer in a given ideal chain model. # Pick a random orientation for the next segment. Semiconducting polymers have promising applications in transistors, light-emitting diodes, and solar cells. # Each polymer consists of N segments of length a, # Output a progress indicator every 100 polymers, # Plot the distribution of Rx as a normalized histogram, # Plot the theoretical probability distribution, Pr, as a function of r, Chapter 9: General Scientific Programming, Chapter 10: General Scientific Programming. We also examined the possibility that additional steric repulsion and apparent thickening could arise from membrane thermal-bending excitations. The Polymer object will also calculate the end-to-end distance, $R$, and will implement a method calc_Rg to calculate and return the polymer's radius of gyration, defined as, $$ The models include ideal chains, Gaussian chains, real chains, wormlike chains, and branched chains. The simplest of these is the random walk or random flight model. # Add on the corresponding displacement vector for this segment. where the mean square position of the segments is $\langle r^2 \rangle = Na^2$. The models treated have been widely used in studying, for example, the phenomena of polymer pinning on a defect line, the behavior of copolymers in proximity to an interface between selective solvents and the DNA denaturation transition. (a) The Variation Of The Average Square End-to-end Distance Of Polymer Conformations With Chain Length N May Be Expressed As (R) - Constant X N? A random coil is a polymer conformation where the monomer subunits are oriented randomly while still being bonded to adjacent units. Drones bodies and parts will delight fans of tiny flying vehicles. Our full kits include all components required to build the model, including hardware. We built models predicting both drug loading efficiency (LE) and loading capacity (LC) using novel descriptors of drug-polymer complexes. The individual bond angle rota-tions represent random variables describing the chain con-figurations, so that when the number of these random vari-ables, corresponding to the possible bond-vectors, gets large rod-like rod-like rod-like Gauche states of torsion angles lead to flexibility in the chain conformation Gaussian chain Polymer chains exhibit in several cases a random chain conformation , i.e. This is termed excluded volume: rather than a polymer chain being a random walk it is a self avoiding walk (SAW). Excluded volume theory- In polymer science , exclusion volume states that one part of the molecule c view the full answer. According to the Halpin–Tsai model, one way to increase the modulus of composites is by increasing the aspect ratio of the filler used. The solid surface is viewed as a singular phase (having zero thickness but finite adsorption capacity) in equilibrium with the solution. a random distribution of T and G states. the excluded volume (self-avoidance) condition is ignored. Under … \Delta z &= a\cos\theta 3 It can next be considered that the lattice model is adaptable to describing a polymer coil. Random selection, when narrowly associated with a simple random sample, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. The theory of the second virial coefficient, A2(M2,T), for dilute polymer solutions is investigated from the point of view, first developed by Zimm, of the Ursell‐Mayer‐like development in terms which successively represent increasing numbers of contacts between molecular subunits of flexible chain molecules. The brighter coloured, larger segments are in the foreground and the … The model has been … The probability density of the span of an N-step adsorbed polymer chain in the direction normal to the solution surface is derived in de Gennes' model for weakly adsorbed chains. For a Gaussian coil the average end to end distance is given by n 1/2 b as for the random walk. All our Peck kits are precision-cut from lightweight balsa on our professional-grade laser cutter. Our factory is specialized in flight model design and manufacture. A random coil is a polymer conformation where the monomer subunits are oriented randomly while still being bonded to adjacent units. To describe the conformation of polymer molecules, a model of a random-flight or freely jointed and volumeless chain is often used as the starting point. Random Flight Model Analysis of Protein-Surfactant Complexes Debasish Saha,1,3 Debes Ray,1 Joachim Kohlbrecher2 and Vinod K. Aswal1, a) 1Solid State … The displacement distribution for a polymer molecule with excluded volume is derived by rejecting the unacceptable configurations available to random‐flight chains. A simple model of a polymer in solution treats its monomer units as totally uncorrelated in position (each monomer unit adopts a random orientation with respect to all the others): this is the random flight model. Polymer Dynamics M. DOI Department of Applied Physics, Nagoya University and S. F. EDWARDS Cavendish Laboratory, University of Cambridge CLARENDON PRESS • OXFORD . Furthermore, several sample conformations of varying Random Walk in an External Force Field, Einstein Relation, Boltzmann Equilibrium, Ornstein-Uhlenbeck Process, Ehrenfest Model. Statistics of Ideal Polymer Chains: Random-Flight Problems 5 3. The polymer consists, at start-up, of 1000 segments of length unity and these are free to rotate through any angle with respect to adjacent segments (random flight model, three dimensional random walk model). The initialization method calls a make_polymer method to populate the segment positions list. See the answer. The idea to model a linear polymer chain as a freely jointed chain which occupies space as a random coil dates back to the 1930th when Kuhn 1,2 defined a polymer chain as having N K rigid links of length l K with no restrictions on the angles between successive bonds. Explain Briefly The Meaning Of "excluded Volume" In Polymer Conformation Theories. We will define a class, Polymer, to describe such a polymer, in which the segment positions are held in a list of (x,y,z) tuples. mass, so this is just the rms position of the segments. What Is Y If The Polymer Chains Are Modeled As Random Flights? P(R) = 4\pi R^2 \left( \frac{3}{2\pi \langle r^2 \rangle}\right)^{3/2}\exp\left(-\frac{3R^2}{2\langle r^2 \rangle}\right), Distribution of the End-to-End Distance and Related Quantities 10 5a. Does The Random-flight Model Of Polymers Account For Excluded Volume? Considering all configurations that the polymer could adopt relative to its centre of mass, one can quantify its size as the radius of gyration, $R_g$: this is the root mean squared distance between all the segments and is also equal to the radius of a hollow sphere with the same moment of inertia as a theoretical random flight polymer. Please be patient and your comment will appear soon. Distribution of the End-to-End Distance and Related Quantities 10 5a. SANS has had major impact on the understanding of polymer conformations, morphology, rheology, thermodynamics, etc. Polymer models Talk given for: Hauptseminar in statistical physics 8/5 2006 Peter Bjødstrup Jensen. To evaluate this hypothesis, we used a random flight theory for the entropic elasticity of polymer loops to model the spectrin network. Molecular weight distributions and concentration regimes of polymer solution are also considered here. This difference is lost in the continuous random flight model of adsorption. The random-flight model assumes there are no long-range interactions: The most obvious long-range interaction is that a chain must avoid itself: two chain segments cannot be in the same place at the same time. Then the length of a fully extended (rod-like) Kuhn polymer chain is . Dongguan Chang'an Flight Model Factory Sales Department was founded in February, 2000 and is located in Chang'an Town, Dongguan City, Guangdong Province. Statistics of Ideal Polymer Chains: Random-Flight Problems 5 3. Rev. In this model, fixed length polymer segments are linearly connected, and all bond and torsion angles are equiprobable. (1 Point) This problem has been solved! The methods used by Fixman in treating excluded volume problems in the random … Drawing an analogy to a random walk, the chapter explains how to calculate various statistical properties of the polymer such as the end‐to‐end distance and the radius of gyration. The two ends of the polymer are marked with black rings. A simple cubic lattice model of the melt of 3-arm star-branched polymers of various length dissolved in a matrix of long linear chains (n, = 800 beads) is studied using a dynamic Monte Carlo method. In spite of the apparent heterogeneity of this list, in mathematical terms, a unified vision emerges. $$ The polymer is drawn in perspective to capture the 3-D structure. # Store it, and update or centre of mass sum. """. In the most basic version of this model, the polymer chain is envisioned as a random walk on a lattice. A Polymer object will be initialized with the values N and a, the number of segments and the segment length respectively. In the following code, a random flight polymer with $N$ segments, each of length $a$, is constructed and displayed. The conformation's name is derived from the idea that, in the absence of specific, stabilizing interactions, a polymer backbone will "sample" all possible … It is noteworthy that the following two different measures of thickness of an adsorbed polymer chain have a different N dependence. Previous question Next question … I love your eclectic work it brings up my mood. Introduction 5 4. Rubber-powered model airplane kits from Peck-Polymers and Golden Age Reproductions.. O ur Peck-Polymers kits include laser-cut parts from light weight balsa wood, c ut with our professional-grade laser cutter.. Our Golden Age Reproductions kits include light weight balsa printwood - the model builder will need to cut out the parts before building their model. monochromated neutron beams or time-of-flight instruments at pulsed neutron sources. Mechanical Behavior of Polymers: Rubber Elasticity, Network Model. Since for a hollow sphere, $I = \frac{2}{3}mr^2$, the inertia ellipsoid corresponding to the radius of gyration (which is a sphere) can also be displayed (here, in red). The random-flight polymer A simple model of a polymer in solution treats it as a sequence of randomly-oriented segments: that is, one for which there is no correlation between the orientation of one segment and any other (this is the so-called random-flight model). The two ends of the polymer are marked with black rings. For example, the mean-square distance of the ends is given by 〈 R 2 〉 = N K l K 2 Then you can pick your starting airport and it will generate several with different flight lengths for you to choose from to fly. Polymer 43:203–222 ... Forsman WC (1968) Matrix methods for determining the dimensions of branched random-flight chains. CG airplanes will fit into simulations, visualizations, advertisements and videos. Approximate Expression for the Bond Probability 16 6. To determine if this is the case for LAR12P, a converged ensemble of The chain in this model is assumed to be phantom-like, i.e. # self.xyz holds the segment position vectors as tuples, # Make our polymer by assigning segment positions, Calculate the segment positions, centre of mass and end-to-end. Because computing accurate conformational statistics for real polymer chains is difficult, various models with differing degrees of simplification have been proposed. (1 Point) This problem has been solved! Just a small unimportant comment : in my case in Python3 the line 8 say hell no ! This implies that a large number of real polymer chains do not obey the simple statistics of random flight … Therefore we propose a random flight model in which the increments are assumed to be exponentially correlated in time. Chapter II. The methods used by Fixman in treating excluded volume problems in the random flight … A simple model of a polymer in solution treats it as a sequence of randomly-oriented segments: that is, one for which there is no correlation between the orientation of one segment and any other (this is the so-called random-flight model). Alternatives to the random flight model were proposed up through the 70's including quasi-nematic ordering, collapsed chain globules, meandering chains, and nodules. R_\mathrm{g} = \sqrt{\frac{1}{N}\sum_{i=1}^N \left(\mathbf{r}_i - \mathbf{r}_\mathrm{CM}\right)^2} This model enables the persistence length of the chain to be defined, a measure of the stiffness of the polymer. In statistical physics, the basic (and highly idealized) model of a ring polymer like bacterial DNA is a closed random flight in 3-space with equal-length steps, often called an equilateral random polygon. Proven-design rubber-powered model airplane kits, known and loved for decades, together with a wide selection of parts and accessories for flying models. Models of single-particle ... field random phase approximation for polymer mixtures are also briefly described. 1,983 aircraft 3D models are available for download on our marketplace. Two different outputs are shown below for two different polymer configurations. large number of configurations for this idealized random walk polymer model [21. And the rigged models models are ready to be imported into game engines and be used for animation or VR/AR projects. \Delta y &= a\sin\theta\sin\phi \\ This is termed excluded volume: rather than a polymer chain being a random walk it is a self avoiding walk (SAW). Additional features include the ability to save your plans, random flight generator, server stats, METAR lookup, and much more. # Add on the corresponding displacement vector for this segment. Keywords Stochastic Differential Equation Wiener Process Shallow Water Equation Random Walk Model Independent Increment These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm … MS mode MS/MS mode INTRODUCTION Interest and research in the polymer industry have increased in recent years for several reasons, including the push towards green chemistry, increasing product complexity, and reduced availability of petroleum products as a feed stock. The Markoff Method for the General Problem of Random Flights 8 5. The set of conformation arrangements of the random flight chain can be considered to be the set of random walks of N steps with a step length of b in an appropriate coordinate system [39,40]. The polymer chain is assumed to consist of n bonds of equal length a, and the direction of each bond is random and independent of the directions of preceding bonds. The polymer can therefore be described by a simple random walk and ideal chain. The polymer consists, at start-up, of 1000 segments of length unity and these are free to rotate through any angle with respect to adjacent segments (random flight model, three dimensional random walk model). # Shift the coordinate system to the centre of mass of the polymer, # Find the rotation matrix and lengths of the semi-principal axes, # Rotate the polymer to align with its principal axes, # The radius of gyration of a theoretical random flight polymer, # Rg is the radius of a hollow sphere with the same moment of inertia as, # the (theoretical) polymer; the ellipsoid of inertia of this theoretical, # polymer is therefore (2/3) times this radius, Chapter 9: General Scientific Programming, Chapter 10: General Scientific Programming, ← Making a Matplotlib plot look better #1, House buyers avoid completing on the 13th of the month →, https://github.com/matplotlib/matplotlib/issues/1077#issuecomment-14006279, The moment of inertia of a random flight polymer. Random walks are key examples of a random processes, and have been used to model a variety of different phenomena in physics, chemistry, biology and beyond. FPLtoIF.com is a free, ad-free, mobile friendly website that assists pilots in copying their flight plan from Flight Aware or SimBrief into Infinite Flight. Die frei bewegliche Kette (englisch freely jointed chain oder ideal chain, auch Gaußkette oder ideales Knäuel) ist das einfachste Modell, womit ein Polymer beschrieben werden kann. Random Walks as Models for Polymer Coils. # Calculate the position of the centre of mass. # Finally, re-centre our polymer on the centre of mass. This was the case up to the 50's when the model began to be questioned.! $$ The polymer is drawn in perspective to capture the 3-D structure. The useful results that arise directly from the freely-jointed chain model of a polymer molecule The program above produces a plot that typically looks like the figure below, suggesting agreement with theory. """ Introduction 5 4. Does The Random-flight Model Of Polymers Account For Excluded Volume? Or even better: You pick your airplane you want to fly. The site identifies missing waypoints and replaces them with GPS coordinates for a complete Infinite Flight plan. The inertia matrix (actually a tensor) of this polymer can be calculated and … Das Modell vernachlässigt Wechselwirkungen zwischen den Monomeren, sodass diese beliebig um ihre beiden Enden rotieren können, was mathematisch einem Random Walk entspricht. Can Excluded-volume Inter- Actions Be Nonlocal, And Why? Growing a random 5-polymer from a random 4 When a cycle forms, a volume-gaining tree is selected proportionately and the corresponding edge deleted; the disk continues to grow. It is found that there is a subtle difference between the set of energy‐weighted random walks generated in the discrete random‐walk model of polymer chain adsorption at the critical energy and the corresponding set of random walks generated in the presence of a reflecting boundary. RSA models of protein adsorption to bare surfaces was extended by modeling the grafted polymers as randomly distributed, preexisting obstacles on the surface. protein may not be completely globular. Expert Answer . Many drug candidates fail therapeutic development because of poor aqueous solubility. Comments are pre-moderated. So I just comment and all work good. Initialize a Polymer object with N segments, each of length a. It is not one specific shape, but a statistical distribution of shapes for all the chains in a population of macromolecules. Show transcribed image text. The inertia matrix (actually a tensor) of this polymer can be calculated and diagonalized to find the principal axes. Kramers Escape Rate From a Trap, Periodic Potentials, Asymmetric Structures, Brownian Ratchets and Molecular Motors (Guest Lecture by Armand Ajdari) J. F. \begin{align*} The Markoff Method for the General Problem of Random Flights 8 5. Approximate Expression for the Bond Probability 16 6. The parallel model provided the longitudinal modulus; (CNTs aligned), while in the calculation with the Halpin–Tsai model a random distribution was assumed. The moment of inertia can be represented as an inertia ellipsoid and this is plotted as a blue wireframe in the code below. are mainly based on the random flight statistics 14-17 but the char-acteristic of the random flight chain is violated by semiflexibil-ity arising from hindrance to internal rotation and structural constraint. Are shown below for two different outputs are shown below for two different polymer configurations for! 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random flight model for polymer 2020 | CommonCrawl |
Disease networks identify specific conditions and pleiotropy influencing multimorbidity in the general population
Visualizing novel connections and genetic similarities across diseases using a network-medicine based approach
Brian Ferolito, Italo Faria do Valle, … Kelly Cho
The multiplex network of human diseases
Arda Halu, Manlio De Domenico, … Amitabh Sharma
An Epidemiological Human Disease Network Derived from Disease Co-occurrence in Taiwan
Yefei Jiang, Shuangge Ma, … Tian-Shyug Lee
Characterisation, identification, clustering, and classification of disease
A. J. Webster, K. Gaitskell, … R. Clarke
Network analysis reveals rare disease signatures across multiple levels of biological organization
Pisanu Buphamalai, Tomislav Kokotovic, … Jörg Menche
Chromatin interactions and expression quantitative trait loci reveal genetic drivers of multimorbidities
Tayaza Fadason, William Schierding, … Justin M. O'Sullivan
Estimating heritability and genetic correlations from large health datasets in the absence of genetic data
Gengjie Jia, Yu Li, … Andrey Rzhetsky
A cross-population atlas of genetic associations for 220 human phenotypes
Saori Sakaue, Masahiro Kanai, … Yukinori Okada
An expanded analysis framework for multivariate GWAS connects inflammatory biomarkers to functional variants and disease
Sanni E. Ruotsalainen, Juulia J. Partanen, … Jukka Koskela
A. Amell1,
A. Roso-Llorach2,3,
L. Palomero4,
D. Cuadras5,
I. Galván-Femenía6,
J. Serra-Musach4,
F. Comellas1,
R. de Cid6,
M. A. Pujana4 &
C. Violán2,3
Genetic association study
Multimorbidity is an emerging topic in public health policy because of its increasing prevalence and socio-economic impact. However, the age- and gender-dependent trends of disease associations at fine resolution, and the underlying genetic factors, remain incompletely understood. Here, by analyzing disease networks from electronic medical records of primary health care, we identify key conditions and shared genetic factors influencing multimorbidity. Three types of diseases are outlined: "central", which include chronic and non-chronic conditions, have higher cumulative risks of disease associations; "community roots" have lower cumulative risks, but inform on continuing clustered disease associations with age; and "seeds of bursts", which most are chronic, reveal outbreaks of disease associations leading to multimorbidity. The diseases with a major impact on multimorbidity are caused by genes that occupy central positions in the network of human disease genes. Alteration of lipid metabolism connects breast cancer, diabetic neuropathy and nutritional anemia. Evaluation of key disease associations by a genome-wide association study identifies shared genetic factors and further supports causal commonalities between nervous system diseases and nutritional anemias. This study also reveals many shared genetic signals with other diseases. Collectively, our results depict novel population-based multimorbidity patterns, identify key diseases within them, and highlight pleiotropy influencing multimorbidity.
Multimorbidity, defined as the co-occurrence of two or more diseases in a given individual, poses a major challenge to quality of care, and emerges as an important issue when considering activity and effort in health systems1,2. Multimorbidity is commonly associated with chronic conditions, but non-chronic or acute diagnoses, such as those related to falls, also contribute to its occurrence3. Chronic diseases are particularly relevant because of their rising prevalence and burden in aging societies, where they incur substantial costs to health care systems. In fact, the economic cost per multimorbid patient is 3–5 times that of non-multimorbid cases4,5. As highlighted by the World Health Organization, chronic diseases have reached epidemic proportions and constitute the leading causes of death in the world6. In Europe, an estimated 50 million people —approximately 7% of the total population— suffer from multimorbidity7. This percentage is even higher (>55%) among the elderly8. Even so, health systems do not meet the needs of multimorbid patients; the structures are typically "disease oriented" and "non-integrative". Thus, care is generally organized around specific medical specialties, an approach that leads to fragmentation, which, in turn, may lead to over-prescription, over-hospitalization, and poor patient satisfaction9,10. Therefore, there is a clear need to improve care for individuals with multimorbidities, but this requires a much more detailed understanding of the trends of disease associations than we currently possess. In addition, there is a need to identify genetic factors influencing multimorbidities, which might then constitute new tools for clinical prevention and monitoring.
To date, the study of age- and gender-dependent disease associations at the population level has mainly focused on chronic1,11 and/or specific12,13 conditions. Broader disease analyses have been performed, but have centered on high-order classifications14, the elderly15, and/or relatively small cohorts2. Network-based approaches have the potential to uncover unexpected relationships between diseases14,15,16,17,18,19,20,21,22. To apply these approaches, systematic and detailed high-quality clinical annotations of a large number of individuals are required. In parallel, collection and analysis of biological samples in the same population can provide the means to identify shared genetic factors among diseases linked to multimorbidity23,24. Here, by constructing and analyzing disease networks from high-quality primary health care data, and by integrating the results with genome-wide association studies (GWASs) of individuals from the same population, we identify key diseases, their cumulative risk trends and genetic factors influencing multimorbidity.
Disease networks built from primary health care data
A dataset from the electronic primary health care records of Catalonia, a Mediterranean region with more than seven million individuals, was analyzed for age- and gender-centered disease network topological properties that may be associated with multimorbidity and/or pleiotropy (Fig. 1A). This primary health dataset, known as SIDIAP-Q, comprises records from the universal coverage health care system and high-quality clinical annotations based on validated scores25,26. Patient diagnoses were based on the International Statistical Classification of Diseases and Related Health Problems, 10th revision (ICD-10)27. A total of 1,749,722 individuals (23.5% of the Catalan population) aged at least 19 years and with two or more open recorded diagnoses between 1st January and 31st December 2010 were grouped by 5-year intervals or strata (from 19–24 to ≥90 years old) and by gender, and included in this study (Supplementary Fig. S1a). To investigate the impact of diseases and multimorbidities that are most relevant to the general population, we only considered diagnoses with a prevalence of ≥1% (Supplementary Table S1) and that were associated with any other disease by a measure of comorbidity strength (hereafter relative risk (RR)15,28) included in the bottom or top five percentiles across the 15 age strata of men and women. These thresholds corresponded to RR estimates of <0.8, which suggests mutually exclusive diseases, or >1.6, which suggests co-occurring or comorbid diseases, respectively, across all the strata (Supplementary Fig. S1b and Supplementary Table S2). The RR estimates were positively correlated (Spearman's correlation coefficients (ρ) = 0.82–0.88, P < 10−16) with the Jaccard index, a statistic frequently used to measure the similarity of sample sets. However, this index is not appropriate for relatively rare observations and cannot distinguish between different directions of association29.
Study design and disease networks. (A) Strategy for the identification of diseases and genetic factors influencing multimorbidity. Network nodes and edges correspond to diseases and relative risks (RRs), respectively, and were constructed using primary health records from the Catalan general population. The human figures were created by Freepik. (B) Distributions of the number of nodes and edges in each main network component across strata and by gender. (C) Exponential decay of cumulative degree (k) distributions of two example disease networks as depicted. (D) Proportions of linked ICD-10 codes that share a clinical chapter; box-plots show the results of 1,000 permutations and the observed value for each stratum network is indicated by a dot. (E) Number of diseases with causal genes/proteins included in the molecular network that revealed at least one disease association with a smaller shortest path than expected at random. The ordered bars indicate the number of disease associations that match this criterion for each disease (ICD-10 codes are indicated on the x-axis). The gray zone indicates diseases that do not match the criterion. A prevalence threshold is also depicted.
For each stratum, a network of morbidities was derived in which nodes represent diseases and edges represent RRs. The main network components included more than 70 nodes or nosological entities, and 300 edges or disease associations (Fig. 1B). Except for the elderly, these components were found to be bigger in women, which is consistent with a higher prevalence of female multimorbidity2,10. The cumulative distributions of the number of edges by nodes (degree (k) distribution) revealed exponential decays (Fig. 1C). This is a similar pattern to that of mortality following emergency medical admission30 and is inversely related to epidemic spread31. In addition, all observed morbidity networks exhibited a predictable property of 'small-world-ness'32 (Supplementary Fig. S2), by which most nodes or diseases can be reached from every other node through a relatively small number of edges33. Therefore, the constructed disease networks are coherent with previous knowledge and reveal expected systems-level features.
Clinical coherence of the disease networks
To assess the clinical coherence of the networks, we performed 1,000 permutations of the associated (based on RRs) ICD-10 codes in each stratum and computed the proportion of code pairs sharing a higher-level clinical classification or chapter; there were 21 of these27. In all strata and for both genders, none of the random sets showed a higher proportion of shared clinical chapters than that of the real networks (Fig. 1D). Next, the clinical coherence of the networks was evaluated using the functional and molecular interactions of the underlying genes and/or proteins (genes/proteins). The ICD-10 codes were linked to the genes/proteins associated to each condition based on the phenotype-genetic associations from the Online Mendelian Inheritance in Man (OMIM)34. We hypothesized that coherent disease associations frequently show relatively small shortest interaction paths between the underlying genes/proteins. Thus, approximately 78% of the diseases with an OMIM annotated gene/protein included in a molecular network showed at least one disease association with a smaller shortest path than randomly expected, and there was no bias with respect to prevalence differences (Fig. 1E). Therefore, the disease networks are also coherent based on higher order clinical annotations and phenotype-genetic associations.
Identification of central diseases
Having established their coherence, we analyzed the networks in order to detect diseases with a major impact on multimorbidity. A modified version of the PageRank35 algorithm was applied to take into account the edge weights indicated by the RRs (see Methods). Thus, 13 and 17 diseases appeared at least four times among the 10 most central diseases across the strata in men and women, respectively (Fig. 2A). Seven diseases (including chronic and non-chronic conditions) were common to both genders and comprised critical diagnoses across different ages, such as "Disorders of adult personality and behavior" (Fig. 2A). Non-chronic, acute conditions, such as injuries and infections, also proved to be central in several strata, building on previous observations in older patients3.
Central diseases and network communities. (A) Diseases emerging as topologically central in men and/or women. The number of appearances (in different strata), the corresponding ages, and the specific condition (chronic or non-chronic) are shown. The dotted lines indicate diseases found to be common to men and women. (B) Disease network for women aged 65–69 years and depicting diseases (ICD-10 codes) identified as central in this gender. The node corresponding to "Diabetes mellitus" (not central) is also indicated (blue font). The node sizes reflect centrality value and their colors indicate communities. Edge thickness is proportional to the magnitude of the RR estimation; black indicates RR > 1.6 and green indicates RR < 0.8. Disease prevalence is shown by font colors as indicated in the inset. (C) Network communities appearing in at least two consecutive strata in men or women. The disease roots of each community are depicted in the insets.
Central nodes commonly show multiple edges linking different "disease communities" (subsequent section). Diseases that are highly prevalent in the population, like "Diabetes mellitus", also have a relatively large number of edges, but these are mainly linked to diseases in the same community (Fig. 2b). Nonetheless, consistent with epidemiological observations36, the strongest association with "Diabetes mellitus" corresponded to "Polyneuropathies and other disorders of the peripheral nervous system" (RR = 3.73, P < 10−16), and this condition emerged as central in this study (Fig. 2A,B). According to their topological feature, deletion of central nodes led to a higher number of network components than that of randomly expected in 3/15 and 12/15 of the male and female disease networks with edges of RRs > 1, respectively. Conversely, no such impacts were observed when central nodes were deleted in networks with edges of RRs < 1 (Supplementary Fig. S3). Collectively, the above data identify chronic and non-chronic conditions with a potential major role in multimorbidity.
Main disease communities
To analyze the patterns of disease aggregations, densely connected sets of nodes or network communities appearing in at least two consecutive strata were identified. The diseases commonly present across the strata comprised the "roots" of the communities. Thus, recognized temporal patterns associated with gender-specific diseases were observed: for instance, cancer-associated communities were identified spanning the 45–64- and 55–74-year-old groups for women and men, respectively (Fig. 2C and Supplementary Table S3). This analysis also highlighted disease communities that may require further health-care efforts based on their sustained presence over time, in particular, a community with root "Injuries and consequences of external causes" in men aged 45–64 years, and a community with root "Mental, behavioral and neurodevelopmental disorders" in women aged 65–89 years (Fig. 2C). Therefore, community-aggregated diseases identify specific multimorbidity patterns, providing a means for following up clustered associations with age.
Unexpected bursts of disease associations leading to multimorbidity
The progression of cumulative disease associations was further analyzed at the level of node degrees. The number of edges (considering only RRs > 1.6) per node was computed across all strata, and nodes with relatively large leaps in their degree (k) were identified; i.e., representing a large increase in the number of associations for a given disease, from a younger to an older stratum. This analysis revealed 19 and 27 nodes in men and women with leaps of k ≥ 10, respectively, and these included 10 diseases common to the two genders (Fig. 3A and Supplementary Table S4). To assess the significance of these multimorbidity bursts, the results were compared with those of 1,000 equivalent random networks in each stratum and gender, preserving the degree distribution and connectedness of each corresponding real network. Remarkably, none of the random networks showed a distribution with a greater or equal number of large-degree leaps than the real networks (one-sided Pempirical < 0.001; Fig. 3B). Four and seven of the 19 and 27 aforementioned diseases, respectively, were previously classified as central, and two were present in both genders: "Complications of surgical and medical care, not elsewhere classified" and "Polyneuropathies and other disorders of the peripheral nervous system" (Figs 2A and 3A). Therefore, particular diseases, some of which also play a central role in networks, act as seeds for multimorbidity.
Multimorbidity bursts. (A) Age-based trajectories of nodes with large degree leaps; ≥10 edges (RRs > 1.6) over time. The left and right panels show results for men and women, respectively. The corresponding diseases are listed below each graph, and their chronic or non-chronic status is also shown. (B) Distribution of connectivity leaps in 1,000 random networks with the same degree distribution and connectedness as that of the real morbidity networks with RRs > 1.6. The y- and x-axes depict the probability and number of nodes with leaps of ≥10 edges, respectively; red arrows indicate the values observed in the real networks.
Most of the bursts (72% (26/36) in men and women) corresponded to chronic conditions acting as seeds (Fig. 3A and Supplementary Table S4). However, the non-chronic diagnoses "Complications of surgical and medical care, not elsewhere classified" and "Falls" also emerged in this analysis in both genders (Fig. 3A). The former condition suggests that prevention of multimorbidity in primary health care should take into account surgical interventions in hospitals. In addition, the identification of "Falls" is consistent with the findings of recent epidemiological studies in the elderly37,38, so monitoring these acute conditions could further improve the management of multimorbidity bursts, particularly in middle-aged women, as suggested by our study (Fig. 2A).
Trajectories of cumulative risks
The results above have shown unexpected bursts of disease associations that may have an important role in the emergence of multimorbidity. However, it remains unknown if there are differential trends of cumulative disease associations among the different types of network nodes. The progressive aggregation of diseases was evaluated by analyzing the trajectories of the sum of all RRs for each disease as a function of age. This analysis was independent of the initially defined RR thresholds and considered all diseases with ≥1% prevalence. While the sum of RRs < 1 (using their inverse value, 1/RR) revealed mostly flat or smoothly decreasing profiles in both genders, substantial increasing trends were observed for summed RRs > 1 (Supplementary Fig. S4). To assess differences in the trends, the 95% confidence interval (CI) estimates of each RR sum distribution were computed; thus, the trends for women and men did not overlap for most of the age groups (Fig. 4A). Women had higher average RR sums, but men, particularly those aged 30–64 years, had a steeper slope (Fig. 4A). A coincidence test indicated that all four distributions (by gender and/or effect) were significantly different (P ≤ 0.001). Remarkably, the global increase of summed RRs > 1 was found to be approximately 60% and 40% in men and women, respectively, further highlighting the relevance of multimorbidity.
Cumulative risk trends. (A) Average and 95% CI of RR sums by gender and age group. The dotted lines indicate slopes significantly different from zero. (B) Average and 95% CI of RR sums of diseases identified as central in the networks or as other, non-central diseases. The arrows indicate the cumulative risk differences between central and non-central diseases in men (60 years) and women (65 years). (C) Graph showing the correlation between the average centrality value of each node across all networks in men, and the difference between the minimum and maximum RR sums of each disease. The linear trend and 95% CI (shaded area) are shown. (D) Average and 95% CI of RR sums of diseases identified as network community roots or other diseases (i.e., non-roots). The arrow indicates the cumulative risk difference between non-root and root diseases in women (65 years). (E) Average and 95% CI of RR sums of diseases identified as having large degree leaps (≥10 edges, and excluding those that are also central) or other diseases. The arrows indicate cumulative risk differences between disease sets with large leaps and no large leaps in men (55 years) and women (60 years).
Next, the trends of the disease sets classified above as central, community roots, or with large degree leaps were analyzed. Consistent with their key role in multimorbidity progression, the central diseases in men and women showed higher RR sums than all other diseases (Pcoincidence ≤ 0.002; Fig. 4B). Again, women had higher sums, but the slopes were steeper in men (Fig. 4B). Building on these observations, analysis of the global correlation between the average centrality of each node across all the networks, and the difference between the minimum and maximum RR sum of each disease across all strata, revealed a positive association in men (ρ = 0.27, P = 0.007; Fig. 4C). Therefore, node centrality in male disease networks is linked to its relative importance in accumulating disease associations with age. The equivalent analysis in female networks did not reveal a significant association, possibly due to the lower minimum-maximum cumulative risk difference (Fig. 4A).
Subsequently, opposite of what was seen for the central diseases, but consistent with the network topology, the diseases identified above as community roots had lower RR sums than did all other diseases, particularly in women (Pcoincidence < 0.001 relative to central; Fig. 4D). However, diseases that are seeds for multimorbidity bursts (excluding those that are also central) also had a higher cumulative risk of comorbidities (Pcoincidence ≤ 0.002 relative to roots; Fig. 4E). These results were corroborated using the cumulative average of RRs for each disease (Supplementary Fig. S5). Therefore, network-based features identify different types of diseases relative to their cumulative risk leading to multimorbidity.
Centrality and pleiotropy linked to causal genes
The identified diseases underlying multimorbidity —particularly those linked to network centrality and/or bursts of disease associations— may be caused by genes that, as a consequence, influence multiple human disorders. To test this hypothesis, we analyzed a curated human disease gene network in which two genes are connected if they are causative of the same disease39. Using this independent dataset and two different measures of network centrality, the causal genes of the central and burst-seed diseases were found to be more central than that of the community-root diseases in both genders (Mann-Whitney P values < 0.001; Fig. 5A). Intriguingly, the causal genes of community-root diseases were also found to be more central than that of the rest of diseases (Mann-Whitney P values < 0.001; Fig. 5A), which further highlights the link of these conditions with major disease aggregations through age (Fig. 2C). Topological analyses of the corresponding gene products in a curated interactome network40 also revealed that all three sets (i.e, central, burst-seed, and community-root) have higher centrality than that of other gene products (Supplementary Fig. S6). In contrast to the gene network results, there were not centrality differences between community-root and central or burst-seed sets using men data, and differences were only marginally significant using women data (Supplementary Fig. S6). This observation might denote a non-lineal relationship between genetic causality and diversity of protein function.
Centrality and pleiotropy linked to causal genes. (A) Graphs showing the distributions of closeness and eigenvector centrality measures for different types of causal genes as indicated in the insets. The results correspond to the analysis of the curated human disease gene network and are shown for men and women disease sets derived from the SIDIAP-Q networks study. The Wilcoxon test P values of the comparisons of distributions are shown. (B) Scatter plots depicting the negative correlations between the gene expression signatures (all genes included) that define diabetic neuropathy (left panel) or undernutrition (right panel) and age at diagnosis of breast cancer. The stage-adjusted linear regression coefficients and their corresponding P values are shown.
As indicated above, one condition emerged as relevant in both the centrality and burst analyses: "Polyneuropathies and other disorders of the peripheral nervous system". This condition had fewer recognized associations with "Malignant neoplasms, stated or presumed to be primary" and "Nutritional anemias" in women (Supplementary Table S4). Following on these observations, the concordance of gene expression alterations underlying the three diseases was assessed41,42,43. Higher overlaps than expected by chance were observed between the gene expression signatures from the three diseases (χ2 P < 0.002). Genes involved in lipid metabolism were found to be common to all three diseases (Supplementary Table S5). By contrast, no significant overlap was found when compared to differentially expressed genes in lung adenocarcinomas44. Furthermore, the expression scores for the signatures characteristic of undernutrition42 and diabetic neuropathy43 were found to be negatively correlated with age at diagnosis of breast cancer (Fig. 5B). Therefore, the central and burst-seed diseases are caused by genes that in turn play a central role in the human disease gene network, and we provide evidence of shared gene expression alterations between diabetic neuropathy and undernutrition that promote breast cancer.
Shared genetic factors among diseases linked to multimorbidity
To further evaluate disease associations at the level of shared genetic factors, a GWAS was performed in the same population as the SIDIAP-Q disease networks study (Genomes for Life)45. This investigation focused on central diseases with detailed clinical definitions and on common diagnoses with more than 200 cases included in the cohort (three and nine diseases, respectively; Supplementary Table S6). The application of a genome-wide association pairwise approach46 revealed that central diseases tended to share, on average, a greater number of significantly associated variants than the nine common diseases (20 vs. 11 significant signals). Subsequently, seven of the 36 possible non-redundant disease pairs showed a higher number of shared variants than that of 100 random GWASs (Fig. 6A). Importantly, these seven pairs corresponded to RRs > 1.5 (P < 10−3) across at least two strata in both genders, which reinforces their epidemiological relevance.
Shared genetic factors among diseases linked to multimorbidity. (A) Matrix depicting pairs of central (blue) and common (green) diseases, and instances with a significant number of shared genetic variants relative to random GWASs (numbers of variants are shown). The corresponding RRs are shown for instances linking central diseases. (B) Distribution of shared genetic variants (also considering those in linkage disequilibrium) among 100 random sets of 31 variants and observed value of shared signals between "Nutritional anemias" and "Diseases of the nervous system". The y- and x-axes depict the probability and number of shared variants, respectively; red arrows indicate the value observed.
Besides expected overlaps (e.g., shared signals between "Diabetes mellitus" and "Disorders of lipoid metabolism" or "Essential (primary) hypertension"), there were shared genetic associations between "Nutritional anemias" and "Diseases of the nervous system" (Fig. 6A), which includes polyneuropathies. Thirty-one significant association signals were detected in this comparison and, notably, three of them were also found to be in linkage disequilibrium (LD, D' > 0.99) with variants previously identified as influencing multiple human traits46,47 (Supplementary Table S7). Most importantly, 17 of the 28 remaining shared signals were found to be in linkage disequilibrium with GWAS results involving one or more other human disorders or traits (Supplementary Table S7). This proportion of 20/31 shared signals was found to be higher than the average proportion of 100 sets of 31 randomly chosen genetic variants (Pempirical ≤ 0.01; Fig. 6B). The neighbor genes of these pleiotropic signals were found to be significantly enriched (false discovery rate (FDR)-adjusted P = 2.1 × 10−7) in loci linked to smoking cessation versus dependence48. Intriguingly, smoking is an established lifestyle factor associated with multimorbidity49,50. Therefore, key disease associations linked to multimorbidity are influenced by shared genetic factors, which in turn may be associated with important lifestyle factors.
None of the 31 signals appeared to be an expression quantitative trait locus (eQTL) when exploring the GTEx database (v6.0)51. However, when variants in LD were considered, the expression of 17 genes may be associated (Supplementary Table S8). Notably, seven of these genes were found to be altered in thyroid tissue, which represents a higher enrichment than expected by chance (χ2 P = 9 × 10−6). This observation might be in concordance with observational studies in animals and humans linking impaired thyroid metabolism to iron-deficiency anemia52. In addition, thyroid deregulation (underactive thyroid) is a risk factor for peripheral neuropathy, which overall provides a tissue-based mechanistic hypothesis for the observed multimorbidity and pleiotropy.
Our results show that specific topological features of disease networks identify conditions with a key role in the emergence and/or progression of multimorbidity in the general population. The causal genes of these key conditions also occupy central positions in the network of human disease genes, which is consistent with their predicted pleiotropic effects. In addition, this study reveals shared genetic factors among diseases linked to certain multimorbidities and, in particular, highlights associations between breast cancer, diabetic neuropathy, and nutritional anemia, and between diseases of the nervous system and nutritional anemias.
Three types of diseases are identified in this study: central, which include chronic and non-chronic conditions, accumulate relatively higher risk of multimorbidity with age in both genders; "community roots", which accumulate less risk, but indicate major disease aggregations with age; and "burst seeds", which nucleate diagnoses for 10 or more conditions in a single individual. In the biomedical scenario, central diseases may be interpreted as those more likely lead to multimorbidity or more likely appear in a given multimorbid patient. In an analogous manner, their causal genes have the potential to influence multiple diseases and, therefore, they may be functionally linked to different molecular process and/or signaling pathways53,54,55. A particular type of central disease with a key role in multimorbidity corresponds to those identified as "burst seeds", which show a sharp accumulation of disease associations. The causal genes of this type of diseases may also harbor pleiotropic effects, but one can speculate that other biological, environmental and/or lifestyle factors critically contribute to the observed burst effect. Finally, the function of causal genes for "community root" diseases may be more specific at the molecular, cellular and/or tissue level.
The observed multimorbidity bursts are generally linked to chronic diseases and, thus, clinical studies of identified seed conditions may be able to improve prevention strategies and health care policies9,10. Nonetheless, two acute conditions ("Complications of surgical and medical care, not elsewhere classified" and "Falls") also emerge as central and mediating bursts, so their integration in prevention could further help improve multimorbidity care, and not only in the elderly37,38. However, there are significant differences in the cumulative risk trends between men and women, which therefore should also be taken into account when preventing and/or managing multimorbidity. "Polyneuropathies and other disorders of the peripheral nervous system" and, again, "Complications of surgical and medical care, not elsewhere classified" appear to be particularly relevant in both genders. The identification of the latter is additional evidence that attention to multimorbidity in primary care should be coordinated with programmed activities in secondary and tertiary care3,56. In contrast to central diseases, network communities provide evidence to detect clustered aggregations across sequential age groups. Thus, community roots should not be the focus of cumulative risk analyses, but they can potentially assist in identifying the most frequent disease aggregations.
Monitoring of individuals diagnosed with diseases identified in this study, in combination with analyses of pleiotropic factors, could potentially reduce the current impact of multimorbidity on health care systems. Crucially, our study shows that the causal genes of central and burst-seed diseases occupy a central position in a genetic network of human disorders, which further endorses their relevance in multimorbidity. Therefore, analyses of these causal genes may be useful for monitoring and/or predicting multimorbidity. Specifically, lipid metabolism appears to be commonly perturbed in breast cancer, diabetic neuropathy, and nutritional alterations, which is also consistent with the proposed causal links between cancer, diabetes, and obesity57. At the germline level, our GWAS in individuals of the same population in which disease networks are studied has identified seven pairs of diseases with a significant number of shared genetic factors. These pairs include "Nutritional anemias" and "Diseases of the nervous system", which are also linked to centrality and bursts in the network analyses. Of note, many of the genetic variants identified in this comparison are in linkage disequilibrium with variants associated with other human traits or diseases46,47, including smoking dependence48. This observation further reinforces the pleiotropic connection between the two diseases and others, and the possibility of identifying markers for estimating and/or preventing the risk of multimorbidity including those conditions. Prospective studies to address these questions may be warranted.
Design, setting and study population
A cross-sectional study was conducted in Catalonia (Spain), a Mediterranean region with 7,434,632 inhabitants, 81% of whom live in urban municipalities (2010 census). The Spanish National Health Service (NHS) provides universal coverage, financed mainly by tax revenue. The Catalan Health Institute (CHI) manages primary health care teams (PHCTs) that serve 5,501,784 patients (274 PHCTs), or 74% of the population; other providers manage the remaining PHCTs. The CHIs Information System for the Development of Research in Primary Care (SIDIAP) contains the coded clinical information recorded in electronic health records by its 274 PHCTs since 2006. A subset of records meeting the highest quality criteria for clinical data (SIDIAP-Q) includes 40% of the SIDIAP population (1,833,125 individuals), attended by 1,365 general practitioners whose data recording scored highest in a validated comparison25. SIDIAP has been shown to be highly representative of the Catalan general population in terms of geography, age and gender distributions according to the official 2010 census. This study included individuals ≥19 years of age and assigned to a PHCT during the period of study (1st January–31st December 2010). The SIDIAP-Q study was approved by the Jordi Gol University Institute for Research Primary Healthcare (IDIAP) ethics committee and the GWAS by the Germans Trias i Pujol Health Sciences Research Institute (IGTP) ethics committee. Regarding SIDIAP and according to Spanish legislation about confidentiality and data protection (Organic Law 15/1999 of 13 December for the Protection of Personal Data), the data included in this database were always anonymized; thus, it was not necessary to ask for informed consent to the participants. All the participants in the GCAT GWAS provided written informed consent. These studies followed national and international regulations for research involving human subjects: Declaration of Helsinki Ethical Principles for Medical Research Involving Human Subjects and Good Research Practice principles and guidelines. The SIDAP-Q data are available upon request and ethics committee approval, and GCAT GWAS data have been deposited in the European Genome-phenome Archive.
Coding and selection of diseases
Diseases are coded in SIDIAP using the ICD-1027. For this study, we selected all active diagnoses recorded in electronic health records as of December 31st 2010, except for R (symptoms, signs, and abnormal clinical and laboratory findings, not elsewhere classified) and Z (factors influencing health status and contact with health services) codes. Non-active diagnoses, identified by the presence of an end date in the records, were excluded from the analysis. These diagnoses cover a broad list of acute diseases for which the system automatically assigns an end date (e.g., 60 days after the initial diagnosis). To facilitate management of the information, the diagnoses were extracted using the 263 blocks (disease categories) in the ICD-10 structure. These are homogeneous categories of very closely related specific diagnoses; for example, hypertensive diseases include "Essential (primary) hypertension, Hypertensive heart disease, Hypertensive renal disease, Hypertensive heart and renal disease, and Secondary hypertension". From the 263 blocks, we excluded the R and Z codes, and 13 codes were not found in SIDIAP-Q, leaving 241 blocks suitable for analysis. To produce consistent and clinically interpretable networks based on binary disease associations, and to avoid inclusion of spurious relationships that could bias the results, we considered only diagnoses with ≥1% prevalence for each of the following age strata: 19–24, 25–29, 30–34, 35–39, 40–44, 45–49, 50–54, 55–59, 60–64, 65–69, 70–74, 75–79, 80–84, 85–89, ≥90 years, and for both genders. This minimum threshold of prevalence led to the analysis of 144 and 141 diseases in men and women, respectively. All patients with two or more coexisting diagnoses recorded on 31st December 2010 were included.
Chronic and non-chronic definition
Each diagnosis was classified using the O'Halloran criteria for chronic conditions in the International Classification for Primary Care-2 (CIAP-2)58. We included all 146 diagnoses considered as chronic diseases by these criteria: i) having a duration that has lasted, or is expected to last, at least six months; ii) having a pattern of recurrence or deterioration; iii) having a poor prognosis; or iv) producing consequences, or sequelae, that have a significant impact on quality of life. The diseases that did not meet these criteria were classified as non-chronic. The ICD-10 codes were mapped to identify chronic and non-chronic diseases. All results were described using these codes.
Relative risk computation and trends
Categorical variables were summarized as frequencies (percentages); normally and non-normally distributed quantitative variables were summarized as means (standard deviations, SDs) and medians (interquartile ranges, IQRs), respectively. The relative risk (RR) was calculated to quantify the strength of disease associations (comorbid if RR > 1 or tending to be mutually exclusive if RR < 1) as previously proposed15,28. The ratio is that of the observed prevalence of patients diagnosed with both diseases to the expectation based on the product of the corresponding disease prevalences. The RR 95% confidence intervals (CIs) and P values were obtained using the methods of Katz59, and Altman and Bland60, respectively. Generalized additive models (GAMs)61 using cubic splines as the smoothing function were fitted to estimate RR sum distributions over age groups, for RR > 1 and RR < 1 associations, stratified by gender. The 95% CI of each distribution was obtained from the standard error of the fitted model. Join-point models (https://surveillance.cancer.gov/help/joinpoint) were used to investigate the trends of RR sum distributions across age groups. Statistical differences in slope were assessed using the annual percent change (APC) test62. To check the similarity of any pair of RR sum distributions tests for parallelism and coincidence63 were conducted. The cumulative distributions of disease associations across age groups were evaluated using the RR and Jaccard index (particularly the 1-Jaccard) estimates, and three similar approaches: 1) by computing the sum of the association estimates for each disease in each stratum and gender; 2) by computing the average of the estimates for each disease in each stratum and gender; and 3) by computing the sum of the estimates for each disease in each stratum and gender, but considering only diseases with ≥1% prevalence and dividing the sum by the number of strata in which a given disease appears. The correlation relative to the network centrality values was computed using, for each disease, the difference between the minimum and maximum of the cumulative estimate across age groups. The centrality values were normalized between 0 and 1, and the average value across all the networks was used for each disease.
Network construction
For each age group and gender (i.e., stratum), a network was built with nodes corresponding to diagnoses matching the criteria detailed above, and edges corresponding to comorbidity if the corresponding RR was included in the top or bottom vigintile of the overall distribution of RRs in a given stratum. These percentiles corresponded to RRs < 0.8 or >1.6 across all strata. The SIDIAP-Q dataset linked the diagnoses in each stratum using the Jaccard index, Jij26. This index accounts for the similarity of two diagnoses di and dj, and takes values between 0 and 1. In parallel, the SIDIAP-Q dataset contained the frequency of the diagnoses, Ni and Nj, and the population number N for each stratum. From these data, RRs were computed as follows:
$$R{R}_{ij}=\frac{[{J}_{ij}({N}_{i}+{N}_{j})/(1+{J}_{ij})]N}{{N}_{i}{N}_{j}}$$
With the criteria of considering diagnosis with prevalence greater than 1%, and discarding disease associations based on their RR percentiles (>5% and <95%), the networks contained between 73 and 111 diagnoses. The number of these diagnoses varied with age and gender, whereby more nodes were generally noted for women and for older age groups.
Small-world-ness
In order to assess the small-world-ness characteristic of the observed morbidity networks, we used the method proposed by Humphries and Gurney32. The approach states that a small-world network fulfills the condition that \({L}_{G}\ge {L}_{{\rm{rand}}}\) and \({{\rm{C}}}_{G}^{{\Delta }}\gg {C}_{{\rm{rand}}}^{{\Delta }}\), where L is the average shortest path length of the network and CΔ is the average clustering coefficient. The small-world-ness SΔ is introduced as follows:
$${S}^{{\Delta }}=\frac{{C}_{G}^{{\Delta }}/{C}_{{\rm{rand}}}^{{\Delta }}}{{L}_{G}/{L}_{{\rm{rand}}}}$$
Therefore, SΔ > 1 corresponds to a small-world network. In this study, we did not consider different weights for the network edges (i.e., all weights had a value of 1). The \({C}_{{\rm{rand}}}^{{\Delta }}\) and Lrand values were, for each network, the average of 1,000 Gn,m random model sample values. The outcomes were SΔ > 1 for all observed networks and SΔ > 2 for the strata younger than 80 years of age.
Node centrality
The PageRank35 algorithm was used to compute node centrality in the networks. This algorithm assigns a weight to each node that ranks its importance among the global set of nodes of the network. A node that is related, either directly or through other nodes, to nodes with a high PageRank value receives a higher weight and is defined as more "central". The PageRank can be considered a variant of the eigenvector and Katz centralities, and overcomes problems like the concentration of most of the centrality on a relatively small number of nodes64. The PageRank value of a node is defined recursively and determined by three main factors: the number of edges it receives and their weight; the number of edges of the neighbors; and the centrality of these neighbors. This ranking algorithm has a probabilistic interpretation using the so-called Google matrix G65. For an undirected positive edge weighted graph, G is defined as follows:
$$G=\alpha W{D}^{-1}+\frac{1-\alpha }{n}J$$
Therefore, α is the damping factor, W is the weighted adjacency matrix of the network, D is the diagonal degree matrix defined by Dij = ∑j |Wij|, and J is the matrix of all ones. The matrix G is a left-stochastic Markov matrix —each column sums to one— and represents random walks in the network. The parameter (1 − α) is the probability of jumping randomly to any node in the Markov chain process without having to follow an edge between the nodes. The PageRank values are the entries of the dominant right eigenvector, which correspond to the steady-state of the Markov chain. The straightforward generalization of PageRank to signed weights, named signed spectral ranking66, raises a problem: G is no longer a stochastic matrix, so the probabilistic interpretation loses meaning. To resolve this limitation, we used a method that considers positive (G+) or negative (G−) weights67 to compute PageRank values for each sub-graph PR+ and PR−, respectively, thereby obtaining the final rank vector as MPR = PR+ − PR−, where MPR stands for the Modified PageRank. The damping parameter is usually assumed to be α ≈ 0.85 for technical and social networks. As there is no established guideline for setting this value, we used α = 0.5 to take into account the fact that nodes represent blocks of diseases. Different values of α might change the order of the ranking, but high-ranked nodes persist. The human disease gene network was built using DisGeNet curated gene-disease associations (version 5.0)39. The distance matrix between all vertices was computed and closeness centrality determined for each vertex as the inverse of the average distance to all other vertices. The eigenvector centrality was computed using the package NetworkX v2.1. All computations were performed using Python v2.7. Similar analyses were performed using the Agile Protein Interactomes DataServer (APID) level 2 dataset, which includes protein interactions proven by two or more experiments40.
Community detection
It is assumed that a community (or clustering) division separates the nodes of the network into groups such that connections are stronger or more frequent within groups than between them. This study took a heuristic approach based on the maximization of modularity, a commonly used community quality measure. Modularity, Q, is a function representing the difference between the total edge weight in sets of the network under study and the total expected weight in the same sets from a random network generated by a given null model:
$$Q=\frac{1}{2m}\sum _{i}\,\sum _{j}\,({A}_{ij}-{P}_{ij}){\delta }_{{\sigma }_{i},{\sigma }_{j}}$$
where m is the number of edges in the network, Aij is the (i, j) element of the adjacency matrix, Pij is the null term, and \({\delta }_{{\sigma }_{i},{\sigma }_{j}}\) is the Kronecker δ between the communities of nodes i and j, that is σi and σj, respectively. With the correct choice of the null model it is possible to incorporate specific features of the network structure. A standard choice is Pij = kikj/2 m, where ki and kj are the degrees of nodes i and j, respectively68. For a weighted signed network, the modularity function Q can also be defined using the appropriate null model. This model should take into account the so-called "resolution limit": modularity optimization might fail to identify small communities. The resolution scale depends on the total size of the network and the interconnectedness of the communities. A possible solution to this problem is to scale the signed null model by introducing parameters γ+ and γ−. The former equation then becomes:
$$Q=\frac{1}{2{w}^{+}+2{w}^{-}}\,\sum _{i}\,\sum _{j}\,[{W}_{ij}-({\gamma }^{+}\frac{{w}_{i}^{+}{w}_{j}^{+}}{2{w}^{+}}-{\gamma }^{-}\frac{{w}_{i}^{-}{w}_{j}^{-}}{2{w}^{-}})]{\delta }_{{\sigma }_{i},{\sigma }_{j}}$$
where W is the signed weighted adjacency matrix of the network,
$$W=[{\tilde{RR}}_{ij}]\in {{\mathbb{R}}}^{n\times n}$$
$${\tilde{RR}}_{ij}=\{\begin{array}{cc}R{R}_{ij} & {\rm{if}}\,R{R}_{ij} > 1\\ -\,1/R{R}_{ij} & {\rm{if}}\,R{R}_{ij} < 1\end{array}$$
\({w}_{i}^{+}\) and \({w}_{i}^{-}\) are signed generalized degrees from
$$\begin{array}{rcl}{w}_{i}^{+} & = & \sum _{j}\,{\rm{\max }}(0,{W}_{ij})\\ {w}_{i}^{-} & = & \sum _{j}\,{\rm{\max }}\,(0,-\,{W}_{ij})\\ {w}_{i} & = & {w}_{i}^{+}-{w}_{i}^{-}\end{array}$$
and the values of γ+ and γ− determine the importance assigned to the null network. Increasing γ+ enables smaller communities to be detected. On the other hand, smaller groups of nodes can be detected by decreasing γ−. A method for estimating the best values of γ+ has recently been described69 and it is extended in this study to estimate γ−. The community configuration σ is obtained by maximizing Q. The number of possible community configurations in a network of n nodes is given by the Bell number, which grows exponentially with n. This is an NP-hard problem70, so heuristic algorithms are required. This study employed a method known as "spin glass community detection"71, an approach from statistical physics and based on the Potts model. In this model, each particle can be in one of several spin states, and the interactions between them determine which particles would prefer to have the same spin state. The analogy links particles with nodes, interactions with edges, and communities with the spin states. One aims to minimize the energy of the system, denoted by the Hamiltonian \( {\mathcal H} \), in order to find the ground state. It is known that the ground state is the most stable configuration of the system, and hence a cohesive community structure. An extension of the spin glass method to signed weighted networks was implemented in python-igraph for use in this study. The Hamiltonian \( {\mathcal H} \), which rewards internal positive and absent negative edges, and penalizes absent internal positive and internal negative edges72, is given as:
$$ {\mathcal H} =-\,\sum _{i}\,\sum _{j}\,[{W}_{ij}-({\gamma }^{+}\frac{{w}_{i}^{+}{w}_{j}^{+}}{2{w}^{+}}-{\gamma }^{-}\frac{{w}_{i}^{-}{w}_{j}^{-}}{2{w}^{-}})]\,{\delta }_{{\sigma }_{i},{\sigma }_{j}}$$
The Hamiltonian \( {\mathcal H} \) and the modularity Q are related by
$$Q=-\,\frac{1}{2{w}^{+}+2{w}^{-}} {\mathcal H} $$
and, consequently, minimizing \( {\mathcal H} \) implies maximizing Q. The algorithm implemented uses a classical simulated annealing method73 to solve the combinatorial problem. This technique can find a good solution, even when there is some noise in the data. Using a probabilistic process, it approximates the global optimum of the given function.
Community independence and roots
To qualitatively rank the degree of independence of a community we used an adaptation of the degree centrality that relies on \({\tilde{R}}_{ij}\) to reward a community for its negative interactions with the other communities and to penalize it for positive interactions. The method is detailed by the following equation, where higher values imply greater independence:
$$I({\sigma }_{k})=\frac{1}{{n}_{k}}\,\sum _{i}\,\sum _{j}\,-\,{\rm{sign}}\,({W}_{ij})\,|{W}_{ij}|{\delta }_{{\sigma }_{k},{\sigma }_{i}}(1-{\delta }_{{\sigma }_{k},{\sigma }_{j}})$$
Therefore, I(σk) accounts for the independence of community σk, nk corresponds to the number of nodes in the community, Wij is the signed weighted adjacency matrix, and \({\delta }_{{\sigma }_{k},{\sigma }_{i}}\) is the Kronecker δ of communities σk and σi. A root was defined by the detection of at least two nodes in a given community across a minimum of two consecutive strata. The three highest ranked/most independent (as defined above) communities in each stratum were analyzed.
Coherence of disease pairs
The biological coherence of the morbidity associations was assessed by analyzing the shortest path distance in a high-quality network of molecular interactions15 between genes and/or proteins (genes/proteins) assigned to diseases, then comparing the results with those of random genes/protein pairs. Causal genes/proteins were assigned on the basis of phenotype-genetic annotations extracted from the OMIM74 database. The OMIM annotations were linked to ICD-10 diagnoses using Metathesaurus included in the Unified Medical Language System (UMLS) version 2015AB75. All the diseases with at least one causal gene/protein were included in the analysis; 111 diseases had at least one causal gene/protein, of which 104 had at least one causal gene/protein represented in the molecular network. For each disease pair present in the observed male or female morbidity networks, we computed the average shortest path between their causal genes/proteins (e.g., shortest path between gene/proteini and gene/proteinj corresponding to the associated diseases i and j, respectively) and compared the result with the average of 1,000 gene/protein pairs for which one of the members was randomly chosen and the other was a defined casual gene/protein. OMIM diseases frequently have more than one causal gene/protein and so we computed the average shortest path between the assigned genes/proteins. The morbidity networks included 1,051 and 1,031 disease associations by RRs > 1.6, and 239 and 206 disease associations by RRs < 0.8 in men and women, respectively, with OMIM-assigned causal genes/proteins. Thus, for each of these associations, we computed the average shortest path and compared the result with that of 1,000 random genes/proteins, thereby obtaining empirical P values. The analysis was performed using the complete interactome dataset compiled by Menche et al.15 or a subset corresponding to interactions with evidence from the literature and binary protein-protein assays.
Degree leaps
To find leaps in the degree (k) of nodes through time, we constructed connectivity trajectories. For each diagnosis, we differentiated between edges corresponding to RRs > 1.6 or RRs < 0.8. Given a diagnosis or node, its connectivity trajectory for RRs > 1.6 was built as follows: for each age stratum, the number of connected edges with RRs > 1.6 was counted and the connectivity trajectory was the result of the number of edges across strata. Therefore, the leaps were defined based on the difference between the maximum and minimum k values of each node. Nodes with no change in their k across the age groups where they appear and nodes with spurious changes (a single change with no continuation in subsequent strata) were not considered in this analysis. To assess the observed distributions of leaps of disease connectivity across age in men and women, we generated random undirected networks that preserved the original node degree distributions and connectedness. The latmio_und function (Brain Connectivity MATLAB Toolbox; https://sites.google.com/site/bctnet/) was used for this analysis. Randomization was carried out using the "rewiring" parameter corresponding to the exact number of nodes in each observed network in the analysis. Thus, 1,000 random networks for each age group and gender were generated and combined (consecutively, one random network from each age group/gender) to obtain 1,000 random distributions of disease connectivity leaps, which were compared with the observed values.
Gene set overlap, gene expression and pathway enrichment
The overlaps with the diabetic neuropathy and undernutrition gene expression signatures were computed using 2 × 2 contingency tables and the χ2 test with Yate's correction, considering an approximate total of 18,500 annotated human genes (the actual number varying by study). Pre-processed and normalized RNAseq data of normal breast tissue and primary breast tumors were taken from The Cancer Genome Atlas (TCGA) repository (Data Access Committee project #11689). A paired t-test was applied to detect differentially expressed genes between the normal tissue and tumors, and in the overlap analysis we only considered the genes corresponding to a false discovery rate (FDR) of <1%. The Reactome enrichment tool76 was used with standard parameters to detect significant pathways with a FDR < 5%. The expression signature scores were computed using the ssGSEA algorithm77 with standard parameters and using all genes included in each signature. The linear correlation analysis between the signature scores and age at diagnosis was adjusted by tumor stage. The association between pleiotropy and smoking cessation/dependence gene targets was based on PubMed enrichment analysis using the DAVID tool78.
GWAS analyses
The GCAT project includes a large prospective cohort from the Catalan general population with ages ranging between 40 and 65 years, baseline epidemiological characterization, and electronic health record-linked data45,79. For this study, we used baseline data at recruitment (2014–2016) for a subset of subjects. The participants (n = 5,459; GCATcore) were genotyped using the Expanded Multi-Ethnic Genotyping Array (MEGAEX) (Illumina). Genotyping was performed at the Genomics Unit IMPPC-IGTP. Extended quality control protocol is available at www.genomesforlife.com/GCATCoreAnalysis. After filtering, 4,988 participants and 1,652,023 genetic variants were included in the analysis. Sexual and mitochondrial chromosomes were discarded as well as autosomal chromosome variants with minor allele frequency (MAF) < 0.01 and AT-CG sites. Imputation used 665,592 (40%) variants and was performed using Shape-IT80 and IMPUTE281 and four reference panels: 1000 Genomes, Genome of the Netherlands, UK10K, and Haplotype Reference Consortium. All variants with imputation correlations <0.7 were removed. The best score was used for those variants present in more than one reference panel. Variant dosage from IMPUTE2 was transformed to binary PLINK82 format by using the "hard-call-threshold 0.1" flag. The final core set was produced by approximately 15 million variants with MAF > 0.001 and 9.5 million variants with MAF > 0.01. Imputation was done at the Barcelona Supercomputing Center (BSC). Clinical conditions were defined from a self-reported questionnaire at baseline; 159 conditions were identified, occurring in 1 to 985 cases, 17 of these were collected by direct query, and some were identified from the open text field query. All reports were curated and mapped to ICD-10 codes. The diagnoses with more than 200 cases included: allergies, arterial hypertension, asthma, depression, dermatitis, hyperlipidemia, migraine, rhinitis, and type II diabetes. The analysis for association signals influencing these diagnoses comprised two consecutive steps: an individual GWAS analysis for each ICD-10-based disease and then a pairwise analysis to detect shared associations. The GWAS summary statistics, with quality control protocols and data are available at the GCAT website, and the raw data have been deposited at the European Genome-phenome Archive83 (access is regulated by GCAT Data Access Committee applications). The analyses were performed using the score test and saddlepoint approximation in the SPAtest R package84. This method accounts for unbalanced case-control designs, as was the case in our study. The 20 first dimensions of the principal component analysis of population substructure, gender and age data were included as covariates in all analyses. The variants with a nominal value of P < 0.05 in any of the considered single disease analyses were selected for pairwise analysis using the GWAS-pw tool46. This tool provides Bayes factor calculations and identifies variants that are shared in pairs of traits. Statistical power is assessed using a log Bayes factor >685 and posterior probability >0.7. In addition, the level of significance of each comparison was inferred empirically from 100 random GWAS-pw tests, based on 10 independent simulated datasets of each pair of conditions. For the comparison between "Nutritional anemias" and "Diseases of the nervous system", to determine the random frequency of variants in linkage disequilibrium (D' > 0.99 and P < 0.05 in the Iberian population of Spain) with GWAS signals of any human trait, we generated 100 random sets of 31 variants (genome version hg19 and minor allele frequency > 0.01) and subsequently computed their degree of linkage disequilibrium in the same 1,000 Genomes population against variants from the GWAS catalog (v1.0.1, 2018-02-28)86 over a range of ±100 kb.
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We thank Marina Pollán and Mariona Pons for their valuable comments on the study. We also wish to thank to all IDIAP members and health care professionals involved in the study of SIDIAP-Q, and the GWAS participants for their contributions. This work was supported by the Carlos III Institute of Health (ISCIII), Ministry of Economy and Competitiveness (MINECO, Health Strategy Action, National Research Program Oriented to Societal Challenges within the Technical, Scientific and Innovation Research National Plan 2013–2016, grants PI12/0042 and PI15/00854; grant MTM2014-60127-P; "Acción de Dinamización" ADE 10/00026; and Network for Prevention and Health Promotion in Primary Health Care (redIAPP), grants RD12/0005/0001 and RD16/0007/001), co-funded with European Union ERDF funds (European Regional Development Fund; FEDER "Una manera de hacer Europa"), and by the Generalitat de Catalunya (SGR 2014-364, 2014-1269, and 2017-449; and CERCA program). R. de Cid was supported by the "Ramón y Cajal" researcher program (RYC-2011-07822).
Department of Mathematics, Technical University of Catalonia, Castelldefels, Barcelona, 08860, Catalonia, Spain
A. Amell & F. Comellas
Jordi Gol University Institute for Research Primary Healthcare (IDIAP Jordi Gol), Barcelona, 08007, Catalonia, Spain
A. Roso-Llorach & C. Violán
Autonomous University of Barcelona, Bellaterra, 08193, Catalonia, Spain
ProCURE, Catalan Institute of Oncology (ICO), Oncobell, Bellvitge Institute for Biomedical Research (IDIBELL), L'Hospitalet del Llobregat, Barcelona, 08908, Catalonia, Spain
L. Palomero, J. Serra-Musach & M. A. Pujana
Statistics Department, Foundation Sant Joan de Déu, Esplugues, 08950, Catalonia, Spain
D. Cuadras
GCAT-Genomes for Life, Germans Trias i Pujol Health Sciences Research Institute (IGTP), Program for Predictive and Personalized Medicine of Cancer (IMPPC), Badalona, 08916, Catalonia, Spain
I. Galván-Femenía & R. de Cid
A. Amell
A. Roso-Llorach
L. Palomero
I. Galván-Femenía
J. Serra-Musach
F. Comellas
R. de Cid
M. A. Pujana
C. Violán
Conceptualization: M.A.P. Data curation: A.A., A.R.-L., L.P., I.G.-F., J.S.-M., R.C. and C.V. Data analysis: A.A., A.R.-L., L.P., D.C., I.G.-F., J.S.-M., F.C., R.C. and M.A.P. Supervision: F.C., R.C., M.A.P. and C.V. Writing draft: M.A.P. Review and editing: A.A., F.C., R.C., M.A.P. and C.V.
Correspondence to R. de Cid, M. A. Pujana or C. Violán.
M.A.P. is recipient of an unrestricted research grant from Roche Pharma for the support of the ProCURE research program of the Catalan Institute of Oncology.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Figures S1-S6
Amell, A., Roso-Llorach, A., Palomero, L. et al. Disease networks identify specific conditions and pleiotropy influencing multimorbidity in the general population. Sci Rep 8, 15970 (2018). https://doi.org/10.1038/s41598-018-34361-3
Multimorbidity
Disease Network
Shared Genetic Factors
Nutritional Anemia
Community Roots
Multimorbidity patterns in old adults and their associated multi-layered factors: a cross-sectional study
Jiao Lu
Yuan Wang
Anle Wei
BMC Geriatrics (2021)
MorbiNet: multimorbidity networks in adult general population. Analysis of type 2 diabetes mellitus comorbidity
Alba Aguado
Ferran Moratalla-Navarro
Victor Moreno | CommonCrawl |
Tin-modified gold-based bulk metallic glasses
Shuo-Hong Wang1 &
Tsung-Shune Chin1,2,3
Gold Bulletin volume 45, pages3–8(2012)Cite this article
Tin was selected as a modifying element in low-gold-content (50 at.%) bulk metallic glasses (BMGs) aiming at developing alloys with cost-effective performance. New gold-based Au–Sn–Cu–Si alloys were fabricated by injection-casting into a copper mold. The as-cast BMG Au50Sn6Cu26Si18 with 18.6-karat gold and a diameter of 1 mm possessed a lower glass transition temperature (T g) of 82°C (355 K), a lower liquid temperature of 330°C (603 K), and a super-cooled liquid region of 31°C. The viscosity range of this BMG Au50Sn6Cu26Si18 was from 108 to 109 Pa s measured at a low applied stress of 13 kPa. To compare the viscosity with different applied stresses, its viscosity clearly increased with applied stress below T g but not so obvious above T g. The low viscosity of this BMG Au50Sn6Cu26Si18 at around 102°C, which is very close to the boiling temperature of water (100°C), rendered easy thermal–mechanical deformation in a boiling water-bath by hand-pressing and tweezers-bending. Such a deformation capability in boiling water is beneficial to the further applications in various fields.
Pure gold is a precious metal and possesses the characteristics of luster, softness, malleability, and ductility. However, in its pure form, it is too soft to be used for monetary exchange and for producing jewelry materials. Thus, hardening of gold by alloying it with Cu, Ag, Ni, or other metals have been a common practice for a long time [1, 2]. Moreover, because of its superior thermal conductivity, excellent electrical conductivity, and high resistance to corrosion, gold has been widely applied in modern industries including IC electronics, aerospace, medicine, and dentistry [3]. Although pure gold and gold-based alloys with higher karats without toxic elements are useful and acceptable for the human body, the existence of grain boundaries in crystalline gold-based alloys is an ongoing concern with biomedical implants.
Metallic glass is amorphous in structure, containing no crystalline anisotropy, dislocation, grain boundary, or crystalline defects [4]. Bulk metallic glasses (BMGs) usually possess unique properties such as high strength and elasticity, increased hardness, good toughness, and excellent resistance to corrosion compared to their crystalline counterparts [5, 6]. Since 1988, a large number of BMG systems have been developed notably those Mg- [7], La- [8], Zr- [9], Fe- [10], Pd- [11], Pt- [12], Ti- [13], Ni- [14], and Ca- [15] alloy systems. These BMGs exhibit large critical glass forming size and high thermal stability and hence reveal new possibilities for industrial applications.
In 1960, Klement et al. [5] synthesized the first amorphous alloy using the binary eutectic Au82Si18 composition, in thickness less than 50 μm. However, its poor glass-forming ability (GFA) has resulted in limited development over the past several decades. Schroers et al. exploited gold-based multi-component BMGs. These Au-based BMGs possessed a low glass transition temperature (T g) of at most 128°C, good thermal stability at ambient temperature, and increased hardness, and they were easier to process. For Au–Ag–Pd–Cu–Si BMGs, the addition of Pd and Ag enhances GFA but they increase Tg [3]. On the other hand, Zhang et al. increased the gold content in Au–Ag–Cu–Si BMGs and resulted in a sharp decrease in T g. The lowest reported T g for an Au-based BMG is 66°C in the high-Au content composition Au70Cu5.5Ag7.5Si17 [16]. The above properties make Au-based BMGs useful for applications such as jewelry, micro-electromechanical systems (MEMS), electronics, nano-molding technologies [17, 18], and dentistry [19].
The purpose of this study is to explore new Au-based BMG composition with cost-effective performance. Composition design is to keep the gold content as low as possible, yet not less than 18 karats. We used tin as an inexpensive alloying element instead of palladium or silver. The performance of major concern is the thermal properties in particular the relationship between viscosity and deformation capability (processing characteristics) of such BMGs. More specifically, our ultimate goal is to develop a new Au-based BMG with low glass transition temperature which will facilitate processing capability to benefit further applications.
In this work, alloy ingots with nominal compositions Au50Sn6Cu26Si18, Au50Sn6Cu24Si20, and Au50Sn9Cu23Si18 were prepared by arc melting the mixtures of pure elements Au, Sn, Cu, and Si with purities of at least 99.9% under vacuum at 10−2 Torr. For homogeneity of the alloys, we melted the ingots six times for each composition. Bulk alloy rods with diameters of 1 mm were fabricated by conventional injection casting into a copper mold under an argon atmosphere. The structure of the as-cast rods was examined by X-ray diffractometry (XRD, Shimadzu XRD-6000) using Cu Kα1 radiation. The thermal properties were studied by a differential scanning calorimetry (DSC; PerkinElmer Diamond DSC) at a fixed heating rate of 20°C/min under flowing argon. To study the processing ability, we used a thermal mechanical analyzer (TMA; PerkinElmer Diamond TMA). The as-cast rods were cut into pieces, each with an aspect ratio of around 2 and polished at both ends. TMA measurements were performed with applied loads of 10 and 50 mN. We raised the temperature of the as-cast rods at a fixed heating rate of 10°C/min starting from room temperature to 200°C. The microstructure was observed on cross sections of the as-cast rods using a transmission electron microscope (TEM; JOEL 2010F) operating at an accelerating potential of 200 kV. Structural identification of the phases was carried out by conventional selected area electron diffraction (SAED).
Figure 1 depicts the X-ray diffraction (XRD) patterns taken from the transverse cross sections of the as-cast 1-mm-diameter rods of Au50Sn6Cu26Si18, Au50Sn6Cu24Si20, and Au50Sn9Cu23Si18 alloys. The XRD results indicate that no obvious diffraction peaks could be identified for the Au50Sn6Cu26Si18 alloy, which shows a fully amorphous structure. The Au50Sn6Cu26Si18 BMG has a GFA capable of casting into amorphous rods of at least 1 mm in diameter and is the easiest to process of all our Au–Sn–Cu–Si quaternary alloys. The new Sn-added alloys show poor GFA when the Cu content decreased from 26 at.% (by being replaced by Si or Sn). This means that the increase in Si content is harmful to the formation of the quaternary Au–Sn–Cu–Si BMG. On the other hand, as Cu is partially substituted by Sn, the GFA also obviously decreases. We thus aimed to further investigate the BMG Au50Sn6Cu26Si18.
X-ray diffraction patterns taken from cross sections of as-cast 1-mm diameter rods of Au50Sn6Cu26Si18, Au50Sn6Cu24Si20, and Au50Sn9Cu23Si18 alloys
The microstructure of the Au50Sn6Cu26Si18 as-cast BMG rod was studied by TEM (Fig. 2) together with a selected area diffraction pattern (SADP). The lack of diffraction spots or rings in the SADP indicates that the structure of the BMG Au50Sn6Cu26Si18 was amorphous even though there were slight contrasts in the matrix.
Typical TEM image and selected area diffraction pattern (SADP) of specimen taken from as-cast 1-mm diameter rod of Au50Sn6Cu26Si18 alloy
A typical DSC curve of the Au50Sn6Cu26Si18 as-cast BMG rod is shown in Fig. 3. The crystallization temperature T x and T g are 113°C and 82°C, respectively. The liquid temperature (T l) of Au50Sn6Cu26Si18 BMG is 330°C. The calculated thermal criteria are the super-cooled liquid region (ΔT x ), 31 K; the reduced glass transition temperature (T rg), 0.59; and the γ-factor 0.403. Although the ΔT x of the Au50Sn6Cu26Si18 BMG was not as high as those reported in the literature [16], it was expected to be potential for easier processing below 100°C in hot water because of low T g and moderate ΔT x . Therefore, we carried out further study on its viscosity at elevated temperatures, as to be delineated later. Table 1 summarizes the thermal parameters of Au-based BMGs measured using the DSC. By comparing the results with those of other Au-based BMGs whose Au content 46–52 at.%, the T g of our Au50Sn6Cu26Si18 BMG is obviously much lower than those of Au-based BMGs containing Pd and Ag. The ΔT x (31°C) of Au50Sn6Cu26Si18 BMG is higher than that of BMG Au46Ag5Cu29Si20 (ΔT x = 25°C) but lower than that of Pd-containing BMG Au49Ag5.5Pd2.3Cu26.9Si16.3 (ΔT x = 58°C). Among these Au-based BMGs, the most well known is the composition Au49Ag5.5Pd2.3Cu26.9Si16.3. Despite its high T g, this BMG has been widely used as the mold material in nano-imprint technology and in MEMS. Therefore, our BMG alloy, which has a lower T g and does not contain Pd and Ag, is more cost-effective in similar applications.
Typical DSC curve of as-cast 1-mm diameter rod of Au50Sn6Cu26Si18 alloy
Table 1 Thermal parameters of several Au-based BMGs found in the literature along with the values determined from this study [3]
Thermal mechanical analysis (TMA) and differential TMA (DTMA) curves of the Au50Sn6Cu26Si18 as-cast BMG rod at an applied load of 50 mN are plotted in Fig. 4. The DTMA curve shows the differentiated results of displacement related to time. To exhibit the relationship between temperature and viscosity several parameters are defined. They are the onset temperature of viscous flow for the initial state (T onset); the temperature corresponding to the lowest viscosity which is the viscosity of semi-steady state (T vs); and the finishing temperature of viscous flow for the full crystallization state (T finish). These are marked on the DTMA curve. From the DTMA curve, T onset, T vs, and T finish are 84°C, 108°C, and 116°C, respectively. The values of T onset and T finish are very close to those of T g (82°C) and T x (113°C), respectively. TMA data and DSC data are almost the same for these two specific temperatures. The viscosity of the Au50Sn6Cu26Si18 BMG can be evaluated according to the Stefan equation [20]:
$$ \eta = \left( {\frac{\sigma }{{3 \dot{\varepsilon } }}} \right){\left( {1 + \frac{{d_0^2}}{{8l_0^2{{\left( {1 + {\varepsilon_{\text{n}}}} \right)}^2}}}} \right)^{{ - 1}}}, $$
where σ is the stress (Pa), \( \dot{\varepsilon} \) is the strain rate (s−1), d 0 is the initial diameter of the specimen, l 0 is the initial length of the specimen, and ε n the engineering strain (nominal strain). The stress is calculated by \( \sigma = \left( {F/{A_0}} \right)\left( {l/{l_0}} \right) \), where F, A 0, and l are the applied loading force (mN), initial cross-sectional area of the specimen, and length after deformation, respectively. Calculated results are plotted in Fig. 5.
TMA and DTMA curves of as-cast Au50Sn6Cu26Si18 BMG rod at the applied load of 50 mN
Estimated viscosity of as-cast Au50Sn6Cu26Si18 BMG, measured under an applied load of a 10 mN (corresponding to 13 kPa) and b 50 mN (65 kPa)
The curves in Fig. 5a and b show that the viscosity values of the Au50Sn6Cu26Si18 BMG are between 108 and 109 Pa s at an applied stress of 13 kPa, and 2 × 108 and 1010 Pa s at an applied stress of 65 kPa. The viscosity of the BMG Au50Sn6Cu26Si18 is in a range similar to that of the BMG Au49Ag5.5Pd2.3Cu26.9Si16.3 (107–109 Pa s) obtained by Tang et al. [17]. However, the lower limit is relatively higher than those of BMGs Mg65Cu25Gd10 and Mg65Cu25Gd10P3 (106–1010 Pa s) obtained by Chang et al. [20]. In addition, we observed that viscosity increases as increasing applied stress from 13 to 65 kPa. These tendencies are consistent with those reported by Chang et al. [21] for the BMG Mg58Cu31Y11. In the literature, Tang et al. reported that the applied stress was in the range of 40–400 kPa or even higher for the BMG Au49Ag5.5Pd2.3Cu26.9Si16.3 [17]. Hence, the applied flow stress of 13 kPa for the BMG Au50Sn6Cu26Si18 is relatively low [21, 22].
In addition, the viscosity value is highly dependent on temperature. In fact, the much higher viscosity measured at 65 kPa is at a temperature much lower than 83°C, which is the TMA glass transition temperature. As the temperature becomes higher than 83°C, the viscosity soon declines to its minimum value. The viscosity decreases from 109 to 108 Pa s for both applied stresses of 13 and 65 kPa. Over this temperature range, the BMG Au50Sn6Cu26Si18 exhibits a typical viscosity of a Newtonian liquid. There exists a subtle difference in minimum viscosity values, 108 Pa s at 13 kPa and 2 × 108 Pa s at 65 kPa, and at different temperatures of the lowest viscosity (T vs), 102°C at 13 kPa and 108°C at 65 kPa. This indicates that the applied stress does influence the measured viscosity to some extent. Above 113°C, the BMG no longer exhibits the viscosity that is characteristic of glassy materials, but instead shows the deformation characteristics of a rigid crystalline solid in responding to applied stress at elevated temperatures. These results indicate that the BMG Au50Sn6Cu26Si18 can have a viscous flow.
According to the result of viscosity, the proposed best temperature for thermal deformation is at 102°C, which is very close to the boiling temperature of water (100°C). Hence, we think it is possible to investigate the deformability of the as-cast Au50Sn6Cu26Si18 BMG, 1 mm in diameter, in boiling water-bath. Figure 6a and b exhibits the results of hand-pressed and tweezers-bent Au50Sn6Cu26Si18 BMG in boiling water-bath. It shows that rod-shaped Au50Sn6Cu26Si18 BMG can be pressed to be flattened, and can be bent to 90–180° repeatedly for several times without breaking or hardening. We did deformation out of the boiling water-bath and the BMG rod soon cracked. These are direct proofs that the BMG alloy deforms by a viscous flow mechanism in boiling water. Such a deformation capability in boiling water, which does not need a temperature controller to maintain isothermal condition, is very useful for application in various fields. These include but not limited to nano-imprint molds, dental prosthesis, jewelry, or MEMS devices.
Deformation capability of as-cast Au50Sn6Cu26Si18 BMG in the boiling water-bath by a hand-pressing and b tweezers-bending
In summary, we explored a cost-effective quaternary gold-based BMG Au50Sn6Cu26Si18 with a gold content of 18.6 karats in this research. This alloy retains a good GFA which renders possibility of inject-casting into BMG rod of at least 1 mm in diameter. This new Au-based BMG is characteristic of low glass transition (82°C), crystallization (113°C), and liquid (330°C) temperatures and a moderate super-cooled liquid region (ΔT x , 31°C). Moreover, the Au50Sn6Cu26Si18 BMG exhibits a viscosity ranging from 108 to 109 Pa s measured under an applied stress as low as 13 kPa. The merit of its significant deformation capability at around 100°C (i.e., at the boiling temperature of water) renders easy and precise deformation without the use of a temperature controller. Based on these outstanding and cost-effective characteristics, the new BMG Au50Sn6Cu26Si18 will find extensive applications in molds for nano-imprinting, dental prosthesis, jewelry, and MEMS devices in the future.
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The authors are grateful to the National Science Council of the Republic of China for sponsoring this research under grant NSC 97-2221-E-035 -011-MY2.
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and the source are credited.
Department of Materials Science and Engineering, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, 30013, Taiwan
Shuo-Hong Wang
& Tsung-Shune Chin
Center for Nanotechnology, Materials Science and Microsystems, National Tsing Hua University, No. 101, Sec-2, Kuang-Fu Road, Hsinchu, 30013, Taiwan
Tsung-Shune Chin
Department of Materials Science and Engineering, Feng Chia University, No. 100, Wenhwa Road, Seatwen District, Taichung, 40724, Taiwan
Search for Shuo-Hong Wang in:
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Correspondence to Tsung-Shune Chin.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wang, S., Chin, T. Tin-modified gold-based bulk metallic glasses. Gold Bull 45, 3–8 (2012) doi:10.1007/s13404-012-0043-z
Issue Date: March 2012
Gold-based alloys
Bulk metallic glasses
Thermal mechanical analysis
Deformation capability | CommonCrawl |
Home Journals MMEP Genetic Algorithm Assisted Support Vector Machine for M-QAM Classification
Genetic Algorithm Assisted Support Vector Machine for M-QAM Classification
Sajjad A. Ghauri* | Mubashar Sarfraz | Nooh Bany Muhammad | Shahrukh Munir
Department of Electrical Engineering, ISRA University, Islamabad 44000, Pakistan
Department of Computer Science & Information Systems, American University of Kuwait, Safat 13034, Kuwait
[email protected]
Automatic modulation classification (AMC) has wide spread applications in today's communication system. AMC has vast applications both in military as well as civilian. In intelligent communication systems such as software defined radios networks and cognitive radio networks, AMC is the most important issue, when there is no prior information about the signal. In this research article, pattern recognition approach has been utilized for classification of M-ARY quadrature amplitude modulated (M-QAM) signals. Higher order cumulants are selected as feature set and Genetic Algorithm assisted Support Vector Machine (SVM) classifier is used for classification of M-QAM signals. The performance of classifier is evaluated on fading channels in the presence of additive white Guassain noise. The classification accuracy is also compared with and without optimized classifier.
automatic modulation classification (AMC), higher order cumulants (HOC), genetic algorithm (GA), M-ARY quadrature amplitude modulated (M-QAM) signal, support vector machine (SVM)
In Cognitive radio (CR) based communications spectrum is automatically sensed and efficiently used [1]. Awareness of wireless radio spectrum, which is the adaptable proposal for spectrum access is dependent on it and is a protuberant characteristic of cognitive radio networks [2]. The conventional communication studies generally focus on making communication systems more reliable, higher power and/or bandwidth efficient, and more secure [3].
One of the essential requirements for a communication system is the security. The two users in communication system don't want their communication to be known to the third user/eavesdropper. In contrast to this, the regularity authority might wish to detect a non-licensed user. The essential step of doing so is to identifying or classifying the modulation scheme of intercepted signal, which is the signature of a transmitter. Such demands also arise in many other military and noncombatant applications such as surveillance, validation of signal, verification, identification of interference, selection of proper demodulation methods in software defined radio (SDR), electronic warfare and threat analysis [4, 5].
AMC is a key element which increases the overall cognitive radio networks performance. The key aim of this research is to empower the receiver in order to identify or classify the signal modulation automatically [6].
In wireless communication systems, multipath fading channel, single carrier transmission method is used which results in corruption of signal. This problem is solved by orthogonal frequency division multiplexing (OFDM). The spectrum is divided into small sub bands then one sub carrier is used for every sub band. So, each of these small band of frequency is transmitted over the flat fading channel and inter symbol interference effect between these small frequency bands is minimized [7].
Furthermore, many different levels of modulation are being used which are dependent on information of channel condition for every sub band. Such kind of method can be identified as adaptive modulation. For instance, IEEE 802.11a is the standard OFDM protocol, have throughput for 64 QAM in the range of 48 Mbps. But the probability of error rises with rise of modulation level. Therefore, high levels of modulation can be utilized by sub carriers having higher SNR values, also the lower levels of modulation can be utilized by low SNR value sub carriers, which result in considerable throughput improvement of a communication system. The adaptive modulation system receivers need to classify the modulation form for each sub carrier so as to choose the demodulation method for each modulation type [8].
This is possible by using a table called bit allocation table (BAT), but this bit allocation table creates an extra overhead, mainly for large numbers of sub carriers as well as small frames of OFDM. The pretty way out for this, is to use AMC on receiver end in order to classify the modulation format for respective sub carrier, thus overall system transmission rate is increased [9].
Figure 1. Maximum likelihood based D.T approaches
In literature, the AMC has been divided into two approaches; Decision Theoretic (DT) Approach and Pattern Recognition (PR) Approach. The DT approach is based on the likelihood function of the received signal. There are several tests exists in the literature: Average likelihood ratio test (ALRT), generalized likelihood ratio test (GLRT), hybrid likelihood ratio test (HLRT), quasi variants of the likelihood test. Figure 1 shows the maximum likelihood based DT approach. The state of art existing work can also be found in ref. [10-21].
Table 1. Summary of features based PR approach
Ref #
2FSK, 4FSK, 8FSK, 16FSK, 32FSK
QPSK, 4FSK, 16QAM
BPSK, QPSK, 16QAM,64QAM
16QAM, 64QAM
Wavelets
2PSK,4PSK,2FSK,4-FSK, 16QAM,
4PSK,8PSK,16QAM,64QAM,256QAM
4QAM, 16QAM, 64QAM
Higher order Cumulants
2PSK-64PSK, 2FSK- 64FSK, 4QAM-64QAM
BPSK, QPSK,8PSK,16-QAM,64-QAM,256-QAM
ASK, PSK, and QAM
BPSK, QPSK, 8PSK, 64QAM and 256QAM
As this research is focusing on the PR approach which is also known as features based approach. The PR approach can be accomplished in two steps;
(i) Parameter extraction & feature selection
(ii) Classification
There are various methods have been proposed in the literature to extract parameters from the received signal and select the number of distinct features from these parameters. Some famous features which have been utilized in the literature are: higher order moments, higher order cumulants, spectral features, cyclic features, Gabor features and wavelet based features [22, 23, 30, 31, 35, 36].
The extracted distinct features are now input to the classifier structure. There are many forms of the classifier structure have been incorporated in the research. Mostly the classifier structure is based on neural network architecture, heuristic computational technique, K-nearest neighbor, Fuzzy C-means. The summary of some of the classifier and features used for the classification are shown in Table 1.
1.1 Contribution of the research article
The contribution is outlined as under: -
Proposed a modulation classification algorithm based on continuation of SVM classifier using HOC's as a feature set. The proposed system has the following benefits:
It provides high accuracy of classification as compared to state of the art existing techniques in literature.
Capable to classify different forms of modulation even in the presence of AWGN noise as well as Rayleigh fading and Rician fading channels.
Feature selection subsystem is based on HOC's and HOM's which is integrated with the proposed SVM and which results in simplified model of classifier.
Performance of classifier is further optimized using one of the evolutionary computing techniques such as Genetic Algorithm (GA).
1.2 Organization of the research article
This research paper is organized as follows: Section I provides the introduction to the problem area and systematic review of the literature along with major contributions. Section II presents the system model and features selected for classification. Section III leads an overview to pattern recognition systems and presents the structure and mechanism of support vector machine as a classifier. Simulation results with optimization and without optimization are incorporated in Section IV, which shows the supremacy of the proposed classifier and it is found that with optimization classification accuracy is much improved. Conclusion and future work is presented in Section V.
2. Signal Model and Selected Features
Figure 2(a) and 2(b) depicts the generalized system model for AMC. The signal is injected into the modulator for modulation subsequently, signal is transmitted over the channel. The additive white Guassain noise (AWGN) is considered throughout the research with different fading channel model i.e. Rayleigh and Rician. At the receiver side, features taken are higher order cumulants (HOC) extracted from the noisy received signal (Figure 3). Once features extracted, then these features are fed into the classifier [36]. The classifier structure is based on support vector machine (SVM) and feed forward back propagation neural network (FFBPNN). After that, the classifier performance is optimized using one of the famous heuristic computational technique i.e. Genetic Algorithm (GA) and particle swarm optimization (PSO). The generalized expression for received signal is given as below:
$r_{n}=s_{n}+g_{n}$ (1)
where, rn is the received baseband signal, gn is the additive white Gaussian Noise, sn is the transmitted signal and is defined as
$s_{n}=\mathrm{K} e^{-i\left(2 \pi f_{0} n T+\theta_{0}\right)} \sum_{j=-\infty}^{j=\infty} S(l) h(n T-j T+\left.\in_{T} T\right)$ (2)
where, S(l) is sequence of symbols at the input that is taken out from the set of M constellations of known symbols and the condition for symbols to be equiprobable is not necessary, K is the signal amplitude, f0 is offset constant of frequency, T is the spacing of symbols, θn is phase jitter which differs from symbol to symbol, h is channel effects and $\in_{T}$ is jitter timing.
Figure 2. Transmitter side of proposed system model
Figure 3. Receiver side for proposed system model
Table 2. Theoretical values of higher order moments and cumulants
QAM2
QAM16
The representation of pth order of Cumulants is same as pth order of moment.
$C_{p q}=\operatorname{cum}[\underbrace{s, \ldots \ldots, S}_{(p-q) \text { terms }}, \quad \underbrace{s^{*}, \ldots . s^{*}}_{(q) \text { terms }}]$ (3)
The nth order Cumulants is the function of the moments order up to n
$\operatorname{cumm}\left[s_{1}, \ldots . ., s_{n}\right]=\sum_{\forall v}(-1)^{q-1}(q-1) ! E\left[\prod_{j \in v 1} s_{j}\right] \ldots E\left[\prod_{j \in v_{q}} s_{j}\right]$ (4)
The features selected for classification of M-QAM signals are as under [37]:
$C_{20}=\mathrm{E}\left[y^{2}(n)\right]=\operatorname{cumm}\{\mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n})\}$ (5)
$C_{21}=\mathrm{E}\left[|y(n)|^{2}\right]=\operatorname{cumm}\left\{\mathrm{y}(\mathrm{n}), y^{*}(n)\right\}$ (6)
$C_{40}=M_{40}-3 M_{20}^{2}=\operatorname{cumm}\{\mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n})\}$ (7)
$C_{41}=M_{40}-3 M_{20} M_{21}=\operatorname{cumm}\left\{\mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n}), y^{*}(n)\right\}$ (8)
$\begin{aligned} C_{42}=M_{42}-\left|M_{20}\right|^{2} &-2 M_{21} = \operatorname{cumm}\left\{\mathrm{y}(\mathrm{n}), \mathrm{y}(\mathrm{n}), y^{*}(n), y^{*}(n)\right\} \end{aligned}$ (9)
$M_{p q}=\mathrm{E}\left[s^{p-q}\left(s^{*}\right)^{q}\right]$ (10)
whereas, p represents the order of moment and s∗ is the complex conjugate of a signal s. C2,0 is the second order cumulants, which is known as the expected value of the square of the received signal also known as the mean. C2,1 is the class of second order cumulants, which represents the expected value of the absolute square of the received signal. C4,0 is the forth order cumulants, with no absolute value of the received signal basically, it is the combination of the M4,0 and M4,1. Similarly C4,1 and C4,2 is the class of 4th order cumulants. The theoretical values of the Cumulants are shown in Table 2.
3. Optimum SVM Classifier
After the features extraction from the noisy signal, these features are now input to the classifier structure. The classifier is based on multi-class SVM. SVM has a solid mathematical model, which can efficiently resolve the construction problem of high dimensional data model in the finite set of samples, and can converge to global best [38].
The SVM basics for solving the best linear hyper plane which could classify all the signals completely. Considered the training data as below:
$\left.\left\{\left(x_{1}, y_{1}\right), x_{2}, y_{2}\right), \ldots\left(x_{i}, y_{i}\right), x \in \operatorname{Rd}, y \in\{+1,-1\}\right\}$ (11)
whereas, xi represent the feature space, yi= +1 means that the signal belongs to the first class, yi= −1 shows that the signal is member of second class. Such kind of data are separated through hyper plane w*x+b = 0, when training data is linearly distinguishable. Then the solution for optimal plane problem is the optimization problem.
Minimize ½ ||$w \|^{2}$, with reference to yi (w.xi + b) ≥ 1, Lagrange multiplier is introduced for solving the quadratic programming problems and the best decision function is obtained by,
$f_{x}=\operatorname{sign}\left[\sum_{i=1}^{n} \alpha_{i} y_{i}\left(x_{i}, x\right)+b\right]$ (12)
whereas, αi is known as Lagrange multiplier. For classification of nonlinear data, SVM make comparison by nonlinearly of training data with high dimensional feature space through its kernel function afterwards it is processed as linear classification. Decision function is given as [39]:
$f_{x}=\operatorname{sign}\left[\sum_{i=1}^{n} \alpha_{i} y_{i} k\left(x_{i}, x\right)+b\right]$ (13)
whereas, k(xi, x)indicates kernel function. The typical kernel functions consist of the radial basis function (RBF):
$\mathrm{K}(\mathrm{x}, \mathrm{y})=\exp \left(-\|\mathrm{x}-\mathrm{y}\|^{2} / 2 \sigma^{2}\right)$ (14)
In short, Modulation classification based on SVM includes the followings steps:
Feature Extraction: Some key features are extracted after which they are converted according to their SVM data format.
Kernel function selection: RBF kernel function is selected.
Kernel function parameter calculation: The best kernel function parameters with cross validation are determined.
Samples training: Sampled signals are trained and classifier model is obtained.
Signals classification: Data are classified according to obtained model in the training phase
To optimize the classification accuracy of SVM based classifier, Genetic Algorithm is used in conjunction with SVM. In this research, the extracted features are optimized in such a way to minimize the mean square error between the theoretical values and original values of the features. The cost function for the GA is mean square error and can be expressed as follows:
$J=\frac{1}{N} \sum_{k=1}^{N} e_{k}(n)^{2}$ (15)
where, J corresponds the mean square error (MSE). The fitness function (FF) for the GA is defined as [40]:
$F F=\frac{1}{1+J} ; \quad 0<F<1$ (16) | CommonCrawl |
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Home Journals I2M Experimental Measurement of Moisture Sorption Isotherms and Isosteric Heat of Palm Hearts (Jomare) Harvested in the Algerian Sahara
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Experimental Measurement of Moisture Sorption Isotherms and Isosteric Heat of Palm Hearts (Jomare) Harvested in the Algerian Sahara
Ahmed Amine Larbi* | Akil Loumani | Ahmed Mediani | Said Bennaceur | Cherif Tigani
Unité de Recherche en Energies renouvelables en Milieu Saharien, URERMS, Centre de Développement des Energies Renouvelables, CDER, Adrar 01000, Alegria
Laboratoire d'Energetique en Zones Arides, Université de Béchar, 08000, Algeria
Corresponding Author Email:
[email protected]
https://doi.org/10.18280/i2m.180310
| Citation
18.03_10.pdf
Mastering the process of drying the palm heart (Jomare) requires knowing their isothermal sorption. The purpose of this work is the experimental determination of the palm heart sorption isotherms. the gravemitric method of saturated saline solutions is used at three different temperatures (40, 50 and 60 °C) with a water activity extending from 0.067 to 0.85. The equilibrium was obtained after about 10 days. Four mathematical models have been used (LANGMUIR, GAB, modified BET and Peleg) to model and predict hygroscopic behavior during drying and storage. After smoothing and optimization of these models on the basis of the statistical processing of the obtained data, the results shows that the GAB and modified BET models best match the sorption isotherms. The isosteric desorption heats for the heart of palm are calculated using the Clausius-Clapeyron equation. Through this study, an expression has been proposed that allow the prediction of the thermodynamic properties of the palm heart. The results of this research can be used to determine characteristic drying curves and to have optimal storage conditions.
solar dryer, palm heart, modelling, hygroscopic, isosteric desorption heats
The significance of wild plants in subsistence farming in the world considered as a food supplement and as a means of survival during times of drought. The utilization of wild plants appears to be more typical and across the board in food insecure regions where an extensive variety of species groups are expended. The Saharian people know about the significance and contributions of wild plants to their day by day consume fewer calories. Wild plant species keep on providing essential vitality and micronutrient needs amid dry spell and social and political turmoil [1]. Compositional information of these plant materials could help in creating technological procedures to make the plant material eatable and more absorbable.
For human use, palms are considered among the most important plant family and ranked at the third position . Various consumable items are acquired from palms, including the date palm fruits products, coconut palm nuts, and different palm oils. Some lesser/underutilized known consumable palm item is palm "Cabbage" or "Palm heart". Albeit most palm items are not accessible industrially, Palm hearts is a noteworthy piece of sustenance industry in Europe and America.
In Africa or more precisely in Algeria, heart of palm is a very widespread species in the south of Algeria. It is located in the central part of the trunk of palms, of whitish color. Several species distinguish from the exploitation way (wild or cultivated).
For several years, little effort has been devoted to the study of palm hearts. Tabora et al. emphasize the use of the palm heart as a vegetable. Tabora chose the three big palms that feed the international palm heart market of the year 1990 to show their economic importance. Salvi et al. considered palm heart as alternative source for human diet and the chemical analyzes of palm heart of Phoenix sylvestris shows the existence of important mineral and vitamin in particular the carbohydrate, crude protein, crude lipid, crude fiber, vitamin B complex, vitamin C, vitamin K, energy and minerals when compared with other commonly consumed. Sylvester et al. has carried out a study on the illegal harvest (consumption and extraction modes) of the palm heart (Geonoma edulis) in the national parks of Costa Rica and concluded with some notion for the protection of the forests of the illegal harvest. Galletti [5] also talked about this illegal trade but in Brazil. The distinction between different species occurs mainly in their method of exploitation (wild or cultivated). There are several standards in the world that determine the codes of practice of of exploitation of the palm heart. The French standard CCP [6] is a norm intended only for wild palms
As vegetables and medicinal herbs are products that do not last long (perishable), their quality depends on several phenomena; harvesting, handling, transportation, storage and marketing . An appropriate examination of these parameters may influence the production of microorganisms . In this way, drying should assume a critical job for the preservation of these items and guarantees their economic and microbiological practicality.
Sorption isotherms are essential parameters for drying and cannot be predicted by theory but must be experimentally established. The sorption isotherm curves give information on the distribution and binding intensity of water molecules as well as their functional availability in biochemical and biological substances. they also provide reliable data on the fluid balance of the product to know its stability range after drying, the sorption isotherms curves provide information to ensure better preservation during storage. This information relates to the optimum moisture content to be achieved during drying, which is close to steady state.
studies on sorption isotherms are carried out by mathematical modeling, eight mathematical models (in the literature) empirical and semi-empirical based on more or less physical bases , describing the correlation of water content with balance with the water activity of the products. among these models: GAB , modified BET , Henderson-Thompson , modified Halsey , modified Oswin , Peleg . GAB is the most widely used model for various foods .
The experimental sorption curves are curves allowing determining the maximum temperature thresholds and the final water contents to serve to optimize the drying conditions of the different products so as to ensure the physicochemical and microbiological stability during the storage. The main focus of this paper is to determine these experimental curves for the palm heart at different temperatures at 30, 40 and 50 °C, and then use different mathematical models to model and predict hygroscopic behavior during drying and storage. The isosteric desorption heats are also presented and calculated by the Clausius-Clapeyron equation. Finally find the expression to predict the thermodynamic properties of the heart of the palm.
2.1 Simple preparation
The research was carried out on young date palms "Gharsa" (see Figure 1) of the area of bouda in the south of Algeria "Adrar", under good conditions in the UREMS laboratory. After extracting the palm heart, preparing the samples for the study. The physico-chemical study leads to a composition shown in Table 1.
Figure 1. Young date palm "gharsa"
Table 1. Physico-chemical composition of the palm heart sample
Dry matter
Mineral matter
Organic material
2.2 Description of the experimental procedure
The procedures for obtaining water sorption isotherms from agri-food products are described in detail by several authors [21, 22]. These procedures include either dynamic methods where the sample is placed in a stream of gas, temperature and humidity held constant, without air agitation, or static procedures (adsorb or desorb) where the sample, is placed in enclosures containing solutions of saturated salts and maintained at constant temperature and relative humidity, until reaching thermodynamic equilibrium. To determine the desorption isotherms of the heart of palm; we used the gravimetric technique of which it is based on the saturated saline solution method. The equilibrium moisture content is calculated for temperatures of 30, 40 and 50 °C.
The experimental device used consists of an oven filled with six jars of saturated saline solutions (KOH, MgCl2 6H2O, K2CO3, NaNO3, KCl and BaCl2 6H2O) which make it possible to obtain a water activity of between 0.07 and 0.89 [23].
Figure 2. Sample preparation for the study of sorption isotherms
These solutions (see Table 2) are prepared by the desolation of a well-studied amount of salt in distilled water. The whole put at a higher temperature to ensure saturation and to arrive at equilibrium [7, 24]. All jars are made of half-liter glass with a tight-fitting lid. Each filled jar respects the ratio of 1/4 saturated saline solution to vacuum. To confirm the saturation of solutions at equilibrium, a layer is kept visible from solid salts.
Each sample weighs 0.07 ± 0.0001 g. All the jars are placed in a tank where the temperature has been well controlled. This operation lasts until the temperature stabilization.
The next step is to put the samples that have been weighed before into pots that contain saturated saline solutions (see Figure 2). After the closing of the jars, the whole will be put in the same tank where the temperature was controlled. The weighing will be daily until the stabilization of the mass with an acceptable error of about 5 %.
$X_{e q}=\frac{m_{\mathrm{ur}}-m_{d}}{m_{d}} \cdot 100$ (1)
The moisture content of the hygroscopic equilibrium product Xeq is calculated by Eq. (1) where mw and md are respectively the drying masses (before and after) [25].
2.3 Statistical criteria of choice of models describing the sorption isotherms
Eight mathematical models are found in the literature to model the sorption isotherms. These models are based on a more or less physical basis [26–28], which describes the relationship between equilibrium moisture content, equilibrium relative humidity and temperature.
Table 2. Standard values of the water activities of the salt solutions used
Water activity
Aw =φ/100
K2CO3
Table 3. Mathematical models used to describe sorption isotherms
Models names
Models equations
GAB (van den berg 1984) [29]
$X_{e q}=\frac{A \cdot B \cdot C \cdot A_{w}}{\left[1-B \cdot A_{w}\right] \cdot\left[1+B \cdot A_{w}+B \cdot C \cdot A_{w}\right]}$
BET modified (Iglesias and chirifie1982) [30]
$X_{e q}=\frac{(A+B T) \cdot C \cdot A_{w}}{\left[1-A_{w}\right] \cdot\left[1-A_{w}+C \cdot A_{w}\right]}$
Peleg model (1993) [16]
$X_{e q}=A \cdot A_{w}^{K 1}+B \cdot A_{w}^{K 2}$
Langmuir (1916) [31]
$X_{e q}=\frac{1}{\left[A+B . A_{w}^{c-1}\right]}$
In this work, we have studied in detail four mathematical models which have been grouped together in Table 3 for the adjustment and modeling of hearts palm sorption isotherms for the three temperatures 30, 40 and 50 °C.
The analysis of the fit and choice of the appropriate model for estimating the model constants from the experimental results is done via two software packages (curve Expert 1.4 and Origin 6). Three statistical parameters for calculating the gap are defined by Eqns. 2, 3 and 4. The correlation coefficient (r), the standard error (S) and the mean relative difference percentage (P).
$S=\sqrt{\frac{\sum_{i=1}^{n_{\max }, \text {data}}\left(X_{e i}-X_{e_{\text {cal}}}\right)}{n_{\text {exp. aata }}-n_{\text {param}}}}$ (2)
$r=\sqrt{1-\frac{\sum\limits_{i=1}^{{{n}_{\exp .data}}}{{{\left( {{X}_{ei}}-{{X}_{{{e}_{cal}}}} \right)}^{2}}}}{\sum\limits_{i=1}^{{{n}_{\exp .data}}}{{{\left( \overset{-}{\mathop{{{X}_{e}}-{{X}_{ei}}}}\, \right)}^{2}}}}}$ (3)
$P \%=\frac{100}{n} \sum_{i=1}^{n_{\text {exp }.d a t a}}\left(\frac{X_{e_{\text {cali}}}-X_{e i}}{X_{e i}}\right)$ (4)
Xecali and Xei are respectively the calculated and the experimental value of equilibrium moisture content, nparam is the parameters number of the particular model.
2.4 Determination of the isosteric heat of sorption
The isosteric heat of sorption is the energy on fixing the water to the substrate, or else additional heat to the heat of vaporization of the pure water that would have to be supplied with the product to dehydrate it, can be determined by an equation derived from the Clausius-Clapeyron equation [18, 32, 33], from the moisture sorption data.
$Q_{s t}=-R\left[\frac{\partial\left(\ln A_{w}\right)}{\partial \frac{1}{T}}\right]$ (5)
Assumed that the isosteric heat of sorption independent of temperature, the integration of this equation gives equation (6):
$\ln \left(a_{w}\right)=-\left(\frac{q_{s t}}{R}\right)\left(\frac{1}{T_{k}}\right)+k$ (6)
The experimental desorption isotherms obtained at 30, 40 and 50 °C are presented in Figure 3. They have a sigmoidal appearance according to the classification of BET, and in a concordance with the behavior of other agri-food products [34]. It is noted that the equilibrium water content increases with the increase in the activity of the water they contain. A bending region observed on sorption isotherms in the range of water activity of 65-80.
Figure 3. desorption isotherms of heart of palm at 30 (a), 40 (b) and 50 °C (c)
According to the lutirature and according to [7, 35], the curves of the sorption isotherms of palm tree can be divided into three zones:
The first zone of the water activity in the range of (13-50) is a zone of a minimal amount of water. This quantity is due to the active sites by the hydrogen bonds in the molecules of the polar groups
The second zone (50-85) is the zone of chemical and biochemical reactions. Under the monolayer is a small amount of water and at a given moment when we have high moisture content the water fills the micropores and macropores.
The third zone (70 to 0.9) where the excess of water in the macrocapillaries causes the creation or the microbial birth.
Table 3 shows the results of the non-linear regression analysis of the palm heart desorption isotherms obtained at 30, 40 and 50 °C. The values of the constants of the models are all included in the table 4. A, B and C are the standard coefficients of each model. r and EST are respectively the correlation coefficient and the standard error.
Table 4. Results of the adjustment of the sorption isotherms
T(°C)
R²
Modified BET
langumier
8 .651
Figures 4 and 5 show the experimental adsorption of the palm hearts at 30 (a), 40 (b) and 50 °C (c) simulated respectively by GAB and BET models. The processing of the obtained data showed us acceptable results for the prediction of the moisture content of the palm heart. The results provided by the two GAB and Peleg models show a good match with the sorption isotherms compared with the other models (low standard error and high correlation coefficient).
Determination of Isosteric Heat of Sorption:
Using Clausius-Clapeyron Eq. 6, the isosteric isotherms of palm core sorption are plotted against ln (aw) versus (1/ Tk) for fixed equilibrium content values. The net isosteric heat of sorption can be calculated at each value of equilibrium moisture content from the slope of the isosteric curves which is equal to (- (qst / R)).
Figure 4. Experimental adsorption isotherms of heart of palm at 30 (a), 40 (b) and 50 °C (c) simulated by the GAB model
Figure 5. Experimental adsorption isotherms of heart of palm at 30 (a), 40 (b) and 50 °C (c) simulated by the BET model
Figure 6. Curves of desorption isosteres for the palm heart
The isosteric curves determined for the palm heart using the Clausius-Clapeyron equation are shown in Figure 6. We have the same remark that [19] concerning the increase in the moisture content of palm heart causes a decrease towards zero absolute values of slopes
The isosteric net sorption heats obtained for different equilibrium water contents were determined using the GAB model in combination with the Eq. 2. The variations of the palm hearts desorption with the water content are shown in Figure 7. They show that isosteric heat is more important at low water content, illustrating the strong bond of water to the substrate, and it becomes negligible in the presence of latent heat at high humidity. The following function used to describe the relationship between isosteric sorption heat and equilibrium water content:
$q_{s t}=\frac{1}{-2.355569+3.576252 X_{e q}-1.400303 X_{e q}^{2}}$ (7)
Figure 7. Isosteric desorption heat as a function of equilibrium water content for the palm heart
The experimental sorption curves have made it possible to determine the maximum temperature thresholds and the final water contents which serve to optimize the drying conditions of the different products, so as to ensure physicochemical and microbiological stability during storage. The curves of the palm heart sorption isotherms were determined experimentally by the static gravimetric method for three temperatures (30, 40 and 50 °C). The experimental results of the palm heart are shown by the sorption isotherms by the three temperatures (40, 50 and 60 °C). These sorption curves are adjusted by four statistical models. The analysis of the statistical parameters shows that the GAB and BET models are the best that describe the set of sorption isotherms and which better represents the relationship between the three parameters; equilibrium water content, water activity and temperature. The net heat of isosteric sorption of palm heart is calculated.
A,B,C,D model coefficients [-]
aw water activity (dimensionless) [-]
K constant [-]
mw mass of wet matter [kg]
md mass of dry matter [kg]
nexp.data number of experimental points [-]
nparam number of parameters of the particular model [-]
P percent average relative deviation [-]
qst net isosteric heat of desorption [J/mol]
R universal gas constant [kJ/mol.K]
r correlation coefficient [-]
S standard error [-]
T temperature [°C]
Tk the absolute temperature [K]
Xecali ith predicted moisture content [kg/kg % d.b]
Xei ith experiment moisture content [kg/kg % d.b]
$X_{e q}$ equilibrium moisture content [kg/kg]
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Assessing coverage of essential maternal and child health interventions using health-facility data in Uganda
Elizabeth M. Simmons1,
Kavita Singh ORCID: orcid.org/0000-0001-8670-22621,2,
Jamiru Mpiima3,
Manish Kumar2 &
William Weiss4,5
Population Health Metrics volume 18, Article number: 26 (2020) Cite this article
Nationally representative household surveys are the gold standard for tracking progress in coverage of life-saving maternal and child interventions, but often do not provide timely information on coverage at the local and health facility level. Electronic routine health information system (RHIS) data could help provide this information, but there are currently concerns about data quality. This analysis seeks to improve the usability of and confidence in electronic RHIS data by using adjustments to calculate more accurate numerators and denominators for essential interventions.
Data from three sources (Ugandan Demographic and Health (UDHS) survey, electronic RHIS, and census) were used to provide estimates of essential maternal (> 4 antenatal care visits (ANC), skilled delivery, and postnatal care visit (PNC)) and child health interventions (diphtheria, pertussis, tetanus, and hepatitis B and Haemophilus influenzae type b and polio vaccination series, measles vaccination, and vitamin A). Electronic RHIS data was checked for quality and both numerators and denominators were adjusted to improve accuracy. Estimates were compared between the three sources.
Estimates of maternal health interventions from adjusted electronic RHIS data were lower than those of the UDHS, while child intervention estimates were typically higher. Adjustment of electronic RHIS data generally improved accuracy compared with no adjustment. There was considerable agreement between estimates from adjusted, electronic RHIS data, and UDHS for skilled delivery and first dose of childhood vaccination series, but lesser agreement for ANC visits and second and third doses of childhood vaccinations.
Nationally representative household surveys will likely continue being the gold standard of coverage estimates of maternal and child health interventions, but this analysis shows that current approaches to adjusting health facility estimate works better for some indications than others. Further efforts to improve accuracy of estimates from RHIS sources are needed.
Nationally representative household surveys are typically considered the gold standard in low- and middle-income countries for tracking progress in the coverage of life-saving maternal, newborn, and child health interventions given concerns about the data quality of countries' routine health information systems (RHIS) [1]. Of note, however, is that these surveys are conducted on a 3 to 5-year cycle and provide estimates at the national and subnational levels, but often neither at the lowest operational nor at the health-facility catchment level [2]. As a result, population-based surveys are not used for real-time (or near real-time) monitoring of health care utilization, service delivery, or health system functioning, especially at the local level [3]. Electronic RHIS data, compiled from health facility (and in cases community based) data, are becoming increasingly important as government health systems decentralize and there is a need for officials at the district and facility levels to have more frequently available data to make financial and managerial decisions and be held accountable for these decisions [4].
The District Health Information Software Version 2 (DHIS2) is a free and open-source software platform used to develop electronic RHIS in over 60 countries in Africa and Asia [5]. The implementation of electronic RHIS using the DHIS2 has led to improvements in both completeness and timeliness of data reporting of health facility data [6, 7]. Despite the benefits of electronic RHIS, there are still problems of completeness and timeliness of reporting, consistency over time, and consistency between facility data and routine health survey data [8, 9]. In order to improve usability of and decision-making around electronic RHIS data, there is a need to improve the estimation of target populations with this data to provide more accurate denominators [10], as well as improve the estimation of intervention coverage to improve numerator estimates.
This analysis will add to the methodology set forth by Maina et al. [1] to improve the usability of health facility data by calculating more accurate numerators and denominators for essential maternal and child interventions in Uganda. We hypothesize that creating more confidence in the accuracy of electronic RHIS data will lead to more use of these data as well as better decision-making around the use of these data. In this analysis, coverage estimates of selected interventions were calculated using unadjusted and adjusted denominators from electronic RHIS data (using the DHIS2 software), and comparisons were made to both nationally representative survey estimates and estimates calculated using census-based population estimates as the denominator.
The DHIS2 was adopted in Uganda at the national level in January 2011 in order to further develop its national, electronic RHIS [11], leading to many improvements in data reporting of health facilities [7]. Uganda has a decentralized health system, with the public, private sectors and donors playing major roles. Its health system is organized in a hierarchical fashion with three levels of health centers, at the village (village health teams), parish (health center II), and subcounty levels (health center III) feeding into a health facility (health center IV) at the subdistrict level. These health facilities then feed into a district-level referral hospital. Each region has an overarching referral hospital as well, with the National Referral Hospitals in Kampala.
The necessary data and statistics for this analysis were obtained from three data sources. The Demographic and Health Surveys (DHS) program has collected and analyzed nationally representative data on population and health, including maternal and child health, in over 90 countries since 1984 [12]. The most recent Uganda Demographic and Health Survey (UDHS) in 2015-2016 provided estimates of coverage of maternal and child interventions at the national and subnational levels in Uganda. This study used UDHS estimates of coverage for each subregion as the gold standard. Data from the DHIS2 software were used to assess estimates of health facility coverage of these interventions at the district level in both 2015 and 2016. We will hereon refer to electronic RHIS data in Uganda as DHIS2 data, as the DHIS2 software was the source of the data. The DHIS2 estimates were aggregated by subregion and in order to provide a comparison with the UDHS data. Then, these data were adjusted to improve the quality of the numerators and denominators. In addition, Uganda has undertaken five population censuses since gaining independence, the most recent of these being the 2014 National Population and Housing Census [13]. These data were used to provide a second, population-based denominator for comparison. Census-based denominators were combined with numerators from the DHIS2 data to calculate census-adjusted estimates of coverage for child interventions for each subregion. We were unable to calculate census-adjusted coverage estimates from DHIS2 for maternal interventions as census data did not include information on the appropriate denominator: expected pregnancies.
For maternal interventions, UDHS-adjusted coverage estimates were calculated by subregion for the following indicators: (i) at least one antenatal care (ANC) visit; (ii) 4 or more ANC visits; (iii) a postnatal care (PNC) visit within 6 days of delivery; and (iv) skilled attendance at birth. Four or more ANC visits were chosen as the focused ANC model of ANC care, which included four visits, was the standard of care at the time of data collection for our analysis [14]. For child health interventions, UDHS- and census-adjusted estimates were calculated by subregion for the following indicators: (i) Bacille Calmette-Guerin (BCG) vaccination; (ii) the three-part diphtheria, pertussis, tetanus, and hepatitis B and Haemophilus influenzae type b (DPT-HepB-Hib) vaccination series; (iii) the three-part polio vaccination series; (iv) measles vaccination; and, (v) receipt of vitamin A. Vitamin A coverage estimates were restricted to children between 6 and 11 months for the UDHS and census data. Children under the age of 1 were used as the age group for DHIS2 for vaccinations, and for vitamin A as it were not possible to disaggregate vitamin A data down to the same level using the DHIS2 software.
Data quality checks
DHIS2 data were checked for quality by looking for district-level outliers in the estimates for selected interventions, and for completeness of facility reporting at the subregion level. The percentage of districts with at least one monthly outlier was calculated for each maternal and child intervention for 2015 and 2016 separately. For each district, the reported monthly number of women or children receiving an intervention was classified as an outlier if its value was more than two standard deviations away from the annual mean of that district. Table 1 shows the percentage of districts with at least one monthly outlier and the number of districts with greater than 1 monthly outlier in 2015 and 2016. For all included interventions, except vitamin A, the percentage of districts with at least one monthly outlier is between 35.9 and 48.4 percent. For vitamin A, over 90% of districts had at least one monthly outlier in 2016.
Table 1 Percentage of districts with at least one monthly outliera and number of districts with > 1 monthly outlier in DHIS2 estimates for the number of women/children receiving each intervention in 2015 and 2016
Reporting rates were calculated for each subregion by dividing the number of facilities submitting monthly reports for 2015-2016 by the total number of expected reports, grouping maternal and child interventions separately. Reporting rates at the subregion level varied from 51 to 95% for maternal interventions and from 54 to 95% for child interventions. Three subregions had reporting rates less than 80% for maternal interventions and two had reporting rates less than 80% for child interventions. Reporting rates are included in Supplementary Table 1, along with other factors used for adjusting DHIS2 numbers.
DHIS2 numerator adjustments
Numerators were first obtained from unadjusted DHIS2 data for the number of women and children reported to have received the individual interventions. The raw numbers for maternal interventions were adjusted for private sector use according to the UDHS as the DHIS2 does not capture all services provided in the private sector. Attending a PNC visit within 6 days of delivery was additionally adjusted for private sector and home use, as over 50% of PNC visits within 6 days of delivery were done either at home or in private facilities in most subregions [15]. Supplementary Table 1 shows adjustment data by subregion and data source. In addition, numerators for skilled delivery and receiving PNC within 6 days were adjusted for twins, based on the twinning rate of Uganda which is 15.4 per 1000 births [16]. The numerators for child interventions were not adjusted for immunizations occurring in the private sector as this information was not available in the UDHS.
DHIS2 denominator adjustments
DHIS2 denominator choices for maternal and child interventions
Denominators from the UDHS and census were left unadjusted. For maternal interventions, the number of women who attended at least one ANC visit was used as the denominator for the three included interventions. The indicator for having at least one ANC visit during pregnancy is high and consistent in Uganda with between 93.6 and 99.8% of pregnant women in each subregion attending their first ANC visit [15], thus providing the best estimate at the appropriate denominator for maternal interventions: expected pregnancies. For child interventions, the number of children receiving BCG is also high across Uganda as between 92.5 and 99.3% of all children receive it and was used as the denominator for the selected interventions [15]. For coverage estimates of the DPT-HepB-Hib and polio vaccination series, we also used the number of children who received the first vaccination in the series as the denominator of the second and third vaccinations, for comparison.
Adjustment for incomplete reporting
DHIS2-based denominators were first adjusted for incomplete reporting with the following equation from Maina et al [1]:
$$ {N}_{adjusted}={N}_{reported}\ast \left(\frac{1}{c}-1\right)\ast k $$
where c is reporting completeness and k is the adjustment factor that represents the expected level of service at the non-reporting facilities. The reporting completeness variables for maternal and child interventions, represented in the above equation by c, are included in Supplementary Table 1. If missing reports are an indication that no services were provided at these facilities during the reporting period then k = 0, but if it is possible that services were provided, but at a lower level than those facilities with complete reports then k is between 0 and 1. As we were uncertain what the appropriate k value was for the Ugandan context, we adjusted the denominators by five k values: 0, 0.25, 0.5, 0.75, and 1.
Adjustment for non-use of services
Next, DHIS2-based denominators were adjusted for non-use of the services. The proportion of women who did not attend at least one ANC visit and children who did not receive BCG, DPT-HebB-Hib1, and Polio 1 were calculated by subregion from the UDHS. The DHIS2-based denominator for each subregion was then inflated by these values (Supplementary Table 1).
Adjustment for stillbirths
Finally, the DHIS2-denominators for maternal interventions were further adjusted to account for stillbirths. Stillbirths cause a change in denominator between the first ANC visit and the 4th ANC visit, delivery, or PNC visit. The stillbirth rate in Uganda is 21 per 1000 [17]. About half of these stillbirths occur in the antepartum period and half during labor and delivery [18]. The denominator for at least 4 ANC visits was deflated by half the stillbirth rate (or 0.0105) and the denominator for skilled delivery and PNC visit was deflated by the full stillbirth rate (0.021) [17].
Calculation of coverage estimates from adjusted-DHIS2 data
Coverage estimates of (a) 4 or more ANC visits, (b) PNC visit within 6 days of delivery, and (c) skilled delivery were calculated by dividing the adjusted DHIS2 numerators by the adjusted-DHIS2 first ANC visit denominators for both 2015 and 2016. Coverage estimates of the (a) DPT-HepB-Hib vaccination series, (b) polio vaccination series, (c) measles vaccination, and (d) receipt of vitamin A were calculated by dividing the DHIS2 numerators (non-adjusted) by the adjusted-DHIS2 BCG denominator within each subregion in 2016. Coverage estimates for the second and third vaccinations of the DPT-HepB-Hib and polio vaccinations series were also calculated by dividing the DHIS2 numerators (non-adjusted) by the adjusted DPT-HepB-Hib1 and Polio1 denominators, respectively. DHIS2 data for PNC within 6 days of delivery and all child interventions were incomplete in 2015 and therefore excluded from the 2015 analysis.
The adjusted maternal intervention coverage estimates from the DHIS2 were then compared with the UDHS and the unadjusted-DHIS2 estimates. The adjusted child intervention coverage estimates from the DHIS2 were compared with the UDHS, the census, and the unadjusted-DHIS2 numbers. The percent difference between the UDHS and DHIS coverage estimates (unadjusted and adjusted) were compared by subregion. The percent of subregions for which there was a difference of less than 10% and less than 20% between the DHIS2 coverage estimates and UDHS was calculated.
Coverage estimates for four or more ANC visits and skilled attendance at birth from the UDHS and the unadjusted and adjusted DHIS2 are shown in Table 2. All adjusted-DHIS2 estimates for maternal interventions are with k = 0 as this value of k consistently produced the most accurate estimates (sensitivity analysis for different values of k is not shown). Unadjusted-DHIS2 estimates are significantly lower than those of the UDHS with adjusted-DHIS2 estimates closer to the UDHS in both 2015 and 2016. The same trend is seen in coverage estimates of skilled delivery, with two exceptions. First, in two subregions both the unadjusted and adjusted DHIS2 estimates in 2015 are significantly higher than those DHIS2 estimates of 2016 (Kampala and South Central). Second, in the Kigezi subregion, the unadjusted-DHIS2 estimates are more similar to the UDHS than those of the adjusted DHIS2.
Table 2 Coverage estimates of at least 4 ANC visits and skilled delivery from UDHS and unadjusted- and adjusted-DHIS2 numbers by subregion for 2015 and 2016
Coverage estimates of having a PNC visit within 6 days of delivery from the UDHS, and the unadjusted and adjusted DHIS2 (with numerators adjusted for both private sector use only and private and home sector use together) are shown in Table 3. Adjusted-DHIS2 estimates are also all produced with a k value of 0 as this k value produced the most comparable estimates to those of the UDHS (sensitivity analysis of different k values not shown). All estimates from the DHIS2 are significantly lower than those of the UDHS.
Table 3 Coverage estimates of PNC visit within 6 days of delivery from UDHS, unadjusted- and adjusted-DHIS2 numbers by subregion in 2015-2016
Figure 1 shows coverage estimates of the DPT-HepB-Hib vaccination series by subregion and the following sources: UDHS, census, unadjusted DHIS2, and adjusted DHIS2 with different k values. Unlike maternal interventions, all k values are shown for child interventions as the k value that produced the most accurate estimate differed by subregion. Census estimates for vaccination coverage are consistently above 100%. In general, adjusted-DHIS2 estimates are closer to that of UDHS than unadjusted estimates and higher than those of the UDHS. Similar results for the polio vaccination series are seen in Fig. 2.
Coverage estimates of the DPT-HepB-Hib vaccination series from UDHS, census, unadjusted- and adjusted-DHIS2 data
Coverage estimates of the polio vaccination series from UDHS, census, unadjusted- and adjusted-DHIS2 data
Figure 3 shows estimates of coverage from UDHS, census, and unadjusted- and adjusted-DHIS2 data of measles and vitamin A by subregion. Unlike the DPT-HepB-Hib and polio vaccination series coverage estimates, the adjusted DHIS2 estimates of measles and vitamin A are significantly higher than those of UDHS in all subregions, but Kampala.
Coverage estimates of the measles and vitamin A from UDHS, census, unadjusted- and adjusted-DHIS2 data
Table 4 shows the percent of subregions for which there was a difference of 10% or less and 20% or less between coverage estimates from the DHIS2 (adjusted and unadjusted) as compared to the UDHS. Denominators for the maternal interventions are adjusted with k = 0, and the numerators are adjusted for private sector use only. Child intervention denominators are adjusted with k = 1 and the vaccination series denominators are based on the first vaccination in the series. For having at least 4 ANC visits, there is moderate agreement between UDHS and adjusted-DHIS2 estimates with 7% of subregions having < 10% difference and 33% of subregions having ≤ 20% in 2015 and 2016. There is no agreement between the UDHS and adjusted-DHIS2 estimates for having a PNC visit within 6 days. Estimates of skilled delivery from UDHS and the adjusted-DHIS2 data compare well, with 40 and 53% of subregions having a ≤ 10% difference and 73 and 87% of subregions having ≤ 20% difference in 2015 and 2016, respectively. For child interventions, there is also frequent agreement between UDHS and adjusted-DHIS2 estimates for first DPT-HepB-Hib and polio vaccinations with 87% of subregions having ≤ 20% difference for both interventions. There is moderate agreement between UDHS and adjusted-DHIS2 estimates for the second in the vaccination series and almost no agreement in coverage estimates for the third dose. There is no agreement in the measles and vitamin A estimates. Across almost all interventions, adjusting the DHIS2 numbers improves agreement.
Table 4 Percent difference in maternal and child interventions from UDHS to unadjusted- and adjusted-DHIS2 estimates
In this study, we compared coverage estimates of essential maternal and child interventions from a nationally representative household survey to those derived from unadjusted- and adjusted-health facility service statistics. We found that there was considerable agreement for the following interventions: skilled attendance at birth; and the first doses of DPT-HepB-Hib and polio vaccinations (polio birth dose not analyzed). There was moderate agreement between adjusted-DHIS2 and UDHS coverage estimates for at least four ANC visits and the second vaccinations in each series. There was no agreement between the adjusted-DHIS2 and the UDHS coverage estimates for the third vaccination dose in each series, for measles vaccination, and for vitamin A. Coverage estimates for child interventions produced using the census also showed no agreement with estimates derived from DHIS2 (adjusted and non-adjusted), and the census-derived estimates were often much higher than 100% (which is usually not possible without unusual levels of visitors coming from other catchment areas for these services).
Other studies have used a similar methodology to improve the accuracy of coverage estimates derived from routinely collected health facility data for key maternal and child health indicators. Similar to our findings, Maina et al. found that similarly adjusted facility-based data in Kenya (also from Kenya's DHIS2) produced coverage estimates for health facility delivery that were similar to the estimates from the Kenya DHS. However, the coverage estimates for at least four ANC visits was lower for the adjusted DHIS2 as compared to the Kenya DHS [1]. Unlike our analysis, Maina et al. calculated adjusted-DHIS2 coverage estimates for the first and third doses in the DPT-HepB-Hib vaccination series that were similar to those of the Kenya DHS [1]. Another study found that health facility data were more accurate for estimating contact indicators, such as ANC, skilled delivery, and PNC, than for estimating health facility indicators that involve the provision of commodities (such as vaccines and vitamin A) [8].
A number of factors could explain the underreporting of maternal interventions in health facility data compared with UDHS data. First, women may seek care from health facilities outside of their subregion [10] or outside of the health facility completely, reducing the accuracy of coverage estimates of maternal and child interventions derived from health facility data alone. Women who travel outside their subregion to receive care could not only explain the lower coverage estimates of health facility data in most subregions but also explain the above 100% coverage reported in health facility data in Kampala. It is also possible that as patients are referred from one facility to another, their records may not follow them, leading to an underreporting of certain services, such as ANC visits. Second, the authors of the Maina et al. analysis raised the idea that the DHS may not be a gold standard for estimates of the number of ANC visits (with possible over-reporting based on respondent recall) and that the truth may be somewhere in between, or perhaps even closer to the adjusted-DHIS2 estimates [1]. Finally, some services might be provided in the private sector, which does not completely report to the DHIS2. The extent to which the private does report to the DHIS2 needs further examination.
There are also possible explanations for the overreporting of child interventions in health facility data compared to that of the UDHS. First, we may not be sufficiently adjusting the denominator for child interventions in order to capture the true population who would be receiving these interventions. Second, there could be pressure to report higher coverage of vaccinations through facility reports than actual vaccinations provided. A study in Uganda found that poor record keeping led to inaccurate immunization records in health facilities [19]. Finally, there could be recall bias in estimates produced by the UDHS making it possible that the true estimate is somewhere between that of the UDHS and health facility data [1]. It is also possible to explain the variation in estimates between subregions. Stockouts could have affected the ability of facilities to provide immunizations and it is unclear the extent to which women and children traveled in order to gain access to essential vaccinations. Stockouts could help explain the variation in vaccination coverage by subregion seen in DHIS2 estimates. For example, a study of Hoima District found stockouts to be a barrier in the provision of immunization services [20].
This study has several strengths. To our knowledge, it is the first study in Uganda to use this methodology to improve coverage estimates of essential maternal and child health interventions using health facility data. The DHIS2 values used to calculate the denominators for DHIS2 estimates (i.e., at least one ANC visit, receipt of BCG vaccination) had consistently high UDHS coverage levels (90% or higher) across Uganda helping to produce an alternative, population-based estimate of the size of key populations in each subregion.
This study also has several limitations. Frequent changes to administrative boundaries could complicate population projections and therefore the denominators of the census and DHIS2 estimates [10]. However, we made our best attempt to define the subregions in a consistent fashion across the three data sources to circumvent this issue by matching newer districts in the DHIS2 and census with subregions in the UDHS in order to avoid double-counting districts toward the denominators. We were also unable to adjust DHIS2 denominators for the migration effect, as these data were unavailable. However, we aggregated the DHIS2 data used in this analysis to the subregional level which should compensate for inter-facility catchment area movement within each subregion. Denominators used for coverage estimates of child health interventions from the DHIS2 were not adjusted for private sector use as this information was not available in the UDHS. The ideal denominator for PNC visits should be expected deliveries, but we were unable to estimate expected deliveries by adjusting expected pregnancies by the rate of miscarriages or abortions.
Nationally representative household surveys will likely continue being the gold standard for population-based coverage estimates of maternal and child health interventions. However, there is increasing demand for more frequent estimates and for estimates that represent smaller areas than national household surveys like the DHS provide. Facility data currently provide more frequent estimates of these interventions at both a national and local levels but the quality of these estimates is suspect, hindering confidence to use these estimates for making appropriate decisions. This and other analyses show that current approaches to adjusting facility-based coverage estimates using population-based sources work better for some indicators than others, and that accuracy of these adjustments vary by country and data source (e.g., survey vs census). Further efforts to improve the accuracy of coverage estimates based on routine health facility data are needed, as well as a better understanding of the conditions when these improvement methods are sufficient and for how long these estimates would be valid.
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. These data are also publicly accessible.
ANC:
BCG:
Bacille Calmette-Guerin
DHIS2:
District Health Information Software Version 2
DHS:
DPT:
Diphtheria, pertussis, tetanus
HepB:
Hib:
PNC:
RHIS:
Routine Health Information System
UDHS:
Uganda Demographic and Health Survey
Maina I, Wanjal P, Soti D, Kipruto H, Droti B, Boerma T. Using health-facility data to assess subnational coverage of maternal and child health indicators, Kenya. Bull World Health Organ. 2017;95:683–94.
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Abouzahr C, Boerma T. Policy and practice health information systems: the foundations of public health; 2005.
Hotchkiss DR, Diana ML, Foreit KGF. How can routine health information systems improve health systems functioning in low and middle-income countries? Assessing the evidence base. Adv Health Care Manag. 2012;12:25–58.
University of Oslo. DHIS2 factsheet: June 2018. Oslo, Norway; 2018.
Mphatswe W, Mate KS, Bennett B, Ngidi H, Reddy J, Barker PM, et al. Improving public health information: a data quality intervention in KwaZulu-Natal, South Africa. Bull World Heal Organ. 2012;90:176–82.
Kiberu VM, Matovu JK, Makumbi F, Kyozira C, Mukooyo E, Wanyenze RK. Strengthening district-based health reporting through the district health management information software system: the Ugandan experience. BMC Med Inform Decis Mak. 2014;14:1–9.
Bhattacharya AA, Umar N, Audu A, Allen E, Schellenberg JRM, Marchant T. Quality of routine facility data for monitoring priority maternal and newborn indicators in DHIS2: a case study from Gombe state, Nigeria. PLoS One. 2019;14:1–21.
Garrib A, Stoops N, Dlamini L, Govender T, Rohde J, Herbst K. An evaluation of the district health information system in rural South Africa. South African Med J. 2008;98:522–49.
Maïga A, Jiwani SS, Mutua MK, Porth TA, Taylor CM, Asiki G, et al. Generating statistics from health facility data: the state of routine health information systems in eastern and southern Africa. BMJ Glob Heal. 2019;4:1–9.
Ministry of Health Uganda. Uganda eHMIS. https://hmis2.health.go.ug/hmis2/dhis-web-commons/security/login.action. Accessed 20 Feb 2020.
ICF. The DHS Program - Team and Partners. https://dhsprogram.com/Who-We-Are/About-Us.cfm. Accessed 11 Feb 2020.
Uganda Bureau of Statistics. National population and housing census 2014: main report. Kampala, Uganda; 2016.
UNDP, UNFPA, WHO, World Bank. WHO Antenatal Care Randomized Trial: manual for the implementation of the new model. Geneva, Switzerland; 2002. https://apps.who.int/iris/bitstream/handle/10665/42513/WHO_RHR_01.30.pdf. Accessed 24 Aug 2020.
Uganda Bureau of Statistics, ICF. Uganda Demographic and Health Survey 2016. Kampala, Uganda and Rockville, Maryland, USA; 2016. www.DHSprogram.com.
Smits J, Monden C. Twinning across the developing world. PLoS One. 2011;6:8–10.
UNICEF. Maternal and newborn health disparities: Uganda. New York: USA; 2020.
Lawn JE, Blencowe H, Waiswa P, Amouzou A, Mathers C, Hogan D, et al. Stillbirths: rates, risk factors, and acceleration towards 2030. Lancet. 2016;387:587–603.
Nsubuga F, Luzze H, Ampeire I, Kasasa S, Toliva OB, Riolexus AA. Factors that affect immunization data quality in Kabarole District, Uganda. PLoS One. 2018;13:e0203747. https://doi.org/10.1371/journal.pone.0203747.
Malande OO, Munube D, Afaayo RN, Annet K, Bodo B, Bakainaga A, et al. Barriers to effective uptake and provision of immunization in a rural district in Uganda. PLoS One. 2019;14:e0212270. https://doi.org/10.1371/journal.pone.0212270.
This work was funded by the United States Agency for International Development (USAID) under the terms of the Data for Impact (D4I) associate award 7200AA18LA00008. The views expressed in this website do not necessarily reflect the views of USAID or the United States Government. We are also grateful for the general support from the Carolina Population Center (P2C Center grant from NIH: P2C HD050924).
Department of Maternal and Child Health, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA
Elizabeth M. Simmons & Kavita Singh
D4I Project, Carolina Population Center, Chapel Hill, NC, USA
Kavita Singh & Manish Kumar
Makerere University, Kampala, Uganda
Jamiru Mpiima
Department of International Health, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD, USA
William Weiss
Public Health Institute, Oakland, CA, USA
Elizabeth M. Simmons
Kavita Singh
KS and WW conceptualized the study and developed the analysis plan. ES conducted the analysis and was the primary writer. JM advised on the data and provided key inputs into the interpretation of the data and findings. All authors read and approved the final manuscript.
Correspondence to Kavita Singh.
Not applicable (this study does not use any individual data and only uses secondary data which has been aggregated at the sub-regional level)
Additional file 1.
: Supplementary Table 1. Adjustment factors for DHIS2 numerators and denominators.
Simmons, E.M., Singh, K., Mpiima, J. et al. Assessing coverage of essential maternal and child health interventions using health-facility data in Uganda. Popul Health Metrics 18, 26 (2020). https://doi.org/10.1186/s12963-020-00236-x
DOI: https://doi.org/10.1186/s12963-020-00236-x
Data adjustments | CommonCrawl |
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STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. 99 to 0 on the left side and 0 to 3. If you want to calculate the p-value, you have to find the integral with lower limit equal to the z-score and upper limit equal to infinity. The z-score provides a simple measure by which different measures can be compared in terms of their deviation from the mean. This is very easy: just stick your Z score in the box marked Z score, select your significance level and whether you're testing a one or two-tailed hypothesis (if you're not sure, go with the defaults), then press the button!. If you want to calculate the z score based on the raw score, mean, and standard deviation, see Z Score Calculator. given something called a z-score. In statistics, the z-score (or standard score) of an observation is the number of standard deviations that it is above or below the population mean. The standard normal distribution has been well-studied, and there are tables that provide areas underneath the curve, which we can then use for applications. What is a Z-score? A z-score shows you the distance between an observed score and the mean in units of standard deviations. Z-score a composite score, summarizing the weighted results of a number of ACCOUNTING RATIOS, that claims to predict a company's likelihood of failure. Appendix Tables A3 T ABLE 3 Binomial Pr obability Distribution C n, r p r q n r This table shows the probability of r successes in n independent trials, each with probability of success p. 45, move along the area in the table and locate the nearest value. You just need to enter a raw score, population mean and standard deviation, and hit "Calculate". Tables • T-11 Table entry for p and C is the critical value t∗ with probability p lying to its right and probability C lying between −t∗ and t∗. The Z score, the number resulting from the calculation, indicates where you fall in the range of body fat measurements. This table covers children between 0 and 60 months of age, but you can modify the table as needed for your project. The Z-Score Calculator uses the following formula: z = (x - μ) / σ. If we normalize the data into a simpler form with the help of z score normalization, then it's very easy to understand by our brains. If it is on the right-hand side, we will have a positive value of z else negative. qThe Z -score is expresses the number of standard deviations the value x is from the mean qA negative Z -score implies that x is to the left of the mean and a positive Z -score implies that x is to the right of the mean Z Score Equation z = x - x s For a score of 83 from the aptitude data set, z = = 1. Return to Statistics Topics. Z-score calculator, p-value from z-table, left tail, right tail, two tail, formulas, work with steps, step by step calculation, real world and practice problems to learn how to find standard score for any raw value of X in the normal distribution. So, here we go. Right Tailed Test. Looking at the table we know P(z <= 3. consider whether z is positive or negative when using the table. Area from a value (Use to compute p from Z) Value from an area (Use to compute Z for confidence intervals). Now, we use the z table to find the percent (or probability) that a person would have an IQ of 120 or less, which is the same as the percent (or probability) that a z score is 1. z scores are assumed to have come from a normal population with mean zero and standard deviation one. Please enter the necessary parameter values, and then click 'Calculate'. 475) is at z=1. 751: 37-0. For George's example we need to use the 2nd table as his test result corresponds to a positive z-score of 0. Z-Scores Can Help Us Understand… How typical a particular score is within bunch of scores. If I have a z-score, I can look up the probability in a distribution table. Altman, who was, at the time, an Assistant Professor of Finance at New York University. Now, therefore, the upper z-score will be z = 1. It is an excel spreadsheet requiring the use of macros but will automatically pull data from the internet and let you calculate the Altman Z Score with this free spreadsheet. Z Score Table- chart value corresponds to area below z score. The z-score corresponding to a left-tail area of 0. 27 for your math score. –z 0 z The area remaining in the tails is. Click for Normal Data Table; National health and nutrition survey (NHANES), CDC/National Center for Health Statistics. This formula allows you to calculate a z-score for any data point in your sample. Most z-tables show the area under the normal curve to the left of z. This page allows you to calculate the probability of chance occurrence of a given z, or to calculate z given a probability Q. To use Table A: ! Find the corresponding z-score. 08 and the standard deviation was 3. STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. Let Z be a standard normal random variable, Use the calculator provided to determine the value of c such that P ( c < Z < c-1. The procedure is not very good for estimating large z-values. Percentiles are provided for Z Scores between -3. 1) calculate the z-score, then find the absolute value of the z-score in column a 2) if the z-score is positive, automatically add 0. If you want to calculate the z score based on the raw score, mean, and standard deviation, see Z Score Calculator. P Value from Z Score Calculator. Determine z-scores for the mitral valve, aortic valve, tricuspid valve, and pulmonic valve using this calculator. You can easily calculate a z-score on a TI-83 calculator or in Excel. There is in depth information on the calculator and the results provided below the form. , z score) that corresponds with Y = 120. Padua Score <4: Low risk of VTE. Since the z-score is a positive number, we look at the Positive Z-Score table for the percent of students who scored within 1. • compute a z-score for a value given that value, its mean, and it's standard deviation. Minitab would calculate a z-score of -1. To convert from an IQ score to a z score, we use the formula: The z score for your IQ of 120 is 1. I have numpy, statsmodel, pandas, and scipy(I think) How do I calculate the z score of a p-value and vice versa? For example if I have a p value of 0. The z- is a N(0, 1) distribution, given by the equation: The area within an interval (a,b) = normalcdf(a,b) = (It is not integrable algebraically. Click on the table below to go to a page from which the table may be printed. If we look this z-score (1. Standard normal distribution table is used to find the area under the f(z) function in order to find the probability of a specified range of distribution. It shows you the percent of population: between 0 and Z (option "0 to Z") less than Z (option "Up to Z") greater than Z (option "Z onwards") It only display values to 0. 5?! With!proportion,!we!can. 96, by the symmetry property of the standard normal distribution. 30 and the column labeled 0. The Altman Z Score. Carry your. To use the table, which is easier than it might look at first sight, we start with our z-score, 0. A Z-Score is a statistical value that tells you how many standard deviations a particular value happens to be from the mean of the entire data set. In A/B Testing terms, all of your visitors are observations, and the Control experience makes up a bell curve. How can I calculate the z score on my casio fx-115ES plus calculator? I need to know for statistics. Features & Benefits Get the exact area: No need to try and interpret hard-to-read z-score tables. 5 down to 1. If he was below the mean, it would be a negative. STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. The given value is significance level. You can learn more about accounting from the following articles – Mortgage Calculator in Excel; Mortgage APR vs Interest Rate – Compare. If data are normally distributed, approximately 95% of the data should have Z-score between -2 and +2. Z is the same size as X. Remember that a z-score is a standard score (also called the standard Gaussian variable) that is calculated by subtracting the mean from a value and dividing the result by the standard deviation: z = (value - mean)/standard deviation. Notice the inequality points to the right. 45, we go to the table and look at where. Normal Distribution Calculations Using Technology All of the calculations that we will do in Math 180A involving the normal distribution can be done using tables. 71 : 2nd -2. We have to find its corresponding confidence level. USING MINITAB INSTEAD OF TABLES As you can see from the table, the answer is 0. This bone mineral density for bone fracture calculator assesses fracture risk in the following five years based on patient age, BMD T score and personal factors. Z tables indicate what percentage of the statistics is under the curve at any given point. How can I calculate the z score on my casio fx-115ES plus calculator? I need to know for statistics. Z score table and calculation z score table t z score table and calculation z score table and calculation Whats people lookup in this blog: Negative Z Score Table Calculator. Enter required numbers for the population mean and the standard deviation and hit Calculate. The idea here is that the values in the table represent area to the left, so if we're asked to find the value with an area of 0. A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. 0 versus ZA=4. \[ p = \Pr(Z z^*) \] How do we find such \(z^*\)? The idea is to a percentile to z score conversion table, which is essentially using a standard normal distribution table. A score that is 30 points below the mean. This data is also used to create dynamic z-score tables. 1) Use the normal distribution table (Table A-2 pp. Z = zscore(X) returns the z-score for each element of X such that columns of X are centered to have mean 0 and scaled to have standard deviation 1. Notice the inequality points to the right. It does not change the shape of the distribution! Raw score does not change into a bell shaped curve when changed into standard scores. PSYCHOMETRIC CONVERSION TABLE Standard Score Percentile Rank Scaled Score ETS Score T-Score Z-Score Description 89 23 Low Average 88 21 425 42 -0. 18 Using the z table in Appendix B, calculate the following percentages for a z score of 1. The Boston Children's Hospital Z-Score Calculator allows for the calcuation of the standard score (z-score) of various regressions based on data gathered over the past 12 years on normal children. What's Next? Calculating Z Score manually and finding the probability in Standard Normal Table is only for you to learn the concept, you can get the value quickly using SPSS and Excel. I do not have the Z values. Corresponding values which are less than the mean are marked with a negative score in the z-table and respresent the area under the bell curve to theContinue Reading. After calculating the standardized score, we need to look up the area (same as probability) using the z-table. The z score is the numerical value which represents how many standard deviations a score is above the mean. of the table shows the proportion of values to the left of each !-value. But why? Let's look at the equations: A z score is the same as a standard score; the number of standard deviations above the mean. Features & Benefits Get the exact area: No need to try and interpret hard-to-read z-score tables. Create a blank Excel worksheet, copy the following table, select cell A1 in your blank Excel worksheet, and then paste the entries so that the table fills cells A1:C24 in your worksheet. The Centers for Disease Control and Prevention (CDC) cannot attest to the accuracy of a non-federal website. z-tables for the standard normal distribution. Get a Z score for normally distributed data using this online calculator. To find the z value for 0. An introductory statistics text for the social sciences. Z score table and calculation z score table t z score table and calculation z score table and calculation Whats people lookup in this blog: Negative Z Score Table Calculator. Z-Score to Percentile Calculator Enter a z-critical value and get the area under the normal curve (a percentage). First, use the Z-table to find the value where the row. Since the z-score is negative and we want the area below that point we will be using the tail area. For negative values just add a "-" sign. Z Score Calculation is a important aspect of six sigma projects. 5 indicates osteoporosis. This concept was adapted to the business and finance world by Dr. 354 std devs above the mean. 5 standard deviations above the mean. The table includes positive z statistics (from 0. Refer the column & row values for z-score. 11 would be in the 14th percentile. Just enter your p-value, which must be between 0 and 1, and then hit the button below. Calculate test statistic 0. The raw score computed is the actual score, or value, obtained. Calculating Z-Score; Calculating Z-Score in SQL. For use in Georgetown University statistics classes: Math-006 and Math-040. 0 versus ZA=4. Use this Z to P calculator to easily convert Z-scores to P-values (one or two-tailed) and see if a result is statistically significant. 49 would be the intersection of 0. 29 (add the 0. First move to the left extreme find the value in the z column. 36, look in the intersecting cell for the row labeled 2. The z-score provides a simple measure by which different measures can be compared in terms of their deviation from the mean. 2 is on page App17. Using the Weight/Height table to calculate the Weight-for-Height/Length percentage or Standard Deviation (Z-score). The Altman Z score was formulated by a professor in 1968 to predict. Calculate Z (two tailed) from Confidence Level To determine the Z value, enter the chosen Confidence Level in the box below and press the Return key or the Calculate button. Extensions and Connections (for all students) Give students the mean and standard deviation of a data set in addition to a z-score, and ask them. Enter a value, population mean and the standard deviation (for the population). You can look up the t-value for a desired level of confidence in a t-distribution table, given the degrees of freedom (n – 1) associated with your sample. You can use our z-score calculator to determine this value for you. Using a Graphing calculator to use a Z-table Finding % given bounds (for a non-standard normal) normalcdf( can be used to give you the % between a lower and upper bound for a non-standard normal (i. ¨ If your Z score is negative, subtract the value in Table A. Coronary Artery Z-Scores. A Z score can be determined by multiplying the Ppk score by 3. Find a critical value in this T value table >>>Click to use a T. Bottom 15% low $$ customers (z-score of the dollar_amt variable less than -1). So, the z score here, z score here is a positive 2. The following questions and answers are from Roger Wimmer's The Research Doctor Archive (with some editing). From the table, ZW=1. If X is a vector, then Z is a vector of z -scores. A Z-score is a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean. I'm using a SQL query to determine the z-score (x - μ / σ) of several columns. First move to the left extreme find the value in the z column. This reference group usually consists of people of the same age and gender; sometimes race and weight are also included. 275 is the cumulative area for the negative zvalue and. Enter mean, standard deviation and cutoff points and this calculator will find the area under normal distribution curve. This is because the z score will be in the nonrejection area. Percentile Rank is the easiest score to help you understand how far the attribute measured is above or below the mean of the normative sample. Altman, who was, at the time, an Assistant Professor of Finance at New York University. Helen, one of the volunteers, scored 74 (X) from maximum possible 120 points. P Value from Z Score Calculator. The calculator will generate a step by step explanation along with the graphic representation of the area you want to find. given something called a z-score. See Z-score for more information. 1BestCsharp blog 7,736,621 views. zscores are given along the 1st column and 1st row. Find a critical value in this T value table >>>Click to use a T. The z-score and actual height measurements are both given underscoring the relationship between the two. There are four ways to obtain the values needed for Z α/2:. Detailed information about what a p-value is, how to interpret it, and the difference between one-sided and two-sided tests of sign. What percentage of scores falls below this z score? b. Cardiac Valve Z-Scores Calculate BSA-adjusted z-scores of the mitral valve, tricuspid valve, aortic valve, and pulmonary valve using data published by Cincinnati Children's Hospital. z scores are assumed to have come from a normal population with mean zero and standard deviation one. 8621, As you can see, both values are identical, no matter if you're using one z score table or another. 8 means it's likely the company is headed for bankruptcy, while companies with scores above 3 are not likely to go bankrupt. The z-score and actual height measurements are both given underscoring the relationship between the two. The most commonly used z-scores are 1. A z-score is a standardized score that describes how many standard deviations an element is from the mean. This online calculator calculates z score from p value. STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. 09 at the end of the -1. Anybody's. Z - score calculator This calculator can be used to find area under standard normal curve $ ( \mu=0 , \sigma=1 )$. This table identifies the area of the body of the z-distribution. If it is on the right-hand side, we will have a positive value of z else negative. As n approaches 30, the t-score associated with a given confidence level approaches the Z score for that confidence level. To find the value associated with a given z-score, you find the first decimal of your z-score on the left or right side and then the 2nd decimal of your z-score across the top or bottom of the table. Bottom 15% low $$ customers (z-score of the dollar_amt variable less than -1). For example, to determine the area under the curve between 0 and 2. The formula may be used to predict the probability that a firm will go into bankruptcy within two years. The easiest way to find the p-value, though, is to use our p-value calculator! Simply type in the value of the z-score and you will have the p-value calculated in no time!. For negative values just add a "-" sign. The normal random variable X from any normal distribution can be transformed into a z score from a standard normal distribution via the following equation: z = (X - μ) / σ. Since the z-score is a positive number, we look at the Positive Z-Score table for the percent of students who scored within 1. These instructions should allow you to do basic statistical procedures at the level of Math-006 on the TI-83. Above this z score b. 64 : 51st : 0. In statistical inference, we are interested to know whether a small sample comes from a population. 0 versus ZA=4. Let x be any number on our bell curve with mean, denoted by mu, and standard deviation denoted by sigma. In the 5th edition, Table B. Background: In statistics, the z score (also called a z-value, standard score, or normal score) is the signed number of standard deviations by which an observation or data is above the mean. Find the value in a look up table of the probability of a z-score of less than. • A T-score below -2. The formula to calculate z-score is. 4505 in our table [Fig-3]. The calculator will generate a step by step explanation along with the graphic representation of the area you want to find and standard normal tables you need to use. This is one version of a z table. 5 (or 50%) to calculate the area that is to the left of the z score. Next lesson. For any given value of z, this page will calculate the respective one-tailed probabilities of —z and +z; the two-tailed probability of±z; and the proportion of the normal distribution falling between —z and +z. The Z-score or standard deviation classification system. Looking at the table we know P(z <= 3. Detailed information about what a p-value is, how to interpret it, and the difference between one-sided and two-sided tests of significance. So, in this case, it would be 0. This reference group usually consists of people of the same age and gender; sometimes race and weight are also included. However, the calculator below can calculate z score for arbitrary p value. Also Z is one of the common yardsticks for measuring process capability is a Z-score. Use to answer all questions which specify a probability and ask for the relevant range of raw or z-scores. Z score table and calculation z score table t z score table and calculation z score table and calculation Whats people lookup in this blog: Negative Z Score Table Calculator. The z-value calculation is based on algorithm 209 from the CACM by D. As z-value increases, the normal table value also increases. Typically in an introductory stats class, you'd use the z-score and look it up in a table and find the probability that way. There are currently several formulas to compute a z-score. Find a critical value in this T value table >>>Click to use a T. Look at the graph. 32% of students scored lesser than Jack. The Z-score scale is linear and therefore a fixed interval of Z-scores has a fixed height difference in cm, or weight difference in kg, for all children of the same age. 1 Answer to Using the z table in Appendix B, calculate the following percentages for a z score of 0. What is % that is less than 50? I get that I can use a calculator like , or figure out the z-score and then use the z-score table, but is there just a straight calculation where I can plug in the above data and get the right answer (2. Corresponding values which are less than the mean are marked with a negative score in the z-table and respresent the area under the bell curve to theContinue Reading. Z-score calculator, p-value from z-table, left tail, right tail, two tail, formulas, work with steps, step by step calculation, real world and practice problems to learn how to find standard score for any raw value of X in the normal distribution. Above this z score b. A Z score can be determined by multiplying the Ppk score by 3. Coronary Artery Z-Scores. Since the z-score is a positive number, we look at the Positive Z-Score table for the percent of students who scored within 1. The z-score, also known as standard score, is a measurement used in statistics. The area to the left of the negative z-score will be the same as the area to the right of the positive z-score. Use this Z to P calculator to easily convert Z-scores to P-values (one or two-tailed) and see if a result is statistically significant. , z score) that corresponds with Y = 120. Practically speaking, Ppk can be determined by dividing the Z score by three. Z Score Calculator Z Score to Percentile Calculator Left Tailed Test. It is a Normal Distribution with mean 0 and standard deviation 1. The Z-score Calculation. given something called a z-score. Student t-values for both one-tailed (right-tail) and two-tailed probabilities will be returned. Use the following format to find a z-score: z = X - μ / σ. ) Finding the Proportion of Data in a Normal Distribution using the Table ex Find the proportion of data that has z-score less than 0. If NORMSDIST(z) returns p, then NORMSINV(p) returns z. ¨ Using Table A. The whole number and the first digit after the decimal point of the z score is displayed in the row and the second digit in the column of the normal distribution table. The z-score and actual height measurements are both given underscoring the relationship between the two. Finding a corresponding probability is fairly easy. Here's how to use the table. I have the z-table imported into excel, How i can look-up values from the table to 2 d. 08 and the standard deviation was 3. This wikiHow teaches you how to calculate a Z score in Microsoft Excel. Where they intersect you will find the decimal expression of the percentage of values that are less than your sample (see example 4). With the z-test calculator, users can easily calculate z-score from the p-value or the vice versa. Carry your. Standard normal distribution table is used to find the area under the f(z) function in order to find the probability of a specified range of distribution. Say we need to look up the probability of. 725 is the cumulative area for the positive z. In this case it is 1. The calculator will generate a step by step explanation along with the graphic representation of the area you want to find and standard normal tables you need to use. 64 : 51st : 0. The z-score of 0. Negative Z score table Use the negative Z score table below to find values on the left of the mean as can be seen in the graph alongside. Selecting two-sided provides the area above Z and below -Z. This table identifies the area of the body of the z-distribution. 67 (if our z-score had more than two decimal places, for example, ours was 0. 75 Low Average 87 19 Low Average 86 18 Low Average 85 16 7 400 40 -1. However, it is recommended that you also learn how to make these calculations using either a graphing calculator or Microsoft Excel. Combinations with z Scores. To calculate a Z score, you need to know the mean (μ) and the standard deviation (σ) of your dataset. Using the R score calculator to calculate your R score: Enter the information required in the fields bellow and press the Enter key or on the button "Add class" to calculate the R score for this class. In the following statistical power of negative z score table to explain the how to calculate the probability value For example, P(0 > Z > 2. P Value Calculator Use this calculator to compute a P value from a Z, t, F, r, or chi-square value that you obtain from a program or publication. z scores are assumed to have come from a normal population with mean zero and standard deviation one. 4 indicates that someone is 1. | CommonCrawl |
DynamicME: dynamic simulation and refinement of integrated models of metabolism and protein expression
Methodology article
Laurence Yang ORCID: orcid.org/0000-0001-6663-76431,
Ali Ebrahim1,
Colton J. Lloyd1,
Michael A. Saunders2 &
Bernhard O. Palsson1,3
BMC Systems Biology volume 13, Article number: 2 (2019) Cite this article
Genome-scale models of metabolism and macromolecular expression (ME models) enable systems-level computation of proteome allocation coupled to metabolic phenotype.
