id stringlengths 1 6 | url stringlengths 16 1.82k | content stringlengths 37 9.64M |
|---|---|---|
11700 | https://estudogeral.uc.pt/bitstream/10316/41835/2/Trabalho%20Final%20Francisco%20Ruas.pdf | Francisco Ruas 1 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 ABSTRACT Carbon monoxide (CO) poisoning is a common cause of death and a public health problem. The authors studied the epidemiology and the postmortem forensic aspects in all the cases with a carboxihemoglobin (COHb) analysis, by reviewing the reports of all the autopsies performed in the Forensic Pathology Department of the Centre Branch of the National Institute of Legal Medicine of Portugal, between 2000 and 2010. The following criteria were analyzed: year, month, season of the year, age, gender, occupation, underlying disease, etiology, source of CO, cause of death, autopsy findings and toxicological analysis results. The relevance of the circumstantial information and the autopsy findings, the evolution during the years of CO poisoning as a cause of death and the challenges of its differential diagnosis, in our practice, are discussed. Between January 2000 and December 2010, 69 COHb analysis were requested in our institution. In approximately 70% of the situations, circumstantial information included a source of CO at the death scene. More than a half of the cases presented thermal lesions. Cherry-red lividity and cherry-red coloration of blood and viscera were found in approximately 30% of the cases. In the studied period, 14 cases (20.3% of all requests) were recorded as CO poisonings. The highest number of cases occurred in 2000 and, for the remainder of the decade, years with no cases followed lower peaks. Winter registered the highest frequency of CO poisonings (53.8%), more frequent in males (69.2%) and in age group of 51-60 year olds (3 cases). 69.2% of the CO poisonings were violent, unintentional deaths and the remainder of the cases were suicides. Fire was the most frequent source of CO (5 cases, 38.5%). Cherry-red lividity was present in 8 (61.5%) CO poisonings and all presented cherry-red blood and viscera coloration, which was related to higher COHb levels (58 ± 23 (SD) % COHb). Suicides also presented higher COHb levels. Older individuals and Francisco Ruas 2 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 those with thermal lesions presented lower levels. Politrauma was the most frequent cause of death among the negative cases (24.4%). It is possible to conclude that forensic aspects of CO poisoning interact in a complex way and differential diagnosis is not straightforward. This study also emphasizes the role played by public prevention campaigns and improvement of heating appliances in reducing the number of unintentional CO poisonings and the importance of preventing urban and forest fires, the major source of CO among us. KEYWORDS: Carbon monoxide poisoning; Forensic practice; Retrospective study Francisco Ruas 3 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 RESUMO As intoxicações por monóxido de carbono (CO) são causa de morte frequente e um problema de saúde pública. Analisámos a epidemiologia e aspetos forenses postmortem em casos com pedido de análise de carboxihemoglobina (COHb), através do estudo dos relatórios de todas as autópsias realizadas no Serviço de Patologia Forense da Delegação do Centro do Instituto Nacional de Medicina Legal, entre 2000 e 2010. Foram avaliados os seguintes critérios: ano, mês, estação do ano, idade, sexo, profissão, comorbilidade, etiologia, fonte de CO, causa de morte, achados de autópsia e resultados de análises toxicológicas. São objeto de discussão, a importância da informação circunstancial e dos achados de autópsia, a evolução da intoxicação por CO enquanto causa de morte e os desafios do seu diagnóstico diferencial na nossa prática forense. Entre Janeiro de 2000 e Dezembro de 2010, foram requisitadas 69 análises de carboxihemoglobina na nossa instituição. Em aproximadamente 70% dos casos, a informação circunstancial incluía referência a uma fonte CO identificada no local da morte. Mais de metade dos casos apresentavam lesões térmicas; o livor carminado e a coloração carminada do sangue e vísceras foram identificados em 30%. Neste período, 14 casos (20,3% de todos os pedidos) foram classificados como intoxicações por CO. O maior número de casos verificou-se no ano 2000 e, na restante década, a anos com números superiores de casos, seguiram-se anos sem quaisquer casos. A maior parte das intoxicações ocorreu no inverno (53,8%) e foram mais frequentes em homens (69,2%) e no grupo etário de 51-60 anos (3 casos). 69,2% das intoxicações (9 casos) foram acidentais e as restantes, suicídios. A fonte de CO mais frequentemente identificada foi um incêndio (38,5%). O livor carminado foi identificado em 61,5% dos casos e todas as intoxicações apresentavam coloração carminada do sangue e vísceras, relacionada com concentrações superiores de COHb (58 ± 23 (DP) %COHb). Francisco Ruas 4 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Também o suicídio se relaciona com concentrações superiores de COHb. Os idosos e os indivíduos com lesões térmicas apresentavam concentrações inferiores. Nos casos com análise negativa, a cause de morte mais frequente foi politrauma (24,4%). É possível concluir que os aspetos forenses da intoxicação por CO interagem de forma complexa e que o seu diagnóstico diferencial não está desprovido de desafios. Os autores dão ênfase ao papel das campanhas de prevenção junto da população e da evolução dos equipamentos de aquecimento, na redução do número de casos acidentais de intoxicação por CO e à importância da prevenção de incêndios florestais e urbanos, a principal fonte de CO entre nós. PALAVRAS-CHAVE: Intoxicações por monóxido de carbono; prática forense; estudo retrospetivo Francisco Ruas 5 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 1. Introduction Carbon monoxide (CO) is a product of the incomplete combustion of organic materials. Natural sources of CO include forest fires and volcanic eruptions but it is mainly produced by human activity . Automobile exhaust fumes, charcoal briquets in confined spaces and defective or improperly ventilated gas heating appliances are common sources of CO. Even though nontoxic natural gas is now the primary source of domestic energy, its combustion with an insufficient supply of oxygen generates CO, with its own deleterious effects . CO also results from heme catabolism and has recently been recognized as a neurotransmitter . Endogenous production and environmental exposure to CO account for baseline levels of carboxihemoglobin (COHb) of less than 1 to 3% in nonsmokers, while smokers may present basal levels up to 10% [1, 4]. CO is odorless, colorless and tasteless. It is not detected by an exposed person and because it has approximately the density of air, it easily spreads in confined spaces . The most generally accepted toxicity mechanism of CO depends on its exceptional affinity for hemoglobin, 200 times greater than that of oxygen, resulting in tissue hypoxia. Today it is recognized that CO may have a direct cellular toxicity [1, 4]. Depending on the concentration of the gas in air, length of exposure and health condition, CO effects range from mild cardiovascular and neurobehavioral symptoms at COHb levels of less than 15 to 20%, to unconsciousness and death. Children, patients with cardiovascular and lung disease and the elderly, are particularly susceptible to CO [4, 5, 6]. In forensic practice, CO-related deaths result mainly from accident or suicide, being homicide rare. Risser and Schneider (1995) report that in 417 CO-related deaths over a ten year study in Vienna, Austria, the majority of cases were accidents, peaking in winter months . These findings are similar to those of Homer et al. (2005) in Cleveland, USA and Ait El Francisco Ruas 6 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Cadi et al. (2009) in Morocco [7, 8]. Scene investigation often provides circumstantial indications of CO poisoning and autopsy usually follows. Characteristic pathological findings include cherry-red or bright pink livor and bright cherry-red coloration of blood, musculature and viscera. According to forensic medicine literature, these changes are consistent with levels of COHb greater than 30% [9, 10]. They are important forensic clues to the cause of death, which is established after determination of COHb levels greater than 50 to 60% in postmortem blood samples. However, establishing the role of CO as the cause of death is not easy and the interpretation of lower or negative COHb levels in victims with concomitant morbidity may be difficult, namely when excluding natural death and delayed CO-related death cases with suggestive circumstances may present with negligible COHb in blood [11, 12]. On the other hand, cherry-red livor may be absent or result from body refrigeration and cyanide poisoning [12, 14]. CO poisoning has recently been drawing more attention in the popular media. Public awareness of the dangers of CO poisoning, along with improved safety of heating and cooking appliances and gas emission controls in transportation may have been responsible for the decline in the number of unintentional CO-related deaths. Nonetheless, CO poisoning remains an important public health problem and a common cause of death in industrialized countries, being involved in over one half of all fatal poisonings worldwide, yearly [4, 5]. The purpose of this study was to investigate the epidemiology and forensic aspects related to CO poisoning in cases with postmortem COHb analysis, from autopsies performed at the Forensic Pathology Department of the Centre Branch of the National Institute of Legal Medicine (NILM) of Portugal between 2000 and 2010. Francisco Ruas 7 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 2. Material studied Autopsy reports of all deaths examined at the Forensic Pathology Department of Coimbra's Delegation of the NILM of Portugal between January 2000 and December 2010 were reviewed. Actual data material consisted of 69 autopsy reports in which there was a reference to a postmortem carboxihemoglobin (COHb) analysis request. COHb analyses were performed at the Forensic Toxicology Department of the Centre Branch of the NILM of Portugal by a molecular absorption spectroscopy methodology , in postmortem blood samples collected by forensic pathologists. Assessment of the examination protocols was conducted by means of a tabular database according to the following criteria: year, month, season of the year (January, February and March were considered winter months; April, May and June, spring months; July, August and September, summer months and October, November and December, autumn months), age, gender, occupation, underlying disease, medico-legal etiology, source of CO, cause of death, autopsy findings (lividity, thermal injuries, coloration of blood and viscera) and toxicological analysis results. These variables were studied and relevant data was submitted to numerical analysis, using SPSS (Statistical Package for Social Sciences). 3. Results 3.1 COHb analysis requests 3.1.1 Year, month and season distribution Between January 2000 and December 2010, 69 postmortem COHb toxicological analysis were requested and performed at the Centre Branch of the NILM. As presented in Fig. 1, the number of COHb analyses requests was highest in 2005, with 15 cases (21.7% of all requests), number very close to the one achieved in 2000, with 14 cases (20.3%). The years 2002 and 2006 presented the lowest number of requests, 2 (2.9%). The month with the Francisco Ruas 8 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 highest number of COHb analysis requests was March (13 cases, corresponding to 18.8%) and winter was the season with the highest number of requests (24 individuals, 34.8% of all requests) (Fig. 1). Fig. 1. COHb analysis requests distribution by year (A), season (B) and month (C). 0 2 4 6 8 10 12 14 16 COHb requests 0 5 10 15 20 25 30 COHb requests 0 2 4 6 8 10 12 14 COHb requests Number of cases Year Number of cases Season of the year Number of cases Month of the year A B C Francisco Ruas 9 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 3.1.2 Gender, age and occupation Distribution of the cases with COHb analysis according to gender and age is presented in Fig. 2. Male individuals represented 63.8 % (44 individuals) of all cases with a COHb analysis request and in 25 female individuals, COHb analysis was requested (corresponding to 36.3% of all individuals with a COHb analysis). Mean age was 55 ± 22 (SD) years. Age group of >70 years had the highest number of COHb analysis requests (22 cases, 31.9%). The age groups of 11-20 years old and <10 years old were the groups with the lowest number of requests (with 1 request - 1.4% - and 2 requests - 2.9%, respectively). In all age groups, except for the two groups with the lowest number of requests, there were more male than female individuals (Fig. 2). Fig. 2. Distribution of the cases with a COHb analysis request according to gender and age group. 0 2 4 6 8 10 12 14 Male Female Number of cases Age group Carbon Monoxide Distribution of the cases with a presented in Fig. 3. In 75.4% of the cases the deceased wasn't, per se, justification for a CO poisoning suspicion, whe was unknown (50.7% of all requests) or because the occupation wasn't associated with an increased risk of CO poisoning (24.
in 13 (18.8%), a pensioner (Fig. 3 Among the cases with a COHb analysis request that weren't associated with an identifiable source of CO, 4 (19%) individuals were occupation was unknown or not associated with an increased risk of CO poisoning. Fig. 3. Distribution of the cases with a a Includes all occupations without an increased risk of CO poisoning and that wouldn't, Pensioner (18.8%) Fireman (5.8%) Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Distribution of the cases with a postmortem COHb analysis request by o of the cases (52 cases), the information about the occupation of , justification for a CO poisoning suspicion, whe 7% of all requests) or because the occupation wasn't associated with an sk of CO poisoning (24.6%). In 4 (5.8%) cases, the individual was a fireman and (Fig. 3). Among the cases with a COHb analysis request that weren't associated with an identifiable source of CO, 4 (19%) individuals were pensioners and in 17 (81%), the occupation was unknown or not associated with an increased risk of CO poisoning. Distribution of the cases with a COHb analysis request by occupation.
Includes all occupations without an increased risk of CO poisoning and that wouldn't, per se, justify an analysis request. 35 4 13 17 Othera (24.6%) Unknown (50. Francisco Ruas 10 COHb analysis request by occupation is (52 cases), the information about the occupation of , justification for a CO poisoning suspicion, whether because it 7% of all requests) or because the occupation wasn't associated with an ual was a fireman and Among the cases with a COHb analysis request that weren't associated with an pensioners and in 17 (81%), the occupation was unknown or not associated with an increased risk of CO poisoning. request by occupation. , justify an analysis request. Unknown (50.7%) Francisco Ruas 11 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 3.1.3 Etiology and source of CO Characterization of the cases with postmortem COHb analysis according to the presence of a source of CO at the death scene and corresponding etiology is described in Table I. Circumstantial information in the autopsy reports indicated that the medico-legal etiology of the death was of natural death in 11 individuals (15.9% of all cases) and in 58 (84.1%) death had a violent etiology. Among the later, 50 individuals (72.5% of all requests) presented circumstantial information that suggested an unintentional death and 6 cases (8.7%) were suicides. Only 2 of the cases (2.9%) were related to homicides (Table I). There was an identified source of CO at the death scene of 48 (69.6%) individuals with a COHb analysis request. 36 individuals (75% of the cases with an identified CO source and 52.2% of all cases with a COHb analysis request) were fire victims. Table I. Characterization of the cases with COHb analysis request according to the source of CO found at the death scene and corresponding medico-legal etiology. Source of CO Analysis requests Etiology Natural death, n (%) Violent, unintentional, n (%) Suicide, n (%) Homicide, n (%) Fire 36 2 (5.6) 33 (91.7) 0 1 (2.8) Charcoal briquettes 2 0 1 (50) 1 (50) 0 Burnt wood can 1 0 0 1 (100) 0 Fire pan 1 0 0 1 (100 0 Defective gas appliance 5 0 3 (60) 2 (40) 0 Fireplace 3 0 3 (100) 0 0 Not present 21 9 (42.9) 10 (47.6) 1(4.8) 1 (4.8) Total 69 11 (15.9) 50 (72.5) 6 (8.7) 2 (2.9) Francisco Ruas 12 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Among the deaths classified as suicides, 5 (83.3% of all suicides with a COHb analysis request) individuals presented an identifiable source of CO. One of the two homicides with a COHb analysis request was a fire victim and the other case had no identifiable source of CO at the death scene, but the individual presented thermal lesions. 3.1.4 Autopsy findings Relevant autopsy findings of the cases with a postmortem COHb analysis request between January 2000 and December 2010 are presented in Table II. 30.4% (21 cases) of the individuals presented cherry-red lividity. In 29% of the cases, corresponding to 20 individuals, lividity was bluish and in 40.6% (28 individuals), lividity wasn't characterized because of skin pigmentation or large extent of thermal lesions. Among the cases in which a source of CO was identified at the death scene, in the highest number of cases (54.2%, corresponding to 26 individuals), lividity couldn't be characterized; on the other hand, in this group, cherry-red lividity was more frequent than bluish lividity (31.3% - 15 cases - vs. 14.6% - 7 cases). Moreover, in the group of requests without an identifiable CO source, bluish livor was more frequent (61.9%, corresponding to 13 cases) but in 6 (28.6%) cases, lividity was cherry-red. Thermal lesions were described in 56.5% (39 cases) of all cases with a postmortem COHb analysis request. Cherry-red coloration of the blood and viscera was found in 21 cases, corresponding to 30.4% of the individuals with a COHb analysis request. This finding was described in 41.7% (20 individuals) of the cases with an identified source of CO. (Table II). In 7 cases (10.1% of all cases with a COHb analysis request), no source of CO, poisoning associated to occupation or suggestive autopsy findings were described. All these cases had a negative postmortem COHb analysis. Francisco Ruas 13 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Table II. Characterization of the cases with COHb analysis request according to the source of CO found at the death scene and corresponding autopsy findings. a Includes fire, charcoal briquettes, burnt wood can, fire pan, defective gas appliance and fireplace. b Refers to the cases in which the lividity was not characteristic, because of skin pigmentation or large extent of thermal lesions. 3.2 Positive analyses 3.2.1 COHb levels according to gender and age Between January 2000 and December 2010, 24 individuals (34.8% of all cases with a postmortem COHb analysis request) had a positive COHb analysis. We verified that in these cases, the lower COHb levels were achieved in the older people (Pearson Correlation Coefficient). Mean % COHb in positive cases was similar in male and female individuals (49 ± 27 vs. 51 ± 16 (SD) % COHb; Student's t-Test for Independent Samples, p=0.952). 3.2.2 COHb levels according to autopsy findings Out of 24 individuals with a positive analysis, 9 cases (corresponding to 37.5%) presented cherry-red lividity (in 2 cases - 8.3%, lividity was bluish and in 13 cases (54.2%), lividity wasn't characterized). We found that in individuals with COHb and characterized lividity, the type of lividity was not dependent on the COHb percentage (Mann-Whitney U Source of CO Analysis requests Autopsy findings Lividity Cherry-red, n (%) Bluish, n (%) Not characterized b, n (%) Thermal lesions n (%) Cherry-red viscera n (%) Soot n (%) Identified a 48 15 (31.3) 7 (14.6) 26 (54.2) 35 (72.9) 20 (41.7) 25 (52.1) Not present 21 6 (28.6) 13 (61.9) 2 (9.5) 4 (19) 1 (4.8) 0 Total 69 21 (30.4) 20 (29) 28 (40.6) 39 (56.5) 21 (30.4) 25 (36.2) Francisco Ruas 14 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Test for Independent Samples, p=0.906). 15 cases (62.5% of all positive requests) presented cherry-red coloration of blood and viscera. We verified that positive cases with cherry-red coloration of blood and viscera had significantly higher COHb levels than the positive cases in which this finding was absent (58 ± 23 (SD) vs. 32 ± 17 (SD) % COHb; Mann-Whitney U Test for Independent Samples, p=0.012). The lowest COHb level in a positive case that presented cherry-red lividity was 3% COHb, which is similar to the lowest COHb level in a positive case that presented cherry-red blood and viscera coloration. The highest level of COHb in a positive case with bluish lividity was 82%. The highest level of COHb in a positive case without cherry-red blood and viscera coloration was 61%. Among the 13 cases with a positive analysis in which lividity wasn't characterized, 6 individuals (46.2%) presented cherry-red blood and viscera coloration. 15 individuals (65.5%) with a positive analysis also presented thermal lesions. In this group, mean % COHb was significantly lower than that of the group of individuals with a positive analysis, without thermal lesions (41 ± 24 vs. 64 ± 18 (SD) % COHb, Mann-Whitney test, p=0.025). 3.2.3 COHb levels according to cause of death Among the positive cases, cause of death was not CO poisoning in 11 cases (45.8% of the positive cases). In this group, 9 individuals (81.8% of the positive cases who did not die due to CO poisoning and 37.5% of all the positive cases) died because of carbonization and 2 individuals (8.3%) because of burns. Mean COHb level in individuals with a positive analysis whose cause of death was not CO poisoning, was of 32 ± 19 (SD) % COHb. The highest COHb level in a positive case whose death was not caused by CO poisoning was 61% COHb. In the group of cases with a positive analysis in a death not caused by CO poisoning, cherry-red lividity was found in only 1 individual (9.1%); 1 individual (9.1%) presented with bluish Francisco Ruas 15 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 lividity and, in the majority of the cases (9 individuals, 81.8%), lividity couldn't be characterized because of the extension of thermal lesions. Also in the group of cases with a positive analysis in a death not caused by CO poisoning, cherry-red blood and viscera coloration was found in 3 individuals (27% of the cases); among the cases with a positive analysis and a cause of death different of CO poisoning in which lividity couldn't be characterized because of thermal lesions, 2 individuals (22.2% of these cases) presented cherry-red blood and viscera coloration. 3.2.4 CO poisonings 3.2.4.1 Year, month and season distribution CO poisoning was the cause of death in 14 cases with a postmortem COHb analysis request between January 2000 and December 2010, representing 20.3% of all cases with an analysis request. This group included 13 cases (92.9% of all CO poisonings) with a positive analysis and 1 case (7.1%) in which the postmortem COHb analysis was negative. Yearly distribution of CO poisonings with a positive analysis is presented in Table III. 2000 was the year with the highest number of fatal CO poisonings with positive COHb analysis (5 cases, 38.8% of all positive CO poisonings). The highest number of positive, fatal CO poisonings, occurred in February, March and May (3 cases in each month, corresponding to 23% of all poisonings with a positive analysis). The remainder occurred, equally, in January, February, April, November and December (1 case in each month, corresponding to 7.7% of the positive poisonings). Winter was the season with the highest number of these cases (7 cases, corresponding to 53.8% of all positive poisonings). 4 cases (30.8%) occurred during spring and 2 (15.8%) during autumn. No cases occurred during summer. Francisco Ruas 16 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 3.2.4.2 Gender, age and occupation As presented in Table III, male individuals account for the great majority of the cases related to CO poisonings with a positive analysis, with 9 individuals (69.2%). Mean age in the group of positive fatal CO poisonings was 42 ± 26 (SD) years. Distribution of positive fatal CO poisonings by age groups is presented in Fig. 4. In 61.5% (8 individuals) of positive CO poisonings, the victim's occupation was unknown. In 2 cases (15.4%), the individual was a pensioner. Student, plumber and waitress accounted for the remaining 3 cases (1 case each). Fig. 4. Distribution of the fatal CO poisonings with positive COHb analysis, by age group. Table III. Yearly distribution of the CO poisonings with positive COHb analysis and corresponding characterization according to gender and etiology of death. Year Positive CO poisonings Gender Male, n (%) Female, n(%) Etiology Violent, unintentional, n (%) Suicide, n (%) 2000 5 4 (80) 1 (20) 5 (100) 0 2001 3 1 (33.3) 2 (66.7) 2 (66.7) 1 (33.3) 2003 1 1 (100) 0 0 1 2008 2010 3 1 3 (100) 0 0 1(100) 1 (33.3) 1 (100) 2 (66.7) 0 Total 13 9 (69.2) 4 (30.8) 9 (69.2) 4 (30.8) 0 1 2 3 4 Positive fatal CO poisonings Age group Number of cases Francisco Ruas 17 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 3.2.4.3 Etiology and source of CO 69.2% (9 cases) of the positive CO poisonings were violent, unintentional deaths. The remainders were suicides (4 cases, corresponding to 30.8% of positive CO poisonings). Considering our population, CO poisoning was the cause of death in all 4 suicides that presented with positive levels of COHb in a postmortem blood sample. The most common CO source among the positive CO poisonings were the fires (5 cases, corresponding to 38.5%). In 3 cases (23.1%), a defective gas appliance was found at the death scene. In 2 cases (15.4%), the sources of CO were burning charcoal briquettes in a confined space. A fireplace (1 case, 7.7%), a can containing burnt wood (1 case, 7.7%) and a fire pan (1 case, 7.7%) were also found among these cases. Source of CO was different in all 4 CO poisoning suicides, having been identified 1 defective gas equipment, 1 can containing burnt wood, 1 fire pan and, in 1 case, charcoal briquettes. 3.2.4.4 Autopsy findings Cherry-red lividity was present in 8 of 13 (61.5%) positive fatal CO poisonings; in 4 (30.8%), lividity wasn't characterized because of skin pigmentation or thermal lesions; in 1 (7.7%) case, lividity was bluish. Nonetheless, all the 13 positive fatal CO poisonings presented cherry-red blood and viscera coloration. 3.2.4.5 Toxicological results Mean % COHb in the cases of individuals with positive COHb analysis who died because of CO poisoning was 65 ± 18 (SD) %. Older individuals with a positive fatal CO poisoning presented lower levels of COHb % (Pearson Correlation Coefficient). Also, COHb levels were significantly higher among positive fatal CO poisonings of suicidal etiology than in violent, unintentional, positive fatal CO poisonings (78 ± 17 vs. 58 ± 15 (SD) % COHb; Francisco Ruas 18 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Kruskal-Wallis Test for Independent Samples , p=0.013). The lowest COHb level found in a CO poisoning with a positive analysis was 42% and 4 (30.8% of the cases) out of the 13 individuals who died due to CO poisoning which presented COHb levels below 50%. Medical history was unknown in these 4 cases and only 1 (25%) presented thermal lesions. None of these 4 individuals was alcohol intoxicated. Analysis for illicit drugs was negative in 1 of these cases (25%) and wasn't performed in 3 cases (75%). The same results were verified for prescription drugs analysis. Moreover, we found that all the 11 positive CO poisonings in which blood alcohol was tested (84.6% of all positive CO poisonings), were not alcohol intoxicated. Illicit drugs were analyzed in 5 cases (38.5%), all with negative results. In all 4 cases with a prescription drug analysis (30.8%), results were negative. 3.3 Negative analyses Between January 2000 and December 2010, 45 individuals (65.2% of all requests) had a negative postmortem COHb analysis. This group included 26 males (57.8%) and 19 females (42.2%).The mean age among these individuals was 60 ± 19 (SD) years. Medico-legal etiology was of natural death in 10 cases (22.2%); violent, unintentional death in 31 cases (68.9%); suicide in 2 cases (4.4%) and homicide in 2 cases (4.4%). Politrauma was the most frequent cause of death among the negative cases (11 cases, 24.4%). Among natural causes of death, acute myocardial infarction was the most frequent, with 4 cases (8.8% of all negatives). In 1 case (2.2%) of an individual with a negative analysis request, CO poisoning was the determined cause of death. This former case, an accident involving a fire, presented cherry-red blood and viscera coloration, but bluish lividity and no thermal lesions. Francisco Ruas 19 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Fig. 5. Cause of death in males and females with a negative COHb analysis. a AMI, acute myocardial infarction. 4. Discussion and conclusion In Portugal, incidence of CO poisoning is 5.86/100000, representing 0.04 and 0.05% of internal medicine admissions in rural and urban areas, respectively . A surveys of fatal cases registered after the fifties was carried out in our country but it also studied the remaining intoxications, thus presenting few conclusions on the characterization of the CO-related deaths analyzed. Moreover, in international publications, studies of this nature reviewed only deaths in which CO poisoning was the cause of death, thus leaving uncharacterized the cases in which there was an unconfirmed suspicion of CO poisoning. Frequently, the population was a specific group of fatal CO-related deaths, selected according to etiology and source of CO [6, 7, 18, 19, 20]. Also, characteristic autopsy findings and their relevance in the differential diagnosis of CO poisoning as a cause of death have only rarely 0 1 2 3 4 5 6 7 8 9 10 Male Female Number of cases Cause of death Francisco Ruas 20 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 been studied in surveys of fatal CO poisonings . Instead, this subject has been addressed in case reports of peculiar CO-related deaths [11, 13, 21] or publications of other natures . By including in our study, not only the deaths caused by CO poisoning, but also those in which the suspicion of CO poisoning was not confirmed (cases with a positive analysis but a different cause of death and cases with a negative analysis), we analyzed the problem of CO poisoning as a cause of death and differential diagnosis in the forensic practice through a broader perspective than that of studies of this nature previously made. The lack of national publications regarding the characterization of fatal CO poisonings and the fact that our study design is unparalleled by similar international publications make our results difficult to discuss taking all variables studied into account but some aspects in these study allow us to point out trends in fatal CO poisoning over the last 11 years and also present, in one hand, a number of circumstantial and autopsy findings that lead us, in our practice, to a CO poisoning suspicion and subsequent COHb analysis request and, on the other hand, the major differential diagnosis of this type of intoxication, among the analyzed cases. We observed that the number of COHb analysis requests peaked in 2000 and 2005 which have been, interestingly, the years with the highest number of forest fires in Portugal, during the decade. In our study, the source of CO most frequently found at the death scene was indeed a fire but the highest number of requests was made in March and, seasonally, during winter, when forest fires are less frequent. Thus, our yearly, seasonal and monthly distribution of COHb analysis requests is probably more directly related to the prevalence of urban fires rather than that of forest fires. Our results are in accordance with the generalized idea that CO poisoning is more frequent during winter months, because of the increased use of heating appliances and diminished house ventilation , confirmed according to our national reality by Sá et al. (2011) . The above considerations also explain the seasonal Francisco Ruas 21 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 distribution of fatal CO poisonings which were, in our study, more frequent during winter months and non-existent during summer, results paralleled by those of other authors [6, 7, 8, 19, 23]. The yearly distribution of COHb requests was not paralleled by the distribution of fatal CO poisoning cases, which were more prevalent in 2000 and never reached so high numbers through the remaining of the decade. Pinho Marques et al. (2002) reported 42 fatal CO poisonings between 1990 and 1999, the decade with the highest number of cases since the fifties, following the overall increase in the number of all intoxications registered . The comparative decrease in the number of fatal CO poisonings between 2000 and 2010 was expected, due to improved treatment (namely the hyperbaric chamber in Matosinhos) , the opening of new medico-legal facilities and the impact of these cases in national public media and prevention campaigns directed to the population, which could also explain that years with more CO poisonings were followed by years with no CO poisonings (Table III). Other authors reported that fatal CO poisonings are more prevalent among males [7, 18, 24], an observation also supported by our own results (Table III). It has been suggested by Nazari et al that men engage in high-risk behaviors such as manual work with fuel-burning tools or appliances and their maintenance, more often than women . In our study, fire was the most common source of CO (Table I) and, according to Blanc and Kuschner (2007), more than one third of all firemen use protection masks less than half the time, during a fire suppression . Moreover, in Portugal, firemen still are mainly male individuals. However, occupation was unknown in the majority of CO poisonings, which makes it difficult to understand the role of occupation in gender differences. The age group with the highest number of cases was between 51 and 60 years-old (Fig. 4), while most authors reported an higher prevalence among the youngest and the eldest [7, 18]. These studies considered unintentional CO-related deaths only and therefore, this Francisco Ruas 22 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 difference may be explained by the particular age distribution of CO-related suicides, described by Risser and Schneider (1995) . Moreover, we found that other age groups, namely those of children (<10), the young (21-30), middle aged (41-50) and the eldest (>70) are also at risk (Fig. 4). The young, despite having higher body resistance, are less careful and preventive and the elderly are at risk because of their limited mobility, poorer economic situation, likelihood of mistaking CO poisoning for fatigue or influenza-like illness, and comorbility . Children are more susceptible to CO poisoning because they have an higher metabolic rate and oxygen uptake . The majority of requests for a COHb analysis were made for deaths of violent, unintentional nature (Table I) and, not unexpectedly, we verified that CO poisonings were mainly unintentional (Table III), which is in accordance to the literature [6, 8, 19]. Interestingly, in a five year study in East Denmark, Theilade (1990) reported an higher relative number of suicides, which is possibly related to the high incidence of suicide in countries of northern Europe . In our study, a request was made in 6 suicides, representing 8.7% of all requests (Table I), an expected percentage considering that Sá et al. (2011) found that, in seven hospitals of northern Portugal, the prevalence of CO-related intoxications resulting from attempted suicide was 10.8% . On the other hand, our percentage of CO-related suicides was 30.8% (Table III). CO intoxications of suicidal nature are more frequently fatal because, as verified in our study and by Risser and Schneider (1995), their COHb levels are usually higher and thereby it is not surprising that the prevalence of suicidal cases is higher when considering fatal cases only. However, our percentage of suicides is also higher than that of other studies of fatal cases [6, 8, 19]. We believe that may be due to the efficiency of prevention measures in reducing the number of unintentional deaths, rather than to an absolute increase in CO-related suicides. Francisco Ruas 23 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 In our study, the source of CO was a heating appliance in all CO-related suicides. Unlike other studies [6, 24], we registered no suicides by inhalation of car exhaust fumes. Studies such as the one of Risser and Schneider (1995), in which this was the most frequent suicide method, refer to the late eighties and early nineties, when catalytic converters were still in development . Most unintentional CO poisonings resulted from fires, as other authors also reported [7, 24]. This result emphasizes, first, the efficiency of safety improvements in heating appliances and public awareness and, secondly, the necessity to further invest in fire prevention, namely through the mass generalization of smoke detector installation in Portuguese homes. In fact, some authors have reported that a working smoke detector reduces the risk of death by fire in the home at least 50% . We found that in most requests, information that a CO source had been identified at the death scene was available and out of these cases, most did not present with characteristic autopsy findings that could suggest the hypothesis of CO poisoning, to the medico-legal expert (Table II), thus confirming the presence of a CO source in the death scene as a putative motivation to request a COHb analysis and the overall importance of circumstantial information in forensic toxicology, as debated by Reys and Santos (1992) . These authors also reported autopsy findings to be a decisive factor for a toxicological analysis request in an important number of cases . Accordingly, we verified that nearly one third of the cases with a COHb analysis request presented with characteristic cherry-red lividity and the same percentage of cases presented cherry-red blood and viscera coloration. Also, these findings were present in some cases in which a source of CO was not identified. In over one half of the cases, the victims presented thermal lesions (Table II). These results allow us to conclude that, in our practice, the presence of a source of CO at the death scene and suggestive autopsy findings concur to a strong suspicion of CO poisoning. The absence of Francisco Ruas 24 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 cherry-red lividity is not, however, conclusive evidence that CO poisoning was not the cause of death. In fact, cherry-red livor was present in only 61,5% of CO poisonings, a percentage much lower than that of Risser et al (1995) . Also unlike the former authors, we found no association between COHb levels and the lividity type. A possible explanation could be that we also considered CO-poisonings caused by fire and therefore, our cases included those in which lividity was not characterizable due to external thermal lesions but that could, otherwise, have presented cherry-red lividity. This is supported by the fact that in our study, cherry-red blood and viscera coloration, not influenced by the above factors, was present in all CO poisonings, including those without cherry-red lividity, and had a strong association with higher COHb levels. On the other hand, we found that cases with a positive analysis and thermal lesions were strongly associated with lower COHb levels, as also reported in other studies [20, 24, 27]. It is known that burns, instead of CO poisoning, is the cause of death in at least thirty percent of fire victims . However, fire victims that were alive when the fire started will likely present positive COHb levels and even in nonfatal concentrations, CO may contribute to death. In fact, death by fire frequently results of a combinations of these causes . Among the CO poisonings in our study, we observed a mean % COHb of 65%, which is in conformity with the widely accepted threshold for fatal cases, 50 or 60% . Older individuals presented lower COHb levels and this was also reported by Risser and Schneider (1995) . It is known that the elderly with underlying cardiovascular disease are less tolerant to CO pathogenicity [6, 20] and this could explain our 4 cases in which CO poisoning was the cause of death but presented with COHb levels under 50%. Disappointingly, no medical history was available in all cases. Blood alcohol, prescription and illicit drug analyses were negative in all CO poisoning cases in which they were performed. These substances have been related to an increased risk Francisco Ruas 25 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 of CO-related accidents, because they impair the decision making capability, and to individuals who ingest large quantities of drugs before committing suicide through CO exposure [20, 28]. On the other hand, Przepyszny and Jenkins (2007), found that the presence of drugs did not change the cause of death in any of their cases . It is not surprising that politrauma was the most common cause of death among our negative cases (Fig. 5). As previously discussed, circumstantial information is a decisive motivation to request a toxicological analysis and many of the fire victims in our study were involved in car accidents and explosions, situations that resulted in fatal craneoencephalic or toracoabdominal lesions but in which CO could have played an important role as the cause of death. The same applies to deaths due to thermal lesions, also an important differential diagnosis among unintentional deaths (Fig. 5). Differential diagnosis with natural death and particularly with cardiovascular disease is also a reality in our forensic practice (Fig. 5). As discussed before, the elderly, more prone to severe cardiovascular disease, are also a group at risk for CO poisoning and cases in which, due to unspecific circumstancial information, absence of cherry-red livor and presence of arteriosclerosis and coronary arteries calcification, differential diagnosis was not straightforward, have been reported by other authors . The fact that delayed death by CO poisoning is consistent with negligible COHb levels may impose further challenges . In summary, we have demonstrated that, in our practice, forensic aspects of CO poisoning interact in a complex way. Establishing CO poisoning as a cause of death thus requires a strong suspicion, based on suggestive circumstantial information and autopsy findings, and levels of COHb higher than 50%. Even when these factors are reunited, differential diagnosis is not straightforward. This study also emphasizes the role played by public prevention campaigns and improvement of heating appliances in reducing the number Francisco Ruas 26 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 of unintentional CO poisonings and the importance of encouraging further prevention measures regarding urban and forest fires, still the major source of CO among us. Francisco Ruas 27 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 AGRADECIMENTOS Agradeço ao Senhor Prof. Doutor Duarte Nuno Vieira, Professor da Faculdade de Medicina da Universidade de Coimbra e Presidente do Instituto Nacional de Medicina Legal, I.P. por, desde o primeiro momento, me ter transmitido o seu interesse por este trabalho, na forma de cordial disponibilidade para nele colaborar. Ao Senhor Prof. Doutor Francisco Corte-Real, Professor da Faculdade de Medicina da Universidade de Coimbra e Diretor da Delegação do Centro do INML, I.P. e co-orientador desta investigação, pela amabilidade com que me recebeu em todas as ocasiões e genuína solicitude perante as exigências e dificuldades surgidas durante este percurso. Ao Serviço de Patologia Forense da Delegação do Centro do INML, I.P. e ao Serviço de Toxicologia Forense da Delegação do Centro do INML, I.P., pelos dados gentilmente disponibilizados e pela forma profissional e diligente como fui recebido. À Senhora Doutora Bárbara Oliveiros, investigadora do Instituto de Biofísica e Biomatemática da Faculdade de Medicina da Universidade de Coimbra - IBILI, pela sua amável colaboração na análise estatística dos dados obtidos. À Senhora Doutora Sandra de Almeida Curado, Técnica Superior de Biblioteca da Biblioteca da Delegação do Centro do INML, I.P., pelo suporte bibliográfico a este trabalho. Ainda e sempre, à Senhora Prof. Doutora Helena Teixeira, Professora da Faculdade de Medicina da Universidade de Coimbra, Diretora do Serviço de Toxicologia Forense da Delegação do Norte do INML, I.P. e orientadora desta tese, pela forma dedicada como abraçou este projeto, em cujo sucesso sempre acreditou e por ter estado presente em todos os momentos deste trajeto conjunto. No final, como em cada ocasião, ter sido seu orientando deixa um sentimento de privilégio. A todos, o meu apreço e gratidão. Francisco Ruas 28 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 References Jaffe F (1997) Pathogenicity of Carbon Monoxide. Am J Forensic Med Pathol 18: 406-410. Gisbert Calabuig JA (2004) Monóxido de Carbono. In: Villanueva Canadas E; Medicina legal y toxicología. 6ª edición; Masson. Leon D, and Rossitza I (2007) Carbon monoxide intoxication: an updated review. J. Neurol. Sci. 262: 122-230. Kao LW, and Nanagas KA (2005) Carbon Monoxide Poisoning. Med Clin N Am 89: 1161-1194. Raub J, Nolf M, and Hampson N (2000) Carbon monoxide poisoning--a public health perspective. Toxicology 145: 141-114. Risser D, and Schneider B (1995) Carbon Monoxide-Related Deaths from 1984 to 1993 in Vienna, Austria. J. Forensic Sci. 40(3): 368-371. Homer C, Engelhart D, and Lavins E (2005) Carbon monoxide-related deaths in a metropolitan county in the USA: an 11-year study. Forensic Sci. Int. 149; 159–165. Ait El Cadi M, Khabbal Y, and Idrissi L (2009) Carbon monoxide poisoning in Morocco during 1999-2007. J Forensic Leg Med 16: 385-387. DiMaio VJ, and DiMaio D (2001) Forensic Pathology. Second Edition; CRC Press. Risser D, and Schneider B (1995) Should Coroners Be Able to Recognize Unintentional Carbon Monoxide-Related Deaths Immediately at the Death Scene? J. Forensic Sci. 40(4): 596-598. Ruszkiewicz A, de Boer B, and Robertson S (1997) Unusual Presentation of Death Due to Carbon Monoxide Poisoning. A Report of Two Cases. Am J Forensic Med Pathol 18(12): 181-184. Francisco Ruas 29 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Dinis-Oliveira R, Carvalho F, and Magalhães T (2010) Postmortem changes in carbon monoxide poisoning. Clin Toxicol 48: 762-763. Carson H, and Esslinger K (2001) Carbon Monoxide Poisoning Without Cherry-Red Livor. Am J Forensic Med Pathol 22(23): 233-235. Bohnert M, Weinmann W, and Stefan P (1999) Spectrophotometric evaluation of postmortem lividity. Forensic Sci. Int. 99: 149-158. Parks J, and Worth H (1985) Carboxyhemoglobin Determination by Second-Derivative Spectroscopy. Clin Chem 31/32, 279-281. Sá M, Rodrigues R, and Moura D (2011) Internamentos por intoxicação com monóxido de carbono em Portugal. Acta Med Port 24: 727-734. Pinho Marques E, Teixeira H, Proença P, Oliveira M, Castanheira A, Barroso M, Margalho C, Ávila S, and Vieira D (2002) Étude rétrospective, depuis les annés 50, des toxiques detectés dans des échantillons biologiques humains, de cadavres autopsiés dans les chefs-slieux appartenant à la corconscription médico-légale de Coimbra. Temas de Medicina Legal, Centro de Estudos de Pósgraduação em Medicina Legal, DN Vieira, A Rebelo, CCordeiro Coord 407–413. Girman J, Chang Y-L, and Liu K-S (1998) Causes of unintentional deaths from carbon monoxide poisonings in California. West J Med 168: 158-165. Sheikhazadi A, Anary S, and Ghadyani M (2010) Nonfire Carbon Monoxide-Related Deaths: A Survey in Tehran, Iran (2002-2006). Am J Forensic Med Pathol 31: 359-363. Popovic V, Atanasijevic T, Nikolic S, and Micic J (2009) Concentration of carbon-monoxide in carbonized bodies - Forensic aspects. Leg Med 11: 318-320. Opeskin K, and Drummer O (1994) Delayed Death Following Carbon Monoxide Poisoning: A Case Report. Am J Forensic Med Pathol 15(11): 36-39. Francisco Ruas 30 Carbon Monoxide Poisoning as a Cause of Death and Differential Diagnosis A Retrospective Study, 2000 to 2010 Wilson R, Saunders P, and Smith G (1998) An epidemiological study of acute carbon monoxide poisoning in the West Midlands. Occup Environ Med 55: 723-728. Nazari J, Dianat I, and Stedmon A (2010) Unintentional Carbon Monoxide Poisoning in Northwest Iran: A 5-year study. J Forensic Leg Med 17: 388-391. Theilade P (1990) Carbon Monoxide Poisoning: Five Years' Experience in a Defined Population. Am J Forensic Med Pathol 11: 219-225. Blanc P, and Kuschner W (2007) Gases & Other Airborne Toxicants. In: Ladou, J.; Current: Occupational & Environmental Medicine; Fourth edition, McGraw Hill. Reys L, and Santos J (1992) Importance of information in forensic toxicology. Am J Forensic Med Pathol 13(11): 33-16. Teige B, Lundevall J, and Fleischer E (1977) Carboxyhemoglobin Concentrations in Fire Victims and in Cases of Fatal Carbon Monoxide Poisoning. Z Rechtsmedizin 80: 17-21. Przepyszny L, and Jenkins A (2007) The Prevalence of Drugs in Carbon Monoxide-Related Deaths: A Retrospective Study, 2000-2003. Am J Forensic Med Pathol 28: 242-248. |
11701 | https://www.tiger-algebra.com/drill/k2%E2%88%9216k=-4/ | Copyright Ⓒ 2013-2025
tiger-algebra.com
This site is best viewed with Javascript. If you are unable to turn on Javascript, please click here.
Solution - Quadratic equations
Other Ways to Solve
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "k2" was replaced by "k^2".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
k^2-16k-(-4)=0
Step by step solution :
Step 1 :
Trying to factor by splitting the middle term
1.1 Factoring k2-16k+4
The first term is, k2 its coefficient is 1 .
The middle term is, -16k its coefficient is -16 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -16 .
| | | | | | | |
--- --- ---
| | -4 | + | -1 | = | -5 | |
| | -2 | + | -2 | = | -4 | |
| | -1 | + | -4 | = | -5 | |
| | 1 | + | 4 | = | 5 | |
| | 2 | + | 2 | = | 4 | |
| | 4 | + | 1 | = | 5 | |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 1 :
Step 2 :
Parabola, Finding the Vertex :
2.1 Find the Vertex of y = k2-16k+4
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ak2+Bk+C,the k -coordinate of the vertex is given by -B/(2A) . In our case the k coordinate is 8.0000
Plugging into the parabola formula 8.0000 for k we can calculate the y -coordinate :
y = 1.0 8.00 8.00 - 16.0 8.00 + 4.0
or y = -60.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = k2-16k+4
Axis of Symmetry (dashed) {k}={ 8.00}
Vertex at {k,y} = { 8.00,-60.00}
k -Intercepts (Roots) :
Root 1 at {k,y} = { 0.25, 0.00}
Root 2 at {k,y} = {15.75, 0.00}
Solve Quadratic Equation by Completing The Square
2.2 Solving k2-16k+4 = 0 by Completing The Square .
Subtract 4 from both side of the equation :
k2-16k = -4
Now the clever bit: Take the coefficient of k , which is 16 , divide by two, giving 8 , and finally square it giving 64
Add 64 to both sides of the equation :
On the right hand side we have :
-4 + 64 or, (-4/1)+(64/1)
The common denominator of the two fractions is 1 Adding (-4/1)+(64/1) gives 60/1
So adding to both sides we finally get :
k2-16k+64 = 60
Adding 64 has completed the left hand side into a perfect square :
k2-16k+64 =
(k-8) • (k-8) =
(k-8)2
Things which are equal to the same thing are also equal to one another. Since
k2-16k+64 = 60 and
k2-16k+64 = (k-8)2
then, according to the law of transitivity,
(k-8)2 = 60
We'll refer to this Equation as Eq. #2.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(k-8)2 is
(k-8)2/2 =
(k-8)1 =
k-8
Now, applying the Square Root Principle to Eq. #2.2.1 we get:
k-8 = √ 60
Add 8 to both sides to obtain:
k = 8 + √ 60
Since a square root has two values, one positive and the other negative
k2 - 16k + 4 = 0
has two solutions:
k = 8 + √ 60
or
k = 8 - √ 60
Solve Quadratic Equation using the Quadratic Formula
2.3 Solving k2-16k+4 = 0 by the Quadratic Formula .
According to the Quadratic Formula, k , the solution for Ak2+Bk+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
k = ————————
2A
In our case, A = 1
B = -16
C = 4
Accordingly, B2 - 4AC =
256 - 16 =
240
Applying the quadratic formula :
16 ± √ 240
k = ——————
2
Can √ 240 be simplified ?
Yes! The prime factorization of 240 is
2•2•2•2•3•5
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 240 = √ 2•2•2•2•3•5 =2•2•√ 15 =
± 4 • √ 15
√ 15 , rounded to 4 decimal digits, is 3.8730
So now we are looking at:
k = ( 16 ± 4 • 3.873 ) / 2
Two real solutions:
k =(16+√240)/2=8+2√ 15 = 15.746
or:
k =(16-√240)/2=8-2√ 15 = 0.254
Two solutions were found :
How did we do?
Why learn this
Terms and topics
Related links
Latest Related Drills Solved
Copyright Ⓒ 2013-2025
tiger-algebra.com |
11702 | https://askfilo.com/user-question-answers-smart-solutions/calculate-the-fourier-number-fo-for-the-rod-in-problem-21-33-3333373131373937 | Question asked by Filo student
Calculate the Fourier number, Fo =αt/ro2, for the rod in problem 21.33. If Fo >0.2, use the one-term approximation for the long cylinder in a trial-and-error procedure to solve Problem 21.33 .
Views: 5,506 students
Updated on: May 13, 2025
Filo tutor solution
Learn from their 1-to-1 discussion with Filo tutors.
Students who ask this question also asked
Views: 5,352
Topic:
Smart Solutions
View solution
Views: 5,818
Topic:
Smart Solutions
View solution
Views: 5,447
Topic:
Smart Solutions
View solution
Views: 5,003
Topic:
Smart Solutions
View solution
| | |
--- |
| Question Text | Calculate the Fourier number, Fo =αt/ro2, for the rod in problem 21.33. If Fo >0.2, use the one-term approximation for the long cylinder in a trial-and-error procedure to solve Problem 21.33 . |
| Updated On | May 13, 2025 |
| Topic | All topics |
| Subject | Smart Solutions |
| Class | Grade 11 |
Are you ready to take control of your learning?
Download Filo and start learning with your favorite tutors right away!
Questions from top courses
Explore Tutors by Cities
Blog
Knowledge
© Copyright Filo EdTech INC. 2025 |
11703 | https://www.physicsforums.com/threads/atwood-machine-deriving-equation-for-acceleration.830772/ | Atwood Machine, deriving equation for acceleration • Physics Forums
Insights Blog-- Browse All Articles --Physics ArticlesPhysics TutorialsPhysics GuidesPhysics FAQMath ArticlesMath TutorialsMath GuidesMath FAQEducation ArticlesEducation GuidesBio/Chem ArticlesTechnology GuidesComputer Science Tutorials
ForumsIntro Physics Homework HelpAdvanced Physics Homework HelpPrecalculus Homework HelpCalculus Homework HelpBio/Chem Homework HelpEngineering Homework Help
Trending
Log inRegister
What's new
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Style variation
SystemLightDark
Contact us
Close Menu
Forums
Homework Help
Introductory Physics Homework Help
Atwood Machine, deriving equation for acceleration
Thread starter mmoe
Start date Sep 3, 2015
TagsAccelerationAtwoodAtwood machinederivingMachine
AI Thread Summary
The discussion focuses on understanding the dynamics of an Atwood Machine, specifically deriving the equations for the acceleration of the masses involved. The user is attempting to show that the acceleration of the movable pulleys is the average of the accelerations of the two lower masses and is seeking guidance on how to proceed with the calculations. They have correctly identified relationships between the distances and accelerations but express uncertainty about the validity of their equations and how to combine them effectively. Clarification is requested on the meaning of certain variables and the reasoning behind the derived equations. The conversation highlights the complexities of applying Newton's laws and conservation principles in solving problems related to Atwood Machines.
Sep 3, 2015
1
mmoe
1 0
Homework Statement
I still don't got the hang on Atwood Machines, and i don't know if I am wrong here or just overthinking it. But i got this one exercise where i am kind of stuck in two of the sub-exercises. So I am hoping there is someone here who could guide me in the right direction.Atwood Machine ( ), with massless strings and pulleys and no friction.
Let the acceleration of the top mass be a 1 and the acceleration for the bottom two a 2 and a 3. Let the movable pulleys have the acceleration a tr.
a) Show that a tr = (a 2 + a 3)/2
b) find the masses' acceleration
Homework Equations
F=ma
The Attempt at a Solution
a) I can see that if i move the top mass a distance y, conservation of string tells me that the movable pulleys will move 1/2y.
y m1 = 1/2y m2
y m1 = 1/2y 2m
y m2 = y 2m
Hence y m1 = (y m2 + y 2m)/2
Two time derivatives gives a 1 = (a 2 + a 3)/2 = a tr
It gives the right answer, but i got a feeling that I've done something that should be quite obvious wrong. b) I find the F=ma for each mass
T-mg = ma 1 (1)
T-mg = ma 2 (2)
t-2mg = ma 3 (3)
i also got these equations from the conservation of string:
a 1=1/2a 2 (4) && a 1=1/2a 3 (5) && a 2=a 3 (6)
Combining (4) and (5) with equations (1),(2) & (3), should give me a 1,a 2,a 3 and T.
The thing is, i really don't know where to start here. Since a2 = a3, you only have to solve for a1 and a2 (correct me if I am wrong). But how would i go forward solving (1) and (2) with (4). I don't know if I am overthinking this way to much or if i have done something wrong, so i come to you kindly asking for directions.
Thanks :)
Discover more
Scientist Networking Platform
Physics Problem Solving
Online tutoring services
Math FAQ
Quantum Mechanics Course
Math
science
Physics Textbooks
Scientific calculators
Physics Articles
Physics news on Phys.org
New adaptive optics system promises sharper gravitational-wave observations
Physics-informed AI learns local rules behind flocking and collective motion behaviors
New perspectives on light-matter interaction: How virtual charges influence material responses
Sep 4, 2015
2
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2024 Award
42,655 10,435
I don't understand some of your equations. What is saying? Are these positions or velocities or accelerations? Why would it be true?
Thread 'I've come across another fluid pressure problem I don't understand'
Neglect gravity other than that of water; neglect friction. F1=ρgsh=1000101(1+4)=50000 N; F1=ρgsh=1000100.14=4000 N; F3=50000-4000=46000 N?
View full post »
By vdance
Friday, 4:31 AM
Replies: 33
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »
By Su6had1p
Friday, 10:41 AM
Replies: 27
Thread 'Minimum mass of a block'
Here we know that if block B is going to move up or just be at the verge of moving up will act downwards and maximum static friction will act downwards Now what im confused by is how will we know " how quickly" block B reaches its maximum static friction value without any numbers, the suggested solution says that when block A is at its maximum extension, then block B will start to move up but with a certain set of values couldn't block A reach...
View full post »
By tellmesomething
Sep 16, 2025
Replies: 22
Similar threads
Why is tension (T) only added to one side of an Atwood Machine?
Oct 31, 2019
Replies 3 Views 2K
Working out the accelerations in a double Atwood pulley system
May 20, 2025
Replies 22 Views 1K
The acceleration of a massless pulley in a double Atwood machine
Aug 4, 2019
Replies 2 Views 2K
Pulley system, calculate acceleration
Jul 16, 2017
Replies 16 Views 5K
Double pulley Atwood machine (with 3 masses)
Apr 22, 2016
2
Replies 97 Views 16K
Question: Solving for Accelerations in an Atwood Machine System
Mar 8, 2020
Replies 16 Views 2K
Solving for Masses in an Atwood Machine
Nov 9, 2023
Replies 8 Views 2K
Atwood's Machine: Solving for Force & Acceleration
Dec 15, 2016
Replies 8 Views 2K
What if check: Am I calculating tension wrong?
Mar 9, 2019
Replies 3 Views 2K
Atwood machine calculation for launching a pebble
Sep 23, 2020
Replies 7 Views 1K
Share:
BlueskyLinkedInShare
Forums
Homework Help
Introductory Physics Homework Help
Hot Threads
M
Rope between inclines with maximum length of hanging middle portion
Started by Meden Agan
May 29, 2025
Replies: 46
Introductory Physics Homework Help
M
Thermally insulated container with alternating series of walls
Started by Meden Agan
Jun 1, 2025
Replies: 49
Introductory Physics Homework Help
Confusion regarding surface tension direction and force balance
Started by palaphys
Apr 4, 2025
Replies: 39
Introductory Physics Homework Help
Torque biomechanics
Started by Naucus
Apr 11, 2025
Replies: 6
Introductory Physics Homework Help
Work Done in slowly pulling a thread
Started by Bling Fizikst
May 10, 2025
Replies: 28
Introductory Physics Homework Help
Recent Insights
InsightsQuantum Entanglement is a Kinematic Fact, not a Dynamical Effect
Started by Greg Bernhardt
Sep 2, 2025
Replies: 11
Quantum Physics
InsightsWhat Exactly is Dirac’s Delta Function? - Insight
Started by Greg Bernhardt
Sep 2, 2025
Replies: 3
General Math
InsightsRelativator (Circular Slide-Rule): Simulated with Desmos - Insight
Started by Greg Bernhardt
Sep 2, 2025
Replies: 1
Special and General Relativity
P
InsightsFixing Things Which Can Go Wrong With Complex Numbers
Started by PAllen
Jul 20, 2025
Replies: 7
General Math
F
InsightsFermat's Last Theorem
Started by fresh_42
May 21, 2025
Replies: 105
General Math
F
InsightsWhy Vector Spaces Explain The World: A Historical Perspective
Started by fresh_42
Mar 13, 2025
Replies: 0
General Math
Change width
Contact
About
Terms
Privacy
Help
RSS
2025 © Physics Forums, All Rights Reserved
Back
Top |
11704 | https://www.longislandpress.com/2025/06/26/united-nations-lake-success/ | Flip through the latest issue
Best of LI
Flip through the latest issue
Best of LI
When the world came to Long Island: Lake Success and the birth of the U.N.
By Christy Hinko
The United Nations turns 80 today (June 26, 1945). Its charter, signed by 50 nations on June 26, 1945, established an organization dedicated to fostering peace, security and human rights. Emphasizing the dignity of all individuals, the charter aimed to prevent war, promote social progress, and uphold international law. Since then, the U.N. has grown to 193 member states, playing an essential role in the world. The 80th anniversary of the charter serves as a reminder of its significance and the commitment of nations to work together for the common good.
Never miss a story
The quiet village of Lake Success played an oversized role in global diplomacy during one of the most pivotal moments in world history. Long before the U.N. found its permanent home along Manhattan’s East River, its earliest days unfolded here—in suburban Long Island.
From 1946 to 1951, Lake Success served as the temporary headquarters of the U.N., the fledgling international organization founded in the aftermath of World War II. The global body, established to foster peace, cooperation and human rights, needed an immediate base of operations while a permanent site was being selected and developed. The village’s serene surroundings and proximity to New York City made it a suitable, if unconventional, location.
The U.N. leased office space at the Sperry Gyroscope Company plant on Marcus Avenue, a sprawling defense industry complex that had expanded rapidly during the war. The building was retrofitted to accommodate international diplomats, secretariat staff and security personnel. What had been a hub for radar and military navigation systems became the birthplace of many formative U.N. debates and resolutions.
Over the next five years, Lake Success witnessed the shaping of key postwar initiatives. The Security Council and the General Assembly held meetings there, establishing early protocols for international peacekeeping, refugee support and nuclear disarmament. It was from Lake Success that the U.N. confronted some of the earliest crises of the Cold War, including tensions in Korea, Palestine and Berlin.
The village, previously known mostly for its private estates and golf clubs, adjusted to its new role on the world stage. Delegates from across the globe filled nearby hotels and homes. Local residents and business owners occasionally crossed paths with diplomats from as far as China, France and India. For a few brief years, the streets of Lake Success were an unlikely but vital center of international affairs.
The arrangement was always intended to be temporary. In 1947, the U.N. selected a site along the East River in Manhattan for its permanent headquarters, thanks in large part to a land donation by John D. Rockefeller, Jr. Construction began shortly afterward and by 1952, the Secretariat building was completed and operations fully relocated to New York City.
Though its moment in the diplomatic spotlight was brief, Lake Success holds a unique distinction as the first operational home of the U.N. The Sperry building, now part of a medical complex, stands as a quiet monument to that historic era.
About the Author
More Great Neck News
Jobs on Long Island
Add your job
View all jobs…
Things to do on Long Island
Post an Event
Long Island History Hunt Long Island
Cruise the Country Side: Wine, Orchards & Craft Beer by Bike East End Bike Tours
“EXPLORATIONS” Marilyn Stevenson Solo Exhibit The Water Mill Museum
Disenchanted! The hit musical Hardscrabble Theater
Podcasts for Beginners Quogue Library Inc
Line Dancing Lucky Strike Bar and Lounge at Port Jeff Bowl
Acoustic Jam Session Connetquot Public Library
Chowdah Chowdown 2025 – benefiting Springs Food Pantry Springs Tavern and Grill
View All Events…
BoLI Spotlight
Entertainment
Latest News
More from our Sister Sites
Ask Beatty: Geraldo Rivera, Sometimes Sorry Just Isn’t Good Enough
NYC Mayor’s Race: Cuomo, Mamdani and Sliwa charge ahead after Eric Adams bows out of contest
«La India», la Princesa de la Salsa, en concierto por primera vez en Long Island
Sag Harbor Home of Famed Filmmaker D.A. Pennebaker Sells in Just One Week |
11705 | https://www.slideserve.com/arleen/carry-propagate-adder | PPT - CARRY PROPAGATE ADDER PowerPoint Presentation, free download - ID:144495
Browse
Recent Presentations
Recent Articles
Content Topics
Updated Contents
Featured Contents
PowerPoint Templates
Create
Presentation
Article
Survey
Quiz
Lead-form
E-Book
Presentation Creator
Pro
Upload
Download Presentation
Discover more
PowerPoint software
Buy presentation templates online
Purchase premium content hosting
Visual content tools
Find survey creation software
E-book creator
Download
1 / 9
Download Download Presentation >>
Discover more
PowerPoint templates
Cloud storage solutions
Infographic creator tool
Professional presentation design services
Graphic design software
Webinar hosting platforms
Presentation sharing service
Discover more
Online learning platforms
Video sharing service
PowerPoint templates
CARRY PROPAGATE ADDER
Sep 05, 2011
• 280 likes • 2.42k Views
CARRY PROPAGATE ADDER. AMIT HINGHER Computational Engineering. Basic Principle of a CPA. Adds two n-bit operands A = (a n-1 ..a 0 ), B=(b n-1 ..b 0 ) and an optional carry-in c in by performing carry propagation
Share Presentation
Embed Code
Link
Download Presentation
arleen + Follow
Download Presentation
CARRY PROPAGATE ADDER
An Image/Link below is provided (as is) to download presentationDownload Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.Content is provided to you AS IS for your information and personal use only. Download presentation by click this link.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.During download, if you can't get a presentation, the file might be deleted by the publisher.