We develop DynamicME, an algorithm enabling time-course simulation of cell metabolism and protein expression. DynamicME correctly predicted the substrate utilization hierarchy on a mixed carbon substrate medium. We also found good agreement between predicted and measured time-course expression profiles. ME models involve considerably more parameters than metabolic models (M models). We thus generate an ensemble of models (each model having its rate constants perturbed), and then analyze the models by identifying archetypal time-course metabolite concentration profiles. Furthermore, we use a metaheuristic optimization method to calibrate ME model parameters using time-course measurements such as from a (fed-) batch culture. Finally, we show that constraints on protein concentration dynamics ("inertia") alter the metabolic response to environmental fluctuations, including increased substrate-level phosphorylation and lowered oxidative phosphorylation.
Overall, DynamicME provides a novel method for understanding proteome allocation and metabolism under complex and transient environments, and to utilize time-course cell culture data for model-based interpretation or model refinement.
Almost 70 years ago, Monod posited that the rate-limiting steps for exponential growth is expected to be distributed over hundreds or thousands of reactions that form an enzymatic reaction network. In the same study, he observed that Escherichia coli cultured in media consisting of two limiting carbon sources underwent two exponential growth phases separated by a short lag phase [1]—the phenomenon he coined diauxie. Today's genome-scale models of E. coli now account for over 2,000 metabolic reactions, and over 4,000 steps involved in the macromolecular expression machinery [2–4]. Consequently, recent studies have been approaching the classic problem of understanding the mechanisms and constraints that govern cellular dynamics armed with a comprehensive view of the genome-scale enzymatic network.
Genome-scale modeling of cell metabolism
Computing the genotype-phenotype relationship is a fundamental challenge for computational biologists. Constraint-based reconstruction and analysis (COBRA) provides one approach for systems-level computation of biological networks using genome-scale biochemical network reconstructions [5]. Flux Balance Analysis (FBA) [6] in particular simulates flux distributions through a metabolic network by optimizing a cellular objective, such as maximizing growth rate subject to physicochemical, regulatory and environmental constraints. COBRA has been used to address a large variety of biological problems [7], and many algorithmic extensions have been developed [8].
Accounting for macromolecular constraints
In an important extension of FBA (FBAwMC), the hierarchy of substrate utilization in mixed carbon media was predicted correctly by imposing intracellular macromolecule crowding constraints [9]. The constraints imposed were based on approximate crowding coefficients for cytosolic enzymes based on estimated molar volume and catalytic efficiency.
Recently, genome-scale reconstructions have expanded significantly with development of integrated models of metabolism and macromolecular expression (ME models) [3, 4, 10–13]. ME models explicitly compute transcription and translation machinery requirements to support metabolic flux distributions. The latest E. coli ME models [11, 12] account for 80% of the proteome by mass and predict the allocation and limitation of proteome toward cellular functions during optimal growth [14]. Therefore, ME models considerably expand the scope of systems-level investigation and computation across multiple biological scales and processes.
Dynamic simulation of cell metabolism and macromolecular composition
Constraint-based models of metabolism have been used in a dynamic simulation framework to investigate by-product secretion [15], diauxic growth [16], transcriptional regulation [17], and metabolic engineering strategies [18, 19]. Recent studies have also incorporated the dynamics of protein expression. For example, temporal resource allocation was studied using a model of the cyanobacterium Synechocystis sp. PCC 6803 [20]. The model consisted of 52 reactions and 50 compounds, and also included coarse-grained reactions for synthesis of macromolecules including ribosome and multiple enzymes. Meanwhile, Waldherr et al. [21] performed a detailed mathematical study on the problem of predicting the dynamics of protein expression and metabolism. They developed Dynamic enzyme-cost FBA (deFBA), which accounts for the dynamics of cell metabolism, biomass production, and biomass composition. The framework accounts also for enzymatic capacity and the cost of their production. The approach could predict dynamic adaptation of enzyme expression from an optimization principle. The method was demonstrated on a core carbon metabolism model of E. coli.
Objectives and outline of this study
Here, we develop a method to simulate the cellular dynamics of metabolism, protein expression, and macromolecular composition in response to environmental changes. We demonstrate a mathematically simple approach with a focus on applying it to a large, comprehensive network. The largest network we simulate consists of 7,027 molecular components (small molecules and macromolecules) and 12,677 reactions involved in metabolic and protein expression processes [2].
The rest of this study is organized as follows. In Methods, we first briefly overview the relevant concepts for computing cell phenotype using a ME model. We then describe our main contribution, the DynamicME procedure. We first derive a simple approach for dynamic simulation using ME models, and extend this procedure to account for protein "inertia" constraints. We then describe the methods used for model parameter sensitivity analysis and model validation used in the rest of the paper. In Results, we apply DynamicME to the case study of batch growth of E. coli on a mixed carbon substrate medium. We then address the challenge of interpreting model simulations when many uncertain parameters are present by generating an ensemble of models with perturbed parameters. These models are analyzed with archetypal analysis to identify prominent time-course metabolite profiles. The overall workflow for this study is shown in Fig. 1.
Schematic of the overall workflow for this study
Growth maximization for ME models
A ME model describes a cell's metabolic and macromolecular state as a vector of n fluxes, \(v\in \mathbb {R}^{n}\) (in mmol/grams dry weight/h) that catalyze biochemical reactions among m components (i.e., small molecules and macromolecules) [2]. To compute the state that maximizes the growth rate μ (in h −1), one solves the following optimization problem (1) [2, 22]:
$$ \begin{aligned} \max_{\mu, v} \quad & \mu \\ \mathrm{subject\ to} \quad & S(\mu) v = 0 \\ & l(\mu) \leq v \leq u(\mu), \end{aligned} $$
where \(S(\mu)\in \mathbb {R}^{m\times n}\) is the stoichiometric matrix, and \(l(\mu)\in \mathbb {R}^{n}\), \(u(\mu) \in \mathbb {R}^{n}\) are the lower and upper flux bounds. These three parameters are functions of μ, for example, due to the hyperbolic relation between growth rate and translation rate, macromolecule dilution, etc. (see [2] for a complete description of these relations). Problem (1) includes constraints in the form of S(μ)v=0, where for any fixed μ, we obtain a linear program. Because our objective function here is to maximize mu, subject to the μ-dependent constraints (S(μ)v=0), a global optimum is found efficiently by bisecting on μ, or using augmented Lagrangian methods [22]. We note that similar optimization problems have also been solved in the context of metabolism and protein expression networks using the Resource Balance Analysis modeling framework (see SI Text E1 in [23] and [24]).
To solve (1), we used the solveME Python module [22]. Specifically, we used bisection (binary search) as in [11] to maximize growth rate to six decimal points. SolveME uses the 128-bit (quad-precision) linear program (LP) and nonlinear program (NLP) solver Quad MINOS 5.6 (qMINOS) [25, 26]. All qMINOS runs were performed with feasibility and optimality tolerances of 10−20. These tight tolerances were used to capture solutions involving fluxes as small as 10−16 mmol/gDW/h and were made possible through the quad-precision capabilities of qMINOS. The models used for this study are available in the Github repositories COBRAme (version 0.0.9) https://github.com/SBRG/ecoli_me_testing and ECOLIme (version 0.0.9) https://github.com/SBRG/ecolime. The COBRAme software [2] was used for building and developing the ME model.
Dynamic simulation using ME models
Our DynamicME implementation extends dynamic FBA (dFBA) [15, 16], which was developed for metabolic models (M-models). We tested two implementations of the DynamicME method: one that does not account for proteome dynamic constraints (protein "inertia") and one that does.
The first implementation assumes that the protein abundances can be adjusted freely between time steps. Also, the uptake rate of a substrate was not made a function of its extracellular concentration. Instead, flux bounds were set to zero if the substrate was depleted (zero concentration), or to a finite value otherwise. Consequently, we did not need to perform a ME-model simulation at every time step. Instead, once exchange (i.e., uptake and secretion) fluxes were computed by the ME-model, the same fluxes were used to compute the extracellular metabolite concentration profile over subsequent time steps. At each time step, DynamicME checked whether a substrate became depleted (fully consumed) or newly available, e.g., by feeding for a fed-batch process or secretion of re-consumable metabolites. If so, a new ME computation was performed with the updated exchange flux bounds. Here, the ME model is capable of selecting the optimal set of metabolites to take up from the medium. The exchange fluxes and growth rate were then updated according to the new optimal solution. These updated values were used to compute biomass and metabolite concentrations. This procedure was repeated until the batch time was reached.
In this first implementation, one can still account for concentration-dependent uptake rates or different feed schedules by performing ME-model simulations at every time step. Furthermore, if additional mechanisms such as growth inhibition by substrates or products are modeled, one should perform ME-model simulations at every time step. The procedure for simulating a batch culture using dynamicME is described in Procedure DynamicME. This implementation of DynamicME is also shown schematically in Fig. 2.
Schematic of the DynamicME procedure. The culture is divided into smaller time steps and extracellular concentrations and biomass are updated at each timepoint. Metabolite exchange fluxes are computed whenever substrate availability changes due to metabolite depletion, feed, or secretion. With each ME simulation, the metabolic flux distribution and proteome composition are also updated
DynamicME with protein inertia constraints
The second implementation accounts for protein abundance at the previous time step (i.e., protein "inertia"). This implementation requires modifying the ME model formulation. Thus, at each time step, we solve the following optimization problem (2):
$$ \begin{aligned} \max_{\mu, v,p, \delta} \quad & \mu \\ \mathrm{s.t.} \quad & S(\mu) v = 0 \\ & v^{\text{form}}_{i} - \mu p_{i} = \delta_{i}, \ \forall i \in {Complex} \\ & \sum_{j\in {CAT}(i)} \frac{v_{ij}}{k^{\text{eff}}_{ij}} \leq p_{i}, \ \forall i \in {Complex} \\ & p_{i} = p^{0}_{i} + \delta_{i} H \\ & l(\mu) \leq v \leq u(\mu) \\ & v_{i}^{\text{form}}\geq 0, \ \forall i \in {Complex} \\ & p_{i} \geq 0, \ \forall i \in {Complex}, \end{aligned} $$
where μ is the growth rate, \(v\in \mathbb {R}^{n}\) the vector of fluxes (metabolic and expression processes), \(v_{i}^{\text {form}} \geq 0\) the flux of protein complex formation reaction for complex i (ComplexFormation reactions in the underlying ME model [2] that convert protein subunits to a complex according to defined complex stoichiometry), \(p \geq 0 \in \mathbb {R}^{k}\) the vector of protein complex concentrations, \(\delta \in \mathbb {R}^{k}\) the vector of protein complex concentration differences, \(p^{0} \in \mathbb {R}^{k}\) the vector of protein complex concentrations at the previous time step, \(S\in \mathbb {R}^{m\times n}\) the stoichiometric matrix that constrains the m components (metabolites and macromolecules), and l, u are the lower and upper flux bounds. H is the time horizon (in hours), which determines the anticipated time window in which to re-allocate the proteome. The value of H need not be the same as the simulation time step when (2) is solved inside the procedure InertiaDynamicME. In this study, we used H=2 hours. Complex is the set of protein complexes that are dynamically constrained, and CAT (i) is the set of reactions that are catalyzed by protein complex (enzyme) i. Note that the original ME network reconstruction accounts for mass balance of all metabolites and macromolecules, including protein complexes. Because we are now allowing accumulation (δi>0) or depletion of complexes (δi<0), we remove from S(μ)v=0 the mass balance constraints on protein complexes in the set Complex, and instead use the constraints \(v_{i}^{\text {form}} - \mu p_{i} = \delta _{i}, \ \forall i \in {Complex}\).
In the protein inertia implementation, we solve Problem (2) at every iteration, rather than only re-solving when environmental conditions change. Re-solving at every iteration is necessary because the intracellular protein abundances are now potentially changing at every iteration, i.e., when δi≠0; therefore, the metabolic fluxes are also subject to change at every iteration, whether or not extracellular conditions are changing. The inertia-constrained dynamicME procedure is described in InertiaDynamicME.
Variable time step procedure (MinTimeStep)
At every time step, we compute the concentration for the next time step based on the current concentration ci for each extracellular metabolite i. If the updated concentration would have become negative, i.e., the time step was too large, we then compute a new time step according to the formula: Δtnew,M= min{ci/(−viX):i=1,…,p}, where p is the number of extracellular metabolites whose concentrations are simulated, and X is the biomass concentration.
Similarly, we also compute a new time step if an intracellular protein concentration is detected to fall below zero with the current time step. In this case, we use the following formula: Δtnew,P= min{pi/(δi):i=1,…,k}. The time step is finally computed as Δt= min{Δt0,Δtnew,M,Δtnew,P}. If the time step Δt differs from Δt0, we re-solve the optimization problem using the updated time step. This way, we ensure that changes to the environment or intracellular concentrations are accounted for in the simulation, regardless of the initial time step.
Model calibration using literature data
A number of adaptive laboratory evolution (ALE) studies have now demonstrated that the proteome of wild-type E. coli is not optimally allocated or efficient for every single nutrient [27–29]. To accurately reflect this wild-type proteome state, we calibrated the model with respect to several known enzymatic features of wild-type E. coli. First, glycerol kinase is known to be significantly less efficient for wild-type compared to ALE endpoints [30]. Second, ALE on lactate minimal medium showed multiple limitations in lactate utilization and enzymes near the phosphoenolpyruvate (PEP) node [27]. Third, respiration is known to have higher proteomic cost than fermentation, leading to acetate overflow [31]. Based on these observations, we calibrated the effective rate constants (keff) (see section below for details). For example, we imposed a realistic turnover rate for isocitrate dehydrogenase based on literature data, effectively increasing proteomic cost for respiration. All calibrated parameters are listed in Additional file 1: Table S1 along with their original and adjusted values. Also, oxygen uptake rate was constrained to −20 mmol/gDW/h to reflect transport limitations not reflected in the proteome cost model.
Sensitivity analysis of dynamic simulations
In the ME model, effective rate constants (keff) relate metabolic flux v to enzyme concentration by the relationship v=keff·e, where e is the enzyme concentration [11]. Precise estimates for these parameters are not available for many reactions and enzymes; therefore, an important step in ME model-based studies has been to assess sensitivity of predictions to these uncertainties [32]. In this study, we investigated the sensitivity of DynamicME predictions to uncertainties in keff. We perturbed keff values from 0.1 to 10 times the nominal values. To avoid exploring the full parameter space consisting of thousands of keff values, we chose relevant pathways and perturbed only these reactions (Additional file 1: Table S1). We generated 200 random samples. Perturbed ME models having good fit to measured metabolite concentration profiles were treated as an ensemble. The exact determination of ensembles is described below.
Archetypal analysis and ensemble of models
Archetypal analysis [33–35] is a dimension-reduction method in which any data point is approximated as a convex combination of the computed archetypes; in turn, each archetype is a convex combination of the data points [33]. Each archetype lies on the convex hull of the data and represents a "pure" phenotype. In our study, we performed archetypal analysis on randomly perturbed samples of model-predicted time-course metabolite concentration profiles (Fig. 3). Thus, each archetype represents a distinct phenotype with a particular substrate utilization hierarchy.
Ensemble model estimation and characterization procedure. (a-c) An ensemble of models with prediction error within a threshold was found by perturbing model parameters (keff). d Archetypal analysis was used to characterize the ensemble where any data point is approximated as a convex combination of archetypes. e The predicted time-course proteome profile from the ensemble with best fit to measured metabolite concentrations was compared with measured time-course expression profiles
To prepare data for archetypal analysis, timepoints and metabolites were collapsed, resulting in a 2D matrix of features (timepoints-and-metabolites) × samples. In archetypal analysis, we then approximate X as X≈ZA, where Z is the matrix of archetypes and A is a matrix of coefficients with the constraints Aij≥0 and \(\sum _{j=1}^{p} A_{ij}=1\) for p archetypes (Fig. 3d). Thus, X is approximated as a convex combination of archetypes. The matrix of archetypes Z is constrained as Z=XB, with the coefficient matrix Bij≥0, \(\sum _{j=1}^{p} B_{ij}=1\) for p archetypes; therefore, the archetypes are constrained to be convex combinations of the data points X.
Once archetypes were determined, the proteome and exchange flux dynamic profiles were also mapped to the archetypes using B. The best number of archetypes was chosen using the elbow method from a scree plot [36] (Additional file 2: Figure S1). Archetypal analysis was performed using the spams Python module [35].
Optimal parameter estimation via metaheuristic optimization
We developed an optimization-based procedure to match time-course concentration profiles by estimating keff values. For optimization, we used a gradient-free metaheuristic method (list-based threshold accepting) [37] because of its efficiency and flexibility. We developed a parallel implementation of this optimization method for increased efficiency (Additional file 3: Figure S2). The implementation allows each parallel node (CPU thread) to choose between following its local search trajectory or restarting the search from the current best solution. This parallel communication was implemented using MPI (Message Passing Interface) via the mpi4py Python module [38]. The objective function was the sum of squared errors between measured and predicted extracellular metabolite concentration profiles.
Model validation using time-series expression profiles
To validate proteome allocation predictions, we computed the time-lagged cross-correlation between simulated and measured time-course proteome profiles. Lagged cross-correlation measures the similarity between two time-series where one lags the other, and has been particularly useful for analyzing time-course expression profiles. For example, it was used to study regulatory interactions of galactose metabolism in E. coli [39]. To compute lagged cross-correlation we used the R function ccf [40].
We obtained microarray hybridization intensity values over time points from Beg et al. [9]. We log2-transformed these values for further analysis. The log2-transformed measurements were compared against simulated protein mass fractions. We define mass fraction in two ways. First, for the ME model without protein inertia constraints, the protein mass fraction \(f_{j} = v^{\text {trsl}}_{j} w_{j}/\sum _{j\in Prot}\left (v^{\text {trsl}} w_{j}\right)\), where \(v_{j}^{\text {trsl}}\) is the translation rate of protein j, wj is its molecular weight, and Prot is the set of all proteins in the ME model. Second, for the ME model with protein inertia constraints, the mass fraction \(f_{j} = p_{j} w_{j} / \sum _{j\in {Prot}} (p_{j} w_{j})\), where pj is the enzyme concentration, which is a variable in this modified ME model.
We first made the time intervals consistent between the measured and simulated expression profiles. To do so, we determined the smallest time interval used (i.e., measured or simulated) for the two profiles and linearly interpolated each profile separately using this time interval. In this study, the time interval was 0.1 h.
To determine the lagged cross-correlation for the entire simulated proteome, we iterated through each lag value, ranging from −1.7 to 1.7 h, and chose the lag corresponding to the highest median cross-correlation across all proteins.
Growth on mixed substrates
When grown on complex media, E. coli uses substrates preferentially or simultaneously, depending on growth conditions [9]. Without additional constraints, FBA may erroneously predict simultaneous uptake of all substrates [9]. FBA with molecular crowding (FBAwMC) improves FBA by adding molecular crowding constraints, and correctly predicted substrate utilization hierarchy under a five-carbon medium [9].
We hypothesized that proteome-limited cellular growth would exhibit a hierarchy of preferential and simultaneous substrate utilization on mixed substrate media. To test this hypothesis, we implemented the DynamicME procedure: namely, time-course simulation of genome-scale integrated models of metabolism and macromolecular expression (ME-models) (Fig. 2). DynamicME extends dynamic FBA (dFBA) [16] to ME-models (see "Methods" section).
Using DynamicME, we simulated cellular growth on the five-carbon mixed substrate media studied by Beg et al. [9] and the simulated metabolite concentration profiles were compared with measurements. To simulate growth on nutrient-excess batch culture, carbon substrate uptake rates were effectively unconstrained (i.e., lower bound =−1000 mmol/gDW/h). Therefore, total proteome limitation became the active constraint rather than nutrient limitation.
DynamicME correctly predicted the majority of substrate uptake hierarchy characteristics, including the single substrate utilization (glucose), mixed utilization, and acetate reconsumption phases observed by Beg et al. [9] (Fig. 4). We found a few differences between simulated and measured profiles. Overall, metabolites were consumed more rapidly than experimentally observed. Also, acetate secretion was lower than measured, and maltose was predicted to be utilized earlier than in experiments.
Predicted substrate uptake hierarchy and proteome allocation. a Predicted time-course metabolite concentration profile. b Predicted time-course proteome mass fraction allocation profile
In the absence of additional constraints, FBA was shown to predict optimal states that accurately reflect ALE (adaptive laboratory evolution) endpoints but may exceed the efficiency of wild-type cells [28, 41–43]. To account for this discrepancy, we implemented a model-calibration procedure to reflect observed metabolic and expression profiles better, as described in the following section.
Model calibration for experimentally consistent concentration time-course profiles
Both the rate and hierarchy of substrate utilization are affected by ME-model parameters. In particular, the effective rate constants keff influence predicted pathway usage [44, 45]. We thus investigated the sensitivity of predicted substrate utilization hierarchy to uncertainty in keff values.
First, we performed 200 random perturbations of keff values and performed DynamicME simulations for the perturbed models. The predictions showed large variations with respect to substrate utilization hierarchy. To aid interpretation, we performed archetypal analysis [33, 34] on the time-course metabolite concentration profiles and identified five archetypes (Fig. 5) as described in "Methods" section. The five archetypes showed considerable variation in substrate utilization hierarchy, reflecting the sensitivity of predictions to uncertainty in keff values. Of the five archetypes, archetype 4 most closely resembled experiments (Fig. 5). The archetypal model correctly predicted the sequence of glucose uptake followed by mixed utilization of maltose, lactate, and galactose, and finally glycerol uptake and acetate re-consumption. The acetate secretion rate was also significantly higher than the initial model and matched measurements better.
Sensitivity and archetypal analysis. Archetypal time-course concentration (a) and proteome mass fraction allocation (b) profiles were computed from simulations with 200 randomly perturbed keff parameters
We also implemented an alternative approach to fit measured concentrations using metaheuristic optimization (Additional files 3: Figure S2 and 4: Figure S3). The optimal profiles were similar to that of archetype 4. Thus, we proceeded with subsequent analyses using archetype 4, which in turn represents an ensemble of experimentally-consistent ME models with differing parameter values.
Predicting time-course proteome allocation
An important novelty of DynamicME is explicit computation of proteome allocation over a time-course simulation. For the mixed substrate medium, DynamicME computed distinct proteome compositions over time, corresponding to the changing metabolic modes (Fig. 4). We compared computed proteome dynamics with measured time-series microarray data [9]. For validation, we used the proteome profile from the most accurate archetype (archetype 4) as determined in the previous section (Fig. 5).
To validate proteome allocation predictions, we computed the lagged cross-correlation [46] between simulated and measured time-course proteome profiles (Fig. 6). The lag time resulting in the highest median cross-correlation across all compared proteins was 1.2 h, indicating that proteome dynamics were faster than measured, which was consistent with metabolite concentration profiles. With this fixed lag time, the median lagged cross-correlation across proteins was 0.64 with values ranging from −0.83 to 0.86 (Fig. 6b).
Lagged cross-correlation of simulated vs. measured expression. a Histograms of lagged cross-correlation values and lag time. b Histogram of cross-correlation values for fixed lag time of 1.2 h. c Functional groups (COGs) of genes with cross-correlations for fixed lag of 1.2 h, which were low (below 0.25) or high (above 0.64, the median)
In addition, certain functional gene sets were predicted better than others. For example, of 138 genes in the COG (Cluster of Orthologous Groups) [47] "Translation, ribosomal structure and biogenesis", DynamicME predicted 74% with high (above 0.64, the median) and 11% with low (below 0.25) cross-correlation, respectively (Fig. 6c). Similarly, "nucleotide transport and metabolism," "amino acid transport and metabolism," and "inorganic ion transport and metabolism" were predicted with high cross-correlation. In contrast, of 69 "Energy production and conversion" genes, 78% had low cross-correlation. Closer inspection of these energy metabolism genes showed that the main discrepancy lay in genes related to oxidative phosphorylation: NADH dehydrogenase, cytochrome oxidase, ATP synthase, and citric acid cycle (Additional file 5: Table S2). The acetate secretion rate for the archetype 4 simulations were lower than measured (Fig. 5a), which was consistent with the discrepancy in gene expression dynamics. We next sought to investigate whether additional constraints could resolve some of these discrepancies.
Effects of protein inertia on dynamic metabolic and protein expression profiles
Next, we investigated the effect of dynamic constraints on intracellular protein abundances. We assume that the protein concentration at a simulation time step depends on the concentration at the previous time step and the rates of synthesis, dilution, and degradation of the protein. Here, we do not account for active protein degradation. We assume that the synthesis rate is constrained by the transcription and translation capacity at that time step, which are computed based on the metabolic and expression network reconstruction and parameters of the underlying ME network used. Dilution rate is determined by growth rate and protein concentration. While not accounted for here, degradation rate depends on the capacity of the proteostasis machinery. Overall, the effect of additional constraints on protein dynamics on the optimal cellular response to environment change is not straightforward to deduce without a network-level model because they are determined by the metabolic and proteomic states of the cell, which change over time.
We hypothesized that the optimal cellular response to changing environments should differ between the scenarios of (a) instantaneous proteome reallocation versus (b) reallocation with dynamic constraints. This hypothesis has been investigated on a coarse-grained model by [48], who showed that proteome adaptation time is theoretically minimized by sequentially synthesizing the set of rate-limiting proteins via an on-off control strategy. Related to this hypothesis is the observation that E. coli expresses proteins that are not needed immediately [13, 32, 49]. This strategy of protein pre-allocation, which enables an increase in these proteins in less time, may provide fitness benefits when alternative carbon sources are encountered [32]. Additionally, when adaptation time is constrained, increased allocation of expression machinery is potentially advantageous to ensure rapid expression—e.g., by allocating a ribosome reserve under feast-famine cycles [13].
To test our hypothesis, we extended DynamicME and implemented proteome dynamic constraints to test the hypothesis above. Our assumptions are as follows. First, we assume a cellular objective of growth rate maximization, (maxμ in (2)). This is the same cellular objective as ME in a static environment and DynamicME without inertia. Second, we assume that under exponential growth on various carbon sources, active protein degradation is negligible compared to dilution. Therefore, in (2) we have a decrease in protein abundance (δi<0) only when dilution rate exceeds complex formation rate, i.e., when \(\mu p_{i} > v_{i}^{\text {form}}\).
Based on these assumptions, we investigated how the proteome dynamics constraints, referred to as protein "inertia", altered dynamic cellular responses to environmental fluctuations.
Protein inertia changes the optimal proteome allocation
The first change due to protein inertia was an overall dampening of protein expression responses, as expected by the additional dependency of protein concentrations on those of the previous time step (2). We also observed two more important effects of inertia constraints: altered proteome allocation and metabolic mode.
First, the proteome composition attained by the end of the batch was itself different, as evidenced by the allocation of protein groups involved in metabolic and expression processes (Fig. 7a-b). Principal components analysis (PCA) also confirmed that while major shifts in proteome allocation occurred at similar time points for both models, their overall directions of change differed considerably (Fig. 7c). A closer examination showed that a major effect of protein inertia constraints was higher investment in cofactor and prosthetic group synthesis very early in the batch culture (Fig. 7b). Specifically, inertia constraints led to higher synthesis of cysteine desulfurase (IscS) and the CyaY protein, which transfer sulfur and Fe(II) groups during iron-sulfur cluster biosynthesis, respectively.
DynamicME simulations with protein dynamics constraints. Mass fraction of protein groups (by metabolic subsystem as in [49]) for baseline (a) and inertia-constrained models (b). (c) Principal component analysis (PCA) of protein concentrations. Percent variance explained is shown in the axis labels. These values were computed using principal components computed from the baseline data. Time points (in hours) are shown next to the markers. (d) Select protein concentrations that differed markedly between models. (e) Extracellular metabolite concentrations simulated by the inertia-constrained model. (f) Select metabolic fluxes that differed markedly between models. GAPD: glyceraldehyde-3-phosphate dehydrogenase. GLCptspp: glucose transport by phosphotransferase system. TPI: triose-phosphate isomerase. FBA: fructose-biphosphate aldolase. ENO: enolase. ATPS4rpp: ATP synthase. NADH16pp: NADH dehydrogenase (ubiquinone). PDH: pyruvate dehydrogenase. PGI: glucose-6-phosphate isomerase. PGK: phosphoglycerate kinase. PFK: phosphofructokinase. NDPK1: nucleoside-diphosphate kinase (ATP:GDP)
Second, and related to this protein expression change was a notable shift in metabolic mode. Inertia-constraints led to lowered oxidative phosphorylation (lower ATP synthase, and NADH dehydrogenase fluxes) and higher substrate-level phosphorylation as evidenced by increased fluxes through phosphoglycerate kinase, phosphofructokinase, and overall higher flux through glycolysis (Fig. 7f). As a result, acetate accumulation was considerably higher than without inertia constraints (Fig. 7e), almost matching the extracellular acetate concentrations observed in experiments by Beg et al. [9].
The predicted alteration in metabolic mode was consistent with the altered proteome allocation. Specifically, pyruvate dehydrgenase (PDH) flux was predicted to increase under inertia constraints (on average 1.8-fold higher than without inertia constraints over all timepoints) (Fig. 7f). The PDH complex consists of three protein subunits (E1, E2, and E3), each having multiple copies [50]. In particular, lipoate moieties are attached to the E2 (AceF) subunit. In turn, lipoate synthesis is catalyzed by lipoyl synthase (LipA), which requires an iron-sulfur cluster. Thus, the increased requirement for lipoate synthase enzymes explains elevated levels of iron-sulfur cluster synthesis proteins IscS and CyaY.
In this study we developed DynamicME, an algorithm for simulating time-course metabolic and proteomic profiles using genome-scale models of metabolism and macromolecular expression (ME-models). We found that DynamicME correctly predicted substrate utilization hierarchy under a five-carbon mixed substrate medium. The biological basis for this hierarchy was proteome-limited cellular growth.
To account for the tendency of constraint-based models, including ME-models, to over-predict metabolic efficiency over wild-type cells, as well as parameter uncertainty, we implemented a model calibration procedure. In this study we focused on perturbing the effective rate constants (keff) to match metabolite concentration profiles better. We arrived at a set of models showing improved prediction of the substrate utilization hierarchy. We note that sensitivity of ME model predictions to keff values has been investigated in several studies including a non-dynamic context [44], in relation to expression of protein groups (or sectors) [49], and for defining a core proteome [45]. However, the sensitivity of the predicted sequence and preference of utilizing mixed carbon substrates over time had not been investigated prior to the present study.
A notable feature of DynamicME is its ability to predict time-course proteome allocation profiles. We observed good agreement between measured and computed time-course expression profiles (median lagged cross-correlation of 0.67). Meanwhile, one subtle difference between measured and predicted time-course profiles was that measured profiles changed less abruptly due to process time constants of transcription and translation dynamics.
Finally, we investigated how proteome dynamic ("inertia") constraints affect the capacity of E. coli to dynamically adjust its proteome allocation, and how this affects metabolism. At first glance, it is not intuitive how the protein inertia constraints (2) would alter the predicted metabolic and proteomic states, other than perhaps a simple smoothing operation of the protein abundances over time. However, because the optimization problem at each time step now accounts for a limited change in protein re-allocation at a future time point, the optimal solution will be quite different from that when the proteome can reallocate freely. Furthermore, because we do not allow active protein degradation by proteases, the only way to decrease protein abundance is by diluting the protein (at a rate exceeding translation), which further limits change of protein abundance. Overall, protein inertia led to higher reliance on substrate-level phosphorylation and reduced oxidative phosphorylation. Coupled to this altered metabolic response was higher requirements for cofactor and prosthetic group biosynthesis, and higher secretion of acetate as a by-product. These altered responses more closely resembled experimental measurements compared with the baseline model. This result reinforces previous studies [48, 51] showing that dynamic protein expression constraints represent a biological phenomenon that is potentially important for determining dynamic cellular states under changing environments.
Computational challenges
One of the challenges for DynamicME, and indeed ME models in general, is the large computational cost compared with metabolic models that do not account for the protein expression network. Without protein inertia constraints, the batch simulation (batch time of 10 h) required approximately 40 min with a simulation time step of 0.1 h. With protein inertia constraints, this batch simulation required 7 h. This increased computational cost for the inertia-constrained model stems from solving the ME model at every time step (i.e., for 100 time steps).
The primary reason for the large computational cost of ME models is that they are ill-conditioned [52]. The reason for ill-conditioning is the wide range in magnitude of coefficients in the stoichiometric matrix. As a result, decision variables take on values ranging 15 orders of magnitude or more [22]. For this reason, ME models are typically solved using quad-precision optimization solvers [25]. Quad-precision solvers require more computational effort than their double-precision counterparts. Some methods have shown that double precision solvers can solve ME models with around 10−6 infeasibility, but this infeasibility tolerance is usually too large for ME models—hence the use of quad-precision. Additionally, ME models include nonlinear constraints as functions of the growth rate but because they are quasi-convex constraints, the models are computed efficiently using bisection [22–24] or augmented Lagrangian methods [22]. Nonetheless, additional solver iterations are required at each time step of DynamicME in order to solve the ME model. These two computational costs are magnified by the larger size of the ME model networks.
Macromolecular expression models and the extension to dynamic allocation
A number of frameworks exist that model cell metabolism and macromolecular allocation at the genome-scale. In addition to the ME framework [2] that provides the reconstructed metabolic and protein expression network for this work, a large number of studies have examined metabolism and macromolecular resource allocation. Please see ref. [13] for a more comprehensive review of such modeling frameworks in non-dynamic contexts, as the scope of this work focuses on the dynamic extension of such models.
A representative method for integrating macromolecule allocation with metabolism (distinct from ME) is resource balance analysis (RBA). RBA extends flux balance analysis [53] with additional constraints and reactions to account for macromolecule synthesis and allocation [54]. A genome-scale RBA model of Bacillus subtilis included 614 reactions and 672 protein-coding genes and modeled cellular processes of metabolism and macromolecular processes (translation, protein folding, ribosome maturation, etc.) [23].
RBA has been used for a number of studies, including estimating in vivo apparent catalytic rates for B. subtilis by integrating proteomics and fluxomics, predicting the hierarchy of using carbon and nitrogen sources, predicting switches between metabolic pathways at the genome-scale, among others [55]. RBA was also used to examine the hierarchy of utilizing multiple carbon sources by integrating combinatorial optimization concepts based on a Boolean formalism [24].
Very recently, a dynamic modeling framework was developed for RBA, called dynamic RBA (dRBA) [56]. The dRBA method was demonstrated on a simplified model of a cell consisting of four fluxes representing conversion of a single substrate into macro-components and a product of interest [56].
In this context, the advancements of our paper are: (i) implemented the dynamic simulation of metabolism and macromolecular expression on a genome-scale ME model having up to 12,677 metabolic and protein expression reactions, (ii) showed that protein inertia (i.e., limitation in change of protein abundance over time due to capped synthesis and dilution rates) can cause a shift in the metabolic mode (lower oxidative phosphorylation, higher acetate overflow), and (iii) we provide the software (publicly on Github) with the aim of broader adoption by the community (see Availability of data and materials).
Future directions
Overall, there is continuing need to develop efficient computational methods for algorithms that utilize ME models, such as dynamic simulations. A number of studies have developed methods for dynamic simulation of integrated models of metabolism and protein expression [20, 23, 24]. Time-scale decomposition and collocation approaches have a rich history in the dynamic modeling domain, and they have been applied to metabolic and expression networks [21]. We hope that future studies will continue to extend such methods for increasingly larger, integrated models of metabolism and protein expression. In particular, the addition of protein inertia constraints significantly increased the computational cost, and this framework may be improved in future studies.
Besides computational methods, DynamicME may be extended further to account for active protein degradation [57] and dynamic stress responses [58]. For example, under conditions of starvation stress starvation [59] or thermal stress [57], active protein degradation becomes important. Furthermore, during the transition between multiple carbon sources, E. coli was shown to up-regulate generic stress response genes [9]. Such extensions would allow the framework to be applied to non-growth phenotypes, where active stress responses including protein homeostasis become important for reallocating the proteome to perform cellular tasks besides biomass synthesis [59].
ME-models compute cellular resource allocation tradeoffs at the proteome scale [13]. This expanded biological scope and predictive capability of ME models is expected to become increasingly useful for biotechnological applications [60]. For example, the metabolic and proteomic burden to the host of expressing biochemical production pathways can be computed explicitly using ME-models. We have shown here that it is furthermore possible to compute how these genome-wide cellular resource dynamics determine transient shifts in metabolic modes under biotechnologically relevant culture conditions: complex media with transient substrate availability. Thus, as ME-models continue to be reconstructed for organisms of biotechnological importance, DynamicME will be a useful approach for analyzing physiological and omics data from cell culture, and for model-aided biotechnological applications that require robust cell factory operation under environmental fluctuations [61]. DynamicME may also be useful for studying protein expression dynamics that are relevant for infectious disease (e.g., persister states [59]), especially by extending the protein inertia procedure to account for active protein degradation, or by utilizing a ME model that includes stress response mechanisms [57, 58].
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This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231.
This work was funded by the National Institute of General Medical Sciences of the National Institutes of Health [awards U01GM102098 and R01GM057089] and the Novo Nordisk Foundation through the Center for Biosustainability at the Technical University of Denmark (NNF10CC1016517). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Department of Bioengineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, 92093, CA, USA
Laurence Yang, Ali Ebrahim, Colton J. Lloyd & Bernhard O. Palsson
Department of Management Science and Engineering, Stanford University, 475 Via Ortega, Stanford, 94305, CA, USA
Michael A. Saunders
Novo Nordisk Foundation Center for Biosustainability, Technical University of Denmark, Kemitorvet 220, Kongens Lyngby, 2800, Denmark
Bernhard O. Palsson
Laurence Yang
Ali Ebrahim
Colton J. Lloyd
LY developed and implemented the procedures and wrote the manuscript. AE and CJL developed the model and edited the manuscript. MAS and BOP edited the manuscript. All authors read and approved the final manuscript.
Correspondence to Laurence Yang.
All data generated or analyzed during this study are included in this published article and its supplementary information files. Additionally, the datasets generated and/or analyzed during the current study are available in the following public repositories: (i) https://github.com/SBRG/dynamicme – the DynamicME Python module and documentation; a Jupyter notebook allowing users to perform DynamicME simulations using the latest version of the E. coli ME-model available from https://github.com/sbrg/cobrame, (ii) https://github.com/SBRG/ecoli_me_testing – the model file used for analyses in this study.
Additional file 1
Table S1. Reactions with perturbed keff. A subset of the genome-scale metabolic network was perturbed with respect to keff values, either manually or randomly. (XLSX 17 kb)
Figure S1. Scree plot for determining number of archetypes. A notable elbow is observed for five archetypes. (PDF 5 kb)
Figure S2. Parameter estimation procedure. We developed a parallel implementation of a metaheuristic optimization procedure. L-TA: list-based threshold accepting algorithm [37]. Variable definitions: Tk, threshold value at iteration k; Tmax, maximum threshold value; Z0, objective value at current solution; Zk, objective value at neighboring solution (generated from current solution) at iteration k. (PDF 189 kb)
Figure S3. Parameter estimation results. The parallel L-TA optimization procedure successfully estimated model parameters that improved consistency with measured concentration profiles. Seven parallel nodes were used here: 1 local and 6 global nodes (see Additional file 3: Figure S2) for explanation of nodes. (PDF 124 kb)
Table S2. Lagged cross-correlation values. Cross-correlation values for fixed lag time of 1.2 h. (XLSX 41 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Yang, L., Ebrahim, A., Lloyd, C.J. et al. DynamicME: dynamic simulation and refinement of integrated models of metabolism and protein expression. BMC Syst Biol 13, 2 (2019). https://doi.org/10.1186/s12918-018-0675-6
Constraint-based modeling
Dynamic simulation
Batch culture
Methods, software and technology | CommonCrawl |
Sequence of series
(Redirected from Serial scheme)
series scheme, serial scheme, triangular array
A double sequence of random variables $ \xi _ {nk} $, $ 1 \leq k \leq k _ {n} $, $ k _ {n} \rightarrow \infty $, $ n \geq 1 $, in which $ \xi _ {n1} \dots \xi _ {nk _ {n} } $( forming the $ n $- th series) are mutually independent for any $ n $. The simplest sequence of series corresponds to the case $ k _ {n} = n $. In the limit theorems of probability theory, a fundamental problem is to determine the limiting behaviour, as $ n \rightarrow \infty $, of the distributions of the random variables
$$ \tag{1 } \eta _ {n} = \ \sum _ { k = 1 } ^ { {k _ n } } \xi _ {nk} . $$
Under suitable conditions the class of limit distributions for such sequences coincides with the class of infinitely-divisible distributions (cf. Infinitely-divisible distribution). Suppose that the sequence of series $ \xi _ {nk} $ satisfies the condition of uniform asymptotic negligibility, that is,
$$ \tag{2 } \max _ {1 \leq k \leq k _ {n} } \ {\mathsf P} \{ | \xi _ {nk} | \geq \epsilon \} \rightarrow 0 $$
as $ n \rightarrow \infty $. Then $ \xi _ {nk} $ is said to form a null sequence of series. The set of distributions that are the limit distributions (in the sense of weak convergence) for the random variables (1), where $ \xi _ {nk} $ is a null sequence of series satisfying the condition of uniform asymptotic negligibility, coincides with the set of infinitely-divisible distributions.
There are conditions for the convergence of the distributions of $ \eta _ {n} $ to a given infinitely-divisible distribution (see [1]). In particular, the condition for convergence to a normal distribution has the following form.
Let $ \xi _ {nk} $ be a sequence of series, and let $ F _ {nk} $ be the distribution function of $ \xi _ {nk} $. For $ \xi _ {nk} $ to satisfy (2), and for the distribution of the sums (2) to be weakly convergent to the normal distribution with parameters $ a $ and $ b $, it is necessary and sufficient that for any fixed $ \epsilon > 0 $ the following conditions hold:
1) $ \sum _ {k = 1 } ^ {k _ {n} } {\mathsf P} \{ | \xi _ {nk} | \geq \epsilon \} \rightarrow 0 $,
2) $ \sum _ {k = 1 } ^ {k _ {n} } \left \{ \int _ {| x | < \epsilon } x ^ {2} dF _ {nk} ( x) - \left ( \int _ {| x | < \epsilon } x dF _ {nk} ( x) \right ) ^ {2} \right \} \rightarrow b ^ {2} $,
3) $ \sum _ {k = 1 } ^ {k _ {n} } \int _ {| x | < \epsilon } x dF _ {nk} ( x) \rightarrow a $.
The study of limit distributions for the normalized partial sums of a sequence of independent random variables is a special case of the study of sequences of series.
As regards sequences of series see also: Infinitely-divisible distribution; Law of large numbers; and Limit theorems. For example, in classical versions of the central limit theorem and the law of large numbers, one considers special cases of sequences of series, formed by random variables
$$ \xi _ {nk} = \ \frac{\xi _ {k} - {\mathsf E} \xi _ {k} }{\sqrt {\sum _ {k = 1 } ^ { n } {\mathsf D} \xi _ {k} } } , $$
$$ \xi _ {nk} = \frac{\xi _ {k} - {\mathsf E} \xi _ {k} }{n} , $$
where the $ \xi _ {k} $ are independent random variables.
[1] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables", Addison-Wesley (1954) (Translated from Russian)
[2] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes", Springer (1969) (Translated from Russian)
[3] V.V. Petrov, "Sums of independent random variables", Springer (1975) (Translated from Russian)
[4] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) pp. Chapt. 1
[a1] M. Loève, "Probability theory" , I , Springer (1977)
Serial scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serial_scheme&oldid=40028
Retrieved from "https://encyclopediaofmath.org/index.php?title=Sequence_of_series&oldid=48672"
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How many different sizes of infinity are there?
It's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?"
My thoughts are that the number of unique cardinalities is equivalent to the number of real numbers, based on the fact that the cardinalities can always be ordered by increasing size. I don't really know how to prove this, though. It's mostly based on intuition, which isn't very reliable when talking about uncountably infinite sets.
I originally asked a somewhat related question at a different (and not math-oriented) forum, and the users there told me that it is not possible to talk about the number of cardinalities without talking about the set of all sets, which forms a paradox. If a set were to contain all of the different sizes of infinity, it would have to contain its own power set, which isn't possible.
However, I'm not completely convinced that it is not possible to talk about a set of all of the cardinalities. Sure, a cardinality represents a size of infinity, but I think that it should be possible to have a set of the cardinalities without having the set actually contain the various infinities. Would this avoid the above paradox?
So, is it possible to measure the number of different sizes of infinity, and what would that size be?
elementary-set-theory infinity
Asaf Karagila♦
PhiNotPiPhiNotPi
$\begingroup$ These three may help: math.stackexchange.com/questions/8013/… math.stackexchange.com/questions/5378/types-of-infinity?lq=1 math.stackexchange.com/questions/39502/… $\endgroup$ – Amzoti Jan 21 '13 at 17:12
$\begingroup$ Also this question. $\endgroup$ – user642796 Jan 21 '13 at 17:20
$\begingroup$ infinitely many $\endgroup$ – AndreasS Jan 21 '13 at 23:16
We don't really talk about "infinities", instead we talk about "cardinalities". Cardinality of the a set is the mathematical way of saying how large it is. Of course infinity could easily just mean $\infty$ which is a formal symbol representing a point larger than all real numbers (but the notion can be transferred to other contexts as well). This is not the same sort of infinity as infinite cardinalities. Infinite cardinalities are a whole other beast, and they are related to set theory (as we measure the size of sets, not the length of an interval).
Cantor's theorem tells us that given a set there is always a set whose cardinality is larger. In particular given a set, its power set has a strictly larger cardinality. This means that there is no maximal size of infinity.
But this is not enough, right? There is no maximal natural numbers either, but there is only a "small amount" of those. As the many paradoxes tell us, the collection of all sets is not a set. It is a proper class, which is a fancy (and correct) way of saying that it is a collection which is too big to be a set, but we can still decide whether or not something is in that collection.
In a similar fashion we can show that the collection of all cardinalities is not a set either. If $X$ is a set of sets, $\bigcup X=\{y\mid\exists x\in X. y\in x\}$ is also a set, and its cardinality is not smaller than that of any $x\in X$. By Cantor's theorem we have that the power set of $\bigcup X$ has an even larger cardinality.
What the above paragraph show is that given a set of cardinals, we can always find a cardinal which is not only not in that set, but also larger than all of those in that set. Therefore the collection of possible cardinalities is not a set.
$\begingroup$ Ok, Asaf, here are some test questions to make this formal: Given a set $X$, are there $X$-many different cardinalities? If not, are there $Y$-many cardinalities for some $Y$ that surjects onto $X$? $\endgroup$ – Andrés E. Caicedo Jan 21 '13 at 17:46
$\begingroup$ @user1729: Actually, there is a Gimel function too. But you can use $\daleth$... :-) $\endgroup$ – Asaf Karagila♦ Aug 20 '14 at 10:48
$\begingroup$ @Andrey: Your comment is very unclear. What does it mean "a set of numbers"? Can you find a natural number which is not in the set of all the natural numbers? Can you find a real number which is not in the set of all the real numbers? What you can do however, is show that for every finite set of natural/rational/real/complex numbers, there is a number which does not lie in that set. Therefore, we can conclude, the set of all natural/rational/real/complex numbers is not finite. $\endgroup$ – Asaf Karagila♦ Apr 7 '15 at 21:54
$\begingroup$ @Andrey: No, it very much does not work with infinite sets. The sum of all the natural numbers is far from a natural number (and it's not $-\frac1{12}$ either!). Sets to classes are like finite subsets of $\Bbb N$ to infinite subsets of $\Bbb N$. $\endgroup$ – Asaf Karagila♦ Apr 9 '15 at 22:31
$\begingroup$ @Andrey: Yes, something like that. Classes are collections of sets which do not have cardinality (because, at least in $\sf ZFC$, having a cardinality means being in bijection with a cardinal which is a set). $\endgroup$ – Asaf Karagila♦ Apr 9 '15 at 22:45
If $A$ is a set, then the power set $P(A)$ is a set of bigger cardinality. If $\{A_i\}_{i\in I}$ is a family of sets, then $P(\bigcup_{i\in I}A_i)$ is a set of bigger cardinality than any of the $A_i$. This allows us to define an infinite set $F(a)$ for each ordinal $a$ such that $a<b$ implies that $F(a)$ has smaller cardinality than $F(b)$. To do so, let
$F(\emptyset)=\mathbb N$,
$F(a)=P(F(b))$ if $a=b+1$ is the successor of $b$,
$F(a)=P(\bigcup_{b<a} F(b))$ if $a$ is a limit ordinal
Now if a set $S$ were able to enumerate all infinite cardinalities, this would give us an injective map from the proper class of all ordinals into this set, which is absurd.
Hagen von EitzenHagen von Eitzen
As the other answers have pointed out (at least within the framework of ZFC), the answer is "proper class many." Let me just point out that this isn't the end of the story.
We can ask whether two proper classes have the same size, by asking about the existence of a definable bijection, etc. In this sense the class of ordinals - or equivalently of cardinalities of well-orderable sets (=initial ordinals) - is actually small: every other proper class surjects onto it! Given a proper class $C$, consider the map $rk: C\rightarrow ON$ sending a set $x\in C$ to the unique $\alpha$ such that $x\in V_{\alpha+1}-V_\alpha$ - that is, its (von Neumann) rank. This is well-defined, and maps $C$ to a cofinal subclass $S$ of $ON$. Now we can "collapse" $S$ onto $ON$ by sending $\alpha\in S$ to $ot(\{\beta\in S: \beta<\alpha\})$; this is surjective. Composing these two maps gives a surjection from $C$ to $ON$.
Meanwhile we can construct models in which there is a proper class $C$ with no surjection from $ON$ (let alone an injection into $ON$): see Joel David Hamkins' answer to https://mathoverflow.net/questions/110799/does-zfc-prove-the-universe-is-linearly-orderable.
Finally, we could ask: can there be a proper class $C$ such that $ON$ does not inject into $C$? This is a subtler question, and class injections are weird things; but I believe the answer is yes via class forcing (I vaguely recall seeing this a long time ago, and it being relatively simple, but I can't remember the details).
The distinction between injection and surjection above might make you think, "But wait a minute! Don't we have the axiom of choice to simplify things?" Indeed we do, but the axiom of choice treats only sets, and at the level of classes "injects into" and "is surjected onto" are still distinct, a priori. We can have an "axiom of choice for classes" (called global choice), if we enlarge our language a bit to talk about classes, and this axiom implies that all proper classes have the same size.
Noah SchweberNoah Schweber
The different sizes of infinities are called transfinities. They can be captured as mathematical sets via formalized definitions. The cardinality (number of) transfinities is not itself a transfinity, it is infinity. It cannot be captured as a set. It cannot be formalized by a mathematical definition. In that sense it is truly infinity which just means an unknowable, indefinable concept.
Borrowing ideas from complexity theory we see that transfinities are compressible whereas infinity is not. Pure randomness is not compressible. The cardinality of transfinities sets is actually pure randomness which is maximum entropy. That's an interesting way to tie in entropy to the concepts of transfinities and their cardinality (number of yransfinities).
Furthermore, mathematically nothingness can be defined as the state of maximum entropy. To get philosophical (and also mathematical) the cardinality of transfinities is both the alpha and the omega.
Nissim LevyNissim Levy
$\begingroup$ -1: not only does this not address the question, it's not remotely correct. No, they're not called transfinities, at least not in set theory - in fact I've never seen the term. Googling for it reveals no scholarly mathematical works on the first page of hits, and the only scholarly works I can find are on "neurophilosophy". As to the content: no, infinite cardinalities need not be definable. The connection to complexity theory makes no sense, and neither does the mention of entropy. To get philosophical, if you're going to get philosophical at least make sure it's actually relevant. $\endgroup$ – Noah Schweber May 24 '15 at 23:00
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Blowup for flat slow manifolds
Kristiansen, Kristian Uldall
In this paper, we present a way of extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimensions, they do appear naturally in certain settings; for example, in (a......) the regularization of piecewise smooth systems by tanh, (b) a particular aircraft landing dynamics model, and finally (c) in a model of earthquake faulting. We demonstrate the approach using a simple model system and the examples (a) and (b)....
Kristiansen, K. U.
In this paper, we present a way of extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimensions, they do appear naturally in certain settings; for example, in (a) the regularization of piecewise smooth systems by \\tanh , (b) a particular aircraft landing dynamics model, and finally (c) in a model of earthquake faulting. We demonstrate the approach using a simple model system and the examples (a) and (b).
Blow-up: A Free Lunch?
Jan Koenderink
Full Text Available We consider operations that change the size of images, either shrinks or blow-ups. Image processing offers numerous possibilities, put at everyone's disposal with such computer programs as Adobe Photoshop. We consider a different class of operations, aimed at immediate visual awareness, rather than pixel arrays. We demonstrate cases of blow-ups that do not sacrifice apparent resolution. This apparent information gain is due to "amodal occlusion.�
Blow-up : A free lunch?
Koenderink, J.J.; Richards, W.; Van Doorn, A.J.
We consider operations that change the size of images, either shrinks or blow-ups. Image processing offers numerous possibilities, put at everyone's disposal with such computer programs as Adobe Photoshop. We consider a different class of operations, aimed at immediate visual awareness, rather than
Blow-up in nonlinear Schroedinger equations. II. Similarity structure of the blow-up singularity
Rypdal, K.; Juul Rasmussen, Jens
invariance and generalizations of the latter. This generalized "quasi-invariance" reveals the nature of the blow-up singularity and resolves an old controversy. Most of the previous work has been done on the cubic nonlinearity. We generalize the results to an arbitrary power nonlinearity....
Heterotic Mini-landscape in blow-up
Bizet, Nana Geraldine Cabo
Localization properties of fields in compact extra dimensions are crucial ingredients for string model building, particularly in the framework of orbifold compactifications. Realistic models often require a slight deviation from the orbifold point, that can be analyzed using field theoretic methods considering (singlet) fields with nontrivial vacuum expectation values. Some of these fields correspond to blow-up modes that represent the resolution of orbifold singularities. Improving on previous analyses we give here an explicit example of the blow-up of a model from the heterotic Mini-landscape. An exact identification of the blow-up modes at various fixed points and fixed tori with orbifold twisted fields is given. We match the massless spectra and identify the blow-up modes as non-universal axions of compactified string theory. We stress the important role of the Green-Schwarz anomaly polynomial for the description of the resolution of orbifold singularities.
Blow-up Mechanism of Classical Solutions to Quasilinear Hyperbolic Systems in the Critical Case
This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.
Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity
Hee Chul Pak
Full Text Available A blow-up time for nonlinear heat equations with transcendental nonlinearity is investigated. An upper bound of the first blow-up time is presented. It is pointed out that the upper bound of the first blow-up time depends on the support of the initial datum.
Finite Time Blowup in a Realistic Food-Chain Model
Parshad, Rana; Ait Abderrahmane, Hamid; Upadhyay, Ranjit Kumar; Kumari, Nitu
We investigate a realistic three-species food-chain model, with generalist top predator. The model based on a modified version of the Leslie-Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. We show that the model exhibits finite time blowup in certain parameter range and for large enough initial data. This result implies that finite time blowup is possible in a large class of such three-species food-chain models. We propose a modification to the model and prove that the modified model has globally existing classical solutions, as well as a global attractor. We reconstruct the attractor using nonlinear time series analysis and show that it pssesses rich dynamics, including chaos in certain parameter regime, whilst avoiding blowup in any parameter regime. We also provide estimates on its fractal dimension as well as provide numerical simulations to visualise the spatiotemporal chaos.
Parshad, Rana
Asymptotic properties of blow-up solutions in reaction-diffusion equations with nonlocal boundary flux
Liu, Bingchen; Dong, Mengzhen; Li, Fengjie
This paper deals with a reaction-diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green's identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.
Numerical study of blow-up in the Davey-Stewartson system
Klein, Christian; Muite, Benson; Roidot, Kristelle
Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow up. We numerically study the long time behavior and potential blow-up of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa solutions. It is shown in this way that both are unstable to blow-up and dispersion, and that blow-up in the Ozawa solution is generic.
Klein, Christian
Blow-up, Global Existence and Persistence Properties for the Coupled Camassa–Holm equations
Zhu Mingxuan
In this paper, we consider the coupled Camassa–Holm equations. First, we present some new criteria on blow-up. Then global existence and blow-up rate of the solution are also established. Finally, we discuss persistence properties of this system.
Transition between extinction and blow-up in a generalized Fisher–KPP model
Hernández-Bermejo, Benito, E-mail: [email protected] [Departamento de Física, Universidad Rey Juan Carlos, Calle Tulipán S/N, 28933, Móstoles, Madrid (Spain); Sánchez-Valdés, Ariel [Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, Calle Tulipán S/N, 28933, Móstoles, Madrid (Spain)
Stationary solutions of the Fisher–KPP equation with general nonlinear diffusion and arbitrary reactional kinetic orders terms are characterized. Such stationary (separatrix-like) solutions disjoint the blow-up solutions from those showing extinction. In addition a criterion for general parameter values is presented, which allows determining the blow-up or vanishing character of the solutions.
Hernández-Bermejo, Benito; Sánchez-Valdés, Ariel
Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant
Jiang, Jie; Wu, Hao; Zheng, Songmu
We investigate blow-up properties for the initial-boundary value problem of a Keller-Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller-Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4,5] from the whole space R3 to the case of bounded smooth domain Ω ⊂R3. Lower global blow-up estimate on ‖ n ‖ L∞ (Ω) is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.
Some blow-up problems for a semilinear parabolic equation with a potential
Cheng, Ting; Zheng, Gao-Feng
The blow-up rate estimate for the solution to a semilinear parabolic equation u=Δu+V(x)|u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].
Perturbational blowup solutions to the compressible Euler equations with damping.
Cheung, Ka Luen
The N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions. In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases [Formula: see text] and [Formula: see text], where [Formula: see text] is the adiabatic constant. With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen. Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points. The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications.
Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations
Novruzov, Emil
This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, ut -uxxt +[ f (u) ] x -[ f (u) ] xxx +[ g (u) + f″/(u) 2 ux2 ] x = 0 that includes the Camassa-Holm equation as well as the hyperelastic-rod wave equation (f (u) = ku2 / 2 and g (u) = (3 - k) u2 / 2) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.
Blow-up boundary regimes for general quasilinear parabolic equations in multidimensional domains
Shishkov, A E; Shchelkov, A G
A new approach (not based on the techniques of barriers) to the study of asymptotic properties of the generalized solutions of parabolic initial boundary-value problems with finite-time blow-up of the boundary values is proposed. Precise conditions on the blow-up pattern are found that guarantee uniform localization of the solution for an arbitrary compactly supported initial function. The main result of the paper consists in obtaining precise sufficient conditions for the singular (or blow-up) set of an arbitrary solution to remain within the boundary of the domain
Improved Longitudinal Blow-up and Shaving in the Booster
Hancock, S
The low-intensity proton beam for p-Pb collisions in the LHC did not come back in the Booster at the beginning of 2013 anything like it had been set up at the end of 2012. In particular there were unexplained intensity fluctuations of ±100%. Although the root cause of the drift in performance was never established, its investigation revealed long-standing issues in the longitudinal plane which, when corrected, allowed single-bunch beams to be delivered with unprecedented reproducibility and control of both intensity and longitudinal emittance. The new approach was adopted for the ion run and subsequently for MDs at higher intensities, where it made possible a robust control of intensity at constant 6D phase space volume. Post-LS1, it may even provide a platform upon which to build a more exotic controlled longitudinal blow-up to generate higher intensity bunches with a flattened line density.
Origins of transverse emittance blow-up during the LHC energy tramp
Kuhn, M; Arduini, G; Kain, V; Schaumann, M; Tomas, R
During LHC Run 1 about 30 % of the potential peak performance was lost due to transverse emittance blow-up through the LHC cycle. Measurements indicated that the majority of the blow-up occurred during the energy ramp. Until the end of LHC Run 1 this emittance blow-up could not be eliminated. In this paper the measurements and observations of emittance growth through the ramp are summarized. Simulation results for growth due to Intra Beam Scattering will be shown and compared to measurements. A summary of investigations of other possible sources will be given and backed up with simulations where possible. Requirements for commissioning the LHC with beam in 2015 after Long Shutdown 1 to understand and control emittance blow-up will be listed.
Controlled Transverse Blow-up of Highenergy Proton Beams for Aperture Measurements and Loss Maps
HÓ§fle, W; Redaelli, S; Schmidt, R; Valuch, D; Wollmann, D; Zerlauth, M
A technique was developed to blow-up transversely in a controlled way high energy proton beams in the LHC. The technique is based on band limited white noise excitation that is injected into the transverse damper feedback loop. The injected signal can be gated to selectively blow-up individual trains of bunches. The speed of transverse blow-up can be precisely controlled. This opens the possibility to perform safely and efficiently aperture measurements and loss maps with high intensity bunch trains well above stored beam energies that are considered to be safe. In particular, lengthy procedures for measurements at top energy, otherwise requiring multiple fills of individual bunches, can be avoided. In this paper, the method is presented and results from beam measurements are discussed and compared with alternative blowup methods.
Blowup with vorticity control for a 2D model of the Boussinesq equations
Hoang, V.; Orcan-Ekmekci, B.; Radosz, M.; Yang, H.
We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a vorticity stretching term and a non-local Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.
On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation
Cortissoz, Jean C.; Montero, Julio A.; Pinilla, Carlos E.
We show a new lower bound on the H .3/2 (T 3 ) norm of a possible blow-up solution to the Navier-Stokes equation, and also comment on the extension of this result to the whole space. This estimate can be seen as a natural limiting result for Leray's blow-up estimates in L p (R 3 ), 3 .5/2 (T 3 ), and give the corresponding extension to the case of the whole space
Description of regional blow-up in a porous-medium equation
Carmen Cortazar
Full Text Available We describe the (finite-time blow-up phenomenon for a non-negative solution of a porous medium equation of the form $$ u_t = Delta u^m + u^m $$ in the entire space. Here $m>1$ and the initial condition is assumed compactly supported. Blow-up takes place exactly inside a finite number of balls with same radii and exhibiting the same self-similar profile.
Pavement Subgrade Performance Study
Zhang, Wei; Ullidtz, Per; Macdonald, Robin
The report describes the second test in the Danish Road Testing Machine (RTM) under the International Pavement Subgrade Performance Study. Pavement response was measured in different layers, and compared to different theroretical values. Performance in terms of plastic strains, rutting...
Concrete pavement joint deterioration.
Concrete pavements are an important part of our national infrastructure. In recent years the relatively small number of reported joints deteriorating prematurely in concrete pavements around Indiana has increased. Changes over the past 45 years in IN...
Permeable pavement study (Edison)
U.S. Environmental Protection Agency — While permeable pavement is increasingly being used to control stormwater runoff, field-based, side-by-side investigations on the effects different pavement types...
Indiana Pavement Preservation Program
Ong, Ghim Ping (Raymond); Nantung, Tommy E.; Sinha, Kumares C.
State highway agencies are facing immense pressure to maintain roads at acceptable levels amidst the challenging financial and economic situations. In recent years, pavement preservation has been sought as a potential alternative for managing the pavement assets, believing that it would provide a cost-effective solution in maintaining infrastructural conditions and meeting user expectations. This study explores the potential of pavement preservation concepts in managing the agency's pavement ...
Longitudinal emittance blowup in the large hadron collider
Baudrenghien, P
The Large Hadron Collider (LHC) relies on Landau damping for longitudinal stability. To avoid decreasing the stability margin at high energy, the longitudinal emittance must be continuously increased during the acceleration ramp. Longitudinal blowup provides the required emittance growth. The method was implemented through the summer of 2010. Band-limited RF phase-noise is injected in the main accelerating cavities during the whole ramp of about 11min. Synchrotron frequencies change along the energy ramp, but the digitally created noise tracks the frequency change. The position of the noise-band, relative to the nominal synchrotron frequency, and the bandwidth of the spectrum are set by pre-defined constants, making the diffusion stop at the edges of the demanded distribution. The noise amplitude is controlled by feedback using the measurement of the average bunch length. This algorithm reproducibly achieves the programmed bunch length of about 1.2ns, at flat top with low bunch-to-bunch scatter and provides a...
Consideration of pavement roughness effects on vehicle-pavement interaction
CSIR Research Space (South Africa)
Steyn, WJvdM
Full Text Available . In this paper the focus is on quantification of the pavement roughness effects on the calculated structural pavement life and the effects of surfacing maintenance on the moving dynamic tyre loads generated by vehicles. A simplified method for calculating... the moving dynamic tyre load population is used together with standard pavement response analysis methods to quantify the effects of pavement surfacing maintenance on roughness and structural pavement life. This method can be used as a pavement management...
Transverse blowup along bunch train caused by electron cloud in BEPC
Liu Yudong; Guo Zhiyuan; Qin Qing; Wang Jiuqing; Zhao Zheng
Electron cloud instability (ECI) may take place in a storage ring when the machine is operated with a multi-bunch positively charged beam. Transverse blowup due to electron cloud has been observed in some machines and is considered to be a major limit factor in the development of high current and high luminosity electron positron colliders. With a streak camera, the transverse blowup along the bunch train was first observed in an experiment at the Beijing Electron-Positron Collider (BEPC) and the simulation results were used to compared with the observation. (authors)
Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source
Pan Zheng
Full Text Available We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source ut=div(|∇um|p−2∇ul+uq, (x,t∈RN×(0,T, where N≥1, p>2 , and m, l, q>1, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.
Blow-up of solutions to the rotation b-family system modeling equatorial water waves
Min Zhu
Full Text Available We consider the blow-up mechanism to the periodic generalized rotation b-family system (R-b-family system. This model can be derived from the f-plane governing equations for the geographical water waves with a constant underlying current in the equatorial water waves with effect of the Coriolis force. When b=2, it is a rotation two-component Camassa-Holm (R2CH system. We consider the periodic R2CH system when linear dispersion is absent (which model is called r2CH system and derive two finite-time blow-up results.
Pavement Testing Facility
Federal Laboratory Consortium — Comprehensive Environmental and Structural AnalysesThe ERDC Pavement Testing Facility, located on the ERDC Vicksburg campus, was originally constructed to provide an...
Asphalt in Pavement Maintenance.
Asphalt Inst., College Park, MD.
Maintenance methods that can be used equally well in all regions of the country have been developed for the use of asphalt in pavement maintenance. Specific information covering methods, equipment and terminology that applies to the use of asphalt in the maintenance of all types of pavement structures, including shoulders, is provided. In many…
Pavement edge treatment.
Four projects were built over two construction seasons using special devices attached to the paving machine that produces a 30 slope on the outside pavement edge instead of the near vertical drop-off common with conventional paving equipment. This ...
Pervious Pavement System Evaluation
Porous pavement is a low impact development stormwater control. The Urban Watershed Management Branch is evaluating interlocking concrete pavers as a popular implementation. The pavers themselves are impermeable, but the spaces between the pavers are backfilled with washed, grade...
Desert Pavement Studies
Haff, Peter
Combining plan view information from aerial photography showing details of stream channels on desert pavement surfaces with process-based erosion models, a high-resolution, synthetic topography DEM...
Haff, P
Combining plan view information from aerial photography showing details of stream channels on desert pavement surfaces with process-based erosion models, a high-resolution, "synthetic topography" DEM...
Exact, rotational, infinite energy, blowup solutions to the 3-dimensional Euler equations
Yuen, Manwai
In this Letter, we construct a new class of blowup or global solutions with elementary functions to the 3-dimensional compressible or incompressible Euler and Navier-Stokes equations. And the corresponding blowup or global solutions for the incompressible Euler and Naiver-Stokes equations are also given. Our constructed solutions are similar to the famous Arnold-Beltrami-Childress (ABC) flow. The obtained solutions with infinite energy can exhibit the interesting behaviors locally. Furthermore, due to divu → =0 for the solutions, the solutions also work for the 3-dimensional incompressible Euler and Navier-Stokes equations. -- Highlights: → We construct a new class of solutions to the 3D compressible or incompressible Euler and Navier-Stokes equations. → The constructed solutions are similar to the famous Arnold-Beltrami-Childress flow. → The solutions with infinite energy can exhibit the interesting behaviors locally.
The blow-up problem for a semilinear parabolic equation with a potential
Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D.
Let [Omega] be a bounded smooth domain in . We consider the problem ut=[Delta]u+V(x)up in [Omega]×[0,T), with Dirichlet boundary conditions u=0 on [not partial differential][Omega]×[0,T) and initial datum u(x,0)=M[phi](x) where M[greater-or-equal, slanted]0, [phi] is positive and compatible with the boundary condition. We give estimates for the blow-up time of solutions for large values of M. As a consequence of these estimates we find that, for M large, the blow-up set concentrates near the points where [phi]p-1V attains its maximum.
Simple countermeasures against the TM110-beam-blowup-mode in biperiodic structures
Euteneuer, H.; Herminghaus, H.; Schoeler, H.
The two fundamental methods of fighting beam blow-up in rf-accelerating-structures are staggered detuning and selective Q-spoiling of their higher order modes. Biperiodic structures offer a very simple way of applying the latter technique of the most dangerous TM 110 -like blowup mode at 1.7 times the accelerating frequency: letting this mode propagate but giving a large gap to the TM 110 -passband. This gap must be positive for electric coupling (f(phi=0) =1.7c. With asymmetric coupling elements between the cavities of a structure, one has a simple tool for staggered detuning: a change of the relative orientation of these elements spreads the resonance frequencies not only of the TM 110 -mode, but of at least all dipole modes. (orig.)
LHC MD 1087: Controlled Longitudinal Emittance Blow-up with Short Bunches
Timko, Helga; Esteban Muller, Juan; Jaussi, Michael; Lasheen, Alexandre; Shaposhnikova, Elena; CERN. Geneva. ATS Department
The aim of the MD was to study the controlled longitudinal emittance blow-up applied during the ramp with bunches that are slightly shorter than operational. Earlier MDs in 2015 have shown that with a short target bunch length, the blow-up is less controlled and a bifurcation of bunch lengths occurs. The presented measurements show that the bifurcation is independent of the presence of the bunch length feedback, pointing towards an intensity-dependent phenomenon, originating from a synchrotron frequency shift with intensity. Accurate measurements of synchrotron frequency shift with intensity are presented as well. The measurements took place between 22nd August 2016, 19:00 and 23rd August 2016, 04:00.
PSB beam longitudinal blow-up by phase modulation with the digital LLRF prototype system
Angoletta, M E; Butterworth, A; Findlay, A; Jaussi, M; Leinonen, P; Molendijk, J; Sanchez-Quesada, J
The PSB will be upgraded to a new, Digital Low-Level RF (DLLRF) system in 2014 at the injectors' restart after LS1. This DLLRF is an evolution of that successfully deployed in LEIR and comprises new hardware, software and implementation strategies. Machine development studies have been carried out in the PSB over recent years with the existing LEIR-style hardware installed in PSB ring four. These studies have allowed testing approaches and validating implementation strategies. This note focuses on a series of MDs carried out during the 2011 run where a new implementation of the longitudinal beam blow-up obtained by phase modulation was tested. Test results and effects on the beam are show for a CNGS-type beam. Finally, an overview is given of the final longitudinal blow-up implementation planned with the new hardware, which will be operationally deployed in 2014.
Blow-up of the quantum potential for a free particle in one dimension
Devillanova, G.; Maddalena, F.; Florio, G.
We derive a non-linear differential equation that must be satisfied by the quantum potential, in the context of the Madelung equations, in one dimension for a particular class of wave functions. In this case, we exhibit explicit conditions leading to the blow-up of the quantum potential of a free particle at the boundary of the compact support of the probability density.
Some problems on non-linear semigroups and the blow-up of integral solutions
Pavel, N.H.
After some introductory remarks, this highly mathematical document considers a unifying approach in the theory of non-linear semigroups. Then a brief survey is given on blow-up of mild solutions from the semilinear case. Finally, the global behavior of solutions to non-linear evolution equations is addressed; it is found that classical results on the behavior of the maximal solution u as t up-arrow tsub(max) hold also for integral solutions
A note on Burgers' equation with time delay: Instability via finite-time blow-up
Jordan, P.M.
Burgers' equation with time delay is considered. Using the Cole-Hopf transformation, the exact solution of this nonlinear partial differential equation (PDE) is determined in the context of a (seemingly) well-posed initial-boundary value problem (IBVP) involving homogeneous Dirichlet data. The solution obtained, however, is shown to exhibit a delay-induced instability, suffering blow-up in finite-time
Controlled longitudinal emittance blow-up using band-limited phase noise in CERN PSB
Quartullo, D.; Shaposhnikova, E.; Timko, H.
Controlled longitudinal emittance blow-up (from 1 eVs to 1.4 eVs) for LHC beams in the CERN PS Booster is currently achievied using sinusoidal phase modulation of a dedicated high-harmonic RF system. In 2021, after the LHC injectors upgrade, 3 eVs should be extracted to the PS. Even if the current method may satisfy the new requirements, it relies on low-power level RF improvements. In this paper another method of blow-up was considered, that is the injection of band-limited phase noise in the main RF system (h=1), never tried in PSB but already used in CERN SPS and LHC, under different conditions (longer cycles). This technique, which lowers the peak line density and therefore the impact of intensity effects in the PSB and the PS, can also be complementary to the present method. The longitudinal space charge, dominant in the PSB, causes significant synchrotron frequency shifts with intensity, and its effect should be taken into account. Another complication arises from the interaction of the phase loop with the injected noise, since both act on the RF phase. All these elements were studied in simulations of the PSB cycle with the BLonD code, and the required blow-up was achieved.
Training development for pavement preservation.
This research project strives to help the Iowa Department of Transportation (DOT) fully achieve the full benefits of pavement : preservation through training on proper selection, design, and application of pavement preservation treatments. In some ca...
Alternative pavements for snowmobile crossings.
Excessive highway pavement wear from snowmobile traffic is a maintenance problem for the New Hampshire Department of : Transportation. The snowmobiles and trail grooming equipment scar and erode the pavement surfaces, eventually creating wide ruts ac...
Status report on pavement management.
This report traces the developments in pavement management in the Virginia Department of Highways and Transportation from the initial efforts in the mid-1970s through early 1984. Included are status reports on pavement management for the interstate, ...
Development of reliable pavement models.
The current report proposes a framework for estimating the reliability of a given pavement structure as analyzed by : the Mechanistic-Empirical Pavement Design Guide (MEPDG). The methodology proposes using a previously fit : response surface, in plac...
Pavement Snow Melting
Lund, John W.
The design of pavement snow melting systems is presented based on criteria established by ASHRAE. The heating requirements depends on rate of snow fall, air temperature, relative humidity and wind velocity. Piping materials are either metal or plastic, however, due to corrosion problems, cross-linked polyethylene pipe is now generally used instead of iron. Geothermal energy is supplied to systems through the use of heat pipes, directly from circulating pipes, through a heat exchanger or by allowing water to flow directly over the pavement, by using solar thermal storage. Examples of systems in New Jersey, Wyoming, Virginia, Japan, Argentina, Switzerland and Oregon are presented. Key words: pavement snow melting, geothermal heating, heat pipes, solar storage, Wyoming, Virginia, Japan, Argentina, Klamath Falls.
Experimental pavement delineation treatments
Bryden, J. E.; Lorini, R. A.
Visibility and durability of materials used to delineate shoulders and medians adjacent to asphalt pavements were evaluated. Materials evaluated were polysulfide and coal tar epoxies, one and two component polyesters, portland cement, acrylic paints, modified-alkyd traffic paint, preformed plastic tape, and thermoplastic markings. Neat applications, sand mortars, and surface treatments were installed in several geometric patterns including cross hatches, solid median treatments, and various widths of edge lines. Thermoplastic pavement markings generally performed very well, providing good visibility under adverse viewing conditions for at least 4 years. Thermoplastic 4 in. wide edge lines appear to provide adequate visibility for most conditions.
Modified pavement cement concrete
Botsman, L. N.; Ageeva, M. S.; Botsman, A. N.; Shapovalov, S. M.
The paper suggests design principles of pavement cement concrete, which covers optimization of compositions and structures at the stage of mixture components selection due to the use of plasticizing agents and air-retaining substances that increase the viability of a concrete mixture. It also demonstrates advisability of using plasticizing agents together with air-retaining substances when developing pavement concrete compositions, which provides for the improvement of physical and mechanical properties of concrete and the reduction of cement binding agent consumption thus preserving strength indicators. The paper shows dependences of the main physical-mechanical parameters of concrete on cement consumption, a type and amount of additives.
Fatigue cracking in road pavement
Mackiewicz, P.
The article presents the problem of modelling fatigue phenomena occurring in the road pavement. The example of two selected pavements shows the changes occurring under the influence of the load in different places of the pavement layers. Attention is paid to various values of longitudinal and transverse strains generated at the moment of passing the wheel on the pavement. It was found that the key element in the crack propagation analysis is the method of transferring the load to the pavement by the tire and the strain distribution in the pavement. During the passage of the wheel in the lower layers of the pavement, a complex stress state arises. Then vertical, horizontal and tangent stresses with various values appear. The numerical analyses carried out with the use of finite element methods allowed to assess the strain and stress changes occurring in the process of cracking road pavement. It has been shown that low-thickness pavements are susceptible to fatigue cracks arising "bottom to top", while pavements thicker are susceptible to "top to bottom" cracks. The analysis of the type of stress allowed to determine the cracking mechanism.
Pavement Performance : Approaches Using Predictive Analytics
Acceptable pavement condition is paramount to road safety. Using predictive analytics techniques, this project attempted to develop models that provide an assessment of pavement condition based on an array of indictors that include pavement distress,...
Optimization and standardization of pavement management processes.
This report addresses issues related to optimization and standardization of current pavement management processes in Kentucky. Historical pavement management records were analyzed, which indicates that standardization is necessary in future pavement ...
A guide for local agency pavement managers
The purpose of this guide is to provide Washington's local agencies with a practical document that will assist pavement managers in understanding the pavement management process and the steps necessary to implement their own pavement management syste...
On blow-up of solutions of the Kuramoto-Sivashinsky equation
Pokhozhaev, S I
The problem of the absence of global solutions of initial-boundary value problems for the Kuramoto-Sivashinsky equation is considered. Sufficient conditions for the absence of global solutions of the problems under consideration are obtained both for bounded and unbounded domains. These conditions imply a priori the blow-up of the solution of the corresponding initial-boundary value problem. The proof uses a generalization of the method of non-linear capacity based on the choice of asymptotically optimal test functions. Bibliography: 20 titles.
Asymptotic analysis of reaction-diffusion-advection problems: Fronts with periodic motion and blow-up
Nefedov, Nikolay
This is an extended variant of the paper presented at MURPHYS-HSFS 2016 conference in Barcelona. We discuss further development of the asymptotic method of differential inequalities to investigate existence and stability of sharp internal layers (fronts) for nonlinear singularly perturbed periodic parabolic problems and initial boundary value problems with blow-up of fronts for reaction-diffusion-advection equations. In particular, we consider periodic solutions with internal layer in the case of balanced reaction. For the initial boundary value problems we prove the existence of fronts and give their asymptotic approximation including the new case of blowing-up fronts. This case we illustrate by the generalised Burgers equation.
Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations
Zhaohui Dai
Full Text Available The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressure p in the homogeneous Besov space Ḃ∞,∞0.
Blow-up criteria for the 3D cubic nonlinear Schrödinger equation
Holmer, Justin; Platte, Rodrigo; Roudenko, Svetlana
We consider solutions u to the 3D nonlinear Schrödinger equation i∂ t u + Δu + |u| 2 u = 0. In particular, we are interested in finding criteria on the initial data u 0 that predict the asymptotic behaviour of u(t), e.g., whether u(t) blows up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for H 1 data) (Duyckaerts–Holmer–Roudenko) if M[u]E[u] ≤ M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u and Q denotes the ground state solution to −Q + ΔQ + |Q| 2 Q = 0. Here we consider the complementary case M[u]E[u] > M[Q]E[Q]. In the first (analytical) part of the paper, we present a result due to Lushnikov, based on the virial identity and the generalized uncertainty principle, giving a sufficient condition for blow-up. By replacing the uncertainty principle in his argument with an interpolation-type inequality, we obtain a new blow-up condition that in some cases improves upon Lushnikov's condition. Our approach also allows for an adaptation to radial infinite-variance initial data that has a conceptual interpretation: for real-valued initial data, if a certain fraction of the mass is contained within the ball of radius M[u], then blow up occurs. We also show analytically (if one takes the numerically computed value of ||Q|| .H 1/2 ) that there exist Gaussian initial data u 0 with negative quadratic phase such that ||u 0 || .H 1/2 .H 1/2 but the solution u(t) blows up. In the second (numerical) part of the paper, we examine several different classes of initial data—Gaussian, super Gaussian, off-centred Gaussian, and oscillatory Gaussian—and for each class give the theoretical predictions for scattering or blow-up provided by the above theorems as well as the results of numerical simulation. We find that depending upon the form of the initial conditions, any of the three analytical criteria for blow-up
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels
Bertozzi, Andrea L; Laurent, Thomas; Carrillo, José A
We consider the multidimensional aggregation equation u t − ∇· (u∇K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717–35), whereas for C 2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K(x) = |x| and some open problems
Geotextiles in Flexible Pavement Construction
Alungbe, Gabriel D.
People everywhere in the developed world regularly drive on paved roads. Learning about the construction techniques and materials used in paving benefits technology and construction students. This article discusses the use of geosynthetic textiles in pavement construction. It presents background on pavements and describes geotextiles and drainage…
Evaluation of airfield pavement evenness
Pietruszewski, Paweł; Poświata, Adam; Wesołowski, Mariusz
The evenness of airfield pavements is one of the basic operating parameters, which characterize them. The evenness determines not only comfort of traffic along an airfield pavement, but also influences the size of dynamic effect on the pavement, hence, the safety of air operations. In addition, the evenness condition changing as a result of dynamic loads, adverse weather conditions or inappropriate airfield pavement construction technology, lead to deviations from the desired condition in the form of longitudinal and transverse unevenness. As a result, systematic and correct performance of tests is a very significant and required factor impacting the improvement of traffic safety on airfield pavements. If the data obtained through the measurements are not sufficiently reliable, they may consequently lead to making incorrect decisions, which can ultimately impact the safety of air operations.
Pavement maintenance scheduling using genetic algorithms
Yang, Chao; Remenyte-Prescott, Rasa; Andrews, John D.
This paper presents a new pavement management system (PMS) to achieve the optimal pavement maintenance and rehabilitation (M&R) strategy for a highway network using genetic algorithms (GAs). Optimal M&R strategy is a set of pavement activities that both minimise the maintenance cost of a highway network and maximise the pavement condition of the road sections on the network during a certain planning period. NSGA-II, a multi-objective GA, is employed to perform pavement maintenance optimisatio...
Beam size blow-up and current loss in the Fermilab Main Ring during storage
Guignard, C.; Month, M.
Observations at Fermilab during storage mode operation show characteristic forms of transverse beam size growth and current loss with time. There are three obvious mechanisms which can produce such blowup. The gas pressure is a source for immediate beam loss by direct nuclear scattering. Protons can also multiple scatter off the orbiting electrons of the gas atoms causing the trasnverse beam size to increase with time. A third mechanism not related to gas pressure is beam growth due to multiple crossing of betatron resonances arising from the synchrotron oscillations of the stored bunches. This simulates a random walk and causes the transverse beam size to grow. This is an attempt to describe the observations with direct nuclear scattering, multiple coulomb scattering, and multiple resonance crossing
Evolution PDEs with nonstandard growth conditions existence, uniqueness, localization, blow-up
Antontsev, Stanislav
This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces, and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.
Accelerated pavement testing of thin RCC over soil cement pavements
Zhong Wu
Full Text Available Three full-scale roller compacted concrete (RCC pavement sections built over a soil cement base were tested under accelerated pavement testing (APT. The RCC thicknesses varied from 102Â mm (4Â in. to 152Â mm (6Â in. and to 203Â mm (8Â in., respectively. A bi-directional loading device with a dual-tire load assembly was used for this experiment. Each test section was instrumented with multiple pressure cells and strain gages. The objective was to evaluate the structural performance and load carrying capacity of thin RCC-surfaced pavements under accelerated loading. The APT results generally indicated that all three RCC pavement sections tested in this study possessed very high load carrying capacity; an estimated pavement life in terms of equivalent single axle load (ESAL for the thinnest RCC section (i.e., RCC thickness of 102Â mm evaluated was approximately 19.2 million. It was observed that a fatigue failure would be the primary pavement distress type for a thin RCC pavement under trafficking. Specifically, the development of fatigue cracking was found to originate from a longitudinal crack at the edge or in the center of a tire print, then extended and propagated, and eventually merged with cracks of other directions. Instrumentation results were used to characterize the fatigue damage under different load magnitudes. Finally, based on the APT performance of this experiment, two fatigue models for predicting the fatigue life of thin RCC pavements were developed. Keywords: Roller compacted concrete, APT, Pavement performance, Non-destructive testing, Fatigue analysis
Nondestructive testing of pavements and pavement bases (a bibliography with abstracts). Report for 1964--Dec 1975
Habercom, G.E. Jr.
Eighty references on nondestructive methods for quality assurance of pavements and pavement bases are presented. Vibration, nuclear activation, radiometry, and acoustic detection are among the various techniques employed
Global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system
Liu, Jingjing; Yin, Zhaoyang
This paper is concerned with global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system. The strategy relies on our observation on new conservative quantities of these systems. Several new global existence results and a new blowup result of strong solutions to the two-component Degasperis- Procesi system and the two-component b-family system are presented by using these new conservative quantities.
Tire-Pavement Friction Characteristics with Elastic Properties of Asphalt Pavements
Miao Yu
Full Text Available The skid-resisting performance of pavement is a critical factor in traffic safety. Recent studies primarily analyze this behavior by examining the macro or micro texture of the pavement. It is inevitable that skid-resistance declines with time because the texture of pavement deteriorates throughout its service life. The primary objective of this paper is to evaluate the use of different asphalt pavements, varying in resilience, to optimize braking performance on pavement. Based on the systematic dynamics of tire-pavement contact, and analysis of the tire-road coupled friction mechanism and the effect of enlarging the tire-pavement contact area, road skid resistance was investigated by altering the elastic modulus of asphalt pavement. First, this research constructed the kinetic contact model to simulate tire-pavement friction. Next, the following aspects of contact behaviors were studied when braking: tread deformation in the tangential pavement interface, actual tire-pavement contact in the course, and the frictional braking force transmitted from the pavement to the tires. It was observed that with improvements in pavement elasticity, the actual tire-pavement contact area increased, which gives us the ability to effectively strengthen the frictional adhesion of the tire to the pavement. It should not be overlooked that the improvement in skid resistance was caused by an increase in pavement elasticity. This research approach provides a theoretical basis and design reference for the anti-skid research of asphalt pavements.
Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Hashira, Takahiro; Ishida, Sachiko; Yokota, Tomomi
This paper deals with the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type in a ball of RN (N ≥ 2). In the case of non-degenerate diffusion, Cieślak-Stinner [3,4] proved that if q > m + 2/N, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if q > m + 2/N (see Ishida-Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when q > m + 2/N.
Extending the Service Life of Pavements
Gschwendt, Ivan
The cost of road construction and expenditures on the maintenance of pavements, i.e., their whole life cost, represents a lot of money. The paper describes a procedure for a pavement management system with degradation models and estimates the length of time for the rehabilitation of an asphalt pavement. Using a theory of pavement mechanics, we calculated the stresses and strains on the layers of two pavement models. High modulus asphalt concrete, an asphalt mix with a high binder content, and an asphalt mix with binder modifications are new road building materials. Prolonging the time for the rehabilitation of pavements is possible.
Pervious Pavement System Evaluation- Abstract
Porous pavement is a low impact development stormwater control. The Urban Watershed Management Branch is evaluating interlocking concrete pavers as a popular implementation. The pavers themselves are impermeable, but the spaces between the pavers are backfilled with washed, gra...
Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals
Yu, Hao; Wang, Wei; Zheng, Sining
This paper considers the two-species chemotaxis system with two chemicals in a smooth bounded domain Ω\\subset{R}2 , subject to the non-flux boundary condition, and χ, ξ, α, β, γ, δ>0 . We obtain a blow-up criterion that if m_1m_2-2π(\\frac{m_1}χβ+\\frac{m_2}ξδ)>0 , then there exist finite time blow-up solutions to the system with m_1:=\\int_Ω u_0(x)dx and m_2:=\\int_Ω w_0(x)dx . When χ=ξ= β=δ=1 , the blow-up criterion becomes m_1m_2-2π(m_1+m_2)>0 , and the global boundedness of solutions is furthermore established with α=γ=1 under the condition that \\max\\{m_1, m_2\\}4π and global boundedness with \\max\\{m_1, m_2\\}Funds for the Central Universities (DUT16LK24).
Quantifying Evaporation in a Permeable Pavement System
Studies quantifying evaporation from permeable pavement systems are limited to a few laboratory studies and one field application. This research quantifies evaporation for a larger-scale field application by measuring the water balance from lined permeable pavement sections. Th...
Dynamic contracting mechanism for pavement maintenance management
Demirel, H.C.; De Ridder, H.A.J.
Technological advances, financial possibilities and changes of demands have increasingly affected the pavement maintenance environment for outsourcing in recent years. This induces complexity in the contracting methods of pavement maintenance activities. Despite the fact that current contracting
Equivalent damage of loads on pavements
Prozzi, JA
Full Text Available This report describes a new methodology for the determination of Equivalent Damage Factors (EDFs) of vehicles with multiple axle and wheel configurations on pavements. The basic premise of this new procedure is that "equivalent pavement response...
Pavement performance evaluation, phase II : data collection.
Phase I and II of this study tested approximately 1500 rehabilitated pavements (asphalt and PCC) : throughout the State. These pavements ranged from 5 to 15 years old and were intended to develop a : snapshot of how various rehabilitations were perfo...
Sustainable concrete pavements : a manual of practice.
Developed as a more detailed follow-up to a 2009 briefing document, Building Sustainable Pavement with Concrete, this guide provides a clear, concise, and cohesive discussion of pavement sustainability concepts and of recommended practices for maximi...
Inverted base pavements: construction and performance
Papadopoulos, Efthymios; Santamarina, Carlos
quality control. This study reviews the extensive South African experience and case histories in the USA. Accumulating evidence suggests that inverted base pavements are a viable alternative and can outperform conventional pavements at a lower cost
An approach to pavement management in Virginia.
The report summarizes the objectives and benefits of formal pavement management systems and outlines an approach believed by the author to be practical for Virginia. The management of Virginia interstate pavements and a proposed random-sampling plan ...
Calibration of PMIS pavement performance prediction models.
Improve the accuracy of TxDOTs existing pavement performance prediction models through calibrating these models using actual field data obtained from the Pavement Management Information System (PMIS). : Ensure logical performance superiority patte...
Appraisal of the Causes of Pavement
pavement failure) or structural (deep-seated pavement failure). ... is primarily due to shallow groundwater level. Adewole et al. ... processing and interpretation of Landsat-7 ETM+ images covering the study area and it's environ for lineaments.
Serviceable pavement marking retroreflectivity levels : technical report.
This research addressed an array of issues related to measuring pavement markings retroreflectivity, factors : related to pavement marking performance, subjective evaluation process, best practices for using mobile : retroreflectometers, sampling pav...
Deterioration of jointed Portland cement concrete pavements.
Information has been gathered regarding the performance of more than 400 lane-miles of jointed PCC interstate pavements located in five construction districts in Virginia. The factors causing pavement deterioration have been identified, the processes...
Miao Yu; Guoxiong Wu; Lingyun Kong; Yu Tang
The skid-resisting performance of pavement is a critical factor in traffic safety. Recent studies primarily analyze this behavior by examining the macro or micro texture of the pavement. It is inevitable that skid-resistance declines with time because the texture of pavement deteriorates throughout its service life. The primary objective of this paper is to evaluate the use of different asphalt pavements, varying in resilience, to optimize braking performance on pavement. Based on the systema...
A Digital System for Longitudinal Emittance Blow-Up in the LHC
JaussI, M; Baudrenghien, P; Butterworth, A; Sanchez-Quesada, J; Shaposhnikova, E; Tuckmantel, J
In order to preserve beam stability with nominal bunch intensity in the LHC, longitudinal emittance blow-up is performed during the energy ramp by injecting phase noise in the main accelerating cavities. The noise spectrum spans a small frequency band around the synchrotron frequency. It is generated continuously in software and streamed digitally into the Digital Signal Processor (DSP) of the Beam Control system where it is added to the pick-up signal of the beam phase loop, resulting in a phase modulation of the accelerating RF. In order to achieve reproducible results, a feedback system, using as input the measured bunch lengths averaged over each ring, controls the strength of the excitation, allowing the operator to simply set a target bunch length. The spectrum of the noise is adjusted to excite the core of the bunch only, extending to the desired bunch length. As it must follow the evolution of the synchrotron frequency through the ramp, it is automatically calculated by the LHC settings management sof...
Impact of pavement conditions on crash severity.
Li, Yingfeng; Liu, Chunxiao; Ding, Liang
Pavement condition has been known as a key factor related to ride quality, but it is less clear how exactly pavement conditions are related to traffic crashes. The researchers used Geographic Information System (GIS) to link Texas Department of Transportation (TxDOT) Crash Record Information System (CRIS) data and Pavement Management Information System (PMIS) data, which provided an opportunity to examine the impact of pavement conditions on traffic crashes in depth. The study analyzed the correlation between several key pavement condition ratings or scores and crash severity based on a large number of crashes in Texas between 2008 and 2009. The results in general suggested that poor pavement condition scores and ratings were associated with proportionally more severe crashes, but very poor pavement conditions were actually associated with less severe crashes. Very good pavement conditions might induce speeding behaviors and therefore could have caused more severe crashes, especially on non-freeway arterials and during favorable driving conditions. In addition, the results showed that the effects of pavement conditions on crash severity were more evident for passenger vehicles than for commercial vehicles. These results provide insights on how pavement conditions may have contributed to crashes, which may be valuable for safety improvement during pavement design and maintenance. Readers should notice that, although the study found statistically significant effects of pavement variables on crash severity, the effects were rather minor in reality as suggested by frequency analyses. Copyright © 2013 Elsevier Ltd. All rights reserved.
Pavement noise measurements in Poland
Zofka, Ewa; Zofka, Adam; Mechowski, Tomasz
The objective of this study is to investigate the feasibility of the On-Board Sound Intensity (OBSI) system to measure tire-pavement noise in Poland. In general, sources of noise emitted by the modern vehicles are the propulsion noise, aerodynamic resistance and noise generated at the tire-pavement interface. In order to capture tire-pavement noise, the OBSI system uses a noise intensity probe installed in the close proximity of that interface. In this study, OBSI measurements were performed at different types of pavement surfaces such as stone mastic asphalt (SMA), regular asphalt concrete (HMA) as well as Portland cement concrete (PCC). The influence of several necessary OBSI measurement conditions were recognized as: testing speed, air temperature, tire pressure and tire type. The results of this study demonstrate that the OBSI system is a viable and robust tool that can be used for the quality evaluation of newly built asphalt pavements in Poland. It can be also applied to generate reliable input parameters for the noise propagation models that are used to assess the environmental impact of new and existing highway corridors.
Geo synthetic-reinforced Pavement systems
Zornberg, J. G.
Geo synthetics have been used as reinforcement inclusions to improve pavement performance. while there are clear field evidence of the benefit of using geo synthetic reinforcements, the specific conditions or mechanisms that govern the reinforcement of pavements are, at best, unclear and have remained largely unmeasured. Significant research has been recently conducted with the objectives of: (i) determining the relevant properties of geo synthetics that contribute to the enhanced performance of pavement systems, (ii) developing appropriate analytical, laboratory and field methods capable of quantifying the pavement performance, and (iii) enabling the prediction of pavement performance as a function of the properties of the various types of geo synthetics. (Author)
ICP experiments more durable pavements
Carta Petrolera
A new asphalts technology that will make more durable the pavement in the highways and roads of the cities of Colombia investigates the Colombian Institute of the Petroleum ICP. The project that will have important incidence in the solution of one of the main problems in the roads of cities like Bogota, is only one of the 35 investigation programs and application of new technologies that with relationship to the sector of the hydrocarbons and its influence branches the ICP advances. The investigation looks for to elevate the current average of useful life of the pavements, with that it would be reached a standard that has the developed countries in this field
Guignard, G.; Month, M.
Observations at Fermilab during the storage mode of operation show characteristic forms of transverse beam size growth and current loss with time. There are three obvious mechanisms which can produce such blowup. The gas pressure is a source for immediate beam loss by direct nuclear scattering. Protons can also multiple Coulomb scatter off the orbiting electrons of the gas atoms causing the transverse beam size to increase with time, t. This effect is therefore also proportional to the gas pressure. A third mechanism not related to the gas pressure is beam growth due to multiple crossing of betatron resonances arising from the synchrotron oscillations of the stored bunches. This simulates a random walk and causes the transverse beam size to grow with √t. An attempt is made to describe the observations with direct nuclear scattering, multiple coulomb scattering and multiple resonance crossing. In addition to the loss rate from direct nuclear scattering, the presence of betatron resonances also contribute to particle loss. In fact this latter effect becomes dominant after the beam size reaches a critical value. This critical size is referred to as the resonance aperture. It is the size at which ''fast'' resonance crossing is no longer valid. The stopband width becomes so large (due both to emittance growth as well as the increase in magnetic field distortions) that particles are locked into the resonance and are extracted to the physical aperture. The model is described in a phenomenological way, and the coefficients involved are estimated. Theoretical curves for transverse beam growth and loss rate are plotted and compared with some measured values. Finally, some general comments are given
Teaching Methodology of Flexible Pavement Materials and Pavement Systems
Mehta, Yusuf; Najafi, Fazil
Flexible pavement materials exhibit complex mechanical behavior, in the sense, that they not only show stress and temperature dependency but also are sensitive to moisture conditions. This complex behavior presents a great challenge to the faculty in bringing across the level of complexity and providing the concepts needed to understand them. The…
Tuning in to pavement radio
Ellis, S.
This article describes a phenomenon known all over Africa, for which there is no really satisfactory term in English but which is summed up in the French term 'radio trottoir', literally 'pavement radio'. It may be defined as the popular and unofficial discussion of current affairs in Africa,
The Concrete and Pavement Challenge
Roman, Harry T.
The modern world is characterized by the extensive use of concrete and asphalt pavement. Periodically, these materials are replaced and the old materials disposed of. In this challenge, students will be asked to develop ways to reuse the old materials. It is important for students to understand how concrete and asphalt are made and applied, as…
Fracture mechanics in pavement design
Denneman, E
Full Text Available . The paper also seeks to promote the use of size independent material properties to characterize pavement materials. Some material properties used in current methods, such as the concepts of strain at break and flexural strength are known to exhibit size...
Minnesota Local Agency Pavement Marking : Mining Existing Data
Pavement marking is important for safety. Maximizing pavement marking performance in terms of increased retroreflectivity, within limited budget constraints, allows agencies to make better decisions toward providing more effective pavement marking pe...
Comparison of winter temperature profiles in asphalt and concrete pavements.
The objectives of this research were to 1) determine which pavement type, asphalt or concrete, has : higher surface temperatures in winter and 2) compare the subsurface temperatures under asphalt and : concrete pavements to determine the pavement typ...
Texas flexible pavements overlays : review and analysis of existing databases.
Proper calibration of pavement design and rehabilitation performance models to : conditions in Texas is essential for cost-effective flexible pavement design. The degree of : excellence with which TxDOTs pavement design models is calibrated will d...
Comparison of accelerated pavement test results with long term pavement behaviour and performance
Jooste, FJ
Full Text Available The aim of this study was to investigate the following:how accelerated pavement testing predictions compare with actual road behaviour and performance the relative influences of load and environmental factors on pavement deterioration and how well...
Calibration of full-scale accelerated pavement testing data using long-term pavement performance data
VdM Steyn, WJ
Full Text Available Accelerated Pavement Testing (APT) has always been conducted with the objective of improving the understanding of real pavements under real traffic and environmental conditions. While APT provides an accelerated view of some of the major structural...
MoDOT pavement preservation research program volume IV, pavement evaluation tools-data collection methods.
The overarching goal of the MoDOT Pavement Preservation Research Program, Task 3: Pavement Evaluation Tools Data : Collection Methods was to identify and evaluate methods to rapidly obtain network-level and project-level information relevant to :...
Verification of mechanistic-empirical design models for flexible pavements through accelerated pavement testing : technical summary.
Midwest States Accelerated Pavement Testing Pooled-Fund Program, financed by the : highway departments of Kansas, Iowa, and Missouri, has supported an accelerated : pavement testing (APT) project to validate several models incorporated in the NCHRP :...
Verification of mechanistic-empirical design models for flexible pavements through accelerated pavement testing.
The Midwest States Accelerated Pavement Testing Pooled Fund Program, financed by the highway : departments of Kansas, Iowa, and Missouri, has supported an accelerated pavement testing (APT) project to : validate several models incorporated in the NCH...
Feasibility of reclaimed asphalt pavement as aggregate in portland cement concrete pavement, phase II : field demonstration.
This research was focused on evaluating the feasibility of using minimally processed reclaimed asphalt pavement (RAP) as : aggregate replacement in concrete pavements. An initial phase of research demonstrated that concretes with up to 50 percent : o...
Feasibility of reclaimed asphalt pavement as aggregate in portland cement concrete pavements.
This research effort was focused on evaluating the feasibility of using minimally processed reclaimed : asphalt pavement (RAP) as aggregate replacement in concrete pavements. This research demonstrated : that concretes with up to 50 percent of the fi...
Advanced Pavement Design: Finite Element Modeling for Rigid Pavement Joints, Report II: Model Development
Hammons, Michael
.... The objective of this research was to obtain data on the response of the ng'id pavement slab-joint-foundation system by conducting laboratory-scale experiments on jointed rigid pavement models...
An integrated pavement data management and feedback system (PAMS) : evaluation of pavement condition rating procedure.
This report describes an evaluation of a method for use in the Highway Needs Study pavement condition rating. The methods by which the Department generates and manages pavement condition data in the overall process of providing a network of highways ...
Existing pavement input information for the mechanistic-empirical pavement design guide.
The objective of this study is to systematically evaluate the Iowa Department of Transportations (DOTs) existing Pavement Management Information System (PMIS) with respect to the input information required for Mechanistic-Empirical Pavement Des...
Integration and Evaluation of Automated Pavement Distress Data in INDOT's Pavement Management System
This study was in two parts. The first part established and demonstrated a framework for pavement data integration. This is critical for fulfilling QC/QA needs of INDOTs pavement management system, because the precision of the physical location re...
Factors Affecting Pavement Surface and Evaluation Rating Accuracy and Variability
Montgomery, Sharlan R; Haddock, John E
The collection of sufficient, accurate, and consistent pavement condition data is essential to an effective pavement management system. Condition data drive a variety of pavement management tasks such as: • Predicting future pavement performance• Identifying current and future maintenance and rehabilitation needs • Estimating budget needs and requirements • Reporting to decision makers • Selecting appropriate pavement management tools Pavement condition data are represented at either th...
Enhancing pavement performance prediction models for the Illinois Tollway System
Laxmikanth Premkumar; William R. Vavrik
Accurate pavement performance prediction represents an important role in prioritizing future maintenance and rehabilitation needs, and predicting future pavement condition in a pavement management system. The Illinois State Toll Highway Authority (Tollway) with over 2000 lane miles of pavement utilizes the condition rating survey (CRS) methodology to rate pavement performance. Pavement performance models developed in the past for the Illinois Department of Transportation (IDOT) are used by th...
MoDOT pavement preservation research program volume III, development of pavement family and treatment performance models.
Pavement performance models describe the deterioration behavior of pavements. They are essential in a pavement management : system if the goal is to make more objective, reliable, and cost-effective decisions regarding the timing and nature of paveme...
Using Tire/Pavement Interface Noise Results to Define Statistically Similar Bituminous Pavements in Massachusetts
Hencken, John; Haas, Edwin; Bennert, Thomas
In October 2010, the Center for Advanced Infrastructure and Transportation (CAIT) utilized the On-Board Sound Intensity Method in Massachusetts to evaluate seven Open-Graded Friction Courses (OGFC) and eight Asphalt Rubber Gap Graded (ARGG) pavements for MassDOT. Since major concerns regarding pavement selection in the northeast revolve around winter conditions and winter maintenance, if noise level similarities could be found between ARGG and OGFC pavements, pavement engineers would be able ...
Pavement Preservation for Elected Officials: The Inside Story of Pavement Deterioration
Olson, Jim
This session is intended to provide elected officials and practitioners with an introduction to the concept of pavement preservation and its applicability within public agencies. Topics include definitions, budget planning, pavement life, road networks, alternative strategies, pavement distresses, guidelines and preservation treatments.
Papadopoulos, Efthymios
Inverted base pavements involve a well-compacted granular aggregate base built between a thin asphalt concrete layer and a cement-treated base. Inverted base pavements can be constructed using conventional equipment and procedures but require proper quality control. This study reviews the extensive South African experience and case histories in the USA. Accumulating evidence suggests that inverted base pavements are a viable alternative and can outperform conventional pavements at a lower cost. Inverted base pavements rely on the complementary interaction between layers. The cement-treated base provides a stiff foundation for efficient compaction and constrains the deformation of the stress-sensitive granular aggregate base. The thin asphalt surface layer deforms as a membrane and develops low tensile stress. Additional large-scale field tests should be conducted to assess the performance of inverted base pavement designs in a wide range of conditions relevant to the USA.
Long Life Pavements; Firmes de larga duracion
Mateos Moreno, A.; Marron Fernandez, J. O.; Perez Ayuso, J.
The existence of long-life pavements is not new; they have been built in Europe and the United States for decades. In fact, the concept arises from the observation of in-service roads; it was verified how particular pavements, initially designed for a 20-year service life, did not seem to have reduced the bearing capacity along the time, and its maintenance necessities had been exclusively focused on the wearing course. The base idea of long-life pavements is the existence of a fatigue threshold below which the damage produced by each load application is, in practice, zero or below the healing potential of the asphalt mix. The use practice of long-life pavements design considers a pavements constituted by three asphalt layers, each one with a very specific role: a wearing course that provides with the surface characteristics, an intermediate course that provides with most of the structural capacity and a base course that provides with the fatigue resistance. Furthermore, one of the particularities is the design against specific distress mechanisms. Maintenance strategy also presents specific particularities for long-life pavements. It is essentially focused on the detection of surface deterioration, and the appropriate and timely repair, before the damage extends beyond the wearing course, putting into risk the structural integrity of the pavement. Nowadays, this new way to conceive the design, the construction and the maintenance of road pavements, constitutes one of the main challenges for pavement engineering worldwide. (Author) 5 refs.
Evaluating life cycle costs of perpetual pavements in China using operational pavement management system
Saud A. Sultan
Full Text Available Highway transportation is considered as vital factor in China's economic growth; many high grade highways have been constructed in China during the last decades. The research and application of perpetual asphalt pavement (PP technology have been deployed in China since 2000. The semi-rigid pavement has been normally considered as typical pavement of high class highways in the design according to the Chinese experience. The objective of this research is to evaluate the performance of different Chinese perpetual pavements using operational pavement management system and to examine its suitability for use in the design and construction of more economical and durable pavements. It has been found that the use of thin asphalt layers over semi-rigid pavement foundation in PP structure will create more sustainable, economical, and durable PP structures in comparison with typical thick asphalt layers PP structures.
Modelling Flexible Pavement Response and Performance
Ullidtz, Per
This textbook is primarily concerned with models for predicting the future condition of flexible pavements, as a function of traffic loading, climate, materials, etc., using analytical-empirical methods.......This textbook is primarily concerned with models for predicting the future condition of flexible pavements, as a function of traffic loading, climate, materials, etc., using analytical-empirical methods....
Numerical Simulation of Tire-Pavement Interaction
Srirangam, S.K.
Good skid resistance of a pavement surface is essential for road safety. Loss of skid resistance can lead to property damage and loss of lives. Ever increasing need of driver safety poses challenges to the highway authorities to evaluate pavement conditions even more precisely under different
Load Response Analysis of Asphalt Pavement
Xiao Zhijun
Full Text Available Based on the filed data of Fuzhou test road, load response regulation of nonlinear pavement structure was analyzed in this paper; Mechanics software was adopted to analyze linear elastic and nonlinear of the pavement structure, analyzed results and road filed data were compared, and determine the correlation between them and the field data.
Analysis and design optimization of flexible pavement
Mamlouk, M.S.; Zaniewski, J.P.; He, W.
A project-level optimization approach was developed to minimize total pavement cost within an analysis period. Using this approach, the designer is able to select the optimum initial pavement thickness, overlay thickness, and overlay timing. The model in this approach is capable of predicting both pavement performance and condition in terms of roughness, fatigue cracking, and rutting. The developed model combines the American Association of State Highway and Transportation Officials (AASHTO) design procedure and the mechanistic multilayer elastic solution. The Optimization for Pavement Analysis (OPA) computer program was developed using the prescribed approach. The OPA program incorporates the AASHTO equations, the multilayer elastic system ELSYM5 model, and the nonlinear dynamic programming optimization technique. The program is PC-based and can run in either a Windows 3.1 or a Windows 95 environment. Using the OPA program, a typical pavement section was analyzed under different traffic volumes and material properties. The optimum design strategy that produces the minimum total pavement cost in each case was determined. The initial construction cost, overlay cost, highway user cost, and total pavement cost were also calculated. The methodology developed during this research should lead to more cost-effective pavements for agencies adopting the recommended analysis methods.
Pervious Pavement System Evaluation-Paper
The use of a pervious pavement can be effective as a low impact development stormwater control. The Urban Watershed Management Branch is evaluating interlocking concrete paver systems as a type of porous pavement. Although the pavers are impermeable, the spaces between the pave...
On the use of blowup to study regularizations of singularities of piecewise smooth dynamical systems in R^3
Kristiansen, Kristian Uldall; Hogan, S. J.
In this paper we use the blowup method of Dumortier and Roussarie, in the formulation due to Krupa and Szmolyan, to study the regularization of singularities of piecewise smooth dynamical systems in R3. Using the regularization method of Sotomayor and Teixeira, we first demonstrate the power of our...... approach by considering the case of a fold line. We quickly extend a main result of Reves and Seara in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when...... the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided...
Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation
Xiaowei An
Full Text Available We consider the following Cauchy problem: -iut=Δu-V(xu+f(x,|u|2u+(W(x⋆|u|2u, x∈�N,t>0, u(x, 0=u0(x,x∈�N, where V(x and W(x are real-valued potentials and V(x≥0 and W(x is even, f(x,|u|2 is measurable in x and continuous in |u|2, and u0(x is a complex-valued function of x. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.
Improving rutting resistance of pavement structures using geosynthetics: an overview.
Mirzapour Mounes, Sina; Karim, Mohamed Rehan; Khodaii, Ali; Almasi, Mohammad Hadi
A pavement structure consists of several layers for the primary purpose of transmitting and distributing traffic loads to the subgrade. Rutting is one form of pavement distresses that may influence the performance of road pavements. Geosynthetics is one type of synthetic materials utilized for improving the performance of pavements against rutting. Various studies have been conducted on using different geosynthetic materials in pavement structures by different researchers. One of the practices is a reinforcing material in asphalt pavements. This paper intends to present and discuss the discoveries from some of the studies on utilizing geosynthetics in flexible pavements as reinforcement against permanent deformation (rutting).
Improving Rutting Resistance of Pavement Structures Using Geosynthetics: An Overview
Sina Mirzapour Mounes
Full Text Available A pavement structure consists of several layers for the primary purpose of transmitting and distributing traffic loads to the subgrade. Rutting is one form of pavement distresses that may influence the performance of road pavements. Geosynthetics is one type of synthetic materials utilized for improving the performance of pavements against rutting. Various studies have been conducted on using different geosynthetic materials in pavement structures by different researchers. One of the practices is a reinforcing material in asphalt pavements. This paper intends to present and discuss the discoveries from some of the studies on utilizing geosynthetics in flexible pavements as reinforcement against permanent deformation (rutting.
Fatigue and rutting lives in flexible pavement
Ahmed Ebrahim Abu El-Maaty Behiry
Full Text Available Flexible pavement is designed based on axle load limits and climatic conditions. The Egyptian code has specified certain load limits that should not be exceeded. The overweight trucks cause severe deterioration to the pavement and thus reduce its life. The study aims at studying the effect of axle load increase, and the variation in pavement modulus, on the overall pavement life. The research uses the BISAR software and the Egyptian environmental and pavement materials conditions to estimate the tensile strains occurring under the asphalt concrete (AC layer and the compressive strains above the subgrade surface. The results revealed that tensile and compressive strain increased with increasing axle loads and decreased with increasing asphalt layer modulus thus the violating trucks should be unloaded when their weights exceed certain limits. Base thickness and subgrade resilient modulus were the key elements which control the equilibrium between fatigue and rutting lives.
Surface Distresses Detection of Pavement Based on Digital Image Processing
Ouyang , Aiguo; Luo , Chagen; Zhou , Chao
International audience; Pavement crack is the main form of early diseases of pavement. The use of digital photography to record pavement images and subsequent crack detection and classification has undergone continuous improvements over the past decade. Digital image processing has been applied to detect the pavement crack for its advantages of large amount of information and automatic detection. The applications of digital image processing in pavement crack detection, distresses classificati...
Laxmikanth Premkumar
Full Text Available Accurate pavement performance prediction represents an important role in prioritizing future maintenance and rehabilitation needs, and predicting future pavement condition in a pavement management system. The Illinois State Toll Highway Authority (Tollway with over 2000 lane miles of pavement utilizes the condition rating survey (CRS methodology to rate pavement performance. Pavement performance models developed in the past for the Illinois Department of Transportation (IDOT are used by the Tollway to predict the future condition of its network. The model projects future CRS ratings based on pavement type, thickness, traffic, pavement age and current CRS rating. However, with time and inclusion of newer pavement types there was a need to calibrate the existing pavement performance models, as well as, develop models for newer pavement types.This study presents the results of calibrating the existing models, and developing new models for the various pavement types in the Illinois Tollway network. The predicted future condition of the pavements is used in estimating its remaining service life to failure, which is of immediate use in recommending future maintenance and rehabilitation requirements for the network. Keywords: Pavement performance models, Remaining life, Pavement management
Assessment of highway pavements using GPR
Plati, Christina; Loizos, Andreas
Highway infrastructure is a prerequisite for a functioning economy and social life. Highways, often prone to congestion and disruption, are one of the aspects of a modern transport network that require maximum efficiency if an integrated transport network, and sustainable mobility, is to be achieved. Assessing the condition of highway structures, to plan subsequent maintenance, is essential to allow the long-term functioning of a road network. Optimizing the methods used for such assessment will lead to better information being obtained about the road and underlying ground conditions. The condition of highway structures will be affected by a number of factors, including the properties of the highway pavement, the supporting sub-base and the subgrade (natural ground), and the ability to obtain good information about the entire road structure, from pavement to subgrade, allows appropriate maintenance programs to be planned. The maintenance of highway pavements causes considerable cost and in many cases obstruction to traffic flow. In this situation, methods that provide information on the present condition of pavement structure non-destructively and economically are of great interest. It has been shown that Ground-Penetrating-Radar (GPR), which is a Non Destructive Technique (NDT), can deliver information that is useful for the planning of pavement maintenance activities. More specifically GPR is used by pavement engineers in order to determine physical properties and characteristics of the pavement structure, information that is valuable for the assessment of pavement condition. This work gives an overview on the practical application of GPR using examples from highway asphalt pavements monitoring. The presented individual applications of GPR pavement diagnostics concern structure homogeneity, thickness of pavement layers, dielectric properties of asphalt materials etc. It is worthwhile mentioning that a number of applications are standard procedures, either
Global warming potential of pavements
Santero, Nicholas J [Department of Civil and Environmental Engineering, 407 McLaughlin Hall, University of California, Berkeley, CA 94720-1712 (United States); Horvath, Arpad, E-mail: [email protected], E-mail: [email protected] [Department of Civil and Environmental Engineering, 215B McLaughlin Hall, University of California, Berkeley, CA 94720-1712 (United States)
Pavements comprise an essential and vast infrastructure system supporting our transportation network, yet their impact on the environment is largely unquantified. Previous life-cycle assessments have only included a limited number of the applicable life-cycle components in their analysis. This research expands the current view to include eight different components: materials extraction and production, transportation, onsite equipment, traffic delay, carbonation, lighting, albedo, and rolling resistance. Using global warming potential as the environmental indicator, ranges of potential impact for each component are calculated and compared based on the information uncovered in the existing research. The relative impacts between components are found to be orders of magnitude different in some cases. Context-related factors, such as traffic level and location, are also important elements affecting the impacts of a given component. A strategic method for lowering the global warming potential of a pavement is developed based on the concept that environmental performance is improved most effectively by focusing on components with high impact potentials. This system takes advantage of the fact that small changes in high-impact components will have more effect than large changes in low-impact components.
Santero, Nicholas J; Horvath, Arpad
Permeable Pavement Research - Edison, New Jersey
This presentation provides the background and summary of results collected at the permeable pavement parking lot monitored at the EPA facility in Edison, NJ. This parking lot is surfaced with permeable interlocking concrete pavers (PICP), pervious concrete, and porous asphalt. ...
PAVEMENT AS CONTINUITY - THE CASE FROM LISBON
Joana Pimenta
To conclude this practical part of the study a small proposal is presented defining guide lines for an intervention regarding the pavement of the study case, in which its different characteristics are and potential analysed.
Characterization and Modeling of Recycled Pavement Sections
Because of its documented cost and environmental benefits, cold central plant recycling (CCPR) has attracted the interest of local and state departments of transportation. In 2015 two test pavement sections with CCPR as base layers were constructed a...
User's guide : pavement marking management system database.
Pavement markings play a critical role in maintaining a safe and efficient driving environment for road users, especially during nighttime conditions. The Texas Department of Transportation (TxDOT) spends millions of dollars each year for installatio...
Noise measurements of highway pavements in Texas.
This report presents the results of noise testing performed on Texas pavements between May of 2006 and the : summer of 2008. Two field test methodologies were used: roadside noise measurement with SPL meters and onvehicle : sound intensity measuremen...
Performance of I-57 recycled concrete pavements.
In 1986-1987 the Illinois Department of Transportation (IDOT) constructed a demonstration project on I-57 near Effingham, Illinois to evaluate the viability : of recycling an existing jointed reinforced concrete pavement for use as its primary aggreg...
Optimization of safety on pavement preservation projects.
To achieve a goal of reducing highway crash fatalities by 4% each year to improve roadway safety, the Georgia Department of Transportation (GDOT) is actively seeking opportunities to incorporate safety improvements into its current pavement preservat...
MoDOT pavement preservation research program.
The following report documents a research project on pavement preservation performed by the Missouri University of Science and : Technology (Missouri S&T) and the University of Missouri-Columbia (UMC) on behalf of the Missouri Department of Transport...
Life cycle assessment of asphalt pavement maintenance.
This study aims at developing a life cycle assessment (LCA) model to quantify the impact of pavement preservation on energy consumption and greenhouse gas (GHG) emissions. The construction stage contains material, manufacture, transportation and plac...
Pervious Pavement System Evaluation- Abstract 1
On-road Bicycle Pavement Markings
Allegheny County / City of Pittsburgh / Western PA Regional Data Center — A mile by mile breakdown of the on-street bicycle pavement markings installed within the City of Pittsburgh. These include bike lanes, shared lane markings...
Alternative pavement designs : Randolph park and ride.
Previous research on alternative pavement type bidding has proven that various treatments : are unique in terms of constructability, material characteristics, and associated performance. : While some treatments may have higher initial costs, it is im...
Evaluation of concrete pavement patching techniques.
This final report presents the results of a study undertaken to improve in concrete pavement patching techniques. Activities included an evaluation of the suitability of the impact hammer and maturity calculations for determining when a patch is read...
Performance evaluation of precast prestressed concrete pavement.
This report describes in detail an experimental investigation of an innovative precast prestressed concrete pavement (PPCP) system used to rehabilitate a 1,000 ft. section of interstate highway located on the northbound lanes of I-57 near Charleston,...
Class C fly ash in pavements.
Portland cement is the most dominant material used in concrete pavements in the state of Nebraska. In order to improve performance, reduce cost, and advance sustainability, a percentage of the Portland cement is replaced with a recycled material know...
Surface sedimentation at permeable pavement systems
Støvring, Jan; Dam, Torben; Jensen, Marina Bergen
Newly installed permeable pavement (PP) systems provide high surface infiltration capacity, but the accumulation of sediments causes a decrease in capacity over time, eventually leading to surface clogging. With the aim of investigating local sedimentation processes and the importance of restorat......Newly installed permeable pavement (PP) systems provide high surface infiltration capacity, but the accumulation of sediments causes a decrease in capacity over time, eventually leading to surface clogging. With the aim of investigating local sedimentation processes and the importance...
Fusing complementary images for pavement cracking measurements
Yao, Ming; Zhao, Zuyun; Xu, Bugao; Yao, Xun
Cracking is a major pavement distress that jeopardizes road serviceability and traffic safety. Automated pavement distress survey (APDS) systems have been developed using digital imaging technology to replace human surveys for more timely and accurate inspections. Most APDS systems require special lighting devices to illuminate pavements and prevent shadows of roadside objects that distort cracks in the image. Most artificial lighting devices are laser based, and are either hazardous to unprotected people or require dedicated power supplies on the vehicle. This study was aimed to develop a new imaging system that can scan pavement surface at highway speed and determine the level of severity of pavement cracking without using any artificial lighting. The new system consists of dual line-scan cameras that are installed side by side to scan the same pavement area as the vehicle moves. Cameras are controlled with different exposure settings so that both sunlit and shadowed areas can be visible in two separate images. The paired images contain complementary details useful for reconstructing an image in which the shadows are eliminated. This paper intends to present (1) the design of the dual line-scan camera system, (2) a new calibration method for line-scan cameras to rectify and register paired images, (3) a customized image-fusion algorithm that merges the multi-exposure images into one shadow-free image for crack detection, and (4) the results of the field tests on a selected road over a long period. (paper)
Nondestructive testing of pavements and pavement bases. (a bibliography with abstracts). Report for 1964-Jan 78
Nondestructive methods for quality assurance of pavements and pavement bases are investigated in these Government-sponsored research reports. Vibration, nuclear activation, radiometry, and acoustic detection are among the various techniques employed. The updated bibliography contains 114 abstracts, 17 of which are new entries to the previous edition
Thermal stability analysis under embankment with asphalt pavement and cement pavement in permafrost regions.
Junwei, Zhang; Jinping, Li; Xiaojuan, Quan
The permafrost degradation is the fundamental cause generating embankment diseases and pavement diseases in permafrost region while the permafrost degradation is related with temperature. Based on the field monitoring results of ground temperature along G214 Highway in high temperature permafrost regions, both the ground temperatures in superficial layer and the annual average temperatures under the embankment were discussed, respectively, for concrete pavements and asphalt pavements. The maximum depth of temperature field under the embankment for concrete pavements and asphalt pavements was also studied by using the finite element method. The results of numerical analysis indicate that there were remarkable seasonal differences of the ground temperatures in superficial layer between asphalt pavement and concrete pavement. The maximum influencing depth of temperature field under the permafrost embankment for every pavement was under the depth of 8 m. The thawed cores under both embankments have close relation with the maximum thawed depth, the embankment height, and the service time. The effective measurements will be proposed to keep the thermal stabilities of highway embankment by the results.
Flexible pavement rehabilitation design based on pavement service life time span left
Gamelyak, I.; Shevchuk, V.
The design of flexible pavement rehabilitation is analysed in terms of durability-cost. A notion of the remaining service life span is described. the model of rehabilitation strategy selection is presented for both design project and operation stages. the results can be used in the pavement management system.
Evaluation of various hot mix asphalt pavement thicknesses over rubblized concrete pavement.
In 1999, the Wisconsin DOT constructed two test sections as part of a rubblized PCC pavement project on I-39 to evaluate whether : increased HMA thickness could extend the pavement system's service life. Design thicknesses of the control and test sec...
Reconsideration of tyre-pavement input parameters for the structural design of flexible pavements
De Beer, Morris
Full Text Available of modern truck tyres. Tyre-pavement contact stress measurements were carried out on a busy national road (N3) in South Africa from the port of Durban to the inland province of Gauteng. The three-dimensional tyre-pavement contact stress regimes of 45 227...
Evaluating the performance of sustainable perpetual pavements using recycled asphalt pavement in China
Full Text Available The vast highways network in China is moving from the phase of construction to the phase of maintenance, and with the introduction of new technique of perpetual pavement in last decade, it is necessary to consider recycling as one of the promising solutions for rehabilitation of old asphalt concrete pavement and ultimately to convert them into perpetual pavements. The aim is to reuse the existing pavement materials for several reasons, mainly to preserve natural resources such as aggregates, and to satisfy economic requirements by reducing the cost of highway construction and rehabilitation. A detailed testing program has been carried out on recycled asphalt pavements materials (RAP to evaluate their mechanical and structural characteristics to be used for the construction and rehabilitation of road pavements. Different types of RAP mixes have been stabilized by Portland cement to find the most suitable one from the point of view of design, construction, economy and environment. The analysis of life cycle costs has been carried out using system analysis and management of pavement program (SAMP5. The analysis of life cycle costs showed that the use of Portland cements with small percentages improves the structural characteristics of recycled asphalt materials to be used as stabilized base pavement layers for new or rehabilitated old road pavements and also for the construction and rehabilitation of perpetual pavements. A large amount of savings in construction and rehabilitation cost has been achieved by the use of stabilized RAP materials in addition to important contributions to the environment and preserving of natural resources.
Long term pavement performance computed parameter : frost penetration
As the pavement design process moves toward mechanistic-empirical techniques, knowledge of seasonal changes in pavement structural characteristics becomes critical. Specifically, frost penetration information is necessary for determining the effect o...
Investigation clogging dynamic of permeable pavement systems using embedded sensors
Permeable pavement is a stormwater control measure commonly selected in both new and retrofit applications. However, there is limited information about the clogging mechanism of these systems that effects the infiltration. A permeable pavement site located at the Seitz Elementary...
Designing Base and Subbase to Resist Environmental Effects on Pavements
MnDOTs current pavement thickness design procedures do not characterize the effects of subgrade soil frost susceptibility. Previous research indicates frost action is the most severe environmental factor on pavement performance. The most accepted ...
Density measurement verification for hot mix asphalt concrete pavement construction.
Oregon Department of Transportation (ODOT) requires a minimum density for the construction of dense-graded hot mix asphalt concrete (HMAC) pavements to ensure the likelihood that the pavement will not experience distresses that reduce the expected se...
Density measurement verification for hot mixed asphalt concrete pavement construction.
Mobile geographic information system solution for pavement condition surveys [summary].
The State Materials Office (SMO) of the Florida : Department of Transportation (FDOT) performs : annual Pavement Condition Surveys (PCS) of : the Departments extensive pavement network. : This work is performed by single-person crews in : inertial...
Evaluation of recycled concrete as aggregate in new concrete pavements.
This study evaluated the use of recycled concrete as coarse aggregate in new concrete pavements. : Recycled concrete aggregate (RCA) produced from demolished pavements in three geographically dispersed locations in Washington state were used to perfo...
Overview of the South African mechanistic pavement design analysis method
Theyse, HL
Full Text Available A historical overview of the South African mechanistic pavement design method, from its development in the early 1970s to the present, is presented. Material characterization, structural analysis, and pavement life prediction are discussed...
Software offers transparent, straightforward assessment of pavement additives : research spotlight.
Adding new materials to pavement layers is a proven technique to : improve performance. Many types of additivesfrom engineered : polymers and acids to recycled pavement, crumb rubber, shingles : and glasshave been used to help construct better ...
Evaluation of microcracking and chemical deterioration in concrete pavements.
The major objective of this research project was to investigate the : chemistry and morphology of portland cement concrete pavements in : Iowa. The integrity of the various pavements was evaluated qualitatively, : based on the presence or absence of ...
Exploration of Machine Learning Approaches to Predict Pavement Performance
Machine learning (ML) techniques were used to model and predict pavement condition index (PCI) for various pavement types using a variety of input variables. The primary objective of this research was to develop and assess PCI predictive models for t...
A pavement management research program for Oregon highways : final report.
An extensive program was developed to measure pavement deflection skid resistance, and rideability throughout Oregon. The data from those "objective" measures were then evaluated for correlations with observed pavement distress and traffic factors. :...
Advanced Automated Detection Analysis and Classification of Cracks in Pavement
Scott, Dennis
Functional Session 5: Pavement Management Moderated by Akyiaa Hosten This presentation was held at the Pavement Evaluation 2014 Conference, which took place from September 15-18, 2014 in Blacksburg, Virginia. Presentation only
Implementation of a pavement management system in Virginia.
The report summarizes the developments in pavement management in the Virginia Department of Transportation through late 1986. Included are discussions of the pavement management process with examples of priority programming, long-range projection of ...
Variability of Pavement Distress Data From Manual Surveys
Daleiden, Jerome
A pavement management research program for Oregon highways : interim report.
This is the first in a series of reports documenting progress on a statewide pavement management research project. The overall project is conducting research into pavement life cycles of different rehabilitation treatment; the cost-effectiveness of e...
Evaluation of Procedures for Backcalculation of Airfield Pavement Moduli
ER D C/ G SL T R -1 5 -3 1 Evaluation of Procedures for Backcalculation of Airfield Pavement Moduli G eo te ch n ic al a n d S tr u...August 2015 Evaluation of Procedures for Backcalculation of Airfield Pavement Moduli Lucy P. Priddy and Carlos R. Gonzalez Geotechnical and...USAF's) airfield pavement structural evaluation procedures. Determining the structural integrity of airfield pavement relies on the analysis of
New Jersey Micro-Surface Pavement Noise Evaluation
Hencken, John; Haas, Edwin; Tulanowski, Michael; Bennert, Thomas
New Jersey DOT is evaluating pavement preservation types for interstate resurfacing as a method to increase network pavement life cycles within depressed budgetary limits. Despite the economic benefits for micro-surface there is concern for the noise quality and pavement noise levels, which will become a significant issue to a greater population as the application increases over an increased area of lane miles throughout the state. Pavement noise research has been conducted on in-service pave...
Precast concrete pavement - systems and performance review
Novak, Josef; Kohoutková, Alena; Křístek, Vladimír; Vodi�ka, Jan
Long-term traffic restrictions belong to the key disadvantages of conventional cast-in-plane concrete pavements which have been used for technical structures such as roads, parking place and airfield pavements. As a consequence, the pressure is put on the development of such systems which have short construction time, low production costs, long-term durability, low maintenance requirements etc.. The paper presents the first step in the development of an entirely new precast concrete pavement (PCP) system applicable to airfield and highway pavements. The main objective of the review of PCP systems is to acquire a better understanding of the current systems and design methods used for transport infrastructure. There is lack of information on using PCP systems for the construction of entirely new pavements. To most extensive experience is dated back to the 20th century when hexagonal slab panels and system PAG were used in the Soviet Union for the military airfields. Since cast-in-situ pavements became more common, the systems based on precast concrete panels have been mainly utilized for the removal of damaged sections of existing structures including roads, highways etc.. Namely, it concerns Fort Miller Super Slab system, Michigan system, Uretek Stitch system and Kwik system. The presented review indicates several issues associated with the listed PCP systems and their applications to the repair and rehabilitation of existing structures. Among others, the type of manufacturing technology, particularly the position of slots for dowel bars, affects the durability and performance of the systems. Gathered information serve for the development of a new system for airfield and highway pavement construction.
0-6673 : improvement in pavement ride, distress, and condition based on different pavement types : [project summary].
Each district is required to develop a 4-year : pavement management plan assigning project : treatment categories for the Pavement : Management Information System (PMIS). : However, there are different interpretations in : the districts about what tr...
Evaluation of current Louisiana flexible pavement structures using PMS data and new M-E pavement design guide.
The proposed study uses the new MEPDG together with data available from LADOTDs Pavement Management System (PMS) and other data sources from LADOTDs main frame to evaluate typical flexible pavement structures currently used and to make changes ...
Effect of age of permeable pavements on their infiltration function
Boogaard, Floris; Lucke, Terry; Beecham, Simon
This study describes field investigations designed to compare the infiltration capacities of 55 permeable pavement systems installed in the Netherlands and in Australia. The ages of the pavements varied from 1 to 12 years. Using infiltrometer testing, the performance of the pavements has been
Pavement sound absorption measurements in the U.S.
In the U.S., the topic of pavement sound absorption in regard to tire-pavement noise has shown increased interest and research over the last several years. Four types of pavement sound absorption measurements with various applications are discussed: ...
Knowledge Discovery and Pavement Performance : Intelligent Data Mining
Miradi, M.
The main goal of the study was to discover knowledge from data about asphalt road pavement problems to achieve a better understanding of the behavior of them and via this understanding improve pavement quality and enhance its lifespan. Four pavement problems were chosen to be investigated; raveling
Terry Lucke; ir. Floris Boogaard; Simon Beecham
Effectiveness of various public private partnership pavement rehabilitation treatments: A big data informatics survival analysis of pavement service life : final report.
Past research efforts have used a wide variety of methodological approaches to analyze pavement performance indicators, pavement rehabilitation treatments, and pavement service life. Using big data informatics methods, the intent of this study is to ...
VALIDITY OF GARBER MODEL IN PREDICTING PAVEMENT CONDITION INDEX OF FLEXIBLE PAVEMENT IN KERBALA CITY
Hussein A. Ewadh
Full Text Available Pavement Condition Index (PCI is one of the important basics in pavement maintenance management system (PMMS, and it is used to evaluate the current and future pavement condition. This importantance in decision making to limit the maintenance needs, types of treatment, and maintenance priority. The aim of this research is to estimate the PCI value for flexible pavement urban roads in the study area (kerbala city by using Garber et al. developed model. Based on previous researches, data are collected for variables that have a significant impact on pavement condition. Data for pavement age (AGE, average daily traffic (ADT, and structural number (SN were collected for 44 sections in the network roads. A field survey (destructive test (core test and laboratory test (Marshall Test were used to determine the capacity of structure layer of pavement (SN. The condition index (CI output from a developed model was compared with the PCI output of PAVER 6.5.7 by using statistical analysis test. The developed model overestimates value of CI rather than PCI estimated from PAVER 6.5.7 due to statistical test to a 95% degree of confidence, (R = 0.771 for 44 sections (arterial and collector.
Study on road surface source pollution controlled by permeable pavement
Zheng, Chaocheng
The increase of impermeable pavement in urban construction not only increases the runoff of the pavement, but also produces a large number of Non-Point Source Pollution. In the process of controlling road surface runoff by permeable pavement, a large number of particulate matter will be withheld when rainwater is being infiltrated, so as to control the source pollution at the source. In this experiment, we determined the effect of permeable road surface to remove heavy pollutants in the laboratory and discussed the related factors that affect the non-point pollution of permeable pavement, so as to provide a theoretical basis for the application of permeable pavement.
Optical microtopographic inspection of asphalt pavement surfaces
Costa, Manuel F. M.; Freitas, E. F.; Torres, H.; Cerezo, V.
Microtopographic and rugometric characterization of surfaces is routinely and effectively performed non-invasively by a number of different optical methods. Rough surfaces are also inspected using optical profilometers and microtopographer. The characterization of road asphalt pavement surfaces produced in different ways and compositions is fundamental for economical and safety reasons. Having complex structures, including topographically with different ranges of form error and roughness, the inspection of asphalt pavement surfaces is difficult to perform non-invasively. In this communication we will report on the optical non-contact rugometric characterization of the surface of different types of road pavements performed at the Microtopography Laboratory of the Physics Department of the University of Minho.
Quantitative analysis of microtubule orientation in interdigitated leaf pavement cells.
Akita, Kae; Higaki, Takumi; Kutsuna, Natsumaro; Hasezawa, Seiichiro
Leaf pavement cells are shaped like a jigsaw puzzle in most dicotyledon species. Molecular genetic studies have identified several genes required for pavement cells morphogenesis and proposed that microtubules play crucial roles in the interdigitation of pavement cells. In this study, we performed quantitative analysis of cortical microtubule orientation in leaf pavement cells in Arabidopsis thaliana. We captured confocal images of cortical microtubules in cotyledon leaf epidermis expressing GFP-tubulinβ and quantitatively evaluated the microtubule orientations relative to the pavement cell growth axis using original image processing techniques. Our results showed that microtubules kept parallel orientations to the growth axis during pavement cell growth. In addition, we showed that immersion treatment of seed cotyledons in solutions containing tubulin polymerization and depolymerization inhibitors decreased pavement cell complexity. Treatment with oryzalin and colchicine inhibited the symmetric division of guard mother cells.
"Full-Scale Testing of Pavement Response"
Ullidtz, Per; Ekdahl, Peter
Three pavement sections in southern Sweden were instrumented in late 1991. Each section have instruments for measuring the strain at the bottom of the asphalt layer, vertical stress and strains in the subgrade and temperatures at different depths. The purpose was to evaluate different theoretical...... methods for determining stresses, strains and deflections in pavement structurers.Falling Weight Deflectometer (FWD) test were done on the test sections, and stresses and strains were measured both under FWD loading and under a rolling wheel load. Different back-analysis procedures were used to derive...
The Inhabitants of the Historical Pavement
and boulevards to kiosks and cigarette packs.This particular excerpt presents a framed view of a quasi-historical pavement in downtown Chicago and the coming and going of people. Some are passing by without paying much attention, for others it is their daily workplace and for some an exotic starting point...... for yet unknown adventures. The pavement and collective history will outlast all of them – but while they are here, they set each other in motion and even the most trivial elements of the street become part of this choreography.Throughout the film, the reciprocal relationship between the individual...
Interim revision of the South African Mechanistic-Empirical pavement design method for flexible pavements
Full Text Available Pavement design methods, in combination with network level management systems must enable road authorities to develop reliable long-term financial plans based on the estimated structural capacity of the road network. Inaccurate design models...
Tyre-pavement interface contact stresses on flexible pavements - quo vadis?
Full Text Available on tyre inflation pressure on this route is also given. The paper also discusses an improved definition of the actual tyre/pavement interaction of modern heavy vehicles and their loads on the current flexible road infrastructure....
Performance studies of continuously reinforced concrete pavement : pavements without transverse steel.
The status of studies of the three continuously reinforced pavement projects near Charlottesville is reported. Of particular interest in these studies are the effects of the elimination of transverse reinforcing steel, the changing characteristics of...
Development of index based pavement performance models for pavement management system (PMS) of LADOTD : tech summary.
A research study was initiated by the Louisiana Department of Transportation and Development (LADOTD) in conjunction with the : Federal Highway Administration (FHWA) to evaluate the overall performance and eff ectiveness of LADOTDs Pavement Manage...
Development of index based pavement performance models for pavement management system (PMS) of LADOTD.
This report focuses on pavement performance and treatment models for Louisiana Department of : Transportation and Development (LADOTD) and is in continuation of Louisiana Transportation : Research Center (LTRC) Report No. 430 Development of Unifor...
Development of Pavement Condition Index Model Using PAVER 6.5.7 for Flexible Pavement Urban Roads in Kerbala City
Hussein Ali Ewadh
Full Text Available The pavement condition index (PCI is an important factor indicating pavement condition of roads. The pavement is one of the basic parts of the road infrastructure. Accordingly the aim of this paper is to develop a pavement condition index model using PAVER 6.5.7 for a flexible pavement urban roads in the Kerbala city center. The development model is depending on the PCI value of 109 sample sections having 57.8 km length, Data collected for pavement distress (type, dimension, and severity were used to find PCI. The filtering and outlier analysis was done by using a polynomial (fourth-degree constrained least squared for statistical process. These processes were done by (PAVER 6.5.7 software. The result of the prediction model of PCI shows that it is valid to be used in the prediction of the condition of pavement for the same family type
Development of New Bituminous Pavement Design Method
The report and work of COST Action 333 sets in place the foundation for a coherent, cost-effective and harmonised European pavement design method. In order to do this, the work programme focused on information gathering, identification of requirements and the selection of the necessary design...
Viscoelastic Pavement Modeling with a Spreadsheet
Levenberg, Eyal
The aim herein was to equip civil engineers and students with an advanced pavement modeling tool that is both easy to use and highly adaptive. To achieve this, a mathematical solution for a layered viscoelastic half-space subjected to a moving load was developed and subsequently implemented...
The Reflective Cracking in Flexible Pavements
Pais Jorge
Full Text Available Reflective cracking is a major concern for engineers facing the problem of road maintenance and rehabilitation. The problem appears due to the presence of cracks in the old pavement layers that propagate into the pavement overlay layer when traffic load passes over the cracks and due to the temperature variation. The stress concentration in the overlay just above the existing cracks is responsible for the appearance and crack propagation throughout the overlay. The analysis of the reflective cracking phenomenon is usually made by numerical modeling simulating the presence of cracks in the existing pavement and the stress concentration in the crack tip is assessed to predict either the cracking propagation rate or the expected fatigue life of the overlay. Numerical modeling to study reflective cracking is made by simulating one crack in the existing pavement and the loading is usually applied considering the shear mode of crack opening. Sometimes the simulation considers the mode I of crack opening, mainly when temperature effects are predominant.
Relationship between road pavement failures, engineering indices ...
This article presents the results of a study carried out to relate the frequency of pavement failures, the engineering indices of the subgrade materials and the underlying geology. The results show a high variability in the indices such as the liquid limit, LL, the plasticity index, PI, the maximum dry density, MDD, the optimum ...
Pavement evaluation using a portable lightweight deflectometer.
The use of Zorn ZFG-3000 portable Lightweight Deflectometer (LWD) in the in-situ : assessment of pavement quality was investigated in this research. A lower load and a : shorter load pulse duration are used in a LWD as compared to a Falling Weight : ...
Lessons learned : pavement marking warranty contract.
In 2012, UDOT implemented a performance-based warranty on a portion of an I-15 pavement marking : project. The awarded contract requested a contractor warranty on the implemented markings for a total : duration of six years. This is the first time th...
Texas perpetual pavements : new design guidelines.
Since 2001, the State of Texas has been designing and constructing perpetual pavements : on some of its heavily trafficked highways where the expected 20-year truck-traffic estimate of : 18 kip ESALs is in excess of 30 million (TxDOT, 2001). To date,...
Cracking of open traffic rigid pavement
Niken Chatarina
Full Text Available The research is done by observing the growth of real structure cracking in Natar, Lampung, Indonesia compared to C. Niken's et al research and literature study. The rigid pavement was done with open traffic system. There are two main crack types on Natar rigid pavement: cracks cross the road, and cracks spreads on rigid pavement surface. The observation of cracks was analyzed by analyzing material, casting, curing, loading and shrinkage mechanism. The relationship between these analysis and shrinkage mechanism was studied in concrete micro structure. Open traffic make hydration process occur under vibration; therefore, fresh concrete was compressed and tensioned alternately since beginning. High temperature together with compression, cement dissociation, the growth of Ca2+ at very early age leads abnormal swelling. No prevention from outside water movement leads hydration process occur with limited water which caused spreads fine cracks. Limited water improves shrinkage and plastic phase becomes shorter; therefore, rigid pavement can't accommodate the abnormal swelling and shrinking alternately and creates the spread of cracks. Discontinuing casting the concrete makes both mix under different condition, the first is shrink and the second is swell and creates weak line on the border; so, the cracks appear as cracks across the road.
Pavement management system for City of Madison.
This project aims to implement a pavement management system (PMS) for the City of Madison using : four specific objectives: 1) build a city-wide GIS database for PMS compatible and incorporable with the : citys GIS system; 2) identify feasible pav...
Effectiveness of weed control methods on pavement
Vermeulen, G.D.; Verwijs, B.R.; Kempenaar, C.
The policy in the Netherlands is to signifiantly reduce the use of herbicides, also on pavements. Existing non-chemical methods to control weeds are much less effective than spot spraying, the usual method at this moment. Therefore, the cost of non-chemical weed management is often estimated to be 4
Permanent deformation and deflection relationship from pavement condition assessment
Fabricio Leiva-Villacorta
Full Text Available The development of permanent deformation in flexible pavements has been a research topic for several decades. Currently there are models included in the structural design of pavements that can predict this type of failure. However, the variables required for the prediction of this distress are complex or difficult to obtain in the field, making its application in pavement evaluation also difficult. Measurement of the deflection of pavement structures by means of non-destructive testing is a technique used to assess the condition of the pavement. This research study seeks to correlate data from deflections of the pavement surface with probable permanent deformation in time. In addition, prediction of the remaining life of the pavement structure using a specified criterion is also analyzed. In order to accomplish these objectives, data acquired from 4 different full scale accelerated pavement test tracks was used to develop a permanent deformation model as a function of deflection, load repetitions and pavement layer thickness. The developed model considered a time series model that incorporates an Auto-regressive parameter of order 1. The proposed model presents an advantage over currently available models because it reduces the required parameters to predict the permanent deformation and/or remaining life in the structure and because these variables can be easily found and updated in a pavement management system. Keywords: HVS, Permanent deformation, Deflections, APT, Time series, Instrumentation
Developing a network-level structural capacity index for structural evaluation of pavements.
The objective of this project was to develop a structural index for use in network-level pavement evaluation to facilitate : the inclusion of the pavements structural condition in pavement management applications. The primary goal of network-level...
On-board sound intensity tire-pavement noise study in North Carolina.
"This research investigated tire-pavement noise on various types of pavements across North Carolina by using On- : Board Sound Intensity (OBSI) method. To mitigate traffic noise, quieter pavement may provide advantages that : noise barriers cannot. T...
Determination of coefficient of thermal expansion effects on Louisiana's PCC pavement design.
With the development of the Mechanistic Empirical Pavement Design Guide (MEPDG) as a new pavement design tool, the : coefficient of thermal expansion (CTE) is now considered a more important design parameter in estimating pavement : performance inclu...
Determination of coefficient of thermal expansion for Portland Cement Concrete pavements for MEPDG Implementation
The Coefficient of Thermal Expansion (CTE) is an important parameter in Portland Cement Concrete (PCC) pavement analysis and design as it is directly proportional to the magnitude of temperature-related pavement deformations throughout the pavement s...
Integration of pavement cracking prediction model with asset management and vehicle-infrastructure interaction models.
Not long after the construction of a pavement or a new pavement surface, various : forms of deterioration begin to accumulate due to the harsh effects of traffic loading : combined with weathering action. In a recent NEXTRANS project, a pavement crac...
Predicting pavement condition index using international roughness index in Washington DC.
A number of pavement condition indices are used to conduct pavement management assessments, two of which are the : International Roughness Index (IRI) and Pavement Condition Index (PCI). The IRI is typically measured using specialized : equipment tha...
A link of full-scale accelerated pavement testing to long-term pavement performance study in the Western Cape Province of South Africa
Anochie-Boateng, Joseph K
Full Text Available of Accelerated Pavement Testing in Pavement Sustainability A Link of Full-Scale Accelerated Pavement Testing to Long-Term Pavement Performance Study in the Western Cape Province of South Africa J. K. Anochie-Boateng W. JvdM Steyn C. Fisher L. Truter...
Molecular mechanisms controlling pavement cell shape in Arabidopsis leaves.
Qian, Pingping; Hou, Suiwen; Guo, Guangqin
Pavement cells have an interlocking jigsaw puzzle-shaped leaf surface pattern. Twenty-three genes involved in the pavement cell morphogenesis were discovered until now. The mutations of these genes through various means lead to pavement cell shape defects, such as loss or lack of interdigitation, the reduction of lobing, gaps between lobe and neck regions in pavement cells, and distorted trichomes. These phenotypes are affected by the organization of microtubules and microfilaments. Microtubule bands are considered corresponding with the neck regions of the cell, while lobe formation depends on patches of microfilaments. The pathway of Rho of plant (ROP) GTPase signaling cascades regulates overall activity of the cytoskeleton in pavement cells. Some other proteins, in addition to the ROPs, SCAR/WAVE, and ARP2/3 complexes, are also involved in the pavement cell morphogenesis.
Assessing Asphalt and Concrete Pavement Surface Texture in the Field
Saad I. Sarsam; Huda N. Al Shareef
The incorporation of safety characteristics into the traditional pavement structural design or in the functional evaluation of pavement condition has not been established yet. The design has focused on the structural capacity of the roadway so that the pavement can withstand specific level of repetitive loading over the design life. On the other hand, the surface texture condition was neither included in the AASHTO design procedure nor in the present serviceability index measurements. The ...
Crushing damage estimation for pavement with lightly cementitious bases
Full Text Available . Department of Transport. Pretoria, South Africa. Litwinowicz and De Beer., 2013. Long term crushing performance of lightly cementitious pavement materials – update to the South African procedures. Road Materials and Pavement Design. Maina, J.W., De... Beer, M. and van Rensburg, Y., 2013. Modelling Tyre-Road Contact Stresses in Pavement Design and Analysis. Proceedings of 32nd Southern Africa transport conference, Pretoria, July 2013. pp 336-345: SAPDM, (2014). http://www.sapdm.co.za/, last...
Dynamic Modeling of Pavements with Application to Deflection Measurements
Madsen, Stine Skov
Pavement surface deflection measurements are the primary means of evaluating the bearing capacity of a pavement. The most common type of device used for measuring pavement surface deflections is the Falling Weight Deflectometer (FWD). However, increasing attention has been given to the Rolling Wheel...... Deflectometer (RWD) type of device due to its ability to measure deflections continuously while driving at traffic speed. To be able to properly interpret deflection measurements from an RWD device, more knowledge about the structural behavior of a pavement when subjected to transient dynamic loads moving...
Hybrid green permeable pave with hexagonal modular pavement systems
Rashid, M A; Abustan, I; Hamzah, M O
Modular permeable pavements are alternatives to the traditional impervious asphalt and concrete pavements. Pervious pore spaces in the surface allow for water to infiltrate into the pavement during rainfall events. As of their ability to allow water to quickly infiltrate through the surface, modular permeable pavements allow for reductions in runoff quantity and peak runoff rates. Even in areas where the underlying soil is not ideal for modular permeable pavements, the installation of under drains has still been shown to reflect these reductions. Modular permeable pavements have been regarded as an effective tool in helping with stormwater control. It also affects the water quality of stormwater runoff. Places using modular permeable pavement has been shown to cause a significant decrease in several heavy metal concentrations as well as suspended solids. Removal rates are dependent upon the material used for the pavers and sub-base material, as well as the surface void space. Most heavy metals are captured in the top layers of the void space fill media. Permeable pavements are now considered an effective BMP for reducing stormwater runoff volume and peak flow. This study examines the extent to which such combined pavement systems are capable of handling load from the vehicles. Experimental investigation were undertaken to quantify the compressive characteristics of the modular. Results shows impressive results of achieving high safety factor for daily life vehicles.
Study on Flexible Pavement Failures in Soft Soil Tropical Regions
Jayakumar, M.; Chee Soon, Lee
Road network system experienced rapid upgrowth since ages ago and it started developing in Malaysia during the colonization of British due to its significant impacts in transportation field. Flexible pavement, the major road network in Malaysia, has been deteriorating by various types of distresses which cause descending serviceability of the pavement structure. This paper discusses the pavement condition assessment carried out in Sarawak and Sabah, Malaysia to have design solutions for flexible pavement failures. Field tests were conducted to examine the subgrade strength of existing roads in Sarawak at various failure locations, to assess the impact of subgrade strength on pavement failures. Research outcomes from field condition assessment and subgrade testing showed that the critical causes of pavement failures are inadequate design and maintenance of drainage system and shoulder cross fall, along with inadequate pavement thickness provided by may be assuming the conservative value of soil strength at optimum moisture content, whereas the exiting and expected subgrade strengths at equilibrium moisture content are far below. Our further research shows that stabilized existing recycled asphalt and base materials to use as a sub-base along with bitumen stabilized open graded base in the pavement composition may be a viable solution for pavement failures.
Crack identification for rigid pavements using unmanned aerial vehicles
Bahaddin Ersoz, Ahmet; Pekcan, Onur; Teke, Turker
Pavement condition assessment is an essential piece of modern pavement management systems as rehabilitation strategies are planned based upon its outcomes. For proper evaluation of existing pavements, they must be continuously and effectively monitored using practical means. Conventionally, truck-based pavement monitoring systems have been in-use in assessing the remaining life of in-service pavements. Although such systems produce accurate results, their use can be expensive and data processing can be time consuming, which make them infeasible considering the demand for quick pavement evaluation. To overcome such problems, Unmanned Aerial Vehicles (UAVs) can be used as an alternative as they are relatively cheaper and easier-to-use. In this study, we propose a UAV based pavement crack identification system for monitoring rigid pavements' existing conditions. The system consists of recently introduced image processing algorithms used together with conventional machine learning techniques, both of which are used to perform detection of cracks on rigid pavements' surface and their classification. Through image processing, the distinct features of labelled crack bodies are first obtained from the UAV based images and then used for training of a Support Vector Machine (SVM) model. The performance of the developed SVM model was assessed with a field study performed along a rigid pavement exposed to low traffic and serious temperature changes. Available cracks were classified using the UAV based system and obtained results indicate it ensures a good alternative solution for pavement monitoring applications.
Incorporating moving dynamic tyre loads in pavement design and analysis
Full Text Available at the University of Pretoria. 4 REAL LIFE TYRE LOADS Characterisation Pavement loading has been shown by various authors to be a dynamic (time-dependent) phenomenon (Divine, 1997; Cebon, 1999). A pavement experiences a vehicle as a moving, time-varying set... frequencies. Body bounce generally dominates the dynamic loading, and is mainly caused by the response of the sprung mass of the vehicle to the pavement roughness. Axle hop becomes more significant at higher vehicle speeds and higher pavement roughnesses...
Proactive pavement asset management with climate change aspects
Zofka, Adam
Pavement Asset Management System is a systematic and objective tool to manage pavement network based on the rational, engineering and economic principles. Once implemented and mature Pavement Asset Management System serves the entire range of users starting with the maintenance engineers and ending with the decision-makers. Such a system is necessary to coordinate agency management strategy including proactive maintenance. Basic inputs in the majority of existing Pavement Asset Management System approaches comprise the actual pavement inventory with associated construction history and condition, traffic information as well as various economical parameters. Some Pavement Management System approaches include also weather aspects which is of particular importance considering ongoing climate changes. This paper presents challenges in implementing the Pavement Asset Management System for those National Road Administrations that manage their pavement assets using more traditional strategies, e.g. worse-first approach. Special considerations are given to weather-related inputs and associated analysis to demonstrate the effects of climate change in a short- and long-term range. Based on the presented examples this paper concludes that National Road Administrations should account for the weather-related factors in their Pavement Management Systems as this has a significant impact on the system outcomes from the safety and economical perspective.
Eroding dipoles and vorticity growth for Euler flows in {{{R}}}^{3}: the hairpin geometry as a model for finite-time blowup
Childress, Stephen; Gilbert, Andrew D.
A theory of an eroding 'hairpin' vortex dipole structure in three-dimensions is developed, extending our previous study of an axisymmetric eroding dipole without swirl. The axisymmetric toroidal dipole was found to lead to maximal growth of vorticity, as {t}4/3. The hairpin is here similarly proposed as a model to produce large 'self-stretching' of vorticity, with the possibility of finite-time blow-up. We derive a system of partial differential equations of 'generalized' form, involving contour averaging of a locally two-dimensional Euler flow. We do not attempt here to solve the system exactly, but point out that non-existence of physically acceptable solutions would most probably be a result of the axial flow. Because of the axial flow the vorticity distribution within the dipole eddies is no longer of the simple Sadovskii type (vorticity constant over a cross-section) obtained in the axisymmetric problem. Thus the solution of the system depends upon the existence of a larger class of propagating two-dimensional dipoles. The hairpin model is obtained by formal asymptotic analysis. As in the axisymmetric problem a local transformation to 'shrinking' coordinates is introduced, but now in a self-similar form appropriate to the study of a possible finite-time singularity. We discuss some properties of the model, including a study of the helicity and a first step in iterating toward a solution from the Sadovskii structure. We also present examples of two-dimensional propagating dipoles not previously studied, which have a vorticity profile consistent with our model. Although no rigorous results can be given, and analysis of the system is only partial, the formal calculations are consistent with the possibility of a finite time blowup of vorticity at a point of vanishing circulation of the dipole eddies, but depending upon the existence of the necessary two-dimensional propagating dipole. Our results also suggest that conservation of kinetic energy as
Microorganism Removal in Permeable Pavement Parking Lots ...
Three types of permeable pavements (pervious concrete, permeable interlocking concrete pavers, and porous asphalt) were monitored at the Edison Environmental Center in Edison, New Jersey for indicator organisms such as fecal coliform, enterococci, and E. coli. Results showed that porous asphalt had much lower concentration in monitored infiltrate compared to pervious concrete and permeable interlocking concrete pavers. Concentrations of monitored organisms in infiltrate from porous asphalt were consistently below the bathing water quality standard. Fecal coliform and enterococci exceeded bathing water quality standards more than 72% and 34% of the time for permeable interlocking concrete pavers and pervious concrete, respectively. Purpose is to evaluate the performance of permeable pavement in removing indicator organisms from infiltrating stormwater runoff.
Pavement Management Systems on a Local Level
Vasquez, Cristian A.
Roads are one of the most valuable infrastructure assets within a community. Proper maintenance of the roadway network can promote the prosperity of a community. In recent times maintenance has become much more difficult with the price of asphalt increasing drastically and city budgets contracting due to the economic downturn. With these conditions, the proper management of an agency's street network is necessary. The use of pavement management systems provides the help needed in the preserva...
Pavement cells and the topology puzzle.
Carter, Ross; Sánchez-Corrales, Yara E; Hartley, Matthew; Grieneisen, Verônica A; Marée, Athanasius F M
D'Arcy Thompson emphasised the importance of surface tension as a potential driving force in establishing cell shape and topology within tissues. Leaf epidermal pavement cells grow into jigsaw-piece shapes, highly deviating from such classical forms. We investigate the topology of developing Arabidopsis leaves composed solely of pavement cells. Image analysis of around 50,000 cells reveals a clear and unique topological signature, deviating from previously studied epidermal tissues. This topological distribution is established early during leaf development, already before the typical pavement cell shapes emerge, with topological homeostasis maintained throughout growth and unaltered between division and maturation zones. Simulating graph models, we identify a heuristic cellular division rule that reproduces the observed topology. Our parsimonious model predicts how and when cells effectively place their division plane with respect to their neighbours. We verify the predicted dynamics through in vivo tracking of 800 mitotic events, and conclude that the distinct topology is not a direct consequence of the jigsaw piece-like shape of the cells, but rather owes itself to a strongly life history-driven process, with limited impact from cell-surface mechanics. © 2017. Published by The Company of Biologists Ltd.
An analysis of pavement heat flux to optimize the water efficiency of a pavement-watering method
Hendel, Martin; Colombert, Morgane; Diab, Youssef; Royon, Laurent
Pavement-watering as a technique of cooling dense urban areas and reducing the urban heat island effect has been studied since the 1990's. The method is currently considered as a potential tool for and climate change adaptation against increasing heat wave intensity and frequency. However, although water consumption necessary to implement this technique is an important aspect for decision makers, optimization of possible watering methods has only rarely been conducted. An analysis of pavement heat flux at a depth of 5 cm and solar irradiance measurements is proposed to attempt to optimize the watering period, cycle frequency and water consumption rate of a pavement-watering method applied in Paris over the summer of 2013. While fine-tuning of the frequency can be conducted on the basis of pavement heat flux observations, the watering rate requires a heat transfer analysis based on a relation established between pavement heat flux and solar irradiance during pavement insolation. From this, it was found that watering conducted during pavement insolation could be optimized to 30-min cycles and water consumption could be reduced by more than 80% while reducing the cooling effect by less than 13%. - Highlights: • The thermal effects of pavement-watering were investigated in Paris, France. • Pavement-watering was found to significantly affect pavement heat flux 5 cm deep. • When insolated, a linear relation was found between heat flux and solar radiation. • Pavement-watering did not alter its slope, but introduced a negative intercept. • Subsequent improvements of the watering period, frequency and rate are proposed
Modeling the Hydrologic Processes of a Permeable Pavement System
A permeable pavement system can capture stormwater to reduce runoff volume and flow rate, improve onsite groundwater recharge, and enhance pollutant controls within the site. A new unit process model for evaluating the hydrologic performance of a permeable pavement system has be...
Permeable Pavement Research at the Edison Environmental Center
There are few detailed studies of full-scale, replicated, actively-used permeable pavement systems. Practitioners need additional studies of permeable pavement systems in its intended application (parking lot, roadway, etc.) across a range of climatic events, daily usage conditio...
Nutrient Infiltrate Concentrations from Three Permeable Pavement Types
While permeable pavement is increasingly being used to control stormwater runoff, field-based, side-by-side investigations on the effects different pavement types have on nutrient concentrations present in stormwater runoff are limited. In 2009, the U.S. EPA constructed a 0.4-ha...
Full scale demonstration of air-purifying pavement
Ballari, M.; Brouwers, H.J.H.
Experiments concerning a full-scale demonstration of air purifying pavement in Hengelo, The Netherlands, are reported. The full width of the street was provided with concrete pavement containing TiO2 over a length of 150 m ("DeNOx street"). Another part of the street, about 100 m, was paved with
Effects of street tree shade on asphalt concrete pavement performance
E.G. McPherson; J. Muchnick
Forty-eight street segments were paired into 24 high-and low-shade pairs in Modesto, California, U.S. Field data were collected to calculate a Pavement Condition Index (PCI) and Tree Shade Index (TSI) for each segment. Statistical analyses found that greater PCI was associated with greater TSI, indicating that tree shade was partially responsible for reduced pavement...
3D Tyre/Road pavement contact stress measurements
Full Text Available The CSIR's proprietary Stress-in-Motion (SIM) measurements provide rational descriptions of 1D, 2D and 3D tyre/road pavement stresses for: Road pavement design testing and evaluation, as well as tyre design, testing and evaluation....
Wet Weather Crater Repair Technologies for Grooved and Smooth Pavements
Dean Geotechnical and Structures Laboratory U.S. Army Engineer Research and Development Center 3909 Halls Ferry Road Vicksburg, MS 39180-6199...ORGANIZATION REPORT NUMBER U.S. Army Engineer Research and Development Center Geotechnical and Structures Laboratory 3909 Halls Ferry Road ...SUBJECT TERMS Crater Concrete Rain and rainfall ADR Grooved pavement Smooth pavement Runoff Runways (Aeronautics) – Maintenance and repair
Soil Properties And Pavement Performance In The Nigerian ...
The geotechnical properties of the sub-structure of the Shagamu-Ore-Benin pavement were studied in an attempt to identifying the quality of construction materials and the pavement performance. The sub-structure soil materials were collected from the field and were taken to the laboratory for particle size analysis, Atterberg ...
Accelerated load testing of geosynthetic base reinforced pavement test sections.
The main objective of this research is to evaluate the benefits of geosynthetic stabilization and reinforcement of subgrade/base aggregate layers in flexible pavements built on weak subgrades and the effect of pre-rut pavement sections, prior to the ...
Effect of training algorithms on neural networks aided pavement ...
Especially, the use of Finite Element (FE) based pavement modeling results for training the NN aided inverse analysis is considered to be accurate in realistically characterizing the non-linear stress-sensitive response of underlying pavement layers in real-time. Efficient NN learning algorithms have been developed and ...
Long-term Metal Performance of Three Permeable Pavements
EPA constructed a 4,000-m2 parking lot surfaced with three permeable pavements (permeable interlocking concrete pavers, pervious concrete, and porous asphalt) on the Edison Environmental Center in Edison, NJ in 2009. Samples from each permeable pavement infiltrate were collected...
Saad I. Sarsam
Full Text Available The incorporation of safety characteristics into the traditional pavement structural design or in the functional evaluation of pavement condition has not been established yet. The design has focused on the structural capacity of the roadway so that the pavement can withstand specific level of repetitive loading over the design life. On the other hand, the surface texture condition was neither included in the AASHTO design procedure nor in the present serviceability index measurements. The pavement surface course should provide adequate levels of friction and ride quality and maintain low levels of noise and roughness. Many transportation departments perform routine skid resistant testing, the type of equipment used for testing varies depending on the preference of each transportation department. It was felt that modeling of the surface texture condition using different methods of testing may assist in solving such problem. In this work, Macro texture and Micro texture of asphalt and cement concrete pavement surface have been investigated in the field using four different methods (The Sand Patch Method, Outflow Time Method, British Pendulum Tester and Photogrammetry Technique. Two different grain sizes of sand have been utilized in conducting the Sand Patch while the Micro texture was investigated using the British Pendulum tester method at wet pavement surface conditions. The test results of the four methods were correlated to the skid number. It was concluded that such modeling could provide instant data in the field for pavement condition which may help in pavement maintenance management.
Highway pavement failure induced by poor soil geotechnical ...
X-ray diffraction studies showed the presence of abundant kaolinite peaks and a subdued goethite peak without any trace of montmorrilonite. The presence of excess fines in the pavement construction materials (soils) contributed to the failure of the highway pavement at this locality. The low CBR value is also a noted cause ...
23 CFR 973.208 - Indian lands pavement management system (PMS).
... 23 Highways 1 2010-04-01 2010-04-01 false Indian lands pavement management system (PMS). 973.208... PROGRAM Bureau of Indian Affairs Management Systems § 973.208 Indian lands pavement management system (PMS... concepts described in the AASHTO's "Pavement Management Guide.� 1 1 "Pavement Management Guide,� AASHTO...
Influence of Pavement on Fatigue Performance of Urban Steel Box Girder Deck
Zheng Zhongyue
Full Text Available Based on spatial finite element analysis method, the Influence of pavement on fatigue performance of orthotropic steel deck was analyzed in terms of pavement system, asphalt pavement stiffness. The result shows that compared with asphalt pavement system, RPC pavement system can not only obviously improve the stress condition of steel bridge deck, but also significantly extend the fatigue life of steel bridge panel; Increasing the stiffness of pavement layer can obviously reduce the stress amplitude of fatigue details, especially for direct contact with the pavement.
The Simulation Realization of Pavement Roughness in the Time Domain
XU, H. L.; He, L.; An, D.
As the needs for the dynamic study on the vehicle-pavement system and the simulated vibration table test, how to simulate the pavement roughness actually is important guarantee for whether calculation and test can reflect the actual situation or not. Using the power spectral density function, the simulation of pavement roughness can be realized by Fourier inverse transform. The main idea of this method was that the spectrum amplitude and random phase were obtained separately according to the power spectrum, and then the simulation of pavement roughness was obtained in the time domain through the Fourier inverse transform (IFFT). In the process, the sampling interval (Δl) was 0.1m, and the sampling points(N) was 4096, which satisfied the accuracy requirements. Using this method, the simulate results of pavement roughness (A~H grades) were obtain in the time domain.
Importance of dowels in transversal joints in concrete pavements
Grosek, Jiri; Chupik, Vladimir; Stryk, Josef; Brezina, Ilja
Concrete pavements are designed for heavy loaded road structures. Their usage brings a number of specific issues. It is necessary to solve them all to ensure that concrete pavements will fulfil their function along the whole design period. One of these issues concerns dowels, which are located in transversal joints. Modelling of load, caused by heavy vehicles, with the use of the finite element method, provides valuable information about the stress condition of concrete pavement. The results of modelling can be verified by measurements or experiments in practice. Dowels and tie bars in jointed unreinforced concrete pavements and the importance of their correct placement, dimensions and material quality on pavement behaviour and lifespan were studied as a part of R&D projects of Technology Agency of the Czech Republic Nos. TA02031195 and TE01020168. The paper presents the experience from the modelling and performed experiments and makes conclusions which are important for the use in practice.
Utilization of Advanced Diagnostic Methods for Texture and Rut Depth Analysis on a Testing Pavement Section
Slabej Martin
Full Text Available Qualitative characteristics of pavement in wide range reflects the pavement serviceability, which is a summary of the characteristics of the pavement, providing a fast, smooth, economical and especially safe driving of motor-vehicles. The target factor of pavement serviceability and safety of roads represents the quality of their surface properties. In the framework of research activities performed in the Research Centre founded under the auspices of University of Žilina, individual parameters of pavement serviceability were monitored by pavement surface scanning. This paper describes the creation of a 3D - road surface model and its analysis and evaluation from the viewpoint of two pavement serviceability parameters - the rut depth and texture. Measurements were performed on an experimental pavement section used contemporary in an Accelerated Pavement Testing experiment. The long-term goal is to ascertain functions predicting degradation of these two pavement serviceability parameters.
Photocatalytic pavement blocks. Air purification by pavement blocks. Final results of the research at BRRC
The use of materials can influence to a large extent the environmental impact of traffic and of road infrastructure. Especially in urban areas, where the risk on smog formation during hot summer days is high, the use of photocatalytic pavement blocks can reduce the air pollution significantly. A project on environmental friendly concrete pavement blocks is conducted at the Belgian Road Research Centre. The use of photocatalytic material in the surface of pavement blocks to obtain air purifying materials is investigated. In contact with light, TiO2 as photocatalyst, is able to reduce the NO and NO2 content in the air, caused by the exhaust of traffic. The efficiency is tested on pavement blocks, but the technique can as well be applied on other road elements (e.g. noise reducing walls, linear elements) or as a coating on new materials or existing structures. At the previous TRA conference in Gotenborgh, Sweden, the principle of photocatalysis was presented. In this paper, emphasis will be put on the final results of the 4-year project obtained in laboratory as well as on site at the Leien of Antwerp (10,000 m{sup 2}). The results indicate a durable efficiency towards NOx reduction, which is in favour for the diminishing of the risk on ozone formation. However, the precise translation from the laboratory towards the site is still in question. The results obtained during the project are discussed in this paper.
The use of materials can influence to a large extent the environmental impact of traffic and of road infrastructure. Especially in urban areas, where the risk on smog formation during hot summer days is high, the use of photocatalytic pavement blocks can reduce the air pollution significantly. A project on environmental friendly concrete pavement blocks is conducted at the Belgian Road Research Centre. The use of photocatalytic material in the surface of pavement blocks to obtain air purifying materials is investigated. In contact with light, TiO2 as photocatalyst, is able to reduce the NO and NO2 content in the air, caused by the exhaust of traffic. The efficiency is tested on pavement blocks, but the technique can as well be applied on other road elements (e.g. noise reducing walls, linear elements) or as a coating on new materials or existing structures. At the previous TRA conference in Gotenborgh, Sweden, the principle of photocatalysis was presented. In this paper, emphasis will be put on the final results of the 4-year project obtained in laboratory as well as on site at the Leien of Antwerp (10,000 m 2 ). The results indicate a durable efficiency towards NOx reduction, which is in favour for the diminishing of the risk on ozone formation. However, the precise translation from the laboratory towards the site is still in question. The results obtained during the project are discussed in this paper
Performance of continuously reinforced concrete pavements volume 5 : maintenance and repair of CRC pavements.
This report is one of a series of reports prepared as part of a recent study sponsored by the Federal Highway Administration (FHWA) aimed at updating the state-of-the-art of the design, construction, maintenance, and rehabiilitation of CRC pavements....
Towards Understanding Tyre-Pavement Contact in APT Research on Flexible Pavements
Full Text Available An important element of Accelerated Pavement Testing (APT) is the selection of appropriate test tyre(s) to be used for APT testing. By far the majority of full scale (or scaled) APT devices uses pneumatic rubber tyres. Tyres differ not only in size...
Pavement noise research : modeling of quieter pavements in Florida, final report, October 19, 2009.
The research on noise created by the tire/pavement interface, while relatively new, has seen considerable interest over the past few years because of the potential benefits and a general desire by the public for quieter highways. The Florida Departme...
Pavement Pre- and Post-Treatment Performance Models Using LTPP Data
Lu, Pan; Tolliver, Denver
This paper determines that pavement performance in International Roughness Index (IRI) is affected by exogenous interventions such as pavement age, precipitation level, freeze-thaw level, and lower level preservation maintenance strategies. An exponential function of pavement age was used to represent pavement IRI performance curves. Moreover, this paper demonstrates a method which calculates short-term post-pavement performance models from maintenance effect models and pre-treatment performa...
Pavement structure mechanics response of flexible on semi-flexible overlay that based on the old cement concrete pavement damage
Jiang Ruinan
Full Text Available The old cement pavement damage status directly affect the design of the paving renovation. Based on the state of the old road investigation, combined with the research data at home and abroad, use the control index that average deflection, deflection value and CBR value to determine the reasonable time to overlay. Draw up the typical pavement structure according to the principle of combination of old cement pavement overlay structure design, and calculated that the tensile stress and shear stress in asphalt layer ,semi-flexible layer and the tensile in the old cement pavement adopting BISA3.0 statics finite element analysis model when modulus in the old road was diminishing. Use the computed result to analyses the influence of old road damage condition the influence of pavement structure.
Identification of Pavement Distress Types and Pavement Condition Evaluation Based on Network Level Inspection for Jazan City Road Network
M Mubaraki
Full Text Available The first step in establishing a pavement management system (PMS is road network identification. An important feature of a PMS is the ability to determine the current condition of a road network and predict its future condition. Pavement condition evaluation may involve structure, roughness, surface distress, and safety evaluation. In this study, a pavement distress condition rating procedure was used to achieve the objectives of this study. The main objectives of this study were to identify the common types of distress that exist on the Jazan road network (JRN, either on main roads or secondary roads, and to evaluate the pavement condition based on network level inspection. The study was conducted by collecting pavement distress types from 227 sample units on main roads and 500 sample units from secondary roads. Data were examined through analysis of common types of distress identified in both main and secondary roads. Through these data, pavement condition index (PCI for each sample unit was then calculated. Through these calculations, average PCIs for the main and secondary roads were determined. Results indicated that the most common pavement distress types on main roads were patching and utility cut patching, longitudinal and transverse cracking, polished aggregate, weathering and raveling, and alligator cracking. The most common pavement distress types on secondary roads were weathering and raveling, patching and utility cut patching, longitudinal and transverse cracking, potholes, and alligator cracking. The results also indicated that 65% of Jazan's main road network has an average pavement condition rating of very good while only 30% of Jazan's secondary roads network has an average pavement condition.
Fatigue Behavior of Modified Asphalt Concrete Pavement
Full Text Available Fatigue cracking is the most common distress in road pavement. It is mainly due to the increase in the number of load repetition of vehicles, particularly those with high axle loads, and to the environmental conditions. In this study, four-point bending beam fatigue testing has been used for control and modified mixture under various micro strain levels of (250 μ�, 400 μ�, and 750 μ� and 5HZ. The main objective of the study is to provide a comparative evaluation of pavement resistance to the phenomenon of fatigue cracking between modified asphalt concrete and conventional asphalt concrete mixes (under the influence of three percentage of Silica fumes 1%, 2%, 3% by the weight of asphalt content, and (changing in the percentage of asphalt content by (0.5% ± from the optimum. The results show that when Silica fumes content was 1%, the fatigue life increases by 17%, and it increases by 46% when Silica fumes content increases to 2%, and that fatigue life increases to 34 % when Silica fumes content increases to 3% as compared with control mixture at (250 μ�, 20°C and optimum asphalt content. From the results above, we can conclude the optimum Silica fumes content was 2%. When the asphalt content was 4.4%, the fatigue life has increased with the use of silica fumes by (50%, when asphalt content was 5.4%, the additives had led to increasing the fatigue life by (69%, as compared with the conventional asphalt concrete pavement.
Pavement cracking measurements using 3D laser-scan images
Ouyang, W; Xu, B
Pavement condition surveying is vital for pavement maintenance programs that ensure ride quality and traffic safety. This paper first introduces an automated pavement inspection system which uses a three-dimensional (3D) camera and a structured laser light to acquire dense transverse profiles of a pavement lane surface when it carries a moving vehicle. After the calibration, the 3D system can yield a depth resolution of 0.5 mm and a transverse resolution of 1.56 mm pixel −1 at 1.4 m camera height from the ground. The scanning rate of the camera can be set to its maximum at 5000 lines s −1 , allowing the density of scanned profiles to vary with the vehicle's speed. The paper then illustrates the algorithms that utilize 3D information to detect pavement distress, such as transverse, longitudinal and alligator cracking, and presents the field tests on the system's repeatability when scanning a sample pavement in multiple runs at the same vehicle speed, at different vehicle speeds and under different weather conditions. The results show that this dedicated 3D system can capture accurate pavement images that detail surface distress, and obtain consistent crack measurements in repeated tests and under different driving and lighting conditions. (paper)
The leaf phenophase of deciduous species altered by land pavements
Chen, Yuanyuan; Wang, Xiaoke; Jiang, Bo; Li, Li
It has been widely reported that the urban environment alters leaf and flowering phenophases; however, it remains unclear if land pavement is correlated with these alterations. In this paper, two popular deciduous urban trees in northern China, ash (Fraxinus chinensis) and maple (Acer truncatum), were planted in pervious and impervious pavements at three spacings (0.5 m × 0.5 m, 1.0 m × 1.0 m, and 2.0 m × 2.0 m apart). The beginning and end dates of the processes of leaf budburst and senescence were recorded in spring and fall of 2015, respectively. The results show that leaf budburst and senescence were significantly advanced in pavement compared to non-pavement lands. The date of full leaf budburst was earlier by 0.7-9.3 days for ash and by 0.3-2.3 days for maple under pavements than non-pavements, respectively. As tree spacing increases, the advanced days of leaf budburst became longer. Our results clearly indicate that alteration of leaf phenophases is attributed to land pavement, which should be taken into consideration in urban planning and urban plant management.
Drainage investigation of surface runoff for highway pavement
Al-adili Aqeel
Full Text Available The aim of this study is to establish the effect of heavy rainfall and the chosen pavement layers on the drainage design, material selection and rutting resistance of the flexible pavement. The test in present study was started with wheel track passing without load and without rain falling on the pavement for a period of time, and it was noticed that no distress appeared on the surface of the pavement. Then, the load is gradually added by using wheel track load of 106 psi for five tests without rain falling and five other tests with gradually increasing rain fall duration and intensity. Deterioration and distresses appeared on the pavement when increasing the wheel track load to (150 psi under high intensity rain and long term duration of rain fall. By increasing the number of days, which is 103 days of study, when the pavement is saturated, the extra amount of the water will runoff. The clogging material which caused a decrease in the water seepage, increases the time of runoff ending. The clogging materials of fine particles that get deposited on the surface of the pavement resulted by passing the wheel track loading and wear & tear of the pavement surface, and other clogging materials such as salt in the water will penetrate to the pavement and seal the voids and decrease its water seepage. The water seepage decreases by increasing number of days, so the amount of the absorbed water decreases by 89% after 71 days of testing for high rain intensity (116 ml/min..
Development of Improved Mechanistic Deterioration Models for Flexible Pavements
Ullidtz, Per; Ertman, Hans Larsen
The paper describes a pilot study in Denmark with the main objective of developing improved mechanistic deterioration models for flexible pavements based on an accelerated full scale test on an instrumented pavement in the Danish Road Tessting Machine. The study was the first in "International...... Pavement Subgrade Performance Study" sponsored by the Federal Highway Administration (FHWA), USA. The paper describes in detail the data analysis and the resulting models for rutting, roughness, and a model for the plastic strain in the subgrade.The reader will get an understanding of the work needed...
Adaptation of AASHTO Pavement Design Guide for Local Conditions
Hajek, J.J. [Applied Research Associates Inc., Toronto, ON (Canada)
The methodology used to adapt the 1993 AASHTO Guide for Design of Pavement Structures to Ontario conditions was described. The guide expresses the effect of traffic loads on pavement performance using the concept of axle Load Equivalency Factors (LEF). LEF is regarded as a pavement damage factor assigned to each specific load and axle configuration. The size of LEF is related to the damage that is expected to occur from a standard load of 80 kN carried by a single axle with dual tires. The factors are summarized to yield the number of Equivalent Single Axle Loads (ESALs) a pavement is expected to sustain during its life. A summary was also provided of the additional work done to prepare for the transition to the proposed mechanistically-based 2002 AASHTO Guide. The paper focused only on the design of flexible pavements in terms of load characterization using equivalent single axle loads along with axle load spectra, below grade and material characterization, plus initial and terminal serviceability and reliability. The AASHTO Guide uses two parameters to deal with design reliability: design reliability level and overall standard deviation. Data collected on Ontario highway pavements and materials was used for assessing the design inputs. Other data was also collected from research and development studies, laboratory experiments, and from a survey of experienced pavement design engineers. The end result was a new grouping of Ontario soils for pavement design, recommended values for the resilient modulus of below grade soils, recommendations for structural layer coefficients for Ontario pavement materials and recommendations for the initial pavement serviceability based on Ontario smoothness specifications. Results of calibration and verification processes indicate that for new flexible pavements, the AASHTO-Ontario model is in good agreement with the observed results. It was recommended that the calibration and verification of the AASHTO-Ontario model should be a
Validation of theoretical models through measured pavement response
mechanics was quite different from the measured stress, the peak theoretical value being only half of the measured value.On an instrumented pavement structure in the Danish Road Testing Machine, deflections were measured at the surface of the pavement under FWD loading. Different analytical models were...... then used to derive the elastic parameters of the pavement layeres, that would produce deflections matching the measured deflections. Stresses and strains were then calculated at the position of the gauges and compared to the measured values. It was found that all analytical models would predict the tensile...
Climate change, energy, sustainability and pavements
Gopalakrishnan, Kasthurirangan; Steyn, Wynand JvdM; Harvey, John
Provides an integrated perspective on understanding the impacts of climate change, energy and sustainable development on transportation infrastructure systems. Presents recent technological innovations and emerging concepts in the field of green and sustainable transportation infrastructure systems with a special focus on highway and airport pavements. Written by leading experts in the field. Climate change, energy production and consumption, and the need to improve the sustainability of all aspects of human activity are key inter-related issues for which solutions must be found and implemented quickly and efficiently. To be successfully implemented, solutions must recognize the rapidly changing socio-techno-political environment and multi-dimensional constraints presented by today's interconnected world. As part of this global effort, considerations of climate change impacts, energy demands, and incorporation of sustainability concepts have increasing importance in the design, construction, and maintenance of highway and airport pavement systems. To prepare the human capacity to develop and implement these solutions, many educators, policy-makers and practitioners have stressed the paramount importance of formally incorporating sustainability concepts in the civil engineering curriculum to educate and train future civil engineers well-equipped to address our current and future sustainability challenges. This book will prove a valuable resource in the hands of researchers, educators and future engineering leaders, most of whom will be working in multidisciplinary environments to address a host of next-generation sustainable transportation infrastructure challenges.
Review of ice and snow runway pavements
Greg White
Full Text Available Antarctica is the highest, driest, coldest, windiest, most remote and most pristine place on Earth. Polar operations depend heavily on air transportation and support for personnel and equipment. It follows that improvement in snow and ice runway design, construction and maintenance will directly benefit polar exploration and research. Current technologies and design methods for snow and ice runways remain largely reliant on work performed in the 1950s and 1960s. This paper reviews the design and construction of polar runways using snow and ice as geomaterials. The inability to change existing snow and ice thickness or temperature creates a challenge for polar runway design and construction, as does the highly complex mechanical behaviour of snow, including the phenomena known as sintering. It is recommended that a modern approach be developed for ice and snow runway design, based on conventional rigid and flexible pavement design principles. This requires the development on an analytical model for the prediction of snow strength, based on snow age, temperature history and density. It is also recommended that the feasibility of constructing a snow runway at the South Pole be revisited, in light of contemporary snow sintering methods. Such a runway would represent a revolutionary advance for the logistical support of Antarctic research efforts. Keywords: Runway, Pavement, Snow, Ice, Antarctic
Gopalakrishnan, Kasthurirangan [Iowa State Univ., Ames, IA (United States). Dept. of Civil, Construction and Environmental Engineering; Steyn, Wynand JvdM [Pretoria Univ. (South Africa). Dept. of Civil Engineering; Harvey, John (ed.) [California Univ., Davis, CA (United States). Dept. of Civil and Environmental Engineering
Provides an integrated perspective on understanding the impacts of climate change, energy and sustainable development on transportation infrastructure systems. Presents recent technological innovations and emerging concepts in the field of green and sustainable transportation infrastructure systems with a special focus on highway and airport pavements. Written by leading experts in the field. Climate change, energy production and consumption, and the need to improve the sustainability of all aspects of human activity are key inter-related issues for which solutions must be found and implemented quickly and efficiently. To be successfully implemented, solutions must recognize the rapidly changing socio-techno-political environment and multi-dimensional constraints presented by today's interconnected world. As part of this global effort, considerations of climate change impacts, energy demands, and incorporation of sustainability concepts have increasing importance in the design, construction, and maintenance of highway and airport pavement systems. To prepare the human capacity to develop and implement these solutions, many educators, policy-makers and practitioners have stressed the paramount importance of formally incorporating sustainability concepts in the civil engineering curriculum to educate and train future civil engineers well-equipped to address our current and future sustainability challenges. This book will prove a valuable resource in the hands of researchers, educators and future engineering leaders, most of whom will be working in multidisciplinary environments to address a host of next-generation sustainable transportation infrastructure challenges.
Application of Conductive Materials to Asphalt Pavement
Hai Viet Vo
Full Text Available Snow-melting pavement technique is an advanced preservation method, which can prevent the forming of snow or ice on the pavement surface by increasing the temperature using an embedded heating system. The main scope of this study is to evaluate the impact of conductive additives on the heating efficiency. The electrical resistivity and thermal conductivity were considered to investigate effects of conductive additives, graphite, and carbon fibers on the snow-melting ability of asphalt mixtures. Also, the distribution of the conductive additives within the asphalt concrete body was investigated by microstructural imaging. An actual test was applied to simulate realistic heating for an asphalt concrete mixture. Thermal testing indicated that graphite and carbon fibers improve the snow-melting ability of asphalt mixes and their combination is more effective than when used alone. As observed in the microstructural image, carbon fibers show a long-range connecting effect among graphite conductive clusters and gather in bundles when added excessively. According to the actual test, adding the conductive additives helps improve snow-melting efficiency by shortening processing time and raising the surface temperature.
The comparisons of computational models of pavement due to the dynamic load act
Valašková Veronika
Full Text Available Finite element method (FEM is frequently used to solve current problems of the structural mechanics in engineering practice. The dynamic interaction of the two systems is one of the topics that can be solved using FEM. To simplify the solution of the interaction, it is possible to divide the system in two independent subsystems, vehicle and pavement. Computing software ADINA, based on the principles of the finite element modelling, was adopted to solve the given problem. The results from the first computational model can be used as the inputs for the second model. To analyse stress state of the pavement with different composition, three specific types of pavements were selected – concrete pavement, semi-rigid pavement and flexible pavement. The pavements represent the typical pavement types used for the regular road structures. This article is focused on the analysis of the stress state and the displacements of the pavement induced by the moving load from the traffic.
An Innovative Cellular Automata Technique for Mapping Cracking Pattern of Airport Pavement
Yin Fucheng
Full Text Available In this study, an innovative cellular automata (CA technique was proposed for mapping cracking pattern of the airport pavement. The CA technique was developed to establish a numerical model describing the effect of boundary condition of pavement on zones (CA cells within the pavement. A state function was used to describe the state values in the cells within the CA lattice. The correction coefficient principle is used as the criterion of zone similarity and the corresponding technique is proposed to find similar zones within and between pavements. Three pavement models, HRS, MRS and LRS, tested in FAA, USA, are set as the base pavements to map the cracking patterns of pavements with different sizes from the base pavements. The mapped cracking patterns of unseen pavements are empirically verified by referring to the relative experimental models.
Cost-Effective Pavement Performance Management of Indiana's Enhanced National Highway System through Strategic Modification of the Pavement Rehabilitation Treatment Trigger Values
Noureldin, Menna; Fricker, Jon D.; Sinha, Kumares C.
Cost-Effective Pavement Performance Management of Indiana's Enhanced National Highway System through Strategic Modification of the Pavement Rehabilitation Treatment Trigger Values Presented during Session 3: Policy and Funding, moderated by Magdy Mikhail, at the 9th International Conference on Managing Pavement Assets (ICMPA9) in Alexandria, VA. Includes conference paper and PowerPoint slides.
Long-term Performance of Granular Bases Including the Effect of Wet-Dry Cycles on Inverted Base Pavement Performance
The main objective of this study was to advance the understanding of alternative pavement designs. In particular, potential techniques such as inverted base pavements (IBP) have increased the importance of granular aggregate bases (GAB) in pavement s...
Metal concentrations from permeable pavement parking lot in Edison, NJ
U.S. Environmental Protection Agency — The U.S. Environmental Protection Agency constructed a 4000-m2 parking lot in Edison, New Jersey in 2009. The parking lot is surfaced with three permeable pavements...
Sustainable Mitigation of Stormwater Runoff Through Fully Permeable Pavement
This report presents the implementation of new design method developed using mechanistic-empirical design approach by University of California Pavement Research Center (UCPRC) through building two test sections at California State University Long Bea...
Evaluation of acceptance strength tests for concrete pavements.
The North Carolina Department of Transportation has used traditionally flexural strength tests for acceptance : testing of Portland cement concrete pavements. This report summarizes a research project implemented to : investigate the feasibility of u...
Effect of friction on rolling tire-pavement interaction.
Accurate modeling of tirepavement contact behavior (i.e., distribution of contact tractions at the : interface) plays an important role in the analysis of pavement performance and vehicle driving safety. : The tirepavement contact is essentiall...
Prediction of thermal behavior of pervious concrete pavements in winter.
Because application of pervious concrete pavement (PCPs) has extended to cold-climate regions of the United States, the safety and : mobility of PCP installations during the winter season need to be maintained. Timely application of salt, anti-icing,...
Pavement evaluation - an international perspective: Fit for purpose?
Wix, Richard
Plenary Session 1 Moderated by Edgar de León Izeppi This presentation was held at the Pavement Evaluation 2014 Conference, which took place from September 15-18, 2014 in Blacksburg, Virginia. Presentation only
Ultra thin continuously reinforced concrete pavement research in south Africa
Perrie, BD
Full Text Available Ultra thin continuously reinforced concrete pavements (UTCRCP), in literature also referred to as Ultra Thin Reinforced High Performance Concrete (UTHRHPC), have been used in Europe successfully as a rehabilitation measure on steel bridge decks...
DURABILITY OF FLEXIBLE PAVEMENTS: A CASE STUDY OF ...
years, ranking, predominant factors affecting pavement durability and the estimate of durability. In this regard .... subgrade soil into the base course and provide the drainage of ..... [3] Oguara T. M. "A management model for road infrastructure ...
Safety performance evaluation of converging chevron pavement markings : final report.
The objectives of this study were (1) to perform a detailed safety analysis of converging chevron : pavement markings, quantifying the potential safety benefits and developing an understanding of the : incident types addressed by the treatment, and (...
sensitivity analysis on flexible road pavement life cycle cost model
of sensitivity analysis on a developed flexible pavement life cycle cost model using varying discount rate. The study .... organizations and specific projects needs based. Life-cycle ... developed and completed urban road infrastructure corridor ...
Ground penetrating radar evaluation of new pavement density.
The objective of this project was to map pavement surface density variations using dielectric : measurements from ground penetrating radar (GPR). The work was carried out as part of an : Asphalt Intelligent Compaction demonstration project on SR 539 ...
Texas pavement preservation center two-year summary report.
Established August 11, 2005, The TPPC serves the broad range of needs of the Texas Department of Transportation (TxDOT), and other agencies within the highway community by promoting awareness of pavement preservation as a feasible and practical maint...
Texas pavement preservation center four-year summary report.
The Texas Pavement Preservation Center (TPPC), in joint collaboration with the Center for Transportation Research (CTR) of the University of Texas at Austin and the Texas Transportation Institute (TTI) of Texas A&M University, promotes the use of pav...
Mobile geographic information system (GIS) solution for pavement condition surveys.
This report discusses the design and implementation of a software-based solution that will improve the data collection processes during the Pavement Condition Surveys (PCS) conducted by the State Materials Office (SMO) of the Florida Department of Tr...
Investigation clogging dynamic of permeable pavement systems using embeded sensors
U.S. Environmental Protection Agency — Permeable pavement is a stormwater control measure commonly selected in both new and retrofit applications. However, there is limited information about the clogging...
Calibration of controlling input models for pavement management system.
The Oklahoma Department of Transportation (ODOT) is currently using the Deighton Total Infrastructure Management System (dTIMS) software for pavement management. This system is based on several input models which are computational backbones to dev...
Pavement subgrade MR design values for Michigan's seasonal changes : appendices.
The resilient modulus (MR) of roadbed soil plays an integral role in the design of pavement systems. Currently, the various regions of the Michigan Department of Transportation (MDOT) use different procedures to determine the MR values. Most of these...
Iowa pavement asset management decision-making framework.
Most local agencies in Iowa currently make their pavement treatment decisions based on their limited experience due primarily to : lack of a systematic decision-making framework and a decision-aid tool. The lack of objective condition assessment data...
Benefits and assessment of annual budget requirements for pavement preservation.
This research identifies methods and best practices that can be used by the Indiana Department of Transportation (INDOT) in : performing various strategies for pavement preservation. It also identifies various methods of calculating the benefits of :...
Evaluation of the use of snowplowable raised pavement markers.
The objective of this study was to evaluate the effectiveness and durability of snowplowable raised pavement markers (RPM) installed on the RPM system in Kentucky. The durability evaluation dealt wit the marker housing. : The data show that continued...
Use of nondestructive evaluation to detect moisture in flexible pavements.
The purpose of this study was to identify the currently available nondestructive evaluation technology that holds the greatest potential to detect moisture in flexible pavements and then apply the technology in multiple locations throughout Virginia....
Evaluation of portable retroreflectometer for use on pavement marking materials.
The subjective rating of the night visibility of pavement marking materials is difficult as it can be influenced by many variables. In an attempt to provide an improved method of determining the night visibility of these materials, a portable retrore...
Tack coat optimization for HMA overlays : accelerated pavement test report.
Interface bonding between hot-mix asphalt (HMA) overlays and Portland cement concrete (PCC) pavements is one : of the most significant factors affecting overlay service life. This study was performed to quantify the effects of HMA type, : tack coat t...
Implementation of warm-mix asphalt mixtures in Nebraska pavements.
The primary objective of this research is to evaluate the feasibility of several WMA mixtures as potential asphalt paving : mixtures for Nebraska pavements. To that end, three well-known WMA additives (i.e., Sasobit, Evotherm, and Advera : synthetic ...
Life cycle and economic efficiency analysis: durable pavement markings.
This project examined the life cycle and economic efficiency of two pavement marking : materials inlaid tape and thermoplastic to find the most economical product for specific : traffic and weather conditions. Six locations in the state of Ma...
Long-Term Field Performance of Pervious Concrete Pavement
Aleksandra Radlińska
Full Text Available The work described in this paper provides an evaluation of an aged pervious concrete pavement in the Northeastern United States to provide a better understanding of the long-lasting effects of placement techniques as well as the long-term field performance of porous pavement, specifically in areas susceptible to freezing and thawing. Multiple samples were taken from the existing pavement and were examined in terms of porosity and unit weight, compressive and splitting tensile strength, and the depth and degree of clogging. It was concluded that improper placement and curing led to uneven pavement thickness, irregular pore distribution within the pervious concrete, and highly variable strength values across the site, as well as sealed surfaces that prevented infiltration.
Development of an Overlay Design Procedure for Composite Pavements
The composite overlay design procedure currently used by ODOT sometimes produces very large overlay thicknesses that are deemed structurally unnecessary, especially for composite pavements already with thick asphalt overlays. This study was initiated...
stabilization of dredged spoils for pavement construction in the niger
Dr. Tse
Natural soils underlying the East-West road are mainly clay and silt of poor quality ... on the dredged soils included particle size distribution, compaction and California ... KEYWORDS: Stabilization, dredge spoil, pavement, Niger Delta, cement.
ANALYSIS OF GROUP MAINTENANCE STRATEGY -ROAD PAVEMENT AND SEWERAGE PIPES-
Tanimoto, Keishi; Sugimoto, Yasuaki; Miyamoto, Shinya; Nada, Hideki; Hosoi, Yoshihiko
Recently, it is critical to manage deteriorating sewerage and road facilities efficiently and strategically. Since the sewerage pipes are mostly installed under road pavement, the works for the replacement of the sewerage pipes are partially common to the works for the road. This means that the replacement cost can be saved by coordinating the timing of the replacements by sewerage pipe and road pavement. The purpose of the study is to develop the model based on Markov decision process to derive the optimal group maintenance policy so as to minimize lifecycle cost. Then the model is applied to case study area and demonstrated to estimate the lifecycle cost using statistical data such as pipe replacement cost, road pavement rehabilitation cost, and state of deterioration of pipes and road pavement.
Recycled tires as coarse aggregate in concrete pavement mixtures.
The reuse potential of tire chips as coarse aggregates in pavement concrete was examined in this research by : investigating the effects of low- and high-volume tire chips on fresh and hardened concrete properties. One concrete : control mixture was ...
Benefit cost models to support pavement management decisions.
A critical role of pavement management is to provide decision makers with estimates of the required budget level to achieve specific steady-state network conditions, and to recommend the best allocation of available budget among competing needs for m...
Fatigue and rutting strain analysis of flexible pavements designed ...
The study was carried out with the layered elastic analysis software EVERSTRESS. Key words: Layered elastic analysis, fatigue, rutting, strains, design, flexible pavement, CBR. ..... Numerical Computational Stresses and Strains in Multi-Layer ...
Inverted base pavements : new field test and design catalogue.
The current economic situation has severely affected the US road infrastructure and funding has become : inadequate for either maintenance or future growth. : The inverted base pavement structure is a promising alternative to achieve high quality roa...
Pavement management system for City of Madison : research brief.
Keeping city streets in good state of repair has been a significant challenge for small communities : where both road repair budgets and staff resources are lacking. To prevent pavement deterioration at : a large scale, local governments need to find...
MDOT Pavement Management System : Prediction Models and Feedback System
As a primary component of a Pavement Management System (PMS), prediction models are crucial for one or more of the following analyses: : maintenance planning, budgeting, life-cycle analysis, multi-year optimization of maintenance works program, and a...
SDDOT'S enhanced pavement management system : visual distress survey manual [2017
In 1993, the South Dakota Department of Transportation initiated the Research Project SD93-14, Enhancement of South Dakotas Pavement Management System. As the Research Project progressed, it was determined that to better evaluate the condition of ...
Pavement management system review : final report, October 2009.
The University of Alabama (UA) researchers worked with Alabama Department of Transportation (ALDOT) managers to investigate various mechanisms to provide quality control of data collection and interpretation for pavements, and to target the data resu...
Assessment of Asphalt Concrete Reinforcement Grid in Flexible Pavements
successfully used as interlayers include asphalt rubber and geotextiles, and the applica- tion of recycling techniques to rework the upper 2–4 in. of the...from a set of weights dropped from increasingly greater predetermined heights onto a rubber buffer system connected to a 12 in. diameter segmented...pavement temperature at depth, IR = the infrared pavement surface temperature (°C), D = the asphalt depth to estimate the temperature (mm), 1
Application of surplus and waste materials in roads pavement making
Mousavi, S. M.; Fazli, A. H.; Rouzmehr, F.
Nowadays there are a lot of problems about surpluses and debris made by humans all around the world. Lots of these surpluses seriously harm our natural environment. Reuse of this kind of materials in other processes like building constructions or pavement help our natural environment in every aspect. Asphalt concrete is the main part of pavements in most parts of the world with an increasing rate of production in need of more ways and roads. In this paper we will provide...
Evaluation of Nontraditional Airfield Pavement Surfaces for Contingency Operations
double, triple seals, and cape seals have been used in various airports in the Australian territories (such as Cocos Island) with triple seals and cape...polymers, or fibers with the natural soil. In stabilizing soil, strength, durability, cohesion, and reduced swelling properties may be improved (UFC...2012. Cocos (Keeling) Island Aircraft pavement refurbishment project. Presentation provided by S. Emery, 5 May 2013. Ellis, C. I. 1979. Pavement
Pavement-Watering for Cooling the Built Environment: A Review
Hendel , Martin
Pavement-watering is being considered by decision-makers in many cities as a means of cooling the built environment and of adapting to rising extreme heat events due to climate change. In this article we review the existing literature on the topic of pavement-watering. We first focus on the methodological choices made in the literature, including study approach and scale, watering methods used as well as how results are analyzed. We then discuss the cooling effects reported, separating micro-...
Evaluation of pavement skid resistance using high speed texture measurement
Jay N. Meegoda
Full Text Available Skid resistance is an important parameter for highway designs, construction, management, maintenance and safety. The purpose of this manuscript is to propose the correlation between skid resistance, which is measured as skid resistance trailer, and mean profile depth (MPD or the macro surface texture, which is measured by vehicle mounted laser, so that highway agencies can predict the skid resistance of pavement without the use of expensive and time consuming skid resistance trailer, which also causes disruption of traffic in use. In this research skid numbers and MPD from 5 new asphalt pavements and 4 old asphalt pavements were collected using a locked wheel skid trailer and a vehicle mounted laser. Using the data collected, a correlation between the skid number (SN40R collected by locked wheel skid tester and the texture data or MPD collected by a vehicle mounted laser operating at highway speeds was developed. The proposed correlation for new pavements was positive for MPD values less than 0.75Â mm to reach a peak SN40R value, then there was a negative correlation as the MPD increases until the MPD value was equal to 1.1Â mm and beyond the MPD value of 1.1Â mm to the maximum value of 1.4Â mm, SN40R value remained almost constant. There were significant data scatter for the MPD value of 0.8Â mm. To explain these results, water film thickness during the friction test was calculated and the critical MPD was defined. The effect of sealed water pool on the SN40R was discussed. The test result showed a similar trend for older asphalt pavements, but with lower SN40R values due to the polishing of pavement micro-texture by traffic. Hence, a reduction factor was proposed for older pavements based on cumulative traffic volume for the above correlation to predict the skid resistance of older pavements.
Influence of pavement condition on horizontal curve safety.
Buddhavarapu, Prasad; Banerjee, Ambarish; Prozzi, Jorge A
Crash statistics suggest that horizontal curves are the most vulnerable sites for crash occurrence. These crashes are often severe and many involve at least some level of injury due to the nature of the collisions. Ensuring the desired pavement surface condition is one potentially effective strategy to reduce the occurrence of severe accidents on horizontal curves. This study sought to develop crash injury severity models by integrating crash and pavement surface condition databases. It focuses on developing a causal relationship between pavement condition indices and severity level of crashes occurring on two-lane horizontal curves in Texas. In addition, it examines the suitability of the existing Skid Index for safety maintenance of two-lane curves. Significant correlation is evident between pavement condition and crash injury severity on two-lane undivided horizontal curves in Texas. Probability of a crash becoming fatal is appreciably sensitive to certain pavement indices. Data suggested that road facilities providing a smoother and more comfortable ride are vulnerable to severe crashes on horizontal curves. In addition, the study found that longitudinal skid measurement barely correlates with injury severity of crashes occurring on curved portions. The study recommends exploring the option of incorporating lateral friction measurement into Pavement Management System (PMS) databases specifically at curved road segments. Copyright © 2012 Elsevier Ltd. All rights reserved.
Pavement Response to Variable Tyre Pressure of Heavy Vehicles
Arshad Ahmad Kamil
Full Text Available In recent years, the effect of overinflated tyre pressure and increased heavy vehicles' axle load on flexible pavements has become a subject of great concern because of the higher stress levels induced and damage caused to road pavements. This paper aims to evaluate the effect of variable tyre inflation pressures (using actual tyre contact/footprint area to determine the responses of flexible pavement. A full scale experiment was conducted on a heavy vehicle with 1:1:2 axle configuration, 10 R 20 tyre size and attached trailer with constant axle load. Measurements were made for actual tyre-pavement contact area. KENPAVE linear elastic program was then used to analyse the effects of the measured actual tyre-pavement contact area and the results was compared using conventional circular tyre contact area. A comparative analysis was then made between the actual contact area and the conventional circular tyre contact area. It was found that high tyre inflation pressure produce smaller contact area, giving more detrimental effect on the flexible pavement. It was also found that the temperature of tyres when the heavy vehicles are operational give less significant impact on tyre inflation pressure for the Malaysian climate.
Effect of speed hump characteristics on pavement condition
Talaat Ali Abdel-Wahed
Full Text Available Speed humps are the most common type of traffic calming devices due to their low cost and easy installation. However, in many Egyptian roads, considerable number of these humps is randomly placed without proper engineering studies and justifications. Deterioration of pavement condition is observed near these humps. This paper presents a case study applied to collect and analyze visual inspection data for the reason of evaluating the impact of speed humps on pavement condition on intercity rural roads. The paper used 52 speed humps located in an intercity two-lane, two-way road that connects two cities, Tahta and Gerga, in Upper Egypt. The total length of this road is about 34Â km. Pavement condition index (PCI, in road sections, near speed humps in the two directions of travel were calculated from the visual inspection measurements. The characteristics of each speed hump (width, height, and distance from preceding hump were measured. Using statistical analyses, the correlations between the pavement conditions and hump characteristics were examined. Regression analysis models were developed to represent the relationships between pavement conditions and hump characteristics. Generally, the results proved that the pavement conditions are greatly influenced by the presence of speed humps and hump characteristics.
Analysis of natural stone block pavements in urban shared areas
Pablo Zoccali
Full Text Available This paper analysed and verified an existing block stone pavement in an urban shared area. Fatigue and rutting verification was performed respectively for bound and unbound pavement materials using analytical curves available in the literature. The commercial finite element (FE software Abaqus® was used to calculate the response of the pavement when subjected to different loading, construction and geometrical configurations (i.e. type of analysis, shape and size of meshes, boundary conditions, and bonding contacts between the pavements layers. At the end of this study, a static model of a structure with hexahedral blocks having sides of 0.02 m, with full bonded layers and restrained horizontal displacements on the model sides, was implemented to evaluate the maximum tensile stress induced in a block when the load is applied at its centre. This analysis highlighted the need for rigorous criteria for a correct design, in order to avoid inappropriate and expensive use of road materials. Keywords: Block pavement, Commercial vehicle loads, Finite element model, Hexagonal Stone block, Pedestrian pavement, Shared area
Urban pavement surface temperature. Comparison of numerical and statistical approach
Marchetti, Mario; Khalifa, Abderrahmen; Bues, Michel; Bouilloud, Ludovic; Martin, Eric; Chancibaut, Katia
The forecast of pavement surface temperature is very specific in the context of urban winter maintenance. to manage snow plowing and salting of roads. Such forecast mainly relies on numerical models based on a description of the energy balance between the atmosphere, the buildings and the pavement, with a canyon configuration. Nevertheless, there is a specific need in the physical description and the numerical implementation of the traffic in the energy flux balance. This traffic was originally considered as a constant. Many changes were performed in a numerical model to describe as accurately as possible the traffic effects on this urban energy balance, such as tires friction, pavement-air exchange coefficient, and infrared flux neat balance. Some experiments based on infrared thermography and radiometry were then conducted to quantify the effect fo traffic on urban pavement surface. Based on meteorological data, corresponding pavement temperature forecast were calculated and were compared with fiels measurements. Results indicated a good agreement between the forecast from the numerical model based on this energy balance approach. A complementary forecast approach based on principal component analysis (PCA) and partial least-square regression (PLS) was also developed, with data from thermal mapping usng infrared radiometry. The forecast of pavement surface temperature with air temperature was obtained in the specific case of urban configurtation, and considering traffic into measurements used for the statistical analysis. A comparison between results from the numerical model based on energy balance, and PCA/PLS was then conducted, indicating the advantages and limits of each approach.
Cloud Impacts on Pavement Temperature in Energy Balance Models
Walker, C. L.
Forecast systems provide decision support for end-users ranging from the solar energy industry to municipalities concerned with road safety. Pavement temperature is an important variable when considering vehicle response to various weather conditions. A complex, yet direct relationship exists between tire and pavement temperatures. Literature has shown that as tire temperature increases, friction decreases which affects vehicle performance. Many forecast systems suffer from inaccurate radiation forecasts resulting in part from the inability to model different types of clouds and their influence on radiation. This research focused on forecast improvement by determining how cloud type impacts the amount of shortwave radiation reaching the surface and subsequent pavement temperatures. The study region was the Great Plains where surface solar radiation data were obtained from the High Plains Regional Climate Center's Automated Weather Data Network stations. Road pavement temperature data were obtained from the Meteorological Assimilation Data Ingest System. Cloud properties and radiative transfer quantities were obtained from the Clouds and Earth's Radiant Energy System mission via Aqua and Terra Moderate Resolution Imaging Spectroradiometer satellite products. An additional cloud data set was incorporated from the Naval Research Laboratory Cloud Classification algorithm. Statistical analyses using a modified nearest neighbor approach were first performed relating shortwave radiation variability with road pavement temperature fluctuations. Then statistical associations were determined between the shortwave radiation and cloud property data sets. Preliminary results suggest that substantial pavement forecasting improvement is possible with the inclusion of cloud-specific information. Future model sensitivity testing seeks to quantify the magnitude of forecast improvement.
On the wave amplitude blow-up in the Berk-Breizman model for nonlinear evolution of a plasma wave driven resonantly by fast ions
Zaleśny, Jarosław; Galant, Grzegorz; Berczyński, Paweł; Berczyński, Stefan; Lisak, Mietek
In this paper the Berk-Breizman (BB) model of plasma wave instability arising on the stability threshold is considered. An interesting although physically unacceptable feature of the model is the explosive behaviour occurring in the regime of small values of the collision frequency parameter. We present an analytical description of the explosive solution, based on a fitting to the numerical solution of the BB equation with the collision parameter equal to zero. We find that the chaotic behaviour taking place for small but non-zero values of the collision parameter is absent in this case; therefore, chaotic behaviour seems to be an independent phenomenon not directly related to the blow-up regime. The time and the velocity dependence of the distribution function are found numerically and plotted to better understand what actually happens in the model. It allows us to obtain a good qualitative understanding of the time evolution of the mode amplitude including the linear growth of the amplitude, reaching its maximum and then decreasing towards the zero value. Nevertheless, we have no satisfactory physical explanation of the amplitude evolution when the amplitude vanishes at some time and then revives but with an opposite phase.
Improve the quality and service life of water-based pavement marking paints on pavements with high-iron aggregates.
White pavement paint marking on airport runways was being discolored by rust-like staining. Discoloration is a critical safety : problem because white paint indicates runways and yellow paint is used for taxiways and aircraft parking. When the white ...
Evaluation and Development of Pavement Scores, Performance Models and Needs Estimates for the TXDOT Pavement Management Information System : Final Report
This project conducted a thorough review of the existing Pavement Management Information System (PMIS) database, : performance models, needs estimates, utility curves, and scores calculations, as well as a review of District practices : concerning th...
Establishing appropriate inputs when using the mechanistic-empirical pavement design guide to design rigid pavements in Pennsylvania.
Each design input in the Mechanistic-Empirical Design Guide (MEPDG) required for the design of Jointed Plain Concrete : Pavements (JPCPs) is introduced and discussed in this report. Best values for Pennsylvania conditions were established and : recom...
0-6683 : develop a pavement project evaluation index to support the 4-year pavement management plan.
A pavement management plan (PMP) identifies : candidate maintenance and rehabilitation (M&R) : projects for a particular roadway network over a : multi-year planning period. The PMP is a living : document because projects are reevaluated and : reprio...
Feasibility of perpetual pavement stage construction in China: A life cycle cost analysis
Zhongyin Guo
Full Text Available The main objective of pavement design and management is to build sustainable pavement structure with minimum costs during its whole life. There are many uncertainties in the process of pavement design pertaining many of its variables, such as future traffic estimation, long time behavior of materials, future weights and types of traveling vehicles, availability of funds etc. Therefore, it is important to apply pavement stage construction technique during the process of pavement design and management to minimize the risk associated with these uncertainties. From the beginning of 2000, the research and application of perpetual asphalt pavement (PP technology has been deployed in China. The semi rigid base for asphalt pavement has been normally considered as typical component of high class highways in the design according to the Chinese experience since 1997. The research objective is to apply pavement stage construction for the evaluation of life cycle costs of Chinese perpetual and traditional semi rigid pavements using operational pavement management system in addition to examine its suitability for design and construction of more economical and durable flexible pavements. It has been found that the stage construction of asphalt layers in PP over semi rigid pavement foundation will create more sustainable and trusted pavement structures in spite of 2–5% increase in present total cost.
Use of lime cement stabilized pavement construction
Kumar, M.A.; Raju, G.V.R.P. [JNTU College of Engineering, Kakinada (India). Dept. of Civil Engineering
Expansive clay is a major source of heave induced structural distress. Swelling of expansive soils causes serious problems and produce damages to many structures. Many research organizations are doing extensive work on waste materials concerning the feasibility and environmental suitability. Fly ash, a waste by product from coal burning in thermal power stations, is abundant in India causing severe health, environmental and disposal problems. Attempts are made to investigate the stabilization process with model test tracks over expansive subgrade in flexible pavements. Cyclic plate load tests are carried out on the tracks with chemicals like lime and cement introduced in fly ash subbase laid on sand and expansive subgrades. Test results show that maximum load carrying capacity is obtained for stabilized fly ash subbase compared to untreated fly ash subbase.
Latex improvement of recycled asphalt pavement
Drennon, C.
The performance of a single unmodified milled recycled asphalt concrete was compared to milled asphalt concrete modified by addition of three types of rubber latex. Latex was added at 2, 3, 5, and 8 percent latex by weight of asphalt in the asphalt concrete. Lattices used were a styrene butadiene (SBR), a natural rubber (NR), an acrylonitrile butadiene (NBR), and four varieties of out of specification SBR lattices. Marshall tests, while indecisive, showed a modest improvement in properties of SBR and NR added material at 3 and 5 percent latex. Addition of NBR latex caused deterioration in Marshall stability and flow over that of control. Repeated load tests were run using the indirect tensile test, analyzed by the VESYS program, which computes life of pavements. Repeated load tests showed improvement in asphalt concrete life when 3 and 5 percent SBR was added. Improvement was also shown by the out of specification SBR.
Standard nomenclature and methods for describing the condition of pavements draft TRH 6
Curtayne, PC
Full Text Available The need for describing the condition of pavements occurs frequently in highway engineering. Accurate descriptions are a prerequisite for establishing procedures with which to evaluate the various aspects of the pavement condition. A variety...
Update to Permeable Pavement Research at the Edison Environmental Center - abstract
Abstract The EPA's Urban Watershed Management Branch (UWMB) has been monitoring the permeable pavement demonstration site at the Edison Environmental Center, NJ since 2010. This site has three different types of permeable pavement including: interlocking concrete permeable pavers...
Development of Pavement Performance Prediction Models for Preservation Treatments : Volume 2
The implementation of a pavement preservation program was initiated in Fiscal Year (FY) 2005 at the Illinois Department of Transportation (IDOT) by appropriating funding for four specific pavement preservation treatments. The types of treatments incl...
Update to permeable pavement research at the Edison Environmental Center - slides
Abstract: The EPA's Urban Watershed Management Branch (UWMB) has been monitoring the permeable pavement demonstration site at the Edison Environmental Center, NJ since 2010. This site has three different types of permeable pavement including: interlocking concrete permeable paver...
Historical performance evaluation of Iowa pavement treatments using data analytics : tech transfer summary.
Evaluate the performance of the most-used pavement treatments in Iowa by considering different parameters such as type of treatment, treatment thickness, traffic, and pavement type : Estimate a service life for each treatment based on the obs...
Modeling of interaction between steel and concrete in continuously reinforced concrete pavements : final report.
Continuously reinforced concrete pavement (CRCP) contains continuous longitudinal reinforcement with no transverse : expansion within the early life of the pavement and can continue to develop cracks in the long-term. The : accurate modeling of CRCPs...
Top-down cracking of rigid pavements constructed with fast setting hydraulic cement concrete
Heath, AC
Full Text Available Jointed plain concrete pavement (JPCP) test sections were constructed using fast setting hydrualic cement concrete (FSHCC) as part of the California accelerated pavement testing program (CAL/APT). Many of the longer slabs cracked under environmental...
Assessment of asphalt concrete reinforcement grid in flexible pavements : final report.
This report investigated the application of accepted methods of pavement structural evaluation to independently assess the potential structural benefit of asphalt geogrid reinforcement of an operational flexible highway pavement. The asphalt interlay...
Nontraditional Fog Seal for Asphalt Pavement : Performance on Shoulder Sections in Minnesota
The pavement engineering community has recently been introduced to a number of nontraditional products intended for uses as surface sealers for bituminous pavements. Many new products use agricultural-based components and little is known regarding th...
Determination of coefficient of thermal expansion effects on Louisiana's PCC pavement design : technical summary report.
The coefficient of thermal expansion (CTE) has been widely considered as a fundamental property of : Portland cement concrete (PCC) pavement but has never played an important role in the thickness design : procedure for PCC pavement until recently. I...
Caltrans accelerated pavement testing (CAL/APT) program - test results: 1993-1996
Nokes, WA
Full Text Available By combining the results of both the HVS and laboratory tests, interpretations of pavement performance are described and recommendations for pavement design and construction have been made to Caltrans....
Engineering properties of stabilized subgrade soils for implementation of the AASHTO 2002 pavement design guide.
A comprehensive laboratory study was undertaken to determine engineering properties of cementitiously stabilized common subgrade soils in Oklahoma for the design of roadway pavements in accordance with the AASHTO 2002 Mechanistic-Empirical Pavement D...
MDOT innovation leading to faster, longer-lasting pavement repairs : research spotlight.
Current methods of patching pavement must evolve to meet increasing mobility demands. : To address this need, MDOT has been testing a new generation of rapid set full-depth : pavement repair materials. Initial results are promising. The new materials...
Historical performance evaluation of Iowa pavement treatments using data analytics : final report.
The pavement network in Iowa has reached a mature state making maintenance and rehabilitation activities more important than new construction. As such, a need exists to evaluate the performance of the pavement treatments and estimate their performanc...
Recycled Portland cement concrete pavements : Part II, state-of-the art summary.
This report constitutes a review of the literature concerning recycling of portland cement concrete pavements by crushing the old pavement and reusing the crushed material as aggregate in a number of applications. A summary of the major projects cond...
Texas flexible pavements and overlays : calibration plans for M-E models and related software.
This five-year project was initiated to collect materials and pavement performance data on a minimum of 100 highway test sections around the State of Texas, incorporating flexible pavements and overlays. Besides being used to calibrate and validate m...
Transition from manual to automatic rutting measurements : effect on pavement serviceability index values.
The Pavement Serviceability Index (pSn is used by New Mexico Department of Transportation : (NMDOT) to express the serviceability level of a pavement section at the network level. The PSI : is calculated with distress ratings (including rutting) and ...
Commonwealth of Pennsylvania Bureau of Aviation pavement evaluation report : final report, March 2010.
As part of the Pennsylvania Department of Transportations (PennDOT) airport pavement management efforts, the Bureau of : Aviation (BOA) retained Applied Pavement Technology, Inc. (APTech), assisted by DY Consultants, to evaluate the condition of t...
Simulation of dynamic traffic loading based on accelerated pavement testing (APT)
Full Text Available The objective of this paper is to introduce the latest Heavy Vehicle Simulator (HVS) technology as part of the South African Accelerated Pavement Testing (APT) efforts, its capabilities and expected impact on road pavement analysis....
Accelerated pavement testing of low-volume paved roads with geocell reinforcement : [technical summary].
The Midwest States Accelerated Pavement Testing Pooled-Fund Program, financed : by the highway departments of Kansas, Iowa, Missouri, and New York, has : supported an accelerated pavement testing (APT) project to study the rehabilitation : of low-vol...
Accelerated testing for studying pavement design and performance (FY 2003) : research summary.
The Midwest States Accelerated Pavement Testing Pooled Fund Program, financed by : the highway departments of Missouri, Iowa, Kansas and Nebraska, has supported an : accelerated pavement testing (APT) project to compare the performance of stabilized ...
Accelerated pavement testing of low-volume paved roads with geocell reinforcement.
The Midwest States Accelerated Pavement Testing Pooled-Fund Program, financed by the highway : departments of Kansas, Iowa, Missouri, and New York, has supported an accelerated pavement testing (APT) project : to study the rehabilitation of low-volum...
Effectiveness of noise barriers installed adjacent to transverse grooved concrete pavement : executive summary report.
In recent years the Ohio Department of Transportation (ODOT) has reconstructed a number of roadways where asphalt pavements were replaced with concrete pavements which were finished with a random transverse grooved surface texture (ODOT specification...
Evaluation of concrete pavements with materials-related distress : appendix B.
An evaluation of cores sampled from six concrete pavements was performed. Factors contributing to pavement distress observed in the field were determined, including expansive alkali-silica reactivity and freeze-thaw deterioration related to poor entr...
Considerations for rigid vs. flexible pavement designs when allowed as alternate bids : technical report.
This report documents the research conducted for the Texas Department of Transportation (TxDOT) : concerning the use of pavement alternates. The scope of the project includes reviewing the state-of-thepractice : in methods used for pavement type sele...
Nondestructive pavement evaluation using finite element analysis based soft computing models.
Evaluating structural condition of existing, in-service pavements constitutes annually a major part of the : maintenance and rehabilitation activities undertaken by State Highway Agencies (SHAs). Accurate : estimation of pavement geometry and layer m...
Alternate bidding strategies for asphalt and concrete pavement projects utilizing life cycle cost analysis (LCCA).
Recent changes in pavement materials costs have impacted the competitive environment relative to the : determination of the most cost effective pavement structure for a specific highway project. In response, State : highway agencies have renewed thei...
A methodology to support the development of 4-year pavement management plan.
A methodology for forming and prioritizing pavement maintenance and rehabilitation (M&R) projects was developed. : The Texas Department of Transportation (TxDOT) can use this methodology to generate defensible and cost-effective : 4-year pavement man...
Evaluation of concrete pavements with materials-related distress : appendix C.
Comparative Study of Flexible Pavement Designs for New Road in Great Britain and Indonesia
Basri, Hanafi
The objective of this article is to compare the design methods for flexible pavements for new roads and overlays between two countries, Great Britain and Indonesia.Pavement design procedures provide a structure that can be accepted in a specific environmental condition and able to satisfy the anticipated traffic loading. There are many factors influencing the thickness design of pavement, such as pavement materials, traffic load and climatic conditions, there is no doubt that the method of pa...
Using a constructive pavement texture index for skid resistance screening
Chia-Pei Chou
Full Text Available Skid resistance measurement is one of the major measurements used to assess road safety. Typical devices used for measuring skid resistance include single point checking type and continuous measurements type. However, conducting periodic full roadway network skid resistance survey can be a relatively time consuming and difficult task because of the limited water carrying capacity used by all kinds of devices, including the continuous types. On the contrary, measurement of pavement surface texture can be conducted easily and continuously by most profilers. This research investigates the relation between pavement skid resistance and surface texture. The focus is on the development of a texture index that can serve as a screening indicator to filter out the high-risk pavement sections for further skid resistance measurement. Two measuring devices, GripTester (GT and National Taiwan University laser profiler (NTU Profiler, were used to collect data on pavement skid resistance and texture in a network of 555 sections, respectively. The pavement texture index of the mean difference of elevation (MDE is calculated at a 1Â mm profile interval, which is a very promising indicator to represent pavement skid resistance. Through a sensitivity study, the MDE threshold of 0.15Â mm is observed to provide a relatively reasonable screening function that filters out 29.01% of the entire surveyed network for further skid resistance inspection. It averted more than 70% skid resistance measurement capacity of the entire network. The accuracy of correct judgment by the selected threshold is about 80%, only 3.78% of roadway sections considered for further inspection of the recommended list are not included. Keywords: Skid resistance, Pavement texture, Macrotexture, Laser profiler, GripTester
Temperature prediction model of asphalt pavement in cold regions based on an improved BP neural network
Xu, Bo; Dan, Han-Cheng; Li, Liang
Highlights: • Pavement temperature prediction model is presented with improved BP neural network. • Dynamic and static methods are presented to predict pavement temperature. • Pavement temperature can be excellently predicted in next 3 h. - Abstract: Ice cover on pavement threatens traffic safety, and pavement temperature is the main factor used to determine whether the wet pavement is icy or not. In this paper, a temperature prediction model of the pavement in winter is established by introducing an improved Back Propagation (BP) neural network model. Before the application of the BP neural network model, many efforts were made to eliminate chaos and determine the regularity of temperature on the pavement surface (e.g., analyze the regularity of diurnal and monthly variations of pavement temperature). New dynamic and static prediction methods are presented by improving the algorithms to intelligently overcome the prediction inaccuracy at the change point of daily temperature. Furthermore, some scenarios have been compared for different dates and road sections to verify the reliability of the prediction model. According to the analysis results, the daily pavement temperatures can be accurately predicted for the next 3 h from the time of prediction by combining the dynamic and static prediction methods. The presented method in this paper can provide technical references for temperature prediction of the pavement and the development of an early-warning system for icy pavements in cold regions.
Integrating Pavement Crack Detection and Analysis Using Autonomous Unmanned Aerial Vehicle Imagery
INTEGRATING PAVEMENT CRACK DETECTION AND ANALYSIS USING AUTONOMOUS UNMANNED AERIAL VEHICLE...protection in the United States. AFIT-ENV-MS-15-M-195 INTEGRATING PAVEMENT CRACK DETECTION AND ANALYSIS USING AUTONOMOUS UNMANNED AERIAL...APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. AFIT-ENV-MS-15-M-195 INTEGRATING PAVEMENT CRACK DETECTION AND ANALYSIS USING AUTONOMOUS
Prediction of IRI in short and long terms for flexible pavements: ANN and GMDH methods
Ziari, H.; Sobhani, J.; Ayoubinejad, J.; Hartmann, Timo
Prediction of pavement condition is one of the most important issues in pavement management systems. In this paper, capabilities of artificial neural networks (ANNs) and group method of data handling (GMDH) methods in predicting flexible pavement conditions were analysed in three levels: in 1 year,
Effects of moving dynamic tyre loads on tyre-pavement contact stresses
Full Text Available The purpose of this paper is to indicate the effect that moving dynamic tyre loads has on the tyre-pavement contact stresses used in pavement analysis. Traditionally tyre loads (in pavement analysis) are modelled as constant loads applied through...
Toward using tire-road contact stresses in pavement design and analysis
Full Text Available pavement design. Four different tire models were used to represent a single tire type in order to demonstrate its effect on road pavement response of a typical South African pavement structure. Only applied vertical stress was used for the analyses...
Evaluation of the MIT-Scan-T2 for non-destructive PCC pavement thickness determination.
The MIT-Scan-T2 device is marketed as a non-destructive way to determine pavement thickness on both : HMA and PCC pavements. PCC pavement thickness determination is an important incentivedisincentive : measurement for the Iowa DOT and contractors. Th...
Evaluation of MIT-SCAN-T2 for Thickness Quality Control for PCC and HMA Pavements
Thickness is currently a pay item for PCC pavements and a quality control item for both PCC and HMA pavements. A change in pavement thickness of 0.5 in. can result in a change of multiple years of service. Current thickness measurements are performed...
Application of the portable pavement seismic analyser (PSPA) for pavement analysis
Full Text Available at the selected locations were compared using a one-way ANOVA and a correlation matrix to analyse the repeatability of the PSPA data, and to determine whether there was a significant difference between the three observations obtained at a specific location... behaviour from pavement layers, the PSPA measurements were conducted at 90° angles at the same location. An ANOVA test and correlation matrix was set up for each point to evaluate the isotropy of the moduli. Unfortunately, no comparative data (such as FWD...
Stormwater quality of spring-summer-fall effluent from three partial-infiltration permeable pavement systems and conventional asphalt pavement.
Drake, Jennifer; Bradford, Andrea; Van Seters, Tim
This study examined the spring, summer and fall water quality performance of three partial-infiltration permeable pavement (PP) systems and a conventional asphalt pavement in Ontario. The study, conducted between 2010 and 2012, compared the water quality of effluent from two Interlocking Permeable Concrete Pavements (AquaPave(®) and Eco-Optiloc(®)) and a Hydromedia(®) Pervious Concrete pavement with runoff from an Asphalt control pavement. The usage of permeable pavements can mitigate the impact of urbanization on receiving surface water systems through quantity control and stormwater treatment. The PP systems provided excellent stormwater treatment for petroleum hydrocarbons, total suspended solids, metals (copper, iron, manganese and zinc) and nutrients (total-nitrogen and total-phosphorus) by reducing event mean concentrations (EMC) as well as total pollutant loadings. The PPs significantly reduced the concentration and loading of ammonia (NH4(+)+NH3), nitrite (NO2(-)) and organic-nitrogen (Org-N) but increased the concentration and loading of nitrate (NO3(-)). The PP systems had mixed performances for the treatment of phosphate (PO4(3-)). The PP systems increased the concentration of sodium (Na) and chloride (Cl) but EMCs remained well below recommended levels for drinking water quality. Relative to the observed runoff, winter road salt was released more slowly from the PP systems resulting in elevated spring and early-summer Cl and Na concentrations in effluent. PP materials were found to introduce dissolved solids into the infiltrating stormwater. The release of these pollutants was verified by additional laboratory scale testing of the individual pavement and aggregate materials at the University of Guelph. Pollutant concentrations were greatest during the first few months after construction and declined rapidly over the course of the study. Copyright © 2014 Elsevier Ltd. All rights reserved.
Coal-tar based pavement sealant toxicity to freshwater macroinvertebrates
Bryer, Pamela J.; Scoggins, Mateo; McClintock, Nancy L.
Non-point-source pollution is a major source of ecological impairment in urban stream systems. Recent work suggests that coal-tar pavement sealants, used extensively to protect parking areas, may be contributing a large portion of the polycyclic aromatic hydrocarbon (PAH) loading seen in urban stream sediments. The hypothesis that dried coal-tar pavement sealant flake could alter the macroinvertebrate communities native to streams in Austin, TX was tested using a controlled outdoor laboratory type approach. The treatment groups were: control, low, medium, and high with total PAH concentrations (TPAH = sum of 16 EPA priority pollutant PAHs) of 0.1, 7.5, 18.4, and 300 mg/kg respectively. The low, medium, and high treatments were created via the addition of dried coal-tar pavement sealant to a sterile soil. At the start of the 24-day exposure, sediment from a minimally impacted local reference site containing a community of live sediment-dwelling benthic macroinvertebrates was added to each replicate. An exposure-dependent response was found for several stream health measures and for several individual taxa. There were community differences in abundance (P = 0.0004) and richness (P < 0.0001) between treatments in addition to specific taxa responses, displaying a clear negative relationship with the amount of coal-tar sealant flake. These results support the hypothesis that coal-tar pavement sealants contain bioavailable PAHs that may harm aquatic environments. - Coal-tar pavement sealants degrade stream invertebrate communities.
Update to Permeable Pavement Research at the Edison ...
The EPA's Urban Watershed Management Branch (UWMB) has been monitoring the permeable pavement demonstration site at the Edison Environmental Center, NJ since 2010. This site has three different types of permeable pavements including interlocking concrete permeable pavers, pervious concrete, and porous asphalt. The permeable pavements are limited to parking spaces while adjacent driving lanes are impermeable and drain to the permeable surfaces. The parking lot is instrumented for continuous monitoring with thermistors and water content reflectometers that measure moisture as infiltrate passes through the storage gallery beneath the permeable pavements into the underlying native soil. Each permeable surface of the parking lot has four lined sections that capture infiltrate in tanks for water quality analyses; these tanks are capable of holding volumes up to 4.1 m3, which represents up to 38 mm (1.5 in.) for direct rainfall on the porous pavement and runoff from adjacent driving lanes that drain into the permeable surface.Previous technical releases concerning the demonstration site focused on monitoring techniques, observed chloride and nutrient concentrations, surface hydrology, and infiltration and evaporation rates. This presentation summarizes these past findings and addresses current water quality efforts including pH, solids analysis, total organic carbon, and chemical oxygen demand. Stormwater runoff continues to be a major cause of water pollution in
A User Friendly Software for Rigid Pavement Design
Aydın Kıcı
Full Text Available Concrete pavements as concrete road slab, appear as a strong alternative for flexible superstructures especially because of their low cost for maintenance and repair and the high performance they show under heavy axle loads. The design of these concrete road slabs is quite different from the traditional concrete and reinforced concrete structures' design. In the design for this kind of pavements, traffic conditions should be defined properly and considered an addition to the concrete and platform properties. There have been designing methods developed based on both experimental and mechanic foundations for concrete pavements. The most important ones of these methods are AASHTO (1993 and PCA (1984. In both design methods, it's significant to know the maximum deflections and maximum strains the vehicles' loads cause on the pavement. The calculation of this maximum deflections and maximum strains can be done via the finite element method or the closed formulas which have been developed by Westergaard. In this study, a user-friendly software has been developed based on AASHTO 1993, PCA 1984 design methods and Westergaard formulas. Thanks to this software, the user who wants to design the concrete pavement as concrete road slab can obtain the essential parameters automatically by entering the required data for the design.
Perpetual pavement – absorbing stress and functional maintenance
Rong Gao
Full Text Available Perpetual Pavement combines the well documented smoothness and safety advantages of asphalt with an advanced, multi-layer paving design process, that with routine maintenance, extends the useful life of a roadway. Perpetual provides long lasting road and smoothness for the construction purposes. This study has the design key points of perpetual pavement based on the idea of life cycle, which has a new direction for the new highway construction, reconstruction and expansion. First, the structure of long life pavement design is studied to analyze the effect of stress absorbing layer. Second, researches on stress absorbing layer from the aspects of raw materials, mix proportion are implemented. Third, the design index of stress absorbing layer is determined by the shear strength test. The results show that the design idea of composite perpetual pavement can be realized by reasonable design of the stress absorbing layer and carrying out the surface functional maintenance can ensure the pavement to avoid structural damage in the operation stage.
Urban evaporation rates for water-permeable pavements.
Starke, P; Göbel, P; Coldewey, W G
In urban areas the natural water balance is disturbed. Infiltration and evaporation are reduced, resulting in a high surface runoff and a typical city climate, which can lead to floods and damages. Water-permeable pavements have a high infiltration rate that reduces surface runoff by increasing the groundwater recharge. The high water retention capacity of the street body of up to 51 l/m(2) and its connection via pores to the surface lead to higher evaporation rates than impermeable surfaces. A comparison of these two kinds of pavements shows a 16% increase in evaporation levels of water-permeable pavements. Furthermore, the evaporation from impermeable pavements is linked directly to rain events due to fast-drying surfaces. Water-permeable pavements show a more evenly distributed evaporation after a rain event. Cooling effects by evaporative heat loss can improve the city climate even several days after rain events. On a large scale use, uncomfortable weather like sultriness or dry heat can be prevented and the urban water balance can be attenuated towards the natural.
Experimental Study on Color Durability of Color Asphalt Pavement
Ning, Shi; Huan, Su
Aiming at the poor Color durability and the lack of research on Color asphalt pavement, spraying an anti-tire trace seal resin emulsion on the surface, a Color durable asphalt pavement was proposed. After long-term rolling and long-term aging test, the Color durability was evaluated by RGB function in Photoshop and trace residue rate formula. Test results proved that the Evaluation method was simple and effective. After long-term rolling, the Color of the road surface tends to a constant value. Spraying the emulsion on the road surface can resist tire traces. After long-term aging test, the resistance to tire traces was increased by 26.6% compared with the conventional type, while the former was 44.1% higher than the latter without long-term aging. The Color durable asphalt pavement can effectively improve the ability of Color asphalt pavement to resist tire traces, and significantly improve the Color durability of Color asphalt pavement.
Automatic Pavement Crack Recognition Based on BP Neural Network
Li Li
Full Text Available A feasible pavement crack detection system plays an important role in evaluating the road condition and providing the necessary road maintenance. In this paper, a back propagation neural network (BPNN is used to recognize pavement cracks from images. To improve the recognition accuracy of the BPNN, a complete framework of image processing is proposed including image preprocessing and crack information extraction. In this framework, the redundant image information is reduced as much as possible and two sets of feature parameters are constructed to classify the crack images. Then a BPNN is adopted to distinguish pavement images between linear and alligator cracks to acquire high recognition accuracy. Besides, the linear cracks can be further classified into transversal and longitudinal cracks according to the direction angle. Finally, the proposed method is evaluated on the data of 400 pavement images obtained by the Automatic Road Analyzer (ARAN in Northern China and the results show that the proposed method seems to be a powerful tool for pavement crack recognition. The rates of correct classification for alligator, transversal and longitudinal cracks are 97.5%, 100% and 88.0%, respectively. Compared to some previous studies, the method proposed in this paper is effective for all three kinds of cracks and the results are also acceptable for engineering application.
Developing Pavement Distress Deterioration Models for Pavement Management System Using Markovian Probabilistic Process
Promothes Saha
Full Text Available In the state of Colorado, the Colorado Department of Transportation (CDOT utilizes their pavement management system (PMS to manage approximately 9,100 miles of interstate, highways, and low-volume roads. Three types of deterioration models are currently being used in the existing PMS: site-specific, family, and expert opinion curves. These curves are developed using deterministic techniques. In the deterministic technique, the uncertainties of pavement deterioration related to traffic and weather are not considered. Probabilistic models that take into account the uncertainties result in more accurate curves. In this study, probabilistic models using the discrete-time Markov process were developed for five distress indices: transverse, longitudinal, fatigue, rut, and ride indices, as a case study on low-volume roads. Regression techniques were used to develop the deterioration paths using the predicted distribution of indices estimated from the Markov process. Results indicated that longitudinal, fatigue, and rut indices had very slow deterioration over time, whereas transverse and ride indices showed faster deterioration. The developed deterioration models had the coefficient of determination (R2 above 0.84. As probabilistic models provide more accurate results, it is recommended that these models be used as the family curves in the CDOT PMS for low-volume roads.
Geo synthetic-reinforced Pavement systems; Sistemas de pavimentos reforzados con geosinteticos
Factors Influencing Stormwater Mitigation in Permeable Pavement
Chun Yan Liu
Full Text Available Permeable pavement (PP is used worldwide to mitigate surface runoff in urban areas. Various studies have examined the factors governing the hydrologic performance of PP. However, relatively little is known about the relative importance of these governing factors and the long-term hydrologic performance of PP. This study applied numerical models—calibrated and validated using existing experimental results—to simulate hundreds of event-based and two long-term rainfall scenarios for two designs of PP. Based on the event-based simulation results, rainfall intensity, rainfall volume, thickness of the storage layer and the hydraulic conductivity of the subgrade were identified as the most influential factors in PP runoff reduction. Over the long term, PP performed significantly better in a relatively drier climate (e.g., New York, reducing nearly 90% of runoff volume compared to 70% in a relatively wetter climate (e.g., Hong Kong. The two designs of PP examined performed differently, and the difference was more apparent in the relatively wetter climate. This study generated insights that will help the design and implementation of PP to mitigate stormwater worldwide.
Mechanochemical Polarization of Contiguous Cell Walls Shapes Plant Pavement Cells.
Majda, Mateusz; Grones, Peter; Sintorn, Ida-Maria; Vain, Thomas; Milani, Pascale; Krupinski, Pawel; Zagórska-Marek, Beata; Viotti, Corrado; Jönsson, Henrik; Mellerowicz, Ewa J; Hamant, Olivier; Robert, Stéphanie
The epidermis of aerial plant organs is thought to be limiting for growth, because it acts as a continuous load-bearing layer, resisting tension. Leaf epidermis contains jigsaw puzzle piece-shaped pavement cells whose shape has been proposed to be a result of subcellular variations in expansion rate that induce local buckling events. Paradoxically, such local compressive buckling should not occur given the tensile stresses across the epidermis. Using computational modeling, we show that the simplest scenario to explain pavement cell shapes within an epidermis under tension must involve mechanical wall heterogeneities across and along the anticlinal pavement cell walls between adjacent cells. Combining genetics, atomic force microscopy, and immunolabeling, we demonstrate that contiguous cell walls indeed exhibit hybrid mechanochemical properties. Such biochemical wall heterogeneities precede wall bending. Altogether, this provides a possible mechanism for the generation of complex plant cell shapes. Copyright © 2017 Elsevier Inc. All rights reserved.
Application of Artificial Intelligence for Optimization in Pavement Management
Reus Salini
Full Text Available Artificial intelligence (AI is a group of techniques that have quite a potential to be applied to pavement engineering and management. In this study, we developed a practical, flexible and out of the box approach to apply genetic algorithms to optimizing the budget allocation and the road maintenance strategy selection for a road network. The aim is to provide an alternative to existing software and better fit the requirements of an important number of pavement managers. To meet the objectives, a new indicator, named Road Global Value Index (RGVI, was created to contemplate the pavement condition, the traffic and the economic and political importance for each and every road section. This paper describes the approach and its components by an example confirming that genetic algorithms are very effective for the intended purpose.
Performance of Kaolin Clay on the Concrete Pavement
Abdullah, M. E.; Jaya, R. P.; Shahafuddin, M. N. A.; Yaacob, H.; Ibrahim, M. H. Wan; Nazri, F. M.; Ramli, N. I.; Mohammed, A. A.
This paper investigates the performance of concrete pavement containing kaolin clay with their engineering properties and to determine the optimum kaolin clay content. The concrete used throughout the study was designed as grade 30 MPa strength with constant water to cement ratio of 0.49. The compressive strength, flexural strength and water absorption test was conducted in this research. The concrete mix designed with kaolin clay as cement replacement comprises at 0%, 5%, 10% and 15% by the total weight of cement. The results indicate that the strength of pavement concrete decreases as the percentage of kaolin clay increases. It also shows that the water absorption increases with the percentage of cement replacement. However, 5% kaolin clay is found to be the optimum level to replace cement in a pavement concrete.
Pavement Subgrade Performance Study in the Danish Road Testing Machine
Ullidtz, Per; Ertman Larsen, Hans Jørgen
Most existing pavement subgrade criteria are based on the AASHO Road Test, where only one material was tested and for only one climatic condition. To study the validity of these criteria and to refine the criteria a co-operative research program entitled the "International Pavement Subgrade...... Performance Study" was sponsored by the FHWA with American, Finnish and Danish partners. This paper describes the first test series which was carried out in the Danish Road Testing Machine (RTM).The first step in this program is a full scale test on an instrumented pavement in the Danish Road Testing Machine....... Pressure gauges and strain cells were installed in the upper part of the subgrade, for measuring stresses and strains in all three directions. During and after construction FWD testing was carried out to evaluate the elastic parameters of the materials. These parameters were then used with the theory...
Recycling of Reclaimed Asphalt Pavement in Portland Cement Concrete
Salim Al-Oraimi
Full Text Available Reclaimed Asphalt Pavement (RAP is the result of removing old asphalt pavement material. RAP consists of high quality well-graded aggregate coated with asphalt cement. The removal of asphalt concrete is done for reconstruction purposes, resurfacing, or to obtain access to buried utilities. The disposal of RAP represents a large loss of valuable source of high quality aggregate. This research investigates the properties of concrete utilizing recycled reclaimed asphalt pavement (RAP. Two control mixes with normal aggregate were designed with water cement ratios of 0.45 and 0.5. The control mixes resulted in compressive strengths of 50 and 33 MPa after 28 days of curing. The coarse fraction of RAP was used to replace the coarse aggregate with 25, 50, 75, and 100% for both mixtures. In addition to the control mix (0%, the mixes containing RAP were evaluated for slump, compressive strength, flexural strength, and modulus of elasticity. Durability was evaluated using surface absorption test.
Determination of Pavement Rehabilitation Activities through a Permutation Algorithm
Sangyum Lee
Full Text Available This paper presents a mathematical programming model for optimal pavement rehabilitation planning. The model maximized the rehabilitation area through a newly developed permutation algorithm, based on the procedures outlined in the harmony search (HS algorithm. Additionally, the proposed algorithm was based on an optimal solution method for the problem of multilocation rehabilitation activities on pavement structure, using empirical deterioration and rehabilitation effectiveness models, according to a limited maintenance budget. Thus, nonlinear pavement performance and rehabilitation activity decision models were used to maximize the objective functions of the rehabilitation area within a limited budget, through the permutation algorithm. Our results showed that the heuristic permutation algorithm provided a good optimum in terms of maximizing the rehabilitation area, compared with a method of the worst-first maintenance currently used in Seoul.
Pavement maintenance optimization model using Markov Decision Processes
Mandiartha, P.; Duffield, C. F.; Razelan, I. S. b. M.; Ismail, A. b. H.
This paper presents an optimization model for selection of pavement maintenance intervention using a theory of Markov Decision Processes (MDP). There are some particular characteristics of the MDP developed in this paper which distinguish it from other similar studies or optimization models intended for pavement maintenance policy development. These unique characteristics include a direct inclusion of constraints into the formulation of MDP, the use of an average cost method of MDP, and the policy development process based on the dual linear programming solution. The limited information or discussions that are available on these matters in terms of stochastic based optimization model in road network management motivates this study. This paper uses a data set acquired from road authorities of state of Victoria, Australia, to test the model and recommends steps in the computation of MDP based stochastic optimization model, leading to the development of optimum pavement maintenance policy.
Rheo-mechanical model for self-healing asphalt pavement
Gömze, A L; Gömze, L N
Examining the rheological properties of different asphalt mixtures at different temperatures, pressures and deformation conditions on the combined rheo-tribometers the authors have found that the generally used Burgers-model doesn't explain the deformation properties of asphalt mixtures and pavements under loading forces and loading pressures. To understand better the rheological and deformation properties of such complex materials like asphalt mixtures and pavements the authors used Malvern Mastersizer X laser granulometer, Bruker D8 Advance X-ray diffractometer, Hitachi TM 1000 Scanning Elektronmicroscope, Tristar 3000 specific surface tester and the combined rheo-tribometer developed and patented by the authors. After the complex investigation of different asphalt mixtures the authors have found a new, more complex rheological model for the asphalts including self-healing asphalt pavements. (paper)
Modified asphalt for pavements; Hosoyo kaishitsu asufuaruto ni tsuite
Tsukagoshi, T. [Nippon Oil Co. Ltd., Yokohama (Japan)
Modified asphalt has been used widely in such applications as countermeasure against rutting, countermeasure against wear caused by tire chains in snowy and cold areas, or bridge deck pavement. Features of various kinds of modified asphalt, standards, and standard properties are introduced. Modified asphalt containing natural asphalt is used for steel plate deck pavement. Semi-blown asphalt is used when emphasis must be given to the countermeasure for flowing resistance of asphalt pavement. Features and standards of asphalt containing rubber, thermoplastic elastomer, and thermoplastic resin are described. Asphalt containing heat-setting resin shows excellent characteristics, which other types of modified asphalt do not possess, in the laboratory resistance test for fatigue, flowing, and wear. Change in the history of modified asphalt in Japan from the initial stage to the present are explained and shown in a table together with time and phenomena, and the change in the production of modified asphalt is shown. 15 refs., 5 figs., 5 tabs.
Pavement system with rubber tire chips in subgrade
Ashtakala, B.; Hoque, A.K.M.M. [Concordia Univ., Montreal, PQ (Canada). Dept. of Civil Engineering
A pavement design method was developed in which shredded rubber tire chips mixed with sand were used as a material for pavement subgrade. Rubber tire chips are highly compressible and produce both elastic and plastic deformations under the application of loads. Sand was added to fill the void between the tire chips and make the mixture a strong material. The design method considered the vertical compressive strain produced by the design life traffic load 18k (80 KN) repetitions. The equivalent thicknesses of the layers above the subgrade corresponding to this vertical compressive strain were determined using contour charts. From this equivalent thickness, the thicknesses for asphalt pavement, base, and sub-base were determined by Odemark`s method. 3 refs., 1 tab., 3 figs.
RAPID INSPECTION OF PAVEMENT MARKINGS USING MOBILE LIDAR POINT CLOUDS
H. Zhang
Full Text Available This study aims at building a robust semi-automated pavement marking extraction workflow based on the use of mobile LiDAR point clouds. The proposed workflow consists of three components: preprocessing, extraction, and classification. In preprocessing, the mobile LiDAR point clouds are converted into the radiometrically corrected intensity imagery of the road surface. Then the pavement markings are automatically extracted with the intensity using a set of algorithms, including Otsu's thresholding, neighbor-counting filtering, and region growing. Finally, the extracted pavement markings are classified with the geometric parameters using a manually defined decision tree. Case studies are conducted using the mobile LiDAR dataset acquired in Xiamen (Fujian, China with different road environments by the RIEGL VMX-450 system. The results demonstrated that the proposed workflow and our software tool can achieve 93% in completeness, 95% in correctness, and 94% in F-score when using Xiamen dataset.
Tactile pavement for guiding walking direction: An assessment of heading direction and gait stability.
Pluijter, Nanda; de Wit, Lieke P W; Bruijn, Sjoerd M; Plaisier, Myrthe A
For maintaining heading direction while walking we heavily rely on vision. Therefore, walking in the absence of vision or with visual attention directed elsewhere potentially leads to dangerous situations. Here we investigated whether tactile information from the feet can be used as a (partial) substitute for vision in maintaining a stable heading direction. If so, participants should be better able to keep a constant heading direction on tactile pavement that indicates directionality than on regular flat pavement. However, such a pavement may also be destabilizing. Thus we asked participants to walk straight ahead on regular pavement, and on tactile pavement (tiles with ridges along the walking direction) while varying the amount of vision. We assessed the effects of the type of pavement as well as the amount of vision on the variability of the heading direction as well as gait stability. Both of these measures were calculated from accelerations and angular velocities recorded from a smartphone attached to the participants trunk. Results showed that on tactile pavement participants had a less variations in their heading direction than on regular pavement. The drawback, however, was that the tactile pavement used in this study decreased gait stability. In sum, tactile pavement can be used as a partial substitute for vision in maintaining heading direction, but it can also decrease gait stability. Future work should focus on designing tactile pavement that does provided directional clues, but is less destabilizing. Copyright © 2015 Elsevier B.V. All rights reserved.
Inverse Analysis of Pavement Structural Properties Based on Dynamic Finite Element Modeling and Genetic Algorithm
Xiaochao Tang
Full Text Available With the movement towards the implementation of mechanistic-empirical pavement design guide (MEPDG, an accurate determination of pavement layer moduli is vital for predicting pavement critical mechanistic responses. A backcalculation procedure is commonly used to estimate the pavement layer moduli based on the non-destructive falling weight deflectometer (FWD tests. Backcalculation of flexible pavement layer properties is an inverse problem with known input and output signals based upon which unknown parameters of the pavement system are evaluated. In this study, an inverse analysis procedure that combines the finite element analysis and a population-based optimization technique, Genetic Algorithm (GA has been developed to determine the pavement layer structural properties. A lightweight deflectometer (LWD was used to infer the moduli of instrumented three-layer scaled flexible pavement models. While the common practice in backcalculating pavement layer properties still assumes a static FWD load and uses only peak values of the load and deflections, dynamic analysis was conducted to simulate the impulse LWD load. The recorded time histories of the LWD load were used as the known inputs into the pavement system while the measured time-histories of surface central deflections and subgrade deflections measured with a linear variable differential transformers (LVDT were considered as the outputs. As a result, consistent pavement layer moduli can be obtained through this inverse analysis procedure.
Numerical Study of Tire Hydroplaning Based on Power Spectrum of Asphalt Pavement and Kinetic Friction Coefficient
Shengze Zhu
Full Text Available Hydroplaning is a driving phenomenon threating vehicle's control stability and safety. It happens when tire rolls on wet pavement with high speed that hydrodynamic force uplifts the tire. Accurate numerical simulation to reveal the mechanism of hydroplaning and evaluate the function of relevant factors in this process is significant. In order to describe the friction behaviors of tire-pavement interaction, kinetic friction coefficient curve of tire rubber and asphalt pavement was obtained by combining pavement surface power spectrum and complex modulus of tread rubber through Persson's friction theory. Finite element model of tire-fluid-pavement was established in ABAQUS, which was composed of a 225-40-R18 radial tire and three types of asphalt pavement covered with water film. Mechanical responses and physical behaviors of tire-pavement interaction were observed and compared with NASA equation to validate the applicability and accuracy of this model. Then contact force at tire-pavement interface and critical hydroplaning speed influenced by tire inflation pressure, water film thickness, and pavement types were investigated. The results show higher tire inflation pressure, thinner water film, and more abundant macrotexture enhancing hydroplaning speed. The results could be applied to predict hydroplaning speed on different asphalt pavement and improve pavement skid resistance design.
Functionality enhancement of industrialized optical fiber sensors and system developed for full-scale pavement monitoring.
Wang, Huaping; Liu, Wanqiu; He, Jianping; Xing, Xiaoying; Cao, Dandan; Gao, Xipeng; Hao, Xiaowei; Cheng, Hongwei; Zhou, Zhi
Pavements always play a predominant role in transportation. Health monitoring of pavements is becoming more and more significant, as frequently suffering from cracks, rutting, and slippage renders them prematurely out of service. Effective and reliable sensing elements are thus in high demand to make prognosis on the mechanical properties and occurrence of damage to pavements. Therefore, in this paper, various types of functionality enhancement of industrialized optical fiber sensors for pavement monitoring are developed, with the corresponding operational principles clarified in theory and the performance double checked by basic experiments. Furthermore, a self-healing optical fiber sensing network system is adopted to accomplish full-scale monitoring of pavements. The application of optical fiber sensors assembly and self-healing network system in pavement has been carried out to validate the feasibility. It has been proved that the research in this article provides a valuable method and meaningful guidance for the integrity monitoring of civil structures, especially pavements.
Finite element analysis of GFRP reinforced concrete pavement under static load
Li, Shiping; Hu, Chunhua
GFRP was more corrosion resistant than traditional reinforced, it is lightweight, high strength thermal expansion coefficient is more close to the concrete and a poor conductor of electromagnetic. Therefore, the use of GFRP to replace the traditional reinforcement in concrete pavement application has excellent practical value. This paper uses ANSYS to establish delamination and reinforcement of Pavement model and analyzed response of GFRP concrete and ordinary concrete pavement structural mechanics on effects of different factors under the action of static. The results showed that under static load, pavement surface layer presented similar changes on stress of surface layer, vertical and horizontal deformation in two kinds of pavement structure, but indicators of GFRP reinforced concrete pavement were obviously better than that of ordinary concrete pavement.
Study on the Vehicle Dynamic Load Considering the Vehicle-Pavement Coupled Effect
The vibration of vehicle-pavement interaction system is sophisticated random vibration process and the vehicle-pavement coupled effect was not considered in the previous study. A new linear elastic model of the vehicle-pavement coupled system was established in the paper. The new model was verified with field measurement which could reflect the real vibration between vehicle and pavement. Using the new model, the study on the vehicle dynamic load considering the vehicle-pavement coupled effect showed that random forces (centralization) between vehicle and pavement were influenced largely by vehicle-pavement coupled effect. Numerical calculation indicated that the maximum of random forces in coupled model was 2.4 times than that in uncoupled model. Inquiring the reason, it was found that the main vibration frequency of the vehicle non-suspension system was similar with that of the vehicle suspension system in the coupled model and the resonance vibration lead to vehicle dynamic load increase significantly.
Functionality Enhancement of Industrialized Optical Fiber Sensors and System Developed for Full-Scale Pavement Monitoring
Huaping Wang
Full Text Available Pavements always play a predominant role in transportation. Health monitoring of pavements is becoming more and more significant, as frequently suffering from cracks, rutting, and slippage renders them prematurely out of service. Effective and reliable sensing elements are thus in high demand to make prognosis on the mechanical properties and occurrence of damage to pavements. Therefore, in this paper, various types of functionality enhancement of industrialized optical fiber sensors for pavement monitoring are developed, with the corresponding operational principles clarified in theory and the performance double checked by basic experiments. Furthermore, a self-healing optical fiber sensing network system is adopted to accomplish full-scale monitoring of pavements. The application of optical fiber sensors assembly and self-healing network system in pavement has been carried out to validate the feasibility. It has been proved that the research in this article provides a valuable method and meaningful guidance for the integrity monitoring of civil structures, especially pavements.
Beneficial use of CFB ash in pavement construction applications
N. Mike Jackson; Scott Schultz; Paul Sander; Lindsay Schopp [Jackson Research Engineers, Inc., Ponte Vedra Beach, FL (United States)
The disposal of ash produced from the combustion of solid fuels has been a major subject of research and product development for many years. An innovative application has recently been employed by JEA to recycle both the bottom ash and fly ash from two new circulating fluidized bed (CFB) boilers as a stabilizer for local sandy soils and pavement base course material. The results of laboratory testing and field applications in the north Florida market area illustrate how this by-product is adding value in pavement and roadway construction applications. 14 refs., 2 figs., 10 tabs.
Calibrating mechanistic-empirical pavement performance models with an expert matrix
Tighe, S.; AlAssar, R.; Haas, R. [Waterloo Univ., ON (Canada). Dept. of Civil Engineering; Zhiwei, H. [Stantec Consulting Ltd., Cambridge, ON (Canada)
Proper management of pavement infrastructure requires pavement performance modelling. For the past 20 years, the Ontario Ministry of Transportation has used the Ontario Pavement Analysis of Costs (OPAC) system for pavement design. Pavement needs, however, have changed substantially during that time. To address this need, a new research contract is underway to enhance the model and verify the predictions, particularly at extreme points such as low and high traffic volume pavement design. This initiative included a complete evaluation of the existing OPAC pavement design method, the construction of a new set of pavement performance prediction models, and the development of the flexible pavement design procedure that incorporates reliability analysis. The design was also expanded to include rigid pavement designs and modification of the existing life cycle cost analysis procedure which includes both the agency cost and road user cost. Performance prediction and life-cycle costs were developed based on several factors, including material properties, traffic loads and climate. Construction and maintenance schedules were also considered. The methodology for the calibration and validation of a mechanistic-empirical flexible pavement performance model was described. Mechanistic-empirical design methods combine theory based design such as calculated stresses, strains or deflections with empirical methods, where a measured response is associated with thickness and pavement performance. Elastic layer analysis was used to determine pavement response to determine the most effective design using cumulative Equivalent Single Axle Loads (ESALs), below grade type and layer thickness.The new mechanistic-empirical model separates the environment and traffic effects on performance. This makes it possible to quantify regional differences between Southern and Northern Ontario. In addition, roughness can be calculated in terms of the International Roughness Index or Riding comfort Index
Bearing capacity evaluation of rubblized concrete pavements
González, M.
Full Text Available The paper presents the findings of a research work performed on a real scale concrete pavement project where Rubblizing technology was used for its structural rehabilitation. Rubblizing may be defined as a fracture technique in which a concrete pavement slab is transformed in a granular base with a very high Modulus. This technique, fractures the concrete slab in angular pieces by using a concentrated dynamic load of low amplitude and high frequency. The research work was based on field study on the rehabilitation of 5 km motorway. The structural evaluations where made, before, during and after one year construction. Measurements and site evaluation where made by using DCP, Light Weight Deflectometer and FWD (on top of asphalt layer and excavating inside pits. The structural capacity of the Rubblized layer was evaluated through theoretical analysis. Because of the anisotropic properties of the Rubblized layer the results are presented using AASHTO structural layer coefficient. The structural layer coefficients recommended are between the range of 0.25 and 0.30 for concrete slabs with thickness grater than 220 mm.El trabajo presenta los resultados de un estudio a escala real de la capacidad estructural de un firme de hormigón rehabilitado utilizando la técnica de Rubblizing. La técnica de Rubblizing ha sido traducida como el pulverizado del firme de hormigón pero, es más bien un efecto combinado de trituración y fracturación de la losa de hormigón en todo su espesor para convertir esta en una base granular de alto módulo. Esta tecnica fractura la losa de hormigón en trozos angulares y entrelazados empleando una carga dinámica concentrada, de baja amplitud y alta frecuencia. La investigación se basó en el estudio de la rehabilitación de 5 km de autopista. Los estudios de la capacidad estructural fueron realizados durante, al término y un año después de la construcción. Para las mediciones y evaluaciones de terreno se utilizó, el
Measuring Clogging with Pressure Transducers in Permeable Pavement Strips
Two issues that have a negative affect on the long term hydrologic performance of permeable pavement systems are surface clogging and clogging at the interface with the underlying soil. Surface clogging limits infiltration capacity and results in bypass if runoff rate exceeds in...
Modeling the Hydrologic Processes of a Permeable Pavement ...
A permeable pavement system can capture stormwater to reduce runoff volume and flow rate, improve onsite groundwater recharge, and enhance pollutant controls within the site. A new unit process model for evaluating the hydrologic performance of a permeable pavement system has been developed in this study. The developed model can continuously simulate infiltration through the permeable pavement surface, exfiltration from the storage to the surrounding in situ soils, and clogging impacts on infiltration/exfiltration capacity at the pavement surface and the bottom of the subsurface storage unit. The exfiltration modeling component simulates vertical and horizontal exfiltration independently based on Darcy's formula with the Green-Ampt approximation. The developed model can be arranged with physically-based modeling parameters, such as hydraulic conductivity, Manning's friction flow parameters, saturated and field capacity volumetric water contents, porosity, density, etc. The developed model was calibrated using high-frequency observed data. The modeled water depths are well matched with the observed values (R2 = 0.90). The modeling results show that horizontal exfiltration through the side walls of the subsurface storage unit is a prevailing factor in determining the hydrologic performance of the system, especially where the storage unit is developed in a long, narrow shape; or with a high risk of bottom compaction and clogging. This paper presents unit
An Assessment of the Models to Predict Pavement Performance
Data collected by the Iowa Department of Transportation (DOT) regarding road conditions across the state of Iowa were used to model pavement condition index (PCI). The data were for calendar year 2013, with the exception of updated PCI values from 20...
Performances of Metal Concentrations from Three Permeable Pavement Infiltrates
The U.S. Environmental Protection Agency designed and constructed a 4000-m2 parking lot in Edison, New Jersey in 2009. The parking lot is surfaced with three permeable pavements: permeable interlocking concrete pavers, pervious concrete, and porous asphalt. Water sampling was con...
Design Basis for Fibre Reinforced Concrete (FRC) Pavements
Bendixen, Søren; Stang, Henrik
-crack opening relationship can beused to descibe the properties of fibre reinforced concrete (FRC) intension and how the stress-crack opening relationship can beapplied in a simple design scheme for pavements. The projectincludes development of design tools, experiments to determine thestress-crack opening...
Monitoring Strategies in Permeable Pavement Systems to Optimize Maintenance Scheduling
As the surface in a permeable pavement system clogs and performance decreases, maintenance is required to preserve the design function. Currently, guidance is limited for scheduling maintenance on an as needed basis. Previous research has shown that surface clogging in a permea...
A review on automated pavement distress detection methods
Coenen, Tom B.J.; Golroo, Amir
In recent years, extensive research has been conducted on pavement distress detection. A large part of these studies applied automated methods to capture different distresses. In this paper, a literature review on the distresses and related detection methods are presented. This review also includes
Coal-tar based pavement sealant toxicity to freshwater macroinvertebrates.
Bryer, Pamela J; Scoggins, Mateo; McClintock, Nancy L
Non-point-source pollution is a major source of ecological impairment in urban stream systems. Recent work suggests that coal-tar pavement sealants, used extensively to protect parking areas, may be contributing a large portion of the polycyclic aromatic hydrocarbon (PAH) loading seen in urban stream sediments. The hypothesis that dried coal-tar pavement sealant flake could alter the macroinvertebrate communities native to streams in Austin, TX was tested using a controlled outdoor laboratory type approach. The treatment groups were: control, low, medium, and high with total PAH concentrations (TPAH = sum of 16 EPA priority pollutant PAHs) of 0.1, 7.5, 18.4, & 300 mg/kg respectively. The low, medium, and high treatments were created via the addition of dried coal-tar pavement sealant to a sterile soil. At the start of the 24-day exposure, sediment from a minimally impacted local reference site containing a community of live sediment-dwelling benthic macroinvertebrates was added to each replicate. An exposure-dependent response was found for several stream health measures and for several individual taxa. There were community differences in abundance (P = 0.0004) and richness (P pavement sealants contain bioavailable PAHs that may harm aquatic environments. Copyright 2009 Elsevier Ltd. All rights reserved.
Nitrogen Transformations in Three Types of Permeable Pavement
In 2009, USEPA constructed a 0.4-ha (1-ac) parking lot at the Edison Environmental Center in Edison, NJ, that incorporated three different permeable pavement types - permeable interlocking concrete pavers (PICP), pervious concrete (PC), and porous asphalt (PA). The driving lanes...
The Edison Environmental Center Permeable Pavement Site - slides
This is a presentation for a second Community Outreach Event called "Chemistry Works!" at West Windsor Public Library on Saturday, November 5th. It will review the permeable pavement research project at the Edison Environmental center. Besides slide persentation, two demo units w...
Bryer, P.J.; Scoggins, M.; McClintock, N.L. [Lamar University, Beaumont, TX (United States). Dept. of Biology
Non-point-source pollution is a major source of ecological impairment in urban stream systems. Recent work suggests that coal-tar pavement sealants, used extensively to protect parking areas, may be contributing a large portion of the polycyclic aromatic hydrocarbon (PAH) loading seen in urban stream sediments. The hypothesis that dried coal-tar pavement sealant flake could alter the macroinvertebrate communities native to streams in Austin, TX was tested using a controlled outdoor laboratory type approach. The treatment groups were: control, low, medium, and high with total PAH concentrations (TPAH = sum of 16 EPA priority pollutant PAHs) of 0.1, 7.5, 18.4, & 300 mg/kg respectively. The low, medium, and high treatments were created via the addition of dried coal-tar pavement sealant to a sterile soil. At the start of the 24-day exposure, sediment from a minimally impacted local reference site containing a community of live sediment-dwelling benthic macroinvertebrates was added to each replicate. An exposure-dependent response was found for several stream health measures and for several individual taxa. There were community differences in abundance (P = 0.0004) and richness (P < 0.0001) between treatments in addition to specific taxa responses, displaying a clear negative relationship with the amount of coal-tar sealant flake. These results support the hypothesis that coal-tar pavement sealants contain bioavailable PAHs that may harm aquatic environments.
Design of semi-rigid type of flexible pavements
Pranshoo Solanki
Full Text Available The primary objective of the study presented in this paper is to develop design curves for performance prediction of stabilized layers and to compare semi-rigid flexible pavement designs between the empirical AASHTO 1993 and the mechanistic-empirical pavement design methodologies. Specifically, comparisons were made for a range of different sections consisting of cementitious layers stabilized with different types and percentages of additives. It is found that the design thickness is influenced by the type of soil, additive, selection of material property and design method. Cost comparisons of sections stabilized with different percentage and type of additives showed that CKD-stabilization provides economically low cost sections as compared to lime- and CFA-stabilized sections. Knowledge gained from the parametric analysis of different sections using AASHTO 1993 and MEPDG is expected to be useful to pavement designers and others in implementation of the new MEPDG for future pavement design. Keywords: Semi-rigid, Mechanistic, Resilient modulus, Fatigue life, Reliability, Traffic
Recyclability of Concrete Pavement Incorporating High Volume of Fly Ash.
Yoshitake, Isamu; Ishida, Takeo; Fukumoto, Sunao
Recyclable concrete pavement was made from fly ash and crushed limestone sand and gravel as aggregates so that the concrete pavement could be recycled to raw materials for cement production. With the aim to use as much fly ash as possible for the sustainable development of society, while achieving adequate strength development, pavement concrete having a cement-replacement ratio of 40% by mass was experimentally investigated, focusing on the strength development at an early age. Limestone powder was added to improve the early strength; flexural strength at two days reached 3.5 MPa, the minimum strength for traffic service in Japan. The matured fly ash concrete made with a cement content of 200 kg/m3 achieved a flexural strength almost equal to that of the control concrete without fly ash. Additionally, Portland cement made from the tested fly ash concrete was tested to confirm recyclability, with the cement quality meeting the Japanese classification of ordinary Portland cement. Limestone-based recyclable fly ash concrete pavement is, thus, a preferred material in terms of sustainability.
Experimental continuously reinforced concrete pavement parameterization using nondestructive methods
L. S. Salles
Full Text Available ABSTRACT Four continuously reinforced concrete pavement (CRCP sections were built at the University of São Paulo campus in order to analyze the pavement performance in a tropical environment. The sections short length coupled with particular project aspects made the experimental CRCP cracking be different from the traditional CRCP one. After three years of construction, a series of nondestructive testing were performed - Falling Weight Deflectometer (FWD loadings - to verify and to parameterize the pavement structural condition based on two main properties: the elasticity modulus of concrete (E and the modulus of subgrade reaction (k. These properties estimation was obtained through the matching process between real and EverFE simulated basins with the load at the slab center, between two consecutive cracks. The backcalculation results show that the lack of anchorage at the sections end decreases the E and k values and that the longitudinal reinforcement percentage provides additional stiffness to the pavement. Additionally, FWD loadings tangential to the cracks allowed the load transfer efficiency (LTE estimation determination across cracks. The LTE resulted in values above 90 % for all cracks.
Measuring Carbon Footprint of Flexible Pavement Construction Project in Indonesia
Hatmoko, Jati Utomo Dwi; Hidayat, Arif; Setiawati, Apsari; Prasetyo, Stefanus Catur Adi
Road infrastructure in Indonesia is mainly dominated by flexible pavement type. Its construction process, however, has raised concerns in terms of its environment impacts. This study aims to track and measure the carbon footprint of flexible pavement. The objectives are to map the construction process in relation to greenhouse gas (GHG) emissions, to quantify them in terms of carbon dioxide equivalents (CO2e) as generated by the process of production and transportation of raw materials, and the operation of plant off-site and on-site project. Data collection was done by having site observations and interviews with project stakeholders. The results show a total emissions of 70.888 tonnes CO2e, consisting of 34.248 tonnes CO2e (48.31%) off-site activities and 36.640 tonnes CO2e (51.687%) on-site activities. The two highest CO2e emissions were generated by the use of plant for asphalt concrete laying activities accounted 34.827 tonnes CO2e (49.130%), and material transportation accounted 24.921 (35.155%). These findings provide a new perspective of the carbon footprint in flexible pavement and suggest the urgent need for the use of more efficient and environmentally friendly plant in construction process as it shows the most significant contribution on the CO2e. This study provides valuable understanding on the environmental impact of typical flexible pavement projects in Indonesia, and further can be used for developing green road framework.
Inferring pavement layer properties from a moving measurement platform
Andersen, Sebastian; Levenberg, Eyal; Andersen, M.B.
was achieved when the correct pavement model and accurate input data were employed. Also, when artificial errors were introduced for assessing sensitivities, only small deviations were generated. Based on these promising outcomes the method will be further studied and developed to support the construction...
Effects of Interlocking and Supporting Conditions on Concrete Block Pavements
Mahapatra, Geetimukta; Kalita, Kuldeep
Concrete Block Paving (CBP) is widely used as wearing course in flexible pavements, preferably under light and medium vehicular loadings. Construction of CBP at site is quick and easy in quality control. Usually, flexible pavement design philosophy is followed in CBP construction, though it is structurally different in terms of small block elements with high strength concrete and their interlocking aspects, frequent joints and discontinuity, restrained edge etc. Analytical solution for such group action of concrete blocks under loading in a three dimensional multilayer structure is complex and thus, the need of conducting experimental studies is necessitated for extensive understanding of the load—deformation characteristics and behavior of concrete blocks in pavement. The present paper focuses on the experimental studies for load transfer characteristics of CBP under different interlocking and supporting conditions. It is observed that both interlocking and supporting conditions affect significantly on the load transfer behavior in CBP structures. Coro-lock block exhibits better performance in terms of load carrying capacity and distortion behavior under static loads. Plate load tests are performed over subgrade, granular sub-base (GSB), CBP with and without GSB using different block shapes. For an example case, the comparison of CBP with conventional flexible pavement section is also presented and it is found that CBP provides considerable benefit in terms of construction cost of the road structure.
High-volume recycled materials for sustainable pavement construction.
The main objective of this research is to evaluate the feasibility of using high-volume recycled materials for concrete production in rigid pavement. The goal was to replace 50% of the solids with recycled materials and industrial by-products. The pe...
Influence Of Geology On Pavement Performance: A Case Study Of ...
Field and laboratory investigations which involved geologic mapping, auger boring, sampling and laboratory testing were carried out to determine the stratigraphy and geotechnical properties of the sub-grade and their relationship to the pavement condition. Field observation indicated that the road may have been based on ...
Road Materials and Pavement Design, 30 May 2017
Full Text Available and Pavement Design Vol. 18, Issue 1 2017 http://www.tandfonline.com/toc/trmp20/18/1?nav=tocList Editorial board EDITORS-IN-CHIEF Hervé Di Benedetto - University of Lyon / ENTPE, Vaulx-en-Velin, France Andrew Charles Collop - De Montfort University...
Evaluation of pavement life cycle cost analysis: Review and analysis
Peyman Babashamsi
Full Text Available The cost of road construction consists of design expenses, material extraction, construction equipment, maintenance and rehabilitation strategies, and operations over the entire service life. An economic analysis process known as Life-Cycle Cost Analysis (LCCA is used to evaluate the cost-efficiency of alternatives based on the Net Present Value (NPV concept. It is essential to evaluate the above-mentioned cost aspects in order to obtain optimum pavement life-cycle costs. However, pavement managers are often unable to consider each important element that may be required for performing future maintenance tasks. Over the last few decades, several approaches have been developed by agencies and institutions for pavement Life-Cycle Cost Analysis (LCCA. While the transportation community has increasingly been utilising LCCA as an essential practice, several organisations have even designed computer programs for their LCCA approaches in order to assist with the analysis. Current LCCA methods are analysed and LCCA software is introduced in this article. Subsequently, a list of economic indicators is provided along with their substantial components. Collecting previous literature will help highlight and study the weakest aspects so as to mitigate the shortcomings of existing LCCA methods and processes. LCCA research will become more robust if improvements are made, facilitating private industries and government agencies to accomplish their economic aims. Keywords: Life-Cycle Cost Analysis (LCCA, Pavement management, LCCA software, Net Present Value (NPV
Time series trends of the safety effects of pavement resurfacing.
Park, Juneyoung; Abdel-Aty, Mohamed; Wang, Jung-Han
This study evaluated the safety performance of pavement resurfacing projects on urban arterials in Florida using the observational before and after approaches. The safety effects of pavement resurfacing were quantified in the crash modification factors (CMFs) and estimated based on different ranges of heavy vehicle traffic volume and time changes for different severity levels. In order to evaluate the variation of CMFs over time, crash modification functions (CMFunctions) were developed using nonlinear regression and time series models. The results showed that pavement resurfacing projects decrease crash frequency and are found to be more safety effective to reduce severe crashes in general. Moreover, the results of the general relationship between the safety effects and time changes indicated that the CMFs increase over time after the resurfacing treatment. It was also found that pavement resurfacing projects for the urban roadways with higher heavy vehicle volume rate are more safety effective than the roadways with lower heavy vehicle volume rate. Based on the exploration and comparison of the developed CMFucntions, the seasonal autoregressive integrated moving average (SARIMA) and exponential functional form of the nonlinear regression models can be utilized to identify the trend of CMFs over time. Copyright © 2017 Elsevier Ltd. All rights reserved.
Evaluation of double drop beads pavement edge lines.
This report presents an evaluation of Double Drop Bead (DDB) edge lines used on ALDOT-maintained highways. It compares DDB to three other pavement marking types in terms of service lives, life-cycle costs, and both dry-night retroreflectivity and wet...
Texas perpetual pavements : experience overview and the way forward.
Since 2001, the State of Texas has been designing and constructing perpetual pavements on some of its heavily : trafficked highways where the expected 20-year truck-traffic estimate of 18 kip ESALs is in excess of 30 million. To : date, there are 10 ...
Recyclability of Concrete Pavement Incorporating High Volume of Fly Ash
Recyclable concrete pavement was made from fly ash and crushed limestone sand and gravel as aggregates so that the concrete pavement could be recycled to raw materials for cement production. With the aim to use as much fly ash as possible for the sustainable development of society, while achieving adequate strength development, pavement concrete having a cement-replacement ratio of 40% by mass was experimentally investigated, focusing on the strength development at an early age. Limestone powder was added to improve the early strength; flexural strength at two days reached 3.5 MPa, the minimum strength for traffic service in Japan. The matured fly ash concrete made with a cement content of 200 kg/m3 achieved a flexural strength almost equal to that of the control concrete without fly ash. Additionally, Portland cement made from the tested fly ash concrete was tested to confirm recyclability, with the cement quality meeting the Japanese classification of ordinary Portland cement. Limestone-based recyclable fly ash concrete pavement is, thus, a preferred material in terms of sustainability. PMID:28793518
Accelerated Performance Testing on the 2006 NCAT Pavement Test Track
The original National Center for Asphalt Technology (NCAT) Pavement Test Track was built in 2000 in Opelika, Alabama where it has served as a state-of-the-art, full-scale, closed-loop accelerated loading facility. The construction, operation, and res...
Use of fiber reinforced concrete for concrete pavement slab replacement.
Unlike ordinary concrete pavement, replacement concrete slabs need to be open to traffic within 24 hours (sooner in : some cases). Thus, high early-strength concrete is used; however, it frequently cracks prematurely as a result of high : heat of hyd...
Evaluation of concrete inlay for continuously reinforced concrete pavement rehabilitation.
In 1996, WisDOT constructed a concrete inlay test section on I43 in Manitowoc County. The existing pavement was CRCP constructed in 1978 and was badly deteriorated with punchouts. In the area of the 2777foot test section, the existing paveme...
Characterization of pavement distress from test pit observation
Mgangira, Martin B
Full Text Available the profiling exercise of an investigation into the causes of the premature failure of a pavement. The field investigation entailed a detailed inspection and recording of the profile exposed in test pits, with particular attention being paid to areas with cracks...
Road Materials and Pavement Design volume 17(2)
Full Text Available Materials and Pavement Design Volume 17, 2016 - Issue 2 Editorial Board Page ebi | Published online: 03 Oct 2016 http://dx.doi.org/10.1080/14680629.2016.1244475 Editors-in-Chief Hervé DI BENEDETTO - University of Lyon/ENTPE, Vaulx-en-Velin, France...
Optimisation of the Crack Pattern in Continuously Reinforced Concrete Pavements
Ren, D.
Recent field investigations on several new Continuously Reinforced Concrete Pavements (CRCP) in Belgium indicate that its crack pattern is characterized by low mean crack spacing along with a high percentage of clusters of closely spaced cracks. Field surveys also indicate that it is difficult to
Common Characteristics of Good and Poorly Performing AC Pavements
This report documents the analysis and findings of a study to identify the site conditions and design/construction features of : flexible pavements that lead to good performance and those that lead to poor performance. Data from the Long Term Pavemen...
Common Characteristics of Good and Poorly Performing PCC Pavements
This report documents the analysis and findings of a study to identify the site conditions and design/construction features of concrete pavements (JPCP, JRCP, CRCP) that lead to good performance and those that lead to poor performance. Data from Long...
Accelerated pavement testing efforts using the heavy vehicle simulator
Du Plessis, Louw
Full Text Available This paper provides a brief description of the technological developments involved in the development and use of the Heavy Vehicle Simulator (HVS) accelerated pavement testing equipment. This covers the period from concept in the late 1960's...
Representative volume element of asphalt pavement for electromagnetic measurements
Terhi Pellinen
Full Text Available The motivation for this study was to investigate the representative volume element (RVE needed to correlate the nondestructive electromagnetic (EM measurements with the conventional destructive asphalt pavement quality control measurements. A large pavement rehabilitation contract was used as the test site for the experiment. Pavement cores were drilled from the same locations where the stationary and continuous Ground Penetrating Radar (GPR measurements were obtained. Laboratory measurements included testing the bulk density of cores using two methods, the surface-saturated dry method and determining bulk density by dimensions. Also, Vector Network Analyzer (VNA and the through specimen transmission configuration were employed at microwave frequencies to measure the reference dielectric constant of cores using two different footprint areas and therefore volume elements. The RVE for EM measurements turns out to be frequency dependent; therefore in addition to being dependent on asphalt mixture type and method of obtaining bulk density, it is dependent on the resolution of the EM method used. Then, although the average bulk property results agreed with theoretical formulations of higher core air void content giving a lower dielectric constant, for the individual cores there was no correlation for the VNA measurements because the volume element seizes deviated. Similarly, GPR technique was unable to capture the spatial variation of pavement air voids measured from the 150-mm drill cores. More research is needed to determine the usable RVE for asphalt.
LADOTD pavement management system (PMS) for project level applications.
To fully address the research needs described in the problem statement, the primary objective of this project is to develop guidelines that provide information on how network level PMS data can be used at a project level in activities of pavement eng...
An assessment of the skid resistance effect on traffic safety under wet-pavement conditions.
Pardillo Mayora, José M; Jurado Piña, Rafael
Pavement-tire friction provides the grip that is required for maintaining vehicle control and for stopping in emergency situations. Statistically significant negative correlations of skid resistance values and wet-pavement accident rates have been found in previous research. Skid resistance measured with SCRIM and crash data from over 1750km of two-lane rural roads in the Spanish National Road System were analyzed to determine the influence of pavement conditions on safety and to assess the effects of improving pavement friction on safety. Both wet- and dry-pavement crash rates presented a decreasing trend as skid resistance values increased. Thresholds in SCRIM coefficient values associated with significant decreases in wet-pavement crash rates were determined. Pavement friction improvement schemes were found to yield significant reductions in wet-pavement crash rates averaging 68%. The results confirm the importance of maintaining adequate levels of pavement friction to safeguard traffic safety as well as the potential of pavement friction improvement schemes to achieve significant crash reductions.
A new structure of permeable pavement for mitigating urban heat island.
Liu, Yong; Li, Tian; Peng, Hangyu
The urban heat island (UHI) effect has been a great threat to human habitation, and how to mitigate this problem has been a global concern over decades. This paper addresses the cooling effect of a novel permeable pavement called evaporation-enhancing permeable pavement, which has capillary columns in aggregate and a liner at the bottom. To explore the efficiency of mitigating the UHI, bench-scale permeable pavement units with capillary columns were developed and compared with conventional permeable pavement. Criteria of capillary capacities of the column, evaporation rates, and surface temperature of the pavements were monitored under simulated rainfall and Shanghai local weather conditions. Results show the capillary column was important in increasing evaporation by lifting water from the bottom to the surface, and the evaporation-enhancing permeable pavement was cooler than a conventional permeable pavement by as much as 9.4°C during the experimental period. Moreover, the cooling effect of the former pavement could persist more than seven days under the condition of no further rainfall. Statistical analysis result reveals that evaporation-enhancing permeable pavement can mitigate the UHI effect significantly more than a conventional permeable pavement. Copyright © 2018 Elsevier B.V. All rights reserved.
The effect of loading time on flexible pavement dynamic response: a finite element analysis
Yin, Hao; Solaimanian, Mansour; Kumar, Tanmay; Stoffels, Shelley
Dynamic response of asphalt concrete (AC) pavements under moving load is a key component for accurate prediction of flexible pavement performance. The time and temperature dependency of AC materials calls for utilizing advanced material characterization and mechanistic theories, such as viscoelasticity and stress/strain analysis. In layered elastic analysis, as implemented in the new Mechanistic-Empirical Pavement Design Guide (MEPDG), the time dependency is accounted for by calculating the loading times at different AC layer depths. In this study, the time effect on pavement response was evaluated by means of the concept of "pseudo temperature.� With the pavement temperature measured from instrumented thermocouples, the time and temperature dependency of AC materials was integrated into one single factor, termed "effective temperature.� Via this effective temperature, pavement responses under a transient load were predicted through finite element analysis. In the finite element model, viscoelastic behavior of AC materials was characterized through relaxation moduli, while the layers with unbound granular material were assumed to be in an elastic mode. The analysis was conducted for two different AC mixtures in a simplified flexible pavement structure at two different seasons. Finite element analysis results reveal that the loading time has a more pronounced impact on pavement response in the summer for both asphalt types. The results indicate that for reasonable prediction of dynamic response in flexible pavements, the effect of the depth-dependent loading time on pavement temperature should be considered.
Summary report on the performance of open graded friction course quieter pavements : I-5 Lynnwood, SR-520 Medina, I-405 Bellevue.
This document summarizes the acoustic properties and pavement performance of three asphalt quieter pavement projects. Each of the projects included open graded friction course pavement built with sections of crumb rubber and polymer modified asphalt ...
Analytical approach for evaluating temperature field of thermal modified asphalt pavement and urban heat island effect
Chen, Jiaqi; Wang, Hao; Zhu, Hongzhou
Highlights: • Derive an analytical approach to predict temperature fields of multi-layered asphalt pavement based on Green's function. • Analyze the effects of thermal modifications on heat output from pavement to near-surface environment. • Evaluate pavement solutions for reducing urban heat island (UHI) effect. - Abstract: This paper aims to present an analytical approach to predict temperature fields in asphalt pavement and evaluate the effects of thermal modification on near-surface environment for urban heat island (UHI) effect. The analytical solution of temperature fields in the multi-layered pavement structure was derived with the Green's function method, using climatic factors including solar radiation, wind velocity, and air temperature as input parameters. The temperature solutions were validated with an outdoor field experiment. By using the proposed analytical solution, temperature fields in the pavement with different pavement surface albedo, thermal conductivity, and layer combinations were analyzed. Heat output from pavement surface to the near-surface environment was studied as an indicator of pavement contribution to UHI effect. The analysis results show that increasing pavement surface albedo could decrease pavement temperature at various depths, and increase heat output intensity in the daytime but decrease heat output intensity in the nighttime. Using reflective pavement to mitigate UHI may be effective for an open street but become ineffective for the street surrounded by high buildings. On the other hand, high-conductivity pavement could alleviate the UHI effect in the daytime for both the open street and the street surrounded by high buildings. Among different combinations of thermal-modified asphalt mixtures, the layer combination of high-conductivity surface course and base course could reduce the maximum heat output intensity and alleviate the UHI effect most.
Application of Finite Layer Method in Pavement Structural Analysis
Pengfei Liu
Full Text Available The finite element (FE method has been widely used in predicting the structural responses of asphalt pavements. However, the three-dimensional (3D modeling in general-purpose FE software systems such as ABAQUS requires extensive computations and is relatively time-consuming. To address this issue, a specific computational code EasyFEM was developed based on the finite layer method (FLM for analyzing structural responses of asphalt pavements under a static load. Basically, it is a 3D FE code that requires only a one-dimensional (1D mesh by incorporating analytical methods and using Fourier series in the other two dimensions, which can significantly reduce the computational time and required resources due to the easy implementation of parallel computing technology. Moreover, a newly-developed Element Energy Projection (EEP method for super-convergent calculations was implemented in EasyFEM to improve the accuracy of solutions for strains and stresses over the whole pavement model. The accuracy of the program is verified by comparing it with results from BISAR and ABAQUS for a typical asphalt pavement structure. The results show that the predicted responses from ABAQUS and EasyFEM are in good agreement with each other. The EasyFEM with the EEP post-processing technique converges faster compared with the results derived from ordinary EasyFEM applications, which proves that the EEP technique can improve the accuracy of strains and stresses from EasyFEM. In summary, the EasyFEM has a potential to provide a flexible and robust platform for the numerical simulation of asphalt pavements and can easily be post-processed with the EEP technique to enhance its advantages.
Prediction of performance and evaluation of flexible pavement rehabilitation strategies
Kang-Won Wayne Lee
Full Text Available Five test sections with different additives and strategies were established to rehabilitate a State-maintained highway more effectively in Rhode Island (RI: control, calcium chloride, asphalt emulsion, Portland cement and geogrid. Resilient moduli of subgrade soils and subbase materials before and after full depth rehabilitation were employed as input parameters to predict the performance of pavement structures using AASHTOWare Pavement ME Design (Pavement ME software in terms of rutting, cracking and roughness. It was attempted to use Level 1 input (which includes traffic full spectrum data, climate data and structural layer properties for Pavement ME. Traffic data was obtained from a Weigh-in-Motion (WIM instrument and Providence station was used for collecting climatic data. Volumetric properties, dynamic modulus and creep compliance were used as input parameters for 19Â mm (0.75 in. warm mix asphalt (WMA base and 12.5Â mm (0.5 in. WMA surface layer. The results indicated that all test sections observed AC top-down (longitudinal cracking except Portland cement section which passed for all criteria. The order in terms of performance (best to worst for all test sections by Pavement ME was Portland cement, calcium chloride, control, geogrid, and asphalt emulsion. It was also observed that all test sections passed for both bottom up and top down fatigue cracking by increasing thickness of either of the two top asphalt layers. Test sections with five different base/subbase materials were evaluated in last two years through visual condition survey and measurements of deflection and roughness to confirm the prediction, but there was no serious distress and roughness. Thus these experiments allowed selecting the best rehabilitation/reconstruction techniques for the particular and/or similar highway, and a framework was formulated to select an optimal technique and/or strategy for future rehabilitation/reconstruction projects. Finally, guidelines for
Recent rubber crumb asphalt pavement projects in Alberta
Soleyman, H.R. [Alberta Univ., Edmonton, AB (Canada). Dept. of Civil and Environmental Engineering; Pulles, B.C.M.; Treleaven, L.B. [EBA Engineering Consultants Ltd., Edmonton, AB (Canada)
Several countries around the world have been using rubber crumb (RC) for pavement applications for more than 20 years. The performance of asphalt pavements is enhanced by the use of recycled tires, which has the added advantage of solving the problem of tire disposal. In 2002, the Tire Recycling Management Association of Alberta (TRMA) and Alberta Transportation (AT) evaluated the feasibility and performance of an overlay using asphalt rubber (AR) pavement in Alberta. A partnership in the project was formed by AT, the City of Calgary, the City of Edmonton, and Strathcona County, where each one paved one section of a road with AR in conjunction with a section using conventional or other asphalt mixture types. The roads selected were: (1) 112 Avenue North West in Calgary, (2) 17 Street and Baseline Road in Strathcona County, (3) 137 Avenue in Edmonton, and (4) highway 630 in the vicinity of North Cooking Lake (Alberta Transportation). Two different AR pavement thicknesses were used to study the effects of overlay thicknesses on the performance of the overlay. The structural and functional factors were monitored before, during and after the construction of all sections. The preliminary evaluation of the application of AR in Alberta was presented, along with a summary of test results and performances of AR pavement sections. A full freeze-thaw cycle is required (Spring 2003) before the results of the analysis of the testing are known. The reduced thickness AR sections performed in a similar manner to those with full thickness AC sections. 10 refs., 5 tabs., 3 figs.
Implementing energy efficient pavements: A socio-economic analysis of the development and implementation of energy efficient pavements with low rolling resistance
Axelsen, Christian; Pettinari, Matteo; Schmidt, Bjarne
for the transportation sector is to make road networks more energy efficient by implementing pavements with low rolling resistance, leading to lower fuel consumption. Through a series of projects focusing on reducing rolling resistance conducted since 2010, the Danish Road Directorate (DRD) has developed a durable......, energy-efficient asphalt pavement. Socio-economic analyses conducted to quantify the benefit to society associated with implementing these asphalt pavements have demonstrated very high benefits. The demonstrated results in terms of durability, energy efficiency and socio-economics have resulted...... in substantial government funding being provided for demonstration trials on 50 kilometers of energy-efficient pavement in 2018. The implementation of energy-efficient pavements will enable Denmark to contribute to the out-of-quota 2030-emission cuts in line with EU regulations....
Research on the characteristics of temperature field of asphalt pavement in seasonal frozen region
Qiao, Jiangang; Liu, Weizheng
The characteristics of climate in seasonal frozen area are low temperature and a large range of temperature variation between day and night in winter. These characteristics often lead to problems of asphalt pavement, especially transverse cracks. To reduce the problems of asphalt pavement, it is necessary to examine the distribution of the temperature range of asphalt pavement. A three-dimensional finite element model was used, taking the SMA asphalt pavement as an example with solid70 and plane55 unit features of ANSYS software. It can obtain the relationship between temperature gradient and time and the relationship between temperature gradient and depth. In addition, a function relation model of stress and time was also established. It can provide a theoretical basis for the prevention and treatment of problems of asphalt pavement in seasonal frozen area. Moreover, it has an important significance for improving asphalt pavement design
Decision of National and Provincial Highway Asphalt Pavement Structure Based on Value Engineering
Yingwei Ren
Full Text Available It is important that decision of asphalt pavement structure requires overall considerations of the performance and financial investment. To have asphalt pavement structure fulfilling good reliability, the asphalt pavement structure decision was researched based on value engineering theory. According to the national and provincial highway investigation data in Shandong Province during the last decade, the asphalt pavement performance attenuation rules of traffic levels and asphalt layer thicknesses were developed, and then the road performance evaluation method was presented. In addition, the initial investments, the costs of road maintenance, and middle-scale repair in a period were analyzed. For the light traffic and medium traffic example, using the value engineering method, the pavement performance and costs of which thickness varies from 6 cm to 10 cm were calculated and compared. It was concluded that value engineering was an effective method in deciding the asphalt pavement structure.
Tests used in Project Level Flexible Pavement rehabilitation studies. Research report
Jordaan, G.J.
This paper gives guidelines for the use of tests and their interpretation in the various phases of flexible pavement rehabilitation design as currently adopted in South Africa. The procedure advocated involves a systematic approach to the investigation, evaluation and analysis of the existing pavement. In the recommended procedure, use is made of past pavement behavior and pavement condition, thereby making possible an early assessment of additional information needed. Much emphasis is placed on the optimum utilization of available resources in designing the best applicable remedy to an existing problem. The paper gives a comprehensive list of criteria for parameters measured on a routine basis on South African roads. The procedure suggested allows the condition of pavements to be expressed in comparable terms. Information on the development and use of some new and specialized tests which are used in pavement rehabilitation is given in an appendix.
Flexibl Pavement Analysis Considering Temperature Profile and Anisotropy Behavior in Hot Mix Asphalt Layer
Choi Joonho
Full Text Available A three Dimensional finite element model (FEM incorporating the anisotropic properties and temperature profile of hot mix asphalt (HMA pavement was developed to predict the structural responses of HMA pavement subject to heavy loads typically encountered in the field. In this study, ABAQUS was adopted to model the stress and strain relationships within the pavement structure. The results of the model were verified using data collected from the Korean Highway Corporation Test Road (KHCTR. The results demonstrated that both the base course and surface course layers follow the anisotropic behavior and the incorporation of the temperature profile throughout the pavement has a substantial effect on the pavement response predictions that impact pavement design. The results also showed that the anisotropy level of HMA and base material can be reduced to as low as 80% and 15% as a result of repeated loading, respectively.
Applications of waste material in the pervious concrete pavement: A review
Shakrani, Shahrul Azwan; Ayob, Afizah; Rahim, Mohd Asri Ab
Pervious concrete pavement is one of the innovative structures designed in order to manage the quantity and quality of urban stormwater for a sustainable development. In general, pervious concrete pavement enables water to permeate through its structure and have a capability to cater dynamic loads at the same time. However, the conventional pervious concrete pavement lacks a superior strength while performing as pavement structure. Thus, an extensive research has been carried out in order to explore the possible materials to be incorporated into the pervious concrete pavement for better physical, structural and mechanical properties. The objectives of this paper are to review the waste materials used in the pervious concrete pavement along with their mechanical, durability and permeability performance.
Experimental studies of the dilution of vehicle exhaust pollutants by environment-protecting pervious pavement.
Liu, Chung-Ming; Chen, Jui-Wen; Tsai, Jen-Hui; Lin, Wei-Shian; Yen, M-T; Chen, Ting-Hao
This study determines whether environment-protecting pervious pavement can dilute pollutants immediately after emissions from vehicle. The turbulence-driven dry-deposition process is too slow to be considered in this aspect. The pavement used is the JW pavement (according to its inventors name), a high-load-bearing water-permeable pavement with patents in over 100 countries, which has already been used for more than 8 years in Taiwan and is well suited to replacing conventional road pavement, making the potential implementation of the study results feasible. The design of this study included two sets of experiments. Variation of the air pollutant concentrations within a fenced area over the JW pavement with one vehicle discharging emissions into was monitored and compared with results over a non-JW pavement. The ambient wind speed was low during the first experiment, and the results obtained were highly credible. It was found that the JW pavement diluted vehicle pollutant emissions near the ground surface by 40%-87% within 5 min of emission; whereas the data at 2 m height suggested that about 58%-97% of pollutants were trapped underneath the pavement 20 min after emission. Those quantitative estimations may be off by +/- 10%, if errors in emissions and measurements were considered. SO2 and CO2 underwent the most significant reduction. Very likely, pollutants were forced to move underneath due to the special design of the pavement. During the second experiment, ambient wind speeds were high and the results obtained had less credibility, but they did not disprove the pollutant dilution capacity of the JW pavement. In order to track the fate of pollutants, parts of the pavement were removed to reveal a micro version of wetland underneath, which could possibly hold the responsibility of absorbing and decomposing pollutants to forms harmless to the environment and human health.
Usage of digital image correlation in assessment of behavior of block element pavement structure
Grygierek, M.; Grzesik, B.; Rokitowski, P.; Rusin, T.