Discover more
Presentation software
Digital marketing tools
Get professional document hosting
Purchase premium content hosting
Get lead form generation tools
Presentation templates
Hire presentation design services
Presentation tools
E N D
Presentation Transcript
CARRY PROPAGATE ADDER AMIT HINGHER Computational Engineering
Basic Principle of a CPA.. • Adds two n-bit operands A = (an-1..a0), B=(bn-1..b0) and an optional carry-in cin by performing carry propagation • Can be implemented as a combinational circuit using n full adders called the Ripple Carry Adder
ARCHITECTURE a) Linear Structureb)Tree Structure
ARITHMETIC EQUATION • 2n cout + S = A + B + cin • 2n cout + Σn-1i=02i si = Σn-1i=0 2iai+ Σn-1i=0 2ibi+ cin = Σn-1i=0 2i (ai + bi) + cin • 2ci+1 + si = ai + bi+ ci ; I = 0,1..n-1 where c0 = cin and cout = cn
LOGICAL EQUATION • gi=ai bi • pi=ai bi • si=pi ci • Ci+1=gi + pi ci ; I = 0,1…n-1 where c0 = cin & cout =cn
Complexity Of CPA !!! • Computation time grows linearly with the operand word length n • Speeding up operation of CPA would require replacement by some faster adder structure • (a) Symbol (b) Ripple Carry implementation of a CPA
CPA(Carry propagate Adder) VsCSA(Carry Save Adder) A four operand adder circuit
Comparison (CPA vs CSA) • The two resulting adder arrays are similar in hardware requirements, logic structure and critical path lengths • Bit arrival time in the CPA is unequal (higher bit arrives later than the lower bits) • Comparatively slow
Why Carry Propagate Adder ? • Performs carry propagation from each bit to higher bit positions • Addition results have to be converted to irredundant integer representation • Does not occupy a significant area of the chip • Less Power Consumption
Discover more
Slide templates
Buy interactive quiz software
Get lead form generation tools
Presentation downloads
Document downloads
Purchase stock presentation images
Related
More by User
#### 4-BIT CARRY LOOK AHEAD ADDER 4-BIT CARRY LOOK AHEAD ADDER. Jaime Cordoba Advisor: Dave Parent May 19, 2004. Agenda. Abstract Introduction Why Simple Theory Back Ground information (Lit Review) Summary of Results Project (Experimental) Details Results Cost Analysis Conclusions. Abstract. 1.78k views • 14 slides
#### Hybrid Conditional Sum/Carry Lookahead Adder. Hybrid Conditional Sum/Carry Lookahead Adder. by Stephen Malchi. Project objective. To design a low power high performance adder Sub micron technology Power is the main concern. Conditional Sum Adder. Generates individual sum and carry bits. 687 views • 13 slides
#### Block Size Optimization in Carry-Skip Adder Block Size Optimization in Carry-Skip Adder. By Lee Hathcock. Carry-Skip Adders. Carry-Skip a quick and dirty method to increase speed over ripple-carry Small hardware / power cost Comparatively large speed increase My approach Use a program written in C++ to generate optimal block sizes. 913 views • 13 slides
#### 4-Bit Arithmetic Logic Unit (ALU) with Carry Look Ahead Adder 4-Bit Arithmetic Logic Unit (ALU) with Carry Look Ahead Adder. Keli Wu Eng Boon Chong Li Li Chun Sum Yeung Advisor: David Parent May 2005. Agenda. Abstract Specifications Introduction ALU Operations Design Flow Design Issues Schematics Lessons Learned Simulation Results 2.22k views • 16 slides
#### 4-bit Carry Look Ahead Adder 4-bit Carry Look Ahead Adder. Samira Sharma Suneera Sharma Advisor: Dave Parent 12/6/04. Agenda. Abstract Introduction Why Simple Theory Summary of Results Project (Experimental) Details Results Cost Analysis Conclusions. Abstract. 4.51k views • 22 slides
#### Carry Save Adder 3.88k views • 8 slides
#### Delays in the ripple carry adder 1. 9. 7. 5. 3. 8. 6. 2. 4. Delays in the ripple carry adder. The diagram below shows a 4-bit adder completely drawn out. This is called a ripple carry adder, because the inputs A 0 , B 0 and CI “ripple” leftwards until CO and S 3 are produced. Ripple carry adders are slow! 1.14k views • 38 slides
#### 4-bit Full-Adder With Ripple Carry 4-bit Full-Adder With Ripple Carry . Adrian Corona Tuessia Ly Ali N. Warriach Advisor: Dave Parent Dec. 6, 2004. Agenda. Abstract Introduction Why Simple Theory Back Ground information (Lit Review) Summary of Results Project (Experimental) Details Results Cost Analysis Conclusions. 1.12k views • 15 slides
#### Half Adder Half Adder . Sec. 3.10 Sec. 4.5, 4.12. Schedule. Test #1: Beginning of March. Outline. Transition from Verilog to Digital Logic Observations Verilog Lesson Application: Half Adder. Observations. Lab submissions are due at the beginning of the next lab, i.e. the following Thursday. 610 views • 27 slides
#### .globl Adder .type Adder, @function Adder: pushq %rbp ////////////////////////////////////////////////////////////////////////// // Adder - code executed by one of the threads ////////////////////////////////////////////////////////////////////////// void Adder(void arg) { int i; for (i = 0; i < LoopCount; i++) { 316 views • 2 slides
#### Carry Skip Adder - with optimization for high performance VLSI(2) SW 2005. Carry Skip Adder - with optimization for high performance. Institute of Applied Microelectronics & Information Technology Kwang Yoal Kim. A 0. B 0. A 1. B 1. A 2. B 2. A 3. B 3. FA. FA. FA. FA. C 0 =C in. C out =C 4. S 0. S 1. S 2. S 3. 505 views • 11 slides
#### Low-Power and Area-Efficient Carry Select Adder on Reconfigurable Hardware Low-Power and Area-Efficient Carry Select Adder on Reconfigurable Hardware. Presented by V.Santhosh kumar , B.Tech ,ECE ,4 th Year, GITAM University . Under the guidance of Mr. M. Raghupathy Assistant Professor Dept. of ECE GITAM University. CONTENTS. ABSTRACT INTRODUCTION 578 views • 23 slides
#### Adder Circuits Adder Circuits. S. Sundar Kumar Iyer. Acknowledgement. Slides taken from which is the web-site of “Digital Integrated Circuit – A Design Perspective” by Rabaey, Chandrakasan, Nicolic. Outline. Background / Basics of Adders Ripple Carry Adder. 953 views • 41 slides
#### Design of an 8-bit Carry-Skip Adder Using Reversible Gates Design of an 8-bit Carry-Skip Adder Using Reversible Gates Vinothini Velusamy, Advisor: Prof. Xingguo Xiong Department of Electrical Engineering, University of Bridgeport, Bridgeport, CT 06604. Abstract. 203 views • 1 slides
Loading...
More Related
#### FOUR BIT CARRY LOOK AHEAD ADDER FOUR BIT CARRY LOOK AHEAD ADDER. SUBMITTED BY: MILAN PATNAIK SHANTHI TENNETI DURGA L NALLARI ADVISOR: PROF. DAVID PARENT DATE : 12-06-2004. Agenda. Abstract Introduction why Theory of operation Project (Experimental) Details Results Cost Analysis Conclusions. Abstract. 518 views • 20 slides
#### Resources for Measurement-Based Quantum Computation Quantum Carry-Lookahead Adder Resources for Measurement-Based Quantum Computation Quantum Carry-Lookahead Adder. Presented in October 21 st , 2008. Agung Trisetyarso Keio University. Abstract. 626 views • 43 slides
#### Adder Adder. Discussion D6.2 Example 17. Full Adder (Appendix I). s i = c i ^ (a i ^ b i ). c i+1 = a i b i + c i (a i ^ b i ). -- Example 17a: 4-bit adder library IEEE; use IEEE.STD_LOGIC_1164. all ; use IEEE.STD_LOGIC_unsigned. all ; entity adder4a is port ( 601 views • 14 slides
#### A 240ps 64b Carry-Lookahead Adder in 90nm CMOS A 240ps 64b Carry-Lookahead Adder in 90nm CMOS. Faezeh Montazeri fmontazeri@ece.ut.ac.ir Advanced VLSI Course Presentation University of Tehran December 2006 Based on : A 240ps 64b Carry-Lookahead Adder in 90nm CMOS Sean Kao, Radu Zlatanovici, Borivoje Nikolić 523 views • 27 slides
#### ADDER, HALF ADDER & FULL ADDER ADDER, HALF ADDER & FULL ADDER. 0 0 1 1 + 0 + 1 + 0 + 1 0 1 1 10. Sum. Carry. Sum. Binary Addition. ADDER. In electronics, an adder is a digital circuit that performs addition of numbers. 4.19k views • 21 slides
#### 4-BIT FAST ADDER (look ahead carry 방식 ) 4-BIT FAST ADDER (look ahead carry 방식 ). 시립인천대학교 전자공학과 960815 정인석 960777 김응기. 순서 1. 4-bit fast adder 개념 설명 2. 4-bit fast adder 의 Diagram 3. P-SPICE 를 이용한 회로도 설계 및 파형 검출 4. Layout 5. 응 용 예 6. 결과분석. 4-Bit fast adder 의 개념 소개. 일반 4bit full-adder 의 Diagram. B3. A2. B2. A1. B1. A0. 1.09k views • 34 slides
#### Optimized Carry Speculative Adder Arithmetic logic units and digital signal processors widely uses adders. It is the most complicated arithmetic circuits in digital electronics. The existing adders suffer from critical path delay, area overhead and power consumption. Speculative adders are designed with variable latency that combines speculation technique along with correction methodology to attain high performance in terms of low area overhead over the existing adders. In speculative adders the sum and carry generation part is separated to reduce the area overhead. Carry Speculative Adder CSPA uses carry predictor circuit to reduce power consumption and to reduce the computational time and it uses error recognition and error correction circuit to find the fault occurred in the partial sum generator and to recover it to get accurate results. B. V. Pavan Kumar | M. Lalitha Bhavani | Y. Himanth "Optimized Carry Speculative Adder" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: Paper URL: 94 views • 4 slides
English
Français
About
Privacy
DMCA
Blog
Contact
© 2025 SlideServe. All rights reserved |
11706 | https://www.yalemedicine.org/conditions/gaucher-disease | Skip to Main Content
Gaucher Disease
Overview
Gaucher disease is a rare, inherited disorder that causes fatty substances to build up in the spleen, liver, and other organs.
Finding a doctor to diagnose Gaucher disease can sometimes be a frustrating experience for patients. They typically go from one specialist to another and get sicker and sicker as the doctors scratch their heads.
At Yale Medicine, we have experts ready to help, and patients come to us from around the country for our expertise in Gaucher disease. It is the most common lyosomal storage disease, a family of inherited metabolic disorders that originate in the lysosomes, or parts of the cell that help with digestion.
“We have an internationally recognized program and a national referral base with incredible models of precision medicine and innovative therapies,” says Yale Medicine’s Pramod K. Mistry, MD, PhD, director of Lyosomal Disease & Inherited Metabolic Liver Disease Program. “People come to us because they have very complicated aspects that no one else can figure out.”
What is Gaucher disease?
Gaucher disease is a rare, inherited disorder where fatty cells build up in areas including the liver, spleen, and bone tissue and marrow. The organs enlarge—sometimes as much as 50 times its normal size for the spleen—and bones are affected, which increases the risk of fracture and severe bone pain (called a “bone crisis”) that requires joint replacement. The disease is caused by the body not having enough of an important enzyme (glucocerebrosidase), which breaks down a fatty chemical (glucocerebroside).
Gaucher disease is divided into three main categories:
Type 1 is the most common form of the disorder in Western countries, constituting approximately 95 percent of patients.
Type 2 is rare and is associated with severe neurological abnormalities and is usually fatal within the first two years of onset.
Type 3 is rare in the United States and Europe, but it is the most common form of the disease around the world. It has the same symptoms as type 1, plus some neurological damage.
Worldwide, Gaucher disease affects 1 in 40,000 people, but its frequency is as high as 1 in 450 people among Jews of Ashkenazi (Eastern European) descent. Although it is present at birth, symptoms can appear at any age.
Research shows that patients with Gaucher disease type 1 may be at increased risk of developing other conditions, usually after age 50. These include:
Parkinson disease
Some cancer types, including multiple myeloma (a blood cancer) and liver cancer
What causes Gaucher disease?
To have Gaucher disease, you must have two mutations of a gene called GBA: one from your mother and one from your father. But you can be a carrier and have just one mutation related to the disorder and not have Gaucher disease.
If both parents are carriers, each pregnancy has a 1 in 4 chance that the baby will have the disease. Today, prenatal testing can be done early in the pregnancy, which people with a family history of the disease may want to consider.
What are the symptoms of Gaucher disease?
Symptoms of Gaucher disease type 1 can differ vastly and can range from severe to none at all. Some signs include:
Swollen belly (from spleen and liver enlargement)
Bone pain and easily fractured bones
Anemia (low red blood cell counts) and fatigue
Bleeding (frequent nosebleeds, gum bleeding) and bruising problems (caused by decrease in blood platelets)
How is Gaucher disease diagnosed?
Because the signs and symptoms of Gaucher disease—including bone pain and anemia—can resemble many other diseases and conditions, it can often go diagnosed for many years. Here are some diagnostics that may be used:
Physical exam: Your doctor can press on your abdomen to check the size of your spleen and liver.
Lab tests: Blood tests can check for levels of the enzyme linked to Gaucher.
Imaging tests: Dual energy X-ray absorptiometry (DXA) uses low-level X-rays to measure bone density. Magnetic resonance imaging (MRI) can show if the spleen or liver is enlarged and if bone marrow has been affected. We also do echocardiograms and EKGs to see if the lungs or heart are involved.
Genetic tests: These can reveal whether you have the disease.
At Yale Medicine, we conduct all of our initial diagnostic and evaluation tests in our medical offices during a single visit.
How is Gaucher disease treated?
Type 1 Gaucher disease, as well as the non-neurological symptoms of type 3, are treatable. Available therapies include:
Enzyme replacement therapy (ERT): This balances the low levels of glucocerebrosidase, which is the underlying cause of Gaucher disease. Patients receive intravenous infusions of the enzyme every two weeks.
Substrate reduction therapy (SRT): This is an oral medication that decreases the rate of formation of glucocerebroside in the body so that excess buildup is reduced.
What is unique about Yale Medicine’s approach to Gaucher disease?
At Yale Medicine, our multidisciplinary team of experts treat all three types of Gaucher disease. We offer the most up-to-date patient care and research.
Yale researchers have been at the forefront of research efforts worldwide. In 2010, Dr. Mistry was the lead author on a study that unraveled some of the biological actions that can trigger the disease, thereby leading the way toward potentially more effective and less expensive treatments.
Yale is also at the forefront of innovative therapies for Gaucher disease. Dr. Mistry led an international study of oral SRT that was published in the Journal of the American Medical Association in 2015 and had since been approved by the FDA.
Yale’s research efforts not only enhance treatment of patients with rare diseases, but they lead to improved understanding of common diseases such as myeloma and Parkinson’s and potential new treatment approaches for them.
“We have patients from as far as Texas and California. We are very active in patient support and building a network,” Dr. Mistry says, adding that helping patients who maybe went years with mysterious symptoms no one could pinpoint is rewarding. “Every time I meet one of these families and see how much suffering they have been through, it moves me.”
We use cookies and similar technologies to make our site work.
We also use optional cookies to measure our site’s performance, personalize content,
assist in our marketing efforts and provide you with the best experience.
You can click on “Manage Cookies” to disable these optional cookies.
Manage Your Cookie Settings
We use different types of cookies to optimize your experience on our website.
You may choose which types of cookies to allow and can change your preferences at any time.
Remember that disabling cookies may effect your experience on the website.
You can learn more about how we use cookies by visiting our
Privacy Policy.
These cookies are necessary for the correct function and cannot be deactivated on our systems.
These cookies allow us to count the number of visits and sources of traffic in order to measure and improve
the performance of our site.
These cookies are put in place by our advertising partners and can be present throughout the entire site. |
11707 | https://pmc.ncbi.nlm.nih.gov/articles/PMC9691874/ | Ultraviolet – Chlorine combined treatment efficiency to eliminate Naegleria fowleri in artificial surf lagoons - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Heliyon
. 2022 Nov 18;8(11):e11625. doi: 10.1016/j.heliyon.2022.e11625
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Ultraviolet – Chlorine combined treatment efficiency to eliminate Naegleria fowleri in artificial surf lagoons
Iñigo Arberas-Jiménez
Iñigo Arberas-Jiménez
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
Find articles by Iñigo Arberas-Jiménez
a,b,c, Ines Sifaoui
Ines Sifaoui
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
Find articles by Ines Sifaoui
a,b,c, María Reyes-Batlle
María Reyes-Batlle
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
f Consorcio Centro de Investigación Biomédica (CIBER) de Enfermedades Infecciosas (CIBERINFEC), Instituto de Salud Carlos III, 28006, Madrid, Spain
Find articles by María Reyes-Batlle
a,b,c,f, Aitor Rizo-Liendo
Aitor Rizo-Liendo
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
Find articles by Aitor Rizo-Liendo
a,b,c, Luis Sancho
Luis Sancho
d CEIT and Tecnun (University of Navarra) Manuel Lardizabal 15, 20018, Donostia, San Sebastián, Spain
Find articles by Luis Sancho
d, Andoni Urruticoechea
Andoni Urruticoechea
e Wavegarden, Instant Sport S.L, Spain
Find articles by Andoni Urruticoechea
e, José E Piñero
José E Piñero
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
f Consorcio Centro de Investigación Biomédica (CIBER) de Enfermedades Infecciosas (CIBERINFEC), Instituto de Salud Carlos III, 28006, Madrid, Spain
Find articles by José E Piñero
a,b,c,f,∗∗, Jacob Lorenzo-Morales
Jacob Lorenzo-Morales
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
f Consorcio Centro de Investigación Biomédica (CIBER) de Enfermedades Infecciosas (CIBERINFEC), Instituto de Salud Carlos III, 28006, Madrid, Spain
Find articles by Jacob Lorenzo-Morales
a,b,c,f,∗
Author information
Article notes
Copyright and License information
a Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias (IUETSPC), Universidad de La Laguna (ULL), Avenida Astrofísico Francisco Sánchez s/n, 38206, La Laguna, Tenerife, Spain
b Departamento de Obstetricia y Ginecología, Pediatría, Medicina Preventiva y Salud Pública, Toxicología, Medicina Legal y Forense y Parasitología, Universidad de La Laguna, Tenerife, Spain
c Red de Investigación Cooperativa en Enfermedades Tropicales (RICET), Spain
d CEIT and Tecnun (University of Navarra) Manuel Lardizabal 15, 20018, Donostia, San Sebastián, Spain
e Wavegarden, Instant Sport S.L, Spain
f Consorcio Centro de Investigación Biomédica (CIBER) de Enfermedades Infecciosas (CIBERINFEC), Instituto de Salud Carlos III, 28006, Madrid, Spain
∗
Corresponding author. jmlorenz@ull.edu.es
∗∗
Corresponding author. jpinero@ull.edu.es
Received 2022 May 11; Revised 2022 Jul 14; Accepted 2022 Nov 10; Collection date 2022 Nov.
© 2022 The Author(s)
This is an open access article under the CC BY license (
PMC Copyright notice
PMCID: PMC9691874 PMID: 36439712
Abstract
Naegleria. fowleri, a protozoa belonging to the free-living amoeba group, is the causative agent of a central nervous system affecting disease that is fatal in more than the 95% of the reported cases. This parasite can be found in warm water bodies such as lakes, rivers or inadequately disinfected swimming pools. On the other hand, chlorination and UV light treatment are two of the most extensively used disinfection methods in recreational water facilities.
In this study the effect of chlorination and UV light on N.fowleri trophozoites was studied in a close water circuit with the aim to assess the efficacy of this disinfection methods in large pools. The obtained results showed that the chlorination was able to decrease the number of viable cells despite the elimination was not totally achieved. Nonetheless, the combination of the UV light with the chlorination allowed the complete removal of the N.fowleri trophozoites from the water in experimental testing conditions.
Keywords: Free-living amoeba, Naegleria fowleri, cove lagoon, UV light, Chlorination, Disinfection
Graphical abstract
Open in a new tab
Highlights
•
Absence of an standardized water treatment protocol to eliminate N.fowleri.
•
Treatment based only on chlorine was ineffective to eliminate N.fowleri.
•
Combined chlorination and UV light treatment was effective to completely eliminate N.fowleri.
•
Combination of chlorine - Uv would be a promising method for water disinfection.
Free-living amoeba; Naegleria fowleri; cove lagoon; UV light; chlorination; disinfection.
1. Introduction
Waters from rivers, lakes and costal used for recreational purposes are classified as recreational water (RW). RW are widely used by people for all manner of recreational activities, including swimming, surfing, white water sports, diving, boating and fishing. Furthermore, those activities were proved to have substantial benefits to health and well-being . Artificial lagoons are nowadays some of the biggest RW bodies existing in the world and companies need to guarantee a safe environment for people enjoying perfect surf waves. Actually, RW users can be exposed to a wide range of microorganisms namely those introduced through human or animal faecal contamination or those naturally present in water, that are found mostly in polluted and unsafe environments . Water's free-living microorganisms are autochthonous to water or once introduced are capable of colonizing the environment .
Among those microorganisms, Free-Living Amoebae (FLA) are widespread in nature and are normal inhabitants of freshwater microbial ecosystems, soil, water and artificial habitats as swimming pools [3, 4, 5]. FLA are considered opportunistic pathogens. In fact, and based on the literature, Acanthamoeba spp, Naegleria fowleri, Balamuthia mandrillaris, Vahlkampfia, Vermamoeba and Sappinia infections have already been reported in humans as well as in animals. On the other hand, FLA can also act as hosts and vectors of pathogenic micro-organisms as bacteria or virus . Only Acanthamoeba spp. and N.fowleri were cited and defined by WHO as quality indicator for non-faecal in recreational water. Naegleria genus are amoebo-flagellates and not as ubiquitous as Acanthamoeba. It was isolated particularly from warm freshwater bodies including manmade lakes and ponds, hot springs, and thermally polluted streams and rivers . Currently, more than 40 species of Naegleria have been reported but only one species, N.fowleri, is pathogenic to humans. The pathogenicity of two other species have been proved in mice after intranasal or intracerebral inoculations .
Outbreaks of recreational water illness have been linked to poor system design and lack in the water treatments process . In order to prevent the spreading of illnesses in RW the pathogen should remain under control and to do so RW should undergo several treatments including coagulation, filtration, disinfection .
Disinfection is a fundamental step in water treatment process by which a microbial hazard is eliminated or inactivated [8, 9]. During this treatment chemical (e.g. chlorination) or physical (e.g. filtration, UV radiation) agent are used to deactivate the pathogen . Chlorination is the most extensively used disinfectant for water treatments due to its low cost, ease to produce, store, transport and use as well as its high oxidizing potential. Usually used as chlorine gas, sodium or calcium hypochlorite, it provides a minimum level of residual disinfectant able to prevent microbial recontamination . Chlorine dioxide have been as well reported by Dupuy et al., (2014) for its efficacy to inhibit three different FLA strains . UV radiation can decrease the microbial charge in the air, on hard surfaces and in thin layer of liquid food and thus by microbial deactivation. It can also eliminate pathogens from potable water and fruit juices . In fact, Yip & Konasewich, (1972) have proved that the UV radiation was effective to eliminate various microorganisms including viruses, bacterial spores and protozoa . Since then, the disinfection of drinking water and wastewater by UV light was investigated in several studies .
Artificial surfing waves lagoons consist of large water bodies where huge amounts of water circulate along a closed circuit and in which there are man made waves. However, no specific protocol is available for the FLA treatment and disinfection of this kind of pools. In the present study, the efficacy of Chlorination and UV radiation treatments separated or combined was conducted against N.fowleri. To assess the efficiency of disinfection methods tested an experimental device to reproduce large pools systems are designed. The cells were analyzed during the time using inverted microscopy. Technologies and operation conditions (free chlorine concentration, Contact Time (CT), UV doses) applied in this study are in the range of the operational conditions of these variables applied in artificial lagoons water management system and though, experimental results and conclusions could be extrapolated to real facilities.
2. Material and methods
2.1. The used amoeba strains
The disinfection methods were evaluated against the type strain of N.fowleri (ATCC 30808™) of the American Type Culture Collection (LG Promochem, Barcelona, Spain). The amoebae were axenically cultured at 37 °C in 2% Bactocasitone medium (Thermo Fisher Scientific, Madrid, Spain) supplemented with 10% (v/v) of foetal bovine serum (FBS), 0.5 mg/ml of streptomycin sulphate and 0.3 μg/ml of penicillin G (Sigma-Aldrich, Madrid, Spain). This N.fowleri strain was cultured in a biological security facility level 3 at the Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias, Universidad de La Laguna as required by the Spanish Government biosafety guidelines for this pathogen.
2.2. Experimental set-up
A recirculating system represented in Figure 1 was set-up to carry out batch test. The system was equipped with UV lamps to provide. Actual UV doses was measured with an UV254 Lightmeter and adjusted to doses in the same range as used in WaveGarden Cove facilities as shown in Table 1. Chlorine was introduced in the chamber. The chlorine concentration was measured using the 4500-Cl G DPD method (APHA, 2005) Contact time adopted was adjusted to similar values as in the WaveGarden Cove facilities. In the same way, UV doses were applied using an UVC lamp in a holding quartz chamber to provide UV doses in the range of WaveGarden Cove facilities.
Figure 1.
Open in a new tab
Experimental setup for chlorination and UV treatment.
Table 1.
Physical-chemical conditions in the present experiment compared to standard disinfection conditions in surf lagoon.
| S | Standard disinfection conditions | Conditions for Chlorine test | Conditions for combined UV + chlorine test |
--- --- |
| Recirculation frequency (hours) | 65 | 0.108 | 0.027 |
| UV dose (mW/cm2.sec) | 60 | 60 | 60 |
| Chlorine dose in pipe (CT) (mg/L·min) | 60 | 52.48 | 52.48 |
| Residual chlorine | 0.25 | ND | ND |
Open in a new tab
2.3. Water pump infection protocol
Initially, the system was filled with 1600 ml of filtered (0.22 μm filter) tap water and the pump was activated, with a flow rate of 150 rpm, so the water was distributed through the circuit. A total of 10 6 amoebae were introduced in the pool with agitation. After 10 min, a sample was collected as negative control. To measure the concentration, 15 ml of water were taken from the pool and then centrifugated to concentrate the amoebae in 100 μl. The counting was carried out in a Neubauer chamber following the manufacturer protocol to obtain the amoebae concentration in the pool. Gibco™ Trypan Blue Solution, 0.4% (Thermo Fisher Scientific, Madrid, Spain) was used to distinguish between viable and non-viable cells.
2.4. Amoebicidal effect evaluation after the water chlorination in a close circuit
To evaluate the effect of the water chlorination against the selected amoebae strain, chlorine was added obtaining a final concentration of 2 mg/L in the chlorination vessel. Finally, samples were collected at 5 and 10 min to measure the number of cells using the same protocol than in the negative control.
2.5. Amoebicidal effect evaluation of the ultraviolet light in a close water circuit
The negative control and the pump filling were performed in the same manner than the chlorination assay. Therefore, the UV lamp was turned on and the water was maintained circulating for 10 min. Afterwards, another sample was collected to finally calculate the cells concentration as it was described in the water pump infection protocol.
2.6. Water chlorination and UV light combination assay
Firstly, the water was chlorinated at a final concentration in the vessel of 2 mg/L and maintained in the chlorination vessel to the defined CT value, afterwards the UV lamp was turned on and the recirculation maintained during 10 min. Finally, a water sample was taken, and the amoebae concentration was calculated as previously described in the 2.3 section.
3. Results
3.1. Amoebicidal effect after water chlorination in a close water circuit
After the amoeba inoculation, a total of 15 mL of water samples from the pool were concentrated after 10 min without treatment as a negative control. The sample was concentrated up to 100 μL to the double counting in the Neubauer chamber (10 3 cells/mL). After the addition of chlorine, we could observe a decrease in the number of cells(Figure 2). However, this reduction in the number of viable trophozoites was not significant and we were still able to count alive trophozoites.
Figure 2.
Open in a new tab
N.fowleri counting after 5 and 10 min of chlorination. Data are represented as means ± standard deviation. Differences between the values were assessed using one-way analysis of variance (ANOVA). Ns: not significant.
These results are in accordance with the few other studies that compare these disinfectants on various genera of FLA [11, 15].
3.2. Amoebicidal effect of the UV light in a close water circuit
The negative control was measuring as it was previously described for the UV effect, obtaining a total of 291 cells/mL after dilution/concentration corrections. Subsequently, the UV assay was performed for 10 min to achieve a 60 mW/cm·sec UV dose, and 15 mL were used for the Neubauer amoebae counting. After 10 min of UV treatment, there were only nonviable amoebae trophozoites in the analysed samples.
3.3. Amoebicidal effect after the combination of UV light and chlorination in a close water circuit
To increase the efficacy of the disinfection method, we decide to combine both UV and chlorination protocols. Therefore, a CT of 52 mg/L·min was applied in the chlorination vessel after that an UV light were applied for 10 min of treatment to complete the 60 mW/cm·sec dose. As a result, we could observe the non-viable cells resulting from this combined treatment (Figure 3).
Figure 3.
Open in a new tab
N.fowleri viable cells counting after the UV light treatment and after the combination of UV light and chlorination. Data are represented as means ± standard deviation.
4. Discussion
N.fowleri, is an opportunistic parasite which causes a severe and acute meningoencephalitis in people who reported previous water exposure. Therefore, since various cases have been recently reported in water parks , indoor swimming pools or surfing parks , the disinfection and maintenance of recreational water facilities are being submitted to a more exhaustive analysis.
In this context, the disinfection and maintenance of recreational water facilities has gained special attention, particularly since various cases have been recently reported in water parks , indoor swimming pools or surfing parks .
UV light and chlorination are two of the most widely used disinfection methods to treat swimming pools and recreational water facilities. Moreover, the combination of both techniques is not only more efficient when killing microorganisms but it is also safer for the swimming pool users since the formation of chlorination by-products is significantly reduced [19, 20]. In fact, the chlorination alone is not able to eliminate some pathogenic microorganisms, such as Cryptosporidium parvum or Giardia lambia . Because of these reasons, the interest in alternative disinfecting methods has increased, been the UV light one of the most popular among them. The UV irradiation inactivates microorganisms by inducing a variety of mutagenic and cytotoxic DNA lesions such as cyclobutane pyrimidine dimers and 6-4 photoproducts formation, as well as DNA strand breaks by interfering the genome integrity [22, 23], therefore it does not produce any harmful chemical products .
The first part of the assay consisted on evaluating the potential of the chlorination (at a final concentration of 2 mg/ml) of the water to eliminate the N.fowleri trophozoites. As it can be illustrated on Figure 2, after 5 min of treatment the number of viable trophozoites was reduced whereas it was not significant. Moreover, the decrease of cells stopped after this time and after 10 min of treatment the number of amoebae was similar, suggesting that the chlorine has an amoebostatic activity. Several studies have reported that the N.fowleri trophozoites are fairly resistant to chlorination [25, 26]. Moreover, the pipe walls of the RW facilities usually contain biofilms which increase the resistance of N.fowleri to chlorination due to the disinfectants consumption by the biofilm and the reduced disinfectant penetration into the biofilm .
On the other hand, when treating the water with both UV light and chlorine, the elimination of viable N.fowleri cells was fully effective showing no viable cells in the circuit. Furthermore, the UV dose administered in this study (60 mW s/cm 2) was also effective against the resistant phase (cyst stage) of N.fowleri according to Sarkar & Gerba . These results demonstrate the great effectiveness of the UV light application in water disinfection, which has been increased in this assay with the chlorination. However, the limitations of the UV light in certain circumstances should also be considered . In recent studies, the control and elimination of organisms by using different nanoparticles has also been reported, so it may be possible to use these new technologies to remove free-living amoebae in aquatic environments too [29, 30, 31].
5. Conclusions
This paper analyses the effect of two disinfection technologies, chlorination and UV radiation, commonly used in recreational waters to remove N.fowleri. The results show significant N.fowleri removal with chlorine doses applied but viable cells still appear in treated water. The use of UV and their combination with similar doses of chlorine achieves a complete N.fowleri removal in the experimental testing conditions.
Declarations
Author contribution statement
Iñigo Arberas-Jiménez; Ines Sifaoui; María Reyes-Batlle; Aitor Rizo-Liendo: Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.
Luis Sancho; Andoni Urruticoechea: Conceived and designed the experiments; Wrote the paper.
José Piñero; Jacob Lorenzo-Morales: Conceived and designed the experiments; Performed the experiments; Contributed reagents, materials, analysis tools or data; Wrote the pape.
Funding statement
This work was funded by projects PI18/01380 from Instituto de Salud Carlos III, Spain and RICET (RD16/0027/0001 project) and PID2019-109476RB-C21 (BIOALGRI) (Spanish Ministry of Science, Madrid, Spain; from Programa Redes Temáticas de Investigación Cooperativa, FIS (Ministerio Español de Salud, Madrid, Spain) and FEDER. Consorcio Centro de Investigación Biomédica (CIBER) de Enfermedades Infecciosas (CIBERINFEC), Instituto de Salud Carlos III, 28006 Madrid, Spain. IAJ (TESIS202001063) and ARL (TESIS2020010054) were funded by Agencia Canaria de Investigación, Innovación y Sociedad de la Información (ACIISI).
Data availability statement
Data will be made available on request.
Declaration of interest's statement
The authors declare no conflict of interest.
Additional information
No additional information is available for this paper.
Contributor Information
José E. Piñero, Email: jpinero@ull.edu.es.
Jacob Lorenzo-Morales, Email: jmlorenz@ull.edu.es.
References
1.Organization W.H. Volume 2. World Health Organization; Geneva: 2006. (Guidelines for Safe Recreational Water Environments). [Google Scholar]
2.Fewtrell L., Kay D. Recreational water and infection: a review of recent findings. Curr. Environ. Heal. Rep. 2015;2(1):85–94. doi: 10.1007/s40572-014-0036-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
3.Najafpoor A.A., Zarrinfar H., Ghaderifar S., Alidadi H., Esmaily H., Hajialilo E., Hosseini Farash B.R., Ahmadi E. Naegleria species population found in pond water of parks in Mashhad city, can the physicochemical factors affect it? MethodsX. 2018;5:1427–1430. doi: 10.1016/j.mex.2018.10.014. [DOI] [PMC free article] [PubMed] [Google Scholar]
4.Ghaderifar S., Najafpoor A.A., Zarrinfar H., Esmaily H., Hajialilo E. Isolation and identification of Acanthamoeba from pond water of parks in a tropical and subtropical region in the Middle East, and its relation with physicochemical parameters. BMC Microbiol. 2018;18(1):139. doi: 10.1186/s12866-018-1301-x. [DOI] [PMC free article] [PubMed] [Google Scholar]
5.Thomas V., McDonnell G., Denyer S.P., Maillard J.Y. Free-living amoebae and their intracellular pathogenic microorganisms: risks for water quality. FEMS Microbiol. Rev. 2010;34(3):231–259. doi: 10.1111/j.1574-6976.2009.00190.x. [DOI] [PubMed] [Google Scholar]
6.Balczun C., Scheid P.L. Free-living amoebae as hosts for and vectors of intracellular microorganisms with public health significance. Viruses. 2017;9(4) doi: 10.3390/v9040065. [DOI] [PMC free article] [PubMed] [Google Scholar]
7.Visvesvara G.S. Pathogenic and opportunistic free-living amoebae. Man. Clin. Microbiol. 2015;(chapter 136):2139–2148. [Google Scholar]
8.Organization W.H. World Health Organization.; 2011. WHO Guide to Ship Sanitation. [PubMed] [Google Scholar]
9.Liviac D., Wagner E.D., Mitch W.A., Altonji M.J., Plewa M.J. Genotoxicity of water concentrates from recreational pools after various disinfection methods. Environ. Sci. Technol. 2010;44(9):3527–3532. doi: 10.1021/es903593w. [DOI] [PubMed] [Google Scholar]
10.Gopal K., Tripathy S.S., Bersillon J.L., Dubey S.P. Chlorination byproducts, their toxicodynamics and removal from drinking water. J.Hazard Mater. 2007;140(1–2):1–6. doi: 10.1016/j.jhazmat.2006.10.063. [pii] [DOI] [PubMed] [Google Scholar]
11.Dupuy M., Berne F., Herbelin P., Binet M., Berthelot N., Rodier M.H., Soreau S., Hechard Y. Sensitivity of free-living amoeba trophozoites and cysts to water disinfectants. Int. J. Hyg Environ. Health. 2014;217(2–3):335–339. doi: 10.1016/j.ijheh.2013.07.007. [DOI] [PubMed] [Google Scholar]
12.Koutchma T. Vol. 2. CRC Press; 2019. (Ultraviolet Light in Food Technology: Principles and Applications). [Google Scholar]
13.Yip, Konasewich D.E., R W. Ultraviolet sterilization of water-its potential and limitations. Water Pollut. Control. 1972;14–18 [Google Scholar]
14.Turtoi M. Ultraviolet light potential for wastewater disinfection. Ann. Food Sci. Technol. 2013;14:153–164. [Google Scholar]
15.Critchley M., Bentham R. The efficacy of biocides and other chemical additives in cooling water systems in the control of amoebae. J.Appl. Microbiol. 2009;106(3):784–789. doi: 10.1111/j.1365-2672.2008.04044.x. [DOI] [PubMed] [Google Scholar]
16.Anjum S.K., Mangrola K., Fitzpatrick G., Stockdale K., Matthias L., Ali I.K.M., Cope J.R., O’Laughlin K., Collins S., Beal S.G., Saccoccio F.M. A case report of primary amebic meningoencephalitis in north Florida. IDCases. 2021;25 doi: 10.1016/j.idcr.2021.e01208. [DOI] [PMC free article] [PubMed] [Google Scholar]
17.Harris G.R., Karmarkar E.N., Quenelle R., Chaille L., Madhok J., Tien V., Gupta J., Jain S., Liu M., Roy S., Narasimhan S., Kimura A., Cope J.R., Ali I.K.M. Death from primary amebic meningoencephalitis after recreational water exposure during recent travel to India—Santa Clara county, California, 2020. Open Forum Infect. Dis. 2021;8(8) doi: 10.1093/ofid/ofab322. [DOI] [PMC free article] [PubMed] [Google Scholar]
18.Hamaty E., Faiek S., Nandi M., Stidd D., Trivedi M., Kandukuri H. A fatal case of primary amoebic meningoencephalitis from recreational waters. Case Rep. Crit. Care. 2020;2020 doi: 10.1155/2020/9235794. [DOI] [PMC free article] [PubMed] [Google Scholar]
19.Wang X., Hu X., Wang H., Hu C. Synergistic effect of the sequential use of UV irradiation and chlorine to disinfect reclaimed water. Water Res. 2012;46(4):1225–1232. doi: 10.1016/j.watres.2011.12.027. [DOI] [PubMed] [Google Scholar]
20.Couto M., Bernard A., Delgado L., Drobnic F., Kurowski M., Moreira A., Rodrigues-Alves R., Rukhadze M., Seys S., Wiszniewska M., Quirce S. Health effects of exposure to chlorination by-products in swimming pools. Allergy. 2021;76(11):3257–3275. doi: 10.1111/all.15014. [DOI] [PubMed] [Google Scholar]
21.Korich D.G., Mead J.R., Madore M.S., Sinclair N.A., Sterling C.R. Effects of ozone, chlorine dioxide, chlorine, and monochloramine on Cryptosporidium parvum oocyst viability. Appl. Environ. Microbiol. 1990;56(5):1423–1428. doi: 10.1128/aem.56.5.1423-1428.1990. [DOI] [PMC free article] [PubMed] [Google Scholar]
22.Cervero-Aragó S., Sommer R., Araujo R.M. Effect of UV irradiation (253.7 Nm) on free Legionella and Legionella associated with its amoebae hosts. Water Res. 2014;67:299–309. doi: 10.1016/j.watres.2014.09.023. [DOI] [PubMed] [Google Scholar]
23.Rastogi R.P., Richa, Kumar A., Tyagi M.B., Sinha R.P. Molecular mechanisms of ultraviolet radiation-induced DNA damage and repair. J.Nucleic Acids. 2010;2010 doi: 10.4061/2010/592980. [DOI] [PMC free article] [PubMed] [Google Scholar]
24.Koca N., Urgu M., Oğul T. In: Technological Approaches for Novel Applications in Dairy Processing. Koca N., editor. 2018. Ultraviolet light applications in dairy processing. London. [Google Scholar]
25.Sarkar Payal, Charles P.,G. Inactivation of Naegleria fowleri by chlorine and ultraviolet light. J.Am. Water Works Assoc. 2012;104(3):E173–E180. [Google Scholar]
26.Cursons R.T., Brown T.J., Keys E.A. Effect of disinfectants on pathogenic free-living amoebae: in axenic conditions. Appl. Environ. Microbiol. 1980;40(1):62–66. doi: 10.1128/aem.40.1.62-66.1980. [DOI] [PMC free article] [PubMed] [Google Scholar]
27.Miller H.C., Wylie J., Dejean G., Kaksonen A.H., Sutton D., Braun K., Puzon G.J. Reduced efficiency of chlorine disinfection of Naegleria fowleri in a drinking water distribution biofilm. Environ. Sci. Technol. 2015;49(18):11125–11131. doi: 10.1021/acs.est.5b02947. [DOI] [PubMed] [Google Scholar]
28.Cope J.R., Murphy J., Kahler A., Gorbett D.G., Ali I., Taylor B., Corbitt L., Roy S., Lee N., Roellig D., Brewer S., Hill V.R. Primary amebic meningoencephalitis associated with rafting on an artificial whitewater river: case report and environmental investigation. Clin. Infect. Dis. 2018;66(4):548–553. doi: 10.1093/cid/cix810. [DOI] [PMC free article] [PubMed] [Google Scholar]
29.Kazemi M., Akbari A., Zarrinfar H., Soleimanpour S., Sabouri Z., Khatami M., Darroudi M. Evaluation of antifungal and photocatalytic activities of gelatin-stabilized selenium oxide nanoparticles. J.Inorg. Organomet. Polym. Mater. 2020;30(8):3036–3044. [Google Scholar]
30.Kazemi M., Akbari A., Sabouri Z., Soleimanpour S., Zarrinfar H., Khatami M., Darroudi M. Green synthesis of colloidal selenium nanoparticles in starch solutions and investigation of their photocatalytic, antimicrobial, and cytotoxicity effects. Bioproc. Biosyst. Eng. 2021;44(6):1215–1225. doi: 10.1007/s00449-021-02515-9. [DOI] [PubMed] [Google Scholar]
31.Rostami N., Alidadi H., Zarrinfar H., Ketabi D., Tabesh H. Interventional effect of nanosilver paint on fungal load of indoor air in a hospital ward. Can. J. Infect Dis. Med. Microbiol. 2021;2021 doi: 10.1155/2021/8658600. [DOI] [PMC free article] [PubMed] [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Data will be made available on request.
Articles from Heliyon are provided here courtesy of Elsevier
ACTIONS
View on publisher site
PDF (468.8 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
Graphical abstract
Highlights
1. Introduction
2. Material and methods
3. Results
4. Discussion
5. Conclusions
Declarations
Contributor Information
References
Associated Data
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
11708 | https://pressbooks.bccampus.ca/introductorygeneralphysics1phys1107/chapter/8-3-conservation-of-momentum/ | Skip to content
Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.
Chapter 8 Linear Momentum and Collisions
8.3 Conservation of Momentum
Summary
Describe the principle of conservation of momentum.
Derive an expression for the conservation of momentum.
Explain conservation of momentum with examples.
Explain the principle of conservation of momentum as it relates to atomic and subatomic particles.
Momentum is an important quantity because it is conserved. Yet it was not conserved in the examples in Chapter 8.2 Impulse and Chapter 8.1 Linear Momentum and Force, where large changes in momentum were produced by forces acting on the system of interest. Under what circumstances is momentum conserved?
The answer to this question entails considering a sufficiently large system. It is always possible to find a larger system in which total momentum is constant, even if momentum changes for components of the system. If a football player runs into the goalpost in the end zone, there will be a force on him that causes him to bounce backward. However, the Earth also recoils —conserving momentum—because of the force applied to it through the goalpost. Because Earth is many orders of magnitude more massive than the player, its recoil is immeasurably small and can be neglected in any practical sense, but it is real nevertheless.
Consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—for example, one car bumping into another, as shown in Figure 1. Both cars are coasting in the same direction when the lead car (labeled m2) is bumped by the trailing car (labeled m1). The only unbalanced force on each car is the force of the collision. (Assume that the effects due to friction are negligible.) Car 1 slows down as a result of the collision, losing some momentum, while car 2 speeds up and gains some momentum. We shall now show that the total momentum of the two-car system remains constant.
Using the definition of impulse, the change in momentum of car 1 is given by
[latex]\boldsymbol{\Delta{\vec{\textbf{p}}}_1=\vec{\textbf{F}}_1\Delta{t},}[/latex]
where $$\boldsymbol{\vec{\textbf{F}}_1}$$ is the force on car 1 due to car 2, and Δt is the time the force acts (the duration of the collision). Intuitively, it seems obvious that the collision time is the same for both cars, but it is only true for objects traveling at ordinary speeds. This assumption must be modified for objects travelling near the speed of light, without affecting the result that momentum is conserved.
Similarly, the change in momentum of car 2 is
[latex]\boldsymbol{\Delta\vec{\textbf{p}}_2=\vec{\textbf{F}}_2\Delta{t},}[/latex]
where $$\boldsymbol{\vec{\textbf{F}}_2}$$ is the force on car 2 due to car 1, and we assume the duration of the collision Δt is the same for both cars. We know from Newton’s third law that $$\boldsymbol{\vec{\textbf{F}}_1=-\vec{\textbf{F}}_2}$$, and so
[latex]\boldsymbol{\Delta\vec{\textbf{p}}_2=-\vec{\textbf{F}}_1\Delta{t}=-\Delta\vec{\textbf{p}}_1.}[/latex]
Thus, the changes in momentum are equal and opposite, and
[latex]\boldsymbol{\Delta\vec{\textbf{p}}_1+\Delta\vec{\textbf{p}}_2=0.}[/latex]
Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is,
[latex]\begin{array}{l} \boldsymbol{\vec{\textbf{p}}_1+\vec{\textbf{p}}_2=\textbf{constant},} \ \boldsymbol{\vec{\textbf{p}}_1+\vec{\textbf{p}}_2=\vec{\textbf{p}}^{\prime}_1+\vec{\textbf{p}}^{\prime}_2,} \end{array}[/latex]
where $$\boldsymbol{\vec{\textbf{p}}^{\prime}_1}$$ and $$\boldsymbol{\vec{\textbf{p}}^{\prime}_2}$$ are the momenta of cars 1 and 2 after the collision. (We often use primes to denote the final state.)
This result—that momentum is conserved—has validity far beyond the preceding one-dimensional case. It can be similarly shown that total momentum is conserved for any isolated system, with any number of objects in it. In equation form, the conservation of momentum principle for an isolated system is written
[latex]\boldsymbol{\vec{\textbf{p}}_{\textbf{tot}}=\textbf{constant},}[/latex]
or
[latex]\boldsymbol{\vec{\textbf{p}}_{\textbf{tot}}=\vec{\textbf{p}}^{\prime}_{\textbf{tot}},}[/latex]
where [latex]\vec{\textbf{p}}_{\textbf{tot}}[/latex] is the total momentum (the sum of the momenta of the individual objects in the system) and [latex]\boldsymbol{\vec{\textbf{p}}^{\prime}_{\textbf{tot}}}[/latex] is the total momentum some time later. (The total momentum can be shown to be the momentum of the center of mass of the system.) An isolated system is defined to be one for which the net external force is zero [latex]\boldsymbol{(\vec{\textbf{F}}_{\textbf{net}}=0).}[/latex]
CONSERVATION OF MOMENTUM PRINCIPLE
[latex]\begin{array}{l} \boldsymbol{\vec{\textbf{p}}_{\textbf{tot}}=\textbf{ constant}} \ \boldsymbol{\vec{\textbf{p}}_{\textbf{tot}}=\vec{\textbf{p}}^{\prime}_{\textbf{tot}}\textbf{ (isolated system)}} \end{array}[/latex]
ISOLATED SYSTEM
An isolated system is defined to be one for which the net external force is zero [latex]\boldsymbol{(\vec{\textbf{F}}_{\textbf{net}}=0).}[/latex]
Perhaps an easier way to see that momentum is conserved for an isolated system is to consider Newton’s second law in terms of momentum, [latex]\boldsymbol{\vec{\textbf{F}}_{\textbf{net}}=\frac{\Delta\vec{\textbf{p}}_{\textbf{tot}}}{\Delta{t}}}.[/latex] For an isolated system, [latex]\boldsymbol{\vec{\textbf{F}}_{\textbf{net}}=0};[/latex] thus, [latex]\boldsymbol{\Delta\vec{\textbf{p}}_{\textbf{tot}}=0},[/latex] and [latex]\vec{\textbf{p}}_{\textbf{tot}}[/latex] is constant.
We have noted that the three length dimensions in nature—x, y, and z—are independent, and it is interesting to note that momentum can be conserved in different ways along each dimension. For example, during projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero and momentum is unchanged. But along the vertical direction, the net vertical force is not zero and the momentum of the projectile is not conserved. (See Figure 2.) However, if the momentum of the projectile-Earth system is considered in the vertical direction, we find that the total momentum is conserved.
Figure 2. The horizontal component of a projectile’s momentum is conserved if air resistance is negligible, even in this case where a space probe separates. The forces causing the separation are internal to the system, so that the net external horizontal force Fx-net is still zero. The vertical component of the momentum is not conserved, because the net vertical force Fy-net is not zero. In the vertical direction, the space probe-Earth system needs to be considered and we find that the total momentum is conserved. The center of mass of the space probe takes the same path it would if the separation did not occur.
The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not.
MAKING CONNECTIONS: TAKE-HOME INVESTIGATION—DROP OF TENNIS BALL AND A BASEBALL
Hold a tennis ball side by side and in contact with a basketball. Drop the balls together. (Be careful!) What happens? Explain your observations. Now hold the tennis ball above and in contact with the basketball. What happened? Explain your observations. What do you think will happen if the basketball ball is held above and in contact with the tennis ball?
MAKING CONNECTIONS: TAKE-HOME INVESTIGATION—TWO TENNIS BALLS IN A BALLISTIC TRAJECTORY
Tie two tennis balls together with a string about a foot long. Hold one ball and let the other hang down and throw it in a ballistic trajectory. Explain your observations. Now mark the center of the string with bright ink or attach a brightly colored sticker to it and throw again. What happened? Explain your observations.
Some aquatic animals such as jellyfish move around based on the principles of conservation of momentum. A jellyfish fills its umbrella section with water and then pushes the water out resulting in motion in the opposite direction to that of the jet of water. Squids propel themselves in a similar manner but, in contrast with jellyfish, are able to control the direction in which they move by aiming their nozzle forward or backward. Typical squids can move at speeds of 8 to 12 km/h.
The ballistocardiograph (BCG) was a diagnostic tool used in the second half of the 20th century to study the strength of the heart. About once a second, your heart beats, forcing blood into the aorta. A force in the opposite direction is exerted on the rest of your body (recall Newton’s third law). A ballistocardiograph is a device that can measure this reaction force. This measurement is done by using a sensor (resting on the person) or by using a moving table suspended from the ceiling. This technique can gather information on the strength of the heart beat and the volume of blood passing from the heart. However, the electrocardiogram (ECG or EKG) and the echocardiogram (cardiac ECHO or ECHO; a technique that uses ultrasound to see an image of the heart) are more widely used in the practice of cardiology.
MAKING CONNECTIONS: CONSERVATION OF MOMENTUM AND COLLISION
Conservation of momentum is quite useful in describing collisions. Momentum is crucial to our understanding of atomic and subatomic particles because much of what we know about these particles comes from collision experiments.
Subatomic Collisions and Momentum
The conservation of momentum principle not only applies to the macroscopic objects, it is also essential to our explorations of atomic and subatomic particles. Giant machines hurl subatomic particles at one another, and researchers evaluate the results by assuming conservation of momentum (among other things).
On the small scale, we find that particles and their properties are invisible to the naked eye but can be measured with our instruments, and models of these subatomic particles can be constructed to describe the results. Momentum is found to be a property of all subatomic particles including massless particles such as photons that compose light. Momentum being a property of particles hints that momentum may have an identity beyond the description of an object’s mass multiplied by the object’s velocity. Indeed, momentum relates to wave properties and plays a fundamental role in what measurements are taken and how we take these measurements. Furthermore, we find that the conservation of momentum principle is valid when considering systems of particles. We use this principle to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei. Figure 3 below illustrates how a particle scattering backward from another implies that its target is massive and dense. Experiments seeking evidence that quarks make up protons (one type of particle that makes up nuclei) scattered high-energy electrons off of protons (nuclei of hydrogen atoms). Electrons occasionally scattered straight backward in a manner that implied a very small and very dense particle makes up the proton—this observation is considered nearly direct evidence of quarks. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale.
Figure 3. A subatomic particle scatters straight backward from a target particle. In experiments seeking evidence for quarks, electrons were observed to occasionally scatter straight backward from a proton.
Section Summary
The conservation of momentum principle is written
[latex]\boldsymbol{\vec{\textbf{p}}_{\textbf{tot}}=\textbf{constant}}[/latex]
or
[latex]\boldsymbol{\vec{\textbf{p}}_{\textbf{tot}}=\vec{\textbf{p}}^{\prime}_{\textbf{tot}}\textbf{ (isolated system)},}[/latex]
[latex]\vec{\textbf{p}}_{\textbf{tot}}[/latex] is the initial total momentum and [latex]\boldsymbol{\vec{\textbf{p}}^{\prime}_{\textbf{tot}}}[/latex] is the total momentum some time later.
An isolated system is defined to be one for which the net external force is zero [latex]\boldsymbol{(\vec{\textbf{F}}_{\textbf{net}}=0)}.[/latex]
During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.
Conservation of momentum applies only when the net external force is zero.
The conservation of momentum principle is valid when considering systems of particles.
Conceptual Questions
1: Professional Application
If you dive into water, you reach greater depths than if you do a belly flop. Explain this difference in depth using the concept of conservation of energy. Explain this difference in depth using what you have learned in this chapter.
2: Under what circumstances is momentum conserved?
3: Can momentum be conserved for a system if there are external forces acting on the system? If so, under what conditions? If not, why not?
4: Momentum for a system can be conserved in one direction while not being conserved in another. What is the angle between the directions? Give an example.
5: Professional Application
Explain in terms of momentum and Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion.
6: Can objects in a system have momentum while the momentum of the system is zero? Explain your answer.
7: Must the total energy of a system be conserved whenever its momentum is conserved? Explain why or why not.
Problems & Exercises
1: Professional Application
Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second having a mass of 110,000 kg and a velocity of -0.120 m/s. (The minus indicates direction of motion.) What is their final velocity?
2: Suppose a clay model of a koala bear has a mass of 0.200 kg and slides on ice at a speed of 0.750 m/s. It runs into another clay model, which is initially motionless and has a mass of 0.350 kg. Both being soft clay, they naturally stick together. What is their final velocity?
3: Professional Application
Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this question be different if the car with the 70-kg passenger had collided with a car that has a mass equal to and is traveling in the opposite direction and at the same speed? Explain your answer.
4: What is the velocity of a 900-kg car initially moving at 30.0 m/s, just after it hits a 150-kg deer initially running at 12.0 m/s in the same direction? Assume the deer remains on the car.
5: A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and the dove’s velocity is 7.00 m/s in the same direction?
Glossary
conservation of momentum principle
: when the net external force is zero, the total momentum of the system is conserved or constant
isolated system
: a system in which the net external force is zero
quark
: fundamental constituent of matter and an elementary particle
Exercises
Problems & Exercises
1: $$\boldsymbol{0.122\textbf{ m/s}}$$
3: In a collision with an identical car, momentum is conserved. Afterwards [latex]\boldsymbol{v_{\textbf{f}}=0}[/latex] for both cars. The change in momentum will be the same as in the crash with the tree. However, the force on the body is not determined since the time is not known. A padded stop will reduce injurious force on body.
5: $$\boldsymbol{22.4\textbf{ m/s}}$$ in the same direction as the original motion
License
x-Douglas College Physics 1107 Fall 2019 Custom Textbook Copyright © August 22, 2016 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.
Share This Book |
11709 | https://www.penguin.co.uk/books/366265/flora-britannica-by-mabeyrichard/9781856193771 | Flora Britannica
Skip to Content
Search
Sign up
Sign up
Menu
Discover
Fiction
Non-fiction
Children's
Penguin Classics
Authors
About us
Penguin Shop
Search
Discover the Penguin books that shaped us
Home
Richard Mabey
Flora Britannica
Flora Britannica
byRichard Mabey
Format:
Hardback
Buy the book from:
Amazon Opens in a new tabBlackwells Opens in a new tabBookshop.org Opens in a new tabFoyles Opens in a new tabHive Opens in a new tabWaterstones Opens in a new tabWHSmith Opens in a new tab
Flora Britannica covers the native and naturalised plants of England, Scotland and Wales, and, while full of fascinating history, is topical and modern. Indeed, Flora Britannica is the definitive contemporary flora, an encyclopaedia of living folklore, a register – a sort of Domesday Book.
It is unique in that it is not a botanical flora but a cultural one – an account of the role of wild plants in social life, arts, custom and landscape. It is also unique in that information has been supplied by the people themselves. Five years of intensive original research have aroused popular interest and ‘grassroots’ involvement on an exceptional scale. People all over Britain – both rural and urban – have been encouraged to record and celebrate the cultural dimensions of their own flora, and to send their memories and anecdotes, observations and regional knowledge to Flora Britannica.
The result is a nationwide record of the popular culture, domestic uses and social meanings of our wild plants. It is both useful and delightful – superbly written by one of the most outstanding English authors on natural history and illustrated with nearly 500 photographs. Including trees and ferns, it covers 1,000 species, many of them in considerable detail. A new flora for the people, Flora Britannica is a testimony to the continuing relationship between nature and human beings, and a celebration that the seasons and the landscape, local character and identity, still matter in Britain.
Read more
A page-turner. He seamlessly weaves together the cultural and social history of wild plants… Undoubtedly the most fascinating book I own
Andy Sturgeon, Gardens Illustrated
About Richard Mabey
Richard Mabey is the father of modern nature writing in the UK. Since 1972 he has written some forty influential books, including the prize-winning Nature Cure, Gilbert White: a Biography, and Flora Britannica. He is a Fellow of the Royal Society of Literature and Vice-President of the Open Spaces Society.
He spent the first half of his life amongst the Chiltern beechwoods, and now lives in Norfolk in a house surrounded by ash trees.
Learn more
Details
Imprint:Chatto & Windus
Published: 07/10/1996
ISBN: 9781856193771
Length: 480 pages
Dimensions: 288mm x 36mm x 230mm
Weight: 2197g
Price: £75.00
You might like...
More from this author
Roger Deakin: A BBC Nature Collection
Turning the Boat for Home Richard Mabey
Birds Britannica Mark Cocker, Richard Mabey
Fencing Paradise Richard Mabey
View more
About us
Our story Opens in a new tab
Publishing houses Opens in a new tab
Social impact Opens in a new tab
Contact us Opens in a new tab
Help and FAQs Opens in a new tab
Work with us
Careers WE'RE HIRING Opens in a new tab
Getting published Opens in a new tab
Partnerships Opens in a new tab
Penguin Ventures Opens in a new tab
Rights and permissions Opens in a new tab
Sales and distribution Opens in a new tab
Stay connected
Follow penguin
Follow ladybird
Follow puffin
Sign up to the Penguin newsletter
For the latest books, recommendations, author interviews and more
Sign up
By signing up, I confirm that I'm over 16. To find out what personal data we collect and how we use it, please visit our Privacy Policy
Penguin Books Limited
APenguin Random HouseCompany.