In diagnostics of existing road pavement structures deflection measurements have fundamental meaning, because of ability to assess present stiffness (bearing capacity) of whole layered construction. During test loading the reaction of pavement structure to applied load is measured in central point or in a few points located along a straight on a 1.5 ÷ 1.8 m distance (i.e. Falling Weight Deflectometer) in similar spacing equal to 20 ÷ 30 cm. Typical measuring techniques are productive and precise enough for most common pavement structures such as flexible, semi-rigid and rigid. It should be noted that in experimental research as well as in pavements in complex stress state, measurement techniques allowing observation of pavement deformation in 3D would have been very helpful. A great example of that type of pavements is a block element pavement structure consisting of i.e. paving blocks or stone slabs. Due to high stiffness and confined ability of cooperation of surrounding block elements, in that type of pavements fatigue life is strongly connected with displacement distribution. Unfortunately, typical deflection measurement methods forefend displacement observations and rotation of single block elements like paving blocks or slabs. Another difficult problem is to carry out unmistakable analysis of cooperation between neighboring elements. For more precise observations of displacements state of block element pavements under a wheel load a Digital Image Correlation (DIC) was used. Application of this method for assessment of behavior of stone slabs pavement under a traffic load enabled the monitoring of deformations distribution and encouraged to formulate conclusions about the initiation mechanism and development of damages in this type of pavement structures. Results shown in this article were obtained in field tests executed on an exploited pavement structure with a surface course made of granite slabs with dimensions 0.5x1.0x0.14 m.
Shengze Zhu; Xiuyu Liu; Qingqing Cao; Xiaoming Huang
Hydroplaning is a driving phenomenon threating vehicle's control stability and safety. It happens when tire rolls on wet pavement with high speed that hydrodynamic force uplifts the tire. Accurate numerical simulation to reveal the mechanism of hydroplaning and evaluate the function of relevant factors in this process is significant. In order to describe the friction behaviors of tire-pavement interaction, kinetic friction coefficient curve of tire rubber and asphalt pavement was obtained by ...
A Research on Low Modulus Distributed Fiber Optical Sensor for Pavement Material Strain Monitoring.
Meng, Lingjian; Wang, Linbing; Hou, Yue; Yan, Guannan
The accumulated irreversible deformation in pavement under repeated vehicle loadings will cause fatigue failure of asphalt concrete. It is necessary to monitor the mechanical response of pavement under load by using sensors. Previous studies have limitations in modulus accommodation between the sensor and asphalt pavement, and it is difficult to achieve the distributed monitoring goal. To solve these problems, a new type of low modulus distributed optical fiber sensor (DOFS) for asphalt pavement strain monitoring is fabricated. Laboratory experiments have proved the applicability and accuracy of the newly-designed sensor. This paper presents the results of the development.
The Road Pavement Condition Index (PCI Evaluation and Maintenance: A Case Study of Yemen
Karim Fareed M. A.
Full Text Available The pavement condition index (PCI is a simple, convenient and inexpensive way to monitor the condition of the surface of roads, identify maintenance and rehabilitation (M-and-R needs as well as ensure that road maintenance budgets are spent wisely. In this research, the pavement condition was evaluated in terms of the surface distresses existing at the time of the field evaluation. The PCI procedure was used in this research because it deals with the subject of pavement distress identification most comprehensively and is based on a sound statistical technique of pavement sampling.
Mechanical performance of porous concrete pavement containing nano black rice husk ash
Ibrahim, M. Y. Mohd; Ramadhansyah, P. J.; Rosli, H. Mohd; Ibrahim, M. H. Wan
This paper presents an experimental research on the performance of nano black rice husk ash on the porous concrete pavement properties. The performance of the porous concrete pavement mixtures was investigated based on their compressive strength, flexural strength, and splitting tensile strength. The results indicated that using nano material from black rice husk ash improved the mechanical properties of porous concrete pavement. In addition, the result of compressive, flexural, and splitting tensile strength was increased with increasing in curing age. Finally, porous concrete pavement with 10% replacement levels exhibited an excellent performance with good strength compared to others.
Iowa pavement asset management decision-making framework : [tech transfer summary].
A structured framework and tool that can reflect local requirements, : practices, and operational conditions would greatly assist local : agencies in making consistent and defensible pavement treatment : selection decisions.
Road Edge of Pavement, Pavement Condition Ratings and Maintenance Designations per Street Centerline. Updated frequently by Public Works Streets Division, Published in 2006, 1:1200 (1in=100ft) scale, City of Asheville Government.
NSGIC Local Govt | GIS Inventory — Road Edge of Pavement dataset current as of 2006. Pavement Condition Ratings and Maintenance Designations per Street Centerline. Updated frequently by Public Works...
Analysing the accuracy of pavement performance models in the short and long terms: GMDH and ANFIS methods
The accuracy of pavement performance prediction is a critical part of pavement management and directly influences maintenance and rehabilitation strategies. Many models with various specifications have been proposed by researchers and used by agencies. This study presents nine variables affecting
Aggregate Toughness/Abrasion Resistance and Durability/Soundness Tests Related to Asphalt Concrete Performance in Pavements
The properties of aggregates used in asphalt concretes are very important to the performance of the pavements in which the asphalt concretes are used. Often pavement distress, such as stripping and rutting, can be traced directly to the aggregates us...
Texas flexible pavements and overlays : year 1 report, test sections, data collection, analyses, and data storage system.
This five-year project was initiated to collect materials and pavement performance data on a minimum of 100 highway test sections around the State of Texas, incorporating both flexible pavements and overlays. Besides being used to calibrate and valid...
Evaluation of long-term pavement performance and noise characteristics of open-graded friction courses : project 2.
This report describes the second of three experimental installations of open-graded friction course (OGFC) quieter pavements designed to reduce the noise generated at the tire/pavement interface. Experimental sections of OGFC were built using a...
Improving the use of crack sealing to asphalt pavement in Louisiana : research project capsule : technology transfer program.
Pavement performance : depends on the effectiveness : and timeliness of : maintenance efforts. : Deferred maintenance : increases the severity of : distresses and leads to a more : rapid decline of a pavements : condition. An effective : maintenan...
Automatic Road Pavement Assessment with Image Processing: Review and Comparison
Sylvie Chambon
Full Text Available In the field of noninvasive sensing techniques for civil infrastructures monitoring, this paper addresses the problem of crack detection, in the surface of the French national roads, by automatic analysis of optical images. The first contribution is a state of the art of the image-processing tools applied to civil engineering. The second contribution is about fine-defect detection in pavement surface. The approach is based on a multi-scale extraction and a Markovian segmentation. Third, an evaluation and comparison protocol which has been designed for evaluating this difficult task—the road pavement crack detection—is introduced. Finally, the proposed method is validated, analysed, and compared to a detection approach based on morphological tools.
Environmental aspects of the implementation of geogrids for pavement optimisation
Kawalec, Jacek; Gołos, Michał; Mazurowski, Piotr
Technological developments in highway construction should not only result in durable, safe and cost-effective solutions for roads and pavements but also, and perhaps above all, lead to solutions that minimise the negative impact of construction on the environment. One of the ways to ensure these requirements are met is to apply technology using geosynthetics. This paper discusses the stabilisation of aggregate with hexagonal geogrids and the benefits - from the point of view of reducing the emission of harmful gases to the atmosphere - which can be realised from this approach, compared with traditional approaches. Solutions for the improvement of weak subgrades and optimisation of the entire pavement structure are discussed, along with the presentation of sample calculations of greenhouse gas emissions, carried out with the use of specialized software related to the construction of the structures in various technologies.
Geogrid reinforced road subgrade influence on the pavement evenness
Šiukš�ius, A.; Vorobjovas, V.; Vaitkus, A.
As a result of increasing geogrid reinforcement applications in the road subgrade, there are number of projects where geogrid reinforcement is used to control road pavement evenness when there are small layers of peat or mud deeper under road construction. For this task geogrid reinforcement application is not documented but widely used in Lithuania for over a decade. This paper evaluates the long term influence of the geogrid reinforced soil influence on the road surface evenness, when the organic soils stratify in the deeper layers of the subgrade. The geological conditions of the investigated sections are reviewed. The experiment methodology and test results are described, which leads to the conclusions and insights how the pavement evenness depend on the geological conditions and its enhancement. The question is raised about the need for including this geogrid application to the normative documentation. Explanation of the problems that are encountered and the need for further research is given.
Application of infrared camera to bituminous concrete pavements: measuring vehicle
Janků, Michal; Stryk, Josef
Infrared thermography (IR) has been used for decades in certain fields. However, the technological level of advancement of measuring devices has not been sufficient for some applications. Over the recent years, good quality thermal cameras with high resolution and very high thermal sensitivity have started to appear on the market. The development in the field of measuring technologies allowed the use of infrared thermography in new fields and for larger number of users. This article describes the research in progress in Transport Research Centre with a focus on the use of infrared thermography for diagnostics of bituminous road pavements. A measuring vehicle, equipped with a thermal camera, digital camera and GPS sensor, was designed for the diagnostics of pavements. New, highly sensitive, thermal cameras allow to measure very small temperature differences from the moving vehicle. This study shows the potential of a high-speed inspection without lane closures while using IR thermography.
A review of nanoclay applications in the pervious concrete pavement
In recent years, the use of nanoclay has received various interests in order to enhance the properties of construction materials which can also be eligible for pavement technology and engineering application. This review paper summarizes the effect of nanoclay as cement replacement and additive to the performance of pervious concrete pavement. The addition of nanoclay to pervious concrete has demonstrated improvements in strength properties such as compressive and flexural strength, durability such as freeze-thaw and chloride penetration resistance, shrinkage, and denser microstructure but at the same time reduced the porosity, permeability and water absorption properties. This enhancement is due to the roles of nanoclay as nanoreinforcements, nanofillers, nucleation site, and reactive pozzolans in order to promote hydration and improve material properties.
AN OPTIMAL MAINTENANCE MANAGEMENT MODEL FOR AIRPORT CONCRETE PAVEMENT
Shimomura, Taizo; Fujimori, Yuji; Kaito, Kiyoyuki; Obama, Kengo; Kobayashi, Kiyoshi
In this paper, an optimal management model is formulated for the performance-based rehabilitation/maintenance contract for airport concrete pavement, whereby two types of life cycle cost risks, i.e., ground consolidation risk and concrete depreciation risk, are explicitly considered. The non-homogenous Markov chain model is formulated to represent the deterioration processes of concrete pavement which are conditional upon the ground consolidation processes. The optimal non-homogenous Markov decision model with multiple types of risk is presented to design the optimal rehabilitation/maintenance plans. And the methodology to revise the optimal rehabilitation/maintenance plans based upon the monitoring data by the Bayesian up-to-dating rules. The validity of the methodology presented in this paper is examined based upon the case studies carried out for the H airport.
Utomo Dwi Hatmoko Jati
Full Text Available Road infrastructure in Indonesia is mainly dominated by flexible pavement type. Its construction process, however, has raised concerns in terms of its environment impacts. This study aims to track and measure the carbon footprint of flexible pavement. The objectives are to map the construction process in relation to greenhouse gas (GHG emissions, to quantify them in terms of carbon dioxide equivalents (CO2e as generated by the process of production and transportation of raw materials, and the operation of plant off-site and on-site project. Data collection was done by having site observations and interviews with project stakeholders. The results show a total emissions of 70.888 tonnes CO2e, consisting of 34.248 tonnes CO2e (48.31% off-site activities and 36.640 tonnes CO2e (51.687% on-site activities. The two highest CO2e emissions were generated by the use of plant for asphalt concrete laying activities accounted 34.827 tonnes CO2e (49.130%, and material transportation accounted 24.921 (35.155%. These findings provide a new perspective of the carbon footprint in flexible pavement and suggest the urgent need for the use of more efficient and environmentally friendly plant in construction process as it shows the most significant contribution on the CO2e. This study provides valuable understanding on the environmental impact of typical flexible pavement projects in Indonesia, and further can be used for developing green road framework.
Initial investigation into development of accelerated pavement evaluation (APE) vehicle
Full Text Available n e n t d e f o r m a t i o n [ m m ] Test 1 Test 2 Test 3 Test 4 Initial indication of performance Indication of performance after traffic application TR-2003/9: Initial investigation into development of APE vehicle. 6 2... Restricted Contract Report Initial Investigation into Development of Accelerated Pavement Evaluation (APE) Vehicle. Author: WJvdM Steyn M de Beer PREPARED FOR: PREPARED BY: CSIR Transportek STEP CSIR Transportek PO...
Razzaghmanesh, Mostafa; Borst, Michael
Permeable pavement is a stormwater control measure commonly selected in both new and retrofit applications. However, there is limited information about the clogging mechanism of these systems that effects the infiltration. A permeable pavement site located at the Seitz Elementary School, on Fort Riley, Kansas was selected for this study. An 80-space parking lot was built behind the school as part of an EPA collaboration with the U.S. Army. The parking lot design includes a permeable interlocking concrete pavement section along the downgradient edge. This study monitored the clogging progress of the pavement section using twelve water content reflectometers and three buried tipping bucket rain gauges. This clogging dynamic investigation was divided into three stages namely pre-clogged, transitional, and clogged. Recorded initial relative water content of all three stages were significantly and negatively correlated to antecedent dry weather periods with stronger correlations during clogged conditions. The peak relative water content correlation with peak rainfall 10-min intensity was significant for the water content reflectometers located on the western edge away from the eastern edge; this correlation was strongest during transition stage. Once clogged, rainfall measurements no longer correlated with the buried tipping bucket rain gauges. Both water content reflectometers and buried tipping bucket rain gauges showed the progress of surface clogging. For every 6 mm of rain, clogging advanced 1 mm across the surface. The results generally support the hypothesis that the clogging progresses from the upgradient to the downgradient edge. The magnitude of the contributing drainage area and rainfall characteristics are effective factors on rate and progression of clogging.
Evaluation of pavement joint sealants :a new testing technique
Abo-Qudais, Saad A.
A new method to evaluate the performance of sealants, used in rigid pavement joints, was developed. A special fixture was designed to transfer cyclic in-line deflection, applied by a testing machine, to cyclic normal and shear deflections on a sealant sandwiched between two-2 x 2 x 2 in. portland cement mortar cubes. The normal and shear deflections were applied simultaneously at a specific ratio controlled by the developed fixture. The new testing method simulates field ...
Novel permeable pavement systems utilising carbon-negative\\ud aggregate
Tota-Maharaj, Kiran; Monrose, John; Hills, Colin
The use of commercially produced Carbon-Negative aggregates from Carbon8 (a British company which applies patented Accelerated Carbonation Technology (ACT) to solidify waste residues producing useful eco-friendly aggregates) is being investigated in the Caribbean islands of Trinidad, Tobago and St. Lucia. Typical construction of the subbase layer of pavements in the Caribbean include layers of virgin aggregate material (gravel, pea gravel) on which the base course layer is located. These mate...
Recent field investigations on several new Continuously Reinforced Concrete Pavements (CRCP) in Belgium indicate that its crack pattern is characterized by low mean crack spacing along with a high percentage of clusters of closely spaced cracks. Field surveys also indicate that it is difficult to significantly reduce the probability of a non-uniform crack pattern - such as closely spaced cracks, meandering, and Y-cracks - by only slightly adjusting the amount of longitudinal steel. Non-unifor...
Kinect, a Novel Cutting Edge Tool in Pavement Data Collection
Mahmoudzadeh, A.; Firoozi Yeganeh, S.; Golroo, A.
Pavement roughness and surface distress detection is of interest of decision makers due to vehicle safety, user satisfaction, and cost saving. Data collection, as a core of pavement management systems, is required for these detections. There are two major types of data collection: traditional/manual data collection and automated/semi-automated data collection. This paper study different non-destructive tools in detecting cracks and potholes. For this purpose, automated data collection tools, which have been utilized recently are discussed and their applications are criticized. The main issue is the significant amount of money as a capital investment needed to buy the vehicle. The main scope of this paper is to study the approach and related tools that not only are cost-effective but also precise and accurate. The new sensor called Kinect has all of these specifications. It can capture both RGB images and depth which are of significant use in measuring cracks and potholes. This sensor is able to take image of surfaces with adequate resolution to detect cracks along with measurement of distance between sensor and obstacles in front of it which results in depth of defects. This technology has been very recently studied by few researchers in different fields of studies such as project management, biomedical engineering, etc. Pavement management has not paid enough attention to use of Kinect in monitoring and detecting distresses. This paper is aimed at providing a thorough literature review on usage of Kinect in pavement management and finally proposing the best approach which is cost-effective and precise.
A. Mahmoudzadeh
Full Text Available Pavement roughness and surface distress detection is of interest of decision makers due to vehicle safety, user satisfaction, and cost saving. Data collection, as a core of pavement management systems, is required for these detections. There are two major types of data collection: traditional/manual data collection and automated/semi-automated data collection. This paper study different non-destructive tools in detecting cracks and potholes. For this purpose, automated data collection tools, which have been utilized recently are discussed and their applications are criticized. The main issue is the significant amount of money as a capital investment needed to buy the vehicle. The main scope of this paper is to study the approach and related tools that not only are cost-effective but also precise and accurate. The new sensor called Kinect has all of these specifications. It can capture both RGB images and depth which are of significant use in measuring cracks and potholes. This sensor is able to take image of surfaces with adequate resolution to detect cracks along with measurement of distance between sensor and obstacles in front of it which results in depth of defects. This technology has been very recently studied by few researchers in different fields of studies such as project management, biomedical engineering, etc. Pavement management has not paid enough attention to use of Kinect in monitoring and detecting distresses. This paper is aimed at providing a thorough literature review on usage of Kinect in pavement management and finally proposing the best approach which is cost-effective and precise.
Development of a fiber optic pavement subgrade strain measurement system
Miller, Craig Emerson
This dissertation describes the development of a fiber optic sensing system to measure strains within the soil subgrade of highway pavements resulting from traffic loads. The motivation to develop such a device include improvements to: (1)all phases of pavement design, (2)theoretical models used to predict pavement performance, and (3)pavement rehabilitation. The design of the sensing system encompasses selecting an appropriate transducer design as well as the development of optimal optical and demodulation systems. The first is spring based, which attempts to match its spring stiffness to that of the soil-data indicate it is not an optimal transducer design. The second transducer implements anchoring plates attached to two telescoping tubes which allows the soil to be compacted to a desired density between the plates to dictate the transducer's behavior. Both transducers include an extrinsic Fabry- Perot cavity to impose the soil strains onto a phase change of the optical signal propagating through the cavity. The optical system includes a low coherence source and allows phase modulation via path length stretching by adding a second interferometer in series with the transducer, resulting in a path matched differential interferometer. A digitally implemented synthetic heterodyne demodulator based on a four step phase stepping algorithm is used to obtain unambiguous soil strain information from the displacement of the Fabry-Perot cavity. The demodulator is calibrated and characterized by illuminating the transducer with a second long coherence source of different wavelength. The transducer using anchoring plates is embedded within cylindrical soil specimens of varying soil types and soil moisture contents. Loads are applied to the specimen and resulting strains are measured using the embedded fiber optic gage and LVDTs attached to the surface of the specimen. This experimental verification is substantiated using a finite element analysis to predict any differences
Configurations Of Tire Pressure On The Pavement For Commercial Vehicles: Calculation Of The 'N' Number And The Consequences On Pavement Performance
Paulo Boulos Filho
Full Text Available Road traffic has a high variety of types of vehicles and cargo transported. One of the main difficulties in pavement design is forecasting the changes in traffic over time and evaluating the comparative deterioration power of the various loads with axle and tire settings and different levels of repetition. The road environment in Brazil, as we know, is the main means of the country's cargo transportation. Throughout the 1990s and 2000s the road modal accounted for over 60% of the total cargo transported in the country leaving the unbalanced Brazilian transport matrix with almost exclusive investments in such means of transportation. According to CNT 2012, which conducted a survey in approximately 45% of the paved network, evaluating the pavement functional conditions, traffic signs and geometry, 46% of the country's road network has some type of deficiency in the pavement. The goal of this work is to contribute to a better understanding of the complex influence of traffic in the design and service life of the pavement. The load pressure of commercial vehicle tires over the pavement has increased in recent decades. A more apparent effect of this increase in tire pressure would be the change in the pavement- tire contact area which in turn may result in an increase in the contact load between tire and pavement and other harmful effects on the structure. This study aims to determine the effect on the calculation of the N number, or the number of solicitations of the standard axle on the road pavement, measuring the overload as a percentage of the legal load indicated by a scale. In short, we calculate how an increase in the overload percentage in comparison with the allowable load will influence the decrease of the calculated durability of the pavement designed for the highway.
GPR used in combination with other NDT methods for assessing pavements in PPP projects
Loizos, Andreas; Plati, Christina
In the recent decades, Public-Private Partnerships (PPP) has been adopted for highway infrastructure procurement in many countries. PPP projects typically take the form of a section of highway and connecting roadways which are to be construction and managed for a given concession period. Over the course of the highway concession period, the private agency takes over the pavement maintenance and rehabilitation duties. On this purpose, it is critical to find the most cost effective way to maintain the infrastructure in compliance with the agreed upon performance measures and a Pavement Management Systems (PMS) is critical to the success of this process. For the prosperous operation of a PMS it is necessary to have appropriate procedures for pavement monitoring and evaluation, which is important in many areas of pavement engineering. Non Destructive Testing (NDT) has played a major role in pavement condition monitoring, assessments and evaluation accomplishing continuous and quick collection of pavement data. The analysis of this data can lead to indicators related to trigger values (criteria) that define the pavement condition based on which the pavement "health" is perceived helping decide whether there is the need or not to intervene in the pavement. The accomplished perception appoints required management activities for preserving pavements in favor not only of the involved highway/road agencies but also of users' service. Amongst NDT methods Ground Penetrating Radar (GPR) seems to be a very powerful toll, as it provides a range of condition and construction pavement information. It can support effectively the implementation of PMS activities in the framework of pavement monitoring and evaluation. Given that, the present work aims to the development and adaptation of a protocol for the use of GPR in combination with other NDT methods, such as Falling Weight Deflectometer (FWD), for assessing pavements in PPP projects. It is based on the experience of Laboratory of
Neuromodulation of Nestmate Recognition Decisions by Pavement Ants.
Andrew N Bubak
Full Text Available Ant colonies are distributed systems that are regulated in a non-hierarchical manner. Without a central authority, individuals inform their decisions by comparing information in local cues to a set of inherent behavioral rules. Individual behavioral decisions collectively change colony behavior and lead to self-organization capable of solving complex problems such as the decision to engage in aggressive societal conflicts with neighbors. Despite the relevance to colony fitness, the mechanisms that drive individual decisions leading to cooperative behavior are not well understood. Here we show how sensory information, both tactile and chemical, and social context-isolation, nestmate interaction, or fighting non-nestmates-affects brain monoamine levels in pavement ants (Tetramorium caespitum. Our results provide evidence that changes in octopamine and serotonin in the brains of individuals are sufficient to alter the decision by pavement ants to be aggressive towards non-nestmate ants whereas increased brain levels of dopamine correlate to physical fighting. We propose a model in which the changes in brain states of many workers collectively lead to the self-organization of societal aggression between neighboring colonies of pavement ants.
Bubak, Andrew N; Yaeger, Jazmine D W; Renner, Kenneth J; Swallow, John G; Greene, Michael J
Ant colonies are distributed systems that are regulated in a non-hierarchical manner. Without a central authority, individuals inform their decisions by comparing information in local cues to a set of inherent behavioral rules. Individual behavioral decisions collectively change colony behavior and lead to self-organization capable of solving complex problems such as the decision to engage in aggressive societal conflicts with neighbors. Despite the relevance to colony fitness, the mechanisms that drive individual decisions leading to cooperative behavior are not well understood. Here we show how sensory information, both tactile and chemical, and social context-isolation, nestmate interaction, or fighting non-nestmates-affects brain monoamine levels in pavement ants (Tetramorium caespitum). Our results provide evidence that changes in octopamine and serotonin in the brains of individuals are sufficient to alter the decision by pavement ants to be aggressive towards non-nestmate ants whereas increased brain levels of dopamine correlate to physical fighting. We propose a model in which the changes in brain states of many workers collectively lead to the self-organization of societal aggression between neighboring colonies of pavement ants.
Reliability-based performance simulation for optimized pavement maintenance
Chou, Jui-Sheng; Le, Thanh-Son
Roadway pavement maintenance is essential for driver safety and highway infrastructure efficiency. However, regular preventive maintenance and rehabilitation (M and R) activities are extremely costly. Unfortunately, the funds available for the M and R of highway pavement are often given lower priority compared to other national development policies, therefore, available funds must be allocated wisely. Maintenance strategies are typically implemented by optimizing only the cost whilst the reliability of facility performance is neglected. This study proposes a novel algorithm using multi-objective particle swarm optimization (MOPSO) technique to evaluate the cost-reliability tradeoff in a flexible maintenance strategy based on non-dominant solutions. Moreover, a probabilistic model for regression parameters is employed to assess reliability-based performance. A numerical example of a highway pavement project is illustrated to demonstrate the efficacy of the proposed MOPSO algorithms. The analytical results show that the proposed approach can help decision makers to optimize roadway maintenance plans. - Highlights: →A novel algorithm using multi-objective particle swarm optimization technique. → Evaluation of the cost-reliability tradeoff in a flexible maintenance strategy. → A probabilistic model for regression parameters is employed to assess reliability-based performance. → The proposed approach can help decision makers to optimize roadway maintenance plans.
EVALUATION OF TIRE RUBBER DISPOSAL IN CONCRETE FOR PAVEMENTS
Rosa Cristina Cecche Lintz
Full Text Available The production of waste by the tire industry has been a growing problem, indicating the need for its reuse. More than thirty million tires are discharged per year in Brazil, where regulation for the environment states that for each four new tires, five unusable ones must be adequately disposed by manufacturers and importers. Paving consumes an extremely large quantity of materials, which can be the source of rational application of waste and rejected materials. Research shows that tire rubber can be added to asphalt, which increases its durability and improves pavement quality and safety conditions by absorbing the rubber elastic properties, and also be used for architectural applications, among others. This study deals with the addition of rubber fibers from tire crushing in concrete for roadway pavements in order to provide proper indication about the alternative material disposal through an evaluation of the mechanical behavior of the modified concrete. Different concrete mixes were produced, within which, part of fine aggregates were substituted by tire rubber and mechanical experiment tests were performed, which show that, due to great resistance losses, the disposal of this alternative material in concrete should be considered for light traffic pavements, with the addition of rubber ranging up to 10% in mass.
Ballari, M.M.; Brouwers, H.J.H.
Highlights: ► The results of a demonstration project for photocatalytic pavement are shown. ► The photocatalytic performance was studied in a street as well as on lab scale. ► The outdoor monitoring was performed in different seasons and weather conditions. ► The NO x concentration was in average 19% lowered by the photocatalytic street. ► Under ideal weather conditions the NO x reduction reached up to 45%. -- Abstract: Experiments concerning a full-scale demonstration of air purifying pavement in Hengelo, The Netherlands, are reported. The full width of the street was provided with concrete pavement containing TiO 2 over a length of 150 m ("DeNO x street�). Another part of the street, about 100 m, was paved with normal paving blocks ("Control street�). The outdoor monitoring was done during 26 days for a period exceeding one year, and measured parameters included traffic intensity, NO, NO 2 and ozone concentrations, temperature, relative humidity, wind speed and direction, and the visible and UV light irradiance. Prior and parallel to these field measurements, the used blocks were also measured in the lab to assess their performance. The NO x concentration was, on average, 19% (considering the whole day) and 28% (considering only afternoons) lower than the obtained values in the Control street. Under ideal weather conditions (high radiation and low relative humidity) a NO x concentration decrease of 45% could be observed
Chou, Jui-Sheng, E-mail: [email protected] [Department of Construction Engineering, National Taiwan University of Science and Technology (Taiwan Tech), 43 Sec. 4, Keelung Rd., Taipei 106, Taiwan (China); Le, Thanh-Son [Department of Construction Engineering, National Taiwan University of Science and Technology (Taiwan Tech), 43 Sec. 4, Keelung Rd., Taipei 106, Taiwan (China)
Roadway pavement maintenance is essential for driver safety and highway infrastructure efficiency. However, regular preventive maintenance and rehabilitation (M and R) activities are extremely costly. Unfortunately, the funds available for the M and R of highway pavement are often given lower priority compared to other national development policies, therefore, available funds must be allocated wisely. Maintenance strategies are typically implemented by optimizing only the cost whilst the reliability of facility performance is neglected. This study proposes a novel algorithm using multi-objective particle swarm optimization (MOPSO) technique to evaluate the cost-reliability tradeoff in a flexible maintenance strategy based on non-dominant solutions. Moreover, a probabilistic model for regression parameters is employed to assess reliability-based performance. A numerical example of a highway pavement project is illustrated to demonstrate the efficacy of the proposed MOPSO algorithms. The analytical results show that the proposed approach can help decision makers to optimize roadway maintenance plans. - Highlights: > A novel algorithm using multi-objective particle swarm optimization technique. > Evaluation of the cost-reliability tradeoff in a flexible maintenance strategy. > A probabilistic model for regression parameters is employed to assess reliability-based performance. > The proposed approach can help decision makers to optimize roadway maintenance plans.
Particulate matter mass concentrations produced from pavement surface abrasion
Fullova Dasa
Full Text Available According to the latest findings particulate matter belong to the most significant pollutants in Europe together with ground-level ozone O3 and nitrogen dioxide NO2. Road traffic is one of the main sources of particulate matter. Traffic volume has unpleasant impact on longevity of the pavements and also on the environment. Vehicle motions cause mechanical wearing of the asphalt pavement surface - wearing course by vehicle tyres. The paper deals with abrasion of bituminous wearing courses of pavements. The asphalt mixtures are compared in terms of mechanically separated particulate matter. The samples of asphalt mixtures were rutted in wheel tracking machine. The particulate matter measurements were performed in laboratory conditions. The experimental laboratory measurements make it possible to sample particulates without contamination from exhaust emissions, abraded particles from vehicles, resuspension of road dust and climate affects. The paper offers partial results of measurements on six trial samples of asphalt mixtures with different composition. It presents particulate matter morphology and the comparison of rutted asphalt samples in terms of PM mass concentrations and chemical composition.
Laboratory Measurements of Particulate Matter Concentrations from Asphalt Pavement Abrasion
Fullová Daša
Full Text Available The issue of emissions from road traffic is compounded by the fact that the number of vehicles and driven kilometres increase each year. Road traffic is one of the main sources of particulate matter and traffic volume is still increasing and has unpleasant impact on longevity of the pavements and the environment. Vehicle motions cause mechanical wearing of the asphalt pavement surface - wearing course by vehicle tyres. The contribution deals with abrasion of bituminous wearing courses of pavements. The asphalt mixtures of wearing courses are compared in terms of mechanically separated particulate matter. The samples of asphalt mixtures were rutted in wheel tracking machine. The particulate matter measurements were performed in laboratory conditions. The experimental laboratory measurements make it possible to sample particulates without contamination from exhaust emissions, abraded particles from vehicles, resuspension of road dust and climate affects. The contribution offers partial results of measurements on six trial samples of asphalt mixtures with different composition. It presents particulate matter morphology and the comparison of rutted asphalt samples in terms of PM mass concentrations and chemical composition.
Comparisons of Faulting-Based Pavement Performance Prediction Models
Weina Wang
Full Text Available Faulting prediction is the core of concrete pavement maintenance and design. Highway agencies are always faced with the problem of lower accuracy for the prediction which causes costly maintenance. Although many researchers have developed some performance prediction models, the accuracy of prediction has remained a challenge. This paper reviews performance prediction models and JPCP faulting models that have been used in past research. Then three models including multivariate nonlinear regression (MNLR model, artificial neural network (ANN model, and Markov Chain (MC model are tested and compared using a set of actual pavement survey data taken on interstate highway with varying design features, traffic, and climate data. It is found that MNLR model needs further recalibration, while the ANN model needs more data for training the network. MC model seems a good tool for pavement performance prediction when the data is limited, but it is based on visual inspections and not explicitly related to quantitative physical parameters. This paper then suggests that the further direction for developing the performance prediction model is incorporating the advantages and disadvantages of different models to obtain better accuracy.
Measuring Carbon Emissions of Pavement Construction in China
Youliang Huang
Full Text Available While various methodologies for quantifying carbon emissions of pavement construction are developed worldwide, adopting and promoting the existing tools to China's market is found fairly challenging due to institutional constraints. Therefore, the objectives of this study are to propose a methodology for measuring carbon emissions of pavement construction compatible with the fixed pricing systems prevalent in China; and develop an automatic tool for carbon estimations. The total carbon emissions are measured by aggregating emissions of energy consumption and materials used along with four stages, namely material manufacture, transportation, construction, and disposal. A set of composite carbon emission factors for energy and materials was calculated based on existing emission factors with the consideration of the boundaries concerned. The quantity of energy and materials used in pavement construction are obtained through bills of quantity and the fixed price system. The database of the emission factors for energy and materials was embedded into a C# based tool, and validated in a real case.
Numerical Simulation of Moving Load on Concrete Pavements
Laj�áková Gabriela
Full Text Available The knowledge of the development with time of the strain and stress states in pavement structures is needed in the solution of various engineering tasks as the design fatigue lifetime reliability maintenance and structure development. The space computing model of the truck TATRA 815 is introduced. The pavement computing model is created in the sense of Kirchhof theory of the thin slab on elastic foundation. The goal of the calculation is to obtain the vertical deflection in the middle of the slab and the time courses of vertical tire forces. The equations of motion are derived in the form of differential equations. The assumption about the shape of the slab deflection area is adopted. The equations of the motion are solved numerically in the environment of program system MATLAB. The dependences following the influence of various parameters (speed of vehicle motion, stiffness of subgrade, slab thickness, road profile on the pavement vertical deflections and the vertical tire forces are introduced. The results obtained from the plate computing model are compared with the results obtained by the FEM analysis. The outputs of the numerical solution in the time domain can be transformed into a frequency domain and subsequently used to solve various engineering tasks.
Study the Relationship between Pavement Surface Distress and Roughness Data
Mubaraki Muhammad
Full Text Available In this paper, pavement sections from the highway connected Jeddah to Jazan were selected and analyzed to investigate the relationship between International Roughness Index (IRI and pavement damage including; cracking, rutting, and raveling. The Ministry of Transport (MOT of Saudi Arabia has been collecting pavement condition data using the Road Surface Tester (RST vehicle. The MOT measures Roughness, Rutting (RUT, Cracking (CRA, raveling (RAV. Roughness measurements are calculated in terms of the International Roughness Index (IRI. The IRI is calculated over equally spaced intervals along the road profile. Roughness measurements are performed at speed between at 80 kilometers per hour. Thus RST vehicle has been used to evaluate highways across the country. The paper shows three relationships including; cracking (CRA verses roughness (IRI, rutting (RUT verses IRI, and raveling (RAV verses IRI. Also, the paper developed two models namely; model relates IRI to the three distress under study, and model relates IRI to ride quality. The results of the analysis claim at 95% confidence that a significant relationship exist between IRI and cracking, and raveling. It's also shown that rutting did not show significant relationship to IRI values. That's leads to conclude that the distresses types: cracking and raveling may possibly be described as ride quality distresses at different level of significant. Rutting distress described as non-ride quality type's distresses.
Determining Surface Infiltration Rate of Permeable Pavements with Digital Imaging
Caterina Valeo
Full Text Available Cell phone images of pervious pavement surfaces were used to explore relationships between surface infiltration rates (SIR measured using the ASTM C1701 standard test and using a simple falling head test. A fiber-reinforced porous asphalt surface and a highly permeable material comprised of stone, rubber and a polymer binder (Porous Pave were tested. Images taken with a high-resolution cellphone camera were acquired as JPEG files and converted to gray scale images in Matlab® for analysis. The distribution of gray levels was compared to the surface infiltration rates obtained for both pavements with attention given to the mean of the distribution. Investigation into the relationships between mean SIR and parameters determined from the gray level distribution produced in the image analysis revealed that mean SIR measured in both pavements were proportional to the inverse of the mean of the distribution. The relationships produced a coefficient of determination over 85% using both the ASTM and the falling head test in the porous asphalt surface. SIR measurements determined with the ASTM method were highly correlated with the inverse mean of the distribution of gray levels in the Porous Pave material as well, producing coefficients of determination of over 90% and Kendall's tau-b of roughly 70% for nonparametric data.
Assessing the role of pavement macrotexture in preventing crashes on highways.
Pulugurtha, Srinivas S; Kusam, Prasanna R; Patel, Kuvleshay J
The objective of this article is to assess the role of pavement macrotexture in preventing crashes on highways in the State of North Carolina. Laser profilometer data obtained from the North Carolina Department of Transportation (NCDOT) for highways comprising four corridors are processed to calculate pavement macrotexture at 100-m (approximately 330-ft) sections according to the American Society for Testing and Materials (ASTM) standards. Crash data collected over the same lengths of the corridors were integrated with the calculated pavement macrotexture for each section. Scatterplots were generated to assess the role of pavement macrotexture on crashes and logarithm of crashes. Regression analyses were conducted by considering predictor variables such as million vehicle miles of travel (as a function of traffic volume and length), the number of interchanges, the number of at-grade intersections, the number of grade-separated interchanges, and the number of bridges, culverts, and overhead signs along with pavement macrotexture to study the statistical significance of relationship between pavement macrotexture and crashes (both linear and log-linear) when compared to other predictor variables. Scatterplots and regression analysis conducted indicate a more statistically significant relationship between pavement macrotexture and logarithm of crashes than between pavement macrotexture and crashes. The coefficient for pavement macrotexture, in general, is negative, indicating that the number of crashes or logarithm of crashes decreases as it increases. The relation between pavement macrotexture and logarithm of crashes is generally stronger than between most other predictor variables and crashes or logarithm of crashes. Based on results obtained, it can be concluded that maintaining pavement macrotexture greater than or equal to 1.524 mm (0.06 in.) as a threshold limit would possibly reduce crashes and provide safe transportation to road users on highways.
Characterizing Pavement Surface Distress Conditions with Hyper-Spatial Resolution Natural Color Aerial Photography
Su Zhang
Full Text Available Roadway pavement surface distress information is critical for effective pavement asset management, and subsequently, transportation management agencies at all levels (i.e., federal, state, and local dedicate a large amount of time and money to routinely evaluate pavement surface distress conditions as the core of their asset management programs. However, currently adopted ground-based evaluation methods for pavement surface conditions have many disadvantages, like being time-consuming and expensive. Aircraft-based evaluation methods, although getting more attention, have not been used for any operational evaluation programs yet because the acquired images lack the spatial resolution to resolve finer scale pavement surface distresses. Hyper-spatial resolution natural color aerial photography (HSR-AP provides a potential method for collecting pavement surface distress information that can supplement or substitute for currently adopted evaluation methods. Using roadway pavement sections located in the State of New Mexico as an example, this research explored the utility of aerial triangulation (AT technique and HSR-AP acquired from a low-altitude and low-cost small-unmanned aircraft system (S-UAS, in this case a tethered helium weather balloon, to permit characterization of detailed pavement surface distress conditions. The Wilcoxon Signed Rank test, Mann-Whitney U test, and visual comparison were used to compare detailed pavement surface distress rates measured from HSR-AP derived products (orthophotos and digital surface models generated from AT with reference distress rates manually collected on the ground using standard protocols. The results reveal that S-UAS based hyper-spatial resolution imaging and AT techniques can provide detailed and reliable primary observations suitable for characterizing detailed pavement surface distress conditions comparable to the ground-based manual measurement, which lays the foundation for the future application
23 CFR Appendix to Subpart F of... - Alternate Method of Determining the Color of Retroreflective Sign Materials and Pavement Marking...
... Retroreflective Sign Materials and Pavement Marking Materials Appendix to Subpart F of Part 655 Highways FEDERAL... Pavement Marking Materials 1. Although the FHWA Color Tolerance Charts depreciate the use of... for Retroreflective Pavement Marking Material With CIE 2° Standard Observer and 45/0 (0/45) Geometry...
Evaluation of MIT-SCAN-T2 for thickness quality control for PCC and HMA pavements : research project capsule.
Thickness is currently a pay item for portland cement concrete (PCC) pavements : and a quality control item for both PCC and hot mix asphalt (HMA) pavements. : A change in pavement thickness of 0.5 in. can result in a reduction of multiple : years of...
Sensitivity of traffic input parameters on rutting performance of a flexible pavement using Mechanistic Empirical Pavement Design Guide
Nur Hossain
Full Text Available The traffic input parameters in the Mechanistic Empirical Pavement Design Guide (MEPDG are: (a general traffic inputs, (b traffic volume adjustment factors, and (c axle load spectra (ALS. Of these three traffic inputs, the traffic volume adjustment factors specifically monthly adjustment factor (MAF and the ALS are widely considered to be important and sensitive factors, which can significantly affect design of and prediction of distress in flexible pavements. Therefore, the present study was undertaken to assess the sensitivity of ALS and MAF traffic inputs on rutting distress of a flexible pavement. The traffic data of four years (from 2008 to 2012 were collected from an instrumented test section on I-35 in Oklahoma. Site specific traffic input parameters were developed. It was observed that significant differences exist between the MEPDG default and developed site-specific traffic input values. However, the differences in the yearly ALS and MAF data, developed for these four years, were not found to be as significant when compared to one another. In addition, quarterly field rut data were measured on the test section and compared with the MEPDG predicted rut values using the default and developed traffic input values for different years. It was found that significant differences exist between the measured rut and the MEPDG (AASHTOWare-ME predicted rut when default values were used. Keywords: MEPDG, Rut, Level 1 inputs, Axle load spectra, Traffic input parameters, Sensitivity
Effects of reclaimed asphalt pavement on indirect tensile strength test of conditioned foamed asphalt mix
Katman, Herda Yati; Norhisham, Shuhairy; Ismail, Norlela; Ibrahim, Mohd Rasdan; Matori, Mohd Yazip
This paper presents the results of Indirect Tensile Strength (ITS) Test for samples prepared with reclaimed asphalt pavement (RAP). Samples were conditioned in water at 25°C for 24 hours prior to testing. Results show that recycled aggregate from reclaimed asphalt pavement performs as well as virgin aggregate. | CommonCrawl |
WHO cone bio-assays of classical and new-generation long-lasting insecticidal nets call for innovative insecticides targeting the knock-down resistance mechanism in Benin
Marius Allossogbe1,2,
Virgile Gnanguenon1,2,
Boulais Yovogan1,2,
Bruno Akinro1,
Rodrigue Anagonou1,2,
Fiacre Agossa1,2,
André Houtoukpe3,
Germain Gil Padonou1,2 &
Martin Akogbeto1,2
Malaria Journal volume 16, Article number: 77 (2017) Cite this article
To increase the effectiveness of insecticide-treated nets (ITN) in areas of high resistance, new long-lasting insecticidal nets (LLINs) called new-generation nets have been developed. These nets are treated with the piperonyl butoxide (PBO) synergist which inhibit the action of detoxification enzymes. The effectiveness of the new-generation nets has been proven in some studies, but their specific effect on mosquitoes carrying detoxification enzymes and those carrying both detoxification enzymes and the knock-down resistance gene in Benin is not well known. Thus, the objective of this study is to evaluate the efficacy of LLINs treated with PBO on multi-resistant Anopheles gambiae s.l.
The study occurred in seven cities in Benin, Abomey, Cotonou, Porto-Novo, Zangnanado, Parakou, Malanville and Tanguiéta, and included ten locations selected on a north–south transect. Mosquito larvae were collected from these sites, and adult females from these larvae were exposed to single-pyrethroid-treated nets (LifeNet, PermaNet 2.0, Olyset Net) and bi-treated nets (PermaNet 3.0 and Olyset Plus) based on their level of resistance and using WHO cone tests following WHO guidelines.
The different LLINs showed 100% mortality of the susceptible laboratory strain Kisumu and the resistant strain Ace-1R Kisumu. However, with the resistant laboratory strain kdr-Kisumu, mortality was low (16–32%) for all LLINs except PermaNet 3.0 (82.9%). The mortality of local strains carrying only the kdr mechanism varied from 0 to 47% for the single-pyrethroid-treated LLINs and 9 to 86% for bi-treated LLINs. With local strains carrying several mechanisms of resistance (kdr + detoxification enzymes), the observed mortality with different LLINs was also low except for PermaNet 3.0, which induced significantly higher mortality, usually greater than 75% (p < 0.001), with multi-resistant strains. The inhibition of the mortalities induced by the LLINs (11–96%) on multi-resistant field populations was similar to the inhibition observed with the laboratory strain carrying only the knock-down resistance mechanism (kdr-Kisumu) (p > 0.05).
This study showed that the new-generation LLINs treated with pyrethroids and PBO showed better efficacy compared to conventional LLINs. Although the addition of PBO significantly increased the mortality of mosquitoes, the significant role of the kdr resistance gene in the low efficacy of LLINs calls for LLIN technology innovation that specifically targets this mechanism.
Malaria is a major public health problem worldwide, and particularly so in Benin. It remains a permanent threat from its high morbidity (214 million) and mortality (438,000). Africa is the most endemic region affected (395,000 deaths per year) [1]. It affects one-fifth of the world population. However, this proportion has decreased significantly by 37% between 2000 and 2015 due to the effect of malaria prevention and treatment methods, including long-lasting insecticidal nets (LLINs), indoor residual spraying of residual insecticides (IRS), chemo-prevention for pregnant women and children, and therapeutic treatment with artemisinin-based combinations.
Among these prevention methods, LLINs have emerged in recent years as a privileged tool to prevent malaria. The insecticides selected by the World Health Organization (WHO) for LLIN treatment are pyrethroids, which have little toxicity to humans, are effective at low doses, are fast acting (knock-down effect) and, along with repellants, have an irritant effect [2]. The Abuja Conference, which brought together all the leaders of Africa and other UN representative states, donors and NGOs in April 2000, gave impetus to a political commitment to the fight against malaria with the use of insecticide treated nets (ITNs) [3]. Efforts are being made to increase accessibility for populations, especially pregnant women and children under five, who are vulnerable to malaria, a major cause of perinatal mortality, low birth weight and maternal anaemia [1].
Several research studies have been conducted and have shown the effectiveness of ITNs in the fight against malaria in Burkina Faso [4], Cameroon [5], Gambia [6,7,8,9], the Democratic Republic of Congo [10], Kenya [11], Ghana [12], Benin [13] and Côte d'Ivoire [14].
However, several studies have shown that Anopheles gambiae s.l. has developed strong resistance to pyrethroids and DDT in Benin, with a very high knock-down resistance frequency of approximately 80% in the urban areas of Cotonou and in rural areas [15,16,17,18,19,20,21,22,23].
Despite this resistance developed by An. gambiae s.l. to pyrethroids, LLINs remain effective in vector resistance areas [24] and provide protection through their mechanical barrier role [25]. However, Asidi et al. [26] showed a decrease in their effectiveness in areas of high resistance of Anopheles in southern Benin. Major developed resistance mechanisms are the targets of modification (kdr resistance and ace-1R) and metabolic resistance (over-expression of detoxification enzymes, oxidases, esterases, GST) [27]. The kdr mutation is associated with pyrethroid and DDT resistance, and ace-1R is associated with organophosphate and carbamate resistance (two classes of insecticides which are not used to treat LLINs) [15, 28].
To increase the effectiveness of ITNs in areas of high resistance, new nets treated with a so-called new-generation of chemicals has been developed. They are treated with a synergist called piperonyl butoxide (PBO). For some LLINs, the PBO is used on all sides of the net (Olyset Plus®). For others, only the upper part of the net is processed (PermaNet® 3.0). The principle of an ITN synergist is to inhibit the action of detoxification enzymes, which will result in increasing the effectiveness of the insecticide against resistant populations of mosquitoes.
Evidence of the efficacy of PermaNet 3.0 has been shown in some studies, particularly in Tanzania [29], but we do not know its specific action on mosquitoes carrying detoxification enzymes and on those carrying both detoxification and kdr mechanisms in West Africa, particularly in Benin. There have been limited data on the bio-efficacy of new-generation LLINs against multi-resistant mosquitoes in Africa in general and particularly in Benin. Thus, the objective of this study is to evaluate the efficacy of long-lasting insecticidal nets (LLINs) treated with PBO on multi-resistant An. gambiae s.l. populations in Benin. It aims to assess the bio-efficacy of LLINs in areas with a high frequency of molecular resistance genes (kdr and ace-1R) and over-expression of detoxification enzymes (oxidases, esterases, GST). The efficacy of the new-generation LLINs against pyrethroid-resistant Anopheles was also compared to that of conventional LLINs.
This study is transversal and compares variability of the efficacy of two different types of LLINs against An. gambiae s.l. carrying kdr resistance mutations and detoxification enzymes in Benin. The two types of LLINs included conventional LLINs only treated with pyrethroids (Olyset Net, LifeNet, and PermaNet 2.0) and a second type of new-generation LLIN treated with pyrethroids and piperonyl butoxide (PBO), which inhibits the action of enzymes, particularly oxidases.
The study was conducted in Benin, a West African country from June 2015 to March 2016. Among the 12 departments of Benin surveyed, seven were selected in this study (Atlantique, Littoral, Oueme, Zou, Borgou, Atacora and Alibori). Priority was given to areas where higher oxidase activity was observed compared to the susceptible strain An. gambiae Kisumu. They were represented by Abomey, Cotonou, Porto-Novo, Zangnanado, Parakou, Malanville and Tanguiéta districts. The assessment of oxidase activity was conducted on 50 An. gambiae s.l. collected from each district using haem-peroxidase assay as described by Brogdon et al. [30].
The larvae of these mosquito populations were collected in different ecological areas (vegetable, urban, rice and cotton areas). The study was also conducted on resistant laboratory strains (kdr-Kisumu and ace-1R-Kisumu).
Study sites
Malanville district is bordered on the north by the Republic of Niger, on the south by Kandi and Segbana districts, on the west by Karimama district and on the east by the Republic of Nigeria. It has an area of 3016 km2 and had a population of 144,843 inhabitants in 2013 (Fig. 1).
Map of Benin showing the study locations
Tanguieta
It is bordered on the north by the Republic of Burkina Faso, on the south by Boukoumbe district, on the east by Kerou, Kouande and Tounkountouna districts and on the west by Materi and Cobly districts. It covers an area of 5456 km2 and had a population of 77,987 inhabitants in 2013 (Fig. 1).
Abomey-Calavi
Abomey-Calavi is bounded on the north by Ze district, on the south by the Atlantic Ocean, on the east by Cotonou and So-Ava districts, and on the west Ouidah and Tori-Bossito districts. It has an area of 539 km2 and had a population of 438,564 inhabitants in 2013 (Fig. 1).
Cotonou is bordered on the North by So-Ava district and Nokoue lake, on the south by the Atlantic Ocean, on the east by Seme-Podji and on the west by Abomey-Calavi district. It has an area of 79 km2 and had a population of 947,917 inhabitants in 2013 (Fig. 1).
Porto-Novo is bounded on the north by Akpro-Missérete and Avrankou districts, on the south by Seme-Podji, on the west by Aguegues district and on the east by Adjarra district. It covers an area of 223,552 km2 and had a population of 318,608 inhabitants in 2013 (Fig. 1).
It is bordered on the north by N'Dali district and on the south, east and west by Tchaourou district; it has an area of 441 km2 and had a population of 213,498 inhabitants in 2013 (Fig. 1).
Zangnanado
This town is bounded on the north by Dassa-Zoume district, on the south by Ouinhi and Zogbodomey districts, on the west by Cove, Zakpota and Djidja districts and on the east by Ketou and Adja-Ouere. It has an area of 540 km2 and had a population of 52,387 inhabitants in 2013 (Fig. 1).
Larvae collection
Bio-efficacy tests were conducted at various selected sites. Such tests required mosquitoes of 2–5 days old, so the larvae were collected. These collections were conducted in the different localities mentioned above. Anopheles gambiae s.l. larvae and pupae were collected from different locations at each site and carried to the insectarium of the Entomological Research Center of Cotonou (CREC), where they were reared to adult stage at a relative humidity of 70–80% and a temperature of 25–30 °C. Female adults aged 2–5 days were used for bio-efficacy tests.
Highlighting resistance mechanisms
Before the bioassays, living and dead mosquito populations kept after susceptibility testing were analyzed by PCR to detect the genotypes of the kdr gene. The detection of kdr mutation L1014F was performed according to the method of Martinez-Torres et al. [31].
For the molecular characterization of insecticide resistance, two molecular markers were used for characterization of the resistance genes, kdr and ace-1R.
Similarly, for the biochemical characterization of resistance mechanisms, biochemical assays were performed to compare the activity levels of mixed function oxidases (MFO), non-specific esterases (NSE) and glutathione S-transferases (GST) according to the protocol described by Hemingway et al. [32] in susceptible Kisumu and field An. gambiae strains. The mosquitoes used for biochemical analysis had not been exposed to insecticides before the biochemical assessment. These enzyme activities were measured using a sample of 50 mosquitoes per site.
Five types of long-lasting insecticidal nets were evaluated in this study. The group of mono-treated LLINs included LifeNet (polypropylene LLIN with fiber coated with 340 mg/m2 ± 25% deltamethrin), Olyset Net (polyethylene LLIN with permethrin incorporated into the fibers at 20 ± 3 g/kg), and PermaNet 2.0 (polyester LLIN with fiber coated with deltamethrin at 55 mg/m2 ± 25%). The group of new-generation LLINs included: Olyset Plus (same characteristics as Olyset Net but with PBO incorporated throughout the LLIN) and PermaNet 3.0 (polyethylene roof with deltamethrin at 2.8 g/kg ± 25% and PBO at 4.0 g/kg ± 25% incorporated into the fibers, and polyester lateral sides with the fibers coated with deltamethrin at 2.8 g/kg ± 25%). All these nets were obtained from local markets. All nets included in the study are rectangular and were selected by type.
Cone test
The cone test is used to assess the effectiveness of an insecticide and its persistence on the net. It was conducted following the WHO protocol. This test aims to compare the behaviour of mosquitoes while in contact with treated mosquito nets without PBO or with PBO.
Cone tests were performed on five types of nets (Olyset Plus, Olyset Net, LifeNet, PermaNet 2.0 and PermaNet 3.0). These tests were carried out using fragments of LLINs (30 cm × 30 cm) cut from five (05) positions on each net. Two standard cones were fixed with a plastic sheet on each of the five (05) screen fragments. For PermaNet 3.0 LLIN, an additional two cones were added on the PBO-containing roof. Five unfed An. gambiae females aged 2–5 days (Kisumu or wild type) were introduced into each cone placed on the LLIN for 3 min. After exposure, the mosquitoes were removed from the cones using a mouth aspirator and then transferred into paper cups and provided 10% sugar solution. Mosquito knock-down was recorded every 5 min for 60 min. A negative control (untreated net) was included in each series of cone tests. After 24 h of observation, mortality post exposure was recorded. No correction of mortality with Abbott's formula was used as mortality in the control was <5%. All these operations were carried out at a temperature of 25 ± 2 °C and a humidity of 70 ± 10%.
According to the WHO, the bio-effectiveness threshold is 95% knock-down and 80% mortality for laboratory mosquitoes; but for resistant field mosquito populations, we used a threshold of 70% knock-down and 50% mortality. Therefore, all nets showing less than 95% knockdown for laboratory mosquitoes and 70% for field mosquitoes after 60 min, or less than 50% mortality for laboratory mosquitoes and 50% for field mosquitoes after 24 h of observation, were considered ineffective. These knock-down thresholds were chosen taking into account the kdr resistance level observed in the country in general (>50%).
The inhibition of mortality induced by resistance mechanisms was estimated using the following equation:
$$ {\text{Inhibition }} = \, 1 - \, \left( {{\text{p}}1 \, /{\text{ p}}2} \right) \, \times \, 100 $$
where p1 = proportion of resistant mosquitoes dead and p2 = proportion of susceptible Kisumu mosquitoes dead.
To determine if there was any significance difference between the outcome variables (knock-down, mortality and inhibition), Poisson regression (for numeric data) and logistic regression (for proportional data) were used. The 50 and 95% knock-down times and their confidence intervals were obtained after log-probit regression using the method described by Finney [33].
Characteristics of the studied mosquito populations
The majority of female mosquitoes were collected and identified morphologically as An. gambiae s.l. The biochemical and molecular analyses indicated that among ten sites, five showed significantly higher oxidase activity than the susceptible strain Kisumu (Table 1). Esterases were significantly expressed in the Tanguieta mosquito population (Table 1). Over-expression of glutathione-S-transferase was observed at four sites (Table 1). However, the allelic frequency of the kdr mutation was high at almost all sites and ranged from 0.03 to 0.93.
Table 1 Biochemical and molecular characteristics of the Anopheles gambiae s.l. populations tested
Knock-down (KD) and mortality of laboratory strains
Figure 2 shows the proportion of laboratory mosquitoes (ace-1R-Kisumu, kdr-Kisumu, and susceptible Kisumu) knocked down after 60 min for each LLIN. The Olyset Plus and PermaNet 3.0 LLINs induced 100% knock-down of An. gambiae Kisumu. The knock-down effect was 96.15% for Olyset, 90.2% for LifeNet and 93.22% for PermaNet 2.0.
Mosquito knock-down at 60 min post-exposure to LLINs
With the ace-1R-Kisumu strain, which carries the acetylcholinesterase-1 resistance gene, there was a knock-down effect greater than 95% for all nets, with 98.11% for LifeNet, 100% for Olyset, 98.18% for Olyset Plus, 97.96% for PermaNet 2.0, and 98.78% for PermaNet 3.0 (Fig. 2).
For the kdr-Kisumu strain (carrying the resistance knock-down), the knock-down effects observed were 89.29% for LifeNet, 63.64 for Olyset Net, 71.43% for Olyset Plus, 45.78 for PermaNet 2.0 and 71.05% for PermaNet 3.0 (Fig. 2).
Kisumu and ace-1R-Kisumu (Fig. 3). With the kdr-Kisumu strain, mortality was 16% for Olyset Net, 26% for PermaNet 2.0, 28% for LifeNet, and 32.1% for Olyset Plus but was more than 82.9% for PermaNet 3.0. Therefore, based on the bio-efficacy threshold set by WHO (80%), PermaNet 3.0 was effective on all laboratory strains, and Olyset Plus was only effective on the susceptible and ace-R1-Kisumu strains (Fig. 3).
Mosquito mortalities 24 h post-exposure to LLINs
Inhibition of mortality conferred by the kdr resistance gene
Comparing the mortality observed with the susceptible Kisumu strain with that of the resistant kdr-Kisumu strain, the inhibition of mortality induced by the kdr gene regarding the effectiveness of LLINs was 84% for Olyset Net, 74% for PermaNet 2.0, 72% for LifeNet, 68% for Olyset Plus and 17% for PermaNet 3.0.
Knock-down (Kd) effect and mortality induced by mosquito nets on local An. gambiae s.l.
Approximately 2819 local An. gambiae s.l. mosquitoes and 889 An. gambiae Kisumu laboratory strain mosquitoes were tested on different types of LLINs. Tables 2 and 3 show the percentage of local strain mosquitoes knocked down after 60 min for LifeNet, Olyset Net, Olyset Plus, PermaNet 2.0, and PermaNet 3.0.
Table 2 Distribution of the knock-down rate observed in localities where there was only one resistance mechanism (kdr)
Table 3 Distribution of the knock-down rate observed in localities where there were several resistance mechanisms (kdr + metabolic resistance)
Knock-down (KD) and mortality induced by the LLINs on mono-resistance mosquito strains
Only PermaNet 3.0, Olyset Plus and LifeNet LLINs showed a knock-down effect greater than 50% at Agblangandan, Vossa, Zangnanado and Malanville (areas of low resistance) (Table 2). These knock-down values varied between 51 and 95%. At Abomey, only PermaNet 3.0 and Olyset Plus LLINs showed a knock-down effect greater than 50%.
PermaNet 3.0 was the only LLIN that showed significantly higher mortality of greater than 50% in all localities where mosquitoes carried only the kdr gene. The average mortality for other types of LLINs tested in these areas varied from 5 to 47% (Table 2). These mortality rates varied from 0 to 14% for Olyset, 7 to 27% for LifeNet, from 9 to 22% for Olyset Plus, from 24 to 47% for PermaNet 2.0 and from 40 to 86% for PermaNet 3.0.
Inhibition of mortality in mono-resistant An. gambiae s.l. strains
The observed inhibition of mortality induced by kdr resistance of local mosquito strains on LLIN effectiveness was 100–86% for Olyset, 92–73% for LifeNet, 53–76% for PermaNet 2.0, 78–91% for Olyset Plus and 14–60% for PermaNet 3.0. These inhibition rates are similar to those observed with the kdr-Kisumu strain (p > 0.05).
Knock-down (KD) and mortality induced by the LLINs on multi-resistant mosquito strains (carrying kdr and biochemical resistance mutations)
In areas with multi-resistance, the knock-down effects observed were also low (Table 3).
At Akron, the percentage of mosquitoes knocked down after 60 min was 31.48% [19.52–45.55] and 74.55% [60.99–85.33] for Olyset Net and Olyset Plus, respectively; 70.49% [57.43–81.84] and 81.71% [71.63–89.38] for PermaNet 2.0 and PermaNet 3.0, respectively, and 30.77% [18.71–45.1] for LifeNet. At Houéyiho, the knock-down effect was 23.08% [12.53–36.84] and 49.15% [35.89–62.5] for Olyset Net and Olyset Plus, respectively; 46.3% [32.62–60.39] and 73.5% [61.46–83.97] for PermaNet 2.0 and PermaNet 3.0, respectively, and 61.11% [46.87–74.08] for LifeNet. It was generally observed that knock-down was significantly higher with Olyset Plus than with Olyset on multi-resistant Akron and Houéyiho strains (p < 0.05). The same observation was made with PermaNet 3.0, whose knock-down was significantly higher than that observed with PermaNet 2.0.
The same observations were made at Ladji, Parakou and Tanguiéta, where the KD induced by Olyset Plus was higher than that of Olyset. Similarly, PermaNet 3.0 (98%) was more effective than PermaNet 2.0 (39%) (Table 3). However, at Tanguieta, only three LLINs were tested. The three types of mosquitoes tested showed a KD effect ≥75%. Overall, in areas where there was high activity of oxidase enzymes associated with the kdr gene, only three LLINs (LifeNet, Olyset Plus, and PermaNet 3.0) showed a KD effect that was generally high. However, the mortality observed in these populations was generally low (Table 3). Only the PermaNet 3.0 LLIN induced significantly higher mortality (p < 0.001) that was generally greater than 75% (Table 3).
Inhibition of mortality in multi-resistant strains
The inhibition of the mortality induced by LLINs observed with strains carrying several resistance mechanisms (compared to the susceptible strain Kisumu) ranged from 60 to 96% for Olyset, 53 to 90.2% for LifeNet, 45 to 86% for PermaNet 2.0, 59 to 76% for Olyset Plus and 11 to 55% for Permanet 3.0. These inhibition rates are similar to those observed with the kdr-Kisumu strains (p > 0.05).
Knock-down time of LLINs on local An. gambiae s.l. strains
The average time estimated for knock-down of 50% of resistant local An. gambiae s.l. populations was significantly shorter with PermaNet 3.0 (12 min) (p < 0.001), followed by Olyset Plus and LifeNet (33 min). However, the time required for 95% of mosquitoes to be knocked down was high for all LLINs. Generally, there was a slower effect with LLINs treated with permethrin (Table 4).
Table 4 Probable time for 50 and 95% knock-down of Anopheles gambiae s.l. per LLIN
This study is one of the first conducted in Benin to compare the response of local malaria vectors in Benin to several LLINs recommended by the WHO. It helps to observe the variation in mortality of vectors submitted to different types of LLINs. This mortality was generally low, especially with LLINs only treated with pyrethroids. Cone tests showed that LLINs treated with piperonyl butoxide and pyrethroids (especially PermaNet 3.0) have optimum efficacy on all strains of An. gambiae s.l. (mono and multi-resistant).
Several studies have shown a decrease in the bio-efficacy of LLINs against local pyrethroid-resistant vectors [34, 35]. The effectiveness of LLINs treated only with deltamethrin (PermaNet 2.0 and LifeNet) was found to be significantly lower compared to that of nets treated with deltamethrin and PBO. The same observation was made with the LLINs treated with permethrin only (Olyset Net) and those treated with permethrin and PBO. However, the effectiveness of LLINs treated with permethrin was generally lower than that of LLINs treated with deltamethrin, with lower mortality and a very slow knock-down time (KDT 50 and 95%) compared to other LLINs. In a recent study conducted in Benin [36], Olyset Plus, treated with permethrin + PBO, demonstrated a higher efficacy than Olyset Net against wild multi-resistant An. gambiae s.l. in experimental huts, as observed in WHO cone tests used in the present study. In south-western Ethiopia [35] and in Uganda [34], a reduced efficacy of mono-treated LLINs was also observed against wild resistant An. gambiae s.l. in comparison with Permanet 3.0 treated with deltamethrin + PBO. The results are similar to those observed in this study. However, these studies did not include Olyset Plus, the second type of new-generation LLINs treated with permethrin + PBO.
The reduced efficacy of LLINs treated with permethrin would be related to the strong resistance of the local vectors to permethrin due to the resistance selection pressures generated by the use of the same class of insecticide for malaria vector control in public health and for pest control in agriculture [16, 17, 23, 37, 38].
The comparison of LLIN bio-efficacy performed in this study provides the necessary information for the selection of appropriate LLINs for mass distribution. The optimal and constant efficacy of PermaNet 3.0 LLINs on all vector populations shows that this combination of deltamethrin and PBO on LLINs is a most successful strategy against pyrethroid resistance in Benin. Variations in the mortality of vectors also showed that certain types of LLINs are more appropriate than others for distribution in specific regions. This is related to the fact that the effectiveness of an LLIN depends on the characteristics of the mosquito population tested and the chemical structure of the molecule (insecticide) used.
The mosquito populations assessed in the present study were characterized by a high frequency of the kdr gene. This high frequency was probably due to the massive use of pyrethroids in agriculture and public health. In some areas, such as Tanguieta, Parakou, Houeyiho, Akron, and Ladji, farmers and gardeners use huge amounts of insecticides to reduce pests in their crops, which explains the presence and strong expression of several resistance mechanisms in the mosquito populations [39, 40]. Over-production of resistance enzymes in these areas would be linked to pressure on mosquito larvae from insecticides used by farmers to protect vegetable crops [41,42,43]. This expression of the kdr resistance gene induced a 17–84% reduction in LLIN efficacy against laboratory strains. These frequencies are similar to those observed in natural populations of An. gambiae s.l. This observation shows that the kdr gene is the main mechanism involved in the reduction of the effectiveness of LLINs. Although detoxification enzymes contribute to resistance, their impact is successfully inhibited by the presence of PBO on new-generation LLINs and the remaining part is more likely related to the presence of kdr gene in the mosquito populations. This also suggests that the search for new molecules or combinations of molecules that target the kdr resistance mechanism should be promoted.
The WHO recommends preventive measures against vector resistance to insecticides [44]. The results of this study therefore constitute important evidence that can guide decision making in the selection and distribution of high efficacy LLINs in specific regions of Benin. The use of LLINs that showed high bio-efficacy against the local vector populations should be encouraged to contribute substantively to reducing the transmission of malaria in Benin.
This study also suggests the need to develop a routine for monitoring the bio-efficacy of LLINs against local malaria vectors for the replacement of ineffective LLINs. However, community studies would be needed to evaluate the epidemiological impact of these LLINs to confirm whether or not the low efficacy observed is followed by a loss of the epidemiological impact of these nets.
Although the important results of this study, it had certain limitations. Strong evaluation would have been possible if tunnel tests were conducted on LLINs that did not meet the criteria of 80% mortality with resistant mosquito strains. In addition, a chemical analysis of the LLINs prior to the start of the study would also have improved the quality of the results. However, all the LLINs demonstrated a good performance with susceptible laboratory stain Kisumu (mortality > 80%), as recommended by WHO [45], and the focus of this study was to demonstrate the important role of resistance mechanisms on LLINs efficacy.
This study showed variable effectiveness of LLINs on An. gambiae s.l. populations from different localities surveyed from north to south in Benin. The new-generation LLINs with pyrethroids and PBO (PermaNet 3.0 and Olyset Plus) showed higher efficacy than conventional LLINs (PermaNet 2.0, LifeNet and Olyset net). However, the strong resistance of local vectors to permethrin suggests that the combination of deltamethrin + PBO is the most appropriate strategy against local vectors in Benin. Although the addition of PBO (targeting many biochemical mechanisms of resistance) significantly increased the mortality of mosquitoes, the significantly high role of the kdr resistance gene in the low efficacy of LLINs calls for LLIN technology innovation that specifically targets this mechanism.
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MA, VG and MCA designed the study, supervised laboratory work, analyzed data and wrote the manuscript. BY, RA, FA and BA conducted field collections, laboratory tests and contributed in the writing of the manuscript. AH and GGP helped in the study design and revising the manuscript. All authors read and approved the final manuscript.
We thank CREC personnel for their technical assistance and collaboration.
Data collected during this study are included in the published article and its additional files.
This work is supported by Faculty of Letters, Arts and Human Sciences of the University of Abomey-Calavi.
Centre de Recherche Entomologique de Cotonou (CREC), Cotonou, Benin
Marius Allossogbe, Virgile Gnanguenon, Boulais Yovogan, Bruno Akinro, Rodrigue Anagonou, Fiacre Agossa, Germain Gil Padonou & Martin Akogbeto
Université d'Abomey-Calavi, Abomey-Calavi, Benin
Marius Allossogbe, Virgile Gnanguenon, Boulais Yovogan, Rodrigue Anagonou, Fiacre Agossa, Germain Gil Padonou & Martin Akogbeto
Medical Care and Development International, Washington, USA
André Houtoukpe
Marius Allossogbe
Virgile Gnanguenon
Boulais Yovogan
Bruno Akinro
Rodrigue Anagonou
Fiacre Agossa
Germain Gil Padonou
Martin Akogbeto
Correspondence to Marius Allossogbe.
Allossogbe, M., Gnanguenon, V., Yovogan, B. et al. WHO cone bio-assays of classical and new-generation long-lasting insecticidal nets call for innovative insecticides targeting the knock-down resistance mechanism in Benin. Malar J 16, 77 (2017). https://doi.org/10.1186/s12936-017-1727-x
Accepted: 07 February 2017
DOI: https://doi.org/10.1186/s12936-017-1727-x
LLINs
Bio-efficacy
Piperonyl butoxide
Resistant mosquitoes | CommonCrawl |
Computer Aided Verification
International Conference on Computer Aided Verification
CAV 2019: Computer Aided Verification pp 97-115 | Cite as
Loop Summarization with Rational Vector Addition Systems
Jake Silverman
Zachary Kincaid
This paper presents a technique for computing numerical loop summaries. The method synthesizes a rational vector addition system with resets (\(\mathbb {Q}\)-VASR) that simulates the action of an input loop, and then uses the reachability relation of that \(\mathbb {Q}\)-VASR to over-approximate the behavior of the loop. The key technical problem solved in this paper is to automatically synthesize a \(\mathbb {Q}\)-VASR that is a best abstraction of a given loop in the sense that (1) it simulates the loop and (2) it is simulated by any other \(\mathbb {Q}\)-VASR that simulates the loop. Since our loop summarization scheme is based on computing the exact reachability relation of a best abstraction of a loop, we can make theoretical guarantees about its behavior. Moreover, we show experimentally that the technique is precise and performant in practice.
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Modern software verification techniques employ a number of heuristics for reasoning about loops. While these heuristics are often effective, they are unpredictable. For example, an abstract interpreter may fail to find the most precise invariant expressible in the language of its abstract domain due to imprecise widening, or a software-model checker might fail to terminate because it generates interpolants that are insufficiently general. This paper presents a loop summarization technique that is capable of generating loop invariants in an expressive and decidable language and provides theoretical guarantees about invariant quality.
The key idea behind our technique is to leverage reachability results of vector addition systems (VAS) for invariant generation. Vector addition systems are a class of infinite-state transition systems with decidable reachability, classically used as a model of parallel systems [12]. We consider a variation of VAS, rational VAS with resets (\(\mathbb {Q}\)-VASR), wherein there is a finite number of rational-typed variables and a finite set of transitions that simultaneously update each variable in the system by either adding a constant value or (re)setting the variable to a constant value. Our interest in \(\mathbb {Q}\)-VASRs stems from the fact that there is (polytime) procedure to compute a linear arithmetic formula that represents a \(\mathbb {Q}\)-VASR's reachability relation [8].
Since the reachability relation of a \(\mathbb {Q}\)-VASR is computable, the dynamics of \(\mathbb {Q}\)-VASR can be analyzed without relying on heuristic techniques. However, there is a gap between \(\mathbb {Q}\)-VASR and the loops that we are interested in summarizing. The latter typically use a rich set of operations (memory manipulation, conditionals, non-constant increments, non-linear arithmetic, etc) and cannot be analyzed precisely. We bridge the gap with a procedure that, for any loop, synthesizes a \(\mathbb {Q}\)-VASR that simulates it. The reachability relation of the \(\mathbb {Q}\)-VASR can then be used to over-approximate the behavior of the loop. Moreover, we prove that if a loop is expressed in linear rational arithmetic (LRA), then our procedure synthesizes a best \(\mathbb {Q}\)-VASR abstraction, in the sense that it simulates any other \(\mathbb {Q}\)-VASR that simulates the loop. That is, imprecision in the analysis is due to inherent limitations of the \(\mathbb {Q}\)-VASR model, rather heuristic algorithmic choices.
One limitation of the model is that \(\mathbb {Q}\)-VASRs over-approximate multi-path loops by treating the choice between paths as non-deterministic. We show that \(\mathbb {Q}\)-VASRS, \(\mathbb {Q}\)-VASR extended with control states, can be used to improve our invariant generation scheme by encoding control flow information and inter-path control dependencies that are lost in the \(\mathbb {Q}\)-VASR abstraction. We give an algorithm for synthesizing a \(\mathbb {Q}\)-VASRS abstraction of a given loop, which (like our \(\mathbb {Q}\)-VASR abstraction algorithm) synthesizes best abstractions under certain assumptions.
Finally, we note that our analysis techniques extend to complex control structures (such as nested loops) by employing summarization compositionally (i.e., "bottom-up"). For example, our analysis summarizes a nested loop by first summarizing its inner loops, and then uses the summaries to analyze the outer loop. As a result of compositionality, our analysis can be applied to partial programs, is easy to parallelize, and has the potential to scale to large code bases.
The main contributions of the paper are as follows:
We present a procedure to synthesize \(\mathbb {Q}\)-VASR abstractions of transition formulas. For transition formulas in linear rational arithmetic, the synthesized \(\mathbb {Q}\)-VASR abstraction is a best abstraction.
We present a technique for improving the precision of our analysis by using \(\mathbb {Q}\)-VASR with states to capture loop control structure.
We implement the proposed loop summarization techniques and show that their ability to verify user assertions is comparable to software model checkers, while at the same time providing theoretical guarantees of termination and invariant quality.
1.1 Outline
This section illustrates the high-level structure of our invariant generation scheme. The goal is to compute a transition formula that summarizes the behavior of a given program. A transition formula is a formula over a set of program variables \(\textsf {Var}\) along with primed copies \(\textsf {Var}'\), representing the state of the program before and after executing a computation (respectively). For any given program P, a transition formula \(\mathbf {TF}[\![P ]\!]\) can be computed by recursion on syntax:1
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A persistent queue and integer model. back_len and front_len models the lengths of the lists front and back; mem_ops counts the number of memory operations in the computation.
$$\begin{aligned} \mathbf {TF}[\![\texttt {x := }\,e ]\!]&\triangleq \text {x}' = e \wedge \bigwedge _{\texttt {y}\ne \texttt {x} \in \textsf {Var}} \texttt {y}' = \texttt {y}\\ \mathbf {TF}[\![\mathbf{if }~ c~ \mathbf{ then }~ P_1 ~\mathbf{ else }~ P_2 ]\!]&\triangleq (c \wedge \mathbf {TF}[\![P_1 ]\!]) \vee (\lnot c \wedge \mathbf {TF}[\![P_2 ]\!])\\ \mathbf {TF}[\![P_1 \texttt {;} P_2 ]\!]&\triangleq \exists X \in \mathbb {Z}{.} \mathbf {TF}[\![P_1 ]\!][\textsf {Var}' \mapsto X] \wedge \mathbf {TF}[\![P_2 ]\!][\textsf {Var} \mapsto X]\\ \mathbf {TF}[\![\mathbf{while }~ c~ \mathbf{ do } ~P ]\!]&\triangleq (c \wedge \mathbf {TF}[\![P ]\!])^\star \wedge (\lnot c[\textsf {Var} \mapsto \textsf {Var}']) \end{aligned}$$
where \((-)^\star \) is a function that computes an over-approximation of the transitive closure of a transition formula. The contribution of this paper is a method for computing this \((-)^\star \) operation, which is based on first over-approximating the input transition formula by a \(\mathbb {Q}\)-VASR, and then computing the (exact) reachability relation of the \(\mathbb {Q}\)-VASR.
We illustrate the analysis on an integer model of a persistent queue data structure, pictured in Fig. 1. The example consists of two operations (enqueue and dequeue), as well as a test harness (harness) that non-deterministically executes enqueue and dequeue operations. The queue achieves O(1) amortized memory operations (mem_ops) in enqueue and queue by implementing the queue as two lists, front and back (whose lengths are modeled as front_len and back_len, respectively): the sequence of elements in the queue is the front list followed by the reverse of the back list. We will show that the queue functions use O(1) amortized memory operations by finding a summary for harness that implies a linear bound on mem_ops (the number of memory operations in the computation) in terms of nb_ops (the total number of enqueue/dequeue operations executed in some sequence of operations).
We analyze the queue compositionally, in "bottom-up" fashion (i.e., starting from deeply-nested code and working our way back up to a summary for harness). There are two loops of interest, one in dequeue and one in harness. Since the dequeue loop is nested inside the harness loop, dequeue is analyzed first. We start by computing a transition formula that represents one execution of the body of the dequeue loop:
$$ \textit{Body}_{\texttt {deq}} = \texttt {back\_len} > 0 \wedge \left( \begin{array}{@{}l@{}l@{}} &{} \texttt {front\_len}' = \texttt {front\_len} + 1\\ \wedge &{} \texttt {back\_len}' = \texttt {back\_len} - 1\\ \wedge &{} \texttt {mem\_ops}' = \texttt {mem\_ops} + 3\\ \wedge &{} \texttt {size}' = \texttt {size} \end{array}\right) $$
Observe that each variable in the loop is incremented by a constant value. As a result, the loop update can be captured faithfully by a vector addition system. In particular, we see that this loop body formula is simulated by the \(\mathbb {Q}\)-VASR \(V_{\texttt {deq}}\) (below), where the correspondence between the state-space of \(\textit{Body}_{\texttt {deq}}\) and \(V_{\texttt {deq}}\) is given by the identity transformation (i.e., each dimension of \(V_{\texttt {deq}}\) simply represents one of the variables of \(\textit{Body}_{\texttt {deq}}\)).
$$\begin{bmatrix}w\\x\\y\\z\end{bmatrix} = \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}{} \texttt {front\_len}\\\texttt {back\_len}\\\texttt {mem\_ops}\\\texttt {size}\\\end{bmatrix}; V_{\texttt {deq}} = \left\{ \begin{bmatrix}w \\ x \\ y \\ z\end{bmatrix} \rightarrow \begin{bmatrix}w + 1 \\ x-1 \\ y + 3 \\ z\end{bmatrix} \right\} . $$
A formula representing the reachability relation of a vector addition system can be computed in polytime. For the case of \(V_{\texttt {deq}}\), a formula representing k steps of the \(\mathbb {Q}\)-VASR is simply
To capture information about the pre-condition of the loop, we can project the primed variables to obtain \(\texttt {back\_len} > 0\); similarly, for the post-condition, we can project the unprimed variables to obtain \(\texttt {back\_len}' \ge 0\). Finally, combining (\(\dagger \)) (translated back into the vocabulary of the program) and the pre/post-condition, we form the following approximation of the dequeue loop's behavior:
$$ \exists k{.}k \ge 0 \wedge \left( \begin{array}{@{}l@{}l@{}} &{} \texttt {front\_len}' = \texttt {front\_len} + k\\ \wedge &{} \texttt {back\_len}' = \texttt {back\_len} - k\\ \wedge &{} \texttt {mem\_ops}' = \texttt {mem\_ops} + 3k\\ \wedge &{} \texttt {size}' = \texttt {size} \end{array}\right) \wedge \left( k> 0 \Rightarrow \left( \begin{array}{@{}l@{}l@{}}&{} \texttt {back\_len} > 0\\ \wedge &{} \texttt {back\_len}' \ge 0)\end{array}\right) \right) . $$
Using this summary for the dequeue loop, we proceed to compute a transition formula for the body of the harness loop (omitted for brevity). Just as with the dequeue loop, we analyze the harness loop by synthesizing a \(\mathbb {Q}\)-VASR that simulates it, \(V_{\texttt {har}}\) (below), where the correspondence between the state space of the harness loop and \(V_{\texttt {har}}\) is given by the transformation \(S_{\texttt {har}}\):
\(\begin{array}{l} \begin{bmatrix}v\\ w\\ x\\ y\\ z\end{bmatrix} = \underbrace{\begin{bmatrix} 0 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 3 &{} 1 &{} 0 &{} 0\\ 1 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1\\ \end{bmatrix}}_{S_{\texttt {har}}} \begin{bmatrix}{} \texttt {front\_len}\\ \texttt {back\_len}\\ \texttt {mem\_ops}\\ \texttt {size}\\ \texttt {nb\_ops}\end{bmatrix}; i{.}e{.}{,} \left( \begin{array}{ll} &{}\texttt {size} = v\\ \wedge &{}\texttt {back\_len} = w\\ \wedge &{}\texttt {mem\_ops}+3\texttt {back\_len} = x\\ \wedge &{}\texttt {back\_len} + \texttt {front\_len} = y\\ \wedge &{}\texttt {nb\_ops} = z\\ \end{array}\right) . \\ V_{\texttt {har}} = \left\{ \underbrace{\begin{bmatrix}v\\ w\\ x\\ y\\ z\end{bmatrix} \rightarrow \begin{bmatrix}v+1\\ w+1\\ x+4\\ y+1\\ z+1\end{bmatrix}}_{\texttt {enqueue}}, \underbrace{\begin{bmatrix}v\\ w\\ x\\ y\\ z\end{bmatrix} \rightarrow \begin{bmatrix}v-1\\ w\\ x+2\\ y-1\\ z+1\end{bmatrix}}_{\texttt {dequeue} \text { fast}}, \underbrace{\begin{bmatrix}v\\ w\\ x\\ y\\ z\end{bmatrix} \rightarrow \begin{bmatrix}v-1\\ 0\\ x+2\\ y-1\\ z+1\end{bmatrix}}_{\texttt {dequeue} \text { slow}} \right\} \end{array}\)
Unlike the dequeue loop, we do not get an exact characterization of the dynamics of each changed variable. In particular, in the slow dequeue path through the loop, the value of front_len, back_len, and mem_ops change by a variable amount. Since back_len is set to 0, its behavior can be captured by a reset. The dynamics of front_len and mem_ops cannot be captured by a \(\mathbb {Q}\)-VASR, but (using our dequeue summary) we can observe that the sum of front_len + back_len is decremented by 1, and the sum of mem_ops + 3back_len is incremented by 2.
We compute the following formula that captures the reachability relation of \(V_{\texttt {har}}\) (taking \(k_1\) steps of enqueue, \(k_2\) steps of dequeue fast, and \(k_3\) steps of dequeue slow) under the inverse image of the state correspondence \(S_{\texttt {har}}\):
$$ \left( \begin{array}{@{}l@{}l@{}} &{} \texttt {size}' = \texttt {size} + k_1 - k_2 - k_3\\ \wedge &{} \left( (k_3 = 0 \wedge \texttt {back\_len}' = \texttt {back\_len} + k_1) \vee (k_3 > 0 \wedge 0 \le \texttt {back\_len}' \le k_1)\right) \\ \wedge &{}\texttt {mem\_ops}' + 3\texttt {back\_len}' = \texttt {mem\_ops} + 3\texttt {back\_len} + 4k_1 + 2k_2 + 2k_3\\ \wedge &{}\texttt {front\_len}' + \texttt {back\_len}' = \texttt {front\_len} + \texttt {back\_len} + k_1 - k_2 - k_3\\ \wedge &{}\texttt {nb\_ops}' = \texttt {nb\_ops} + k_1 + k_2 + k_3 \end{array}\right) $$
From the above formula (along with pre/post-condition formulas), we obtain a summary for the harness loop (omitted for brevity). Using this summary we can prove (supposing that we start in a state where all variables are zero) that mem_ops is at most 4 times nb_ops (i.e., enqueue and dequeue use O(1) amortized memory operations).
The syntax of \(\exists \text {LIRA}{}\), the existential fragment of linear integer/rational arithmetic, is given by the following grammar:
$$\begin{aligned} s,t \in \textsf {Term}\,&{:}{:}{=} \,c \mid x \mid s + t \mid c \cdot t\\ F,G \in \textsf {Formula}\,&{:}{:}{=}\, s < t \mid s = t \mid F \wedge G \mid F \vee G \mid \exists x \in \mathbb {Q}{.} F \mid \exists x \in \mathbb {Z}{.} F \end{aligned}$$
where x is a (rational sorted) variable symbol and c is a rational constant. Observe that (without loss of generality) formulas are free of negation. \(\exists \text {LRA}{}\) (linear rational arithmetic) refers to the fragment of \(\exists \text {LIRA}{}\) that omits quantification over the integer sort.
A transition system is a pair \((S, \rightarrow )\) where S is a (potentially infinite) set of states and \(\rightarrow \subseteq S \times S\) is a transition relation. For a transition relation \(\rightarrow \), we use \(\rightarrow ^*\) to denote its reflexive, transitive closure.
A transition formula is a formula \(F(\mathbf {x},\mathbf {x}')\) whose free variables range over \(\mathbf {x} = x_1,...,x_n\) and \(\mathbf {x}' = x_1',...,x_n'\) (we refer to the number n as the dimension of F); these variables designate the state before and after a transition. In the following, we assume that transition formulas are defined over \(\exists \text {LIRA}{}\). For a transition formula \(F(\mathbf {x},\mathbf {x}')\) and vectors of terms \(\mathbf {s}\) and \(\mathbf {t}\), we use \(F(\mathbf {s},\mathbf {t})\) to denote the formula F with each \(x_i\) replaced by \(s_i\) and each \(x_i'\) replaced by \(t_i\). A transition formula \(F(\mathbf {x},\mathbf {x}')\) defines a transition system \((S_{F}, \rightarrow _{F})\), where the state space \(S_{F}\) is \(\mathbb {Q}^{n}\) and which can transition \(\mathbf {u} \rightarrow _{F} \mathbf {v}\) iff \(F(\mathbf {u},\mathbf {v})\) is valid.
For two rational vectors \(\mathbf {a}\) and \(\mathbf {b}\) of the same dimension d, we use \(\mathbf {a} \cdot \mathbf {b}\) to denote the inner product \(\mathbf {a} \cdot \mathbf {b} = \sum _{i=1}^d a_ib_i\) and \(\mathbf {a} * \mathbf {b}\) to denote the pointwise (aka Hadamard) product \((\mathbf {a} * \mathbf {b})_i = a_i b_i\). For any natural number i, we use \(\mathbf {e}_i\) to denote the standard basis vector in the ith direction (i.e., the vector consisting of all zeros except the ith entry, which is 1), where the dimension of \(\mathbf {e}_i\) is understood from context. We use \(I_n\) to denote the \(n \times n\) identity matrix.
A rational vector addition system with resets (\(\mathbb {Q}\) -VASR) of dimension d is a finite set \(V{} \subseteq \{0,1\}^d \times \mathbb {Q}^d\) of transformers. Each transformer \((\mathbf {r},\mathbf {a}) \in V{}\) consists of a binary reset vector \(\mathbf {r}\), and a rational addition vector \(\mathbf {a}\), both of dimension d. \(V{}\) defines a transition system \((S_{V{}}, \rightarrow _{V{}})\), where the state space \(S_{V{}}\) is \(\mathbb {Q}^d\) and which can transition \(\mathbf {u} \rightarrow _{V{}} \mathbf {v}\) iff \(\mathbf {v} = \mathbf {r}* \mathbf {u} + \mathbf {a}\) for some \((\mathbf {r},\mathbf {a}) \in V{}\).
A rational vector addition system with resets and states (\(\mathbb {Q}\)-VASRS) of dimension d is a pair \(\mathcal {V}= (Q, E)\), where Q is a finite set of control states, and \(E \subseteq Q \times \{0,1\}^d \times \mathbb {Q}^d \times Q\) is a finite set of edges labeled by (d-dimensional) transformers. \(\mathcal {V}\) defines a transition system \((S_{\mathcal {V}}, \rightarrow _{\mathcal {V}})\), where the state space \(S_{\mathcal {V}}\) is \(Q \times \mathbb {Q}^n\) and which can transition \((q_1, \mathbf {u}) \rightarrow _{\mathcal {V}} (q_2, \mathbf {v})\) iff there is some edge \((q_1, (\mathbf {r},\mathbf {a}), q_2) \in E\) such that \(\mathbf {v} = \mathbf {r}* \mathbf {u} + \mathbf {a}\).
Our invariant generation scheme is based on the following result, which is a simple consequence of the work of Haase and Halfon:
Theorem 1
([8]). There is a polytime algorithm which, given a d-dimensional \(\mathbb {Q}\)-VASRS \(\mathcal {V}{} = (Q, E)\), computes an \(\exists \text {LIRA}\) transition formula \(\textit{reach}(\mathcal {V}{})\) such that for all \(\mathbf {u},\mathbf {v} \in \mathbb {Q}^d\), we have \((p, \mathbf {u}) \rightarrow _{\mathcal {V}{}}^* (q, \mathbf {v})\) for some control states \(p, q \in Q\) if and only if \(\mathbf {u} \rightarrow _{\textit{reach}(\mathcal {V}{})} \mathbf {v}\).
Note that \(\mathbb {Q}\)-VASR can be realized as \(\mathbb {Q}\)-VASRS with a single control state, so this theorem also applies to \(\mathbb {Q}\)-VASR.
3 Approximating Loops with Vector Addition Systems
In this section, we describe a method for over-approximating the transitive closure of a transition formula using a \(\mathbb {Q}\)-VASR. This procedure immediately extends to computing summaries for programs (including programs with nested loops) using the method outlined in Sect. 1.1.
The core algorithmic problem that we answer in this section is: given a transition formula, how can we synthesize a (best) abstraction of that formula's dynamics as a \(\mathbb {Q}\)-VASR? We begin by formalizing the problem: in particular, we define what it means for a \(\mathbb {Q}\)-VASR to simulate a transition formula and what it means for an abstraction to be "best."
Let \(A = (\mathbb {Q}^n,\rightarrow _A)\) and \(B = (\mathbb {Q}^m,\rightarrow _B)\) be transition systems operating over rational vector spaces. A linear simulation from A to B is a linear transformation \(S : \mathbb {Q}^{m \times n} \) such that for all \(\mathbf {u},\mathbf {v} \in \mathbb {Q}^n\) for which \(\mathbf {u} \rightarrow _A \mathbf {v}\), we have \(S\mathbf {u} \rightarrow _B S\mathbf {v}\). We use \(A \Vdash _S B\) to denote that S is a linear simulation from A to B.
Suppose that \(F(\mathbf {x},\mathbf {x}')\) is an n-dimensional transition formula, \(V{}\) is a d-dimensional \(\mathbb {Q}\)-VASR, and \(S : \mathbb {Q}^{d \times n}\) is linear transformation. The key property of simulations that underlies our loop summarization scheme is that if \(F \Vdash _S V{}\), then \(\textit{reach}(V{})(S\mathbf {x},S\mathbf {x}')\) (i.e., the reachability relation of \(V{}\) under the inverse image of S) over-approximates the transitive closure of F. Finally, we observe that simulation \(F \Vdash _S V{}\) can equivalently be defined by the validity of the entailment \(F \,\models \,\gamma (S,V{})\), where
$$ \gamma (S,V{}) \triangleq \bigvee _{(\mathbf {r}, \mathbf {a}) \in V{}} S\mathbf {x}' = \mathbf {r} * S\mathbf {x} + \mathbf {a}$$
is a transition formula that represents the transitions that \(V{}\) simulates under transformation S.
Our task is to synthesize a linear transformation S and a \(\mathbb {Q}\)-VASR \(V{}\) such that \({F \Vdash _S V{}}\). We call a pair \((S,V{})\), consisting of a rational matrix \(S \in \mathbb {Q}^{d \times n}\) and a d-dimensional \(\mathbb {Q}\)-VASR \(V{}\), a \(\mathbb {Q}\)-VASR abstraction. We say that n is the concrete dimension of \((S,V{})\) and d is the abstract dimension. If \(F \Vdash _S V{}\), then we say that \((S,V{})\) is a \(\mathbb {Q}\)-VASR abstraction of F. A transition formula may have many \(\mathbb {Q}\)-VASR abstractions; we are interested in computing a \(\mathbb {Q}\)-VASR abstraction \((S,V{})\) that results in the most precise over-approximation of the transitive closure of F. Towards this end, we define a preorder \(\preceq \) on \(\mathbb {Q}\)-VASR abstractions, where \((S^1,V{}^1) \preceq (S^2,V{}^2)\) iff there exists a linear transformation \(T \in \mathbb {Q}^{e\times d}\) such that \(V{}^1 \Vdash _T V{}^2\) and \(TS^1 = S^2\) (where d and e are the abstract dimensions of \((S^1,V{}^1)\) and \((S^2,V{}^2)\), respectively). Observe that if \((S^1,V{}^1) \preceq (S^2,V{}^2)\), then \(\textit{reach}(V{}^1)(S^1\mathbf {x},S^1\mathbf {x}') \,\models \, \textit{reach}(V{}^2)(S^2\mathbf {x},S^2\mathbf {x}')\).
Thus, our problem can be stated as follows: given a transition formula F, synthesize a \(\mathbb {Q}\)-VASR abstraction \((S,V{})\) of F such that \((S,V{})\) is best in the sense that we have \((S,V{}) \preceq (\widetilde{S},\widetilde{V{}})\) for any \(\mathbb {Q}\)-VASR abstraction \((\widetilde{S},\widetilde{V{}})\) of F. A solution to this problem is given in Algorithm 1.
Algorithm 1 follows the familiar pattern of an AllSat-style loop. The algorithm takes as input a transition formula F. It maintains a \(\mathbb {Q}\)-VASR abstraction (S, V) and a formula \(\varGamma \), whose models correspond to the transitions of F that are not simulated by (S, V). The idea is to build (S, V) iteratively by sampling transitions from \(\varGamma \), augmenting (S, V) to simulate the sample transition, and then updating \(\varGamma \) accordingly. We initialize (S, V) to be \((I_n, \emptyset )\), the canonical least \(\mathbb {Q}\)-VASR abstraction in \(\preceq \) order, and \(\varGamma \) to be F (i.e., \((I_n,\emptyset )\) does not simulate any transitions of F). Each loop iteration proceeds as follows. First, we sample a model M of \(\varGamma \) (i.e., a transition that is allowed by F but not simulated by \((S,V{})\)). We then generalize that transition to a set of transitions by using M to select a cube C of the DNF of F that contains M. Next, we use the procedure described in Sect. 3.1 to compute a \(\mathbb {Q}\)-VASR abstraction \(\hat{\alpha }(C)\) that simulates the transitions of C. We then update the \(\mathbb {Q}\)-VASR abstraction (S, V) to be the least upper bound of (S, V) and \(\hat{\alpha }(C)\) (w.r.t. \(\preceq \) order) using the procedure described in Sect. 3.2 (line 7). Finally, we block any transition simulated by the least upper bound (including every transition in C) from being sampled again by conjoining \(\lnot \gamma (S,V)\) to \(\varGamma \). The loop terminates when \(\varGamma \) is unsatisfiable, in which case we have that \(F \Vdash _S V{}\). Theorem 2 gives the correctness statement for this algorithm.
Given a transition formula F, Algorithm 1 computes a simulation S and \(\mathbb {Q}\)-VASR V such that \(F \Vdash _S V\). Moreover, if F is in \(\exists \text {LRA}{}\), Algorithm 1 computes a best \(\mathbb {Q}\)-VASR abstraction of F.
The proof of this theorem as well as the proofs to all subsequent theorems, lemmas, and propositions are in the extended version of this paper [20].
3.1 Abstracting Conjunctive Transition Formulas
This section shows how to compute a \(\mathbb {Q}\)-VASR abstraction for a consistent conjunctive formula. When the input formula is in \(\exists \text {LRA}\), the computed \(\mathbb {Q}\)-VASR abstraction will be a best \(\mathbb {Q}\)-VASR abstraction of the input formula. The intuition is that, since \(\exists \text {LRA}{}\) is a convex theory, a best \(\mathbb {Q}\)-VASR abstraction consists of a single transition. For \(\exists \text {LIRA}{}\) formulas, our procedure produces a \(\mathbb {Q}\)-VASR abstract that is not guaranteed to be best, precisely because \(\exists \text {LIRA}{}\) is not convex.
Let C be consistent, conjunctive transition formula. Observe that the set \({\textit{Res}_C \triangleq \{ \langle \mathbf {s}, a \rangle : C \,\models \, \mathbf {s} \cdot \mathbf {x'} = a\}}\), which represents linear combinations of variables that are reset across C, forms a vector space. Similarly, the set \(\textit{Inc}_C = \{ \langle \mathbf {s}, a \rangle : C \,\models \,\mathbf {s} \cdot \mathbf {x'} = \mathbf {s} \cdot \mathbf {x} + a\}\), which represents linear combinations of variables that are incremented across C, forms a vector space. We compute bases for both \(\textit{Res}_C\) and \(\textit{Inc}_C\), say \(\{ \langle \mathbf {s}_1, a_1 \rangle , ..., \langle \mathbf {s}_m,a_m \rangle \}\) and \(\{ \langle \mathbf {s}_{m+1}, a_{m+1} \rangle , ..., \langle \mathbf {s}_d,a_d \rangle \}\), respectively. We define \(\hat{\alpha }(C)\) to be the \(\mathbb {Q}\)-VASR abstraction \(\hat{\alpha }(C) \triangleq (S, \{(\mathbf {r}, \mathbf {a})\})\), where
$$\begin{aligned} S \triangleq \begin{bmatrix} \mathbf {s}_1\\\vdots \\\mathbf {s}_d\end{bmatrix} \quad \mathbf {r}\triangleq [\underbrace{0 \cdots 0}_{m\text { times}} \overbrace{1 \cdots 1}^{(d-m)\text { times}}\!\!\!]\ \quad \mathbf {a}\triangleq \begin{bmatrix} a_1\\ \vdots \\ a_d\end{bmatrix}. \end{aligned}$$
Let C be the formula \(x' = x + y \wedge y' = 2y \wedge w' = w \wedge w = w + 1 \wedge z' = w\). The vector space of resets has basis \(\{\langle \begin{bmatrix}0&0&-1&1\end{bmatrix}, 0 \rangle \}\) (representing that \(z - w\) is reset to 0). The vector space of increments has basis \(\{\langle \begin{bmatrix}1&-1&0&0\end{bmatrix}, 0 \rangle , \langle \begin{bmatrix}0&0&1&0\end{bmatrix}, 0 \rangle , \langle \begin{bmatrix}0&0&-1&1\end{bmatrix}, 1 \rangle \}\) (representing that the difference \(x - y\) does not change, the difference \(z - w\) increases by 1, and the variable w does not change). A best abstraction of C is thus the four-dimensional \(\mathbb {Q}\)-VASR
$$ V= \left\{ \left( \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix}0 \\ 0 \\0 \\ 1\end{bmatrix}\right) \right\} , S = \begin{bmatrix}0&0&-1&1\\1&-1&0&0\\0&0&1&0\\ 0&0&-1&1\end{bmatrix}. $$
In particular, notice that since the term \(z - w\) is both incremented and reset, it is represented by two different dimensions in \(\hat{\alpha }(C)\).
For any consistent, conjunctive transition formula C, \(\hat{\alpha }(C)\) is a \(\mathbb {Q}\)-VASR abstraction of C. If C is expressed in \(\exists \text {LRA}\), then \(\hat{\alpha }(C)\) is best.
3.2 Computing Least Upper Bounds
This section shows how to compute least upper bounds w.r.t. the \(\preceq \) order.
By definition of the \(\preceq \) order, if \((S,V{})\) is an upper bound of \((S^1,V{}^1)\) and \((S^2,V{}^2)\), then there must exist matrices \(T^1\) and \(T^2\) such that \(T^1S^1 = S = T^2S^2\), \(V^1 \Vdash _{T^1} V\), and \(V^2 \Vdash _{T^2} V\). As we shall see, if (S, V) is a least upper bound, then it is completely determined by the matrices \(T^1\) and \(T^2\). Thus, we shift our attention to computing simulation matrices \(T^1\) and \(T^2\) that induce a least upper bound.
In view of the desired equation \(T^1S^1 = S = T^2S^2\), let us consider the constraint \(T^1S^1 = T^2S^2\) on two unknown matrices \(T^1\) and \(T^2\). Clearly, we have \(T^1S^1 = T^2S^2\) iff each \((T^1_i,T^2_i)\) belongs to the set \(\mathcal {T} \triangleq \{ (\mathbf {t}^1, \mathbf {t}^2) : \mathbf {t}^1S^1 = \mathbf {t}^2S^2 \}\). Observe that \(\mathcal {T}\) is a vector space, so there is a best solution to the constraint \(T^1S^1 = T^2S^2\): choose \(T^1\) and \(T^2\) so that the set of all row pairs \((T^1_i,T^2_i)\) forms a basis for \(\mathcal {T}\). In the following, we use \(\textit{pushout}(S^1,S^2)\) to denote a function that computes such a best \((T^1, T^2)\).
While \(\textit{pushout}\) gives a best solution to the equation \(T^1S^1 = T^2S^2\), it is not sufficient for the purpose of computing least upper bounds for \(\mathbb {Q}\)-VASR abstractions, because \(T^1\) and \(T^2\) may not respect the structure of the \(\mathbb {Q}\)-VASR \(V^1\) and \(V^2\) (i.e., there may be no \(\mathbb {Q}\)-VASR V such that \(V^1 \Vdash _{T^1} V\) and \(V^2 \Vdash _{T^2} V\)). Thus, we must further constrain our problem by requiring that \(T^1\) and \(T^2\) are coherent with respect to \(V^1\) and \(V^2\) (respectively).
Let \(V\) be a d-dimensional \(\mathbb {Q}\)-VASR. We say that \(i,j \in \{1,...,d\}\) are coherent dimensions of \(V\) if for all transitions \((\mathbf {r},\mathbf {a}) \in V{}\) we have \(r_i = r_j\) (i.e., every transition of \(V{}\) that resets i also resets j and vice versa). We denote that i and j are coherent dimensions of \(V{}\) by writing \(i \equiv _{V{}} j\), and observe that \(\equiv _{V{}}\) forms an equivalence relation on \(\{1,...,d\}\). We refer to the equivalence classes of \(\equiv _{V{}}\) as the coherence classes of \(V{}\).
A matrix \(T \in \mathbb {Q}^{e \times d}\) is coherent with respect to \(V{}\) if and only if each of its rows have non-zero values only in the dimensions corresponding to a single coherence class of \(V{}\).
For any d-dimensional \(\mathbb {Q}\)-VASR \(V{}\) and coherence class \(C = \{c_1,...,c_k\}\) of \(V{}\), define \(\varPi _{C}\) to be the \(k \times d\) dimensional matrix whose rows are \(\mathbf {e}_{c_1}, ..., \mathbf {e}_{c_k}\). Intuitively, \(\varPi _{C}\) is a projection onto the set of dimensions in C.
Coherence is a necessary and sufficient condition for linear simulations between \(\mathbb {Q}\)-VASR in a sense described in Lemmas 1 and 2.
Lemma 1
Let \(V{}^1\) and \(V{}^2\) be \(\mathbb {Q}\)-VASR (of dimension d and e, respectively), and let \(T \in \mathbb {Q}^{e \times d}\) be a matrix such that \(V{}^1 \Vdash _T V{}^2\). Then T must be coherent with respect to \(V{}^1\).
Let \(V{}\) be a d-dimensional \(\mathbb {Q}\)-VASR and let \(T \in \mathbb {Q}^{e \times d}\) be a matrix that is coherent with respect to \(V{}\) and has no zero rows. Then there is a (unique) e-dimensional \(\mathbb {Q}\)-VASR \(\textit{image}(V{},T)\) such that its transition relation \(\rightarrow _{\textit{image}(V{},T)}\) is equal to \(\{(T\mathbf {u},T\mathbf {v}) : \mathbf {u} \rightarrow _{V} \mathbf {v}\}\) (the image of \(V{}\)'s transition relation under T). This \(\mathbb {Q}\)-VASR can be defined by:
$$\textit{image}(V{},T) \triangleq \{ (T \boxtimes \mathbf {r}, T\mathbf {a}) : (\mathbf {r},\mathbf {a}) \in V{} \} $$
where \(T \boxtimes \mathbf {r}\) is the reset vector \(\mathbf {r}\) translated along T (i.e., \((T \boxtimes \mathbf {r})_i = r_j\) where j is an arbitrary choice among dimensions for which \(T_{ij}\) is non-zero—at least one such j exists because the row \(T_i\) is non-zero by assumption, and the choice of j is arbitrary because all such j belong to the same coherence class by the assumption that T is coherent with respect to \(V{}\)).
Let \(V{}\) be a d-dimensional \(\mathbb {Q}\)-VASR and let \(T \in \mathbb {Q}^{e \times d}\) be a matrix that is coherent with respect to \(V{}\) and has no zero rows. Then the transition relation of \(\textit{image}(V{},T)\) is the image of \(V{}\)'s transition relation under T (i.e., \(\rightarrow _{\textit{image}(V{},T)}\) is equal to \(\{(T\mathbf {u},T\mathbf {v}) : \mathbf {u} \rightarrow _{V} \mathbf {v}\}\)).
Finally, prior to describing our least upper bound algorithm, we must define a technical condition that is both assumed and preserved by the procedure:
A \(\mathbb {Q}\)-VASR abstraction \((S,V{})\) is normal if there is no non-zero vector \(\mathbf {z}\) that is coherent with respect to \(V{}\) such that \(\mathbf {z}S = 0\) (i.e., the rows of S that correspond to any coherence class of \(V{}\) are linearly independent).
Intuitively, a \(\mathbb {Q}\)-VASR abstraction that is not normal contains information that is either inconsistent or redundant.
We now present a strategy for computing least upper bounds of \(\mathbb {Q}\)-VASR abstractions. Fix (normal) \(\mathbb {Q}\)-VASR abstractions \((S^1,V{}^1)\) and \((S^2,V^2{})\). Lemmas 1 and 2 together show that a pair of matrices \(\widetilde{T}^1\) and \(\widetilde{T}^2\) induce an upper bound (not necessarily least) on \((S^1,V{}^1)\) and \((S^2,V^2{})\) exactly when the following conditions hold: (1) \(\widetilde{T}^1S^1 = \widetilde{T}^2S^2\), (2) \(\widetilde{T}^1\) is coherent w.r.t. \(V^1\), (3) \(\widetilde{T}^2\) is coherent w.r.t. \(V^2\), and (4) neither \(\widetilde{T}^1\) nor \(\widetilde{T}^2\) contain zero rows. The upper bound induced by \(\widetilde{T}^1\) and \(\widetilde{T}^2\) is given by
$$ \textit{ub}(\widetilde{T}^1, \widetilde{T}^2) \triangleq (\widetilde{T}^1 S^1, \textit{image}(V{}^1,\widetilde{T}^1) \cup \textit{image}(V{}^2,T^2)) . $$
We now consider how to compute a best such \(\widetilde{T}^1\) and \(\widetilde{T}^2\). Observe that conditions (1), (2), and (3) hold exactly when for each row i, \((\widetilde{T}^1_i, \widetilde{T}^2_i)\) belongs to the set
$$ \mathcal {T} \triangleq \{ (\mathbf {t}^1, \mathbf {t}^2) : \mathbf {t}^1S^1 = \mathbf {t}^2S^2 \wedge \mathbf {t}^1 \textit{coherent w.r.t.}~ V^1 \wedge \mathbf {t}^1 \textit{coherent w.r.t.}~ V^2 \} . $$
Since a row vector \(\mathbf {t}^i\) is coherent w.r.t. \(V^i\) iff its non-zero positions belong to the same coherence class of \(V^i\) (equivalently, \(\mathbf {t}^i = \mathbf {u}\varPi _{C^i}\) for some coherence class \(C^i\) and vector \(\mathbf {u}\)), we have \(\mathcal {T} = \bigcup _{C^1,C^2} \mathcal {T}(C^1,C^2)\), where the union is over all coherence classes \(C^1\) of \(V^1\) and \(C^2\) of \(V^2\), and
$$ \mathcal {T}(C^1,C^2) \triangleq \{ (\mathbf {u}^1\varPi _{C^1}, \mathbf {u}^2\varPi _{C^2}) : \mathbf {u}^1\varPi _{C^1}S^1 = \mathbf {u}^2\varPi _{C^2}S^2 \} . $$
Observe that each \(\mathcal {T}(C^1,C^2)\) is a vector space, so we can compute a pair of matrices \(T^1\) and \(T^2\) such that the rows \((T^1_i, T^2_i)\) collectively form a basis for each \(\mathcal {T}(C^1,C^2)\). Since \((S^1,V{}^1)\) and \((S^2,V^2{})\) are normal (by assumption), neither \(T^1\) nor \(T^2\) may contain zero rows (condition (4) is satisfied). Finally, we have that \(\textit{ub}(T^1,T^2)\) is the least upper bound of \((S^1,V{}^1)\) and \((S^2,V^2{})\). Algorithm 2 is a straightforward realization of this strategy.
Let \((S^1,V{}^1)\) and \((S^2,V{}^2)\) be normal \(\mathbb {Q}\)-VASR abstractions of equal concrete dimension. Then the \(\mathbb {Q}\)-VASR abstraction \((S,V{})\) computed by Algorithm 2 is normal and is a least upper bound of \((S^1,V{}^2)\) and \((S^2,V{}^2)\).
4 Control Flow and \(\mathbb {Q}\)-VASRS
In this section, we give a method for improving the precision of our loop summarization technique by using \(\mathbb {Q}\)-VASRS; that is, \(\mathbb {Q}\)-VASR extended with control states. While \(\mathbb {Q}\)-VASRs over-approximate control flow using non-determinism, \(\mathbb {Q}\)-VASRSs allow us to analyze phenomena such as oscillating and multi-phase loops.
An oscillating loop and its representation as a \(\mathbb {Q}\)-VASR and \(\mathbb {Q}\)-VASRS.
We begin with an example that demonstrates the precision gained by \(\mathbb {Q}\)-VASRS. The loop in Fig. 2a oscillates between (1) incrementing variable i by 1 and (2) incrementing both variables i and x by 1. Suppose that we wish to prove that, starting with the configuration \(x = 0 \wedge i = 1\), the loop maintains the invariant that \(2x \le i\). The (best) \(\mathbb {Q}\)-VASR abstraction of the loop, pictured in Fig. 2b, over-approximates the control flow of the loop by treating the conditional branch in the loop as a non-deterministic branch. This over-approximation may violate the invariant \(2x \le i\) by repeatedly executing the path where both variables are incremented. On the other hand, the \(\mathbb {Q}\)-VASRS abstraction of the loop pictured in Fig. 2c captures the understanding that the loop must oscillate between the two paths. The loop summary obtained from the reachability relation of this \(\mathbb {Q}\)-VASRS is powerful enough to prove the invariant \(2x \le i\) holds (under the precondition \(x = 0 \wedge i = 1\)).
4.1 Technical Details
In the following, we give a method for over-approximating the transitive closure of a transition formula \(F(\mathbf {x},\mathbf {x}')\) using a \(\mathbb {Q}\)-VASRS. We start by defining predicate \(\mathbb {Q}\)-VASRS, a variation of \(\mathbb {Q}\)-VASRS with control states that correspond to disjoint state predicates (where the states intuitively belong to the transition formula F rather than the \(\mathbb {Q}\)-VASRS itself). We extend linear simulations and best abstractions to predicate \(\mathbb {Q}\)-VASRS, and give an algorithm for synthesizing best predicate \(\mathbb {Q}\)-VASRS abstractions (for a given set of predicates). Finally, we give an end-to-end algorithm for over-approximating the transitive closure of a transition formula.
A predicate \(\mathbb {Q}\) -VASRS over \(\mathbf {x}\) is a \(\mathbb {Q}\)-VASRS \(\mathcal {V}{} = (P, E)\), such that each control state is a predicate over the variables \(\mathbf {x}\) and the predicates in P are pairwise inconsistent (for all \(p \ne q \in P\), \(p \wedge q\) is unsatisfiable).
We extend linear simulations to predicate \(\mathbb {Q}\)-VASRS as follows:
Let \(F(\mathbf {x},\mathbf {x}')\) be an n-dimensional transition formula and let \(\mathcal {V}{} = (P, E)\) be an m-dimensional \(\mathbb {Q}\)-VASRS over \(\mathbf {x}\). We say that a linear transformation \(S : \mathbb {Q}^{m \times n}\) is a linear simulation from F to \(\mathcal {V}{}\) if for all \(\mathbf {u},\mathbf {v} \in \mathbb {Q}^n\) such that \(\mathbf {u} \rightarrow _F \mathbf {v}\), (1) there is a (unique) \(p \in P\) such that \(p(\mathbf {u})\) is valid (2) there is a (unique) \(q \in P\) such that \(q(\mathbf {v})\) is valid, and (3) \((p, S\mathbf {u}) \rightarrow _{\mathcal {V}{}} (q, S\mathbf {v})\).
Let \(\mathcal {V}{}^1 = (P^1, E^1)\) and \(\mathcal {V}{}^2 = (P^2, E^2)\) be predicate \(\mathbb {Q}\)-VASRSs over \(\mathbf {x}\) (for some \(\mathbf {x}\)) of dimensions d and e, respectively. We say that a linear transformation \(S : \mathbb {Q}^{e \times d}\) is a linear simulation from \(\mathcal {V}{}^1\) to \(\mathcal {V}{}^2\) if for all \(p^1,q^1 \in P^1\) and for all \(\mathbf {u},\mathbf {v} \in \mathbb {Q}^d\) such that \((p^1, \mathbf {u}) \rightarrow _{\mathcal {V}{}^1} (q^1, \mathbf {v})\), there exists (unique) \(p^2, q^2 \in P^2\) such that (1) \((p^2, S\mathbf {u}) \rightarrow _{\mathcal {V}{}^2} (q^2,S\mathbf {v})\), (2) \(p^1 \,\models \,p^2\), and (3) \(q^1 \,\models \, q^2\).
We define a \(\mathbb {Q}\)-VASRS abstraction over \(\mathbf {x} = x_1,...,x_n\) to be a pair \((S,\mathcal {V}{})\) consisting of a rational matrix \(S \in \mathbb {Q}^{d \times n}\) and a predicate \(\mathbb {Q}\)-VASRS of dimension d over \(\mathbf {x}\). We extend the simulation preorder \(\preceq \) to \(\mathbb {Q}\)-VASRS abstractions in the natural way. Extending the definition of "best" abstractions requires more care, since we can always find a "better" \(\mathbb {Q}\)-VASRS abstraction (strictly smaller in \(\preceq \) order) by using a finer set of predicates. However, if we consider only predicate \(\mathbb {Q}\)-VASRS that share the same set of control states, then best abstractions do exist and can be computed using Algorithm 3.
Algorithm 3 works as follows: first, for each pair of formulas \(p, q \in P\), compute a best \(\mathbb {Q}\)-VASR abstraction of the formula \(p(\mathbf {x}) \wedge F(\mathbf {x},\mathbf {x}') \wedge q(\mathbf {x}')\) and call it \((S_{p,q},V{}_{p,q})\). \((S_{p,q},V{}_{p,q})\) over-approximates the transitions of F that begin in a program state satisfying p and end in a program state satisfying q. Second, we compute the least upper bound of all \(\mathbb {Q}\)-VASR abstractions \((S_{p,q},V{}_{p,q})\) to get a \(\mathbb {Q}\)-VASR abstraction (S, V) for F. As a side-effect of the least upper bound computation, we obtain a linear simulation \(T_{p,q}\) from \((S_{p,q},V_{p,q})\) to (S, V) for each p, q. A best \(\mathbb {Q}\)-VASRS abstraction of \(F(\mathbf {x},\mathbf {x}')\) with control states P has S as its simulation matrix and has the image of \(V_{p,q}\) under \(T_{p,q}\) as the edges from p to q.
Given an transition formula \(F(\mathbf {x},\mathbf {x}')\) and control states P over \(\mathbf {x}\), Algorithm 3 computes the best predicate \(\mathbb {Q}\)-VASRS abstraction of F with control states P.
We now describe iter-VASRS (Algorithm 4), which uses \(\mathbb {Q}\)-VASRS to over-approximate the transitive closure of transition formulas. Towards our goal of predictable program analysis, we desire the analysis to be monotone in the sense that if F and G are transition formulas such that F entails G, then iter-VASRS(F) entails iter-VASRS(G). A sufficient condition to guarantee monotonicity of the overall analysis is to require that the set of control states that we compute for F is at least as fine as the set of control states we compute for G. We can achieve this by making the set of control states P of input transition formula \(F(\mathbf {x},\mathbf {x}')\) equal to the set of connected regions of the topological closure of \(\exists \mathbf {x}'{.}F\) (lines 1–4). Note that this set of predicates may fail the contract of abstract-VASRS: there may exist a transition \(\mathbf {u} \rightarrow _F \mathbf {v}\) such that \(\mathbf {v} \,\not \models \, \bigvee P\) (this occurs when there is a state of F with no outgoing transitions). As a result, \((S,\mathcal {V}) =\) abstract-VASRS(F, P) does not necessarily approximate F; however, it does over-approximate \(F \wedge \bigvee P(\mathbf {x}')\). An over-approximation of the transitive closure of F can easily be obtained from \(\textit{reach}(\mathcal {V})(S\mathbf {x},S\mathbf {x}')\) (the over-approximation of the transitive closure of \(F \wedge \bigvee P(\mathbf {x}')\) obtained from the \(\mathbb {Q}\)-VASRS abstraction (\(S,\mathcal {V}\))) by sequentially composing with the disjunction of F and the identity relation (line 6).
Precision Improvement. The abstract-VASRS algorithm uses predicates to infer the control structure of a \(\mathbb {Q}\)-VASRS, but after computing the \(\mathbb {Q}\)-VASRS abstraction, iter-VASRS makes no further use of the predicates (i.e., the predicates are irrelevant in the computation of \(\textit{reach}(\mathcal {V})\)). Predicates can be used to improve iter-VASRS as follows: the reachability relation of a \(\mathbb {Q}\)-VASRS is expressed by a formula that uses auxiliary variables to represent the state at which the computation begins and ends [8]. These variables can be used to encode that the pre-state of the transitive closure must satisfy the predicate corresponding to the begin state and the post-state must satisfy the predicate corresponding to the end state. As an example, consider the Fig. 2 and suppose that we wish to prove the invariant \(x \le 2i\) under the pre-condition \(i = 0 \wedge x = 0\). While this invariant holds, we cannot prove it because there is counter example if the computation begins at \(i\%2 == 1\). By applying the above improvement, we can prove that the computation must begin at \(i\%2 == 0\), and the invariant is verified.
5 Evaluation
The goals of our evaluation is the answer the following questions:
Are \(\mathbb {Q}\)-VASR sufficiently expressive to be able to generate accurate loop summaries?
Does the \(\mathbb {Q}\)-VASRS technique improve upon the precision of \(\mathbb {Q}\)-VASR?
Are the \(\mathbb {Q}\)-VASR/\(\mathbb {Q}\)-VASRS loop summarization algorithms performant?
We implemented our loop summarization procedure and the compositional whole-program summarization technique described in Sect. 1.1. We ran on a suite of 165 benchmarks, drawn from the C4B [2] and HOLA [4] suites, as well as the safe, integer-only benchmarks in the loops category of SV-Comp 2019 [22]. We ran each benchmark with a time-out of 5 min, and recorded how many benchmarks were proved safe by our \(\mathbb {Q}\)-VASR-based technique and our \(\mathbb {Q}\)-VASRS-based technique. For context, we also compare with CRA [14] (a related loop summarization technique), as well as SeaHorn [7] and UltimateAutomizer [9] (state-of-the-art software model checkers). The results are shown in Fig. 3.
The number of assertions proved correct using \(\mathbb {Q}\)-VASR is comparable to both SeaHorn and UltimateAutomizer, demonstrating that \(\mathbb {Q}\)-VASR can indeed model interesting loop phenomena. \(\mathbb {Q}\)-VASRS-based summarization significantly improves precision, proving the correctness of 93% of assertions in the svcomp suite, and more than any other tool in total. Note that the most precise tool for each suite is not strictly better than each of the other tools; in particular, there is only a single program in the HOLA suite that neither \(\mathbb {Q}\)-VASRS nor CRA can prove safe.
CRA-based summarization is the most performant of all the compared techniques, followed by \(\mathbb {Q}\)-VASR and \(\mathbb {Q}\)-VASRS. SeaHorn and UltimateAutomizer employ abstraction-refinement loops, and so take significantly longer to run the test suite.
Experimental results.
Compositional Analysis. Our analysis follows the same high-level structure as compositional recurrence analysis (CRA) [5, 14]. Our analysis differs from CRA in the way that it summarizes loops: we compute loop summaries by over-approximating loops with vector addition systems and computing reachability relations, whereas CRA computes loop summaries by extracting recurrence relations and computing closed forms. The advantage of our approach is that is that we can use \(\mathbb {Q}\)-VASR to accurately model multi-path loops and can make theoretical guarantees about the precision of our analysis; the advantage of CRA is its ability to generate non-linear invariants.
Vector Addition Systems. Our invariant generation method draws upon Haase and Halfon's polytime procedure for computing the reachability relation of integer vector addition systems with states and resets [8]. Generalization from the integer case to the rational case is straightforward. Continuous Petri nets [3] are a related generalization of vector addition systems, where time is taken to be continuous (\(\mathbb {Q}\)-VASR, in contrast, have rational state spaces but discrete time). Reachability for continuous Petri nets is computable polytime [6] and definable in \(\exists \text {LRA}{}\) [1].
Sinn et al. present a technique for resource bound analysis that is based on modeling programs by lossy vector addition system with states [21]. Sinn et al. model programs using vector addition systems with states over the natural numbers, which enables them to use termination bounds for VASS to compute upper bounds on resource usage. In contrast, we use VASS with resets over the rationals, which (in contrast to VASS over \(\mathbb {N}\)) have a \(\exists \text {LIRA}{}\)-definable reachability relation, enabling us to summarize loops. Moreover, Sinn et al.'s method for extracting VASS models of programs is heuristic, whereas our method gives precision guarantees.
Affine and Polynomial Programs. The problem of polynomial invariant generation has been investigated for various program models that generalize \(\mathbb {Q}\)-VASR, including solvable polynomial loops [19], (extended) P-solvable loops [11, 15], and affine programs [10]. Like ours, these techniques are predictable in the sense that they can make theoretical guarantees about invariant quality. The kinds invariants that can be produced using these techniques (conjunctions of polynomial equations) is incomparable with those generated by the method presented in this paper (\(\exists \text {LIRA}{}\) formulas).
Symbolic Abstraction. The main contribution of this paper is a technique for synthesizing the best abstraction of a transition formula expressible in the language of \(\mathbb {Q}\)-VASR (with or without states). This is closely related to the symbolic abstraction problem, which computes the best abstraction of a formula within an abstract domain. The problem of computing best abstractions has been undertaken for finite-height abstract domains [18], template constraint matrices (including intervals and octagons) [16], and polyhedra [5, 24]. Our best abstraction result differs in that (1) it is for a disjunctive domain and (2) the notion of "best" is based on simulation rather than the typical order-theoretic framework.
This style of analysis can be extended from a simple block-structured language to one with control flow and recursive procedures using the framework of algebraic program analysis [13, 23].
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September 2017, 22(7): 2857-2877. doi: 10.3934/dcdsb.2017154
The stabilized semi-implicit finite element method for the surface Allen-Cahn equation
Xufeng Xiao 1, , Xinlong Feng 2,, and Jinyun Yuan 3,
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, China
Departamento de Matemática, Universidade Federal do Paraná, Centro Politácnico, Curitiba 81531-980, PR, Brazil
* Corresponding author: Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, P.R. China
Received June 2016 Revised March 2017 Published May 2017
Fund Project: The first author is supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2016006) and the Graduate Student Research Innovation Program of Xinjiang (No. XJGRI2015009). The second author is supported by the NSF of Xinjiang Province (No.2016D01C058), NCET-13-0988, and the NSF of China (No. 11671345,11271313). The third author is supported by CAPES (No. 88881.068004/2014.01) and CNPq (No. 300326/2012-2,470934/2013-1, INCT-Matemática) of Brazil
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Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.
Keywords: Surface Allen-Cahn equation, surface finite element method, stabilized semi-implicit scheme, operator splitting method, error estimate.
Mathematics Subject Classification: Primary:65M60, 76T99;Secondary:65M12.
Citation: Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154
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Figure 1. Simulation of phase separation on sphere by SSI1 with $\delta t=5\times10^{-4}$.
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Figure 2. Simulation of phase separation on sphere by SSI1 with $\delta t=10^{-4}$.
Figure 5. Non-dimensional discrete total energy line of SSI1 (a) and SSI2 (b) with $\delta t=5\times10^{-4}$(blue) and $\delta t=10^{-4}$(red) on sphere.
Figure 6. Simulation of phase separation on torus by OS1 with $\delta t=5\times10^{-4}$.
Figure 7. Simulation of phase separation on torus by OS1 with $\delta t=10^{-4}$.
Figure 8. Non-dimensional discrete total energy curves of OS1 with different with $\delta t=5\times10^{-4}$(blue) and $\delta t=10^{-4}$(red) (a). And the side view of the solution of OS1 at t=0.05 with $\delta t=5\times10^{-4}$ on torus (b).
Figure 9. Solutions of phase separation on torus by OS2 with different $\delta t$.
Figure 10. Comparison between the SSI1, SSI2, OS1 and OS2 for the simulation of mean curvature flow on sphere with $\delta t=5\times10^{-4}$ (a) and $\delta t=10^{-4}$ (b).
Figure 11. Simulation of motion of a circle on sphere by SSI2 with $\delta t=10^{-4}$.
Figure 12. Comparison between the SSI1, SSI2, OS1 and OS2 for the simulation of mean curvature flow on hyperboloid with $\delta t=5\times10^{-4}$ (a) and $\delta t=10^{-4}$ (b).
Figure 13. Simulation of motion of a circle on hyperboloid by OS1 with $\delta t=10^{-4}$.
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Xufeng Xiao Xinlong Feng Jinyun Yuan | CommonCrawl |
Local Well-posedness and Persistence Property for the Generalized Novikov Equation
Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation
February 2014, 34(2): 821-841. doi: 10.3934/dcds.2014.34.821
A nonlinear diffusion problem arising in population genetics
Peng Zhou 1, , Jiang Yu 1, and Dongmei Xiao 2,
Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China, China
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
Received September 2012 Revised April 2013 Published August 2013
In this paper we investigate a nonlinear diffusion equation with the Neumann boundary condition, which was proposed by Nagylaki in [19] to describe the evolution of two types of genes in population genetics. For such a model, we obtain the existence of nontrivial solutions and the limiting profile of such solutions as the diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Our results show that as $d\rightarrow0$, the location of nontrivial solutions relative to trivial solutions plays a very important role for the existence and shape of limiting profile. In particular, an example is given to illustrate that the limiting profile does not exist for some nontrivial solutions. Moreover, to better understand the dynamics of this model, we analyze the stability and bifurcation of solutions. These conclusions provide a different angle to understand that obtained in [17,21].
Keywords: Nonlinear diffusion equation, existence, bifurcation., limiting profile, stability.
Mathematics Subject Classification: Primary: 35K61, 35B40; Secondary: 92D1.
Citation: Peng Zhou, Jiang Yu, Dongmei Xiao. A nonlinear diffusion problem arising in population genetics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 821-841. doi: 10.3934/dcds.2014.34.821
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Peng Zhou Jiang Yu Dongmei Xiao | CommonCrawl |
European Journal of Applied Physiology and Occupational Physiology
August 1979 , Volume 41, Issue 4, pp 233–245 | Cite as
Aerobic performance of female marathon and male ultramarathon athletes
C. T. M. Davies
M. W. Thompson
The aerobic performance of thirteen male ultramarathon and nine female marathon runners were studied in the laboratory and their results were related to their times in events ranging in distance from 5 km to 84.64 km.
The mean maximal aerobic power output (VO2 max) of the men was 72.5 ml/kg·min compared with 58.2 ml/kg·min (p<0.001) in the women but the O2 cost (VO2) for a given speed or distance of running was the same in both sexes. The 5 km time of the male athletes was closely related to their VO2 max (r=−0.85) during uphill running but was independent of relative power output (%VO2 max). However, with increasing distance the association of VO2 max with male athletic performance diminished (but nevertheless remained significant even at 84.64 km), and the relationship between VO2 max and time increased. Thus, using multiple regression analysis of the form:
$$\begin{gathered} 42.2 km (marathon) time (h) = 7.445 - 0.0338 \dot V{\text{O}}_{{\text{2 max}}} ({\text{ml/kg }} \cdot {\text{ min}}) \hfill \\ - 0.0303\% \dot V{\text{O}}_{{\text{2 max}}} (r = 0.993) \hfill \\ \end{gathered} $$
$$\begin{gathered} 84.64 {\text{km (London}} - {\text{Brighton) time (h) = 16}}{\text{.998 }} - {\text{ 0}}{\text{.0735 }}\dot V{\text{O}}_{{\text{2 max}}} \hfill \\ ({\text{ml/kg }} \cdot \min ) - 0.0844\% \dot V{\text{O}}_{{\text{2 max}}} (r = 0.996) \hfill \\ \end{gathered} $$
approximately 98% of the total variance of performance times could be accounted for in the marathon and ultramarathon events. This suggests that other factors such as footwear, clothing, and running technique (Costill, 1972) play a relatively minor role in this group of male distance runners. In the female athletes the intermediate times were not available and they did not compete beyond 42.2 km (marathon) distance but for this event a similar association though less in magnitude was found with VO2 max (r=−0.43) and %VO2 max (= −0.49). The male athletes were able to sustain 82% VO2 max (range 80–87%) in 42.2 km and 67% VO2 max (range 53–76%) in 84.64 km event. The comparable figure for the girls in the marathon was 79% VO2 max (ranges 68–86%). Our data suggests that success at the marathon and ultramarathon distances is crucially and (possibly) solely dependent on the development and utilisation of a large VO2 max.
Aerobic performance Athletes, marathon, ultramarathon
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1.MRC Environmental Physiology UnitLondon School of Hygiene and Tropical MedicineLondonEngland
Davies, C.T.M. & Thompson, M.W. Europ. J. Appl. Physiol. (1979) 41: 233. https://doi.org/10.1007/BF00429740
Accepted 09 March 1979
DOI https://doi.org/10.1007/BF00429740
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Chinese Physics C> In Press> Article
New Geiger-Nuttall law for two-proton radioactivity
Hong-Ming Liu 1 ,
You-Tian Zou 1 ,
Xiao Pan 1 ,
Jiu-Long Chen 1 ,
Biao He 2 ,
Xiao-Hua Li 1,3,4,,
School of Nuclear Science and Technology, University of South China, Hengyang 421001, China
College of Physics and Electronics, Central South University, Changsha 410083, China
Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, Hengyang 421001, China
Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, Changsha 410081, China
In the present work, a two-parameter empirical formula is proposed, based on the Geiger-Nuttall law, to study two-proton ($ 2p $ ) radioactivity. Using this formula, the calculated $ 2p $ radioactivity half-lives are in good agreement with the experimental data as well as with calculated results obtained by Goncalves et al. [Phys. Lett. B 774, 14 (2017)] using the effective liquid drop model (ELDM), Sreeja et al. [Eur. Phys. J. A 55, 33 (2019)] using a four-parameter empirical formula, and Cui et al. [Phys. Rev. C 101: 014301 (2020)] using a generalized liquid drop model (GLDM). In addition, this two-parameter empirical formula is extended to predict the half-lives of 22 possible $ 2p $ radioactivity candidates with $ 2p $ radioactivity released energy $ Q_{2p} > 0 $ , obtained from the latest evaluated atomic mass table AME2016. The predicted results are highly consistent with those obtained using other theoretical models such as the ELDM, GLDM and four-parameter empirical formula.
two-proton radioactivity ,
Geiger-Nuttall law ,
empirical formula
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Hong-Ming Liu 1,
You-Tian Zou 1,
Xiao Pan 1,
Jiu-Long Chen 1,
Biao He 2,
Corresponding author: Xiao-Hua Li, [email protected]
1. School of Nuclear Science and Technology, University of South China, Hengyang 421001, China
2. College of Physics and Electronics, Central South University, Changsha 410083, China
3. Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, Hengyang 421001, China
4. Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, Changsha 410081, China
Received Date: 2020-06-29
Available Online: 2021-02-01
Abstract: In the present work, a two-parameter empirical formula is proposed, based on the Geiger-Nuttall law, to study two-proton ($ 2p $ ) radioactivity. Using this formula, the calculated $ 2p $ radioactivity half-lives are in good agreement with the experimental data as well as with calculated results obtained by Goncalves et al. [Phys. Lett. B 774, 14 (2017)] using the effective liquid drop model (ELDM), Sreeja et al. [Eur. Phys. J. A 55, 33 (2019)] using a four-parameter empirical formula, and Cui et al. [Phys. Rev. C 101: 014301 (2020)] using a generalized liquid drop model (GLDM). In addition, this two-parameter empirical formula is extended to predict the half-lives of 22 possible $ 2p $ radioactivity candidates with $ 2p $ radioactivity released energy $ Q_{2p} > 0 $ , obtained from the latest evaluated atomic mass table AME2016. The predicted results are highly consistent with those obtained using other theoretical models such as the ELDM, GLDM and four-parameter empirical formula.
In recent years, there has been much work focused on nuclei beyond the proton drip line, due to the fact that they show new phenomena that cannot be found in stable nuclei [1-3]. This includes the two-proton ($ 2p $) radioactivity phenomenon, which was predicted by Zel'dovich [4] and Goldansky [5, 6] in the 1960s. From the perspective of pairing energy, the two protons emitted in a $ 2p $ radioactivity process should be simultaneous emission from the ground state of a radioactive nucleus beyond the drip line. However, there is no agreement on whether the two protons are simultaneously emitted as two indepedent protons or as a "diproton emission" similar to the emission of a 2He-like cluster from the mother nucleus. Obviously, for odd-proton-number (odd-Z) nuclei, proton radioactivity is the predominant decay mode. For even-proton-number (even-Z) nuclei lying near the proton drip line, the $ 2p $ radioactivity phenomenon may occur due to the effect of proton pairing [7]. Experimentally, the probability of the $ 2p $ decay width of 16Ne and 12O was reported in 1978 [8]. Later, the ground-state true $ 2p $ radioactivity of 45Fe was observed at the Grand Acc$ \acute{\rm{e}} $$ \rm{l} $$ \acute{\rm{e}} $rateur National d'Ions Lourds (GANIL) [9] and Gesellschaft f$ \ddot{\rm{u}} $r Schwerionenforschung (GSI) [10], respectively. In 2005, the $ 2p $ radioactivity of 54Zn was detected at GANIL [11], followed by the $ 2p $ radioactivity of 48Ni [12]. In 2007, the $ 2p $ radioactivity of 19Mg was revealed by tracking the decay products [13]. Recently, the $ 2p $ emission of 67Kr was observed in an experiment with the BigRIPS separator [14].
Theoretically, various models have been proposed to investigate 2p radioactivity, including the direct decay model [15-21], the simultaneous versus sequential decay model [22], the diproton model [23, 24], and the three-body model [25-28]. Using an R-matrix formula, B. A. Brown et al. reproduced the $ 2p $ radioactivity half-lives of 45Fe [29]. Following this, using the continuum shell model, J. Rotureau et al. microcosmically described the $ 2p $ radioactivity in 45Fe, 48Ni and 54Zn [30]. In 2017, M. Goncalves et al. used the effective liquid drop model (ELDM) to calculate the half-lives of $ 2p $ radioactive nuclei [31]. Their calculated results can reproduce the experimental data well [32, 33]. Furthermore, based on the ELDM, they predicted the $ 2p $ radioactivity half-lives of 33 nuclei with $ 2p $ radioactivity released energy $ Q_{2p} > 0 $, obtained from the latest evaluated atomic mass table AME2016 [34, 35]. In 2019, Sreeja et al. proposed a four-parameter empirical formula to study the $ 2p $ radioactivity half-lives [36]. These parameters were obtained by fitting the predicted results from Goncalves et al. [33]. Their calculated results agree well with the known experimental data. Recently, Cui et al. studied the $ 2p $ radioactivity of the ground state of nuclei based on a generalized liquid drop model (GLDM) [37], in which the $ 2p $ radioactivity process is described as a pair particle preformed near the surface of the parent nucleus penetrating the barrier between the cluster and daughter nucleus. In this view, $ 2p $ radioactivity shares a similar theory to barrier penetration with different kinds of charged particle radioactivity, such as $ \alpha $ decay, cluster radioactivity, proton radioactivity and so on [38-43]. In our previous work [44], based on the Geiger-Nuttall (G-N) law [45], we proposed a two-parameter empirical formula for a new G-N law for proton radioactivity, which can be treated as an effective tool to study proton radioactivity. Therefore, whether the G-N law can be extended to study $ 2p $ radioactivity or not is an interesting topic. In this work, a two-parameter analytic formula, which is related to the $ 2p $ radioactivity half-life $ T_{1/2} $, $ 2p $ radioactivity released energy $ Q_{2p} $, the charge of the daughter nucleus $ Z_d $, and the orbital angular momentum l taken away by the two emitted protons, is proposed to study $ 2p $ radioactivity.
This article is organized as follows. In the next section, the theoretical framework for the new G-N law is described in detail. In Section III, the detailed calculations, discussion and predictions are provided. In Section IV, a brief summary is given.
II. THEORETICAL FRAMEWORK
In 1911, Geiger and Nuttall found there is a phenomenological relationship between the $ \alpha $ decay half-life $ T_{1/2} $ and the decay energy $ Q_{\alpha} $. This relationship is the so-called Geiger-Nuttall (G-N) law. It is expressed as:
$ {\rm{log_{10}}}{T}_{1/2} = {a}\,{Q_{\alpha}}^{-1/2} + {b}, $
where a and b represent the two isotopic chain–dependent parameters of this formula. Later, the G-N law was widely applied to study the half-lives of $ \alpha $ decay [38, 46-48], cluster radioactivity [49-51] and proton radioactivity [52-54]. However, relative to $ \alpha $ decay and cluster radioactivity, the proton radioactivity half-life is more sensitive to the centrifugal barrier. This means that the linear relationship between the half-life of the proton radioactivity and the released energy $ Q_p $ only exists for proton-radioactive isotopes with the same orbital angular momentum l taken away by the emitted proton [44, 52, 54]. Similarly, the $ 2p $ radioactivity half-life may also depend strongly on the $ 2p $ radioactivity released energy $ Q_{2p} $ and the orbital angular momentum l taken away by the two emitted protons. Recently, considering the contributions of $ Q_{2p} $ and the orbital angular momentum l to the $ 2p $ radioactivity half-life, Sreeja et al. put forward a four-parameter empirical formula to study the $ 2p $ radioactivity half-lives, which is expressed as [36]
$ {\rm{log_{10}}}{T}_{1/2} = (({a}\times {l}) + {b})\,Z_{d}^{\,0.8}\,{Q_{2p}}^{-1/2} + (({c}\times {l}) + {d}), $
where a = 0.1578, b = 1.9474, $ c = -1.8795 $, and $ d = -24.847 $ denote the adjustable parameters, which are obtained by fitting the calculated results of the ELDM [33]. Their calculated results can reproduce the known experimental data well.
In our previous work [44], considering the contributions of the daughter nuclear charge $ Z_d $ and the orbital angular momentum l taken away by the emitted proton to the proton radioactivity half-life, we proposed a two-parameter empirical formula for a new G-N law for proton radioactivity. This formula is written as:
$ {\rm{log_{10}}}{T}_{1/2} = {a_{\beta}}\,(Z_{d}^{\,0.8}+{l\,}^{\beta})\,{{{Q}}_{p}}^{-1/2} + {b_{\beta}}, $
where $ a_{\beta} = 0.843 $ and $ b_{\beta} = -27.194 $ are the fitted parameters. The exponent on the orbital angular momentum l taken away by the emitted proton, $ \beta $ , is 1, which is obtained by fitting 44 experimental data points of proton radioactivity in the ground state and isomeric state. Combined with the work from Sreeja et al. [36, 54] and our previous work [44], it is interesting to examine whether or not a two-parameter form of the empirical formula is suitable to investigate $ 2p $ radioactivity. In this work, because there are no experimental data for $ 2p $ radioactive nuclei with orbital angular momentum $ l \ne 0 $, we choose the experimental data for true $ 2p $ radioactive nuclei (19Mg, 45Fe, 48Ni, 54Zn and 67Kr) with l = 0, and the predicted $ 2p $ radioactivity half-lives of 7 nuclei with $ l \ne 0 $ (1 case with l = 1, 4 cases with l = 2 and 2 cases with l = 4) are extracted from Goncalves et al. [33].
First, for the $ \beta $ value describing the effect of l on the $ 2p $ radioactivity half-life, we choose the $ \beta $ value corresponding to the smallest standard deviation $ \sigma $ between the database and the calculated $ 2p $ radioactivity half-lives as the optimal value, with $ \beta $ varying from 0.1 to 0.5. The relationship between the $ \sigma $ and $ \beta $ values is shown in Fig. 1. It is clear that $ \sigma $ is smallest when $ \beta $ is equal to 0.25. Comparing with the $ \beta $ value of Eq. (3) reflecting the effect of l on the proton radioactivity half-life, this $ \beta $ value is smaller. The reason may be that the reduced mass $ \mu $ of a proton-radioactive nucleus is smaller than that of a $ 2p $-radioactive nucleus, leading to the contribution of the centrifugal barrier to the half-life of the $ 2p $-radioactive nucleus being smaller. Correspondingly, the values of parameters a and b are given as:
Figure 1. (color online) Relationship between the standard deviation $\sigma$ and the value of β.
$ a = 2.032,\,\;\; b = 26.832, $
Then, we can obtain a final formula, which can be written as:
$ {\rm{log_{10}}}{T}_{1/2} = 2.032\,(Z_{d}^{\,0.8}+{l}^{\,0.25})\,{Q_{2p}}^{-1/2} - 26.832. $
III. RESULTS AND DISCUSSION
The primary aim of this work is to verify the feasibility of using Eq. (5) to investigate $ 2p $ radioactivity. The calculated logarithmic half-lives of $ 2p $-radioactive nuclei are listed in the seventh column of Table 1. Meanwhile, for comparison, the calculated results using GLDM, ELDM and a four-parameter empirical formula are shown in the fourth to sixth column of this table, respectively. In Table 1, the first three columns denote the $ 2p $-radioactive nucleus, the experimental $ 2p $ radioactivity released energy $ Q_{2p} $ and the logarithmic experimental $ 2p $ radioactivity half-life $ {\rm{log}}_{10}T_{1/2}^{\rm{exp}} $, respectively. For quantitative comparisons between the calculated $ 2p $ radioactivity half-lives using our empirical formula and the experimental results, the last column gives the logarithm of errors between the experimental $ 2p $ radioactivity half-lives and those calculated using our empirical formula $ {\rm{log_{10}}}HF = {\rm{log}}_{10}{{T}_{1/2}^{\,\rm{exp}}} -{\rm{log}}_{10}{{T}_{1/2}^{\,\rm{cal}}} $. From this table, it can be seen that for the true $ 2p $-radioactive nuclei 19Mg, 45Fe, 48Ni, 54Zn and 67Kr ($ Q_p < 0 $, $ Q_{2p} > 0 $), most values of $ {\rm{log_{10}}}HF $ are between -1 and 1. Particularly, for the cases of 48Ni, with $ Q_{2p} $ = 1.290, and 45Fe, with $ Q_{2p} $ = 1.154, the values of $ {\rm{log_{10}}}HF $ are 0.07 and 0.24, indicating our calculated results can reproduce the experimental data well. As for the sequential or pseudo-$ 2p $-radioactive nuclei 6Be, 12O and 16Ne ($ Q_p > 0 $, $ Q_{2p} > 0 $), the values of $ {\rm{log_{10}}}HF $ for 6Be and 16Ne are relatively large. Likewise, the differences between the experimental data and the calculated $ 2p $ radioactivity half-lives using GLDM, ELDM and the four-parameter empirical formula are more than three orders of magnitude. This may be due to the limitations of the early experimental equipment, resulting in the measured decay widths of these $ 2p $ radioactivity nuclei not being accurate enough. It would be helpful to measure the experimental $ 2p $ half-lives of these nuclei again in the future. In the case of 12O, the values of $ {\rm{log_{10}}}HF $ are small, implying that our formula may also be suitable for studying pseudo-$ 2p $-radioactive nuclei which have relatively accurate experimental data.
Nucleus $ Q_{2p}^{\rm{\,exp}} $ /MeV $ {\rm{log_{10}}}{T}_{1/2}^{\exp} $ /s $ {\rm{log_{10}}}{T}_{1/2}^{{\rm{GLDM}}} $ /s [37] $ {\rm{log_{10}}}{T}_{1/2}^{{\rm{ELDM}}} $ /s [33] $ {\rm{log_{10}}}{T}_{1/2} $ /s [36] $ {\rm{log_{10}}}{T}_{1/2}^{{\rm{This}}\;{\rm{work}}}$ /s $ {\rm{log_{10}}} $ HF
6Be 1.371 [55] $ -20.30 $ [55] $ -19.37 $ $ -19.97 $ $ -21.95 $ $ -23.81 $ 3.51
12O 1.638 [56] $ >-20.20 $ [56] $ -19.17 $ $ -18.27 $ $ -18.47 $ $ -20.17 $ $ >-0.03 $
1.820 [8] $ -20.94 $ [8] $ -20.94 $ – $ -18.79 $ $ -20.52 $ $ -0.42 $
1.790 [57] $ -20.10 $ [57] $ -20.10 $ – $ -18.74 $ $ -20.46 $ $ 0.36 $
16Ne 1.33 [8] $ -20.64 $ [8] $ -16.45 $ – $ -15.94 $ $ -17.53 $ $ -3.11 $
1.400 [59] $ -20.38 $ [59] $ -16.63 $ $ -16.60 $ $ -16.16 $ $ -17.77 $ $ -2.61 $
19Mg 0.750 [13] $ -11.40 $ [13] $ -11.79 $ $ -11.72 $ $ -10.66 $ $ -12.03 $ $ 0.63 $
45Fe 1.100 [10] $ -2.40 $ [10] $ -2.23 $ – $ -1.25 $ $ -2.21 $ $ -0.19 $
1.140 [9] $ -2.07 $ [9] $ -2.71 $ – $ -1.66 $ $ -2.64 $ $ 0.57 $
1.210 [60] $ -2.42 $ [60] $ -3.50 $ – $ -2.34 $ $ -3.35 $ $ 0.93 $
1.154 [12] $ -2.55 $ [12] $ -2.87 $ $ -2.43 $ $ -1.81 $ $ -2.79 $ $ 0.24 $
48Ni 1.350 [12] $ -2.08 $ [12] $ -3.24 $ – $ -2.13 $ $ -3.13 $ $ 1.05 $
54Zn 1.480 [11] $ -2.43 $ [11] $ -2.95 $ $ -2.52 $ $ -1.83 $ $ -2.81 $ $ 0.38 $
1.280 [62] $ -2.76 $ [62] $ -0.87 $ – $ -0.10 $ $ -1.01 $ $ -1.75 $
67Kr 1.690 [14] $ -1.70 $ [14] $ -1.25 $ $ -0.06 $ 0.31 $ -0.58 $ $ -1.12 $
Table 1. Comparison of the experimental data for $ 2p $-radioactive nuclei with different theoretical models (GLDM, ELDM, the four-parameter empirical formula of Ref. [36] and our empirical formula. Experimental data are taken from the corresponding references.
To further test the feasibility of our empirical formula, we also use Eq. (5) to predict the $ 2p $ radioactivity half-lives of 22 nuclei with $ 2p $ radioactivity released energy $ Q_{2p} > 0 $. The $ Q_{2p} $ values are taken from the latest evaluated atomic mass table AME2016 and shown in the second column of Table 2. In this table, the first and third columns give the $ 2p $ radioactivity candidates and the angular momentum l taken away by the two emitted protons, respectively. For a benchmark, the predicted results using GLDM, ELDM and the four-parameter empirical formula, extracted from Refs. [37], [33] and [36] respectively, are also listed in this table. We can clearly see that for $ l\ne 0 $, the predicted results using our empirical formula are closer to those predicted using ELDM than those predicted using the four-parameter empirical formula. Most of the predicted results are of the same order of magnitude. As an example, in the cases of 28Cl (60As), the predicted $ 2p $ radioactivity half-lives using ELDM, the four-parameter empirical formula and our empirical formula are $ -12.95 $ ($ -8.68 $), $ -14.52 $ ($ -10.84 $) and $ -12.46 $ ($ -8.33 $), respectively. This implies that our empirical formula is also suitable for studying nuclei with orbital angular momentum $ l\neq0 $. In the case of $ l = 0 $, the predicted $ 2p $ radioactivity half-lives using our empirical formula are in good agreement with those from GLDM and ELDM. To further demonstrate the significant correlation between the $ 2p $ radioactivity half-lives $ T_{1/2} $ and the $ 2p $ radioactivity released energies $ Q_{2p} $, based on Eq. (5), we plot the quantity $ [{\rm{log_{10}}}{T}_{1/2} + 26.832]/(Z_{d}^{0.8}+l^{\,0.25}) $ as a function of $ Q_{2p}^{-1/2} $ in Fig. 2. In this figure, there is an obvious linear dependence of $ {\rm{log}}_{10}T_{1/2} $ on $ Q_{2p} $$ ^{-1/2} $, while the contributions of charge number $ Z_{d} $ and orbital angular momentum l on the $ 2p $ radioactivity half-lives are removed.
Nucleus $ Q_{2p} $ /MeV l $ {\rm{log_{10}}}{T}_{1/2}^{{\rm{GLDM}}} $ /s [37] $ {\rm{log_{10}}}{T}_{1/2}^{{\rm{ELDM}}} $ /s [33] $ {\rm{log_{10}}}{T}_{1/2} $ /s [36] $ {\rm{log_{10}}}{T}_{1/2}^{{\rm{This}}\;{\rm{work}}} $ /s
22Si 1.283 0 $ -13.30 $ $ -13.32 $ $ -12.30 $ $ -13.74 $
26S 1.755 0 $ -14.59 $ $ -13.86 $ $ -12.71 $ $ -14.16 $
34Ca 1.474 0 $ -10.71 $ $ -9.91 $ $ -8.65 $ $ -9.93 $
36Sc 1.993 0 $ -11.74 $ $ -10.30 $ $ -11.66 $
38Ti 2.743 0 $ -14.27 $ $ -13.56 $ $ -11.93 $ $ -13.35 $
39Ti 0.758 0 $ -1.34 $ $ -0.81 $ $ -0.28 $ $ -1.19 $
40V 1.842 0 $ -9.85 $ $ -8.46 $ $ -9.73 $
42Cr 1.002 0 $ -2.88 $ $ -2.43 $ $ -1.78 $ $ -2.76 $
47Co 1.042 0 $ -0.11 $ $ 0.21 $ $ -0.69 $
49Ni 0.492 0 $ 14.46 $ $ 14.64 $ $ 12.78 $ $ 12.43 $
56Ga 2.443 0 $ -8.00 $ $ -6.42 $ $ -7.61 $
58Ge 3.732 0 $ -13.10 $ $ -11.74 $ $ -9.53 $ $ -10.85 $
59Ge 2.102 0 $ -6.97 $ $ -5.71 $ $ -4.44 $ $ -5.54 $
60Ge 0.631 0 $ 13.55 $ $ 14.62 $ $ 12.40 $ $ 12.04 $
61As 2.282 0 $ -6.12 $ $ -4.74 $ $ -5.85 $
10N 1.3 1 $ -17.64 $ $ -20.04 $ $ -18.59 $
28Cl 1.965 2 $ -12.95 $ $ -14.52 $ $ -12.46 $
32K 2.077 2 $ -12.25 $ $ -13.46 $ $ -11.55 $
62As 0.692 2 $ 14.52 $ $ 13.83 $ 14.18
52Cu 0.772 4 $ 9.36 $ $ 8.62 $ 8.74
60As 3.492 4 $ -8.68 $ $ -10.84 $ $ -8.33 $
Table 2. Comparison of calculated $ 2p $ radioactivity half-lives using GLDM, ELDM, the four-parameter empirical formula from Ref. [36] and our empirical formula. The $ 2p $ radioactivity released energy $ Q_{2p} $ and orbital angular momentum l taken away by the two emitted protons are taken from Ref. [33].
Figure 2. (color online) The linear relationship between the quantity $[{\rm{log_{10}}}{T}_{1/2} + 26.832]/(Z_{d}^{0.8}+l^{\,0.25})$ and $Q_{2p}$ for the database used to fit the parameters of Eq. (5).
IV. SUMMARY
In this work, considering the contributions of the charge of the daughter nucleus $ Z_{d} $ and the orbital angular momentum l taken away by the two emitted protons, a two-parameter empirical formula of a new Geiger-Nuttall law is proposed for studying $ 2p $ radioactivity. Using this formula, the experimental data of the true $ 2p $-radioactive nuclei can be reproduced well. Meanwhile, it is found that the calculated results using our empirical formula are agreement with those from GLDM, ELDM and the four-parameter empirical formula. Moreover, using our formula, the half-lives of possible $ 2p $ radioactivity candidates are predicted. These predicted results may provide theoretical help for future experiments.
We would like to thank X. -D. Sun, J. -G. Deng, and J. -H. Cheng for useful discussions.
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MathOverflow Meta
What are some important but still unsolved problems in mathematical logic?
In the past, first-order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of main controversial axioms were established by forcing method.
I wonder if there still exist some "natural" questions in mathematical logic that are still unsolved? Or is it the case that most of the major questions have been already answered?
I'd love to know about some important, but still unsolved problems that puzzle logicians and why would the young logician\mathematician care about those? (that is, Whey they are important?)
I'm not an expert in logic (nor in any other mathematical field, I'm undergraduate) but I'm interested in logic so I would like to know about the current problems that logicians face and what are the trends of research in the discipline nowdays and what type of problems people are trying to solve.
I know that logic is a vast term which includes many sub-disciplines: model theory, proof theory, set theory, recursion theory, higher-order logics , non-classical logics, modal logics, algebraic logic and many others. So feel free to tell us about problems form whichever topic you would love to.
lo.logic soft-question big-list open-problems
Fawzy Hegab
$\begingroup$ You might want to look at Saharon Shelah's "Logical dreams," although this is more about set theory than logic. arxiv.org/abs/math/0211398 He also wrote a sequel, "Reflecting on logical dreams," in the book Interpreting Goedel (ed. Juliette Kennedy). $\endgroup$ – Timothy Chow Dec 27 '15 at 19:53
$\begingroup$ I believe that this question may be much too broad to answer sensibly. The field of mathematical logic is sweepingly broad, and even small parts of it, say, large cardinal set theory, have a dozen or more huge research programs going on, each with dozens or more open questions that are framing current work. And it is the same with most other topics in logic that you mention. Every one of them has numerous major and minor open questions on which people are hard at work, hundreds of questions altogether. A proper answer to the question will fill a book! $\endgroup$ – Joel David Hamkins Dec 27 '15 at 20:30
$\begingroup$ @JoelDavidHamkins, I agree but I think that dividing the question to many sub-questions posting each one in a new thread will be annoying to the community so I preferred to post it in one thread here. Do you think that If I want more details on a particular topic (say higher order logics), then should I post a new question? $\endgroup$ – Fawzy Hegab Dec 28 '15 at 22:36
$\begingroup$ The current question is hopelessly broad. I think that much more focussed questions would get more useful and illuminating answers: What are the main open questions guiding research in the Turing degrees? What are the main open questions for cardinal characteristics of the continuum? For forcing axioms? Concerning the very largest large cardinals? What are the main open questions in o-minimality? In models of arithmetic? And so on. Let me add that the suggestion in the current post that perhaps "all the major open questions [in mathematical logic] have been already answered" is frankly absurd. $\endgroup$ – Joel David Hamkins Dec 28 '15 at 22:53
$\begingroup$ @Thomas That doesn't really seem relevant - amongst the Millenium Problems, only P vs. NP is conceivably a logic problem, and it's already listed in none's answer below. $\endgroup$ – Noah Schweber Dec 30 '15 at 19:12
Yes, there are several. Here's a few which I personally care about (described in varying amounts of precision). This is not meant to be an exhaustive list, and reflects my own biases and interests.
I am focusing here on questions which have been open for a long amount of time, rather than questions which have only recently been raised, in the hopes that these are more easily understood.
MODEL THEORY
The compactness and Lowenheim-Skolem theories let us completely classify those sets of cardinalities of models of a first-order theory; that is, sets of the form $$\{\kappa: \exists \mathcal{M}(\vert\mathcal{M}\vert=\kappa, \mathcal{M}\models T)\}.$$ A natural next question is to count the number of models of a theory of a given cardinality. For instance, Morley's Theorem shows that if $T$ is a countable first-order theory which has a unique model in some uncountable cardinality, then $T$ has a unique model of every uncountable cardinality (this is all up to isomorphism of course).
Surprisingly, the countable models are much harder to count! Vaught showed that if $T$ is a (countable complete) first-order theory, then - up to isomorphism - $T$ has either $\aleph_0$, $\aleph_1$, or $2^{\aleph_0}$-many countable models. Vaught's Conjecture states that we can get rid of the weird middle case: it's either $\aleph_0$ or $2^{\aleph_0}$. In case the continuum hypothesis holds, VC is vacuously true; but in the absence of CH, very little is known. VC is known for certain special kinds of theories (see e.g. Vaught's conjecture for partial orders and http://link.springer.com/article/10.1007%2FBF02760651) and a counterexample to VC is known to have some odd properties, including odd computability-theoretic properties (https://math.berkeley.edu/~antonio/papers/VaughtEquiv.pdf), but the conjecture is wide open.
NOTE: VC can be rephrased as a "countable/perfect" dichotomy, in which case it is not trivially true if CH holds and is in fact forcing invariant; see e.g. How do we know if Vaught's Conjecture is Absolute?.
PROOF THEORY
If $T$ is a strong enough reasonable theory, we can define the proof-theoretic ordinal of $T$; roughly, how much induction is necessary to prove that $T$ is consistent. For instance, the proof-theoretic ordinal of $PA$ is $$\epsilon_0=\omega^{\omega^{\omega^{...}}}.$$ Proof-theoretic ordinals have been calculated for a variety of systems reaching up to (something around) $\Pi^1_2$-$CA_0$, a reasonably strong fragment of second-order arithmetic which is in turn a very very small part of ZFC. It seems unfair, based on this, to list "finding the proof-theoretic ordinal of ZFC" as one of these problems, based on how far away it is; but "find ordinals for stronger theories" is an important program.
See e.g. Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?.
COMPUTABILITY THEORY
I believe the oldest open problem in computability theory is the automorphism problem. In Turing's 1936 paper, he introduced - in addition to the usual Turing machine - the oracle Turing machine (or o-machine). This is a Turing machine which is equipped with "extra information" in the form of a (fixed arbitrary) infinite binary string. Oracle machines allow us to compare the non-computability of sets of natural numbers: we write $A\le_T B$ if an oracle machine equipped with $B$ can compute $A$. This yields a partial ordering $\mathcal{D}$, the Turing degrees. Initially the Turing degrees were thought to be structurally simple; for instance, it was conjectured (I believe by Shoenfield) that the partial order is "very homogeneous" (there were many different conjectures).
As it turned out, however, the exact opposite happens: the Turing degrees have surprisingly rich structure. See e.g. http://www.jstor.org/stable/2270693?seq=1#page_scan_tab_contents for an early example of this by Feiner, and http://www.pnas.org/content/76/9/4218.full.pdf for a later one by Shore. Indeed, currently the general belief is that $\mathcal{D}$ is rigid, and it has been shown (see e.g. https://math.berkeley.edu/~slaman/papers/IMS_slaman.pdf, Theorem 4.30) that $Aut(\mathcal{D})$ is at most countable. The automorphism problem is exactly the question of determining $Aut(\mathcal{D})$; I don't have a reference as to when it was first stated, but I vaguely recall the date 1955.
We can also ask about "local" degree structures - e.g., the partial order of the c.e. degrees, or the degrees below $0'$ - and there are interesting connections between the local and global pictures.
Another structural question about the Turing degrees is what sort of natural operations on Turing degrees exist. For instance, there is the Turing jump, and its iterates; but these seem to be the only natural ones. Martin's conjecture states that indeed, every "reasonable" increasing function on the Turing degrees is "basically" an iterate of the Turing jump; MC has a few different forms, for instance "all Borel functions . . ." or "In $L(\mathbb{R})$ . . .". See e.g. https://math.berkeley.edu/~slaman/talks/vegas.pdf.
An important theme in set theory is the development of canonical models for extensions of ZFC. The first example is Goedel's $L$, which has a number of nice properties: a well-understood structure, a "minimality" property, and a canonical (in particular, foring-invariant) definition. We can ask whether similar models exist for ZFC + large cardinals: e.g. is there a "core" model for ZFC + "There is a measurable cardinal"? This is the inner model program, and has been developed extensively. Surprisingly, there is an end in sight: in an appropriate sense, if a canonical inner model for ZFC + "There is a supercompact cardinal" can be constructed, then this inner model will in fact capture all the large cardinal properties of the universe.
I am breezing past a truly gargantuan amount of detail here, but the picture is roughly accurate. See e.g. http://www.math.uni-bonn.de/ag/logik/events/young-set-theory-2011/Slides/Grigor_Sargsyan_slides.pdf for more details, as well as the recent presentation https://www.youtube.com/watch?v=MFDVN7UEUSg&list=PLTn74Qx5mPsQlRpBE5OnxMdN3R1d3DLUO&index=4 by Woodin.
SET THEORIES
When someone says "set theory," they usually mean ZFC-style set theory. But this isn't necessarily so; there are alternative set theories. As far as I know, the oldest open consistency problem here is whether Quine's NF - an alternative to ZFC - is consistent. Seemingly small variations of NF are known to be consistent, relative to very weak theories, but these proofs dramatically fail to establish the consistency of NF. Recently Gabbay (http://arxiv.org/abs/1406.4060) and Holmes (http://math.boisestate.edu/~holmes/holmes/basicfm.pdf) proposed proofs of Con(NF); my understanding is that Gabbay has withdrawn his proof, and Holmes' proof has not been evaluated by the community (it is quite long and intricate).
FINITE MODEL THEORY
For a first-order sentence $\varphi$, let the spectrum of $\varphi$ be the set of sizes of finite models of $\varphi$: $$\operatorname{Spec}(\varphi)=\{n: \exists\mathcal{M}(\vert\mathcal{M}\vert=n, \mathcal{M}\models\varphi)\}.$$ We can ask what sets of natural numbers are spectra of sentences; in particular, the finite spectrum problem (see the really lovely paper http://www.diku.dk/hjemmesider/ansatte/neil/SpectraSubmitted.pdf) asks whether the complement of a spectrum is also a spectrum. It is known, for example, that the complement of the spectrum of a sentence not using "$=$" is a spectrum (http://www.inf.u-szeged.hu/actacybernetica/edb/vol07n2/pdf/Ecsedi-Toth_1985_ActaCybernetica.pdf).
There is a complexity theory connection here: a set is a spectrum iff it is in NEXP. So the finite spectrum problem asks, "Is $NEXP=coNEXP$?"
We can also ask about spectra for non-first-order sentences.
ABSTRACT MODEL THEORY
Abstract model theory is the study of logics other than first-order. The classic text is "Model-theoretic logics" edited by Barwise and Feferman; see (freely available!) https://projecteuclid.org/euclid.pl/1235417263. The field began (arguably) with Lindstrom's Theorem, which showed that there is no "reasonable" logic stronger than first-order logic which satisfies both the Compactness and Lowenheim-Skolem properties.
Shortly after Lindstrom's result, attention turned towards Craig's interpolation theorem, a powerful result in proof theory (see https://math.stanford.edu/~feferman/papers/Harmonious%20Logic.pdf). Feferman, following Lindstrom, asked whether there is a reasonable logic stonger than first-order which satisfies compactness and the interpolation property. As far as I know, this question - and many weaker versions! - are still completely open.
I believe this is by far the youngest question in this answer.
share|cite
Noah Schweber
$\begingroup$ I was planning to write this as a separate post but since your answer already mentions Turing degrees, maybe I should not. May I suggest Martin's conjecture as an important open problem relevant to your list? $\endgroup$ – Burak Dec 27 '15 at 19:58
$\begingroup$ @Burak I completely forgot Martin's conjecture! I've added a little bit on it; feel free to make a more detailed answer. $\endgroup$ – Noah Schweber Dec 27 '15 at 20:01
$\begingroup$ If you're bringing up inner model theory, why not link to Woodin's notes (and lecture videos) from the recent YST in Jerusalem? $\endgroup$ – Asaf Karagila Dec 27 '15 at 22:25
$\begingroup$ @AsafKaragila Because I didn't know about them. $\endgroup$ – Noah Schweber Dec 27 '15 at 22:43
$\begingroup$ @NoahSchweber Thank you very much for the extensive well-written answer! I appreciate your effort very much. If you know something about the open problems on higher order logics or non classical logics, I would love to know about them. $\endgroup$ – Fawzy Hegab Dec 28 '15 at 19:13
The modern logic (and foundational mathematics in general) of the 20th century gave us many important things: Russell's type theory, Zermelo-Fraenkel's set theory, meta-theorems about first order logic, including completeness and incompletness phenomena, model theory, and computability theory. Logic expanded into and around mathematics.
In the summer of 2014 at the Vienna Summer of Logic a thousand logicians, if not more, attended a plethora of events that were divided into two parts:
Mathematical Logic hosted 4 conferences, 2 workshops, and one competition.
Logic in Computer Science hosted 8 conferences, $\aleph_0$ workshops, Computer Logic Olympic Games, and a summer school.
It is safe to say that computer science logic has grown bigger than its older sister.
If you are looking for exciting new developments and challenges in logic, turn attention to computer science. But the challenges there are rarely hard mathematical nuts to be cracked. Instead, they are about development of new and strange kinds of logic, about algorithms and decision procedures that serve the needs of computer science, about design of formal systems that is suitable for computer implementation, and so on.
Logic of the early 20th century was a reaction to a conceptual and methodological crisis in mathematics. Logic of the early 21st century is the tool for conquering the newly discovered land of computer science. Now is a good time to be an Edison or a Tesla of logic, and a little less a Cantor or a Russell. (It's always good to be a Gödel or a Turing, of course.)
Andrej Bauer
I do not know whether it is logic or number theory, but Hilbert's 10th problem over rationals ("is there an algorithm to decide whether a given polynomial equation $f(x_1,\dots,x_n)=0$ with rational coefficients have a rational solution?") is still open, on the contrast with negative solution to equations over integers (DPRM theorem.)
Fedor Petrov
$\begingroup$ I think it's definitely in both; see qcpages.qc.cuny.edu/~rmiller/slides/IMS2015.pdf and math.wisc.edu/~lempp/conf/ams15/rmiller.pdf. In particular, it's a question about decidability (so definitely part of logic), and number theory seems deeply relevant (this is actually the first context I saw Bunyakovsky's Conjecture!). $\endgroup$ – Noah Schweber Dec 27 '15 at 21:11
$\begingroup$ Positive answer would probably have nothing to do with logic (as positive answers for degrees 1 and 2 do not have), but the answer is believed to be negative. $\endgroup$ – Fedor Petrov Dec 28 '15 at 10:17
If we're going to include computer science, the big kahuna is of course P vs NP, first proposed by Gödel in a now-famous letter to von Neumann in the 1950s. A fair amount of the existing progress on it has been made by logicians.
Furthermore, the problem can be stated in a particularly logical way: "Is existential second-order logic able to describe languages (of finite linearly ordered structures with nontrivial signature) that first-order logic with least fixed point cannot?"
Woodin's omega conjecture. See this pdf
This is perhaps closer to computational complexity than to logic, but there are many open problems in propositional proof complexity that are analogous to, and have close connections to, well-known open problems in computational complexity. For example, it is widely believed that there are tautologies whose proof in a Frege system is necessarily exponential in size; however, so far, no superpolynomial lower bounds have been proven.
Timothy Chow
A recent list of open problems in Reverse Mathematics is:
Antonio Montalbán, Open questions in reverse mathematics, Bulletin of Symbolic Logic 17:3 (2011), 431-454. Preprint
A slightly older list is from Steve Simpson: Open problems in Reverse Mathematics (1999)
I will mention one specific and tantalizing problem: the strength of Hindman's theorem. Upper and lower bounds for this theorem were first established by Blass, Hirst, and Simpson in a joint paper from 1987. Since then, there have been several papers on the ultrafilters that are relevant to the proof, but the overall bounds have been only slightly improved, for example by
Henry Towsner, A Simple Proof and Some Difficult Examples for Hindman's Theorem, Notre Dame Journal of Formal Logic, 53(1): 53-65. 2012. Preprint
Hindman's theorem is of interest as a purely combinatorial result, but also because it is so closely related to results of topological dynamics such as the Auslander--Ellis theorem.
Carl Mummert
The paper "One Hundred and Two Problems in Mathematical Logic" by Harvey Friedman is an article that lists 102 problems in mathematical logic.
These problems were selected in the form of statements "at least as likely as their negations". The problems were open as of 1973, and the article has a postscript from September 1974 with news of three of the problems being solved.
Matt F.
$\begingroup$ Obviously a lot has changed since then; in a once-over, I noted two problems with positive solutions since publication: #25 ("the first-order theory of any nonabelian free group is decidable", by Kharlampovich and Myasnikov), and #38 ("$HA^\omega$ with extensionality and the full axiom of choice is a conservative extension of $HA$", by Beeson and others). I'm sure others would note many other solved problems on the list. $\endgroup$ – Matt F. Jan 11 '19 at 3:32
$\begingroup$ The absence of $P=NP$ from the list is interesting. The problem had been formulated by Cook as of 1971, but perhaps was not obviously a problem "in mathematical logic" as of the first draft of Friedman's article in July 1973. The logical description of NP in terms of existential second-order logic had only just been proved, in Ronald Fagin's thesis, submitted at Berkeley in June 1973. $\endgroup$ – Matt F. Jan 11 '19 at 3:36
$\begingroup$ The absence of P=NP is due to how everyone knows in their hearts that $P\neq NP$ though no one can prove it. Friedman was looking for problems that are likely to be solved in either direction. $\endgroup$ – Joseph Van Name Mar 18 '19 at 22:38
$\begingroup$ that's not what Friedman said in the paper, which is why I edited the answer. $\endgroup$ – Matt F. Mar 19 '19 at 2:23
One of the very interesting open problems in Logic (and CS) I think:
Is the field of constructible numbers known to be decidable?
Saeed Salehi
$\begingroup$ I have an admittedly very naive question: How is the real number whose constructibility is of interest defined? $\endgroup$ – Joel Adler Sep 12 '18 at 8:37
$\begingroup$ @JoelAdler What do you mean? The problem asks if there exists an algorithm that can tell whether any given first-order sentence in the language of rings is true when interpreted in the field of all constructible numbers. It does not involve any specific real number. $\endgroup$ – Emil Jeřábek supports Monica Sep 12 '18 at 8:59
$\begingroup$ @EmilJerabek Thanks a lot for your comment, this settles my confusion. I thought it was about deciding whether there was an algorithm deciding the constructibility of a real number, which is decidable. $\endgroup$ – Joel Adler Sep 12 '18 at 21:53
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Sojourn time of almost semicontinuous integral-valued processes in a fixed state
D. V. Gusak
Let $\xi(t)$ be an almost lower semicontinuous integer-valued process with the moment generating function of the negative part of jumps $\xi_k : \textbf{E}[z^{\xi_k} / \xi_k < 0] = \frac{1 − b}{z − b},\quad 0 ≤ b < 1.$ For the moment generating function of the sojourn time of $\xi(t)$ in a fixed state, we obtain relations in terms of the roots $z_s < 1 < \widehat{z}_s$ of the Lundberg equation. By passing to the limit $(s → 0)$ in the obtained relations, we determine the distributions of $l_r(\infty)$.
Gusak, D. V. "Sojourn Time of Almost Semicontinuous Integral-Valued Processes in a Fixed State". Ukrains'kyi Matematychnyi Zhurnal, Vol. 63, no. 8, Aug. 2011, pp. 1021-9, https://umj.imath.kiev.ua/index.php/umj/article/view/2782. | CommonCrawl |
Boundedness in a class of duffing equations with oscillating potentials via the twist theorem
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Asymptotic behavior of solutions to a model system of a radiating gas
January 2011, 10(1): 193-207. doi: 10.3934/cpaa.2011.10.193
Asymptotic behavior for solutions of some integral equations
Yutian Lei 1, and Chao Ma 2,
School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States
Received January 2010 Revised June 2010 Published November 2010
In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of the Hardy-Littlewood-Sobolev type in $R^n$
$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,
$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $
We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.
Keywords: Integral equations, weighted Hardy-Littlewood-Sobolev inequality., singularities, asymptotic analysis.
Mathematics Subject Classification: Primary: 45E10, 45G0.
Citation: Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure & Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
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Yutian Lei Chao Ma | CommonCrawl |
Signal Processing Meta
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How can I decompose a signal into square waves?
I am dealing with signals that are a superposition of different square waves with different amplitudes and phases. Normally, one would decompose a signal into sine waves with help of the Fourier transform, but in this particular case a decomposition into square waves would be much more effective. A Fourier transform would produce a very complicated spectrum, while a square wave decomposition should give just a few clear lines.
I know that such a decomposition is possible. In fact, I could use any periodic function as a basis for the decomposition and this is mentioned in many texts on the subject. But I could never find a formula or an explicit example for a decomposition into a non-sinusoidal basis.
My approach to decompose a signal consisting of the $N$ samples $x_k$, was to use a DFT-like formula $$ u_k = \sum_{n=0}^{N-1} x_n \, \mathcal{R}_k(n)$$ where $\mathcal{R}_k$ is a real-valued square wave with a frequency $k$ times the base frequency. But this is certainly not complete, since I do not obtain any phase information for the constituent square waves, and I couldn't invert the procedure.
How can I decompose my signals into square waves with well-defined amplitude and phase?
discrete-signals
signal-analysis
signal-synthesis
SentrySentry
$\begingroup$ a serious decomposition would start by finding (or defining) a base of N signal vectors, which would span the signal vector space of your interest. Then you would youse an inner product measure to compute the coefficients of those signal decompositions in terms of base vectors. $\endgroup$
– Fat32
$\begingroup$ Fat32 is right: you want to be sure the signals you're interested in are spanned by the set of square waves you've choses. In general you'll also want the basis to be orthonormal. $\endgroup$
– MBaz
$\begingroup$ "But this is certainly not complete, since I do not obtain any phase information for the constituent square waves" : In a Fourier transform for a single frequency you need two reals(or one complex) coefficients, the first one is the result of the convolution with a cosine and the second one with a sine (which is just a $\frac{\pi}{2}$ shifted cosine). So I guess that for squares and for a given period $T$ you also need to decompose on a $\frac{T}{2}$ shifted square wave. $\endgroup$
– agemO
What is described in the question is very near the Discrete Wavelet Transform (DWT) with the use of the Haar Wavelet.
The DWT decomposes a signal into a sum of dilated and translated orthogonal functions that do not necessarily have to be trigonometric. The DWT does not transform a signal from the time domain to a frequency domain but to a scale space where the "time" dimension is preserved. The Haar wavelet is effectively just one period of a square wave and due to its dilation and replication as the transform progresses it would appear as occurring at different frequencies. For more information on the link between decomposition level and frequency please see this link
Another transform that might be of help here, is the Walsh-Hadamard transform which does exactly that, decompose a signal into a sum of square waveforms which are orthogonal (Please note the sequency there as well).
For a brief example that seems to be near what you are after, please see this link
A_AA_A
$\begingroup$ I vote for Walsh! $\endgroup$
– rrogers
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Domain-wall excitations in the two-dimensional Ising spin glass
Hamid Khoshbakht, Martin Weigel
School of Computing, Electronics and Maths
The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10\,000\times 10\,000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and $d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.
https://doi.org/10.1103/PhysRevB.97.064410
Copyright © 2018 by American Physical Society. All rights reserved.
Individual articles may be downloaded for personal use; users are forbidden to reproduce, republish, redistribute, or resell any materials from this journal in either machine-readable form or any other form without permission of the APS or payment of the appropriate royalty for reuse. DOI: https://doi.org/10.1103/PhysRevB.97.064410
cond-mat.dis-nn
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physics.comp-ph
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Khoshbakht, H., & Weigel, M. (2018). Domain-wall excitations in the two-dimensional Ising spin glass. Physical Review B, 97, [064410]. https://doi.org/10.1103/PhysRevB.97.064410
Domain-wall excitations in the two-dimensional Ising spin glass. / Khoshbakht, Hamid; Weigel, Martin.
In: Physical Review B, Vol. 97, 064410, 16.02.2018.
Khoshbakht, H & Weigel, M 2018, 'Domain-wall excitations in the two-dimensional Ising spin glass', Physical Review B, vol. 97, 064410. https://doi.org/10.1103/PhysRevB.97.064410
Khoshbakht H, Weigel M. Domain-wall excitations in the two-dimensional Ising spin glass. Physical Review B. 2018 Feb 16;97. 064410. https://doi.org/10.1103/PhysRevB.97.064410
Khoshbakht, Hamid ; Weigel, Martin. / Domain-wall excitations in the two-dimensional Ising spin glass. In: Physical Review B. 2018 ; Vol. 97.
@article{c6693cd175ba4e5daa1127c85fa22279,
title = "Domain-wall excitations in the two-dimensional Ising spin glass",
abstract = "The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10\,000\times 10\,000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and $d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.",
keywords = "cond-mat.dis-nn, cond-mat.stat-mech, physics.comp-ph, quant-ph",
author = "Hamid Khoshbakht and Martin Weigel",
note = "Copyright {\textcopyright} 2018 by American Physical Society. All rights reserved. Individual articles may be downloaded for personal use; users are forbidden to reproduce, republish, redistribute, or resell any materials from this journal in either machine-readable form or any other form without permission of the APS or payment of the appropriate royalty for reuse. DOI: https://doi.org/10.1103/PhysRevB.97.064410 Copyright {\textcopyright} and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. ",
doi = "10.1103/PhysRevB.97.064410",
T1 - Domain-wall excitations in the two-dimensional Ising spin glass
AU - Khoshbakht, Hamid
AU - Weigel, Martin
N1 - Copyright © 2018 by American Physical Society. All rights reserved. Individual articles may be downloaded for personal use; users are forbidden to reproduce, republish, redistribute, or resell any materials from this journal in either machine-readable form or any other form without permission of the APS or payment of the appropriate royalty for reuse. DOI: https://doi.org/10.1103/PhysRevB.97.064410 Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.
N2 - The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10\,000\times 10\,000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and $d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.
AB - The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10\,000\times 10\,000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and $d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.
KW - cond-mat.dis-nn
KW - cond-mat.stat-mech
KW - physics.comp-ph
KW - quant-ph
U2 - 10.1103/PhysRevB.97.064410
DO - 10.1103/PhysRevB.97.064410 | CommonCrawl |
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The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics
Asked 10 years, 11 months ago
In this question, Orbicular made the following comment to Feb7 and my own answers;
Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" insight to physics. The reason is that they only have toy models, all of which are unphysical (in the sense that they predict things which differ from real world measurements). Furthermore even the toy models are usually extremely complicated, killing most expectations to get a "real" model (which is not toyish).
First, I want to thank Orbicular for pointing this out, as it is something that I 'kinda' knew, but often forget. The purpose of this question, is to ask for a deeper explanation, either from Orbicular or someone else. In particular
to what degree does Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry influence/assist 'real' models and actual physics related to the physical world?
I don't wish for this question to turn into a debate about whether or not these maths will later be applied in some beautiful stringy-quantum-symmetry theory; I would much rather it be some explanation of the real use of these things. Specifically, I am interested in hearing about the use of Quantum Groups and their representations to Physicists along with some thoughts on the actual usefulness of the results in NC Algebraic Geometry of those articles I posted over here. Another particularly interesting subject I would like to hear about is the usefulness of commutative algebraic geometry in physics.
Some things I have found
Just two references that I have found that at least address these things to some degree are Peter Woit's lecture notes on Representation Theory, and in Shawn Majid's book on Quantum Groups he discusses some definite physical motivation for studying quantum groups.
qa.quantum-algebra noncommutative-geometry ag.algebraic-geometry big-picture mp.mathematical-physics
122 silver badges33 bronze badges
B. BischofB. Bischof
$\begingroup$ Representation theory per se is ubiquitous in physics and chemistry. They usually abusively call it "group theory". See mathoverflow.net/questions/11784/… for starters. $\endgroup$ – Steve Huntsman Feb 8 '10 at 20:11
$\begingroup$ I am familiar with the more classical representation theory applications to physics as in particle physics. I should have been more clear that I am referring to representation of quantum groups and the like. $\endgroup$ – B. Bischof Feb 8 '10 at 20:14
$\begingroup$ @BB: OK, then consider the Yang-Baxter equation en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation $\endgroup$ – Steve Huntsman Feb 8 '10 at 20:18
$\begingroup$ I am under the impression that the YB equation first arose in physics, and that quantum group were first invented in connection to quantum integrable systems. But I don't know any references, nor any definite facts, hence this is a comment, not an answer. $\endgroup$ – Theo Johnson-Freyd Feb 8 '10 at 21:09
$\begingroup$ I think this is a great question but wonder why the same people who have perniciously closed other broad foundational questions haven't jumped on this? Based on previous criteria, shouldn't it at least be a community wiki? $\endgroup$ – Ian Durham Feb 8 '10 at 21:12
Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.
The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.
The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.
The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.
More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.
Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.
Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab. In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)
José Figueroa-O'FarrillJosé Figueroa-O'Farrill
$\begingroup$ thanks so much for this answer. I will need some time to look it over, but the first reading was very interesting . $\endgroup$ – B. Bischof Feb 9 '10 at 2:42
$\begingroup$ Your comments concerning classical representation theory are true, obviously. Please keep in mind that the question concerned "the use of Quantum Groups and their representations to Physicists", so Wigner's and Gell-Mann's results do not apply. Neither supersymmetry nor string theory have been observed up to this point. On the contrary, new (questionable) hypotheses have been introduced (like superpartners and additional space-time dimensions). These hypotheses introduce new degrees of freedom, making the observability (and falsifiability) at least more difficult! $\endgroup$ – Orbicular Feb 9 '10 at 8:10
$\begingroup$ For the record, I believe that one of the simplest (if not the simplest) quantum system with supersymmetry is a non-relativistic spin-1/2 particle confined in a 1-d harmonic potential. This system can most likely be realized experimentally by placing an electron in a special magneto-optical trap. $\endgroup$ – Igor Khavkine Feb 12 '10 at 9:54
$\begingroup$ I think it depends on what you mean by supersymmetry. If you mean simply that the spectrum arranges itself into representations of a Lie superalgebra, then sure. But then there are (approximate) supersymmetries in nuclear physics already. I am slightly more conservative and insist that for a system to be supersymmetric, the superalgebra ought to be a "spacetime" superalgebra (or in statistical mechanical systems a "euclidean" superalgebra). This is the case for the Josephson junction, where you have a conformal superalgebra acting conformally on the two-dimensional space. $\endgroup$ – José Figueroa-O'Farrill Feb 12 '10 at 10:26
The (apparent, supposed) mathematics behind the fractional quantum hall effect involves the TQFT invariants coming from representations of quantum groups at roots of unity.
Edit: some links for further reading:
From String Nets to Nonabelions
A class of P,T-invariant topological phases of interacting electrons
Topological quantum computation
Scott Morrison♦Scott Morrison
$\begingroup$ I appreciate your answer as it is exactly what I am interested in working on. I would love it if you could possibly expound on your answer a bit. If you don't want to but you know some links, that would also be greatly appreciated. If you don't know any that's fine too, thanks again!!! $\endgroup$ – B. Bischof Feb 9 '10 at 5:15
$\begingroup$ A reference you may find interestings, even if they don't use noncommutative geometry so much as C*-algebras: Scattering theory of topological insulators and superconductors by I. C. Fulga, F. Hassler, A. R. Akhmerov arxiv.org/abs/1106.6351. $\endgroup$ – Terry Loring Mar 31 '12 at 17:55
Connes and Chamseddine have applied NCG to particle physics directly and made predictions for the Higgs mass. (See, e.g., here.) I would say this counts as "actual physics". Whether or not their predictions will survive is another issue.
Steve HuntsmanSteve Huntsman
$\begingroup$ I have to point out that the model of Connes and Chamseddine is based on the standard model, i.e. they use it. Now they get a different expected Higgs mass (it is higher I think). Now suppose the LHC does not find the Higgs at the energies predicted by the (classical) standard model. Then the standard model has to be wrong (and a lot of physics redone). Connes and C. would consider that their triumph, even though the "wrong theory" gave the correct answer. A colleague of mine pointed out that Connes probably knows best how terribly far away he is from a physically reasonable theory. $\endgroup$ – Orbicular Feb 8 '10 at 20:16
$\begingroup$ Thank you for that link, it sounds interesting and this is the sort of answer I was hoping for. $\endgroup$ – B. Bischof Feb 8 '10 at 20:16
$\begingroup$ Orbicular: the framework of Connes and Chamseddine only admits some of the Lagrangians that would be acceptable in quantum field theory, and one of these admissible Lagrangians is the standard model. The observation that it is admissible is what physicists mean when they say that they are "using it." I consider this a predictive model (although, as Connes would admit, it is still very far from a useful theory of physics). $\endgroup$ – Peter Shor Jul 31 '10 at 13:19
$\begingroup$ Orbicular: The prediction of Connes and Chamseddine for the Higgs mass is only valid in the "great desert" scenario where there are no new particles with masses on the same order as the Higgs, and the existence of dark matter casts doubt on this scenario. The standard model does not actually predict a mass for the Higgs. Only some Higgs masses are consistent with the standard model, and Connes and Chamseddine's prediction falls squarely within this range. However, recent experiments seem to indicate that the Higgs mass is considerably lighter than Connes and Chamseddine's predictions. $\endgroup$ – Peter Shor Jul 31 '10 at 13:20
The most straightforward example of application of quantum groups to real physics is integrable spin chains. For instance, XXX-1/2 spin chain has excitations which transform under fundamental rep of $\mathfrak{su}(2)$. Let us now consider the quantum deformation of this symmetry $U_q(\mathfrak{su}(2))$. It turns out that the system you'll get is the anisotropic XXZ spin chain where the deformation parameter $q$ is related to the anisotropy parameter of the spin chain.
You can think of much more examples for different algebras. But I agree, this is a rare occasion that modern math comes into physics life.
Peter KoroteevPeter Koroteev
The Chern numbers, Spin Chern numbers and so forth in condensed matter physics are very important in understanding topological insulators. There are many ways to compute this invariants, and some of them come straight out of noncommutative geometry.
See "Disordered topological insulators: a non-commutative geometry perspective" by Emil Prodan, in Journal of Physics A: Mathematical and Theoretical, 44(2011), 113001.
Terry LoringTerry Loring
I might be wrong, but as far as I remember, in his ICM paper Drinfeld provides some motivation from physics for quantum groups.
Anton LyubininAnton Lyubinin
$\begingroup$ It is related with the Peter's answer above. The origin of quantum groups are in papers by L.D. Faddeev's school on quantum spin chains. $\endgroup$ – Alexander Chervov Apr 1 '12 at 14:42
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Decoding a binary linear code given its generator matrix
Let $C$ be the binary linear code with the following generator matrix
$G= \begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix}$
I need to decode the received word $r$, $\begin{pmatrix} 1 & 1 & 0 &0 & 0 &1 &1\end{pmatrix}$.
I am really strugggling with this. I know a message $m$ is encoded as $mG$.
So $mG$=$\begin{pmatrix} 1 & 1 & 0 &0 & 0 &1 &1\end{pmatrix}$.
I have tried doing this by eye but couldn't find a single $m$ that I could multiply $G$ by to give me $r.$ Is there another method? Also I know that there must be $2^4=16$ codewords and when I wrote them out this one wasn't in them? Not sure if that helps or maybe I did that wrong?
matrices coding-theory error-correcting-codes
harry55harry55
If you don't want to check all possibilities, you can use row operations to convert four columns of the matrix (of your own choosing) to the identity matrix, which will enable for very quick decoding. The code generated is exactly the same.
In more details, each row operation corresponds to multiplying the generator matrix from the left by some invertible matrix. In all, the effect of row operations is to multiply the original generator matrix by some invertible matrix from the left, say $H = AG$ is the new generator matrix. Since $A$ is invertible, both generator matrices generate the same code. If $r = xH$ then $r = (xA)G$, so if you decode $r$ for $H$, you can also decode it for $G$ very easily.
In your case, the row operations result in the following matrix: $$ \begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix} $$
To decode $\begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 1 & 1 \end{bmatrix}$, we simply add the first two rows to get the vector $\begin{bmatrix} 1 & 1 & 0 & 0 & 1 & 0 & 1 \end{bmatrix}$. This shows that the original vector doesn't belong to the code.
Yuval FilmusYuval Filmus
$\begingroup$ I have followed what you said by using row operations to convert four columns of the matrix (of your own choosing) to the identity matrix and got: $$G= \begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{bmatrix}$$ But I'm not quite sure where to go from here? $\endgroup$ – harry55 Feb 27 '17 at 13:51
$\begingroup$ Well, this makes it very easy to find out exactly which vector in the code starts with $1100$: this vector must be the sum of the first two rows. $\endgroup$ – Yuval Filmus Feb 27 '17 at 13:56
$\begingroup$ I don't quite follow. Summing the first two rows gives me $\begin{pmatrix} 1 & 1 & 0 &0 & 0 &1 &1\end{pmatrix}$ which is in fact $r$ but still can't see what $m$ would give me that $\endgroup$ – harry55 Feb 27 '17 at 14:06
$\begingroup$ How is the first row $1011100$? This would not give columns of the identity matrix $\endgroup$ – harry55 Feb 27 '17 at 14:16
$\begingroup$ This is exact decoding. If you manipulated the matrix correctly, then $r$ must have been a codeword after all. If it's not a codeword, then you must have made some mistake when manipulating the matrix. Row operations just don't change the code generated by the matrix. $\endgroup$ – Yuval Filmus Feb 27 '17 at 14:47
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How to construct a random error-correcting code (its generator matrix) according to the code parameters? | CommonCrawl |
Biarticular muscles are most responsive to upper-body pitch perturbations in human standing
Christian Schumacher ORCID: orcid.org/0000-0001-9363-36561,2,
Andrew Berry ORCID: orcid.org/0000-0001-7075-86752,
Daniel Lemus ORCID: orcid.org/0000-0002-0633-47522,
Christian Rode3,
André Seyfarth1 &
Heike Vallery2
Scientific Reports volume 9, Article number: 14492 (2019) Cite this article
Balancing the upper body is pivotal for upright and efficient gait. While models have identified potentially useful characteristics of biarticular thigh muscles for postural control of the upper body, experimental evidence for their specific role is lacking. Based on theoretical findings, we hypothesised that biarticular muscle activity would increase strongly in response to upper-body perturbations. To test this hypothesis, we used a novel Angular Momentum Perturbator (AMP) that, in contrast to existing methods, perturbs the upper-body posture with only minimal effect on Centre of Mass (CoM) excursions. The impulse-like AMP torques applied to the trunk of subjects resulted in upper-body pitch deflections of up to 17° with only small CoM excursions below 2 cm. Biarticular thigh muscles (biceps femoris long head and rectus femoris) showed the strongest increase in muscular activity (mid- and long-latency reflexes, starting 100 ms after perturbation onset) of all eight measured leg muscles which highlights the importance of biarticular muscles for restoring upper-body balance. These insights could be used for improving technological aids like rehabilitation or assistive devices, and the effectiveness of physical training for fall prevention e.g. for elderly people.
Dealing with typical perturbations (e.g. pushing, stumbling or walking on uneven ground) comprises coordinating multiple joints1. For this, biarticular muscles (that span two joints) might play a key role. In contrast to monoarticular muscles (spanning one joint), biarticular muscles contribute strongly to the leg force that acts perpendicular to the leg axis2. This seems to make biarticular muscles especially suitable for postural control because the perpendicular component of leg force regulates the angular momentum.
In accordance with this, in static experiments humans mainly used biarticular thigh muscles to control the direction of the Ground Reaction Force (GRF)3,4,5,6. This ability might also be exploited to achieve stable walking by directing the GRF towards a point above the Center of Mass (CoM)7,8,9,10. Further, humans responded quickly with hamstring activity to a perturbation of angular momentum in a stumbling experiment11. Simulations and robotic demonstrators revealed the potential of biarticular structures (e.g. springs or muscles) to stabilise the trunk during walking and generate appropriate leg swing motions12,13,14,15. However, experimental evidence for the actual use of biarticular muscles for upper-body balance in humans is still missing. Based on their ability to generate appropriate combinations of required hip and knee torques4,5,6,16, we hypothesised that biarticular thigh muscles would react strongly to a perturbation of upper body angular momentum during quiet standing.
Perturbing a system and investigating its response is a standard method to analyse the system's dynamics17. To study human balance strategies, unexpected and specific mechanical perturbations such as surface translations (Fig. 1a) have frequently been used to study the human balance response to slipping17,18,19,20. Recently, research groups have also applied pulls and pushes at the hip or at the shoulder21,22,23,24,25,26,27 to resemble other common perturbation scenarios (Fig. 1b). By generating a horizontal force, these systems perturb balance of the body as a whole28. Pushes/pulls and surface movement perturbations require corrections of angular momentum, i.e. keeping the CoM within the base of support and restoring the upright upper-body orientation. Thus, such perturbations necessitate a complex response of the neuro-musculoskeletal system, which might involve multiple response mechanisms. This might complicate or hamper the interpretation of the role of biarticular muscles for upper-body balance.
Exemplary perturbation types: Schematic visualisation of Center of Mass (CoM), leg, and upper-body kinematics (transparent: before perturbation, solid: after perturbation). (a) Back and forth surface translations result in a whole-body balance perturbation by producing first an acceleration followed by a deceleration. (b) Force pushes or pulls can result in irregular body postures with small upper body perturbations. (c) Torque perturbation: the generated torque on the upper body (curved red arrow) is equivalent to a force pair (straight red arrows) with zero net horizontal force. This type of perturbation results mainly in rotational acceleration of the upper body. Minimal CoM excursions occur due to the muscular coupling of the upper body to the leg. (d) The Angular Momentum Perturbator (AMP) creates external torques by rotating a spinning flywheel (angular momentum H, purple arrow) around a perpendicular gimbal axis (here: longitudinal, angular velocity \(\dot{{\boldsymbol{\gamma }}}\), green arrow, initial gimbal position γ0). The created torque τAMP is exerted in the direction perpendicular to both, the rotation of the spinning flywheel and the gimbal axis, and rotates together with the gimbal (red arrow).
For the first time, we propose to exert a pure torque, or force couple, on the upper body. Applying a pure torque without a translational (horizontal force) component entails that the whole-body CoM position is only minimally influenced and reduces artefacts of cross-talk between different balance strategies. For this purpose, we use an Angular Momentum Perturbator (AMP)29. This system opens a new category in the framework of Shirota et al.30 and enables a specifically targeted perturbation (Fig. 1c), making it a suitable tool to study upper-body balance response strategies while maintaining a natural leg axis alignment. The system is portable – the subject can move freely without being bound to a treadmill or a frame construction – and is capable of generating powerful torque bursts.
In this study, we contribute to the ongoing research of human motor control during non-stepping balance recovery by examining the muscular response of major leg muscles for two major perturbation directions: (1) positive torque perturbations resulting in a forward upper-body pitch and (2) negative torque perturbations resulting in a backward pitch. Compared to other commonly used perturbation types, the AMP produced perturbations with a distinct upper-body pitch and only a small shift in CoM position, allowing us to assess the contribution of different leg muscles (using electromyographic data (EMG)) for restoring the upper-body balance in near-absence of whole-body balance corrections.
Angular Momentum Perturbator (AMP)
The AMP is worn like a backpack and contains a control moment gyroscope, an actuator that exerts torques by manipulating the angular momentum of an internal flywheel. This flywheel is mounted to a motorised gimbal frame (Fig. 1d), which enables the flywheel to be reoriented with respect to the wearer.
For the present study, we modified the AMP prototype described previously29 to reduce its mass. After replacing the gimbal motor and transmission, the system weighed 16 kg and was capable of exerting a maximum torque of 53 Nm.
The AMP exerts torques on the wearer (τAMP) by changing the direction or magnitude of the angular momentum of the flywheel (H), described in detail by e.g. Schaub et al.31. Changing the direction by rotating the gimbal or rotating the trunk produces gyroscopic torques proportional to the angular velocities of the gimbal (\(\dot{{\boldsymbol{\gamma }}}\)) and trunk of the wearer (ω = [ωx, ωy, ωz], using the frame definitions of Fig. 1d). When exploiting this gyroscopic effect, the gimbal motor applies a torque (τg) in the gimbal axis to modulate \(\dot{{\boldsymbol{\gamma }}}\); however, in doing so, this changes the magnitude of H and is experienced by the wearer as an opposite reaction torque in the same axis. The total torque is thus the sum of these components:
$${{\boldsymbol{\tau }}}_{{\rm{A}}{\rm{M}}{\rm{P}}}(t)=-\,\dot{{\boldsymbol{H}}}\,(t)=\mathop{\underbrace{-\,(\dot{{\boldsymbol{\gamma }}}(t)+{\boldsymbol{\omega }}(t))\times {\boldsymbol{H}}(t)}}\limits_{{\rm{g}}{\rm{y}}{\rm{r}}{\rm{o}}{\rm{s}}{\rm{c}}{\rm{o}}{\rm{p}}{\rm{i}}{\rm{c}}\,{\rm{e}}{\rm{f}}{\rm{f}}{\rm{e}}{\rm{c}}{\rm{t}}}\,-\mathop{\underbrace{{{\boldsymbol{\tau }}}_{g}(t)}}\limits_{{\rm{g}}{\rm{i}}{\rm{m}}{\rm{b}}{\rm{a}}{\rm{l}}\,{\rm{m}}{\rm{o}}{\rm{t}}{\rm{o}}{\rm{r}}}.$$
By design, the gyroscopic torque is generally much larger than the gimbal torques (τg) to allow the use of a relatively small gimbal motor and reduce power requirements. Since this gyroscopic torque depends not only on the controlled gimbal velocity (\(\dot{{\boldsymbol{\gamma }}}\)), but also on the angular velocity of the wearer ω, it is in general only partially controllable. However, in the present study, since ω is relatively small (especially during quiet standing), the uncontrolled gyroscopic torques were typically at least one order of magnitude smaller than those due to \(\dot{{\boldsymbol{\gamma }}}\) and were mostly considered negligible.
When generating controlled gyroscopic torques by rotating the gimbal, the direction of τAMP simultaneously changes with the gimbal angle (γ). In a human-fixed frame (unit vectors \(({\hat{{\bf{e}}}}_{x},{\hat{{\bf{e}}}}_{y},{\hat{{\bf{e}}}}_{z})\) attached to frame (x, y, z), Fig. 1d), the total perturbation torque τAMP consists of components in all three directions:
$$\begin{array}{ccc}{{\boldsymbol{\tau }}}_{{\rm{A}}{\rm{M}}{\rm{P}}}(t) & = & -\mathop{\underbrace{{\tau }_{t}(t)\,\cos \,(\gamma (t)){\hat{{\boldsymbol{e}}}}_{x}}}\limits_{{\rm{p}}{\rm{i}}{\rm{t}}{\rm{c}}{\rm{h}}\,{\rm{c}}{\rm{o}}{\rm{m}}{\rm{p}}{\rm{o}}{\rm{n}}{\rm{e}}{\rm{n}}{\rm{t}}}-\mathop{\underbrace{{\tau }_{t}(t)\,\sin \,(\gamma (t)){\hat{{\boldsymbol{e}}}}_{y}}}\limits_{{\rm{r}}{\rm{o}}{\rm{l}}{\rm{l}}\,{\rm{c}}{\rm{o}}{\rm{m}}{\rm{p}}{\rm{o}}{\rm{n}}{\rm{e}}{\rm{n}}{\rm{t}}}+\mathop{\underbrace{({\omega }_{t}(t)H(t)-{\tau }_{g}(t)){\hat{{\boldsymbol{e}}}}_{z}}}\limits_{{\rm{y}}{\rm{a}}{\rm{w}}\,{\rm{c}}{\rm{o}}{\rm{m}}{\rm{p}}{\rm{o}}{\rm{n}}{\rm{e}}{\rm{n}}{\rm{t}}}\\ & \approx & -\dot{\gamma }(t)\,H\cos \,(\gamma (t)){\hat{{\boldsymbol{e}}}}_{x}-\dot{\gamma }(t)\,H\sin \,(\gamma (t)){\hat{{\boldsymbol{e}}}}_{y}+({\omega }_{x}(t)\,H\cos \,(\gamma (t))-{\tau }_{g}(t)){\hat{{\boldsymbol{e}}}}_{z},\end{array}$$
where τt(t) = (\(\dot{{\boldsymbol{\gamma }}}\)(t) + ωz(t))H(t) ≈ \(\dot{{\boldsymbol{\gamma }}}\)(t)H(t) and ωt(t) = ωx(t)cos(γ(t)) + ωy(t)sin(γ(t)) ≈ ωx(t)cos(γ(t)). All non-bold variables indicate the signed scalar magnitudes of the vector quantities defined previously, and the magnitude of the flywheel angular momentum was approximately constant throughout all experiments (H(t) ≈ H).
To generate a torque of a specific magnitude and direction requires inversion of Eq. (2) to produce a reference gimbal motion, described in general by Berry et al.32. For discrete open-loop perturbations, this inversion is simplified, and, for a given initial gimbal angle (γ0) and desired torque profile, the final angle (γf) can be computed in advance. To produce a torque primarily in the sagittal plane (around the x-axis) and limit the component in the frontal plane (around the y-axis), the range of gimbal rotation was constrained and γ0 was chosen such that γ ≈ 0 when the magnitude of τAMP was maximal (Fig. 1d).
The perturbation torque τAMP was selected to be a symmetric trapezoidal profile (Fig. 2 inset), consisting of a peak torque of 60 Nm with a rise time, hold time, and fall time each of 100 ms. This shape reflects the finite ability of the gimbal motor to accelerate or decelerate rotation of the gimbal structure to produce the gyroscopic effect. A gimbal motor torque (τg) of approximately 12 Nm was necessary to generate the desired perturbations. Because the roll (frontal) torque components alternated sign throughout the perturbation, all perturbations were repeated with the gimbal inverted (rotated by 180°) to also generate the opposite pattern (of roll torque component) with similar pitch torque component for comparison.
AMP generated (a) positive and (b) negative torque perturbations resulting in forward and backward pitch of the upper body, respectively. Change in upper-body pitch angle of one subject (mean and standard deviation from the last 5 trials × 2 gimbal configurations) with respect to the initial upper-body posture (top left). The inset shows the desired (grey, 100 rise, hold and fall time) and actual pitch perturbation torque profile (orange/green: mean and SD). Change in sagittal CoM (grey: mean and SD) and CoP position (black: mean and SD) of the same subject with respect to the initial positions (top right). Exemplary EMG of the same subject (bottom panel): filtered and normalised signals of all trials (grey) and one individual response of mono- (yellow) and biarticular muscles (red).
Experimental protocol
The experimental protocol consisted of three successive sets of measurements involving different settings. In the first setting, the subjects were asked to stand still for 60 seconds with arms crossed over the chest and without wearing the perturbator ('Unloaded Standing'). In the second setting, they repeated the same task but wore a safety harness and the AMP with spinning flywheel, but no active torque perturbations ('Loaded Standing'). In the final setting of standing, a series of 48 trials were executed involving 40 trials of active torque perturbation of the AMP ('Perturbation Trials') and 8 trials which lacked a perturbation ('Control Trials'). The 40 Perturbation Trials consisted of 4 conditions with 10 repetitions each: positive and negative torque directions with both original (γ0 = 0°) and inverted (γ0 = 180°) initial gimbal orientations. Conditions of positive and negative torque direction (inducing forward and backward upper-body pitch, respectively) were chosen to investigate a potential direction dependency. Since the AMP generates a complex perturbation that acts in multiple planes (mainly sagittal but also in the frontal plane), the initial gimbal position was altered to investigate potential side effects of the roll torque component. Due to relatively long gimbal re-positioning times before every trial (to reach the initial gimbal position), a blocked protocol was preferred over a fully randomised protocol ('Block 1' with original and 'Block 2' with inverted gimbal position). This was done to reduce the measurement time and subject fatigue. Within the two blocks, both the direction and timing of the active perturbations were randomised. Subject were given time to rest in between measurements. For referencing, subjects conducted ten additional walking trials (distance approx. 7 m) at preferred walking speeds (1.20 ± 0.14 ms−1, Mean ± SD) without any additional loads.
To prevent fall-related injury during the perturbations, a safety harness was used at all times when the AMP was worn. The harness was attached to the Rysen body weight support system (Motekforce Link, Amsterdam, The Netherlands), which is capable of actively detecting and arresting falling motions of the subject33. To avoid vertical unloading forces during the measurements, the Rysen system was lowered and locked in a position where the harness straps were slack during both quiet and perturbed standing, but could still prevent falls if balance could not be successfully recovered.
Data collection and processing
Seventeen healthy adult subjects (two female) participated in the study. All subjects volunteered to participate in the study in summer 2018 and gave written informed consent in advance of the experiment. The experimental protocol was approved by and performed in accordance with the relevant guidelines and regulations of the Human Research Ethics Committee of the Delft University of Technology (Project ID: 350). Before the experiment, all participants gave their consent and filled in the revised Waterloo Footedness Questionnaire34, which was used to assess limb dominance regarding stabilisation tasks.
We collected EMG, kinetic, and kinematic data of the subject. All measurement devices including the data logging of the AMP were synchronised by a manual trigger signal. Before processing the data, one female subject was excluded from the data analysis due to repetitive stepping responses during the experiments. A further five (male) subjects were excluded due to missing EMG data or missing marker data that impeded a calculation of the CoM. Only the data of the remaining eleven subjects (one female), of age 34 ± 14 years (Mean ± SD), weight 74.8 ± 12.5 kg, and height 1.81 ± 0.08 m, were considered for further analysis.
Measured data
Sixteen surface EMG electrodes (Trigno, Delsys Inc., Natick, USA) were used to record at 2000 Hz the electrical activity of relevant leg muscles. The set of electrodes were placed on the following muscles of each leg: tibialis anterior (TA), soleus (SOL), gastrocnemius lateralis (GAS), vastus lateralis (VL), rectus femoris (RF), biceps femoris long head (HAM), tensor fasciae latae (TFL) and gluteus maximus (GLU) (see Supplementary Fig. S1). To ensure a good electrical connectivity, the skin was prepared following the SENIAM recommendations35. After attachment, electrode locations were checked for voluntary muscle signals and low noise values. The raw EMG was band-pass-filtered with cut-off frequencies of 20 Hz (high-pass) and 450 Hz (low-pass), further rectified and low-pass filtered at 50 Hz. For each muscle and subject, the filtered EMG signals were normalised by the mean filtered background activity recorded during unloaded level walking (WEMG), and expressed as a percentage of WEMG.
Individual GRF of both legs were measured at a frequency of 1000 Hz (3rd-order analogue Butterworth low-pass filter with 500 Hz cut-off frequency) using two force plates (9260AA3, Kistler Holding AG, Winterthur, Switzerland), each for a single leg. These were combined to compute the Whole-body Center of Pressure (CoP).
An inertial measurement unit (IMU) (MPU-9250, InvenSense, San Jose, USA) within the AMP and a motion capture system (Qualisys, Gothenburg, Sweden) were used to collect kinematic data (200 Hz) of the subject and the AMP. Nineteen reflective markers were placed at relevant body locations (see Supplementary Fig. S1): the tragion (TRA), 7th cervial vertebrae (C7), acromion (ACR), greater trochanter (GTR), lateral femoral condyle (LFC), fibulare (FIB), lateral melleoius (LM), calcaneus (CAL), 1st metatarsal head (MT1), and 5th metatarsal head (MT5). The CoM of the AMP was estimated by suspending the AMP from a single attachment point, and measuring the point of intersection of the vertical axis as the attachment point was changed. Four addition markers were placed on the rigid and static frame of the AMP such that their mean position resembled the CoM of the AMP. All kinematic data was upsampled to 1000 Hz by linear interpolation and filtered using a zero-lag 4th-order low-pass filter with a cut-off frequency of 6 Hz.
Computing outcome measures
Since the unintended roll torque components created only slight opposing asymmetric behaviours with similar magnitude in both conditions of initial gimbal positions ('Block 1' vs. 'Block 2', see Supplementary Figs S9–S14), both sets were merged for comprehensibility reasons. Outcome measures, such as mean and SD of kinematic and kinetic data, were calculated from both blocks together.
Mean muscle stimulation of 'Unloaded Standing' and 'Loaded Standing' was calculated for a 20 s period of quiet standing. The mean signal of the preceding 500 ms prior to perturbation onset (quiet standing) was used as 'Pre-Perturbation' activation. From this signal, the mean of each muscle and subject of the last 10 trials per block ('Block 1' and 'Block 2') was compared against the condition of 'Loaded Standing' to evaluate muscle activity changes due to anticipation.
In order to account for temporal variability of the muscular activity for all 'Perturbation Trials', three time intervals were chosen to evaluate the appearance of medium-latency response (RI1: from 100 ms to 150 ms) and long-latency responses (RI2: from 170 ms to 250 ms, RI3: from 270 ms to 350 ms) that involve contributions from supra-spinal centres or poly-synaptic reflex responses36,37,38. The relative reflex response of each muscle was defined as the difference between the mean EMG in the response intervals and corresponding mean EMG in the 'Pre-Perturbation' interval of the same trial, each expressed as a percentage of WEMG. As adaptation processes of EMG responses have been found to settle after about 5 trials39, we considered only the last 5 of the 10 trials per condition, in order to reduce the effects of adaptation processes. We used the Grubbs' test (with a significance level of α = 0.05) to identify and remove unphysiological EMG values (outliers), stemming from e.g. physical collisions of the hip belt and hip muscle electrodes. To evaluate the muscular reflexes, we first computed the mean relative reflex response of all analysed trials per subjects (last 5 of the 10 trials per condition, combining 'Block 1' and 'Block 2') before performing the statistical analysis. Outcome measures of pooled data are reported by grand means and SD of the averaged subject data. Data analysis was done using Matlab 2016b (The MathWorks Inc., Natick, USA).
We compared different joint angles and activity levels between 'Unloaded Standing' and 'Loaded Standing' as well as 'Loaded Standing' and 'Pre-Perturbation' conditions. First we evaluated the normality of the residuals by the Shapiro-Wilk test. If they were normally distributed we applied a paired two-sided t-test. In other cases we tested for differences with the non-parametric two-sided Wilcoxon signed-rank test. All comparisons were performed using a 5% significance level.
For comparisons of the relative reflex response, a repeated measures analysis of variance (rmANOVA) with between ('Direction', 'Side' and 'Muscles') and within subject factors ('Response Intervals') was performed. Shapiro-Wilk tests confirmed normality of the data. If Mauchly's test for sphericity revealed that homogeneity of the data was not given, the Greenhouse-Geisser correction was used. Comparisons were performed using a 5% significance level. If the rmANOVA showed significant interaction effects, post-hoc tests were computed with the Tukey's Honest Significant Difference procedure to adjust for multiple comparisons.
We perturbed upper-body posture during quiet standing by applying external torques – instead of forces – created by a new type of perturbation device, the AMP. This was done to specifically investigate the role of biarticular leg muscles used to control upper-body balance.
The outcome measures of both conditions of initial gimbal positions ('Block 1' and 'Block 2') were merged before computing statistics. The results will thus be presented for both main perturbation directions: positive torque perturbation (τAMP > 0) with forward upper-body pitch and negative torque perturbation (τAMP < 0) with backward upper-body pitch. In the following, we will focus on the results of the left leg (the dominant leg for stabilisation of all subjects) as leg behaviour (see Supplementary Figs S12–S14 and Table S2) and response characteristics ('Side' main effect of rmANOVA: F(1,314) = 0.18, p = 0.676) were similar for both sides.
AMP produced dynamic and reproducible torque perturbations
To verify that the AMP generated appropriate perturbation profiles, the calculated gyroscopic torque (based on measured gimbal angular velocity \(\dot{{\boldsymbol{\gamma }}}\)) was compared to the desired profile (Fig. 2 insets). For both perturbation directions, the AMP was able to generate repeatable perturbations that closely resembled the shape of the desired profile. The rise and fall dynamics were tracked accurately, but the realised peak torque (53 Nm) fell lower than the target of 60 Nm. For both perturbation directions, the small standard deviation (SD) of the measured profile confirms a consistent perturbation generation over multiple trials (Fig. 2 insets).
Torque perturbations resulted in specific upper-body pitch perturbations
In order to study the resulting posture of the subjects, we evaluated upper-body pitch (sagittal-plane rotation), whole-body CoM and CoP (top panels in Fig. 2a). Following the positive torque perturbation, the upper-body segment pitched forward with a peak rotation of 15° to 17° with respect to the initial posture before perturbation onset. The peak pitch was reached after 300 ms to 400 ms after perturbation onset, and about 1000 ms to 1500 ms were needed to recover the initial posture. This response was accompanied by a backwards movement of the hip joint (see Supplementary Fig. S4). In contrast to the substantial upper-body rotation, the CoM moved forward by only about 1 cm to 2 cm. The CoM progression started after 400 ms. The mean CoP moved anterior by 4 cm to reach its peak after 500 ms. The subjects' mechanical response within the first 1000 ms consisted of a distinct early upper-body pitch, mainly due to hip flexion within first 500 ms (see Supplementary Fig. S2), followed by a delayed, but subtle CoM forward sway.
For negative torque perturbations, the resultant backwards pitch of the upper body (top panels in Fig. 2b) was smaller (approx. −12°) compared to the forward perturbations and appeared at similar timing (300 ms to 400 ms after perturbation onset). During upper-body pitch, the hip joint moved forward (see Supplementary Fig. S4). Again delayed, the CoM moved posterior to a peak of about −2 cm from the initial position. A distinct CoP pattern was measured: for the time of backward acceleration of the upper body, the CoP moved forward before quickly moving backwards to around −5 cm. In the backwards condition, the upper-body rotation was not achieved through hip extension (remained in initial posture), but by knee bending such that upper body and thigh segments pivoted together around the knee joint (see Supplementary Fig. S2).
Influence of AMP weight, noise, and vibrations
Next, we tested the extent to which the weight, noise, and vibrations of the AMP influenced the initial posture and the EMG. Wearing the AMP led to a more bent hip (−6.2°, t(10) = −5.528, p < 0.001) and more extended knee (2.1°, t(10) = 3.652, p = 0.002) while no difference was found for the ankle joint (see Supplementary Table S1). By comparing EMG data of the two conditions of quiet standing 'Unloaded Standing' and 'Loaded Standing' changes of muscle stimulation were identified (Fig. 3). The additional weight, noise, and vibration of the rotating flywheel of the AMP resulted in significant reductions of muscular activity in GLU (unloaded: 78 ± 28% WEMG, loaded: 57 ± 24% WEMG, Z = −2.934, p = 0.003) and HAM (unloaded: 108 ± 75% WEMG, loaded: 62 ± 34% WEMG, Z = −2.490, p = 0.013) of the left leg – reflecting the reduced demand of muscle generated hip extension torque to maintain similar upper-body postures – as well as VL of the right leg (see Supplementary Table S2). No other significant differences were found.
Influence of AMP artefacts and prestimulation activity. Boxplots of muscular activity of the left leg in 'Unloaded Standing' (UL), 'Loaded Standing' (L) and the mean 'Pre-Perturbation' activation of the last 10 trials per block ('Block 1' and 'Block 2') for all 11 subjects. Reported EMGs (WEMG) are the mean intervals of the filtered EMG signals normalised by the muscle's mean activity during walking (see Methods). Results are presented with significant (*p < 0.05) comparisons of the paired t-test or two-sided Wilcoxon signed-rank test.
Prestimulation of monoarticular hip muscles to prepare for perturbations
We compared the muscular activity in the 'Loaded Standing' and the 'Pre-Perturbation' activation to assess if subjects increased muscular activity in anticipation of perturbation, despite the random timing and direction (Fig. 3). Significant changes of the monoarticular hip muscles GLU of the left leg (increased by: 8 ± 13% WEMG, Z = 2.134% WEMG, p = 0.033). From all other muscles, only TFL in the right leg showed a significant reduction (see Supplementary Table S2).
Biarticular thigh muscles were major contributors to recover upper-body balance
We next tested which muscles showed the highest response activity to identify the most reactive muscles. For both perturbations, early EMG responses within the first 100 ms after the perturbation were not found. In the analysed response intervals, both perturbation directions evoked different muscular responses ('Direction' main effect: F(1,314) = 173.8, p < 0.001).
For positive torque perturbation with forward upper-body pitch, we found the highest muscle activity levels for the biarticular thigh muscle HAM, the biarticular ankle muscle GAS and the monoarticular ankle muscle SOL. HAM response activity started already in the mid-latency response window (RI1) and lasted until RI3 with up to 366 ± 271% WEMG. GAS (up to 128 ± 61% WEMG) and SOL (up to 101 ± 35% WEMG) responded later in RI2 or RI3. By this, HAM was found to increase its activity (in RI1 and RI2) by more than double of its mean activity during walking (Fig. 4a) which was significantly higher than in all other muscles (see Supplementary Table S3).
Reflex activity of mono- (yellow) and biarticular muscles (red) for (a) positive and (b) negative torque perturbations in different response intervals. Grand mean and SD of averaged relative reflex responses of the left leg (last 5 trials per condition × 2 gimbal configurations) for all 11 subjects. Relative reflex activity in each response interval is computed with respect to 'Pre-Perturbation' and normalised by the muscle's mean activity during walking (see Methods).
In negative perturbations, the biarticular thigh muscle RF, the monoarticular ankle flexor TA and the monoarticular knee extensor VL were found to have the highest EMG amplitude with respect to their mean activity during normal walking. RF activity was high in all response intervals (up to 866 ± 341% WEMG). RF, TA (up to 706 ± 435% WEMG) and VL (up to 582 ± 178% WEMG) reached the highest response in RI2. TFL activity with a maximum of 428 ± 257% WEMG in RI2 increased by half of the RF increase. Since 'Pre-Perturbation' activity was similar for TFL, RF and VL, RF was also the muscle with the strongest and earliest increase in activity (Fig. 4b). Compared to all other muscles, RF reflex activity was significantly stronger than in all other muscles (see Supplementary Table S3). From a lower activity (24 ± 9% WEMG) before the perturbation onset, TA increased by 678 ± 433% WEMG in RI2, but only showed low increase in RI1.
Summarising these results, biarticular thigh muscles (HAM and RF) were found to have the highest activity in the earliest response interval (RI1), the highest overall amplitude with respect to their mean activity during walking and the highest increase compared to unperturbed standing (significantly higher than all other measured muscles). Ankle joint muscles like GAS, SOL or TA contributed moderately with delayed increase, mainly in RI2 and RI3. In contrast to biarticular thigh muscles, monoarticular hip muscles GLU and TFL increased only slightly or moderately in response to the evoked upper-body pitch deflection.
Previous research identified beneficial features of biarticular muscles that could potentially contribute to recover balance after postural perturbations4,6,40,41. Experimental evidence that highlights the specific contribution of biarticular muscles to upper-body balance is however missing. To fill this gap of knowledge, this study examined the muscular response of major leg muscles for recovering from artificial torque perturbations to the upper body. For the first time, an Angular Momentum Perturbator (AMP) was used to directly apply reproducible torques to the upper body during quiet standing without a translational (e.g. horizontal foreward/backward force) component of other typical perturbations like pushes or pulls. By analysing the mechanical response of the subjects and EMG of major leg muscles, we found that
Biarticular thigh muscles HAM and RF showed the strongest increase in activity in response to upper-body balance perturbations of all measured leg muscles;
AMP torques resulted in nearly isolated upper-body postural alignment perturbations.
Biarticular muscles and upper-body balance
In our study, biarticular thigh muscles showed the strongest reflex responses. Perturbing the upper-body challenged the motor control system to re-erect the upper-body. The positive perturbation created forward upper-body pitch (about 17°). Restoring the original body posture requires hip extension torques. These would cause knee overextension which must be counteracted by knee flexion torques. The biarticular HAM muscle group delivers simultaneous hip extension and knee flexion torques. This might explain our finding that the HAM showed the strongest EMG increase of all measured muscles. The negative perturbation led to backwards upper-body pitch (about −12°). Conversely, to re-erect the upper-body, hip flexion and knee extension torques are required. The biarticular RF muscle group delivers simultaneous hip flexion and knee extension torques. Correspondingly, the strongest EMG response was found for the biarticular RF.
Our results confirmed our hypothesis of a strong involvement of biarticular thigh muscles for restoring upper-body balance. This result agrees well with the previously suggested torque-based (net hip minus knee torque; extension torques are positive) control scheme of biarticular muscles for joint coordination in tasks involving opposing changes in hip and knee torques. For example, high HAM and RF activity was found during GRF direction manipulations in isometric and isokinetic conditions4,6, load-lifting42,43, cycling44 as well as in the swing leg during walking and running16. In contrast to these studies involving planned movements, our experiment targeted perturbation recovery. Studies in this area involving surface translations and force pushes/pulls17,18,19,20,21,22,23,24,25,26,27 show that, among other muscles, biarticular muscles react to compensate for these perturbations. Our experiments refined these findings by providing evidence for a strong reactive involvement of biarticular thigh muscles to compensate torque perturbations applied at the upper body. This can most clearly be observed for the positive torque perturbation.
The positive and the negative perturbation generated different responses that might be related to joint range-of-motion constraints. While the positive perturbation lead to strong hip flexion (about 16°) and only small knee extension (about 2°, see Supplementary Fig. S2), the negative perturbation generated moderate hip extension (about 6°) and strong knee flexion (about 14°, see Supplementary Fig. S2). From a straight configuration as reference, the range of motion of the hip is approximately 120° flexion and only 10° hip extension45. The knee constraint is even more strict with approximately 140° flexion and 2° extension45. While our positive perturbation induced mainly upper body forward pitch, the negative perturbation resulted in a backwards pitch of the upper body and the thigh segments together. Thus, the asymmetric perturbation responses reflect the joint range-of-motion constraints.
The observed asymmetric perturbation responses induced different torque requirements. In the case of the positive perturbation, biarticular HAM activity seems to be sufficient to re-erect the upper body. For negative perturbations, in addition to pronounced RF activity we observed increases in activity of monoarticular knee extensors (VL) and to a lesser extent of the functionally monoarticular hip flexor muscles (TFL). It seems that despite its ability to provide for simultaneous knee extension and hip flexion torques, RF may not be able to meet the torque requirement at these joints. RF has approximately half of the physiological cross-sectional area and thus half the force capacity of HAM46. In addition, it has a lower lever arm at the hip than HAM47,48, which may explain the increase in activity of the functionally monoarticular hip flexor TFL. The different capacity of the biarticular HAM and RF muscles to generate joint torques may stem from an adaptation towards their daily use, reflecting a dominating use of HAM in walking and running due to a forward lean of the upper body49. For less intense negative torque perturbations, we expect to see disproportionally less pronounced knee flexion (reducing the kinematic difference between positive and negative perturbations) and reduced activity of monoarticular muscles. RF torques might then be sufficient to re-erect the upper body.
During perturbations in both directions, different mechanisms led to opposing CoM and CoP patterns. In positive perturbations, the upper body pitched forward about the hip. During this early response, extension hip torques were generated by increased HAM activity. The hip coupling to the leg (and subsequently to the ground) resulted in a delayed acceleration of the CoM in anterior direction (after 400 ms). The CoP travel is due to ankle extension torques provided by the SOL to counteract the forward movement of the CoM. During the backward perturbation, the upper body rotated backwards by extensive bending at the knee joint. This induced a pronounced eccentric stretch of the pre-activated SOL resulting in an instantaneous increase in muscle force50. This explains the initial anterior CoP travel (within the first 300 ms to 400 ms). This, however, is a critical condition for balance, because both the required hip flexion (to realign the upper body) and the ankle extension accelerate the CoM backwards51. To counter this, immediate deactivation of the SOL and activation of the TA would be required which we also observed in the EMG responses (RI3 in Fig. 4b). Further proof of these interpretations requires modelling and simulation of the neuro-muscle-skeletal system.
Reported EMG signals might contain cross-talk from other muscle groups. For example, the RF might have been contaminated by activity from the vastus intermedius52. Using needle electrodes might be a more reliable alternative. However, due to their invasive nature, the intramuscular electrodes require technical expertise, usually more time for preparing the subject and can create discomfort53,54, which in turn might influence the behaviour of the subject. By following the SENIAM recommendations35 for skin preparation and electrode placement, and by utilising appropriate surface electrodes55,56, we aimed at reducing the influence of cross-talk to a minimum.
Our experiments show that the biarticular thigh muscles strongly relate to postural upper-body control in standing. We speculate that these muscles could also play a key role for upper-body balance in walking. During walking, GRFs intersect in a point above the CoM which helps to stabilise the angular momentum of the whole body similar to a physical pendulum7,9. However, upper body stability is not achieved in this way, because the resulting forces acting on the upper body intersect below the CoM of the upper body10. Here, biarticular muscles can help to regulate the angular momentum of the upper body by applying torques via the stance leg2,12,42,57,58,59,60 or the swing leg14,15,16. The stabilising function of biarticular structures for the upper body during walking was demonstrated in a dynamically walking robot13. Further perturbation experiments will target the role of biarticular muscles for upper-body balance in walking.
Anticipatory and reactive control strategies
When mechanical perturbations are applied, humans use different strategies to prepare or counteract the effect of the perturbation17,61,62. For instance, when a perturbation is expected or predicted, activity of antagonistic muscles can be increased simultaneously (co-contraction) in order to build up joint impedance before the perturbation onset63. This zero-delay mechanism is governed by feed-forward commands of supra-spinal centres and usually called 'prestimulation'50,64. Especially when knowledge about the perturbation (e.g. intensity, timing or direction) is rare, joint impedance can help to preserve the body posture65.
In our experiment, an anticipatory co-contraction was not found. Only the monoarticular hip extensor GLU of the left leg showed an increased activity level in anticipation of the perturbation. In the right leg, the hip flexor TFL reduced its prestimulation. Explanations of this could be the influence of feed-forward commands50, e.g. limb dominance, or an unbalanced loading of both legs caused by the natural (lateral) sway51 that was also present in our study (see Supplementary Fig. S7). It is also likely that subjects adapted their prestimulation throughout the progress of the experiment. However, the results of Welch and Ting66, which found only modest changes of prestimulation in response to surface translations and only one to two trials with converging behaviour towards a preferred prestimulation activity, suggest that the influence of training-induced adaptation or habituation on prestimulation activity should be small.
Reactive responses ('reflexes') can (with a certain delay) respond to the perturbation by processing sensory feedback with only minimal or no prior knowledge17,67,68. We found that the temporal organisation of observed muscular responses remained (with only some exceptions) similar throughout the three response intervals. However, muscular responses were only found after a delay of 100 ms after the perturbation onset. Similar delays were found for surface translations19,20, while other (more sudden) upper-body perturbations resulted in shorter delays of e.g. 30 ms to 80 ms69,70. Since we measured an initial change of upper-body pitch 50 ms after onset, RI1 responses can be considered medium-latency while RI2 and RI3 denote long-latency responses37. Previous studies71,72 suggest that short-latency responses (not observed in our experiment) are induced by joint errors (e.g. stretch reflexes) and long-latency responses are the result of a task-level feedback. While these long-latency responses have been found to undergo modulations of task- and context-specific constraints36,67,73,74,75,76,77,78, they also reflect the inter-segmental coupling of joints79,80,81,82. For instance, perturbations to one joint can evoke responses in muscles that only act on unperturbed joints18,82. Such patterns were associated with pre-defined coordination patterns (e.g. muscle synergies) that are triggered by a certain stimulus and generate purposeful responses in the context of spartio-temporal or task-specific constraints71,73,83 also reflecting the requirements of everyday activities, such as controlling inertial effects of e.g. accelerated, neighbouring segments79,82. While reactive and anticipatory control strategies are usually combined19,66,84,85,86, our results show that biarticular muscles mainly follow a reactive control strategy.
Such long-latency responses can also undergo adaptation processes71. While the additional loading (the weight of the AMP) resembled a task that all subjects faced previously (e.g. when wearing a heavy backpack), pure torque perturbations are rare during activities of daily living. It is thus unlikely that subjects were familiar with the type of perturbation, resulting in adaptation or habituation effects from updating an internal model of the AMP and the applied torques. Prior information about the perturbation, learning, or training has been found to guide the selection of long-latency responses towards a strategy that also reduces muscular activity19,39,66,87. Also, previous studies found a wide variety of adaptation processes, ranging from approximately 5 trials39 to more than 50 trials87, so it is difficult to conclude that adaptation had ceased during the 10 trials per condition in our experiment. However, to reduce influences of learning on the reflex responses66, we considered only the last 5 of the 10 trials per condition. Still, we cannot exclude further training or learning of subjects. Nevertheless, given also other factors, such as an accelerated rate of fatigue during load carriage, the number of discarded trials must be kept small for practicality.
Gyroscopic perturbation generation
The AMP is an example of 'reactionless actuation', in which torques are exerted by exchanging angular momentum with the actuator itself, rather than by exerting forces against the ground or an inertially-fixed object (e.g. an immobile robot transmitting forces via linkages or cables). This principle has both theoretical and practical benefits for perturbation experiments.
Specificity of a perturbation is difficult to achieve. Force-controlled systems are generally preferred over position-controlled systems due to their absence of kinematic constraints and more natural responses88,89. While it is, in principle, possible to generate a pure torque with a pair of opposing forces, it is, in practice, difficult to synchronise these forces and track the location of the CoM in real-time. Reactionless actuation is attractive because both synchronisation and alignment of the equivalent force couple happens inherently.
A practical benefit of reactionless actuation is that, since there is no necessity to exert forces against an inertially-fixed environment, the actuator can be entirely self-contained and even take the form of a wearable backpack, as in the AMP. This enables the AMP to be portable and suitable for overground experiments, and to be combined with other systems (e.g. treadmills, other perturbators) and measurement apparatus (e.g. motion capture, force plates) with minimal interference. To study other balance mechanisms in the future, wearable reactionless actuators might be placed on other parts of the body to perturb specific limbs. For various other (assistive) applications, such actuators have already been described for placement on the arms90,91, legs92,93, and distributed across the body94.
The gyroscopic torque vector rotates along with the gimbal creating – together with the desired pitch perturbation – roll torque components in the frontal plane (Eq. (2)). While the gimbal initial angle and total rotation were chosen to reduce the magnitude of this roll component, it was nevertheless present. In additional conditions in which the gimbal was rotated by 180°, the same pitch perturbations were produced but with the opposite roll components. From this, it was established that a lateral trunk lean did occur (up to 8°, see Supplementary Fig. S9), and that the load shifted from one leg to the other, but that this did not appear with asymmetric magnitudes or dependent of the side of dominance (see Supplementary Figs S10 and S11).
The frontal balancing of the upper body is mainly achieved by an (un-)loading mechanism of hip abductors61,95 in which the torque generated by gravity is compensated by hip abductor activity to regulate angular momentum and ensure frontal upper-body balance96,97,98. Since the AMP also generated roll torque components, it is likely that also frontal balance control strategies were used. This would probably result in altered hip muscle activities (depending on the corresponding perturbation torque) as found by previous multidirectional perturbation studies when applying surface transitions38,99,100,101,102 and rotations102,103. Given that each leg individually controls the CoP61, cross-talk from the contralateral leg to ipsilateral ankle joint muscles should be small. For the hip, however, neuronal coupling of both legs – next to their mechanical interaction – e.g. by interneurons is very likely. Still, muscular responses did not appear to differ (see Supplementary Figs S12–S14) allowing us to merge both conditions of inverted roll but similar pitch perturbations.
To generate a rotation of the gimbal wrt. to the body frame of the subjects, a peak gimbal torque of 12 Nm was necessary. This torque created an angular acceleration (Eq. (2)) that resulted in upper-body yaw rotations of 3° to 5° (see Supplementary Fig. S8). This could especially influence GLU and TFL activity, as both muscles also function to generate a leg rotation in the transversal plane (wrt. to the pelvis). However, since the observed upper-body yaw was small compared to our main perturbation direction, such influence is expected to be small. A future version of the AMP may use instead two or more smaller actuators to allow both the magnitude and direction of the torque vector to be controlled simultaneously, thereby eliminating torques in unintended directions104 and allowing multi-directional perturbation studies38,83.
A drawback of the selected actuation principle is that the mass of the actuator is borne by the test subject and cannot be placed externally. The prototype AMP in this study weighed 16 kg, or 15% to 29% of the participant's body weight. Other studies of standing with similar backpack load showed tendencies of increasing activity for anterior and reductions for posterior muscles groups. In particular, non-significant increases of VL, RF and reduced HAM activity as well as increased CoP sway were found105,106. In our study, significant reductions in muscle activity were only seen in GLU, HAM of the left leg and VL of the right leg. Still, we cannot exclude other influences of e.g. modulation of muscular stiffness or related effects on balance. Also, an increased hip flexion (by 6.2°), more extended knee (by 2.1°) and increased CoP sway were found (see Supplementary Table S1 and Fig. S7). The additional weight makes the balance control more challenging106. When being perturbed, it is likely that the added moment of inertia of the AMP helped subjects to maintain the initial upper-body posture. The influence of muscle fatigue should be small, since subjects had sufficient time to rest. Additionally, also the generated vibrations or the noise could result in changed muscle activity, e.g. by stimulating muscle-spindle proprioception107,108 or psychological factors like attention or anxiety109,110,111,112.
In future generations of the AMP it will be possible to reduce the mass substantially by (i) using lightweight components, (ii) scaling down the maximum possible perturbation magnitude, (iii) increasing the diameter of the flywheel, or (iv) rotating the flywheel faster. Lightweight reactionless actuators that are wearable90,91,94,113 or handheld114,115 have been recently developed, and the technology has even been demonstrated on the sub-gram scale116. However, for the same perturbation magnitude, we estimate that it is currently feasible to construct a AMP weighing approximately half the one used in this study.
This study investigated the specific role of biarticular muscles in dealing with upper-body pitch perturbations in human standing. Our main finding was that biarticular thigh muscles (hamstring and rectus femoris) showed the strongest response of all measured major leg muscles. Our results indicate a reactive control for biarticular thigh muscles in line with a previously suggested torque control strategy considering the difference between hip and knee torques. To the knowledge of the authors, this was the first study on human subjects providing experimental evidence that biarticular muscles play a key role in reactive upper-body balance control.
In order to focus on upper-body balance while reducing method-dependent artefacts (arising e.g. from inter-segmental couplings or global perturbations), a new type of perturbation was applied by the Angular Momentum Perturbator (AMP). Complementing existing perturbation devices, the generated torque perturbation at the trunk provides a new methodology for studying human response strategies.
Future research will involve applying mechanical and neuromechanical models20,46,51,117,118 and/or data-driven approaches119 to increase our understanding of underlying control mechanisms as well as classification of user intention, e.g. for controlling wearable robotic devices like the AMP.
A deeper understanding of postural balance control of the upper body and involved muscle responses will facilitate the design of robotic systems for assistance or rehabilitation and improve the effectiveness of physical training for fall prevention e.g. for elderly people.
Processed datasets of the AMP, kinematic, kinetic, electromyographic and anthropometric data that were used in this study are available in the 4TU.ResearchData repository120.
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C.S. is supported by the German Academic Research Service (DAAD) by a Short-Term Doctorate Scholarship and by a Short Term Scientific Mission (STSM) of the COST Action CA16116 'Wearable Robots for augmentation, assistance or substitution of human motor functions'. A.B., D.L. and H.V. are supported by the Netherlands Organisation for Scientific Research (NWO) Innovational Research Incentives Scheme Vidi grant 14865, and the USA Department of Health and Human Services NIDILRR grants 90RE5010 (formerly H133E120010) 'Machines Assisting Recovery from Stroke and Spinal Cord Injury for Reintegration into Society (MARS3)' and 90REGE0005 'Collaborative Machines Enhancing Therapies (COMET)'. We also acknowledge support by the German Research Foundation and the Open Access Publishing Fund of Technische Universität Darmstadt. The authors would also like to thank Saher Jabeen, Patricia Baines, Michiel Plooij, and Jaap Harlaar for their help during the experiments and for fruitful scientific discussions.
Lauflabor Locomotion Laboratory, Institute of Sport Science, Centre for Cognitive Science, Technische Universität Darmstadt, Darmstadt, 64289, Germany
Christian Schumacher & André Seyfarth
Delft Biorobotics Lab, BioMechanical Engineering, Delft University of Technology, Delft, 2628 CD, The Netherlands
Christian Schumacher, Andrew Berry, Daniel Lemus & Heike Vallery
Friedrich-Schiller-Universität Jena, Institute of Zoology and Evolutionary Research, Jena, 07743, Germany
Christian Rode
Christian Schumacher
Andrew Berry
Daniel Lemus
André Seyfarth
Heike Vallery
C.S., A.B., D.L., A.S. and H.V. contributed conception and design of the study; C.S. organised the database; C.S. and C.R. performed the statistical analysis; C.S. wrote the first draft of the manuscript; C.S., A.B., C.R. wrote sections of the manuscript. All authors contributed to revision of the manuscript, and read and approved the submitted version.
Correspondence to Christian Schumacher.
Schumacher, C., Berry, A., Lemus, D. et al. Biarticular muscles are most responsive to upper-body pitch perturbations in human standing. Sci Rep 9, 14492 (2019). https://doi.org/10.1038/s41598-019-50995-3
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On the role of financial aid in a default episode
Gabriel Cuadra1,
Manuel Ramos-Francia1 &
Santiago Garcia-Verdu ORCID: orcid.org/0000-0002-3425-61301
Our aim is to explore the role of financial aid in a default episode. To that end, we develop a dynamic stochastic quantitative model of sovereign default featuring fiscal policy, endogenous financial aid and risk-averse foreign lenders. After calibrating the model, we feed output shocks into the model to show that it captures some of the most salient features of the fiscal and debt situation in Argentina during the 1998–2002. This underscores the economic nature of the decision to default and the role that official aid could have taken in avoiding such an event. In effect, given the economic challenges endured by Argentina, a full-fledged default took place. In addition, we discuss a number of policy implications associated with financial aid programs aimed at preventing sovereign default episodes.
In the academic literature, there is some consensus on the negative effects of sovereign default episodes. For instance, De Paoli et al. (2011), and Yeyati and Panizza (2011) find that sovereign defaults have an adverse impact on economic growth. The literature has underscored several transmission channels as sovereign defaults affect economic activity. For example, Fuentes and Saravia (2010) find that foreign direct investment tends to fall after a default event. Moreover, other papers have documented that default episodes commonly lead to higher borrowing costs, and at times, to lose access to international financial markets. Borensztein et al. (2009) find that the cost of borrowing is significantly higher immediately after a default episode (relative to the cost prior to the default). Arteta et al. (2008) document that foreign credit to the private sector declines after a sovereign default episode. Cruces and Trebesch (2013), and Richmond et al. (2009) find that debt restructuring can have a significant and lasting impact on access to foreign financing. In short, sovereign defaults are costly.
However, for a government, there are benefits related to defaulting on its outstanding debt obligations. The economic literature has highlighted such benefits, particularly so, in papers that model sovereign default events based on a willingness to pay approach (Eaton and Gersovitz (1981)). In this context, an important issue for the international financial community, particularly so for multilateral institutions, has been the implementation of financial aid programs aimed at changing the incentives to default that a given country could face. Such an implementation has as an objective to prevent a sovereign default event. More generally, some economies have had (implicit or explicit) multilateral support when they have found themselves in financial difficulties. Other economies have had to live with the uncertainty of whether they could obtain such a support. As an important instance of the latter situation, we have the Argentina in the early 2000s. In such a case, given the substantial debt burden and the slowdown of economic activity, a full-fledged default took place. Critically, we believe that the choice to default is largely the dire consequence of an economic cost–benefit analysis, which depends on the macroeconomic conditions and constraints faced by each country, particularly so, its debt level and economic growth prospects, among others.
To gain a better understanding of the key features explaining the decision to opt for financial aid or for default by a sovereign, this paper puts forward a dynamic stochastic quantitative model of a small open economy with endogenous fiscal policy, financial aid, and sovereign default. This model illustrates the main trade-offs that a government faces when considering such a decision. Relatedly, we explore the extent to which financial support (official aid) by a multilateral institution to Argentina could have had prevented the referred default episode.
Our paper builds on and contributes to the literature on quantitative models of sovereign debt. Aguiar and Gopinath (2006), and Arellano (2008) were the first papers to develop quantitative models of sovereign default. They based their models on the theoretical contributions of Eaton and Gersovitz (1981). The model put forward in this paper shares key features with that of Cuadra et al. (2010), who develop a model of sovereign debt and fiscal policy to explain the pro-cyclical nature of fiscal policy in emerging market economies, and that of Arellano (2008) and Lizarazo (2013), who propose a stochastic discount factor to account for risk-averse foreign lenders.
Other studies in the literature have incorporated conditional financial support into sovereign default models. For example, Boz (2011) ponders a model in which the government has access to credit from the private sector and from international financial institutions, which in this paper we refer to as multilateral institution. Each type of creditor offers a different debt contract. He then goes on to analyze how the government allocates its borrowing needs between these two types of creditors, and when it is optimal to default on its debt to the private sector. On their part, Fink and Scholl (2016) propose a model in which the government has access to loans from a multilateral institution. However, this institution imposes fiscal conditions that restrict public expenditure. Since public expenditure and private consumption are substitutes, the government responds by reducing the tax rate. This, in turn, increases private consumption. That is, financial support leads to a private consumption rise.Footnote 1 In contrast, in our model, a multilateral institution provides financial aid directly and imposes fiscal conditions that restrict the amount that government can borrow. Hence, in our case, domestic absorption unambiguously falls, in line with the data.
More generally, several contributions have used a mixture of theoretical and empirical methods to assess the extent to which a highly indebted country needs to adjust its fiscal policy to stabilize its debt to GDP ratio. Berrittella et al. (2015) use a DSGE model to estimate the tax rate necessary to stabilize public debt in several countries. They find that, on average, it is close to 50%. Furthermore, they find that tax adjustments are important to mitigate short-term risks arising from high debt, while growth is necessary to ensure long-term debt sustainability. Trachanas and Katrakilidis (2013) find evidence that governments tend to increase taxes after they take spending decisions. In other words, they find evidence on the "spend and tax" hypothesis rather than on the "tax and spend" one. In an interesting contribution, Tielens et al. (2014) use a vector autoregressive model to assess the impact that Eurobonds could have on debt dynamics in Portugal, Ireland, and Greece. They argue that Eurobonds can improve budgetary sustainability by absorbing macroeconomic shocks and by decreasing uncertainty over debt forecasts. In turn, the theoretical model of Cheng et al. (2017) shows that, absent the role of lender of last resort for the central bank in a monetary union, banking regulation and government deposit guarantees may not be able to prevent the occurrence of twin banking and sovereign crises.
Finally, there are contributions that study the effectiveness of bailout programs in ending or, at least, attenuating government debt crises. Li et al. (2015), for instance, study the history of IMF's bailouts. They question the effectiveness of the austerity packages linked to the bailouts. They assert that the design of the packages tends to overlook the recipient country's characteristics, and for the most part, focuses on short-term solutions that are often damaging in the long-term.
In our model, a multilateral institution can provide conditional financial aid by transferring resources to the domestic government, and imposing certain fiscal conditions. In particular, it restricts the amount of resources that the government can borrow from foreign lenders. In turn, the government has the option to accept or decline the aid. In other words, there are both benefits and costs to such a decision. The model captures them with the following features. A parameter accounts for the size of the transfer, modeled as the fraction of the country's outstanding debt. The other is a restriction on the debt's size that the country can take on the next period. We see this as a way of instilling some fiscal discipline. It is worth mentioning that when a government is close to defaulting, it would be trying to reduce its leverage. The referred restriction should increase the pace at which a government would try to do so. Of course, ex ante, the restriction does not necessarily need to hold. This reduces the default probability, and consequently, lowers its borrowing costs, partly because investors know that fiscal discipline is, quite commonly, part of a financial aid package. This underscores the relevance of aid by multilateral institutions.
As a caveat, we assume that the government can commit to fiscal discipline in the implementation of financial aid. This presupposes that there is no lack of commitment principle in this specific case. Such an assumption is not as strong as it may initially appear. We think that this is the case since the implementation of the financial aid package and fiscal discipline take place in the same period. In contrast, the commitment issue in the general sovereign debt setup involves several periods. Accordingly, the latter case does not preclude the lack of commitment principle in the general sovereign debt setup.
We calibrate our model to Argentine data. It is able to account for several business cycle's empirical regularities, such as, the countercyclical nature of interest rate spreads and the highly pro-cyclical character of domestic absorption. We then feed the model with a sequence of output shocks that resemble the dynamics of economic activity in Argentina from 1993 to 2002. The model is able to replicate the default episode of 2002. Then, we use our model as a laboratory with which we assess whether a financial aid program would have been useful to prevent the default event. Our quantitative framework suggests that financial aid could have sufficiently reduced the incentives to default, for instance, as reflected in the lower interest rate premium.
More generally, one should consider that the difficulties faced by Argentina are multifaceted, to say the least. They entail a plethora of historical, cultural, institutional, political, or even other economic aspects that we do not account for in the model. Of course, a single model attempting to account for all of them would quickly become intractable. Thus, our model inevitable abstracts from several aspects, which could have been present in sovereign debt crises and could be, nonetheless, pertinent. Yet, we believe that our model captures key elements and essential economic incentives that policy makers face during a crisis. In particular, it emphasizes that under certain conditions, which we consider quite sensible, the decision to default on their sovereign debt might be the optimal one. It, moreover, highlights the role of financial aid provided by a multilateral institution in preventing a full-fledged sovereign default episode.
The rest of the paper proceeds as follows. The second section presents the sovereign debt model. The following section has the calibration and the numerical exercise. It also illustrates the quantitative implications of the model. The last section has some key concluding remarks.
Model economy
In our model, fiscal policy and the option of receiving financial aid are endogenous, in an otherwise standard sovereign default setup.Footnote 2 There are three types of agents: households, government, and foreign lenders. Our interest in including fiscal policy is to highlight two features. First, only the government can issue sovereign debt and can engage in conditional financial support programs from multilateral institutions. Second, in general, in a crisis the government ultimately absorbs private debts.
The representative household has preferences over the expected value of the stream of its consumption in each period:
$$E_{0} \mathop \sum \limits_{t = 0}^{\infty } \beta^{t} \frac{{\left( {C_{t}^{\alpha } G_{t}^{1 - \alpha } } \right)^{1 - \sigma } }}{1 - \sigma },$$
where the subjective discount factor is \(\beta \in (0,1)\) and σ is the coefficient of relative risk aversion. Households derive utility from private consumption \(C_{t}\) and public expenditure \(G_{t}\). We consider a Cobb–Douglas aggregator, where the parameter α captures the private consumption weight. The households' output \(Y_{t}\) follows a Markov process, with conditional density function \(Q(Y_{t + 1} |Y_{t} )\) where \(Y_{t}\) has values belonging to a set \(\varUpsilon\). Footnote 3 Private consumption plus public expenditure is equal to output. The government taxes output. Households take public expenditure and taxation as given. Thus, they consume according to the following equation.
$$C_{t} = \left( {1 - T_{t} } \right)Y_{t} .$$
The government maximizes households' utility by issuing one-period bonds, taxing households' output, and financing public expenditure.Footnote 4 Conditional on being in good credit standing, it decides between paying the outstanding debt, receiving financial aid, and defaulting on it. To take its decision, the government compares the costs and benefits associated with each of these three options, as we explain later in detail. We note that receiving financial aid entails paying the outstanding debt in a given period. In other words, one cannot default and then opt for financial aid.
If the government decides on paying the outstanding debt, it maintains its access to external financing. Consequently, it would be able to issue new debt. The government then decides how much to borrow, the level of public expenditure and, in tandem, the tax rate. On the other hand, if the government opts to default on its debt, it would face exclusion from international financial markets. Such an exclusion would take place for a random number of periods. In addition, the economy would suffer an output loss.
When the country is in financial autarchy, the government only chooses the public expenditure and the tax rate. The benefits and costs of these two options are as follows. If it honors its debt, the government maintains access to international financial markets. However, it has to pay the debt service. Conversely, if it defaults, it can consume what it was going to allocate to pay its debt and its service. Yet, it would face financial autarchy, and in tandem, suffer an output loss.
In addition, in our model, the government can opt for financial aid. We think of it as an option that the government has only when having access to financial markets. If it chooses to receive financial aid, it will incur in both benefits and costs, which we operationalize, respectively, as a direct transfer of resources (financial aid) and as a restriction on the government debt next period, which we broadly interpret as fiscal discipline. Importantly, financial aid lasts only for one period in the model, afterwards, the government has again to decide whether to pay its debt, receive financial aid once more or, perhaps, default. One can interpret such a financial aid as support provided by a multilateral institution, which enacts some restrictions to the indebted government.
One can express this problem as a dynamic program. The state variables are output Y and debt B. The government decides between honoring its debt, receiving financial aid, and defaulting on its debt, by comparing the value function associated with honoring its outstanding debt obligations VC(B, Y) with that of receiving financial aid Vf(B, Y) and with that of the value function Vd(Y) which corresponds to defaulting.Footnote 5 The following value function will be useful to describe the government's problem:
$$V_{0} \left( {B,Y} \right) = { \hbox{max} }\left\{ {V^{c} \left( {B,Y} \right),V^{f} (B,Y),V^{d} \left( Y \right)} \right\}.$$
Accordingly, we define the decision function D as follows:
$$\begin{aligned} D(B,Y) &= 2\;{\text{if}}\; V^{c} \left( {B,Y} \right) > \hbox{max} \{ V^{f} (B,Y),V^{d} \left( Y \right)\} ; \hfill \\ D(B,Y) &= 1\;{\text{if}}\; V^{f} \left( {B,Y} \right) > \hbox{max} \{ V^{c} (B,Y),V^{d} \left( Y \right)\} ;\;{\text{and}}, \hfill \\ D(B,Y) &= 0\;{\text{otherwise}}. \hfill \\ \end{aligned}$$
This allows us to define the repayment (R), financial aid (F), and default (D) sets, which depend on the level of debt, as follows:
$$\begin{aligned} R\left( B \right)& = \{ Y \in \varUpsilon {:}\; D\left( {Y,B} \right) = 2\} ; \hfill \\ F\left( B \right) &= \{ Y \in \varUpsilon {:}\; D\left( {Y,B} \right) = 1\} ;{\text{and}}, \hfill \\ D\left( B \right) &= \{ Y \in \varUpsilon {:}\; D\left( {Y,B} \right) = 0\} . \hfill \\ \end{aligned}$$
The government maximizes the representative household's utility subject to her budget constraint and its own budget constraint. When the government honors its debts and does not receive financial aid, its debt choices are not restricted. Thus, the problem, when the government has access to credit markets, is as follows:
$$\begin{aligned} V^{c} \left( {B,Y} \right) &= { \hbox{max} }_{{T, G, B^{'} }} \left\{ {\frac{{\left( {C^{\alpha } G^{1 - \alpha } } \right)^{1 - \sigma } }}{1 - \sigma } + \beta \mathop \sum \limits_{{Y^{'} }} V_{0} \left( {B^{'} ,Y^{\prime}} \right)Q\left( {Y^{\prime} |Y} \right)} \right\} \\ &\quad {\text{s}} . {\text{t}} .\\ &\qquad G + B = TY + q\left( {B^{'} ,Y} \right)B^{'} , \\ &\qquad C = (1-T)Y , \\ \end{aligned}$$
where the price of the bond is q(B′, Y), B′ is foreign debt next period, and Y is output in the current period.Footnote 6
When the government opts for financial aid, it has to satisfy a borrowing restriction. We assume that the option of receiving financial aid is only available when output is low, specifically, when output is smaller than its mean value.Footnote 7 We motivate this by the fact that, absent financial aid, the probability of default is decreasing in the level of output. Hence, a default is less of a concern if the economy in question is performing relatively well in terms of economic growth. In this case, the government's problem is:
$$\begin{aligned} V^{f} \left( {B,Y} \right) &= { \hbox{max} }_{{T, G, B^{'} }} \left\{ {\frac{{\left( {C^{\alpha } G^{1 - \alpha } } \right)^{1 - \sigma } }}{1 - \sigma } + \beta \mathop \sum \limits_{{Y^{'} }} V_{0} \left( {B^{'} ,Y'} \right)Q\left( {Y' |Y} \right)} \right\} \\ &\quad s.t. \\ &\quad G + B = TY + F\left( B \right) + q\left( {B^{'} ,Y} \right)B^{'} , \\ &\quad B^{'} \le B, \\ &\quad C = \left( {1 - T} \right)Y \\ \end{aligned}$$
$$F\left( B \right) = \left\{ \begin{aligned} \delta B\quad {\text{if}}\;Y < E(Y) \hfill \\ 0\quad {\text{if}}\;Y \ge E\left( Y \right) \hfill \\ \end{aligned} \right.$$
and δB accounts for the transfer of resources (financial aid). In light of the transfer, the government has now, in principle, more resources to finance public expenditure. However, it also faces a restriction B′ ≤ B on the future evolution of its debt. The government cannot borrow from foreign lenders more than what it has borrowed in the previous period. One can see such a restriction as some sort of fiscal discipline. In the period after receiving financial aid, once more, it faces international credit markets, possibly without the referred aid.
If the government defaults, financial markets exclude the country and it suffers an exogenous output loss. The output in autarky is equal to h(Y), which accounts for the cost on output after defaulting. One can motivate its presence by costs not directly associated with the default, e.g., a loss in productivity (e.g., see Alonso-Ortiz et al. 2017). We provide a precise definition of h(Y) below. Accordingly, the output tax is the only source to finance public expenditure. In this case, the problem is:
$$\begin{aligned} V^{d} \left( Y \right)& = { \hbox{max} }_{{T_{d} ,G_{d} }} \left\{ {\frac{{\left( {C_{d}^{\alpha } G_{d}^{1 - \alpha } } \right)^{1 - \sigma } }}{1 - \sigma } + \beta \mathop \sum \limits_{{Y^{'} }} [\mu V_{0} \left( {0,Y^{'} } \right) + (1 - \mu )V^{d} \left( {Y'} \right)]Q\left( {Y' |Y} \right)} \right\} \\ &\quad{\text{s}} . {\text{t}} .\\ &\qquad G_{d} = T_{d} h\left( Y \right), \\ &\qquad C_{d} = \left( {1 - T_{d} } \right)h\left( Y \right). \\ \end{aligned}$$
Once excluded from the international credit markets, in the next period, it regains access to them with exogenous probability μ. If it regains access, it would do so with no debt, i.e., B = 0.Footnote 8
In addition, foreign lenders are risk-averse and have access to an international risk-free bond with interest rate \(r_{f}\). Assuming no-arbitrage, the equilibrium bond price is determined by:
$$q(B',Y) = \mathop \sum \limits_{{Y^{ '} \in R\left( {B^{ '} } \right)\mathop \cup \nolimits F(B^{ '} )}} {\text{SDF}}(Y^{ '} ,Y){\text{Q}}\left( {Y ' |Y} \right)$$
where the stochastic discount factor (SDF) is given by \({\text{SDF}} \left( {Y^{'} ,Y} \right) = (1 + r_{\text{f}} )^{ - 1} - \lambda (\log Y^{'} - \rho_y \log Y)\), as in Arellano (2008). The coefficient λ measures the foreign lenders' degree of risk aversion. Lenders are more risk-averse the larger λ is (λ ≥ 0). In fact, one can interpret it as a coefficient of relative risk aversion (CRRA).Footnote 9 We note that the covariance between the output and the SDF is different from zero. The states of nature where there is no default depend directly on B′ through the repayment sets.
Additionally, consider the cases in which the government would opt for financial aid. One can divide them into two. There are some in which, under the absence of financial aid the government would not default. There are other cases in which, under the absence of financial aid, the government would default. Of course, one can think of normative, and perhaps some, political economy arguments to minimize the former and maximize the latter.
On a related note, the debt restriction B′ ≤ B might or might not bind when the government decides to accept financial aid. In effect, if the government were close to defaulting, it would be trying to reduce its debt level independently of such a restriction. It is thus in the multilateral institution's interest to procure a binding restriction, which would instill fiscal discipline and yet, in tandem, consider the trade-offs against the financial aid benefits.Footnote 10
For the model economy, a recursive equilibrium is:
A set of value functions \(V_{0}\),\(V^{c}\),\(V^{d}\) and \(V^{f}\) for the government.
A policy function for the household's consumption.
Policy functions for government´s financial aid and default decision D, optimal asset holdings B′, optimal government expenditures G and optimal tax rates T.
A bond price function q(B′, Y).
such that:
Given the government and the bond price function, the household policies for consumption satisfy the household's budget constraint; and,
Given the bond price function q and the optimal policies for households, the government's value functions V0, Vc,Vf, and Vd, and the policy functions D, B′, G and T solve (3), (5), (6) and (7).
Calibration and numerical exercise
We calibrate our model and then perform a numerical exercise to analyze the role of financial aid in preventing a sovereign default episode.Footnote 11 To that end, we first discuss the values of the parameters used in such an exercise. Second, we briefly analyze how changes in the parameters' values related to financial aid affect the incentives to pay, to opt for financial aid, or not to pay in the model. Third, we simulate the model and present several of its key statistics. Finally, we choose a sequence of output shocks that match the evolution of the Argentinian economic activity during the years prior to its sovereign default episode of 2002, and assess the model economy's dynamics. Below, we describe the precise way in which we do this. It is worth mentioning that we have used a yearly frequency.
Value of the parameters
The values of the parameters used in the model's solution are comparable to those utilized in the literature of sovereign default quantitative models (e.g., see Aguiar and Gopinath (2006), Arellano (2008), Cuadra et al. (2010)). We choose other parameters to emulate the most salient empirical features of the Argentinian economy. Table 1 presents their specific values and, if applicable, their sources.
Table 1 Parameter values
The parameter σ, the coefficient of relative risk aversion, is set equal to two, a standard value in the literature (e.g., see Aguiar and Gopinath (2006)). The subjective discount factor (β) is set equal to 0.81, in line with Arellano (2008).Footnote 12 During the period of study, the public expenditure to private consumption ratio in Argentina was, on average, around 0.20. A Cobb–Douglas utility functions and a private consumption weight α equal to 0.83 helps to match this fact.Footnote 13
The parameter \(\mu\) corresponds to the exogenous probability of reentering international financial markets after a default episode. It is set equal to 0.1. This value implies that a country expects to be able to regain access to financial markets 10 years after defaulting.Footnote 14
There is output loss in autarky.Footnote 15 We capture this in the model with the function h(Y), as in Arellano (2008). Specifically, we assume that a default episode entails an output cost of the following formFootnote 16:
$$\begin{aligned} h\left( Y \right) & = \left( {1 - \theta } \right)E\left( Y \right)\quad {\text{if}}\;Y > \left( {1 - \theta } \right)E\left( Y \right), \\ h(Y) &= Y\quad {\text{if}}\;Y \le \left( {1 - \theta } \right)E\left( Y \right). \\ \end{aligned}$$
The default cost parameter θ is set equal to 0.03, as in Arellano (2008).
We assume that the cyclical component of the GDP follows an AR(1) process:
$$z_{t} = \rho_{z} z_{t - 1} + e_{t}$$
where \(e_{t}\) is i.i.d. \(N(0,\sigma_{z}^{2} )\).
The parameters of the stochastic process for output ρz and σz are set to match the autocorrelation and standard deviation of the cyclical component of the Argentinian GDP.Footnote 17 To solve the dynamic program, the AR(1) process is approximated by a discrete Markov chain.Footnote 18 Then, we solve the model numerically using the value function iteration algorithm.Footnote 19
In terms of the financial aid, we first consider as a benchmark scenario without financial aid, that is, the financial aid parameter δ is set equal to zero. Subsequently, we assess which value of this parameter would have prevented the Argentinean sovereign default episode of 2002. As for the stochastic discount factor, following Lizarazo (2013), we have set λ = 5, which measures the foreign lenders' risk aversion.Footnote 20 As explained, one can interpret the parameter λ as a coefficient of relative risk aversion. The risk-free rate rf is equal to 0.04 (i.e., 4.00%), a standard value for annual data in the business cycles' literature.
Financial aid entails benefits and costs, as initially explained. Specifically, increasing the financial aid parameter augments the incentives to opt for it. In effect, the government would be getting additional resources. Financial aid, everything else being equal, reduces the debt burden. That is, the aid increases the amount of resources available to the economy. Accordingly, defaulting on the outstanding debt becomes less attractive than opting for financial aid. On the other hand, a stricter fiscal discipline lessens the incentives to opt for financial aid.Footnote 21 This implies a tighter constraint on the debt that the government can take next period. The institution providing financial aid balance its costs and benefits to reduce the possibility of a default scenario and, in tandem, to mitigate moral hazard. Of course, in our model, we are not explicitly accounting for moral hazard.
We simulate the model to assess how well it matches the business cycle's features of Argentina. The first column of Table 2 shows a number of statistics for such an economy. We have considered annual data from 1980 to 2014. Output and absorption are in logs, and we express the fiscal balance as a fraction of GDP.Footnote 22 The interest rate spreads correspond to the difference between the interest rate of Argentina and the U.S.Footnote 23 For calibration, we filter all time series with the Hodrick–Prescott filter with the standard value for its parameter when using annual data. The second column of Table 2 shows the corresponding business cycle's statistics for the simulated model. These statistics depend on the average values of 100,000 simulations. Overall, the model is able to match several statistics. In the data and in the model, absorption is more volatile than output, while fiscal balance is less volatile than output. In addition, it is able to match the high pro-cyclical nature of absorption, as well as the counter-cyclical behavior of interest rate spreads.
Table 2 Calibration statistics
Having solved and calibrated the model, we choose a sequence of output shocks such that its path is as close as possible to the path of the cyclical component of the GDP estimated for Argentina.Footnote 24 Figure 1 shows such a path from 1993 to 2002 and the one matched with the model economy. As can be seen, economic activity was increasing from 1995 to 2000. Nonetheless, in 2001, it began contracting. Given this exogenous output path, we consider the policy functions to solve for the model economy's dynamics. In what follows, we discuss the economy's dynamics when there is no financial aid.
(Source: World Bank)
In our simulation, period 1 stands for 1993, 2 for 1994,…, and 10 for 2002. At the outset, the economy faces a negative shock in period 3 (which corresponds to the Tequila Crisis in 1995) and the interest rate spread increases at the time. Then, from period 4 to period 6 (1996–1998), the economy faces a sequence of positive shocks. The favorable economic performance leads to an interest rate spread equal to zero. Thereafter, in period 7, the economy starts to experience a sequence of negative output shocks and, thus, the expansion eventually comes to a halt. From that moment, foreign lenders demand a risk premium to keep on lending to the government, which increases the interest rate spread. Figure 2 depicts both the output level and the interest rate spread in the model economy. The negative correlation between these two variables is strong (Fig. 2). Subsequently, the fall in economic activity in period 10 (2002) is of such a magnitude that it is optimal for the government to default on its financial obligations.
(Source: Model simulation)
GDP vs. spread model economy (%)
We next compare our results with the developments of the Argentinian economy in the period prior to 2002. Figure 3 exhibits the interest rate spreads of Argentina and those generated by the model. As the figure indicates, in the early years, an increase in the spread seems to be associated with the 1995 slowdown. In subsequent years, the spreads pointed to a very low sovereign risk. This could relate to the economic expansion observed until 1998 (Fig. 1). However, after economic activity began faltering, interest rates spreads started to rise, a dynamic akin to that based on the model. In this way, the model is able to replicate the correlation between output and interest rate spreads as well as the sovereign default episode, which took place in Argentina in 2002.
(Source: Model and database of Neumayer and Perri (2005))
Interest rate spreads (%)
We next simulate the model with several values for the parameter δ to assess whether the government would have not defaulted. Based on this exercise, we determine that a value of 0.05 (5% of the outstanding debt) would have been sufficient to prevent the default episode. As we know, the financial aid did not materialize at the time. Moreover, any value higher than 5% would had implied that the multilateral institution would be assigning to the financial aid program an amount of resources greater than the one necessary to avoid a default event.
Given the lackluster economic environment and the cost of paying the outstanding debt, in period 9 (i.e., 2001) and afterwards the government opts to take on financial aid. In period 9, although taking financial aid is the optimal decision, the value of honoring the outstanding debt without receiving financial aid is higher than the value of defaulting. Figure 4 illustrates this point. In effect, it shows the value functions for the output registered in period 9. We observe that for the level of debt that the government had to pay in that period (B*), the value function of opting for financial aid is higher than the one corresponding to honoring the outstanding debt. In turn, the latter is higher than the one corresponding to default.
Value Functions (Source: Model simulation)
As a result, one can see financial aid as a direct subsidy for this specific period. We depict this situation in region 3 in Fig. 5. However, in period 10 (i.e., 2002), while opting for financial aid is the optimal decision, the value of defaulting is higher than the value of paying back the outstanding debt obligations without taking financial aid. That is, there are cases in which a full-fledged default episode would have taken place had it not been for the presence of financial aid. In other words, in our counterfactual experiment, financial aid is crucial to avert a sovereign default episode. This situation in region 2 is depicted in Fig. 5.
Value functions
To illustrate this feature of the model, Fig. 5 depicts the value functions associated with honoring the outstanding debt obligations, receiving financial aid and defaulting. These value functions assume a fixed value of output. One then can identify four regions. In region 1 (R1), the value of defaulting is higher than the other two value functions (i.e., paying and financial aid), therefore, defaulting on the outstanding debt is the optimal solution. In region 2 (R2), the value of receiving financial aid is higher than the other two value functions (i.e., defaulting and paying). In addition, the value of defaulting is higher than the value of honoring the outstanding debt. Accordingly, in this region, financial aid prevents a default episode. In region 3 (R3), the value of receiving financial aid is higher than the value of paying back debt and the latter is higher than the value of defaulting. Therefore, financial aid works as a subsidy. In region 4 (R4), the value of honoring the outstanding debt is higher than the other two value functions (i.e., defaulting and financial aid). This has as a repercussion that the agent will not default regardless of official aid.
Official aid, by changing the relative prices of the decision to default, reduces the incentives to do so. Given the non-contingent debt's size, official aid reduces the cost of its service in terms of the foregone consumption in the absence of such aid. Under such circumstances, the benefits of defaulting (preventing a fall in current consumption) decrease relative to the costs of doing so (going to autarky). In this context, a less generous transfer (financial aid), everything else constant, increases the incentives to default. As mentioned, if we consider a value of 0% instead of, say, 5% for the financial aid parameter, the government defaults on its debt obligations in period 10 (i.e., 2002). In contrast, financial aid, in the form of a resource transfer and fiscal discipline, reduces the incentives to default. In turn, a lower probability of default leads to a lower interest rate spread, and consequently, borrowing costs would fall. Figure 6 illustrates how financial aid reduces the interest rate spreads. Specifically, it shows the interest rate spreads for the model, with and without financial aid. In the latter case, the interest rate spreads are visibly higher.
Source: Model simulation
Interest rate spread model economy
To analyze the impact of financial aid on the level of sovereign debt, Fig. 7 shows the dynamics of sovereign debt for the model without financial aid and the model with financial aid equal to 5%. With financial aid, everything else being constant, the government can support higher levels of sovereign debt. As mentioned, the presence of financial aid contributes to reducing the incentives to default. Since foreign lenders know this, they are willing to lend to the government at lower interest rates and, thus, one observes greater levels of sovereign debt.Footnote 25
Debt model economy (% of GDP)
In general, countries that find themselves in financial distress might reach a point where it is in their own interest to cease debt payments. Their decisions would account for the reputational costs and the lost access to international markets. More formally, if a country reaches a point in which the level of utility obtained under default is higher than the level of utility it gets from paying its debt's service, it will then be a rational economic decision to default. Nonetheless, the possibility of receiving financial aid can certainly contribute to avoid a sovereign default episode.
Key policy implications
One of our main aims has been to explore the implications of financial aid. At the heart of its provision, there is the lack of an effective enforcement mechanism. Thus, a multilateral institution should provide financial aid along with some fiscal discipline, in lieu of an enforcement mechanism, which in the international scenario would be impossible to have. As mentioned, financial aid changes the relative prices that a sovereign faces when confronting the possibility of defaulting. In an ideal world, its provision should avoid a default, have no economic distortions, and contribute toward an economic recovery. However, the model underscores that the presence of financial aid does not assure that the sovereign will avoid a default. Moreover, under some contingencies, it may introduce economic distortions. Prominently, a multilateral institution could have provided financial aid when there was no real need.
In this context, it is illustrative to discuss some of the more specific policy implications. Although we could do this from one of various perspectives, but that of a multilateral institution providing the aid is the most useful. We then retake some of the results in our model to provide a richer context to our discussion. We have underscored the existence of four regions in the debt space as function of the sovereign actions, which we list next (Fig. 5).
The sovereign will default regardless of the presence of financial aid.
It will opt for the financial aid and a default would have taken place under its absence.
It will take the financial aid, but no default would occur under its absence.
It will not default regardless of the existence of financial aid.
The most relevant regions are those denoted by two and three. One can then see region 3 as a distortion and region 2 where the financial aid makes a difference. The shape and size of these regions depend on the design of the implementation of financial aid. Policy wise, the key is to strike a balance between financial aid and fiscal discipline to minimize 3 and maximize 2.
In our case, we have introduced financial aid as a discount on the sovereign debt. In turn, we introduce fiscal discipline as a constraint that limits government debt. While both are linear restrictions, the former is on a stock and the latter, on a flow, providing some flexibility to its implementation.Footnote 26 As a polar case, a burdensome discipline will make the sovereign shed any financial aid, making regions 2 and 3 empty sets. Accordingly, we focus on feasible levels of financial aid.
As Fig. 5 shows, given a level of output, debt determines in which of the four regions the sovereign is. First, low levels of debt correspond to region 4 in which the sovereign honors its financial obligations and does not opt for financial aid. In this region, the debt burden is manageable and the risk premium is close or equal to zero. Under such circumstances, the sovereign has sufficient incentives to borrow, taking advantage of the low interest rates. Financial aid is not sufficiently attractive since it would impose a borrowing constraint. However, as the sovereign debt increases, the risk premium and, consequently, the interest rate tend also to rise. Thus, financing public expenditure and private consumption through borrowing becomes less appealing. Thus, the borrowing constraint associated with receiving financial aid from the multilateral institution would have more slackness. At the same time, the amount of debt is not sufficiently high to trigger a default episode. Under these circumstances, the sovereign would be located in region 3 and the economic distortion would increase. If debt continues to grow, its burden would increase and defaulting would become more attractive. In this setting, financial aid would prevent a default episode (region 2). Finally, for high enough levels of debt, the sovereign would default irrespective of the presence of financial aid (region 1).
Once we have analyzed the sovereign's incentives in each region, it can be useful to ponder some policy measures that may help minimizing the size of region 3. A key question then is how to change the relative prices to achieve this. For tractability reasons, we have modeled financial aid as a fraction of current debt. However, as our previous analyses illustrate, offering financial aid for low levels of debt will not make a significant difference (region 4). In addition, relative prices must change in region 3 to make financial aid less attractive. One can achieve this by reducing the amount of financial aid. Based on the previous analysis, one way to do this is as follows. Instead of setting the amount of financial aid as a constant fraction of current debt, such a fraction can be an increasing function of debt. In this manner, for relatively low levels of debt the resources that a country would receive as financial aid would be lower. Then, for relatively high levels of debt, financial aid would be proportionally greater. Of course, a multilateral institution would have to face the challenge of calibrating the amount of financial aid.
To gain a better understanding of the key economic trade-offs and the dynamics of some macroeconomic variables in a (potentially) defaulting economy, we have developed a dynamic stochastic quantitative model of sovereign debt with endogenous financial aid. In our model, the government can opt for aid or default on its debt obligations, thereby generating endogenous interest rate spreads.
In the quantitative analysis, we calibrate the model to match key features of the Argentinian economy. The model accounts for several business cycle's stylized facts, such as the countercyclical nature of interest rate spread and the highly pro-cyclical nature of domestic absorption. Moreover, we feed the model with a sequence of output shocks that resembles the dynamics of economic activity in Argentina before the default episode in early 2000s. The model predicts a default in 2002. Then, we have used the model as a laboratory to assess the amount of financial aid that would have prevented the default event.
Overall, the model illustrates how the introduction of financial aid changes the relative prices that the sovereign faces when dealing with the possibility of defaulting on its debt obligations. Ideally, the presence of financial aid should prevent a default event and minimize economic distortions. However, the model highlights that financial aid does not always ensure that the sovereign will pay its debts. What is more, in some cases, it may introduce economic distortions. In particular, there may be situations in which a multilateral institution grants financial aid to a country that would have repaid its debts in any case. By underscoring the presence of these distortions, the model highlights that a multilateral institution should consider them when designing financial aid programs. How to design the above goes beyond the scope of this paper. However, we have outlined some ideas that may be useful to address this and other related issues.
Finally, although we have not included the exchange rate as part of our model, we think that its consideration could be a useful line of research (e.g., see Da-Rocha et al. 2013). We next briefly discuss the possible role of financial aid when the government defaults and cannot commit to maintain the exchange rate. Accounting for the exchange rate regime leads to the modification of the trade-offs face by the government if it knows a priori that it cannot commit to maintaining the exchange rate. On the one hand, it will more likely accept the financial aid. In effect, a depreciated exchange rate will imply that it will have fewer resources to pay its debt in the future. On the other hand, a depreciation could hasten the economy's recovery if a default does take place. Thus, the government might have fewer incentives to take the financial aid. Having rejected the financial aid, it defaults, but given the depreciation, net exports increase and the current account adjusts, which could prompt a swifter recovery.
What is more, the effect on absorption is ambiguous since public expenditure declines but private consumption increases.
Our model without fiscal policy and financial aid is similar to that of Arellano (2008).
In the calibration of the model, we first estimate an AR(1) model for the cyclical component of output. Then, we discretize it to obtain the Markov chain. We later explain this procedure in more detail.
Hatchondo and Martínez (2009) and Chatterjee and Eyigungor (2012) have put forward sovereign default models having long-term debt bonds. The analytical challenge in both cases is to keep the number of state variables low while introducing such a type of bonds. Hatchondo and Martínez (2009) model accounts for a higher and more volatile interest rate, while Chatterjee and Eyigungor (2012) for a higher debt to output ratio and spreads, as observed in the data.
As explained, the country's debt is set to zero if the agent decides to default. Hence, the value function corresponding to default is not a function of B. In other words, it is constant with respect to debt.
The timing convention is as follows. The government starts period t with debt B, observes output y, and decides debt for the next period B′. It does not observe output y′ until the next period.
In the numerical exercises, we consider the cyclical component of output, so an endowment lower than the mean value of output corresponds to a negative output gap.
Note that we are assuming there is a full-fledged default, i.e., the government does not pay any of its debt. In practice; however, a credit event in which, for example, the government misses a payment, one might consider it a default (technically). In the model, we do not consider credit events, only full-fledged defaults.
To see this, note that under a CRRA the SDF is given by \(\beta { \exp }( - \lambda \Delta c_{t + 1} )\) where \(\Delta c_{t + 1} = { \log }(C_{t + 1} /C_{t} )\), which is approximately \((1 + r_{f} )^{ - 1} - \lambda \Delta c_{t + 1}\). The working assumption in Arellano (2008) is that the innovations in consumption growth coincide with those of (the log of) output.
Some aspects of the functional form of financial aid that are worth mentioning. As said, financial aid is only available when the output is below its mean. Thus, if is it above its mean, financial aid is zero, regardless of the debt value. Thus, suppose it is below its mean and consider the following cases. First, if debt is significantly large, the government will default without taking any financial aid. Second, if debt is very large, the government will take the financial aid. Else, a default would take place. Third, if debt is large, the government will take the financial aid. However, in its absence, there would be no default. Fourth, if debt is moderate or zero, the government will not take financial aid. In all cases, the lower the output level, the more possibilities there are that it will opt for financial aid.
The model abstracts from some factors affecting the dynamics of sovereign debt, such as the role of banks.
Arellano (2008) explores the Argentinian case with quarterly data. She uses a value of 0.95, which is equivalent to 0.81 with annual data. For the subjective discount factor with annual data, values that are more standard fall within the 0.95–0.99 range. However, one drawback of sovereign default quantitative models is that they require a higher level of impatience (i.e., a smaller β) to generate default in equilibrium.
The public expenditure to private consumption ratio in the model is (1 − α/α).
According to Benjamin et al. (2009), a process of sovereign debt restructuring takes, on average, 8 years.
Rose and Andrew (2005) provide a rationale for the loss of output when countries face debt crises.
As explained in Arellano (2008), this specification extends the range of debt values that carries positive default premiums, which allows the model to generate higher interest rate spreads, all else being equal.
To estimate the cyclical component of GDP (z), we considered the log of the Argentinian GDP and the Hodrick–Prescott filter.
The fact that we use an AR(1) process to approximate the cyclical component of output (z) allows us a direct conversion to a Markov chain model in discrete time. This facilitates solving the model numerically. Once we have the discretized values of z, we consider \(Y = \exp (z)\) Thus, in the numerical exercises, we only work with positive values.
To implement the value function algorithm, we discretize the state space with a grid. The choice of its dimension is such that the results are robust. In other words, they do not change when the dimension of the grid is increased. A finer grid allows for results that are more accurate. In the numerical solution of the model, we consider a grid of 1000 values for debt and a grid of 25 values for output.
As explained, we use Lizarazo (2013) smaller coefficient as it is closer to an accepted CRRA value.
We can think of other schemes to introduce financial aid and the requirement of fiscal discipline, which would to different modeling implications. For example, an alternative would be the following: if the government opts for financial aid, it would face a restriction on the future evolution a debt as a fraction of output: B′/Y′ ≤ B/Y. This restriction is equivalent to the one proposed in our paper when output is equal to its mean value, since we have normalize it to one. When output is at its mean value and there are negative shocks to output, that is it begins to take values lower than its mean value, then the restriction becomes stricter than the one we use and, accordingly, the incentives to opt for financial aid decline. In particular, a fall in output, everything else being constant, would increase the debt output ratio. To keep such a ratio constant, the amount of debt issued must be even lower. However, the solution of the model, with this alternative specification for fiscal discipline might be intractable, mostly because at the time of choosing B′, the value of y′ is unknown.
The national accounts data are from the World Bank and cover the period from 1980 to 2016. In turn, we obtained the fiscal balance data from the IMF. They cover the period from 1993 to 2016. The interest rate data are from the database of Neumayer and Perri (2005).
The US interest rate corresponds to the rate of 12 month T-Bills.
Hence, when mentioning GDP or economic activity, we are referring to the cyclical component of GDP.
Recall that, in our model, a one-period non-contingent bond is considered.
Other ways of introducing financial aid can be non-linear, and have similar implications.
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MRF conceived the paper. GC, MRF, and SGV drafted the manuscript; GC collected the data and did the initial coding; SGV extended the model and the code; GC revised the code. GC, MRF, and SGV revised the draft. All authors read and approved the final manuscript.
The opinions in this paper are those of the authors and do not necessarily reflect those of Banco de México. We would like to thank two anonymous referees for their comments.
Banco de Mexico, Mexico City, Mexico
Gabriel Cuadra
, Manuel Ramos-Francia
& Santiago Garcia-Verdu
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Correspondence to Santiago Garcia-Verdu.
Cuadra, G., Ramos-Francia, M. & Garcia-Verdu, S. On the role of financial aid in a default episode. Lat Am Econ Rev 27, 9 (2018) doi:10.1186/s40503-018-0060-2
Sovereign default | CommonCrawl |
Chupakhin, Oleg Nikolaevich
Total publications: 21
Scientific articles: 21
This page: 316
Abstract pages: 2258
Full texts: 6
Member of the Russian Academy of Sciences
Doctor of chemical sciences
Birth date: 9.06.1934
http://www.mathnet.ru/eng/person74390
1. E. V. Nosova, S. Achelle, G. N. Lipunova, V. N. Charushin, O. N. Chupakhin, "Functionalized benzazines as luminescent materials and components for optoelectronics", Usp. Khim., 88:11 (2019), 1128–1178 ; Russian Chem. Reviews, 88:11 (2019), 1128–1178
2. S. A. Vakarov, D. A. Gruzdev, G. L. Levit, V. P. Krasnov, V. N. Charushin, O. N. Chupakhin, "Synthesis of enantiomerically pure 2-aryloxy carboxylic acids and their derivatives", Usp. Khim., 88:10 (2019), 1063–1080 ; Russian Chem. Reviews, 88:10 (2019), 1063–1080
3. K. V. Savateev, E. N. Ulomsky, I. I. Butorin, V. N. Charushin, V. L. Rusinov, O. N. Chupakhin, "Azoloazines as $\mathrm{A}_{2a}$ receptor antagonists. Structure–activity relationship", Usp. Khim., 87:7 (2018), 636–669 ; Russian Chem. Reviews, 87:7 (2018), 636–669
4. G. N. Lipunova, E. V. Nosova, V. N. Charushin, O. N. Chupakhin, "Synthesis and antitumour activity of 4-aminoquinazoline derivatives", Usp. Khim., 85:7 (2016), 759–793 ; Russian Chem. Reviews, 85:7 (2016), 759–793
5. I. S. Kovalev, D. S. Kopchuk, G. V. Zyryanov, V. L. Rusinov, O. N. Chupakhin, V. N. Charushin, "Organolithium compounds in the nucleophilic substitution of hydrogen in arenes and hetarenes", Usp. Khim., 84:12 (2015), 1191–1225 ; Russian Chem. Reviews, 84:12 (2015), 1191–1225
6. G. V. Zyryanov, D. S. Kopchuk, I. S. Kovalev, E. V. Nosova, V. L. Rusinov, O. N. Chupakhin, "Chemosensors for detection of nitroaromatic compounds (explosives)", Usp. Khim., 83:9 (2014), 783–819 ; Russian Chem. Reviews, 83:9 (2014), 783–819
7. A. V. Shchepochkin, O. N. Chupakhin, V. N. Charushin, V. A. Petrosyan, "Direct nucleophilic functionalization of $\mathrm{C(sp^2)-H}$ bonds in (hetero)arenes by electrochemical methods", Usp. Khim., 82:8 (2013), 747–771 ; Russian Chem. Reviews, 82:8 (2013), 747–771
8. G. N. Lipunova, T. G. Fedorchenko, O. N. Chupakhin, "Verdazyls: synthesis, properties, application", Usp. Khim., 82:8 (2013), 701–734 ; Russian Chem. Reviews, 82:8 (2013), 701–734
9. T. I. Gorbunova, V. I. Saloutin, O. N. Chupakhin, "Chemical methods of transformation of polychlorobiphenyls", Usp. Khim., 79:6 (2010), 565–586 ; Russian Chem. Reviews, 79:6 (2010), 511–530
10. E. V. Shchegolkov, Ya. V. Burgart, O. G. Khudina, V. I. Saloutin, O. N. Chupakhin, "2-(Het)arylhydrazono-1,3-dicarbonyl compounds in organic synthesis", Usp. Khim., 79:1 (2010), 33–64 ; Russian Chem. Reviews, 79:1 (2010), 31–61
11. N. A. Itsikson, Yu. Yu. Morzherin, A. I. Matern, O. N. Chupakhin, "Receptors for anions", Usp. Khim., 77:9 (2008), 803–816 ; Russian Chem. Reviews, 77:9 (2008), 751–764
12. A. I. Matern, V. N. Charushin, O. N. Chupakhin, "Progress in the studies of oxidation of dihydropyridines and their analogues", Usp. Khim., 76:1 (2007), 27–45 ; Russian Chem. Reviews, 76:1 (2007), 23–40
13. I. N. Egorov, G. V. Zyryanov, V. L. Rusinov, O. N. Chupakhin, "Asymmetric induction in the nucleophilic addition in the series of aromatic azines", Usp. Khim., 74:12 (2005), 1176–1192 ; Russian Chem. Reviews, 74:12 (2005), 1073–1087
14. N. A. Itsikson, G. V. Zyryanov, O. N. Chupakhin, A. I. Matern, "Heteroditopic receptors", Usp. Khim., 74:8 (2005), 820–829 ; Russian Chem. Reviews, 74:8 (2005), 747–755
15. A. G. Pokrovskii, T. N. Il'icheva, S. K. Kotovskaya, S. A. Romanova, V. N. Charushin, O. N. Chupakhin, "Фторированные производные бенз[4,5]имидазо[1,$2-b$][1,3]тиазолов — ингибиторы репродукции вируса кори", Dokl. Akad. Nauk, 398:3 (2004), 412–414
16. O. N. Chupakhin, D. G. Beresnev, "Nucleophilic attack on the unsubstituted carbon atom of azines and nitroarenes as an efficient methodology for constructing heterocyclic systems", Usp. Khim., 71:9 (2002), 803–818 ; Russian Chem. Reviews, 71:9 (2002), 707–720
17. S. G. Perevalov, Ya. V. Burgart, V. I. Saloutin, O. N. Chupakhin, "(Het)aroylpyruvic acids and their derivatives as promising building blocks for organic synthesis", Usp. Khim., 70:11 (2001), 1039–1058 ; Russian Chem. Reviews, 70:11 (2001), 921–938
18. V. I. Saloutin, Ya. V. Burgart, O. N. Chupakhin, "Fluorine-containing 2,4-dioxo acids in the synthesis of heterocyclic compounds", Usp. Khim., 68:3 (1999), 227–239 ; Russian Chem. Reviews, 68:3 (1999), 203–214
19. D. N. Kozhevnikov, V. L. Rusinov, O. N. Chupakhin, "1,2,4-Triazine <i>N</i>-oxides and their annelated derivatives", Usp. Khim., 67:8 (1998), 707–722 ; Russian Chem. Reviews, 67:8 (1998), 633–648
20. V. N. Charushin, O. N. Chupakhin, "The Reactions of Azines on Treatment with 1,3-Bifunctional Nucleophiles", Usp. Khim., 53:10 (1984), 1648–1674 ; Russian Chem. Reviews, 53:10 (1984), 956–970
21. O. N. Chupakhin, I. Ya. Postovskii, "Nucleophilic Substitution of Hydrogen in Aromatic Systems", Usp. Khim., 45:5 (1976), 908–937 ; Russian Chem. Reviews, 45:5 (1976), 454–468
I. Ya. Postovsky Institute of Organic Synthesis, Ural Branch of Russian Academy of Sciences, Ekaterinburg
Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Chemical Technological Institute, Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg | CommonCrawl |
Flocking and line-shaped spatial configuration to delayed Cucker-Smale models
DCDS-B Home
Bifurcations in an economic model with fractional degree
doi: 10.3934/dcdsb.2020367
Strong convergence rates for markovian representations of fractional processes
Philipp Harms
Department of Mathematical Stochastics, University of Freiburg, Germany
Received February 2019 Revised August 2020 Published December 2020
Fund Project: The author gratefully acknowledges support in the form of a Junior Fellowship of the Freiburg Institute of Advances Studies
Many fractional processes can be represented as an integral over a family of Ornstein–Uhlenbeck processes. This representation naturally lends itself to numerical discretizations, which are shown in this paper to have strong convergence rates of arbitrarily high polynomial order. This explains the potential, but also some limitations of such representations as the basis of Monte Carlo schemes for fractional volatility models such as the rough Bergomi model.
Keywords: Fractional processes, Markovian representation, Monte Carlo simulation, numerical discretization, strong convergence rates, rough Bergomi model.
Mathematics Subject Classification: Primary: 60G22; Secondary: 60G15, 65C05, 91G60.
Citation: Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020367
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Figure 1. Volterra Brownian motion of Hurst index $ H\in(0,1/2) $ can be represented as an integral $ W^H_t = \int_0^\infty Y_t(x) x^{-1/2-H}dx $ over a Gaussian random field $ Y_t(x) $. The smoothness of the random field in the spatial dimension $ x $ allows one to approximate this integral efficiently using high order quadrature rules
Figure 2. Dependence of the approximations on the number $ n $ of quadrature intervals and the Hurst index $ H $. Left: varying the number $ n\in\{2,5,10,20,40\}$ = of quadrature intervals with fixed parameters $ H = 0.1 $, $ m = 5 $. Right: varying the Hurst index $ H\in\{0.1,0.2,0.3,0.4\}$ = with fixed parameters $ n = 40 $, $ m = 5 $
Figure 3. The upper bound $ 2Hm/3 $ on the convergence rate established in Remark 6.2 for $ m $-point interpolatory quadrature closely matches the numerically observed one (here: at $ t = 1 $, computed analytically from the covariance functions of the Gaussian processes $ W^H $ and $ W^{H,n} $). Left: relative error $ e = \|W^H_1-W^{H,n}_1\|_{L^2(\Omega)}/\|W^H_1\|_{L^2(\Omega)} $ for $ m\in\{2,3,\dots,20\}$ = with $ H = 0.1 $. Right: slopes of the lines in the left plot (dots) and predicted convergence rate (line)
Table 1. Complexity of several numerical methods for sampling a fractional process $ (W^H_{i/k})_{i\in\{1,\dots,k\}} $ with Hurst index $ H\in(0,1/2) $ at $ k $ equidistant time points
Method Structure Error Complexity
Cholesky Static 0 $ k^3 $
Hosking, Dieker [28,17] Recursive 0 $ k^2 $
Dietrich, Newsam [18] Static 0 $ k\log k $
Bennedsen, Lunde, Pakkanen [12] Recursive $ k^{-H} $ $ k \log k $
Carmona, Coutin, Montseny [15] Recursive $ \epsilon $ $ k\epsilon^{-3/(4H)} $
This paper Recursive $ \epsilon $ $ k\epsilon^{-1/r} $ for $ r\in(0,\infty) $
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2019 Impact Factor: 1.27 | CommonCrawl |
Using single nucleotide variations in single-cell RNA-seq to identify subpopulations and genotype-phenotype linkage
Single-cell RNA-seq variant analysis for exploration of genetic heterogeneity in cancer
Erik Fasterius, Mathias Uhlén & Cristina Al-Khalili Szigyarto
A multicenter study benchmarking single-cell RNA sequencing technologies using reference samples
Wanqiu Chen, Yongmei Zhao, … Charles Wang
A multi-center cross-platform single-cell RNA sequencing reference dataset
Xin Chen, Zhaowei Yang, … Charles Wang
Combinatory use of distinct single-cell RNA-seq analytical platforms reveals the heterogeneous transcriptome response
Yukie Kashima, Ayako Suzuki, … Yutaka Suzuki
Joint analysis of heterogeneous single-cell RNA-seq dataset collections
Nikolas Barkas, Viktor Petukhov, … Peter V. Kharchenko
Tools for the analysis of high-dimensional single-cell RNA sequencing data
Yan Wu & Kun Zhang
A single-cell and single-nucleus RNA-Seq toolbox for fresh and frozen human tumors
Michal Slyper, Caroline B. M. Porter, … Aviv Regev
Single-cell RNA sequencing technologies and bioinformatics pipelines
Byungjin Hwang, Ji Hyun Lee & Duhee Bang
Fully-automated and ultra-fast cell-type identification using specific marker combinations from single-cell transcriptomic data
Aleksandr Ianevski, Anil K. Giri & Tero Aittokallio
Olivier Poirion1,
Xun Zhu1,2,
Travers Ching1,2 &
Lana X. Garmire ORCID: orcid.org/0000-0003-1672-69173
Computational models
Despite its popularity, characterization of subpopulations with transcript abundance is subject to a significant amount of noise. We propose to use effective and expressed nucleotide variations (eeSNVs) from scRNA-seq as alternative features for tumor subpopulation identification. We develop a linear modeling framework, SSrGE, to link eeSNVs associated with gene expression. In all the datasets tested, eeSNVs achieve better accuracies than gene expression for identifying subpopulations. Previously validated cancer-relevant genes are also highly ranked, confirming the significance of the method. Moreover, SSrGE is capable of analyzing coupled DNA-seq and RNA-seq data from the same single cells, demonstrating its value in integrating multi-omics single cell techniques. In summary, SNV features from scRNA-seq data have merits for both subpopulation identification and linkage of genotype-phenotype relationship.
Characterization of phenotypic diversity is a key challenge in the emerging field of single-cell RNA-sequencing (scRNA-seq). In scRNA-seq data, patterns of gene expression (GE) are conventionally used as features to explore the heterogeneity among single cells1,2,3. However, GE features are subject to a significant amount of noises4. For example, GE might be affected by batch effect, where results obtained from two different runs of experiments may present substantial variations5, even when the input materials are identical. Additionally, the expression of particular genes varies with cell cycle6, increasing the heterogeneity observed in single cells7. To cope with these sources of variations, normalization of GE is usually a mandatory step before downstream functional analysis7. Even with these procedures, other sources of biases still exist, e.g., dependent on read depth, cell capture efficiency and experimental protocols etc.
Single-nucleotide variations (SNVs) are genetic alterations of one single base occurring in specific cells as compared to the population background. SNVs may manifest their effects on gene expression by cis and/or trans effect8,9.The disruption of the genetic stability, e.g. increasing number of new SNVs, is known to be linked with cancer evolution10,11. A cell may become the precursor of a subpopulation (clone) upon gaining a set of SNVs. Considerable heterogeneity exists not only between tumors but also within the same tumor12,13. Therefore, investigating the patterns of SNVs provides means to understand tumor heterogeneity.
In single cells, SNVs are conventionally obtained from single-cell exome-sequencing and whole-genome sequencing approaches14. The resulting SNVs can then be used to infer cancer cell subpopulations15,16. In this study, we propose to obtain useful SNV-based genetic information from scRNA-seq data, in addition to the GE information. Rather than being considered the by-products of scRNA-seq, the SNVs not only have the potential to improve the accuracy of identifying subpopulations compared to GE, but also offer unique opportunities to study the genetic events (genotype) associated with gene expression (phenotype)17,18. Moreover, when the coupled DNA- and RNA-based single-cell sequencing techniques become mature, the computational methodology proposed in this report can be adopted as well19.
Here we first built a computational pipeline to identify SNVs from scRNA-seq raw reads directly. We then constructed a linear modeling framework to obtain filtered, effective, and expressed SNVs (eeSNVs) associated with gene expression profiles. In all the datasets tested, these eeSNVs show better accuracies at retrieving cell subpopulation identities, compared to those from gene expression (GE). Moreover, when combined with cell entities into bipartite graphs, they demonstrate improved visual representation of the cell subpopulations. We ranked eeSNVs and genes according to their overall significance in the linear models and discovered that several top-ranked genes (e.g., HLA genes) appear commonly in all cancer scRNA-seq data. In summary, we emphasize that extracting SNV from scRNA-seq analysis can successfully identify subpopulation complexity and highlight genotype–phenotype relationships.
SNV calling from scRNA-seq data
We implemented a pipeline to identify SNVs directly from FASTQ files of scRNA-seq data, following the SNV guideline of GATK (Supplementary Figure 1). We applied this pipeline to five scRNA-seq cancer datasets (Kim20, Ting21, Miyamoto22, Patel23, and Chung24 see Methods), and tested the efficiency of SNV features on retrieving single cell groups of interest. These datasets vary in tissue types, origins (Mouse or Human), read lengths and map-ability (Table 1). They all have pre-defined cell types (subclasses), providing useful references for assessing the performance of a variety of clustering methods used in this study.
Table 1 Summary of scRNA-seq datasets used in this study
We evaluated the GATK SNV calling pipeline using several approaches. First, we estimated the true positive rates of the SNV calling pipeline at different depths of scRNA-seq reads. For this we performed a simulation experiment by artificially introducing 50,000 random SNVs in the exonic regions of hg19 and measured recovery of these SNVs using our pipeline on Kim dataset. The true positive rates monotonically increase with read depth. For as few as 4 read-depth, the pipeline achieves on average over 50% true positive rate and increases to 68% true positive rate when read depth is more than 6 (Fig. 1a). This accuracy is in line with what was reported from bulk cell RNA-seq25. The false positive rate is consistently <0.1, and the median reaches below 0.05 when the read depth is >6 (Fig. 1b). We compared the SNV call results from GATK to those from another SNV caller FreeBayes26 and obtained similar results (Supplementary Figure 2A, B). Additionally, we conducted simulation experiment on a new non-cancer 10X Genomic dataset, and obtained comparable true positive rates (Supplementary Figure 2C, D). Moreover GATK shows better performance than FreeBayes in the 10X dataset. We thus opted to use GATK to call SNVs for the remainder of the report, given its popularity and performance.
The performance measurements of GATK SNV calling and SSrGE pipelines. a, b Performance measurement of GATK SNV calling pipeline. Box plots of true positive rate (a) and false positive rate (b) with respect to the read depth at the called SNV position. The rates are calculated from GATK SNV calling pipeline, using hg19 reference genome to align modified scRNA-seq reads from a subset of 20 cells from the Kim dataset, which were introduced 50,000 random artificial mutations the exonic region of the reads. Error bars represent standard deviation. c, d Comparisons of importance the different types of features in SSrGE models, with respect to the ranking, in Miyamoto dataset (c) and Kim dataset (d). The scores of the SNVs and CNVs correspond to the sum of the coefficients inferred by the SSrGE models. The gene score is the sum of the SNVs scores for a particular gene. Blue: CNV feature; Red: eeSNV feature; Green: gene feature
Using SSrGE to detect eeSNVs in scRNA-seq data
To link the relationship between SNV and GE, we developed a method called Sparse SNV inference to reflect Gene Expression (SSrGE), as detailed in Methods. In addition to SNV, we also optionally considered the effect of CNVs on gene expression, since copy number variation (CNV) may contribute to gene expression variation as well. Similar to gene-based association method PrediXscan17, SSrGE uses SNVs and additionally optionally CNVs as predictors to fit a linear model for gene expression, under LASSO regularization and feature selection27. We choose LASSO rather than elasticNet for penalization, so that the list of resulting eeSNVs is short (Supplementary Figure 3). These eeSNVs serve as refined descriptive features for subsequent subpopulation identification. To directly pinpoint the contributions of SNVs relevant to protein-coding genes, we used the SNVs residing between transcription starting and ending sites of genes as the inputs. We further assessed the relative contributions of eeSNVs and CNVs to gene expression and found that the coefficients of the CNVs are significantly lower than those of eeSNVs(Fig. 1c, d). The ranks of the top genes with and without CNVs in the SSrGE models are not statistically different overall, as the Kendall-Tau correlation scores28 are close to 1 with p-values = 0 (Kendall-Tau test).
Additionally, SNV genotypes and allelic specific level gene expression may also complicate the relationships between eeSNVs and gene expression. Therefore, we further calibrated SSrGE model by considering SNV genotype and allelic specific gene expression. We used QUASAR29 to estimate the SNV genotypes (Supplementary Table 1), and the allelic specific gene expression using the SNV genotype. We rebuilt individual SSrGE models using only the SNVs from a particular genotype and allelic-specific gene expression, and then merged the eeSNV weights from related SSrGE models together to obtain a final ranking of eeSNVs. The new rankings are not statistically different compared to the previous approach (Supplementary Table 1). The Kendall-Tau scores, which evaluate the similarities between the re-calibrated model and the original model have p-values = 0 (Kendall-Tau test) in all data sets.
Lastly, to quantitatively evaluate if the eeSNVs obtained from SSrGE are truly significant, we designed a simulation pipeline (Methods). The pipeline creates random binary matrices of SNVs for n simulated cells, which are connected to the matrices of gene expression. The SNVs present in the simulated cell have probabilities to modify gene expression of the genes positively or negatively. We used various levels of noise to perturb the GE and the SNV matrices. We compared the ranks of top genes identified by SSrGE to the expected impact of each gene provided by the simulation. The inferred top ranked genes using SSrGE have monotonic and positive correlations with those set by the simulation (Supplementary Figure 4A). These correlations are all significant (p-value « 0.05, Kendall-Tau test), independently of the alpha and the level of noise used, confirming the value of SSrGE model. Moreover, to simulate the patterns of dropout in the data, we also introduced two other parameters, one for random dropout or biased dropout toward both lowly expressed genes, and other for dropout rate relative to cell, gene or reads (Methods). We observed that SSrGE performs well on all dropout models (Supplementary Figure 4B). Thus, SSrGE method is validated to generate eeSNVs that are truly important.
eeSNVs are better than gene counts at finding subpopulations
We measured the performance of SNVs and gene expression (GE) to identify subpopulations on the five datasets, using five clustering approaches (Fig. 2). These clustering approaches include two dimension reduction methods, namely principal component analysis (PCA)30 and factor analysis (FA)31, followed by either K-means or the hierarchical agglomerative method (agglo) with WARD linkage32. We also used a recent algorithm SIMLR designed explicitly for scRNA-seq data clustering and visualization33. To evaluate the accuracy of obtained subpopulations in each dataset, we used the metric of adjusted mutual information (AMI) over 30 bootstrap runs, from the optimal a parameters (Supplementary Data 1). These optimal parameters were estimated by testing different a values for each dataset and each clustering approach (Supplementary Figure 5). As shown in Fig. 2, eeSNVs are better features to retrieve cancer cell subpopulations compared to GE, independent of the clustering methods used. Among the clustering algorithms, SIMLR tends to be a better choice using eeSNV features.
Comparison of clustering accuracy using eeSNV and gene expression (GE) features. a–e Bar plots comparing the clustering performance using eeSNV vs. gene expression (GE) as features, over five different clustering strategies and five datasets, each with its own pre-defined classes as the truth measure: a Kim dataset, b Ting dataset, c Chung dataset, d Miyamoto dataset, and e Patel dataset. Y-axis is the adjusted mutual information (AMI) obtained across 30 bootstrap runs (mean ± s.d.). Error bars represent standard deviation. *P < 0.05, **P < 0.01, and ***P < 0.001 (paired t-test). f Heatmap of the rankings among different methods and datasets as shown in a–e
Additionally, we computed the Adjusted Rand index (ARI)34 and V-measure35, two other metrics for modularity measurements (Methods) and obtained similar trends (Supplementary Figure 6). Similar to AMI, ARI is a normalized metric against random chance, and evaluates the number of correct pairs obtained. On the other hand, V-measure combines the homogeneity score, which measures the homogeneity of reference classes in the obtained clusters, and the completeness score, which measures the homogeneity of obtained clusters within the reference classes. Due to the high number of small homogenous clusters obtained for the Miyamoto dataset, we observed higher V-measure scores, compared to AMI and ARI results (Supplementary Figure 6).
Visualization of subpopulations with bipartite graphs
Bipartite graphs are an efficient way to describe the binary relations between two different classes of objects. We next represented the presence of the eeSNVs into single cell genomes with bipartite graphs using ForceAtlas2 algorithm36. We drew an edge (link) between a cell node and a given eeSNV node whenever an eeSNV is detected. The results show that bipartite graph is a robust and more discriminative alternative (Fig. 3), comparing to PCA plots (using GE and eeSNVs) as well as SIMLR (using GE). For Kim dataset, bipartite graph separates the three classes perfectly. However, gene-based visualization approaches using either PCA or SIMLR have misclassified data points. For Ting data, the eeSNV-cell bipartite graph gives a clear visualization of all six different subgroups of single cells. Other three approaches have more exaggerated separations among the same mouse circulating tumor cells (CTC) subgroup MP (orange color), but mix some other subpopulations (e.g., GM, MP, and TuGMP groups). Miyamoto dataset is the most difficult one to visualize among the four datasets, due to its high number (24) of reference classes and heterogeneity among CTCs. Bipartite graphs are not only able to condense the whole populations, but also separate subpopulations (e.g., the orange colored PC subpopulation) much better than the other three methods.