© 1995 –2025 Penguin Books Ltd. Registered number: 861590 England.Registered office: One Embassy Gardens, 8 Viaduct Gardens, London, SW11 7BW, UK.
Privacy policy Opens in a new tab
Cookies policy Opens in a new tab
Cookie settings Opens in a new tab
Modern slavery statement Opens in a new tab
Accessibility Opens in a new tab
Product recalls Opens in a new tab
Terms & conditions Opens in a new tab
Pay gap reports Opens in a new tab
Industry commitment to professional behaviour Opens in a new tab
You're in control
We use cookies and other technologies on this site which are essential to make our website work. If you accept all cookies this means we can personalise your experience and, for example, recommend books you'll love! You can manage your cookie settings at any time. For more info, see ourCookie Policy
Cookie Settings Strictly Necessary Cookies Accept All Cookies
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Allow All
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Strictly Necessary Cookies Confirm My Choices |
11710 | https://thecuriousastronomer.wordpress.com/2014/10/21/derivation-of-the-surface-area-of-a-sphere/ | thecuriousastronomer
Life, the Universe, and everything
Derivation of the surface area of a sphere
21/10/2014 by RhEvans
In this blog, I used polar coordinates to derive the well-known expression for the area of a circle, . In today’s blog, I will go from 2 to 3-dimensions to derive the expression for the surface area of a sphere, which is . To do this, we need to use the 3-dimensional equivalent of polar coordinates, which are called spherical polar coordinates.
Let us imagine drawing a line in 3-D space of length into the positive part of the coordinate system. We will draw this line at an angle above the x-y (horizontal) plane, and at an angle to the y-z (vertical) plane (see the figure below). When we drop a vertical line from our point onto the x-y plane it has a length , as shown in the figure below.
We then increase the angle by a small amount , and increase the angle by a small amount . As the figure shows, the small surface element which is thus created is just multiplied by , so .
Using spherical polar coordinates, the area element on the surface of a sphere is given by .
.
Using spherical polar coordinates, the area element on the surface of a sphere is given by .
To find the surface area of the sphere, we need to integrate this area element over the entire surface of the sphere. Therefore, we keep and we vary . We can go from a (the negative to the positive ), and from a (one complete rotation about the z-axis on the x-y plane), so we have
so, the total surface area of a sphere is
as required. In a future blog I will use spherical polar coordinates to derive the volume of a sphere, where will no longer be constant as it is here.
Share this:
Related
Posted in mathematics, Physics, Science | Tagged Calculus, Integration, Polar coordinates, sphere, spherical polar coordinates, surface area of a sphere | 8 Comments
8 Responses
[…] In this blog, I derived the expression for the surface area of a sphere, . In today’s blog, I will derive the expression for the volume of a sphere. Actually, once one has understood how to derive the surface area of a sphere using spherical polar coordinates, deriving the volume is pretty straight forward. It only involves one extra step, and that is to create a volume element with the same surface area that we had before, but with a thickness , and to integrate over in addition to integrating over . […]
How did you work out the lengths of the sides of the small area to be deltaTheta r and deltaPhi rcos(theta)
Look at the diagram carefully. From the definition of angles expressed in radians, you should be able to see that the length of one side is . If you can see why this is, the length of the other side being follows the same logic.
It isn’t a cos in this formula r²cos(Teta)d(Teta)d(Phi). It is a sin.
If you look more carefully at my diagram, I think that you’ll find that it is a cosine. I have defined theta as the angle between the horizontal (x-y) plane and the radius vector.
Then your Theta, when you’re at (0,0, Zmax) is equal to Pi/2 ?
Theta, as I have defined it, would go from -pi/2 to +pi/2 (the -z direction to the +z direction). You can see this in the limits of the integral for the theta part of the integral (we integrate over both theta and phi).
How can I used line integral to over a circular surface to find the surface area for 3-D sphere equation?
Comments RSS
Leave a comment Cancel reply
Δ
Recent Posts
Archives
Categories
Meta
Follow Blog via Email
Enter your email address to follow this blog and receive notifications of new posts by email.
Email Address:
Follow
Blog at WordPress.com.
WPThemes. |
11711 | https://byjus.com/maths/magnitude-of-a-vector/ | What is the Magnitude of a Vector?
As we know, the vector is an object which has both the magnitude as well as direction. To find the magnitude of a vector, we need to calculate the length of the vector. Quantities such as velocity, displacement, force, momentum, etc. are vector quantities. But speed, mass, distance, volume, temperature, etc. are scalar quantities. The scalar has the only magnitude, whereas the vectors have both magnitude and direction.
The magnitude of a vector formula is used to calculate the length for a given vector (say v) and is denoted as |v|. So basically, this quantity is the length between the initial point and endpoint of the vector. To calculate the magnitude of the vector, we use the distance formula, which we will discuss here.
Magnitude of a Vector Formula
Suppose, AB is a vector quantity that has magnitude and direction both. To calculate the magnitude of the vector
(\begin{array}{l}\vec{AB}\end{array} )
, we have to calculate the distance between the initial point A and endpoint B. In XY – plane, let A has coordinates (x0, y0) and B has coordinates (x1, y1). Therefore, by distance formula, the magnitude of vector
(\begin{array}{l}\vec{AB}\end{array} )
, can be written as;
|
(\begin{array}{l}{\vec{AB}}\end{array} )
| =
(\begin{array}{l}\sqrt{(x_1 – x_0)^2 + (y_1 – y_0)^2}\end{array} )
Now if the starting point is at (x, y) and the endpoint is at the origin, then the magnitude of a vector formula becomes;
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2 + y^2}\end{array} )
Direction of a vector
The direction of a vector is nothing but the measurement of the angle which is made with the horizontal line. One of the methods to find the direction of the vector
(\begin{array}{l}\vec{AB}\end{array} )
is;
tan α = y/x; endpoint at 0.
Where x is the change in horizontal line and y is the change in a vertical line.
Or tan α =
(\begin{array}{l}\frac{y_1 – y_0}{x_1 – x_0}\end{array} )
; where (x0 , y0) is initial point and (x1 , y1) is the endpoint.
Problems on Magnitude of a Vector
Problem 1: Find the magnitude of the vector
(\begin{array}{l}\vec{AB}\end{array} )
whose initial point, A is (1, 2) and endpoint, B is (4, 3).
Solution: Given, A is (1, 2) and B is (4, 3) as the initial point and endpoint respectively.
Therefore, x0 = 1 & y0 = 2 and x1 = 4 & y1 = 3
Using distance formula,
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(x_1 – x_0)^2 + (y_1 – y_0)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(4 – 1)^2 + (3 – 2)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{3^2 + 1^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{9 + 1}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Now if the starting point is at (x, y) and the endpoint is at the origin, then the magnitude of a vector formula becomes;
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2 + y^2}\end{array} )
Direction of a vector
The direction of a vector is nothing but the measurement of the angle which is made with the horizontal line. One of the methods to find the direction of the vector
(\begin{array}{l}\vec{AB}\end{array} )
is;
tan α = y/x; endpoint at 0.
Where x is the change in horizontal line and y is the change in a vertical line.
Or tan α =
(\begin{array}{l}\frac{y_1 – y_0}{x_1 – x_0}\end{array} )
; where (x0 , y0) is initial point and (x1 , y1) is the endpoint.
Problems on Magnitude of a Vector
Problem 1: Find the magnitude of the vector
(\begin{array}{l}\vec{AB}\end{array} )
whose initial point, A is (1, 2) and endpoint, B is (4, 3).
Solution: Given, A is (1, 2) and B is (4, 3) as the initial point and endpoint respectively.
Therefore, x0 = 1 & y0 = 2 and x1 = 4 & y1 = 3
Using distance formula,
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(x_1 – x_0)^2 + (y_1 – y_0)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(4 – 1)^2 + (3 – 2)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{3^2 + 1^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{9 + 1}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Direction of a vector
The direction of a vector is nothing but the measurement of the angle which is made with the horizontal line. One of the methods to find the direction of the vector
(\begin{array}{l}\vec{AB}\end{array} )
is;
tan α = y/x; endpoint at 0.
Where x is the change in horizontal line and y is the change in a vertical line.
Or tan α =
(\begin{array}{l}\frac{y_1 – y_0}{x_1 – x_0}\end{array} )
; where (x0 , y0) is initial point and (x1 , y1) is the endpoint.
Problems on Magnitude of a Vector
Problem 1: Find the magnitude of the vector
(\begin{array}{l}\vec{AB}\end{array} )
whose initial point, A is (1, 2) and endpoint, B is (4, 3).
Solution: Given, A is (1, 2) and B is (4, 3) as the initial point and endpoint respectively.
Therefore, x0 = 1 & y0 = 2 and x1 = 4 & y1 = 3
Using distance formula,
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(x_1 – x_0)^2 + (y_1 – y_0)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(4 – 1)^2 + (3 – 2)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{3^2 + 1^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{9 + 1}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{(4 – 1)^2 + (3 – 2)^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{3^2 + 1^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{9 + 1}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{3^2 + 1^2}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{9 + 1}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{9 + 1}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
|
(\begin{array}{l}\vec{AB}\end{array} )
| =
(\begin{array}{l}\sqrt{10}\end{array} )
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
(\begin{array}{l}\vec{v}\end{array} )
= xi + yj + zk
|
(\begin{array}{l}\vec{AB}\end{array} )
| ≈ 3.2
The magnitude of
(\begin{array}{l}\vec{AB}\end{array} )
≈ 3.2
If we have to calculate the magnitude of a 3d vector, then there would be three coordinates points and we can represent the magnitude of the vector
(\begin{array}{l}\vec{v}\end{array} )
as;
|
(\begin{array}{l}\vec{v}\end{array} )
| =
(\begin{array}{l}\sqrt{x^2+y^2+z^2}\end{array} )
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
(\begin{array}{l}\vec{OP}=\sqrt{OA^2 +AB^2 + BP^2}\end{array} )
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Let us consider a 3d object.
In the figure, you can see the magnitude of the vector is represented by the vector
(\begin{array}{l}\vec{V}\end{array} )
(blue line), which is the length of the arrow representing
(\begin{array}{l}\vec{V}\end{array} )
. So, the length of the OP is the magnitude of
(\begin{array}{l}\vec{V}\end{array} )
. To calculate the magnitude, we have to use Pythagoras theorem here, for the triangles, OBP and OAB.
In triangle OBP (1)
OP2 = OB2 + BP2
And in triangle OAB (2)
OB2 = OA2 + AB2
From equation 1 and 2 we get,
OP2 = OA2+ AB2 + BP2
Therefore, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Problem 2Find the magnitude of a 3d vector 2i + 3j + 4k.
Solution: We know, the magnitude of a 3d vector xi + yj + zk =
(\begin{array}{l}\sqrt{x^2 +y^2 + z^2}\end{array} )
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Therefore, the magnitude of a 3d vector 2i + 3j + 4k =
(\begin{array}{l}\sqrt{2^2 +3^2 + 4^2}\end{array} )
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
=
(\begin{array}{l}\sqrt{4 + 9 + 16}\end{array} )
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
=
(\begin{array}{l}\sqrt{29}\end{array} )
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Hence, the magnitude of a 3d vector 2i + 3j + 4k ≈ 5.38
Download BYJU’S – The Learning App and enjoy learning with us by getting interactive and fun videos.
| | |
--- |
| Related Links | |
| Scalar And Vectors | Components Of A Vector |
| Vector And Scalar Quantities | Multiplication Of Vectors And Scalar Quantity |
| Difference Between Scalar and Vector | Scalar And Vector Products |
Comments
Leave a Comment Cancel reply
Your Mobile number and Email id will not be published. Required fields are marked
Request OTP on
Voice Call
Website
Post My Comment
Register with BYJU'S & Download Free PDFs
Register with BYJU'S & Watch Live Videos |
11712 | https://sat.blog.targettestprep.com/sat-statistics-questions/ | Menu
SAT Statistics Questions
Jeff Miller January 5, 2024
| |
| Getting your Trinity Audio player ready... |
Last Updated on May 7, 2024
If you haven’t taken a statistics course, you might be terrified at the prospect of having to face SAT questions on the subject. Before you let fear take over, though, realize that there are many statistical concepts that you have already learned! And, as for the ones you haven’t, well, they can be mastered with just a bit of extra study time. In this article, we’ll give you a solid review of the basics and a preview of the more challenging statistics questions that you might encounter on the SAT.
Here are the topics we’ll cover:
An Overview of SAT Math Topics
The Average (Arithmetic Mean)
Example 1: Average — Missing Value
Example 2: Average — Variable Values
The Weighted Average
The Weighted Average Formula
Example 3: Weighted Average
The Median
Median Mini-Example 1
Median Mini-Example 2
Example 4: Median — Large Data Set
The Mode
Example 5: Mode
The Range
Example 6: Range
The Standard Deviation
Standard Deviation Facts
Example 7: Standard Deviation
Summary: SAT Statistics Problems
What’s Next?
Before we look at some practice questions, let’s take a look at the subtopics that we’ll encounter in SAT statistics.
Earn a Higher SAT Math Score Start Studying With TTP Today! TRY OUR SAT MATH COURSE FOR FREE!
An Overview of SAT Math Topics
On the digital SAT, there have been significant changes from the pencil-and-paper test. First, the number of questions in the Math section has been reduced from 58 to 44. Second, students may use a calculator in both modules of the Math section. Third, the test-maker has implemented what is called “adaptive scoring.” This means that the difficulty of Math module 2 is based on your performance in Math module 1. Module 2 could be harder, easier, or about the same as module 1.
There are four major categories of SAT math questions. Let’s look at each category and list the math topics that are covered in each:
Algebra: linear equations, linear functions, systems of linear equations, and linear inequalities
Advanced Math: nonlinear equations, systems of nonlinear equations, nonlinear functions
Problem Solving and Data Analysis (PSDA): ratios, proportions, percentages, data distributions, measures of center and spread, data interpretation, scatterplots, probability, statistical significance, margin of error, evaluating statistical claims
Geometry and Trigonometry: area, volume, lines, angles, triangles, right triangle trigonometry, circles
Questions from the first two categories (Algebra and Advanced Math) each constitute about 35% of SAT math questions. The third and fourth categories (Problem Solving and Data Analysis, and Geometry and Trigonometry) each constitute about 15% of SAT math questions.
We see that statistics questions are in the Problem Solving and Data Analysis (PSDA) category, and the bulk of questions in that category are statistics questions. Thus, we can expect to see as many as 6 statistics questions on the SAT. This is sufficient reason to continue reading this article and learning the facts and concepts presented!
KEY FACT:
You can expect to see as many as 6 statistics questions in the SAT Math section.
Let’s start with the average.
The Average (Arithmetic Mean)
On the SAT, you will see the terms “average” and “arithmetic mean” used interchangeably. So, to calculate the average or the arithmetic mean, you’ll use the basic formula:
average = sum of values / number of values
Here is a basic arithmetic mean question:
What is the average of Erica’s test scores of 80, 62, 74, and 76?
We use the formula for the average:
average = sum of values / number of values
average = (80 + 62 + 74 + 76) / 4
average = 292 / 4 = 73
KEY FACT:
The formula for the average, or arithmetic mean, is average = sum of values / number of values.
Let’s look at a more challenging example.
Example 1: Average — Missing Value
Five basketball players scored points in last night’s game. The average number of points scored was 16. Jeremiah, Winston, Lonnie, and Morris scored 7, 19, 30, and 12 points, respectively. How many points did Alfred, the fifth player, score?
12
13
16
17
Solution:
We know that the average of the 5 players is 16. To find the sum of the points scored by all five players, we can use the formula for the average:
average = sum of values / number of values
16 = sum / 5
80 = sum
We see that the sum of the 5 players’ scores is equal to 80. If we let x represent Alfred’s unknown score, we can express the sum as:
80 = 7 + 19 + 30 + 12 + x
We can then solve for x:
80 = 68 + x
12 = x
We see that Alfred scored 12 points.
Answer: A
In this next example, the values contain variables. We can still use the formula for the average.
Example 2: Average — Variable Values
The arithmetic mean of (2x + 5), (x + 2), 3, and x is 8. What is the value of x?
2
4.5
5.5
6
Solution:
We have 4 terms with an arithmetic mean (average) of 8. Let’s use the formula for the average:
average = sum of values / number of values
8 = [(2x + 5) + (x + 2) + 3 + x] / 4
32 = 4x + 10
22 = 4x
5.5 = x
Answer: C
Earn a Higher SAT Math Score Start Studying With TTP Today! TRY OUR SAT MATH COURSE FOR FREE!
Let’s now consider the weighted average.
The Weighted Average
The SAT doesn’t always test you on the easiest version of a concept. Rather than being asked to compute a simple average, you might be given a question that tests you on the weighted average. Let’s see how this works.
Consider the following 6 values: 1, 1, 7, 7, 7, 7.
If we were asked to calculate the average, we could use the average formula:
average = sum of values / number of values
average = (1 + 1 + 7 + 7 + 7 + 7) / 6 = 30 / 6 = 5
Another way of computing the average would be to note that there are two 1s and four 7s. We could then use multiplication to find the sum:
average = sum of values / number of values
average = [(2 x 1) + (4 x 7)] / 6 = (2 + 28) / 6 = 5
The Weighted Average Formula
When we have repeated values and we use multiplication to make the arithmetic easier, we are calculating the weighted average, or weighted mean. We can use the following formula:
weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
The formula tells us to multiply each value by its frequency, then add the products, and finally divide by the number of values (the sum of the frequencies).
Consider the following question:
During a fund drive, six donors gave $50 each, three donors gave $100 each, and four donors gave $175 each. What was the average donation?
Solution:
We can use the weighted average formula:
weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
weighted average = [(6)(50) + (3)(100) + (4)(175)] / (6 + 3 + 4)
weighted average = (300 + 300 + 700) / 13
weighted average = 1300 / 13 = 100
The average donation is $100.
KEY FACT:
Weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
Let’s consider a more challenging example.
Example 3: Weighted Average
A team of ten wildlife biologists counted the number of sea turtle nests at various locations in the Southeast, and the average number of nests reported was 420. Five biologists reported 350 nests each and four biologists reported 500 nests each. How many nests were reported by the tenth biologist?
400
420
435
450
Solution:
Let’s let x = the number of nests reported by the tenth biologist. We can then use the weighted average formula:
weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
420 = [(5)(350) + (4)(500) + (1)(x)] / (10)
4200 = (1750 + 2000 + x)
4200 = 3750 + x
x = 450
Answer: D
The Median
Another measure of the center of a set is the median. The median is found by ordering the values in a set and finding the middle value. The following mini-examples illustrate how to manually determine the value of the median.
Median Mini-Example 1
What is the median of the set {6, 10, 13, 1, 4, 3, 18}?
Solution:
We first list the values in ascending order.
1 3 4 6 10 13 18
Because there are 7 values (an odd number), we know that the median is the middle value, with three values below it and three values above it. Thus, the median is 6.
If a set has an even number of values, there is no single middle value. As a result, we must calculate the average of the two middle values.
Median: Mini-Example 2
What is the median of the set {20, 1, 6, 14, 13, 9}?
Solution:
First, we put the values in ascending order:
1 6 9 13 14 20
Because we have an even number of values, there is no single middle value. So, we must calculate the average of the two middle values to determine the median:
Median = (9 + 13) / 2 = 22 / 2 = 11
The median is 11.
The technique we just performed is fast and easy if we have a small data set. But if our set contains a large number of values, this technique is time-consuming. We need an easier method for determining which value is the middle one.
Below is a shortcut formula for finding which value is the median:
Position of median = (n + 1) / 2
In this formula, n = the number of values. Be sure to note that this formula does not compute the value of the median; instead, it indicates which value is the median. Let’s look at an example question that uses this formula to identify the median.
Example 4: Median — Large Data Set
Members of the school band sold tickets to an upcoming concert. Eight students sold 12 tickets each, fourteen students sold 20 tickets each, ten students sold 25 tickets each, and five students sold 50 tickets each. What is the median number of tickets sold by the band members?
9
19
20
29
Solution:
First, we note that the number of students is 8 + 14 + 10 + 5 = 37. Thus, n = 37. We can use the formula for the position of the median to determine the number of tickets that is the median value.
Position of median = (n + 1) / 2 = (37 + 1) / 2 = 38 / 2 = 19
The median is the 19th value in the list when the values are ordered.
We see that 8 students (the 1st through the 8th student) sold 12 tickets each. The next 14 students, who sold 20 tickets each, include the 9th through the 22nd student. Since the median is the 19th value, or the number of tickets sold by the 19th student, we see that this student sold 20 tickets.
Answer: C
KEY FACT:
For large data sets, it may be easier to determine the position of the median by using the formula: position of median = (n + 1) / 2.
The Mode
In a set of values, the mode is the one that occurs most often. The mode is sometimes confusing because a set can have no mode, one mode, or multiple modes. Let’s look at how this works.
What is the mode of the set {7, 5, 2, 1, 4, 7, 4, 4}?
Here, the mode is 4. It occurs three times.
What is the mode of the set {1, 12, 9, 2, 3, 7, 15}?
Here, there is no mode: no value occurs more often than any other.
What is the mode of the set {4, 4, 2, 11, 6, 7, 11, 7}?
Here, there are three modes: 4, 7, and 11. Each occurs twice.
KEY FACT:
A set can have no mode, one mode, or multiple modes.
On the SAT, you will probably not be asked such basic questions about the mode as those posed above. Let’s look at a question about the mode that we might encounter on the actual exam, using a scenario that we encountered earlier in this article.
Example 5: Mode
Members of the school band sold tickets to an upcoming concert. Eight students sold 12 tickets each, fourteen students sold 20 tickets each, ten students sold 25 tickets each, and five students sold 50 tickets each. What is the mode of this distribution?
12
14
20
25
Solution:
If we listed the numbers of tickets sold by the 37 students, we would have:
12, 12, 12, 12, 12, 12, 12, 12, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50.
We see that 20 occurs more often than any other value, so 20 is the mode.
Rather than listing the numbers individually, it is more efficient to note that the value with the greatest frequency is the mode. From the original problem, we are told that fourteen students sold 20 tickets each. Thus, since 14 is the greatest frequency, we know that 20 is the mode.
Answer: C
The mean, median, and mode are the three main measures of the center of a set because they focus on the middle of a distribution. Let’s now consider what are called “measures of dispersion,” which describe how the data are spread around the center of a set. The two main measures of spread are the range and the standard deviation.
Earn a Higher SAT Math Score Start Studying With TTP Today! TRY OUR SAT MATH COURSE FOR FREE!
The Range
The range is the difference between the greatest value and the least value in a data set. A small range generally indicates that the values are relatively close to the center, whereas a large range generally indicates that the values are more widely spread around the center.
KEY FACT:
Range = highest value – lowest value
Let’s do a practice problem.
Example 6: Range
The quarterback of a high school football team completed 16, 12, 22, and 10 passes, respectively, in the first four games of the season. After the fifth game, the team statistician stated that the range of the number of completed passes for the season had doubled, compared to the range of just the first four games. How many passes did the quarterback complete during the fifth game?
14
24
34
44
Solution:
First, let’s calculate the range for the first four games of the season:
Range = highest value – lowest value
Range = 22 – 10 = 12
We know that after the fifth game, the range of the number of passes completed doubled. Thus, 24 is the range after the fifth game.
We know that the new range is 24 and the lowest value is 10. If we let x = the number of completed passes during the fifth game, we have:
Range = highest value – lowest value
24 = x – 10
34 = x
Answer: C
The last statistical measure is also a measure of the spread of a set of data: the standard deviation.
The Standard Deviation
The standard deviation is another way to measure the spread of values in a set. More specifically, the standard deviation measures how far individual data points are from the mean of the set. It is more widely used by statisticians than the range.
The SAT tests you on limited aspects of the standard deviation. For example, you will not be required to calculate the actual value of the standard deviation, but you must have a basic knowledge of its properties.
KEY FACT:
The standard deviation is a measure of how far individual data points are from the average of the set.
Standard Deviation Facts
Several facts about the standard deviation can be used to help us answer SAT questions:
If all the values in a set are the same, the standard deviation is 0.
The standard deviation can never be negative.
The farther away the values are from the mean, the greater the standard deviation.
TTP PRO TIP:
Know the basic standard deviation facts.
Example 7: Standard Deviation
Consider the following three sets:
Set A: {8, 9, 10, 13}
Set B: {10, 10, 10, 10}
Set C: {0, 0, 20, 20}
Which of the following statements must be true?
I. The averages of the three sets are equal
II. The standard deviations of all three sets are positive
III. Set C has the greatest standard deviation.
I and II only
I and III only
II and III only
I, II, and III
Solution:
Let’s consider each statement individually.
Statement I
The mean of Set A is (8 + 9 + 10 + 13) / 4 = 40 / 4 = 10.
The mean of Set B is (10 + 10 + 10 + 10) = 40 / 4 = 10.
The mean of Set C is (0 + 0 + 20 + 20) / 4 = 40 / 4 = 10.
Statement I is true.
Statement II
Recall that if all the values in a set are equal, the standard deviation is 0. Thus, the standard deviation of Set B, in which all values are equal to 10, is 0, which is not a positive number.
Statement II is not true.
Statement III
We have already established that the standard deviation of Set B is 0. Now, if we compare the values in Set A to its mean of 10 and the values in Set C to its mean of 10, we see that the individual values in Set C are much farther from the mean of 10 than the values in Set A are. Thus, Set C has the greatest standard deviation.
Statement III is true
Answer: B
Summary: SAT Statistics Problems
In this article, we covered six statistics SAT concepts that you need to know, and we have provided practice problems for each.
The measures of central tendency include the average (or arithmetic mean), the weighted average, the median, and the mode. Measures of spread include the range and the standard deviation. Being familiar with these basic statistical measures will aid you significantly in answering statistics questions on the SAT quickly and accurately.
There are additional SAT statistics review topics, including probability, box plots, linear regression, sampling methods, statistical significance, and margin of error, that were not covered here. But you can access them, and all SAT topics, in our SAT self-study course.
What’s Next?
Statistics is just one of the 20+ topics in the SAT Math section. You want a great score on the exam, so it’s important to spread your time, energy, and focus over all the topics. And you can read our article about improving your math score on the SAT to get some useful score-enhancing tips for getting a great score.
Good luck!
Earn a Higher SAT Math Score Start Studying With TTP Today! TRY OUR SAT MATH COURSE FOR FREE!
Share
You May Also Like...
Digital SAT Math Questions
Hardest SAT Math Questions
Linear Function Questions on the SAT
About The Author
Jeff
Jeffrey Miller is the head SAT instructor for Target Test Prep. Jeff has more than 14 years of experience in the business of helping students with low SAT scores hurdle the seemingly impossible and achieve the scores they need to get into the top 20 colleges in the world. Jeff has cultivated many successful college graduates through his SAT instruction, and will be a pivotal resource for many more to follow.
Leave a Reply Cancel Reply
Share
Tweet
WhatsApp
Share |
11713 | https://physics.stackexchange.com/questions/52837/when-do-i-apply-significant-figures-in-physics-calculations | Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
When do I apply Significant figures in physics calculations?
Ask Question
Asked
Modified 4 years, 6 months ago
Viewed 20k times
2
$\begingroup$
I'm a little confused as to when to use significant figures for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.
Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$
Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$
Should I round $125.664~\mathrm{m}/\mathrm{s}$ to $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two?
error-analysis
estimation
Share
Improve this question
edited Jun 18, 2013 at 18:06
Qmechanic♦
222k5252 gold badges636636 silver badges2.6k2.6k bronze badges
asked Feb 2, 2013 at 0:15
ScottScott
17322 gold badges22 silver badges44 bronze badges
$\endgroup$
3
1
$\begingroup$ mathematical-physics probably isn't the right tag for this. But I don't know what is, mathematics isn't appropriate either. $\endgroup$
tpg2114
– tpg2114
2013-02-02 00:23:04 +00:00
Commented Feb 2, 2013 at 0:23
$\begingroup$ @tpg2114 I agree, I couldn't find an appropriate tag for this question. $\endgroup$
Scott
– Scott
2013-02-02 00:25:54 +00:00
Commented Feb 2, 2013 at 0:25
$\begingroup$ Related: physics.stackexchange.com/q/32078/2451 and links therein. $\endgroup$
Qmechanic
– Qmechanic ♦
2013-06-18 18:06:16 +00:00
Commented Jun 18, 2013 at 18:06
Add a comment |
4 Answers 4
Reset to default
8
$\begingroup$
You should always find an answer that is a formula, and then only apply significant figures once you get to the one final step of substituting your numbers back into the problem in place of variables.
Avoid multiple intermediate steps of substituting numbers at all costs. Not only will this save your pencil a lot of work, but it will also cause your answer to be more accurate, as rounding errors can pile up, even when using a calculator.
Share
Improve this answer
answered Feb 2, 2013 at 0:20
Zo the RelativistZo the Relativist
42.3k22 gold badges8080 silver badges147147 bronze badges
$\endgroup$
4
$\begingroup$ So the correct answer would just be 130 m/s then, right? $\endgroup$
Scott
– Scott
2013-02-02 00:23:42 +00:00
Commented Feb 2, 2013 at 0:23
$\begingroup$ @Scotty Yes, if the 500 and 50 given to you each had at least 2 sig figs. Some conventions are "50" has one sig fig (so your answer should round to 100) and "50." (note the decimal) is more precise. You should check with your instructor/textbook for what the conventions are. $\endgroup$
user10851
– user10851
2013-02-02 00:37:05 +00:00
Commented Feb 2, 2013 at 0:37
1
$\begingroup$ This is good advice, but sometimes it makes sense to, e.g., write down an intermediate result so that you can check whether it's reasonable, because it's of some interest on its own, or because you're doing calculations in your head or on paper without your calculator handy. (Or for those of us old enough to remember slide rules ...) In this situation, you need to keep enough extra sig figs to avoid accumulating significant rounding errors. If there are 10 arithmetic operations, then we expect that roughly 1 extra sig fig is needed. If 100 operations, 2 extra sig figs, etc. $\endgroup$
user4552
– user4552
2013-06-19 00:50:43 +00:00
Commented Jun 19, 2013 at 0:50
$\begingroup$ The other thing to point out to the OP is the reason for sig fig rules. The reason is to avoid miscommunicating to other people the precision of your result, e.g., in the CIA fact book that gives the population of Nigera to 8 sig figs. If intermediate steps aren't being communicated to anyone else, then their sig figs aren't an issue. $\endgroup$
user4552
– user4552
2013-06-19 00:52:07 +00:00
Commented Jun 19, 2013 at 0:52
Add a comment |
1
$\begingroup$
Keep precision all the way through to the number you report and then truncate accordingly at the end.
Share
Improve this answer
answered Feb 2, 2013 at 0:20
tpg2114tpg2114
16.9k66 gold badges5555 silver badges8787 bronze badges
$\endgroup$
1
1
$\begingroup$ No, this is silly. What if your intermediate result is $\sqrt{2}$? Are you going to write down infinitely many sig figs, or keep infinitely many sig figs in the memory of your calculator? If you're going to record intermediate numerical results, then it makes sense to keep enough extra sig figs (typically 1 or 2) to keep from accumulating significant rounding errors. $\endgroup$
user4552
– user4552
2013-06-19 00:46:45 +00:00
Commented Jun 19, 2013 at 0:46
Add a comment |
0
$\begingroup$
I have been taught and continue to teach that you should record (and use) the answers to your intermediate calculations (and/or conversions) with one extra significant digit beyond the number of significant digits that will be in your final answer.
Share
Improve this answer
answered Oct 4, 2018 at 23:33
NickiNicki
11
$\endgroup$
2
$\begingroup$ I don't think the question is about intermediate calculations, but about how to express the final result. Right? $\endgroup$
AccidentalFourierTransform
– AccidentalFourierTransform
2018-10-04 23:53:52 +00:00
Commented Oct 4, 2018 at 23:53
$\begingroup$ What is the reason that you think this rule is correct? Just stating the rule without a reason isn't very helpful. This rule is not correct in some cases. For example, if you have a complicated calculation with hundreds of intermediate numerical steps, then one extra sig fig might be insufficient. $\endgroup$
user4552
– user4552
2018-10-05 01:27:10 +00:00
Commented Oct 5, 2018 at 1:27
Add a comment |
0
$\begingroup$
Just revisited this in chemistry this year. You actually have to apply the correct number of significant figures based on the rules of operations (addition/subtraction or multiplication/division) as you perform each calculation in a problem with multiple calculations. The purpose of using significant figures is not to get accurate results or results closest to what a calculator would get. The point of using significant figures is not to mislead the reader or person following your work in thinking that you used exact numbers when in fact you did not. So the more steps you have in a problem, the further your answer will be from the answer a calculator would derive. If you apply the correct number of significant figures in every step, you are accurately accounting for what you actually MEASURED (this is why they are applied to measured numbers, not exact numbers). And if youre sitting somewhere computing the problem, you did not measure anything. We also do not just chose to use an extra significant figure or two but have to stick to the rules of operation. ie- least amount of significant figures for multiplication and division and lowest decimal place for addition and subtraction.
Hope the link above with consistent explanations and videos helps.
Share
Improve this answer
answered Mar 31, 2021 at 9:23
KimKim
1
$\endgroup$
2
$\begingroup$ So above, the correct work to show would be 6283 (4 sig fig)/ 50s $\endgroup$
Kim
– Kim
2021-03-31 09:27:38 +00:00
Commented Mar 31, 2021 at 9:27
$\begingroup$ So above, the correct work to show would be 6283 (4 sig fig) / 50s (2 sig fig)= 125.66, which becomes 130 ( 2 sig fig). This also turns out to be the same answer even if you dont use the appropriate significant figures in the first step but that wont always be the case so its best to practice the right way so that you always get the questions right on an exam, if your professor uses the answers from the textbook and doesnt edit them or formulate problems without using the correct rules. $\endgroup$
Kim
– Kim
2021-03-31 09:33:45 +00:00
Commented Mar 31, 2021 at 9:33
Add a comment |
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
error-analysis
estimation
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Linked
Significant figure rules
1 What is the correct way to handle significant figures when calculating compound uncertainties?
Significant Figures
Related
1 Significant Figures in Physics
1 # significant figures in calculations?
1 Do Significant Figures apply with numbers without units?
1 Significant figures in calculations
0 When to use Significant Figures?
3 Determining the appropriate number of significant figures to report in least squares analyses
2 Significant figures in simple calculations
Hot Network Questions
How do you emphasize the verb "to be" with do/does?
Is this commentary on the Greek of Mark 1:19-20 accurate?
Are there any alternatives to electricity that work/behave in a similar way?
Can a GeoTIFF have 2 separate NoData values?
Copy command with cs names
Can Monks use their Dex modifier to determine jump distance?
The geologic realities of a massive well out at Sea
is the following argument valid?
Numbers Interpreted in Smallest Valid Base
Singular support in Bezrukavnikov's equivalence
How can blood fuel space travel?
Lingering odor presumably from bad chicken
Calculating the node voltage
Quantizing EM field by imposing canonical commutation relations
How random are Fraïssé limits really?
Traversing a curve by portions of its arclength
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
I'm having a hard time intuiting throttle position to engine rpm consistency between gears -- why do cars behave in this observed way?
Determine which are P-cores/E-cores (Intel CPU)
Is the cardinality of a set equal to the cardinality of the set of all smaller cardinalities?
How to start explorer with C: drive selected and shown in folder list?
Verify a Chinese ID Number
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
Data lost/Corrupted on iCloud
more hot questions
Question feed |
11714 | https://miamioh.ecampus.com/schaums-outline-laplace-transforms-1st/bk/9780070602311?srsltid=AfmBOoosGIdW3-0Fk0xA33JobkNY0UM73MVzDGobEE9qn-DmLsHmVhMK | Schaum's Outline of Laplace Transforms
Skip Navigation
MIAMI UNIVERSITY OFFICIAL BOOKSTORE
Login/Sign Up
Home
Miami Alumni
Miami Athletics
MENU
Shop
Textbooks
Campus Locations
Login/Sign Up
Back|
Campus Locations
Home
Home
Miami Alumni
Miami Alumni
Miami Athletics
Miami Athletics
Back|
Shop
Clothing
Clothing
Accessories
Accessories
Gifts
Gifts
Graduation
Graduation
Supplies
Supplies
Back|
Clothing
Kids
Men
Sweatshirts
Women
View All
Back|
Accessories
For You
For Your Car
For Your Home
For Your Pet
For Your Tech
View All
Back|
Gifts
Artwork
Cooking Essentials
Games
Gift Wraps
Holiday
Home Decor
Mascot
Office Decor
Outdoor/Recreation
View All
Back|
Graduation
Graduation Gear
Graduation Gifts
View All
Back|
Supplies
Art Supplies
For Your Office
Medical Supplies
Office Supplies
School Supplies
View All
Kids
Back|
Kids
Bibs
Bottoms
Dresses
Headwear
Hoodies
Matching Sets
Onesies
Shirts
Sweatshirts
Men
Back|
Men
Bottoms
Footwear
Hoodies
Jerseys
Mens Apparel
Outerwear
Polos
Shirts
Sweatshirts
T-Shirts
Sweatshirts
Back|
Sweatshirts
Women
Back|
Women
Bottoms
Dresses
Headwear
Hoodies
Outerwear
Pants
Shirts
Sweatshirts
Undergarments
View All Clothing >
Miami Merger
Myaamia Heritage Collection
Alumni Collection
Miami Regionals
Custom Items
Miami Cradle of Coaches collection
For You
Back|
For You
Backpacks
Bags
Buttons
Drinkware
Fan Gear
Flags
Hair Accessories
Headwear
ID Holders
Jewelry
Keychains
Knitwear
Lanyards
Lapel Pins
Pennants
Socks
Ties
Umbrellas
Wallets
For Your Car
Back|
For Your Car
Decals
License Plates
For Your Home
Back|
For Your Home
Banners
Blankets
Cooking Essentials
Decals
Drinkware
Flags
Frames
Home Decor
Magnets
Pillows
Signs
Stickers
For Your Pet
Back|
For Your Pet
Beds
Bowls
Charms
Clothes
Toys
For Your Tech
Back|
For Your Tech
Computer Accessories
Phone Accessories
View All Accessories >
Artwork
Back|
Artwork
Ornaments
Wall Art
Cooking Essentials
Back|
Cooking Essentials
Food
Games
Back|
Games
Balls
Puzzles
Gift Wraps
Back|
Gift Wraps
Gift Bags
Holiday
Back|
Holiday
Ornaments
Stocking
Home Decor
Back|
Home Decor
Candles
Mascot
Back|
Mascot
Plushies
Office Decor
Back|
Office Decor
Desk Accessories
Outdoor/Recreation
Back|
Outdoor/Recreation
Tailgate
View All Gifts >
Graduation Gear
Back|
Graduation Gear
Caps and Gowns
Hoods
Stoles
Study Abroad Sashes
Tassel
Tassels
Graduation Gifts
Back|
Graduation Gifts
Diploma Frames
Yard Signs
View All Graduation >
Art Supplies
Back|
Art Supplies
Art Supply Products
For Your Office
Back|
For Your Office
Desk Accessories
Pens
Medical Supplies
Back|
Medical Supplies
Kits
Office Supplies
Back|
Office Supplies
Padfolios
School Supplies
Back|
School Supplies
Folders
Notebooks
Planners
View All Supplies >
Erin Condren
Shopping Cart (0)
Shop
Textbooks
Search
Shopping Cart (0)
Write a Review
Schaum's Outline of Laplace Transforms
bySpiegel, Murray
Edition: 1st
ISBN13: 9780070602311
ISBN10: 007060231X
Format: Paperback
Pub. Date: 1965-06-22
Publisher(s): McGraw Hill
Other versions by this Author
This Item Qualifies for Free Shipping! Excludes marketplace orders.
List Price: ~~$26.25~~
Rent Book
Select for Price
Add to Cart
There was a problem. Please try again later.
Buy New
Arriving Soon. Will ship when available.
$25.00
Add to Cart
Used Book
We're Sorry
Sold Out
eBook
We're Sorry
Not Available
Buy from our Marketplace starting at $3.39
How Marketplace Works:
This item is offered by an independent seller and not shipped from our warehouse
Item details like edition and cover design may differ from our description; see seller's comments before ordering.
Sellers much confirm and ship within two business days; otherwise, the order will be cancelled and refunded.
Marketplace purchases cannot be returned to eCampus.com. Contact the seller directly for inquiries; if no response within two days, contact customer service.
Additional shipping costs apply to Marketplace purchases. Review shipping costs at checkout.
Currently unavailable
Summary
This book proveds a clear, concise explanation of Laplace transforms, or operational calculus--now necessary knowledge for engineers, physicists, mathematicians and other scientists. Definitions, principles and theorems are illustrated and reinforced by hundreds of fully worked out problems. Also included are chapters on complex variable theory and Fourier series and integrals, so important to discussions of the complex inversion formula.
Table of Contents
The Laplace Transform.
The Inverse Laplace Transform.
Applications to Differential Equations.
Applications to Integral and Difference Equations.
Complex Variable Theory.
Fourier Series and Integrals.
The Complex Inversion Formula.
Applications to Boundary-Value Problems.
Appendix A: Table of General Properties of Laplace Transforms.
Appendix B: Table of Special Laplace Transforms.
Appendix C: Table of Special Functions.
We are currently experiencing difficulties. Please try again later.
We are currently experiencing difficulties. Please try again later.
An electronic version of this book is available through VitalSource.
This book is viewable on PC, Mac, iPhone, iPad, iPod Touch, and most smartphones.
By purchasing, you will be able to view this book online, as well as download it, for the chosen number of days.
Digital License
You are licensing a digital product for a set duration. Durations are set forth in the product description, with "Lifetime" typically meaning five (5) years of online access and permanent download to a supported device. All licenses are non-transferable.
More details can be found here.
A downloadable version of this book is available through the eCampus Reader or compatible Adobe readers.
Applications are available on iOS, Android, PC, Mac, and Windows Mobile platforms.
Please view the compatibility matrix prior to purchase.
We are currently experiencing difficulties. Please try again later.
TEXTBOOKS
TEXTBOOKS
Order Textbooks
Sell Textbooks
My Account
Gift Cards
Price Match Opens in New Tab
Help
APPAREL
APPAREL
Men
Women
Kids
GIFTS & ACCESSORIES
GIFTS & ACCESSORIES
All Gifts
All Accessories
LOCATION
LOCATION
Brick and Ivy Campus Store
701 E Spring St
Oxford, OH 45056
513-529-2600
HOURS
HOURS
M-F: 8:30AM - 5PM
SAT: 10AM - 4PM
ABOUT US
ABOUT US
Terms & Conditions
Privacy Policy
Accessibility
CONTACT US
Copyright
1995 - 2025
Download PDF Reader Opens in New Tab
Please wait while the item is added to your cart...
We are currently experiencing difficulties. Please try again later. |
11715 | https://math.gordon.edu/ntic/ntic/chapter-linear-congruences.html | NTIC Linear Congruences
Skip to main content
Number Theory:In Context and Interactive
Karl-Dieter Crisman
ContentsIndex
Search Book
close
Search Results:
No results.
PrevUpNext
Front Matter
Colophon
About the Author
Dedication
Acknowledgements
To Everyone
To the Student
To the Instructor
1 Prologue
1.1 A First Problem
1.2 Review of Previous Ideas
1.2.1 Well-Ordering
1.2.2 Induction
1.2.3 Divisibility
1.3 Where are we going?
1.4 Exercises
1.5 Using Sage for Interactive Computation
2 Basic Integer Division
2.1 The Division Algorithm
2.1.1 Statement and examples
2.1.2 Proof of the Division Algorithm
2.1.3 Uses of the division algorithm
2.2 The Greatest Common Divisor
2.3 The Euclidean Algorithm
2.4 The Bezout Identity
2.4.1 Backwards with Euclid
2.4.2 Proving the final characterization
2.4.3 Other gcd questions
2.4.4 Relatively prime
2.5 Exercises
3 From Linear Equations to Geometry
3.1 Linear Diophantine Equations
3.1.1 If c is not a multiple of gcd(a,b)
3.1.2 If a or b is zero
3.1.3 If c=gcd(a,b)
3.1.4 If c is a nontrivial multiple of the gcd
3.2 Geometry of Equations
3.3 Positive Integer Lattice Points
3.3.1 Solution ideas
3.3.2 Toward the full solution
3.4 Pythagorean Triples
3.4.1 Definition
3.4.2 Characterizing Pythagorean triples
3.4.2.1 Preliminaries
3.4.2.2 An intricate argument
3.4.2.3 The punch line
3.4.3 Areas of Pythagorean triangles
3.4.3.1 Which areas are possible?
3.4.3.2 Which areas are square?
3.5 Surprises in Integer Equations
3.6 Exercises
3.7 Two facts from the gcd
4 First Steps with Congruence
4.1 Introduction to Congruence
4.2 Going Modulo First
4.3 Properties of Congruence
4.4 Equivalence classes
4.5 Why modular arithmetic matters
4.5.1 Starting to see further
4.5.2 Taking powers
4.6 Toward Congruences
4.7 Exercises
5 Linear Congruences
5.1 Solving Linear Congruences
5.2 A Strategy For the First Solution
5.3 Systems of Linear Congruences
5.3.1 Introducing the Chinese Remainder Theorem
5.3.2 The inverse of a number
5.4 Using the Chinese Remainder Theorem
5.4.1 Constructing simultaneous solutions
5.4.2 A theoretical but highly important use of CRT
5.5 More Complicated Cases
5.5.1 Moduli which are not coprime
5.5.2 The case of coefficients
5.5.3 A practical application
5.6 Exercises
6 Prime Time
6.1 Introduction to Primes
6.1.1 Definitions and examples
6.1.2 Prime fun
6.2 To Infinity and Beyond
6.2.1 Infinite primes
6.2.2 The sieve of Eratosthenes
6.3 The Fundamental Theorem of Arithmetic
6.3.1 Preliminaries and statement
6.3.2 Proof of the FTA
6.4 First consequences of the FTA
6.5 Applications to Congruences
6.5.1 Factoring the modulus
6.5.2 Moduli that are prime powers
6.6 Exercises
7 First Steps With General Congruences
7.1 Exploring Patterns in Square Roots
7.2 From Linear to General
7.2.1 Combining solutions
7.2.2 Prime power congruences
7.3 Congruences as Solutions to Congruences
7.4 Polynomials and Lagrange’s Theorem
7.5 Wilson’s Theorem and Fermat’s Theorem
7.5.1 Wilson’s Theorem
7.5.2 Fermat’s Little Theorem
7.6 Epilogue: Why Congruences Matter
7.7 Exercises
7.8 Counting Proofs of Congruences
7.8.1 Counting motivation
7.8.2 The combinatorial proofs
8 The Group of Integers Modulo n
8.1 The Integers Modulo n
8.1.1 Definition
8.1.2 Visualization
8.1.3 Inverses
8.2 Powers
8.2.1 Returning to visualizing
8.3 Essential Group Facts for Number Theory
8.3.1 Step-by-step notions to the definition
8.3.1.1 Sets
8.3.1.2 Binary operations
8.3.1.3 Closed operations
8.3.1.4 Associative operations
8.3.1.5 Identity
8.3.1.6 Inverses
8.3.2 What is a group?
8.3.3 Properties of groups we will need
8.3.3.1 Solutions to equations
8.3.3.2 Inverses of product
8.3.3.3 Finite groups
8.3.3.4 Order of a group
8.3.3.5 Order of an element
8.3.3.6 The connection
8.3.3.7 Cyclic groups
8.3.3.8 Abelian groups
8.4 Exercises
9 The Group of Units and Euler’s Function
9.1 Groups and Number Systems
9.1.1 Solving linear equations – again
9.1.2 A new group
9.1.2.1 The group of units
9.1.2.2 More facts and examples
9.2 The Euler Phi Function
9.2.1 Euler’s theorem
9.3 Using Euler’s Theorem
9.3.1 Inverses
9.3.2 Using Euler’s theorem with the CRT
9.4 Exploring Euler’s Function
9.5 Proofs and Reasons
9.5.1 Computing prime powers
9.5.2 Multiplicativity
9.5.3 Addition Formula
9.5.4 Even more questions
9.6 Exercises
9.7 The Conductor, solved
10 Primitive Roots
10.1 Primitive Roots
10.1.1 Definition
10.1.2 Two characterizations
10.1.3 Finding primitive roots
10.2 A Better Way to Primitive Roots
10.2.1 A useful lemma
10.2.2 Using the test lemma
10.3 When Does a Primitive Root Exist?
10.3.1 Primitive roots of powers of two
10.3.2 Two important lemmas
10.3.2.1 How the lemmas work
10.3.2.2 How the lemmas (don’t) fail
10.4 Prime Numbers Have Primitive Roots
10.5 A Practical Use of Primitive Roots
10.5.1 Finding a higher root
10.5.2 Discrete logarithms
10.6 Exercises
10.7 All the Primitive Roots
11 An Introduction to Cryptography
11.1 What is Cryptography?
11.1.1 Encoding and decoding
11.2 Encryption
11.2.1 Simple ciphers
11.2.2 Decryption and inverses
11.2.3 Getting more sophisticated
11.2.4 Linear algebra and encryption
11.2.5 Asymmetric key cryptography
11.3 A Modular Exponentiation Cipher
11.3.1 The Diffie-Hellman method
11.3.2 A bigger example
11.3.3 Recap
11.3.4 A brief warning
11.4 An Interesting Application: Key Exchange
11.4.1 Diffie-Hellman Key Exchange
11.4.2 In the Middle
11.5 RSA Public Key
11.5.1 The background
11.5.2 The practice of RSA
11.5.3 Why RSA works
11.6 RSA and (Lack Of) Security
11.6.1 Beating the man in the middle
11.6.2 A cautionary tale
11.6.3 The explanation
11.6.4 A solution
11.7 Other applications
11.7.1 Secret sharing
11.8 Exercises
12 Some Theory Behind Cryptography
12.1 Finding More Primes
12.1.1 Fermat primes
12.1.2 Primes from Fermat numbers
12.1.3 Mersenne primes
12.1.4 Primes from Mersenne numbers
12.2 Primes – Probably
12.2.1 Pseudoprimes
12.2.2 Prime impostors, and how to avoid them
12.3 Another Primality Test
12.3.1 Another pattern
12.3.2 Miller’s test
12.4 Strong Pseudoprimes
12.5 Introduction to Factorization
12.5.1 Factorization and the RSA
12.5.2 Trial division
12.5.3 Starting in the middle
12.6 A Taste of Modernity
12.6.1 The Pollard Rho algorithm
12.6.2 More factorization
12.7 Exercises
13 Sums of Squares
13.1 Some First Ideas
13.1.1 A first pattern
13.1.2 Geometry
13.1.3 Connections to some very old mathematics
13.2 At Most One Way For Primes
13.3 A Lemma About Square Roots Modulo n
13.4 Primes as Sum of Squares
13.4.1 A useful plot
13.4.2 Primes which are sums of squares
13.4.3 Visualizing the proof
13.4.4 Finishing the proof
13.5 All the Squares Fit to be Summed
13.6 A One-Sentence Proof
13.7 Exercises
14 Beyond Sums of Squares
14.1 A Complex Situation
14.1.1 A new interpretation
14.1.2 Revisiting the norm
14.1.3 A different approach to sums of squares
14.2 More Sums of Squares and Beyond
14.2.1 Summing more squares
14.2.2 Beyond squares
14.2.3 Waring’s problem
14.3 Related Questions About Sums
14.4 Exercises
15 Points on Curves
15.1 Rational Points on Conics
15.1.1 Rational points on the circle
15.1.2 Parametrization in general
15.1.3 When curves don’t have rational points
15.2 A tempting cubic interlude
15.3 Bachet and Mordell Curves
15.3.1 Verifying points don’t exist
15.3.2 More on Mordell
15.4 Points on Quadratic Curves
15.4.1 Transforming conic sections
15.4.2 More conic sections
15.5 Making More and More and More Points
15.5.1 Toward integer points
15.5.2 A surprising application
15.6 The Algebraic Story
15.6.1 Computing the hyperbola
15.6.2 Yet more number systems
15.6.3 The general solution (any n)
15.7 Exercises
16 Solving Quadratic Congruences
16.1 Square Roots
16.1.1 Recalling existing answers
16.1.2 Finding more answers
16.2 General Quadratic Congruences
16.2.1 Completing the square solves our woes
16.3 Quadratic Residues
16.3.1 Some definitions
16.3.2 First try for new square roots
16.3.3 Some history
16.4 Send in the Groups
16.4.1 Quadratic residues form a group
16.4.2 Quadratic residues connect to primitive roots
16.5 Euler’s Criterion
16.6 Introducing the Legendre Symbol
16.7 Our First Full Computation
16.8 Exercises
17 Quadratic Reciprocity
17.1 More Legendre Symbols
17.2 Another Criterion
17.2.1 Laying the foundation
17.2.2 Getting the new criterion
17.2.3 The final form
17.3 Using Eisenstein’s Criterion
17.4 Quadratic Reciprocity
17.4.1 The theorem
17.4.2 Why is this theorem different from all other theorems?
17.4.2.1 What does it mean?
17.4.2.2 What does it do?
17.4.2.3 The Jacobi symbol
17.5 Some Surprising Applications of QR
17.5.1 Factoring, briefly
17.5.2 Primality testing
17.5.3 Yes, even cryptography
17.5.4 Solving equations
17.5.5 Artin’s conjecture
17.6 A Proof of Quadratic Reciprocity
17.6.1 Re-enter geometry
17.6.2 Proving proper parity
17.6.3 Postlude
17.7 Exercises
18 An Introduction to Functions
18.1 Three Questions for Euler phi
18.1.1 Formulas
18.1.2 Relations
18.1.3 Summation (and limits)
18.2 Three Questions, Again
18.2.1 Formulas
18.2.2 Relations
18.2.3 Limits (and summation)
18.3 Exercises
19 Counting and Summing Divisors
19.1 Exploring a New Sequence of Functions
19.2 Conjectures and Proofs
19.2.1 Prime powers
19.2.2 Multiplicativity
19.2.3 A very powerful lemma
19.3 The Size of the Sum of Divisors Function
19.4 Perfect Numbers
19.4.1 A perfect definition and theorem
19.4.2 Speculation and more terminology
19.4.3 The abundancy index
19.4.4 Amicable Numbers
19.5 Odd Perfect Numbers
19.5.1 Are there odd perfect numbers?
19.5.2 The abundancy index and odd perfect numbers
19.5.3 Even more about odd perfect numbers, if they exist
19.6 Exercises
20 Long-Term Function Behavior
20.1 Sums of Squares, Once More
20.1.1 Errors, not just limits
20.1.2 Landau notation
20.2 Average of Tau
20.2.1 Beginnings
20.2.2 Heuristics for tau
20.2.3 Using sums to get closer
20.2.4 But Big-Oh isn’t enough
20.3 Digging Deeper and Finding Limits
20.3.1 Moving toward a proof
20.3.2 Getting a handle on error
20.3.3 The end of the story
20.4 Heuristics for the Sum of Divisors
20.4.1 Numbers instead of points
20.4.2 Order calculations and more
20.5 Looking Ahead
20.6 Exercises
21 The Prime Counting Function
21.1 First Steps
21.1.1 A funky formula
21.1.2 A very low bound
21.1.3 Knowledge from nowhere
21.2 Some History
21.2.1 The first really accurate estimate and errors
21.2.2 Exploring L i
21.3 The Prime Number Theorem
21.3.1 Stating the theorem
21.3.2 Chebyshev’s contributions
21.4 A Slice of the Prime Number Theorem
21.4.1 Functions to know
21.4.2 Getting a formula with sleights of hand
21.4.3 Finish the proof
21.5 Exercises
22 More on Prime Numbers
22.1 Prime Races
22.1.1 Infinitude of types of primes
22.1.2 Back to bias
22.1.3 Other prime races
22.2 Sequences and Primes
22.2.1 Primes in sequences
22.2.2 Sequences in primes
22.3 Types of Primes
22.3.1 Twin primes
22.3.2 Heuristics for twin primes
22.3.3 Other types of primes
22.4 Exercises
23 New Functions from Old
23.1 The Moebius Function
23.1.1 Möbius mu
23.1.2 A formula
23.1.3 Another definition
23.2 Inverting Functions
23.2.1 Some useful notation
23.2.2 Proof of Moebius inversion
23.3 Making New Functions
23.3.1 First new functions
23.3.2 More new functions
23.4 Generalizing Moebius
23.4.1 The monoid of arithmetic functions
23.4.2 Bringing in group structure
23.4.3 More dividends from structure
23.5 Exercises
24 Infinite Sums and Products
24.1 Products and Sums
24.1.1 Products
24.1.2 Products that are sums
24.2 The Riemann Zeta Function
24.2.1 A fundamental function
24.2.2 Motivating the Zeta function
24.2.3 Being careful
24.3 From Riemann to Dirichlet and Euler
24.3.1 Dirichlet series
24.3.2 Euler products
24.4 Multiplication
24.5 More series and convergence
24.5.1 A series for Euler phi and a general theorem
24.5.2 A missing step: Convergence of Dirichlet series
24.6 Four Facts
24.6.1 Random integer lattice points
24.6.2 Dirichlet for the absolute Moebius
24.6.3 The prime harmonic series
24.6.4 The average value of Euler phi
24.7 Exercises
25 Further Up and Further In
25.1 Taking the PNT Further
25.2 Improving the PNT
25.3 Toward the Riemann Hypothesis
25.3.1 Zeta beyond the series
25.3.2 Zeta on some lines
25.4 Connecting to the Primes
25.4.1 Connecting to Moebius
25.5 Connecting to Zeta
25.5.1 Turning the golden key
25.5.2 Detailing the connections
25.6 Connecting to Zeros
25.6.1 Where are the zeros?
25.6.2 Analyzing the connection
25.7 The Riemann Explicit Formula
25.8 Epilogue
25.9 Exercises
Back Matter
A List of Sage notes
B List of Historical Remarks
C Notation
D List of Figures
E References and Further Resources
E.1 Introduction to the References
E.2 General References
E.3 Proof and Programming References
E.4 Specialized References
E.5 Historical References
E.6 Other References
E.7 Useful Articles
Index
Chapter 5 Linear Congruences
There are many questions one can ask of the integers, and in the preceding material we have already encountered many, especially those asking for solutions of simple equations in one or two variables.
One can ask very similar questions (and many more) about the integers modulo .n. So we will focus on congruences, which are simply equations modulo n (see Definition 4.6.3). To exemplify this, consider the following similar ideas:
2 x+3 y=5 (solutions are pairs of integers)
(mod 2 x+3 y≡5(mod 7) (solutions would be pairs of equivalence classes[x],[y] modulo 7)
(mod 2 x+3 y≡5(mod n) for any particular n (solutions would be triplets ,[x],[y],n, since it would depend on n)
Try comparing solutions to these by hand; what is similar about them, what is not?
In one sense the latter problems are a big improvement in the level of difficulty. For instance, in the second one you just have to try x,y from 0 to 6 (the least nonnegative residues) in the congruence 2 x+3 y≡5 (mod 7).
On the other hand, if the third congruence was modulo ,n=10 100, that would be less desirable, especially if the techniques for Z proved not to be useful with a congruence.
Finally, if we slapped an x 2 in the middle of the congruence, it might very hard indeed to solve quickly. So in this chapter, we will stay focused on the simplest case, of the analogue to linear equations, known as linear congruences (of one variable). This includes systems of such congruences (see Section 5.3).
5.1 Solving Linear Congruences
5.2 A Strategy For the First Solution
5.3 Systems of Linear Congruences
5.4 Using the Chinese Remainder Theorem
5.5 More Complicated Cases
5.6 Exercises
Summary:Linear Congruences
In this chapter we begin the process of shifting from solving equations as ‘sentences for equality’ to solving congruences as ‘sentences for congruence’. We start with the simplest context, linear congruences.
In Proposition 5.1.1 and Proposition 5.1.3 we have a full characterization of solutions to the basic linear congruence a x≡b (mod n).
To use the previous section in situations where a solution exists, we need Strategies that work for simplifying congruences. The cancellation propositions 5.2.6 and 5.2.7 are key tools.
It is an ancient question as to how to solve systems of linear congruences, and the Chinese Remainder Theorem is the prime tool for this. We also introduce The Inverse of a Number in this section.
In the next section we then make this explicit in Algorithm 5.4.1, and practice it. In the future the corollary Proposition 5.4.5 will prove very useful.