Comparison of clustering visualization using eeSNV and gene expression (GE) features. a Bipartite graphs using eeSNVs and cells as two groups of nodes. An edge between a cell and an eeSNV represents the presence of the eeSNV within that cell. b Principle component analysis (PCA) results using GE as features of the cells. c PCA results using eeSNVs as features of the cells. d SIMILR results using GE as the input
Characteristics of eeSNVs
In SSrGE, regularization parameter a is the only tuning variable, controlling the sparsity of the linear models and influences the number of eeSNVs. We next explored the relationship between eeSNVs and a (Fig. 4). For every dataset, increasing the value of a decreases the number of selected eeSNVs overall (Fig. 4a), as well as the average number of eeSNVs associated with every expressed gene (Fig. 4b). The optimal a depends on the clustering algorithm and the dataset used (Supplementary Data 1 and Supplementary Figure 5). Increasing the value of a expands the proportion of eeSNVs that have annotations in human dbSNP138 database, indicating a higher true positive rate of SNVs compared to that prior to SSrGE filtering (Fig. 4c). Additionally, increasing a increases the average number of cells sharing the same eeSNVs (Fig. 4d), corresponding to the decreasing number of eeSNVs (Fig. 4b). Note the slight drop in the average number of cells sharing the same eeSNVs in Kim data when a > 0.6, this is due to overpenalization (e.g., a = 0.8 yields only 34 eeSNVs).
Characteristics of the eeSNVs. X-axis: the regularization parameter a values used by LASSO penalization in the SSrGE models. And the Y-axes are: a Log10 transformation of the number of eeSNVs. b The average number of eeSNVs per gene. c The proportion of SNVs with dbSNP138 annotations (human datasets). d The average number of cells sharing eeSNVs
Cancer relevance of eeSNVs
Following the simulation results, we ranked the different eeSNVs and the genes for the five datasets, from SSrGE models (Supplementary Data 2). We found that eeSNVs from multiple genes in human leukocyte antigen (HLA) complex, such as HLA-A, HLA-B, HLA-C, and HLA-DRA, are top ranked in all four human datasets (Table 2 and Supplementary Data 2). HLA is a family encoding the major histocompatibility complex (MHC) proteins in human. Beta-2-microglobulin (B2M), on the other hand, is ranked 7th and 45th in Ting and Patel datasets, respectively (Table 2). Unlike HLA that is present in human only, B2M encodes a serum protein involved in the histocompatibility complex MHC that is also present in mice. Other previously identified tumor driver genes are also ranked top by SSrGE, demonstrating the impact of mutations on cis-gene expression (Table 2 and Supplementary Data 2). Notably, KRAS, previously linked to tumor heterogeneity (Kim et al.37), is ranked 13th among all eeSNV containing genes (Supplementary Data 2). AR and KLK3, two genes reported to show genomic heterogeneity in tumor development in the original study22, are ranked 6th and 19th, respectively. EGFR, the therapeutic target in Patel study with an important oncogenic variant EGFRvIII (Patel et al.23), is ranked 88th out of 4225 genes. Therefore, genes top-ranked by their eeSNVs are empirically validated.
Table 2 A list of interested genes highly ranked
Next we conducted more systematic investigation to identify KEGG pathways enriched in each dataset, using these genes as the input for DAVID annotation tool38 (Fig. 5a). The pathway-gene bipartite graph illustrates the relationships between these genes and enriched pathways (Fig. 5b). As expected, Antigen processing and presentation pathway stands out as the most enriched pathway, with the sum −log10 (p-value) of 15.80 (Fig. 5b). Phagosome is the second most enriched pathway in all four data sets, largely due to its members in HLA families (Fig. 5b). Additionally, pathways related to cell junctions and adhesion (focal adhesion and cell adhesion molecules CAMs), protein processing (protein processing in endoplasmic reticulum and proteasome), and PI3K-AKT signaling pathway are also highly enriched with eeSNVs (Fig. 5a).
Gene and KEGG pathways enriched with eeSNVs in the five scRNA-seq datasets. a Bipartite graph using significant KEGG pathways and genes enriched with eeSNVs as nodes. An edge exists between a significant pathway and a gene if this gene is part of the pathway. Genes of each data set is represented with a distinct color. The size of the nodes reflects the gene and the pathway scores. The gene scores are computed by SSrGE and the pathway scores are the sum of the gene scores linked for each pathway. b KEGG pathways enriched within the top 100 genes based on eeSNV contributions in the five datasets. Pathways are sorted by the sum of the −log10 (p-value) of each dataset, in the descending order
Heterogeneity markers using eeSNVs
We display the potential of eeSNV as heterogeneity markers via pseudo-time reconstruction and heatmap, using Kim dataset (Fig. 6a, b). We built a Minimum Spanning Tree, similarly to the Monocle algorithm39, to reconstruct the pseudo-time ordering of the single cells. The graphs beautifully capture the continuity among cells, from the primary to metastasized tumors (Fig. 6a). Moreover, it highlights ramifications inside each of the subgroups, demonstrating the intra-group heterogeneity. On the contrary, pseudo-time reconstruction using GE showed much less complexity and more singularity (Supplementary Figure 7). As a confirmation, hierarchical clustering of eeSNV heatmp also shows almost perfect separation of the three subgroups (Fig. 6b). Next, we used our method to identify eeSNVs specific to each single-cell subgroup and ranked the genes according to these eeSNVs. We compared the characteristics of the metastasis cells to primary tumor cells. Two top-ranked genes identified by the method, CD44 (1st) and LPP (2nd), are known to promote cancer cell dissemination and metastasis growth after genomic alteration40,41,42,43 (Supplementary Data 2). Other top-ranked genes related to metastasis are also identified, including LAMPC2 (7th), HSP90B1 (14th), MET (44th), and FN1 (52th). As expected, Pathways in Cancer are the top-ranked pathway enriched with mutations (Fig. 6b). Additionally, Focal Adhesion, and Endocytosis pathways are among the other significantly mutated pathways, providing new insights on the mechanistic difference between primary and metastasized RCC tumors (Fig. 6c).
Heterogeneity revealed by Kim dataset. a Pseudo-time ordering reconstruction of the different subgroups: single cells from PDX primary tumor (green), patient metastasis (blue) and PDX metastasis (red). The eeSNVs are obtained with a = 0.6. The tree is inferred using the MST algorithm on Pearson's correlation-based distance matrix. b Heatmap of the cells (row) and eeSNVs (column). c Bipartite graph using KEGG pathways (orange color) and genes enriched with significant eeSNVs (green color) as two set of nodes. The significant eeSNVs are inferred from the metastasized cells, compared to the primary tumor cells. The size of the nodes reflects the gene scores (given by SSrGE) and the pathway scores (sum of the gene scores). Lighter green indicates genes with a lower rank
Another application is to explore the potential of eeSNVs to separate different cell types within the same individual. Toward this, we extended the same analysis on the two patients BC03 and BC07 from the Chung dataset, who have primary and metastasized tumor cells as well as infiltrating immune cells. Again, bipartite graphs and minimum spanning trees-based visualization illustrate clear separations of tumor (primary and metastasized) cells from immune cells (Supplementary Figure 8). Furthermore, the top-ranked genes relative to the metastasis subgroups (BC03M and BC07M) present some similarities with those found in Kim dataset (Supplementary Data 3). Strikingly, CD44 is also top-ranked (23rd) among the significant genes of BC07M. Similarly, HSP90B1 is top-ranked as the 63rd and 51st most important genes, in BC03M and BC07M, respectively.
Integrating DNA- and RNA-seq data in the same single cells
Coupled DNA-seq and RNA-seq measurements from the same single cell are the new horizon of single-cell genomics. To demonstrate the potential of SSrGE in integrating DNA and RNA data, we downloaded a public single cells data, which have DNA methylation and RNA-seq records from the same hepatocellular carcinoma (HCC) single cells (Hou dataset)44. We then inferred SNVs from the aligned reduced representation bisulfite sequencing (RRBS) reads (see Methods), and used them to predict the scRNA-seq data from the same samples. Given the fact that SNVs are heterozygous among tumor and normal cells, and that a small fraction of genes harboring eeSNVs are subject to CNV, we included both the percentages of SNVs as well as CNVs as additional predictive variables in the SSrGE model besides SNV features. Interestingly, the identified eeSNVs can clearly separate normal hepatocellular cells from cancer cells and highlight the two cancer subtypes identified in the original study (Fig. 7). Pseudo-time ordering shows an early divergence between the two previously assumed subtypes (Fig. 7b). This observation is confirmed by hierarchical clustering of eeSNV based heatmap (Fig. 7c). A simplified version of SSrGE model, where only SNV features were considered as predictors for gene expression, shared 92% eeSNVs as those in Fig. 7a, and achieved almost identical separations between normal hepatocellular cells and cancer cells. This confirms the earlier observation that eeSNVs are much more important predictive features, compared to CNVs (Fig. 1c, d).
Heterogeneity revealed by eeSNVs from multi-omics single cell HCC (Hou) dataset. Normal cells are colored green and HCC tumor cells are colored bright (subpopulation I) or dark (subpopulation II) red. a Bipartite-graph representation using the single cells and eeSNVs from RRBS reads as two sets of nodes. b Pseudo-time ordering reconstruction of the HCC cells, using eeSNVs in from RRBS. c Heatmap of the cells (row) and eeSNVs (column)
We postulated that a considerable part of bisulfite reads was aligned with methylation islands associated with gene promoter regions. Thus, we annotated eeSNVs within 1500 bp upstream of the transcription starting codon, and obtained genes with these eeSNVs, which are significantly prevalent in certain groups. When comparing HCC vs. normal control cells, two genes PRMT2, SULF2 show statistically significant mutations in HCC cells (p-values < 0.05, Fisher's exact test). Downregulation of PRMT2 was previously associated with breast cancer45, SULF2 was known to be upregulated in HCC and promotes HCC growth46.
Using GE to accurately analyze scRNA-seq data has many challenges, including technological biases such as the choice of the sequencing platforms, the experimental protocols and conditions. These biases may lead to various confounding factors in interpreting GE data5. SNVs, on the other hand, are less prone to these issues given their binary nature. In this report, we demonstrate that eeSNVs extracted from scRNA-seq data are ideal features to characterize cell subpopulations. Moreover, they provide a means to examine the relationship between eeSNVs and gene expression in the same scRNA-seq sample.
The process of selecting eeSNVs linked to GE allows us to identify representative genotype markers for cell subpopulations. We speculate the following reasons attributed to the better accuracies of eeSNVs compared to GE. First, eeSNVs are binary features rather than continuous features like GE. Thus, eeSNVs are more robust at separating subpopulations. We have noticed that SNVs are less affected by batch effects (Supplementary Figure 9). Secondly, LASSO penalization works as a feature selection method and minimizes the spurious SNVs (false positive) from the filtered set of eeSNVs. Thirdly, since eeSNVs are obtained from the same samples as scRNA-seq data, they are more likely to have biological impacts, and this is supported by the observation that they have high prevalence of dbSNP annotations.
A small number of eeSNVs can be used to discriminate distinct single-cell subpopulations, as compared to thousands of genes that are normally used for scRNA-seq analyses. Taking advantage of the eeSNV-GE relationship, a very small number of top eeSNVs still can clearly separate cell subpopulations of the different datasets (e.g., 8 eeSNV features have decent separations for Kim dataset). Moreover, our SSrGE package can be easily parallelized and process each gene independently. It has the potential to scale up to very large datasets, well poised for the new wave of scRNA-seq technologies that can generate thousands of cells at one time47. One can also easily rank the eeSNVs and the genes harboring them, for the purpose of identifying robust eeSNVs as genetic markers for a variety of cancers.
The eeSNV based approach to identify subpopulations can be generalized to other non-cancer tissues and scRNA-seq data generated from various platforms. Additionally, we analyzed nine datasets, including two non-cancer 10X Genomics datasets with a large number of cells, four non-cancer datasets (GSE79457, GSE81903, GSE71485, and GSE80232), and three cancer single-cell datasets (GSE69405, GSE81383, and GSE65253). Bipartite graphs with eeSNVs and cells were able to separate the single-cell subpopulations based on their genotype (Supplementary Figure 10A–I).
SSrGE uses an accumulative ranking approach to select eeSNVs linked to the expression of a particular gene. Mainly, HLA class I genes (HLA-A, HLA-B, and HLA-C) are top-ranked for the three human datasets, and they contribute to antigen processing and presentation pathway, the most enriched pathways of the four datasets. HLA has amongst the highest polymorphic genes of the human genome48, and the somatic mutations of genes in this family occurred in the development and progression of various cancers49,50. eeSNVs of HLA genes could be used as fingerprints to identify the cellular state of the cancer cells, and lead to better separation between the primary (pRCC, green) and metastatic cells (mRCC, red and blue) compared to GE of HLA genes (Supplementary Figure 11). B2M, another gene with top-scored eeSNVs in Ting and Patel datasets, is also known to be a mutational hotspot51. It is immediately linked to immune response as tumor cell proliferation49,51. Many other top-ranked genes, such as SPARC, were reported to be driver genes in the original studies of the different dataset. Thus, it is reasonable to speculate that SSrGE is capable of identifying some driver genes. Another possibility is that some of the eeSNVs reflect aberrant splicing of genes such as the HLA family, which are regularly found in deregulated cancer cells52. Nevertheless, SSrGE may miss some driver mutations due to the incomplete DNA coverage of scRNA-seq reads. Also, its primary goal is to identify a minimal set of eeSNV features by LASSO penalization but in case of correlated features, LASSO may select one of those highly correlated SNV features that correspond to GE.
We have showed with strong evidence that eeSNVs unveil inter- and intra-tumor cells heterogeneities better than gene expression count data obtained from the same RNA-seq reads. Reconstructing the pseudo-time ordering of cancer cells from the same tumor (Kim dataset) displays branching even inside primary tumor and metastasis subgroups, which gene expression data are unable to do. We identified genes enriched with SNVs specific to the metastasis, which were not reported in the original HCC single cell study44. Most interestingly, we showed that eeSNVs can also be retrieved from RRBS reads in a multi-omics single-cell HCC dataset, a twist from their original purpose of single-cell DNA methylation. Again, genes ranked by eeSNVs from RRBS reads only differentiate normal from cancer cells but also the different cancer subtypes. We identified several genes that are significant in either HCC or HCC subgroup, whose promoters are highly impacted by eeSNVs. Thus, we have demonstrated that our method is on the fore-front to analyze data generated by new single-cell technologies extracting multi-omics from the same cells44,53.
Bipartite graphs are a natural way to visualize eeSNV-cell relationships. We have used force-directed graph drawing algorithms involve spring-like attractive forces and electrical repulsions between nodes that are connected by edges. This approach has the advantage to reveal outlier single cells, with a small set of eeSNVs, compared to those distance-based approaches. Moreover, the bipartite representation also reveals directly the relationship between single cells and the eeSNV features. Contrary to dimension reduction approaches such as PCA that requires linear transformation of features into principle components, bipartite graphs preserve all the binary information between cell and eeSNV. Graph analysis software such as Gephi36 or Cytoscape54 can be utilized to explore the bipartite relationships in an interactive manner.
In this manuscript, we demonstrated the efficiency of using eeSNVs for cell subpopulation identification over multiple datasets. eeSNVs are excellent genetic markers for intra-tumor heterogeneity and may serve as genetic candidates of new treatment options. We also have developed SSrGE, a linear model framework that correlates genotype (eeSNV) and phenotype (GE) information in scRNA-seq data. Moreover, we have showed the capacity of SSrGE in analyzing multi-omics data from the same single cells, obtained from the most cutting-edge genomics techniques55,37. Our method has the great promise as part of routine analyses in scRNA-seq pipelines56, as well as multi-omics single-cell integration projects.
scRNA-seq datasets
All six datasets were downloaded from the NCBI Gene Expression Omnibus (GEO) portal57.
Kim dataset (accession GSE73121) contains three cell populations from matched primary and metastasis tumor from the same patient20. Patient Derivated Xenographs (PDX) were constructed using cells from the primary Clear Cell Renal Cell Carcinoma (PDX-pRCC) tumor and from the lung metastasic tumor (PDX-mRCC). Also, metastatic cells from the patient (Pt-mRCC) were sequenced.
Patel dataset (accession GSE57872) contains five glioblastoma cell populations isolated from five individual tumors from different patients (MGH26, MGH28, MGH29 MGH30, and MGH31) and two gliomasphere cell lines, CSC6 and CSC8, used as control23.
Miyamoto dataset (accession GSE67980) contains 122 CTCs from Prostate cancer from 18 patients, 30 single cells derived from 4 different cancer cell lines: VCaP, LNCaP, PC3, and DU145, and 5 leukocyte cells from a healthy patient (HD1)22. A total of 23 classes (18 CTC classes + 4 cancer cell lines + 1 healthy leukocyte cell lines) were obtained.
Ting dataset (subset of accession GSE51372) contains 75 CTCs from Pancreatic cancer from 5 different KPC mice (MP2, MP3, MP4, MP6, and MP7), 18 CTCs from two GFP-lineage traced mice (GMP1 and GMP2), 20 single cells from one GFP-lineage traced mouse (TuGMP3), 12 single cells from a mouse embryonic fibroblast cell line (MEF), 12 single cells from mouse white blood (WBC) and 16 single cells from the nb508 mouse pancreatic cell line (nb508)21. KPC mice have uniform genetic cancer drivers (Tp53, Kras). Due to their shared genotype, we merged all the KPC CTCs into one single reference class. CTCs from GMP1 did not pass the QC test and were dismissed. CTCs from GMP2 mice were labeled as GMP. Finally, 6 reference classes were used: MP, nb508, GMP, TuGMP, MEF, and WBC.
Hou dataset (accession GSE65364) contains 25 hepatocellular carcinoma single cells (Ca) extracted from the same patient and 6 normal liver cells (HepG2) obtained from the adjacent normal tissue of another HCC patient44. The 32 cells were sequenced using scTrio-seq in order to obtain reads from both RNA-seq and reduced representation bisulfate sequencing (RRBS). The authors highlighted that one of the Ca cells (Ca_26) was likely to be a normal cell, based on CNV measurements, and thus we discarded this cell. We used the RRBS reads to infer the SNVs. We use gene expression data provided by the authors to construct a GE matrix. For controls, we used the bulk genome of all the RNA-seq and RRBS reads of the HepG2 group.
Chung dataset (accession GSE75688) contains 549 single cells from primary breast cancer and lymph nodes metastases, extracted from 11 patients (BC01-11) of distinct molecular subtypes. BC01-02 are estrogen receptor positive (ER+); BC04-06 are human epidermal growth factor receptor 2 positive (HER+); BC03 is double positive (ER+ and HER+); BC07-11 are triple negative breast cancer (TNBC)24. Only BC03 and BC07 presented cells extracted from lymph nodes metastases (BC03M and BC07M). Additionally, the dataset contains a large part of infiltrating tumor cells. Following the original analytical procedure of the original study24, we performed an unsupervised clustering analysis to separate the cancer from the immune cells. We first reduced the dimension with a PCA analysis and then used a Gaussian mixture model to infer the clusters. We obtained a total of 372 cancer cells and 177 immune cells.
SNV detection using scRNA-seq data
The SNV detection pipeline using scRNA-seq data follows the guidelines of GATK (http://gatkforums.broadinstitute.org/wdl/discussion/3891/calling-variants-in-rnaseq). It includes four steps: alignment of spliced transcripts to the reference genome (hg19 or mm10), BAM file preprocessing, read realignment and recalibration, and variant calling and filtering (Supplementary Figure 1)58.
Specifically, FASTQ files were first aligned using STAR aligner59, using mm10 and hg19 as reference genomes for mouse and human datasets, respectively. The BAM file quality check was done by FastQC60, and samples with lower than 50% of unique sequences were removed (default of FastQC). Also, samples with more than 20% of the duplicated reads were removed by STAR. Finally, samples with insufficient reads were also removed, if their reads were below the mean minus two times the standard deviation of the entire single-cell population. The summary of samples and reads filtered by these steps is listed in Table 1. Raw gene counts Xj were estimated using featureCounts61, and normalized using the logarithmic transformation:
$$f(X_j) = {\mathrm{log}}_2\left( {1 + X_j.\frac{{10^9}}{{(G_j.R)}}} \right)$$
where Xjis the raw expression of gene j, R is the total number of reads and Gj is the length of the gene j. BAM files were pre-processed and reordered using Picard Tools (http://broadinstitute.github.io/picard/), before subject to realignment and recalibration using GATK tools62. SNVs are then calculated and filtered using the GATK tools HaplotypeCaller and VariantFiltration using default parameters.
Additionally, we used Freebayes26 with default parameters to infer the SNVs, as alternative to the HaplotypeCaller and VariantFiltration softwares. The SNV calling results between the two callers are very similar (Supplementary Figure 2).
SNV detection using RRBS data
We first aligned the RRBS reads on the hg19 reference genome using the Bismark software63. We then processed the bam files using all the preprocessing steps as described in the SNV detection using scRNA-seq data section (i.e., Picard Preprocessing, Order reads, Split reads and Realignments), except the base recalibration step. Finally, we called the SNVs using the BS-SNPer software (default setting)64. The details are the following.
–minhetfreq 0.1 # Threshold of frequency for calling heterozygous SNP
–minhomfreq 0.85 # Threshold of frequency for calling homozygous SNP
–minquali 15 # Threshold of base quality
–mincover 10 # Threshold of minimum depth of covered reads
–maxcover 1000 # Threshold of maximum depth of covered reads
–minread2 2 # Minimum mutation reads number
–errorate 0.02 # Minimum mutation rate
–mapvalue 20 # Minimum read mapping value
10X genomic data preprocessing
We downloaded the fastq files of two data sets: bone marrow mononuclear cells (BMMC) of 2 individuals and peripheral blood mononuclear cells (PBMC), from the 10X Genomic website (https://support.10xgenomics.com/single-cell-gene-expression/datasets). We de-multiplexed the reads using the BC tags extracted with the pysam library (http://pysam.readthedocs.io/en/latest/api.html). For bipartite graph display, we used a subset of 2000 BMMC cells and 3300 PBMC cells with the highest number of reads in each dataset, respectively.
SNV simulation
We created a modified version of the hg19 reference genome by introducing 50,000 random mutations in the exonic region of the genes. To introduce a new mutation, we weighted each exon depending on its base length and selected one randomly and proportional to its weight. We realigned the scRNA-seq reads from a subset of 20 cells from the Kim dataset with the introduced new mutations, using the same SNV detection pipelines described above. We used BedTools65 to compute the read depth of each mutation.
SNV annotation
To annotate human SNV datasets, dbSNP138 from the NCBI Single Nucleotide Polymorphism database66 and reference INDELs from 1000 genomes (1000_phase1 as Mills_and_1000G_gold_standard)67 were used. To annotate the mouse SNV dataset, dbSNPv137 for SNPs and INDELs were downloaded from the Mouse Genomes Project of the Sanger Institute, using the following link: ftp://ftp-mouse.sanger.ac.uk/REL-1303-SNPs_Indels-GRCm38/68. The mouse SNP databases were sorted using SortVcf command of Picard Tools in order to be properly used by Picard Tools and GATK.
SSrGE package to correlate eeSNVs to gene expression
For each dataset, we denote MSNV and MGE as the SNV and gene expression matrices, respectively. MSNV is binary (MSNVc,s∈{0,1}) indicating the presence/absence of SNV s in cell c. MGEc,s is the log transformed gene expression value of the gene g in cell c. Copy number variation (CNV) MCNV, can be added as an additional optional predictor in SSrGE. We computed CNV for each gene g in each cell c using the online platform Ginkgo68. For the Hou dataset, we inferred CNV using the same approach described by the authors44. We removed any SNV present in less than k = 3 cells, as it offers the best tradeoff between accuracy and SSrGE running time. We also filtered SNVs associated with genes having normalized expression value below 2.0. We also discarded genes with normalized expressions below 2.0 and expressed in less than 10 cells from SSrGE analysis. For each gene g, we applied a sparse linear regression using LASSO to identify Wg, the linear coefficient associated to SNV, as well as Wcnv, the coefficient associated to the CNV of g (if CNV was considered). The objective function for SSrGE to minimize is:
$$\begin{array}{*{20}{c}}{\rm{min}}_{{W_{g + {\mathrm{CNV}}}}} \left({\frac{1}{{2n}}\left\Vert {M_{{\mathrm{SNV}} + {\mathrm{CNV}}g}.W_{g + {\mathrm{CNV}}g}^T - M_{{\mathrm{GE}} \ast ,g}} \right\Vert_2^2 + \alpha .\left\Vert {W_{g + {\mathrm{CNV}}g}} \right\Vert_1} \right)\end{array}$$
where α is the regularization parameter, MSNV+CNVg represents the MSNV matrix with an additional column that corresponds to the CNV of the gene g for each cell. In this configuration, Wg+CNV has one more column too, corresponding to the weight associated with the CNV. An SNV was considered as eeSNV when Wg(s)≠0.
When CNV is not considered in SSrGE, the objective function is simplified as:
$$\begin{array}{*{20}{c}}{\rm{min}}_{{W_{g}}}\left( {\frac{1}{{2n}}\left\Vert {M_{\mathrm{SNV}}.W_g^T - M_{\mathrm{GE} \ast ,g}} \right\Vert_2^2 + \alpha .\left\Vert {W_g} \right\Vert_1} \right)\end{array}$$
Inference of SNV genotype and allele-specific SSrGE calibration
We used the package QuASAR (https://github.com/piquelab/QuASAR) to identify the genotype of each SNV29. For each dataset, we collected the SNVs and constructed an n × 3 matrix using the number of reads mapping to the reference allele, the alternate allele, or neither of the alleles. We then fit this matrix with QuASAR to estimate the true genotype of each SNV. We then estimated the allelic gene expression for each cell by multiplying the normalized gene counts with the fraction of the SNV of a particular genotype. To calibrate SSrGE model with allele expression, we first fit an SSrGE model for each genotype using the allele-specific SNVs and gene expression as inputs. We then merged the eeSNVs and weights inferred for each model into a final model.
Ranking of eeSNVs and genes
SSrGE generates coefficients of eeSNVs for each gene, as a metric for their contributions to the gene expression. The score of an eeSNV is given by the sum of its weights over all genes:
$${\mathrm {score}}_{\mathrm {eeSNV}} = \mathop {\sum }\limits_ g \left| {W_g^T\left( s \right)} \right|$$
Each gene also receives a score according to its associated eeSNVs:
$${\mathrm {score}}_{{\mathrm {gene}}_ {g}} = \mathop {\sum }\limits_{{\mathrm {eeSNV}} \in {\mathrm {gene}}_{g}} {\mathrm {score}}_{{\mathrm {eeSNV}}}$$
In practice, we first obtained eeSNVs using a minimum filtering of α = 0.1, before using these two scores above to rank eeSNVs and the genes.
Ranking of eeSNVs and genes for a subpopulation
For a given single-cell subpopulation p, an eeSNV is defined as specific to the subpopulation p when it has a significantly higher frequency in p than in any other subpopulation. For each eeSNV we took only the subset of cells expressing the gene g associated with the eeSNV. We then computed the Fisher's exact test to compare the presence of the eeSNV between single-cells inside and outside p. We considered an eeSNV significant for p-value < 0.05. p′ designates the subset of cells from p expressing g. The score of an eeSNV for p is given by:
$${\mathrm {score}}_{{\mathrm {eeSNV}}}^p = \frac{{\left| {\left\{ {{\mathrm {cell}}\,{\mathrm {with}}\,{\mathrm {eeSNV}}|{\mathrm {cell}} \in p\prime } \right\}} \right|}}{{\left| p\prime \right|}}{\mathrm {score}}_{{\mathrm {eeSNV}}}$$
The score of a given gene g for p is thus given by:
$${\mathrm {score}}_{{\mathrm {gene}}_{g}}^p = \mathop {\sum }\limits_{{\mathrm {eeSNV}}\, \in \,{\mathrm {gene}}_{g}} {\mathrm {score}}_{{\mathrm {eeSNV}}}^p$$
To rank eeSNVs from the promoter regions of the RRBS reads in Hou dataset, we applied a similar methodology: we annotated the eeSNVs within 1500 bp upstream of genes' starting codon regions.
Perturbation-model based simulation to quantitatively assess SSrGE
We first simulated an interaction table which gives an interaction score (between −1 and 1) for each gene-SNV pair, denoted as interaction g v for gene g and SNV v. This interaction score follows a mixed normal distribution, with two normal distribution components. These two distributions are named the inhibiting component (centered at −0.5), and the enhancing component (centered at 0.5). The type of interaction determines the weighting factor. We define a cis-interaction if the SNV is located within the gene and a trans-interaction otherwise. A cis-interaction has equal weights on the inhibiting and enhancing components. A trans-interaction has significantly larger weight on the inhibiting component, as previous studies found cis interaction is more likely to be inhibitive.
We then simulated the unperturbed expression matrix using Splatter69, with parameters estimated from the Kim dataset. We also simulated the SNV matrix using random shuffle of the SNV matrix extracted from the same dataset. We then applied the interactions to the unperturbed expression. For a gene g in a sample s, the expression level GEgs was perturbed to GE'gs using the following formula:
$${\mathrm{GE}}_{{gs}}^\prime = {\mathrm{GE}}_{{gs}}{\mathrm{exp}}^{\left( {\mathop {\sum }\limits_v {\mathrm{snv}}_v.{\mathrm{interaction}}_{{gv}}} \right)}$$
This created the perturbed expression matrix. Then, we added drop-outs to the perturbed expression matrix using different strategies, in order to create the final observed expression matrix. To mimic dropouts, we introduced two variables: one for dropout bias (either bias toward lower expressed genes or randomly), and the other for dropout rate dependency (cell-based, gene-based, or read-based). We did not consider dropout bias towards highly expressed genes, as it contradicts observations documented earlier70. Random dropout (no bias) removes the dependency on cell, gene, or reads. We also added random noises to the SNV matrix (following a Bernoulli distribution) to generate the final observed SNV matrix.
Pseudo-time ordering reconstruction
To estimate the trajectory of cell evolvement, we adopted the following procedure, motivated by the method described earlier39. We first constructed the following distance matrix to reflect the Pearson's correlation between each pair of cells: Di,j = 1−Correlation(Samplei,Samplej)
Subpopulation clustering algorithms
We combined two-dimension reduction algorithms: PCA30 and FA31 with two popular clustering approaches: the K-means algorithm71 and agglomerative hierarchical clustering (agglo) with WARD linkage32. We also used SIMLR, a recent algorithm specifically tailored to cluster and visualize scRNA-seq data, which learns the similarity matrix from subpopulations33. Similar to the original SIMLR study, we used the embedding of the cells produced by the algorithm to apply K-means algorithm.
PCA and FA were performed using their corresponding implementation in Scikit-Learn (sklearn)72. For PCA, FA, and SIMLR, we used various input dimensions D [2, 3, 5, 10, 15, 20, 25, 30] to project the data. To cluster the data with K-means or the hierarchical agglomerative procedure, we used a different cluster numbers N (2 to 80) to obtain the best clustering results from each dataset. We computed accuracy metrics for each (D, N) pair and chose the combination that gives the overall best score. Between the two clustering methods, K-means was the implementation of sklearn package with the default parameter, and hierarchical clustering was done by the AgglomerativeClustering implementation of sklearn, using WARD linkage.
Validation metrics
To assess the accuracy of the obtained clusters, we used three metrics: Adjusted Mutual Information (AMI), Adjusted Rand Index (ARI) and V-measure34,35. These metrics compare the obtained clusters C to some reference classes K and generate scores between 0 and 1 for AMI and V-measure, and between −1 and 1 for ARI. A score of 1 means perfect match between the obtained clusters and the reference classes. For ARI, a score below 0 indicates a random clustering.
AMI normalizes Mutual Information (MI) against chances34. The Mutual Information between two sets of classes C and K is equal to: \({\mathrm{MI}}({{C}},{{K}}) = \mathop {\sum }_{i = 1}^{|{{C}}|} \mathop {\sum }_{{{j}} = 1}^{|{{K}}|} {P}\left( {{{i}},{{j}}} \right){\mathrm{log}}\left( {\frac{{{P}\left( {{{i}},{{j}}} \right)}}{{{P}\left( {{i}} \right){P}\prime \left( {{j}} \right)}}} \right)\), where P(i) is the probability that an object from C belongs to the class i, P′(j) is the probability that an object from K belongs to class j, and P(i, j) is the probability that an object are in both class i and j. AMI is equal to: \({\mathrm{AMI}}(C,K) = \frac{{{\mathrm{MI}}(C,K) - {E}({\mathrm{MI}}(C,K))}}{{\max \left( {\{ H\left( C \right),H(K)\} } \right) - {E}({\mathrm{MI}}(C,K))}}\), where H(C) and H(K) designates the entropy of C and K.
Similar to AMI, ARI normalizes RI against random chances: \({\mathrm{ARI}} = \frac{{{\mathrm{RI}} - {E}({\mathrm{RI}})}}{{\max \left( {{\mathrm{RI}}} \right) - {E}({\mathrm{RI}})}}\)34. Rand Index (RI) was computed by: \({\mathrm{RI}} = \frac{{{a} + {b}}}{{{C}_2^{{n}_{{\mathrm{sample}}}}}}\), where a is the number of con-concordant sample pairs in obtained clusters C and reference classes K, whereas b is the number of dis-concordant samples.
V-measure, similar to F-measure, calculates the harmonic mean between homogeneity and completeness. Homogeneity is defined as \(1 - \frac{{H(C|K)}}{{H(C)}}\), where H(C|K) is the conditional entropy of C given K. Completeness is the symmetrical of homogeneity: \(1 - \frac{{H(K|C)}}{{H(K)}}\).
Graph visualization
The different datasets were transformed into GraphML files with Python scripts using iGraph library73. Graphs were visualized using GePhi software36 and spatialized using ForceAtlas274, a specific graph layout implemented into the GePhi software.
Pathway enrichment analysis
We used the KEGG pathway database to identify pathways related to specific genes75. We first selected the top 100 genes for each dataset, according to the ranks given by SSrGE. We then selected genes scored with significant eeSNVs for the metastasis cells from Kim dataset. We used DAVID 6.8 functional annotation tool to identify significant pathways amongst these genes38. We used the default significance value (adjusted p-value threshold of 0.10). Significant pathways are represented as a bipartite graph using Gephi: Nodes are either genes or pathway and the size of each nodes represent the score of the genes or, in the case of pathways, the sum of the scores of the genes linked to the pathways. We used the same methodology to infer significant pathways of cancer cells, compared to normal cells, from Hou dataset. However, we used all the genes ranked rather than only the significant genes, since only few genes are found to be significant for cancer cells.
Code availability
The SNV calling pipeline and SSrGE are available through the following GitHub project: https://github.com/lanagarmire/SSrGE.
The datasets used in this study are all publicly available and are accessible from the NCBI Gene Expression Omnibus (GEO) portal. Notably, the six main datasets have the following GEO accession numbers: GSE73121, GSE57872, GSE67980, GSE51372, GSE65364, and GSE75688. In addition, we used seven other datasets from the NCBI GEO portal with the following accession numbers: GSE79457, GSE69405, GSE81383, GSE81903, GSE80232, GSE65253, GSE71485. Moreover, we used three public datasets downloaded from 10X Genomics and available at the following urls: https://support.10xgenomics.com/single-cell-gene-expression/datasets/1.1.0/frozen_bmmc_healthy_donor1, https://support.10xgenomics.com/single-cell-gene-expression/datasets/1.1.0/frozen_bmmc_healthy_donor2, and http://cf.10xgenomics.com/samples/cell-exp/1.1.0/frozen_pbmc_b_c_50_50/frozen_pbmc_b_c_50_50_web_summary.html. Finally, all the data generated for this study, such as gene expression/eeSNV matrices, are available from the corresponding author on reasonable request.
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This research was supported by grants K01ES025434 awarded by NIEHS through funds provided by the trans-NIH Big Data to Knowledge (BD2K) initiative (www.bd2k.nih.gov), P20 COBRE GM103457 awarded by NIH/NIGMS, R01 LM012373 awarded by NLM, R01 HD084633 awarded by NICHD and Hawaii Community Foundation Medical Research Grant 14ADVC-64566 to L.X. Garmire. We acknowledge K. Chaudhary and John Breck Yunits for manuscript proofreading.
Epidemiology Program, University of Hawaii Cancer Center, Honolulu, HI, 96813, USA
Olivier Poirion, Xun Zhu & Travers Ching
Molecular Biosciences and Bioengineering Graduate Program, University of Hawaii at Manoa, Honolulu, HI, 96822, USA
Xun Zhu & Travers Ching
Department of Computational Medicine and Bioinformatics, Building 520, 1600 Huron Parkway, Ann Arbor, MI, 48109, USA
Lana X. Garmire
Olivier Poirion
Xun Zhu
Travers Ching
L.G. envisioned this project. O.P. implemented the project and conducted genomics analysis, X.Z. and T.C. helped on implementation. O.P. and L.G. wrote the manuscript. All authors have read and agreed on the manuscript.
Correspondence to Lana X. Garmire.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Description of Additional Supplementary Files
Poirion, O., Zhu, X., Ching, T. et al. Using single nucleotide variations in single-cell RNA-seq to identify subpopulations and genotype-phenotype linkage. Nat Commun 9, 4892 (2018). https://doi.org/10.1038/s41467-018-07170-5
Received: 02 March 2017
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doi: 10.3934/ipi.2021072
Euler equations and trace properties of minimizers of a functional for motion compensated inpainting
Riccardo March 1, and Giuseppe Riey 2,,
Istituto per le Applicazioni del Calcolo, CNR, Via dei Taurini 19, 00185 Roma, Italy
Dipartimento di Matematica e Informatica - Università della Calabria, Via Pietro Bucci, 31B, 87036 Rende (CS), Italy
*Corresponding author: Giuseppe Riey
Received October 2020 Revised September 2021 Early access December 2021
Fund Project: G.R. has been supported by the Italian PRIN Research Project 2017 "Qualitative and quantitative aspects of nonlinear PDE"
We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [17] as the relaxation of a modified version of the functional proposed in [16]. The functional is defined on vectorial functions of bounded variations, therefore we also get the Euler equations holding on the singular sets of minimizers, highlighting in particular the conditions on the jump sets. Such conditions are expressed by means of traces of geometrically meaningful vector fields and characterized as pointwise limits of averages on cylinders with axes parallel to the unit normals to the jump sets.
Keywords: Calculus of variations, Euler equations, Functions of bounded variations, Video inpainting, Optical flow estimation.
Mathematics Subject Classification: Primary: 49Q20, 49K20, 68T45; Secondary: 26B30.
Citation: Riccardo March, Giuseppe Riey. Euler equations and trace properties of minimizers of a functional for motion compensated inpainting. Inverse Problems & Imaging, doi: 10.3934/ipi.2021072
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Riccardo March Giuseppe Riey | CommonCrawl |
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Complexity of generating non-uniform random variates
What can we say about the complexity of generating (negative) binomial and (negative) hypergeometric random variates? In particular, it is possible to generate (negative) binomial and (negative) hypergeometric variates in (expected) constant time (i.e. independent of the distribution parameters)?
There is quite a bunch of literature; however, it's hard to understand a lot of the papers. Moreover, I found some statements that seem contradictory to me (probably due to wrong understanding).
For example, Stadlober (or similar here) mentions a "generalization [of the ratio-of-uniforms approach] to any unimodal discrete distribution". The ratio-of-uniforms approach has been called uniformly fast, which is a synonym for constant time, I suppose (?). That would mean that we can generate random variates for each discrete distribution in (expected) constant time.
However, in another paper, I found the following theorem: On a RAM with word size $w$, any algorithm sampling a geometric random variate $Geo(p)$ with parameter $p \in (0,1)$ needs at least expected runtime $\Omega(1 + \log(1/p)/w)$.
Apparently, this means that it is not possible to generate negative binomial variates in time independent of the distribution parameters.
time-complexity sampling random-number-generator
$\begingroup$ It's very unlikely that "uniformly fast" means "constant time". "Constant time" is such a widely understood and widely known phrase that nobody's going to invent a new name for it (either because they don't know that there's already a standard name or because they think there's something wrong with the standard name). Another indication that it doesn't mean "constant time" is that you've found some papers that claim that it can be done "uniformly fast" and some that say it can't be done in constant time! $\endgroup$
$\begingroup$ @DavidRicherby From the first paper/link: "The acceptance/rejection approach of Von Neumann leads to uniformly fast algorithms, i.e., algorithms with bounded execution time over the defined parameter range." I'm not quite sure how to interpret this sentence (that's why I wrote " I suppose"), but to me it seems like the running time is bounded (i.e. does not grow infinitely, i.e. is constant in O-notation) no matter what distribution parameters we choose. $\endgroup$
$\begingroup$ @DavidRicherby By now, I'm relatively sure that uniformly fast means constant time, since Lemma 4.11 in nrbook.com/devroye/Devroye_files/chapter_ten.pdf states that an algorithm is uniformly fast, and in the proof they show O(1). $\endgroup$
$\begingroup$ @DavidRicherby And actually, the second paper mentioned in the question does allow constant time variate generation, as long as the word size is logarithmic in the mean of the distribution. I guess it's a common assumption in algorithm analysis that all the numbers "we deal with" fit into a single word, isn't it? $\endgroup$
$\begingroup$ OK -- seems very strange to make up a new term for "constant time" but you've read the papers and I haven't! And, yes, numbers fitting in a word is a common assumption, e.g., in the word RAM model as you mention. $\endgroup$
The binomial, negative binomial, and hypergeometric distributions are discrete distributions. Knuth and Yao (1976) gave complexity results for discrete distributions in general. Given a stream of i.i.d. unbiased random bits, any algorithm that samples from a discrete distribution will require at least—$$b = -\sum_{x\in\Omega} \mathbb{P}(X=x)*\log_2(\mathbb{P}(X=x))$$random bits on average to sample a variate $X$ from that distribution (where $\Omega$ is the distribution's support), and they gave an algorithm that comes within 2 bits of this lower bound (and showed that coming within 2 bits is the best possible for any algorithm).
The case of continuous distributions such as the normal distribution is more complicated, but Devroye and Gravel gave complexity results for sampling a continuous variate to a desired accuracy, given a stream of i.i.d. unbiased random bits, using Wasserstein distance as the accuracy metric.
I also want to note one more thing. If the distribution of $X$ has a continuous and bounded quantile function (inverse cumulative distribution function) $F^{-1}$, I suspect that the bit complexity of sampling $X$ also depends on the modulus of continuity of $F^{-1}$, call it $\omega(h)$. In this sense, Lipschitz continuous quantile functions (where $\omega(h) = O(h)$) are the simplest cases of such functions and in that case it's simple to describe an algorithm to sample $X$ to a desired accuracy. For such functions, the algorithm uses—$$ceil(\ln(\max(1, L))-\ln(\epsilon))/\ln(\beta))$$random digits, where $L$ is the Lipschitz constant, $\epsilon$ is the desired accuracy, and $\beta$ is the digit base (such as 2 for binary). I suspect that with other moduli of continuity, the complexity will be as follows: $$ceil(-\ln(\omega^{-1}(\epsilon))/\ln(\beta)).$$
Knuth, Donald E. And Andrew Chi-Chih Yao. "The complexity of nonuniform random number generation", in Algorithms and Complexity: New Directions and Recent Results, 1976.
Devroye, L., Gravel, C., "Random variate generation using only finitely many unbiased, independently and identically distributed random bits", arXiv:1502.02539v6 [cs.IT], 2020.
Peter O.Peter O.
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Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4
Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.
Click here if solved 157
An Example of a Matrix that Cannot Be a Commutator
Let $I$ be the $2\times 2$ identity matrix.
Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.
7 Problems on Skew-Symmetric Matrices
Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.
(a) Prove that $A+B$ is skew-symmetric.
(b) Prove that $cA$ is skew-symmetric for any scalar $c$.
(c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric.
(d) Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix.
(e) Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix.
(f) Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$.
(g) Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$.
Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
\[\mathbf{v}_1=\begin{bmatrix}
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
a \\
\end{bmatrix}\] be vectors in $\R^3$.
Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.
Two Matrices are Nonsingular if and only if the Product is Nonsingular
An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.
(a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.
(b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:
The matrix $B$ is nonsingular.
The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)
A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors
Let $A$ be a singular $n\times n$ matrix.
\[\mathbf{e}_1=\begin{bmatrix}
\vdots \\
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}
\end{bmatrix}\] be unit vectors in $\R^n$.
Prove that at least one of the following matrix equations
\[A\mathbf{x}=\mathbf{e}_i\] for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.
The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$.
Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.
Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.
Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$
For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.
(a) $A=\begin{bmatrix}
2& 1
\end{bmatrix}$.
(b) $A=\begin{bmatrix}
The Transpose of a Nonsingular Matrix is Nonsingular
Let $A$ be an $n\times n$ nonsingular matrix.
Prove that the transpose matrix $A^{\trans}$ is also nonsingular.
If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$
Let $N$ be a normal subgroup of a group $G$.
Suppose that $G/N$ is an infinite cyclic group.
Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.
Construction of a Symmetric Matrix whose Inverse Matrix is Itself
Let $\mathbf{v}$ be a nonzero vector in $\R^n$.
Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$.
Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by
\[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where $I$ is the $n\times n$ identity matrix.
Prove that $A$ is a symmetric matrix and $AA=I$.
Conclude that the inverse matrix is $A^{-1}=A$.
The Range and Null Space of the Zero Transformation of Vector Spaces
Let $U$ and $V$ be vector spaces over a scalar field $\F$.
Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$.
(a) Prove that $T:U\to V$ is a linear transformation.
(Hence, $T$ is called the zero transformation.)
(b) Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.
If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial
Let $x, y$ be generators of a group $G$ with relation
xy^2=y^3x,\tag{1}\\
yx^2=x^3y.\tag{2}
Prove that $G$ is the trivial group.
Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism
Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula
\end{bmatrix} \,\right)=\begin{bmatrix}
x_1+3x_2-2x_3 \\
2x_1+3x_2 \\
x_2-x_3
Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.
Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
1 & 3 & -2 \\
\end{bmatrix}$
-1 &-3 &2 \\
The Sum of Cosine Squared in an Inner Product Space
Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$.
Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$.
Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$.
Prove that
\[\cos ^2\theta_1+\cdots+\cos^2 \theta_n=1.\]
Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors
Consider the $2\times 2$ matrix
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$.
(a) Find the characteristic polynomial of the matrix $A$.
(b) Find the eigenvalues of the matrix $A$.
(c) Determine the eigenvectors corresponding to each of the eigenvalues of $A$.
Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent
By calculating the Wronskian, determine whether the set of exponential functions
\[\{e^x, e^{2x}, e^{3x}\}\] is linearly independent on the interval $[-1, 1]$.
A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$
An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that
$AB=I$, and
$BA=I$,
where $I$ is the $n\times n$ identity matrix.
If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted by $A^{-1}$.
In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition.
So if we know $AB=I$, then we can conclude that $B=A^{-1}$.
Let $A$ and $B$ be $n\times n$ matrices.
Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix.
Prove that $BA=I$, and hence $A^{-1}=B$.
Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$
Let $A$ be an $n\times n$ nonsingular matrix with integer entries.
Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.
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Basics of network science with Networkx - 1
First instalment in the exploratory data analysis of graphs
By Vamshi Jandhyala in data science python
Definition of a network
A network $G$ has two parts, a set of $N$ elements, called nodes or vertices, and a set of $L$ pairs of nodes, called links or edges. The link $(i,j)$ joins the nodes $i$ and $j$. A network can be undirected or directed. A directed network is also called a digraph. In directed networks, links are called directed links and the order of the nodes in a link represents the direction: the link $(i,j)$ goes from the source node $i$ to the target node $j$. In undirected networks, all links are bi-directional and the order of the two nodes in a link does not matter. A network can be unweighted or weighted. In a weighted network, links have associated weights: the weighted link $(i,j,w)$ between nodes $i$ and $j$ has weight $w$. A network can be both directed and weighted, in which case it has directed weighted links.
Each network is characterized by the total number of nodes $N$ and the total number of links $L$. We call $N$ the size of the network because it identifies the number of distinct elements composing the system.
There are several other classes of networks. A network might have more than one type of node. In a bipartite network, there are two groups of nodes such that links only connect nodes from different groups and not nodes from the same group. A network might have multiple types of links, in which case it is called a multiplex network.
Subnetworks
A subnetwork (or subgraph) is obtained by selecting a subset of the nodes and all of the links among these nodes. The abundance of certain types of subnetworks and their properties is important in the characterization of real networks. As an example, a clique is a complete subnetwork: a subset of nodes all linked to each other. A special type of subnetwork is the ego network of a node, which is the subnetwork consisting of the chosen node — called the ego — and its neighbors.
Network Representations
Adjacency Matrix
The adjacency matrix is an $N\times N$ matrix in which each element represents the link between the nodes indexed by the corresponding row and column. Element $a_{ij}$ of the adjacency matrix represents the link between nodes $i$ and $j$. $a_{ij}=1$ if $i$ and $j$ are adjacent, $a_{ij}=0$ otherwise.
Adjacency List
Adjacency matrix representation is not efficient for storing real networks, which are typically large and sparse. The required storage space grows like the square of the network size, but if the network is sparse, most of this space is wasted storing zeros (non-existing links). With large sparse networks, a more compact network representation is the adjacency list, a data structure that stores the list of neighbors for each node. Adjacency lists represent sparse networks efficiently because the non-existing links are ignored; only the existing links (non-zero values of the adjacency matrix) are considered.
We only store the list of edges in the graph.
Density and Sparsity
The maximum number of links in a complete undirected network = ${N \choose 2}$ = $N(N-1)/2$. The maximum number of links in a complete directed network = $N(N-1)$. The maximum number of links in a complete bi-partite network = $N_1N_2$ where there are $N_1$ nodes of one colour and $N_2$ nodes of the second colour.
The fraction of possible links that actually exist, is called the density of the network. A complete network has maximal density: one. However, the actual number of links is typically much smaller than the maximum, as most pairs of nodes are not directly connected to each other. Therefore the density is often much, much smaller than one — by orders of magnitude in most real-world, large networks. This is called sparsity.
The density of a undirected network is given by $d = 2L/L_{max} = 2L/N(N-1)$. The density of a directed network is given by $d = L/L_{max} = L/N(N-1)$.
We say that the network is sparse if the number of links grows proportionally to the number of nodes ($L \sim N$), or even slower. If instead the number of links grows faster, e.g. quadratically with network size ($L \sim N^2$), then we say that the network is dense.
Undirected Networks
The degree of a node is its number of links, or neighbors. A node with no neighbours is called a singleton. The average degree of a network is denoted by $\overline{k}=\frac{\sum_ik_i}{N}$ where $k_i$ is the degree of node $i$. Since each link contributes to the degree of two nodes in an undirected network, we have $\sum_ik_i=2L$. From the definition of density for an undirected network we have
$$ \overline{k}=\frac{\sum_ik_i}{N} = \frac{2L}{N} = d(N-1) $$
The maximum possible degree of a node is $k_{max}=N-1$.
Directed Networks
When we consider the degree of a node in a directed network, we have to think of incoming and outgoing links separately. The number of incoming links, or predecessors, of node $i$ is called the in-degree and denoted by $k_i^{in}$. The number of outgoing links, or successors, of node $i$ is called the out-degree and denoted by $k_i^{out}$.
The average in-degree for a directed network is given by
$$ \overline{k^{in}}=\frac{\sum_ik^{in}_i}{N} $$
The average out-degress for a directed network can be similarly defined.
Weighted Networks
The weighted degree, or strength of a node in an undirected network, is the sum of the weights of its links. Similarly, we can define in-strength and out-strength in the case of a directed weighted network.
The weighted degree, or strength, of node $i$ in an undirected weighted network is denoted by
$$ s_i = \sum_j w_{ij} $$
where $w_{ij}$ is the weight of the link between nodes $i$ and $j$. We assume $w_{ij}=0$ if there is no link between $i$ and $j$. We can analogously generalize in-degree and out-degree to in-strength and out-strength in a directed weighted network:
$$ s^{in}i = \sum_j w{ij} \\
s^{out}i = \sum_j w{ij} $$
where $w_{ij}$ is the weight of the directed link from $i$ to $j$.
Homophily
Often, nodes that are connected to each other in a social network tend to be similar in their features: for example, relatives may live near each other, and friends may have similar interests. The technical name of this property is assortativity. Because of assortativity, we are able to make predictions about a person's qualities by inspecting their neighbors.
Multiple factors may be responsible for the presence of assortativity in social networks. One possibility is that if people are similar in some way, they are more likely to select each other and become connected. This property is captured by a popular proverb: "birds of a feather flock together." Its technical name is homophily.
The converse mechanism is when people who are friends become more similar over time, through the process of social influence. It is difficult to separate the causes of assortativity — does similarity induce links, or do links induce similarity?
Assortativity
Assortativity based on degree is called degree assortativity or degree correlation: this occurs when high-degree nodes tend to be connected to other high-degree nodes, while low-degree nodes tend to have other low-degree nodes as neighbors. Networks with this property are called assortative.
There are two ways to measure the degree assortativity of a network, both based on measuring the correlation between degrees of neighbor nodes. We say that two variables are positively (negatively) correlated if larger values of one variable tend to correspond to larger (smaller) values of the other. Pearson's correlation coefficient is a common way to measure correlation; it takes values in $[-1,1]$, with $0$ meaning no correlation and $\pm 1$ meaning perfect positive/negative correlation.
One measure of network assortativity is the assortativity coefficient, defined as the Pearson correlation between the degrees of pairs of linked nodes. When the assortativity coefficient is positive, the network is assortative, and when it is negative, the network is disassortative.
The second method is based on measuring the average degree of the neighbors of node $i$:
$$ k_nn(i) = \frac{1}{k_j} \sum_j a_{ij} $$
where $a_{ij}=1$ if $i$ and $j$ are neighbors, and $0$ otherwise. We then define the k-nearest-neighbors function $\overline{k_{nn}}(k)$ for nodes of a given degree $k$ as the average of $k_{nn}(i)$ across all nodes with degree $k$. If $\overline{k_{nn}}(k)$ is an increasing function of $k$, then high-degree nodes tend to be connected to high-degree nodes, therefore the network is assortative; if $\overline{k_{nn}}(k)$ decreases with $k$, the network is disassortative.
The properties of assortativity are useful in the field of epidemiology, since they can help understand the spread of disease or cures. For instance, the removal of a portion of a network's vertices may correspond to curing, vaccinating, or quarantining individuals or cells. Since social networks demonstrate assortative mixing, diseases targeting high degree individuals are likely to spread to other high degree nodes. Alternatively, within the cellular network—which, as a biological network is likely dissortative—vaccination strategies that specifically target the high degree vertices may quickly destroy the epidemic network.
Paths and Distances
The path is the sequence of links traversed. The number of links in a path is called the path length. There may be multiple paths between the same two nodes. These paths may have different lengths, and may or may not share some common links. In directed paths, we must comply with link directions. A cycle is a special path that can be traversed to go from one node back to itself. A simple path never goes through the same link more than once.
The natural distance measure between two nodes is defined as the minimum number of links that must be traversed in a path connecting the two nodes. Such a path is called the shortest path, and its length is called the shortest-path length. There may be multiple shortest paths between two nodes; obviously they all must have the same length.
In many networks, link weights express a measure of similarity or intensity of interaction between two connected nodes. We may then be interested in finding paths with large weights. A common approach is to transform the weights into distances by taking the inverse (one divided by the weight), so that a large weight corresponds to a short distance. Then the problem becomes equivalent to finding short-distance paths.
By using the shortest-path length as a measure of distance among nodes, it is possible to define aggregate distance measures for an entire network: the average shortest-path length (or simply average path length) is obtained by averaging the shortest-path lengths across all pairs of nodes. The diameter of the network is instead the maximum shortest-path length across all pairs of nodes (i.e. the length of the longest shortest path in the network).
The average path length of an undirected, unweighted network as
$$ \overline{l} = \frac{\sum_{i,j} l_{ij}}{N \choose 2} $$
where $l_{ij}$ is the shortest-path length between nodes $i$ and $j$, and $N$ is the number of nodes. The sum is over all pairs of nodes, and we divide by the number of pairs to compute the average.
In the directed network case the definition is analogous, but the distance $l_{ij}$ is based on the shortest directed path between $i$ and $j$, and each pair of nodes is considered twice for paths in both directions:
$$ \overline{l} = \frac{\sum_{i,j}l_{ij}}{N(N-1)} $$
The diameter of a network is defined as
$$ l_{max} = \max_{i,j} l_{ij} $$
Connectedness and Components
A network is connected if you can reach any node from any other node by following a path along links and intermediate nodes). If a network is not connected, we say that it is disconnected; it is composed of more than one connected components or simply components. A component is a subnetwork containing one or more nodes, such that there is a path connecting any pair of these nodes, but there is no path connecting them to other components. The largest connected component in many real networks includes a substantial portion of the network and is called the giant component. In a connected network, the giant component coincides with the entire graph.
To measure network distance when a network is disconnected. One way is to consider only the nodes in the giant component. Another approach is to average the distance only across pairs of nodes in the same component, but considering all components. To calculate the diameter of a disconnected network, one can calculate the diameter of each component and then take the maximum.
In the directed case, we have to pay attention to link directions when determining whether a node can be reached from another. We can of course ignore the link directions and treat the links as if they were undirected. In this case we refer to the components as weakly connected.
We say that a directed network is strongly connected if it is a single strongly connected component. A directed network is weakly connected if it is a single weakly connected component.
Trees are a special class of undirected, connected networks such that the deletion of any one link will disconnect the network into two components. The number of links in a tree is $N-1$. They have no cycles. Trees are hierarchical. You can pick any node in a tree and call it a root. Each node in a tree is connected to a parent node (toward the root) and to one or more children nodes (away from the root). The exceptions are the root, which has no parent, and the so-called leaves of the tree, which have no children.
The average path length, characterizes how close or far we expect nodes to be in a network. Intuitively, in a grid-like network like road networks and power grids, paths can be long. Is this typical of many real-world networks? Let us start by considering a few social networks, in which this question has been explored extensively.
It turns out that pretty much all social networks have very short paths among nodes - which gives rise to the popular notion of "small world" and "six degrees of separation". Consequently, the number of friends of friends out there is much, much larger than we think.
We say that the average path length is short when it grows very slowly with the size of the network. We can express slow growth mathematically by saying that the average path length scales logarithmically with the size of the network.
Short paths that obey this kind of relationship are found across social networks, including academic collaborations, actor networks, networks of high-school friends, and online social networks such as Facebook. As it turns out, short paths are a ubiquitous feature in almost all real-world networks; grid-like networks are among the few exceptions.
In a social network, if Alice and Bob are both friends of Charlie's, they are also likely to be friends of each other. In other words, there is a good chance that a friend of my friend is also my friend. This translates into the presence of many triangles in the network. The connectivity among the neighbors of the nodes is an important feature of the local structure of the network because it captures how tightly knit, or clustered, the nodes are.
The clustering coefficient of a node is the fraction of pairs of the node's neighbors that are connected to each other. This is the same as the ratio between the number of triangles that include the node, and the maximum number of triangles in which the node could participate.
The clustering coefficient of node $i$ is formally defined as
$$ C(i) = \frac{\tau(i)}{\tau_{max}(i)} = \frac{\tau(i)}{k_i \choose 2} = \frac{2\tau(i)}{k_i(k_i-1)} $$
where $\tau(i)$ is the number of triangles involving $i$. The maximum possible number of triangles for $i$ is the number of pairs formed by its $k_i$ neighbors. Note that $C(i)$ is only defined if the degree $k_i>1$ due to the terms $k_i$ and $k_{i-1}$ in the denominator: a node must have at least two neighbors for any triangle to be possible.
The clustering coefficient of the entire network is the average of the clustering coefficients of its nodes:
$$ C = \frac{\sum_{i;k_i>1}C(i)}{N_{k>1}} $$
Nodes with degree $k<2$ are excluded when calculating the average clustering coefficient.
A low clustering coefficient (near zero) means that the network has few triangles, while a high clustering coefficient (near one) means that the network has many triangles. Social networks have a large clustering coefficient; a significant portion of all possible triangles are present. A simple mechanism explains the abundance of triangles in social networks: we meet people through shared contacts, thus closing triangles. This mechanism, is called triadic closure. Online social networks make suggestions based on triadic closure. For example, Facebook recommends "people you may know" based on common friends, and Twitter recommends accounts followed by your friends (whose accounts you follow). These recommendations result in high clustering.
Triangle count and clustering coefficient have been shown to be useful as features for classifying a given website as spam, or non-spam, content. Clustering coefficient has been used to investigate the community structure of Facebook's social graph, where they found dense neighbourhoods of users in an otherwise sparse global graph. Clustering coefficient has been proposed to help explore thematic structure of the web, and detect communities of pages with a common topic based on the reciprocal links between them.
Python code using networkx
# Create Graphs
G = nx.Graph()
G.add_node(1)
G.add_edge(1,2)
G.add_nodes_from([3,4,5])
G.add_edges_from([(2,3), (3,4),(3,5)])
nx.draw(G,with_labels=True)
# List nodes, links and neighbors of a given node
print(G.number_of_nodes())
print(G.number_of_edges())
print(G.nodes())
print(G.edges())
print(list(G.neighbors(3)))
# Subgraphs
nx.draw(nx.subgraph(G, (2,3,4,5)), with_labels=True)
# Density
print(nx.density(G))
# Degree
print(G.degree(2)) # returns the degree of node 2
print(G.degree()) # returns the degree of all nodes of G
# Network representation
## Adjacency Matrix
print("graph\n",nx.adjacency_matrix(G)) # graph
## Adjacency List
for n,neighbors in G.adjacency():
for number,link_attributes in neighbors.items():
print('(%d, %d)' % (n,number))
## Edge List
nx.write_edgelist(G, "file.edges")
G2 = nx.read_edgelist("file.edges") # G2 same as G
# Assortativity
assort_a = nx.attribute_assortativity_coefficient(G, category)
assort_n = nx.numeric_assortativity_coefficient(G, quantity)
import scipy.stats
knn_dict = nx.k_nearest_neighbors(G)
k, knn = list(knn_dict.keys()), list(knn_dict.values())
r, p_value = scipy.stats.pearsonr(k, knn)
r,p_value
# Paths and distances
print("Path between 1 and 2 exists? ", nx.has_path(G, 1, 2))
print("Shortest path between 1 and 5 ", nx.shortest_path(G, 1, 5))
print("Shortest path length between 1 and 5 ", nx.shortest_path_length(G,1,5))
print("Shortest path from 1 to other nodes ",nx.shortest_path(G, 1))
print("Shortest path lengths from 1 to other nodes ",nx.shortest_path_length(G, 1))
print("Shortest paths for all pairs of nodes ",nx.shortest_path(G)) # all pairs
print("Shortest path lengths for all pairs of nodes ",list(nx.shortest_path_length(G))) # all pairs
print("Average shortest path length", nx.average_shortest_path_length(G))
# Connectedness and components
nx.is_connected(G)
comps = sorted(nx.connected_components(G), key=len, reverse=True)
nodes_in_giant_comp = comps[0]
GC = nx.subgraph(G, nodes_in_giant_comp)
nx.is_connected(GC)
nx.is_strongly_connected(D)
nx.is_weakly_connected(D)
list(nx.weakly_connected_components(D))
list(nx.strongly_connected_components(D))
# Trees
nx.is_tree(S)
# Clustering
nx.triangles(G) # dict node -> no. triangles
nx.clustering(G, 2) # clustering coefficient of node
nx.clustering(G) # dict node -> clustering coeff.
nx.average_clustering(G) # network's clustering coeff.
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Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics
General initial data for a class of parabolic equations including the curve shortening problem
May 2020, 40(5): 2987-3011. doi: 10.3934/dcds.2020158
Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data
Keyan Wang 1,, and Yao Xiao 2,
School of Mathematics,, Shanghai University of Finance and Economics, Shanghai 200433, China
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
* Corresponding author: Keyan Wang
Fund Project: The first author is supported by NSFC grant No. 11971290.
In this paper, we prove that for the ill-prepared initial data of the form
$ \begin{equation} \nonumber u_0^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^3(x_\epsilon))^T,\quad x_\epsilon = (x_h, \epsilon x_3)^T, \end{equation} $
the Cauchy problem of the incompressible Navier-Stokes equations on
$ \mathbb{R}^3 $
is locally well-posed for all
$ \epsilon > 0 $
, provided that the initial velocity profile
$ v_0 $
is analytic in
$ x_3 $
but independent of
$ \epsilon $
Keywords: Local well-posedness, Navier-Stokes equations, ill-prepared.
Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 35Q30.
Citation: Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158
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HTML views (72)
Keyan Wang Yao Xiao | CommonCrawl |
Kuz'min, Aleksei Sergeevich
Total publications: 26
Scientific articles: 23
This page: 2232
Abstract pages: 9944
Full texts: 3978
References: 452
Doctor of physico-mathematical sciences
http://www.mathnet.ru/eng/person8967
https://mathscinet.ams.org/mathscinet/MRAuthorID/334316
1. A. S. Kuzmin, V. T. Markov, A. A. Mikhalev, A. V. Mikhalev, A. A. Nechaev, "Cryptographic algorithms on groups and algebras", Fundam. Prikl. Mat., 20:1 (2015), 205–222 ; J. Math. Sci., 223:5 (2017), 629–641
2. A. S. Kuzmin, V. I. Nozdrunov, "Relationship between the coefficients of polynomials over $\mathrm{GF}(2^n)$ and weights of Boolean functions represented by them", Prikl. Diskr. Mat., 2014, 4(26), 28–46
3. A. S. Kuzmin, A. A. Nechaev, "Reconstruction of a linear recurrence of maximal period over a Galois ring from its highest coordinate sequence", Diskr. Mat., 23:2 (2011), 3–31 ; Discrete Math. Appl., 21:2 (2011), 145–178
4. A. S. Kuzmin, G. B. Marshalko, "Reconstruction of linear recurrent sequence over prime residue ring from its image. II", Mat. Vopr. Kriptogr., 2:2 (2011), 81–93
5. A. S. Kuz'min, G. B. Marshalko, A. A. Nechaev, "Reconstruction of linear recurrent sequence over prime residue ring from its image", Mat. Vopr. Kriptogr., 1:2 (2010), 31–56
6. A. S. Kuz'min, V. T. Markov, A. A. Nechaev, V. A. Shishkin, A. B. Shishkov, "Bent and Hyper-bent Functions over a Field of $2^l$ Elements", Probl. Peredachi Inf., 44:1 (2008), 15–37 ; Problems Inform. Transmission, 44:1 (2008), 12–33
7. A. S. Kuz'min, A. A. Nechaev, V. A. Shishkin, "Parameters of (hyper-)bent functions over the field of $2^l$ elements", Tr. Diskr. Mat., 11:1 (2008), 47–59
8. A. S. Kuz'min, A. A. Nechaev, V. A. Shishkin, "Bent and hyper-bent functions over the finite field", Tr. Diskr. Mat., 10 (2007), 97–122
9. A. S. Kuz'min, V. T. Markov, A. A. Nechaev, A. B. Shishkov, "Approximation of Boolean functions by monomial functions", Diskr. Mat., 18:1 (2006), 9–29 ; Discrete Math. Appl., 16:1 (2006), 7–28
10. A. S. Kuz'min, V. L. Kurakin, A. A. Nechaev, "Almost uniform linear recurrent sequences over Galois rings and $QF$-modules of characteristic 4", Tr. Diskr. Mat., 5 (2002), 103–158
11. A. S. Kuz'min, V. L. Kurakin, A. A. Nechaev, "Structural properties of linear recurrent sequences over Galois rings and quasi-Frobenius modules of charasteristic 4", Tr. Diskr. Mat., 4 (2001), 91–128
12. O. V. Kamlovskii, A. S. Kuz'min, "Bounds for the number of occurrences of elements in a linear recurring sequence over a Galois ring", Fundam. Prikl. Mat., 6:4 (2000), 1083–1094
13. A. A. Nechaev, A. S. Kuz'min, V. L. Kurakin, "Structural, analitical and statistical properties of linear and polylinear recurrent sequences", Tr. Diskr. Mat., 3 (2000), 155–194
14. O. V. Kamlovskii, A. S. Kuz'min, "Distribution of elements on cycles of linear recurrent sequences over Galois rings", Uspekhi Mat. Nauk, 53:2(320) (1998), 149–150 ; Russian Math. Surveys, 53:2 (1998), 392–393
15. A. S. Kuz'min, V. L. Kurakin, A. A. Nechaev, "Properties of linear and polylinear recurrent sequences over Galois rings. I", Tr. Diskr. Mat., 2 (1998), 191–222
16. A. A. Nechaev, A. S. Kuz'min, V. T. Markov, "Linear codes over finite rings and modules", Fundam. Prikl. Mat., 3:1 (1997), 195–254
17. A. S. Kuz'min, V. L. Kurakin, A. A. Nechaev, "Pseudorandom and polylinear sequences", Tr. Diskr. Mat., 1 (1997), 139–202
18. A. S. Kuz'min, "Polynomials of maximal period over primary residue rings", Fundam. Prikl. Mat., 1:2 (1995), 549–551
19. A. S. Kuz'min, A. A. Nechaev, "Linearly representable codes and the Kerdock code over an arbitrary Galois field of characteristic 2", Uspekhi Mat. Nauk, 49:5(299) (1994), 165–166 ; Russian Math. Surveys, 49:5 (1994), 183–184
20. A. S. Kuz'min, "Lower estimates for the ranks of coordinate sequences of linear recurrent sequences over primary residue rings of integers", Uspekhi Mat. Nauk, 48:3(291) (1993), 193–194 ; Russian Math. Surveys, 48:3 (1993), 203–204
21. A. S. Kuz'min, A. A. Nechaev, "Linear recursive sequences over Galois rings", Uspekhi Mat. Nauk, 48:1(289) (1993), 167–168 ; Russian Math. Surveys, 48:1 (1993), 171–172
22. A. S. Kuz'min, "The distribution of elements on cycles of linear recurrents over rings of residues", Uspekhi Mat. Nauk, 47:6(288) (1992), 213–214 ; Russian Math. Surveys, 47:6 (1992), 219–221
23. A. S. Kuz'min, A. A. Nechaev, "Construction of noise-resistant codes by means of linear recurrences over Galois rings", Uspekhi Mat. Nauk, 47:5(287) (1992), 183–184 ; Russian Math. Surveys, 47:5 (1992), 189–190
24. M. M. Glukhov, M. V. Zaicev, A. S. Kuzmin, V. N. Latyshev, V. T. Markov, A. V. Mikhalev, A. A. Mikhalev, I. A. Chubarov, "Alexandr Alexandrovich Nechaev (7.8.1945–14.11.2014)", Fundam. Prikl. Mat., 20:1 (2015), 3–7 ; J. Math. Sci., 223:5 (2017), 495–497
25. A. S. Kuz'min, "Information on the Workshop CTCrypt-2014", Mat. Vopr. Kriptogr., 6:2 (2015), 5
26. S. M. Buravlev, M. M. Glukhov, A. M. Zubkov, A. P. Kovalenko, A. S. Kuzmin, V. N. Latyshev, A. I. Markov, V. N. Markov, Yu. I. Medvedev, V. G. Mikhailov, A. V. Mikhalev, A. A. Nechaev, B. A. Pogorelov, Yu. V. Prokhorov, V. N. Sachkov, "V. L. Kurakin (13.05.1966 – 14.08.2010)", Mat. Vopr. Kriptogr., 2:3 (2011), 111–112
Academy of Cryptography of Russian Federation
Moscow State Technical University of Radioengineering, Electronics and Automation
Centre for New Information Technologies, Lomonosov Moscow State University | CommonCrawl |
Earth and Environmental Sciences (128)
flm;turbulent flows;turbulent convection
Journal of Fluid Mechanics (128)
Ryan Test (128)
Turbulent temperature fluctuations in a closed Rayleigh–Bénard convection cell
Yin Wang, Xiaozhou He, Penger Tong
We report a systematic study of spatial variations of the probability density function (PDF) $P(\unicode[STIX]{x1D6FF}T)$ for temperature fluctuations $\unicode[STIX]{x1D6FF}T$ in turbulent Rayleigh–Bénard convection along the central axis of two different convection cells. One of the convection cells is a vertical thin disk and the other is an upright cylinder of aspect ratio unity. By changing the distance $z$ away from the bottom conducting plate, we find the functional form of the measured $P(\unicode[STIX]{x1D6FF}T)$ in both cells evolves continuously with distinct changes in four different flow regions, namely, the thermal boundary layer, mixing zone, turbulent bulk region and cell centre. By assuming temperature fluctuations in different flow regions are all made from two independent sources, namely, a homogeneous (turbulent) background which obeys Gaussian statistics and non-uniform thermal plumes with an exponential distribution, we obtain the analytic expressions of $P(\unicode[STIX]{x1D6FF}T)$ in four different flow regions, which are found to be in good agreement with the experimental results. Our work thus provides a unique theoretical framework with a common set of parameters to quantitatively describe the effect of turbulent background, thermal plumes and their spatio-temporal intermittency on the temperature PDF $P(\unicode[STIX]{x1D6FF}T)$ .
Convective heat transfer along ratchet surfaces in vertical natural convection
Hechuan Jiang, Xiaojue Zhu, Varghese Mathai, Xianjun Yang, Roberto Verzicco, Detlef Lohse, Chao Sun
We report on a combined experimental and numerical study of convective heat transfer along ratchet surfaces in vertical natural convection (VC). Due to the asymmetry of the convection system caused by the asymmetric ratchet-like wall roughness, two distinct states exist, with markedly different orientations of the large-scale circulation roll (LSCR) and different heat transport efficiencies. Statistical analysis shows that the heat transport efficiency depends on the strength of the LSCR. When a large-scale wind flows along the ratchets in the direction of their smaller slopes, the convection roll is stronger and the heat transport is larger than the case in which the large-scale wind is directed towards the steeper slope side of the ratchets. Further analysis of the time-averaged temperature profiles indicates that the stronger LSCR in the former case triggers the formation of a secondary vortex inside the roughness cavity, which promotes fluid mixing and results in a higher heat transport efficiency. Remarkably, this result differs from classical Rayleigh–Bénard convection (RBC) with asymmetric ratchets (Jiang et al., Phys. Rev. Lett., vol. 120, 2018, 044501), wherein the heat transfer is stronger when the large-scale wind faces the steeper side of the ratchets. We reveal that the reason for the reversed trend for VC as compared to RBC is that the flow is less turbulent in VC at the same $Ra$ . Thus, in VC the heat transport is driven primarily by the coherent LSCR, while in RBC the ejected thermal plumes aided by gravity are the essential carrier of heat. The present work provides opportunities for control of heat transport in engineering and geophysical flows.
Control of light gas releases in ventilated tunnels
L. Jiang, M. Creyssels, G. R. Hunt, P. Salizzoni
The release of buoyant harmful gases within enclosed spaces, such as tunnels and corridors, may engender specific health, industrial and transportation risks. For safety, a simple ventilation strategy for these spaces is to impose a flow along the tunnel, whose velocity is defined as 'critical', that confines the front of harmful buoyant gases immediately downstream of the source of emission. Determining the critical velocity as a function of the geometrical and dynamical conditions at the source is a fundamental fluid mechanics problem which has yet to be elucidated; this problem concerns the dynamics of non-Boussinesq releases relating to large differences between the densities of the buoyant and the ambient fluids. We have investigated this problem theoretically, by means of a simplified model of a top-hat plume in a cross-flow, and in complementary experiments by means of tests in a reduced-scale ventilated tunnel, examining releases from circular sources. Experimental results reveal: (i) the existence of two flow regimes depending on the plume Richardson number at the source $\unicode[STIX]{x1D6E4}_{i}$ , one for momentum-dominated releases, $\unicode[STIX]{x1D6E4}_{i}\ll 1$ , and a second for buoyancy-dominated releases, $\unicode[STIX]{x1D6E4}_{i}\gg 1$ , with a smooth transition between the two; and (ii) the presence of relevant non-Boussinesq effects only for momentum-dominated releases. All these features can be conveniently predicted by the plume-based model, whose validity is, strictly speaking, limited to releases issuing from 'small' sources in 'weak' ventilation flows. Analytical solutions of the model are generally in good agreement with the experimental data, even for values of the governing parameters that are beyond the range of validity for the model. The solutions aid to clarify the effect of the source radius, and reveal interesting behaviours in the limits $\unicode[STIX]{x1D6E4}_{i}\rightarrow 0$ and $\unicode[STIX]{x1D6E4}_{i}\rightarrow \infty$ . These findings support the adoption of simplified models to simulate light gas releases in confined ventilated spaces.
Pseudophase change effects in turbulent channel flow under transcritical temperature conditions
Kukjin Kim, Jean-Pierre Hickey, Carlo Scalo
Journal: Journal of Fluid Mechanics / Volume 871 / 25 July 2019
We have performed direct numerical simulations of compressible turbulent channel flow using R-134a as a working fluid in transcritical temperature ranges ( $\unicode[STIX]{x0394}T=5$ , 10 and 20 K, where $\unicode[STIX]{x0394}T$ is top-to-bottom temperature difference) at supercritical pressure. At these conditions, a pseudophase change occurs at various wall-normal locations within the turbulent channel from $y_{pb}/h=-0.23$ ( $\unicode[STIX]{x0394}T=5$ K) to 0.89 ( $\unicode[STIX]{x0394}T=20$ K), where $h$ is the channel half-height and $y=0$ the centreplane position. Increase in $\unicode[STIX]{x0394}T$ also results in increasing wall-normal gradients in the semi-local friction Reynolds number. Classical, compressible scaling laws of the mean velocity profile are unable to fully collapse real fluid effects in this flow. The proximity to the pseudotransitioning layer inhibits turbulent velocity fluctuations, while locally enhancing the temperature and density fluctuation intensities. Probability distribution analysis reveals that the sheet of fluid undergoing pseudophase change is characterized by a dramatic reduction in the kurtosis of density fluctuations, hence becoming thinner as $\unicode[STIX]{x0394}T$ is increased. Instantaneous visualizations show dense fluid ejections from the pseudoliquid viscous sublayer, some reaching the channel core, causing positive values of density skewness in the respective buffer layer region (vice versa for the top wall) and an impoverishment of the turbulent flow structure population near pseudotransitioning conditions.
Turbulent Rayleigh–Bénard convection in an annular cell
Xu Zhu, Lin-Feng Jiang, Quan Zhou, Chao Sun
Journal: Journal of Fluid Mechanics / Volume 869 / 25 June 2019
Published online by Cambridge University Press: 29 April 2019, R5
Print publication: 25 June 2019
We report an experimental study of turbulent Rayleigh–Bénard (RB) convection in an annular cell of water (Prandtl number $Pr=4.3$ ) with a radius ratio $\unicode[STIX]{x1D702}\simeq 0.5$ . Global quantities, such as the Nusselt number $Nu$ and the Reynolds number $Re$ , and local temperatures were measured over the Rayleigh range $4.2\times 10^{9}\leqslant Ra\leqslant 4.5\times 10^{10}$ . It is found that the scaling behaviours of $Nu(Ra)$ , $Re(Ra)$ and the temperature fluctuations remain the same as those in the traditional cylindrical cells; both the global and local properties of turbulent RB convection are insensitive to the change of cell geometry. A visualization study, as well as local temperature measurements, shows that in spite of the lack of the cylindrical core, there also exists a large-scale circulation (LSC) in the annular system: thermal plumes organize themselves with the ascending hot plumes on one side and the descending cold plumes on the opposite side. Near the upper and lower plates, the mean flow moves along the two circular branches. Our results further reveal that the dynamics of the LSC in this annular geometry is different from that in the traditional cylindrical cell, i.e. the orientation of the LSC oscillates in a narrow azimuthal angle range, and no cessations, reversals or net rotation were detected.
$Nu\sim Ra^{1/2}$ scaling enabled by multiscale wall roughness in Rayleigh–Bénard turbulence
Xiaojue Zhu, Richard J. A. M. Stevens, Olga Shishkina, Roberto Verzicco, Detlef Lohse
In turbulent Rayleigh–Bénard (RB) convection with regular, mono-scale, surface roughness, the scaling exponent $\unicode[STIX]{x1D6FD}$ in the relationship between the Nusselt number $Nu$ and the Rayleigh number $Ra$ , $Nu\sim Ra^{\unicode[STIX]{x1D6FD}}$ can be ${\approx}1/2$ locally, provided that $Ra$ is large enough to ensure that the thermal boundary layer thickness $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ is comparable to the roughness height. However, at even larger $Ra$ , $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ becomes thin enough to follow the irregular surface and $\unicode[STIX]{x1D6FD}$ saturates back to the value for smooth walls (Zhu et al., Phys. Rev. Lett., vol. 119, 2017, 154501). In this paper, we prevent this saturation by employing multiscale roughness. We perform direct numerical simulations of two-dimensional RB convection using an immersed boundary method to capture the rough plates. We find that, for rough boundaries that contain three distinct length scales, a scaling exponent of $\unicode[STIX]{x1D6FD}=0.49\pm 0.02$ can be sustained for at least three decades of $Ra$ . The physical reason is that the threshold $Ra$ at which the scaling exponent $\unicode[STIX]{x1D6FD}$ saturates back to the smooth wall value is pushed to larger $Ra$ , when the smaller roughness elements fully protrude through the thermal boundary layer. The multiscale roughness employed here may better resemble the irregular surfaces that are encountered in geophysical flows and in some industrial applications.
Thermal forcing and 'classical' and 'ultimate' regimes of Rayleigh–Bénard convection
Charles R. Doering
The fundamental challenge to characterize and quantify thermal transport in the strongly nonlinear regime of Rayleigh–Bénard convection – the buoyancy-driven flow of a horizontal layer of fluid heated from below – has perplexed the fluid dynamics community for decades. Rayleigh proposed controlling the temperature of thermally conducting boundaries in order to study the onset of convection, in which case vertical heat transport gauges the system response. Conflicting experimental results for ostensibly similar set-ups have confounded efforts to discriminate between two competing theories for how boundary layers and interior flows interact to determine transport through the convecting layer asymptotically far beyond onset. In a conceptually new approach, Bouillaut, Lepot, Aumaître and Gallet (J. Fluid Mech., vol. 861, 2019, R5) devised a procedure to radiatively heat a portion of the fluid domain bypassing rigid conductive boundaries and allowing for dissociation of thermal and viscous boundary layers. Their experiments reveal a new level of complexity in the problem suggesting that heat transport scaling predictions of both theories may be realized depending on details of the thermal forcing.
Resolved and subgrid dynamics of Rayleigh–Bénard convection
Riccardo Togni, Andrea Cimarelli, Elisabetta De Angelis
Journal: Journal of Fluid Mechanics / Volume 867 / 25 May 2019
In this work we present and demonstrate the reliability of a theoretical framework for the study of thermally driven turbulence. It consists of scale-by-scale budget equations for the second-order velocity and temperature structure functions and their limiting cases, represented by the turbulent kinetic energy and temperature variance budgets. This framework represents an extension of the classical Kolmogorov and Yaglom equations to inhomogeneous and anisotropic flows, and allows for a novel assessment of the turbulent processes occurring at different scales and locations in the fluid domain. Two relevant characteristic scales, $\ell _{c}^{u}$ for the velocity field and $\ell _{c}^{\unicode[STIX]{x1D703}}$ for the temperature field, are identified. These variables separate the space of scales into a quasi-homogeneous range, characterized by turbulent kinetic energy and temperature variance cascades towards dissipation, and an inhomogeneity-dominated range, where the production and the transport in physical space are important. This theoretical framework is then extended to the context of large-eddy simulation to quantify the effect of a low-pass filtering operation on both resolved and subgrid dynamics of turbulent Rayleigh–Bénard convection. It consists of single-point and scale-by-scale budget equations for the filtered velocity and temperature fields. To evaluate the effect of the filter length $\ell _{F}$ on the resolved and subgrid dynamics, the velocity and temperature fields obtained from a direct numerical simulation are split into filtered and residual components using a spectral cutoff filter. It is found that when $\ell _{F}$ is smaller than the minimum values of the cross-over scales given by $\ell _{c,min}^{\unicode[STIX]{x1D703}\ast }=\ell _{c,min}^{\unicode[STIX]{x1D703}}Nu/H=0.8$ , the resolved processes correspond to the exact ones, except for a depletion of viscous and thermal dissipations, and the only role of the subgrid scales is to drain turbulent kinetic energy and temperature variance to dissipate them. On the other hand, the resolved dynamics is much poorer in the near-wall region and the effects of the subgrid scales are more complex for filter lengths of the order of $\ell _{F}\approx 3\ell _{c,min}^{\unicode[STIX]{x1D703}}$ or larger. This study suggests that classic eddy-viscosity/diffusivity models employed in large-eddy simulation may suffer from some limitations for large filter lengths, and that alternative closures should be considered to account for the inhomogeneous processes at subgrid level. Moreover, the theoretical framework based on the filtered Kolmogorov and Yaglom equations may represent a valuable tool for future assessments of the subgrid-scale models.
Ablation of sloping ice faces into polar seawater
Mainak Mondal, Bishakhdatta Gayen, Ross W. Griffiths, Ross C. Kerr
Journal: Journal of Fluid Mechanics / Volume 863 / 25 March 2019
Print publication: 25 March 2019
The effects of the slope of an ice–seawater interface on the mechanisms and rate of ablation of the ice by natural convection are examined using turbulence-resolving simulations. Solutions are obtained for ice slopes $\unicode[STIX]{x1D703}=2^{\circ }{-}90^{\circ }$ , at a fixed ambient salinity and temperature, chosen to represent common Antarctic ocean conditions. For laminar boundary layers the ablation rate decreases with height, whereas in the turbulent regime the ablation rate is found to be height independent. The simulated laminar ablation rates scale with $(\sin \unicode[STIX]{x1D703})^{1/4}$ , whereas in the turbulent regime it follows a $(\sin \unicode[STIX]{x1D703})^{2/3}$ scaling, both consistent with the theoretical predictions developed here. The reduction in the ablation rate with shallower slopes arises as a result of the development of stable density stratification beneath the ice face, which reduces turbulent buoyancy fluxes to the ice. The turbulent kinetic energy budget of the flow shows that, for very steep slopes, both buoyancy and shear production are drivers of turbulence, whereas for shallower slopes shear production becomes the dominant mechanism for sustaining turbulence in the convective boundary layer.
Direct numerical simulation of backward-facing step flow at $Re_{\unicode[STIX]{x1D70F}}=395$ and expansion ratio 2
A. Pont-Vílchez, F. X. Trias, A. Gorobets, A. Oliva
Backward-facing step (BFS) constitutes a canonical configuration to study wall-bounded flows subject to massive expansions produced by abrupt changes in geometry. Recirculation flow regions are common in this type of flow, driving the separated flow to its downstream reattachment. Consequently, strong adverse pressure gradients arise through this process, feeding flow instabilities. Therefore, both phenomena are strongly correlated as the recirculation bubble shape defines how the flow is expanded, and how the pressure rises. In an incompressible flow, this shape depends on the Reynolds value and the expansion ratio. The influence of these two variables on the bubble length is widely studied, presenting an asymptotic behaviour when both parameters are beyond a certain threshold. This is the usual operating point of many practical applications, such as in aeronautical and environmental engineering. Several numerical and experimental studies have been carried out regarding this topic. The existing simulations considering cases beyond the above-mentioned threshold have only been achieved through turbulence modelling, whereas direct numerical simulations (DNS) have been performed only at low Reynolds numbers. Hence, despite the great importance of achieving this threshold, there is a lack of reliable numerical data to assess the accuracy of turbulence models. In this context, a DNS of an incompressible flow over a BFS is presented in this paper, considering a friction Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}$ ) of 395 at the inflow and an expansion ratio 2. Finally, the elongation of the Kelvin–Helmholtz instabilities along the shear layer is also studied.
Transition to the ultimate regime in a radiatively driven convection experiment
Vincent Bouillaut, Simon Lepot, Sébastien Aumaître, Basile Gallet
Journal: Journal of Fluid Mechanics / Volume 861 / 25 February 2019
Published online by Cambridge University Press: 04 January 2019, R5
We report on the transition between two regimes of heat transport in a radiatively driven convection experiment, where a fluid gets heated up within a tunable heating length $\ell$ in the vicinity of the bottom of the tank. The first regime is similar to that observed in standard Rayleigh–Bénard experiments, the Nusselt number $Nu$ being related to the Rayleigh number $Ra$ through the power law $Nu\sim Ra^{1/3}$ . The second regime corresponds to the 'ultimate' or mixing-length scaling regime of thermal convection, where $Nu$ varies as the square root of $Ra$ . Evidence for these two scaling regimes has been reported in Lepot et al. (Proc. Natl Acad. Sci. USA, vol. 115, 2018, pp. 8937–8941), and we now study in detail how the system transitions from one to the other. We propose a simple model describing radiatively driven convection in the mixing-length regime. It leads to the scaling relation $Nu\sim (\ell /H)Pr^{1/2}Ra^{1/2}$ , where $H$ is the height of the cell and $Pr$ is the Prandtl number, thereby allowing us to deduce the values of $Ra$ and $Nu$ at which the system transitions from one regime to the other. These predictions are confirmed by the experimental data gathered at various $Ra$ and $\ell$ . We conclude by showing that boundary layer corrections can persistently modify the Prandtl number dependence of $Nu$ at large $Ra$ , for $Pr\gtrsim 1$ .
Roughness effects in turbulent forced convection
M. MacDonald, N. Hutchins, D. Chung
We conducted direct numerical simulations of turbulent flow over three-dimensional sinusoidal roughness in a channel. A passive scalar is present in the flow with Prandtl number $Pr=0.7$ , to study heat transfer by forced convection over this rough surface. The minimal-span channel is used to circumvent the high cost of simulating high-Reynolds-number flows, which enables a range of rough surfaces to be efficiently simulated. The near-wall temperature profile in the minimal-span channel agrees well with that of the conventional full-span channel, indicating that it can be readily used for heat-transfer studies at a much reduced cost compared to conventional direct numerical simulation. As the roughness Reynolds number, $k^{+}$ , is increased, the Hama roughness function, $\unicode[STIX]{x0394}U^{+}$ , increases in the transitionally rough regime before tending towards the fully rough asymptote of $\unicode[STIX]{x1D705}_{m}^{-1}\log (k^{+})+C$ , where $C$ is a constant that depends on the particular roughness geometry and $\unicode[STIX]{x1D705}_{m}\approx 0.4$ is the von Kármán constant. In this fully rough regime, the skin-friction coefficient is constant with bulk Reynolds number, $Re_{b}$ . Meanwhile, the temperature difference between smooth- and rough-wall flows, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$ , appears to tend towards a constant value, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ . This corresponds to the Stanton number (the temperature analogue of the skin-friction coefficient) monotonically decreasing with $Re_{b}$ in the fully rough regime. Using shifted logarithmic velocity and temperature profiles, the heat-transfer law as described by the Stanton number in the fully rough regime can be derived once both the equivalent sand-grain roughness $k_{s}/k$ and the temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ are known. In meteorology, this corresponds to the ratio of momentum and heat-transfer roughness lengths, $z_{0m}/z_{0h}$ , being linearly proportional to the inner-normalised momentum roughness length, $z_{0m}^{+}$ , where the constant of proportionality is related to $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ . While Reynolds analogy, or similarity between momentum and heat transfer, breaks down for the bulk skin-friction and heat-transfer coefficients, similar distribution patterns between the heat flux and viscous component of the wall shear stress are observed. Instantaneous visualisations of the temperature field show a thin thermal diffusive sublayer following the roughness geometry in the fully rough regime, resembling the viscous sublayer of a contorted smooth wall.
Characterization of wind-shear effects on entrainment in a convective boundary layer
Armin Haghshenas, Juan Pedro Mellado
Journal: Journal of Fluid Mechanics / Volume 858 / 10 January 2019
Print publication: 10 January 2019
Direct numerical simulations are used to characterize wind-shear effects on entrainment in a barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. We consider weakly to strongly unstable conditions $-z_{enc}/L_{Ob}\gtrsim 4$ , where $z_{enc}$ is the encroachment CBL depth and $L_{Ob}$ is the Obukhov length. Dimensional analysis allows us to characterize such a sheared CBL by a normalized CBL depth, a Froude number and a Reynolds number. The first two non-dimensional quantities embed the dependence of the system on time, on the surface buoyancy flux, and on the buoyancy stratification and wind velocity in the free atmosphere. We show that the dependence of entrainment-zone properties on these two non-dimensional quantities can be expressed in terms of just one independent variable, the ratio between a shear scale $(\unicode[STIX]{x0394}z_{i})_{s}\equiv \sqrt{1/3}\unicode[STIX]{x0394}u/N_{0}$ and a convective scale $(\unicode[STIX]{x0394}z_{i})_{c}\equiv 0.25z_{enc}$ , where $\unicode[STIX]{x0394}u$ is the velocity increment across the entrainment zone, and $N_{0}$ is the buoyancy frequency of the free atmosphere. Here $(\unicode[STIX]{x0394}z_{i})_{s}$ and $(\unicode[STIX]{x0394}z_{i})_{c}$ represent the entrainment-zone thickness in the limits of weak convective instability (strong wind) and strong convective instability (weak wind), respectively. We derive scaling laws for the CBL depth, the entrainment-zone thickness, the mean entrainment velocity and the entrainment-flux ratio as functions of $(\unicode[STIX]{x0394}z_{i})_{s}/(\unicode[STIX]{x0394}z_{i})_{c}$ . These scaling laws can also be expressed as functions of only a Richardson number $(N_{0}z_{enc}/\unicode[STIX]{x0394}u)^{2}$ , but not in terms of only the stability parameter $-z_{enc}/L_{Ob}$ .
The anatomy of large-scale motion in atmospheric boundary layers
G. G. Katul
The atmospheric boundary layer is the level of the atmosphere where all human activities occur. It is a layer characterized by its turbulent flow state, meaning that the velocity, temperature and scalar concentrations fluctuate over scales that range from less than a millimetre to several kilometres. It is those fluctuations that make dispersion of pollutants and transport of heat, momentum as well as scalars such as carbon dioxide or cloud-condensation nuclei efficient. It is also the layer where a 'hand-shake' occurs between activities on the land surface and the climate system, primarily due to the action of large energetic swirling motions or eddies. The atmospheric boundary layer experiences dramatic transitions depending on whether the underlying surface is being heated or cooled. The existing paradigm describing the size and energetics of large-scale and very large-scale eddies in turbulent flows has been shaped by decades of experiments and simulations on smooth pipes and channels with no surface heating or cooling. The emerging picture, initiated by A. A. Townsend in 1951, is that large- and very large-scale motions appear to be approximated by a collection of hairpin-shaped vortices whose population density scales inversely with distance from the boundary. How does surface heating, quintessential to the atmospheric boundary layer, alter this canonical picture? What are the implications of such a buoyancy force on the geometry and energy distribution across velocity components in those large eddies? How do these large eddies modulate small eddies near the ground? Answering these questions and tracking their consequences to existing theories used today to describe the flow statistics in the atmospheric boundary layer are addressed in the work of Salesky & Anderson (J. Fluid Mech., vol. 856, 2018, pp. 135–168). The findings are both provocative and surprisingly simple.
Velocity and acceleration statistics in rapidly rotating Rayleigh–Bénard convection
Hadi Rajaei, Kim M. J. Alards, Rudie P. J. Kunnen, Herman J. H. Clercx
Journal: Journal of Fluid Mechanics / Volume 857 / 25 December 2018
Print publication: 25 December 2018
Background rotation causes different flow structures and heat transfer efficiencies in Rayleigh–Bénard convection. Three main regimes are known: rotation unaffected, rotation affected and rotation dominated. It has been shown that the transition between rotation-unaffected and rotation-affected regimes is driven by the boundary layers. However, the physics behind the transition between rotation-affected and rotation-dominated regimes are still unresolved. In this study, we employ the experimentally obtained Lagrangian velocity and acceleration statistics of neutrally buoyant immersed particles to study the rotation-affected and rotation-dominated regimes and the transition between them. We have found that the transition to the rotation-dominated regime coincides with three phenomena; suppressed vertical motions, strong penetration of vortical plumes deep into the bulk and reduced interaction of vortical plumes with their surroundings. The first two phenomena are used as confirmations for the available hypotheses on the transition to the rotation-dominated regime while the last phenomenon is a new argument to describe the regime transition. These findings allow us to better understand the rotation-dominated regime and the transition to this regime.
Unsteady turbulent line plumes
Andrew J. Hogg, Edward J. Goldsmith, Mark J. Woodhouse
The unsteady ascent of a buoyant, turbulent line plume through a quiescent, uniform environment is modelled in terms of the width-averaged vertical velocity and density deficit. It is demonstrated that for a well-posed, linearly stable model, account must be made for the horizontal variation of the velocity and the density deficit; in particular the variance of the velocity field and the covariance of the density deficit and velocity fields, represented through shape factors, must exceed threshold values, and that models based upon 'top-hat' distributions in which the dependent fields are piecewise constant are ill-posed. Numerical solutions of the nonlinear governing equations are computed to reveal that the transient response of the system to an instantaneous change in buoyancy flux at the source may be captured through new similarity solutions, the form of which depend upon both the ratio of the old to new buoyancy fluxes and the shape factors.
Buoyancy effects on large-scale motions in convective atmospheric boundary layers: implications for modulation of near-wall processes
S. T. Salesky, W. Anderson
A number of recent studies have demonstrated the existence of so-called large- and very-large-scale motions (LSM, VLSM) that occur in the logarithmic region of inertia-dominated wall-bounded turbulent flows. These regions exhibit significant streamwise coherence, and have been shown to modulate the amplitude and frequency of small-scale inner-layer fluctuations in smooth-wall turbulent boundary layers. In contrast, the extent to which analogous modulation occurs in inertia-dominated flows subjected to convective thermal stratification (low Richardson number) and Coriolis forcing (low Rossby number), has not been considered. And yet, these parameter values encompass a wide range of important environmental flows. In this article, we present evidence of amplitude modulation (AM) phenomena in the unstably stratified (i.e. convective) atmospheric boundary layer, and link changes in AM to changes in the topology of coherent structures with increasing instability. We perform a suite of large eddy simulations spanning weakly ( $-z_{i}/L=3.1$ ) to highly convective ( $-z_{i}/L=1082$ ) conditions (where $-z_{i}/L$ is the bulk stability parameter formed from the boundary-layer depth $z_{i}$ and the Obukhov length $L$ ) to investigate how AM is affected by buoyancy. Results demonstrate that as unstable stratification increases, the inclination angle of surface layer structures (as determined from the two-point correlation of streamwise velocity) increases from $\unicode[STIX]{x1D6FE}\approx 15^{\circ }$ for weakly convective conditions to nearly vertical for highly convective conditions. As $-z_{i}/L$ increases, LSMs in the streamwise velocity field transition from long, linear updrafts (or horizontal convective rolls) to open cellular patterns, analogous to turbulent Rayleigh–Bénard convection. These changes in the instantaneous velocity field are accompanied by a shift in the outer peak in the streamwise and vertical velocity spectra to smaller dimensionless wavelengths until the energy is concentrated at a single peak. The decoupling procedure proposed by Mathis et al. (J. Fluid Mech., vol. 628, 2009a, pp. 311–337) is used to investigate the extent to which amplitude modulation of small-scale turbulence occurs due to large-scale streamwise and vertical velocity fluctuations. As the spatial attributes of flow structures change from streamwise to vertically dominated, modulation by the large-scale streamwise velocity decreases monotonically. However, the modulating influence of the large-scale vertical velocity remains significant across the stability range considered. We report, finally, that amplitude modulation correlations are insensitive to the computational mesh resolution for flows forced by shear, buoyancy and Coriolis accelerations.
The energy flux spectrum of internal waves generated by turbulent convection
Louis-Alexandre Couston, Daniel Lecoanet, Benjamin Favier, Michael Le Bars
We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ( $Ra\in [4\times 10^{7},10^{9}]$ ) and large stratification (Brunt–Väisälä frequencies $f_{N}\gg f_{c}$ , where $f_{c}$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like $k_{\bot }^{4}\,f^{-13/2}$ for weakly damped waves (with $k_{\bot }$ and $f$ the waves' horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with $z$ , the distance from the convective region, like $z^{-13/8}$ .
Data-driven reduced modelling of turbulent Rayleigh–Bénard convection using DMD-enhanced fluctuation–dissipation theorem
M. A. Khodkar, Pedram Hassanzadeh
Journal: Journal of Fluid Mechanics / Volume 852 / 10 October 2018
Published online by Cambridge University Press: 06 August 2018, R3
Print publication: 10 October 2018
A data-driven model-free framework is introduced for the calculation of reduced-order models (ROMs) capable of accurately predicting time-mean responses to external forcings, or forcings needed for specified responses, e.g. for control, in fully turbulent flows. The framework is based on using the fluctuation–dissipation theorem (FDT) in the space of a limited number of modes obtained from dynamic mode decomposition (DMD). Use of the DMD modes as the basis functions, rather than the commonly used proper orthogonal decomposition modes, resolves a previously identified problem in applying FDT to high-dimensional non-normal turbulent flows. Employing this DMD-enhanced FDT method ( $\text{FDT}_{DMD}$ ), a linear ROM with horizontally averaged temperature as state vector is calculated for a 3D Rayleigh–Bénard convection system at a Rayleigh number of $10^{6}$ using data obtained from direct numerical simulation. The calculated ROM performs well in various tests for this turbulent flow, suggesting $\text{FDT}_{DMD}$ as a promising method for developing ROMs for high-dimensional turbulent systems.
Confined inclined thermal convection in low-Prandtl-number fluids
Lukas Zwirner, Olga Shishkina
Published online by Cambridge University Press: 10 July 2018, pp. 984-1008
Any tilt of a Rayleigh–Bénard convection cell against gravity changes the global flow structure inside the cell, which leads to a change of the heat and momentum transport. Especially sensitive to the inclination angle is the heat transport in low-Prandtl-number fluids and confined geometries. The purpose of the present work is to investigate the global flow structure and its influence on the global heat transport in inclined convection in a cylindrical container of diameter-to-height aspect ratio $\unicode[STIX]{x1D6E4}=1/5$ . The study is based on direct numerical simulations where two different Prandtl numbers $Pr=0.1$ and 1.0 are considered, while the Rayleigh number, $Ra$ , ranges from $10^{6}$ to $10^{9}$ . For each combination of $Ra$ and $Pr$ , the inclination angle is varied between 0 and $\unicode[STIX]{x03C0}/2$ . An optimal inclination angle of the convection cell, which provides the maximal global heat transport, is determined. For inclined convection we observe the formation of two system-sized plume columns, a hot and a cold one, that impinge on the opposite boundary layers. These are related to a strong increase in the heat transport. | CommonCrawl |
Home » Introductory Workshop: Algebraic Topology
Introductory Workshop: Algebraic Topology January 27, 2014 - January 31, 2014
January 27, 2014 over 5 years ago
To apply for Funding you must register by:
October 27, 2013 over 5 years ago
Parent Program:
Mathematical Sciences Research Institute, Berkeley CA
Organizers Teena Gerhardt (Michigan State University), Jesper Grodal (University of Copenhagen), Kathryn Hess (École Polytechnique Fédérale de Lausanne (EPFL)), LEAD Michael Hill (University of California, Los Angeles)
Show List of Speakers
Mark Behrens (Massachusetts Institute of Technology)
Julie Bergner (University of Virginia)
Andrew Blumberg (University of Texas, Austin)
Michael Ching (Amherst College)
Thomas Church (Stanford University)
Christopher Douglas (University of Oxford)
William Dwyer (University of Notre Dame)
Soren Galatius (Stanford University)
Lars Hesselholt (Nagoya University)
Michael Hopkins (Harvard University)
Po Hu (Wayne State University)
Daniel Isaksen (Wayne State University)
Nitya Kitchloo (Johns Hopkins University)
Tyler Lawson (University of Minnesota Twin Cities)
Bob Oliver (Université de Paris XIII (Paris-Nord))
Oscar Randal-Williams (University of Cambridge)
John Rognes (University of Oslo)
Brooke Shipley (University of Illinois at Chicago)
Constantin Teleman (University of California, Berkeley)
Craig Westerland (University of Minnesota Twin Cities)
Algebraic topology is a rich, vibrant field with close connections to many branches of mathematics. This workshop will describe the state of the field, focusing on major programs, open problems, exciting new tools, and cutting edge techniques. The introductory workshop serves as an overview to the overlying programmatic theme. It aims to familiarize graduate students, postdocs, and non-experts to major and new topics of the current program. Though the audience is expected to have a general mathematical background, knowledge of technical terminology and recent findings is not assumed.
Algebraic topology is a rich, vibrant field with close connections to many branches of mathematics. This workshop will describe the state of the field, focusing on major programs, open problems, exciting new tools, and cutting edge techniques.
The introductory workshop serves as an overview to the overlying programmatic theme. It aims to familiarize graduate students, postdocs, and non-experts to major and new topics of the current program. Though the audience is expected to have a general mathematical background, knowledge of technical terminology and recent findings is not assumed.
Keywords and Mathematics Subject Classification (MSC)
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To apply for funding, you must register by the funding application deadline displayed above.
Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.
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MSRI has preferred rates at the Hotel Shattuck Plaza, depending on room availability. Guests can call the hotel's main line at 510-845-7300 and ask for the MSRI- Mathematical Science Research Institute discount. To book online visit this page (the MSRI rate will automatically be applied).
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MSRI has a preferred rates at Easton Hall and Gibbs Hall, depending on room availability. Guests can call the Reservations line at 510-204-0732 and ask for the MSRI- Mathematical Science Research Inst. rate. To book online visit this page, select "Request a Reservation" choose the dates you would like to stay and enter the code MSRI (this code is not case sensitive).
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Show Schedule, Notes/Handouts & Videos
Dwyer notes 5.15 MB application/pdf
Goodwillie's calculus of functors
The calculus of homotopy functors provides a systematic way to approximate a given functor (say from based spaces to spectra) by so-called `polynomial' functors. Each functor F that preserves weak equivalences has a `Taylor tower' (analogous to the Taylor series of ordinary calculus) which in turn is built from homogeneous pieces that are classified by certain `derivatives' for F. I will review this material and consider the problem of how the Taylor tower of F can be reconstructed from its derivatives. We will discuss some important examples built from mapping spaces. Then. if time permits, I will us this approach to give a classification of analytic functors from based spaces to spectra and try to describe some connections to the Goodwillie-Weiss manifold calculus
Ching notes 5.37 MB application/pdf
Morita theory in stable homotopy
: I will give an introduction to Morita theory in stable homotopy theory. This will include the context of differential graded algebra and of spectral algebra. I will also survey some recent related results.
Shipley Notes 4.73 MB application/pdf
MSRI/Evans Lecture: Topological cyclic homology (Evans Hall, UC Berkeley)
UC Berkeley, 60 Evans Hall
Topological cyclic homology is a topological refinement of Connes' cyclic homology. It was introduced twenty-five years ago by Bökstedt-Hsiang-Madsen who used it to prove the K-theoretic Novikov conjecture for discrete groups all of whose integral homology groups are finitely generated. In this talk, I will give an introduction to topological cyclic homology and explain how results obtained in the intervening years lead to a short proof of this result in which the necessity of the finite generation hypothesis becomes transparent. In the end I will explain how one may hope to remove this restriction and discuss number theoretic consequences that would ensue.
Higher categories and algebraic K-theory
In this talk, I will given an overview of recent work on formulating the structural properties of algebraic K-theory using the framework for studying homotopical categories provided by the development of higher category theory
Blumberg notes 7.14 MB application/pdf
Towards explicit models for higher K-theories
Douglas notes 6.1 MB application/pdf
Models for homotopical higher categories
Models for homotopical categories, known as $(\infty,1)$-categories, are now fairly well-developed. Comparisons are in place between them, and there is a wide variety of applications ranging from topology to algebra to algebraic geometry. Generalizing them to models for homotopical higher categories, or $(\infty,n)$-categories, requires substantially greater technicality. Much work has been done in this direction, due to recent work of many authors, but the full picture is still very much work in progress. In this talk we will review different models for $(\infty,1)$-categories and discuss the number of ways they are being generalized to models for $(\infty,n)$-categories, as well as the known and conjectured comparisons between them
Bergner notes 5.81 MB application/pdf
Computations in the stable homotopy groups of spheres
I will give an overview of the rich structure in the stable homotopy groups of spheres, both giving a summary of what we do know, and what we don't know.
Behrens notes 15 MB application/pdf
Computations in motivic homotopy theory
We will begin with an introduction to stable motivic homotopy theory. Then we will use the motivic version of the Adams spectral sequence to compute stable motivic homotopy groups. We will discuss a number of open questions concerning these computations. Along the way, we will encounter some new results about classical stable homotopy groups and equivariant stable homotopy groups.
Isaken notes 5.26 MB application/pdf
Oliver notes 6.22 MB application/pdf
Equivariant homotopy and localization
This will be a general talk on equivariant homotopy theory
Hopkins notes 6.25 MB application/pdf
Homotopy theory of Kac-Moody groups
The homotopy theory of compact Lie groups is very well understood by now. The rich structure of these groups (for example: existence and uniqueness of maximal tori, corresponding Weyl groups etc.) may be exploited to classify these groups. This classification even extends to homotopical versions of these groups known as p-compact groups. In the last few decades a beautiful new class of (non-compact) topological groups has been constructed. These are known as Kac-Moody groups and they share most of the structure that compact Lie groups admit. Kac-Moody groups have been shown to be relevant in mathematical physics and further investigation by several mathematicians (including the speaker) seems to suggest that Kac-Moody groups are surprisingly amenable to homotopical techniques. This makes these groups prime candidates for study from the standpoint of homotopy theory
Kitchloo notes 6.22 MB application/pdf
Topological automorphic forms
: Topological modular forms and its generalizations are objects in stable homotopy that realize a connection to 1-dimensional formal group laws. In this talk I'll describe how this perspective on stable homotopy has emerged, and how using algebraic geometry is providing us with a library of new objects in homotopy theory
Lawson notes 6.37 MB application/pdf
Representation stability and applications to homological stability
I will give an introduction to the theory of representation stability, through the lens of its applications in homological stability. I'll focus on three applications: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; and stability for twisted homology such as H_i( GL_n(R); R^n ), where the coefficients change along with the groups.
Church notes 7.78 MB application/pdf
Stability of moduli spaces of manifolds
I will first discuss some fairly classical homological stability phenomena: spaces of 0-manifolds (e.g. configuration spaces), and spaces of 2-manifolds (e.g. Riemann's moduli space). I will explain the general method, introduced by Quillen, for proving these stability theorems, and then explain some recent work with Søren Galatius which proves such a stability theorem for moduli spaces of 2n-dimensional manifolds (for n > 2).
R-W Notes 5.96 MB application/pdf
Stable homology of moduli spaces of manifolds
MSRI:
I will discuss recent joint work with Oscar Randal-Williams, aimed at calculating the cohomology of BDiff(W) and related spaces, where W is a smooth 2n-dimensional manifold, Diff(W) is the topological group of diffeomorphisms of W, and BDiff(W) is its classifying space. Surprisingly, the cohomology ring turns out to be partially independent of W through a range of degrees (homological stability). In this talk, I will discuss how infinite loop spaces can be used to describe the cohomology in this stable range.
Galatius notes 5.95 MB application/pdf
Loop Groups, TQFTs and algebraic geometry
In this survey talk, I will shall the role that loop groups have played in the construction of topological gauge theories in dimensions 2 and possibly 3.
Teleman notes 21.1 MB application/pdf
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Search results for: E. MacDonald
Items from 1 to 20 out of 361 results
Search for a heavy pseudoscalar boson decaying to a Z and a Higgs boson at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
A. M. Sirunyan, A. Tumasyan, W. Adam, F. Ambrogi, more
The European Physical Journal C > 2019 > 79 > 7 > 1-27
A search is presented for a heavy pseudoscalar boson $$\text {A}$$ A decaying to a Z boson and a Higgs boson with mass of 125$$\,\text {GeV}$$ GeV . In the final state considered, the Higgs boson decays to a bottom quark and antiquark, and the Z boson decays either into a pair of electrons, muons, or neutrinos. The analysis is performed using a data sample corresponding to an integrated luminosity...
Search for supersymmetry in final states with photons and missing transverse momentum in proton-proton collisions at 13 TeV
The CMS collaboration, A. M. Sirunyan, A. Tumasyan, W. Adam, more
Journal of High Energy Physics > 2019 > 2019 > 6 > 1-34
Abstract Results are reported of a search for supersymmetry in final states with photons and missing transverse momentum in proton-proton collisions at the LHC. The data sample corresponds to an integrated luminosity of 35.9 fb−1 collected at a center-of-mass energy of 13 TeV using the CMS detector. The results are interpreted in the context of models of gauge-mediated supersymmetry breaking. Production...
Search for the associated production of the Higgs boson and a vector boson in proton-proton collisions at s $$ \sqrt{s} $$ = 13 TeV via Higgs boson decays to τ leptons
Abstract A search for the standard model Higgs boson produced in association with a W or a Z boson and decaying to a pair of τ leptons is performed. A data sample of proton-proton collisions collected at s $$ \sqrt{s} $$ = 13 TeV by the CMS experiment at the CERN LHC is used, corresponding to an integrated luminosity of 35.9 fb−1. The signal strength is measured relative to the expectation...
Evaluation of linkage disequilibrium, population structure, and genetic diversity in the U.S. peanut mini core collection
Paul I. Otyama, Andrew Wilkey, Roshan Kulkarni, Teshale Assefa, more
BMC Genomics > 2019 > 20 > 1 > 1-17
Background Due to the recent domestication of peanut from a single tetraploidization event, relatively little genetic diversity underlies the extensive morphological and agronomic diversity in peanut cultivars today. To broaden the genetic variation in future breeding programs, it is necessary to characterize germplasm accessions for new sources of variation and to leverage the power of genome-wide...
Search for a low-mass τ−τ+ resonance in association with a bottom quark in proton-proton collisions at s $$ \sqrt{s} $$ = 13 TeV
Abstract A general search is presented for a low-mass τ−τ+ resonance produced in association with a bottom quark. The search is based on proton-proton collision data at a center-of-mass energy of 13 TeV collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of 35.9 fb−1. The data are consistent with the standard model expectation. Upper limits at 95% confidence level...
Search for supersymmetry in events with a photon, jets, $$\mathrm {b}$$ b -jets, and missing transverse momentum in proton–proton collisions at 13$$\,\text {Te}\text {V}$$ Te
A search for supersymmetry is presented based on events with at least one photon, jets, and large missing transverse momentum produced in proton–proton collisions at a center-of-mass energy of 13$$\,\text {Te}\text {V}$$ Te . The data correspond to an integrated luminosity of 35.9$$\,\text {fb}^{-1}$$ fb-1 and were recorded at the LHC with the CMS detector in 2016. The analysis characterizes signal-like...
Combined measurements of Higgs boson couplings in proton–proton collisions at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
Combined measurements of the production and decay rates of the Higgs boson, as well as its couplings to vector bosons and fermions, are presented. The analysis uses the LHC proton–proton collision data set recorded with the CMS detector in 2016 at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te , corresponding to an integrated luminosity of 35.9$${\,\text {fb}^{-1}} $$ fb-1 . The combination is based...
Combinations of single-top-quark production cross-section measurements and |fLVVtb| determinations at s $$ \sqrt{s} $$ = 7 and 8 TeV with the ATLAS and CMS experiments
The ATLAS collaboration, M. Aaboud, G. Aad, B. Abbott, more
Abstract This paper presents the combinations of single-top-quark production cross-section measurements by the ATLAS and CMS Collaborations, using data from LHC proton-proton collisions at s $$ \sqrt{s} $$ = 7 and 8 TeV corresponding to integrated luminosities of 1.17 to 5.1 fb−1 at s $$ \sqrt{s} $$ = 7 TeV and 12.2 to 20.3 fb−1 at s $$ \sqrt{s} $$ = 8 TeV. These combinations...
Measurement of inclusive very forward jet cross sections in proton-lead collisions at s N N $$ \sqrt{s_{\mathrm{NN}}} $$ = 5.02 TeV
Abstract Measurements of differential cross sections for inclusive very forward jet production in proton-lead collisions as a function of jet energy are presented. The data were collected with the CMS experiment at the LHC in the laboratory pseudorapidity range −6.6 < η < −5.2. Asymmetric beam energies of 4 TeV for protons and 1.58 TeV per nucleon for Pb nuclei were used, corresponding to a...
Measurement of the energy density as a function of pseudorapidity in proton–proton collisions at $$\sqrt{s} =13\,\text {TeV} $$ s=13TeV
A measurement of the energy density in proton–proton collisions at a centre-of-mass energy of $$\sqrt{s} =13$$ s=13 $$\,\text {TeV}$$ TeV is presented. The data have been recorded with the CMS experiment at the LHC during low luminosity operations in 2015. The energy density is studied as a function of pseudorapidity in the ranges $$-\,6.6<\eta <-\,5.2$$ -6.6<η<-5.2 and $$3.15<|\eta...
Measurement of the $${\mathrm {t}\overline{\mathrm {t}}}$$ tt¯ production cross section, the top quark mass, and the strong coupling constant using dilepton events in pp collisions at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
A measurement of the top quark–antiquark pair production cross section $$\sigma _{\mathrm {t}\overline{\mathrm {t}}} $$ σtt¯ in proton–proton collisions at a centre-of-mass energy of 13$$\,\text {Te}\text {V}$$ Te is presented. The data correspond to an integrated luminosity of $$35.9{\,\text {fb}^{-1}} $$ 35.9fb-1 , recorded by the CMS experiment at the CERN LHC in 2016. Dilepton events ($$\mathrm...
Search for vector-like quarks in events with two oppositely charged leptons and jets in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te}\text {V} $$ s=13Te
A search for the pair production of heavy vector-like partners $$\mathrm {T}$$ T and $$\mathrm {B}$$ B of the top and bottom quarks has been performed by the CMS experiment at the CERN LHC using proton–proton collisions at $$\sqrt{s} = 13\,\text {Te}\text {V} $$ s=13Te . The data sample was collected in 2016 and corresponds to an integrated luminosity of 35.9$$\,\text {fb}^{-1}$$ fb-1 . Final states...
Measurements of the pp → WZ inclusive and differential production cross sections and constraints on charged anomalous triple gauge couplings at s $$ \sqrt{s} $$ = 13 TeV
Abstract The WZ production cross section is measured in proton-proton collisions at a centre-of-mass energy s $$ \sqrt{s} $$ = 13 TeV using data collected with the CMS detector, corresponding to an integrated luminosity of 35.9 fb−1. The inclusive cross section is measured to be σtot(pp → WZ) = 48.09 − 0.96+ 1.00 (stat) − 0.37+ 0.44 (theo) − 2.17+ 2.39 (syst) ± 1.39(lum) pb, resulting in...
Search for nonresonant Higgs boson pair production in the b b ¯ b b ¯ $$ \mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}} $$ final state at s $$ \sqrt{s} $$ = 13 TeV
Abstract Results of a search for nonresonant production of Higgs boson pairs, with each Higgs boson decaying to a b b ¯ $$ \mathrm{b}\overline{\mathrm{b}} $$ pair, are presented. This search uses data from proton-proton collisions at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb−1, collected by the CMS detector at the LHC. No signal is observed, and...
Search for contact interactions and large extra dimensions in the dilepton mass spectra from proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV
Abstract A search for nonresonant excesses in the invariant mass spectra of electron and muon pairs is presented. The analysis is based on data from proton-proton collisions at a center-of-mass energy of 13 TeV recorded by the CMS experiment in 2016, corresponding to a total integrated luminosity of 36 fb−1. No significant deviation from the standard model is observed. Limits are set at 95% confidence...
Measurement of the top quark mass in the all-jets final state at $$\sqrt{s}=13\,\text {TeV} $$ s=13TeV and combination with the lepton+jets channel
A top quark mass measurement is performed using $$35.9{\,\text {fb}^{-1}} $$ 35.9fb-1 of LHC proton–proton collision data collected with the CMS detector at $$\sqrt{s}=13\,\text {TeV} $$ s=13TeV . The measurement uses the $${\mathrm {t}\overline{\mathrm {t}}}$$ tt¯ all-jets final state. A kinematic fit is performed to reconstruct the decay of the $${\mathrm {t}\overline{\mathrm {t}}}$$ tt¯ system...
Search for resonant production of second-generation sleptons with same-sign dimuon events in proton–proton collisions at $$\sqrt{s} = 13\,\text {TeV} $$ s=13TeV
A search is presented for resonant production of second-generation sleptons ($$\widetilde{\mu } _{\mathrm {L}}$$ μ~L , $$\widetilde{\nu }_{\mu }$$ ν~μ ) via the R-parity-violating coupling $${\lambda ^{\prime }_{211}}$$ λ211′ to quarks, in events with two same-sign muons and at least two jets in the final state. The smuon (muon sneutrino) is expected to decay into a muon and a neutralino (chargino),...
Search for resonant t t ¯ $$ \mathrm{t}\overline{\mathrm{t}} $$ production in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV
Abstract A search for a heavy resonance decaying into a top quark and antiquark t t ¯ $$ \left(\mathrm{t}\overline{\mathrm{t}}\right) $$ pair is performed using proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV. The search uses the data set collected with the CMS detector in 2016, which corresponds to an integrated luminosity of 35.9 fb−1. The analysis considers three exclusive...
Search for excited leptons in ℓℓγ final states in proton-proton collisions at s = 13 $$ \sqrt{\mathrm{s}}=13 $$ TeV
Abstract A search is presented for excited electrons and muons in ℓℓγ final states at the LHC. The search is based on a data sample corresponding to an integrated luminosity of 35.9 fb−1 of proton-proton collisions at a center-of-mass energy of 13 TeV, collected with the CMS detector in 2016. This is the first search for excited leptons at s $$ \sqrt{s} $$ = 13 TeV. The observation is consistent...
Search for dark matter produced in association with a Higgs boson decaying to a pair of bottom quarks in proton–proton collisions at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
A search for dark matter produced in association with a Higgs boson decaying to a pair of bottom quarks is performed in proton–proton collisions at a center-of-mass energy of 13$$\,\text {Te}\text {V}$$ Te collected with the CMS detector at the LHC. The analyzed data sample corresponds to an integrated luminosity of 35.9$$\,\text {fb}^{-1}$$ fb-1 . The signal is characterized by a large missing transverse...
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Achieving Ultra-Wideband and Elevated Temperature Electromagnetic Wave Absorption via Constructing Lightweight Porous Rigid Structure
Special issue on EM Wave Functional Materials
Zibao Jiao1,2,
Wenjun Huyan1,2,
Feng Yang1,2,
Junru Yao1,2,
Ruiyang Tan3,
Ping Chen3,
Xuewei Tao4,
Zhengjun Yao1,2,
Jintang Zhou1,2 &
Peijiang Liu1
Nano-Micro Letters volume 14, Article number: 173 (2022) Cite this article
Constructing a porous carbon fiber/polymethacrylimide (CP) structure for acquiring promising electromagnetic absorption performance and withstanding both elevated temperature and high strength in a low density.
The absorption bandwidth of CP composite can reach ultra-wideband absorption of 14 GHz at room temperature and even cover the whole X-band at 473 K.
The lightweight of the CP composite with a density of only 110 mg cm−3 coupled with high compressive strength of 1.05 MPa even at 453 K.
Realizing ultra-wideband absorption, desirable attenuation capability at high temperature and mechanical requirements for real-life applications remains a great challenge for microwave absorbing materials. Herein, we have constructed a porous carbon fiber/polymethacrylimide (CP) structure for acquiring promising microwave absorption performance and withstanding both elevated temperature and high strength in a low density. Given the ability of porous structure to induce desirable impedance matching and multiple reflection, the absorption bandwidth of CP composite can reach ultra-wideband absorption of 14 GHz at room temperature and even cover the whole X-band at 473 K. Additionally, the presence of imide ring group in polymethacrylimide and hard bubble wall endows the composite with excellent heat and compressive behaviors. Besides, the lightweight of the CP composite with a density of only 110 mg cm−3 coupled with high compressive strength of 1.05 MPa even at 453 K also satisfies the requirements in engineering applications. Compared with soft and compressible aerogel materials, we envision that the rigid porous foam absorbing material is particularly suitable for environmental extremes.
Avoid the common mistakes
Electromagnetic (EM) wave absorbing materials can eliminate the EM pollution produced by EM equipment during operation and therefore offer important potential in the field of military stealth and 5G [1,2,3,4,5,6,7,8]. In the quest for better EM compatibility, the development trend of new electronic instruments is multiple and wide frequency band [9,10,11,12,13]. Thus, there is an urgency for high-quality microwave absorbing materials with wide band absorption to eliminate the excess EM pollution generated during equipment operation [14,15,16,17,18]. Previous research work mainly focused on the control of micromorphology and the optimal combination of dielectric and magnetic constitutions [19,20,21,22,23,24,25,26]. However, considering the frequently used range of 2–18 GHz, realizing an effective absorption bandwidth (EAB) of more than 10 GHz is still an unachievable challenge.
Researchers found that constructing porous structures is more conducive to microwave absorption (MA) promotion compared with using solid particles under the same amount [27,28,29,30,31,32]. Freeze–drying method is the most common way for constructing porous structures in terms of aerogel and foam sponge. For example, Liu et al. [33] reported a simple freeze–drying technology to prepare porous rGO aerogels with a wide EAB up to 7.47 GHz. Zhang and his colleges [34] prepared ultra-light graphene foam by hydrothermal method, and the EAB can reach amazing 12 GHz. It is well accepted that the impedance matching characteristic of porous structure is very close to air, thus most of EM waves propagate in the absorbent material rather than be reflected. Further, the porous structure also gives rise to multiple scattering and multiple reflection of EM wave, which is beneficial to the extension of transmission route and subsequent conversion of EM energy into heat energy. At some point, construction of a 3D conductive network through porous structure is the key to expanding perspectives on ultralight and high-performance broadband MA materials. However, the ultra-wide porous absorbing material is still facing two critical issues. First of all, such ultra-wideband absorbing performance is readily achieved at room temperature. Nevertheless, considering the various applicabilities toward a high-temperature (up to 473 K) or harsh environment, absorbing frequency band usually requires to cover a specified region such as but not limited to full band coverage in military X-band [35,36,37]. Secondly, the porous structure of aerogel is soft and compressible to some extent that is a lack of ability to withstand certain compression. When subjected to pressure, the interior structure and size of aerogel proceed to change, then resulting in an uncontrollable reversal of absorbing behavior. In view of these facts, it is necessary and urgent to explore a novel kind of porous foam with excellent EM attenuation, superior heat resistance, and great compression resistance both at room and high temperature.
The polymethacrylimide (PMI) foam features a rigid structure with temperature resistance up to 220 °C and close cell content of more than 90%. Moreover, its uniform cross-linked pore wall structure affords outstanding structural stability and excellent mechanical properties [38]. Without considering other aspects, PMI foam exhibits higher compressive strength and stiffness than other polymeric foams at the same density, not to mention its high heat resistance [39, 40]. These fascinating features of lightweight PMI foam meet the ever-increasing needs of aerospace for lightweight and high strength. Besides, employment of PMI greatly reduces the overall weight of structural parts and thus allows the simplicity of mechanical structure design. Relying on the high heat resistance (up to 453 K) and pressure resistance (about 0.6 MPa), the fabrication of PMI is therefore applicable to autoclave co-curing process, and mainly adopts acrylonitrile and acrylic acid as monomer [41]. Although utilization of acrylonitrile and methacrylic acid as the main monomer reduces the overall cost of PMI products, the reaction side-effect that is intense exotherms of polymerization and even explosion caused by the difference of reactivity ratios of the two monomers is inevitable. This results in incomplete reaction and considerable residual molecules. More importantly, the inorganic fillers, especially carbon-based materials, undergo degeneration or oxidation procedure, causing irreversible deterioration of functionality. Consequently, maintaining the processing reliability of PMI whilst retaining other beneficial functions of inorganic fillers is still an enormous challenge.
In this work, we have designed a kind of carbon fiber/PMI (CP) composite by mixing carbon fibers (CFs) to the monomers and subsequently conducting water-bath polymerization. Through precisely controlling the amounts of CF absorbents, rational modulation of EM properties can be readily gained, therefore allowing to realize significant EM attenuation capability. Ultra-wideband absorption can also be gained under thinner thickness by using a genetic algorithm and multilayer design. Meanwhile, the excellent heat resistance and compressive strength of CP composite allow the EM attenuation effectiveness to cover whole X-band at high temperature (473 K). Conceptually, broadening the EAB of EM absorbing foams with rigid porous structure no matter at room temperature or elevated temperature provides a new idea for the design of novel absorbing materials.
Experimental Section
Methyl acrylonitrile was purchased from Hunan Huateng Pharmaceutical Co., Ltd., China. Methacrylic acid, light magnesium oxide, methyl methacrylate, isopropanol, tert butyl methacrylate, azobisisobutyronitrile, benzoyl peroxide and white carbon black were purchased from Macklin. Short, cut CF was purchased from Shanghai Liso Composite Material Technology Co., Ltd., China.
Synthesis of CP Composites with Different CF Contents
Methacrylic acid, methyl acrylonitrile, light magnesium oxide, methyl methacrylate, isopropanol, tert-butyl methacrylate, azobisisobutyronitrile and benzoyl peroxide were added to a three-neck bottle, stirred for 30 min to ensure that the entire system is uniform. Then added white carbon black and started to stir quickly for 30 min until the whole system becomes viscous. Then the short cut CF was added, and stirring was continued for 15 min. The above viscous liquid was poured into a mold and then reacted in a water bath at 50 °C for 4 days to obtain prepolymers. The prepolymers were freely foamed in a 240 °C oven for 1 h to obtain different contents of CP foams.
Characterization and Measurement
The microstructures of CP composites were observed by scanning electron microscope (SEM, Hitachi, S4800). The changes in groups during synthesis were analyzed using an infrared spectrometer (Bruker, Vector-33). The crystal structure of the composites was evaluated by X-ray diffraction (XRD, Bruker, D8 Advance). An X-ray photoelectron spectroscope (XPS) was used to analyze the changes in chemical states of the structures (Thermo Scientific, ESCALAB 250XI). A thermogravimetric analyzer (TGA) was used to characterize the heat resistance of the material, and N2 atmosphere was used during the test (NETZSCH, STA409PC). The EM parameters of the CP composites at room temperature were tested using a vector network analyzer (CeYear, 3672B) via a coaxial method, and the test frequency band was 2–18 GHz. The sample size was 7.00 mm in outer diameter, 3.04 mm in inner diameter and 2 mm in thickness. The EM parameters at high temperature were tested using waveguide method in the range of 8.2–12.4 GHz, and the sample size was 22.8 × 10.1 × 2 mm3. The reflection loss curve of the multilayer structures was measured via space method, and the sample size was 300 × 300 × 30 mm3.
Analysis of Microstructure of CP Composites
The schematic diagram of the synthesis steps of CP is shown in Fig. 1a. The samples with mass fraction of CF in monomer (methacrylic acid and methacrylonitrile) of 0, 1, 2, 3, and 4% were denoted as CP-0, CP-1, CP-2, CP-3, and CP-4, respectively. After water bath treatment, methacrylic acid and methacrylonitrile polymerized into long-chain prepolymer. Afterward, the adjacent carboxyl group (–COOH) reacted with the cyanide group (–CN) to afford the heat-resistant imide ring during the foaming process. Fourier transform infrared spectroscopy (FT-IR) spectra illustrate the formation of imide ring structure (Fig. S1b), in which the peaks at 2239 and 1699 cm−1 are corresponding to the stretching vibrations of carbonyl group (–C=O) and cyanide group (–CN), respectively, and the peak at 1223 cm−1 relates to the complete vibration peak of C–N bond.
a Schematic illustration of the synthesis process for CP composites, SEM images of b CP-1, c CP-2, e CP-3, f CP-4, d photograph of CP-4 placed on the stamen and g photograph of a person stand on CP-4
The morphologies of samples are presented in Fig. 1b–g, the order of which follows the above species of various CF contents. All CP composites exhibit a typical closed porous structure, whilst CF uniformly and evenly disperses in PMI interior. Excellent impedance matching is the first important step in EM absorption so that incident waves can unimpededly transmit on the absorber surface, which facilitates further EM attenuation. The porous structures of the CP composites offering substantial air are conducive to the impedance matching, so that substantial EM waves can enter the materials. At the same time, as the EM wave arrives the interior of materials, the porous structure will stimulate multiple scattering that is also advantageous to consume EM waves. When the amount of CF is less than 2 wt%, the size of bubble structures is relatively uniform and about 150–200 µm. However, as the content of CF increases, some defects and nonfoaming phenomena appear in the bubbles. Two reasons may be put forward: On the one hand, with the increase in CF, a few of unwanted agglomerations generate owing to van der Waals force and settlement effect during foaming process, which minimize the interface energy and thus hinders the solidification and foaming process. On the other hand, excessive amounts of CFs will break the bubble hole during the foaming process and further suppress cell expansion, as illustrated in Fig. 1e–f. These unpleasant defects and nonfoaming phenomena are unfavorable to either EM absorption or mechanical robustness. Therefore, the amount of CF was not further increased in this work. Owing to the poriferous and rigid structures, CP composites show a lightweight and high-strength character, as shown in Fig. 1d, g. The density of CP-4 is evaluated to be only 110 mg cm−3, and such a small density allows it to be placed on the stamen without damaging. In particular, it can also withstand the pressure of 80 kg without generating indentation. This lightweight and high-strength structure indicates a high application value in the aviation and aerospace fields.
In addition, the CP composites show excellent heat resistance. As can be seen from Fig. S1a, there is almost no thermal weight loss before 360 °C, an indicative of good thermal stability of the CP composites. This excellent heat resistance enables their application in the field of high-temperature MA. As the amount of CF increases, the heat resistance of the CP composites gradually increases, and the thermal deformation temperature of CP-4 remarkably improves and reaches 371.7 °C. Such excellent heat resistance is mainly derived from the imide ring groups on the molecular chain of PMI, which is of powerful help to the rigidity of the molecular chain. This strategy toward the improvement of the MA performance by constructing a porous structure does not change the chemical group, crystal structure and chemical state of the material (Fig. S1b–d).
Analysis of Microwave Absorbing Properties of Single Layer CP Composites
EM parameters of the material have been measured via a coaxial method, and the relevant EM wave absorption performance was calculated in accordance with transmission line theory. The RL value of the reflection loss can be expressed by the following formula:
$$ R_{\text{L}} = 20 {\text{log}}\left| \Gamma \right| = 20{\text{ log}}\left| {\frac{{Z_{{{\text{in}}}} - 1}}{{Z_{{{\text{in}}}} + 1}}} \right| $$
$$ Z_{{{\text{in}}}} = \sqrt {\frac{{\mu_{r} }}{{\varepsilon_{r} }}} \tan h\left[ {j\left( {\frac{2\pi fd}{c}} \right)\sqrt {\mu_{r} \varepsilon_{r} } } \right] $$
where, RL is the reflection loss (dB), Zin is the normalized input impedance, εr is the complex dielectric constant, μr is the complex permeability, f is the frequency of EM wave, d is the thickness of absorbing material, and c is the propagation speed of EM wave in vacuum.
The reflection losses at different frequencies and thicknesses are clearly shown in a two-dimensional contour map (Fig. 2a–d). The smaller the value of RL is, the more EM wave energy is consumed. We regard −10 dB as the boundary to indicate the effective EM wave absorption. The frequency range with RL less than −10 dB is defined as the EAB, which is marked with dotted line in Fig. 2a–d. The peak of maximum RL value moving toward a lower frequency with the increase in thickness can be explained by λ/4 equation:
$$ d = \frac{{\lambda_{0} }}{4} \cdot \frac{1}{{\sqrt {\varepsilon_{{\text{r}}} \mu_{{\text{r}}} } }}\left( {2l - 1} \right) = \frac{c}{{4 \cdot f_{{\text{m}}} }} \cdot \frac{1}{{\sqrt {\varepsilon_{{\text{r}}} \mu_{{\text{r}}} } }}\left( {2l - 1} \right) $$
Two-dimensional RL map of a CP-1, b CP-2, c CP-3, d CP-4, maximum reflection loss value and EAB at different thicknesses of e CP-1, f CP-2, g CP-3, h CP-4
where, d is the thickness of the absorbing material at the maximum RL value, λ is the wavelength and fm is the frequency. It is clear that the thickness, frequency and electromagnetic parameters have relevance on the maximum RL value. Because no magnetic material is used in this work, μr remains unchanged, εr increases with the amount of CF increases (Fig. S3). In addition, the d and fm are the thickness and frequency of the absorbing material at the maximum RL value, while c is constant. Under the same thickness, the fm is directly proportion to the square root of εr. Accordingly, εr increases with the amount of CF increases, the RLmin value shifts to lower frequencies. The detailed EM absorption data of all samples are summarized in Fig. 2a–f. Undoubtedly, the CP composites show excellent MA properties. Even though the amount of CF is extremely low, an EAB greater than 10 GHz is achieved. More precisely, when the amount of CF is further increased, the EAB of CP-3 and CP-4 can reach as high as 14 GHz. The absorbing property of the composite material basically improves with the increase in CF under the same thickness. These findings suggest that the excellent absorbing performance is attributed to the synergetic behaviors of multiple mechanisms including the suited impedance matching, multiple reflections of porous structure, great polarization loss between components, and conductive loss of CF itself.
Aiming at elaborating the advancement and application of the present absorbing materials, RL curves and the comparison of CP composites with other absorbers reported previously are exhibited in Fig. 3. The MA capacity of the CP-3 and CP-4 composites based on porous structure can cover the whole X and Ku bands and most of the C-band, suggesting a distinguished EAB with the 87.5% test range. In view of that, the CP composites fully exhibit advancement and practicability toward ultralight and ultra-wideband absorbing application compared with the porous structures [34, 42,43,44,45,46] and resin matrix composites [47,48,49] reported previously (Fig. 3b).
a RL curves of CP composites with the thickness of 4 cm, b comparison with the MA properties of other absorbers reported previously
The λ/4 equation corresponds to interference in electromagnetic wave consumption. In the transmission line calculation model, the absorbing material is coated on the metal base plate by default. When the incident electromagnetic wave perpendicularly penetrates into the absorbing material and reaches the metal base plate, the corresponding reflection phenomenon will occur. During the period, interference behavior is present as the reflected electromagnetic wave encounters the incident electromagnetic wave, and the thickness d should satisfy Formula (3). With the increase in material thickness, multiple interference absorption peaks are identified. As shown in Fig. S2, the peaks of the reflection loss curves of CP composites at the thickness of 4 cm located on the λ/4 curves, indicating the well fit with the λ/4 model [50, 51]. The interference absorption peak is symbolized by sharp and strong absorption. With the increase in carbon fiber, these kinds of sharp peaks decrease, especially for CP-3 and CP-4, mainly because of the interference behavior and their own attenuation ability. The EM loss capacity of CP composite with the increase in carbon fiber dosage is exhibited in Fig. S4b. The loss capability enhancement validates the fact that enormous electromagnetic waves entering the material are basically consumed rather than reflected. As a result, interference will not occur. The reflection loss curve thus becomes relatively smooth with almost no sharp peaks [52]. This phenomenon is more prominent at high frequencies since EM wave with a shorter wavelength at high-frequency region is preferable to be consumed under the same material thickness. There still exists the partial interference after the electromagnetic wave which is not completely consumed at low frequency is reflected back. Thus, the reason for the degradation of the RLmin of CP-4 is that the reflection loss is mainly due to its own loss capacity of electromagnetic waves.
The absorbing properties of the absorbers heavily depend on their own EM parameters. To deeply investigate the source of absorbing properties, the EM parameters are analyzed. Given that the magnetic ingredients are not used in composite materials, only the dielectric behavior is studied when discussing the EM parameters. Accordingly, only polarization and conductive losses were present in the CP composites. With the increase in CF content, both the real part and the imaginary part of dielectric parameters show a rising trend (Fig. S2). The dielectric real part increases from 2.2 to 4.2, and the dielectric imaginary portion rises from 0.4 to 1.8, respectively. The protrusion of the imaginary part of the dielectric curve indicates the existence of multiple polarization relaxation processes. With the increase in the amount of CF, the heterogeneous interface between the carbon fiber and the PMI foam increases, which shows the increase in the imaginary part of the dielectric [53, 54]. In Fig. S3b, the protrusion of CP-4 composite material at 3–4 GHz is more obvious than that of other composite materials. In general, the dielectric loss positive cut value represents the dielectric loss capability of the absorbing material. In Fig. S3c, the maximum dielectric loss tangent of CP-4 can reach about 0.5, which implies the super dielectric loss ability of CP-4 composite. In order to provide comprehensive views of the dielectric loss, the imaginary part of dielectric (ε″) is divided into polarization loss (εp″) and conductive loss (εc″), which can be calculated using the following formula in accordance with Debye relaxation theory:
$$ \varepsilon_{c} ^{\prime\prime} = \frac{{\upsigma }}{{{\upomega }\varepsilon_{0} }} $$
$$ \varepsilon_{p}^{^{\prime\prime}} = \frac{{\varepsilon_{s} - \varepsilon_{\infty } }}{{1 + {\upomega }^{2} {\uptau }^{2} }}\omega \tau = {{\varepsilon^{\prime\prime}}} - \varepsilon_{c} ^{\prime\prime} $$
where σ is the electrical conductivity, \(\upomega \) is the angular frequency, ε0 is the permittivity of vacuum, and τ is the period of relaxation. CP-4 is analyzed in accordance with the above equation, as shown in Fig. S3d. With the increase in frequency, εc″ decreases rapidly, whereas εp″ increases gradually. Concretely, εc″ is dominant in the low frequency domain (2–3.8 GHz), whereas εp″ is dominant in the high frequency domain (3.8–18 GHz). This features evidences that when the frequency is relatively high, the steering of the dipole cannot keep up with the EM change, resulting in the relaxation process. In the EM field, as the direction in the EM field changes constantly, the positive and negative charges of the electron cloud are constantly separated, and thus polarization loss is generated.
When the EM wave is incident on the surface of the absorbent material, a portion of the EM wave enters the inside of the material, and other EM waves are reflected in the free space. The reflected EM wave is unable to be consumed, so the primary condition for the consumption of EM wave is to facilitate as much EM wave into the material as possible. The index representing the EM wave entering into the material is normalized input impedance. The larger the value is, the better the impedance matching characteristic of material is. Whether the EM wave entering the absorbing material can be consumed depends on the attenuation coefficient of the material itself. The normalized wave impedance matching coefficient Z and attenuation coefficient α can be expressed by the following formula:
$$ Z = \frac{{Z_{1} }}{{Z_{0} }} = \left| {\sqrt {\frac{{\mu_{r} }}{{\varepsilon_{r} }}} } \right| $$
$$ {\upalpha } = \frac{\sqrt 2 \pi f}{c} \times \sqrt {\left( {\mu^{\prime\prime}\varepsilon^{\prime\prime} - \mu ^{\prime}\varepsilon ^{\prime\prime}} \right) + \sqrt {\left( {\mu^{\prime\prime}\varepsilon^{\prime\prime} - \mu ^{\prime}\varepsilon ^{\prime}} \right)^{2} + \left( {\mu^{\prime}\varepsilon^{\prime\prime} + \mu ^{\prime\prime}\varepsilon ^{\prime}} \right)^{2} } } $$
As mentioned above, the construction of a porous structure is conducive to improving the impedance matching characteristics. Even for the CP-4 composites with the worst impedance matching, most of the Z values are above 0.5 (Fig. S3a), which are significantly better than those of the common microwave absorber/paraffin composite. The strategy of improving the impedance matching of microwave absorbing materials by constructing porous structures is feasible and effective. Such excellent impedance matching characteristics enable ultrawide frequency absorption. Obviously, with the increase in the amount of CF, the attenuation ability of microwave absorbing materials to EM waves is enhanced. For CP-3 and CP-4 composites, their attenuation coefficients reach the maximum values of as high as 68 and 91, respectively (Fig. S3b). The cooperative role of impedance matching and attenuation capacity enables CP-3 and CP-4 to achieve ultrawide effective absorption of about 14 GHz.
Multilayer Structure Design and Characterization
Through multilayer design optimization, the performance of each layer can be complementary, and the EAB can be further expanded at the same thickness. Alternatively, the total thickness of the multilayer material can be further reduced at certain absorbing properties. Genetic algorithm is the most effective method to design multilayer absorbing materials. In this study, EM parameters of each layer absorbing material are regarded as database resources, the total thickness is limited to 3 cm, the number of layers of multiple layers is less than three layers, effective absorption in the range of 4–18 GHz is considered convergence conditions, and optimization is carried out using a genetic algorithm. When the CP-1 material with the thickness of 0.4 cm is used in the upper layer, and CP-4 material with the thickness of 2.6 cm is used in the lower layer, the optimized simulation results of the reflection loss suggest that an effective absorption of 4–18 GHz (14 GHz) is achieved, as shown in the purple dotted line in Fig. 4a. In comparison, the EABs of the single-layer CP-1 and CP-4 materials at the same thickness are only 7.0 and 12.9 GHz, respectively. We may deduce that the multilayer design can maximize the absorbing effect of the absorbent material, and excellent absorbing properties can be achieved at a thinner thickness. For comparison, the real MA performance of objective CP-4 is tested via space method. The sample integrated by the 30 × 30 × 0.4 cm3 CP-1 material and the 30 × 30 × 2.6 cm3 CP-4 material has been fabricated through the vacuum bag pressing process to fulfill the measure requirement. As shown in Fig. 4, the measured absorbing performance is basically consistent with the simulation result, and the position of absorption peak is slightly offset, which may be related to the size error of the material in the preparation process. To analyze the absorption mechanism of the multilayer structure, CST Microwave Studio is used to conduct in-depth research on the electric field intensity and power loss distribution of each absorption peak frequency point, as depicted in Fig. 4b–d. The maximum value of electric field intensity mainly concentrates at the first layer and between the two layers of the multilayer structure, which is caused by interference of multiple λ/4. From the power loss image, the power loss mainly concentrates on the surface of the second layer, where the EM wave energy is consumed in large quantities. As for the first layer, the power loss is relatively low, implying that the first layer is mainly a matching layer and more EM waves are allowed to enter the interior of the multilayer structure. Besides, as the frequency increases, the power loss becomes increasingly concentrated on the upper and surface of the second layer. This may be considered that the wavelength λ of EM waves becomes small as the frequency increases, and the EM wave can be consumed within a shorter transmission path after the EM wave enters the structure.
a Results of multilayer optimization design, simulation of electric field intensity and power loss at b 4.7 GHz, c 8.3 GHz and d 12.2 GHz
Analysis of Absorbing Mechanism
Based on the above results discussions and simulation analysis, the schematic diagram of the absorbing mechanism of CP composite is shown in Fig. 5. On the macroscale, the impedance gradient structure is conducive to enhanced impedance matching. The dielectric parameters of materials with low dielectric coefficient are closer to those of air, which can be used as a matching layer to facilitate EM waves to enter the structure. The porous structure of CP composites, which contain a large amount of air, rendering a better impedance matching characteristic compared with conventional absorbing materials [55, 56]. This strategy to improve the MA performance by constructing a porous structure is an important first step to broaden EAB. Simultaneously, multiple reflections occur when EM waves enter the porous structure, which improve the travel distance of EM waves in the structure, resulting the EM waves being consumed in a shorter time. From the microscopic point of view, the consumption of EM wave mainly comes from the conductive loss, the polarization relaxation of dipole, the polarization relaxation of interface and the defect induced polarization. Under the action of an external electric field, electrons of CF will move and cause weak current, resulting in conductive loss. The 3D network forms a local conductive network, which is highly advantageous for the formation of conductive loss. Additionally, the existence of polar functional groups such as carbonyl radical contributes to the dipoles. Under the action of an alternating electric field, the rotation of dipoles cannot keep up with the speed of EM, which leads to dipole polarization. Dipole polarization relaxation basically occurs in a high-frequency range, which is mainly caused by the faster turning speed of the electric field at the high frequency. Dipole polarization consumes electromagnetic waves in the form of thermal motion of the dipole under the action of an applied alternating electric field. In addition, CF and PMI are two components with different dielectric properties. When additional EM fields are applied, interface polarization occurs between the interfaces of the two components. These two substances are structured capacitor-like and collect positive and negative charges at the ends of the capacitor. When the direction of the applied EM field changes constantly, the positive and negative charges between the two plates also change constantly, thus forming interface polarization. More importantly, grain boundary as a kind of planar defect is commonly found in absorbers with multiple components. The existence of defect sites on interfaces of different components for accommodating their lattice mismatch. In this regard, both interfacial polarization and defect-induced polarization can lead to EM wave loss.
Schematic diagram of the absorbing mechanism of CP composite
Analysis of High Temperature Mechanical and Microwave Absorbing Properties
Aside from the ultra-wideband absorption, exhibiting excellent absorbing properties are required for absorbing materials when encountering high-temperature harsh environments. The changes in dielectric parameters of the CP-4 composite at different temperatures are shown in Fig. 6a–b. As the temperature increases, the curve shape of the real and imaginary dielectric parts remains substantially unchanged, but the curve position is shifted upward. The tangent value of dielectric loss changes minimally on the whole and shows a slight trend of decreasing (Fig. 6c). With the increase in temperature, the dielectric loss ability gradually decreases, and the absorbing ability of composite materials also presents a trend of declining. The variation in dielectric parameters can be explained by the following formula in Debye relaxation theory:
$$ \varepsilon ^{\prime} = \varepsilon_{\infty } + \frac{{\varepsilon_{s} + \varepsilon_{\infty } }}{{1 + \omega^{2} \tau^{2} }} $$
$$ \varepsilon = \frac{{\varepsilon_{s} - \varepsilon_{\infty } }}{{1 + \omega^{2} \tau^{2} }}\omega \tau + \frac{\sigma }{{2\pi f\varepsilon_{0} }} $$
a Dielectric real part, b dielectric imaginary part, c dielectric loss tangent value, d RL curves at different temperature, e compressive strength and f typical stress–strain curves of CP-4 with different temperatures
When the temperature of the absorbing material changes, the greatest influence is the time corresponding to the polarization relaxation. According to Eqs. (8) and (9), with the increase in temperature, the thermal kinetic energy of the activated molecule increases, and the faster the process tends to steady state, the shorter the relaxation time is. Therefore, the dielectric real part increases with the increase in temperature. Similarly, the temperature is positively correlated with conductivity. The increase in temperature will activate substantial free electrons, resulting in an increase in electrical conductivity, and an increase in the value of the dielectric imaginary part. According to Eq. (1), RL values under different thicknesses are calculated. The thickness of composite material that can cover the whole X band increases with the increase in temperature. As the temperature increases from 323 to 473 K, the corresponding thickness increases from 2.5 to 3.3 cm, which suggests a slight decrease in absorption performance.
If the absorbing material has both absorbing function and excellent mechanical properties, it can play a role in the integration of structure and function in aerospace applications, so that more space can be saved for other functions. A rigid foam is generally formed by co-curing process to prepare a sandwich structure. High temperature and pressure are required in the curing process. The maximum curing temperature can reach 453 K, and the curing pressure can reach 0.6 MPa. With the increase in temperature from 300 to 453 K, the compressive strength of the CP-4 composite decreases from 2.85 to 1.05 MPa (Fig. 6d). At 453 K, the compressive strength retention ratio of CP-4 is 36.8%, which is still higher than that at 0.6 MPa. The imide ring structure endows the composite with rigidity, so that it can maintain dimensional stability during curing process. At the same time, in engineering applications, CP-4 composite materials can also play the role of absorbing wave and bearing pressure in high-temperature and harsh environments. Typical stress–strain curves show that CP-4 exhibits good toughness (Fig. 6e). After yielding, CP-4 has no brittle fracture, instead of maintaining a high compressive capacity in a long strain range. In engineering applications, this feature provides a guarantee for structural security.
In conclusion, CP composites with different CF contents were prepared by free radical copolymerization. The MA properties of the CP composites were tested through experiments and simulations to verify the feasibility of the porous structure to broaden the EAB. Notably, by constructing a porous structure, the EAB of the CP composites can reach an excellent ultra-wideband of 14 GHz in the range of 2–18 GHz. Meanwhile, the compressive strength of the CP composites is 1.05 MPa at 453 K, which fully meets the conditions for preparing sandwich structures in aeronautics and astronautics engineering applications. Moreover, the CP composites can still cover the X band (8.2–12.4 GHz) commonly used in military stealth at 473 K. They meet the requirements of lightweight and high strength in the field of ultra-wide frequency absorption and high-temperature environments.
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This work is supported by the National Natural Science Foundation of China (52172091 and 52172295) and the Interdisciplinary Innovation Foundation for Graduates (Nanjing University of Aeronautics and Astronautics, No. KXKCXJJ202009).
Open access funding provided by Shanghai Jiao Tong University.