In the last section there are several more advanced topics which we briefly mention to inspire readers, but do not pursue – notably, Qin’s solution for the situation when we have Moduli which are not coprime.
There are once again many Exercises, but it is worth mentioning that this is a chapter where making up your own congruences (or systems of congruences) is a great way to get extra practice.
PrevTopNext |
11716 | https://algocademy.com/blog/mastering-bitmask-dynamic-programming-a-comprehensive-guide/ | Mastering Bitmask Dynamic Programming: A Comprehensive Guide
In the world of competitive programming and technical interviews, particularly for top tech companies like FAANG (Facebook, Amazon, Apple, Netflix, Google), mastering advanced algorithmic techniques is crucial. One such powerful technique that often appears in complex problem-solving scenarios is Bitmask Dynamic Programming. This blog post will dive deep into this concept, exploring its fundamentals, applications, and providing practical examples to enhance your coding skills.
Table of Contents
Introduction to Bitmask Dynamic Programming
Understanding the Basics
When to Use Bitmask DP
Implementing Bitmask DP
Real-world Examples and Problem Solving
Optimization Techniques
Practice Problems
Conclusion
1. Introduction to Bitmask Dynamic Programming
Bitmask Dynamic Programming is an advanced algorithmic technique that combines the power of dynamic programming with bitwise operations. It’s particularly useful when dealing with problems involving subsets, permutations, or state representations where the number of elements is relatively small (typically less than 32).
The core idea behind Bitmask DP is to use bits to represent the state of a problem, where each bit corresponds to a particular element or condition. This representation allows for efficient storage and manipulation of states, leading to optimized solutions for otherwise complex problems.
2. Understanding the Basics
2.1 Bitwise Operations Refresher
Before diving into Bitmask DP, it’s essential to have a solid understanding of basic bitwise operations. Here’s a quick refresher:
AND (&): Returns 1 if both bits are 1, otherwise 0
OR (|): Returns 1 if at least one bit is 1, otherwise 0
XOR (^): Returns 1 if bits are different, otherwise 0
NOT (~): Inverts all bits
Left Shift (<<): Shifts bits to the left, filling with 0
Right Shift (>>): Shifts bits to the right, filling with 0 (unsigned) or the sign bit (signed)
2.2 Bitmask Representation
In Bitmask DP, we use an integer to represent a set of elements. Each bit in the integer corresponds to the presence (1) or absence (0) of an element in the set. For example, if we have a set of 5 elements, the bitmask 10110 (22 in decimal) represents the subset containing the 2nd, 3rd, and 5th elements.
2.3 Common Bitmask Operations
Here are some common operations used in Bitmask DP:
Check if the i-th bit is set: mask & (1 << i)
Set the i-th bit: mask | (1 << i)
Unset the i-th bit: mask & ~(1 << i)
Toggle the i-th bit: mask ^ (1 << i)
Get the number of set bits: __builtin_popcount(mask) (GCC) or implement your own
3. When to Use Bitmask DP
Bitmask DP is particularly useful in scenarios where:
The problem involves subsets or permutations of a small set of elements (typically < 32).
You need to keep track of multiple states or conditions simultaneously.
The problem requires exploring all possible combinations efficiently.
The state space is too large for a straightforward DP approach but can be compressed using bits.
Common problem types that benefit from Bitmask DP include:
Traveling Salesman Problem (TSP) and its variants
Set covering problems
Assignment problems
Subset sum and partitioning problems
Graph problems involving small sets of vertices
4. Implementing Bitmask DP
The general approach to implementing Bitmask DP involves the following steps:
Define the state representation using bitmasks
Initialize the DP table
Define the base cases
Implement the state transitions
Iterate through all possible states
Retrieve the final answer
Let’s look at a simple example to illustrate these steps:
Problem: Subset Sum
Given a set of integers and a target sum, determine if there exists a subset of the given set with a sum equal to the target sum.
```
// C++ implementation of Subset Sum using Bitmask DP
include
include
using namespace std;
bool subsetSum(vector& nums, int target) {
int n = nums.size();
vector dp(1 << n, false);
dp = true; // Empty subset has sum 0
for (int mask = 0; mask < (1 << n); mask++) {
for (int i = 0; i < n; i++) {
if (mask & (1 << i)) {
int prevMask = mask ^ (1 << i);
if (dp[prevMask]) {
dp[mask] |= (dp[prevMask] && nums[i] <= target);
target -= nums[i];
}
}
}
}
return dp[(1 << n) - 1] && target == 0;
}
int main() {
vector nums = {3, 34, 4, 12, 5, 2};
int target = 9;
cout << (subsetSum(nums, target) ? "True" : "False") << endl;
return 0;
}
```
In this implementation:
We use a bitmask to represent subsets of the given set.
The DP table dp stores whether a subset sum is possible for each bitmask.
We iterate through all possible subsets (bitmasks) and update the DP table.
The final answer is stored in dp[(1 << n) - 1], which represents the full set.
5. Real-world Examples and Problem Solving
Let’s explore a more complex problem that showcases the power of Bitmask DP in real-world scenarios.
Problem: Traveling Salesman Problem (TSP)
Given a set of cities and the distances between each pair of cities, find the shortest possible route that visits each city exactly once and returns to the starting city.
```
include
include
include
using namespace std;
const int INF = INT_MAX / 2;
int tsp(vector>& dist) {
int n = dist.size();
vector> dp(1 << n, vector(n, INF));
// Base case: start from city 0
dp = 0;
// Iterate through all subsets of cities
for (int mask = 1; mask < (1 << n); mask++) {
for (int u = 0; u < n; u++) {
if (mask & (1 << u)) {
for (int v = 0; v < n; v++) {
if (v != u && (mask & (1 << v))) {
dp[mask][u] = min(dp[mask][u], dp[mask ^ (1 << u)][v] + dist[v][u]);
}
}
}
}
}
// Find the minimum cost to return to city 0
int finalMask = (1 << n) - 1;
int minCost = INF;
for (int u = 1; u < n; u++) {
minCost = min(minCost, dp[finalMask][u] + dist[u]);
}
return minCost;
}
int main() {
vector> dist = {
{0, 10, 15, 20},
{10, 0, 35, 25},
{15, 35, 0, 30},
{20, 25, 30, 0}
};
cout << "Minimum cost of TSP: " << tsp(dist) << endl;
return 0;
}
```
In this TSP implementation:
We use a 2D DP table where dp[mask][u] represents the minimum cost to visit all cities in the bitmask and end at city u.
The bitmask represents the set of cities visited.
We iterate through all subsets of cities and compute the minimum cost for each state.
Finally, we find the minimum cost to return to the starting city.
This solution demonstrates how Bitmask DP can efficiently solve the NP-hard Traveling Salesman Problem for small to medium-sized inputs.
6. Optimization Techniques
While Bitmask DP is already an optimization technique, there are ways to further improve its performance:
6.1 Memoization
Instead of using a bottom-up approach, you can implement a top-down approach with memoization. This can be more intuitive and may avoid unnecessary computations.
6.2 Bit Manipulation Tricks
Use efficient bit manipulation techniques to speed up operations:
Use __builtin_popcount() (GCC) for counting set bits
Use mask & - mask to get the least significant set bit
Use mask & (mask - 1) to turn off the least significant set bit
6.3 State Compression
If the number of elements is larger than what can fit in a single integer, consider using multiple integers or a custom bitset implementation.
6.4 Parallel Processing
For very large problem sizes, consider parallelizing the computation of independent subproblems.
7. Practice Problems
To master Bitmask DP, practice is key. Here are some problems you can try:
Maximum Students Taking Exam (LeetCode)
Kefa and Dishes (Codeforces)
Courier (SPOJ)
Matching (AtCoder)
Elevator Rides (CSES)
These problems cover a range of difficulties and applications of Bitmask DP, helping you build confidence and intuition for when and how to apply this technique.
8. Conclusion
Bitmask Dynamic Programming is a powerful technique that combines the efficiency of bitwise operations with the problem-solving capabilities of dynamic programming. By mastering this approach, you’ll be better equipped to tackle complex algorithmic challenges in competitive programming and technical interviews, especially for top tech companies.
Remember that like any advanced technique, proficiency in Bitmask DP comes with practice. Start with simpler problems and gradually work your way up to more complex scenarios. As you gain experience, you’ll develop an intuition for when Bitmask DP can be applied to optimize your solutions.
Keep in mind that while Bitmask DP is incredibly useful, it’s not always the best solution. Always consider the problem constraints and requirements before deciding on your approach. Sometimes, a simpler solution or a different algorithm might be more appropriate.
As you continue your journey in algorithmic problem-solving, don’t forget to explore other advanced techniques as well. The combination of Bitmask DP with other algorithms and data structures can lead to even more powerful and efficient solutions.
Happy coding, and may your bits be ever in your favor!
Our interactive tutorials and AI-assisted learning will help you master problem-solving skills and teach you the algorithms to know for coding interviews.
Links
Privacy Policy
Terms of Service
© 2025 AlgoCademy Blog. All rights reserved. |
11717 | https://stats.stackexchange.com/questions/4125/regression-proof-that-the-point-of-averages-x-y-lies-on-the-estimated-regressi | distributions - Regression Proof that the point of averages (x,y) lies on the estimated regression line - Cross Validated
Join Cross Validated
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Cross Validated helpchat
Cross Validated Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Regression Proof that the point of averages (x,y) lies on the estimated regression line
Ask Question
Asked 14 years, 11 months ago
Modified4 years, 11 months ago
Viewed 35k times
This question shows research effort; it is useful and clear
14
Save this question.
Show activity on this post.
How do you show that the point of averages (x,y) lies on the estimated regression line?
distributions
regression
proof
self-study
Share
Share a link to this question
Copy linkCC BY-SA 2.5
Cite
Improve this question
Follow
Follow this question to receive notifications
edited Nov 2, 2010 at 13:50
csgillespie
13k 9 9 gold badges 63 63 silver badges 91 91 bronze badges
asked Nov 2, 2010 at 1:22
Justin MeltzerJustin Meltzer
387 1 1 gold badge 3 3 silver badges 8 8 bronze badges
8
What have you done so far?user28 –user28 2010-11-02 01:30:33 +00:00 Commented Nov 2, 2010 at 1:30
I don't know where to start.Justin Meltzer –Justin Meltzer 2010-11-02 01:38:40 +00:00 Commented Nov 2, 2010 at 1:38
I know the simple regression line is y=B0 + B1x Justin Meltzer –Justin Meltzer 2010-11-02 01:39:03 +00:00 Commented Nov 2, 2010 at 1:39
1 And I guess by the average x and y is E(x) and E(y), not sure how to link that to the regression line though.Justin Meltzer –Justin Meltzer 2010-11-02 01:40:20 +00:00 Commented Nov 2, 2010 at 1:40
The regression line is the line that minimizes the sum of squared errors. Knowing that, and a basic knowledge of calculus, find the values of B0 and B1 that minimize that sum of squared errors. The rest requires a little bit of high school level algebra.Christopher Aden –Christopher Aden 2010-11-02 02:06:22 +00:00 Commented Nov 2, 2010 at 2:06
|Show 3 more comments
1 Answer 1
Sorted by: Reset to default
This answer is useful
22
Save this answer.
Show activity on this post.
To get you started: y¯=1/n∑y i=1/n∑(y^i+ϵ^i)y¯=1/n∑y i=1/n∑(y^i+ϵ^i) then plug in, how the y^i y^i are estimated by the x i x i and you're almost done.
EDIT: since no one replied, here the rest for sake of completeness: the y^i y^i are estimated by y^i=β^0+β^1 x i 1+…+β^n x i n+ϵ^i y^i=β^0+β^1 x i 1+…+β^n x i n+ϵ^i, so you get y¯=1/n∑β^0+β^1 x i 1+…+β^n x i n y¯=1/n∑β^0+β^1 x i 1+…+β^n x i n (the ϵ^i ϵ^i sum to zero) and finally:
y¯=β^0+β^1 x¯1+…+β^n x¯n y¯=β^0+β^1 x¯1+…+β^n x¯n. And that's it: The regression line goes through the point (x¯,y¯)(x¯,y¯)
Share
Share a link to this answer
Copy linkCC BY-SA 2.5
Cite
Improve this answer
Follow
Follow this answer to receive notifications
edited Nov 5, 2010 at 13:59
answered Nov 2, 2010 at 15:33
psjpsj
1,101 8 8 silver badges 12 12 bronze badges
1
4 nice answer! just one note: in your edit, you defined y i^=β 0^+β 1^x i 1+...+β n^x i n+ϵ i^y i^=β 0^+β 1^x i 1+...+β n^x i n+ϵ i^. Actually, y i^y i^ should not include the residual ϵ i^ϵ i^, which you had already included inside the summation ∑(y i^+ϵ i^)∑(y i^+ϵ i^) in your original answer.Roberto Tatis Muvdi –Roberto Tatis Muvdi 2020-10-02 09:30:54 +00:00 Commented Oct 2, 2020 at 9:30
Add a comment|
Protected question. To answer this question, you need to have at least 10 reputation on this site (not counting the association bonus). The reputation requirement helps protect this question from spam and non-answer activity.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
distributions
regression
proof
self-study
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Linked
0How do you prove that OLS Estimator must pass through mean of X and Y
1Does the OLS regression line pass through the mean of ALL covariates?
5Why does the total least squares line in 2D pass through the average across all data points?
2Why does centering/standardizing the regressor make the p value for the intercept zero?
2Given correlation and linear relationship between x and y find a linear relationship with the variables switchrd
1Sample error terms relationship
Related
1How to prove that averaging averages of different partitions of a dataset produces the same overall average
3Proof: Linear Regression
0Can residuals be calculated from N-point moving averages or just the regression line? Also, what is the standard way to determine regression line?
1Show regression line passes through points (X 1,Y 2¯),(X 2,Y 2¯)(X 1,Y 2¯),(X 2,Y 2¯)
2What is the difference between least squares line and the regression line?
3When will point (x¯x¯, y¯y¯) not go through the regression line?
2Math behind the position of the point in linear regression with respect to the line?
Hot Network Questions
Explain answers to Scientific American crossword clues "Éclair filling" and "Sneaky Coward"
How to convert this extremely large group in GAP into a permutation group.
I have a lot of PTO to take, which will make the deadline impossible
Discussing strategy reduces winning chances of everyone!
Do we need the author's permission for reference
Implications of using a stream cipher as KDF
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
How to start explorer with C: drive selected and shown in folder list?
Does a Linux console change color when it crashes?
Why does LaTeX convert inline Python code (range(N-2)) into -NoValue-?
Passengers on a flight vote on the destination, "It's democracy!"
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
Origin of Australian slang exclamation "struth" meaning greatly surprised
Do sum of natural numbers and sum of their squares represent uniquely the summands?
how do I remove a item from the applications menu
For every second-order formula, is there a first-order formula equivalent to it by reification?
In Dwarf Fortress, why can't I farm any crops?
в ответе meaning in context
How to rsync a large file by comparing earlier versions on the sending end?
How to home-make rubber feet stoppers for table legs?
Why are LDS temple garments secret?
Program that allocates time to tasks based on priority
How can the problem of a warlock with two spell slots be solved?
How do you emphasize the verb "to be" with do/does?
more hot questions
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Cross Validated
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
11718 | https://www.chegg.com/homework-help/questions-and-answers/nilsson-riedel-electric-circuits-10th-edition-c2015-tktbk2-pdf-adobe-acrobat-reader-dc-fil-q23855123 | Solved Nilsson & Riedel -Electric Circuits 10th Edition | Chegg.com
Skip to main content
Books
Rent/Buy
Read
Return
Sell
Study
Tasks
Homework help
Understand a topic
Writing & citations
Tools
Expert Q&A
Math Solver
Citations
Plagiarism checker
Grammar checker
Expert proofreading
Career
For educators
Help
Sign in
Paste
Copy
Cut
Options
Upload Image
Math Mode
÷
≤
≥
o
π
∞
∩
∪
√
∫
Math
Math
Geometry
Physics
Greek Alphabet
Engineering
Electrical Engineering
Electrical Engineering questions and answers
Nilsson & Riedel -Electric Circuits 10th Edition c2015 tktbk2 pdf - Adobe Acrobat Reader DC File Edit View Window Help Ho ECES43-Fall201 ECES43-Fall201 ECES43-Fall201 ECES43-Fall201- ECES43-Fall201.. Nilsson & Ried." ECES41 Lecture- O Sign In × 132 (156 of 821) ence node S Export PDF c) Compare the choice of r and (b). Which is better, a 4.18 Use the
Your solution’s ready to go!
Our expert help has broken down your problem into an easy-to-learn solution you can count on.
See Answer See Answer See Answer done loading
Question: Nilsson & Riedel -Electric Circuits 10th Edition c2015 tktbk2 pdf - Adobe Acrobat Reader DC File Edit View Window Help Ho ECES43-Fall201 ECES43-Fall201 ECES43-Fall201 ECES43-Fall201- ECES43-Fall201.. Nilsson & Ried." ECES41 Lecture- O Sign In × 132 (156 of 821) ence node S Export PDF c) Compare the choice of r and (b). Which is better, a 4.18 Use the
Solve Problem 4.18 please
Show transcribed image text
There are 3 steps to solve this one.Solution Share Share Share done loading Copy link Step 1 4.18)Given that,
The circuit diagram is shown below, Explanation: The current flowing in the 100 ohm resistor is io View the full answer Step 2 UnlockStep 3 UnlockAnswer Unlock Previous questionNext question
Transcribed image text:
Nilsson & Riedel -Electric Circuits 10th Edition c2015 tktbk2 pdf - Adobe Acrobat Reader DC File Edit View Window Help Ho ECES43-Fall201 ECES43-Fall201 ECES43-Fall201 ECES43-Fall201- ECES43-Fall201.. Nilsson & Ried." ECES41 Lecture- O Sign In × 132 (156 of 821) ence node S Export PDF c) Compare the choice of r and (b). Which is better, a 4.18 Use the node-voltage method to calculate the PSPICE power delivered by the dependent voltage source in ULTISIMthe circuit in Fig. P4.18 Adobe Export PDF Convert PDF Files to Word or Excel Online Fiqure P4.22 25 V Select PDF File Figure P4.18 Nilsson & R.15 txtbk2.pdfX 10Ω 30Ω Convert to 2 A 50 Ω Microsoft Word ("docx) Document Language English (U.S) Change 160 V :100 Ω 20 Ω 4.23 Use the node-voltage method Convert 4.19 a) Use the node-voltage method to find the totPCE in the circuit in Fig. P4.23 MULTISIM power developed in the circuit in Fig. P4.19. PSPICE Figure P4.23 MULTISIM Create PDF b) Check your answer by finding the total power 800 Ω absorbed in the circuit Edit PDF Figure P4.19 Comment 1250 Ω 200 Ω 80 Ω 40 Ω Combine Files 20 mA 2500込 50 V 50Ω Store and share files in the Document Cloud Learn More 8.97 x 11.34 in OType here to search 9/22/2017
Not the question you’re looking for?
Post any question and get expert help quickly.
Start learning
Chegg Products & Services
Chegg Study Help
Citation Generator
Grammar Checker
Math Solver
Mobile Apps
Plagiarism Checker
Chegg Perks
Company
Company
About Chegg
Chegg For Good
Advertise with us
Investor Relations
Jobs
Join Our Affiliate Program
Media Center
Chegg Network
Chegg Network
Busuu
Citation Machine
EasyBib
Mathway
Customer Service
Customer Service
Give Us Feedback
Customer Service
Manage Subscription
Educators
Educators
Academic Integrity
Honor Shield
Institute of Digital Learning
© 2003-2025 Chegg Inc. All rights reserved.
Cookie NoticeYour Privacy ChoicesDo Not Sell My Personal InformationGeneral PoliciesPrivacy PolicyHonor CodeIP Rights
Do Not Sell My Personal Information
When you visit our website, we store cookies on your browser to collect information. The information collected might relate to you, your preferences or your device, and is mostly used to make the site work as you expect it to and to provide a more personalized web experience. However, you can choose not to allow certain types of cookies, which may impact your experience of the site and the services we are able to offer. Click on the different category headings to find out more and change our default settings according to your preference. You cannot opt-out of our First Party Strictly Necessary Cookies as they are deployed in order to ensure the proper functioning of our website (such as prompting the cookie banner and remembering your settings, to log into your account, to redirect you when you log out, etc.). For more information about the First and Third Party Cookies used please follow this link.
More information
Allow All
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Sale of Personal Data
[x] Sale of Personal Data
Under the California Consumer Privacy Act, you have the right to opt-out of the sale of your personal information to third parties. These cookies collect information for analytics and to personalize your experience with targeted ads. You may exercise your right to opt out of the sale of personal information by using this toggle switch. If you opt out we will not be able to offer you personalised ads and will not hand over your personal information to any third parties. Additionally, you may contact our legal department for further clarification about your rights as a California consumer by using this Exercise My Rights link.
If you have enabled privacy controls on your browser (such as a plugin), we have to take that as a valid request to opt-out. Therefore we would not be able to track your activity through the web. This may affect our ability to personalize ads according to your preferences.
Targeting Cookies
[x] Switch Label label
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Reject All Confirm My Choices
mmmmmmmmmmlli mmmmmmmmmmlli mmmmmmmmmmlli |
11719 | https://blog.csdn.net/weixin_53630942/article/details/124002457 | 解构计数问题：兼选与全选的学生人数-CSDN博客
博客
下载
学习
社区
GitCode
InsCodeAI
会议
搜索
AI 搜索
登录
登录后您可以：
复制代码和一键运行
与博主大V深度互动
解锁海量精选资源
获取前沿技术资讯
立即登录
会员·新人礼包
消息
历史
创作中心
创作
容斥问题
最新推荐文章于 2024-10-20 14:34:12 发布
@Adzc于 2022-04-06 22:49:40 发布
阅读量560收藏
点赞数
CC 4.0 BY-SA版权
分类专栏：数学算法文章标签：数学
版权声明：本文为博主原创文章，遵循CC 4.0 BY-SA版权协议，转载请附上原文出处链接和本声明。
本文链接：
算法 同时被 2 个专栏收录
43 篇文章
订阅专栏
数学
3 篇文章
订阅专栏
解题原则：在计数时，需做到不重不漏，重复的需要减掉，遗漏的需要补上。
常用公式:
I=A+B+C-只 属于两种的-2三种都属于的+三种都不属于的
既
I=A+B+C-AB-AC-BC+ABC+三种都不属于的
例题：某专业有若干学生，有甲乙丙三门选修课。有40人选甲，36人选乙，30人选丙；兼选甲、乙的28人，兼选甲、丙的26人，兼选乙、丙的24人，甲乙丙都选的20人。三门课都没选的2人，该专业共有学生多少人？
40+36+30-28-26-24+20+2=50
同时：兼、顾
22+16+25-8-6-x =42
x=7
2.
189+152+135-130-269+44=252
确定要放弃本次机会？
福利倒计时
: :
立减 ¥
普通VIP年卡可用
立即使用
@Adzc
关注关注
0点赞
踩
0 收藏 觉得还不错? 一键收藏
0评论
分享复制链接 分享到 QQ 分享到新浪微博 扫一扫
举报 举报
专栏目录
单价数量和总价的公式_「行测笔记」数量关系方法精讲（四）
weixin_29463345的博客
01-04 3539
【知识点】周期余数：1.题型特征：出现循环或周期，问第/过N个(天、年)。是循环分布的，比如 AB CD AB CD……。注意区分“第”和“过”，二者是有区别的。2.补例1：1月1号是星期一，问1月份第16天(1月16号)是星期几？答：16天可以挨个数，但如果是第300天就不能数，可以按照整周期考虑，将 16 天按照 7 天一个周期去数，16/7=2……2，因此第 16 天等价于第 2 天(后边的余数)。...
【数论】容 斥 问题 教程 + 符号含义 + 公式（c + +）
ptyz306的博客
09-06 785
比如我们平常考试，我们会统计有几个语文及格，有几个 数学 及格，比如5个语文及格，2个 数学 及格，当然了，也会有大学霸两科都及格，比如1个人语文 数学 都及格，那我们班上一共有几个人呢？这张图片中，有两个集合，一个是语文及格，一个是 数学 及格，而∪这个符号，表示的是这张图中被红线画到的地方，注意，这里图中的这样两个集合被画到的地方，叫做并集，而∪个符号，叫做并。|A|+|B|+|CA∩BB∩CC∩A|这部分，和上一条公式一样，问题 是为什么要加上|A∩B∩C|？再来考考你，|A∪B∪C|是什么意思？
参与评论 您还未登录，请先 登录 后发表或查看评论
c语言 容 斥 原理,超级干货 - 容 斥 原理大集合!!!
9-13
A.1人B.2人 C.3人D.4人 【答案】B 【解析】本题为三集合 容 斥 原理公式1。设三门课均为选的人数为x,根据三集合公式A + B + C - AB - BC - AC + AB C=总个数 - 都不满足的个数,可得40 + 36 + 30 - 28 - 26 - 24 + 20=50 - X,解得X=2。因此选择B选项。 【例3】某乡镇举行运动会,共有长跑、跳远和短跑三个项目。参加长跑...
【CodeForces 611C】一维 + 二维前缀和 | 容 斥 原理 | 动态规划 | E_cod...
9-20
同理,dp[r][c - 1] == A + ac + C,dp[r - 1][c - 1] == A 将上一行的三个等式带入我们的状态转移方程第一步的右边,得到右边(即dp[r - 1][c] + dp[r][c - 1] - dp[r - 1][c - 1])==A + ab + B + A + ac + C - A==A + B + C + ab + ac 再看这个结果和 dp[r][c] 差什...
容 斥 原理简述
bdn_nbd的博客
10-18 363
容 斥 原理简述
公务员考试 容 斥 问题 借鉴.pdf
12-06
1. 三者 容 斥 问题 计算： 这类 问题 涉及三个互不相交的集合，如甲、乙、丙的投篮情况。在解决这类 问题 时，若不能直接通过文氏图得出关系，我们需要利用集合间的数量关系建立等式求解。例如，如果已知甲乙共投进48...
透彻理解联合索引(a,b,c)联合索引 _ab c中 ac 行吗
9-24
透彻理解联合索引(a,b,c) 本文解析了索引在数据库中的作用,重点讲解了如何利用‘最左前缀’原则理解联合索引(如B + 树结构),以及为何范围查询会影响索引使用。理解了这些,能帮助开发者更高效地设计数据库查询策略。 用到索引的有a,ab,ab c,ac 因为优化器会自动调整and前后的顺序,所以ba,cba,bca,ca都会用到索...
组合计数公式
9-20
综上,公式:A + B + C − A B − A C − B C + A B C A + B + C - AB - AC - BC + AB CA + B + C−AB−AC−BC + AB C
五年级奥数 容 斥 问题 PPT学习教案.pptx
10-09
【五年级奥数 容 斥 问题】是小学 数学 竞赛中的一种重要题型，主要涉及的是统计和组合的概念。容 斥 原理，又称包涵排 斥 原理，是解决这类 问题 的基础。它的核心思想是，在计数时，若两个或多个集合有交集，则在计算总数时要...
公务员考试数量关系之三集合 容 斥 问题.docx
02-23
### 公务员考试数量关系之三集合 容 斥 问题 #### 一、三集合 容 斥 问题 概述 在公务员考试的数量关系部分，经常会遇到涉及多个集合的题目，其中以三集合 容 斥 问题 最为常见且重要。这类 问题 的核心在于理解和运用 容 斥 原理...
验证四个事件概率的加法公式 用卡诺图形式的表格加加减减_四个事件的...
9-25
包含事件最多的重叠面积的块数,比如中间的4代表同时包含 AB CD的面积的块数,但它也同时属于A,B,C和D,对于最下面的3,它处于的最精确的位置是 AC D,准确的说是 AB'CD,不过不管它,此时中间的4的位置也可以是 AC D,但从重叠次数多的面积开始写可以避免这个 问题,所以最下面的3就代表 AC D包含面积重叠的块数,而不是 AC 或者...
线性代数(第二章:矩阵)_线性代数ae=a在书上哪说的
9-27
矩阵相等:所有元素对应相等,A = B 。 3)矩阵的运算 A + B = B + A (A + B)·C = AC + BC A·(B + C) = AB + AC A·(BC) = (AB)C k·(A + B) = kA + kB (kA)·(lB) = kl AB AE = A , EA = A OA = O , AO = O ...
四年级奥数 容 斥 问题 PPT学习教案.pptx
10-10
这个例子很好地展示了 容 斥 原理在处理组合 问题 中的作用。 接下来，我们通过几个练习题来进一步加深对 容 斥 原理的理解。练习一要求我们计算同时在语文和 数学 上都获得优秀成绩的学生数。我们可以先计算语文优秀的学生数...
公务员考试 容 斥 问题 分享.pdf
02-23
公务员考试 容 斥 问题 分享.pdf
判断三角形_2a2 - 2b2 - 4c2 + ab + 4bc - 2 ac=0
8-22
∴a2 + b2 + c2 - ab - bc - ac=0,∴2a2 + 2b2 + 2c2 - 2 ab - 2bc - 2 ac=0 即(a - b)2 +(b - c)2 +(c - a)2=0,∴a - b=b - c=c - a=0,故a=b=c,∴△AB C是等边三角形。 五、利用三角形的面积 例5:设△AB C的三条高线之和等于此三角形三个角平分线的交点到一边的距离的9倍,则△AB C是等边三角形。
一招搞定线性代数_线性代数第七版知识点总结
9-24
( 1) A 为 m × n 阶矩阵,则 r(A)≤ min (m,n) ( 2) r(A±B)≤ r ( A)±( B) ( 3) r(AB)≤ min{r ( A),r(B)} ( 4) r(kA)=r(A)(k≠0) ( 5) r(A)=r (AC)(C 是一个可逆矩阵) ( 7)设 A 是 m×n 阶矩阵, B 是 n×s 矩阵, AB=O,则 r ( A) + r(B...
【离散 数学】容 斥 原理
行码棋
01-09 3446
容 斥 定理详解及经典习题
容 斥 原理的两种证明
bdn_nbd的博客
02-27 2777
容 斥 原理的两种证明方法
容 斥 原理基础
weixin_61426225的博客
02-08 1694
至少喜欢一门的人：A + B + C - AB - AC—BC + AB C=20 + 30 + 25 - 10 - 7 - 8 + 2=52 有60 - 52=8个同学一门都不喜欢。至少喜欢一门 20 + 30 - 10=40 都不喜欢 50 - 40=10。再将上面的课程门数进一步扩展为3门，问题 变为。同时喜欢语文 数学 的人有10人。将上述 问题 抽象用 数学 的形式表示。同时喜欢 AB 的人有10人。同时喜欢 AB C的人有2人。同时喜欢 AC 的人有7人。同时喜欢BC的人有8人。喜欢语文的人有20人。喜欢 数学 的人有30人。二进制枚举超集的写法。
HDOJ4135 Co - prime - - - 容 斥 原理
。
11-08 199
题目链接 m.hdu.edu.cn/showproblem.php?pid=4135 Problem Description Given a number N, you are asked to count the number of integers between A and B inclusive which are relatively prime to N. Tw...
（数学）容 斥 原理 最新发布
miss_ben的博客
10-20 3503
容 斥 原理保姆级教程，涉及到组合 数学
容 斥 原理——经典例题（组合 数学） 热门推荐
weixin_43908980的博客
03-14 2万+
[一.容 斥 原理 就是人们为了不重复计算重叠部分，想出的一种不重复计算的方法。 先来认识一下这两个符号：与（如图） 蓝色的圈就是c1c2，红色的圈围起来的就是c1c2 二.例题：组合 数学 1.题目 1.1.题目描述 八是个很有趣的数字啊。八=发，八八=爸爸，88=拜拜。当然最有趣的还是8用二进制表示是1000。怎么样，有趣吧。当然题目和这些都没有关系。 某个人很无聊，他想找出a,b...
【组合 数学】容 斥&鸽巢原理
恭仔的博客
12-05 5223
组合 数学 容 斥 原理
容 斥 原理详解
菜鸡的博客
04-13 1万+
翻译：vici@cust 对 容 斥 原理的描述 容 斥 原理是一种重要的组合 数学 方法，可以让你求解任意大小的集合，或者计算复合事件的概率。 描述 容 斥 原理可以描述如下： 要计算几个集合并集的大小，我们要先将所有单个集合的大小计算出来，然后减去所有两个集合相交的部分，再加回所有三个集合相交的部分，再减去所有四个集合相交的部分.........依此类推，一...
关于我们
招贤纳士
商务合作
寻求报道
400-660-0108
kefu@csdn.net
在线客服
工作时间 8:30-22:00
公安备案号11010502030143
京ICP备19004658号
京网文〔2020〕1039-165号
经营性网站备案信息
北京互联网违法和不良信息举报中心
家长监护
网络110报警服务
中国互联网举报中心
Chrome商店下载
账号管理规范
版权与免责声明
版权申诉
出版物许可证
营业执照
©1999-2025北京创新乐知网络技术有限公司
@Adzc
博客等级
码龄5年
71 原创54 点赞 123 收藏 37 粉丝
关注
私信
热门文章
proxyBeanMethods 4675
人机互动版猜拳游戏（采用OOP面向对象思想，运用Spring IoC方式实现 3926
蓝桥 小明的游戏1 博弈论 nim 3337
蓝桥 小明的游戏 反nim 博弈论 3231
蓝桥 小明的衣服 哈夫曼树，贪心，优先队列 3169
分类专栏
C++6篇
工具1篇
SpringBoot4篇
idea2篇
日常bug1篇
java2篇
数据库1篇
fisco-baco2篇
区块链日常bug1篇
SSM9篇
算法43篇
JS2篇
线性代数1篇
数学3篇
排列组合1篇
展开全部收起
上一篇： 错位重排 下一篇： 公约数与公倍数
最新评论
Dockerfile中yum install 无法使用的问题
CSDN-Ada助手:云原生入门 技能树或许可以帮到你：
Spingboot 多模块引入第三方jar包
cngm110:测试了一下，您这个方法打包后，在web-inf/lib/里面是缺少应取的第三方jar的的。
webase-front 与java 原生交互 关于 JSON的报错
Yokkkkk:请问报错这个：{"result":"{\"code\":201151,\"data\":null,\"errorMessage\":\"Invalid method set , supported functions are: []\"}"}怎么解决呢 set方法确认定义了，智能合约中的参数和后端也是对应的
文件操作
CSDN-Ada助手:Java 技能树或许可以帮到你：
蓝桥 最大获利
米花男团:你是怎么样怪写的？
大家在看
三数之和问题详解：五种语言实现与深度解析
智慧城市的核心逻辑与发展趋势：从数字治理到城市智能体 557
夸克网盘免费扩容，新用户轻松领取1TB免费空间！一步一步教你如何操作!
Python + PyTorch：从科研到工业落地的全场景实战指南，解锁深度学习核心应用
运用Gitee创建代码仓库和管理代码的详细步骤及部分问题的解决办法（入门级教学，含所需全套资源） 812
最新文章
Dockerfile中yum install 无法使用的问题
文件操作
结构体
2024年 5篇
2023年 5篇
2022年 61篇
2021年 1篇
上一篇： 错位重排 下一篇： 公约数与公倍数
分类专栏
C++6篇
工具1篇
SpringBoot4篇
idea2篇
日常bug1篇
java2篇
数据库1篇
fisco-baco2篇
区块链日常bug1篇
SSM9篇
算法43篇
JS2篇
线性代数1篇
数学3篇
排列组合1篇
展开全部收起
登录后您可以享受以下权益：
免费复制代码
和博主大V互动
下载海量资源
发动态/写文章/加入社区
×立即登录
评论
被折叠的 条评论 为什么被折叠?到【灌水乐园】发言
查看更多评论
添加红包
祝福语
请填写红包祝福语或标题
红包数量
个
红包个数最小为10个
红包总金额
元
红包金额最低5元
余额支付
当前余额 3.43 元 前往充值 >
需支付：10.00 元
取消 确定
成就一亿技术人!
领取后你会自动成为博主和红包主的粉丝 规则
hope_wisdom
发出的红包
实付 元
使用余额支付
点击重新获取
扫码支付
钱包余额 0
抵扣说明：
1.余额是钱包充值的虚拟货币，按照1:1的比例进行支付金额的抵扣。
2.余额无法直接购买下载，可以购买VIP、付费专栏及课程。
余额充值
确定 取消
举报
选择你想要举报的内容（必选）
内容涉黄
政治相关
内容抄袭
涉嫌广告
内容侵权
侮辱谩骂
样式问题
其他
原文链接（必填）
请选择具体原因（必选）
包含不实信息
涉及个人隐私
请选择具体原因（必选）
侮辱谩骂
诽谤
请选择具体原因（必选）
搬家样式
博文样式
补充说明（选填）
取消
确定
AI助手
AI 搜索 智能体 AI 编程 AI 作业助手
下载APP 程序员都在用的中文IT技术交流社区 公众号 专业的中文 IT 技术社区，与千万技术人共成长 视频号 关注【CSDN】视频号，行业资讯、技术分享精彩不断，直播好礼送不停！客服返回顶部 |
11720 | https://www.youtube.com/watch?v=Ulk21amITts | Calculus 2 (203) -5.2 - Area using Horizontal Slices
Mathematics Marianopolis College
609 subscribers
2 likes
Description
342 views
Posted: 12 Mar 2021
"This video was created by mathematics teachers at Marianopolis College. It follows along with notes that you can download free here :
It covers the following items :
5.2 - Area using Horizontal Slices
Proposition 5.2.1
Method : Area using horizontal slices
Example 5.2.1"
Transcript:
welcome back in the last video we talked about how to use vertical slices and step one was to make sure that your curves are labeled as y equals a function of x if they're not you could isolate y if you really want to use vertical slices but in the next section in section 5.2 we'll see what to do if you don't want to isolate y but you have x isolated instead all right so area using horizontal slices for a certain region it is easier to divide the region using horizontal rectangles and horizontal slices for example consider a region bounded by curve x equals a function of y so let's put that one here x equals a function of y and on the other side another one of these right and let's say we want to look at this region right here all right if you have a region like this then the more efficient way of actually splitting it up into slices is not to use vertical slices that would keep getting cut off and you'd have to split it into tons and tons of different parts and so the easiest way to split it up into thin sticks is to use horizontal sticks so horizontal slices all right so this is a horizontal size and so their width of one slice let's be let's take this one it would be h of y minus j of y notice that bigger numbers are this way and so it's going to be the one on the right that's bigger so it's going to be right minus left now the height is very very thin it's a small change it's a small change in y so i'm going to call it d y and so the area of one slice will be h of y minus j of y or right minus left times d y okay so it's the same idea as what we've done before it's just that now the roles of x and y are flipped right so the area of the region bounded by x equals h of y and x equals j of y between y equals c and y equals d is the integral from c to d of h of y minus j of y d y so now y is my variable and i'm assuming that h of y is the bigger one of the two for the entire interval um the more helpful way of thinking of this is to think of it as right minus left d y alright so if you want to use horizontal slices you need to first make sure that your curves are labeled as x equals a function of y again if it isn't the case you can isolate x then you find all intersection points algebraically between each pair of consecutive intersection points find left and right by plugging in a point between the these intersections and then you can give a rough sketch of the region and finally you set up the integral and you evaluate it and you need to split the integral up whenever right and left switch it's the same principle but just flipped on its side all right let's try example 5.2.1 i want to find the area enclosed by x equals y squared minus 4y and x equals minus y squared plus 2y alright so i have two curves that are described as x equals something so i'm going to use i'm going to use horizontal slices if you wanted to use vertical slices here you'd have to isolate y and that's kind of complicated and we'll do it in an example just to show you how complicated it is here i i actually get two different functions of x for each one of these curves so that would involve four different function of x all right so don't need to do the first step it's already done and then i'm going to take i want to take algebra to use to find the intersection points so i need x equals y squared minus 4y equals minus y squared plus 2y don't hesitate to stop and do whatever part you feel comfortable i'm going to send everything to one side to get a quadratic equals 0. then i'm going to factor whatever's in common i'll get y minus 3 left and so we get y equals 0 or y equals 3. if y equals 0 x equals y squared minus 4y equals 0 as well and if y equals 3 then y squared minus 4y that's 9 minus 12 minus 3. all right let's put those in sorry let's put those in the graph all right so we have 0 0 right here and then x is minus 3. this is a bit confusing right we're used to having x here and y here it's flipped so x is minus 3 and y is 3. so 1 2 3. here's my second point oh sorry minus three three all right i wanna figure out which of the two parabolas is right and which one is left i need to plug in a value between the two remember y is my variable so i'm going to plug in a value of y um i could plug in y equals 1. that's fine that's between the two so at y equals one the first x is one minus four it's minus three and the second one is minus one plus two is one all right so let me use a choice of color here if i use blue here that's the first one at one it's at minus three and if i use green for the other one at one it's at one all right so i don't really need to know more than this i'm just gonna sketch it course in this case since there are parabolas i could actually get a closer picture by finding the vertices of but really all i need is this right so that's my region let me label the curve that's going to help me so this one we said was x equals um minus y squared plus 2y and this one is y squared minus 4y all right perfect we need the integral from c to d um right minus left d y um i haven't talked about c and d much these are c would be the smallest value of y for the whole region and so in this case that's the lowest value of y so c is zero and then d would be the biggest of y for the whole region and so here i'm sorry d would be 3. so i get the integral from 0 to 3 right is minus y squared plus 2y left sorry right so right is on this curve and left is this curve so left would be y squared minus 4y d y all right for most problems we'll just ask you to set it up so this would be enough if you want to integrate it i recommend simplifying first you'll get -2 y squared plus 6 y d y i actually integrated it for you and got 9 if you want to check that you can do it yourself go ahead |
11721 | https://stackoverflow.com/questions/51233769/integer-ceilsqrtx | Stack Overflow
About
For Teams
Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers
Advertising Reach devs & technologists worldwide about your product, service or employer brand
Knowledge Solutions Data licensing offering for businesses to build and improve AI tools and models
Labs The future of collective knowledge sharing
About the company Visit the blog
Collectives¢ on Stack Overflow
Find centralized, trusted content and collaborate around the technologies you use most.
Learn more about Collectives
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Integer ceil(sqrt(x))
Ask Question
Asked
Modified 7 years, 2 months ago
Viewed 5k times
6
The answer gives the following code for computing floor(sqrt(x)) using just integers. Is it possible to use/modify it to return ceil(sqrt(x)) instead? Alternatively, what is the preferred way to calculate such value?
Edit: Thank you all so far and I apologise, I should have make it more explicit: I was hoping there is more "natural" way of doing this that using floor(sqrt(x)), possibly plus one. The floor version uses Newton's method to approach the root from above, I thought that maybe approaching it from below or similar would do the trick.
For example the answer even provides how to round to nearest integer: just input 4x to the algorithm.
math
integer
square-root
Share
Improve this question
edited Jul 8, 2018 at 16:59
minmaxminmax
asked Jul 8, 2018 at 16:00
minmaxminmax
52333 silver badges1010 bronze badges
Add a comment |
2 Answers 2
Reset to default
7
If x is an exact square, the ceiling and the floor of the square root are equal; otherwise, the ceiling is one more than the square root. So you could use (in Python),
result = floorsqrt(x) if result result != x: result += 1
Modifying the code you linked to is not a good idea, since that code uses some properties of the Newton-Raphson method of calculating the square root. Much theory has been developed about that method, and the code uses that theory. The code I show is not as neat as modifying your linked code but it is safer and probably quicker than making a change in the code.
Share
Improve this answer
answered Jul 8, 2018 at 16:36
Rory DaultonRory Daulton
22.7k77 gold badges4545 silver badges5151 bronze badges
Comments
0
You can use this fact that:
floor(x) = (ceil(x) - 1) if x \not \in Z else ceil(x)
Hence, check if N is in the form 2^k, the code is the same, and if it is not, you can -1 the result of the current code.
Share
Improve this answer
answered Jul 8, 2018 at 16:10
OmGOmG
19k1313 gold badges6969 silver badges9696 bronze badges
Comments
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
math
integer
square-root
See similar questions with these tags.
The Overflow Blog
The history and future of software development (part 1)
Getting Backstage in front of a shifting dev experience
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
New and improved coding challenges
New comment UI experiment graduation
Policy: Generative AI (e.g., ChatGPT) is banned
Linked
82 Writing your own square root function
Related
0 php sqrt function that returns an integer?
14 Calculate Nth root with integer arithmetic
5 sqrt() of int type in C
82 Integer square root in python
Finding a square root using only integers
Integer functions of square and square root
How do I the calculate the sqrt of a natural or rational number in coq?
square numbers Java
Integer Square root of a number
0 How can I find a square root in simplest form given an integer? (Java)
Hot Network Questions
Why do universities push for high impact journal publications?
Is direct sum of finite spectra cancellative?
Lingering odor presumably from bad chicken
How long would it take for me to get all the items in Bongo Cat?
Switch between math versions but without math versions
How to design a circuit that outputs the binary position of the 3rd set bit from the right in an 8-bit input?
The altitudes of the Regular Pentagon
How do you emphasize the verb "to be" with do/does?
Exchange a file in a zip file quickly
How to use \zcref to get black text Equation?
Change default Firefox open file directory
Who is the target audience of Netanyahu's speech at the United Nations?
Another way to draw RegionDifference of a cylinder and Cuboid
Why is the definite article used in “Mi deporte favorito es el fútbol”?
how do I remove a item from the applications menu
Is it ok to place components "inside" the PCB
How can blood fuel space travel?
How do trees drop their leaves?
What happens if you miss cruise ship deadline at private island?
What meal can come next?
How to use cursed items without upsetting the player?
Proof of every Highly Abundant Number greater than 3 is Even
Are there any world leaders who are/were good at chess?
Verify a Chinese ID Number
more hot questions
Question feed |
11722 | https://www.wikidoc.org/index.php/Logarithm | Logarithm
Jump to navigation Jump to search
In mathematics, the logarithm of a given number to a given base is the power or exponent to which the base must be raised in order to produce the given number.
For example, the logarithm of 1000 to the common base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.
The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,
An important feature of logarithms is that they reduce multiplication to addition, by the formula:
That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers. The use of logarithms to facilitate complex calculations was a significant motivation in their original development.
1 Properties of the logarithm
2 The logarithm as a function
3 Logarithm of a complex number
4 Group theory
5 Bases
5.1 Other notations
5.2 Change of base
6 Uses of logarithms
6.1 Science
6.2 Exponential functions
6.3 Easier computations
7 Related operations
7.1 Cologarithms
7.2 Antilogarithms
8 Calculus
9 Series for calculating the natural logarithm
10 Computers
11 Generalizations
12 History
13 Tables of logarithms
14 See also
15 References
16 External links
Properties of the logarithm
Main article: List of logarithmic identities
When x and b are restricted to positive real numbers, logb(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers.
The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:
which by taking logarithms becomes
A related property is reduction of exponentiation to multiplication. Using the identity:
it follows that c to the power p (exponentiation) is:
or, taking logarithms:
In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = bproduct.
Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. Logarithms make lengthy numerical operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis (see Bode plot).
The logarithm as a function
Though logarithms have been traditionally thought of as arithmetic sequences of numbers corresponding to geometric sequences of other (positive real) numbers, as in the 1797 Britannica definition, they are also the result of applying an analytic function. The function can therefore be meaningfully extended to complex numbers.
The function logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) in standard usage refers to a function of the form logb(x) in which the base b is fixed and so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the inverse function of the exponential function bx. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.
The base can also be a complex number; the evaluation of the log is just slightly more complicated in this case. See imaginary base.
Logarithm of a complex number
When the base b is real and z is a complex number, say z = x + i y, the logarithm of z is found easily by putting z in polar form that is, z = (x2 + y2)1/2 exp (i tan−1 (y / x) ). If the base of the logarithm is chosen as e , that is, using loge (denoted by ln and called the natural logarithm), the logarithm becomes:
This evaluation uses the properties of all logarithms (see above), regardless of choice of base: logb (c d ) = logb (c ) + logb (d ) and its generalization to arbitrary products logb bz = z. Because the inverse tangent is a multiple valued function of its argument, the logarithm of a complex number is not unique either. See article on complex logarithm.
Group theory
From the pure mathematical perspective, the identity
is fundamental in two senses. First, the remaining three arithmetic properties can be derived from it. Furthermore, it expresses an isomorphism between the multiplicative group of the positive real numbers and the additive group of all the reals.
Logarithmic functions are the only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers.
Bases
The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828... and 2. When "log" is written without a base (b missing from logb), the intent can usually be determined from context:
natural logarithm (loge, ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
common logarithm (log10 or simply log; sometimes lg) in various engineering fields, especially for power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify hand calculations
binary logarithm (log2; sometimes lg, lb, or ld) in information theory and computer-related fields
indefinite logarithm when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation.
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.
Other notations
The notation "ln(x)" invariably means loge(x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline:
Mathematicians generally understand both "ln(x)" and "log(x)" to mean loge(x) and write "log10(x)" when the base-10 logarithm of x is intended. Calculus textbooks will occasionally write "lg(x)" to represent "log10(x)".
Many engineers, biologists, astronomers, and some others write only "ln(x)" or "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, sometimes in the context of computing, log2(x).
On most calculators, the LOG button is log10(x) and LN is loge(x).
In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."
Some people use Log(x) (capital L) to mean log10(x), and use log(x) with a lowercase l to mean loge(x).
The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
A notation frequently used in some European countries is the notation blog(x) instead of logb(x).
This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.
As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley. As of 2005, many mathematicians have adopted the "ln" notation, but most use "log".
In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and popularized by Donald Knuth. However, lg(x) is also sometimes used for the common log, and lb(x) for the binary log. In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm. In German, lg(x) also denotes the base 10 logarithm, while sometimes ld(x) or lb(x) is used for the base 2 logarithm.
The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, which suggests these notations:
The notation "ln(x)" means loge(x);
The notation "lg(x)" means log10(x);
The notation "lb(x)" means log2(x).
As the difference between logarithms to different bases is one of scale, it is possible to consider all logarithm functions to be the same, merely giving the answer in different units, such as dB, neper, bits, decades, etc.; see the section Science and engineering below. Logarithms to a base less than 1 have a negative scale, or a flip about the x axis, relative to logarithms of base greater than 1.
Change of base
While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually loge and log10). To find a logarithm with base b, using any other base k:
Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:
Uses of logarithms
Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation bn = x, b can be determined with radicals, n with logarithms, and x with exponentials. See logarithmic identities for several rules governing the logarithm functions.
Science
Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list.
In chemistry, the negative of the base-10 logarithm of the activity of hydronium ions (H3O+, the form H+ takes in water) is the measure known as pH. The activity of hydronium ions in neutral water is 10−7 mol/L at 25 °C, hence a pH of 7.
The bel (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. The Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), equal to 0.1 bel, is more commonly used. The neper is a similar unit which uses the natural logarithm of a ratio.
The Richter scale measures earthquake intensity on a base-10 logarithmic scale.
In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.
In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.
In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.
In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using comparison can require time proportional to the product N × log N. Similarly, base-2 logarithms are used to express the amount of storage space or memory required for a binary representation of a number—with k bits (each a 0 or a 1) one can represent 2k distinct values, so any natural number N can be represented in no more than (log2 N) + 1 bits.
Similarly, in information theory logarithms are used as a measure of quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log2 N bits.
In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.
Many types of engineering and scientific data are typically graphed on log-log or semilog axes, in order to most clearly show the form of the data.
In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data does not meet the assumption of normality.
Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents (hundredths of an equally-tempered semitone). The interval between two notes in cents is the base-21/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). In MIDI, notes are numbered on the semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI).
Exponential functions
One way of defining the exponential function ex, also written as exp(x), is as the inverse of the natural logarithm. It is positive for every real argument x.
The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by
The antilogarithm function is another name for the inverse of the logarithmic function. It is written antilogb(n) and means the same as bn.
Easier computations
Logarithms can be used to replace difficult operations on numbers by easier operations on their logs (in any base), as the following table summarizes. In the table, upper-case variables represent logs of corresponding lower-case variables:
| Operation with numbers | Operation with exponents | Logarithmic identity |
---
| | | |
| | | |
| | | |
| | | |
These arithmetic properties of logarithms make such calculations much faster. The use of logarithms was an essential skill until electronic computers and calculators became available. Indeed the discovery of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of computers in the 20th century because it made feasible many calculations that had previously been too laborious.
As an example, to approximate the product of two numbers one can look up their logarithms in a table, add them, and, using the table again, proceed from that sum to its antilogarithm, which is the desired product. The precision of the approximation can be increased by interpolating between table entries. For manual calculations that demand any appreciable precision, this process, requiring three lookups and a sum, is much faster than performing the multiplication. To achieve seven decimal places of accuracy requires a table that fills a single large volume; a table for nine-decimal accuracy occupies a few shelves. Similarly, to approximate a power cd one can look up log c in the table, look up the log of that, and add to it the log of d; roots can be approximated in much the same way.
One key application of these techniques was celestial navigation. Once the invention of the chronometer made possible the accurate measurement of longitude at sea, mariners had everything necessary to reduce their navigational computations to mere additions. A five-digit table of logarithms and a table of the logarithms of trigonometric functions sufficed for most purposes, and those tables could fit in a small book. Another critical application with even broader impact was the slide rule, an essential calculating tool for engineers. Many of the powerful capabilities of the slide rule derive from a clever but simple design that relies on the arithmetic properties of logarithms. The slide rule allows computation much faster still than the techniques based on tables, but provides much less precision, although slide rule operations can be chained to calculate answers to any arbitrary precision.
Related operations
Cologarithms
The cologarithm of a number is the logarithm of the inverse of said number, meaning cologb(x)=logb(1/x)= - logb(x).
Antilogarithms
The antilogarithm is the logarithmic inverse of the logarithm, meaning that the antilogb(logb(x))=x. Thus, setting by=x implies that logb(x)=y. By taking the antilogb of both sides, antilogb(logb(x))=antilogby, thus x=antilogby. Therefore, by=antilogby.
Calculus
The natural logarithm of a positive number x can be defined as
The derivative of the natural logarithm function is
By applying the change-of-base rule, the derivative for other bases is
The antiderivative of the natural logarithm ln(x) is
and so the antiderivative of the logarithm for other bases is
See also: Table of limits, list of integrals of logarithmic functions.
Series for calculating the natural logarithm
There are several series for calculating natural logarithms. The simplest, though inefficient, is:
: when
To derive this series, start with ()
Integrate both sides to obtain
Letting and thus , we get
A more efficient series is
for z with positive real part.
To derive this series, we begin by substituting −x for x and get
Subtracting, we get
Letting and thus , we get
For example, applying this series to
we get
and thus
\frac{1}{7 \cdot 1000000} + \cdots \right )
where we factored 1/10 out of the sum in the first line.
For any other base b, we use
Computers
Most computer languages use log(x) for the natural logarithm, while the common log is typically denoted log10(x). The argument and return values are typically a floating point (or double precision) data type.
As the argument is floating point, it can be useful to consider the following:
A floating point value x is represented by a mantissa m and exponent n to form
Therefore
Thus, instead of computing we compute for some m such that 1 ≤ m < 2. Having m in this range means that the value is always in the range . Some machines use the mantissa in the range and in that case the value for u will be in the range In either case, the series is even easier to compute.
To compute a base 2 logarithm on a number between 1 and 2 in an alternate way, square it repeatedly. Every time it goes over 2, divide it by 2 and write a "1" bit, else just write a "0" bit. This is because squaring doubles the logarithm of a number.
The integer part of the logarithm to base 2 of an unsigned integer is given by the position of the left-most bit, and can be computed in O(n) steps using the following algorithm:
int r = 0; while( (x >> r) != 0){ r++; } return r-1; // returns -1 for x==0, floor(log2(x)) otherwise
}
However, it can also be computed in O(log n) steps by trying to shift by powers of 2 and checking that the result stays nonzero: for example, first >>16, then >>8, ... (Each step reveals one bit of the result)
Generalizations
The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus of z, arg(z) is the argument, and i is the imaginary unit; see complex logarithm for details.
The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bn = x, where b and x are elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in public key cryptography.
The logarithm of a matrix is the inverse of the matrix exponential.
It is possible to take the logarithm of a quaternions and octonions.
A double logarithm, , is the inverse function of the double exponential function. A super-logarithm or hyper-4-logarithm is the inverse function of tetration. The super-logarithm of x grows even more slowly than the double logarithm for large x.
For each positive b not equal to 1, the function logb (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.
History
The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston, in Scotland. (Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier.) Early resistance to the use of logarithms was muted by Kepler's enthusiastic support and his publication of a clear and impeccable explanation of how they worked.
Their use contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, they were used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides the utility of the logarithm concept in computation, the natural logarithm presented a solution to the problem of quadrature of a hyperbolic sector at the hand of Gregoire de Saint-Vincent in 1647.
At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: Template:Polytonic (logos) meaning proportion, and Template:Polytonic (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers they represent, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.
Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10−7 = 0.999999 (Bürgi chose r = 1 + 10−4 = 1.0001). Napier's original logarithms did not have log 1 = 0 but rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 107(1 − 10−7)L. Since (1 − 10−7)107 is approximately 1/e, this makes L/107 approximately equal to log1/e N/107.
Tables of logarithms
Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.
In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals.
Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error." An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.
François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.
Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1700s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.
See also
List of logarithm topics
List of logarithmic identities
Logarithmic scale
Natural logarithm
Common logarithm
Complex logarithm
Imaginary-base logarithm
Indefinite logarithm
Iterated logarithm
Logarithmic units
Discrete logarithm
Zech's logarithms
Logarithm of a matrix
Log-normal distribution
Decibel
Equal temperament
Richter magnitude scale
pH
References
Template:Wiktionarypar
↑ In general, x and b both can be complex numbers; see Kwok below, and imaginary-base logarithms.
↑ Yue Kuen Kwok (2002). Applied complex variables for scientists and engineers. Cambridge MA: Cambridge University Press. p. p. 102. ISBN 0521004624.CS1 maint: Extra text (link)
↑ See e (mathematical constant)
↑ ""Mathematisches Lexikon" at Mateh_online.at".
↑ 5.0 5.1 Gullberg, Jan (1997). Mathematics: from the birth of numbers. W. W. Norton & Co. ISBN 039304002X.
↑ ""Common Logarithm" at MathWorld".
↑ B. N. Taylor (1995). "Guide for the Use of the International System of Units (SI)". NIST Special Publication 811, 1995 Edition. US Department of Commerce.
↑ | Wooster Woodruff B, Smith David E: "Academic Algebra", page 360. Ginn & Company, 1902
↑ Handbook of Mathematical Functions, National Bureau of Standards (Applied Mathematics Series no.55), June 1964, page 68.
↑ Much of the history of logarithms is derived from The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions, by James Mills Peirce, University Professor of Mathematics in Harvard University, 1873.
↑ (section "Astronomical Tables")
↑ Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872.
↑ English Cyclopaedia, Biography, Vol. IV., article "Prony."
External links
| |
| Wikimedia Commons has media related to . |
Explaining Logarithms
Logarithm on MathWorld
Jost Burgi, Swiss Inventor of Logarithms
Translation of Napier's work on logarithms
Logarithms - from The Little Handbook of Statistical Practice
Algorithm for determining Log values for any base
ar:لوغاريتم bs:Logaritam bg:Логаритъм ca:Logaritme cs:Logaritmus da:Logaritme de:Logarithmus el:Λογάριθμος eo:Logaritmo fa:لگاریتم gl:Función logaritmo hi:लघुगणक ko:로그 hr:Logaritam io:Logaritmo id:Logaritma ia:Logarithmo is:Logri it:Logaritmo he:לוגריתם la:Logarithmus lv:Logaritms lt:Logaritmas hu:Logaritmus ms:Logaritma nl:Logaritme no:Logaritme simple:Logarithm sk:Logaritmus sl:Logaritem sr:Логаритам fi:Logaritmi sv:Logaritm th:ลอการิทึม uk:Логарифм
Retrieved from "
Categories:
CS1 maint: Extra text
Pages with script errors
Pages with broken file links
Commons category link from Wikidata
Logarithms
Elementary special functions
Scottish inventions
Cookies help us deliver our services. By using our services, you agree to our use of cookies.
Navigation menu |
11723 | https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/06%3A_Conic_Sections/6.04%3A_Equations_of_Circles/6.4.01%3A_Circles_Centered_at_the_Origin | 6.4.1: Circles Centered at the Origin - K12 LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
6.4: Equations of Circles
Unit 6: Conic Sections
{ }
{ "6.4.01:_Circles_Centered_at_the_Origin" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "6.4.02:_Circles_Not_Centered_at_the_Origin" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "6.01:_Ellipses" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "6.02:_Parabolas" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "6.03:_Hyperbolas" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "6.04:_Equations_of_Circles" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "6.05:_Conic_Sections_and_Degenerate_Conics" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Sun, 27 Mar 2022 16:57:00 GMT
6.4.1: Circles Centered at the Origin
14764
14764
admin
{ }
Anonymous
Anonymous User
2
false
false
[ "article:topic", "program:ck12", "authorname:ck12", "license:ck12", "source@ ]
[ "article:topic", "program:ck12", "authorname:ck12", "license:ck12", "source@ ]
Search site Search Search Go back to previous article
Username Password Sign in
Sign in
Sign in
Forgot password
Expand/collapse global hierarchy
1. Home
2. Bookshelves
3. Mathematics
4. Analysis
5. Unit 6: Conic Sections
6. 6.4: Equations of Circles
7. 6.4.1: Circles Centered at the Origin
Expand/collapse global location
6.4.1: Circles Centered at the Origin
Last updated
Save as PDF
Page ID 14764
( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}})
( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{#1}}} )
( \newcommand{\id}{\mathrm{id}}) ( \newcommand{\Span}{\mathrm{span}})
( \newcommand{\kernel}{\mathrm{null}\,}) ( \newcommand{\range}{\mathrm{range}\,})
( \newcommand{\RealPart}{\mathrm{Re}}) ( \newcommand{\ImaginaryPart}{\mathrm{Im}})
( \newcommand{\Argument}{\mathrm{Arg}}) ( \newcommand{\norm}{\| #1 \|})
( \newcommand{\inner}{\langle #1, #2 \rangle})
( \newcommand{\Span}{\mathrm{span}})
( \newcommand{\id}{\mathrm{id}})
( \newcommand{\Span}{\mathrm{span}})
( \newcommand{\kernel}{\mathrm{null}\,})
( \newcommand{\range}{\mathrm{range}\,})
( \newcommand{\RealPart}{\mathrm{Re}})
( \newcommand{\ImaginaryPart}{\mathrm{Im}})
( \newcommand{\Argument}{\mathrm{Arg}})
( \newcommand{\norm}{\| #1 \|})
( \newcommand{\inner}{\langle #1, #2 \rangle})
( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\AA}{\unicode[.8,0]{x212B}})
( \newcommand{\vectorA}{\vec{#1}} % arrow)
( \newcommand{\vectorAt}{\vec{\text{#1}}} % arrow)
( \newcommand{\vectorB}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}})
( \newcommand{\vectorC}{\textbf{#1}})
( \newcommand{\vectorD}{\overrightarrow{#1}})
( \newcommand{\vectorDt}{\overrightarrow{\text{#1}}})
( \newcommand{\vectE}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} )
( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}})
( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{#1}}} )
(\newcommand{\avec}{\mathbf a}) (\newcommand{\bvec}{\mathbf b}) (\newcommand{\cvec}{\mathbf c}) (\newcommand{\dvec}{\mathbf d}) (\newcommand{\dtil}{\widetilde{\mathbf d}}) (\newcommand{\evec}{\mathbf e}) (\newcommand{\fvec}{\mathbf f}) (\newcommand{\nvec}{\mathbf n}) (\newcommand{\pvec}{\mathbf p}) (\newcommand{\qvec}{\mathbf q}) (\newcommand{\svec}{\mathbf s}) (\newcommand{\tvec}{\mathbf t}) (\newcommand{\uvec}{\mathbf u}) (\newcommand{\vvec}{\mathbf v}) (\newcommand{\wvec}{\mathbf w}) (\newcommand{\xvec}{\mathbf x}) (\newcommand{\yvec}{\mathbf y}) (\newcommand{\zvec}{\mathbf z}) (\newcommand{\rvec}{\mathbf r}) (\newcommand{\mvec}{\mathbf m}) (\newcommand{\zerovec}{\mathbf 0}) (\newcommand{\onevec}{\mathbf 1}) (\newcommand{\real}{\mathbb R}) (\newcommand{\twovec}{\left[\begin{array}{r}#1 \ #2 \end{array}\right]}) (\newcommand{\ctwovec}{\left[\begin{array}{c}#1 \ #2 \end{array}\right]}) (\newcommand{\threevec}{\left[\begin{array}{r}#1 \ #2 \ #3 \end{array}\right]}) (\newcommand{\cthreevec}{\left[\begin{array}{c}#1 \ #2 \ #3 \end{array}\right]}) (\newcommand{\fourvec}{\left[\begin{array}{r}#1 \ #2 \ #3 \ #4 \end{array}\right]}) (\newcommand{\cfourvec}{\left[\begin{array}{c}#1 \ #2 \ #3 \ #4 \end{array}\right]}) (\newcommand{\fivevec}{\left[\begin{array}{r}#1 \ #2 \ #3 \ #4 \ #5 \ \end{array}\right]}) (\newcommand{\cfivevec}{\left[\begin{array}{c}#1 \ #2 \ #3 \ #4 \ #5 \ \end{array}\right]}) (\newcommand{\mattwo}{\left[\begin{array}{rr}#1 \amp #2 \ #3 \amp #4 \ \end{array}\right]}) (\newcommand{\laspan}{\text{Span}{#1}}) (\newcommand{\bcal}{\cal B}) (\newcommand{\ccal}{\cal C}) (\newcommand{\scal}{\cal S}) (\newcommand{\wcal}{\cal W}) (\newcommand{\ecal}{\cal E}) (\newcommand{\coords}{\left{#1\right}_{#2}}) (\newcommand{\gray}{\color{gray}{#1}}) (\newcommand{\lgray}{\color{lightgray}{#1}}) (\newcommand{\rank}{\operatorname{rank}}) (\newcommand{\row}{\text{Row}}) (\newcommand{\col}{\text{Col}}) (\renewcommand{\row}{\text{Row}}) (\newcommand{\nul}{\text{Nul}}) (\newcommand{\var}{\text{Var}}) (\newcommand{\corr}{\text{corr}}) (\newcommand{\len}{\left|#1\right|}) (\newcommand{\bbar}{\overline{\bvec}}) (\newcommand{\bhat}{\widehat{\bvec}}) (\newcommand{\bperp}{\bvec^\perp}) (\newcommand{\xhat}{\widehat{\xvec}}) (\newcommand{\vhat}{\widehat{\vvec}}) (\newcommand{\uhat}{\widehat{\uvec}}) (\newcommand{\what}{\widehat{\wvec}}) (\newcommand{\Sighat}{\widehat{\Sigma}}) (\newcommand{\lt}{<}) (\newcommand{\gt}{>}) (\newcommand{\amp}{&}) (\definecolor{fillinmathshade}{gray}{0.9})
Circles Centered at the Origin
Circles Centered at the Origin
Finding the Equation of a Circle
Examples
Example 1
Example 2
Example 3
Example 4
Example 5
Review
Answers for Review Problems
Vocabulary
Image Attributions
Circles Centered at the Origin
You draw a circle that is centered at the origin. You measure the diameter of the circle to be 32 units. Does the point (\ (14,8)) lie on the circle?
Circles Centered at the Origin
Until now, your only reference to circles was from geometry. A circle is the set of points that are equidistant (the radius) from a given point (the center). A line segment that passes through the center and has endpoints on the circle is a diameter.
Now, we will take a circle and place it on the x−y plane to see if we can find its equation. In this concept, we are going to place the center of the circle on the origin.
Finding the Equation of a Circle
Step 1: On a piece of graph paper, draw an x−y plane. Using a compass, draw a circle, centered at the origin that has a radius of 5. Find the point (3,4) on the circle and draw a right triangle with the radius as the hypotenuse.
Step 2: Using the length of each side of the right triangle, show that the Pythagorean Theorem is true.
Step 3: Now, instead of using (3, 4), change the point to (x, y) so that it represents any point on the circle. Using r to represent the radius, rewrite the Pythagorean Theorem.
The equation of a circle, centered at the origin, is (\ x^{2}+y^{2}=r^{2}), where r is the radius and (x, y) is any point on the circle.
Let's find the radius of (\ x^{2}+y^{2}=16) and graph.
To find the radius, we can set (\ 16=r^{2}), making (\ r=4). (\ r) is not -4 because it is a distance and distances are always positive. To graph the circle, start at the origin and go out 4 units in each direction and connect.
Now, let's find the equation of the circle with center at the origin and passes through (\ (−7,−7)).
Using the equation of the circle, we have: (\ (-7)^{2}+(-7)^{2}=r^{2}). Solve for (\ r^{2}).
(\ \begin{aligned}
(-7)^{2}+(-7)^{2} &=r^{2} \
49+49 &=r^{2} \
98 &=r^{2}
\end{aligned})
So, the equation is (\ x^{2}+y^{2}=98). The radius of the circle is (\ r=\sqrt{98}=7 \sqrt{2}).
Finally, let's determine if the point (\ (9,−11)) is on the circle (\ x^{2}+y^{2}=225).
Substitute the point in for x and y and see if it equals 225.
(\ \begin{aligned}
9^{2}+(-11)^{2} &=225 \
81+121 & \stackrel{?}{=} 225 \
202 & \neq 225
\end{aligned})
The point is not on the circle.
Examples
Example 1
Earlier, you were asked to determine if the point (14 ,8) lies on the circle that is centered at the origin and has a diameter of 32 units.
Solution
From this lesson, you know that the equation of a circle that is centered at the origin is (\ x^{2}+y^{2}=r^{2}), where (\ r) is the radius and (\ (x,y)) is any point on the circle.
With the point (\ (14,8)), (\ x=14) and (\ y=8). We are given the diameter, but we need the radius. Recall that the radius is half the diameter, so the radius is (\ \frac{32}{2}=16).
Plug these values into the equation of the circle. If they result in a true statement, the point lies on the circle.
(\ \begin{array}{r}
x^{2}+y^{2}=r^{2} \
14^{2}+8^{2} \stackrel{?}{=} 16^{2} \
196+64 \stackrel{?}{=} 256 \
260 \neq 256
\end{array})
Therefore the point does not lie on the circle.
Example 2
Graph and find the radius of (\ x^{2}+y^{2}=4).
Solution
(\ r=\sqrt{4}=2)
Example 3
Find the equation of the circle with a radius of (\ 6 \sqrt{5}).
Solution
Plug in (\ 6 \sqrt{5}) for (\ r) in (\ x^{2}+y^{2}=r^{2})
(\ \begin{array}{l}
x^{2}+y^{2}=(6 \sqrt{5})^{2} \
x^{2}+y^{2}=6^{2} \cdot(\sqrt{5})^{2} \
x^{2}+y^{2}=36 \cdot 5 \
x^{2}+y^{2}=180
\end{array})
Example 4
Find the equation of the circle that passes through (5, 8).
Solution
Plug in (5, 8) for x and y, respectively.
(\ \begin{aligned}
5^{2}+8^{2} &=r^{2} \
25+64 &=r^{2} \
89 &=r^{2}
\end{aligned})
The equation is (\ x^{2}+y^{2}=89)
Example 5
Determine if (\ (−10, 7)) is on the circle (\ x^{2}+y^{2}=149).
Solution
Plug in (\ (−10, 7)) to see if it is a valid equation.
(\ \begin{aligned}
(-10)^{2}+7^{2} &=149 \
100+49 &=149
\end{aligned})
Yes, the point is on the circle.
Review
Graph the following circles and find the radius.
(\ x^{2}+y^{2}=9)
(\ x^{2}+y^{2}=64)
(\ x^{2}+y^{2}=8)
(\ x^{2}+y^{2}=50)
(\ 2 x^{2}+2 y^{2}=162)
(\ 5 x^{2}+5 y^{2}=150)
Write the equation of the circle with the given radius and centered at the origin.
(\ 14)
(\ 6)
(\ 9 \sqrt{2})
Write the equation of the circle that passes through the given point and is centered at the origin.
(\ (7,−24))
(\ (2,2))
(\ (−9,−10))
Determine if the following points are on the circle, (\ x^{2}+y^{2}=74).
(\ (−8,0))
(\ (7,−5))
(\ (6,−6))
Challenge In Geometry, you learned about tangent lines to a circle. Recall that the tangent line touches a circle at one point and is perpendicular to the radius at that point, called the point of tangency.
The equation of a circle is (\ x^{2}+y^{2}=10) with point of tangency (\ (−3,1)).
Find the slope of the radius from the center to (\ (−3,1)).
Find the perpendicular slope to (a). This is the slope of the tangent line.
Use the slope from (b) and the given point to find the equation of the tangent line.
Repeat the steps in #16 to find the equation of the tangent line to (\ x^{2}+y^{2}=34) with a point of tangency of (\ (3,5)).
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 10.3.
Vocabulary
| Term | Definition |
--- |
| center | The center of a circle is the point that defines the location of the circle. All points on the circle are equidistant from the center of the circle. |
| Circle | A circle is the set of all points at a specific distance from a given point in two dimensions. |
| Diameter | Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius. |
| Equation of a Circle | If the center of a circle is (0, 0), then the equation of the circle is of the form (\ x^{2}+y^{2}=r^{2}), where (\ r) is the radius. |
| Radius | The radius of a circle is the distance from the center of the circle to the edge of the circle. |
Image Attributions
[Figure 1]
Credit: Tony Webster;Robert Pernett
Source:
This page titled 6.4.1: Circles Centered at the Origin is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.
LICENSED UNDER
Back to top
6.4: Equations of Circles
6.4.2: Circles Not Centered at the Origin
Was this article helpful?
Yes
No
Recommended articles
Article typeSection or PageAuthorCK12LicenseCK-12OER program or PublisherCK-12
Tags
source@
© Copyright 2025 K12 LibreTexts
Powered by CXone Expert ®
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org. |
11724 | https://www.science-gate.com/IJAAS/Articles/2019/2019-6-11/1021833ijaas201911006.pdf | International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 Contents lists available at Science-Gate International Journal of Advanced and Applied Sciences Journal homepage: 35 Impact of hydrodynamic forces on a static vehicle at varying Froude numbers under partial submergence and subcritical flow conditions Syed Muzzamil Hussain Shah, Zahiraniza Mustaffa , Khamaruzaman Wan Yusof Department of Civil and Environmental Engineering, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak, Malaysia A R T I C L E I N F O A B S T R A C T Article history: Received 29 April 2019 Received in revised form 31 August 2019 Accepted 2 September 2019 Roads are often the first assets affected by inundations which usually ends up as a serious hazard to the traffic. A vehicle, when submerged in floodwaters, is affected by several hydrodynamic forces, namely frictional, buoyancy, lift and drag forces. An understanding of the relevant forces involved is necessary to attempt to characterize the stability thresholds of vehicles in floodwaters. With that regards, a series of flume experiments were conducted on a static scaled model vehicle, Volkswagen Scirocco (1:24) which ensured the similarity laws. To assess the limiting thresholds that could lead to the stability failure modes, the vehicle was controlled to be partially submerged under the subcritical flow conditions. The vehicle was placed at the same domain with different orientations, namely 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315° and 360° for each run to reduce inconsistencies in the test data. Based on the varying hydraulic variables assessed from the experimental runs, the hydrodynamic forces were theoretically estimated. Later, the relation between the hydrodynamic forces at varying Froude numbers was determined. From the study outcomes, an inverse relation of Froude number with respect to buoyancy force was noticed. On the other hand, an increment in the lift force slightly increased the Froude number. Similarly, a positive trend between the drag and frictional forces with respect to the Froude number was witnessed. Keywords: Floodwaters Hydrodynamic forces Froude number Static vehicle Partial submergence Subcritical flow © 2019 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license ( 1. Introduction Flood related deaths are increasingly associated with people perishing in vehicles that become unstable while driving through floodwaters (Smith et al., 2019). Today, most of the major settlements of townships in many parts of the world are located along floodplains. With the continuing increase in urbanisation coupled with associated land development and encroachment activities, the natural form of the floodplains has significantly altered which often leads to the excessive flooding. Moreover, due to inadequate storm drainage system and the low-lying nature of these areas, the developed floodplains have been flooded frequently by many disastrous flooding events which have resulted in potential hazard risks to human life and Corresponding Author. Email Address: zahiraniza@utp.edu.my (Z. Mustaffa) Corresponding author's ORCID profile: 2313-626X/© 2019 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license ( much damage has often occurred to the property and infrastructure (Teo et al., 2012a). Floodwater is essentially a powerful component, strong enough to even move vehicles at the lowest hydraulic parameters (Martínez‐Gomariz et al., 2018). A vehicle’s stability would be compromised when the hydraulic variables exceed a certain limit, similar to the instability of pedestrians. However, in case of vehicles, characteristics like weight, aerodynamic shape, ground clearance and sealing capacity determines the level of stability in floodwaters (Shah et al., 2019). In the simplest terms, the damage and danger that floodwaters might cause to a vehicle could be related to the force of the flood flows as it travels down a floodplain. This force is generally described in terms of flow velocity and water depth (v×y) (Smith, 2015). To investigate the hydraulic variables, namely flow velocity (v) and water depth (y) that could lead to the stability failure of a static vehicle in floodwaters, laboratory physical experiments were conducted in the hydraulic flume located at University Technology PETRONAS (UTP), Malaysia. The vehicle was placed at the same domain with different orientations, namely 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315° and 360° for each run to Shah et al/International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 36 reduce inconsistencies in the test data. The data points measured from the experimental investigation were then used to theoretically assess the hydrodynamic forces namely frictional, buoyancy, lift and drag forces. Later, the relation of these forces with respect to the Froude number has been conversed. 2. Background theory Moving water, particularly floodwaters, create hydrodynamic forces. To assess the stability thresholds of vehicles in floodwaters, an understanding of the relevant forces, namely buoyancy, lift, drag and frictional forces is necessary to characterize the stability failure modes. 2.1. Buoyancy force (𝑭𝑩) Buoyancy force is an upward force exerted by a fluid that opposes the weight of the immersed object. In case of vehicles, when the buoyancy force exceeds vehicle weight, the vehicle starts to float and is carried away by the flow. This force can be expressed as: 𝐹𝐵 = 𝜌𝑔𝑉 (1) where, ρ is the density of water, g is the acceleration due to gravity and V is the submerged volume of the vehicle (Shu et al., 2011). 2.2. Lift force (𝑭𝑳) The lift force is a component of a force which is flowing past the surface of a body and acts perpendicular to the incoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction (Chien and Wan, 1999). The lift force can be expressed as: 𝐹𝐿= 1 2 𝜌𝐶𝐿𝐴𝐿𝑣2 (2) where, CL is the lift coefficient, AL is the acting area by the lift force which is given by AL= (lc × bc) as shown in Fig. 1, where lc and bc are the length and width of the vehicle, respectively, and v is the flow velocity (Xia et al., 2011). Fig. 1: Acting area by the lift force (AL) 2.3. Drag force (𝑭𝑫) Drag force can be defined as resistance to moving through a fluid (Poirot, 2012). In fluid dynamics, drag acts opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force relies on the fluid density, drag coefficient, area of changing momentum and fluid velocity. It can be expressed as: 𝐹𝐷= 1 2 𝜌𝐶𝐷𝐴𝐷𝑣2 (3) where, CD is the drag coefficient, AD is the submerged area projected normal to the flow and v is the flow velocity (Teo et al., 2012b). 2.4. Frictional force (𝑭𝑹) Frictional force varies depending on the state of brakes being applied. However, once the vehicle is lifted off from the surface, the frictional force becomes zero. The general formula for the friction force can be expressed as: 𝐹𝑅=µ𝐹𝐺 (4) where, μ is the friction coefficient and FG is the net weight of the vehicle (Teo et al., 2012b). The friction coefficient is simply a material property that relates to the two bodies involved and can be best estimated experimentally (Teo et al., 2012a). 3. Laboratory experimental setup Laboratory physical experiments were conducted in the hydraulic flume of UTP. The flume is 10 m long, 0.45 m deep and 0.3 m wide, provided with a bed that can be adjusted to the desired slopes. Since the study was performed on a flat surface, the bed slope was kept at 0°. Today’s vehicle on roads are different in design from the past, where new improvements in these vehicle design have taken into consideration. Therefore, in the current study a modern vehicle, Volkswagen Scirocco R with the geometric scale ratio of 1:24 was selected. The specifications of the prototype and model are shown in Fig. 2. Experiments were conducted to observe the limiting thresholds that would lead to the possibility of stability failure. With that regards, the discharge in the flume was adjusted gradually by means of valve, whereas the water depth was controlled through the vertical movement of wooden plank served as a sluice gate downstream of the flooded vehicle. As the flow approached the flooded vehicle, the velocity and water depth varied along the channel, therefore a certain section located at one vehicle length upstream of the flooded vehicle was assumed where the velocity and water depth were measured. This section was specified as representative of flow parameters acting on the vehicle (Xia et al., 2011). The model vehicle was placed at the same domain with different Shah et al/International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 37 orientations for each run to reduce inconsistencies in the test data. A point gauge attached to a roller (located at the top of the flume) was used to determine the water depths, whereas the approaching velocity was measured by using a portable velocity meter. A compact multi line laser was also used to profoundly observe the instability of vehicle as it was uncertain to observe the vehicle movement in the water through naked eyes. Further, the instability points were visually recorded by means of a camcorder. The detailed schematic diagram of the experimental setup is shown in Fig. 3. Fig. 2: Exterior dimensions (a) Prototype and (b) Model (1: 24) Fig. 3: Laboratory experimental setup 4. Theoretical assessment of hydrodynamic forces Recall that for the estimation of buoyancy force (FB), the submerged volume of the vehicle at different water depths (V), water density (ρ) and acceleration due to gravity (g) are required. Since ρ and g remains constant, therefore to estimate the buoyancy force, the submerged volume of the vehicle at different water depths was needed. Within that context, the chassis of the vehicle was precisely designed on scale (1:24) using AutoCAD. A two-dimensional design was initially prepared. The file was then saved as ‘.dwg’ format. It was then transferred to SolidWorks for further modification where the model was enhanced and extruded to prepare a complete three-dimensional chassis of the car. In contrast to the complex design of the chassis, the ring-shaped tire was simple to design as the diameter and width of the tire were known. Thus, the submerged volume of the tires at different water depths was separately determined. Herein the study was performed on a flat roadway condition under the subcritical flow conditions. Therefore, it was straight forward to estimate the submerged volume of the car at the required water depths as it equally covered the vehicle chassis. Prior to crop the chassis at the desired depth, the ground clearance of the vehicle was also taken into consideration and it was ensured that the water depth was estimated exactly from the point where the tires were in contact to the ground. This file was then saved as ‘.igs’ format. Lastly, the saved file was then transferred to ANSYS (Static Structural) and the submerged volume of the vehicle at different water depths was obtained. Once the submerged volume of the chassis was known, it was then accumulated with the submerged volume of the tires for the estimation of buoyancy force. The lift force (FL) is a component of a vertical force which is flowing past the surface of a body that acts perpendicular to the direction of incoming flow. This contrasts with the drag force which is the component of the horizontal force that acts parallel to the direction of incoming flow. Recall that for the estimation of lift force, water density (ρ), lift coefficient (CL), vertically projected area of the vehicle which is equivalent to the cross-sectional area of the car base (AL) and the flow velocity are required. The lift coefficient can be determined if the distribution of the dynamic pressure and shear force Shah et al/International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 38 on the entire body are known. Pressure and shear forces are difficult to obtain along a surface for a non-geometry body either theoretically or experimentally and thus can only be obtained numerically (Salleh, 2009). Since ρ remains constant, therefore the value of CL was obtained from the numerical simulations conducted on a similar car-type to observe the contribution of lift force to the incipient motion of a partly submerged flooded vehicle (Arrighi et al., 2015). In the current study, the proposed CL value was considered as a reference value due to two main reasons: (i) both the studies were carried under partial submergence and (ii) the geometry of the city car considered for numerical simulations tallied well to the model vehicle tested in this study. Thus, the lift coefficient used to determine the lift force was selected as 0.45. Lastly, the plan area of the vehicle, AL was considered when the water depth approaching the vehicle chassis exceeded the ground clearance. Once the water depth reached the bottom of the chassis, the vertically projected area of the vehicle remained constant and further increment in the water depth did not affect the cross-sectional area. The drag force (FD) on the vehicle is usually caused by the incoming flow which if exceeds the frictional force between the tires and the ground surface causes sliding instability failure. Unlike buoyancy force, which mainly relies on the submerged volume of the vehicle (V), the drag force depends on the submerged area projected normal to the flow (AD), drag coefficient (CD) and the approaching flow velocity (v). Therefore, the influence of drag force varies with vehicle position and the varying combination of hydraulic variables, i.e., (v·y) values. Herein ρ was taken as 1000 kg/m3, CD was set to 1.1 or 1.15 depending on the depth of the floodwater level with respect to vehicle chassis (Xia et al., 2011) and v was experimentally determined at different water depths using the velocity meter. However, to estimate AD, the submerged volume of vehicle (V) was fractionated by the chassis width (bc) at different water depths. The friction force (FR) concerns to the net weight of the vehicle (FN) and the friction coefficient (μ). FN is equivalent to vehicle weight in dry condition minus the vertical pushing force, namely buoyancy and lift forces, whereas μ depends on the tires condition and the roughness of the ground. For every increment in the water depth, the net weight of the vehicle was obtained once the buoyancy and lift forces were known. Moreover, for this study (static condition), the friction coefficient was assumed to be 0.3, which was confirmed almost adequate for the all the surface types as proposed by Bonham and Hattersley (1967). 5. Results and discussions Flows in the flumes can be categorized as free surface flow which are generally driven both by gravity and inertia. Therefore, Froude number is the best correlation required to analyse the flow. On the other hand, Reynolds number deals with the relationship between frictional and inertial forces. For various reasons, the requirements posed by the Froude and Reynolds numbers cannot be satisfied simultaneously. It is then necessary to decide which the dominant forces are. Typically, free surface flows are governed by gravity forces, whereas Reynolds number effects (viscous drag) are most likely to become more significant at low Reynolds number (Mustaffa, 2004). However, under this section, an attempt has been made to study the relation of varying hydrodynamic forces with respect to the Froude number. At first, the relation of buoyancy force at varying Froude numbers has been discussed as shown in Fig. 4. The relationship of buoyancy force and Froude number is very straight forward. From the graph, it can be inferred that with the increasing Froude number, a decrease in the buoyancy force was noticed. Fig. 4: Buoyancy force computed at varying Froude numbers The buoyancy force solely relies on the submerged volume which varies with the water depth, whereas Froude number is inversely proportional to it. For instance, at the lowest Froude number, the impact of buoyancy force was found maximum, whereas when the Froude number reached the maximum value, the buoyancy impact was found nominal. Based on these findings, it can be Shah et al/International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 39 concluded that Froude number is inversely proportional to the buoyancy force. Likewise, the impact of buoyancy force, the impact of lift force relative to the Froude numbers has also been computed as shown in Fig. 5. Fig. 5: Lift force computation at varying Froude numbers The vertically projected area of the car, AL remains constant as the floodwater depth reaches the vehicle chassis. Therefore, further increment in the water depth does not affect the plan area (AL). On the other hand, the flow velocity is proportional to both lift force and the Froude number. Therefore, from the graph, it can be inferred that an increment in the lift force slightly increased the Froude number. Thus, a positive relation between the lift force and Froude number has been witnessed. Drag force relies on the submerged area projected normal to the flow (AD), which varies based on the level of submergence, position of vehicle towards the direction of incoming flow and the intensity of the flow at which it travels (v). A pattern of gradual fluctuations in the drag force was noticed with an increase in the water depth. This highlights that drag force does not necessarily raise with the increasing water depth. However, referring to vehicle orientations, it can be seen that the influence of drag force was comparatively lower at the following orientations, namely 0°, 360° and 180°, as at these positions, the front and the rear ends of the vehicle were facing the incoming flow which provided smaller region to the drag force to take effect. However, the impact of drag force on the vehicle positioned at different angles, relative to Froude numbers has been further illustrated in Fig. 6. Fig. 6: Drag force computation at varying Froude numbers From the above graph, it can be inferred that below the critical water depth, the influence of velocity was more dominant in contrast to shallow depths. For instance, at majority of the orientations, it was noticed that the increasing Froude number increased the impact of drag force on the vehicle. Though, it relies both on the varying combinations of (v·y) values, but the graph concludes that at shallow depths, the flow velocity was more dominant to generate drag. Further, as highlighted earlier that drag varies with the vehicle orientation, thus, it can be seen that the influence of drag force differed nearly at the same Froude number for the vehicle at different orientations. For instance, at 0°, the impact Shah et al/International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 40 of drag force was noticed to be 0.23 N, when the value of Froude number reached 0.82. On the other hand, at 180°, the impact of drag force was noticed to be 0.15 N, for the same Froude number. The buoyancy force takes effect immediately when a body is immersed in water and its impact becomes more intense as the water level goes up. This is suggested to be the major reason for the reduction in the frictional force under the given circumstances. For instance, the maximum friction force between the vehicle tires and ground surface was found at the lowest water depth, i.e., 0.030 m, whereas at the maximum level (below critical water depth), i.e., 0.042 m, its influence was effectively reduced. Fig. 7 shows the relevance of friction force with respect to the Froude numbers so that the significance between the two can be studied. Fig. 7: Friction force computation at varying Froude numbers The graph displays that the friction force is in proportion to the Froude number. It is straightforward because Froude number is inversely proportional to the water depth. So, for each increment in the water depth, the friction force is decreased and so does the Froude number. For instance, the impact of frictional force was found negligible at low Froude numbers, whereas it successively increased with the increasing Froude numbers. 6. Conclusion The main findings of the study have highlighted that: (i) buoyancy force solely relies on the submerged volume which varies with the water depth, whereas the Froude number is inversely proportional to it. Thus, an inverse relation between the two was noticed, (ii) the lift force is concerned with the vehicle’s vertical area of projection which is equivalent to the cross-sectional area of the car base (AL) and the flow velocity (v). Notably, the affected area (AL) does not vary once the water level reaches the chassis of the car. Thus, a positive relation between the lift force and Froude number has been witnessed, (iii) below the critical water depth, the influence of velocity was more dominant in contrast to shallow depths. For instance, at majority of the orientations, it was noticed that the increasing Froude number increased the impact of drag force on the vehicle. Though, it relies both on the varying combinations of (v·y) values, but the outcomes conclude that at shallow depths, the flow velocity was more dominant to generate the drag. Thus, a positive trend between the two was noticed and (iv) lastly, as the water depth goes up, the contact of the tires with the ground reduces. Since Froude number is inversely proportional to the water depth, therefore, a positive relation between the Froude number and friction force was noticed. Acknowledgment This research was supported by University Technology PETRONAS (UTP) Internal Grant (URIF 0153AAG24), the Prototype Fund Grant (Cost Centre: 015PBA–008) and the Technology Innovation Program (Grant No.: 10053121) funded by the Ministry of Trade, Industry and Energy (MI, Korea). Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest. References Arrighi C, Alcèrreca-Huerta JC, Oumeraci H, and Castelli F (2015). Drag and lift contribution to the incipient motion of partly submerged flooded vehicles. Journal of Fluids and Structures, 57: 170-184. Bonham AJ and Hattersley RT (1967). Low level causeways. Technical Report No. 100, Water Research Laboratory, University of New South Wales, Kensington, Australia. Chien N and Wan Z (1999). Mechanics of sediment transport. American Society of Civil Engineers Press, Reston, USA. Shah et al/International Journal of Advanced and Applied Sciences, 6(11) 2019, Pages: 35-41 41 Martínez‐Gomariz E, Gómez M, Russo B, and Djordjević S (2018). Stability criteria for flooded vehicles: A state‐of‐the‐art review. Journal of Flood Risk Management, 11(S2): S817-S826. Mustaffa Z (2004). An experimental investigation of the hydraulics of street inlets. M.Sc. Thesis, University of Alberta, Edmonton, Canada. Poirot S (2012). Handbook of fluid dynamics and fluid hydronomics: Fundamentals and applications. Auris Reference, London, UK. Salleh MKA (2009). Simulation and analysis drag and lift coefficient between sedan and hatchback car. Ph.D. Dissertation, Universiti Malaysia Pahang, Gambang, Malaysia. Shah SMH, Mustaffa Z, Martinez-Gomariz E, Kim DK, and Yusof KW (2019). Criterion of vehicle instability in floodwaters: Past, present and future. International Journal of River Basin Management: 1-41. Shu C, Xia J, Falconer RA, and Lin B (2011). Incipient velocity for partially submerged vehicles in floodwaters. Journal of Hydraulic Research, 49(6): 709-717. Smith G (2015). Expert opinion: Stability of people, vehicles and buildings in flood water. Technical Report 2015/11, Water Research Laboratory, Allambie Heights, Australia. Smith GP, Modra BD, and Felder S (2019). Full‐scale testing of stability curves for vehicles in flood waters. Journal of Flood Risk Management. Teo FY, Falconer RA, Lin B, and Xia J (2012a). Investigations of hazard risks relating to vehicles moving in flood. The Journal of Water Resources Management, 1(1): 52-66. Teo FY, Xia J, Falconer RA, and Lin B (2012b). Experimental studies on the interaction between vehicles and floodplain flows. International Journal of River Basin Management, 10(2): 149-160. Xia J, Teo FY, Lin B, and Falconer RA (2011). Formula of incipient velocity for flooded vehicles. Natural Hazards, 58(1): 1-14. |
11725 | https://bmcpregnancychildbirth.biomedcentral.com/articles/10.1186/s12884-024-06898-x | Laparotomy versus laparoscopy for the treatment of adnexal torsion during pregnancy | BMC Pregnancy and Childbirth | Full Text
Your privacy, your choice
We use essential cookies to make sure the site can function. We also use optional cookies for advertising, personalisation of content, usage analysis, and social media.
By accepting optional cookies, you consent to the processing of your personal data - including transfers to third parties. Some third parties are outside of the European Economic Area, with varying standards of data protection.
See our privacy policy for more information on the use of your personal data.
Manage preferences for further information and to change your choices.
Accept all cookies
Skip to main content
Advertisement
Search
Explore journals
Get published
About BMC
Login
Menu
Explore journals
Get published
About BMC
Login
Search all BMC articles
Search
BMC Pregnancy and Childbirth
Home
About
Articles
Collections
Submission Guidelines
Join the Editorial Board
Submit manuscript
Laparotomy versus laparoscopy for the treatment of adnexal torsion during pregnancy
Download PDF
Download PDF
Research
Open access
Published: 30 October 2024
Laparotomy versus laparoscopy for the treatment of adnexal torsion during pregnancy
Zhenyu Zhang1,
Yajuan Zhang1,
Hanlin Fu1,2&
…
Ruixia Guo1,2
Show authors
BMC Pregnancy and Childbirthvolume 24, Article number:714 (2024) Cite this article
934 Accesses
Metrics details
Abstract
Background
Adnexal torsion (AT) is a rare emergency complication during pregnancy. Increasing evidence implies that operative laparoscopy for adnexal torsion performed during pregnancy could be safe and feasible. We procured and evaluated the surgical and obstetric outcomes between laparoscopy and laparotomy to determine the optimal approach for treating AT during pregnancy.
Methods
This was a retrospective study involving telephone questionnaire on adnexal torsion during pregnancy that occurred between July 2012 and July 2023 in the First Affiliated Hospital of Zhengzhou University. The study cohort included 155 pregnant women who underwent laparotomy or laparoscopic surgery. The clinical characteristics, surgical interventions, postoperative pathology and pregnancy outcomes were analyzed.
Results
A total of 102 patients were treated by laparoscopy, and 53 patients were treated by laparotomy. Compared with the laparotomy group, the laparoscopy group had significantly less blood loss (17.5 ml vs. 20.0 ml, p = .004), a lower incidence of delayed incision healing (1.0% vs. 11.3%, p = .011), and a shorter hospital stay (5.0 days vs. 8.0 days, p< .001). There was no significant difference between the two groups in terms of obstetric outcomes, including preterm delivery, miscarriage rate, birth weight, delivery gestation, cesarean delivery rate, or neonatal intensive care unit admission. All the pathological findings were benign except for one case of borderline mucinous cystadenoma. The most common pathological types were luteal cysts in the laparoscopy group and mature teratomas in the laparotomy group.
Conclusion
In this retrospective study, compared with laparotomy, laparoscopy yielded successful outcomes, with less surgical bleeding, less delayed wound healing, and shorter hospital stays. Laparoscopy could be a promising approach for diagnosing and treating AT during pregnancy.
Peer Review reports
Introduction
Adnexal torsion occurs when the ovary and fallopian tube twist on the axis created between the infundibulopelvic ligament and the utero-ovarian ligament. It generally involves only the ovary and rarely involves only the fallopian tube. It can occur in patients of any age but is not commonly found in women during pregnancy. Pregnancy is a risk factor for AT, with an estimated incidence of 1–5:10,000 of all pregnancies [1, 2].
Early diagnosis and prompt treatment of AT are critical for preserving ovarian tissue and preserving future function. However, the diagnosis of AT during pregnancy is often difficult due to its nonspecific signs and symptoms. Changes in the position of the ovaries and fallopian tubes with an enlarged uterus increase the difficulty of diagnosis. Misdiagnosis or delayed treatment can lead to ovarian necrosis, abortion, premature delivery and other risks. The clinical incidence of AT during pregnancy is low, and surgical treatment is mostly limited to small sample studies [3, 4].
In addition, laparoscopy during pregnancy requires consideration of specific intraoperative surgical risks as well as the pregnancy’s obstetric outcomes. Rottenstreich et al. demonstrated via logistic regression analysis that the performance of laparoscopy prior to 8 weeks of gestation was an independent factor associated with miscarriage. Pregnancy status and an enlarged uterus may affect treatment decisions. Due to the enlargement of the uterus, both traditional laparotomy and laparoscopic surgery are limited by the abdominal space. Bassi et al. reported that only 16% of pregnant women underwent laparoscopy, and 9% of them underwent conversion to open surgery.
The present study aimed to compare the surgical and obstetric outcomes of laparotomy and laparoscopic surgery and to evaluate the efficacy of laparoscopic surgery for treating AT during pregnancy.
Materials and methods
This was a retrospective cohort study in one medical center in China after approval by the institutional review board. The electronic medical records of patients who were hospitalized between July 2012 and July 2023 at The First Affiliated Hospital of Zhengzhou University were reviewed. Through retrospective case analysis, the keywords “torsion of ovary”, “adnexal torsion”, “pregnancy”, “torsion of ovarian cyst” and “torsion of fallopian tube” were used as search keywords. The number of pregnant women hospitalized during this period was 99,126, and a total of 159 pregnant women were diagnosed with AT by surgery, with an overall incidence of 0.16/100. Four patients who underwent cesarean section and adnexal cyst resection in late pregnancy were not included. The indications for surgery included sudden-onset abdominal pain and persistent masses on sonography. All laparoscopic surgeries were performed under general endotracheal anesthesia, whereas laparotomy was performed under regional anesthesia. The patients were divided into 2 groups according to the surgical method used: 53 patients were included in the traditional laparotomy group, and 102 patients were included in the laparoscopy group.
Clinical, ultrasound, surgical and pathological data were retrospectively retrieved from the medical records. The clinical information included the patient’s age, parity, mode of conception, gestational age, symptoms and signs, surgical data, postoperative pathology and pregnancy outcome. The operative time was calculated from the beginning to the end of anesthesia. We defined delayed-healing wounds as healing of the abdominal incision for more than seven days. Pregnancy outcome follow-up included the miscarriage rate, time of termination of pregnancy, method of termination of pregnancy, fetal weight, etc. Sonographic data were collected retrospectively from the last examination reports. The maximum diameter of the three sections was used as the reference data. The gestational trimesters were classified as follows: (1) first trimester, 12 weeks or before; (2) second trimester, 13–27 weeks; and (3) third trimester, 28 weeks and above.
Follow-up was conducted by medical records inquiry and telephone follow-up. The telephone questionnaire was administered by a physician using a standard script, which was uploaded to the supplement file. After providing verbal informed consent, the participants were asked about their pregnancy outcomes (live birth or abortion) and gestational age of delivery, etc.
The data analysis was carried out using SPSS 26.0 statistical analysis software (IBM Inc., Chicago, IL). Normally distributed continuous variables are reported as the mean ± standard deviation; nonnormally distributed variables are reported as the median and interquartile range. Categorical variables are reported as counts and proportions. Univariate analyses, including the chi-square test or Fisher’s exact test, were used to compare categorical variables. Continuous variables were tested by unpaired Student’s t tests or Mann‒Whitney tests, depending on the distribution of the continuous variables. Statistical significance was indicated at P< .05.
Results
Their demographic and clinical characteristics are described in Table1. There were no significant differences between the two groups in terms of age, number of pregnancies, size of cysts, history of pelvic surgery or assisted reproductive techniques. The gestational age at onset differed between the two groups: in the laparoscopy group, 83.3% of women were in the first trimester, 16.7% were in the second trimester, while in the laparotomy group, 22.6% of women were in the first trimester, 35.8% were in the second trimester and 41.5% were in the third trimester.
Table 1 Clinic characteristics of the two groups
Full size table
The operative characteristics of the patients in the study cohort are presented in Table2. There was no difference in the number of adnexal twists between the two groups. Most torsion cases were managed by removal of the involved adnexal cyst. If ovarian necrosis was severe, the affected adnexa should be removed concurrently. There was no significant difference between the two groups in terms of surgical type, including salpingo-oophorectomy, oophorectomy and cystectomy. However, the amount of blood loss, length of hospital stay and incidence of delayed-healing wounds in the laparotomy group were significantly greater than those in the laparoscopy group (P< .05). Elsewhere, the median gestational age at surgery was 10.1 weeks (range: 8.1–13.0) in the laparoscopy group and 24.4 weeks (range: 14.4–38.0) in the laparotomy group.
Table 2 Surgical details of the two groups
Full size table
The pathological characteristics are shown in Table3. The main pathological types were luteal cysts in the laparoscopy group and mature teratomas in the laparotomy group. In the laparoscopy group, detorsion alone was performed in three patients, and cyst aspiration was performed in one patient with a history of OHSS. In 18 patients, tissue ischemia and necrosis were badly damaged, and the pathological types could not be distinguished. All the pathological findings were benign except for 1 patient with borderline mucinous cystadenoma. The most common histopathological diagnosis was benign cystic teratoma (47.8%, 22/46); the second most common histopathological diagnosis was mesosalpinx cyst (17.4%, 8/46) in the laparotomy group; the most common histopathological diagnosis was corpus luteum cyst (48.9%, 43/88); and the second most common histopathological diagnosis was benign cystic teratoma (15.9%, 14/88) in the laparoscopy group.
Table 3 Pathologic diagnoses (%) of the two groups
Full size table
Obstetric outcomes were evaluated in 101 women in the laparoscopy group and 51 women in the laparotomy group (Table4). Progesterone therapy or inhibition of uterine contractions was routinely given to patients before 34 weeks after surgery, and uterine contractions were no longer inhibited after 34 weeks. Ultrasound was routinely performed before discharge to determine the presence of fetal heartbeats. All patients recovered well after surgery, without thrombosis, massive bleeding, or sepsis. Among these patients, two (2.3%) in the laparoscopy group and two (3.8%) in the laparotomy group were lost to follow-up after surgery. Artificial abortion was performed in one patient in the laparotomy group and 13 patients in the laparoscopy group because of an unplanned pregnancy. There were no significant differences between the two groups in terms of the rate of miscarriages, mode of pregnancy termination, gestational delivery, birth weight, neonatal intensive care unit, or rate of preterm delivery. A total of 7 miscarriages occurred in the laparoscopic group, 6 of which occurred within the first postoperative week. A total of 2 abortion in the laparotomy group, one was occurred within the first week after surgery.
Table 4 Obstetric outcome of the two groups
Full size table
Discussion
In the present study, we compared different surgical approaches for treating AT during pregnancy over a 10-year period in our hospital, and 155 patients were included. The present article was based on a large study comparing precise obstetric outcomes between laparotomy and laparoscopy. The results indicated that laparoscopic surgery for AT during pregnancy had better surgical outcomes, shorter lengths of hospital stay, less surgical bleeding and delayed-healing wounds than did laparotomy. No significant differences were observed between the groups in terms of fetal loss, preterm delivery, birth weight, delivery mode or neonatal intensive care unit admission.
The most common clinical symptom of torsion is intermittent, sudden-onset abdominal pain associated with nausea and vomiting. The diagnosis of AT is more challenging during pregnancy because these clinical symptoms are also typical for uterus rupture, placental abruption, urethral or renal colic, and hemolysis, elevated liver enzymes, low platelets (HELLP) syndrome. Transabdominal ultrasound is the preferred method for the examination of AT . It forms the foundation of diagnostic evaluation due to its ability to directly and rapidly evaluate both ovarian anatomy and perfusion. In our study, ultrasound was first performed in all 155 (100%) patients, and the most common ultrasound finding was an enlarged adnexal cyst with an average diameter of more than 7 cm in both groups. Ultrasonography findings suggestive of ovarian torsion include unilateral ovarian enlargement and ovarian edema with a coiled vascular pedicle (referred to as the “whirlpool sign”). The “whirlpool sign”, which is the main hallmark seen on imaging, is usually observed . Nevertheless, it was previously reported that Doppler ultrasonography was normal in 60% of patients despite the presence of ovarian torsion, and a normal ultrasound examination cannot exclude this diagnosis. Djavadian et al. demonstrated that in 15% of 38 patients, the initial diagnosis was incorrect from the time of hospitalization to surgery.
AT is a surgical diagnosis. Daykan et al. reported that surgery during pregnancy for suspected AT was not a risk factor for preterm labor and did not lead to adverse pregnancy or neonatal outcomes. Considering maternal and fetal safety, surgery should be performed in a timely manner when the patient is diagnosed. Thus, the uterus is obviously enlarged during the middle and late periods, and the space available for surgery is limited. The evidence thus far suggests that laparoscopy can be performed during any trimester of pregnancy, including the third trimester [12, 13]. In terms of the near-term effects, the first thing is to reduce the spontaneous abortion rate of surgical intervention. Wilasrusmee et al. reported in a meta-analysis that the risk of spontaneous abortion and preterm delivery was significantly greater among pregnant women with suspected appendicitis in the laparoscopy group than among those who underwent laparotomy. Rottenstreich et al. demonstrated that the use of a laparoscopic procedure during the early stages of pregnancy was associated with increased rates of miscarriage. However, many reports had discussed the relative safety of laparoscopy compared with that of laparotomy. Similar to previous studies [4, 11, 15], no significant difference in fetal loss or preterm labor was observed between the two groups in our study. Cagino et al. concluded that laparoscopy did not increase the risk of spontaneous abortion or preterm delivery based on a systematic review and meta-analysis of large samples. However, unlike other results originating from adnexal masses, which have precluded comparisons between laparoscopy and laparotomy. Our study focused on the comparison of surgical methods in cases of AT, which was unique. Doğan et al. reported a case of heterotopic pregnancy treated with transvaginal natural orifice transluminal endoscopic surgery (vNOTES), which was a less invasive alternative to laparoscopy. Perhaps someday vNOTES may be used in cases of AT during pregnancy.
The potential effects of CO2 insufflation on the fetus have also led to apprehension over its use. Because CO2 exchange occurs with intraperitoneal insufflation, there has been concern about the effects of acidosis . Sun B et al. reported that a CO2 pneumoperitoneum duration of less than one hour had no effect on the growth hormone (GH)–insulin-like growth factor I (IGF-I) axis, while a longer duration of CO2 pneumoperitoneum progressively inhibited the GH–IGF-I axis. This meant that maximizing surgical time could reduce the effects of CO2 pneumoperitoneum. If the torsion mass is malignant, it is not practical to complete laparoscopic surgery for a long time.
In addition, we found that the probability of delayed incision healing of more than seven days in the laparotomy group was significantly greater than that in the laparoscopy group. All six cases of delayed incision healing occurred in the third trimester of pregnancy, and two of them were sutured twice. Postoperative uterine contractions may repeatedly irritate the wound and affect its healing.
Regarding the long-term effects of surgery, Daykan et al. demonstrated that there were no significant differences in birth weight, gestational duration, intrauterine growth restriction, infant death, or fetal malformation between patients who underwent laparoscopy and those who underwent laparotomy for singleton pregnancies. In our study, we compared delivery gestation, delivery mode, and newborn weight between the two groups, and there were no significant differences. This suggests that neither laparoscopy nor laparotomy affects long-term pregnancy outcomes.
The occurrence of cancer during pregnancy is an uncommon event, occurring in approximately 1.26 of 1000 pregnancies . A meta-analysis revealed that the incidence of ovarian cancer in patients with AT was 1% . When surgery is required for known or suspected malignancies, laparotomy is a better approach for optimal visualization to minimize postoperative uterine irritability and contractions [22, 23]. The vast majority of AT tumors are benign, with only one borderline mucinous tumor in the laparoscopy group in this study. Conservative treatment includes only untwisting the adnexa and aspirating any associated cyst . Thus, we conducted detorsion alone in three patients and cyst aspiration in one patient with a history of OHSS. Corpus luteum cysts were more common in the laparoscopy group than in the laparotomy group. Torsion of the corpus luteum cyst most commonly occurs during the first trimester because it usually disappears by 12 to 16 weeks.
Our study also had a few limitations. First, this was a retrospective study in one center, and bias due to incomplete data might have created this study. Second, the gestation weeks at surgery of the two groups were not at the same baseline level because surgeons were more inclined to choose laparotomy in middle-late pregnancy.
Conclusions
AT is a rare, frequently misdiagnosed condition during pregnancy that is associated with a high rate of morbidity. Once AT is suspected, laparoscopy is a minimally invasive diagnostic method. On the basis of our retrospective study, laparoscopy for treating AT during pregnancy may be more beneficial than laparotomy. Further prospective comparisons of pregnant women with confirmed AT during pregnancy are needed to validate our results.
Data availability
All data generated or analyzed during this study are included in this published article.
Abbreviations
AT:
Adnexal torsion
IVF:
Assisted reproductive technique
OHSS:
Ovarian hyperstimulation syndrome
HELLP:
Hemolysis, elevated liver enzymes, low platelets syndrome
NICU:
Neonatal Intensive Care Unit
References
Asfour V, Varma R, Menon P. Clinical risk factors for ovarian torsion. J Obstet Gynaecol. 2015;35(7):721–5.
CASPubMedGoogle Scholar
Bridwell RE, Koyfman A, Long B. High risk and low prevalence diseases: ovarian torsion. Am J Emerg Med. 2022;56:145–50.
ArticlePubMedGoogle Scholar
Halimeh R, Tomassian S, El Hage M, Metri N, Bersaoui M, Daou R, Anastasiadis E. Laparoscopic Adnexal Detorsion in a 20-Week Pregnant Patient: A Case Report and Literature Review. Case reports in obstetrics and gynecology 2019, 2019:1093626.
Dvash S, Pekar M, Melcer Y, Weiner Y, Vaknin Z, Smorgick N. Adnexal Torsion in pregnancy managed by Laparoscopy is Associated with favorable obstetric outcomes. J Minim Invasive Gynecol. 2020;27(6):1295–9.
ArticlePubMedGoogle Scholar
Rottenstreich M, Rotem R, Hirsch A, Moran I, Ben-Shushan A, Armon S, Grisaru-Granovsky S, Rottenstreich A. Maternal and perinatal outcomes following laparoscopy for suspected adnexal torsion during pregnancy: a multicenter cohort study. Arch Gynecol Obstet. 2020;302(6):1413–9.
ArticlePubMedGoogle Scholar
Bassi A, Czuzoj-Shulman N, Abenhaim HA. Effect of pregnancy on the management and outcomes of ovarian torsion: a Population-based Matched Cohort Study. J Minim Invasive Gynecol. 2018;25(7):1260–5.
ArticlePubMedGoogle Scholar
Ssi-Yan-Kai G, Rivain AL, Trichot C, Morcelet MC, Prevot S, Deffieux X, De Laveaucoupet J. What every radiologist should know about adnexal torsion. Emerg Radiol. 2018;25(1):51–9.
ArticlePubMedGoogle Scholar
Adnexal Torsion in Adolescents. ACOG Committee Opinion No, 783. Obstet Gynecol. 2019;134(2):e56–63.
ArticleGoogle Scholar
Sasaki KJ, Miller CE. Adnexal torsion: review of the literature. J Minim Invasive Gynecol. 2014;21(2):196–202.
ArticlePubMedGoogle Scholar
Djavadian D, Braendle W, Jaenicke F. Laparoscopic oophoropexy for the treatment of recurrent torsion of the adnexa in pregnancy: case report and review. Fertil Steril. 2004;82(4):933–6.
ArticlePubMedGoogle Scholar
Daykan Y, Bogin R, Sharvit M, Klein Z, Josephy D, Pomeranz M, Arbib N, Biron-Shental T, Schonman R. Adnexal Torsion during pregnancy: outcomes after Surgical Intervention—A retrospective case-control study. J Minim Invasive Gynecol. 2019;26(1):117–21.
ArticlePubMedGoogle Scholar
Machado NO, Grant CS. Laparoscopic appendicectomy in all trimesters of pregnancy. JSLS: J Soc Laparoendoscopic Surg. 2009;13(3):384–90.
Google Scholar
Chohan L, Ramirez MM, Wray CJ, Kilpatrick CC. Laparoscopic management of fallopian tube torsion at 35 weeks of gestation: case report. J Minim Invasive Gynecol. 2011;18(3):390–2.
ArticlePubMedGoogle Scholar
Wilasrusmee C, Sukrat B, McEvoy M, Attia J, Thakkinstian A. Systematic review and meta-analysis of safety of laparoscopicversusopen appendicectomy for suspected appendicitis in pregnancy. Br J Surg. 2012;99(11):1470–8.
ArticleCASPubMedGoogle Scholar
Koo YJ, Kim HJ, Lim KT, Lee IH, Lee KH, Shim JU, Yoon SN, Kim JR, Kim TJ. Laparotomy versus laparoscopy for the treatment of adnexal masses during pregnancy. Aust N Z J Obstet Gynaecol. 2012;52(1):34–8.
ArticlePubMedGoogle Scholar
Cagino K, Li X, Thomas C, Delgado D, Christos P, Acholonu U Jr. Surgical Management of Adnexal Masses in pregnancy: a systematic review and Meta-analysis. J Minim Invasive Gynecol. 2021;28(6):1171–82.
ArticlePubMedGoogle Scholar
Doğan K, Kaya C, Ilıman DE, Karaca İ, Cengiz H. A spontaneous heterotopic pregnancy presenting with acute abdomen treated with natural orifice transluminal endoscopic surgery procedure: Case report. J Turkish Soc Obstetric Gynecol. 2020;16(4):271–3.
ArticleGoogle Scholar
Hunter JG, Swanstrom L, Thornburg K. Carbon dioxide pneumoperitoneum induces fetal acidosis in a pregnant ewe model. Surg Endosc. 1995;9(3):272–7.
ArticleCASPubMedGoogle Scholar
Sun B, Fu A, Wang R, Zhang Y. Influence of carbon dioxide pneumoperitoneum on the growth hormone-insulin-like growth factor I axis in mid- and late-pregnancy rats. J Obstet Gynaecol Res. 2015;41(9):1394–8.
ArticleCASPubMedGoogle Scholar
Esposito G, Franchi M, Dalmartello M, Scarfone G, Negri E, Parazzini F, La Vecchia C, Corrao G. Obstetric and neonatal outcomes in women with pregnancy associated cancer: a population-based study in Lombardy, Northern Italy. BMC Pregnancy Childbirth. 2021;21(1):31.
ArticlePubMedPubMed CentralGoogle Scholar
Oltmann SC, Fischer A, Barber R, Huang R, Hicks B, Garcia N. Pediatric ovarian malignancy presenting as ovarian torsion: incidence and relevance. J Pediatr Surg. 2010;45(1):135–9.
ArticlePubMedGoogle Scholar
Korenaga TK, Tewari KS. Gynecologic cancer in pregnancy. Gynecol Oncol. 2020;157(3):799–809.
ArticleCASPubMedPubMed CentralGoogle Scholar
Cavaco-Gomes J, Jorge Moreira C, Rocha A, Mota R, Paiva V, Costa A. Investigation and Management of Adnexal Masses in Pregnancy. Scientifica 2016, 2016:3012802.
Shwyiat R, Taso OA, Al-Edwan F, Khreisat B, Al-Dubees A. Retrospective analysis of patients with surgically proven ovarian torsion, our experience. J Family Med Prim Care. 2023;12(4):637–43.
ArticlePubMedGoogle Scholar
Download references
Acknowledgements
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Department of Obstetrics and Gynecology, The First Affiliated Hospital of Zhengzhou University, Zhengzhou, 450052, Henan, China
Zhenyu Zhang,Yajuan Zhang,Hanlin Fu&Ruixia Guo
Henan province engineering research center for gynecologic oncology fertility protection, Zhengzhou, 450052, Henan, China
Hanlin Fu&Ruixia Guo
Authors
1. Zhenyu ZhangView author publications Search author on:PubMedGoogle Scholar
2. Yajuan ZhangView author publications Search author on:PubMedGoogle Scholar
3. Hanlin FuView author publications Search author on:PubMedGoogle Scholar
4. Ruixia GuoView author publications Search author on:PubMedGoogle Scholar
Contributions
R.X.G conceived and designed the study, analyzed the data, wrote the paper. ZH.Y.ZH contributed materials, analyzed the data, wrote the paper. Y.J.ZH collected, and wrote the paper. H.l.F contributed analysis tools, analyzed the data and wrote the paper. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Ruixia Guo.
Ethics declarations
Ethics approval and consent to participate
This study was approved by the Ethics Committee of the First Affiliated Hospital of Zhengzhou University (No. 2023-KY-0840-002; approved on Sep. 5, 2023). The institutional ethics committee did not require written informed consent because the data were obtained anonymously from medical records, and verbal informed consent was obtained from all patients by a telephone questionnaire.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Supplementary Material 1
Supplementary Material 2
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, 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 you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. 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
Reprints and permissions
About this article
Cite this article
Zhang, Z., Zhang, Y., Fu, H. et al. Laparotomy versus laparoscopy for the treatment of adnexal torsion during pregnancy. BMC Pregnancy Childbirth24, 714 (2024).
Download citation
Received: 12 July 2024
Accepted: 14 October 2024
Published: 30 October 2024
DOI:
Share this article
Anyone you share the following link with will be able to read this content:
Get shareable link
Sorry, a shareable link is not currently available for this article.
Copy to clipboard
Provided by the Springer Nature SharedIt content-sharing initiative
Keywords
Laparoscopy
Laparotomy
Adnexal torsion
Pregnancy outcomes
Download PDF
Sections
References
Abstract
Introduction
Materials and methods
Results
Discussion
Conclusions
Data availability
Abbreviations
References
Acknowledgements
Funding
Author information
Ethics declarations
Additional information
Electronic supplementary material
Rights and permissions
About this article
Advertisement
Asfour V, Varma R, Menon P. Clinical risk factors for ovarian torsion. J Obstet Gynaecol. 2015;35(7):721–5.
CASPubMedGoogle Scholar
Bridwell RE, Koyfman A, Long B. High risk and low prevalence diseases: ovarian torsion. Am J Emerg Med. 2022;56:145–50.
ArticlePubMedGoogle Scholar
Halimeh R, Tomassian S, El Hage M, Metri N, Bersaoui M, Daou R, Anastasiadis E. Laparoscopic Adnexal Detorsion in a 20-Week Pregnant Patient: A Case Report and Literature Review. Case reports in obstetrics and gynecology 2019, 2019:1093626.
Dvash S, Pekar M, Melcer Y, Weiner Y, Vaknin Z, Smorgick N. Adnexal Torsion in pregnancy managed by Laparoscopy is Associated with favorable obstetric outcomes. J Minim Invasive Gynecol. 2020;27(6):1295–9.
ArticlePubMedGoogle Scholar
Rottenstreich M, Rotem R, Hirsch A, Moran I, Ben-Shushan A, Armon S, Grisaru-Granovsky S, Rottenstreich A. Maternal and perinatal outcomes following laparoscopy for suspected adnexal torsion during pregnancy: a multicenter cohort study. Arch Gynecol Obstet. 2020;302(6):1413–9.
ArticlePubMedGoogle Scholar
Bassi A, Czuzoj-Shulman N, Abenhaim HA. Effect of pregnancy on the management and outcomes of ovarian torsion: a Population-based Matched Cohort Study. J Minim Invasive Gynecol. 2018;25(7):1260–5.
ArticlePubMedGoogle Scholar
Ssi-Yan-Kai G, Rivain AL, Trichot C, Morcelet MC, Prevot S, Deffieux X, De Laveaucoupet J. What every radiologist should know about adnexal torsion. Emerg Radiol. 2018;25(1):51–9.
ArticlePubMedGoogle Scholar
Adnexal Torsion in Adolescents. ACOG Committee Opinion No, 783. Obstet Gynecol. 2019;134(2):e56–63.