College of Materials Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People's Republic of China
Zibao Jiao, Wenjun Huyan, Feng Yang, Junru Yao, Zhengjun Yao, Jintang Zhou & Peijiang Liu
Key Laboratory of Material Preparation and Protection for Harsh Environment (Nanjing University of Aeronautics and Astronautics), Ministry of Industry and Information Technology, Nanjing, 211100, People's Republic of China
Zibao Jiao, Wenjun Huyan, Feng Yang, Junru Yao, Zhengjun Yao & Jintang Zhou
School of Electronic Science and Engineering, Nanjing University, Nanjing, 210023, People's Republic of China
Ruiyang Tan & Ping Chen
School of Materials Science and Engineering, Nanjing Institute of Technology, Nanjing, 211167, People's Republic of China
Xuewei Tao
Zibao Jiao
Wenjun Huyan
Feng Yang
Junru Yao
Ruiyang Tan
Ping Chen
Zhengjun Yao
Jintang Zhou
Peijiang Liu
Correspondence to Zhengjun Yao, Jintang Zhou or Peijiang Liu.
Supplementary file1 (PDF 980 kb)
Jiao, Z., Huyan, W., Yang, F. et al. Achieving Ultra-Wideband and Elevated Temperature Electromagnetic Wave Absorption via Constructing Lightweight Porous Rigid Structure. Nano-Micro Lett. 14, 173 (2022). https://doi.org/10.1007/s40820-022-00904-7
Porous structure
EM wave absorption | CommonCrawl |
Quiver varieties and finite dimensional representations of quantum affine algebras
Author: Hiraku Nakajima
Journal: J. Amer. Math. Soc. 14 (2001), 145-238
MSC (2000): Primary 17B37; Secondary 14D21, 14L30, 16G20, 33D80
DOI: https://doi.org/10.1090/S0894-0347-00-00353-2
Published electronically: October 2, 2000
View in AMS MathViewer
Abstract: We study finite dimensional representations of the quantum affine algebra ${\mathbf {U}}_q(\widehat {\mathfrak {g}})$ using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
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Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 17B37, 14D21, 14L30, 16G20, 33D80
Retrieve articles in all journals with MSC (2000): 17B37, 14D21, 14L30, 16G20, 33D80
Hiraku Nakajima
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
MR Author ID: 248505
Email: [email protected]
Received by editor(s): December 9, 1999
Received by editor(s) in revised form: July 10, 2000
Additional Notes: The author was supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan, and National Science Foundation Grant #DMS 97-29992. | CommonCrawl |
Retrieval of radiative and microphysical properties of clouds from multispectral infrared measurements
Hironobu Iwabuchi ORCID: orcid.org/0000-0002-9311-85981,
Masanori Saito1,
Yuka Tokoro1,
Nurfiena Sagita Putri1 &
Miho Sekiguchi2
Satellite remote sensing of the macroscopic, microphysical, and optical properties of clouds are useful for studying spatial and temporal variations of clouds at various scales and constraining cloud physical processes in climate and weather prediction models. Instead of using separate independent algorithms for different cloud properties, a unified, optimal estimation-based cloud retrieval algorithm is developed and applied to moderate resolution imaging spectroradiometer (MODIS) observations using ten thermal infrared bands. The model considers sensor configurations, background surface and atmospheric profile, and microphysical and optical models of ice and liquid cloud particles and radiative transfer in a plane-parallel, multilayered atmosphere. Measurement and model errors are thoroughly quantified from direct comparisons of clear-sky observations over the ocean with model calculations. Performance tests by retrieval simulations show that ice cloud properties are retrieved with high accuracy when cloud optical thickness (COT) is between 0.1 and 10. Cloud-top pressure is inferred with uncertainty lower than 10 % when COT is larger than 0.3. Applying the method to a tropical cloud system and comparing the results with the MODIS Collection 6 cloud product shows good agreement for ice cloud optical thickness when COT is less than about 5. Cloud-top height agrees well with estimates obtained by the CO2 slicing method used in the MODIS product. The present algorithm can detect optically thin parts at the edges of high clouds well in comparison with the MODIS product, in which these parts are recognized as low clouds by the infrared window method. The cloud thermodynamic phase in the present algorithm is constrained by cloud-top temperature, which tends not to produce results with an ice cloud that is too warm and liquid cloud that is too cold.
Clouds play a vital role in regulating the Earth's radiation budget, through shortwave cooling and longwave warming effects (Ramanathan et al. 1989). The cloud radiative effects depend on the type of cloud, and thus, the radiation budget is controlled by the occurrence of various types of clouds (Hartmann et al. 1992), which complicates our understanding of cloud roles in the climate system. In particular, the radiative effects of ice clouds are not well understood, partly because the optical properties of ice clouds are not well quantified (Baran 2009), which is a major source of uncertainty in ice cloud representations in global climate models. There are discrepancies in satellite observation climatology of ice clouds, and improvement of ice cloud processes is still a challenge (e.g., Waliser et al. 2009). Climatology and spatial and temporal variations of clouds on various scales are also important to understand cloud response and feedback in climate systems. Satellite remote sensing can provide constraints for global cloud properties that are useful for developing cloud parameterizations. Macroscopic, microphysical, and optical properties are generally used in satellite remote sensing of clouds. There are specialized methods for each property, including cloud fraction, cloud-top properties (temperature/pressure/height), cloud thermodynamic phase, cloud optical thickness (COT), and cloud-particle effective radius (CER).
There are two passive remote sensing methods that are commonly used for cloud optical and microphysical properties: infrared (IR) window (split-window) (Inoue 1985; Parol et al. 1991; Giraud et al. 1997) and visible/shortwave IR (VIS/SWIR) bispectral (Nakajima and King 1990) approaches. IR window cloud retrieval is suitable for optically thin high clouds with COT of 0.1–5 (e.g., Garnier et al. 2012), whereas the VIS/SWIR method is suitable for optically thick clouds with COT greater than 1 (Nakajima and King 1990; Platnick et al. 2003). We have developed an IR method to retrieve COT and CER by using the 8.5, 11, and 12 μm bands of the moderate resolution imaging spectroradiometer (MODIS) onboard the Aqua satellite (Iwabuchi et al. 2014). In this method, inversion was based on the optimal estimation method (Rodgers 2000), which simultaneously fits the physics model to measurements and diagnoses rigorous uncertainties and retrieval quality. The optimal estimation method has been used widely for cloud remote sensing (Cooper et al. 2003; Heidinger and Pavolonis 2009; Watts et al. 2011; Walther and Heidinger 2012; Poulsen et al. 2012; Sourdeval et al. 2013; 2015; Wang et al. 2016). In a previous work (Iwabuchi et al. 2014), cloud retrieval was applied only to the ice phase cloud, and the a priori cloud-top temperature (CTT) was independently estimated by the CO2 slicing technique (Menzel et al. 2008) in the MODIS operational product. Thus, the retrieval was strongly constrained by cloud-top prior information and affected by the CTT accuracy in the MODIS product. Because the CTT retrieval itself can depend on COT and microphysical properties, the overall retrieval performance can be obtained if the cloud-top height (CTH), COT, and effective radius are retrieved simultaneously from the window and absorption bands.
In addition, the cloud thermodynamic phase is important because liquid and ice clouds play different roles in regulating the Earth's radiation budget and hydrological cycle. Although cloud retrieval using passive sensors usually assumes single-layer ice or liquid clouds, it leads to substantial errors in estimated cloud optical and microphysical properties if there is a multilayer cloud system or if the assumed cloud phase is wrong (Davis et al. 2009). Recent studies using active remote sensing from CloudSat and Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellites have obtained a globally averaged multilayered cloud occurrence of 25–28 % (Li et al. 2015). A cloud analysis algorithm should include methods for detection and property retrieval of multilayered cloud systems and determination of the cloud phase.
In this paper, an optimal estimation-based cloud retrieval algorithm is presented, where COT, CER, cloud-top pressure (CTP), and surface temperatures are simultaneously retrieved from measurements in ten thermal IR (TIR) bands of MODIS including the window and CO2 and water vapor absorption bands. Combined use of TIR bands enables the cloud thermodynamic phase to be distinguished and allows the method to be used for multilayer clouds, as previous pioneering studies suggest. The cloud retrieval algorithm is developed as part of the Integrated Cloud Analysis System (ICAS), which we develop in this study. This paper is organized as follows. The "Methods" section describes the source data used for cloud analysis, the cloud retrieval algorithm, the forward model, and the measurement and model errors, which are thoroughly quantified by model-to-model and model-to-observation comparisons. In the "Results and Discussion" section, retrieval errors are evaluated based on retrieval simulations in idealized cases, to understand the advantages and limitations of the algorithm. The algorithm is applied to a MODIS granule, and the retrieved cloud properties are compared with the MODIS Collection 6 (C6) operational product. The conclusion is given in the "Conclusions" section.
The measurement data used in this study are from the level 1B product of MODIS onboard the Aqua satellite. MODIS has a swath of 2330 km, and a granule every 5 min covers an area of 2330 × 2030 km. TIR bands have a 1 km resolution, and the ten TIR bands are summarized in Table 1. In addition to bands 29, 31, and 32 in the atmospheric window, the bands used include ozone absorption band 30 (9.6 μm), water vapor absorption bands 27 and 28, and carbon dioxide absorption bands 32–36. The spectral radiance of each band is converted to the brightness temperature (BT). The band mean Planck function for temperature T is defined as
Table 1 Characteristics of MODIS bands used in this study
$$ \overline{B}(T)=\frac{{\displaystyle {\int}_0^{\infty }}B\left(\lambda, T\right)\phi \left(\lambda \right)d\lambda }{{\displaystyle {\int}_0^{\infty }}\phi \left(\lambda \right)d\lambda }, $$
where B is the Planck function, \( \lambda \) is the wavelength, and \( \phi \) is the response function of each MODIS band. The band mean Planck function is precalculated for different temperatures, and the Akima interpolation (Akima 1970) is used to calculate the function or its inverse function, the BT, from the look-up table.
The source data used in ICAS are summarized in Table 2. Meteorological field data, including temperature and ozone and water vapor mixing ratios, are obtained by interpolation in the space-time domain from the Modern-Era Retrospective analysis for Research and Applications (MERRA) product, IAU 3D assimilated state on pressure, which has a horizontal resolution of 1.25 ° × 1.25 °, 42 pressure levels, and a time interval of 3 h. The MERRA product is the atmosphere re-analysis product of the National Aeronautics and Space Administration, Goddard Earth Observing System Model, Version 5 (GEOS-5) with its atmospheric data assimilation system (Rienecker et al. 2011). Concentrations of carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O) are global monthly mean values provided from the World Data Center for Greenhouse Gases of the World Meteorological Organization Global Atmosphere Watch program (Tsutsumi et al. 2009).
Table 2 Summary of MODIS operational product data used in the retrieval algorithm
Sea surface temperature data are from the MODIS 8 day mean level 3 product that is based on the TIR split window method (Brown et al. 1999). The root-mean-square error (RMSE) of SST by the split window method is evaluated as 0.35 K. Sea surface emissivity is determined by using the Fresnel equations for a flat sea surface based on the complex refractive index and the satellite zenith angle. The effects of a rough surface, including the effects of multiple reflection and wind direction, are sufficiently small for our purposes when the satellite zenith angle is 60 ° or less (Masuda 2012). The complex refractive index of seawater is synthesized from that of pure water based on Downing and Williams (1975) with a correction for the salinity effect based on Friedman (1969).
The land surface temperature is from the MODIS land 8 day mean level 3 product (MYD11C2), which is based on the day–night algorithm (Wan and Li 1997). For each day and night satellite overpass, the 8 day mean values are available in the product. In the present study, the land surface temperature is temporarily interpolated by considering the diurnal variation. The RMSE of the land surface temperature is less than 1 K (Wan et al. 2004; Wang et al. 2008). The land surface emissivity is from the baseline-fit emissivity database (BFED) monthly mean product (Seemann et al. 2008). Spectral interpolation is used to infer land surface emissivity in the MODIS bands, assuming that the emissivity is linear to the wavelength as recommended by the BFED documentation. The RMSE of land surface emissivity in the BFED is 0.01 or less in the IR window region and about 0.015–0.025 in the other TIR bands.
Forward model
A physics-based forward model is developed and used in the cloud retrieval algorithm. The forward model takes auxiliary data for the atmospheric profile and background surface properties mentioned above, and it computes the BTs and their partial derivatives with respect to several atmospheric and surface variables. The radiative transfer is calculated by using the correlated k-distribution (CKD) method with six quadrature points for each band. The optimization method of Sekiguchi and Nakajima (2008) is used to determine the CKD coefficients from line-by-line radiative transfer calculations with the HITRAN2012 database (Rothman et al. 2013) and the continuum absorption model (Mlawer et al. 2012). Modeled gas species include water vapor, carbon dioxide, ozone, nitrous oxide, carbon monoxide, and methane.
The bulk optical properties of clouds are precalculated and tabulated for ice and liquid clouds with different particle size distributions and ice crystal habit distributions considering the spectral response function of MODIS spectral bands. In the forward model calculation, the optical properties are interpolated with respect to the CER from the look-up table by using the Akima interpolation. The optical properties of water droplets are computed by the Lorenz-Mie theory. The optical properties of ice particles are obtained from a database published by Yang et al. (2013), who used a combination of the discrete dipole approximation and the improved geometrical optics method for randomly oriented ice crystals of various shapes. Several models of particle habit distribution are incorporated into the model, including solid column, plate, column aggregate, the general habit mixture (Baum et al. 2011; Cole et al. 2013), and the two-habit model (Liu et al. 2014), with different degrees of surface roughness. In the present study, the column aggregate model with very rough surfaces is used because it is assumed in obtaining the MODIS C6 cloud product. TIR measurements are not strongly sensitive to the ice habit assumptions (Cooper et al. 2006).
Radiative transfer in a plane-parallel multilayered atmosphere is solved by the two-stream approximation (Nakajima et al. 2000) with the delta-M method (Wiscombe 1977). Solutions of the two-stream approximation are upward and downward irradiances at layer boundaries, from which the radiances at the top of the atmosphere in arbitrary directions can be calculated. The radiative transfer equation for a single homogeneous layer is written as
$$ \mu \frac{dI\left(\tau, \mu \right)}{d\tau }=I\left(\tau, \mu \right)-\left[\frac{\varpi }{2\pi}\left\{{F}^{\downarrow}\left(1-\frac{3}{2}\mathit{\mathsf{g}}\mu \right)+{F}^{\uparrow}\left(1+\frac{3}{2}\mathit{\mathsf{g}}\mu \right)\right\}+\left(1-\varpi \right)B\left(\tau \right)\right] $$
where I(τ,μ) is radiance at optical depth τ from the top of the layer in a direction with μ = cosθ for view zenith angle θ, ϖ is the single-scattering albedo, g is the asymmetry factor, and F ↑ and F ↓ are upward and downward irradiances, respectively. The second term on the right-hand side of (2) is the radiative source function, J(τ,μ), which is here approximated to be linear to τ, as
$$ J\left(\tau, \mu \right)=a\left(\mu \right)\tau +b\left(\mu \right), $$
where a and b are coefficients determined by the source function values at the layer boundaries. Thus, upward radiance emergent from this layer in the μ direction at the top of layer with optical depth Δτ is analytically solved as
$$ {I}_{\mathrm{top}}\left(\mu \right)={I}_{\mathrm{bot}}\left(\mu \right){e}^{-\frac{\varDelta \tau }{\mu }}-\left(a\left(\mu \right)\varDelta \tau +a\left(\mu \right)\mu +b\left(\mu \right)\right){e}^{-\frac{\varDelta \tau }{\mu }}+a\left(\mu \right)\mu +b\left(\mu \right) $$
Total radiance at the top of atmosphere is computed by the sum of components emergent from all atmospheric layers and the background surface. Band mean radiance calculated by integration over the CKD terms is converted to the BT.
The error of this approximate radiative transfer model is evaluated by comparing the model with an accurate model based on the discrete ordinate method for a variety of atmosphere and cloud states. Correction formulae based on a cubic polynomial for BT bias are developed for each band. After the bias correction, the RMSE reaches a maximum of 0.3 K in band 29, where the scattering effect is strong compared with other TIR bands. The two-stream approximation enables fast calculations, whereas the errors from the radiative transfer approximation are sufficiently small. For cloud retrieval, uncertainties in atmospheric profile and background surface properties are a major source of errors in the forward model.
Figure 1 shows the BT and BT differences (BTDs) at the split window band, calculated for liquid and ice clouds with a CTT of 247 K in a tropical atmosphere with a sea surface temperature of 300 K. As suggested by prior studies, a combination of multiple bands in the window region of 8–13 μm allows the CER to be inferred. Measurements in these bands are sensitive to clouds with a COT of 0.05–20 and an effective radius of 3–100 μm for ice clouds. The COT is defined at a wavelength of 550 nm throughout this paper. The water phase (liquid/ice) is moderately important to the spectral differences in BTs in the split window. Absorption by ice and liquid particles becomes stronger at wavelengths longer than 11 μm, although the ice and liquid phases have different spectral dependences of absorption, which means that the cloud thermodynamic phase can be determined from these bands.
Sensitivities of the split window bands to COT, effective particle radius, and cloud thermodynamic phase. Theoretical relationships of BTDs with COT (a, b) and BT (c, d) in the 11 μm band. Calculations are shown for different effective particle radii for ice and liquid clouds with the same CTTs of 247 K in a typical tropical atmosphere with sea surface temperature of 300 K. The two-habit model from Liu et al. (2014) is assumed for ice particles
Retrieval algorithm
The optimal estimation method (Rodgers 2000) is used to solve an inverse problem. The method fits the forward model to the measurement under constraints by an a priori probability distribution of the state vector in the forward model. Defining state vector x, measurement vector y with the BTs in the MODIS TIR bands as elements, and the model parameter vector b, the problem to be solved is written as
$$ \mathbf{y}=\mathbf{F}\left(\mathbf{x},\mathbf{b}\right)+\mathbf{e}, $$
where F is the forward model, and e = y – F(x,b) is a measurement–model error vector. A cost function is given by
$$ J\left(\mathbf{x}\right)={\left[\mathbf{y}-\mathbf{F}\left(\mathbf{x},\mathbf{b}\right)\right]}^{\mathrm{T}}{S}_{\mathrm{e}}^{-1}\left[\mathbf{y}-\mathbf{F}\left(\mathbf{x},\mathbf{b}\right)\right]+{\left[\mathbf{x}-{\mathbf{x}}_{\mathrm{a}}\right]}^{\mathrm{T}}{S}_{\mathrm{a}}^{-1}\left[\mathbf{x}-{\mathbf{x}}_{\mathrm{a}}\right], $$
where S a is an error covariance matrix of the a priori x a, and S e is a measurement–model error covariance matrix. The Levenberg-Marquardt method is used to obtain a minimized J, at which the solution converges. The final value of J is the retrieval cost, which represents the degree of fit between the model and measurement. The criterion that J is sufficiently small with an optimal solution is set as J < 2m, where the m is a number of the observation vector elements. A feature of the optimal estimation is that the uncertainty of the solution can be diagnosed quantitatively with an error covariance matrix. In addition, diagnostics of the estimation quality, such as the degree of freedom for signal (DOFS) and the information content, are obtained.
The cloud inversion is tried first with a single-layer cloud. If an optimal solution is obtained, the single-layer assumption is accepted. Otherwise, an inversion with a double-layer assumption is tried. A double-layer cloud solution is accepted if J is smaller than that of the single-layer assumption and the COT of the upper cloud is less than 8. This is because TIR measurements lose sensitivity to the lower cloud under the double-layer cloud assumption if the upper COT is more than about 8. The state vector includes cloud properties such as cloud water path (CWP), CER, CTP, and background surface temperature in single-layer cloud cases. With nonlinearity in mind, logarithms of CTP, CWP, and CER are elements of the state vector. The top-pressure of the lower cloud is inferred in double-layer clouds, instead of background surface temperature, as a similar double-layer cloud retrieval is proposed by Watts et al. (2011).
The cloud layer is assumed to be composed of either liquid droplets or ice particles. The ice (liquid) phase is accepted if an optimal solution is obtained with CTT below a threshold temperature, T f = –38 °C (above T m = 0 °C). If the CTT is between T f and T m, or an optimal solution is not obtained, then inversion with the other cloud phase assumptions are tried. Final judgment of cloud phase is to select a lower cloud phase cost function, R, of
$$ R=\frac{P_1}{V_1}+\frac{P_2}{V_2}+\frac{P_3}{V_3}+\frac{J}{J_{\mathrm{opt}}}, $$
where weighting coefficients are set as V 1 = 8 K2, V 2 = 302 K2, V 3 = 0.32, and J opt = 2m. As shown in Fig. 1, the split window bands are sensitive to the cloud phase. Thus, P 1 is the sum of squares of BTD between the observation and model at the split window bands at wavelengths of 8.5, 11, and 12 μm
$$ {P}_1={\left({\mathrm{BTD}}_{8.5-11,\mathrm{o}\mathrm{b}\mathrm{s}}-{\mathrm{BTD}}_{8.5-11,\mathrm{m}\mathrm{d}\mathrm{l}}\right)}^2+{\left({\mathrm{BTD}}_{11-12,\mathrm{o}\mathrm{b}\mathrm{s}}-{\mathrm{BTD}}_{11-12,\mathrm{m}\mathrm{d}\mathrm{l}}\right)}^2, $$
where subscripts "obs" and "mdl" denote the observation and model, respectively. Because the CTT is a main factor that prescribes cloud phase, P 2 increases with increasing deviation of CTT (T top) from a critical temperature, T c,ice = –15 °C for ice and T c,liq = –23 °C for liquid, as
$$ {P}_2=\left\{\begin{array}{c}\hfill {\left[ \min \left(0,{T}_{\mathrm{top}}-{T}_{\mathrm{c},\mathrm{l}\mathrm{i}\mathrm{q}}\right)\right]}^2\kern0.75em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{liquid}\hfill \\ {}\hfill {\left[ \max \left(0,{T}_{\mathrm{top}}-{T}_{\mathrm{c},\mathrm{ice}}\right)\right]}^2\kern0.75em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{i}\mathrm{ce}\hfill \end{array}\right.\kern1.25em . $$
However, CTT is uncertain when CTP retrieval is uncertain. Thus, P 3 in Eq. (7) is the error variance of the logarithm of CTP. Finally, an index of the cloud phase, Q, is calculated as
$$ Q=1+\frac{R_{\mathrm{liq}}^2}{R_{\mathrm{liq}}^2+{R}_{\mathrm{ice}}^2}, $$
which has a value between 1 and 2: Q is 1–1.5 for liquid and 1.5–2 for ice. If the cloud phase costs for liquid and ice phase assumptions have similar values, then Q is nearly 1.5, which means that cloud phase determination is ambiguous.
Assumptions and prior information
As shown by Cooper et al. (2003) and Garrett et al. (2009), explicit representation of cloud top and base boundaries is important in the TIR cloud retrieval. In this study, the cloud base pressure is parameterized using an empirical formula. Sassen and Comstock (2001) and Sassen et al. (2008) showed that the geometrical thickness of the cirrus with COT less than 3 is 1–3 km with a global mean of 2 km based on ground and space-based lidar observations. Sassen and Comstock (2001) and Veglio and Maestri (2011) showed that an optically thicker cirrus cloud becomes geometrically thicker. In the present algorithm, the geometric thickness, H (m), is represented by a function of CWP, W (kg m–2),
$$ H=\left\{\begin{array}{c}\hfill {B}_{\mathrm{liq}}+{A}_{\mathrm{liq}}\sqrt{W/{W}_{\mathrm{liq}}}\kern0.75em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{liquid}\ \mathrm{cloud}\hfill \\ {}\hfill {B}_{\mathrm{ice}}+{A}_{\mathrm{ice}}\sqrt{W/{W}_{\mathrm{ice}}}\kern0.75em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{ice}\ \mathrm{cloud}\ \hfill \end{array}\right.\kern1em , $$
B liq = 20 m, B ice = 20 m,
A liq = 400 m, A ice = 2000 m,
W liq = 0.06 kg m–2, W ice = 0.02 kg m–2.
The cloud base pressure is determined from H and atmospheric temperature and pressure profiles.
The prior information about the cloud properties and background surface temperature is given in Table 3, where T' sfc is the sea or land surface temperature obtained from the MODIS level 3 product, W is CWP, and r e is CER. It is assumed that the surface temperature RMSEs are set to include the uncertainty from daily diurnal variations. The a priori probability distributions of CWP have large dispersions to under-constrain the CWP retrieval. In contrast, the a priori CER has a relatively small standard deviation because CERs obtained from passive remote sensing are in limited ranges (Hong et al. 2007; Wang et al. 2016). S a is an almost diagonal matrix, with weak correlation between the CWP and CER with a correlation coefficient of 0.25. In the inversion, cloud properties are limited to prescribed ranges of realistic values. Lower and upper limits of COT are set because the TIR measurements lose sensitivity in cases with very small and large COTs.
Table 3 A priori information and prescribed ranges of the elements of the solution vector
Because the temperature ranges at which liquid and ice clouds exist are known a priori, the a priori and variance of CTP are determined by considering the vertical distribution of the air temperature profile. It is assumed that a liquid (ice) cloud is not present with CTT lower than –40 °C (higher than 2 °C). In addition, the a priori CTT (T a) and CTP range are assumed as follows.
For liquid clouds, P flz < P top < 0.96P sfc, T a = 5 °C
For ice clouds, 0.9P trp < P top < P mlt, T a = –55 °C
P frz and P mlt are pressures at an air temperature of –40 and 2 °C, respectively, and P tpp is a tropopause pressure. The standard deviation of the a priori CTP on a logarithmic scale, σ lnP , is determined to cover the CTP ranges as
$$ {\sigma}_{\ln P}=0.7\times \max \left[\left| \ln \left({P}_a/{P}_{\min}\right)\right|,\left| \ln \left({P}_a/{P}_{\max}\right)\right|\right], $$
where P a is a priori CTP, and P min and P max are the lower and upper limits of CTP, respectively, determined as previously mentioned.
Measurement and model errors
Observations and models may have bias and noise-like error components arising from various error sources. Model errors include (1) error due to radiative transfer approximations, (2) errors from the representation of atmosphere with a finite number of atmospheric layers, (3) errors in the sea or land surface emissivity, (4) uncertainty in atmospheric temperature and gas concentration profiles, (5) error from assuming the cloud base pressure, (6) uncertainty of the ice habit model and particle size distribution, and (7) error from the vertical and horizontal heterogeneity of the clouds. Each error component may depend on the state of the atmosphere and the background surface, which make it complicated to quantify the error covariance matrix appropriately.
Simple assumptions can be made about several error components. The RMSE of sea surface temperature is assumed as 0.7 K in the inversion by considering daily and diurnal variations in sea surface temperature and possible differences between clear-sky and cloudy cases. According to the observations of Newman et al. (2005), the RMSE of sea surface emissivity due to the uncertainty of seawater optical constants is estimated to be approximately 0.001 at satellite zenith angles of less than 60 °. Over land, the surface temperature and emissivity in cloudy cases are likely to differ significantly from those in clear-sky cases, although the magnitude is uncertain. The RMSE of land surface temperature is assumed as 3 K in this study, although precise quantification is required in the future. BFED land surface emissivity product (Seemann et al. 2008) is created by using the MODIS land surface emissivity product. The error covariance matrix of the land surface emissivity is constructed considering the MODIS product error and the BFED modeling error documented in the literature.
The measurement–model error covariance matrix is divided into two components, as
$$ {S}_{\mathrm{e}}={K}_{\mathrm{b}}{S}_{\mathrm{b}}{K}_{\mathrm{b}}^{\mathrm{T}}+{S}_{\mathrm{e},\mathrm{off}}. $$
The first and second terms on the right-hand side are online and offline calculation terms, respectively. The online term is calculated from the error covariance matrix, S b, and the Jacobian matrix, K b, for the model parameters. K b is calculated in the forward model. Model parameters included in the online calculation term are the background surface emissivity for each band and cloud base pressure. For the cloud base pressure error, the standard deviation of the logarithm of the cloud base pressure is assumed to be approximately proportional to the geometrical thickness of the cloud, which is approximately proportional to ln(P bas/P top), as
$$ \sigma ={S}_{\mathrm{bas}} \ln \left(\frac{P_{\mathrm{bas}}}{P_{\mathrm{top}}}\right), $$
where factor S bas is 0.2. The offline calculation term is divided into three components as
$$ {S}_{\mathrm{e},\mathrm{off}}={S}_{\mathrm{e},\mathrm{R}\mathrm{T}\mathrm{M}}+{S}_{\mathrm{e},\mathrm{atm}}+{S}_{\mathrm{e},\mathrm{noise}}. $$
The right-hand side contains the errors from radiative transfer approximation, S e,RTM, atmospheric profile uncertainty, S e,atm, and measurement noise, S e,noise. S e,RTM is small, as previously described.
To estimate several error components in the measurements and the model, BTs from daytime clear-sky pixel measurements over the ocean are compared with those from the model. Because the RMSE of sea surface temperature is as small as 0.7 K, errors in atmospheric data and model approximations and assumptions are evaluated by the comparison. The clear-sky area is determined based on the cloud mask data in the MODIS Product (Ackerman et al. 1998). If an area about 30 km2 is composed of only the "confidently clear" pixels, then the center area of about 20 km2 is used for the model-measurement comparison. The covariance matrix of measurement noise, S e,noise, is estimated from the variance and covariance of measurement–model differences within a 20 km2 area. The maximum noise is 0.4 K at MODIS band 27, and the noise is less than 0.25 K in the window bands. Figure 2 shows comparisons of the clear-sky BTs. Each data point represents mean values over a 20 km2 segment. Water vapor absorption bands tend to exhibit larger differences between model and measurement as uncertainties in temperature and water vapor amount increase model errors. From this comparison, the systematic difference (bias) between the observations and model calculations is evaluated for each MODIS band, and then the biases are removed from the forward model used in the subsequent analyses.
Comparison of observed and simulated BTs for clear-sky pixels over the ocean. Results are shown for MODIS band 27 (water vapor absorption band) and band 31 (window band). Scatterplots of observed BTs (a, b) and differences in the observation from the simulation (c, d). Red lines in a and b denote fit lines with a slope constrained to be 1. Red lines in c and d denote biases. Each data point represents mean values over a 20 km2 segment
The atmospheric profile error is estimated from the covariance matrix of measurement–model differences obtained in the oceanic clear-sky comparison. The vertical distribution of error patterns of temperature and water vapor, and ozone mixing ratios are determined by fitting the simulated error covariance matrix to the observations. The best estimate of the error covariance matrix is shown in Fig. 3 along with that obtained from observation. By using the estimated atmospheric profile errors, S e,atm for cloudy cases is evaluated by model simulations under a variety of atmospheric conditions. The error in cloudy cases strongly depends on CTT, because a major source of atmospheric profile error in cloudy cases is the error in the amount of water vapor above the cloud top. For lower CTTs, the amount of water vapor above the cloud top generally tends to be smaller in various atmospheric profiles. Thus, S e,atm for cloudy cases is tabulated in five CTT ranges from 200 to 300 K with an interval of 20 K. The results for high and low CTTs are shown in Fig. 3. The error covariance matrix with high CTT is close to that for clear-sky cases. The errors decrease at the water vapor absorption bands for lower CTT.
Model error covariance matrices for atmospheric profile errors. a Estimation from comparison of clear-sky observations over the ocean to model calculations. b, c, d Estimations from Monte Carlo simulation of estimated errors in atmospheric profiles, for b clear sky, c cloudy sky with CTT of 280–300 K, and d cloudy sky with CTT of 280–300 K. Each panel shows an image of error covariance matrix. The row and columns denote ten MODIS bands
Because not all error sources are included in Eq. (13), initial tests show that the model does not fit the measurements well if Eq. (13) is used directly in the cloud property inversion. The uncertainty due to the horizontal and vertical heterogeneity in clouds and the uncertainty in the optical properties of ice particles from the ice habit model are not included in Eq. (13). These uncertainties are difficult to quantify; however, based on by trial and error, we artificially set the diagonal elements of the error covariance matrix obtained from Eq. (13) as 20 % larger.
Retrieval error evaluation by simulations
The errors and performance of cloud retrieval are tested by retrieval simulations. Measurement signals with errors are simulated by the forward model calculations for perturbed atmospheric and surface states with random noise that obey the error covariance matrices assumed above. Retrieval errors are evaluated by comparing the retrieved cloud properties from the noise-superimposed measurement signals with the initial values. This methodology is identical to that used by Iwabuchi et al. (2014). For each state, a series of 1000 retrieval simulations are performed to evaluate the mean bias error and the RMSE. The satellite viewing zenith angle ranges from 0 ° to 60 °, and a tropical atmosphere is assumed with a sea surface temperature of 300 K.
Figure 4 shows results for ice clouds with CTT = 221 K. The retrieval bias and RMSE, correct cloud phase discrimination rate, and DOFS are shown by color scales as functions of initially assumed values (truth values in this test) of COT and CER. The rate of the optimal cloud retrievals for all trials is also computed for each cloud state. The retrieval errors presented are evaluated only for optimal retrievals with correct cloud phase identification. The estimation of CTP is accurate with a small bias of within ±10 % for almost all cloud states tested here. CTP RMSE is less than 20 % for optically thick clouds with COT of 0.3 or more. When the initial COT is 0.1–10 and CER is 3–60 μm, the CER and COT retrieval biases are generally less than 15 %, RMSE is less than 30 %, and DOFS is close to 3. For very thin and very thick clouds, the sensitivity of the forward model to CER is low, which explains the CER retrievals close to the a priori values and the low DOFS. Optimal cloud retrievals are obtained for 100 % of clouds with COT of 0.1. Optically thinner clouds are not retrieved because they are excluded in the retrieval algorithm as clear sky. With the exception of cases with very optically thin clouds, cloud phase determination is correct with a correct identification rate of about 100 %.
Retrieval test results by retrieval simulations for ice clouds over the ocean in a tropical atmosphere. Results are shown for CTT of 221 K and different COT and CER. MBE, RMSEs, and DOFS are evaluated from retrievals with optimal solutions and correct cloud phase determination. Optimal cloud retrieval rate and correct cloud phase retrieval rate is also evaluated. For each cloud state, 1000 retrievals from noise-superimposed simulated measurements are attempted
Figure 5 shows the results for liquid clouds with CTT = 277 K. Because the CTT is closer to the background surface temperature than for ice clouds, the retrieval performance for liquid cloud is worse than for ice clouds, resulting in lower DOFS. The CTP bias and RMSE are very small over a wide range of COT and CER, which is better than for ice clouds. This is mainly because a liquid cloud is geometrically thinner than an ice cloud. Use of a weakly absorbing CO2 band (MODIS band 33) might improve the estimation of the top of low clouds. Both CER and COT bias are within ±30 % when the COT is 0.3–20 and CER is 4–20 μm. The percentage of correct cloud phase identification is about 100 % in most cases shown here. When both the COT and CER are small, cloud phase identification is not correct with a score of about 50 %. This is because COT is limited to be more than 0.25 for liquid clouds.
Retrieval performance is tested at various CTTs. The initial assumptions, as the truth in this test, about cloud phase are as follows. There is exclusively an ice cloud for CTT ≤ –38 °C, exclusively a liquid cloud for CTT ≥ 0 °C, and the ice and liquid cloud fractions vary linearly with CTT between –38 and 0 °C. CERs are assumed to be typical values of 13 μm for liquid clouds and 32 μm for ice clouds. Figure 6 shows the test results. The optimal cloud retrieval rate is approximately 100 % for clouds that are optically thick enough. Very optically thin clouds are not retrieved because they are identified as clear sky in the retrieval algorithm. Cloud phase discrimination is accurate for CTT lower than –38 °C or higher than 0 °C, except for very optically thin cases. Cloud phase determination is difficult when CTT is between the two critical temperatures, particularly for COT of less than 0.5. CTP of optically thin (COT <0.3) clouds also has a problem with biases. Low sensitivity to CTP means that CTP retrievals tend to be close to the a priori values. The poor estimation accuracy of the CTP is observed for CTT around 200 and 265 K. However, the CTP can be retrieved accurately for optically thick clouds for any CTTs. The CTP bias is less than 10 % for COT >0.3, and the CTP RMSE is less than about 10 % for COT >0.5. For the retrieval error of COT, the bias is less than 15 %, and the RMSE is less than 30 % for high clouds with COT = 0.1–10. The COT retrieval error generally increases with increasing CTT. However, CER retrieval shows good performance over wide ranges of CTT and COT in this test, where CER truths are assumed to be close to the a priori values.
Retrieval test results by retrieval simulations for clouds with different COT and CTT. It is assumed that occurrence of the ice phase increases with decreasing CTT from 0 to –40 °C. CER is assumed to be 13 μm for liquid clouds and 32 μm for ice clouds. MBE, RMSEs, and DOFS are evaluated from retrievals with optimal solutions and correct cloud phase determination. For each cloud state, 1000 retrievals from noise-superimposed simulated measurements are attempted
It would be desirable to clarify the benefits of using ten TIR bands to retrieve cloud macrophysical and microphysical properties simultaneously. In the CO2 slicing technique for CTP retrieval, cloud effective temperature is estimated assuming that the cloud is isothermal and the cloud emissivities in the two neighboring bands are identical. Cloud emissivities in the split window bands can be obtained by using this cloud temperature estimate. Heidinger et al. (2015) presented a simple, efficient two-stage method for retrieving COT and CER from the cloud emissivities in the split window. First, the macrophysical properties (CTP/CTH/CTT) are retrieved, followed by the optical and microphysical properties. Iwabuchi et al. (2014) also used the CTT obtained from the CO2 slicing technique to help estimate the optical and microphysical properties from the split window bands of MODIS. In the present study, we test a surrogate method for CTP retrieval, instead of the CO2 slicing technique itself. MODIS bands 35 and 36 are used to retrieve CTP in the optimal estimation framework, with CER strictly constrained at the a priori value for ice clouds. The two bands are sensitive to the upper troposphere. In the MODIS cloud product, CTPs of most high clouds in the tropics are retrieved by using those two bands. The fixed CER assumption is reflected in the fixed spectral dependence of the optical properties of the ice particles, which is a better assumption than identical cloud emissivities at the two bands. Figure 7 shows the CTP retrieval errors for ice clouds with a CER of the a priori value (25 μm). The two-band retrieval is compared with the ten-band retrieval with the same assumptions about the measurement and model errors. The simultaneous retrievals from using all ten bands outperform those based on using only the two CO2 bands, and the ten-band retrieval produces more certain estimates of CTP. If the assumed cloud temperature is highly uncertain in the second stage of retrieval of COT and CER, it is likely that the COT and CER estimates will be more erroneous. The simultaneous retrieval of macrophysical and microphysical properties can provide better consistency between the physics model and measurements for all bands.
CTP retrieval errors estimated by retrieval simulation for ice clouds with different CTTs. The CTTs are a 204.5 K and b 234.5 K. In this simulation, CTP are retrieved from MODIS bands 35 and 36 (yellow) or all bands (blue), with CER strictly constrained at the a priori value for ice clouds. Mean bias error and standard deviation of error are denoted by diamond markers and error bars, respectively. For visibility, results for bands 35 and 36 are shifted slightly along the horizontal axis
Application to tropical cloud systems
As an initial test, the results obtained from the present retrieval algorithm in ICAS are compared with the MODIS C6 cloud products with a 1 km resolution. The MODIS granule in this case is acquired on April 1, 2007, at 3:55 UTC, which is the same as the case in Iwabuchi et al. (2014). The region is located in the ocean to the north of New Guinea. An area of about 200 × 1200 km is selected to illustrate the comparison results, covering the CloudSat/CALIPSO ground track (Fig. 8). The CTH obtained from ICAS is compared with the CloudSat/CALIPSO cloud mask product developed by Kyushu University (Hagihara et al. 2010). Because CloudSat/CALIPSO observes the Earth in a nadir view, a parallax correction is applied to the ICAS retrieval to compare the datasets coherently. Furthermore, the parallax-corrected ICAS retrieval is regridded into the CloudSat/CALIPSO track by using nearest neighbor interpolation. The evaluated pixels are only pixels with optimal retrieval. In the region of interest, there are about 1868 collocated pixels between ICAS (with optimal retrieval) and the radar–lidar product. More than 90 % of the pixels are detected as cloudy pixels by both products, whereas about 2 % are detected as clear-sky pixels. ICAS incorrectly assigns about 1.5 % of pixels as cloudy and misses about 0.5 % of cloudy pixels in the radar–lidar product, although collocation is not certain owing to the parallax.
A test case of tropical cloud systems over the ocean north of New Guinea. The MODIS granule is taken at 3:55–4:00 UTC on April 1, 2007. a, b False color image with band combinations of MODIS bands. a Reflectances in bands 6, 2, and 31 and b BTD between bands 29 and 31 and BTs of 32 and 31 for the red, green, and blue channels. Yellow lines in a and b denote the ground track of CloudSat and CALIPSO satellite. c Comparison of retrieved CTHs with cloud mask data obtained from CloudSat radar and CALIPSO lidar measurements. Blue denotes clouds detected by radar only (RO), green by lidar only (LO), yellow by both radar and lidar (RAL), and orange denotes missing data (Miss). The scattered crosses show the location of the ICAS cloud top at the respective longitude, where black crosses represent cloud top of the first layer and red crosses represent cloud top of the second layer
Figure 8 shows that the radar–lidar product detects many pixels with a high cloud top. The upper clouds cover wide areas, and parts of the left and right sides of the figures are covered with thick clouds. Middle-level clouds are present in the middle of the figures. The MERRA atmospheric profile shows that the CTT of the middle cloud is about –20 °C. The upper part of the high clouds probably consists of small ice particles and it can be detected only by lidar (green). The CTH obtained from ICAS tends to be lower than that of the radar–lidar product and close to the cloud top detected by cloud radar (blue and yellow). The top height of ICAS varies from 5 to 17 km, whereas the radar–lidar product has a more uniform top height with an altitude of around 15 km. Similar to the ICAS cloud top, the cloud top detected by radar is more variable than the cloud top detected by lidar. This is an expected limitation of ICAS because previous studies have shown that IR measurements are not sensitive to very optically thin clouds, which can be sensed only by lidar and not by radar (Watts et al. 2011). ICAS detects more pixels with a cloud top above 15 km compared with radar–lidar products. Some pixels correspond to CTHs that are higher than the lidar measurements (around region C). This is probably an erroneous retrieval because the ICAS results are influenced primarily by the uncertainty of the atmospheric profile, and possibly by the ice habit assumption and vertical inhomogeneity within the cloud, yielding an ICAS cloud top near the tropopause that is too high compared with the lidar measurement. Red crosses in Fig. 8c denote the cloud top of the lower cloud of the double-layer cloud retrieval in ICAS. Several parts of the second layer top in ICAS match the cloud top of the third cloud layer in the radar–lidar profile, although there are parts that deviate greatly. The cloud tops of the upper cloud layer in double-layer cases are well estimated, similar to single-layer cloud cases. ICAS misses most multilayer clouds at longitudes of 146.35 °–146.45 ° (region A), where the upper first and second cloud layers are detected only by lidar. ICAS wrongly identified these pixels as single-layer cloud and retrieved CTHs between the first and second cloud layers. A similar problem occurs at longitudes of 146.1 °–146.2 ° (region B). In these cases, the uppermost cloud is too optically thin, and ICAS cannot identify the upper cloud in a multilayer cloud system, probably because the retrieval with the single-layer cloud assumption has an optimal solution with clouds at the wrong height. These results suggest that the algorithm requires re-examination and improvement.
Figure 9 shows a comparison of the cloud properties in ICAS and C6. In C6, the VIS/SWIR method is used for cloud optical and microphysical properties (COT and CER), and the CO2 slicing method and the IR window method are used to estimate the cloud top in C6. Two methods are used for cloud thermodynamic phase determination in the MODIS 1 km products. One is a method based on the BT and BTD in the TIR bands supplemented by using the cloud emissivity ratios (TIR method; Baum et al. 2012). The other is a method used in the retrieval of daytime optical properties of clouds (shortwave algorithm; Marchant et al. 2016), in which cloud phase is determined from "votes" from several tests based on the BTs of the TIR window bands, CER values determined from different combinations of multiple VIS and SWIR bands, CTT, and the water vapor absorption band at 1.38 μm. There is a high correlation between COTs in ICAS and C6, although the COT in ICAS tends to be smaller and limited to less than 20. ICAS works well for retrieval of thin clouds. In C6, COT (CER) retrieval is not available for thin clouds with COT of less than about 0.5 (Eq. (1)). The retrieval in ICAS is available for many pixels of optically thin, high clouds. As shown in the previous subsection, CER of optically thick clouds has a large uncertainty in ICAS because of low sensitivity. In the optically thin parts at the edges of the upper clouds, the CTH in C6 is low, whereas ICAS CTH has high values. In the CO2 slicing method, C6 and ICAS agree well. In the IR window method, C6 shows low clouds with CTH of 0–3 km, whereas the CTH is 10–15 km in ICAS, resulting in significant differences as large as 10 km or more. ICAS retrieval has more ice cloud pixels (Fig. 9i–k), particularly at optically thin edges of high clouds. Over the central area of the images, thin upper layer clouds overlap over the middle-level cloud, as seen in the radar–lidar profile. In these areas, many pixels are determined as ice phases in ICAS, whereas CTHs are in the middle of the thin upper cloud and middle cloud. The shortwave algorithm in C6 shows a liquid phase for these pixels, and the phase is undetermined in the TIR method in C6, indicating difficulty in determining unique cloud phases in multilayer cloud systems.
Comparison of retrieved cloud properties between ICAS and MODIS products. CTH in a ICAS and b MODIS. c CTH method used in the MODIS product, showing pixels for the IR window and CO2 slicing methods and clear-sky pixels. COT in d ICAS and e MODIS. CER in f ICAS and g MODIS. Cloud thermodynamic phase estimated in ICAS. h The phase index defined in Eq. (10), and i binary results for the cloud phase. Ice/liquid binary results are based on the original cloud phase index data that has continuous values from 1 to 2. Cloud thermodynamic phase estimated in the MODIS product: j shortwave (SW) algorithm and k TIR algorithm. Gray pixels in k represent the mixed phase
Statistical comparisons for the collocated pixels are made for a full granule that covers a 2000 km2 region. Figure 10 shows the CTH comparison between ICAS and C6. In C6, CO2 slicing is applied to the high clouds, and the CTH retrieval agrees well with the CTH in ICAS. In the optically thin ice clouds, CTH in C6 tends to be higher than the ICAS estimates (Fig. 10c). If the IR window method is used in C6, the pixels are covered by upper clouds in ICAS, whereas they are covered by low clouds in C6. When the comparison is limited to ice cloud retrieval in ICAS and IR window retrieval in C6, the ICAS–C6 difference in CTH tends to increase with decreasing COT. As seen in the spatial distribution, the optically thin parts at the edges of upper clouds are treated as lower clouds in C6. These results suggest that ICAS can capture the CTH of optically thin upper layer clouds well.
Comparison of CTHs obtained from ICAS and the MODIS product. The occurrence frequency distributions are shown. Results are shown for pixels retrieved by the a CO2 slicing method and b IR window method in the MODIS product. CTH differences plotted against COT for ice clouds in ICAS retrieval for the c CO2 slicing method and d IR window method in the MODIS product
Figure 11 shows results for the COT and CER for ice clouds. COT shows good agreement when the COT is less than about 6. For thick clouds, TIR measurements lose their sensitivity to the lower part of cloud systems, whereas the visible reflectance is sensitive to the total column COT. Many pixels with multilayer clouds are probably included in the results presented here. The COT from the TIR measurements should be considered as the COT of upper clouds in multilayer cloud systems. This explains why the C6 COT is often larger than ICAS COT, particularly when COT is larger than about 6. In contrast, CERs in ICAS and C6 exhibit significant dispersion, showing a weak correlation with a correlation coefficient of 0.24. The correlation is higher with a coefficient of 0.31 for comparing ice clouds with COTs in a range of 1–6, where both TIR and VIS/SWIR algorithms have a high sensitivity to CER.
Comparison of COTs and CERs obtained from ICAS and the MODIS product. Results for a COT and b CER of ice clouds, shown as the occurrence frequency distribution
A simple comparison of the cloud thermodynamic phase on a pixel-by-pixel basis does not make sense because different products retrieve or assume different CTTs in the cloud phase determination. We also compare the temperature dependence of the ice/liquid phase occurrence fraction. Figure 12 shows the ice cloud fraction as function of CTT. ICAS and C6 ice results are plotted against CTT for the products. The ice fraction in ICAS decreases from 1 to 0 over a temperature range of –40 to –0 °C. The ice fraction from the C6 TIR algorithm is similar to ICAS for CTTs lower than –5 °C. In the C6 TIR algorithm, there are ice phase pixels even with CTT above 0 °C. However, it is important to check that the absolute number of such pixels is small because most pixels are covered by ice clouds in the results from all the algorithms. In contrast, for the shortwave algorithm (dotted line), the ice fraction is much lower than those in the ICAS and C6 TIR algorithms. The cloud phase in the shortwave algorithm is determined to explain the VIS/SWIR reflectance measurements well. In multilayer cloud systems with optically thin ice clouds over optically thick liquid clouds, shortwave reflectances could be sensitive to the lower cloud. Thus, the cloud phase determination can differ from that from TIR measurement, which is sensitive to the upper cloud. In the C6 shortwave algorithm result, the ice fraction changes abruptly at a CTT of about –33 °C (~240 K). According to Marchant et al. (2016), the CTT test uses a threshold at 240 K to "vote" for the ice clouds. This could result in the discontinuity of the ice fraction. As shown previously, cloud phase inference in ICAS is based on the consistency between the model and measurements under the constraints given by a priori knowledge about the relationship between CTT and the phase, relying on BTDs in the split window with large weights. Thus, the cloud phase determination in ICAS is constrained strongly by the cloud phase dependence on the CTTs, which guarantees that ice clouds that are too warm and liquid clouds that are too cold are not retrieved. However, the strong constraint on the cloud phase is a limitation of our algorithm because a small amount of information about the cloud phase comes from measurements. Baum et al. (2012) showed that in their algorithm refinement for MODIS C6, using cloud emissivity ratios between the split window bands substantially improves the inference of the ice cloud phase, especially for optically thin ice clouds. In future work, cloud phase inference could be improved by using this index.
Ice cloud occurrence fraction for ICAS and MODIS C6 SW and TIR algorithms. The ice fraction is shown as a function of CTT. The MODIS results are calculated from the numbers of pixels of ice, liquid, and mixed phase clouds, not including undetermined phase pixels
An optimal estimation-based cloud retrieval algorithm has been developed to estimate the optical and physical properties of clouds simultaneously from measurements of several TIR bands. A major source of modeling errors is uncertainties in atmospheric profiles, which are usually difficult to quantify. In this study, they are assessed by direct comparison of the clear-sky observations over the ocean with the model calculations. This type of model-measurement comparison is important for developing cloud retrieval algorithms. A feature of the present algorithm is that the COT and the CTH is retrieved well for optically thin clouds by simultaneously fitting the model to the measurement in multiple TIR bands. Although the cloud top inferred from TIR measurement fluctuates with the cloud top from the cloud radar profile, the topmost parts of clouds seen only in the lidar profile are not detected well by TIR measurements. Compared with MODIS C6 operational products, COT of less than 5 agrees well, although CER deviates greatly. CTH estimates agree well for optically thick clouds when the MODIS product is based on the CO2 slicing method, whereas there is significant disagreement in CTHs between the present study and C6 products for optically thin clouds at the cloud edges. In the present algorithm, the determination of the cloud thermodynamic phase is strongly constrained by a priori knowledge about cloud phase dependence on the CTTs. It guarantees that ice clouds that are too warm and water clouds that are too cold are not retrieved; however, more statistical verification of the temperature dependence should be performed by increasing the number of cases.
The present algorithm will be used in studies with observations from the Himawari-8 satellite, a Japanese next-generation geostationary meteorological satellite, which has been operated by the Japan Meteorological Agency since 7 July 2015 and carries a visible-to-IR imager with greatly improved radiometric, spectral, spatial, and temporal resolution (Bessho et al. 2016). The development strategy used in this study will be used to create an algorithm for Himawari-8. The algorithm will have several modifications to accommodate the different spectral bands and will have improvements to the multilayer cloud retrieval and cloud phase discrimination. CALIPSO lidar measurements are suitable for retrieving optically thin cloud and reliable cloud phase discrimination (Hu et al. 2009). Using depolarization lidar comparison on a global scale, would help determine the performance of the TIR-based algorithm. In the future, further comparison of collocated data from different cloud products will be performed to characterize respective strengths and limitations of different methods.
BFED:
Baseline-fit emissivity database
Brightness temperature
BTD:
Brightness temperature difference
C6:
Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation
Cloud-particle effective radius
CKD:
Correlated k-distribution
COT:
Cloud optical thickness
CTH:
Cloud-top height
CTP:
Cloud-top pressure
CTT:
Cloud-top temperature
DOFS:
Degree of freedom for signal
ICAS:
Integrated Cloud Analysis System
MBE:
Mean bias error
MODIS:
Moderate resolution imaging spectroradiometer
RMSE:
Root-mean-square error
SWIR:
Shortwave infrared
TIR:
Thermal infrared
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The authors are grateful to Prof. Hajime Okamoto of Kyushu University, Japan, for providing the cloud mask data made from CloudSat/CALIPSO data and Dr. Shuichiro Katagiri of Kyushu University, Japan, for the valuable comments during this study.
The MODIS data were obtained from the NASA websites.
This work was promoted and supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 25287117.
Data will not be shared because the main results of this paper are development of cloud retrieval technique and described fully in this paper.
HI proposed the topic, conceived and designed the study, and conducted major parts of the study. M. Saito collaborated with the corresponding author in the development of the inversion module and carried out the evaluations of forward model and retrieval errors. YT collaborated with the corresponding author in the development and evaluation of the forward model and carried out the analysis using the MODIS data. NSP carried out the comparison analysis using CloudSat/CALIPSO product data. M. Sekiguchi developed the CKD model. All authors read and approved the final manuscript.
Center for Atmospheric and Oceanic Studies, Graduate School of Science, Tohoku University, 6-3 Aoba, Aramakiaza, Aoba-ku, Sendai, Miyagi, 980-8578, Japan
Hironobu Iwabuchi
, Masanori Saito
, Yuka Tokoro
& Nurfiena Sagita Putri
The Graduate School of Marine Science and Technology, Tokyo University of Marine Science and Technology, 2-1-6 Etchujima, Koto-ku, Tokyo, 135-8533, Japan
Miho Sekiguchi
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Correspondence to Hironobu Iwabuchi.
Iwabuchi, H., Saito, M., Tokoro, Y. et al. Retrieval of radiative and microphysical properties of clouds from multispectral infrared measurements. Prog. in Earth and Planet. Sci. 3, 32 (2016) doi:10.1186/s40645-016-0108-3
Effective particle radius
Ice cloud
Optimal estimation method
Satellite remote sensing
2. Atmospheric and hydrospheric sciences | CommonCrawl |
Tag: arxiv
Who dreamed up the primes=knots analogy?
Published May 16, 2011 by lievenlb
One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding
$\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$
embeds this circle as a knot in a 3-dimensional simply connected manifold which, after Perelman, has to be $S^3$. For more see the what is the knot associated to a prime?-post.
In recent months new evidence has come to light allowing us to settle the genesis of this marvelous idea.
1. The former consensus
Until now, the generally accepted view (see for example the 'Mazur-dictionary-post' or Morishita's expository paper) was that the analogy between knots and primes was first pointed out by Barry Mazur in the middle of the 1960's when preparing for his lectures at the Summer Conference on Algebraic Geometry, at Bowdoin, in 1966. The lecture notes where later published in 1973 in the Annales of the ENS as 'Notes on etale cohomology of number fields'.
For further use in this series of posts, please note the acknowledgement at the bottom of the first page, reproduced below : "It gives me pleasure to thank J.-P. Serre for his vigorous editing and his suggestions and corrections, which led to this revised version."
Independently, Yuri I. Manin spotted the same analogy at around the same time. However, this point of view was quickly forgotten in favor of the more classical one of viewing number fields as analogous to algebraic function fields of one variable. Subsequently, in the mid 1990's Mikhail Kapranov and Alexander Reznikov took up the analogy between number fields and 3-manifolds again, and called the resulting study arithmetic topology.
2. The new evidence
On december 13th 2010, David Feldman posted a MathOverflow-question Mazur's unpublished manuscript on primes and knots?. He wrote : "The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur. I'm not absolutely sure this is the one I mean, but in his paper, Analogies between group actions on 3-manifolds and number fields, Adam Sikora cites B. Mazur, Remarks on the Alexander polynomial, unpublished notes."
Two months later, on february 15th David Feldman suddenly found the missing preprint in his mail-box and made it available. The preprint is now also available from Barry Mazur's website. Mazur adds the following comment :
"In 1963 or 1964 I wrote an article Remarks on the Alexander Polynomial [PDF] about the analogy between knots in the three-dimensional sphere and prime numbers (and, correspondingly, the relationship between the Alexander polynomial and Iwasawa Theory). I distributed some copies of my article but never published it, and I misplaced my own copy. In subsequent years I have had many requests for my article and would often try to search through my files to find it, but never did. A few weeks ago Minh-Tri Do asked me for my article, and when I said I had none, he very kindly went on the web and magically found a scanned copy of it. I'm extremely grateful to Minh-Tri Do for his efforts (and many thanks, too, to David Feldman who provided the lead)."
The opening paragraph of this unpublished preprint contains a major surprise!
Mazur points to David Mumford as the originator of the 'primes-are-knots' idea : "Mumford has suggested a most elegant model as a geometric interpretation of the above situation : $\pmb{spec}(\mathbb{Z}/p\mathbb{Z})$ is like a one-dimensional knot in $\pmb{spec}(\mathbb{Z})$ which is like a simply connected three-manifold."
In a later post we will show that one can even pinpoint the time and place when and where this analogy was first dreamed-up to within a few days and a couple of miles.
For the impatient among you, have a sneak preview of the cradle of birth of the primes=knots idea…
Art and the absolute point (2)
Last time we did recall Manin's comparisons between some approaches to geometry over the absolute point $\pmb{spec}(\mathbb{F}_1)$ and trends in the history of art.
In the comments to that post, Javier Lopez-Pena wrote that he and Oliver Lorscheid briefly contemplated the idea of extending Manin's artsy-dictionary to all approaches they did draw on their Map of $\mathbb{F}_1$-land.
So this time, we will include here Javier's and Oliver's insights on the colored pieces below in their map : CC=Connes-Consani, Generalized torified schemes=Lopez Pena-Lorscheid, Generalized schemes with 0=Durov and, this time, $\Lambda$=Manin-Marcolli.
Durov : romanticism
In his 568 page long Ph.D. thesis New Approach to Arakelov Geometry Nikolai Durov introduces a vast generalization of classical algebraic geometry in which both Arakelov geometry and a more exotic geometry over $\mathbb{F}_1$ fit naturally. Because there were great hopes and expectations it would lead to a big extension of algebraic geometry, Javier and Oliver associate this approach to romantism. From wikipedia : "The modern sense of a romantic character may be expressed in Byronic ideals of a gifted, perhaps misunderstood loner, creatively following the dictates of his inspiration rather than the standard ways of contemporary society."
Manin and Marcolli : impressionism
Yuri I. Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$ and Matilde Marcolli in Cyclotomy and endomotives develop a theory of analytic geometry over $\mathbb{F}_1$ based on analytic functions 'leaking out of roots of unity'. Javier and Oliver depict such functions as 'thin, but visible brush strokes at roots of 1' and therefore associate this approach to impressionism. Frow wikipedia : 'Characteristics of Impressionist paintings include: relatively small, thin, yet visible brush strokes; open composition; emphasis on accurate depiction of light in its changing qualities (often accentuating the effects of the passage of time); common, ordinary subject matter; the inclusion of movement as a crucial element of human perception and experience; and unusual visual angles.'
Connes and Consani : cubism
In On the notion of geometry over $\mathbb{F}_1$ Alain Connes and Katia Consani develop their extension of Soule's approach. A while ago I've done a couple of posts on this here, here and here. Javier and Oliver associate this approach to cubism (a.o. Pablo Picasso and Georges Braque) because of the weird juxtapositions of the simple monoidal pieces in this approach.
Lopez-Pena and Lorscheid : deconstructivism
Torified varieties and schemes were introduced by Javier Lopez-Pena and Oliver Lorscheid in Torified varieties and their geometries over $\mathbb{F}_1$ to get lots of examples of varieties over the absolute point in the sense of both Soule and Connes-Consani. Because they were fragmenting schemes into their "fundamental pieces" they associate their approach to deconstructivism.
Another time I'll sketch my own arty-farty take on all this.
Art and the absolute point
In his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$ Yuri I. Manin sketches and compares four approaches to the definition of a geometry over $\mathbb{F}_1$, the elusive field with one element.
He writes : "Preparing a colloquium talk in Paris, I have succumbed to the temptation to associate them with some dominant trends in the history of art."
Remember that the search for the absolute point $\pmb{spec}(\mathbb{F}_1)$ originates from the observation that $\pmb{spec}(\mathbb{Z})$, the set of all prime numbers together with $0$, is too large to serve as the terminal object in Grothendieck's theory of commutative schemes. The last couple of years have seen a booming industry of proposals, to the extent that Javier Lopez Pena and Oliver Lorscheid decided they had to draw a map of $\mathbb{F}_1$-land.
Manin only discusses the colored proposals (TV=Toen-Vaquie, M=Deitmar, S=Soule and $\Lambda$=Borger) and compares them to these art-history trends.
Toen and Vaquie : Abstract Expressionism
In Under $\pmb{spec}(\mathbb{Z})$ Bertrand Toen and Michel Vaquie argue that geometry over $\mathbb{F}_1$ is a special case of algebraic geometry over a symmetric monoidal category, taking the simplest example namely sets and direct products. Probably because of its richness and abstract nature, Manin associates this approach to Abstract Expressionism (a.o. Karel Appel, Jackson Pollock, Mark Rothko, Willem de Kooning).
Deitmar : Minimalism
Because monoids are the 'commutative algebras' in sets with direct products, an equivalent proposal is that of Anton Deitmar in Schemes over $\mathbb{F}_1$ in which the basic affine building blocks are spectra of monoids, topological spaces whose points are submonoids satisfying a primeness property. Because Deitmar himself calls this approach a 'minimalistic' one it is only natural to associate to it Minimalism where the work is stripped down to its most fundamental features. Prominent artists associated with this movement include Donald Judd, John McLaughlin, Agnes Martin, Dan Flavin, Robert Morris, Anne Truitt, and Frank Stella.
Soule : Critical Realism
in Les varietes sur le corps a un element Christophe Soule defines varieties over $\mathbb{F}_1$ to be specific schemes $X$ over $\mathbb{Z}$ together with a form of 'descent data' as well as an additional $\mathbb{C}$-algebra, morally the algebra of functions on the real place. Because of this Manin associates to it Critical Realism in philosophy. There are also 'realism' movements in art such as American Realism (o.a. Edward Hopper and John Sloan).
Borger : Futurism
James Borger's paper Lambda-rings and the field with one element offers a totally new conception of the descent data from $\mathbb{Z}$ to $\mathbb{F}_1$, namely that of a $\lambda$-ring in the sense of Grothendieck. Because Manin expects this approach to lead to progress in the field, he connects it to Futurism, an artistic and social movement that originated in Italy in the early 20th century.
Lists 2010 : StackExchange sites
Published December 27, 2010 by lievenlb
One of the trends of 2010 was the proliferation of StackExchange sites. I guess by now most of us visit MathOverflow along with the arXiv daily. But, there are plenty of other StackExchange sites around that may be of interest to the mathematics-community :
Mathematics somewhat less high-brow than MathO.
Physics still in the beta-phase (see below)
TeX for TeX and LaTeX-lovers
iPad 4 edu for those who want to use their iPad in the classroom
"Opening a StackExchange site is damn hard. First you have to find at least 60 people interested in the site. Then, when this limit is reached, a large amount of people (in the hundreds, but it really depends on the reputation of each participant) must commit and promise to create momentum for the site, adding questions and answers. When this amount is reached, the site is open and stays in closed beta for seven days. During this time, the committers have to enrich the site so that the public beta (which starts after the first seven days) gets enough hits and participants to show a self-sustained community." (quote from ForTheScience's StackExchange sites proliferation, this post also contains a list of StackExchange-projects in almost every corner of Life)
The site keeping you up to date with StackExchange proposals and their progress is area51. Perhaps, you want to commit to some of these proposals
Math Errata Database
Blogging and the Blogosphere
or simply browse around area51 until you find the ideal community for you to belong to…
math & manic-depression, a Faustian bargain
Published July 15, 2010 by lievenlb
In the wake of a colleague's suicide and the suicide of three students, Matilde Marcolli gave an interesting and courageous talk at Caltech in April : The dark heart of our brightness: bipolar disorder and scientific creativity. Although these slides give a pretty good picture of the talk, if you can please take the time to watch it (the talk starts 44 minutes into the video).
Courageous because as the talk progresses, she gives more and more examples from her own experiences, thereby breaking the taboo surrounding the topic of bipolar mood disorder among scientists. Interesting because she raises a couple of valid points, well worth repeating.
We didn't can see it coming
We are always baffled when someone we know commits suicide, especially if that person is extremely successful in his/her work. '(S)he was so full of activity!', 'We did not see it coming!' etc. etc.
Matilde argues that if a person suffers from bipolar mood disorder (from mild forms to full-blown manic-depression), a condition quite common among scientists and certainly mathematicians, we can see it coming, if we look for the proper signals!
We, active scientists, are pretty good at hiding a down-period. We have collected an arsenal of tricks not to send off signals when we feel depressed, simply because it's not considered cool behavior. On the other hand, in our manic phases, we are quite transparent because we like to show off our activity and creativity!
Matilde tells us to watch out for people behaving orders-of-magnitude out of their normal-mode behavior. Say, someone who normally posts one or two papers a year on the arXiv, suddenly posting 5 papers in one month. Or, someone going rarely to a conference, now spending a summer flying from one conference to the next. Or, someone not blogging for months, suddenly flooding you with new posts…
As scientists we are good at spotting such order-of-magnitude-out-behavior. So we can detect friends and colleagues going through a manic-phase and hence should always take such a person serious (and try to offer help) when they send out signals of distress.
Mood disorder, a Faustian bargain
The Faust legend :
"Despite his scholarly eminence, Faust is bored and disappointed. He decides to call on the Devil for further knowledge and magic powers with which to indulge all the pleasures of the world. In response, the Devil's representative Mephistopheles appears. He makes a bargain with Faust: Mephistopheles will serve Faust with his magic powers for a term of years, but at the end of the term, the Devil will claim Faust's soul and Faust will be eternally damned."
Mathematicians suffering from mood disorder seldom see their condition as a menace, but rather as an advantage. They know they do their best and most creative work in short spells of intense activity during their manic phase and take the down-phase merely as a side effect. We fear that if we seek treatment, we may as well loose our creativity.
That is, like Faust, we indulge the pleasures of our magic powers during a manic-phase, knowing only too well that the devilish depression-phase may one day claim our life or mental sanity… | CommonCrawl |
Clustering Algorithm Considering Sensor Node Distribution in Wireless Sensor Networks
Boseon Yu* , Wonik Choi** , Taikjin Lee* and Hyunduk Kim***
Corresponding Author: Hyunduk Kim*** ([email protected])
Boseon Yu*, Sensor System Research Center, Korea Institute of Science and Technology, Seoul, Korea, [email protected]
Wonik Choi**, School of Information Communication Engineering, Inha University, Incheon, Korea, [email protected]
Taikjin Lee*, Sensor System Research Center, Korea Institute of Science and Technology, Seoul, Korea, [email protected]
Hyunduk Kim***, Policy Planning Agency, Chungnam Techno Park, Cheonan, Korea, [email protected]
Received: December 8 2015
Revision received: May 3 2016
Abstract: In clustering-based approaches, cluster heads closer to the sink are usually burdened with much more relay traffic and thus, tend to die early. To address this problem, distance-aware clustering approaches, such as energy-efficient unequal clustering (EEUC), that adjust the cluster size according to the distance between the sink and each cluster head have been proposed. However, the network lifetime of such approaches is highly dependent on the distribution of the sensor nodes, because, in randomly distributed sensor networks, the approaches do not guarantee that the cluster energy consumption will be proportional to the cluster size. To address this problem, we propose a novel approach called CACD (Clustering Algorithm Considering node Distribution), which is not only distance-aware but also node density-aware approach. In CACD, clusters are allowed to have limited member nodes, which are determined by the distance between the sink and the cluster head. Simulation results show that CACD is 20%–50% more energy-efficient than previous work under various operational conditions considering the network lifetime.
Keywords: CACD , Clustering , EEUC , Node Distribution , WSN
In wireless sensor networks, sensor nodes are generally not rechargeable or replaceable since they are deployed in large numbers; furthermore, the deployed areas are usually disaster areas that are difficult to reach. Thus, considering that sensor nodes are usually powered by limited energy resources, improving the lifetime of the network has always been a primary objective of wireless sensor networks.
Many efficient routing protocols have been proposed to extend the lifetime of wireless sensor networks [1-5]. Clustering-based approaches have especially attracted a great deal of interest because these approaches can achieve high energy efficiency and, at the same time, increase the network scalability. Typically, in clustering-based approaches, cluster heads are elected by local competitions. Usually the highest-energy nodes in the neighborhood are elected as cluster heads. Then, each nonhead node picks a cluster head with which to form a cluster. After clustering is finished, cluster heads aggregate the sensed data from the member nodes and forward it to the sink [6-8]. To optimize the transmission energy consumption, cluster heads eliminate duplicates and redundancies before the actual transmission.
Clustering-based approaches usually employ a multi-hop communication model, which is proven to be more efficient than direct communication model, to deliver the sensed data from the data sources to the sink. However, in such approaches, cluster heads closer to the sink tend to deplete their energy resources much faster than the other cluster heads due to equal clustering. More specifically, cluster heads placed closer to the sink are usually involved in many more traffic relays, and thus, they tend to deplete their energy resources much faster than other cluster heads (hot-spot problem). This may result in partitioning the sensor network as well as reducing the sensing coverage. To address this hot-spot problem, energy-efficient unequal clustering (EEUC) has been proposed in [9]. In EEUC, the cluster size decreases as the distance from the cluster head to the sink decreases. As a result, in EEUC, clusters closer to the sink have smaller cluster sizes than the other clusters. More specifically, EEUC places more, smaller clusters closer to the sink so that they consume less energy for sensing, aggregating, and intra-cluster communications. This unequal clustering may result in fewer burdens on specific cluster heads as well by selecting different routes for delivering data. However, EEUC is less effective for sensor networks with random distributions. EEUC achieves longer lifetimes by making a tradeoff between cluster energy consumption (i.e., sensing and intra-cluster transmissions) and energy consumption for inter-cluster communications. However, EEUC does not guarantee that a smaller-sized cluster will have a smaller node density in a randomly distributed sensor network. Placing small clusters in a crowded area may make EEUC less effective. To address this problem, we propose a novel approach, called CACD (clustering algorithm considering node distribution) that adjusts the size of each cluster considering the node density. Furthermore, CACD automatically adjusts the node density of each cluster on a per-cluster energy-consumption basis. In CACD, we first compute the optimized number of cluster heads for a given sensor network and then, adjust the size and the node density of each cluster according to the distance between the sink and the cluster head. Simultaneously, we estimate the energy level that each cluster requires for sensing and intra-cluster communication to allow for further adjustments. In addition, to keep the sensor network balanced, we partially reconstruct the cluster structures unlike previous methods that rebuild cluster structures at the beginning of each round. The reminder of this paper is organized as follows. In Section 2, we briefly review previous work for wireless sensor network. Section 3 gives the description of the proposed CACD approach. In Section 4, performance evaluation and analysis are given by simulation results. Finally, we conclude in Section 5.
2.1 LEACH and LEACH-C
Low-energy adaptive clustering hierarchy (LEACH) is the first clustering-based approach for wireless sensor networks [10]. LEACH operates on a round-to-round basis and elects a set of cluster heads by a localized competition at the beginning of each round. To evenly distribute the energy dissipation among all the nodes, LEACH rotates the cluster heads among the nodes. In LEACH, each round has two phases: the set-up phase and the steady-state phase as shown in Fig. 1. At the beginning of the setup phase, each node decides whether or not to be a cluster head with a certain possibility, pi(t):
[TeX:] $$p _ { i } ( t ) = \left\{ \begin{array} { l } { \frac { C N } { N - C N * \left( r \bmod \frac { N } { C N } \right) } , c _ { i } ( t ) = 1 } \\ { 0 , C _ { i } ( t ) = 0 } \end{array} \right.$$
where N is the total number of nodes, k is the number of clusters and r is the number of rounds that have passed. Ci(t) = 0, if node i has been a cluster head in most recent (r mod N/k) rounds and 1 otherwise. Thus, Eq. (1) ensures that all nodes are cluster heads the same number of times.
Round composition of LEACH.
After the cluster head election, each cluster head for the current round broadcasts an ADV message using the CSMA (carrier sense multiple access) protocol to the rest of the nodes. Each non-cluster head node that receives the ADV message decides the cluster to which it belongs for the round, according to the strength of the ADV message. After each non-cluster head node has decided on its cluster, it informs its cluster head that it will be a member of the cluster by sending a Join-REQ message. Then, the cluster head sets up a TDMA (time division multiple access) schedule and broadcasts the schedule to the member nodes. In steady-state phase, sensor nodes send their data to the cluster heads during their assigned time slots. To prevent wasted energy, sensor nodes turn off their radio except during their allocated time slots. Cluster heads aggregate the sensed data and forward it to the sink using the CDMA module. Unfortunately, LEACH does not guarantee well-established cluster structures. Since LEACH elects the cluster heads according to residual energy, its cluster structure is highly dependent on the distribution of the sensor nodes. To address this problem, a centralized LEACH clustering algorithm (LEACH-C) has been proposed. However, because LEACH-C assumes that the sink knows the locations of the sensor nodes, this approach is more expensive and requires more transmission cost to form a cluster structure.
2.2 EEUC
Clustering-based algorithms for wireless sensor networks generally use a multi-hop communication model to forward data to the sink. This indicates that cluster heads closer to the sink are involved in many more traffic relays, and thus, tend to deplete their energy resources much faster than other cluster heads. To address this problem, EEUC proposes the unequal clustering scheme shown in Fig. 2. EEUC adjusts the cluster size based on the distance between a cluster head and the sink, such that clusters closer to the sink are smaller. More specifically, EEUC decides the cluster size by Eq.(2).
[TeX:] $$S _ { i } \cdot R _ { C O M P } = \left( 1 - c \frac { d _ { \max } - d \left( S _ { i } , \sin k \right) } { d _ { \max } - d _ { \min } } \right) R _ { C O M P } ^ { 0 }$$
where Rcomp0 is the pre-defined maximum cluster size. dmax and dmin denote the maximum and minimum distances between the sink and a cluster head, respectively. d(si, sink) indicates the distance between the cluster head,si and the sink, and c is a constant coefficient between 0 and 1. According to Eq. (2), the size of a cluster varies from (1-c)Rcomp0 to Rcomp0. In EEUC, clusters closer to the sink consume less energy for activities inside the clusters (i.e. sensing, aggregating and intra-cluster communications) and thus can afford more traffic relays. However, EEUC becomes less effective for randomly distributed sensor networks because it does not guarantee that a cluster with a smaller cluster size has a smaller node density for such sensor networks. It is obvious that small clusters with a large node density make the unequal clustering-based approach less effective.
EEUC (unequal clustering scheme).
EEUC.
In addition, for wireless sensor networks with random distributions, EEUC may have clusters that are not involved in any traffic relays due to their small size, as shown in Fig. 3. In Fig. 3, clusters depicted by the dashed line are not involved in any data transmissions and thus the cluster heads will consume much less energy. To extend the network lifetime, it is more desirable for such clusters to be larger. This motivates us to design a protocol with partial re-construction. In our protocol, clusters that are involved in fewer inter-cluster communications are allowed to have more member nodes for optimizations. A more detailed explanation is given in Subsection 3.3.
3. The CACD Scheme
The CACD approach proposed in this paper has three phases: set-up, data-transmit, and re-construct. In the set-up phase, CACD elects a set of cluster heads, and each non-cluster head node chooses a cluster head to form a cluster. Similar to EEUC, CACD adjusts the node density of each cluster according to its distance from the sink. Sensor nodes transmit data to their cluster heads, and then the cluster heads aggregate and forward the data to the sink, using a multi-hop communication model in the data-transmit phase. In the re-construct phase, if certain clusters consume much more energy during the set-up and data-transmit phases, CACD balances the energy consumption by partially reconstructing those clusters. Fig. 4 shows how each round of CACD is organized. Please note that after re-construct phase, we omit for the next several rounds the set-up phase that is quite costly in order to fully utilize the well-established cluster structure as shown in Fig. 4. In Fig. 4, set-up phase is replaced with re-construct phase in round 2 and for the following rounds, the set-up phase is omitted and thus, the result of re-construct phase is maintained. By maintaining a well-established cluster structure and omitting the expensive set-up phase, CACD preserves more energy for inter-cluster communication. More operational details will be described in the next subsections. Without a loss of generality, we assume the following properties about the sensor network:
• The sink is stationary and has unlimited energy resources.
• Each sensor node is capable of measuring its distance from the sink according to the strength of receiving signal.
• Sensor nodes are stationary and have similar capabilities (processing, communication, energy resources).
• Sensor nodes can control their transmission power level according to the distance.
Time-line of CACD.
3.1 Set-Up Phase
In the set-up phase, CACD elects a set of cluster heads and then, forms clusters with the remaining nodes. CACD employs the hybrid energy-efficient distributed (HEED) cluster-head selection mechanism [11]. With a given initial percentage of cluster heads, Cprob, which is set to 5%, HEED rotates the cluster heads based on their residual energy. Eq. (3) is given for cluster head selection.
[TeX:] $$C H _ { p r o b } = C _ { p r o b } * \frac { E _ { \text {residual} } } { E _ { \text {max} } },$$
where CHCHprob indicates the probability of becoming a cluster head. Eresidual and Emax are the estimated remaining energy and the maximum energy of a node, respectively. After the cluster-head election, CACD forms clusters using the ADV and Join messages. The sink that overhears the ADV message estimates the distance between a cluster head and the sink according to the signal strength and assigns proper cluster sizes and member nodes to the cluster heads. With this information, each cluster head forms a cluster. The number of member nodes assigned to a cluster follows Eq. (4).
[TeX:] $$O S N = \frac { N } { C N } + P,$$
where, optimal sensor node (OSN) is the number of sensor nodes that should be included in a cluster while N and CN are the number of sensor nodes and cluster heads for the sensor network, respectively. P is a constant to adjust the node density of each cluster according to the distance from the sink. For effective assignment of P, CACD follows Eq. (5).
[TeX:] $$P = \frac { d \left( s _ { i } , \sin k \right) } { F } * v,$$
where, d(si, sink) is the distance between a cluster head, si, and the sink. F indicates the size of the sensor field, while v is a user-defined constant to assign the proper number of member nodes to a cluster. If a cluster has more member nodes than OSN, the cluster is partitioned or passes member nodes to neighboring clusters to build a more energy-efficient cluster structure. We define the threshold, M, for cluster partitioning as follows:
[TeX:] $$M = O S N * 1.5$$
Figs. 5 and 6 give a running example of a cluster that requires partitioning. As shown in Fig. 5, cluster B has more sensing nodes than M (=9); thus, cluster B should be partitioned to balance the energy dissipation. For partitioning, the two most-distant nodes are elected as new cluster heads and the new cluster heads form clusters using the ADV and Join messages, as mentioned above. Fig. 6 shows the result of the partitioning procedure. In Fig. 6, cluster B is partitioned into two clusters B' and D. As shown in the running example, CACD balances the energy consumption of each cluster by limiting the maximum number of member nodes in each cluster. Even though the partitioning procedure generates more energy-efficient cluster structures, the procedure can be expensive. Accordingly, CACD applies a simply node transmission procedure to the clusters that have member nodes more or less than OSN for further optimization. In order for all clusters to have as many member nodes as their OSN, those clusters with member nodes more or less than their OSN send OVER or LESS message, respectively to the neighboring cluster heads. The OVER message and LESS message have information on excess and shortage of member nodes. If a cluster with more member nodes (CO) and a cluster with less member node (CL) are one-hop neighbor, some member nodes of CO that are close to CL are moved to CL so that all clusters have member nodes as many as their OSN can, Please note that the maximum number of member nodes varies according to the distance from the sink. Thus, the results of CACD may be similar to that of EEUC in that the clusters close to the sink tend to be small. However, CACD guarantees more energy-efficient cluster structures even for crowded areas, by limiting the maximum number of nodes in a cluster, which varies according to the distance from the sink.
Unbalanced cluster structure (before split).
Balanced cluster structure (after split).
3.2 Data-Transmit Phase
After finishing the set-up phase, cluster heads create TDMA schedules according to the number of member nodes, and pass the schedules to their member nodes. Then, the member nodes transfer their data to the cluster heads during their assigned time slots. After the cluster heads aggregate the data, they forward it to the sink, using CDMA to prevent data collisions. When delivering the data, if the distance between the sending and receiving nodes is less than d0, CACD uses the Friis free-space model. Otherwise, the two-ray ground propagation model is used.
3.3 Re-construct Phase
During the set-up and data-transmit phases, the cluster heads estimate the energy consumed for building cluster structures and for aggregating and transmitting data to the sink (Eo). In the re-construct phase, cluster heads send Eo to the sink, which estimates the average energy cost (AEC) and re-construct energy cost (REC) of each cluster. Then, the sink sends the AEC and REC to the cluster heads. REC is a threshold for triggering the re-construct phase.
[TeX:] $$A E C = \frac { E _ { \text {total} } } { C N ^ { \prime } }$$
[TeX:] $$R E C = A E C * 1.5$$
where Etotal indicates the total energy consumption of the sensor network and CN′ is the number of cluster heads. If a cluster head (Ct) has Eo larger than REC, Ct broadcasts a reconstruct message to its neighboring cluster heads to re-construct the cluster. Please note that in Eq. (8), the constant coefficient 1.5 can be varied by simulation conditions and thus should be carefully decided by experimental results. If cluster heads (Cr), whose Eo is less than AEC, exist, the member nodes of Ct that are close to Cr become member nodes of Cr. Then, Cr recreates the TDMA schedule and sends it to the member nodes. After the re-construct phase, the set-up phase is omitted for the next k round to utilize this reconstructed cluster structure, as shown in Fig. 4. A more detailed explanation for the best setting of k will be given in Section 4. If all neighboring clusters have Eo larger than AEC, the re-construct phase is aborted and CACD begins the next round so that the entire cluster structure can be re-built in the set-up phase.
CACD algorithm
The above algorithm gives a sketch of CACD. In line 1, we elect the cluster heads according to the mechanism used in HEED. Then, CACD builds a cluster structure in the remainder of the algorithm. While building the cluster structure, a cluster whose member nodes are more than M is partitioned in lines 5-7. When a cluster head has more member nodes than OSN, the cluster head finds clusters that are able to have more member nodes, and passes its member nodes to them in lines 8-18. In line 19, each cluster decides whether to trigger the re-construction phase. If Eo > REC, the cluster head triggers re-construct phase by broadcasting a reconstruct message in line 20 and waits for the cluster-extend message. If the cluster head receives the cluster-extend message, the re-construct phase is triggered in line 22. Otherwise, CACD simply moves to the next round so that the entire cluster structure is re-built in the set-up phase in line 24. If a cluster whose Eo is less than AEC receives a reconstruction messages, it broadcasts a cluster-extend message and triggers the re-construction phase in lines 26-31.
4. Performance Evaluation and Analysis
In this section, we evaluate the performance of our approach, CACD by comparing it against LEACH, HEED, and EEUC. We use Network Simulator 2 (NS2; http://www.isi.edu/nsnam/ns) for simulations and, without a loss of generality, the energy consumption model used in the simulations is the first-order radio model (Eq. (9)), which is also used for LEACH and EEUC.
[TeX:] $$E _ { T X } ( t , d ) = E _ { \text {elec} } * t + \varepsilon _ { a m p } * t * d ^ { n } \\ = \left\{ \begin{array} { l } { E _ { e l e c } * t + \varepsilon _ { f s } * t * d ^ { 2 } , d < d _ { 0 } } \\ { E _ { e l e c } * t + \varepsilon _ { m p } * t * d ^ { 4 } , d > d _ { 0 } } \end{array} \right.$$
[TeX:] $$E _ { R X } ( t , d ) = E _ { e l e c } * t$$
ETX in Eq. (9) indicates the energy consumed by transmitting data. Eelec represents the energy consumed for processing one bit of data, while t and d represent the data size and the transmission distance, respectively. εamp is the energy required to amplify the signal. When d < d0, we use εfs, which is from the Friis free space model. Otherwise, εamp from the two-ray ground propagation model is used. ERX represents the energy consumed when receiving data. The required d0 here is defined by Eq. (10).
[TeX:] $$d _ { 0 } = \frac { 4 \pi \sqrt { L } h _ { r } h _ { t } } { \lambda }$$
where L is the system loss factor, which is not related to the wave propagation while λ is the wave length of the carrier signal. ht and hr are the heights above the ground of the transmitting and receiving antennas, respectively.
L = 1 indicates no loss in the system hardware. ht = hr = 1.5 m, and λ is related to the carrier frequency. If we use a carrier frequency of 914 MHz, λ is as follows:
[TeX:] $$\lambda = \frac { 3 * 10 ^ { 8 } } { 914 * 10 ^ { 6 } } = 0.328$$
Using these parameters, do = 86.2 m. The simulation parameters are given in Table 1. All of the alternatives are compared under the same setup, and we use the scenario generator of NS2 to generate randomly deployed sensor nodes, as shown in Fig. 7.
Random deployment of sensor nodes.
The impact of k on the network lifetime of CACD.
The first set of our simulation is to find the best value for k (Section 3.3). By omitting the set-up phase for k rounds after the re-construct phase, CACD preserves more energy for intra-cluster communication and thus can extend the lifetime of the sensor networks. In Fig. 8, we count the rounds until the first node dies, while k varies from 1 to 10. As shown in the figure, CACD shows poor performance when k is less than 3. This is because, with such small k values, CACD cannot fully utilize the benefit of the re-construct phase. More specifically, during the re-construct phase, CACD produces a more energy-efficient cluster structure, but consumes extra energy in the process. Thus, in order to compensate for the extra energy and preserve energy for inter-cluster communications, CACD omits the set-up phase and maintains the result of the re-construct phase for the next k rounds. However, Fig. 8 shows that CACD cannot achieve these goals with such small k values. In addition, CACD becomes less effective when k is greater than 8. This is because maintaining a cluster structure for such a long period causes a rapid depletion of the cluster heads' energy resources. This rapid depletion makes our approach less effective, as shown in Fig. 8. As a result, in order to optimize the CACD's performance, we omit the set-up phase for the next 6 rounds after re-construct phase, which is the best setting of k, as shown in Fig. 8.
The next set of our simulation is to study the effectiveness of CACD. We evaluate and compare the energy consumption of each sensor node as shown in Fig. 9. Fig. 9 shows the average energy consumption of each round for the four alternatives. CACD is up to 3 times more energy efficient than LEACH, and up to 1.6 times more energy efficient than HEED and EEUC. LEACH's poor performance is because it does not take into account the node density and the size of each cluster. Furthermore, the direct communication between each cluster head and the sink highly contributes to the poor performance.
HEED outperforms LEACH mainly due to the multi-hop communication model for forwarding data, but may suffer from the hot-spot problem since cluster heads closer to the sink are burdened with forwarding data [12]. This hot-spot problem accelerates the energy consumption of neighboring nodes and, in turn, may result in partitioning the sensor network as well as reducing the sensing coverage, which is undesirable for wireless sensor networks.
As shown in Fig. 9, EEUC consumes less energy than HEED due to its distance-aware clustering approach, but may become less effective for a wireless sensor network with a random distribution. CACD extends EEUC by considering the node density of each cluster and successively reduces the energy consumption of each round, as shown in Fig. 9.
Average energy consumption per round.
Fig. 10.
The number of alive sensor nodes over time.
Fig. 10 shows the number of sensor nodes that are still alive over the simulation time of the four alternatives. CACD clearly improves the lifetime of the network over LEACH, HEED, and EEUC.
When forming cluster structures, other alternatives do not consider the node density and energy consumption of each cluster and may create unbalanced cluster structures. These unbalanced cluster structures may waste unnecessary energy while processing and transmitting data. In contrast, CACD forces a cluster to have fewer member nodes than OSN, which varies according to the distance from the sink. This property of CACD guarantees more balanced energy load over the sensor network, even if the sensor network has a random distribution.
The last set of our experiments is to observe the tradeoff resulting from keeping the cluster structures. Fig. 11 shows energy consumptions of cluster heads on CACD and HEED. EEUC selects cluster heads considering the energy level of each node as well but we compare CACD only with HEED because detailed selection algorithm is not given in the paper. As shown in the figure, CACD reports that cluster heads consume more energy than HEED until 2 ROUND due to set-up phase and re-construct phase but after the round, CACD consumes less energy since CACD maintains the cluster structure for 7 rounds. The cluster heads maybe more burdened in the CACD but the stable cluster structure minimizes the burdens and makes CACD more energy efficient considering the overall energy consumption.
Average energy consumption of each cluster over ROUNDs.
In this paper, we proposed the CACD approach that aimed to prolong the lifetime of wireless sensor networks, which are usually powered by limited energy resources. The CACD approach creates cluster structures that can balance energy consumption, even for randomly distributed wireless sensor networks, by considering the node density and the energy consumption of each cluster. Analysis and performance evaluations showed that the CACD approach is up to three times more energy-efficient than previous methods.
As future work, we plan to extend our approach by exploring the cooperative communication model [13,14] to create a more energy-efficient cluster structure.
Boseon Yu
He received the M.S. and Ph.D. degree in School of Information and Communication Engineering from Inha University, Incheon, Korea in 2010 and 2015, respectively. From September 2015 to now, he has been working as a post-doctoral researcher for Sensor System Research Center in Korea Institute Science and Technology, Korea. His research interests include database system, GPU and sensor network system.
Wonik Choi
He is an Associate Professor in the School of Information and Communication Engineering at Inha University, where he runs Ubiquitous Computing Lab. He received his Ph.D. in Computer Engineering from Seoul National University, Korea. His research interests include spatio-temporal databases, sensor network topology and GIS/LBS. He was a visiting scholar in the school of Computer Science at Georgia Institute of Technology in 2012. He is a member of IEEE since 2006.
Taikjin Lee
He received the B.S. and Ph.D. degrees in School of Mechanical and Aerospace Engineering in Seoul National University, Seoul, Korea in 2001 and 2008, respectively. In 2008, he was with the School of Mechanical and Aerospace Engineering, Seoul National University where he was a postdoctoral fellow. Since 2010, he has been with Korea Institute of Science and Technology as a senior researcher. The areas of interests are indoor navigation system, pattern recognition, and sensor network.
Hyunduk Kim
He received M.S. and Ph.D. degrees in School of Information and Communication Engineering from Inha University in 2009 and 2014, respectively. From August 2014 to March 2016, he worked as a post-doctoral researcher for Sensor System Research Center in Korea Institute Science and Technology, Korea. He is now with the Policy Planning Agency in the Chungnam Techno Park as a Senior Researcher, since April 2016. His current research interests include WSN routing protocol, cooperative communication, wireless mobile communication, bio-sensor device and indoor LBS.
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Parameter Value
Network coverage (m) (0, 0)–(200, 200)
Sink location (m) (-100, 100)–(100, 100)
Initial energy (J) 2
t (bit) 4,000
do (m) 86.2
Eelec (nJ/bit) 50
Ɛfs (pJ/bit/m2) 10
Ɛmp (pJ/bit/m4) 0.0013
EDA (nJ/bit/signal) 5
C 0.5
Initial cluster head rate (%) 5
P 0.05 | CommonCrawl |
Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon
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The Liouville theorems for elliptic equations with nonstandard growth
November 2015, 14(6): 2393-2409. doi: 10.3934/cpaa.2015.14.2393
Symmetry of solutions to semilinear equations involving the fractional laplacian
Lizhi Zhang 1,
School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007
Received February 2015 Revised July 2015 Published September 2015
Let $0<\alpha<2$ be any real number. Let $\Omega\subset\mathbb{R}^n$ be a bounded or an unbounded domain which is convex and symmetric in $x_1$ direction. We investigate the following Dirichlet problem involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega, (1)\\ u(x)\equiv0, & \qquad x\notin\Omega. \end{array}\right. \end{equation}
Applying a direct method of moving planes for the fractional Laplacian, with the help of several maximum principles for anti-symmetric functions, we prove the monotonicity and symmetry of positive solutions in $x_1$ direction as well as nonexistence of positive solutions under various conditions on $f$ and on the solutions $u$. We also extend the results to some more complicated cases.
Keywords: Dirichlet problem, semi-linear elliptic equation, maximum principle for anti-symmetric functions, nonexistence of positive solutions, narrow region principle, a direct method of moving planes, the fractional Laplacian, decay at infinity., symmetry, Monotonicity.
Mathematics Subject Classification: Primary: 35S15, 35B06, 35J6.
Citation: Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
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Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
Sibei Yang, Dachun Yang, Wenxian Ma. Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022006
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Lizhi Zhang | CommonCrawl |
US-00-10 and AMV-00-10
To be published Physical Review D.
Evolution of the Yukawa coupling constans and seesaw operators in the universal seesaw model
The general features of the evolution of the Yukawa coupling constants and seesaw operators in the universal seesaw model with det$M_F=0$ are investigated. Especially, it is checked whether the model causes bursts of Yukawa coupling constants, because in the model not only the magnitude of the Yukawa coupling constant $(Y_L^u)_{33}$ in the up-quark sector but also that of $(Y_L^d)_{33}$ in the down-quark sector is of the order of one, i.e., $(Y_L^u)_{33} \sim (Y_L^d)_{33} \sim 1$. The requirement that the model should be calculable perturbatively puts some constraints on the values of the intermediate mass scales and $\tan\beta$ (in the SUSY model).
uuencoded Latex file
Can the SO(10) Model with Two Higgs Doublets Reproduce the Observed Fermion Masses?
K.Matsuda, Y.Koide, T.Fukuyama
It is usually considered that the SO(10) model with one 10 and one 126 Higgs scalars cannot reproduce the observed quark and charged lepton masses. Against this conventional conjecture, we find solutions of the parameters which can give the observed fermion mass spectra. The SO(10) model with one 10 and one 120 Higgs scalars is also discussed.
Physical Review D63, 037301 (2001).
Bimaximal Neutrino Mixing in a Zee-type Model with Badly Broken Flavor Symmetry
Yoshio Koide and Ambar Ghosal
A Zee-type neutrino mass matrix model with a badly broken horizontal symmetry SU(3)_H is investigated. By putting a simple ansatz on the symmetry breaking effects of SU(3)_H for transition matrix elements, it is demonstrated that the model can give a nearly bimaximal neutrino mixing with the ratio $\Delta m^2_{solar}/\Delta m^2_{atm} \simeq \sqrt{2} m_e/m_{\mu}=6.7 \times 10^{-3}$, which are in excellent agreement with the observed data. In the near future, the lepton-number violating decay $Z\to \mu^\pm \tau^\mp$ will be observed
To be published in the Proceedings of the workshop on "Neutrino Oscillations and their Origin", at Fuji-Yoshida, Feb., 2000, Universal Academy Press, Tokyo, 2000
Large Angle Neutrino Mixing and Pseudo-Dirac Neutrinos
A possibility that the observed large angle neutrino mixing is caused by a set of pseudo-Dirac neutrinos is discussed. Since the recent Superkamiokande data disfavor to the large $\nu_\mu$-$\nu_s$ mixing, a model without a large $\nu_\mu$-$\nu_s$ mixing is also discussed.
Phys.Lett. B488 (2000) 344-350 .
Democratic Universal Seesaw Model with Three Light Sterile Neutrinos
Based on the ``democratic" universal seesaw model, where mass matrices $M_f$ of quarks and leptons f_i (f=u,d,\nu,e; i=1,2,3) are given by a seesaw form M_f\simeq -m_L M_F^{-1} m_R, and m_L and m_R are universal for all the fermion sectors, and the mass matrices M_F of hypothetical heavy fermions F_i have a democratic structure, a possible neutrino mass matrix is investigated. In the model, there are three sterile neutrinos \nu_{iR} which mix with the active neutrinos \nu_{iL} with \theta \sim 10^{-2} and which are harmless for cosmological constraints. The atmospheric, solar and LSND neutrino data are explained by the mixings \nu_{\mu L} \leftrightarrow \nu_{\tau L}, \nu_{eL} \leftrightarrow \nu_{eR} and \nu_{eL} \leftrightarrow \nu_{\mu L}, respectively. The model predicts that \Delta m^2_{solar}/\Delta m^2_{atm} \simeq m_e/\sqrt{m_{\mu} m_{\tau}} and \Delta m_{LSND}^2/ \Delta m^2_{atm} \simeq (1/4)\sqrt{m_{\mu}/m_e} (\Delta m^2_{solar} \simeq 3\times 10^{-6} eV^2 and \Delta m_{LSND}^2 \simeq 0.5 eV^2 for \Delta} m^2_{atm} \simeq 2.2\times 10^{-3} eV^2), and \sin^{2}2\theta_{atm} \simeq 1 and \sin^{2} 2\theta_{LSND} \simeq 4m_e/m_{\mu} \simeq 0.02.
Quark and Lepton Mass Matrices with a Cyclic Permutation Invariant Form
As an attempt to give an unified description of quark and lepton mass matrices M_f, the following mass matrix form is proposed: the form of the mass matrices are invariant under a cyclic permutation (f_1 \rightarrow f_2, f_2 \rightarrow f_3, f_3 \rightarrow f_1) among the fermions f_i. The model naturally leads to the maximal mixing between \nu_\mu and \nu_\tau, and with an additional ansatz, it leads to the well-satisfied relations |V_{us}| \simeq \sqrt{m_d/m_s} and |V_{cb}| \simeq \sqrt{m_d/m_b}. \\
US-00-04-02
A Two Parameter Texture of Nearly Bi-maximal Neutrino Mixing
Ambar Ghosa
We propose a texture of three generation Majorana-type neutrino mass matrix in terms of only two parameters which gives rise to nearly bi-maximal mixing angles. We also demonstrate an explicit realization of such type of neutrino mass-matrix in the context of an $SU(2)_L$$\times$$U(1)_Y$ model due to higher dimensional mass terms through the inclusion of discrete $Z_3$$\times$$Z_4$ symmetry and two extra singlet Higgs fields.
An Exact Solution of the Randall-Sundrum Model and the Mass Hierarchy Problem
An exact solution of the Randall-Sundrum model for a simplified case (one wall) is obtained. It is given by the $1/k^2$-expansion (thin wall expansion) where $1/k$ is the thickness of the domain wall. The vacuum setting is done by the 5D Higgs potential and the solution is for a family of the Higgs parameters. The mass hierarchy problem is examined. Some physical quantities in 4D world such as the Planck mass, the cosmological constant, and fermion masses are focussed. Similarity to the domain wall regularization used in the chiral fermion problem is pointed out. We propose and examine the possibility that the 4D massless chiral fermion bound to the domain wall in the 5D world can be regarded as the real 4D fermions such as neutrinos, quarks and other leptons. | CommonCrawl |
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Effects of Deletion of Genes Encoding Fe-Only Hydrogenase of Desulfovibrio vulgaris Hildenborough on Hydrogen and Lactate Metabolism
Brant K. J. Pohorelic, Johanna K. Voordouw, Elisabeth Lojou, Alain Dolla, Jens Harder, Gerrit Voordouw
Brant K. J. Pohorelic
Department of Biological Sciences, University of Calgary, Calgary, Alberta T2N 1N4, Canada
Johanna K. Voordouw
Elisabeth Lojou
BIP-IBSM-CNRS 13402 Marseilles Cedex 20, France
Alain Dolla
Jens Harder
Max Planck Institute for Marine Microbiology, D-28359 Bremen, Germany
Gerrit Voordouw
Department of Biological Sciences, University of Calgary, Calgary, Alberta T2N 1N4, CanadaBIP-IBSM-CNRS 13402 Marseilles Cedex 20, FranceMax Planck Institute for Marine Microbiology, D-28359 Bremen, Germany
For correspondence: [email protected]
DOI: 10.1128/JB.184.3.679-686.2002
The physiological properties of a hyd mutant of Desulfovibrio vulgaris Hildenborough, lacking periplasmic Fe-only hydrogenase, have been compared with those of the wild-type strain. Fe-only hydrogenase is the main hydrogenase of D. vulgaris Hildenborough, which also has periplasmic NiFe- and NiFeSe-hydrogenases. The hyd mutant grew less well than the wild-type strain in media with sulfate as the electron acceptor and H2 as the sole electron donor, especially at a high sulfate concentration. Although the hyd mutation had little effect on growth with lactate as the electron donor for sulfate reduction when H2 was also present, growth in lactate- and sulfate-containing media lacking H2 was less efficient. The hyd mutant produced, transiently, significant amounts of H2 under these conditions, which were eventually all used for sulfate reduction. The results do not confirm the essential role proposed elsewhere for Fe-only hydrogenase as a hydrogen-producing enzyme in lactate metabolism (W. A. M. van den Berg, W. M. A. M. van Dongen, and C. Veeger, J. Bacteriol. 173:3688–3694, 1991). This role is more likely played by a membrane-bound, cytoplasmic Ech-hydrogenase homolog, which is indicated by the D. vulgaris genome sequence. The physiological role of periplasmic Fe-only hydrogenase is hydrogen uptake, both when hydrogen is and when lactate is the electron donor for sulfate reduction.
Sulfate-reducing bacteria of the genus Desulfovibrio contain the genes for several hydrogenases, including the hynBA and hysBA genes for the NiFe- and NiFeSe-hydrogenases, the hydAB genes for Fe-only hydrogenase, and the hndABCD genes for NADP-reducing hydrogenase. The first three enzymes are translocated by the twin-arginine translocation (tat) system and are thus either periplasmic or membrane bound with the active site facing the periplasm. Only the NADP-reducing hydrogenase has been shown elsewhere to be cytoplasmic (11). Desulfovibrio vulgaris strain Hildenborough has the Fe-only hydrogenase as well as the NiFe- and NiFeSe-hydrogenases. The former is a soluble periplasmic enzyme, whereas the latter two are membrane bound. Searching of the database for the D. vulgaris genome at http://www.tigr.org indicated that D. vulgaris has an HndD homolog but lacks the hndA, hndB, and hndC genes. D. vulgaris is thus unlikely to have a cytoplasmic, NADP+-reducing hydrogenase.
The hydAB genes encode the 46-kDa α and the 10-kDa β subunit of Fe-only hydrogenase from D. vulgaris (25). The structures of Fe-only hydrogenase have been determined recently both for CpI, the cytoplasmic enzyme from Clostridium pasteurianum (16), and for the periplasmic enzyme from Desulfovibrio desulfuricans (14). The sequences of the α and β subunits of the periplasmic enzyme in D. desulfuricans and D. vulgaris form a contiguous, single polypeptide of 60 kDa in CpI. The splitting of the sequence into two polypeptides in Desulfovibrio spp. is for export: the β subunit has a long twin-arginine-type signal sequence for this purpose (24). The function of CpI in the fermentative metabolism of C. pasteurianum is to reoxidize reduced ferredoxin, using protons as the electron acceptor to produce H2 (2). A similar function has been proposed elsewhere for Fe-only hydrogenase in lactate metabolism by D. vulgaris (23). Reduction of the Fe-only hydrogenase content by expression of hydAB antisense RNA reduced the growth rate and growth yield of D. vulgaris in lactate- and sulfate-containing medium. Observation of a reduced H2 burst in the initial stages of growth on this medium also pointed to a decreased H2 production activity.
Desulfovibrio spp. are the only microorganisms with a periplasmic Fe-only hydrogenase. If the physiological function of this enzyme is indeed in H2 production, then the question arises why it is not simply cytoplasmic as in Clostridium spp. A cytoplasmic, H2-producing hydrogenase contributes to the transmembrane pH gradient, whereas a periplasmic, H2-producing hydrogenase consumes this gradient. Because Fe-only hydrogenase is the most abundant periplasmic hydrogenase in D. vulgaris (10), one expects it to also catalyze hydrogen uptake at least under certain metabolic conditions. We have now constructed a hydAB mutant with a modification of the sacB replacement mutagenesis method first used to mutate the dcrA gene of D. vulgaris (7). The physiological properties of the resulting hyd mutant are compared with those of the wild-type strain.
Growth of bacteria.All bacterial strains used are listed in Table 1. Tryptone-yeast extract medium was used for growth of Escherichia coli strains at 37°C (27). D. vulgaris strains were stored in Postgate's medium B and grown in medium C and on solid medium E (17) at 32°C in a mixed-gas atmosphere (5% [vol/vol] H2, 10% CO2, and 85% N2) as described previously (7, 8). Antibiotics kanamycin (50 μg/ml) and chloramphenicol (CHL) (10 μg/ml) were added, when appropriate. Media B, C, and E contain 1 g of yeast extract/liter. D. vulgaris strains were also grown on defined BT media without yeast extract (4–6). Growth in these media was under conditions of exchange with the mixed-gas atmosphere of the anaerobic hood. BT media were made by combining, per 100 ml, 90 ml of minimal salt solution, 1.25 ml of trace element solution, 1 ml of CaCl2 solution, 1 ml of sodium lactate solution (for BT lactate-sulfate medium) or 1 ml of sodium acetate solution (for BT H2-sulfate medium; acetate and CO2 serve as the carbon source, and H2 serves as sole electron donor in this medium), 0.5 to 5 ml of 1 M Na2SO4, and 3 to 6 ml of water. All these were formulated as described before (6), except for the minimal salt solution. Because sulfate was a variable in the growth studies, it was removed from the minimal salt solution, which contained, per liter, 1.65 g of NH4Cl, 0.9 g of KH2PO4, and 0.36 g of MgCl2 · 6H2O at pH 7. Cultures of 100 ml were grown in 300-ml nephelometer (Erlenmeyer) flasks (Bellco Glass Inc., Vineland, N.J.) equipped with a 1.2- by 13-cm sidearm and closed with a sponge cap allowing gas exchange. Cell densities were measured with a Klett-Summerson colorimeter installed in the anaerobic chamber (Forma Scientific, Marietta, Ohio) or with a Shimadzu spectrophotometer at 600 nm. A cell density of 147 Klett units corresponded to an optical density at 600 nm (OD600) of 1, which in turn corresponded to a cell dry weight of 0.162 g/liter (6). BT medium plates were prepared by adding 15 g of agar per liter of medium.
Bacterial strains, primers, vectors and plasmids used
Growth in 500-ml closed serum bottles, containing 250 ml of headspace consisting of 90% (vol/vol) N2 and 10% (vol/vol) CO2, was in defined Widdel-Pfennig (WP) medium. Use of BT medium in this closed culture system led to formation of significant amounts of precipitated sulfides due to buildup of H2S, which interfered with the OD600 determination. WP media were formulated according to the work of Widdel and Bak (28) using 38 mM lactate and 28 mM sulfate.
Plasmid construction and replacement mutagenesis.All plasmids and vectors used are listed in Table 1. The 4,678-bp HindIII-EcoRI insert from plasmid pHV15 was ligated to pNOT19, digested with HindIII and EcoRI, to give plasmid pHV15Not. pHV15Not was digested with SacII (nucleotide [nt] 1789) and MstII (nt 3796) to delete a 2-kb fragment containing the hydAB genes. The digested plasmid was end repaired with Klenow polymerase and deoxynucleoside triphosphates and ligated in the presence of BamHI linkers (5′-CCGGATCCGG) to give plasmid pΔHydAB. A 1.4-kb BamHI fragment from plasmid pUC19Cm, containing the cat gene, was ligated into the BamHI site of plasmid pΔHydAB to give plasmid pΔHydABcat. Plasmid pΔHydABcat was next digested with restriction endonuclease NotI and ligated with the 4.5-kb NotI fragment from pMOB2, containing the oriT locus and the sacBR genes, to give pΔHydAB-CTB. A map of this 11.2-kb plasmid is shown in Fig. 1.
Maps of the integration plasmid pΔHydAB-CTB and the DNA region containing the wild-type and mutated hydAB locus (WT and Hyd100, respectively). Positions of restriction sites for BamHI (B), EcoRI (E), HindIII (H), KpnI (K), MstII (M), NotI (N), PstI (P), SalI (S), and SacII (Sc) are shown. The 2-kb SacII-MstII fragment, containing the hydAB genes, in the wild-type strain was replaced by a 1.4-kb BamHI fragment containing the cat gene in D. vulgaris Hyd100.
E. coli S17-1(pΔHydAB-CTB) was conjugated with D. vulgaris, and single-crossover integrants were selected on medium E plates with CHL and kanamycin (7). Southern blot analysis was used to verify the integration of pΔHydAB-CTB into the D. vulgaris chromosome. A 2-kb upstream region probe (Fig. 1, up region) was prepared by digesting pHV15 with HindIII and SacII and agarose gel electrophoresis to isolate the 2-kb fragment and radioactive labeling with the random hexamer procedure using [α-32P]dCTP. The probe was hybridized with the blots under highly stringent conditions (18, 27).
Gene replacement was achieved by growing a mapped, single-crossover integrant, either D. vulgaris Hyd 3-2 or D. vulgaris Hyd 3-7, in 5 ml of medium C with CHL, followed by growth in 5 ml of BT H2-sulfate medium with CHL. Aliquots (50 to 100 μl) of the latter culture were then plated on BT H2-sulfate medium plates containing CHL and 5% (wt/vol) sucrose. Colonies, appearing after 3 to 4 weeks, were grown in 5 ml of medium C with CHL. DNA was isolated from these cultures and used to test their genotype by Southern blotting, yielding D. vulgaris Hyd100 for further study.
Analysis of the D. vulgaris Hyd100 phenotype.For immunoblotting, the Hyd100 and wild-type strains were grown in 5 ml of medium C to stationary phase. Cells were pelleted and suspended in 250 μl of water. An equal volume of sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) incubation buffer (8) was then added, and the samples were boiled immediately. Aliquots (30 μl; ca. 100 μg of protein) were loaded onto 12.5% (wt/vol) polyacrylamide gels together with molecular mass markers and a known amount of purified Fe-only hydrogenase. SDS-PAGE was performed according to the method of Laemmli (9). The separated proteins were electroblotted onto nitrocellulose (22). The blots were blocked with gelatin, incubated with a 2,000-fold dilution of anti-HydA antibodies for 4 h and then with a 2,000-fold dilution of the anti-rabbit, alkaline phosphatase-conjugated secondary antibody for 2 h, and then stained by incubation with nitroblue tetrazolium and 5-bromo-4-chloro-3-indolyl-β-d-phosphopyranoside (BCIP).
Hydrogenase activity in the wild-type and Hyd100 strains was qualitatively evaluated by PAGE under nondenaturing conditions using the Phast System (Amersham Pharmacia Biotech). Following culturing of D. vulgaris Hyd100 and wild type in 100 ml of medium C to stationary phase, cells were pelleted and broken with a French press (590 MPa). After centrifugation for 20 min at 4,000 × g, a volume of supernatant containing 1 μg of total protein was loaded onto a PhastGel gradient gel of 8 to 25% polyacrylamide (pH 8.3). A 1-μl sample (2.5 μM) of pure Fe-only hydrogenase from D. vulgaris Hildenborough was also loaded. The gel was stained for hydrogenase activity according to the method of Ackrell et al. (1).
Measurement of hydrogenase activity by CV.Enzyme activities were derived from enzyme-catalyzed electrochemical reactions (21) as measured by cyclic voltammetry (CV) An electrode modified with whole cells or cell fractions of D. vulgaris wild-type or Hyd100 strains produced a methyl viologen (MV)-mediated catalytic current (Ik) for evolution or consumption of H2. The catalytic scheme for H2 evolution can be summarized as follows: $$mathtex$$\[\mathrm{MV}^{2{+}}_{\mathrm{electrode}}{+}\mathrm{e}^{{-}}\ {\leftrightarrow}\ \mathrm{MV}^{{\cdot}{+}}_{\mathrm{electrode}}\]$$mathtex$$ $$mathtex$$\[\mathrm{MV}^{{\cdot}{+}}_{\mathrm{electrode}}\ {\leftrightarrow}\ \mathrm{MV}^{{\cdot}{+}}_{\mathrm{cell}}\]$$mathtex$$ $$mathtex$$\[\mathrm{MV}^{{\cdot}{+}}_{\mathrm{cell}}{+}\mathrm{H}_{2}\mathrm{ase}_{\mathrm{ox}}\ {\rightarrow}\ \mathrm{MV}^{2{+}}_{\mathrm{cell}}{+}\mathrm{H}_{2}\mathrm{ase}_{\mathrm{red}}\]$$mathtex$$ $$mathtex$$\[\mathrm{H}_{2}\mathrm{ase}_{\mathrm{red}}{+}\mathrm{H}^{{+}}\ {\rightarrow}\ \mathrm{H}_{2}\mathrm{ase}_{\mathrm{ox}}{+}1/2\mathrm{H}_{2}\]$$mathtex$$ $$mathtex$$\[\mathrm{MV}^{2{+}}_{\mathrm{cell}}\ {\rightarrow}\ \mathrm{MV}^{2{+}}_{\mathrm{electrode}}\]$$mathtex$$ CV measurements were carried out using an EGG 263A potentiostat modulated by EGG PAR M270 software. Cyclic voltammograms were recorded at a sweep rate of 20 mV s−1 at room temperature. A three-electrode system consisting of a Metrohm Ag-AgCl-NaCl-saturated reference electrode, a platinum wire auxiliary electrode, and the glassy carbon membrane electrode (GCME) as the working electrode was used throughout. All potentials were relative to the Ag-AgCl-NaCl-saturated reference electrode. Potential versus the standard H2 electrode can be obtained by adding 0.210 V. For each H2 evolution measurement, the solution (100 mM acetate [pH 5.6], 0.1 mM MV2+) was flushed with high-purity nitrogen, whereas for each H2 uptake measurement the solution (100 mM Tris-HCl [pH 8.5], 0.1 mM MV2+) was flushed with H2 gas. Prior to use, the GCME (A = 0.07 cm2 from Tokay) was first polished with 0.05-μm alumina slurry. A 2-μl aliquot of either whole cells or cellular fraction suspensions of known protein concentration, prepared as described below, was then dropped onto the surface of the glassy carbon electrode. After the solvent had evaporated under N2 flow, the electrode was pressed against a square piece (1 by 1 cm) of dialysis membrane (Visking; molecular weight cutoff, 12,000 to 16,000; 30 μm thick). A rubber ring was fitted around the electrode body so that a uniform thin layer existed between the electrode surface and the membrane. The membrane electrode was then placed in the electrochemical cell, The dialysis membrane prevented loss of cells or protein molecules from the electrode surface, whereas small substrate and product molecules freely diffused through the membrane. The limiting current Ik was independent of the sweep rate, as expected for a catalytic process.
For sample preparation, all buffers were deaerated by flushing with argon. Cultures of D. vulgaris wild-type and Hyd100 strains, grown in 10 ml of medium C for 13 h, were centrifuged (5,700 × g, 15 min), and the cell pellets were resuspended in 1 ml of 0.1 M Tris-HCl-0.15 M NaCl (pH 7.6) and recentrifuged. The cell pellets were then resuspended in 1 ml of 0.01 M Tris-HCl, pH 7.6. For preparation of subcellular fractions, 50-ml cultures in medium C were centrifuged and the pellets were resuspended in 0.5 ml of 0.1 M Tris-HCl-0.1 M EDTA (pH 9.0) and incubated at 37°C for 30 min. After centrifugation (5,700 × g, 15 min, 4°C), periplasmic fractions were collected as the supernatants. These were concentrated and desalted by ultrafiltration on Centricon YM10 (Millipore) filters using 0.01 M Tris-HCl, pH 7.6. The pellets were resuspended in 4 ml of 0.01 M Tris-HCl, pH 7.6, and then passed twice through a French press unit at 590 MPa. After centrifugation (5,700 × g, 60 min, 4°C), the supernatant was centrifuged at 125,000 × g (12 h, 4°C). The cytoplasmic fractions were collected as the supernatants, and the membrane fractions were collected as the pellets from the latter centrifugation. The pellets were washed once by resuspension in 4 ml of 0.01 M Tris-HCl, pH 7.6, and centrifugation (125,000 × g, 12 h, 4°C). The membrane fractions were then resuspended in 500 μl of 0.01 M Tris-HCl buffer, pH 7.6. Purity of periplasmic and cytoplasmic fractions was verified by spectroscopic determination of the presence of cytochrome c3 and of desulfoviridin, respectively.
Analytical procedures.H2 concentrations in the headspace of serum bottles containing cultures in WP media were analyzed by gas chromatography using a Shimadzu GC-8A gas chromatograph with a Molecular Sieve 5A column (80 to 100; 0.125 in. by 2 m; 32 ml of N2/min at 40°C) and an RGD2 reduction gas detector (Trace Analytical). Headspace samples of 1 ml were withdrawn and appropriately diluted in closed serum bottles filled with air. A 1-ml diluted sample was then withdrawn with a gastight syringe and injected into the gas chromatograph. H2 concentrations were calculated by comparing the peak area for the diluted sample with that of an H2 standard (20 ppm in helium).
Molecular biology reagents and (bio)chemicals.DNA manipulation enzymes were obtained from Pharmacia or Roche Molecular Biochemicals. [α-32P]dCTP (3,000 Ci/mmol; 10 mCi/ml) was from ICN Biomedicals. Deoxyoligonucleotide primers P173-r and P174-f (Table 1) were obtained from University Core DNA Services of the University of Calgary. Rabbit antiserum generated against HydA purified from D. vulgaris was as described before (26). The chromogenic substrates nitroblue tetrazolium and BCIP, as well as anti-rabbit and anti-mouse immunoglobulin G-alkaline phosphatase conjugates, were from Promega. Nitrogen and mixed gas were from Praxair Canada. MV dichloride was purchased from Aldrich. All other chemicals were of reagent grade and used as received.
Construction of an hydAB deletion mutant.Conjugation of E. coli S17-1(pΔHydAB-CTB) and D. vulgaris gave rise to single-crossover integrants in which the suicide plasmid recombined with the chromosome through either the upstream (Fig. 1, up region) or the downstream (Fig. 1, down region) homologous regions. Digestion of chromosomal DNA of these and of the wild type with EcoRI and hybridization of a Southern blot with the upstream region probe resulted in hybridizing fragments of 7.2 and/or 4.8 and/or 2.5 kb, as expected based on the maps shown in Fig. 1. It appeared impossible to achieve the second gene-replacing crossover by growing single-crossover integrants in medium C or medium E with CHL and sucrose. In view of this lack of success in rich media (both medium C and medium E contain 1 g of yeast extract per liter), we decided to attempt BT H2-sulfate medium. Growth in this medium is slow (μ = 0.050 h−1 [6]), and colonies emerged on solid BT H2-sulfate medium containing CHL and sucrose only after 3 to 4 weeks of incubation at 32°C. However, when these were subsequently cultured in liquid BT H2-sulfate medium with CHL or medium C with CHL a large fraction (50 to 100%) was found to have the required genotype both by Southern blot analysis and by PCR using primers P173-r and P174-f. One of these, D. vulgaris Hyd100, was selected for further study.
Immunoblotting and native PAGE.Comparison of whole-cell extracts following SDS-PAGE and immunoblotting indicated the presence of the 46-kDa α subunit of Fe-only hydrogenase in D. vulgaris wild type (Fig. 2B, lane 2) and its absence in the Hyd100 strain (Fig. 2B, lane 3). Native gel electrophoresis also showed definitively that active Fe-only hydrogenase was present in the wild type but not in the Hyd100 strain. The membrane-bound NiFe- and NiFeSe-hydrogenases did not give an activity band under the conditions used. Some hydrogenase activity remained in the wells (Fig. 2C). In order to accurately evaluate the remaining hydrogenase activity in Hyd100 compared to that in wild-type cells, H2 uptake and evolution were quantitated in both whole cells and subcellular fractions using an electrochemistry approach.
Electrophoretic comparison of D. vulgaris wild-type and Hyd100 strains. (A) Gel stained with Coomassie blue following SDS-PAGE. Lane M, molecular mass markers; from top to bottom, 94, 67, 43, and 30 kDa; lane 1, 1 μg of purified Fe-only hydrogenase; lane 2, D. vulgaris wild type, 50 μg of protein; lane 3, D. vulgaris Hyd100, 50 μg of protein. (B) Immunoblot of gel as in panel A incubated with a polyclonal antiserum specific for the 46-kDa α subunit of Fe-only hydrogenase. (C) Native gel stained for hydrogenase activity. Lane 2, cell extract from D. vulgaris wild type, 4 μg of protein; lane 3, cell extract from D. vulgaris Hyd100, 4 μg of protein.
H2 uptake and production activity by bioelectroanalysis.CV curves for measurement of H2 production and consumption activities are shown in Fig. 3A and B. Figure 3C indicates a reversible electrochemical wave with a midpoint potential of −630 mV, representing the MV2+-MV·+ electrochemical couple. This signal turned into an S-shaped pH-dependent wave when using the D. vulgaris cell-modified GCME (Fig. 3A and B). The value of the limiting current Ik is directly proportional to the enzyme activity in the linear part of the curve of Ik versus protein concentration. This is illustrated in Fig. 3D for whole cells of the D. vulgaris wild-type and Hyd100 strains. It appeared that Ik first linearly increased with protein concentration and then leveled off as protein concentration was increased further. Estimates of the specific hydrogenase activity (relative units) were obtained by dividing the limiting current Ik by the current obtained at the GCME with MV2+ alone and by the protein concentration. The results are summarized in Table 2, which shows that the specific hydrogenase activity of the Hyd100 strain is about 50% less than that of the wild-type strain in both H2 uptake and H2 evolution reactions. A large difference was observed in the periplasmic fraction, which is in agreement with the deletion of the gene encoding periplasmic Fe-only hydrogenase in the Hyd100 strain. On the other hand, no significant changes were observed in either the cytoplasmic or the membrane fractions (Table 2).
Determination of hydrogenase activities with CV. (A to C) Cyclic voltammograms obtained with a GCME. (A) D. vulgaris cell-modified GCME (1 μg of protein) in N2-saturated 0.1 M Na-acetate buffer, pH 5.6, with 100 μM MV2+; (B) D. vulgaris cell-modified GCME (1 μg of protein in H2-saturated 0.1 M Tris-HCl buffer [pH 8.5] with 100 μM MV2+); (C) bare GCME under same conditions as in panel A. (D) Limiting catalytic current Ik plotted against protein concentration (milligrams per milliliter) for experiments with cell-modified GCME under the same conditions as for panel A. Data are for whole cells of D. vulgaris wild-type (closed circles) and Hyd100 (open circles) strains.
Specific hydrogenase activities derived from CV experiments on whole cells or subcellular fractions prepared from D. vulgaris wild-type and Hyd100 strains
Comparison of growth yields of the wild-type and Hyd100 strains in BT media.Growth of D. vulgaris Hyd100 colonies on solid BT H2-sulfate medium already indicated that Fe-only hydrogenase is not critically important for H2 uptake. Colonies of the wild-type and Hyd100 strains also grew similarly on medium E plates, in which lactate serves as the electron donor for sulfate reduction. The Hyd100 strain is thus capable of using both lactate and H2 as the electron donor for sulfate reduction.
Growth rates and yields were compared quantitatively in BT media in which either H2 or lactate and H2 served as the electron donor for sulfate reduction. H2 was in principle never limiting, because growth was in an atmosphere containing 5% (vol/vol) H2 under conditions of gas exchange. These culture conditions allow a specific growth rate (μ) of 0.050 h−1 and a molar growth yield, Ysulfate, of 5.1 g of cells mol−1 with H2 (6), similar to values observed by Badziong and Thauer (3) for the same medium. Growth studies were carried out at different sulfate concentrations (5 to 50 mM). Growth rates of D. vulgaris wild-type and Hyd100 strains were similar. However, differences in final cell densities were found (Fig. 4). Final cell densities were constant (Fig. 4A) or tended to display a maximum (Fig. 4B). The average of four determined values at or near the maximum (Fig. 4B; 70 to 100 h) was taken as the final cell density in these cultures. Comparison of final cell densities as a function of sulfate concentration indicated that the hyd mutant grew less well than did the wild-type strain when only H2 was available, especially at a high sulfate concentration (Fig. 5A). Statistical analysis of the data indicated that the ratio (R) of final cell densities of the wild-type and Hyd100 strains was R = (1.10 ± 0.16) for sulfate concentrations from 5 to 20 mM and R = (1.30 ± 0.15) for sulfate concentrations from 25 to 50 mM. The limiting slope at low sulfate concentration was the same for both strains and indicated a Ysulfate of 4.4 g of cells mol−1 for H2 as electron donor, in reasonable agreement with previously reported values. With lactate (38 mM) and hydrogen as the electron donor, no differences in final cell density between the two strains were found at sulfate concentrations of 0 to 15 mM [R = (0.98 ± 0.10)]. The final cell density increased linearly with sulfate concentration in this range (Fig. 5B). Above 15 mM sulfate, this increase ceased abruptly, and final cell densities decreased with increasing sulfate concentration. Because the transition point is close to 19 mM sulfate, where the culture goes from sulfate to lactate limitation, one can conclude that lactate serves as the main electron donor in the cultures with 5 to 15 mM sulfate. The data in this range indicate a Ysulfate of 6.2 g of cells mol−1 for lactate as the electron donor. Above 19 mM sulfate, the wild-type strain grew to a higher final cell density than did the Hyd100 strain [Fig. 5B, R = (1.13 ± 0.08)]. The reasons why final cell densities decrease in cultures with increasing sulfate concentration in the range 20 to 50 mM are not clear.
Growth of D. vulgaris wild-type (□) and Hyd100 (◊) strains in BT medium. (A) H2 as sole electron donor for sulfate reduction; 40 mM sulfate. (B) Lactate (38 mM) and H2 as electron donor for sulfate reduction; 30 mM sulfate.
Final cell density (OD600) for cultures of D. vulgaris wild-type (□) and Hyd100 (◊) strains in BT medium as a function of the sulfate concentration. All data points are for single cultures, except where the vertical bars indicate the spread of data obtained for duplicate cultures. Cultures were with H2 only (A) or with lactate (38 mM) and H2 as the electron donor for sulfate reduction (B). Lactate becomes limiting at sulfate concentrations above 19 mM (↓). Final cell densities for wild type were (13 ± 8)% larger than for Hyd100 cells at 25 to 50 mM sulfate. The downward-sloping lines drawn are best fits to the data in this range of sulfate concentrations and reflect this difference in cell density.
Comparison of growth and hydrogen production in a closed culture system.Cultures of the wild-type and Hyd100 strains in 500-ml serum bottles containing 250 ml of WP medium (38 mM lactate and 28 mM sulfate) and 250 ml of a 90% (vol/vol) N2 and 10% (vol/vol) CO2 headspace were compared. The Hyd100 strain grew to a significantly lower cell density than did the wild type (Fig. 6A) while transiently producing significantly larger amounts of hydrogen (Fig. 6B). The levels of hydrogen indicated in Fig. 6B were reproducible in two other experiments (data not shown). The fact that the absence of Fe-only hydrogenase leads to increased headspace hydrogen concentrations points to a role of this enzyme in hydrogen consumption, not hydrogen production during lactate metabolism.
Growth of wild-type and hyd mutant strains in WP medium containing lactate (38 mM), sulfate (28 mM), and a headspace of 10% (vol/vol) CO2 and 90% N2. Plotted as a function of time (hours) are cell density (OD600) (A) and H2 concentration in the headspace (parts per million) (B).
The genes for Fe-only hydrogenase were the first to be cloned and sequenced for D. vulgaris (25, 26). However, the isolation of a deletion mutant proved to be problematic. van den Berg et al. (23) were unsuccessful in their attempts to construct a mutant and resorted to antisense RNA expression technology to reduce hydAB gene expression. Following transfer of pWB1151, a broad-host-range plasmid that constitutively expresses hydAB antisense mRNA, a two- to threefold-reduced content of Fe-only hydrogenase of D. vulgaris was found. This decreased both growth rates and growth yields when lactate (70 mM) was the electron donor for sulfate (53 mM) reduction. The strain with reduced Fe-only hydrogenase content showed a reduced H2 concentration in the headspace compared to that for the wild type. This H2 was transient and was eventually all used for sulfate reduction. The effects of reducing Fe-only hydrogenase content on growth with H2 as electron donor were not reported by these authors.
We have reduced expression of Fe-only hydrogenase to zero (Fig. 2), but we cannot confirm the phenotype suggested by van den Berg et al. (23). With lactate as the electron donor and H2 in the headspace, there was no difference in growth yield between the Hyd100 and wild-type strains when sulfate was limiting (Fig. 5B; 5 to 20 mM). Because lactate (38 mM) was in excess (two lactate molecules reduce one sulfate molecule), it served as the main electron donor for sulfate reduction under these conditions. At sulfate concentrations in excess of available lactate, the wild-type strain reached a higher final cell density than did the Hyd100 strain (Fig. 5B, 20 to 50 mM), possibly indicating more efficient H2 metabolism. Indeed, when H2 was the sole electron donor for sulfate reduction, larger differences in final cell density were seen, especially at high sulfate concentrations (Fig. 5A). When strains were cultured on lactate- and sulfate-containing media, in the absence of H2 (Fig. 6) the hyd mutant was found to generate more, not less, H2 than that generated by the wild-type strain. Therefore, we cannot support the conclusion (23) that Fe-only hydrogenase has an important role in H2 production from lactate when sulfate is in excess. Instead, the data in Fig. 5 and 6 suggest a role of Fe-only hydrogenase in H2 uptake under these conditions. It should be pointed out that the culture conditions are not completely comparable, as van den Berg et al. (23) used rich medium, containing 1 g of yeast extract per liter, whereas we used defined medium in which lactate was the only organic molecule present. The strains used by van den Berg et al. and in this study are identical.
Reduced efficiency of growth on H2 has also been found by Malki et al. (12), who described the effects of single and double mutations in hydrogenase genes of Desulfovibrio fructosovorans. This organism has hyn, hyd, and hnd genes, encoding a periplasmic NiFe-, a periplasmic Fe-only, and a cytoplasmic NADP+-reducing hydrogenase, respectively. The latter is absent from D. vulgaris. Strains with deletions in either the hnd or the hyn gene, as well as a strain with deletions in both the hyn and hnd genes, were still able to grow on H2 as the sole electron donor for sulfate reduction, which was credited to the presence of the remaining Fe-only hydrogenase. The double mutant also had reduced growth efficiency when lactate or pyruvate was used as the electron donor for sulfate reduction.
The data in Fig. 6B indicate that hydrogen production and consumption are an important feature of the use of lactate as the electron donor for sulfate reduction. However, the hydrogen-producing hydrogenase is not the periplasmic Fe-only hydrogenase, as we have shown. A good candidate is a homolog of the Ech-hydrogenase from Methanosarcina barkeri, which appears to be present in the D. vulgaris genome. This enzyme consists of six subunits (EchA to EchF), which together form a membrane-bound hydrogenase of which the active site faces the cytoplasm. The enzyme from M. barkeri has been purified previously as a six-subunit complex following its solubilization from membranes (13). The function ascribed to Ech-hydrogenase, based on biochemical studies of the purified enzyme, is to reoxidize reduced ferredoxin (Fdred) formed during oxidative conversion of acetyl coenzyme A to methyl-tetrahydrosarcinapterin (CH3-H4SPT) and CO2. Ech-hydrogenase reoxidizes Fdred and uses the electrons together with cytoplasmic protons for hydrogen production. The produced hydrogen may then be captured by an externally located NiFe-hydrogenase of M. barkeri, with the electrons being used for cytoplasmic reduction reactions. This hydrogen cycling mechanism contributes to the proton gradient formed across the membrane by M. barkeri, when it metabolizes acetate to methane.
An H2 cycling mechanism was first proposed for Desulfovibrio spp. by Odom and Peck (15). However, a cytoplasmic hydrogenase has so far not been identified. The Ech-hydrogenase homolog of D. vulgaris is encoded by an operon with the same gene order as that in the ech operon from M. barkeri. All subunits have the corresponding sequences from M. barkeri as their closest homologs. Thus, D. vulgaris is likely to have a very similar cytoplasmic, energy-conserving Ech-hydrogenase. Fdred is formed during lactate metabolism by D. vulgaris in the oxidative conversion of pyruvate to acetyl coenzyme A and CO2, catalyzed by pyruvate-ferredoxin oxidoreductase. Fdred may be reoxidized by the Ech-hydrogenase homolog, with the H2 formed by this enzyme diffusing across the membrane to be captured by the periplasmic Fe-only and nickel-containing hydrogenases. The absence of the main periplasmic hydrogenase results in less efficient hydrogen capture and thus in higher hydrogen headspace concentrations. These are transient due to the presence of excess sulfate under the conditions of the experiment shown in Fig. 6.
Why is H2 production and subsequent capture by the Hyd100 mutant bioenergetically less efficient than the proposed H2 cycling catalyzed by the wild-type strain? One factor to consider is that thermodynamically the energy content of H2 per mole is proportional to the logarithm of its concentration. During lactate metabolism, a high H2 concentration may be maintained in the periplasm. Yet, little escapes due to the high periplasmic hydrogenase activity of wild-type D. vulgaris (Fig. 6B). In the case of the hyd mutant, H2 escaping into the headspace is considerably diluted. Although this headspace H2 is used for sulfate reduction, its availability at a reduced concentration leads to less energy conservation. This could be one of the factors contributing to the large difference in cell densities seen in Fig. 6A and could also explain why the presence of 5% (vol/vol) H2 in the headspace reduces this difference (Fig. 5B). Another factor that may contribute to the larger difference in final cell densities between the cultures in Fig. 6A than between those in Fig. 5B is the buildup of H2S, but this remains to be investigated.
This work was supported by a Research Grant from the Natural Science and Engineering Research Council of Canada to G. Voordouw, who also acknowledges the support of a fellowship from the Hanse Wissenschaftskolleg in Bremen, Germany, and of a visiting professorship of the CNRS in Marseille, France. A. Dolla was supported by a NATO postdoctoral fellowship during part of this work. A database of the D. vulgaris Hildenborough genome was searched at the website of The Institute for Genomic Research at http://www.tigr.org. Sequencing of this genome is financially supported by the U.S. Department of Energy.
We thank Marie-Claire Durand, Mireille Bruschi, Daniela Lange, Ramona Appel, Christina Probian, Ralf Rabus, and Fritz Widdel for discussions and technical assistance. We thank Reiner Hedderich for drawing our attention to the presence of Ech-hydrogenase in the D. vulgaris genome.
Received 25 June 2001.
Accepted 31 October 2001.
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Journal of Bacteriology Feb 2002, 184 (3) 679-686; DOI: 10.1128/JB.184.3.679-686.2002
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Sequenced Visit
Given a set of locations the robot should visit all the locations in sequence.
$\mathcal{F} (l_1 \wedge \mathcal{F}(l_2 \wedge \ldots \mathcal{F}(l_n)))$ , where ($l_1, l_2, \ldots$ are location propositions)
Note that the pattern is general and consider the case in which a robot can be in two locations at the same time. For example, a robot can be in an area of a building indicated as l1 (e.g., area 01) and at the same time in a room of the area indicated as l2 (e.g., room 002) at the same time. If the topological intersection of the considered locations is empty, then the robot cannot be in two locations at the same time and the transitions labeled with both l1 and l2 cannot be fired.
This problem is also indicated in literature as reach, cover, or search, meaning that a robot should reach, cover or search a set areas or points, one following the other. This sequence may reflect e.g., a traversal strategy. Note that given two areas a and b that must be visited in sequence by a robot r , the sequence pattern does not forbid robot r to visit b before visiting a, but only requires the robot to visit b after and a while a must be visited.
Locations $l_1$, $l_2$, $l_3$ must be covered following this sequence. The trace $l_1 \rightarrow l_4 \rightarrow l_3 \rightarrow l_1 \rightarrow l_4 \rightarrow l_2 \rightarrow (l_{\# \setminus 3})^\omega$ violates the mission since $l_3$ does not follow $l_2$. The trace $l_1 \rightarrow l_3 \rightarrow l_1 \rightarrow l_2 \rightarrow l_4 \rightarrow l_3 \rightarrow (l_{\#})^\omega$ satisfies the mission requirement.
Sequenced Visit and Avoidance often go together. Avoidance patterns are used to require robots to avoid obstacles as e.g. they guard an area. Triggers can also be used in combination with the Sequenced Visit pattern to specify conditions upon which Visit should start or stop.
The Sequenced Visit pattern generalizes the Ordered visit patterns that further constrain how locations are visited.
Yoo et al. Kress-Gazit et al. and formulate an LTL mission specification that ensures that a set of areas are visited in sequence. Specifically, an LTL mission specification is provided for the following mission requirement: visit a set of area $l1, l2 \ldots ln$ by ensuring that $l1$ is visited after (but not necessarily only after) area $l2$.
$\overset{n}{\underset{i=1}{\bigwedge}} \forall \mathcal{F} (l_i)$
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Plant hairy roots enable high throughput identification of antimicrobials against Candidatus Liberibacter spp.
Sonia Irigoyen ORCID: orcid.org/0000-0003-0647-48081,
Manikandan Ramasamy1,
Shankar Pant1 nAff9,
Prakash Niraula1,
Renesh Bedre1,
Meena Gurung1,
Denise Rossi1,
Corinne Laughlin1,
Zachary Gorman ORCID: orcid.org/0000-0001-8180-60032,
Diann Achor3,
Amit Levy3,4,
Michael V. Kolomiets2,
Mamoudou Sétamou5,
Ismael E. Badillo-Vargas1,6,
Carlos A. Avila ORCID: orcid.org/0000-0002-1969-47061,7,
Michael S. Irey8 &
Kranthi K. Mandadi ORCID: orcid.org/0000-0003-2986-40161,2
Nature Communications volume 11, Article number: 5802 (2020) Cite this article
Agricultural genetics
Biotic
A major bottleneck in identifying therapies to control citrus greening and other devastating plant diseases caused by fastidious pathogens is our inability to culture the pathogens in defined media or axenic cultures. As such, conventional approaches for antimicrobial evaluation (genetic or chemical) rely on time-consuming, low-throughput and inherently variable whole-plant assays. Here, we report that plant hairy roots support the growth of fastidious pathogens like Candidatus Liberibacter spp., the presumptive causal agents of citrus greening, potato zebra chip and tomato vein greening diseases. Importantly, we leverage the microbial hairy roots for rapid, reproducible efficacy screening of multiple therapies. We identify six antimicrobial peptides, two plant immune regulators and eight chemicals which inhibit Candidatus Liberibacter spp. in plant tissues. The antimicrobials, either singly or in combination, can be used as near- and long-term therapies to control citrus greening, potato zebra chip and tomato vein greening diseases.
Fastidious and obligate phytopathogens cause devastating diseases in several food and commodity crops: for example, the phloem-limited bacteria Candidatus Liberibacter spp. cause severe yield losses in the Solanaceae [potato (Solanum tuberosum L.), tomato (S. lycopersicum L.), pepper (Capsicum annuum L.), and tobacco (Nicotiana tabacum L.)] and Rutaceae (Citrus spp.) families1. Potato zebra chip (ZC) disease, putatively caused by Candidatus Liberibacter solanacearum (CLso), is transmitted by an insect vector, the potato psyllid [Bactericera cockerelli (Šulc); Hemiptera: Triozidae]2. Since its discovery in 1994, ZC has been documented in several commercial potato-growing regions of the United States, Mexico, Central America, Australia, and New Zealand, causing yield losses of up to 94%3,4. Similarly, citrus greening or huanglongbing (HLB), putatively caused by Candidatus Liberibacter asiaticus (CLas), is the most devastating disease threatening citrus production worldwide; its insect vector is the Asian citrus psyllid (Diaphorina citri Kuwayama [Hemiptera: Liviidae])5. Between 2006 and 2011, HLB caused losses of more than US$4.5 billion in the US state of Florida alone6. CLas and CLso share some commonalities, such as their unculturable nature, genomic attributes, and lifestyles as psyllid-vectored and phloem-limited bacteria7,8,9,10. These and other diseases caused by fastidious plant pathogens are major threats to global agriculture and food security.
A challenge in identifying therapies against CLas and CLso is our inability to culture or maintain them outside their plant and insect hosts. Over 99% of all microorganisms are estimated to be unculturable in the laboratory11. Attempts have been made to customize artificial growth media and culture conditions to culture such microbes, including specialized modifications to growth medium composition and growth conditions tailored to the particular microbe based on what we know about its natural growth environment (e.g., temperature, pH, incubation time, inoculum size, and CO2/O2 ratio), coculturing with other microbes, and growth in biofilms or other simulated environments11,12,13,14,15,16,17,18,19,20,21. Although these approaches have allowed some success in mono- or cocultures as means to propagate fastidious pathogens but they have provided little contribution to downstream applications such as the screening of antimicrobials against target pathogens, and therefore are not broadly utilized. Because of the constraints that exist in culturing fastidious pathogens in defined media and evaluating potential therapies, we explored alternative strategies.
Hairy roots are neoplastic growth that forms on plant tissues after wounding and infection with the soil bacterium Rhizobium rhizogenes (previously Agrobacterium rhizogenes)22,23. R. rhizogenes introduces its root-inducing transfer DNA (Ri T-DNA), encoding the root locus (rolA, rolB, rolC, and rolD) genes, into the plant genome. Expression of rol genes in planta modulates auxin homeostasis and induce hairy root initiation and proliferation in vitro; thus, hairy root cultures do not require the addition of phytohormones. Hairy roots are anatomically and metabolically similar to normal roots, and possess intact xylem and phloem vasculature connected to the source explant24,25,26. They constitute a valuable tool in plant functional biology, biotechnology, metabolic engineering, molecular pharming, and studies of root–rhizosphere interactions26,27,28,29. Classical microbiological techniques developed early in the 19th century, cultured animal and mammalian viruses in host cells, tissues, and embryonated eggs30,31,32,33,34,35,36. In a similar manner, we hypothesized that plant hairy roots could be suitable for propagating fastidious pathogens.
In this work, we show that hairy roots support the accumulation of fastidious plant bacteria, Candidatus Liberibacter spp. Furthermore, using the microbial hairy roots, we conduct antimicrobial screening leading to the discovery of multiple genetic and chemical therapies that effectively suppress Candidatus Liberibacter spp. in plant tissues.
Hairy roots support propagation of fastidious pathogens
Since Candidatus Liberibacter spp. (CLso and CLas) are phloem-limited pathogens, we tested whether plant hairy roots, which possess intact vasculature24,25,26, could sustain their growth. Hairy roots are anatomically and metabolically similar to normal roots and can be induced in a variety of plants using R. rhizogenes24,25,26. To test our hypothesis, we generated ex vivo and in planta aerial hairy roots from CLso- and CLas-infected potato, tomato and citrus explant tissues (Fig. 1a, b and Supplementary Fig. 1). We used R. rhizogenes harboring a binary T-DNA vector with a green fluorescent protein (GFP) marker to induce hairy root growth. Fluorescence microscopy analysis confirmed the production and authenticity of the transgenic hairy roots, which were distinguished from the adventitious roots by their GFP signal (Fig. 1c, d). The transformation efficiency of the hairy root induction, defined by the percentage of GFP-positive hairy roots among all roots generated, was ~70–90% in potato and tomato and ~50–70% in citrus.
Fig. 1: Candidatus Liberibacter solanacearum and Candidatus Liberibacter asiaticus in potato and citrus hairy roots.
a, b Hairy roots (indicated by arrows) from Candidatus Liberibacter solanacearum (CLso)-infected potato (a) and Candidatus Liberibacter asiaticus (CLas)-infected citrus explants (b). c, d Visual confirmation of GFP expression in hairy roots by fluorescence microscopy. Scale bars, 1 cm. e, f Detection of CLso and CLas in the hairy roots by PCR amplification of diagnostic markers specific to CLso (16S rDNA) and CLas (rplk04/J5 and RNR). GFP, rolB, and rolC encoded on the Ti and Ri plasmids, respectively, and co-transformed into the hairy roots, were used as additional markers for hairy root authenticity; RPL2 and GAPC2 are endogenous potato and citrus genes, respectively, used as genomic DNA controls for PCR. "L" and "HR" indicate leaf and hairy root samples, respectively. "+" indicates positive controls used for the respective PCR amplifications. g, h Temporal growth curves of CLso and CLas in the hairy roots. The Ct values of CLso and CLas in hairy root samples collected at different days in propagation are plotted, whereas the approximate genome equivalents (GE) per nanogram of root genomic DNA are indicated above the bar graph columns. Error bars represent ± standard error of mean (n = 4 and n = 3 for g and h, respectively). Uncropped raw agarose gel images used to prepare e and f are presented in Supplementary Fig. 10. Source data underlying Fig. 1g, h are provided as a Source Data file.
The genes, rolB and rolC, encoded by the Ri T-DNA of R. rhizogenes, were amplified using a polymerase chain reaction (PCR) and used as additional markers to confirm the transformation of the hairy roots (Fig. 1e, f). PCR amplification using diagnostic markers specific to CLso (16S rDNA)37,38 and CLas (A2/J5 and RNR)39,40 readily detected these pathogens in the infected potato and citrus hairy roots, but not in the uninfected hairy roots (Fig. 1e, f). Similar results were obtained with CLso in a second host, tomato (Supplementary Fig. 3). The identity of CLso and CLas in the hairy roots was confirmed by Sanger sequencing and Basic Local Alignment Search Tool (BLAST) analysis of the respective PCR amplicons for similarity to the NCBI nucleotide sequence database (Supplementary Fig. 8). Copy number and bacterial growth curve estimation by quantitative (q) PCR analysis further showed that CLso and CLas could be maintained in the potato and citrus hairy roots for at least 28 and 120 days, respectively (Fig. 1g, h).
Next, we performed transmission electron microscopy (TEM) to visualize CLso/CLas and probe the anatomy of the hairy roots. Unlike PCR/Sanger sequencing or immuno-labeling approaches, TEM alone is not a confirmatory test to identify pathogens. Unfortunately, immuno-labeling is technically challenging to perform for CLso/CLas detection, primarily due to limitations of existing antibodies (only available for CLas) and the low-titers and highly variable nature of CLso/CLas accumulation in infected plant tissues41. Thus far, CLas immuno-labeling was primarily successful in insect-vector42. Nevertheless, our TEM analysis showed multiple rounds- and bacilliform-shaped bacteria-like cells in the infected potato and citrus hairy roots, but not in healthy hairy roots (Supplementary Fig. 9), along with notable symptoms of vascular deterioration and collapse, presumably caused by CLso/CLas43,44. The morphology and the vascular degeneration phenotypes were reminiscent of those reported for CLas-infected citrus43,44. Furthermore, the lack of other microbes in healthy tissues suggests that the normal microbiome is well-below the detection limits of TEM hence the bacteria-like cells observed in the infected tissues are associated with CLso/CLas infection. Combined with the Sanger-sequencing confirmatory analysis, our results sufficiently demonstrate the presence of CLso/CLas in the hairy roots and suggest that the CLso/CLas can be maintained in plant hairy roots like normal infected plant roots.
Genetic evaluation using microbial hairy roots
Hairy roots are functional plant tissues that mimic the pathogen's native host root environment. Hence, we leveraged them to conduct functional genetic screening to identify therapeutics that can inhibit Candidatus Liberibacter spp. Because CLas and CLso are closely related to each other7,8,9,10, we focused mainly on CLso-hairy potato roots for functional genetic screening due to the faster growth of potato hairy roots compared to citrus hairy roots. First, we evaluated whether gain-of-function genetic studies could be performed using the microbial hairy root system. Nonexpresser of pathogenesis-related gene 1 (NPR1) from Arabidopsis thaliana (AtNPR1 and AT1G64280)45 is a well-known broad-spectrum immune modulator that is effective against diverse pathogens. NPR1 is a receptor of the defense hormone salicylic acid (SA) and functions as a transcriptional activator of pathogenesis-related (PR) genes in plant defense and systemic acquired resistance (SAR) responses. We set out to evaluate whether any NPR1 ortholog(s) in Solanaceae could confer tolerance to Candidatus Liberibacter spp. By phylogenetic analysis (Supplementary Fig. 2) with AtNPR1 as a reference, we identified an orthologous gene from tomato (SlNPR1) and cloned the coding sequences into a binary vector containing a GFP reporter gene (Fig. 2a). Then we transformed each NPR1 construct and an empty vector control containing only GFP into unchallenged (healthy) or CLso-infected (CLso) explant tissue for hairy root induction using R. rhizogenes (Fig. 2b)46,47,48,49. Finally, we performed molecular diagnostics on three biological replicates of each type of hairy roots, 30 days after transformation, to confirm the presence of GFP and NPR1 expression in the respective hairy roots (Fig. 2c, Supplementary Fig. 3), followed by a quantitative polymerase chain reaction (qPCR)-based estimation of the CLso titers (Fig. 2d). The heterologous expression of SlNPR1 significantly (p ≤ 0.01) decreased CLso titers (>90%) in the hairy roots compared to the hairy roots transformed with the empty vector (Fig. 2).
Fig. 2: Screening and identification of antimicrobial genes that confer tolerance to Candidatus Liberibacter spp.
a Schematic of a typical transgene overexpression construct with a gene and a GFP marker cassette (included to serve as a visual marker for construct integration in the hairy roots). b Representative images of healthy and CLso-potato hairy roots at 30 days post transformation (i.e., 30 dpt) with control (GFP alone) or with putative disease-resistance genes from Arabidopsis (AtNPR1) and tomato (SlNPR1). c Visualization of the hairy roots under a fluorescence microscope. Scale bars, 1 cm. d Quantification of relative CLso titers in transformed hairy roots at 30 dpt, compared to control (GFP alone). Error bars represent ± standard error of mean (n = 3). p Values were calculated by two-sample t test (one-tailed). The experiment was independently repeated two times, and all attempts of replication were successful. e Relative expression of PR1-like and PR3-like and WRKY6-like genes in AtNPR1- and SlNPR1-expressing hairy roots. Error bars represent ± standard error of mean (n = 3). p Values were calculated by two-sample t test (one-tailed) relative to respective control samples. f Quantification of salicylic acid (SA) in the transformed hairy roots showed lower SA accumulation in AtNPR1 and SlNPR1 overexpressors. Error bars represent ± standard error of mean (n = 4). p Values were calculated by two-sample t test (one-tailed). g Evaluation of eight putative antimicrobial peptides in CLso-potato and CLas-citrus hairy roots. Relative CLso and CLas titers were calculated from five biological replicates. Error bars represent ± standard error of mean (n = 5). p Values were calculated by two-sample t test (one-tailed) relative to infected controls. The experiment was independently repeated two times, and all attempts of replication were successful. Source data underlying Fig. 2d–f are provided as a Source Data file.
To determine the mechanism whereby NPR1 suppressed Candidatus Liberibacter spp. in potato, we monitored the changes in the downstream expression of two PR genes (PR-1 like and PR-3 like). Expression of these PR genes was significantly higher in both SlNPR1- and AtNPR1-expressing hairy roots under healthy conditions than in empty-vector-transformed controls (Fig. 2e). These expression levels were further greatly amplified upon CLso infection in both SlNPR1- and AtNPR1-expressing hairy roots when compared to vector-alone control (Fig. 2e). Expression of a WRKY transcription factor (WRKY6-like) involved in NPR1-mediated PR gene activation50,51 was also strongly induced in SlNPR1- and AtNPRI-expressing hairy roots (Fig. 2e). Compared to healthy hairy roots, CLso-infected hairy roots had significantly higher levels of free SA (Fig. 2f), as determined by liquid chromatography–mass spectrometry (LC–MS/MS). However, despite the activation of downstream defense-related genes, SA levels were significantly lower in SlNPR1- and AtNPR1-expressing hairy roots than in empty vector controls, under both healthy and CLso-infected conditions (Fig. 2f).
In addition to evaluating plant immune modulators, we tested the efficacy of eight putative broad-spectrum antimicrobial peptides (AMPs) isolated from spinach (Spinacia oleracea L.) that show some inhibitory activity against other Gram-positive and Gram-negative bacteria52,53,54. Briefly, we cloned the coding sequences of each peptide into the binary vectors and transformed them into the CLso-potato and CLas-citrus explant tissues for hairy root induction and analyzed them in a manner similar to NPR1 evaluation (Fig. 2b–d). The results of the antimicrobial assays showed that five of the eight AMPs (denoted AMP1–AMP5) significantly (p ≤ 0.05 or 0.01) reduced CLso and CLas titers (~70–98%) in the hairy roots compared to the controls transformed with the empty vector containing GFP alone (Fig. 2g). Conversely, AMP6 and AMP8 appear to slightly increase CLso and CLas titers, possibly by inhibiting other unknown competitive microbes present in the hairy roots.