ArticleGoogle Scholar
Sasaki KJ, Miller CE. Adnexal torsion: review of the literature. J Minim Invasive Gynecol. 2014;21(2):196–202.
ArticlePubMedGoogle Scholar
Djavadian D, Braendle W, Jaenicke F. Laparoscopic oophoropexy for the treatment of recurrent torsion of the adnexa in pregnancy: case report and review. Fertil Steril. 2004;82(4):933–6.
ArticlePubMedGoogle Scholar
Daykan Y, Bogin R, Sharvit M, Klein Z, Josephy D, Pomeranz M, Arbib N, Biron-Shental T, Schonman R. Adnexal Torsion during pregnancy: outcomes after Surgical Intervention—A retrospective case-control study. J Minim Invasive Gynecol. 2019;26(1):117–21.
ArticlePubMedGoogle Scholar
Machado NO, Grant CS. Laparoscopic appendicectomy in all trimesters of pregnancy. JSLS: J Soc Laparoendoscopic Surg. 2009;13(3):384–90.
Google Scholar
Chohan L, Ramirez MM, Wray CJ, Kilpatrick CC. Laparoscopic management of fallopian tube torsion at 35 weeks of gestation: case report. J Minim Invasive Gynecol. 2011;18(3):390–2.
ArticlePubMedGoogle Scholar
Wilasrusmee C, Sukrat B, McEvoy M, Attia J, Thakkinstian A. Systematic review and meta-analysis of safety of laparoscopicversusopen appendicectomy for suspected appendicitis in pregnancy. Br J Surg. 2012;99(11):1470–8.
ArticleCASPubMedGoogle Scholar
Koo YJ, Kim HJ, Lim KT, Lee IH, Lee KH, Shim JU, Yoon SN, Kim JR, Kim TJ. Laparotomy versus laparoscopy for the treatment of adnexal masses during pregnancy. Aust N Z J Obstet Gynaecol. 2012;52(1):34–8.
ArticlePubMedGoogle Scholar
Cagino K, Li X, Thomas C, Delgado D, Christos P, Acholonu U Jr. Surgical Management of Adnexal Masses in pregnancy: a systematic review and Meta-analysis. J Minim Invasive Gynecol. 2021;28(6):1171–82.
ArticlePubMedGoogle Scholar
Doğan K, Kaya C, Ilıman DE, Karaca İ, Cengiz H. A spontaneous heterotopic pregnancy presenting with acute abdomen treated with natural orifice transluminal endoscopic surgery procedure: Case report. J Turkish Soc Obstetric Gynecol. 2020;16(4):271–3.
ArticleGoogle Scholar
Hunter JG, Swanstrom L, Thornburg K. Carbon dioxide pneumoperitoneum induces fetal acidosis in a pregnant ewe model. Surg Endosc. 1995;9(3):272–7.
ArticleCASPubMedGoogle Scholar
Sun B, Fu A, Wang R, Zhang Y. Influence of carbon dioxide pneumoperitoneum on the growth hormone-insulin-like growth factor I axis in mid- and late-pregnancy rats. J Obstet Gynaecol Res. 2015;41(9):1394–8.
ArticleCASPubMedGoogle Scholar
Esposito G, Franchi M, Dalmartello M, Scarfone G, Negri E, Parazzini F, La Vecchia C, Corrao G. Obstetric and neonatal outcomes in women with pregnancy associated cancer: a population-based study in Lombardy, Northern Italy. BMC Pregnancy Childbirth. 2021;21(1):31.
ArticlePubMedPubMed CentralGoogle Scholar
Oltmann SC, Fischer A, Barber R, Huang R, Hicks B, Garcia N. Pediatric ovarian malignancy presenting as ovarian torsion: incidence and relevance. J Pediatr Surg. 2010;45(1):135–9.
ArticlePubMedGoogle Scholar
Korenaga TK, Tewari KS. Gynecologic cancer in pregnancy. Gynecol Oncol. 2020;157(3):799–809.
ArticleCASPubMedPubMed CentralGoogle Scholar
Cavaco-Gomes J, Jorge Moreira C, Rocha A, Mota R, Paiva V, Costa A. Investigation and Management of Adnexal Masses in Pregnancy. Scientifica 2016, 2016:3012802.
Shwyiat R, Taso OA, Al-Edwan F, Khreisat B, Al-Dubees A. Retrospective analysis of patients with surgically proven ovarian torsion, our experience. J Family Med Prim Care. 2023;12(4):637–43.
ArticlePubMedGoogle Scholar
BMC Pregnancy and Childbirth
ISSN: 1471-2393
Contact us
Submission enquiries: bmcpregnancyandchildbirth@biomedcentral.com
General enquiries: ORSupport@springernature.com
Read more on our blogs
Receive BMC newsletters
Manage article alerts
Language editing for authors
Scientific editing for authors
Policies
Accessibility
Press center
Support and Contact
Leave feedback
Careers
Follow BMC
BMC Twitter page
BMC Facebook page
BMC Weibo page
By using this website, you agree to our Terms and Conditions, Your US state privacy rights, Privacy statement and Cookies policy. Your privacy choices/Manage cookies we use in the preference centre.
© 2025 BioMed Central Ltd unless otherwise stated. Part of Springer Nature. |
11726 | https://www.sciencedirect.com/science/article/pii/036013239190067L | Calculation of the steady-state heat transfer through a slab-on-ground floor - ScienceDirect
Typesetting math: 100%
Skip to main contentSkip to article
Journals & Books
Access throughyour organization
Purchase PDF
Search ScienceDirect
Article preview
Abstract
References (14)
Cited by (51)
Building and Environment
Volume 26, Issue 4, 1991, Pages 405-415
Calculation of the steady-state heat transfer through a slab-on-ground floor
Author links open overlay panel B.R.Anderson†
Show more
Add to Mendeley
Share
Cite
rights and content
Abstract
An analytic formula is derived for the steady-state heat transfer through a slab-on-ground floor, with and without thermal insulation, based on the solution of the fundamental heat transfer equation with relevant boundary conditions.
So far as the building dimensions are concerned, the heat loss depends only on a “characteristic length” of the floor, defined as the floor area divided by half the exposed perimeter, a result which greatly simplifies the expression of heat losses and their evaluation for a building of arbitrary shape. The heat transfer is restricted by the thickness of the external walls, surface resistance and by any applied thermal insulation. The influence of these factors is combined appropriately so as to permit the straightforward evaluation of the thermal transmittance of the floor for any wall thickness and insulation thickness.
Access through your organization
Check access to the full text by signing in through your organization.
Access through your organization
Recommended articles
References (14)
A.E. Delsante et al.
Application of Fourier transforms to periodic heat flow into the ground under a building
Int. J. Heat. Mass Transfer (1983)
A.E. Delsante
Theoretical calculations of the steady-state heat losses through a slab-on-ground floor
Bldg. Envir. (1988)
H.S. Carslaw et al.
Conduction of heat in solids
(1959)
H.H. Macey
Heat loss through a solid floor
J. Inst. Fuel (1949)
CIBSE Guide Section A3
Thermal properties of building structures
(1980)
C.-E. Hagentoft, Lund Institute of Technology. Private communication...
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions and tablets. Dover Publications Inc., New...
There are more references available in the full text version of this article.
Cited by (51)
Thermal performance of non air-conditioned buildings with vaulted roofs in comparison with flat roofs
2006, Building and Environment Show abstract Vaulted or domed roofs have been frequently adopted by builders and architects throughout the Middle East and other hot dry areas. However, the thermal performance of such buildings under hot dry climatic conditions has rarely been quantitatively studied. In this paper, a detailed finite element model for the investigation of the thermal performance of non air-conditioned buildings with vaulted roofs (VR) is suggested based on two-dimensional unsteady heat transfer in such roofs and solar geometry. This model allows a comparison of the thermal performance of non air-conditioned buildings with a VR and a flat roof (FR) under different climatic conditions. Results obtained by numerical calculation show that, irrespective of building type the VRs are applied to, buildings with a VR have lower indoor temperatures as compared to those with a FR. The reason is that such roofs dissipate more heat than a FR does by convection and thermal radiation at night due to the enlarged curved surfaces. This implies that such roof forms are suitable for buildings located in hot dry regions but not for those located in hot humid areas, and reasonably explains why curved roofs have been extensively adopted by builders and architects in the hot dry areas in the past. However, with the decrease in the half rim angle of a VR, the difference of indoor thermal condition between a VR and a FR building becomes small and insignificant. Results also indicate that the indoor air temperature is slightly influenced by the half rim angle θ 0 and the orientation φ v of the VR. To be effective to create a favorable thermal condition inside buildings with a VR under hot dry climatic conditions, the half rim angle of a VR should be θ 0>50°, instead of θ 0<5 0∘, which is the optimal half rim angle of a VR of air-conditioned buildings, as found by the present authors in a previous study.
### Modelling floor heating systems using a validated two-dimensional ground-coupled numerical model
2005, Building and Environment Show abstract This paper presents a two-dimensional simulation model of the heat losses and temperatures in a slab on grade floor with floor heating which is able to dynamically model the floor heating system. The aim of this work is to be able to model, in detail, the influence from the floor construction and foundation on the performance of the floor heating system. The ground-coupled floor heating model is validated against measurements from a single-family house. The simulation model is coupled to a whole-building energy simulation model with inclusion of heat losses and heat supply to the room above the floor. This model can be used to design energy efficient houses with floor heating focusing on the heat loss through the floor construction and foundation. It is found that it is important to model the dynamics of the floor heating system to find the correct heat loss to the ground, and further, that the foundation has a large impact on the energy consumption of buildings heated by floor heating. Consequently, this detail should be in focus when designing houses with floor heating.
### The influence of soil moisture transfer on building heat loss via the ground
2004, Building and Environment Show abstract In this paper, the influence of soil moisture transfer on building heat loss via the ground is investigated by comparing fully coupled simulations with linear thermal simulations. The observed influences of coupling are (1) the larger amplitude of surface temperature, (2) the variation of thermal conductivity with moisture content, and (3) the advection of sensible heat by liquid transfer. In a parameter study, it is shown that these conclusions hold for a variety of climates, soils and foundation constructions. However, given the current accuracy level of standard methods, the integration of coupling effects in these calculation methods cannot be defended.
### On thermal performance of an improved roof pond for cooling buildings
2004, Building and Environment Show abstract In this study, a detailed simulation model is developed for the investigation and analysis of a roof pond with gunny bags (RPWGB) floating on the water surface which was suggested and experimentally investigated in a previous study of the present authors. The model is based on the newly proposed empirical correlation of water evaporation rate from a wetted surface to the ambient air proposed by the present authors, and takes into account the response of buildings as a whole to evaporative cooling. Results by simulations show that, regardless of the type of buildings the technique is applied to, RPWGB has a better cooling performance in terms of the indoor air temperature and heat flux through the roof into pond as compared to a roof covered with wetted gunny bags, which had been widely considered to be one of the most efficient roof cooling techniques. The reason for its higher efficiency is the thermal stratification inside the pond. The simulated results also reveal that the optimal water depth in the RPWGB depends on the roof type in general, and it is about 20 cm for a concrete roof and is about 5 cm for metal-decked roofs.
### Ground heat transfer effects on the thermal performance of earth-contact structures
2000, Renewable Sustainable Energy Reviews Show abstract A review of ground heat transfer effects on the thermal performance of earth contact structures is presented. The fundamental heat transfer processes relevant to the problem are described along with methods of determining thermal properties of soils. An overview of the many analytical, semi-analytical and numerical methods available to solve the heat transfer problem is also provided, followed by a brief summary of design guides. The review also considers the influence of changes in ground water content on the heat transfer properties of soils. A description of the processes that give rise to changes in ground water conditions is provided. The bulk thermal conductivity of a soil is shown to be strongly related to its water content. An overview of methods of analysing changes in soil moisture content is then presented. Methods of estimating the relevant hydraulic properties of soils are also considered. The final part of the review provides a brief outline of the theoretical approach required to analyse coupled heat and moisture migration in soils. Notwithstanding the fact that there are many practical design tools available, it appears that further work is necessary to clarify the circumstances in which more sophisticated analysis is warranted. Recent studies indicate that geometric simplification can lead to quite significant errors in heat loss calculation. Full three-dimensional treatment appears to be necessary in some cases. Thermal properties of soils vary according to the properties and proportions of the constituent phases (air/water/solid). Soil moisture content variations occur naturally or as a result of anthropogenic activity. The influence of such variations on the thermal conductivity of the ground is significant. The review outlines some simplified methods of accommodating this feature of the ground heat transfer problem. However, this aspect of the problem appears to need further consideration.
### Dynamic calculation of thermal bridges
1997, Energy and Buildings Show abstract Thermal bridges play a relatively large role in heat loss with increasing thermal insulation level of building envelope. In this paper, multi-dimensional dynamic models are set up to predict the thermal performances of thermal bridges using a frequency response method. The thermal bridge is divided into finite meshes where the nodal temperatures are expressed in the form of complex quantities that are related to amplitude and phase lag of the variations of the temperature. By considering the heat balance for the nodes, the real and imaginary parts of the nodal temperatures can be solved simultaneously. A PC computer program has been developed to calculate the dynamic responses of temperatures and heat flows within thermal bridges, and has been utilized for thermal bridges such as a light weight metal stud wall, a heavy weight wall—intermediate floor junction and a wall foundation with ground coupling.
View all citing articles on Scopus
†
Building Research Establishment, Scottish Laboratory, Kelvin Road, East Kilbride, G75 0RZ.
View full text
Copyright © 1991 Published by Elsevier Ltd.
Recommended articles
Efficient yellow-green organic light-emitting diodes based on sublimable cationic iridium complexes
Dyes and Pigments, Volume 130, 2016, pp. 1-8 Fuli Zhang, …, Bin Zhai
### Performance indicators matrix as a methodology for energy management in municipal water services
Journal of Cleaner Production, Volume 125, 2016, pp. 108-120 Margarida Ribau Teixeira, …, Luís M.Nunes
### Actualités réglementaires et jurisprudentielles pour les établissements et professions de la psychiatrie (Avril et mai 2024)
Santé mentale et Droit, Volume 24, Issue 4, 2024, pp. 734-742 Haji Safar
### Considerations involved in elective generator replacement of implantable cardioverter-defibrillators: balancing risks, benefits, and individual factors
Revista Española de Cardiología (English Edition), Volume 77, Issue 1, 2024, pp. 2-5 Manuel Martínez-Sellés, …, Tomás Datino
### Comparative Analysis of Simplified Daylight Glare Methods and Proposal of a new Method Based on the Cylindrical Illuminance
Energy Procedia, Volume 78, 2015, pp. 699-704 Santiago Torres, Valerio R.M. Lo Verso
### In situ assessment of the fabric and energy performance of five conventional and non-conventional wall systems using comparative coheating tests
Building and Environment, Volume 109, 2016, pp. 68-81 Eshrar Latif, …, Pete Walker
Show 3 more articles
About ScienceDirect
Remote access
Contact and support
Terms and conditions
Privacy policy
Cookies are used by this site.Cookie settings
All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply.
We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. You can manage your cookie preferences using the “Cookie Settings” link. For more information, see ourCookie Policy
Cookie Settings Accept all cookies
Cookie Preference Center
We use cookies which are necessary to make our site work. We may also use additional cookies to analyse, improve and personalise our content and your digital experience. For more information, see our Cookie Policy and the list of Google Ad-Tech Vendors.
You may choose not to allow some types of cookies. However, blocking some types may impact your experience of our site and the services we are able to offer. See the different category headings below to find out more or change your settings.
You may also be able to exercise your privacy choices as described in our Privacy Policy
Allow all
Manage Consent Preferences
Strictly Necessary Cookies
Always active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work.
Cookie Details List
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site.
Cookie Details List
Contextual Advertising Cookies
[x] Contextual Advertising Cookies
These cookies are used for properly showing banner advertisements on our site and associated functions such as limiting the number of times ads are shown to each user.
Cookie Details List
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Confirm my choices |
11727 | https://www.quora.com/Why-does-a-common-mercury-thermometer-consist-of-a-large-volume-filled-mercury-bulb-attached-to-a-capillary-tube | Why does a common mercury thermometer consist of a large volume filled mercury bulb attached to a capillary tube? - Quora
Something went wrong. Wait a moment and try again.
Try again
Skip to content
Skip to search
Sign In
Physics
Temperature Gauge
Capillary Action
Measurement Units
Mercury Thermometer
Thermometers
Measurements
Temperature Measurements
Capillary Dynamics
5
Why does a common mercury thermometer consist of a large volume filled mercury bulb attached to a capillary tube?
All related (32)
Sort
Recommended
Hrishav
B.Tech in Chemical Engineering, Indian Institute of Technology, Bombay (IITB)
·9y
the bulb of a common thermometer is a sphere. sphere has the property that it has highest surface area to volume ratio. so it takes less time to attain the temperature at which it is kept. moreover the capillary tube magnifies the expansion of mercury for easier interpretition of rise or fall in temperature. hope this helps..
Upvote ·
9 7
Promoted by JetBrains
JetBrains
The drive to develop
·4y
How can I get the GoLand IDE for free?
There are several ways to get access to GoLand for free or with minimal expense:
Participate in the Early Access Program. EAP builds let you try out the latest features for free. Please be aware that these builds are not fully tested and might be unstable. Just like the releases, EAPs happen three times a year, each lasting for 8-9 weeks. All EAP builds are valid for 30 days from their date of release. You can learn more about it here.
Take advantage of special offers. Just like all JetBrains IDEs, GoLand is available for free to students and open-source contributors. There are also free license
Continue Reading
There are several ways to get access to GoLand for free or with minimal expense:
Participate in the Early Access Program. EAP builds let you try out the latest features for free. Please be aware that these builds are not fully tested and might be unstable. Just like the releases, EAPs happen three times a year, each lasting for 8-9 weeks. All EAP builds are valid for 30 days from their date of release. You can learn more about it here.
Take advantage of special offers. Just like all JetBrains IDEs, GoLand is available for free to students and open-source contributors. There are also free licenses for user groups and other special offers. Read the details on this page.
Extended trial for teams. If your team wants to try GoLand, you can request a 60-day free trial for an unlimited number of users. You can apply here.
If you’re not sure about buying a yearly subscription, try a monthly one first. You can cancel at any time.
Finally, keep in mind that there is a 20% discount for the second year and a 40% discount for the third year onwards. For individuals, it means that you’ll have to pay only a few US dollars a month for the third year and beyond. Check out prices for individuals and organizations.
Upvote ·
999 101
9 3
Related questions
More answers below
Why do we use a cylindrical bulb instead of a spherical bulb in mercury thermometers?
Why does mercury not attach to the capillary tube of a thermometer?
Why are digital thermometers replaced with mercury thermometers?
Why is there a vacuum above the mercury in the capillary tube of a laboratory thermometer?
What are the benefits of Mercury Thermometer?
Garion Hall
Leading adult web site owner and operator since 2000 (www.abbywinters.com). · Author has 2.3K answers and 20.9M answer views
·9y
Related
Why do Mercury thermometers have a bulb?
The bulb is a vessel in which to store mercury that is not being used to indicate a temperature on the scale.
Continue Reading
The bulb is a vessel in which to store mercury that is not being used to indicate a temperature on the scale.
Upvote ·
Peter Hand
Lives in Las Vegas, NV · Upvoted by
Malcolm Sargeant
, Degree level applied chemistry + 20yr experience in corrosion prevention and water treatment and
Jerzy Michał Pawlak
, PhD in High Energy Physics (experimental) · Author has 10.8K answers and 58M answer views
·7y
Related
What color is the mercury in a mercury thermometer?
It’s a sad thing that the young people of today have probably never seen a mercury thermometer. That’s because of the overblown hysteria surrounding the metal, such that if a student breaks one in a school lab the entire school has to be evacuated and the lab gutted.
It is bright silver.
I couldn’t tell you how many hundreds of times I’ve had one of these toxic instruments of instant death inserted under my tongue as a child. It was such fun when one got broken and we’d chase the tiny balls of mercury around the floor trying to get them to amalgamate into one blob. But somehow my generation surv
Continue Reading
It’s a sad thing that the young people of today have probably never seen a mercury thermometer. That’s because of the overblown hysteria surrounding the metal, such that if a student breaks one in a school lab the entire school has to be evacuated and the lab gutted.
It is bright silver.
I couldn’t tell you how many hundreds of times I’ve had one of these toxic instruments of instant death inserted under my tongue as a child. It was such fun when one got broken and we’d chase the tiny balls of mercury around the floor trying to get them to amalgamate into one blob. But somehow my generation survived.
Upvote ·
999 360
99 34
Assistant
Bot
·
Sep 6
A common mercury thermometer uses a large mercury-filled bulb connected to a narrow capillary tube because that geometry converts small temperature-driven volume changes into easily readable length changes while keeping the device compact, accurate, and mechanically simple. Key reasons:
Volume-to-length amplification
Mercury’s thermal expansion is volumetric. A relatively large bulb contains most of the mercury so a small fractional volume change ΔV (from ΔT) produces a much larger change in column height Δh when forced into a narrow capillary: Δh ≈ ΔV / A, where A is the capillary cross-secti
Continue Reading
A common mercury thermometer uses a large mercury-filled bulb connected to a narrow capillary tube because that geometry converts small temperature-driven volume changes into easily readable length changes while keeping the device compact, accurate, and mechanically simple. Key reasons:
Volume-to-length amplification
Mercury’s thermal expansion is volumetric. A relatively large bulb contains most of the mercury so a small fractional volume change ΔV (from ΔT) produces a much larger change in column height Δh when forced into a narrow capillary: Δh ≈ ΔV / A, where A is the capillary cross-sectional area. Smaller A → larger Δh for the same ΔV, improving resolution and readability.
Sensitivity and resolution
The bulb maximizes the amount of mercury that expands per degree, increasing ΔV. The capillary’s small area converts that into a long, thin column, so a small temperature change produces a visibly long displacement.
Fast thermal response with stability balance
A bulb large enough to contain sufficient mercury balances sensitivity and thermal inertia: big enough to give measurable expansion, small enough to respond reasonably quickly to temperature changes. Thin capillaries have negligible thermal mass compared with the bulb.
Linear, repeatable scale and calibration
The uniform cross section of the capillary produces a nearly linear relationship between column height and volume change (h ∝ V for constant A), simplifying calibration and the use of evenly spaced scale markings.
Visibility and containment
A narrow tube produces a high surface-to-length contrast so the meniscus position is easily seen against a scale. Mercury’s cohesive, reflective column in a glass capillary yields a sharp, visible meniscus. The bulb contains the bulk of mercury safely; the tube only needs a small continuous column.
Practical manufacturing and sealing
Glass bulbs and capillaries are straightforward to blow and seal. Filling a single large reservoir connected to a capillary is simpler and more reliable than trying to fabricate a long, uniform bulb-shaped reservoir.
Reduction of errors from thermal expansion of glass
Because most mercury sits in the bulb, the relative effect of glass expansion of the tube is smaller on the reading; calibration can compensate more easily.
Combined, these factors produce a thermometer that magnifies minute thermal expansion into an easy-to-read vertical displacement, while remaining compact, robust, and simple to manufacture.
Upvote ·
Related questions
More answers below
Is it safe to have a little bit of mercury touch you when broken from a thermometer?
What are the parts of a mercury thermometer?
Does the mercury in a clinical thermometer in a capillary tube not contract once removed?
What happens if a mercury thermometer breaks?
What are the disadvantages of a mercury thermometer?
John Walker
Retired science teacher. BA in Natural Sciences. · Author has 1.5K answers and 3.4M answer views
·Updated 8y
Related
Why is mercury used in thermometers?
Liquid in glass thermometers are the cheapest kind of thermometer, and are also easy to use. Mercury is easily the best liquid to use, for 5 important reasons.
It is very reflective, so it's easy to see and to read accurately.
It doesn't wet the glass, so you don't get inaccurate readings if the temperature is falling.
It is a metal, so it's a good conductor of heat. This means that it reacts quickly to changes of temperature.
It expands evenly with temperature, so a linear scale can be used with quite a high degree of accuracy.
There's a large range of temperature for which it is a
Continue Reading
Liquid in glass thermometers are the cheapest kind of thermometer, and are also easy to use. Mercury is easily the best liquid to use, for 5 important reasons.
It is very reflective, so it's easy to see and to read accurately.
It doesn't wet the glass, so you don't get inaccurate readings if the temperature is falling.
It is a metal, so it's a good conductor of heat. This means that it reacts quickly to changes of temperature.
It expands evenly with temperature, so a linear scale can be used with quite a high degree of accuracy.
There's a large range of temperature for which it is a liquid. However, it freezes at - 39 Celsius, so Mercury cannot be used below this temperature. It boils at 365 Celsius, which is high enough for most purposes.
Colored water is no good because it's a poor conductor, it wets the glass, it expands unevenly with temperature, and there is only a limited range of temperatures for which it is a liquid, 0 - 100 Celsius.
Cheaper, and for temperatures below -39 C, colored ethanol is used. This freezes at - 144 C, but it boils at 78 C, making it of no use at high temperatures. It expands quite evenly with temperature, but otherwise it has the same defects as water. To get an accurate reading, it should be vertical and you have to give it plenty of time before you take a reading.
Upvote ·
999 196
99 20
9 7
Sponsored by Hudson Financial Partners
Do I need someone to manage my wealth?
If your net worth is high wealth management might be for you, let's schedule a quick call and find out!
Contact Us
9 1
Virgil Debique
Works at Hiden Isochema · Author has 146 answers and 83.1K answer views
·7y
Related
Why does a thermometer have a larger volume of liquid at the bulb?
If you mean the bulb at the bottom, this is so there is enough to fill the thermometer should the temperature increase enough. If the bulb was smaller, then thermometer would not be able to reach high temperatures as it would run out of liquid.
Upvote ·
Vern Shellman
Long time interest in science. · Author has 5.6K answers and 16.7M answer views
·4y
Related
Why is the bulb of a clinical thermometer not quite full of mercury at room temperature?
If you noticed, the scale on that thermometer doesn’t go down to zero. It probably bottoms out around 90 degrees. That means that you need to heat it up to at least 90 degrees to get it moving. And at room temperature there’s probably a bit of a gap since it has to expand a lot. Even if it’s something other than mercury it’s the same concept.
Continue Reading
If you noticed, the scale on that thermometer doesn’t go down to zero. It probably bottoms out around 90 degrees. That means that you need to heat it up to at least 90 degrees to get it moving. And at room temperature there’s probably a bit of a gap since it has to expand a lot. Even if it’s something other than mercury it’s the same concept.
Upvote ·
9 2
Sponsored by H&R Block
Late on Expat taxes? Avail Amnesty Now.
US expats: Amnesty ending soon! Catch up on taxes and avoid heavy penalties now.
Sign Up
99 26
Yashkumar Sanariya
Chemical Engineer from NIT, Surat
·8y
Related
Why is mercury used in thermometers?
Mercury is shiny liquid metal with high co-efficient of expansion. Keeping high-profile words aside, to put in simple terms,
It’s shiny. You can easily see the level of mercury inside a glass thermometer. Imagine using water or any other translucent liquid.
It’s sensitive. Heating even on a small scale causes significant expansion in it’s volume which can be made clearly visible on a marked scale.
It’s hard to boil. The boiling point of mercury is nearly 674.06 degrees Fahrenheit. So using it in college laboratory or hospitals really doesn’t convert the mercury to vapors.
It’s non-sticky. The prop
Continue Reading
Mercury is shiny liquid metal with high co-efficient of expansion. Keeping high-profile words aside, to put in simple terms,
It’s shiny. You can easily see the level of mercury inside a glass thermometer. Imagine using water or any other translucent liquid.
It’s sensitive. Heating even on a small scale causes significant expansion in it’s volume which can be made clearly visible on a marked scale.
It’s hard to boil. The boiling point of mercury is nearly 674.06 degrees Fahrenheit. So using it in college laboratory or hospitals really doesn’t convert the mercury to vapors.
It’s non-sticky. The property of mercury to not stick on glass is a main reason to use it. Unlike water, droplets of mercury never stick on the glass walls of the thermometer,
Thanks for A2A RAVI KUMAR VAALMKI
Upvote ·
99 32
9 5
Zuraiz Ghouri
FSc in Pre-Medical, Punjab Group of Colleges, Rahim Yar Khan · Author has 79 answers and 327.1K answer views
·8y
Related
How do mercury thermometers work?
A mercury thermometer has a bulb with a capillary tube attached to it surrounded by glass tube casing on which degrees are mentioned .
SO, when its bulb having mercury filled in it ,come in touch with something hotter or colder or even having moderate temperature ,the mercury in the capillary tube attached to it rises or falls according to the temperature of the body in contact with its bulb.
Continue Reading
A mercury thermometer has a bulb with a capillary tube attached to it surrounded by glass tube casing on which degrees are mentioned .
SO, when its bulb having mercury filled in it ,come in touch with something hotter or colder or even having moderate temperature ,the mercury in the capillary tube attached to it rises or falls according to the temperature of the body in contact with its bulb.
Upvote ·
9 9
Sponsored by Dell Technologies
Built for what's next.
Do more, faster with Dell AI PCs powered by #IntelCoreUltra processors & ready for Windows 11.
Learn More
99 10
Ken Gedling
Teacher. Artist. Parent. Human. · Author has 551 answers and 509.8K answer views
·7y
Related
What color is the mercury in a mercury thermometer?
Mercury thermometer: Liquid silver. Alcohol thermometer: Clear red liquid.
One time at the high school I worked I came across the principal and a secretary holding one of the lab thermometers - broken.
Principal: Do we have to evacuate the school?
Me: Oh, don’t worry. It’s an alcohol thermometer. It’s harmless.
Secretary: I think we should call 9–1–1. They can send a fire truck who can tell us what to do.
Principal: But, I think we should evacuate until we have the all clear.
Me: It’s an alcohol thermometer. It’s harmless.
Secretary: I think we have to call 9–1–1 or CDC. Mercury is very poisonous a
Continue Reading
Mercury thermometer: Liquid silver. Alcohol thermometer: Clear red liquid.
One time at the high school I worked I came across the principal and a secretary holding one of the lab thermometers - broken.
Principal: Do we have to evacuate the school?
Me: Oh, don’t worry. It’s an alcohol thermometer. It’s harmless.
Secretary: I think we should call 9–1–1. They can send a fire truck who can tell us what to do.
Principal: But, I think we should evacuate until we have the all clear.
Me: It’s an alcohol thermometer. It’s harmless.
Secretary: I think we have to call 9–1–1 or CDC. Mercury is very poisonous and can get in the air!
Principal: We’ll have to use the phone-tree to call the parents. I should call the county for more directions.
Me: IT’S. AN. ALCOHOL. THERMOMETER. IT’S. HARMLESS.
By this time I was hoping that they did call 9–1–1. Hopefully they would close the school for a few days. But, somehow, the principal realized I was trying to say something and the Crisis Was Averted.
Upvote ·
9 9
9 2
Vivek Bhardwaj
Former Senior Infra Mgmt Analyst at Xerox (company) (2014–2015) · Upvoted by
Stijn Hommes
, I have a BSc in Chemistry from Utrecht University
·10y
Related
Why is mercury used in thermometer why not any other liquid?
Mercury is used in thermometers because it has high coefficient of expansion. Hence, the slightest change in temperature is notable when it's used in a thermometer.
also, mercury has a freezing point of -38.83 °C and boiling point at 356.7 °C, this gives a very big range to measure temperatures for may general purposes, also it does not stick to(wet) the capillary wall of the thermometers, and its easy to obtain in its elemental form.
It was used very much in thermometers, but now a days it is being phased out as it is very harmful to humans in it vapor form.
now for some science talk :)
Thermal e
Continue Reading
Mercury is used in thermometers because it has high coefficient of expansion. Hence, the slightest change in temperature is notable when it's used in a thermometer.
also, mercury has a freezing point of -38.83 °C and boiling point at 356.7 °C, this gives a very big range to measure temperatures for may general purposes, also it does not stick to(wet) the capillary wall of the thermometers, and its easy to obtain in its elemental form.
It was used very much in thermometers, but now a days it is being phased out as it is very harmful to humans in it vapor form.
now for some science talk :)
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature, through heat transfer. higher the coefficient of thermal expansion, more sensitive is the material to change in temperature.
The coefficient of thermal expansion describes how the size of an object changes with a change in temperature. Specifically, it measures the fractional change in size per degree change in temperature at a constant pressure. Several types of coefficients have been developed: volumetric, area, and linear. Which is used depends on the particular application and which dimensions are considered important.
higher the coefficient of thermal expansion, more sensitive is the material to change in temperature. So its good to measure the small changes in human body temperatures.
Upvote ·
99 46
9 2
9 1
Jozef Mitros
Retired Electrical Engineer, Ph.D. · Author has 7.2K answers and 4.6M answer views
·4y
Related
A temperature rise of 20°C causes mercury to expand and rise 8 cm in a thermometer tube. If the tube cross sectional area is doubled, the mercury will rise by what cm?
The change in mercury volume Δ V is proportional to the temperature increase Δ T according to the formula: Δ V = βV Δ T. ΔT =20°C .The relation between how much mercury rises in the thermometer Δh and the volume increase ΔV is following
ΔV = AΔh
Therefore if the tube cross sectional area is doubled A₂ = 2A₁ then the mercury rise will be twice smaller because ΔV is the same ΔV = A₁Δh₁ = A₂Δh₂
A₁ 8 = A₂Δh₂
A₁ 8 = 2A₁Δh₂
Δh₂ = 4 cm
Continue Reading
The change in mercury volume Δ V is proportional to the temperature increase Δ T according to the formula: Δ V = βV Δ T. ΔT =20°C .The relation between how much mercury rises in the thermometer Δh and the volume increase ΔV is following
ΔV = AΔh
Therefore if the tube cross sectional area is doubled A₂ = 2A₁ then the mercury rise will be twice smaller because ΔV is the same ΔV = A₁Δh₁ = A₂Δh₂
A₁ 8 = A₂Δh₂
A₁ 8 = 2A₁Δh₂
Δh₂ = 4 cm
Upvote ·
9 6
Eashaan Godbole
Student of science · Author has 628 answers and 2M answer views
·7y
Related
Why capillary of mercury falls and the same mercury in barometer rises?
TL;DR:
Barometer - Pressures
Capillary - Surface energies
Explanation:
These are two different phenomena. Let us start with the simpler case.
Barometer case:
Mercury is pushed up the barometer tube due to the atmospheric pressure acting on the mercury reservoir below. Note that the tube is closed at the top and inverting it causes the formation of an almost vacuum inside the tube above the mercury.
So, the hydrostatic pressure due to the mercury column is equal to the atmospheric pressure (or whatever pressure is being measured).
Capillary case: (assuming glass capillary)
The capillary is open at both
Continue Reading
TL;DR:
Barometer - Pressures
Capillary - Surface energies
Explanation:
These are two different phenomena. Let us start with the simpler case.
Barometer case:
Mercury is pushed up the barometer tube due to the atmospheric pressure acting on the mercury reservoir below. Note that the tube is closed at the top and inverting it causes the formation of an almost vacuum inside the tube above the mercury.
So, the hydrostatic pressure due to the mercury column is equal to the atmospheric pressure (or whatever pressure is being measured).
Capillary case: (assuming glass capillary)
The capillary is open at both ends - one end is dipped in the mercury. When mercury goes inside the capillary, it has to “touch” the capillary from the inside - formation of a mercury-glass interface takes place. The formation of an interface requires energy and any system tends to minimise its overall potential energy.
In case of a mercury-glass interface, the interface formation is disfavoured and lesser area of contact between t he mercury and the glass capillary is preferred. Thus, the mercury dips down. The increase in the potential energy due to the very small increase in height of the mercury reservoir due to the small dip in the mercury level of the capillary equals the interfacial energy saved due to the decrease in the area of contact.
For contrast, water rises in the capillary because the water-glass interface formation is favoured. In this case, the increase in potential energy due to rise of water is balanced by the decrease in potential energy due to interface formation, i.e. increase in area of contact.
PS: I know that the explanation for the capillary case is a bit rambly, but re-read it carefully if necessary.
Your response is private
Was this worth your time?
This helps us sort answers on the page.
Absolutely not
Definitely yes
Upvote ·
9 7
Related questions
Why do we use a cylindrical bulb instead of a spherical bulb in mercury thermometers?
Why does mercury not attach to the capillary tube of a thermometer?
Why are digital thermometers replaced with mercury thermometers?
Why is there a vacuum above the mercury in the capillary tube of a laboratory thermometer?
What are the benefits of Mercury Thermometer?
Is it safe to have a little bit of mercury touch you when broken from a thermometer?
What are the parts of a mercury thermometer?
Does the mercury in a clinical thermometer in a capillary tube not contract once removed?
What happens if a mercury thermometer breaks?
What are the disadvantages of a mercury thermometer?
What is limit of mercury thermometer?
How accurate is a mercury thermometer?
What is the typical outside diameter of a mercury thermometer?
Are all mercury thermometers accurate when used properly?
Why is a digital thermometer better than a mercury thermometer?
Related questions
Why do we use a cylindrical bulb instead of a spherical bulb in mercury thermometers?
Why does mercury not attach to the capillary tube of a thermometer?
Why are digital thermometers replaced with mercury thermometers?
Why is there a vacuum above the mercury in the capillary tube of a laboratory thermometer?
What are the benefits of Mercury Thermometer?
Is it safe to have a little bit of mercury touch you when broken from a thermometer?
Advertisement
About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press ·
© Quora, Inc. 2025 |
11728 | https://research.jku.at/en/publications/high-mobility-si-and-ge-structures | High-mobility Si and Ge structures - JKU & KUK Research Portal
Skip to main navigation
Skip to search
Skip to main content
JKU & KUK Research Portal Home
Help & FAQ
English
Deutsch
Home
Research units
Profiles
Research output
Projects
Activities
Datasets
Prizes
Press/Media
Search by expertise, name or affiliation
High-mobility Si and Ge structures
Friedrich Schäffler
Department of Semiconductor Physics
Institute of Semiconductor and Solid State Physics
Research output: Contribution to journal › Review article › peer-review
Overview
| Original language | English |
| Pages (from-to) | 1515-1549 |
| Number of pages | 35 |
| Journal | Semiconductor Science and Technology |
| Volume | 12 |
| Issue number | 12 |
| DOIs | |
| Publication status | Published - Dec 1997 |
Fields of science
103 Physics, Astronomy
103009 Solid state physics
103011 Semiconductor physics
103017 Magnetism
103018 Materials physics
103040 Photonics
202032 Photovoltaics
210006 Nanotechnology
Access to Document
10.1088/0268-1242/12/12/001
Other files and links
Link to publication in Scopus
Cite this
APA
Author
BIBTEX
Harvard
Standard
RIS
Vancouver
Schäffler, F. (1997). High-mobility Si and Ge structures. Semiconductor Science and Technology, 12(12), 1515-1549.
Schäffler, Friedrich. / High-mobility Si and Ge structures. In: Semiconductor Science and Technology. 1997 ; Vol. 12, No. 12. pp. 1515-1549.
@article{c84d238045134fdfb3bd37d5b88c74da,
title = "High-mobility Si and Ge structures",
author = "Friedrich Sch{\"a}ffler",
year = "1997",
month = dec,
doi = "10.1088/0268-1242/12/12/001",
language = "English",
volume = "12",
pages = "1515--1549",
journal = "Semiconductor Science and Technology",
issn = "0268-1242",
publisher = "IOP Publishing",
number = "12",
}
Schäffler, F 1997, 'High-mobility Si and Ge structures', Semiconductor Science and Technology, vol. 12, no. 12, pp. 1515-1549.
High-mobility Si and Ge structures. / Schäffler, Friedrich.
In: Semiconductor Science and Technology, Vol. 12, No. 12, 12.1997, p. 1515-1549.
Research output: Contribution to journal › Review article › peer-review
TY - JOUR
T1 - High-mobility Si and Ge structures
AU - Schäffler, Friedrich
PY - 1997/12
Y1 - 1997/12
UR -
U2 - 10.1088/0268-1242/12/12/001
DO - 10.1088/0268-1242/12/12/001
M3 - Review article
SN - 0268-1242
VL - 12
SP - 1515
EP - 1549
JO - Semiconductor Science and Technology
JF - Semiconductor Science and Technology
IS - 12
ER -
Schäffler F. High-mobility Si and Ge structures. Semiconductor Science and Technology. 1997 Dec;12(12):1515-1549. doi: 10.1088/0268-1242/12/12/001
Powered by Pure, Scopus&Elsevier Fingerprint Engine™
All content on this site: Copyright © 2025 JKU & KUK Research Portal, its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply
We use cookies to help provide and enhance our service and tailor content. By continuing you agree to the use of cookies
Log in to Pure
Impressum
About web accessibility
Report vulnerability
Contact us |
11729 | https://www.youtube.com/watch?v=1PYyM8daBaQ | Exponential equations unknown in exponent (mathdou) Grade 10 and older
Dou Kleefstra
14100 subscribers
15 likes
Description
2045 views
Posted: 9 Mar 2021
Many equations with unknown exponents are solved
2 comments
Transcript:
Intro good day in this video we're going to solve these equations and a few more all the equations have a variable in the air so we have a unknown exponent that we have to solve later in life you will use logs to solve it but now we're not allowed to do that we will show you a method how you can solve them Solving for x in all the equations we have to solve for x let's do the first two the first one says 2 to the power of x equals 8 we have to solve for x we prime factorize the 8 becomes 2 to the power of 3 and now because the bases are the same and we have an equation we can drop the basis final answer x equals to 3. the next equation looks like this because we have a divide over here and our bases are the same we will minus the exponents so you write down the base once and you minus the exponents from each other this one i also want it to become a two i need my bases to be the same so we write the 1 as 2 to the power of 0. now the bases are the same that 2x minus x gives me x so x minus 1 equals to 0 plus 1 on both sides we get x equals to 1. remember you should check your answer with a calculator you take this 1 you substitute it back in there and there and see if the left-hand side equals the right-hand side next equation looks like this we have to solve for x convert this to exponents keep the left hand side the square root of 3 can be written as 3 to the power of half and that 3 has this exponent as 1. because the bases are the same and i'm dividing i will minus the exponents that'll give me 3 to the power of a half minus 1 then the right hand becomes 3 to the power of minus a half then we have 3 to the power of x plus 1 equals 3 to the power of minus half because my bases are the same i can drop them we write x plus one equals minus a half minus one on both sides and we get x equals to minus one and a half please stop the video and get your head around this one the next equation looks like this we have to solve for x this nine we change to three square the cube root becomes to the power of a third here the square root of three i make it three to the power of a half keep the two x as an exponent times the 2 and the third we get two thirds in the air we times the half and the 2x we get an x in the air as exponents of course the bases are the same i may drop the threes final answer x equals to two over three or two thirds next equation looks like this we have to solve for x we change the third to exponential way of writing three to the power of minus one times that one with that x and that one with that minus one then it looks like this this one over twenty-seven becomes three to the power of minus three twenty-seven is three cubed and then becomes three to the minus three because these two bases are the same and i'm multiplying i add the exponents so we get 3 to the power of x minus 2 equals 3 to the power of minus 3. because the bases are the same we can drop the bases x minus 2 equals to minus 3 plus 2 on both sides so x equals to minus one please stop the video and get your head around this one try and keep your weights together on the next one we have to solve for x naught comma three seven five we can write as 375 over a thousand if you simplify that it'll become three-eighths possible way from here on is to prime factorize all the big numbers we keep the three the eight becomes two cubed the nine becomes the three squared and the 64 becomes 2 to the power of 6. now we can see that this and this looks related so i need the base to look the same on this side going backwards on exponent law 3 you can take out a power of 2 then this will be the same as this and then this and this looks the same because this base and this base are the same you may drop the bases then x equals to 2. please stop the video and get your brain around this one the next equation looks like this and we have to solve for x the first thing we can do is to take this to this side the 8 becomes 2 to the power of 3 so that becomes 2 to the power of minus 3x if you times that 3 with that minus x that 2 weights that 4 we can make a 2 to the power of 2 or 2 squared we can multiply that 2 with that x and that minus 1. so this is what the equation looks like now because we are multiplying the bases are the same we add the exponents we get 2 to the power of minus 3x equals to the power of 2x minus 1. that 1 minus 2 is minus 1. of course the bases are the same i may drop the bases then we have minus 3x equals to 2x minus 1. we minus 2x on both sides to get the x's to one side we get minus 5x equals to -1 then we divide by minus five on both sides we get x equals to one fifth please stop the video and get your brain around this one here we also have to solve for x because we don't have any laws for exponents adding and subtracting we have to find another way to solve this problem we can write down 3 to the power of x then we ask 3 to the power of x times what gives us 3 to the power of x which is 1. then we ask 3 to the power of x times what will give me 3 to the power of x again which is times 1 again and 3 to the power of x times what will give me 3 to the power of x again which is 1 again think backwards this times this will give me that this times this will give me that and this times this will give me that that gives us 3 times 3 to the power of x is this because we are timesing and the bases are the same we add the exponents we take that one plus x and we write it so now our bases are the same so we drop the bases one plus x equals three x minus x on both sides then we get one equals to two x divide by 2 on both sides to get rid of this 2 and we get a half equals x the value of x is half some people understand that this 3x plus 3x plus 3x is 3 of them like we could always have a k plus a k plus a k would give us three k's the same way we can say this plus this plus this is three times this if you can get your head around that and then we just go down the same way please stop the video and get your brain around this the next equation looks like this we have to solve for x we change the one to three to the power of zero because the bases are the same i might drop the bases we get x squared minus four equals to zero plus four on both sides to get the x alone to get rid of the square we square root both sides and remember to write x equals to plus or minus the square root of 4. final answer x equals to minus 2 or x equals 2 plus 2. the next equation looks like this we have to solve for x we prime factorize the 27 so that becomes 3 cubed times the 3 with the x squared and the 3 with the x and we end up with this here we prime factorize the 9 that becomes 3 squared because we are multiplying here and the bases are the same we write down the base ones and we add the exponents now because the bases are the same i may drop the bases then the equation will look like this now we have to solve for x we minus 3x squared on both sides then we have 3x equals to 2. divide by 3 on both sides we get x equals to 2 over 3 or 2 thirds stop the video get your brain around this one let's look at the last two equations it still says solve for x if you see a situation like this with plus and minuses and that exponent is twice as big as that one we will factorize this we'll take this and you will write 3 to the power of x and 3 to the power of x because 3 to the power of x times 3 to power of x will give us 3 to the power of 2x almost like this becomes k square k and no k please look at the video that says solve equations without using k so like we said we have 3 2 power of x over here and 3 2 power of x over here to get 27 we write 3 times 9 because we want to get back to 6. because there's a minus here we need our signs to be different because there's a plus the biggest number will get the plus we put the plus there and the minus there now we'll get two answers coming out of here get three to the power of x equals to three that comes out of here and three two over x plus nine becomes three to the out of x equals to minus nine because these two bases are the same i can just write x equals to 1. on this side we'll write no solution if we have a power with a positive over there the answer can't be a negative please stop the video and get your head around this remember to look at the video that says solving equations without the k method right the last Bonus question question is a bonus question it still says solve for x in this case it's impossible to get the bases to be the same so what we can do is we can keep the left hand side and on the right hand side we can take a common two out of that exponent like this that then gives us 4 to the power of x minus 3 equals 9 to the power of x minus 3. now we can see that the exponents look the same the only way we can have a 4 to the power of something equal to 9 to the power of something if there's something is equal to 0 because 4 to the power of 0 is 1 and 9 to the power of 0 is 1 and that means then they are equal let me just say this x minus 3 is equal to 0 and we plus 3 on both sides we get x equals to 3. if you take that 3 and you substitute it back in there you're going to get 4 to the power of 3 minus 3 which is 4 to the power of 0. take that 3 substitute it back in there 2 times 3 is 6 minus 6 is 0 that gives you 4 to the power of 0 equals 3 to the power of 0 and that will be true you may send a message to this number if you want to arrange online lesson in mathematics please indicate whether you liked or disliked the video and subscribe to the channel enjoy exponents |
11730 | https://artofproblemsolving.com/wiki/index.php/Trigonometric_identities?srsltid=AfmBOor1o4ntaT96SYe-M-wCuLX6VZHUQaDT5bzTxDxHi77YHRkD-q7u | Art of Problem Solving
Trigonometric identities - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Trigonometric identities
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Trigonometric identities
In trigonometry, trigonometric identities are equations involving trigonometric functions that are true for all input values. Trigonometric functions have an abundance of identities, of which only the most widely used are included in this article.
Contents
[hide]
1 Pythagorean identities
2 Angle addition identities
3 Double-angle identities
3.1 Cosine double-angle identity
4 Half-angle identities
5 Product-to-sum identities
6 Sum-to-product identities
7 Other identities
7.1 Triple-angle identities
7.2 Even-odd identities
7.3 Conversion identities
7.4 Euler's identity
7.5 Miscellaneous
8 Resources
9 See also
Pythagorean identities
The Pythagorean identities state that
Using the unit circle definition of trigonometry, because the point is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, . To derive the other two Pythagorean identities, divide by either or and substitute the respective trigonometry in place of the ratios to obtain the desired result.
Angle addition identities
The trigonometric angle addition identities state the following identities:
There are many proofs of these identities. For the sake of brevity, we list only one here.
Euler's identity states that . We have that By looking at the real and imaginary parts, we derive the sine and cosine angle addition formulas.
To derive the tangent addition formula, we reduce the problem to use sine and cosine, divide both numerator and denominator by , and simplify. as desired.
Double-angle identities
The trigonometric double-angle identities are easily derived from the angle addition formulas by just letting . Doing so yields:
Cosine double-angle identity
Here are two equally useful forms of the cosine double-angle identity. Both are derived via the Pythagorean identity on the cosine double-angle identity given above.
In addition, the following identities are useful in integration and in deriving the half-angle identities. They are a simple rearrangement of the two above.
Half-angle identities
The trigonometric half-angle identities state the following equalities:
The plus or minus does not mean that there are two answers, but that the sign of the expression depends on the quadrant in which the angle resides.
Consider the two expressions listed in the cosine double-angle section for and , and substitute instead of . Taking the square root then yields the desired half-angle identities for sine and cosine. As for the tangent identity, divide the sine and cosine half-angle identities.
Product-to-sum identities
The product-to-sum identities are as follows:
They can be derived by expanding out and or and , then combining them to isolate each term.
Sum-to-product identities
Substituting and into the product-to-sum identities yields the sum-to-product identities.
Other identities
Here are some identities that are less significant than those above, but still useful.
Triple-angle identities
All of these expansions can be proved using trick and perform the angle addition identities. Same for and for .
Even-odd identities
The functions , , , and are odd, while and are even. In other words, the six trigonometric functions satisfy the following equalities:
These are derived by the unit circle definitions of trigonometry. Making any angle negative is the same as reflecting it across the x-axis. This keeps its x-coordinate the same, but makes its y-coordinate negative. Thus, and .
Conversion identities
The following identities are useful when converting trigonometric functions.
All of these can be proven via the angle addition identities.
Euler's identity
Euler's identity is a formula in complex analysis that connects complex exponentiation with trigonometry. It states that for any real number , where is Euler's constant and is the imaginary unit. Euler's identity is fundamental to the study of complex numbers and is widely considered among the most beautiful formulas in math.
Similar to the derivation of the product-to-sum identities, we can isolate sine and cosine by comparing and , which yields the following identities:
They can also be derived by computing and . These expressions are occasionally used to define the trigonometric functions.
Miscellaneous
These are the identities that are not substantial enough to warrant a section of their own.
Resources
Table of trigonometric identities
List of Trigonometric Identities
See also
Trigonometry
Trigonometric substitution
Proofs of trig identities
Retrieved from "
Category:
Trigonometry
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
11731 | https://math.stackexchange.com/questions/2074238/how-to-calculate-cos-pi-4-and-sin-pi-4 | trigonometry - How to calculate $\cos(\pi/4)$ and $\sin(\pi/4)$? - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
How to calculate cos(π/4)cos(π/4) and sin(π/4)sin(π/4)? [closed]
Ask Question
Asked 8 years, 9 months ago
Modified7 years, 11 months ago
Viewed 1k times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
Closed. This question is off-topic. It is not currently accepting answers.
This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level.
Closed 8 years ago.
Improve this question
I need to calculate sin(π 4)sin(π 4) and cos(π 4)cos(π 4)
only by using:
1) cos(a+b)=cos(a)cos(b)−sin(a)sin(b)cos(a+b)=cos(a)cos(b)−sin(a)sin(b)
2) sin(a+b)=sin(a)cos(b)+cos(a)sin(b)sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
3) cos(a)2+sin(a)2=1 cos(a)2+sin(a)2=1
I am stuck and don't know how to start. I know what the values are, but I have no clue how to calculate them using the three identities. Any hints will be appreciated.
trigonometry
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Oct 6, 2017 at 14:21
user9464
asked Dec 28, 2016 at 0:38
de_dustde_dust
595 4 4 silver badges 13 13 bronze badges
5
5 Do you have any givens? For example, can you use that s i n(π 2)=1 s i n(π 2)=1?cool.coolcoolcool –cool.coolcoolcool 2016-12-28 00:40:02 +00:00 Commented Dec 28, 2016 at 0:40
Also this is not really calculus or real-analysis, avoid putting non-applicable tags.cool.coolcoolcool –cool.coolcoolcool 2016-12-28 00:41:35 +00:00 Commented Dec 28, 2016 at 0:41
@cool.coolcoolcool: I don't think I have any givens. Just the three identities.de_dust –de_dust 2016-12-28 00:43:18 +00:00 Commented Dec 28, 2016 at 0:43
1 @de_dust That doesn't sound right. You must have been allowed to use the fact that sin(π/2)=1 sin(π/2)=1.gowrath –gowrath 2016-12-28 00:44:45 +00:00 Commented Dec 28, 2016 at 0:44
4 It is impossible to answer the question without knowing for what a a the value sin a sina or cos a cosa is assumed known.user9464 –user9464 2016-12-28 00:44:47 +00:00 Commented Dec 28, 2016 at 0:44
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
In the sum of angle theorems, let a=b a=b so that
cos(2 a)=cos 2(a)−sin 2(a)cos(2 a)=cos 2(a)−sin 2(a)
By the last identity, notice that
cos 2(a)−sin 2(a)=2 cos 2(a)−1 cos 2(a)−sin 2(a)=2 cos 2(a)−1
cos 2(a)−sin 2(a)=1−2 sin 2(a)cos 2(a)−sin 2(a)=1−2 sin 2(a)
Now let a=π/4 a=π/4 and use known values.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Dec 28, 2016 at 0:41
Simply Beautiful ArtSimply Beautiful Art
76.8k 13 13 gold badges 134 134 silver badges 301 301 bronze badges
2
Perhaps one should add that "known values" here are assumed to be cos(π/2)=0 cos(π/2)=0?user9464 –user9464 2016-12-28 00:49:33 +00:00 Commented Dec 28, 2016 at 0:49
@Jack One should figure that they need that as they are doing the problem, and it shouldn't be a super big deal (I'd really hope it isn't a problem)Simply Beautiful Art –Simply Beautiful Art 2016-12-28 00:50:40 +00:00 Commented Dec 28, 2016 at 0:50
Add a comment|
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
trigonometry
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
2Finding the derivatives of sin(x) and cos(x)
1Proving the identity: sin 3 x+sin x=2 sin 2 x cos x sin3 x+sinx=2 sin2 x cosx
1Why is sin(α+β)=sin α cos β+sin β cos α sin(α+β)=sinα cosβ+sinβ cosα
1Calculate cos(x/3)cos(x/3) from sin(x)sin(x) and cos(x)cos(x)
28Calculate 1 sin(x)+1 cos(x)1 sin(x)+1 cos(x) if sin(x)+cos(x)=7 5 sin(x)+cos(x)=7 5
4Trigonometric functions -- sin - cos - tan - Differences, actual applications,
3How to simplify sin 3 A−cos 3 A sin A+cos A+1 sin3 A−cos3 A sinA+cosA+1?
2If cos x+cos y+cos z cos(x+y+z)=sin x+sin y+sin z sin(x+y+z)=T cosx+cosy+cosz cos(x+y+z)=sinx+siny+sinz sin(x+y+z)=T, then T=cos(x+y)+cos(x+z)+cos(z+x)T=cos(x+y)+cos(x+z)+cos(z+x)
2How to derive from scratch the identities sin 2 x=t 2 1+t 2 sin 2x=t 2 1+t 2, cos 2 x=1 1+t 2 cos 2x=1 1+t 2, sin x cos x=t 1+t 2 sinx cosx=t 1+t 2, where t=tan x t=tanx?
Hot Network Questions
Vampires defend Earth from Aliens
Is it safe to route top layer traces under header pins, SMD IC?
Making sense of perturbation theory in many-body physics
alignment in a table with custom separator
Weird utility function
Should I let a player go because of their inability to handle setbacks?
Xubuntu 24.04 - Libreoffice
Can you formalize the definition of infinitely divisible in FOL?
Does the curvature engine's wake really last forever?
I have a lot of PTO to take, which will make the deadline impossible
How to fix my object in animation
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
Why include unadjusted estimates in a study when reporting adjusted estimates?
What happens if you miss cruise ship deadline at private island?
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
What's the expectation around asking to be invited to invitation-only workshops?
Is existence always locational?
Do we declare the codomain of a function from the beginning, or do we determine it after defining the domain and operations?
In Dwarf Fortress, why can't I farm any crops?
With line sustain pedal markings, do I release the pedal at the beginning or end of the last note?
Do we need the author's permission for reference
Change default Firefox open file directory
Interpret G-code
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
more hot questions
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
11732 | http://hyperphysics.phy-astr.gsu.edu/hbase/Newt.html | | | |
--- |
| | Index |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | |
--- |
| Newton's First Law Newton's First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. It may be seen as a statement about inertia, that objects will remain in their state of motion unless a force acts to change the motion. Any change in motion involves an acceleration, and then Newton's Second Law applies. The First Law could be viewed as just a special case of the Second Law for which the net external force is zero, but that carries some presumptions about the frame of reference in which the motion is being viewed. The statements of both the Second Law and the First Law here are presuming that the measurements are being made in a reference frame which is not itself accelerating. Such a frame is often referred to as an "inertial frame". The statement of these laws must be generalized if you are dealing with a rotating reference frame or any frame which is accelerating. Newton's First Law contains implications about the fundamental symmetry of the universe in that a state of motion in a straight line must be just as "natural" as being at rest. If an object is at rest in one frame of reference, it will appear to be moving in a straight line to an observer in a reference frame which is moving by the object. There is no way to say which reference frame is "special", so all constant velocity reference frames must be equivalent. | IndexMass on string example. Newton's laws concepts. |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | | |
---
| Centripetal Force Example The string must provide the necessary centripetal force to move the ball in a circle. If the string breaks, the ball will move off in a straight line. The straight line motion in the absence of the constraining force is an example of Newton's first law. The example here presumes that no other net forces are acting, such as horizontal motion on a frictionless surface. The vertical circle is more involved. | | | Centripetal force calculation | | Index |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | |
--- |
| | Index Newton's laws concepts. |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | | |
---
| Newton's Second Law Newton's Second Law as stated below applies to a wide range of physical phenomena, but it is not a fundamental principle like the Conservation Laws. It is applicable only if the force is the net external force. It does not apply directly to situations where the mass is changing, either from loss or gain of material, or because the object is traveling close to the speed of light where relativistic effects must be included. It does not apply directly on the very small scale of the atom where quantum mechanics must be used. Data can be entered into any of the boxes below. Specifying any two of the quantities determines the third. After you have entered values for two, click on the text representing the third to calculate its value. | | | Limitations of Newton's Second Law | | Index Second Law ConceptsVisualization example. |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | |
--- |
| Newton's Second Law IllustrationNewton's 2nd Law enables us to compare the results of the same force exerted on objects of different mass. | Index |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | | | | |
--- ---
| Newton's Third Law Newton's third law: All forces in the universe occur in equal but oppositely directed pairs. There are no isolated forces; for every external force that acts on an object there is a force of equal magnitude but opposite direction which acts back on the object which exerted that external force. In the case of internal forces, a force on one part of a system will be countered by a reaction force on another part of the system so that an isolated system cannot by any means exert a net force on the system as a whole. A system cannot "bootstrap" itself into motion with purely internal forces - to achieve a net force and an acceleration, it must interact with an object external to itself. | | | Without specifying the nature or origin of the forces on the two masses, Newton's 3rd law states that if they arise from the two masses themselves, they must be equal in magnitude but opposite in direction so that no net force arises from purely internal forces. | Newton's third law is one of the fundamental symmetry principles of the universe. Since we have no examples of it being violated in nature, it is a useful tool for analyzing situations which are somewhat counter-intuitive. For example, when a small truck collides head-on with a large truck, your intuition might tell you that the force on the small truck is larger. Not so! | | | Small truck,large truck | | | | Example of Newton's Third Law | | IndexExample Newton's laws concepts. |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back |
| | |
--- |
| Newton's Third Law Example Newton's third law can be illustrated by identifying the pairs of forces which are involved in supporting the blocks on the spring scale. Presuming that the blocks are supported and at equilibrium, then the net force on the system is zero. All the forces occur in Newton's third law pairs. | Index |
| | | | --- | | HyperPhysics Mechanics | R Nave | | Go Back | |
11733 | https://www.media4math.com/library/math-example-decimal-concepts-adding-decimals-tenths-regrouping-example-14 | Persistent Menu Library
If you are a subscriber, please log in.
Display Title Math Example--Decimal Concepts--Adding Decimals to the Tenths (With Regrouping)--Example 14
Display Title
Math Example--Decimal Concepts--Adding Decimals to the Tenths (With Regrouping)--Example 14
Topic
Adding Decimals
Description
This example showcases the addition of 0.7 and 0.6 using a number line and place value columns. Students are guided to add the digits in the tenths place, then regroup by carrying 1 to the ones place, resulting in a sum of 1.3. The visual representation helps reinforce the concept of regrouping in decimal addition.
Adding decimals is an essential skill that builds upon students' understanding of place value and whole number addition. This collection of examples provides a variety of scenarios to help students understand the regrouping process when adding decimals. By using visual aids like number lines and place value charts, students can better grasp the concept of carrying over to the ones place when the sum of tenths is 1 or greater.
Multiple worked-out examples are crucial for students to fully grasp the concept of adding decimals with regrouping. Each example reinforces the steps involved, helping students recognize patterns and develop problem-solving strategies. By presenting different combinations of decimals, students can better understand when and how to apply regrouping, ultimately building their confidence in working with decimal numbers.
For a complete collection of math examples related to Adding Decimals click on this link: Math Examples: Adding Decimals Collection.
| | |
--- |
| Common Core Standards | CCSS.MATH.CONTENT.5.NBT.B.7 |
| Grade Range | 4 - 6 |
| Curriculum Nodes | Arithmetic • Decimals • Add and Subtract Decimals • Numbers and Patterns • Place Value |
| Copyright Year | 2021 |
| Keywords | place value, decimals, adding decimals |
Math Examples
The Math Examples Library
Media4Math has a huge collection of instructional math examples covering a wide range of math topics. This collection of resources is made up images that you can easily incorporate into your lesson plans.
To see the complete collection of Math Examples, click on this link.
In answer to the question, "Why so many examples?": Math concepts are best learned through multiple examples of increasing complexity. For example, in learning about the slope formula, students need to see examples covering these scenarios:
But this only covers points in the first quadrant. Students also need to see examples in the other quadrants, to see the impact of negative coordinates on the formula. Plus, students need to see points on two separate coordinates.
As you can see, just with this one topic, there are dozens of possible examples to cover in order for students to get a better understanding of the nuances of working with the slope formula. Textbooks and other curriculum solutions provide a handful of examples, but hardly a comprehensive. This is where Media4Math comes in. We provide significantly more examples, each providing a detailed solution. In the case of the slope formula, click on this link to see the full set of math examples on the slope formula.
Below is a listing of the categories of math examples covered. To see the complete library of Math Examples, click on this link.
The Evidence Basis for Multiple Examples
The What Works Clearinghouse has a number of evidence-based Practice Guides that focus on math practices that are associated with effective teaching. For example the Practice Guide entitled Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students has one recommendation involving showing students solved problems in order for them to better understand abstract algebraic concepts. Watch the video below to learn more about this approach.
Having students study solved problems allows them to analyze and internalize the mathematical reasoning involved. This is what the Math Examples Library attempts to do: Solve enough problems for students to gain a firm understanding of the underlying concepts.
The other two recommendations in the practice guide involve having students look at the mathematical structure from the standpoint of multiple representations of mathematical quantities, as well as choosing from different strategies for solving problems. The Math Examples Library also has a wealth of examples that show multiple representations and different approaches to solving math problems.
Footer menu
Follow
© Media4Math. All rights reserved |
11734 | https://artofproblemsolving.com/wiki/index.php/Projective_geometry_(simplest_cases)?srsltid=AfmBOopmV7u5deE6OCb0R3p4saUOA3TyEKqGlbWCqW5tNXzXnLiDFrb3 | Art of Problem Solving
Projective geometry (simplest cases) - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Projective geometry (simplest cases)
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Projective geometry (simplest cases)
Projective geometry contains a number of intuitively obvious statements that can be effectively used to solve some Olympiad mathematical problems. We explore only central projection.
Useful simplified information
Let two planes and and a point not lying in them be defined in space. To each point of plane we assign the point of plane at which the line intersects this plane. We want to find a one-to-one mapping of plane onto plane using such a projection.
We are faced with the following problem. Let us construct a plane containing a point and parallel to the plane Let us denote the line along which it intersects the plane as No point of the line has an image in the plane Such new points are called points at infinity.
To solve it, we turn the ordinary Euclidean plane into a projective plane. We consider that the set of all points at infinity of each plane forms a line. This line is called the line at infinity. The plane supplemented by such line is called the projective plane, and the line for which the central projection is not defined is called (in Russian tradition) the exceptional line of the transformation. We define the central projection as follows.
Let us define two projective planes and and a point
For each point of plane we assign either:
the point of plane at which line intersects
or a point at infinity if line does not intersect plane
We define the inverse transformation similarly.
A mapping of a plane onto a plane is called a projective transformation if it is a composition of central projections and affine transformations.
Properties of a projective transformation
A projective transformation is a one-to-one mapping of a set of points of a projective plane, and is also a one-to-one mapping of a set of lines.
The inverse of a projective transformation is projective transformation. The composition of projective transformations is a projective transformation.
Let two quadruples of points and be given. In each quadruple no three points lie on the same line: Then there exists a unique projective transformation that maps to to to to
There is a central projection that maps vertices of any quadrilateral to vertices of the square.
There is a central projection that maps a circle to a circle, and a chosen interior point of the first circle to the center of the second circle. This central projection maps the polar of the chosen point to the line at infinity.
The relationships of segments belonging to lines parallel to the exceptional line are the same for images and preimages.
The double (anharmonic) ratio for given four points on a line (or on a circle) is a number It is the only projective invariant of a quadruple of such points.
Contents
1 Projection of a circle into a circle
2 Butterfly theorem
3 Sharygin’s Butterfly theorem
4 Semi-inscribed circle
5 Fixed point
6 Set of lines
7 Set of lines in circle and triangle
8 Projection of any triangle into regular one
9 Some properties of an equilateral triangle
10 Invariant for points on circles
11 Two generated triangles
12 Sphere and two points
13 Projecting non-convex quadrilateral into rectangle
14 Projecting convex quadrilateral into square
15 Two lines and two points
16 Crossing lines
17 Convex quadrilateral and point
18 Theorem on doubly perspective triangles
19 Complete quadrilateral theorem
20 Medians crosspoint
21 Six segments
22 Sines of the angles of a quadrilateral
23 Triangle and line
24 Polar and point
Projection of a circle into a circle
Let a circle with diameter and a point on this diameter be given.
Find the prospector of the central projection that maps the circle into the circle and the point into point - the center of
Solution
Let be the center of transformation (perspector) which is located on the perpendicular through the point to the plane containing Let be the diameter of and plane is perpendicular to
Spheres with diameter and with diameter contain a point , so they intersect along a circle
Therefore the circle is a stereographic projection of the circle from the point
That is, if the point lies on , there is a point on the circle along which the line intersects
It means that is projected into under central projection from the point
is antiparallel in
is the symmedian.
Corollary
Let The inverse of a point with respect to a reference circle is
The line throught in plane of circle perpendicular to is polar of point
The central projection of this line to the plane of circle from point is the line at infinity.
vladimir.shelomovskii@gmail.com, vvsss
Butterfly theorem
Let be the midpoint of a chord of a circle through which two other chords and are drawn; and intersect chord at and correspondingly.
Prove that is the midpoint of
Proof
Let point be the center of
We make the central projection that maps the circle into the circle and the point into the center of
Let's designate the images points with the same letters as the preimages points.
Chords and maps into diameters, so maps into rectangle and in this plane is the midpoint of
The exceptional line of the transformation is perpendicular to so parallel to
The relationships of segments belonging to lines parallel to the exceptional line are the same for images and preimages. We're done! .
vladimir.shelomovskii@gmail.com, vvsss
Sharygin’s Butterfly theorem
Let a circle and a chord be given. Points and lyes on such that Chords and are drawn through points and respectively such that quadrilateral is convex.
Lines and intersect the chord at points and
Prove that
Proof
Let us perform a projective transformation that maps the midpoint of the chord to the center of the circle . The image will become the diameter, the equality will be preserved.
Let and be the points symmetrical to the points and with respect to line the bisector
Denote (Sharygin’s idea.)
is cyclic
is cyclic
points and are collinear.
Similarly points and are collinear.
We use the symmetry lines and with respect and get in series
symmetry and with respect
symmetry and CB with respect
symmetry and with respect
vladimir.shelomovskii@gmail.com, vvsss
Semi-inscribed circle
Let triangle and circle centered at point and touches sides and at points and be given.
Point is located on chord so that
Prove that points and (the midpoint are collinear.
Proof
Denote point on line such that
Therefore line is the polar of
Let us perform a projective transformation that maps point to the center of
Image is the point at infinity, so images and are parallel.
Image is diameter, so image is midpoint of image and image is midpoint of image
so image is parallel to the line at infinity and the ratio is the same as ratio of images.
vladimir.shelomovskii@gmail.com, vvsss
Fixed point
Let triangle and circle centered at point and touches sides and at points and be given.
The points and on the side are such that
The cross points of segments and with form a convex quadrilateral
Point lies at and satisfies the condition
Prove that
Proof
Let us perform a projective transformation that maps point to the center of
Image is the point at infinity, so images and are parallel. The plane of images is shown, notation is the same as for preimages.
Image is diameter image is parallel to the line at infinity, so in image plane
Denote is rectangle, so
vladimir.shelomovskii@gmail.com, vvsss
Set of lines
Let points at the line at the line at the line be given.
Denote circle
Prove that lines and are concurrent.
Proof
According the Pascal theorem in case shown in diagram points and are collinear at Pascal line.
Note that lines
Let point be the pole of Pascal line.
Let us perform a projective transformation that maps point to the center of
Then image of is parallel to image of image of is parallel to image of , image of is parallel to image of
As shown in (Set of parallel lines) images of lines and are concurrent.
Therefore lines and are concurrent.
vladimir.shelomovskii@gmail.com, vvsss
Set of lines in circle and triangle
Let points at the line at the line at the line be given. Denote circle
Prove that lines and are concurrent.
Proof
Let us consider the inscribed hexagon shown in diagram. According the Pascal theorem points and are collinear at Pascal line. Note that lines
The Pascal line has no common point with so the pole of this line is the inner point of the circle.
Let us perform a projective transformation that maps this pole to the center of
Image of is the point at infinity, so image is parallel to image Similarly, images and are parallel to images and
As shown in (Set of lines in triangle) images of lines and are concurrent. Therefore lines and are concurrent.
vladimir.shelomovskii@gmail.com, vvsss
Projection of any triangle into regular one
Find a projective transformation that maps the given triangle into a regular one, and its inscribed circle into a circle.
Solution
Any point of tangency of the circle and line and any crosspoint of the lines are invariants of any projective transformation. Therefore, the Gergonne point of preimage maps into Gergonne point of image.
We make transformation which maps the Gergonne point of given triangle into the center if the incircle. According with Lemma, given triangle maps into regular one.
Lemma
Let the Gergonne point of coincide with the incenter of this triangle. Prove that is regular.
Proof
The inradius connect the incenter and point of tangency, bisector connect vertex and incenter, Gergonne point belong the line connect vertex and point of tangency, so these objects lie at the same line.
The radius is perpendicular to the side at the points of tangency, which means that the bisector coincide with the altitude of the triangle. The axial symmetry with respect to bisector maps one side of triangle to another, the base side is perpendicular to the bisector so symmetric sides are equal. Applying symmetry with respect to another bisector, we find that all three sides are equal and the triangle is regular.
vladimir.shelomovskii@gmail.com, vvsss
Some properties of an equilateral triangle
Let an equilateral triangle be given. Denote the incircle of
Points are located on so that ordered triplets of points are collinear.
Find and
Prove that quadrilateral is cyclic, is tangent to and symmetric to with respect
Proof
The circumradius is twice the inradius From considerations of symmetry we conclude that is regular, so
where is the golden ratio:
Let be the angle of rotation of the spiral similarity that maps into By applying the Law of Sines on we get is A-bisector of So is the midpoint Similarly,
Corollary
Lines and are concurrent.
vladimir.shelomovskii@gmail.com, vvsss
Invariant for points on circles
Let a circle set of points on it, and a point outside plane be given. Let the central projection on some plane from be the circle Prove that Proof
Denote - the diameter perpendicular to
Projection on is the diameter so bisectors and lies in the plane perpendicular to
Therefore there is the sphere which contains and
The power of a point with respect sphere is
The plane cross sphere by circle, so quadrilateral is cyclic. Corollaryvladimir.shelomovskii@gmail.com, vvsss
Two generated triangles
Let triangle be given. Denote the incircle of . Points are located on so that ordered triplets of points are collinear.
Prove that:
Lines and are concurrent.
Lines and are concurrent.
Proof
We make transformation which maps the incircle into incircle and the Gergonne point of given triangle into the center of this incircle.
The image of is a regular triangle shown in (Some properties of an equilateral triangle).
All statements are projective invariants, they are true for images, which means they are also true for preimages.
Corollary
To build a configuration, we construct points and point Line cross at points and
vladimir.shelomovskii@gmail.com, vvsss
Sphere and two points
Let a sphere and points and be given in space. The line does not has the common points with the sphere. The sphere is inscribed in tetrahedron
Prove that the sum of the angles of the spatial quadrilateral (i.e. the sum does not depend on the choice of points and
Proof
Denote points of tangency and faces of (see diagram),
It is known that Similarly, The sum not depend on the choice of points and
vladimir.shelomovskii@gmail.com, vvsss
Projecting non-convex quadrilateral into rectangle
Let a non-convex quadrilateral be given. Find a projective transformation of points into the vertices of rectangle.
Solution
WLOG, point is inside the
Let and be the rays, be any point on segment Planes and are perpendicular, planes and are parallel, so image is line at infinity and is rectangle.
Let's paint the parts of the planes and that maps into each other with the same color.
maps into (yellow).
Green infinite triangle between and maps into where plane is parallel to plane
Blue infinite quadrilateral between and with side maps into quadrilateral
Therefore inner part of quadrilateral maps into external part of rectangle For example maps into where is the intersection of planes and
Note that the lines through pairs of points (for example, ) maps into the corresponding lines, and the intersection point of and maps into the center of the rectangle.
vladimir.shelomovskii@gmail.com, vvsss
Projecting convex quadrilateral into square
Let be a convex quadrilateral with no parallel sides.
Find the projective transformation of into the square if the angle between the planes and is given. This angle is not equal to or
Solution
Denote
Let be the point satisfying the conditions
The locus of such points is the intersection circle of spheres with diameters and
Let be the perspector and the image plane be parallel to plane We use the plane contains so image
Then image is the line at infinity, point is point at infinity, so images (line ) and (line ) are parallel to
Similarly point is the point at infinity, so images is the rectangle.
Point is the point at infinity, so Point is the point at infinity, so
is the square.
Let be such point that
The angle between and plane is the angle we can choose. It is equal to the angle between planes and
vladimir.shelomovskii@gmail.com, vvsss
Two lines and two points
Let lines and intersecting at point , a point not lying on any of these lines, and points and on line be given. Find the locus of points
Solution
Let be the point such that be the midpoint Let us prove that the points and are collinear.
The quadrilateral is convex. We make the projective transformation of into the square.
Then line is the line at infinity, so image is the midpoint of image image is the center of the square.
Therefore images and are parallel and points and are collinear.
vladimir.shelomovskii@gmail.com, vvsss
Crossing lines
Let a convex quadrilateral be given.
Denote Prove that lines and are collinear.
Solution
The quadrilateral is convex.
We make the projective transformation of into the square.
Then image of the line is the line at infinity, image of is the center of the square.
Images of and are parallel, so image is the midpoint of the image Similarly images of and are midpoints of the square sides.
Therefore images and are parallel, they are crossed at the point in infinity witch lyes at the line at infinity, that is at
Convex quadrilateral and point
Let a convex quadrilateral and an arbitrary point be given,
Prove that lines and are concurrent.
Proof
The quadrilateral is convex. We make the projective transformation of into the square. Then image of the line is the line at infinity, images of and are parallel. Similarly
We use the Cartesian coordinate system with
Then
So line is line is line is
These lines contain point
Therefore preimages of and are concurrent in preimage of the point .
Theorem on doubly perspective triangles
Let two triangles and be given. Let the lines and be concurrent at point and the lines and be concurrent at point
Prove that the lines and are concurrent (the theorem on doubly perspective triangles).
Proof
WLOG, the quadrilateral is convex.
We make the projective transformation of into the square.
Then image of the line contains point is the line at infinity, images of and are parallel. Similarly
We use the Cartesian coordinate system with So the line is line is line is
These lines contain point
Therefore preimages of and are concurrent in point .
vladimir.shelomovskii@gmail.com, vvsss
Complete quadrilateral theorem
Let points no three of which are collinear, be given. Prove that
Proof
We make the projective transformation of the vertices of into vertices of the square. Then image of the point is the point at infinity, image of is the center of the square, images of and are parallel, so for images and
The double ratio is the projective invariant of a quadruple of collinear points so the equality also holds for the preimages.
Medians crosspoint
Let a convex quadrilateral and line in common position be given (points and not belong sides and diagonals are not parallel to
Denote Denote and midpoints of and respectively.
Prove that lines and are collinear.
Proof
Let the angle be fixed and the line moves in a plane parallel to itself.
Then the line on which the median of the triangle lies is also fixed. Similarly, lines and are fixed. Denote
Let moves in a plane parallel to itself to the position where
It is known (Six segments) that
After some simple transformations one can get
Six segments
Let a convex quadrilateral and line in common position be given (points and not belong sides and diagonals are not parallel to Denote Prove that
Proof
By applying the law of sines, we get:
(see Sines of the angles of a quadrilateral)
Sines of the angles of a quadrilateral
Let a convex quadrilateral be given. Prove that
Proof
By applying the law of sines, we get:
vladimir.shelomovskii@gmail.com, vvsss
Triangle and line
Let triangle and line be given.
Denote
Let and be the points on such that
Prove that points are collinear.
Proof
Under projective transformation line from point to points and maps into points and so Similarly,
We multiply these three equations and get: We use the Menelaus' theorem for and a transversal line and get: We make reduction of fractions, we take into account the given condition and get: Therefore in accordance with the Menelaus' theorem for points and are collinear.
Polar and point
Let triangle with circumcircle and point be given.
Let points be the projections of points on from
Let point be the projection of point onto line from point Define points and similarly.
Prove that if then points and are collinear.
Solution
Case 1. Points and are outside the circle Under projective transformation which maps the line into an infinite line, and into itself.
In this case, the pole of the line maps into the center of , and the points and maps into infinitely distant points in perpendicular directions.
In the plane of images of the triple of lines and are parallel among themselves and perpendicular to each other, as shown in the diagram.
Let us consider points and which are symmetrical to and with respect respectively.
Therefore,
Similarly
So points and are collinear. Another proof in Triangle and perpendiculars .
Corollary
Circles with diameters and have two common points ( Radical axis).
Case 2. Point is inside the circle
We use the projective transformation which maps the point into the center and into itself.
In the plane of images of the triple of segments are the diameters, are the parallel lines as shown in the diagram.
Denote In and by applying the law of sines, we get Similarly
So points and are collinear.
Case 3. Points is outside the circle point is inside the circle We use the projective transformation which maps the point into the center and into itself. Point maps into infinity. Proof is the same as in Case 2.
vladimir.shelomovskii@gmail.com, vvsss
Retrieved from "
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
11735 | https://en.wiktionary.org/wiki/common_logarithm | Jump to content
Search
Contents
Beginning
1 English
1.1 Noun
1.1.1 Coordinate terms
1.1.2 Translations
common logarithm
Eesti
Ελληνικά
Magyar
Malagasy
日本語
Polski
தமிழ்
Tiếng Việt
中文
Entry
Discussion
Read
Edit
View history
Tools
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Print/export
Create a book
Download as PDF
Printable version
In other projects
Appearance
From Wiktionary, the free dictionary
English
[edit]
English Wikipedia has an article on:
common logarithm
Wikipedia
Noun
[edit]
common logarithm (plural common logarithms)
(mathematics) The logarithm in base 10.
: Synonyms: Briggs logarithm, Briggsian logarithm
Coordinate terms
[edit]
natural logarithm
Translations
[edit]
| |
| German: Zehnerlogarithmus (de) m Greek: κοινός λογάριθμος (el) m (koinós logárithmos) Japanese: 常用対数 (ja) Polish: logarytm dziesiętny (pl) m Spanish: logaritmo común m Spanish: logaritmo común m |
Retrieved from "
Categories:
English lemmas
English nouns
English countable nouns
English multiword terms
en:Mathematics
en:Functions
Hidden categories:
Pages with entries
Pages with 1 entry
Translation table header lacks gloss
Entries with translation boxes
Terms with German translations
Terms with Greek translations
Terms with Japanese translations
Terms with Polish translations
Terms with Spanish translations
common logarithm
Add topic |
11736 | https://hturan.com/writing/complex-numbers-glsl | ← Harley Turan•April 2022
Visualizing Complex Numbers Using GLSL
Intrigued by this series of tweets by Keenan Crane (and this exploration by Varun Vachhar), I spent a recent Sunday morning dusting off high school memories of complex numbers in order to write some pretty shaders.
Keenan’s tweet showed a few formulas visualized in 2D space, and I was immediately faced with a couple of questions:
What does any of this mean?
How do I make art using it?
Hopefully this post helps break down what’s going on here (or at least to the best of my understanding), and how we can use what we learn to play with and visualize complex numbers.
GLSL, or the OpenGL Shading Language, is a graphics programming language that can be used to program visual effects. Shaders are both incredibly versatile and also conceptually simple — you write a set of instructions, and those instructions are run for every pixel on the screen. The result of those instructions tells the pixel what color it should display as.
Whilst conceptually simple, shaders can be really complex to work with. Instead of thinking “I want to make the 62nd pixel across and the 38th pixel down be red” (as seen in my seminal work Red Pixel, 2019), we need to think in mathematical terms about the shape of functions in 2D space. Lots of trigonometry and obscure functions to make shapes. The Book of Shaders is a lovely introduction.
The visuals we'll be creating today are all really just vizualising math — drawn on a flat screen and given colors. Zach Lieberman (of the School for Poetic Computation and MIT Media Lab) put this well recently:
So! Math…
To start with, let’s talk about complex numbers, because they’re fun and weird.
At a high level, Complex numbers are numbers that are split into two parts — real and imaginary. They are represented in the form a+bi, where a and b are real numbers, and i is the imaginary unit (representing −1, an impossible number).
Just like me, GLSL doesn't really understand complex numbers, so our job here is to represent them in a way that we're both familiar with.
Luckily, as a and b are real numbers, we can do real number things with them! They can be represented in GLSL as regular floating point values (float), and stored in a floating-point vec2 vector (just a set of two floats). By representing our complex number like this, we can perform complex arithmetic on it and have GLSL able to follow along.
Also, as with anything we can represent with two values, we can then visualize these values on a 2D surface (known as the "complex plane" when dealing with complex numbers). On the complex plane, the horizontal x-axis represents a, our real component (Re, the real axis), and the vertical y-axis represents b, our imaginary component (Im, the imaginary axis).
Now that I've written some LATEX and it’s given me the false confidence that I know what I’m talking about, let’s break down the formula in Keenan’s first tweet above:
Im
(
log
(
z
−
q
z
−
p
))
Here’s what we're working with —
z is our complex number (we'll be using it as our x and y coordinates for each point on the plane, or our uv.xy if you're already familiar with shader programming)
p and q are both arbitrary points on the plane between which we'll draw a line
log is the complex logarithm
Im is the imaginary part of our complex number (you'll also often see Re for the real part).
What we're doing here is effectively plotting the complex logarithm between two points on the complex plane. It‘s easy for me to write that as if I know what it means or what we’ll see when we do the math, but I’m confident it’ll look good regardless. Let‘s press on!
Right now, we have a complex plane (our GLSL canvas), we have some complex numbers we can put on the plane (our pixels and points), and we have the ability to perform arithmetic on them (the instructions we’ll write in GLSL). Let’s write some shader code!
glsl
vec2 uv = (gl_FragCoord. xy - 0.5 u_resolution. xy) / min(u_resolution. y, u_resolution. x); vec2 z = uv;
Here, gl_FragCoord is the xy coordinate of the fragment (the pixel we're working on), and u_resolution is the resolution of our canvas we'll be rendering to. What we're doing here is setting our bounds — rather than working with values matching the resolution (say, 500, 500 for a 500px square canvas), we're taking our pixel coordinate and mapping it to the range -0.5, 0.5 (with some aspect ratio correction thrown in using min).
As mentioned before, we'll be using z in math terms as our uv in shader terms, meaning that for our purposes, each pixel on the screen is now represented as a complex number.
Let’s also define our static points p and q between which we'll draw a line:
glsl
```
// Lower-left point
vec2
p
=
vec2
(
-
0.25
,
-
0.25
)
;
// Upper-right point
vec2
q
=
vec2
(
0.25
,
0.25
)
;
```
And with all the parameters now defined, we can convert our formula Im(log(z−qz−p)) to GLSL:
glsl
```
// Divide z-p by z-q using complex division
vec2
division
=
cx_div
(
(
z
-
p
)
,
(
z
-
q
)
)
;
// Calculate the log of our division
vec2
log_p_over_q
=
cx_log
(
division
)
;
// Extract the imaginary number
float
imaginary
=
log_p_over_q
.
y
;
```
And there we have it — a float value for our shader to paint to the screen.
Our imaginary value gives us everything we need, but it‘s just a float. Shaders output four numbers per pixel (red, green, blue, and alpha — RGBA), and we just have the one. To make things a little more visually-interesting, we can use Inigo Quilez’s procedural palette function to add some color to our current grayscale value:
glsl
```
// From
vec3
palette
(
in
float
t
,
in
vec3
a
,
in
vec3
b
,
in
vec3
c
,
in
vec3
d
)
{
return
a
+
b
cos
(
2.
PI
(
c
t
+
d
)
)
;
}
```
This function takes in a float value — which for us is will be the imaginary part of our logarithm — along with four vec3s that can be altered endlessly to produce beautiful output.
With all of these pieces combined (and some over-complicated tone-mapping using our palette function) we get the following:
glsl
```
version
300
es
precision
mediump
float
;
out
vec4
fragColor
;
uniform
vec2
u_resolution
;
uniform
float
u_time
;
define
PI
3.141592653589793
define
cx_div
(
a
,
b
)
vec2
(
(
(
a
.
x
b
.
x
+
a
.
y
b
.
y
)
/
(
b
.
x
b
.
x
+
b
.
y
b
.
y
)
)
,
(
(
a
.
y
b
.
x
-
a
.
x
b
.
y
)
/
(
b
.
x
b
.
x
+
b
.
y
b
.
y
)
)
)
vec2
as_polar
(
vec2
z
)
{
return
vec2
(
length
(
z
)
,
atan
(
z
.
y
,
z
.
x
)
)
;
}
vec2
cx_log
(
vec2
a
)
{
vec2
polar
=
as_polar
(
a
)
;
float
rpart
=
polar
.
x
;
float
ipart
=
polar
.
y
;
if
(
ipart
>
PI
)
ipart
=
ipart
-
(
2.0
PI
)
;
return
vec2
(
log
(
rpart
)
,
ipart
)
;
}
vec3
palette
(
in
float
t
,
in
vec3
a
,
in
vec3
b
,
in
vec3
c
,
in
vec3
d
)
{
return
a
+
b
cos
(
0.38
2.
PI
(
c
t
+
d
)
)
;
}
void
main
(
)
{
vec2
uv
=
(
gl_FragCoord
.
xy
-
0.5
u_resolution
.
xy
)
/
min
(
u_resolution
.
y
,
u_resolution
.
x
)
;
vec2
z
=
uv
;
float
angle
=
sin
(
u_time
/
5.
)
2.
PI
;
float
length
=
.2
;
// Spin our points in a circle of radius length
float
c
=
cos
(
angle
)
;
float
s
=
sin
(
angle
)
;
vec2
p
=
vec2
(
s
length
,
c
length
)
;
vec2
q
=
vec2
(
s
-
length
,
c
-
length
)
;
// Divide z-p by z-q using complex division
vec2
division
=
cx_div
(
(
z
-
p
)
,
(
z
-
q
)
)
;
// Calculate the log of our division
vec2
log_p_over_q
=
cx_log
(
division
)
;
// Extract the imaginary number
float
imaginary
=
log_p_over_q
.
y
/
PI
;
vec3
col
=
palette
(
imaginary
,
vec3
(
0.50
,
.52
,
0.53
)
,
vec3
(
.46
,
.32
,
.35
)
,
vec3
(
.82
,
.84
,
.65
)
,
vec3
(
0.53
,
0.23
,
0.22
)
)
;
fragColor
=
vec4
(
col
,
1.0
)
;
}
```
As with any shader, by jamming a bunch of random numbers into the many float variables we have here (and by making some tweaks to how our values are rendered across .xy screen space), we can get output like the following —
Beautiful! Who would imagine.
If you’ve got time and patience, I’ve got a little more math for you.
Keenan‘s original tweet actually showed a slightly more advanced representation of complex numbers, showing meromorphic functions (functions where certain points shoot off up to infinity) —
When we take a look at this function:
Im
(
log
(
∑
k
0
n
b
k
z
k
∑
k
0
m
a
k
z
k
))
a
0
,
…
,
a
m
,
b
0
,
…
,
b
n
∈
C
We recognize some old friends — log, Im, z — but also some new faces:
a and b are a set of arbitrary points (complex numbers) we'll be using to draw with.
m and n are just the number of points we'll be showing here, the total counts for a and b
For those not familiar with sigma notation, ∑k=0m just means "sum the following function m times starting at k"
That sigma statement in this case can be written out longhand as a0+a1z+⋯+amzm. In fact, if we set our limits m and n to 3, we get —
Im
(
log
(
b
0
+
b
1
z
+
b
2
z
2
+
b
3
z
3
a
0
+
a
1
z
+
a
2
z
2
+
a
3
z
3
))
What this formula represents is the imaginary part of the complex logarithm of the ratio of two complex polynomials.
Okay, I know. That was a lot of math. I’m sorry.
With our a and b still representing x and y points on the complex plane, we can scatter some points around the complex plane and, like before, represent this formula in GLSL:
glsl
```
version
300
es
precision
mediump
float
;
out
vec4
fragColor
;
uniform
vec2
u_resolution
;
uniform
float
u_time
;
define
PI
3.1415926535897932384626433832795
define
cx_mul
(
a
,
b
)
vec2
(
a
.
x
b
.
x
-
a
.
y
b
.
y
,
a
.
x
b
.
y
+
a
.
y
b
.
x
)
define
cx_div
(
a
,
b
)
vec2
(
(
(
a
.
x
b
.
x
+
a
.
y
b
.
y
)
/
(
b
.
x
b
.
x
+
b
.
y
b
.
y
)
)
,
(
(
a
.
y
b
.
x
-
a
.
x
b
.
y
)
/
(
b
.
x
b
.
x
+
b
.
y
b
.
y
)
)
)
define
cx_sin
(
a
)
vec2
(
sin
(
a
.
x
)
cosh
(
a
.
y
)
,
cos
(
a
.
x
)
sinh
(
a
.
y
)
)
define
cx_cos
(
a
)
vec2
(
cos
(
a
.
x
)
cosh
(
a
.
y
)
,
-
sin
(
a
.
x
)
sinh
(
a
.
y
)
)
vec2
cx_tan
(
vec2
a
)
{
return
cx_div
(
cx_sin
(
a
)
,
cx_cos
(
a
)
)
;
}
vec2
cx_log
(
vec2
a
)
{
float
rpart
=
sqrt
(
(
a
.
x
a
.
x
)
+
(
a
.
y
a
.
y
)
)
;
float
ipart
=
atan
(
a
.
y
,
a
.
x
)
;
if
(
ipart
>
PI
)
ipart
=
ipart
-
(
2.0
PI
)
;
return
vec2
(
log
(
rpart
)
,
ipart
)
;
}
vec2
as_polar
(
vec2
z
)
{
return
vec2
(
length
(
z
)
,
atan
(
z
.
y
,
z
.
x
)
)
;
}
vec2
cx_pow
(
vec2
v
,
float
p
)
{
vec2
z
=
as_polar
(
v
)
;
return
pow
(
z
.
x
,
p
)
vec2
(
cos
(
z
.
y
p
)
,
sin
(
z
.
y
p
)
)
;
}
float
im
(
vec2
z
)
{
return
(
(
atan
(
z
.
y
,
z
.
x
)
/
PI
)
+
1.0
)
0.5
;
}
vec3
pal
(
in
float
t
,
in
vec3
a
,
in
vec3
b
,
in
vec3
c
,
in
vec3
d
)
{
return
a
+
b
cos
(
2.
PI
(
c
t
+
d
)
)
;
}
// Define our points
vec2
a0
=
vec2
(
0.32
,
-
0.45
)
;
vec2
a1
=
vec2
(
-
0.49
,
-
0.32
)
;
vec2
a2
=
vec2
(
-
0.31
,
0.38
)
;
vec2
a3
=
vec2
(
-
0.12
,
0.04
)
;
vec2
b0
=
vec2
(
-
0.71
,
0.53
)
;
vec2
b1
=
vec2
(
0.01
,
0.23
)
;
vec2
b2
=
vec2
(
-
0.24
,
0.31
)
;
vec2
b3
=
vec2
(
-
0.01
,
-
0.42
)
;
void
main
(
)
{
// Set up our imaginary plane
vec2
uv
=
(
gl_FragCoord
.
xy
-
0.5
u_resolution
.
xy
)
/
min
(
u_resolution
.
y
,
u_resolution
.
x
)
;
vec2
z
=
uv
2.
;
// Calculate the sum of our first polynomial
vec2
polyA
=
a0
+
cx_mul
(
a1
,
z
)
+
cx_mul
(
a2
,
cx_pow
(
z
,
2.0
)
)
+
cx_mul
(
a3
,
cx_pow
(
z
,
3.0
)
)
;
// Calculate the sum of our second polynomial
vec2
polyB
=
b0
+
cx_mul
(
b1
,
z
)
+
cx_mul
(
b2
,
cx_pow
(
z
,
2.
)
)
+
cx_mul
(
b3
,
cx_pow
(
z
,
3.
)
)
;
// Calculate the ratio of our complex polynomials
vec2
result
=
cx_div
(
polyA
,
polyB
)
;
float
imaginary
=
cx_log
(
result
)
.
y
;
float
col
=
(
imaginary
/
PI
)
;
fragColor
=
vec4
(
pal
(
col
,
vec3
(
.52
,
.45
,
.61
)
,
vec3
(
.40
,
.42
,
.31
)
,
vec3
(
.26
,
.30
,
.35
)
,
vec3
(
.15
,
.4
,
.4
)
)
,
1.0
)
;
}
```
By putting some wild values in, we can get some wild images out —
I don’t know about you, but I think Red Pixel, 2019 can go straight in the trash.
I find GLSL (and shaders in general) to be a fascinating medium. It is fundamentally a medium that can represent anything that can be pictured on a two-dimensional plane — 3D models, fractals, paintings, poetry, ink, photographs. If you can see it, you can create a shader to represent it. But more than this, if you can't see it, you can create a shader to represent it.
There are a lot of numbers out there.
Go see them.
❋
References
Fork Meromorphi mla 166 by mla
Fork Meromorphic Poly by akohdr
Complex Log x 3 by mla
The 5 ways to visualize complex functions by Mathemaniac
Domain Coloring Plot by Jim Fowler |
11737 | https://www.health.harvard.edu/heart-health/mitral-valve-prolapse-a-to-z | Mitral Valve Prolapse - Harvard Health
Harvard Health PublishingHarvard Health Publishing
HarvardHealthOnline+
Search
Customer ServiceContent LicensingAbout Us
Log In
Open mobile menu
Free HealthBeat Signup
Shop
SubscriptionsSpecial Health ReportsOnline Courses
Pay Subscription Bill
Common ConditionsStaying HealthyResourcesBlogCOVID-19 Updates
Close menu
HarvardHealthOnline+
Main Content
Common Conditions
Staying Healthy
Resources
Blog
Helpful Links
Customer Service
About Us
Content Licensing
Free Healthbeat Signup
Subscriptions
Special Health Reports
Online Courses
Pay Subscription Bill
Search
Search
Close
Recent Articles
5 timeless habits for better health What are the symptoms of prostate cancer? Is your breakfast cereal healthy? When pain signals an emergency: Symptoms you should never ignore Does exercise give you energy? Acupuncture for pain relief: How it works and what to expect How to avoid jet lag: Tips for staying alert when you travel Biofeedback therapy: How it works and how it can help relieve pain Best vitamins and minerals for energy Should you take probiotics with antibiotics?
1/10
Heart Health
Mitral valve prolapse
July 22, 2025
Reviewed by Robert H. Shmerling, MD, Senior Faculty Editor, Harvard Health Publishing; Editorial Advisory Board Member, Harvard Health Publishing
Share
Share this page to FacebookShare this page to XShare this page via EmailCopy this page to clipboard
Print This Page Click to Print
What is mitral valve prolapse?
Mitral valve prolapse is a malfunction of the heart's mitral valve.
Normally, the mitral valve opens when the heart relaxes, allowing blood to flow from the upper part of the heart (the left atrium) into the lower part (the left ventricle). When the heart contracts to pump blood out of the left ventricle to the rest of the body, the mitral closes to prevent blood from returning to the left atrium.
With mitral valve prolapse, however, a slight deformity of the mitral valve prevents the valve from closing normally. This appears as an abnormal floppiness, or prolapse, of the valve. The result is that small amounts of blood leak back into the left atrium, with very little effect on the heart's overall ability to pump blood.
In some people, the leak worsens to create a significant backflow of blood into the left atrium. This is called mitral regurgitation. People with severe mitral regurgitation can develop symptoms such as shortness of breath, fatigue, and leg swelling. In severe cases, heart surgery is needed to repair the damaged valve.
You might also be interested in...
Workout Workbook
The numbers are shocking. Just three out of 10 American adults are active enough to stay healthy and fit, and nearly four out of 10 admit they aren’t active at all, despite reams of research proving that exercise is a powerful preventive, and sometimes an antidote, for disability and illness. Which side of this divide are you standing on — and why? Maybe you’re not sure what blend of exercise is best. Or perhaps you know exactly what you need to do, but your workouts have become so humdrum that it’s harder than ever to dig up the energy to do them. Do you want to jump-start your sputtering exercise program? Or are you stuck on a plateau and wishing you could kick it up to the next level?
LEARN MORE
In most people with mitral valve prolapse, the cause is unknown. However, in a small number of patients, mitral valve prolapse may be related to another medical condition, such as an inherited abnormality in the way the body produces collagen (connective tissue) or rheumatic heart disease (a rare complication of strep throat).
Health experts estimate that mitral valve prolapse currently affects up to 3% of people in the United States. In some studies, the condition is more common in women than in men. The condition can run in families, suggesting that the condition can be inherited.
Symptoms of mitral valve prolapse
Most people with mitral valve prolapse do not have any symptoms, and the condition is discovered only when a doctor hears a certain clicking sound with or without a heart murmur during a routine physical examination. In the past, mitral valve prolapse was blamed for symptoms such as chest pains, palpitations, and lightheadedness. Although a person with mitral valve prolapse can also have these types of symptoms, mitral valve prolapse is not the cause.
People who have mitral regurgitation that continues to get worse can have symptoms of congestive heart failure (fatigue, shortness of breath, leg swelling).
Diagnosing mitral valve prolapse
Your doctor will review your medical history, including any history of heart disease, excessive fatigue, leg swelling, or shortness of breath.
You might also be interested in...
Starting to Exercise
What can improve your mood, boost your ability to fend off infection, and lower your risk for heart disease, diabetes, high blood pressure, and colon cancer? The answer is regular exercise. It may seem too good to be true, but it's not. Hundreds of studies demonstrate that exercise helps you feel better and live longer. Starting to Exercise answers many important questions about physical activity. It will also help guide you through starting and maintaining an exercise program that suits your abilities and lifestyle.
Learn More
You and your doctor may not suspect that you have mitral valve prolapse until your doctor hears a sound called a systolic click with the stethoscope during the cardiac portion of a routine physical examination. The systolic click is an abnormal heart sound related to the malfunction of the mitral valve in mitral valve prolapse. Some people also have a heart murmur, an abnormal heart sound caused by turbulent blood flow.
Your doctor may suspect mitral valve prolapse if he or she hears the typical click. To confirm the diagnosis and assess its severity, he or she may order an echocardiogram. This is a painless test that uses sound waves to map out your heart structure. If significant mitral regurgitation is seen on the echocardiogram, a cardiac MRI may be recommended. The MRI can provide more detailed information to help direct treatment and assess prognosis.
If you have symptoms such as chest pain, dizziness, or palpitations, further tests may be needed to determine the cause.
Expected duration of mitral valve prolapse
Mitral valve prolapse is a lifelong condition. However, most people with the condition do not have any symptoms.
Preventing mitral valve prolapse
In most cases, mitral valve prolapse cannot be prevented.
Treating mitral valve prolapse
If you have mitral valve prolapse but do not have any symptoms, then you will not need treatment. In the past, your doctor may have recommended antibiotics prior to certain procedures and dental appointments, but current guidelines from the American Heart Association indicate that antibiotics are no longer recommended for most patients with mitral valve prolapse. Some people who have mitral valve prolapse with chest pains or palpitations take a beta blocker medication such as propranolol (Inderal), metoprolol (Lopressor), or atenolol (Tenormin) to treat symptoms.
You might also be interested in...
Cardio Exercise
Do you get winded climbing stairs? Have trouble keeping up with your kids or grandkids? Wish that you had more energy to power through your day? If so, it’s a good indicator that you need to either start doing some type of cardio exercise, or else kick your current routine up a notch. Even if you can sprint down the block with ease, you’re not off the hook. Whether you’re young, old, male, female, healthy, ill, pregnant, or disabled, you should be doing some form of regular cardio exercise.
Learn More
If mitral valve prolapse develops into severe mitral regurgitation, the abnormal mitral valve may need to be repaired or replaced surgically.
When to call a professional
Call your doctor immediately if you have palpitations or an abnormal pulse, especially if these symptoms have also made you feel lightheaded or faint. Also call your doctor immediately if you have chest pain at any age.
Prognosis
Most people with mitral valve prolapse never have symptoms, and the condition never significantly affects lifestyle or life expectancy. When surgery is performed to reconstruct or replace the mitral valve, it is successful in about 95% of cases.
Additional info
American Heart Association (AHA)
National Heart, Lung, and Blood Institute (NHLBI)
About the Reviewer
Robert H. Shmerling, MD, Senior Faculty Editor, Harvard Health Publishing; Editorial Advisory Board Member, Harvard Health Publishing
Dr. Robert H. Shmerling is the former clinical chief of the division of rheumatology at Beth Israel Deaconess Medical Center (BIDMC), and is a current member of the corresponding faculty in medicine at Harvard Medical School.… See Full Bio
View all posts by Robert H. Shmerling, MD
Share
Share this page to FacebookShare this page to XShare this page via EmailCopy this page to clipboard
Print This Page Click to Print
Disclaimer:
As a service to our readers, Harvard Health Publishing provides access to our library of archived content. Please note the date of last review or update on all articles.
No content on this site, regardless of date, should ever be used as a substitute for direct medical advice from your doctor or other qualified clinician.
Recent Articles
5 timeless habits for better health What are the symptoms of prostate cancer? Is your breakfast cereal healthy? When pain signals an emergency: Symptoms you should never ignore Does exercise give you energy? Acupuncture for pain relief: How it works and what to expect How to avoid jet lag: Tips for staying alert when you travel Biofeedback therapy: How it works and how it can help relieve pain Best vitamins and minerals for energy Should you take probiotics with antibiotics?
1/10
Related Topics
Staying Healthy ### Can the shingles vaccine protect my heart? Heart Health ### Cardiac rehab appears to help people with atrial fibrillation Heart Health ### Alcohol and heart health: A complex relationship
Heart Disease
Free Healthbeat Signup
Get the latest in health news delivered to your inbox!
Sign Up
Footer
Harvard Health Publishing
FacebookTwitterLinkedinYouTube
My Account
Customer Service
Log in
Order Now
Online Learning Courses
Digital Subscriptions
Special Health Reports
Print Subscriptions
More
About Us
Permissions
Content Licensing
Topics
Trademark Notice
© 2025 Harvard Health Publishing® of The President and Fellows of Harvard College
Do not sell my personal information | Privacy Policy and Terms of Use
Scroll To Top
Close
Thanks for visiting. Don't miss your FREE gift.
25 Gut Health Hacks is yours absolutely FREE when you sign up to receive health information from Harvard Medical School.
Sign up to get tips for living a healthy lifestyle, with ways to lessen digestion problems…keep inflammation under control…learn simple exercises to improve your balance…understand your options for cataract treatment…all delivered to your email box FREE.
I want to get healthier
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Close
Health Alerts from Harvard Medical School
And a FREE copy of the new report 25 Gut Health Hacks!
Sign up to receive HealthBeat emails from Harvard Health Publishing and get helpful tips and guidance for ways to lessen digestion problems… keep inflammation under control… learn simple exercises to improve your balance… understand your options for cataract treatment… all delivered to your email inbox FREE. PLUS, you'll get the latest news on medical advances and breakthroughs from Harvard Medical School experts, and special offers on content from Harvard Health Publishing.
I want to get healthier
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Close
Stay on top of latest health news from Harvard Medical School.
Plus, get a FREE copy of 25 Gut Health Hacks.
Sign me up
Close
Stay on top of latest health news from Harvard Medical School.
Plus, get a FREE copy of 25 Gut Health Hacks.
Sign me up
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply. |
11738 | https://blog.42yeah.is/rendering/cuda/2024/06/08/adaptiflux.html | Paper Talk: Topology-guided accelerated vector field streamline visualization | Of Shaders & Triangles
A semi-curated blog of computer graphics and rendering.
Paper Talk: Topology-guided accelerated vector field streamline visualization
2024-06-08 12:30:00 +0800
PERMALINK
A year ago I submitted the paper Topology-guided accelerated vector field streamline visualization to The Visual Computer, which has since been accepted. My paper is now available online here. Please don’t use LibGen or Sci-Hub to download my paper, cuz that would be naughty :)
Before we begin, massive thanks to Marek Fiser’s blog post and Yilun Yang’s previous work, both of which I drew inspiration (and code) from, without which this paper wouldn’t have been possible.
Abstract
In the paper, I proposed an adaptive method for locating planar critical points during streamline and streamtube tracings. New seed points are dynamically added on-the-fly and traced iteratively. See it work in action here:
Notice anything different? The left one is uniformly seeded, and the right one is seeded with the adaptive planar critical point method. Compared to the left one, our method dynamically spawns new seed points when critical points were discovered, where the vector field has the most significant changes. This results in much richer streamlines and streamtubes, capable of producing more information with the same amount of seed points. For example, more streamlines trace the turbulence underneath the upper drafts.
We also made a complete vector field visualization system for this, called AdaptiFlux. It supports various vector field visualization modes, including lines, arrow glyphs, streamlines and streamtubes, with our method implemented. Here’s a screenshot:
Introduction
To trace streamlines in a vector field, initial “seeds”, or “seed points”, must be specified. Those seeds are then shifted according to the underlying vector field direction and velocity, drawing out the streamlines in the process. The fancy name for the above process is called Line Integral Convolution (LIC) - because to draw the line, we have to move very little delta steps, effectively performing an integration. Think about a camera pointing at the starry night with crazy long exposure.
Since initial seeds must be provided, it’s quite easy to then know the final visualization result will be highly dependent on the seeding location. Here’s the delta wing scene visualized using two sets of initial seeds, with each of them being slightly different. This is stolen directly from Marek’s blog:
However, sometimes the streamlines traced won’t be good even with a relatively good initial streamline placement. Some important twists and turns only happens later down the line, and how can we capture that if we’ve already spent all the seeds at the beginning? And how do we even find these twists and turns, if possible?
Introducing the our adaptive seeding method. In a massive nutshell, suppose we have a seed quota of N=200. the algorithm first conserves half of them to be used later, then trace the initial 100 seeds. During the initial trace, we detect if these seeds are falling into critical points. And if they do, a seed point explosion event occurs, where new seed points are created around the critical region. Those new seed points are then traced during the second iteration, and so on. The tracing doesn’t stop until:
No new seed points are created during the last iteration.
The seed point quota is used up (that is, N seeds have already been traced.)
When Should We Explode?
How do we determine when to trigger an explosion event though? Well, we split it into two categories. First, we sample the vector field under point p. If v(p)=0, that means we are approaching a 3D critical point. A simple rationale for this is that the storm is always quitest in its center. 3D critical points tend to be quite complex, with the vectors around it being possible to be pointing in all sorts of directions. Like this:
This is only a cross section of a conventional critical point. Though, we can already see its quite messy around the edges. When we encounter a 3D critical point, we just generate new seed points around the old one randomly in a uniform sphere, and hope for the best.
The second category, which is also the main contribution of our paper, is a planar critical point. In a planar critical point, v(p) is not required to be 0; au contraire, they are supposed to be non-zero vectors pointing in arbitrary direction. The catch? The vectors around point p, if projected onto a plane, must satisfy (or approach) 2D critical point rules.
Take a look at the point in the center. Though technically not a critical point, it still kinda is because the vectors around it is…well, swirling around it. Look, finding proper words for these are hard. But hopefully, you know what I mean. This is quite useful because planar critical points can help us capture what 3D critical points can’t. Even better, we can use make use of the 2D critical point types to place new seed points so that the traced result can better reflect the critical region.
If a planar critical point type is determined to be a source, sink, or saddle point, new seed points are generated in a circular fashion; this is because circular placement can best reflect the overall topology shape of the region. In contrast, new seeds for centers, attracting foci, and repelling foci are placed in a linear fashion. This makes the most out of all the seed points, maximizing the amount of information:
How to Find Planar Critical Points?
“Well that’s good and all,” I hear you say, “but can you even locate these planar critical points guv’nor?” Well first, I haven’t heard the word “guv’nor” in ages. Second, its quite simple actually. Since planar critical points cannot be zero, it must be pointing to some sort of direction. This direction can act as a normal vector, so that the right vector r and the up vector u can be:
{r=v(p)×a[v(p)]||v(p)×a[v(p)]||u=r×v(p)||r×v(p)||
Where a[v(p)] is an arbitrary up vector for v(p). Then, ‖N‖ neighbors are taken surrounding this plane, in a circular fashion. The sample points are quite like the “circular explosion” in the figure above:
N i=p+cos(2 π i−1‖N‖)r+sin(2 π i−1‖N‖)u
An alignment metric μ p is evalated by computing the normalized dot product of v(p)and v(N i):
μ p=∑i=1‖N‖|norm[v(p)]T norm[v(N i)]|‖N‖
If they align poorly, as in, μ p is under a certain threshold, then we determine that we have encountered a planar critical point. In ideal circumstances, μ p should equal to 0 because all its neighbors are perpendicular to v(p); but obviously this is quite difficult to happen.
In any case, once its planar critical point status was determined, we need to find out its critical point type next by finding out the eigenvalues for its 2D planar Jacobian matrix, shown in Table 1. To do this, we first find the matrix U to transform v(p) to local space (or tangent space) - so that we only have to deal with 2D vectors. That is,
U v(p)‖v(p)‖=[0 0 1]
U can be solved by imaging v(p) is rotated from [0 0 1] and then reverse that rotation:
{θ=arccos(v(p)y)ϕ={arccos(v(p)z sinθ),v(p)≥0 arccos(−v(p)z sinθ)+π,otherwise U=[cosϕ cosθ sinϕ sinθ cosϕ 0−sinθ cosθ−sinϕ cosθ cosθ cosϕ sinθ cosϕ]T
We can now calculate the vector gradients along r and u:
{Δ r v(p)=v(p)−v(p−ε r)ε Δ u v(p)=v(p)−v(p−ε u)ε
And plug them into the Jacobian matrix:
J p=[U Δ r v(p)x U Δ r v(p)y U Δ u v(p)x U Δ u v(p)y]
Now that we have the Jacobian matrix, notice that sources, sinks and saddle points have real eigenvalues, while centers, attracting foci, and repelling foci do not. That means we don’t really need to find out the actual eigenvalues - we just need to find out if they have real eigenvalues. Recall that we only have two new seed placement strategies: circular, which corresponds to the former three, while linear corresponds to the latter.
With the aforementioned information, we can actually do a little trick to accelerate computation. Take a look at A quick trick for computing eigenvalues from 3Blue1Brown (and yes he’s my favorite math YouTuber) - especially around 4:28, where he showed the quick formula for finding out the eigenvalues:
λ 1,λ 2=m±m 2−p
Where m=J p 1,1+J p 2,2 2 and p=det(J p). Notice that we can only have real roots when m 2−p>=0. This can be used to quickly determine the critical point type without ever needing to solve the eigenvalues (though, to be frank, is not that far off), and, in turn, determine the critical point type, and the seed point spawning strategy.
Stramline Simplification
After tracing the streamlines, a simple method is used to de-clutter the them - based on the global distortion value. Its just a kinda fancy name for the ratio between the actual line length and the distance between streamline starting point and ending point:
T(L)=‖L‖‖L end−L begin‖
Streamlines are removed if T(L) is too low, or in other words, if they are too straight. Here’s how the decluttered streamlines look like. Mostly straight lines are removed to again, prioritize showing interesting streamlines first.
… And this is how we improve upon the original streamline tracing methods. This can be a direct plug in for other initial streamline seeding strategies as well; since our method is adaptive in nature, it could be plugged into, say, spherical seeding, and it could still be able to produce overall better streamlines. Think of it as a plug-in for the existing initial seeding strategies.
AdaptiFlux
Finally, let’s take a look at the aforementioned vector field visualization program AdaptiFlux. Because I have extended the debugger so much during the implementation it became a full-fledged CUDA-accelerated vector field visualizer. So, let’s take a look at what it’s capable of.
Four visualization methods: Lines, arrow glyphs, streamlines, and streamtubes.
Contains the full implementation of all proposed methods.
A real time FPS monitor is available right within the system.
Debugging features: seed point visualization visualizer, camera pose save/retrieve, and more.
Two initial seeding strategies: linear and spherical.
All visualizations are real time; tweak the parameters, and they are updated instantaneously. The rendering procedure is done by OpenGL/CUDA interoperation.
The source code of AdaptiFlux is available at Here are some extra goodies:
Vector field datasets used in the paper.
Uncompressed paper figures.
Conclusion
So, in conclusion, we defined a new sort of critical point in 3D, planar critical point, which we find by calculating an alignment metric around the neighbor. Once its found, the planar critical point type is determined based on the eigenvalues of the Jacobians, which we optimized a little due to the fact that we don’t actually need to evaluate the eigenvalues. The critical point type is used to determine new seed placement locations, so that we can maximize the amount of information and decrease repetitive streamlines. Finally, we made a vector field visualization system.
Obviously, quite a lot of things can be improved upon. For example, there are waaay too much parameters right now; in the future, maybe some form of machine learning can help decrease that and improve that? The implementation code quality is also something to be improved, as I was still learning CUDA as of this paper (well, actually I still am). However, seeing that I am about to graduate, I may not have time to work on this anymore (not to mention this is not a particularly good paper). So I guess there’s that.
The reason I write this blog post is because I think that accessible knowledge, not only in the form of, well, being free, but also easier to read, and therefore to learn, is important. And I always have trouble reading academic papers. That’s why I try to write a blog post on my own paper: hopefully when someone stumbles upon my paper one day, they will Google it online, and then they’ll encounter this. And hopefully the more relaxed explanation is easier to understand.
And that’s it! Whew. First blog post in 6 months! I was extremely busy the last 6 months due to school work and what not, but I am glad to be back. Stay tuned for more <3
Comments
A valid, non-empty username is required.
Comment
Loading comments +
Copyleft 2023 42yeah.