Genome editing using microbial hairy roots
Next, we evaluated whether the microbial hairy root system could be used to perform genome-editing studies. For proof-of-concept, we evaluated the editing of a stably integrated GFP gene in a genetically engineered potato. To test this hypothesis, we generated constructs containing Cas9 alone (control) or Cas9–sgGFP. We then used these to transform healthy or CLso-containing explant tissues for hairy root induction, as described above (Fig. 3a, b). Fluorescence microscopy imaging of the transformed hairy roots showed a clear loss of GFP fluorescence (from the transgene) (Fig. 3c, d) in hairy roots transformed with Cas9–sgGFP, but not in Cas9 only (control). Amplicon sequencing of the target site confirmed successful editing had occurred in Cas9–sgGFP hairy roots, but not in Cas9 only roots. Frequencies of indels detected were ~86 and 100% in the healthy and CLso hairy roots, respectively (Fig. 3c–f). In addition, we sought to edit an endogenous gene, NPR3, in potato. Unlike NPR1, its homologs NPR3 and NPR4 function antagonistically to suppress plant defense in Arabidopsis, and their levels and activities are tightly regulated via complex biochemical interactions in order to maintain defense homeostasis55,56,57. Therefore, it seemed possible that reducing endogenous NPR3 activity in Solanaceae could impart plant tolerance for CLso. To test this hypothesis, we identified the potato ortholog of AtNPR3 (StNPR3) and generated constructs containing Cas9 alone (control) or Cas9–sgNPR3. We then used these to transform healthy or CLso-containing explant tissues for hairy root induction, as described above (Supplementary Fig. 4a, b). Amplicon sequencing of the target site confirmed that successful editing had occurred in the Cas9–sgNPR3 hairy roots, but not the Cas9 control hairy roots (Supplementary Fig. 4c), albeit at a lower rate of ~40%, which was not unexpected given the heterozygous autotetraploid nature of potato (2n = 4x = 48), which means there are four copies of each endogenous gene. Nevertheless, when we estimated the CLso levels of three biological replicates at 30 days post transformation using qPCR, we found that the Cas9–sgNPR3 hairy roots had significantly (p ≤ 0.01) lower CLso titers (>90%) than the control hairy roots (Supplementary Fig. 4d). The suppression of CLso titers was strongly associated with induced expression of NPR1 and several defense marker genes (WRKY6-like, PR-1 like, and PR-3 like) in the Cas9–sgNPR3-edited hairy roots (Supplementary Fig. 4e). Similar activation of plant defenses was observed in Theobroma cacao wherein CRISPR editing of ~27% alleles of NPR3 was sufficient to enhance resistance to Phytophthora tropicalis58. The observed resistance even when partial NPR3 alleles are removed could be due to the systemic nature of the SAR response, whereby a defense signal originated from the NPR3-edited cells is perpetuated to non-edited cells, that ultimately limits the pathogen accumulation. Alternatively, it is possible that native NPR3 activity is affected by gene dosage or haploinsufficiency59,60,61 and/or dominant-negatively inhibited by the truncated/aberrant NPR3 proteins resulting from the edited NPR3 alleles62. Regardless of the exact mechanisms of the NPR3 action, these results offer proof-of-concept for conducting genome-editing experiments using the microbial hairy root system and uncover NPR3 as a potential target for CRISPR editing to gain resistance to Candidatus Liberibacter spp.
Fig. 3: Evaluation of genome editing in microbial hairy roots.
a Typical CRISPR–Cas9 gene editing construct. The underlined sequence is the target DNA site used in the sgRNA. b–d Transformation of Cas9 alone (control) and Cas9–sgGFP targeting stably expressed GFP transgene in healthy and Candidatus Liberibacter solanacearum (CLso) hairy roots. The loss of GFP fluorescence in Cas9–sgGFP, but not in Cas9 alone, indicates successful editing of the GFP transgene. Scale bars, 1 cm. The experiment was independently repeated two times, and all attempts of replication were successful. e, f Amplicon sequencing confirmed gene editing in the target site (GFP), as indicated by presence of indels in the Cas9–sgGFP hairy roots, but not in Cas9-alone hairy roots. Frequencies of indels detected were ~86 and 100% in the healthy and CLso hairy roots, respectively. Uncropped raw agarose gel images used to prepare Supplementary Fig. 3b are presented in Supplementary Fig. 11.
High throughput in vitro therapeutic screening
The use of trans/cis-genic-based or genome-editing strategies is a stable, long-term solution to combat diseases; however, it is often a precarious one, owing to the tremendous regulatory burden associated with agricultural use of genetically modified crops, even those produced by CRISPR/Cas. Thus, we also sought to find short- to intermediate-term solutions, and we therefore conducted high-throughput screening of small-molecule inhibitors among a diverse collection of bioactive compounds approved for use in United States and other countries (Microsource Discovery Systems, Inc.)63,64. For this, we developed an in vitro multi-well plate assay using the microbial hairy roots, which is scalable from 12- to 48- to 96-well layout. Briefly, we generated microbial hairy roots with CLso and CLas, transferred them to a multi-well plate containing growth medium to maintain the hairy roots, and then added small molecules of interest to the assay plates and incubated the hairy roots for 72 h (Fig. 4a). We performed a primary screening with ~220 compounds in CLso hairy roots, using two biological replicates per compound (10 µM) along with untreated or solvent-alone controls (DMSO, 0.1% v/v). By molecular diagnostics, we identified nine molecules that inhibited CLso titers by >50–70% when compared to the untreated controls (Supplementary Dataset 1). We retested these nine molecules using five biological replicates, alongside a reference antibiotic, tetracycline (TC, 250 ppm), that was previously reported to moderately inhibit CLas in planta65. We also screened the efficacy of the nine compounds (25 µM) and tetracycline (TC, 500 ppm) in CLas hairy roots. Seven out of the nine test compounds, as well as tetracycline, consistently showed statistically significant (p ≤ 0.05 or 0.01) inhibition of CLso (Fig. 4b). Four compounds showed statistically significant (p ≤ 0.05 or 0.01) inhibition of CLas, on par with tetracycline (Fig. 4c). Three compounds inhibited both CLso and CLas (#3, #8 and #9) (Fig. 4b, c). In general, the active compounds appear to possess diverse structures and bioactive profiles (Fig. 4d–m): three of them were indexed as antibacterial, whereas others were classified as antifungal, antiprotozoal, hemostatic, antihistaminic, antidiabetic, and mydriactic drugs targeting various biological processes. Note that since the retested nine molecules were originally prescreened in CLso-potato hairy roots (Supplementary Data 1), it is very likely that further re-screening of the entire library using CLas-citrus hairy roots would identify additional hits.
Fig. 4: High-throughput screening and identification of small molecules that confer tolerance to Candidatus Liberibacter spp.
a–c Multi-well microbial hairy root culture plates were used to conduct high-throughput screening of ~220 compounds (Supplementary Dataset 1). The efficacy data for nine selected hit compounds were re-assayed at 10 and 25 µM, respectively, in a, b CLso and a, c CLas hairy roots. Untreated (UT) and tetracycline (TC)-treated hairy roots (at 250 and 500 ppm, respectively, for the CLso and CLas assays) were used as positive and negative controls. The bacterial titers were estimated by qPCR after 72 h of treatment with each compound and plotted relative to those of untreated samples (set to 100%). Error bars represent ± standard error of mean (n = 5). p Values were calculated by two-sample t test (one-tailed) relative to untreated samples. d–m Chemical structures of d tetracycline and the nine hits: e aminocaproic acid (#1), f carbinoxamine maleate (#2), g chloroxylenol (#3), h chlorpropamide (#4), i chlortetracycline (#5), j cinoxacin (#6), k cortisone acetate (#7), l duartin (#8), and m cyclopentolate hydrochloride (#9). The structures of the different chemical compounds were retrieved from the ChemSpider database (http://www.chemspider.com/) (last accessed on 15 October 2020). Source data underlying Fig. 4b, c are provided as a Source Data file.
Besides screening small molecules, we also tested whether the in vitro hairy root bioassay setup could be used for evaluating small peptides (~50 aa). For this, we tested the efficacy of the two AMPs (AMP2 and AMP5; Fig. 2g) by direct vacuum infiltration of the recombinant peptides (5 and 10 µg/ml) into the CLas-citrus microbial hairy roots, in a manner similar to the small molecules (Fig. 4a). Molecular diagnostics performed after 72 h of treatment revealed that both AMPs showed statistically significant (p ≤ 0.05 or 0.01) dose-dependent inhibition of CLas (Supplementary Fig. 5) compared to untreated controls.
Delivery of small molecules into the plant phloem, especially in hardy perennial citrus trees, is a major challenge. The most practical and common approach is foliar spraying; however, this was shown to be relatively ineffective in citrus as a means to deliver bactericides such as oxytetracycline to inhibit CLas65. Multiple research groups are evaluating alternative approaches, such as trunk injection and/or nanoparticle-based systems65,66. Potato plants are relatively easier to handle and are amenable to foliar spraying techniques. Thus, to determine the in planta efficacy of the three molecules that showed good inhibitory activity against both CLso and CLas (#3, #8, and #9) in the hairy root assays (Fig. 4), we used the CLso-potato plant system. First, we repeated efficacy assays in CLso-potato microbial hairy roots, testing a broader range of concentrations (0, 5, 10, 25, and 50 µM) (Supplementary Fig. 6) to establish a general dose-response curve. Based on these results, 10 and 25 µM concentrations were chosen for in planta foliar spraying experiments. Plants were sprayed with each compound twice a week, alongside untreated plants and tetracycline (250 ppm)-treated plants as negative and positive controls, respectively. Disease progression was monitored for 28 days post infection (dpi), by which time untreated plants showed the typical foliar disease symptoms such as chlorosis, necrosis, leaf curling and wilting, and were quite close to dying (Fig. 5a). In contrast, potatoes sprayed with any of the three test molecules showed a lower level of disease symptoms that was also dose-dependent, i.e., plants sprayed with 25 µM showed the least symptoms, followed by those sprayed with 10 µM and untreated plants, and the level of protection was on par with that from tetracycline treatment (Fig. 5a). The attenuated symptoms also correlated with lowered CLso titers in the respective treatment groups compared to untreated controls (Fig. 5b). Together, these experiments demonstrate that the identified compounds can inhibit Candidatus Liberibacter spp. in planta, and the efficacy results conform with the results of the hairy root bioassays.
Fig. 5: In planta application of small molecules confer tolerance to Candidatus Liberibacter spp. in potatoes.
a Visual symptoms of plants at 28 days post infection (dpi). Tetracycline (TC, 250 ppm) and three selected small molecules (#3, #8, and #9; see Fig. 4) were sprayed on the plants at two concentrations (10 and 25 µM) as indicated. Disease symptoms of chlorosis, necrosis, leaf curling, and wilting were monitored. b CLso titers were determined by qPCR at 28 dpi. All titers are plotted relative to those of untreated samples, which were set to 100%. Error bars represent ± standard error of mean (n = 4). p Values were calculated by two-sample t test (one-tailed) relative to untreated samples. Source data underlying Fig. 5b are provided as a Source Data file.
A recent consensus study by the US National Academy of Sciences (NAS) on the status of the citrus greening research and development concluded that there is no single effective therapy to fight the disease67. This holds not only for citrus greening but also for other insect-vectored diseases caused by Candidatus Liberibacter spp.1. The NAS report also suggested deploying a combination of therapies and management practices to combat the disease54,67,68,69,70. However, a major challenge in finding therapies has been the unculturable nature of Candidatus Liberibacter spp., which makes conventional gene- or chemical-therapy evaluation laborious and time-consuming. For instance, evaluating the efficacy of an AMP using either stable transformation or transient viral vectors (e.g., tobacco rattle virus and citrus tristeza virus) followed by disease challenges in mature plants could take from 6 months to several years71,72,73. Similarly, screening of small-molecule inhibitors of active ingredients by foliar spraying, by stem injection, or by grafting using mature plants, is inherently variable due to the challenges associated with delivering the inhibitors to the bacteria that reside unevenly within the plant vasculature65. Furthermore, the latter approaches are laborious and time-consuming, and hence not suitable for high-throughput studies65,68,69. Researchers have also resorted to using distantly related microbes as surrogates74,75,76,77. However, these results may or may not be reliable or extrapolatable to the target pathogen, owing to differences in pathosystems and virulence mechanisms (or lack thereof).
In a manner similar to the classical culture strategies for mammalian viruses in host cells, tissues, and embryonated eggs30,31,32,33,34,35,36, our experiments demonstrate that plant hairy roots can be used for ex vivo propagation of fastidious pathogens like Candidatus Liberibacter spp. We hypothesize similar strategies using insect-vector cell lines could be used for propagating Candidatus Liberibacter spp. Although this is not an axenic culture system, the microbial hairy root system is highly versatile and enables faster genetic and chemical therapeutic screening, which led to the identification of multiple potential short- and long-term therapies to control CLso and CLas. As short-term therapies, we identified eight bioactive compounds that are quite diverse in their chemical structures and modes of action (Fig. 4, Supplementary Dataset 1). Toward long-term solutions, we identified six broad-spectrum AMPs and one positive and one negative plant defense regulator (SlNPR1 and StNPR3, respectively), that inhibit Candidatus Liberibacter spp. in plant tissues. All these, either individually or in combinations, could be further deployed in the host plants to develop robust resistance against these phytopathogens.
We also show that the microbial hairy root system can be used to obtain mechanistic insights into gene function. Characterization of SlNPR1 and AtNPR1 function in potato hairy roots revealed that NPR1 confers tolerance to CLso by activating PR gene expression, consistent with its known role as a transcriptional co-activator in other plant–microbe interactions. Interestingly, SlNPR1 and AtNPR1-expressing hairy roots accumulate SA to a lesser degree than empty vector controls. There are multiple scenarios that could explain this. First, it is possible that SA accumulation in NPR1 overexpressors is directly (positively) correlated to the levels of CLso. Second, SA-mediated signaling could be far more potent in the NPR1 overexpressors than in the controls, and thus less SA might be needed to mediate the defense responses. The second hypothesis posits that in empty vector controls, NPR1 concentration/activity is less than optimal for triggering SA-mediated defenses, and so more SA is produced to compensate for the relatively lower amounts of NPR1. Third, a negative feedback loop in the NPR1 overexpressors could suppress SA levels to maintain defense homeostasis. Although we did not find any previous reports in which SA levels in NPR1-overexpressing plants were determined, studies of Arabidopsis npr1 mutants and other SA biosynthesis mutants indicate that NPR1 participates in negative feedback regulation of SA biosynthesis78,79,80,81,82. Recently, Wang et al.57 also showed no concomitant increase in the levels of SA despite activation of two SA-related genes (PR1 and PAL1) in maize roots colonized by Trichoderma83. Thus, the results here support a model wherein NPR1 mediates PR gene activation to inhibit CLso in potato and negatively regulates SA accumulation to prevent toxic build-up and/or to maintain defense homeostasis78,79,80,81,82,83.
A salient feature of the microbial hairy root system is that the efficacy assays are conducted in living (root) tissues that not only mimic the native environment of the pathogen but also encompass features that underpin native plant–microbe interactions, aspects that cannot be replicated in axenic culturing systems. A growing number of studies have indicated the pivotal roles of roots, root–rhizosphere interactions and root–microbiome composition in plant health and disease, including citrus health and HLB progression84,85,86,87,88,89,90. For instance, in the majority of presymptomatic citrus trees, roots appear to be the first site of CLas detection, preceding leaves86. CLas also is more abundant and more consistently detectable, in roots than in shoots, making root a preferred source tissue in which to perform disease diagnostics tests. Furthermore, roots act as the reservoir for CLas between periods of foliar flushing86 and to endure the seasonal fluctuations of high temperatures in summer months84. Relatively few groups have studied CLso root infections, but the levels and distribution of CLso are greater in roots than shoots, much as for CLas91. Given the significance of roots to CLas and CLso pathology, we posit that the hairy-root-based bioassays described here are more biologically relevant than surrogate pathogen and/or axenic culture systems74,75,76. This is also underscored by reproducible screening results with Arabidopsis NPR1, and a broad-spectrum bactericide (tetracycline) (Figs. 2 and 4), which inhibited Candidatus Liberibacter spp. in the microbial hairy root bioassays, much as they did in planta (Fig. 5)65,73. Moreover, hairy root bioassays circumvent challenges associated with inefficient delivery of compounds into the phloem of whole plants that can result in inconclusive results65. For all of these reasons, this approach is very well suited both for primary screening of potential antimicrobial therapies and for narrowing down potential hits.
The hairy-root propagation and antimicrobial screening approaches are relatively straightforward and do not require specialized equipment or apparatus11,12,13,14,15,16,17,18,19. The only prerequisite is that the host plant or variety should be amenable to R. rhizogenes-mediated hairy root transformation92. Depending on the host plant, the microbial hairy roots can be maintained for ~6 weeks (e.g., CLso-potato hairy roots) or >12 months (e.g., CLas-citrus hairy roots) (Fig. 1g, h). In our experience, this is mainly determined by the lifestyle of the host plant (annual vs. perennial), as well as its degree of tolerance for the pathogen. The latter is relevant because the hairy roots are living tissue matrices, and they eventually succumb to pathogen pressure and show vascular degeneration symptoms (Supplementary Fig. 9) in a manner similar to the host plant roots43,44. Nevertheless, the hairy root propagation timeframes we have demonstrated are more than sufficient to accomplish various antimicrobial screening experiments.
Another advantage of using the microbial hairy root system is that the screening can be accomplished four to six times faster than with conventional approaches68,69,71,72. For instance, small molecules or peptides can be screened using in vitro hairy root assays in ~72 h (Fig. 4), vs. several weeks to months when using conventional in planta approaches (Fig. 5). For evaluating larger proteins, in size range of NPR1, or those forming biochemical complexes as in CRISPR–Cas9, we suggest using the genetic-based hairy root transformation approach (Fig. 2 and Supplementary Fig. 4). Caution must also be exercised to ensure that the small peptides being tested are appropriately folded and/or have the necessary post-translational modifications when produced via synthetic or recombinant techniques. Nevertheless, evaluation of genes/CRISPR targets using genetic constructs still can be accomplished in ~30 days in potatoes and ~90–120 days in citrus after transformation vs. several months to years, respectively, for conventional in planta techniques, thus facilitating faster screening and discovery of therapies.
In conclusion, microbial hairy roots offer a versatile and robust system for the discovery of multiple therapies effective against fastidious pathogens. We suggest that small-molecule inhibitors, AMPs, and genes discovered in this study, either individually or in combination, can be further utilized in planta to control potato ZC and citrus greening diseases.
Plant and insect maintenance and disease challenges
Potato (Solanum tuberosum L. var. Atlantic) and tomato (Solanum lycopersicum L. var. Lance) plants were propagated in a professional growth mix (Metro-Mix 360, Sun Gro Horticulture, MA) and maintained in a growth chamber at 21–22 °C, with a 14-h/10-h light/dark photoperiod and 50% relative humidity. Bactericera cockerelli (Šulc) psyllid colonies that were either CLso positive (haplotype B, the most virulent and prevalent haplotype) or CLso negative were collected in Texas, USA38, and maintained in nylon mesh cages (Bugdorm, Taiwan) in the laboratory. The presence and haplotype of CLso in the psyllid colonies were periodically confirmed using PCR, with primers specific to the single sequence repeat (SSR) loci of CLso (SSR-F and SSR-R; Supplementary Table 1)93. To generate CLso-infected plants, 1-month-old potato and tomato plants were challenged with ~20 psyllids per plant harboring CLso. The psyllids were released onto the plants in an enclosed organza bag tied to a lower branch with fully expanded leaves and were allowed to feed for 1 week. Subsequently, the bags and psyllids were removed from the plant, and the unchallenged upper branches were sampled periodically using PCR, as described below, to confirm the systemic spread of CLso in the plants, which typically could be detected at around 14 days post challenge.
Microbial hairy root production and molecular diagnostics
Both ex vivo and in planta approaches were evaluated to generate potato and tomato hairy roots using CLso-infected plants as an inoculum. In the ex vivo approach, branches were excised from either healthy or CLso-infected plants 21 days post challenge and then dipped into a fresh culture of R. rhizogenes (American Type Culture Collection strain 15834 or 43056; OD600 = 0.5)47,48,49 followed by vacuum infiltration for ~30 min at 30 psi. All explants were subsequently transferred to the inert vermiculite matrix in a humidity dome tray with frequent misting to maintain high humidity. In the in planta approach, R. rhizogenes (OD600 = 0.5) were directly inoculated into the stems of live plants using a syringe, after which the stems were wrapped with foil to maintain humidity and placed in a growth chamber. To generate the CLas-citrus hairy roots, CLas-positive branches, confirmed using PCR, were collected from either HLB-graft-infected or naturally infected citrus trees (Rio Red grapefruit [C. × aurantium L. var. racemosa (Risso & Poit.)] or sour orange [C. × aurantium L.]) maintained in the research plots of Texas A&M University Kingsville Citrus Center, Weslaco, TX, USA (26°10′00.1″N 97°57′27.7″W). Alternatively, uninfected citrus explants from glasshouse-grown trees could be used for hairy root induction, followed by exposure to the Asian citrus psyllid (Diaphorina citri) to transmit CLas, although it takes longer to passage CLas into hairy roots by the later approach. In either approach, for hairy root induction citrus explants were inoculated with R. rhizogenes46,47,48,49. Briefly, budwood tissues (~0.5 cm diameter, ~5 cm length) with a single node were excised, and the cut ends were immersed in a solution of R. rhizogenes (OD600 = 0.5), followed by vacuum infiltration for ~30 min at ~30 psi. All explants were subsequently transferred to vermiculite media in a humidity dome tray. The explants were frequently misted with water to maintain high humidity, which is critical for explant viability and hairy root recovery.
Typical hairy root formation in potato and citrus begins approximately 14 and 30 days after transformation, respectively. Upon emergence the authenticity of hairy roots was confirmed using GFP visualization with fluorescence microscopy (Olympus Corporation, Japan). The hairy root transformation efficiency was calculated using the following formula: number of GFP-positive roots/total number of roots × 100. The transformation of the hairy roots was further confirmed using PCR amplification of either the GFP or rolB/rolC genes encoded on the binary and Ri T-DNA plasmids, respectively, which were transformed into the hairy roots. The Solanaceae RPL2 and citrus GAPC2 endogenous genes were used as the PCR controls. The presence of CLso in the infected hairy roots were assessed by amplification of CLso 16s rDNA using specific primers (OA2-F/OI2c-R and Lso-F/HLB-R)37,38, whereas CLas was estimated using specific markers for genes encoding ribosomal protein (rplA, A2-F/J5-R) and ribonucleotide reductase β-subunit gene (nrdB, RNR-F/RNR-R)39,40. Sanger sequencing and BLAST analysis further confirmed the identity of CLso and CLas PCR amplicons (Supplementary Fig. 8). The temporal patterns of CLso and CLas accumulation in the hairy roots were assessed by determining the titers at various days in propagation. CLso samples were collected at 0, 7, 14, 21, and 28 days, whereas CLas samples were collected at 0, 30, 90, 120, and 180 days (Fig. 1g, h). To estimate the copy number, standard curves for CLso and CLas were generated and regression equation was used to determine the genome equivalents (GEs), as described below.
Estimation of CLso and CLas copy number using standard curves
Two separate standard curves were established to estimate the copy number and GE of CLso and CLas present in the potato and citrus hairy roots, respectively (Supplementary Fig. 7). Briefly, a 76-bp amplicon corresponding to the 16S rDNA of CLso37 and an 81-bp amplicon corresponding to the nrdB gene of CLas40 were synthesized and cloned into the pUC57 vector (GenScript, NJ, USA)37. This plasmid was used as a template to generate the standard curves. Copy number of these targets was determined from the size of the amplified fragment (76 bp for CLso and 81 bp for CLas) and the size of the vector in which the fragment was cloned (2716 + 76 = 2792 bp for CLso and 2716 + 81 = 2797 bp for CLas). The molar mass of the plasmid was estimated in Daltons using Eq. 1
$${\mathrm{g}}/{\mathrm{mol}} = \left( {{\mathrm{bp}}\,{\mathrm{size}}\,{\mathrm{of}}\,{\mathrm{plasmid}} + {\mathrm{insert}}} \right) \times (330\,{\mathrm{Da}} \times 2\,{\mathrm{nucleotides}}/{\mathrm{bp}}).$$
The weight of a single copy of the plasmid was estimated using Eq. 2
$${\mathrm{g}}/{\mathrm{molecule}} = {\mathrm{Weight}}\,{\mathrm{in}}\,{\mathrm{Daltons}}\left( {{\mathrm{g}}/{\mathrm{mol}}} \right)/6.02 \times 10^{23}.$$
Finally, the number of the molecules in a qPCR reaction was determined using the formula: molecules/μL = concentration of plasmid (g/μL)/weight of the one molecule (g/molecule). For the standard curve determination, the plasmid was diluted 10-fold to obtain ~8.17 × 109 to 8.17 × 100 copies and was spiked into 50 ng of DNA extracted from healthy hairy roots for further qPCR analysis. The qPCR analysis was performed as described in the material and methods. The log10 value of the copy number was plotted against the threshold cycle (Ct) to establish a standard curve (Supplementary Fig. 7). The standard curve regression equations 3 and 4
$$y = - 3.7698x + 39.428,$$
were used to calculate CLso and CLas copy numbers, respectively. Since three copies of the 16S rRNA gene37 and five copies of nrdB gene40 were reported per bacterial genome, to determine the final GE in a test sample, the copy number was divided by total number of copies in each bacterium. The copy number shown in Fig. 1g, h are reported as number of bacterial cells present per 1 ng of hairy root extracted DNA37.
Healthy and CLso/CLas-infected hairy roots were surface sterilized with 2% bleach, followed by five washes with sterile distilled water. Hairy-root tips (~3 cm) were excised and fixed with 3% (v/v) glutaraldehyde in 0.1 M Sorenson phosphate buffer (pH 7.2) for overnight at 4 °C. Fixed roots were washed in Sorenson phosphate buffer, and post-fixed in 2% osmium tetroxide (w/v) in the same buffer for 4 h at room temperature94. All samples were washed in Sorenson phosphate buffer and dehydrated in 10% acetone (v/v), followed by infiltration and embedding in Spurr's resin over 3 days. Embedded roots were sectioned (100 nm) and mounted on a formvar-coated copper grids (200 mesh), and stained with 2% aqueous uranyl acetate (w/v) and Reynolds lead citrate. Samples were examined under a FEI Morgagni 268 transmission electron microscope (FEI, Netherlands).
SA quantification
One hundred milligrams of hairy root tissue was mixed with 500 μL phytohormone extraction buffer (1-propanol/water/HCl [2:1:0.002 vol/vol/vol]) containing 500 nM of the internal standard, d6-SA (Sigma-Aldrich, St. Louis, MO, USA). Zirconia beads were added to the samples, and they were homogenized at 6000 rpm for 1 min at room temperature in a Precellys 24 homogenizer (Bertin Technologies, Montigny-le-Bretonneux, France). The samples were then agitated for 30 min at 4 °C under darkness. Afterwards, 500 μL dichloromethane was added to each sample and the samples were again agitated for 30 min at 4 °C in darkness before being centrifuged at 17,000g for 5 min. The lower organic layer of each sample was transferred to a glass vial for evaporation under nitrogen gas. Samples were resuspended in 150 μL methanol, transferred to 1.5-mL microcentrifuge tubes and stored in a −20 °C freezer overnight. The samples were then centrifuged at 17,000g for 2 min to pellet any debris, and then 90 µL of the supernatant of each sample was transferred into an autosampler vial for LC–MS/MS analysis. An Ascentis Express C-18 Column (3 cm × 2.1 mm, 2.7 µm) (Sigma-Aldrich, St. Louis, MO, USA) connected to an API 3200 LC–MS/MS (Sciex, Framingham, MA, USA) using electrospray ionization with multiple reaction monitoring was used to detect SA [m/z = 140.8/96.9 (Q1/Q3)] and d6-SA [m/z = 137/92.9 (Q1/Q3)]. The injection volume was 10 μL, and the 500 μL min−1 mobile phase consisted of Solution A (0.2% acetic acid in water) and Solution B (0.2% acetic acid in acetonitrile) with a gradient consisting of (time—%B): 0.5–10%, 1.0–20%, 21.0–70%, 24.6–100%, 24.8–10%, 29–stop. SA was quantified by comparison against the isotopically labeled internal standard, d6-SA (St. Louis, MO, USA).
Antimicrobial efficacy testing using microbial hairy roots
The phylogenetic analysis of Arabidopsis, potato, and tomato NPR genes was performed using the Maximum Likelihood statistical method and Jones–Taylor–Thornton substitution model using MEGAX95. For the transformation of the antimicrobial genes and CRISPR constructs into the CLso-potato or CLas-citrus hairy roots, binary vectors containing either AtNPR1-, SlNPR1-, or StNPR3-CRISPR–Cas9 constructs or the appropriate negative controls (empty vector/GFP alone) were transformed into R. rhizogenes and used for the induction of hairy roots using pre-infected explant source material, as described above. About 30 days (for CLso-potato) and 90 days (CLas-citrus) after the transformation, the hairy roots were visually screened for GFP under a fluorescence dissection microscope. Three independent biological replicates, comprising randomly selected GFP-positive hairy roots, were collected from the controls and treatments, flash-frozen in liquid nitrogen and stored at −80 °C until use for molecular diagnostics.
High-throughput screening of small molecules was performed using ~220 compounds from the Spectrum bioactives collection (Microsource Discovery Systems, Inc., CT, USA)63,64. Primary screening was performed with two biological replicates of each compound (10 µM in aqueous DMSO, 0.1% v/v), along with untreated and solvent-alone negative controls (DMSO, 0.1% v/v) against CLso. Briefly, ~200 mg of the CLso-containing hairy roots were surface sterilized with 2% bleach and then washed five times with sterile distilled water. The hairy roots were transferred into multi-well plates containing Gamborg's B-5 medium with 1% sucrose. The compounds were then added to the plate, and the medium was vacuum infiltrated at ~20 psi to facilitate the penetration of the molecules into the hairy root matrices. The hairy roots were incubated on a 3D platform rotator (Thermo Fisher Scientific, Waltham, MA, USA) at 60 rpm in the dark at 21 °C for 72 h. The hairy root tissues were sampled after 72 h, flash-frozen in liquid nitrogen and stored at −80 °C in preparation for further processing. The most effective compounds identified in the primary screening were retested at both at 10 and 25 µM, using five biological replicates each, against both CLso and CLas, along with tetracycline as a positive control. The structures of the different chemical compounds were retrieved from the ChemSpider database (http://www.chemspider.com/) (last accessed on 15 October 2020). Direct protein infiltration assays were performed in the same manner as for small molecules, but with purified recombinant peptides (Shifa Biomedical Corp., PA, USA) corresponding to AMP2 and AMP5.
In planta efficacy trials
Three-week-old potato (var. Atlantic) plants were challenged with CLso-containing psyllids. Briefly, five psyllids carrying CLso were restricted to the lowest branch of each plant and contained using an organza drawstring bag tied around the branch. After 1 week, the bags containing psyllids were clipped off the plant to end inoculation and prevent the escape of the newly emerging psyllids and/or nymphs. All the challenged plants were treated with either 0 (untreated), 10 or 25 µM of the selected compounds and tetracycline (250 ppm) twice a week for 4 weeks. To enhance penetration into leaf tissues, all the compounds were mixed with an R-11 nonionic surfactant, which improves the activity and efficacy of the spray application by reducing surface tension and increasing penetration into the plant surface. Four biological replicate plants were included for each treatment. The visual disease symptoms on the plants were monitored weekly, and tissue samples were collected (as described above) at 28 dpi for molecular diagnostics as described below.
Plasmid design and amplicon sequencing
To obtain the full-length NPR1 coding sequences, the total RNA was extracted from 3-week-old Arabidopsis (A. thaliana; ecotype Col-0) and tomato seedlings and used for cDNA synthesis with SuperScript III Reverse Transcriptase (Thermo Fisher Scientific, Waltham, MA, USA), according to the manufacturer's instructions. The Arabidopsis NPR1 (1,782-bp) (AT1G64280.1) and tomato NPR1 (1731 bp) coding sequences (Solyc07g040690.2.1) were amplified with Phusion DNA polymerase (New England Biolabs, Ipswich, MA, USA) from the template cDNA using primers containing the EcoRI restriction site at the 5ʹ end of the forward primer and a BamHI site at the 3ʹ end of the reverse primer (Supplementary Table 1). The PCR products were then digested with EcoRI/BamHI and cloned into a binary vector containing the GFP reporter gene (pBIN-mGFP) under the control of a double-enhanced CaMV 35S promoter (DE35S-P) and the 35S terminator (35S-T). All the nucleotide sequences were confirmed using Sanger DNA sequencing before being transformed into R. rhizogenes to generate hairy roots.
For the CRISPR experiments, multiple single guide RNA (sgRNA) targets were designed for the GFP and NPR3 gene sequences (PGSC0003DMG401000923) using the CRISPR-P design toolset96. The sequences of the top sgRNA hits are provided in Fig. 3a and Supplementary Fig. 4a. The respective sgRNA targets were produced as synthetic oligos (Integrated DNA Technologies, Inc., IA, USA) with the appropriate overhang sequences. Each pair of oligonucleotides was annealed and phosphorylated. The pChimera vector containing the Arabidopsis U6-26 promoter97 was linearized using BbsI, and the spacer was ligated between the two BbsI sites using the annealed oligonucleotides. The customized RNA chimera was then subcloned into pBIN-HcoCas9:mGFP98 to obtain a functional Cas9 expression construct for targeted mutagenesis (Fig. 3a). The Cas9 vector carrying sgNPR3 or the sgGFP spacer sequence and Cas9 were independently transferred into R. rhizogenes for the generation of the hairy roots.
To detect indels in the transgenic hairy roots, a 500-bp target region in NPR3 or GFP was amplified using PCR with gene-specific primers (Supplementary Table 1) and Phusion Polymerase (New England Biolabs). The PCR amplicons were cleaned using DNA PCR clean-up kits (Zymo Research, Irvine, CA, USA) and sequenced using Sanger sequencing (Eton Bioscience, Inc., San Diego, CA, USA). The percentage of indels in the transgenic hairy roots was analyzed using the Inference of CRISPR Edits tool99.
To confirm that the transgene was expressed in the transformed hairy roots, total RNA was isolated from the transformed materials using the Direct-zol RNA MiniPrep Plus kit (Zymo Research) according to manufacturer's instructions and then subjected to reverse transcription PCR. First-strand cDNAs were synthesized from 2.0 μg total RNA using the Superscript III First-Strand Synthesis System (Thermo Fisher Scientific), following the manufacturer's instructions. Subsequently, PCR analysis was performed using a ProFLex PCR System (Thermo Fisher Scientific) to validate gene expression in the hairy roots. The tomato and potato RIBOSOMAL PROTEIN L2 (RPL2) genes (accessions Solyc10g006580.2 and PGSC0003DMG400025015, respectively) and citrus GLYCERALDEHYDE-3-PHOSPHATE DEHYDROGENASE C2 (GAPC2, XM_006483974.2) were used as endogenous reference genes for the normalization and estimation of the relative expression levels. The primer pairs used are listed in Supplementary Table 1 and targeted the following: the 5′ and 3′ untranslated regions (UTRs) of Tomato Etch Virus (TEV) flanking the NPR1 genes (TEV-F and TEV-R) (Fig. 2a), tomato RPL2 (RPL2-F and RPL2-R)100, potato RPL2 (StRPL2-F and StRPL2-R), and citrus GAPC2 (GAPC2-F and GAPC2-R)101.
To determine the effects of small molecules, transgenes and the CRISPR–Cas9 constructs on the CLso and CLas titers within the hairy roots, qPCR analysis was performed using DNA isolated from the various samples. DNA was isolated from the citrus hairy roots from ~200 mg of tissue homogenized using a Precellys 24 homogenizer (MO BIO Laboratories, Carlsbad, CA, USA)102. The quality and quantity of DNA were determined using a NanoDrop 1000 Spectrophotometer (Thermo Fisher Scientific) and agarose gel electrophoresis. Approximately, 50 ng total DNA was used for the qPCR analysis, which was performed on a CFX384™ Real-Time PCR Detection System (Bio-Rad Laboratories) using iTaq™ universal SYBR® Green supermix (Bio-Rad Laboratories), following the manufacturer's instructions. The CLso 16S rDNA-specific primers (OA2-F/OI2c-R and Lso-F/HLB-R) were used to detect CLso37,38, whereas primers for the CLas genes encoding ribosomal protein (rplA, A2-F/J5-R) and ribonucleotide reductase β-subunit (nrdB, RNR-F/RNR-R) were used to detect CLas39,40 (Supplementary Table 1). A dissociation or melting curve analysis was performed to confirm the lack of nonspecific amplification and primer-dimers. For the antimicrobial treatments, the relative CLso titers were estimated using the 2−ΔΔCt method103. The CLso Ct values were normalized to RPL2 to correct for concentration differences in the DNA templates among the samples and were plotted relative to the level in the control (DMSO-treated) CLso hairy root samples, which was set to 100%. Student's t test was used to determine statistically significant (p ≤ 0.05 or 0.01) differences among the controls and treatments.
Data analysis and statistics
All the data analysis, statistics and graphs were displayed as described in the figure legends using Microsoft Excel software (version 2009) and Python bioinfokit package (https://github.com/mandadi-lab/bioinfokit).
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
The data supporting the findings of this study are available in the Supplementary Information Files. A reporting summary for this article is available as a Supplementary Information file. The datasets and/or plant materials generated and analyzed during the current study are available from the corresponding author upon request. Source data are provided with this paper.
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This study was supported in part by funds from a Foundation for Food and Agricultural Research New Innovator Award (2018-534299), by USDA-NIFA-CDRE (2018-70016-28198) to K.K.M., and by Texas A&M AgriLife Research Insect-Vectored Disease Seed Grants (124190-96210 to K.K.M. and 114185-96180 to C.A.A.). The authors acknowledge support of several citrus and commodity crop boards and stakeholders: Texas Citrus Pest and Disease Management Corporation and Texas Citrus Mutual (TX), Wonderful Citrus (TX), U.S. Citrus (TX), Rio Farms (TX), J&D Produce (TX), J. W. Farms (TX), Southern Gardens Citrus (FL), Philip Rucks Nursery (FL) and Citrus Research Board (CA). The authors acknowledge Carissa Villareal, Victoria Garza, Esmeralda Mendez, Ninfa Ramos, Victoria Mora and Rolando Mireles (Texas A&M AgriLife Research, Weslaco, TX) for their technical support with citrus, potato and psyllid propagation and diagnostics. The authors also thank Keerthi Rathore, Karen Scholthof, Herman Scholthof (Texas A&M University, College Station, TX), Will Cody (Stanford University, CA) and Holger Puchta (Karlsruhe Institute of Technology, Germany) for kindly providing various reagents and plasmid resources.
Shankar Pant
Present address: Agricultural Research Service, US Department of Agriculture, Stillwater, OK, USA
Texas A&M AgriLife Research and Extension Center, Weslaco, TX, USA
Sonia Irigoyen, Manikandan Ramasamy, Shankar Pant, Prakash Niraula, Renesh Bedre, Meena Gurung, Denise Rossi, Corinne Laughlin, Ismael E. Badillo-Vargas, Carlos A. Avila & Kranthi K. Mandadi
Department of Plant Pathology and Microbiology, Texas A&M University, College Station, TX, USA
Zachary Gorman, Michael V. Kolomiets & Kranthi K. Mandadi
Citrus Research and Education Center, University of Florida, Lake Alfred, FL, USA
Diann Achor & Amit Levy
Department of Plant Pathology, University of Florida, Gainesville, FL, USA
Amit Levy
Texas A&M University-Kingsville, Citrus Center, Weslaco, TX, USA
Mamoudou Sétamou
Department of Entomology, Texas A&M University, College Station, TX, USA
Ismael E. Badillo-Vargas
Department of Horticultural Sciences, Texas A&M University, College Station, TX, USA
Carlos A. Avila
Southern Gardens Citrus, Clewiston, FL, USA
Michael S. Irey
Sonia Irigoyen
Manikandan Ramasamy
Prakash Niraula
Renesh Bedre
Meena Gurung
Denise Rossi
Corinne Laughlin
Zachary Gorman
Diann Achor
Michael V. Kolomiets
Kranthi K. Mandadi
K.K.M. and S.I. developed the concept and designed the experiments. S.I., M.R., S.P., P.N., M.G., R.B., D.R., C.L., Z.G., and D.A. performed the experiments and conducted the data analysis. S.I., A.L., M.K., M.S., I.B., C.A., M.S.I., and K.K.M. supervised the study. S.I., M.R., and K.K.M. prepared the paper and all authors reviewed the paper.
Correspondence to Kranthi K. Mandadi.
The Texas A&M University System has filed patent applications for the microbial hairy root system [15/353,645 and 16/54,178, status pending]. K.K.M., S.I., M.R., S.P., and P.N. are co-inventors on the related disclosures and patents. Southern Gardens Citrus (Clewiston, FL) has licensed rights in these patents and patent applications from The Texas A&M University System. A disclosure of invention for the compounds screened using this system and a U.S. provisional patent application (63031962) has been submitted. Southern Gardens Citrus also provided matching funds to the Foundation for Food and Agricultural Research New Innovator Award (2018-534,299) to K.K.M. Co-author M.S.I. (Director of Research and Business Development, Southern Gardens Citrus) contributed to the study design for screening antimicrobial peptides, data analysis, and preparation of the manuscript. The remaining authors declare no competing interests.
Peer review information Nature Communications thanks Ed Stover, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Peer Review File
Description of Additional Supplementary Files
Supplementary Data1
Source data
Irigoyen, S., Ramasamy, M., Pant, S. et al. Plant hairy roots enable high throughput identification of antimicrobials against Candidatus Liberibacter spp.. Nat Commun 11, 5802 (2020). https://doi.org/10.1038/s41467-020-19631-x | CommonCrawl |
How do I accurately count the integers(1-1000) that are not divisible by 3,4,5,6?
I have the general algorithm here that my teacher gave us( see full at http://i.imgur.com/pbzQb.png) )
To count we just divide, correct?
like - 1000/3 = 333 ?
What is the sigma notation used here(it's for principle of inclusion, exclusion I thiink)? Also, I am wondering about the boundary cases: near 1000.
Update: I am close I think i can do this :)
combinatorics divisibility
AdelAdel
$\sum_{i=1}^4 N_i$ is shorthand for $N_1+N_2+N_3+N_4$.
Similarly, $\sum_{i,j=1}^4N_{ij}$ means write down all the numbers of the form $N_{ij}$ where $i$ and $j$ take on the values from 1 to 4, and add them up. However, that's not really what you want: you really want $\sum_{1\le i\lt j\le4}N_{ij}$, which is $N_{12}+N_{13}+N_{14}+N_{23}+N_{24}+N_{34}$.
EDIT: Actually, it appears from your $N_{3,4,5,6}$ that what you really want is for your sums to go from 3 to 6, not from 1 to 4, e.g., you want $\sum_{i=3}^6N_i=N_3+N_4+N_5+N_6$.
Gerry MyersonGerry Myerson
The number of multiples of $k$ that are less than or equal to some integer $n$ is $\left\lfloor\frac{n}k\right\rfloor$, the largest integer $\le\frac{n}k$. For instance, the number of multiples of $3$ less than or equal to $10$ is $\left\lfloor\frac{10}3\right\rfloor=3$: they are $3,6$, and $9$. Thus, in your problem $N_3=\left\lfloor\frac{1000}3\right\rfloor=333$, $N_4=\left\lfloor\frac{1000}4\right\rfloor=250$, and so on. There's no problem with numbers near $1000$.
The calculation that you're trying to do is indeed an inclusion-exclusion calculation, but your notation needs to be fixed. If $N_3,N_4,N_5$, and $N_6$ are respectively the numbers of multiples of $3,4,5$, and $6$ less than or equal to $1000$, then you don't want $$\sum_{i=1}^4N_i\;:$$ that means $N_1+N_2+N_3+N_4$, and you don't even have an $N_1$ or $N_2$. You want $$\sum_{i=3}^6N_i=N_3+N_4+N_5+N_6\;.$$ This is standard summation notation.
The idea of the inclusion-exclusion calculation is that a first approximation to $N$, the number of multiples of $3,4,5$, or $6$ less than or equal to $1000$, can be obtained by adding together the number of multiples of $3$, of $4$, of $5$, and of $6$: $$N\approx\sum_{i=3}^6N_i=N_3+N_4+N_5+N_6\;.\tag{1}$$ Of course this is a very bad approximation, since these categories overlap a lot, as in your Venn diagram. For instance, $15$ is both a multiple of $3$ and a multiple of $5$, so $(1)$ counts it twice. Therefore we count the numbers that are multiples of both $3$ and $5$ and subtract that number to correct for the double-counting in $(1)$. Specifically, if $N_{j,k}$ is the number of positive integers less than or equal to $1000$ that are divisible by both $j$ and $k$, we want to subtract $N_{j,k}$ from the total in $(1)$. There are actually six such pairs; their counts are $N_{3,4},N_{3,5},N_{3,6},N_{4,5},N_{4,6}$, and $N_{5,6}$. Using summation notation we can represent the sum of these numbers as $$\sum_{3\le j<k\le 6}N_{j,k}=N_{3,4}+N_{3,5}+N_{3,6}+N_{4,5}+N_{4,6}+N_{5,6}\;,$$ and we want to subtract this from $(1)$ to get a second, better approximation:
$$N\approx\sum_{k=3}^6N_k-\sum_{3\le j<k\le 6}N_{j,k}\;.\tag{2}$$
Unfortunately, it turns out that we've overcorrected: a number like $60$ that is a multiple of $3,4$, and $5$, say is counted $3$ times in $(1)$, and both counted and removed $(3)$ times in $(2)$, so it isn't counted at all in $(2)$.
To correct for this, we count the numbers that are multiples of three of the divisors $3,4,5$, and $6$ and add them back in. There are four such combinations, whose counts are $N_{3,4,5},N_{3,4,6},N_{3,5,6}$, and $N_{4,5,6}$. Their sum is
$$\sum_{3\le i<j<k\le 6}N_{i,j,k}=N_{3,4,5}+N_{3,4,6}+N_{3,5,6}+N_{4,5,6}\;,$$
and we get our third approximation by adding it back in:
$$N\approx\sum_{k=3}^6N_k-\sum_{3\le j<k\le 6}N_{j,k}+\sum_{3\le i<j<k\le 6}N_{i,j,k}\;.\tag{3}$$
Once again it turns out that we've overcorrected, so we subtract $N_{3,4,5,6}$, the number that are multiples of $3,4,5$, and $6$. At this stage we have an exact count:
$$N=\sum_{k=3}^6N_k-\sum_{3\le j<k\le 6}N_{j,k}+\sum_{3\le i<j<k\le 6}N_{i,j,k}-N_{3,4,5,6}\;.\tag{4}$$
Of course if we'd really been thinking, we could have ignored the multiples of $6$ from the start: every multiple of $6$ is already a multiple of $3$, so there's no point counting it separately. Thus, the inclusion-exclusion formula could have been reduced to $$N=\Big(N_3+N_4+N_5\Big)-\Big(N_{3,4}+N_{3,5}+N_{4,5}\Big)-N_{3,4,5}\;.$$
Brian M. ScottBrian M. Scott
$\begingroup$ That every multiple of 6 is a multiple of 2 seems to be irrelevant; that every multiple of 6 is a multiple of 3 is the reason we can ignore multiples of 6, right? $\endgroup$ – Gerry Myerson May 10 '12 at 1:56
$\begingroup$ @Gerry: Yep. For some reason I was thinking that $2$ was in the divisor set when I wrote that. Thanks; fixed. $\endgroup$ – Brian M. Scott May 10 '12 at 1:58
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Networked and Autonomous Model-scale Vehicles for Experiments in Research and Education
04/17/2020 ∙ by Patrick Scheffe, et al. ∙ 0 ∙ share
This paper presents the μCar, a 1:18 model-scale vehicle with Ackermann steering geometry developed for experiments in networked and autonomous driving in research and education. The vehicle is open source, moderately costed and highly flexible, which allows for many applications. It is equipped with an inertial measurement unit and an odometer and obtains its pose via WLAN from an indoor positioning system. The two supported operating modes for controlling the vehicle are (1) computing control inputs on external hardware, transmitting them via WLAN and applying received inputs to the actuators and (2) transmitting a reference trajectory via WLAN, which is then followed by a controller running on the onboard Raspberry Pi Zero W. The design allows identical vehicles to be used at the same time in order to conduct experiments with a large amount of networked agents.
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A demonstration video of this work is available at https://youtu.be/aH1Q8AKXmUs.
The vehicle software, bill of materials and a production tutorial is referenced from our website http://cpm.embedded.rwth-aachen.de.
Research on networked and autonomous vehicles is ongoing since multiple decades. When new methods are developed, the necessity of testing them arises. This can be done with little effort in simulation as in naumann2018 . The meaningfulness of results in simulation is restricted, as only aspects of reality that are modeled are considered. More meaningful are experiments in true scale, but those require a high effort and are expensive, especially when testing methods on networked vehicles, as multiple test platforms are required. Midway between those options, methods can be tested on scaled testbeds. In scaled experiments, many challenges of the true-scale problem are apparent, e.g. communication delays and losses, synchronization problems or actuator dynamics. Another benefit compared to the true-scale experiment is that setting up the experiment is simpler and quicker, which allows for rapid development cycles.
The curriculum at a university should prepare students for research in networked and autonomous vehicles. This includes for example the design of algorithms for embedded hardware, designing controllers for nonlinear systems, or coupling of networked agents for collision avoidance. Seeing an algorithm one has developed running in an experiment fills students with enthusiasm about learning concepts of control by applying it to the CPM system. The modified model-scale vehicle proposed in this paper enables those experiments.
This paper is structured as follows.Section 2 compares model-scale vehicles with Ackermann steering geometry from literature. Section 3 describes how we transform a model-scale race car to a networked and autonomous vehicle with off-the-shelf components, excluding a printed circuit board. The lab environment in which the vehicles operate is sketched in section 4. In section 5, examples are given to show in what form the vehicles can be used in control education.
2 Existing platforms
In the last decade, a number of model-scale testbeds have been developed. In paull2017 , 15 platforms for education and research with a cost lower than 300[$] are compared. These differ from the model-scale vehicle we present, as they are wheeled differential drive platforms or platforms with slip-stick forwards motion.
In table 1, an overview of recently developed model-scale vehicles with Ackermann steering geometry is given. Having a scaling factor of 1:43 and 1:24 respectively, the ORCA Racer liniger2014 and the Cambridge Minicar hyldmar2019 are smaller than the vehicle presented in this work. The ORCA Racer is based on the Kzosho dnano RC race car, but substitutes its original board with a custom PCB. This board features an ARM Cortex-M4 processor, Bluetooth communication and an IMU. The vehicles are designed to receive externally computed control inputs via Bluetooth, and apply these inputs with an onboard LLC. The Cambridge Minicar is based on the CMJ RC Cars Range Rover Sport. Its controlled by a Raspberry Pi Zero W. These vehicles are controlled by sending externally computed control inputs via broadband radio.
The BARC from gonzales2016 , the MIT Racecar from karaman2017 and the F1/10 from okelly2019 share the scale of 1:10. The mechanical base for all three vehicles is a Traxxas rally car. At this size, the vehicles are capable of carrying more computational power and more sensors additionally to an IMU. In the BARC, 4 rotary encoders are installed for speed measurement and a camera is mounted. Optionally, it is possible to install a lidar and a GNSS receiver. The HLC and main computing unit is an ODROID-XU4, the LLC, i.e. sensor read and actuator control, is performed with an Arduino Nano. The setup of the MIT Racecar and the F1/10 is similar. The speed is given by a VESC electronic speed controller, and optional sensors include a 3D stereo cameras and a lidar. The main computing element is the Nvidia Jetson Tegra X1. The greater computing power and additional sensors allow for onboard autonomy. This is also a reason why these setups cost around 1000[$]. At the scale of 1:10, a lot of space is required for indoor experiments on cooperative driving with multiple vehicles. Due to the cost and the size of the platforms, indoor experiments with a large amount of vehicles are difficult.
At the largest scale of 1:5, the GATech Auto-Rally from williams2016 and the IRT buggy from reiter2014 and reiter2017 are designed for outdoor experiments. The Auto-Rally is equipped with two forward facing cameras, a Lord Microstrain 3DM-GX4-25 IMU, a GNSS receiver, and wheel speed sensors. The computational power is provided by an Intel quad-core i7 processor, 16GB RAM, and an Nvidia GTX-750ti graphics card. With this elaborate hardware setup, the Auto-Rally is used for aggressive driving. The IRT buggy is designed for versatile use. It shares the separation of HLC and LLC in two hardware components with the BARC. Sensors include a GNSS-sensor, an IMU, and two rotary encoders at the rear wheels. Its modular setup allows for other sensors such as a lidar. Similar to the ORCA Racer, this platform is not open source.
ETHZ ORCA Racer
Cambridge Minicar
µCar
MIT Racecar
GATech AutoRally
IRT buggy
Table 1: Recent model-scale Ackermann-steering platforms
The larger model-scale vehicles are equipped with sensors and computing power to allow autonomy. The µCar, as well as the ORCA Racer and the Cambridge Minicar are reliant on the interaction with a lab environment. This lab environment provides the positioning of the vehicles and therefore substitutes the GNSS of the real world experiment. In the case of the Cambridge Minicar, this is done with an OptiTrack motion capture system that requires multiple cameras, while the lab environment of the ORCA Racer only uses one camera, similar as our CPM lab. In contrast to those two labs, in addition to the option of sending control inputs to the vehicle, a trajectory following mode exists, where an onboard controller determines the control inputs necessary to follow a given trajectory.
3 Vehicle setup
Figure 1: The µCar, a 1:18 model-scale vehicle
The model-scale vehicle presented here is shown in fig. 1. It is an Ackermann-steered, non-holonomic mobile robot in the scale of 1:18 compared to a typical passenger vehicle. Its length is 220mm, its width 107mm, its height 70mm, its wheelbase L=$150mm$ and its weight is 500g. The vehicle has a maximum speed of 3.7m/s. The power consumption in standby (without steering or acceleration) is 250mW. In experiments, the battery powers the car for about five hours. table 2 lists the components used in the model-scale vehicle. The cost calculation refers to an order of 20 vehicles, as a single PCB would cost 45€, but ordering a panel cluster with 20 PCB on one board reduces the price of a unit to 15€. Assembling a vehicle takes one person around six hours of time.
Cost [€]
XRAY M18 Pro Mechanical platform 170
Gens ace 3500mAh LiPo Battery 30
NF113LG-011 Motor 15
Hitec D89MW Servo 50
PCB Board 15
Raspberry Pi Zero W MLC 18
8GB SD Card Memory 7
ATmega2560 LLC 12
Pololu VNH5019 Motor Driver 23
DeboSens BNO055 IMU 34
Eletronic Parts 21
SUM 395
Table 2: Components used in the µCar; cost rounded to the next integer
Using an off-the-shelf mechanical platform allows for a quick start in building a networked and autonomous model-scale vehicle. We use the mechanical components from the XRAY M18 PRO LiPo. It is a 1:18 micro car that is designed for holding a battery, a servo motor for steering and a motor for propulsion. The motor drives all four wheels as the shaft is connected to each one with differentials. The minimum turning radius given by the mechanical design is approximately 0.3m.
The vehicle's hardware architecture is illustrated in fig. 2. A Raspberry Pi Zero W takes the role of the MLC on the vehicle. It is responsible for the communication via WLAN with the HLC, as described in section 4, and for clock synchronization using the NTP. Additionally, the MLC fuses the sensor data to obtain accurate localization. The MLC also supplies the LLC with control inputs. This is either realized by forwarding control inputs received via WLAN, or by running a controller for trajectory following as described in the next paragraph. The tasks on the Raspberry are repeated in a frequency of 50Hz, i.e. a time interval of 20ms.
In order to ensure the most individual and adaptable handling of the vehicle, we designed a custom PCB connecting the components.. This PCB serves as an interface between the actuators, sensors and control electronics. The PCB with its components is shown in fig. 3. This PCB embeds an ATmega2560 microcontroller with a 16MHz clock rate. This microcontroller represents the LLC, reading the sensor data and applying the control inputs to the actuators. The hardware separation in MLC and LLC introduces a hierarchical architecture, which creates a hardware abstraction layer. Even in the case the MLC is changed, the interface to the hardware will stay the same.
In a frequency of 50Hz, the MLC and the LLC exchange information via SPI. The MLC provides the control inputs, while the LLC returns the sensor readings. A TXB0104 bidirectional voltage-level translator was installed for level adaptation of the SPI bus. The 3.3V SPI level of the Raspberry is converted into a 5V SPI signal for the ATmega.
The IMU is a DeboSens BNO055 and provides the required sensor data using a 9-DOF sensor. The ATmega microcontroller can retrieve this data via the two wire I2C bus.
The motor driver board VNH5019 drives the single brushed DC motor of the vehicle via an integrated H-Bridge. The ATmega controls the engine driver via a PWM signal with a frequency of up to 20kHz. A current sensing output provides the ATmega with a signal which is proportional to the current applied to the motor. The power source is a 2000mAh LiPo battery which provides a 7.4V voltage. This voltage is directly fed to the motor driver unit. Since the Raspberry and all the other components (except the motor driver unit) are specified to 5V or 3.3V respectively, the voltage is reduced by an NCP1117 LDO voltage regulator. To protect the LiPo battery as well as the electronic components a battery protection circuit was inserted.
Three Hall-effect sensors mounted on a separate odometer board measure the motor shaft rotation. A diametrically polarized magnet is attached to the motor shaft in order to make the rotational motion of the axis electrically visible. With this setup, it is possible to distinguish six different motor angles per rotation. The digital signals of the Hall sensors are directly transmitted to three I/Os of the ATmega, which translates the signals into rotation ticks.
Four LEDs are installed on the vehicle, which are also connected to the odometer board and controlled by the ATmega. The outer three LEDs are used for positioning with an IPS, while the inner one communicates the vehicle's ID.
Mid-level controller
Low-level controller
ATmega2560
Figure 2: Vehicle hardware architecture
Odometer board
connector slot
Figure 3: The PCB on the vehicle with several components installed
The vehicles can operate in the two different modes (1) external control and (2) trajectory following. If a trajectory is provided to the vehicle, the MLC determines the control inputs to follow that trajectory. The trajectory is provided as a list of tuples (t,x,y,vx,vy)
. Usually, a trajectory point is understood to be a tuple of time and position. The controller needs reference trajectory points at controller-specific points in time. If the time step between trajectory points is assumed to be larger than the control time step, the MLC interpolates the trajectory to determine a sensible reference point. By fixing the derivative of the trajectory in each point, the MLC is enabled to interpolate between trajectory points with cubic Hermite splines. Additionally, if new reference trajectory points are transmitted, the MLC-internal reference trajectory will not change. If the MLC receives control inputs, it switches to directly applying those to the actuators. This behavior allows for manual control of the vehicle with a gamepad or a keyboard for example. It is also possible to compute control inputs depending on the vehicle state and reference trajectory externally and send those via WLAN.
4 Environment: CPM lab
Figure 4: CPM lab overview: vehicles communicate via WLAN with their respective computers and the IPS.
As mentioned earlier, the vehicles are used for experiments in a lab environment as visualized in fig. 4, which we call CPM lab. This lab provides a driving area of about 4.5$m$×4$m$. Communication between the vehicles and this environment is established through DDS RTI Connext DDS. An IPS provides the vehicles with their pose (position and orientation) with a worst-case accuracy of 3.25cm and 2.25°. A camera detects the position of the three LEDs on the vehicle. These LEDs define a vehicle's pose due to their arrangement on the vehicle in a non-equilateral triangle. The vehicle corresponding to a detected pose is identified with a signal code sent by the fourth LED on the vehicle as shown in kloock2020
. Additionally, a reference trajectory or the actuator inputs for the vehicles are sent via WLAN. The vehicle returns its current state, which includes the estimated pose as well as sensor readings and actuator commands.
5 The vehicles in control education
The vehicle's hierarchical architecture allows students to work at different levels of abstraction.
It is possible to learn the basics of embedded programming when working with the LLC (the ATmega2560). At this level, students need to understand MCU data sheets in order to determine how to read sensors and control actuators correctly in C-code.
At the level of the MLC (the Raspberry Pi Zero W), tasks like trajectory control or sensor fusion can be tackled. Measurements of multiple sensors need to be fused for vehicle localization in the proposed setup, which reflects the real world application. The IPS provides absolute positioning, but its measurement data is transmitted to the vehicle via WLAN, which makes the measurements relatively slow and also unreliable. On the other hand, onboard sensors like the IMU and the odometer are fast and accurate for short distances, but need a reference. A controller for trajectory following can be implemented as simple as a PID-controller, or more advanced as a mpc. The µCarcurrently uses mpc for trajectory following. Restrictions by the limited computation power of the MLC still apply, which motivates efficient algorithms and a programming language like C++.
On the highest abstraction level, ideas can be developed on an external PC with programming languages common in optimization (e.g. MATLAB, Python). It is possible to work on trajectory planners as well as on external controllers for the vehicles, depending on which mode of operation one wishes to use.
The modularity allows to focus on one specific part of networked and autonomous vehicles. It is possible to provide necessary interfaces with working components, so the content to be taught can be chosen freely and appropriately.
A basis for many control tasks is an appropriate model of the system. A system model is useful for e.g. simulation or controller design. The purpose of the model defines its requirements. For simulation, the goal might be to represent the system as truthfully as possible, while for a controller using mpc the ability for fast computation might be necessary. Since having a system model is the prerequisite of many aspects in control, we show an example of how a model for the model-scale vehicles can be obtained. The goal of this endeavor is to illustrate how the vehicles might serve as a platform to control engineering education.
5.1 Vehicle dynamics model
In this example, we aim for a model that is suitable for mpc of a vehicle's pose and velocity on embedded hardware. The model needs to be simple enough for quick computation, while accurate enough for predicting the states. We propose a kinematic bicycle model with some added terms to account for various errors.
The model has the states x and inputs u
x =(xyψv)T (1)
u =(mdu)T,
where x and y are the x- and y-position respectively, ψ is the yaw angle, v the speed at the vehicle rear axle, m the dimensionless motor command, d the dimensionless steering command and u the battery voltage. The battery voltage is of course not an input set by the controller, but one that affects the system dynamics.
ℓr
(x,y)
Figure 5: Kinematic bicycle model of the vehicle
The model used to describe the vehicle's dynamics is a non-linear kinematic bicycle model according to rajamani2011 . Similar to alrifaee2017 , it is assumed that no slip occurs on the front and rear wheels, and no forces act on the vehicle. The velocity dynamics are described with a PT1 behavior, which results in the following equations
˙x= v⋅√1+(ℓrL⋅tanδ)2⋅cos(ψ+β) (2)
˙y= v⋅√1+(ℓrL⋅tanδ)2⋅sin(ψ+β)
˙ψ= v⋅1L⋅tanδ
˙v= −1Tv⋅v+KvTv⋅v%in(m,u)
β= tan−1(ℓrLtanδ).
The model variables are illustrated in fig. 5. ℓr is the distance from the rear axle to the vehicle's reference point, L is the distance between front and rear axle, δ is the steering angle which is related to the steering command d, Kv and Tv are the gain and time constant of the velocity's PT1 behavior and vin is the input velocity, which is modelled as a function of the motor command m and the battery voltage u. The change of the vehicle's x- and y-position is dependent on the velocity vc at the vehicle's reference point. From the fact that the angular velocity ˙ψ is equal at every point of the vehicle, we get
˙ψ=vcRc=vR, (3)
where Rc and R are the radii of the circular movement at the vehicle center and rear axle respectively. With (3) and Pythagoras' theorem we obtain
vc=v⋅√1+(ℓrL⋅tanδ)2. (4)
In order to simplify computational tasks on the model, we can approximate some terms with Taylor series at the point δ=0. The side slip angle β due to steering is approximated with a first-order Taylor series
β(δ)=ℓrL⋅δ+O(δ3). (5)
Equation (4) is simplified with a second-order Taylor approximation:
vc=v⋅(1+(ℓrL)2⋅δ2+O(δ4)). (6)
Now substituting the model's variables with parameters and introducing some parameters to account for various inaccuracies, the parameterized bicycle model is given by:
˙x= p1⋅v⋅(1+p2⋅(d+p9)2) (7)
⋅cos(ψ+p3⋅(d+p9)+p10)
˙y= p1⋅v⋅(1+p2⋅(d+p9)2)
⋅sin(ψ+p3⋅(d+p9)+p10)
˙ψ= p4⋅v⋅(d+p9)
˙v= p5⋅v+(p6+p7⋅u)⋅sign(m)⋅|m|p8.
An extra parameters introduced is p1, which compensates the calibration error between IPS speed and odometer speed. p2 and p3 substitute the model parameters in (6) and (5) respectively. p4 takes care of the model parameter 1L as well as the conversion of steering command to steering angle. p5 substitutes −1Tvin the velocity's PT1 model. The steady state velocity is modeled as a power function, where the constant factor is represented by p6 and the exponent by p8. In order to avoid the trouble that comes with negative bases and real exponents, the absolute value of the motor command m is used as the base and the sign of m is multiplied. As the motor strength depends on the battery voltage, we added it as a multiplying factor with the parameter p7. p9 is an extra parameter introduced to correct steering misalignment, while p10 accounts for a yaw calibration error in the IPS.
This is an end-to-end, grey-box model for the vehicle dynamics. The model parameters are not measured directly, but optimized to best fit the vehicle behavior as shown in section 5.3.
5.2 Model discretization
The model is discretized with the explicit Euler method, as follows:
xk+1=xk+Δt⋅f(xk,uk,p). (8)
Here, f is obtained from the continuous vehicle dynamics model (7). This discretization is chosen for its simplicity and computational efficiency. Measurements are taken in time intervals of Δt=$0.02s$. This short time interval compensates partly for the inaccuracies introduced by the method used, and the discretization is included in the parameter identification.
5.3 Parameter identification
Since the dynamics of nonholonomic vehicles are nonlinear, model identification procedures for nonlinear systems need to be used. Identifying the vehicle dynamics can be achieved by formulating the task as an optimal parameter estimation problem. The optimization tries to find a set of model parameters that best reproduce the measurement data. A measurement vector at timestep
k contains:
^xk=(^xk^yk^ψk^vk)T. (9)
Here, ^x and ^y is the IPS x- and y-position respectively, ^ψ is the IPS yaw angle and ^v the odometer speed.
The optimization problem is then given as
minimizexjk,p nexperiments∑j=1ntimesteps% ∑k=0E(xjk−^xjk) (10)
subject to xjk+1=xjk+Δt⋅f(xjk,ujk,p)k=0,\mathellipsis,(ntimesteps−1)j=1,\mathellipsis,nexperiments,
where ujk are the measured inputs, f is the discrete vehicle model as in (8), p is the vector of model parameters p1 to p10, Δt is a constant timestep of 0.02s and E is the error penalty function. Since the vehicle pose lives in SE(2), an adequate error metric needs to be used. We used a weighted quadratic error function and accounted for the period of 2π in the yaw error function using sin2(Δψ/2).
This kind of optimization problem is not well suited for identifying the delay times. The optimization problem is therefore solved multiple times for combinations of delay times in an outer loop. The delays that create the lowest objective value are taken as the solution.
Figure 6: Driven trajectory for measurement data collection
The measurement data used in the parameter optimization is shown in fig. 6. This data is sliced into parts of 100 consecutive data points, i.e. time intervals of 2s, which are fed to the optimization problem as experiments. The resulting parameters are
p=( 1.00 −0.14 0.20 3.56 −2.19 (11)
−9.73 2.52 1.32 0.03 −0.01 ).
The delays identified are 1 timestep for the IPS data, 0 timesteps for the local measurement information and 5 timesteps for the motor and steering actuation.
This paper presented how a regular RC race car can be transformed to a networked and autonomous vehicle with mainly off-the-shelf components. The vehicles are used for teaching in multiple courses at RWTH Aachen University at the moment. We are eager to see the impact of applying concepts on real control systems on the students' learning experience.
Currently, a fleet of 20 vehicles is being built up. This should enable students and researchers alike to perform various experiments on networked and autonomous driving in moderately large scale networked systems.
Appendix A Required Demonstrator Space
The 1:18 model-scale vehicles will be presented with a reduced lab environment. For that, we need
space of about $1m$×$1.5m$ and
power outlets.
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Nanoscale Research Letters
Nano Express
High Fill Factors of Si Solar Cells Achieved by Using an Inverse Connection Between MOS and PN Junctions
Liang-Xing Wang1, 2,
Zhi-Quan Zhou1,
Tian-Ning Zhang3,
Xin Chen3 and
Ming Lu1Email author
Nanoscale Research Letters201611:453
Fill factors (FFs) of ~0.87 have been obtained for crystalline Si (c-Si) solar cells based on Ag front contacts after rapid thermal annealing. The usual single PN junction model fails to explain the high FF result. A metal/oxide/semiconductor (MOS) junction at the emitter is found to be inversely connected to the PN one, and when its barrier height/e is close to the open-circuit voltage of the solar cell, very high FF is obtainable. In this work, although the open-circuit voltage (<580 mV) is not high here, the efficiency of c-Si solar cell still reaches the state-of-the-art value (>20 %) due to the high FF achieved.
Si solar cell
Fill factor
Ag contact
MOS barrier
Photoelectric conversion efficiency (η) of solar cell is determined by the product of open-circuit voltage (V OC), short-circuit current density (J SC), and fill factor (FF); therefore, achieving high V OC and J SC as well as FF is crucial for getting high η [1, 2]. Considering the large abundance of Si on the earth's crust and the hyper amount of investment input in the Si solar cell industry, new approaches to enhance V OC, J SC, and/or FF for further increasing Si solar cell efficiency are always in demand. To our knowledge, FFs for the mostly efficient single crystalline- or c-Si solar cells reported so far are basically no larger than 0.835, for instance, for the PERL-type Si solar cell, FF = 0.828 (V OC = 706 mV, J SC = 42.7 mA/cm2, η = 25.0 %) [3], for the IBC-type Si solar cell, FF = 0.830 (V OC = 730 mV, J SC = 41.2 mA/cm2, η = 25.0 %) [4], for the HIT-type Si solar cell, FF = 0.835 (V OC = 738 mV, J SC = 40.8 mA/cm2, η = 25.1 %) [5], and for the HIT + IBC type Si solar cell, FF = 0.827 (V OC = 740 mV, J SC = 41.8 mA/cm2, η = 25.6 %) [6]. This work reports our finding of high FF (~0.87) and analyses of its origin based on an inverse connection of metal/oxide /semiconductor (Ag/SiO2/Si) or MOS and PN junctions.
The substrate of the Si solar cell is P-type c-Si < 100 > wafer (two-sided polish, 10 × 10 × 0.2 mm3 in size, 1–5 Ω · cm). First, the Si wafer was degreased and ultrasonically cleaned and then dipped in diluted HF (1 %). Then, it was placed in the boiling NaOH solution with concentration of 1 g/L at 90 °C for 30 min and rinsed in deionized water and blown dry to texture the surface for antireflection. A phosphorous paste was deposited on the front surface of Si, followed by annealing the wafer at temperature of 900 °C for 20 min in a tube furnace in nitrogen with purity of 99.999 % to form PN junction. A 60-nm-thick SiO2 was evaporated by means of electron beam heating onto the top surface of now N+ Si emitter for surface passivation [7] in a home-made vacuum chamber with a base pressure less than 1 × 10−4 Pa. At the rear of the substrate, a 20-nm-thick Al2O3 was evaporated also by means of electron beam heating for rear passivation [7]. A 1.0-μm-thick Ag grid was then deposited onto the SiO2 passivation layer as the front contact by resistance heating in another home-made vacuum chamber with a base pressure less than 5 × 10−3 Pa. A 1.0-μm-thick Al layer was deposited onto the Al2O3 passivation layer as the rear contact by resistance heating in the same system. Finally, the whole device was annealed in nitrogen atmosphere at 510 °C for 2 min followed by rapid thermal annealing (RTA) at 700 °C for 1 s. For RTA, we firstly pushed the sample from one end of the furnace tube to its middle within 3 s. The variation of temperature was from room temperature to 700 °C. Then, the sample was kept in the middle of the tube for 1 s, followed by being pulled out from the middle to the end of the tube within 3 s. All the processes of device fabrication were performed in a clean room. Its degree of cleanness was 100; that means, the number of dust particle with ϕ ≥ 0.5 μm was less than 3500/m3. The surface morphology was measured on a scanning electron microscope (SEM) (Philips, XL 30). The reflectance spectra were obtained with a spectrophotometer (Perkin-Elemer Lambda 900). The photovoltaic (PV) parameters of solar cells were measured on a solar simulator (Oriel/Newport, model 94023A) under 1-sun AM1.5G condition. In order to confirm the observed results of the high FF, the PV measurements were independently conducted on other two different solar simulators (Oriel/Newport, model 94043A, and Oriel/Newport, model 94023A). The external quantum efficiency (EQE) of the solar cell was acquired on a QE system of Oriel/Newport.
Figure 1 shows the schematic structure of c-Si solar cell prepared in this work as described above. Figure 2a gives the Si surface image after texturing. The pyramid-like nanostructures with average height of ~2 μm help to trap the incident light [8]. Figure 2b gives the measured EQE spectrum of the textured c-Si solar cell we prepared as shown in Fig. 1, which is termed cell A, and that of a plane c-Si solar cell. The surface reflectance spectra of Si before and after surface texturing are also plotted. It is seen that after surface texturing, the reflectance decreases significantly, and the lower reflectance yields higher EQE of the c-Si solar cell.
Schematic drawing of c-Si solar cell in this work
a SEM image of Si surface after texturing. b Measured external quantum efficiency (EQE) and surface reflectance spectra of Si before and after surface texturing
In Fig. 3a, a J-V (current density-voltage) curve of c-Si solar cell A is shown. Also shown are the J-V curves of the same solar cell before the final annealing, which is termed cell A−, and that was over-annealed (RTA at 700 °C for 30 s), which is termed cell A+. The PV parameters of the three cells are provided in Table 1. For cell A, V OC = 572 mV, J SC = 40.8 mA/cm2, FF = 0.869, and η = 20.3 %. For cells A− and A+, the PV parameters all deteriorate severely, especially FF and η. Note that the obtained FF (0.869) is considerably larger than the highest experimental one (≤0.835) reported [3–6].
a J-V curves of cell A, those without annealing (A−), and over-annealed (A+). b J-V curves of cell B measured at Fudan University, those measured at SARI (B (SARI)) and SITP (B (SITP))
PV parameters of the c-Si solar cells
V OC (mV)
J SC (mA/cm2)
η (%)
A (700 °C, 1 s)
A− (no annealing)
A+ (over-annealing)
B (SARI)
B (SITP)
aMeasured at Fudan University
bMeasured at Shanghai Advanced Research Institute (SARI)
cMeasured at Shanghai Institute of Technical Physics (SITP)
In order to examine whether a charging effect of capacitance exists during the measurement, the J-V curve has been measured by scanning the voltage either from the lower voltage side to the higher one, or from the higher voltage side to the lower one, or to and fro repeatedly. They all remained the same. Hence, no charging effect exists.
Figure 3b gives the J-V curves of another solar cell prepared in a similar way as cell A, which is termed cell B. Its PV parameters are also listed in Table 1, which are quite close to those of cell A. Hence, the c-Si solar cell as shown in Fig. 3a is reproducible. Figure 3b also presents the J-V curves of the same cell B as measured independently on other two different solar simulators in two different affiliations, which are termed cells B (SARI) and B (SITP), respectively. Here, SARI stands for Shanghai Advanced Research Institute of Chinese Academy of Sciences, and SITP is for Shanghai Institute of Technical Physics of Chinese Academy of Sciences. All these curves are nearly the same. Therefore, the high FF achieved cannot arise from instrumental errors. The PV parameters of the two newly measured J-V curves of cell B in Fig. 3b are given in Table 1, too.
We now investigate the origin of the very high FF. Firstly the J-V curve of solar cell A or B is analyzed with the usual single PN junction solar cell model as described below [9],
$$ J={J}_{\mathrm{ph}}-{J}_0\cdot \left[ \exp \left(\frac{V_{\mathrm{PN}}+J\cdot \mathrm{R}\mathrm{s}}{nVt}\right)-1\right]-\frac{V_{\mathrm{PN}}+J\cdot \mathrm{R}\mathrm{s}}{\mathrm{Rp}} $$
where V PN is the voltage across the PN junction. Vt = kT/e, T is absolute temperature, and e is electron charge. J 0 means dark current density and J ph is photocurrent density. Rs and Rp are serial and parallel resistances, respectively. For the ideal situation when no recombination current exists in the depletion zone, n, the ideal factor, equals 1.0 and Rs = 0 and Rp = ∞. In reality, n = 1~2.
The experimental J-V curve of cell A in Fig. 3a is then fitted with Eq. (1). It is found that only when n = 0.56 can the calculated curve fit the experimental one well. However, the n obtained here (0.56) is far less than 1.0, which is unphysical according to its definition. Recently, Richter et al. [10] have simulated the PV parameters of c-Si solar cell with the model of single PN junction. According to their simulations, it is unlikely to have the very high FF for the relatively low V OC as here (<600 mV). Therefore, the model of Eq. (1) is unable to explain the result of very high FF here.
We notice that the c-Si solar cell is actually composed of a PN junction and two metal/oxide/semiconductor or MOS junctions (Ag/SiO2/N+ Si and Al/Al2O3/P-Si). If the barrier heights of the two MOS junctions are very low, or if they are in Ohmic contacts with the PN junction, the J-V curves of Si solar cell can still be described by Eq. (1) [11]. This is surely not the case according to the analysis above. In fact, the Ag/SiO2/N-Si junction is inversely connected to the PN one; that is, the internally built-in fields of the two junctions are reverse in direction; while, Al/Al2O3/P-Si junction would be either forwardly connected to the PN one; that is, the built-in fields of the two junctions are the same in direction or be in Ohmic contact with the PN junction considering the P-type feature of the substrate [9–11]. The J-V dependence of MOS junction cell can be described by Eq. (2) in the following [9, 11–17],
$$ J={J}_s\cdot \exp \left(\frac{V_{\mathrm{MOS}}}{\mathrm{ns}\cdot \mathrm{V}\mathrm{t}}\right)\cdot \left[1- \exp \left(-\frac{V_{\mathrm{MOS}}}{\mathrm{Vt}}\right)\right] $$
whereJ s = A* ⋅ T 2 ⋅ exp(−φ B /Vt). A * means effective Richardson constant. ϕ B is the barrier height. ns is ideal factor of the MOS junction. For N-Si, A * = 112 Acm−2K−2, and for P-Si, A * = 32 Acm−2K−2 [17].
Figure 4 depicts the simulated J-V curve of one PN junction solar cell, as calculated with Eq. (1) by using PV parameters of V OC = 572 mV, J SC = 40.8 mA/cm2, n = 1.06, Rs = 0.11 Ω · cm2, and Rp = 8552 Ω · cm2. The PV parameters selected are taken from the fitting results as will be explained in the following. The J-V curve of the MOS junction solar cell was calculated with Eq. (2) by using the same J SC (40.8 mA/cm2), with ns = 1.21, and ϕ B = 0.505 eV, which are also from the following fitting results. For the case of forward connection, or the model of PN + MOS,
Simulated J-V curves of one PN junction solar cell. PN junction forwardly connected to MOS (PN + MOS) and PN junction inversely connected to MOS (PN-MOS)
$$ V={V}_{\mathrm{PN}}+{V}_{\mathrm{MOS}} $$
and for the case of inverse connection, or the model of PN-MOS,
$$ V={V}_{\mathrm{PN}}-{V}_{\mathrm{MOS}} $$
To simulate the J-V curve of PN + MOS or PN-MOS, a numerical calculation procedure was used; that is, starting from J = J SC down to J = 0, for each point of J (the step of J was chosen 0.1 mA/cm2), both V PN and V MOS were calculated by Eqs. (1) and (2), respectively, combining all the other selected PV parameters. Then, Eqs. (3) and (4) were applied to get the total voltage across the solar cell for the models of forward and inverse connections, respectively. The resulting J-V curves of PN + MOS and PN-MOS are given in Fig. 4, too. It is clear that for the forward connection, FF even becomes lower; while, for the inverse connection, higher FF is available. It should be pointed out that it is only when ϕ B /e of the MOS junction is not far from V OC of the solar cell, which is the case during the simulation of Fig. 4, could a high FF be obtained. In fact, we have simulated the relation between FF and ϕ B . It is found that when ϕ B /e is very close to or even equals V OC, FF larger than 0.9 can be obtained.
We then use the model of PN-MOS to fit the J-V curve of cell A with Eqs. (1) and (2). The fitting curve has been plotted in Fig. 3a. At this moment, the derived PV parameters are n = 1.06, ns = 1.21, ϕ B = 0.505 eV, Rs = 0.11 Ω · cm2, and Rp = 8552 Ω · cm2. The values of n and ns are now rational, and ϕ B /e is close to V OC (although not very close to 0.572 V), which is consistent with the inference derived above.
It needs to be pointed out that all the PV parameters including FF are very sensitive to the annealing conditions, as have been illustrated in Fig. 3a. If there is no annealing or the annealing is insufficient, the barrier height and the serial resistance would become larger [18–20], which could severely degrade the performance of solar cell. On the other hand, if the annealing is overdone, Ag will easily penetrate through SiO2 and into the emitter zone or even into the PN junction, then the whole device would be ruined. The RTA is mainly used to modulate the barrier height of MOS junction, so a proper RTA is crucial to get a very high FF. On the other hand, the thickness of SiO2 also influences FF as well as V OC and J SC, via influencing Ag diffusion and barrier height of the MOS junction. An optimized combination of annealing and SiO2 thickness has been performed here to achieve a high efficiency.
In summary, we have obtained high FFs (~0.87) of c-Si solar cells by using Ag contact as front electrode after proper annealing and rapid thermal annealing treatments. Our model analysis indicates that the very high FF is caused by the inverse connection of MOS (Ag/SiO2/Si) and the PN junctions, with the MOS junction possessing a barrier height/e close to V OC. By using this approach, c-Si solar cells with efficiencies >20 % are achieved due to their high FFs, although their open-circuit voltages are not high (<580 mV) here. It is expected that further improvements on increasing open-circuit voltage and meanwhile remaining the very high FF could lead to further higher efficiency.
Electron charge
EQE:
External quantum efficiency
FFs:
Fill factors
J 0 :
Dark current density
J ph :
Photocurrent density
J SC :
Short-circuit current density
MOS:
Metal/oxide/semiconductor
n :
The ideal factor of the PN junction
ns:
Ideal factor of the MOS junction
Rp:
Parallel resistance
Serial resistances
RTA:
Rapid thermal annealing
SARI:
Shanghai Advanced Research Institute
SEM:
SITP:
Shanghai Institute of Technical Physics
T :
Absolute temperature
V OC :
Open-circuit voltage
η :
Conversion efficiency
ϕ B :
The barrier height
This work is supported by the National Basic Research Program of China (973 Program), Grant No. 2012CB934303, and the National Natural Science Foundation of China under Grant Nos. 61275178 and 51472051.
LXW prepared all the samples and measured the SEM and I-V data. ZQZ measured the QE data and I-V data. TNZ and XC measured the reflectance data and I-V data. ML designed the experiments and wrote the manuscript. All authors read and approved the final manuscript.
Department of Optical Science and Engineering, and Shanghai Ultra-Precision Optical Manufacturing Engineering Center, Fudan University, Shanghai, 200433, China
Thin Film Optoelectronic Technology Center, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China
Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, 200083, China
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3 edition of Holomorphic curves in symplectic geometry found in the catalog.
Published 1994 by Birkhäuser in Basel, Boston .
Symplectic manifolds.,
Holomorphic functions.
Statement Michèle Audin, Jacques Lafontaine, editors.
Series Progress in mathematics ;, v. 117, Progress in mathematics (Boston, Mass) ;, v. 117.
Contributions Audin, Michèle., Lafontaine, J. 1944 Mar. 10-
LC Classifications QA649 .H65 1994
ISBN 10 0817629971, 3764329971
August 4 to 9, at the American Institute of Mathematics, Palo Alto, California. organized by Yasha Eliashberg and John Etnyre. This workshop, sponsored by AIM and the NSF, will be devoted to the development of holomorphic curve techniques in contact geometry and advent of holomorphic curve techniques in contact topology, as exemplified in Symplectic Field Theory (SFT), .
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Holomorphic curves in symplectic geometry Download PDF EPUB FB2
This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings : Hardcover.
Holomorphic Curves in Symplectic Geometry. Editors (view affiliations) Michèle Audin; Jacques Lafontaine; Book. Citations; Applications of pseudo-holomorphic curves to symplectic topology.
Basic symplectic geometry. Front Matter. Pages PDF. An introduction to symplectic geometry. Augustin Banyaga. Pages Symplectic and. Holomorphic Curves in Symplectic Geometry. Editors: Audin, Michele, Lafontaine, Jacques (Eds.) Introduction Applications of pseudo-holomorphic curves to symplectic topology.
Pages Services for this Book. Download Product Flyer Download High-Resolution Cover. Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.- 3 Proofs of the symplectic rigidity results.- 4 What is in the book and what is not.- 1: Basic symplectic geometry.- I An introduction to.
Deals with the pseudo-holomorphic curve methods in symplectic geometry. This book contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of. an invaluable reference for users. but for a first stab at j-holomorphic curves and applications in symplectic topology one might want to try the little version instead (same authors, title contains the words "quantum cohomology") - which as a bonus is available for free on at least one of the authors' website (i think).Cited by: The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as Brand: Springer International Publishing.
The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of \(J\)-holomorphic curves in four-dimensional manifolds.
From symplectic geometry to symplectic topology 10 Contact geometry and the Weinstein conjecture 13 Symplectic fillings of contact manifolds 19 Chapter 2. Local properties 23 Almost complex manifolds and J-holomorphic curves 23 Compatible and tamed almost complex structures 27 Linear Cauchy-Riemann type operators File Size: 1MB.
For a more Lie-group focused account, you can try Robert Bryant's lectures on Lie groups and symplectic geometry which are available online here.
In the final lecture he describes the h-principle and others ideas of Gromov in symplectic geometry, like pseudo-holomorphic curves. The second half of the book then extends this program in two complementary directions: (1) a gentle introduction to Gromov-Witten theory and complete proof of the classification of uniruled symplectic 4-manifolds; and (2) a survey of punctured holomorphic curves and their applications to questions from 3-dimensional contact topology, such as.
The school, the book This book is based on lectures given by the authors of the various chapters in a three week long CIMPA summer school, held in Sophia-Antipolis (near Nice) in July The first week was devoted to the basics of symplectic and Riemannian geometry (Banyaga, Audin, Lafontaine, Gauduchon), the second was the technical one (Pansu, Muller, Duval, Lalonde and Sikorav).
The. From symplectic geometry to symplectic topology 10 Contact geometry and the Weinstein conjecture 13 Symplectic fillings of contact manifolds 19 Chapter 2. Fundamentals 25 Almost complex manifolds and J-holomorphic curves 25 Compatible and tame almost complex structures 29 Linear Cauchy-Riemann type operators 40 Cited by: 3 Pseudo-holomorphic curves The aim of this part is to study some of the important properties of the pseudo-holomorphiccurves.
Definition(Pseudo-holomorphiccurve). Let(M,J) beanalmostcomplexmanifold. A J-holomorphic curve in M is a smooth map σfrom a Riemann surface (i.e a surface withacomplexstructure)(S,j) to(M,J) suchthat: Tσ j= J TσFile Size: KB. J-holomorphic Curves and Symplectic Topology (2nd) Dusa McDuff, Dietmar Salamon.
Categories: Holomorphic Curves in Symplectic Geometry. Birkhäuser Basel. Jacques Lafontaine, Michèle Audin (auth.) You can write a book review and share your experiences.
Other readers will always be interested in your opinion of the books you've read. Bloggat om Holomorphic Curves in Symplectic Geometry Innehållsförteckning Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.- 3 Proofs of the symplectic rigidity results.- 4 What is in the book.
Pseudo-holomorphic curves Almost complex and symplectic geometry. An almost complex structure on a manifold Mis a bundle endomorphism J: TM → TM with square −idT M. In other words, Jmakes TM into a complex vector bundle and we have the canonical decomposition TM⊗R C = T 1,0 M ⊕T 0,1 M = TM⊕TM into real and imaginary parts.
From symplectic geometry to symplectic topology 10 Contact geometry and the Weinstein conjecture 13 Symplectic fillings of contact manifolds 19 Chapter 2. Fundamentals 25 Almost complex manifolds and J-holomorphic curves 25 Compatible and tame almost complex structures 29 Linear Cauchy-Riemann type operators 41 So, what's a J-holomorphic curve?Well, as the Preface to the first edition of the book under review states, it goes back to a paper by Mikhail Gromov, titled "Pseudo-holomorphic curves in symplectic manifolds," and on p.3 of the book McDuff and Salamon give its definition as a (j,J)-holomorphic mapping from a Riemann surface (with j being — well, what else.
— its j-invariant) to. In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann uced in by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic particular, they lead to the Gromov–Witten.
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate ctic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floer theory as a whole.
Holomorphic curves in symplectic geometry, –, Progress in Mathematics, Birkhäuser, Basel, The goal of the program is to explore different aspects of the theory of holomorphic curves and their interaction.
A special accent will be made on applications to Symplectic geometry in low-dimensional topology. The book can also serve as an introduction to current work in symplectic topology: there are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum : $ Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton's equation.
It is related to the theory of dynamical systems and - via holomorphic curves – to algebraic geometry. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic. Part 1.
Elementary symplectic geometry 7 Chapter 2. Symplectic linear algebra 9 1. Basic facts 9 2. Complex structure 13 Chapter 3. Symplectic differential geometry 17 1.
Moser's lemma and local triviality of symplectic differential geometry 17 2. The groups Ham and Di f f. 21 Chapter 4. More Symplectic differential Geometry: Reduction and File Size: KB. Holomorphic Curves, Planar Open Books and Symplectic Fillings A MINICOURSE by Chris Wendl The overarching theme of this minicourse will be the properties of pseudoholomorphic curves and their use in proving global results about symplectic or contact manifolds based on more "localized" information.
$\begingroup$ It's unclear to me what you are looking for here, or what you know already. The question of how many holomorphic curves there are in a given homology class (with constraints possibly) is given by Gromov-Witten invariants. One possible option seems to be taking critial points off.
But then the immersion is not proper and the proof of the monotonicity formula seems to use properness. For example, the proof in the book "holomorphic curves in symplectic geometry" uses a compactly supported vector field. An holomorphic symplectic manifold X is a kähler manifold X with a holomorphic non degenerate closed form σ ∈ H0(X,Ω2 X) An irreducible holomorphic symplectic manifold X is compact and H0(X,Ω∗ X) = C[σ] (eq.
X is simply connected and = H0(X,Ω2 X)). Calabi-Yau: projective mfds with H0(X,Ω∗ X) = C+Cω, where ω is a generator File Size: KB. Pseudoholomorphic curves. Differential geometry -- Symplectic geometry, contact geometry -- Symplectic manifolds, general.
Differential geometry -- Symplectic geometry, contact geometry -- Gromov-Witten invariants, quantum cohomology, Frobenius manifolds. computing $\mathcal{M}(A, J_1)$ using techniques from algebraic geometry.
An important aspect of Gromov's use of almost complex structure and (pseudo)holomorphic. Intersection theory has played a prominent role in the study of closed symplectic 4-manifolds since Gromov's famous paper on pseudoholomorphic curves, leading to myriad beautiful rigidity results that are either inaccessible or not true in higher dimensions.
Siefring's recent extension of the. 4. Symplectic fibrations 5. Hofer's geometry of Ham(M, ω) 6. C0-Symplectic topology and Hamiltonian dynamics.
Part II. Rudiments of Pseudoholomorphic Curves: 7. Geometric calculations 8. Local study of J-holomorphic curves 9. Gromov compactification and stable maps Fredholm theory Applications to symplectic topology.
References Index. Download Citation | Lectures on Holomorphic Curves in Symplectic and Contact Geometry | This is a set of expository lecture notes created originally for a graduate course on holomorphic curves Author: Chris Wendl.
The main focus of this workshop will be on holomorphic curve techniques in low-dimensional topology and symplectic geometry. The workshop is a part of the FRG: Collaborative Research: Topology and Invariants of Smooth 4-Manifolds. It is funded by NSF Focused Research Grant DMS J$-holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics.
Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds. Read "Symplectic Topology and Floer Homology: Volume 1, Symplectic Geometry and Pseudoholomorphic Curves" by Yong-Geun Oh available from Rakuten Kobo.
Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology Brand: Cambridge University Press.
Symplectic geometry on moduli spaces of J-holomorphic curves J. Coffey, L. Kessler, and A. Pelayo´ Abstract Let (M;!) be a symplectic manifold, and a compact Riemann surface. We define a 2-form. S i() on the space S i() of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations.
of J-holomorphic curves and symplectic topology has now become routine and causes no confusion. This book begins with a sixteen-page overview of the subject of symplectic topology, the theory of J-holomor-phic curves, and its applications to symplectic topology, algebraic File Size: KB.Symplectic 4-manifolds The group of symplectomorphisms Hofer geometry Distinguishing symplectic structures Chapter Gluing The gluing theorem Connected sums of J-holomorphic curves Weighted norms Cutoff functions Construction of the gluing map J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 – 28, Since their introduction by Gromov [4] in the mid's J-holomorphic curves have been one of the most widely used tools in symplectic geometry, leading to the formulation of various theories (Gromov-Witten invariants, quantum co-Author: Janko Latschev.
dixsept.club - Holomorphic curves in symplectic geometry book © 2020 | CommonCrawl |
Time-scaling transformation for optimal control problem with time-varying delay
On evolution quasi-variational inequalities and implicit state-dependent sweeping processes
Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes
Ahmed Aghriche 1, , Radouane Yafia 2,, , M. A. Aziz Alaoui 3, and Abdessamad Tridane 4,
Ibn Zohr University, CST Campus Universitaire Ait Melloul, Agadir, Morocco
Laboratory of Analysis, Geometry and Applications (LAGA), Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Campus Universitaire, BP 133, Kenitra, Morocco
Normandie Univ, France; ULH, LMAH, F-76600 Le Havre; FR-CNRS-3335, ISCN, 25 rue Ph. Lebon 76600, Le Havre, France
Department of Mathematical Sciences, United Arab Emirates University Al Ain, Abu Dhabi, United Arab Emirates
* Corresponding author: Radouane Yafia
Received November 2018 Revised April 2019 Published December 2019
Figure(14) / Table(2)
Aedes aegypti (Ae. aegypti: mosquito) is a known vector of several viruses including yellow fever, dengue, chikungunya and zika. In the current paper, we present a delayed mathematical model describing the dynamics of Ae. aegypti. Our model is governed by a system of three delay differential equations modeling the interactions between three compartments of the Ae. aegypti life cycle (females, eggs and pupae). By using time delay as a parameter of bifurcation, we prove stability/switch stability of the possible equilibrium points and the existence of bifurcating branch of small amplitude periodic solutions when time delay crosses some critical value. We establish an algorithm determining the direction of bifurcation and stability of bifurcating periodic solutions. In the end, some numerical simulations are carried out to support theoretical results..
Keywords: Ae. aegypti, DDE, stability, Hopf bifurcation, direction of bifurcation.
Mathematics Subject Classification: Primary: 39A05, 92D25; Secondary: 92D30, 39A23, 39A28.
Citation: Ahmed Aghriche, Radouane Yafia, M. A. Aziz Alaoui, Abdessamad Tridane. Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020194
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Figure 1. Schematic representation describing the interaction between females ($ F $), eggs ($ E $) and pupae ($ P $) of Ae. aegypti population
Figure 2. Surfaces representing the effect of parameters (left $ \gamma $ and $ \alpha $) and (right $ \sigma $ and $ \alpha $) on the variations of the population reproduction number $ R $
Figure 3. Surfaces representing the effect of parameters(left $ \beta $ and $ \alpha $) and (right $ \mu_{F} $ and $ \alpha $) on the variations of the population reproduction number $ R $
Figure 4. Surfaces representing the effect of parameters (left $ \mu_{E} $ and $ \alpha $) and (right $ \mu_{P} $ and $ \alpha $) on the variations of the population reproduction number $ R $
Figure 5. Surfaces representing the effect of parameters (left $ \mu_{P} $ and $ \mu_{F} $) and (right $ \mu_{P} $ and $ \beta $) on the variations of the population reproduction number $ R $
Figure 6. Surface representing the effect of parameters $ \mu_{P} $ and $ \gamma $ on the variations of the population reproduction number $ R $
Figure 7. Stability of $ E_0 $ for $ \tau = 0 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right) and non existence of $ E_1 $ for $ \mu_F = 5 $ and $ R = 0.269 $
Figure 8. Instability of $ E_0 $ and stability of $ E_1 $ for $ \tau = 0 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right) with $ R = 8.43 $
Figure 9. Instability of $ E_0 $ and stability of $ E_1 $ for $ \tau = 5 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)
Figure 10. Mikhailov hodograph indicating the stability of $ E_1 $ for $ \tau = 5 $ (left) and instability of $ E_1 $ for $ \tau = 50 $ (right). Note that, The steady state $ E_1 $ is stable if the curve crosses imaginary axis above zero (at all crossing points)
Figure 11. Periodic solution bifurcated from the steady state $ E_1 $ for $ \tau = \tau_0 = 10.35 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)
Figure 12. Periodic solution bifurcated from the steady state $ E_1 $ for $ \tau = \tau_0+\epsilon = 10.85 $ with $ \epsilon = 0.5 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)
Figure 13. Chaotic solution bifurcated from the steady state $ E_1 $ for $ \tau = \tau_0+\epsilon = 11.15 $ with $ \epsilon = 0.8 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)
Figure 14. Existence of pair purely imaginary roots (left). The branches of bifurcation (diagram of bifurcation), branch $ 2 $ (blue line) is the branch of the periodic orbits that arise from the Hopf point, branch $ 3 $ (red line) is the branch of period doubling bifurcation point in branch $ 2 $ and branch $ 4 $ (green line) is the branch of period doubling bifurcation point in branch $ 3 $ (right). Note that, the branch $ 1 $ is the steady state $ E_1 $ with amplitude $ 0 $
Table 1. Parameters estimation
parameter value reference
$ \gamma $ $ 0.90 $ Assumed
$ \alpha $ $ 0 .6 $ Assumed
$ \sigma $ $ 4 $ [29]
$ \beta $ $ 0.4 $ [29]
$ \mu_F $ $ 0.16 $ Assumed
$ \mu_E $ $ 0.15 $ Assumed
$ \mu_P $ $ 0.01 $ Assumed
$ k $ $ 500 $ Assumed
Table 2. The sensitivity indices of the population reproduction number $ R $
Parameter Sensitivity index Index at parameters value
$ \gamma $ $ \frac{\mu_{E}}{\gamma+\mu_{E}} $ $ +0.1428 $
$ \alpha $ $ \frac{\mu_{P}}{\alpha+\mu_{P}} $ $ +0.366 $
$ \sigma $ $ +1 $ $ +1 $
$ \beta $ $ +1 $ $ +1 $
$ \mu_F $ $ -1 $ $ -1 $
$ \mu_E $ $ -\frac{\mu_{E}}{\gamma+\mu_{E}} $ $ -0.1428 $
$ \mu_P $ $ -\frac{\mu_{P}}{\gamma+\mu_{P}} $ $ -0.336 $
Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731
Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
Ahmed Aghriche Radouane Yafia M. A. Aziz Alaoui Abdessamad Tridane | CommonCrawl |
Items from 121 to 140 out of 37,223 results
Algebraic Characterization of the Local Craig Interpolation Property
Zalán Gyenis
Bulletin of the Section of Logic > 2018 > 47 > 1
The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka– Németi and Sain.
Domination Parameters of a Graph and its Complement
Wyatt J. Desormeaux, Teresa W. Haynes, Michael A. Henning
Discussiones Mathematicae Graph Theory > 2017 > 38 > 1 > 203-215
A dominating set in a graph G is a set S of vertices such that every vertex in V (G) \ S is adjacent to at least one vertex in S, and the domination number of G is the minimum cardinality of a dominating set of G. Placing constraints on a dominating set yields different domination parameters, including total, connected, restrained, and clique domination numbers. In this paper, we study relationships...
Inverse Problem on the Steiner Wiener Index
Xueliang Li, Yaping Mao, Ivan Gutman
Discussiones Mathematicae Graph Theory > 2017 > 38 > 1 > 83-95
The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) =∑u,v∈V (G) dG(u, v), where dG(u, v) is the distance (the length a shortest path) between the vertices u and v in G. For S ⊆ V (G), the Steiner distance d(S) of the vertices of S, introduced by Chartrand et al. in 1989, is the minimum size of a connected subgraph of G whose vertex set contains S. The k-th...
The Distance Magic Index of a Graph
Aloysius Godinho, Tarkeshwar Singh, S. Arumugam
Let G be a graph of order n and let S be a set of positive integers with |S| = n. Then G is said to be S-magic if there exists a bijection ϕ : V (G) → S satisfying ∑x∈N(u) ϕ(x) = k (a constant) for every u ∈ V (G). Let α(S) = max{s : s ∈ S}. Let i(G) = min α(S), where the minimum is taken over all sets S for which the graph G admits an S-magic labeling. Then i(G) − n is called the distance magic index...
Upper Bounds for the Strong Chromatic Index of Halin Graphs
Ziyu Hu, Ko-Wei Lih, Daphne Der-Fen Liu
Discussiones Mathematicae Graph Theory > 2017 > 38 > 1 > 5-26
The strong chromatic index of a graph G, denoted by χ′s(G), is the minimum number of vertex induced matchings needed to partition the edge set of G. Let T be a tree without vertices of degree 2 and have at least one vertex of degree greater than 2. We construct a Halin graph G by drawing T on the plane and then drawing a cycle C connecting all its leaves in such a way that C forms the boundary of...
Cubic Graphs with Total Domatic Number at Least Two
Saieed Akbari, Mohammad Motiei, Sahand Mozaffari, Sina Yazdanbod
Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two.
Core Index of Perfect Matching Polytope for a 2-Connected Cubic Graph
Xiumei Wang, Yixun Lin
For a 2-connected cubic graph G, the perfect matching polytope P(G) of G contains a special point [...] xc=(13,13,…,13) $x^c = \left( {{1 \over 3},{1 \over 3}, \ldots ,{1 \over 3}} \right)$ . The core index ϕ(P(G)) of the polytope P(G) is the minimum number of vertices of P(G) whose convex hull contains xc. The Fulkerson's conjecture asserts that every 2-connected cubic graph G has six perfect matchings...
Fibonacci and Telephone Numbers in Extremal Trees
Urszula Bednarz, Iwona Włoch
In this paper we shall show applications of the Fibonacci numbers in edge-coloured trees. In particular we determine the successive extremal graphs in the class of trees with respect to the number of (A, 2B)-edge colourings. We show connections between these numbers and Fibonacci numbers as well as the telephone numbers.
Hereditary Equality of Domination and Exponential Domination
Michael A. Henning, Simon Jäger, Dieter Rautenbach
We characterize a large subclass of the class of those graphs G for which the exponential domination number of H equals the domination number of H for every induced subgraph H of G.
Rainbow Vertex-Connection and Forbidden Subgraphs
Wenjing Li, Xueliang Li, Jingshu Zhang
A path in a vertex-colored graph is called vertex-rainbow if its internal vertices have pairwise distinct colors. A vertex-colored graph G is rainbow vertex-connected if for any two distinct vertices of G, there is a vertex-rainbow path connecting them. For a connected graph G, the rainbow vertex-connection number of G, denoted by rvc(G), is defined as the minimum number of colors that are required...
Cores, Joins and the Fano-Flow Conjectures
Ligang Jin, Eckhard Steffen, Giuseppe Mazzuoccolo
The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on...
On the Totalk-Domination in Graphs
Sergio Bermudo, Juan C. Hernández-Gómez, José M. Sigarreta
Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a...
Bounding the Openk-Monopoly Number of Strong Product Graphs
Dorota Kuziak, Iztok Peterin, Ismael G. Yero
Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ, and let k ∈ {1 − ⌈δ/2⌉, . . . , ⌊δ/2⌋} be an integer. Given a set M ⊂ V, a vertex v of G is said to be k-controlled by M if [...] δM(v)≥δG(v)2+k $\delta _M (v) \ge {{\delta _G (v)} \over 2} + k$ , where δM(v) represents the number of neighbors of v in M and δG(v) the degree of v in G. A set M is called an open k-monopoly...
Matchings Extend to Hamiltonian Cycles in 5-Cube
Fan Wang, Weisheng Zhao
Ruskey and Savage asked the following question: Does every matching in a hypercube Qn for n ≥ 2 extend to a Hamiltonian cycle of Qn? Fink confirmed that every perfect matching can be extended to a Hamiltonian cycle of Qn, thus solved Kreweras' conjecture. Also, Fink pointed out that every matching can be extended to a Hamiltonian cycle of Qn for n ∈ {2, 3, 4}. In this paper, we prove that every matching...
Sharp Upper Bounds on the Clar Number of Fullerene Graphs
Yang Gao, Heping Zhang
The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30.
A Note on the Interval Function of a Disconnected Graph
Manoj Changat, Ferdoos Hossein Nezhad, Henry Martyn Mulder, N. Narayanan
In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.
Requiring that Minimal Separators Induce Complete Multipartite Subgraphs
Terry A. McKee
Complete multipartite graphs range from complete graphs (with every partite set a singleton) to edgeless graphs (with a unique partite set). Requiring minimal separators to all induce one or the other of these extremes characterizes, respectively, the classical chordal graphs and the emergent unichord-free graphs. New theorems characterize several subclasses of the graphs whose minimal separators...
The Graphs Whose Permanental Polynomials Are Symmetric
Wei Li
The permanental polynomial [...] π(G,x)=∑i=0nbixn−i $\pi (G,x) = \sum\nolimits_{i = 0}^n {b_i x^{n - i} }$ of a graph G is symmetric if bi = bn−i for each i. In this paper, we characterize the graphs with symmetric permanental polynomials. Firstly, we introduce the rooted product H(K) of a graph H by a graph K, and provide a way to compute the permanental polynomial of the rooted product H(K). Then...
Power Domination in Knödel Graphs and Hanoi Graphs
Seethu Varghese, A. Vijayakumar, Andreas M. Hinz
In this paper, we study the power domination problem in Knödel graphs WΔ,2ν and Hanoi graphs [...] Hpn . We determine the power domination number of W3,2ν and provide an upper bound for the power domination number of Wr+1,2r+1 for r ≥ 3. We also compute the k-power domination number and the k-propagation radius of [...] Hp2 .
Bounds on the Locating Roman Domination Number in Trees
Nader Jafari Rad, Hadi Rahbani
A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = {v ∈ V : f(v) = i} for i = 0, 1, 2. An RDF f = (V0, V1, V2) is...
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Identify and describe outcomes
Certain, Impossible and In-Between
Results from an experiment (Investigation)
Likelihood
Events that impact on others
Events that can/can't happen with others
Range of Probabilities
Outcomes of Chance Experiments
Probabilities as Fractions
Describing Chance
Sample Spaces
Probabilities as Rational Number
Experimental Probability I
Conducting Experiments (Investigation)
Experimental and Expected Frequencies
Outcomes of experiments
Experimental Probability II
Theoretical Probability
Can you win at the casino? (Investigation)
Simple Venn Diagrams
Generating data for probability analysis (Investigation)
Venn Diagrams and Two Way Tables
Rolling ball experiment (Investigation)
Complementary Events
Describing and/or events
Sweet Probability (Investigation)
Relative frequencies of And/Or events
Replacement and non-replacement probabilities
Using Frequency tables and graphs to estimate probabilities
Expectation and Fair Value
Mixed Questions on Probability
Two Way Tables
Frequency Trees
Identifying Independent and Dependent Events
Is it fair? (Investigation)
Independent Events
Dependent Events
Probability with words (Investigation)
Replacement and non-replacement probabilities with dice and ducks (Investigation)
Probability of Equally Likely Outcome
Mutually Exclusive and Non-Mutually Exclusive Events
Mixed Probability and Sample Spaces
Conditional Probability - Sample Spaces
Conditional Probability - Combinations
Probability Venn Diagrams - Conditional, Independent and Mutually Exclusive events
Venn Diagrams and Two-Way Tables - Word Problems
Mathematical Notations for Probability
Independent vs Dependent Events
Life is full of different events. You may walk or drive to school, you may take your lunch or buy it from the canteen. In probability, we break events up into two groups: independent and dependent events.
Independent events are events that are not affected by any other event. For example, flipping a coin multiple times is an independent event because each flip is an isolated event. You have an even chance of getting a head or a tail each time and this is not affected by what you flip before.
In contrast, dependent events are those that are affected by previous events. For example, if you have a bag full of different coloured marbles, the probability of selecting a particular colour will change depending on the colour of the marble that was previously selected and taken out.
Mutually Inclusive vs Mutually Exclusive
Mutually exclusive events are events that can't occur at the same time. For example, you can't turn left and right at the same time!
Mutually inclusive events are not as common as mutually exclusive events. Basically, it means that one event cannot occur without the other.
The best way to visualise mutually inclusive/exclusive is in a Venn diagram. The Venn diagram below is showing two exclusive events (A can happen or B can happen) and one inclusive event (the green part- A and B).
Let's look at a more concrete example with numbers. Some numbers are even ($2,4,20$2,4,20 etc), which can be written in the dark green circle. Some numbers are multiples of three ($3,9,15$3,9,15 etc) which can be written in the light green circle. These are mutually exclusive events. What if I asked you for an even that is also a multiple of three? Now we need to think of numbers that are mutually inclusive (ie. satisfy both conditions) such as $6,12,18$6,12,18 and so on. These are written in the space where the circles overlap.
Conditional probability is concerned with multiple events and is similar to mutually inclusive events. The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. In other words, we calculate the probability of B, given that A has occurred.
Careful!
A lot of conditional probability questions involve replacement:
"With replacement" means the two events remain independent (ie. the probabilities don't change)
"Without replacement" means that the two events are dependent (the chances change depending on the previous selection)
Tree diagrams are often used to display conditional events and make our probability calculations easier.
Let's look at an example of a conditional event that can be displayed in a tree diagram.
In a netball game, there are two people who can shoot - the goal shooter (GS) and the goal attack (GA). Let's say the GS in a team attempts the shots $60%$60% of the time and scores $70%$70% of the goals that they attempt, while the GA attempts to shoot $40%$40% of the time and scores $60%$60% of the goals that they attempt.
The blue lines represent the "branches" and we write each possible outcome at the end of each branch (in this case, who takes the shot and whether the goal is scored). The probability of each outcome is written along the branch.
So what is the probability of the GA scoring?
To work out this conditional probability, we multiply along the branches, considering the probability of the goal attack attempting the shot (GA) and then scoring (goal).
So, $\text{P(GA Score)}=0.4\times0.6$P(GA Score)=0.4×0.6$=$=$0.24$0.24. This means that the goal attack will successfully score $24%$24% of the time in a match.
What if we want to know the probability of either person scoring a goal?
Well, as you can see in the diagram below, there are two possible ways this combination can occur: 1) the goal shooter can score or 2) the goal attack can score.
$P(goal)$P(goal) $=$= $0.6\times0.7+0.4\times0.6$0.6×0.7+0.4×0.6
$=$= $0.42+0.24$0.42+0.24
$=$= $0.66$0.66
That means, the team scores $66%$66% of the time. If you do the same process for the probability of the team missing, you'll see that it's $34%$34%. So we know that we have added everything up correctly because scoring and missing are complementary events and $P(goal)+P(miss)=0.66+0.34$P(goal)+P(miss)=0.66+0.34$=$=$1$1.
The probabilities of all your outcomes should add up to $1$1.
Multiply the probabilities along the branches.
There are four cards marked with the numbers $2$2, $5$5, $8$8, and $9$9. They are put in a box. Two cards are selected at random one after the other without replacement to form a two-digit number.
Draw a tree diagram to illustrate all the possible outcomes.
How many different two-digit numbers can be formed.
What is the probability of obtaining a number less than $59$59?
What is the probability of obtaining an odd number?
What is the probability of obtaining an even number?
What is the probability of obtaining a number greater than $90$90?
What is the probability that the number formed is divisible by 5?
A pile of playing cards has $4$4 diamonds and $3$3 hearts. A second pile has $2$2 diamonds and $5$5 hearts. One card is selected at random from the first pile, then the second.
Construct a tree diagram of this situation with the correct probability on each branch.
What is the probability of selecting two hearts ?
The Venn diagram shown shows the number of students in a school playing the sports of Rugby League, Rugby Union, both or neither.
What is the probability that a student selected at random plays Rugby League?
What is the probability that a student selected at random plays Rugby Union?
What is the probability that a student selected at random plays Rugby League only?
What is the probability that a student selected at random does not play Rugby League?
What is the probability that a student selected randomly does not play Rugby Union?
What is the probability that a student chosen at random plays Rugby Union only? | CommonCrawl |
The bifurcation of interfacial capillary-gravity waves under O(2) symmetry
A note on a superlinear and periodic elliptic system in the whole space
July 2011, 10(4): 1165-1181. doi: 10.3934/cpaa.2011.10.1165
The inverse Fueter mapping theorem
Fabrizio Colombo 1, , Irene Sabadini 2, and Frank Sommen 3,
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano
Politecnico di Milano, Dipartimento di Matematica, Via Bonardi, 9, 20133 Milano, Italy
Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent, Belgium
Received January 2010 Revised November 2010 Published April 2011
In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function $f$ of the form $f=\alpha+\underline{\omega}\beta$ (where $\alpha$, $\beta$ satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function $\bar{f}=A+\underline{\omega}B$ (where $A,B$ satisfy the Vekua's system) given by $\bar{f}(x)=\Delta^{\frac{n-1}{2}}f(x)$ where $\Delta$ is the Laplace operator in dimension $n+1$. In this paper we solve the inverse problem: given an axially monogenic function $\bar{f}$ determine a slice monogenic function $f$ (called Fueter's primitive of $\bar{f}$ such that $\bar{f}=\Delta^{\frac{n-1}{2}}f(x)$. We prove an integral representation theorem for $f$ in terms of $\bar{f}$ which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution $f$ of the equation $\Delta^{\frac{n-1}{2}}f(x)=\bar{f} (x)$ in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.
Keywords: slice monogenic functions, axially monogenic function, Fueter mapping theorem in integral form, inverse Fueter mapping theorem in integral form., Cauchy-Riemann equations, Vekua's system, Fueter's primitive.
Mathematics Subject Classification: Primary: 30G3.
Citation: Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
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Fabrizio Colombo Irene Sabadini Frank Sommen | CommonCrawl |
Mechanics of Advanced Materials and Modern Processes
Linear and non-linear vibration analysis of moderately thick isosceles triangular FGPs using a triangular finite p-element
SA. Belalia1Email author
Mechanics of Advanced Materials and Modern Processes20173:4
Published: 1 February 2017
The geometrically non-linear formulation based on Von-Karman's hypothesis is used to study the free vibration isosceles triangular plates by using four types of mixtures of functionally graded materials (FGMs - AL/AL2O3, SUS304/Si3N4, Ti- AL-4V/Aluminum oxide, AL/ZrO2). Material properties are assumed to be temperature dependent and graded in the thickness direction according to power law distribution.
A hierarchical finite element based on triangular p-element is employed to define the model, taking into account the hypotheses of first-order shear deformation theory. The equations of non-linear free motion are derived from Lagrange's equation in combination with the harmonic balance method and solved iteratively using the linearized updated mode method.
Results for the linear and nonlinear frequencies parameters of clamped isosceles triangular plates are obtained. The accuracy of the present results are established through convergence studies and comparison with results of literature for metallic plates. The results of the linear vibration of clamped FGMs isosceles triangular plates are also presented in this study.
The effects of apex angle, thickness ratio, volume fraction exponent and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are studied. The results obtained in this work reveal that the physical and geometrical parameters have a important effect on the non-linear vibration of FGMs triangular plates.
The mixtures effect of Ceramic-Metal
Linear and Non-linear vibration
Moderately thick FGM plates
p-version of finite element method
In recent years, the geometrically non-linear vibration of functionally graded Materials (FGMs) for different structures has acquired great interest in many researches. In 1984, The concept of the FGMs was introduced in Japan by scientific researchers (Koizumi 1993; Koizumi 1997). FGMs are composite materials which are microscopically inhomogeneous. The mechanical properties of FGMs are expressed with mathematical functions, and assumed to vary continuously from one surface to the other.
Since the variation of mechanical properties of FGM is nonlinear, therefore, studies based on the nonlinear deformation theory is required for these type of materials. Many works have studied the static and dynamic nonlinear behavior of functionally graded plates with various shapes. The group of researchers headed by (Reddy and Chin 1998; Reddy et al. 1999; Reddy 2000) have done a lot of numerical and theoretical work on FG plates under several effects (thermoelastic response, axisymmetric bending and stretching, finite element models, FSDT-plate and TSDT-plate). Woo & Meguid (2001) analyzed the nonlinear behavior of functionally graded shallow shells and thin plates under temperature effects and mechanical loads. The analysis of nonlinear bending of FG simply supported rectangular plates submissive to thermal and mechanical loading was studied by (Shen 2002). (Huang & Shen 2004) applied the perturbation technique to nonlinear vibration and dynamic response of FG plates in a thermal environment. Chen (2005) investigated the large amplitude vibration of FG plate with arbitrary initial stresses based on FSDT. An analytical solution was proposed by Woo et al. 2006 to analyzed the nonlinear vibration of functionally graded plates using classic plate theory. Allahverdizadeh et al. (2008a, 2008b) have studied the non-linear forced and free vibration analysis of circular functionally graded plate in thermal environment. The p-version of the FEM has been applied to investigate the non-linear free vibration of elliptic sector plates and functionally graded sector plates by (Belalia & Houmat 2010; 2012). Hao et al. 2011 analyzed the non-linear vibration of a cantilever functionally graded plate based on TSDT of plate and asymptotic analysis and perturbation method. Duc & Cong 2013 analyzed the non-linear dynamic response of imperfect symmetric thin sandwich FGM plate on elastic foundation. Yin et al. 2015 proposed a novel approach based on isogeometric analysis (IGA) for the geometrically nonlinear analysis of functionally graded plates (FGPs). the same approach (IGA) and a simple first-order shear deformation plate theory (S-FSDT) are used by Yu et al. 2015 to investigated geometrically nonlinear analysis of homogeneous and non-homogeneous functionally graded plates. Alinaghizadeh & Shariati 2016, investigated the non-linear bending analysis of variable thickness two-directional FG circular and annular sector plates resting on the non-linear elastic foundation using the generalized differential quadrature (GDQ) and the Newton–Raphson iterative methods.
The p-version FEM has many advantages over the classic finite element method (h-version), which includes the ability to increase the accuracy of the solution without re-defining the mesh (Han & Petyt 1997; Ribeiro 2003). This advantage is suitable in non-linear study because the problem is solved iteratively and the non-linear stiffness matrices are reconstructed throughout each iteration. Using the p-version with higher order polynomials, the structure is modeled by one element while satisfying the exactitude requirement. In p-version, the point where the maximum amplitude is easy to find it as there is a single element, contrary to the h-version this point must be sought in every element of the mesh which is very difficult. The advantages of the p-version mentioned previously, make it more powerful to the nonlinear vibration analysis of plates. So far, no work has been published to the study of linear and nonlinear vibration of FGMs isosceles triangular plate by using the p-version of FEM.
In the present work, the non-linear vibration analysis of moderately thick FGMs isosceles triangular plates was investigated by a triangular finite p-element. The shape functions of triangular finite p-element are obtained by the shifted orthogonal polynomials of Legendre. The effects of rotatory inertia and transverse shear deformations are taken into account (Mindlin 1951). The Von-Karman hypothesis are used in combination with the harmonic balance method (HBM) to obtained the motion equations. The resultant equations of motion are solved iteratively using the linearized updated mode method. The exactitude of the p-element is investigated with a clamped metallic triangular plate. Comparisons are made between current results and those from published results. The effects of thickness ratio, apex angle, exponent of volume fraction and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are also studied.
Consider a moderately thick isosceles triangular plate with the following geometrical parameters thickness h, base b, height a and apex angle β (Fig. 1). The triangular p-element is mapped to global coordinates from the local coordinates ξ and η. The differential relationship between the two coordinates systems is given as a function of the Jacobian matrix ( J ) by
Geometry of isosceles triangular plate
$$ \left\{\begin{array}{c}\hfill \frac{\partial }{\partial \xi}\hfill \\ {}\hfill \frac{\partial }{\partial \eta}\hfill \end{array}\right\}=\boldsymbol{J}\left\{\begin{array}{c}\hfill \frac{\partial }{\partial x}\hfill \\ {}\hfill \frac{\partial }{\partial y}\hfill \end{array}\right\} $$
where J is given by
$$ \boldsymbol{J}=\left[\begin{array}{cc}\hfill \frac{\partial x}{\partial \xi}\hfill & \hfill \frac{\partial y}{\partial \xi}\hfill \\ {}\hfill \frac{\partial x}{\partial \eta}\hfill & \hfill \frac{\partial y}{\partial \eta}\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill b\hfill & \hfill 0\hfill \\ {}\hfill b/2\hfill & \hfill b/2 tg\left(\frac{\beta}{2}\right)\hfill \end{array}\right] $$
In first-order shear deformation plate theory, the displacements (u, v and w) at a point with coordinate (x, y, z) from the median surface are given as functions of midplane displacements (u 0 , v 0 , w) and independent rotations (θ x and θ y ) about the x and y axes as
$$ \begin{array}{c}\hfill u\;\left( x, y, z, t\right)={u}_0\left( x, y, t\right)+ z{\theta}_y\left( x, y, t\right)\hfill \\ {}\hfill v\;\left( x, y, z, t\right)={v}_0\left( x, y, t\right)- z{\theta}_x\left( x, y, t\right)\hfill \\ {}\hfill w\;\left( x, y, z, t\right)= w\;\left( x, y, t\right)\hfill \end{array} $$
The in-plane displacements (u, v) and out-of-plane displacements (w, θ x and θ y ) will be expressed using the p-version FEM as
$$ \left\{\begin{array}{c}\hfill u\hfill \\ {}\hfill v\hfill \end{array}\right\}=\left[\begin{array}{cc}\hfill N\left(\xi, \eta \right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill N\left(\xi, \eta \right)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {q}_u\hfill \\ {}\hfill {q}_v\hfill \end{array}\right\} $$
$$ \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill w\hfill \\ {}\hfill {\theta}_y\hfill \end{array}\hfill \\ {}\hfill {\theta}_x\hfill \end{array}\right\}=\left[\begin{array}{ccc}\hfill N\left(\xi, \eta \right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill N\left(\xi, \eta \right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill N\left(\xi, \eta \right)\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\} $$
where q u , q v are the vectors of generalized in-plane displacements, q w , \( {q}_{\theta_y} \) and \( {q}_{\theta_x} \) are the vectors of generalized transverse displacement and rotations, respectively, N(ξ, η) are the hierarchical shape functions of triangular p-element (Belalia & Houmat 2010).
Using FSDT of plate in combination with Von-Karman hypothesis, the nonlinear strain–displacement relationships are expressed as
$$ \left\{\varepsilon \right\}=\left\{{\varepsilon}^L\right\}+\left\{{\varepsilon}^{NL}\right\} $$
where the linear and the non-linear strains can be expressed as,
$$ \left\{{\varepsilon}^L\right\}=\left\{\begin{array}{c}\hfill {\varepsilon}_P^L\hfill \\ {}\hfill 0\hfill \end{array}\right\}+\left\{\begin{array}{c}\hfill z{\varepsilon}_b\hfill \\ {}\hfill {\varepsilon}_s\hfill \end{array}\right\}\kern1em \mathrm{and}\kern1em \left\{{\varepsilon}^{NL}\right\}=\left\{\begin{array}{c}\hfill {\varepsilon}_P^{NL}\hfill \\ {}\hfill 0\hfill \end{array}\right\} $$
the components of linear and the non-linear strains given in Eq. (7) are defined as
$$ \left\{{\varepsilon}_P^L\right\}=\left\{\begin{array}{c}\hfill \frac{\partial u}{\partial x}\hfill \\ {}\hfill \frac{\partial v}{\partial y}\hfill \\ {}\hfill \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\hfill \end{array}\right\},\kern1em \left\{{\varepsilon}_b\right\}=\left\{\begin{array}{c}\hfill \frac{\partial {\theta}_y}{\partial x}\hfill \\ {}\hfill -\frac{\partial {\theta}_x}{\partial y}\hfill \\ {}\hfill \frac{\partial {\theta}_y}{\partial y}-\frac{\partial {\theta}_x}{\partial x}\hfill \end{array}\right\} $$
$$ \left\{{\varepsilon}_s\right\}=\left\{\begin{array}{c}\hfill \frac{\partial w}{\partial x}+{\theta}_y\hfill \\ {}\hfill \frac{\partial w}{\partial y}-{\theta}_x\hfill \end{array}\right\}\kern1em \left\{{\varepsilon}_P^{NL}\right\}=\left\{\begin{array}{c}\hfill \frac{1}{2}{\left(\frac{\partial w}{\partial x}\right)}^2\hfill \\ {}\hfill \frac{1}{2}{\left(\frac{\partial w}{\partial y}\right)}^2\hfill \\ {}\hfill \frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\hfill \end{array}\right\} $$
The differential relationship used in Eqs. 8–9 is obtained by inversing Eq. 1 as
$$ \left\{\begin{array}{c}\hfill \frac{\partial }{\partial x}\hfill \\ {}\hfill \frac{\partial }{\partial y}\hfill \end{array}\right\}={\boldsymbol{J}}^{-1}\left\{\begin{array}{c}\hfill \frac{\partial }{\partial \xi}\hfill \\ {}\hfill \frac{\partial }{\partial \eta}\hfill \end{array}\right\} $$
The strain energy E S and kinetic energy E K of the functionally graded moderately thick plate can expressed as
$$ {E}_S=\frac{1}{2}{\displaystyle \iint}\left[{\left\{{\varepsilon}_p\right\}}^{\mathrm{T}}\left[{\mathrm{A}}_{\mathrm{ij}}\right]\left\{{\varepsilon}_p\right\}+{\left\{{\varepsilon}_p\right\}}^{\mathrm{T}}\left[{\mathrm{B}}_{\mathrm{ij}}\right]\left\{{\varepsilon}_b\right\}\right.+{\left\{{\varepsilon}_b\right\}}^T\left[{B}_{ij}\right]\left\{{\varepsilon}_p\right\}+{\left\{{\varepsilon}_b\right\}}^T\left[{D}_{ij}\right]\left\{{\varepsilon}_b\right\}+\left.{\left\{{\varepsilon}_s\right\}}^T\left[{S}_{ij}\right]\left\{{\varepsilon}_s\right\}\right] dxdy $$
$$ {E}_K=\frac{1}{2}{\displaystyle \iint}\left[{I}_1\left({\left(\frac{\partial u}{\partial t}\right)}^2+{\left(\frac{\partial v}{\partial t}\right)}^2+{\left(\frac{\partial w}{\partial t}\right)}^2\right)+{I}_3\left({\left(\frac{\partial {\theta}_x}{\partial t}\right)}^2+{\left(\frac{\partial {\theta}_y}{\partial t}\right)}^2\right)\right] dxdy $$
where [A ij ], [B ij ] and [D ij ], are extensional, bending-extensional and bending stiffness constants of the FG plate and are given by
$$ \left[{A}_{ij},{B}_{ij},{D}_{ij}\right]={\displaystyle \underset{-\frac{h}{2}}{\overset{+\frac{h}{2}}{\int }}}{Q}_{ij}\left(1, z,{z}^2\right) d z\kern0.75em \left( i, j=1,2,6\right) $$
$$ \left[{S}_{ij}\right]= k{\displaystyle \underset{-\frac{h}{2}}{\overset{+\frac{h}{2}}{\int }}}{Q}_{ij} d z\kern0.75em \left( i, j=4,5\right) $$
where k is a shear correction factor and is equal to π2/12
$$ {Q}_{11}={Q}_{22}=\frac{E(z)}{1-{\nu}^2(z)}\kern1.12em {Q}_{12}=\nu (z){Q}_{11}\kern0.62em {Q}_{44}={Q}_{55}={Q}_{66}=\frac{E(z)}{2\left(1+\nu (z)\right)} $$
$$ \left({I}_1,{I}_3\right)={\displaystyle \underset{- h/2}{\overset{+ h/2}{\int }}}\rho (z)\left(1, z \mathit{^2}\right) d z $$
The material properties E(z),ν(z), and ρ(z) of the functionally graded triangular plate assumed to be graded only in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents which is expressed a
$$ E(z)=\kern0.75em \left({E}_c-{E}_m\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^n+{E}_m $$
$$ \nu (z)=\kern0.75em \left({\nu}_c-{\nu}_m\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^n+{\nu}_m $$
$$ \rho (z)=\kern0.75em \left({\rho}_c-{\rho}_m\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^n+{\rho}_m $$
where c and m index designate the ceramic and the metal, respectively, n is the exponent of the volume fraction (n ≥ 0), z is the thickness coordinate variable, E elastic modulus, ρ mass density, h is the thickness of the plate and ν is the Poisson's ratio. The bottom layer of the functionally graded triangular plate is fully metallic material and the top layer is fully ceramic material. The constants of material for four types of FGMs considered in this study (AL/AL2O3,SUS304/Si3N4, Ti-6AL-4 V/Aluminum oxide, AL/ZrO2) are shown in Table 1.
Mechanical properties of FGMs components Yang et al. (2003) and Zhao et al. (2009)
E (109 N/m2)
ρ (kg/m3)
Aluminium (Al)
Alumina (Al2O3)
Stainless steel SUS304
Silicon nitride Si3N4
Ti-6AL-4 V
Zirconia (ZrO2)
Inserting Eqs. (11–12) in Lagrange's equations the equations of free motion are obtained as:
$$ \left[\overline{\boldsymbol{M}}\right]\left\{\begin{array}{c}\hfill {\ddot{q}}_u\hfill \\ {}\hfill {\ddot{q}}_v\hfill \end{array}\right\}+\left[\overline{\boldsymbol{K}}\right]\left\{\begin{array}{c}\hfill {q}_u\hfill \\ {}\hfill {q}_v\hfill \end{array}\right\}+\left[\overset{\smile }{\boldsymbol{K}}+\widehat{\boldsymbol{K}}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\}=0 $$
$$ \left[\boldsymbol{M}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {\ddot{q}}_w\hfill \\ {}\hfill {\ddot{q}}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {\ddot{q}}_{\theta_x}\hfill \end{array}\right\}+\left[\tilde{\boldsymbol{K}}+\boldsymbol{K}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\}+\left[2\widehat{\boldsymbol{K}}+\overset{\smile }{\boldsymbol{K}}\right]\left\{\begin{array}{c}\hfill {q}_u\hfill \\ {}\hfill {q}_v\hfill \end{array}\right\}=0 $$
The vector of generalized displacement in free motion will be given as
$$ \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {q}_w\hfill \\ {}\hfill {q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {q}_{\theta_x}\hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\} cos\left(\omega t\right)= Qcos\left(\omega t\right) $$
By neglecting the in-plane inertia, and taking into account the effects of the transverse shear deformation and inertia of rotation. Inserting Eqs. (20) and (22) into Eq. (21) and applying the HB-method, the final equation of free motion are of the form
$$ \left[-\omega {}^2\boldsymbol{M}+\boldsymbol{K}-{\overset{\smile }{\boldsymbol{K}}}^{T}{\overline{\boldsymbol{K}}}^{-1}\overset{\smile}{\boldsymbol{K}}\right] \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\} +\frac{3}{4}\left[\begin{array}{ccc}\hfill \widetilde{\boldsymbol{K}}-2{\widehat{\boldsymbol{K}}}^T{\overline{\boldsymbol{K}}}^{-1}\widehat{\boldsymbol{K}}\hfill & \hfill \kern.5em 0\hfill & \hfill \kern.5em 0\hfill \\ {}\hfill 0\hfill & \hfill \kern2.7em 0\hfill & \hfill \kern.5em 0\hfill \\ {}\hfill 0\hfill & \hfill \kern2.7em 0\hfill & \hfill \kern.5em 0\hfill \end{array}\right] \left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\}=0 $$
Where M is the out-of-plane inertia matrices, \( \overline{\boldsymbol{K}} \), K and \( \overset{\smile }{\boldsymbol{K}} \) are the extension, bending and coupled extension-rotation linear stiffness matrices, \( \tilde{\boldsymbol{K}} \) and \( \widehat{\boldsymbol{K}} \) represent the nonlinear stiffness matrices. These matrices are given in Appendix A.
The system of equations given in Eq. (23) are solved iteratively using the linearized updated mode method. This method needs two type of amplitudes, the first is the specific amplitude which depends on the plate thickness, the second is the maximum amplitude to be calculated for each iteration. The new system of equations is solved using any known technique with an accuracy of around (e.g.10−5).
The maximum amplitude w max is evaluated as
$$ {w}_{max}=\left[\begin{array}{ccc}\hfill N\left({\xi}_0,{\eta}_0\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {Q}_w\hfill \\ {}\hfill {Q}_{\theta_y}\hfill \end{array}\hfill \\ {}\hfill {Q}_{\theta_x}\hfill \end{array}\right\}\kern2.5em \left( i=1,2,\dots \left( p+1\right)\left( p+2\right)/2\right) $$
Study of convergence and comparison for linear vibration
In this part a convergence and comparison study is made for the linear vibration of clamped metallic isosceles triangular plates to validate the current formulation and methods proposed.
Table 2 shows the convergence of the first three frequencies parameter \( \Omega =\omega b{}^2\sqrt{\rho h/ D} \) of metallic clamped isosceles triangular plate (β = 90°) for the three following different thickness ratio (h/b = 0.05, 0.1 and 0.15). The convergence of results can be accelerated by increasing the polynomial order p from 6 to 11. To validate the accuracy of the present solution, a comparison, listed in Table 3, is made between the present results and the results of p-version Ritz method (Liew et al. 1998) of first three linear frequency parameters for metallic clamped isosceles triangular plate, the geometric parameters of this plate are taken (β = 30°, 60° and 90°) for apex angle and (h/b = 0.05, 0.1 and 0.15) for thickness ratio. From this table, it can be found that the present results are in good agreement with the published results. From this table, it can be found that the present results are in good agreement with the published results.
Convergence of the first three linear frequency parameters for clamped metallic isosceles triangular plate (β = 90°)
h/b
Comparison of the first three linear frequency parameters for clamped metallic isosceles triangular plate
Liew et al. (1998)
Linear vibration of FGMs isosceles triangular plate
This part of study present the linear free vibration of thick FGMs isosceles triangular plates designed by four different mixtures (FGM 1: AL/AL2O3, FGM 2: SUS304/Si3N4, FGM 3: Ti-6AL-4 V/Aluminum oxide and FGM 4: AL/ZrO2). Tables 4, 5, 6, 7 display the first three linear frequency parameters \( {\Omega}_{\mathrm{L}}=\omega b{}^2\sqrt{12{\rho}_m\left(1-\nu \mathit{^2}\right)/{E}_m} \) for a clamped FGMs isosceles triangular plate, three apex angles (β = 30°, 60° and 90°) and three thickness ratio (h/b = 0.05, 0.1 and 0.15) are considered. The exponent of volume fraction vary from 0 to ∞ and it takes the values presented in tables. The results presented in this section comes to enrich the results of literatures. The tables visibly show that the linear frequency parameters is proportional to the angle and thickness and inversely proportional to the volume fraction exponent. For the triangular plate with apex angle (β = 60°), it is noted that the second and third modes are double modes for cases purely metal or purely ceramic, but varied the volume fraction exponent there is a small spacing between the two modes, the maximum spacing is the round of n = 1.
The first three linear frequency parameters of clamped FG AL/AL2O3 isosceles triangular plate
ΩL1
The first three linear frequency parameters of clamped FG SUS304/Si3N4 isosceles triangular plate
The first three linear frequency parameters of clamped FG Ti-6AL-4 V/Aluminum oxide isosceles triangular plate
The first three linear frequency parameters of clamped FG AL/ZrO2 isosceles triangular plate
Non-linear vibration of isosceles triangular FG-plate
The investigation of the effects of the FGM mixtures, volume fraction exponent, thickness ratio, apex angle and boundary conditions on the hardening behavior are investigated in this part. The resultant backbone curves which shows the change in the nonlinear-to-linear frequency ratio ΩNL/ΩL according to maximum amplitude-to-thickness ratios |w max |/h are plotted in Figs. 2, 3, 4, 5 for clamped FG isosceles triangular plate. In Fig. 2, four different mixtures of FGM (FGM 1: AL/AL2O3, FGM 2: SUS304/Si3N4, FGM 3: Ti-6AL-4 V/Aluminum oxide and FGM 4: AL/ZrO2) are considered for volume fraction exponent n = 0.5. The thickness ratio and the apex angle of FG isosceles triangular plate are taken respectively as h/b = 0.1, β = 60°. The effect of apex angle and thickness on the backbone curve for the first mode of the functionally garded AL/AL2O3 clamped triangular plate with (β = 60°) and n =1 are presented in Figs. 3, 4. The effects of mixtures, thickness ratio and apex angle are clearly shown on the plot of these figures. The plots clearly show that if the thickness and angle increases the effects of the hardening behavior increases automatically. Also, the nonlinear vibration of the triangular plate with mixture FGM 4 presents the greatest hardening behavior compared to others mixtures of FGM.
Material mixtures effects on the fundamental backbone curves for clamped FG triangular plate (β = 60°, h/b = 0.1, n = 0.5)
The thickness effects on the fundamental backbone curves for clamped FG AL/AL2O3 isosceles triangular plate (β = 60° and n =1)
The apex angle effects on the fundamental backbone curves for clamped FG AL/AL2O3 isosceles triangular plate (h/b = 0.1and n =1)
The boundary conditions effects on the fundamental backbone curves for FG AL/AL2O3 isosceles triangular plate (β = 90°, h/b = 0.05 and n =1)
The boundary conditions effects on the fundamental backbone curves for FG AL/AL2O3 isosceles triangular plate are investigated in Fig. 5. Four different boundary conditions are considered in this part of study SSS, CSS, SCC and CCC (S: simply supported edge and C : clamped edge). The volume fraction exponent, thickness ratio and the apex angle of FG isosceles triangular plate are taken respectively as n =1, h/b = 0.05 and β = 90°. The figure clearly show that the FG plate with simply supported boundary conditions presents a more accentuated hardening behavior than the other boundary conditions. It is noted that the hardening effect increases when the plate becomes more free (SSS) and decreases as the plate becomes more fixed (CCC), this difference in the results is due to the rotation of the edges.
The variation of frequency ratio ΩNL/ΩL according to volume fraction exponent for clamped isosceles triangular plate with four different mixtures of FGMs is shown in Fig. 6. The exponent of volume fraction take values from 0 to 20 and maximum amplitude-to-thickness ratios take three values |w max |/h = 0.6, 0.8 and 1. The geometric parameters of the plate are (β = 90°) and h/b =0.1. Noted that the shape of the graph is similar for three values of the maximum amplitude-to-thickness ratios of this fact and to understand the phenomenon and good interpretation, Fig. 7 plot only the results of the largest value of the maximum amplitude |w max |/h = 1. It can be seen for volume fraction exponent which varied between n = 0 to n = 4 the hardening effect is maximum for the first mixture (AL/AL2O3), for values n ≥ 4 the second mixture (which SUS304/Si3N4) presents the greatest hardening effect. For third and fourth mixtures (Ti-6AL-4 V/Aluminum oxide and AL/ZrO2) the shape of the two curves are parallel with superiority of the values obtained for the fourth mixture FGM 4. Note that the peak of the hardening behavior for four curves is obtained for volume fraction exponent n = 1, at which corresponds to a linear variation of constituent materials of the mixture. By comparing the spacing between curves FGM1 (Al/Al2O3) and FGM4 (Al/ZrO2) we see clearly the influence of physical properties of the two ceramic (Al2O3 and ZrO2) on hardening behavior. This influence is not due to metal (Al) since the same metal is used in both mixtures.
Material mixtures effects on the variation of the nonlinear-to-linear fundamental frequency ratio with the volume fraction exponent for clamped FG isosceles triangular plate (h/b = 0.1, β = 90°)
Material mixtures effects on the variation of the nonlinear-to-linear fundamental frequency ratio with the volume fraction exponent for clamped FG isosceles triangular plate (|w max |/h = 1, h/b = 0.1, β = 90°)
Figures 8, 9, 10 shows the normalized non-linear fundamental mode shape of isosceles triangular plate for four different mixtures of FGM along the line passes through the point of maximum amplitude (ξ 0, η 0). The mode shape are normalized by dividing by their own maximum displacement. Three apex angles and thickness ratio of FG plate are considered (β = 30°, 60° and 90°), (h/b = 0.05) respectively, volume fraction exponent n = 1 and the maximum amplitude |w max |/h = 1. It can see from these graphs that the displacement is maximum for the FGM 2 (SUS304/Si3N4) then comes FGM3 (Ti-6Al-4 V/Aluminum oxide) with a percentage of displacement 83% of maximum displacement, FGM 1 (AL/AL2O3) with 72% and lastly FGM 4 (AL/ZrO2) with 64%. The normalized non-linear of second and third modes shape of isosceles triangular plates for the same mixtures used early are plotted in Figs. 11, 12, respectively. The geometric parameters used are h/b = 0.05, β = 90° and |w max |/h = 0.8. It can be seen from this plot the effect of mixtures on normalized non-linear first three fundamental mode shape of isosceles triangular plate. This is due to fact that the composition of mixtures contribute to various in-plane forces in the isosceles triangular plate.
Section of normalized non-linear fundamental mode shapes of FG isosceles triangular plate : a) along of ξ; b) alone of η (β = 30°, n = 1, h/b = 0.05)
Section of normalized non-linear fundamental mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 60°, n = 1, h/b = 0.05)
Section of normalized non-linear second mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 90°, n = 1, h/b = 0.05)
Section of normalized non-linear third mode shapes of FG isosceles triangular plate: a) along of ξ; b) alone of η (β = 90°, n = 1, h/b = 0.05)
The non-linear free vibration of moderately thick FGMs clamped isosceles triangular plates was analyzed by a triangular p-element. The material properties of the functionally graded triangular plate assumed to be graded only in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents. The shape functions of triangular finite p-element are obtained by the shifted orthogonal polynomials of Legendre. The components of stiffness and mass matrices were calculated using numerical integration of Gauss-Legendre. The equations of motion are obtained from Lagrange's equation in combination with the harmonic balance method (HBM). Results for linear and non-linear frequency for the lowest three modes of FGMs clamped isosceles triangular plates were obtained. The parametric studies show that the boundary conditions have a great influence on the shape of the backbone curves, the hardening spring effect decreases for clamped FG plate. For simply supported FG plate and by increasing thickness ratio and sector angle of FG plates the hardening spring effect increases. A increase in the volume fraction exponent produces a variation in the hardening spring effect with an increasing part and another decreasing part, the peak in the curves of the nonlinear-to-linear fundamental frequency ratio FG triangular plate is obtained around of n = 1 at which the hardening behavior is maximum, and is obtained for AL/AL2O3 FG plate. This value of volume fraction exponent corresponds to equal mixtures of metal and ceramic in the composition of the FG plate. Not only the hardening behavior is influenced by this mixture but the non-linear mode shape of FG isosceles triangular plate is also influenced.
$$ {\overline{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{cc}\hfill {\overline{K}}_{2\alpha -1,2\beta -1}\hfill & \hfill {\overline{K}}_{2\alpha -1,2\beta}\hfill \\ {}\hfill {\overline{K}}_{2\alpha, 2\beta -1}\kern1em \hfill & \hfill {\overline{K}}_{2\alpha, 2\beta}\kern1em \hfill \end{array}\right] $$
(A.1)
$$ {\mathbf{K}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {K}_{3\alpha -2,3\beta -2}\kern1em \hfill & \hfill {K}_{3\alpha -2,3\beta -1}\kern1em \hfill & \hfill {K}_{3\alpha -2,3\beta}\hfill \\ {}\hfill {K}_{3\alpha -1,3\beta -2}\kern1em \hfill & \hfill {K}_{3\alpha -1,3\beta -1}\kern1em \hfill & \hfill {K}_{3\alpha -1,3\beta}\hfill \\ {}\hfill {K}_{3\alpha, 3\beta -2}\kern1em \hfill & \hfill {K}_{3\alpha, 3\beta -1}\kern1em \hfill & \hfill {K}_{3\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
$$ {\mathbf{M}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {M}_{3\alpha -2,3\beta -2}\kern1em \hfill & \hfill {M}_{3\alpha -2,3\beta -1}\kern1em \hfill & \hfill {M}_{3\alpha -2,3\beta}\hfill \\ {}\hfill {M}_{3\alpha -1,3\beta -2}\kern1em \hfill & \hfill\ {M}_{3\alpha -1,3\beta -1}\kern1em \hfill & \hfill {M}_{3\alpha -1,3\beta}\hfill \\ {}\hfill {M}_{3\alpha, 3\beta -2}\kern1em \hfill & \hfill {M}_{3\alpha, 3\beta -1}\kern1em \hfill & \hfill {M}_{3\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
$$ {\widehat{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {\widehat{K}}_{2\alpha -1,3\beta -2}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha -1,3\beta -1}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha -1,3\beta}\hfill \\ {}\hfill {\widehat{K}}_{2\alpha, 3\beta -2}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha, 3\beta -1}\kern1em \hfill & \hfill {\widehat{K}}_{2\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
$$ {\overset{\smile }{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {\overset{\smile }{K}}_{2\alpha -1,3\beta -2}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha -1,3\beta -1}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha -1,3\beta}\hfill \\ {}\hfill {\overset{\smile }{K}}_{2\alpha, 3\beta -2}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha, 3\beta -1}\kern1em \hfill & \hfill {\overset{\smile }{K}}_{2\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
$$ {\tilde{\mathbf{K}}}_{\alpha, \beta}=\left[\begin{array}{ccc}\hfill {\tilde{K}}_{3\alpha -2,3\beta -2}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -2,3\beta -1}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -2,3\beta}\hfill \\ {}\hfill {\tilde{K}}_{3\alpha -1,3\beta -2}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -1,3\beta -1}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha -1,3\beta}\hfill \\ {}\hfill {\tilde{K}}_{3\alpha, 3\beta -2}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha, 3\beta -1}\kern1em \hfill & \hfill {\tilde{K}}_{3\alpha, 3\beta}\kern1em \hfill \end{array}\right] $$
The non-zero elements of the matrices M, K, \( \overline{\mathrm{K}} \), \( \widehat{\mathrm{K}} \), \( \overset{\smile }{\mathrm{K}} \) and \( \tilde{\mathrm{K}} \) are expressed as
$$ {M}_{3\;\alpha -2,3\beta -2}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}{I}_1{N}_{\alpha}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {M}_{3\;\alpha -1,3\beta -1}={M}_{3\;\alpha, 3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}{I}_3{N}_{\alpha}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha -2,3\beta -2}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;\left({A}_{44}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+{A}_{55}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha -2,3\beta -1}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{44}\frac{\partial {N}_{\alpha}}{\partial \eta}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
(A.10)
$$ {K}_{3\alpha -2,3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{55}\frac{\partial {N}_{\alpha}}{\partial \xi}{N}_{\beta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha -1,3\beta -2}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{44}{N}_{\alpha}\frac{\partial {N}_{\beta}}{\partial \eta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha -1,3\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+ k\;{A}_{44}{N}_{\alpha}{N}_{\beta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha -1,3\beta}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathrm{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha, 3\beta -2}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }} k\;{A}_{55}{N}_{\alpha}\frac{\partial {N}_{\beta}}{\partial \eta}\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha, 3\beta -1}=-{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {K}_{3\alpha, 3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({D}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+{D}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}+ k\;{A}_{55}{N}_{\alpha}{N}_{\beta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overline{K}}_{2\alpha -1,2\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overline{K}}_{2\alpha -1,2\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overline{K}}_{2\alpha, 2\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial y}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overline{K}}_{2\alpha, 2\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({A}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}+{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ \begin{array}{l}{\widehat{K}}_{2\alpha -1,3\beta -2}=\frac{1}{2}{\displaystyle \sum_{\delta =1}^r\left({\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;\left({A}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \xi}+{A}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \eta}\right.}}\right.}\\ {}\kern4em +\left.2{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \eta}\right)\;\left.\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta \right)\kern0.24em {Q}_{3\delta -2}\end{array} $$
$$ \begin{array}{l}{\widehat{K}}_{2\alpha, 3\beta -2}=\frac{1}{2}{\displaystyle \sum_{\delta =1}^r\left({\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;}}\left({A}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \eta}+{A}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \xi}\right.\right.}\\ {}\kern3em +\left.2{A}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \eta}\right)\;\left.\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta\;\right)\kern0.24em {Q}_{3\delta -2}\end{array} $$
$$ {\overset{\smile }{K}}_{2\alpha -1,3\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{12}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}-{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overset{\smile }{K}}_{2\alpha -1,3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}+{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overset{\smile }{K}}_{2\alpha, 3\beta -1}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}-{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ {\overset{\smile }{K}}_{2\alpha, 3\beta}={\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\left({B}_{12}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \xi}+{B}_{66}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \eta}\right)\;}}\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta $$
$$ \begin{array}{l}{\tilde{K}}_{3\alpha -2,3\beta -2}=\frac{1}{2}{\displaystyle \sum_{\delta =1}^r{\displaystyle \sum_{\gamma =1}^r\left({\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{1-\xi}{\int }}\;}}\left({A}_{11}\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \xi}\frac{\partial {N}_{\gamma}}{\partial \xi}\right.\right.}}\\ {}\kern10em +{A}_{22}\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \eta}\frac{\partial {N}_{\gamma}}{\partial \eta}\kern5em \\ {}\kern10em +\left({A}_{12}\right.+\left.2{A}_{66}\right)\frac{\partial {N}_{\alpha}}{\partial \xi}\frac{\partial {N}_{\beta}}{\partial \xi}\frac{\partial {N}_{\delta}}{\partial \eta}\frac{\partial {N}_{\gamma}}{\partial \eta}\\ {}\kern10em \left.+\left({A}_{12}\right.+\left.2{A}_{66}\right)\frac{\partial {N}_{\alpha}}{\partial \eta}\frac{\partial {N}_{\beta}}{\partial \eta}\frac{\partial {N}_{\delta}}{\partial \xi}\frac{\partial {N}_{\gamma}}{\partial \xi}\right)\;\left.\left|\mathbf{J}\right|\;\mathrm{d}\xi\;\mathrm{d}\eta\;\right)\kern0.24em {Q}_{3\delta -2}{Q}_{3\gamma -2}\end{array} $$
The author declare no significant competing financial, professional or personal interests that might have influenced the performance or presentation of the work described in this manuscript.
Laboratory of Computational Mechanics, Faculty of Technology, Department of Mechanical Engineering, University of Tlemcen, B.P. 230, Tlemcen 13000, Algeria.
Faculty of Technology, Department of Mechanical Engineering, University of Tlemcen, B.P. 230, Tlemcen, 13000, Algeria
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Siegel–Veech transforms are in $ \boldsymbol{L^2} $(with an appendix by Jayadev S. Athreya and Rene Rühr)
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William A. Veech's publications
2019, 14: v-xxv. doi: 10.3934/jmd.2019v
Bill Veech's contributions to dynamical systems
Giovanni Forni 1, , Howard Masur 2, and John Smillie 3,
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Received February 10, 2019 Published February 2019
Citation: Giovanni Forni, Howard Masur, John Smillie. Bill Veech's contributions to dynamical systems. Journal of Modern Dynamics, 2019, 14: v-xxv. doi: 10.3934/jmd.2019v
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Figure 1. Rauzy diagram from Veech's personal notes, June 21, 1977
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Giovanni Forni Howard Masur John Smillie | CommonCrawl |
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