▲ |
11739 | https://ebaanz.org/wp-content/uploads/2021/05/EBAANZ-Donor-Selection-Guidelines-2019.pdf | GUIDELINES FOR THE MEDICAL SELECTION OF OCULAR TISSUE DONORS EDITION 3.0 AUGUST 2019 EDITED BY GRAEME POLLOCK GUIDELINES FOR THE MEDICAL SELECTION OF OCULAR TISSUE DONORS. EDITION 3.0 © AUGUST 2019: Edited by Graeme Pollock graemeap@unimelb.edu.au For further information, please contact: info@ebaanz.org or visit www.ebaanz.org About EBAANZ The Eye Bank Association of Australia & New Zealand (EBAANZ) is a not-for-profit organisation, and the peak body for eye donation and transplantation services in Australia and New Zealand (ANZ). EBAANZ is dedicated to helping restore sight, provide national and international leadership, develop standards for eye banking, and advocate for the eye banking sector by promoting the unique requirements of eye banks, and facilitating the sharing of information and expertise amongst EBAANZ members. Special Thanks EBAANZ would like to thank the Centre for Eye Research Australia, the University of Melbourne, and the Victorian Lions Clubs for their financial and collegial support towards the development of this framework, and our volunteer Steering Committee, Peer Comentators and External Policy Advisors for their invaluable expertise and support. Contents Introduction ................................ i Changes from Version 2.1 August 2016. ......................... iii Guidelines for the Medical Selection of Eye Donors (A-Z) .... 1 Acanthamoeba (ocular infection): ................................................... 1 Accident: .................................... 1 Achondroplasia: ......................... 1 Acne: .......................................... 1 Acne - Treatment: ...................... 2 Acne Rosacea: ............................ 2 Acupuncture: .............................. 2 Addiction and Drug Abuse: ........ 3 African Trypanosomiasis (Sleeping Sickness): .................... 4 Age: ............................................ 4 Age-related Macular Degeneration (AMD): ................. 4 AIDS: ........................................... 4 Alchoholism:............................... 4 Allergy: ....................................... 4 Alternative Therapies: ................ 5 Amyotrophic lateral sclerosis (Motor Neurone Disease): ......... 5 Anaemia: .................................... 5 Angelman Syndrome: ................. 5 Animal Derived Tissue Products: 6 Ankylosing Spondylitis: .............. 6 Anthrax: ...................................... 6 Antibiotic Therapy: ..................... 6 Antidepressant Therapy: ............ 6 Antifungal Therapy:.................... 7 Antiviral Therapy: ....................... 7 Arthritis: ..................................... 7 Arthropod Borne Encephalitis (Arbovirus infection):................. 7 Australian Bat Lyssavirus: .......... 8 Autoimmune Disorders: ............ 8 Avodart Therapy: ....................... 9 Babesiosis: ................................. 9 Bacteraemia: ............................. 9 Barmah Forest Virus: ................. 9 Basal Cell Carcinoma: ................ 9 BCG/BCG Immunisation: ........... 9 Bilharzia: .................................. 10 Binswanger's disease: .............. 10 Bipolar Disorder: ...................... 10 Bisexual and Homosexual – Female: .................................... 10 Bisexual – Male: ....................... 10 Bisexual – Female partners of Bisexual male: .......................... 10 Bites - Human: ......................... 10 Bites and Scratches – Bat: ....... 11 Bites and Scratches – Non-human, non-bat mammal:....... 11 Bites and Stings – Envenomation: ................................................. 11 Blind Donor: ............................. 11 Blood Components and Blood Products:.................................. 11 Body Piercing: .......................... 11 Bone Graft: .............................. 12 Borreliosis (Lyme Disease): ...... 12 Brain Surgery: .......................... 12 Brain Tumour: .......................... 12 Breast Lump: ............................ 12 Bronchitis – Acute: .................. 12 Bronchitis – Chronic: ............... 12 Brucellosis (Undulant Fever): ... 12 Cancer: ..................................... 13 Cannabis: .................................. 13 Cardiomyopathy: ..................... 13 Cardiovascular Disease: ........... 13 Cataract: ................................... 13 Central Nervous System Disease: ................................................. 14 Cervical Carcinoma in Situ: ...... 15 Chagas’ Disease (South American Trypanosomiasis): .................... 15 Chicken Pox - Herpes Zoster (Varicella Zoster): ..................... 15 Chikungunya Virus: .................. 16 Chlamydia: ............................... 16 Cholera Immunisation: ............ 17 Chronic Fatigue Syndrome: ...... 17 Cirrhosis: .................................. 17 Coeliac Disease: ....................... 18 Congenital Disorders: ............... 18 Colitis: ...................................... 18 Congo Fever (Crimean Fever): . 18 Contact Lenses: ........................ 19 Contact with Infectious Disease: ................................................. 19 Contagious Pustular Dermatitis (Orf virus): ................................ 19 Cortisone Tablets: .................... 19 Coxsackie Virus: ....................... 19 Creutzfeldt-Jakob Disease: ....... 19 Crimean Fever: ......................... 19 Crohn’s Disease: ....................... 19 Cytomegalovirus: ..................... 20 Death from Unknown Cause: ... 20 Dementia: ................................ 21 Dengue Fever: .......................... 23 Depression: .............................. 23 Dermatitis: ............................... 23 Diabetes Insipidus: ................... 23 Diabetes Mellitus: .................... 24 Diabetic Retinopathy: .............. 24 Diphtheria: ............................... 24 Diphtheria Immunisation: ........ 24 Diptheria Tetanus Immunisation: ................................................. 25 Diptheria Tetanus Pertussis Immunisation: .......................... 25 Disease of Unknown Aetiology:25 Diverticulosis: ........................... 25 Down Syndrome:...................... 25 Drowning: ................................. 25 Drug Abuse: .............................. 25 Drug Treatment:....................... 25 Dry Eye Syndrome: ................... 25 DTP Immunisation (Triple Antigen): ................................... 25 Duodenal Ulcer: ....................... 25 Dutasteride (Avodart): ............. 25 Ear Piercing: ............................. 26 Ebola Fever: .............................. 26 Eczema: .................................... 26 Ehlers-Danlos Syndrome: ......... 26 Electrolysis: .............................. 26 Elliptocytosis: ........................... 27 Emphysema: ............................. 27 Encephalitis: ............................. 27 Endocarditis: ............................ 27 Epilepsy: ................................... 27 Episcleritis: ............................... 27 Extended-Spectrum Beta-lactamase (ESBL): ..................... 27 Eye Disease: ............................. 28 Eye Drops: ................................ 28 Fever: ....................................... 28 Filariasis: .................................. 29 Finasteride (Proscar): .............. 29 Fits: .......................................... 29 Foreign Travel: ......................... 29 Fungal Infection: ...................... 29 G6PD Deficiency (Glucose-6-phosphate dehydrogenase deficiency): .............................. 29 Gall Bladder Disease: ............... 29 Gastrointestinal Disease: ......... 29 Gastric Ulcer: ........................... 30 Genital Herpes Infection: ........ 30 Genital Warts:.......................... 30 German Measles: ..................... 30 Giardiasis: ................................ 30 Gilbert Syndrome: ................... 30 Glandular Fever: ...................... 30 Glasses (Spectacles): ................ 31 Glaucoma:................................ 31 Goitre: ...................................... 31 Gonorrhoea: ............................ 31 Gout: ........................................ 32 Grand Mal: ............................... 32 Granuloma Inguinale: .............. 32 Grave’s Disease: ...................... 32 Growth Hormone: ................... 32 Guillain-Barre Syndrome: ........ 32 Haemotological Disease: ......... 33 Haemochromatosis: ................ 33 Haemoglobin Disorders: .......... 33 Haemolytic Anaemia: .............. 34 Haemophilus Influenzae Type B Immunisation: ......................... 34 Hand, Foot and Mouth Disease: ................................................. 34 Hashimoto’s Disease: .............. 34 Hendra Virus (Henipavirus): .... 34 Hepatitis: .................................. 35 Hepatitis A - Infection: ............. 36 Hepatitis A – Sexual Partner of Confirmed Case: ....................... 36 Hepatitis A – Person Sharing Home:....................................... 36 Hepatitis A – Immunisation: .... 36 Hepatitis B - Infection: ............. 36 Hepatitis B – History of Infection: ................................................. 37 Hepatitis B – Current or Former Sexual Partners of infected Individuals: ............................... 37 Hepatitis B – Current or Former Sexual Partners of Person who has recovered from Hepatitis B: ................................................. 38 Hepatitis B – Person Sharing Home with a person with Hepatitis B infection: ............... 38 Hepatitis B – Immunisation: .... 38 Hepatitis C: ............................... 38 Hepatitis C – Current and Former sexual partners of HCV positive individuals: ............................... 39 Hepatitis C – Person currently or formerly Sharing Home: ........... 39 Hepatitis E: ............................... 39 Hereditary Elliptocytosis: ......... 39 Hereditary Spherocytosis: ........ 40 Herpes – Genital: ..................... 40 Herpes – Ocular: ...................... 40 Herpes – Oral: .......................... 41 Herpes Simplex: ....................... 41 Herpes Zoster: .......................... 42 HIV - Infection: ......................... 42 HIV- Current or Former Sexual Partners of Confirmed Case: .... 42 Hodgkin Lymphoma (Disease): 42 Homosexual – Female: ............. 42 Homosexual – Male: ................ 42 Hormone Replacement Therapy: ................................................. 43 HTLV Infection: ......................... 43 HTLV- Former and Current Sexual Partners of Confirmed Case: .... 43 HTLV- Persons currently or formerly sharing a home with a Confirmed Case: ....................... 43 Huntington Disease: ................. 43 Hyperthyroidism: ..................... 44 Hypothyroidism:....................... 44 Immune Thrombocytopenia Pupura (Idiopathic Thrombocytopenia Pupura (ITP)): ................................................. 44 Immunisation – Live: ................ 44 Immunisation – Non-Live: ........ 45 Immunosuppression: ............... 46 Infection – Acute: ..................... 46 Infection – Chronic: .................. 47 Infection – General: ................. 47 Infection – Tropical: ................. 48 Infectious Diseases – Contact with: ......................................... 48 Inflammatory Bowel Disease: .. 49 Inflammatory Eye Disease: ...... 49 Influenza Immunisation: .......... 49 Inherited Diseases: ................... 49 Inoculation Injury: .................... 49 Irritable Bowel Syndrome: ....... 49 Japanese Encephalitis: ............. 49 Japanese Encephalitis Immunisation: .......................... 50 Jaundice: .................................. 50 Kala-Azar: ................................. 50 Keratoconjunctivitis Sicca: ....... 50 Kidney Disease: ........................ 51 Kidney Recipient: ..................... 51 Klinefelter’s Syndrome: ............ 51 Laser Treatment (Surgery): ...... 52 Lassa Fever: ............................. 52 Legionnaire’s Disease: ............. 52 Leishmaniasis: .......................... 52 Leptospirosis: ........................... 53 Lesbian: .................................... 53 Leukaemia: .............................. 54 Listeriosis (Listeria infection): .. 54 Lyme Disease: .......................... 55 Lyssavirus: ................................ 55 Lymphogranuloma Venereum: 55 Lymphoma: .............................. 55 Macular Degeneration:............ 55 Malaria: ................................... 55 Malignancy: ............................. 56 Malignant (Disseminated) Melanoma: .............................. 56 Marburg Fever: ........................ 56 Marfan’s Syndrome: ................ 57 Measles:................................... 57 Measles Immunisation: ........... 57 Measles Mumps Rubella (MMR) Immunisation: ......................... 57 Measles Rubella Immunisation: ................................................. 57 Meniere’s Disease: .................. 57 Meningitis: ............................... 57 Meningococcal Meningitis Immunisation: ......................... 58 Mental Health Problems: ........ 58 Middle East Respiratory Syndrome coronavirus (MERS-CoV): ........................................ 59 Motor Neuron(e) Disease: ....... 60 MRSA (Methicillin Resistant Staphylococcus Aureus): ......... 60 Multiple Resistant Organisms (MRO), Multiple Drug Resistance (MDR): ..................................... 60 Multiple Sclerosis: ................... 61 Mumps:.................................... 61 Mumps Immunisation: ............. 61 Murray Valley encephalitis virus: ................................................. 61 Muscular Dystrophy: ................ 61 Myasthenia Gravis: .................. 61 Myelodysplastic Syndrome: ..... 62 Myeloproliferative Disease (Syndrome): ............................. 62 Myocarditis: ............................. 62 Myxoedema: ............................ 62 Needle-Stick Injury: .................. 63 Nephritis: ................................. 63 Neurofibromatosis: .................. 63 Neurosurgery: .......................... 63 Nonsteroidal Anti-Inflammatory Drugs (NSAID): ......................... 63 Ocular Surgery: ........................ 63 Onchocerciasis (River Blindness): ................................................. 63 Orf: ........................................... 63 Organ Recipient: ...................... 63 Osteoarthritis: .......................... 63 Osteogenesis Imperfecta: ........ 63 Osteomalacia: .......................... 64 Osteomyelitis: .......................... 64 Osteoporosis: ........................... 64 Ovarian Cyst: ............................ 64 Paget’s Disease of Bone: .......... 64 Paratyphoid: ............................. 64 Parkinson’s Disease:................. 64 Peptic Ulcer: ............................. 65 Pericarditis: .............................. 65 Peritonitis: ................................ 65 Permanent Makeup: ................ 66 Pituitary Extract – Human: ....... 66 Plague (Yersinia Pestis): ........... 66 Plasma Dilution: ....................... 67 Platelet Disorder: ..................... 67 Pleurisy: .................................... 67 Pneumococcal Immunisation: .. 67 Pneumonia: .............................. 67 Poisoning: ................................. 68 Poliomyelitis: ............................ 68 Polio (Poliomyelitis) Immunisation: .......................... 68 Polycythaemia: ......................... 68 Prion Associated Diseases: ....... 69 Prisons & Psychiatric Institutions: ................................................. 70 Proctitis: ................................... 70 Progressive Multifocal Leukoencephalaopathy: ........... 70 Psoriasis: .................................. 71 Psychiatric Problems: ............... 71 Pyelonephritis: ......................... 71 Pyruvate Kinase Deficiency: ..... 72 Q Fever: .................................... 72 Rabies: ...................................... 72 Radiation Therapy: ................... 73 Raynaud’s Syndrome: .............. 73 Refractive Error (Vision): .......... 73 Refractive Surgery: ................... 74 Renal Disease: .......................... 75 Respiratory Disease:................. 75 Retinitis Pigmentosa: ............... 75 Reyes Syndrome:...................... 75 Rheumatic Fever: ..................... 76 Rheumatoid Arthritis: .............. 76 River Blindness: ........................ 76 Ross River Fever: ...................... 76 Rubella – Acute Infection: ........ 76 Rubella – Contact with: ............ 77 Rubella – Congenital: ............... 77 Rubella Immunisation: ............. 77 Salpingitis: ............................... 77 Sanfilippo Syndrome: .............. 77 Sarcoidosis: .............................. 78 SARS (Severe Acute Respiratory Syndrome): .............................. 78 Schistosomiasis: ....................... 78 Scleritis: ................................... 79 Sexually Transmitted Diseases - Infection: ................................. 79 Sexually Transmitted Disease – Sexual Partner: ........................ 79 Sex Worker: ............................. 80 Shingles: ................................... 80 Sicca: ........................................ 80 Sickle-Cell Disease: .................. 80 Sideroblastic Anaemia: ............ 80 Skin Cancer: ............................. 80 Skin Disease: ............................ 81 Sjögren's syndrome: ................ 81 Sleeping Sickness: .................... 81 Smallpox Immunisation: .......... 81 South American Trypanosomiasis (and risk of):............................. 82 Spectacles: ............................... 82 Spherocytosis: ......................... 82 Squamous Cell Carcinoma: ...... 82 Steroid Therapy: ...................... 82 Subacute sclerosing panencephalitis (SSPE): ........... 82 Syphilis: .................................... 83 Syphilis – Sexual Contact: ........ 83 Systemic Lupus Erythematosus: ................................................. 83 Tattoo: ..................................... 84 Temporal Arteritis: .................. 84 Tetanus Immunisation: ............ 84 Thalassaemia: .......................... 84 Thrombocytosis: ...................... 84 Thyroid Disease: ....................... 85 Tick-Borne Encephalitis: ........... 85 Tissue and Organ Recipients: ... 86 Tissue Products / Tissue grafts (Animal derived): ..................... 86 Toxoplasmosis: ......................... 87 Transfusion: ............................. 87 Transgender Individuals: .......... 88 Trauma: .................................... 88 Travel: ...................................... 89 Trisomy 21 (Down’s Syndrome): ................................................. 89 Trypanosoma Cruzi Infection: .. 89 Tuberculosis: ............................ 89 Turner’s Syndrome: ................. 90 Typhoid: ................................... 90 Typhoid Immunisation (Injected): ................................................. 90 Typhoid Immunisation (Oral): .. 90 Ulcerative Colitis: ..................... 90 Vasculitis: ................................. 91 Viral Haemorrhagic Fever: ....... 91 Vitiligo: ..................................... 92 Von Recklinghausen’s Disease: 92 West Nile Virus: ....................... 92 Whooping Cough: .................... 92 Wilson’s Disease: ..................... 93 Xenotransplantation: ............... 93 Yaws: ........................................ 93 Yellow Fever: ............................ 94 Zieve's syndrome: .................... 94 Zika Fever (Virus): .................... 94 Zollinger–Ellison syndrome: ..... 95 i Introduction Purpose These guidelines are intended as a reference document to assist the medical selection of eye donors for transplantation purposes. It is hoped that it will be useful as • a teaching tool for new eye donor coordinators (in particular) • a reference for other organ donor and tissue donor coordinators who may be involved the referral of a potential eye donor • a guideline to assist in the interpretation and application of standards applicable to ocular tissue for transplantation Although written using definitive terminology, this document is a guideline and should not be considered a definitive standard but rather as a helpful starting point in determining medical suitability for eye donation (explanations and additional information are therefore kept deliberately brief). It should be read in conjunction with applicable regulatory body and professional associations’ standards. (In Australia this includes the applicable standards dictated by the Therapeutic Goods Administration and in Australia and New Zealand the Eye Bank Association of Australia and New Zealand Medical and Quality Standards). These standards are considered mandatory minimal standards within their jurisdictions, and their application has been incorporated within these guidelines. These guidelines are only for eye donor selection. The advice given may not be applicable (and may be in conflict) to the standards and guidelines that are applicable for other forms of donation. The guidelines also seek to expand on specific eye conditions (eg. cataract, acanthamoeba) and provide direction on the suitability of the corneas for specific keratoplasty procedures. Acknowledgements and Reference Documents The style and content of the document is highly derivative of the United Kingdom Tissue Donor Selection Guidelines Deceased Donors (TDSG-DD) Edition 203. (available at Many of the guidelines are those of the United Kingdom Blood Transfusion Service and the National Institute for Biological Standards and Control. The Guidelines are also informed by the Eye Bank Association of Australia and New Zealand Medical and Quality Standards, the Eye Bank Association of America Medical Standards, the European Eye Bank Association Minimum Medical Standards, the Therapeutic Good Administration Therapeutic Goods Orders 85 and 88, and the Australian Red Cross Blood Service Guidelines for the Selection of Blood Donors. General Principles Eye/corneal donors are selected to ensure that their tissue is unlikely to harm any recipient. Where there is recognised risk to the recipient the guidelines must be followed. ii The immediate responsibility is with the trained and qualified Eye Donation Professional who must ensure that the donor fulfils the respective selection guidelines. When it is not clear from these guidelines if an individual donation is acceptable, no tissue should be used without further discussion with the Eye Bank’s Medical Director. The following terms are used throughout the A-Z listing: Including Lists any other terms which may be covered by the Guideline Definition Where additional clarity is required, a definition is provided Action This indicates the recommended action that should be taken. If an Action statement is included in a listing it may include the term(s): • Must Not Donate -The donor must not donate if any of the statements apply to them, unless an “accept” statement clearly applies. Often the exclusion will depend on time related factors. • Accept – The donation is acceptable for transplant purposes. It may also provide reasons why a donor may be permitted to donate (especially if preceded by a “do not donate” statement. The statements are conditional. All statements that must be fulfilled come before the final statement that they may be accepted. If the donor fulfils these requirements, as well as others that apply, then they can be accepted. • Refer to Medical Director -This is used when there may be a need for further discussion or expert opinion regarding a specific case. Where the guideline states “Cornea-only” this refers to the use (transplant) not the donation. Thus “cornea-only” indicates that corneas may be used for transplant but not the sclera for the same donor. Advisory Although a condition is not excluded by any Medical Standards it may be possible to reduce any further risk by following the advisory. See if Relevant Is used when an A-Z entry may or may not need to be consulted. This will depend on the information at hand regarding the donor. See Means that the specified A-Z entry should be consulted Additional Information This provides background information on the disease or condition (and provides information as to why a particular “Action” may be required. Additional TGA Criteria Where the TGA have criteria that are in addition to those of the EBAANZ Medical Standards, or where further clarity of explanation is required, the TGA criteria are listed here (in blue). Australian Eye Banks must adhere to these additional criteria. iii Use of the A-Z Listing Any medical condition, or possible contraindication to donation elicited at any point in the donation process (eg. written history, physicians’ history, medical and lifestyle interview) should be managed according to the A-Z listing. It is recognised that the listing is not comprehensive of all possible diseases, conditions or signs and symptoms that may arise during a specific donation. Therefore donations must not be accepted from donors that exhibit health risks that are not listed in this guidance without referral to and acceptance by the Medical Director. Revisions and Up-to-Date Releases These guidelines will be regularly reviewed to ensure that they are contemporary with world-best-practice, changing standards, improved knowledge of the disease process and with developments in technology, diagnosis and prognosis. To assist in the further development and content of these guidelines, including notification of error, omissions and suggestions for improvements, the users are encouraged to contact the author. Document Control Extensive revisions of this document are known as Editions. Changes involving additions, minor revision and corrections are known as Revisions (beginning at 1 for each Edition). Thus for example: Edition 1, Revision 3 will be known shorthand as “Guidelines for the Selection of Eye Donors 1.3”. Release date of each Revision will be listed on the document. Changes from Version 2.1 August 2016. Addiction and Drug Abuse Additional information provided on UK approach to individual risk assessment for those injecting 3-12 months period. Bi-sexual male Reduced deferral period from 4 to 3 months following MSM if NAT performed. Body Piercing Reduction to 3 months exclusion if NAT performed. Six months without NAT (serology only). Ebola Fever Now referral to Viral Haemorrhagic Fever Hepatitis A Reduction in deferral period from 12 to 6 months after recovery Hepatitis B Rewrite and simplification of all sections. Acceptability of sexual partners and house sharers if more than 3 months since last contact - if less than 3 months acceptable if donor shown to be immunised against HBV Hepatitis C Inclusion of information on treatment. Accept current or former sexual partners if more than 3 months since last sexual contact. Hepatitis E Reduction in deferral period from 12 to 6 months after recovery. Removal of household and sexual contacts guidelines. Update of additional information to reflect new knowledge. iv HIV Acceptance of sexual partners of HIV infected individuals if 3 months since last sexual contact (down from 4 months) Homosexual Male Reduced deferral period from 4 to 3 months following MSM if NAT performed HTLV infection Introduced sections on current or former sexual partners and Current and former persons sharing a home. Inoculation Injury Reduced deferral from 4 to 3 months from inoculation if NAT performed Marburg Fever Now referral to Viral Haemorrhagic Fever Poisoning New addition Prostitution Changed entry to Sex Worker Reyes syndrome Added that now acceptable is more than 3 months after recovery Sex Change Removed and replaced with Transgender Individuals Sex Worker Introduced entry to replace “Prostitution”. Reduced deferral period from 4 to 3 months with NAT. Defined practice. Viral Haemorrhagic Fever New addition to encompass all viral haemorrhagic disease. Ebola and Marburg are now referred to here. Changed individual affected to permanent deferral. Contacts and travel have been changed and clarified. Additional information on these conditions added. Return to Table of Contents 1 Guidelines for the Medical Selection of Eye Donors (A-Z) Acanthamoeba (ocular infection): Action Must not donate if past or active infection. Additional Information Acanthamoeba keratitis is a rare disease in which amoebae invade the cornea of the eye. It may result in permanent visual impairment or blindness. In the developed world it is nearly always associated with contact lens use, as Acanthamoeba can survive in the space between the lens and the eye. Accident: Includes Trauma. Action Accept if no eye involvement. See if relevant Transfusion. Additional Information If there is significant trauma, in particular penetrating trauma – an open wound or other source of infection is a risk for tissues becoming contaminated. Blunt chest trauma can result in damage to cardiovascular tissue. Note that if trauma has resulted in severe blood loss, the validity of any blood sample must be checked due to the possibility of plasma dilution of the donor. Achondroplasia: Action Accept. Additional Information Achondroplasia is a common cause of dwarfism. It occurs as a sporadic mutation in approximately 80% of cases (associated with advanced paternal age) or may be inherited as an autosomal dominant genetic disorder. People with achondroplasia have abnormal structural bone, so structural bone is not suitable for donation. Acne: Action Accept if no ocular surface disease. See if relevant Acne – treatment. Additional Information Secondary infection of acne is usually obvious with swelling and redness of affected spots. There is a risk of bacteria entering the blood. This could be a serious threat to anybody receiving tissues other than ocular as the bacteria can multiply to dangerous levels. For corneas preserved by organ culture (but not hypothermic) there is an opportunity to detect contaminating bacteria in the tissue and it should be safe to donate, but if there is no ocular involvement hypothermic storage would also present no significant risk. Secondary infection of the lid margin (blepharitis) on its own should not preclude eye donation, but donations must not be taken if there is also ocular surface disease. Return to Table of Contents 2 Acne - Treatment: Action Accept for corneas even if treated with Etretinate (Tigason), Acitretin (Neotigason), Isotretinoin (Roaccutane) or Alitretinoin (Toctino). Additional Information Etretinate (Tigason), Acitretin (Neotigason), Isotretinoin (Roaccutane) and Alitretinoin (Toctino) can cause birth defects in babies exposed to them while inside the womb. It is important to allow time for the drug to be cleared from the donor. It takes longer to clear some drugs than others. However, it is not considered to accumulate in corneas, nor present a risk to a corneal recipient. Etretinate was approved by the FDA in 1986 to treat severe psoriasis. It was subsequently removed from the market in the late 1990s due to the high risk of birth defects. It has an extremely long half-life and all tissues (other than corneas) must not be donated. Acitretin and Isotretinoin can cause birth defects but have a lower half-life than Etretinate. Tissues other than corneas should not be donated if less than 24 months from last dose of Acitretin and 4 weeks from last dose of Isotretinoin. Acne Rosacea: Action Accept if no ocular involvement. Additional Information Rosacea, or acne rosacea, is a type of non-contagious skin inflammation that typically affects the face. The small surface blood vessels (capillaries) of the skin enlarge, giving the appearance of a permanent flush. The forehead, cheeks and chin may develop yellow-headed pimples. Unlike acne, rosacea does not scar. In severe cases of rosacea, the nose can become reddened and enlarged (rhinophyma). The condition tends to appear between the ages of 30 and 50 years, initially with frequent flushing. Over time, this flushing becomes permanent as the capillaries enlarge and pustules begin to form. The symptoms tend to worsen with advancing age. The cause is unknown and there is no permanent cure. Complications include Ocular Rosacea: Red (due to telangiectasias), dry, irritated or gritty, eyes and eyelids. Watery eyes. Eyelids often develop cysts. Some other symptoms include foreign body sensations, itching, burning, stinging, and sensitivity to light. Eyes can become more susceptible to infection. Blurry vision and loss of vision can occur. Acupuncture: Action Accept, if performed with sterile single use needles. No deferral period required. Must not donate if a) Less than six months from completing treatment or four months if NAT HCV is performed when single-use sterile needles cannot be confirmed b) The condition for which treatment was given is not acceptable Additional TGA Criteria Six month and four month exclusion periods are mandated by the TGA. Return to Table of Contents 3 Additional Information Acupuncture needles that have been reused have passed infection from person to person. Australia In 2012 the Chinese Medicine Board of Australia (CMBA) was created, and in 2013 established an interim accreditation standard for the profession in partnership with the Australian Health Practitioner Regulation Agency. The legislation put in place stipulates that only practitioners who are state-registered may use the following titles: Acupuncture, Chinese Medicine, Chinese Herbal Medicine, Registered Acupuncturist, Registered Chinese Medicine Practitioner, and Registered Chinese Herbal Medicine Practitioner. The aim of registration was to protect the public from the risks of acupuncture by ensuring a high baseline level of competency and education of registered acupuncturists, enforcing guidelines regarding continuing professional education and investigating complaints of practitioner conduct. The Infection prevention and control guidelines for acupuncture practice (available at stipulate that only sterile single use needles are to be used. Therefore In practice in Australia, there should be no concerns or deferral periods following acupuncture. New Zealand Currently the acupuncture industry is self-regulated. There are a few organisations offering voluntary membership including New Zealand Register of Acupuncture Incorporated and New Zealand Acupuncture Standards Authority Incorporated. These organisations in their code of practice (available at stipulate the use of sterile single use needles. Addiction and Drug Abuse: Action Must not donate: Persons who have ever performed intravenous, intramuscular or subcutaneous non-prescribed drug use (even if only once) within the previous six months (or 4 months if HCV NAT is performed) or where history or examination gives cause to suspect non-medical intravenous drug use. This includes body-building drugs. May be acceptable if injected drugs were prescribed by the donor’s physician for a condition that would not lead to exclusion. Previous use of non-parenteral drugs does not necessarily require exclusion. Additional TGA Criteria The TGA mandates a five year exclusion period. Additional Information Injecting drug use has been linked with the passing on of many infections, including hepatitis and HIV. This guidelines recommendation of six month exclusion period is a rational approach based on the worst case scenario “window periods” for hepatitis and HIV testing. It is also internally consistent with other risk of infectious disease exclusion periods. The TGA mandate for Australian Eye Banking of five years appears to be based on the theoretical risk of recidivism in addiction and drug abuse. UK and Netherlands approaches have changed towards individual assessment of risk if injected drugs of addition within past 3-12 months by a “designated officer” if the risk of acquiring an infectious disease may be outweighed by the risk of delaying a transplant. UK will also accept if the donor has not injected with other non-prescription drugs (other than drugs of addition), such as bodybuilding drugs or injectable tanning agents within the past 3 months. Return to Table of Contents 4 African Trypanosomiasis (Sleeping Sickness): Action Accept for corneas but not sclera. Additional Information Corneas are avascular and therefore there is considered to be no risk of transmitting a protozoal infection. Human African trypanosomiasis, sleeping sickness, African lethargy, or Congo trypanosomiasis] is a parasitic disease of humans and other animals, caused by protozoa of the species Trypanosoma brucei and transmitted by the tsetse fly. Age: Action Lower age limit for donation of eye tissue for transplantation is two years. The upper age limit is left to the discretion of the Medical Director, since no definite relationship has been established between the quality of donor tissue and advanced age. Additional Information Low age problems relate to both the technical difficulties in using infant tissue and the complication of post-operative myopia. The extreme thinness of the infant cornea means that it is likely to fold upon itself during handling. This, combined with the small diameter of the cornea, creates problems during the donation surgery, trephining, and during placement and suturing in the host bed. The myopic shift after keratoplasty with infant donor corneas has been related to the steep curvature of these corneas. Age-related Macular Degeneration (AMD): Action Accept. Additional Information Age-related macular degeneration (AMD) is a medical condition which usually affects older adults and results in a loss of vision in the centre of the visual field (the macula) because of damage to the retina. It occurs in "dry" and "wet" forms. It is a major cause of blindness and visual impairment in older adults (>50 years). It does not affect the cornea or suitability for transplant. AIDS: See HIV. Alchoholism: Action Accept. See if relevant Cirrhosis Additional Information Of relevance to non-ocular tissue donation - If nutrition is poor the quality of bone is likely to be poor and not suitable for donation. Allergy: Action Accept. See if relevant Steroid therapy Return to Table of Contents 5 Alternative Therapies: Action Accept For therapies involving penetration by needles - accept, if performed with sterile single use needles. No deferral period required. Must not donate if a) Therapies involving the penetration by needles - less than six months from completing treatment or four months if NAT HCV is performed b) The condition for which treatment was given is not acceptable See if relevant Acupuncture Amyotrophic lateral sclerosis (Motor Neurone Disease): Action Must not donate See Motor Neurone Disease Additional Information Amyotrophic Lateral Sclerosis (ALS) – also referred to as Motor Neurone Disease (MND), as Lou Gehrig's disease in the United States– is a debilitating disease with varied aetiology characterised by rapidly progressive weakness, muscle atrophy and fasciculations, muscle spasticity, difficulty speaking (dysarthria), difficulty swallowing (dysphagia), and difficulty breathing (dyspnea). ALS is the most common of the five motor neuron diseases. The precise cause of ALS is still not known. Where no family history of the disease is present – i.e., in around 90% of cases – there is no known cause for ALS. Approximately 20% of Familial ALS has been associated with mutations in the gene that produces the Cu/Zn superoxide dismutase (SOD1) enzyme. Anaemia: Action Accept if underlying cause of anaemia (or iron deficiency) is not a reason to exclude See if relevant Haemogloblin Disorders Malignancy If treated with blood components or products: Transfusion Angelman Syndrome: Action Consult with Medical Director on likely risk of pathology to cornea. Additional Information It is associated (20-80%) with strabismus and/or hypo-pigmented eyes (and skin) but there is no suggestion of direct congenital effects on the cornea. Angelman syndrome is a neuro-genetic disorder characterized by severe intellectual and developmental disability, sleep disturbance, seizures, jerky movements (especially hand-flapping), frequent laughter or smiling, and usually a happy demeanour. AS is a classic example of genomic imprinting in that it is caused by deletion or inactivation of genes on the maternally inherited chromosome 15 while the paternal copy, which may be of normal sequence, is imprinted and therefore silenced. The sister syndrome, Prader-Willi syndrome, is caused by a similar loss of paternally inherited genes and maternal imprinting. Return to Table of Contents 6 Animal Derived Tissue Products: See Tissue Products / Tissue Grafts (Animal Derived) Ankylosing Spondylitis: Action Accept if no ocular involvement. Note: Condition may be associated with higher risk of uveitis or iritis. Additional Information Ankylosing spondylitis, previously known as Bechterew's disease (or syndrome) and Marie-Strümpell disease, is a chronic inflammatory disease of the axial skeleton with variable involvement of peripheral joints and nonarticular structures. AS is a form of spondyloarthritis, a chronic, inflammatory arthritis where immune mechanisms are thought to have a key role. It mainly affects joints in the spine and the sacroiliac joint in the pelvis. About one third to 40% of people with spondylitis will experience anterior uveitis or iritis at least once. AS can also affect the heart valves and the aorta. See Autoimmune Disorders Anthrax: Action Infection: Must not donate until six months post-recovery Contact/Exposure: Must not donate until six months after last exposure Additional Information Anthrax is an acute disease caused by the bacterium Bacillus anthracis. Most forms of the disease can be lethal. Some forms of the disease respond well to antibiotic treatment. Bacillus anthracis can form dormant endospores that are able to survive in harsh conditions for decades or even centuries. Such spores can be found on all continents, even Antarctica. When spores are inhaled, ingested, or come into contact with a skin lesion on a host, they may become reactivated and multiply rapidly. It most commonly affects the skin through direct contact with the infected material such as animal hides. If spores have been inhaled there is no evidence that there is any spread to the bloodstream until the person has developed signs of infection. In the UK it is considered safe to accept exposed donors provided they have not shown signs of infection, even if they have been given prophylactic antibiotics. Antibiotic Therapy: Action Accept if disorder being treated is not a reason to exclude Additional Information Treatment with antibiotics is not of itself a reason for deferral but the reason for the treatment may be. See Infection – General Antidepressant Therapy: See Mental Health Problems Return to Table of Contents 7 Antifungal Therapy: Action Accept, if on local therapy only for chronic superficial fungal infections. Must not donate if treatment is for an unresolved systemic fungal infection. Additional Information Fungaemia is the presence of fungi or yeasts in the blood. It is most commonly caused by Candida species, but can be caused by other fungi as well, including Saccharomyces, Aspergillus and Cryptococcus. It is most commonly seen in immunosuppressed or immunocompromised patients with severe neutropenia, oncology patients, or in patients with intravenous catheters. Organ culture of corneas will not necessarily eliminate the risk of transmission of a fungal infection to the recipient’s eye causing a fungal endophthalmitis. See Infection – General Antiviral Therapy: See Infection – General Arthritis: See if relevant Ankylosing Spondylitis Autoimmune Disorders Osteoarthritis Psoriasis Rheumatoid Arthritis Arthropod Borne Encephalitis (Arbovirus infection): Action Accept for corneas but not sclera only after clinical opinion that the infection has resolved. See if relevant Barmah Forest Virus Chikungunya Virus Dengue Fever Japanese encephalitis virus Murray Valley encephalitis virus Ross river virus Tick-borne encephalitis West Nile virus Yellow Fever virus Return to Table of Contents 8 Additional Information Arbovirus is a term used to refer to a group of viruses that are transmitted by arthropod vectors. The word arbovirus is an acronym (ARthropod-BOrne virus). Symptoms of arbovirus infection generally occur 3–15 days after exposure to the virus and last 3 or 4 days. The most common clinical features of infection are fever, headache and malaise, but encephalitis and haemorrhagic fever may also occur. Arboviruses maintain themselves in nature by going through a cycle between a host, an organism that carries the virus, and a vector, an organism that carries and transmits the virus to other organisms. Vectors are commonly mosquitoes, ticks and sandflies. Person-to-person transmission of arboviruses is not common, but can occur. Blood transfusions, organ transplantation and the use of blood products can transmit arboviruses if the virus is present in the donor's blood or organs. Corneas are avascular and therefore there is considered to be low or no risk of transmitting an infection. Common arboviruses include: Murray Valley encephalitis virus, Ross River virus, Barmah Forest virus Tick-borne encephalitis virus, Japanese encephalitis virus, Yellow fever virus, West Nile virus. (of immediate relevance to Australia). Australian Bat Lyssavirus: Action Must not donate See Bites and Scratches – Bat Additional Information Australian bat lyssavirus (ABLV) is a zoonotic virus closely related to rabies virus. It was first identified in a 5-month old juvenile Black Flying Fox collected near Ballina in northern New South Wales, Australia in January 1995 during a national surveillance program for the recently identified Hendra virus. ABLV is distributed throughout Australia in a variety of bat species which are believed to be the primary reservoir for the virus. There have been three confirmed cases of ABLV in humans, all of them fatal. The first occurred in November 1996 when an animal carer was scratched by a Yellow-bellied Sheath-tailed Bat. Onset of a rabies-like illness occurred 4–5 weeks following the incident, with death twenty days later. In August 1996, a woman in Queensland was bitten on the finger by a flying fox while attempting to remove it from a child it had landed on. After a 27 month incubation a rabies-like illness developed. The condition worsened after hospital admission and she died 19 days after the onset of illness. In December 2012 an eight-year-old boy was bitten or scratched by a bat in north Queensland. He became ill three weeks later, and died on 22 February 2013. Rabies vaccine and immunoglobulin are effective in prophylactic and therapeutic protection from ABLV infection. Since the emergence of the virus, rabies vaccine is administered to individuals with a heightened risk of exposure and vaccine and immunoglobulin are provided for post exposure treatment. ABLV is one of four zoonotic viruses discovered in Pteropid bats since 1994, the others being Hendra virus, Nipah virus and Menangle virus. Of these, ABLV is the only virus known to be transmissible to humans directly from bats without an intermediate host. Autoimmune Disorders: Action Accept if no ocular involvement See Is there an entry for the condition? Return to Table of Contents 9 See if relevant If treated with immunoglobulin or plasma exchange: Immunosuppression Transfusion Avodart Therapy: See Dutasteride Therapy Babesiosis: Action Accept for corneas but not sclera Additional Information Corneas are avascular and therefore there is considered to be no risk of transmitting a protozoal infection. Babesiosis is a malaria-like parasitic disease caused by infection with Babesia, a genus of protozoal piroplasms. After trypanosomes, Babesia is thought to be the second most common blood parasite of mammals. However, human babesiosis is uncommon, but reported cases have risen because of expanded medical awareness. Bacteraemia: Action Must not donate unless corneas are preserved by organ culture Additional Information Septicaemia (and thus bacteraemia) is a contraindication in a prospective corneal donor because of the risk of pathogenic organisms being harboured in the conjunctiva, surviving the antibiotics used throughout the retrieval and buttoning process, and eventually leading to an endophthalmitis in the recipient’s eye. The only exception to this is if the corneas are to be placed in organ culture for preservation. The microbiological surveillance during organ culture allows the exclusion prior to transplantation of those corneas that may be harbouring organisms. Barmah Forest Virus: Action Must not donate if within 12 months of last infection or recurrence See Arthropod Borne Encephalitis (Arbovirus infection) Additional Information Barmah Forest virus is a virus currently found only in Australia. Although there is no specific treatment for infection with the Barmah Forest virus, the disease is non-fatal and most infected people recover. The Barmah forest virus causes similar symptoms as the Ross River virus. The virus can only be transmitted to humans by bites from infected mosquitos. Direct contact with an infected person or animal does not cause infection. The virus is hosted mainly by marsupials, especially possums, kangaroos and wallabies. Basal Cell Carcinoma: Action Accept unless ocular area is involved Additional Information Although basal cell carcinoma (BCC) is a form of cancer it only spreads locally. As it does not spread by the blood stream it is not generally considered a risk to people receiving donated material. BCG/BCG Immunisation: Action Accept if more than 4 weeks after inoculation Must not donate if a) The inoculation site has not yet healed b) Less than 4 weeks after inoculation. Return to Table of Contents 10 Additional Information Bacillus Calmette–Guérin (BCG) is a vaccine against tuberculosis that is prepared from a strain of the attenuated (weakened) live bovine tuberculosis bacillus, Mycobacterium bovis. As a live bacteria vaccine, by four weeks, the infection caused by the inoculation should have been controlled. If the wound has not healed it is possible that there may still be infection present. Bilharzia: See Schistosomiasis Binswanger's disease: See Dementia Bipolar Disorder: See Mental Health Problems Bisexual and Homosexual – Female: Action Accept Additional information There is no evidence that there is an increased risk of sexually transmitted infections in bisexual or homosexual females compared to heterosexual females. Bisexual – Male: Action Must not donate if any oral or anal sex with another man in the past six months (even if a condom or other protective was used) Additional TGA Criteria The TGA mandates 12 month exclusion for increased risk sexual practice but does not define it Additional information Men who have had sex with other men have a higher chance of having an undiagnosed communicable disease infection. Therefore on an epidemiological population basis, male to male sex is considered high-risk behaviour. If one considers having a good knowledge of an individual’s behaviour, then male to male sex may not be considered high-risk behaviour (e.g. in the instance of a long-term monogamous relationship). The suggested six month exclusion period is consistent with that of other high-risk for infectious disease exclusions, based on “worse-case scenario” of hepatitis and HIV window periods. 6 months or 4 months if HCV NAT performed Bisexual – Female partners of Bisexual male: Action Must not donate if any oral or anal sex with another man in the past six months by the male partner was determined to be high-risk behaviour. Additional TGA Criteria The TGA mandates 12 month exclusion for increased risk sexual practice but does not define it See also Bisexual – Male, Additional information Bites - Human: Action Must not donate less than six months from receiving bite, or four months if NAT HCV is performed Additional TGA Criteria The six month exclusion period for these increased risks (mucosal splash with blood, needle stick injury, tattoo, body piercing) is mandated by the TGA Additional Information Human bites are considered to have the same risk as exposure to the following risks of acquiring a blood borne transmissible infection: mucosal splash with blood, needle stick injury, tattoo, body piercing Return to Table of Contents 11 Bites and Scratches – Bat: Action Must not donate if less than 12 months since the bat bite or scratch - in any country (including Australia and New Zealand) See Rabies Australian Bat Lyssavirus Additional Information Worldwide, bats are considered a risk for rabies infections. In Australia they are also considered a risk for lyssavirus (which is a rhabdovirus like rabies and caused 3 deaths in Queensland) and Hendra virus (4 deaths in Queensland) Bites and Scratches – Non-human, non-bat mammal: Action Must not donate if potential donor was bitten or scratched in a rabies endemic area in the past 12 months (if the donor was NOT pre-immunised against rabies) Accept if the donor was bitten or scratched in a rabies endemic area and had been pre-immunised against rabies See Rabies Additional Information In essence, rabies is considered endemic in any area outside of Australia, New Zealand and the British Isles. Bites and Stings – Envenomation: Action Must not donate if potential donor died as a result of the envenomation Must not donate if venom has directly contacted the eyes or peri-orbital area Additional Information Venom is composed of hundreds to thousands of different proteins and enzymes, all serving a variety of purposes, such as interfering with a prey's cardiac system or increasing tissue permeability so that venom is absorbed faster. Depending on the family or species, venom can be a combination of many toxins, including cytotoxins, haemotoxins, neurotoxins, and myotoxins. Relatively little is known about the protein structure of the venom of Australian creatures - including reptiles, arachnids, cephalopods, fish and cnidarian. Little is known about the distribution of toxins throughout the body and the accumulation and retention of toxins. Effects on the body are as varied as the venoms, and venoms also vary within a species. Therefore the risk to the recipient of receiving a dose of venom toxin from their transplant, or receiving tissue damaged by the venom, is largely unknown. Blind Donor: Action Accept if no corneal disorder. If necessary discuss with Medical Director. Additional Information All other tissues should be suitable unless there is a contraindication to the disease causing the blindness (eg. River Blindness). Blood Components and Blood Products: See Transfusion Body Piercing: Includes Permanent and Semi-permanent Makeup Action Must not donate if less than six months after last piercing or four months if NAT HCV performed Return to Table of Contents 12 Additional TGA Criteria The six month exclusion period for these increased risks (mucosal splash with blood, needle stick injury, tattoo, body piercing) is mandated by the TGA Additional Information Piercing has passed infection from person to person. Waiting six months (four with NAT HCV) ensure that the infections tested for will be picked up Bone Graft: See Tissue and Organ Recipients Borreliosis (Lyme Disease): See Infection – Acute Additional Information Lyme disease (Lyme borreliosis) is an infectious disease caused by at least three species of bacteria belonging to the genus Borrelia transmitted to humans by the bite of infected ticks belonging to a few species of the genus Ixodes ("hard ticks"). Identified in Europe and North America its presence in Australia or New Zealand is largely unknown Brain Surgery: See Neurosurgery Brain Tumour: See Neurosurgery Malignancy Breast Lump: See If related to malignancy, see Malignancy Bronchitis – Acute: See Infection – Acute Bronchitis – Chronic: See if relevant Infection – General Steroid Therapy Brucellosis (Undulant Fever): Action Must not donate Return to Table of Contents 13 Additional Information Brucellosis, also called Bang's disease, Crimean fever, Gibraltar fever, Malta fever, Maltese fever, Mediterranean fever, rock fever, or undulant fever is a highly contagious zoonosis caused by ingestion of unsterilized milk or meat from infected animals or close contact with their secretions. Transmission from human to human, through sexual contact or from mother to child, is rare but possible. It is uncommon in Australia and New Zealand. Brucella are small, gram-negative, non-motile, non-spore-forming, rod shaped (coccobacilli) bacteria. They function as facultative intracellular parasites causing chronic disease, which usually persists for life. The duration of the disease can vary from a few weeks to many months or even years. In the first stage of the disease, septicaemia occurs and leads to the classic triad of undulant fevers, sweating (often with characteristic smell, likened to wet hay), and migratory arthralgia and myalgia. With combination antibiotic therapy, most individuals recover in 2 to 3 weeks. Untreated, however, the infection may progress and increase in severity and also affect new tissues. Brucellosis can take a chronic form, with periods of illness alternating with periods of no symptoms. Approximately 10% of individuals may have a relapse, even after treatment is completed. This disease has a low mortality rate (lower than 2%); the most likely cause of death is endocarditis caused by Brucella melitensis. Cancer: See Malignancy Cannabis: See Addiction and Drug Abuse Cardiomyopathy: Action Must not donate if: a) Associated with extra-ocular muscle paresis e.g. Kearns-Sayre Syndrome b) Not recovered from a possible viral cause of the cardiomyopathy If a bacterial infective process is thought to be involved it may be preferable to preserve the corneas by organ culture. Additional Information Kearns–Sayre syndrome (abbreviated KSS) also known as oculocraniosomatic disease or Oculocraniosomatic neuromuscular disease is a mitochondrial myopathy with a typical onset before 20 years of age. KSS is a more severe syndromic variant of chronic progressive external ophthalmoplegia, a syndrome that is characterised by isolated involvement of the muscles controlling eyelid movement (levator palpebrae, orbicularis oculi), and those controlling eye movement (extra-ocular muscles). This results in ptosis and ophthalmoplegia respectively. Cardiovascular Disease: Action Accept, but consider those entries under “See if relevant”. See if relevant Cardiomyopathy Endocarditis Myocarditis Cataract: Includes Cataract Surgery Return to Table of Contents 14 Action Accept Additional Information A cataract is a clouding of the lens inside the eye which leads to a decrease in vision. It is the most common cause of blindness and is conventionally treated with surgery. Cataract by itself (untreated) will not affect the cornea. Surgical removal of the lens (cataract) and its replacement with an intraocular lens (IOL) has the potential to affect the cornea and make it slightly less likely that the cornea will be suitable for transplantation. However, eye bank assessment will determine if there has been any significant effect on the cornea and, if so, the cornea may still be suitable for anterior lamellar corneal transplantation. The common surgery today which is less likely to affect the cornea that past surgery is Phacoemulsification, or phaco. A small incision is made on the side of the cornea, the clear, dome-shaped surface that covers the front of the eye. A small probe is inserted into the eye. This device emits ultrasound waves that soften and break up the lens so that it can be removed by suction. Most cataract surgery today is done by phacoemulsification, also called "small incision cataract surgery." Extracapsular cataract extraction (ECCE), consists of removing the lens manually, but leaving the majority of the capsule intact. Extracapsular extraction is less frequently performed than phacoemulsification but can be useful when dealing with very hard cataracts or other situations where emulsification is problematic. Intracapsular cataract extraction (ICCE) is now rarely performed. The lens and surrounding capsule are removed in one piece through a large incision while pressure is applied to the vitreous membrane. Central Nervous System Disease: Action Must not donate if: a) Dementia b) History of CNS disease of possible infective origin (e.g. multiple sclerosis and Creutzfeldt-Jakob disease) c) Neurodegenerative conditions of unknown aetiology (e.g. Parkinson’s disease) Accept: a) Individuals who have had Bell's palsy more than four weeks ago and have discontinued any treatment for the condition for at least seven days, even if they have residual paralysis. b) If a definite diagnosis of transient global amnesia has been made. See if relevant Neurosurgery Prion Associated Diseases Rabies Return to Table of Contents 15 Additional Information Often the exact cause of a degenerative brain condition only becomes known after death. For this reason, when there is any doubt as to the underlying cause of a brain condition, it is considered safest not to accept a donation. It is thought that degenerative brain disease in the form of vCJD has been transmitted by blood transfusion. Transient global amnesia is a temporary and isolated disorder of memory. Affected individuals are usually over 50 years of age and there is an association with migraine. There is no association with cerebrovascular disease. Cervical Carcinoma in Situ: Action Accept – see Malignancy Chagas’ Disease (South American Trypanosomiasis): Action Accept for corneas but not sclera Additional Information Chagas disease, also called South American trypanosomiasis, is a tropical parasitic disease caused by the flagellate protozoan Trypanosoma cruzi. T. cruzi is commonly transmitted to humans and other mammals by an insect vector, the blood-sucking "kissing bugs" of the subfamily Triatominae. The disease may also be spread through blood transfusion and organ transplantation, and ingestion of food contaminated with parasites. Infection is very common in many parts of South or Central America and is often symptomless. It can be passed from an infected mother to her unborn baby and by transfusion. The insect that passes the infection on is only common in rural areas and the greater time that an individual has spent living in housing conditions with thatched roofs or mud lined walls which harbour the insect vector, the greater their risk of becoming infected. Camping or trekking in the jungle in South or Central America (including Southern Mexico) is not considered of high enough risk to merit exclusion for other tissues. Donations (other than eye) will defer: a) if the potential donor (or their Mother) was born in South America, Central America (including southern Mexico) b) if ever had a transfusion in South America or Central America (including Mexico) c) has lived or worked in rural subsistence farming communities in these countries for 4 weeks or more. As corneas are avascular there is not considered to be a risk of transmitting protozoal infections. Chicken Pox - Herpes Zoster (Varicella Zoster): Action Do not donate unless infection has completely resolved See Infection – Acute Herpes – Ocular Infectious Disease – Contact with Return to Table of Contents 16 Additional Information Chickenpox, also known as varicella, is a highly contagious disease caused by the initial infection with varicella zoster virus (VZV). It is an airborne disease which spreads easily through the coughs and sneezes of an infected person. It may be spread from one to two days before the rash appears until all lesions have crusted over. It may also spread through contact with the blisters. Those with shingles may spread chickenpox to those who are not immune through contact with the blisters. Symptoms usually last five to ten days. Complications may occasionally include pneumonia, inflammation of the brain, or bacterial infections of the skin among others. The disease is often more severe in adults (especially men) than children. Symptoms begin ten to twenty one days after exposure to the virus. In 2013 chickenpox resulted in 7,000 deaths globally – down from 8,900 in 1990. Death occurs in about 1 per 60,000 cases. Non-immune pregnant women and those with a suppressed immune system are at highest risk of serious complications. Chikungunya Virus: Definition Chikungunya endemic areas are shown and updated in the United Kingdom’s Geographical Disease Risk Index ( The Centers for Disease Control and Prevention (CDC) also have valuable current information on disease outbreaks by geography ( Action Must not donate if a) It is less than six months from a donor’s return from a chikungunya endemic area and the donor has been diagnosed or has had symptoms suggestive of chikungunya. b) In other cases, it is less than four weeks from a donor’s return from a chikungunya endemic area. Accept if six months after their return from an affected area. This may be reduced to four weeks if they have neither symptoms nor evidence of infection See if relevant Arthropod Borne Encephalitis (Arbovirus infection) Malaria South American Trypanosomiasis Risk Infection – Tropical Additional Information Chikungunya virus (CHIKV) is an arthropod-borne virus, of the genus Alphavirus that is transmitted to humans by virus-carrying Aedes mosquitoes. There have been recent breakouts of CHIKV associated with severe illness to death. Most commonly it causes arthritis (typically in the knee, ankle and small joints of the extremities), high fever, and a macropapular rash. It is geographically widespread but has reached epidemic proportions in parts of India and islands in the Indian Ocean since 2005. It is known to be spread by blood in symptomatic cases and on theoretical grounds could be spread by transfusion and transplantation of organs and tissues from people with pre-symptomatic or asymptomatic disease. It is spread by the same mosquitos as dengue fever. The problem can vary both in relation to geography and time of year, so it is not possible to state areas from which donors need to be deferred and dates of disease activity. Take note of the UK Index and CDC notifications. Chlamydia: Action Must not donate if current infection Return to Table of Contents 17 See Infection - Acute Lymphogranuloma Venereum Additional Information Chlamydia infection is a common sexually transmitted infection (STI) in humans caused by the bacterium Chlamydia trachomatis. The term Chlamydia infection can also refer to infection caused by any species belonging to the bacterial family Chlamydiaceae. C. trachomatis is found only in humans. Chlamydia is a major infectious cause of human genital and eye disease. Chlamydia conjunctivitis or trachoma was once the most important cause of blindness worldwide. The infection can be spread from eye to eye by fingers, shared towels or cloths, coughing and sneezing and eye-seeking flies. Cholera Immunisation: See Immunisation- Non-Live Chronic Fatigue Syndrome: Action Consider possible underlying causes and differential diagnoses. Additional Information Chronic fatigue syndrome (CFS) is the common name for a group of debilitating medical conditions characterised by persistent fatigue and other specific symptoms that lasts for a minimum of six months. The fatigue is not due to exertion, not significantly relieved by rest, and is not caused by other medical conditions. CFS may also be referred to as myalgic encephalomyelitis (ME), post-viral fatigue syndrome (PVFS), chronic fatigue immune dysfunction syndrome (CFIDS), or by several other terms. Biological, genetic, infectious and psychological mechanisms have been proposed, but the etiology of CFS is not understood and it may have multiple causes. The most commonly used diagnostic criteria and definition of CFS for research and clinical purposes were published by the United States Centers for Disease Control and Prevention (CDC). The CDC recommends the following three criteria be fulfilled: 1. A new onset (not lifelong) of severe fatigue for six consecutive months or greater duration which is unrelated to exertion, is not substantially relieved by rest, and is not a result of other medical conditions. 2. The fatigue causes a significant reduction of previous activity levels. 3. Four or more of the following symptoms that last six months or longer: • Impaired memory or concentration • Post-exertional malaise, where physical or mental exertions bring on "extreme, prolonged exhaustion and sickness" • Unrefreshing sleep • Muscle pain (myalgia) • Pain in multiple joints (arthralgia) • Headaches of a new kind or greater severity • Sore throat, frequent or recurring • Tender lymph nodes (cervical or axillary) Cirrhosis: Action Accept. If related to malignancy, see Malignancy. Return to Table of Contents 18 See if relevant Alcoholism Autoimmune Disorders Malignancy Coeliac Disease: Action Accept Coeliac disease is an autoimmune disorder of the small intestine that occurs in genetically predisposed people of all ages. Symptoms include pain and discomfort in the digestive tract, chronic constipation and diarrhoea, failure to thrive (in children), anaemia and fatigue. Coeliac disease is caused by a reaction to gliadin, a prolamin (gluten protein) found in wheat, and similar proteins found in the crops of the tribe Triticeae (which includes other common grains such as barley and rye). Congenital Disorders: Action Must not donate if the congenital disorder is high-risk for ocular disease, dysplasia, dystrophy, malformation or a metabolic disorder that may affect normal ocular function. Accept if it can be firmly established (through ophthalmic examination during the person’s lifetime) that there is no ocular dysplasia or malformation. See Achondroplasia Down Syndrome Angelman Syndrome Marfan Syndrome Sanfilippo Syndrome Turners Syndrome Wilsons Disease Ehlers-Danlos Syndrome Additional Information A congenital disorder, or congenital disease, is a condition existing at birth and often before birth, or that develops during the first month of life, regardless of causation. A congenital disorder may be the result of genetic abnormalities, the intrauterine (uterus) environment, errors of morphogenesis, infection, epigenetic modifications on a parental germline, or a chromosomal abnormality. Subtle changes in the cornea may be difficult to detect after death and donation (e.g. keratoconus) and therefore exclusion should be on the basis of the risk of the congenital disorder being associated with ocular manifestations. Colitis: See if relevant Infection – General Inflammatory Bowel Disease Malignancy Congo Fever (Crimean Fever): Action Must not donate if less than 12 months following recovery. Return to Table of Contents 19 Additional Information Crimean–Congo hemorrhagic fever (CCHF) is a widespread tick-borne viral disease, a zoonosis of domestic animals and wild animals that may affect humans. The pathogenic virus, especially common in East and West Africa, is a member of the Bunyaviridae family of RNA viruses. Clinical disease is rare in infected mammals, but commonly severe in infected humans, with a 30% mortality rate. Outbreaks of illness are usually attributable to handling infected animals or people. Contact Lenses: Action Accept if no underlying disease process, infection or corneal scarring. Must not donate if related to or involved in an underlying disease e.g. keratoconus, infection. Contact with Infectious Disease: See Infectious Disease – Contact with Contagious Pustular Dermatitis (Orf virus): See Infection – Acute Additional Information Orf is an exanthaemous disease caused by a parapox virus and occurring primarily in sheep and goats. It is also known as contagious pustular dermatitis, infectious labial dermatitis, ecthyma contagiosum, thistle disease and scabby mouth. Orf virus is zoonotic. It causes a purulent-appearing papule locally and generally no systemic symptoms. Infected locations can include the finger, hand, arm, face. The papule may persist for 7 to 10 weeks and spontaneously resolves. It is an uncommon condition and may be difficult to diagnose. There have been no reported cases of human to human contraction. While Orf is usually a benign self-limiting illness, it can be progressive and even life-threatening in the immune-compromised host. Cortisone Tablets: See Steroid Therapy Coxsackie Virus: See Hand, Foot and Mouth Disease Creutzfeldt-Jakob Disease: See Prion Associated Diseases Crimean Fever: See Congo Fever Crohn’s Disease: Action Accept if eyes are unaffected (Note below) See Inflammatory Bowel Disease Return to Table of Contents 20 Additional Information Crohn's disease, also known as Crohn syndrome and regional enteritis, is a type of inflammatory bowel disease that may affect any part of the gastrointestinal tract from mouth to anus, causing a wide variety of symptoms. It primarily causes abdominal pain, diarrhoea (which may be bloody if inflammation is at its worst), vomiting (can be continuous), or weight loss but may also cause complications outside the gastrointestinal tract such as anaemia, skin rashes, arthritis, inflammation of the eye, tiredness, and lack of concentration. Crohn's disease is caused by interactions between environmental, immunological and bacterial factors in genetically susceptible individuals. The cause of this condition is not fully understood and may include infection. Lesions caused by the disease can increase the risk of bacteria entering the blood stream. Therefore it may be relevant to only accept eyes if the corneas are to be preserved by organ culture. There is a chance that other organs and tissues may not be accepted for donation. Cytomegalovirus: Action Accept See Infection – General Additional Information Human cytomegalovirus is a species of virus that belongs to the viral family known as Herpesviridae or herpesviruses. It is typically abbreviated as HCMV and is alternatively known as Human herpesvirus 5 (HHV-5). After infection, HCMV has an ability to remain latent within the body over long periods. Although they may be found throughout the body, HCMV infections are frequently associated with the salivary glands. HCMV infection is typically unnoticed in healthy people, but can be life-threatening for the immune-compromised, such as HIV-infected persons, organ transplant recipients, or new born infants. The majority of the population are infected with HCMV. It presence is not considered relevant for eye donation as corneal recipients are not immune-compromised like organ recipients. Death from Unknown Cause: Action Accept if there is nothing to suggest that a possible cause of death or any other co-morbidities are contraindications to donation. Must not donate in all other instances Return to Table of Contents 21 Dementia: Action Must not donate if: a) Rapidly progressive dementia. e.g. CJD b) Slowly progressive dementia that is not Vascular dementia e.g. Alzheimer's disease, Frontotemporal lobar degeneration, Dementia with Lewy bodies. Accept: a) Fixed cognitive impairment due to traumatic brain injury, stroke, past infections, alcohol dementia. Accept with caution: b) Slowly progressive dementia due to vascular disease e.g. multi-infarct dementia, Binswanger disease. Accept with caution because slowly progressive dementia due to vascular disease and slowly progressive dementia due to Alzheimer’s disease, can be almost impossible to tell apart clinically and are often misdiagnosed. Indeed they often occur together. A definitive diagnosis and how that diagnosis was arrived at should be required. For example Binswanger disease, whilst a vascular dementia, mimics Alzheimer’s disease and is difficult to definitively diagnose (even with MRI) until brain histology is done after death. Additional Information Dementia is a serious loss of global cognitive ability in a previously unimpaired person, beyond what might be expected from normal aging. It may be static, the result of a unique global brain injury, or progressive, resulting in long-term decline due to damage or disease in the body. Although dementia is far more common in the geriatric population (about 5% of those over 65 are said to be involved), it can occur before the age of 65, in which case it is termed "early onset dementia". Dementia is not a single disease, but a non-specific syndrome (i.e., set of signs and symptoms). Affected cognitive areas can be memory, attention, language, and problem solving. Normally, symptoms must be present for at least six months to support a diagnosis. Cognitive dysfunction of shorter duration is called delirium. Some of the most common forms of dementia are: Alzheimer's disease, vascular dementia, frontotemporal dementia, semantic dementia, dementia with Lewy bodies and Binswanger disease (also known as subcortical leukoencephalopathy). Fixed cognitive impairment: Various types of brain injury may cause irreversible but fixed cognitive impairment. Traumatic brain injury may cause generalized damage to the white matter of the brain, or more localized damage (as also may neurosurgery). A temporary reduction in the brain's supply of blood or oxygen may lead to hypoxic-ischemic injury. Strokes (ischemic stroke, or intracerebral, subarachnoid, subdural or extradural haemorrhage) or infections (meningitis and/or encephalitis) affecting the brain, prolonged epileptic seizures and acute hydrocephalus may also have long-term effects on cognition. Excessive alcohol use may cause alcohol dementia, Wernicke's encephalopathy and/or Korsakoff's psychosis. Return to Table of Contents 22 Slowly progressive dementia: Dementia that begins gradually and worsens progressively over several years is usually caused by neurodegenerative disease—that is, by conditions that affect only or primarily the neurons of the brain and cause gradual but irreversible loss of function of these cells. Causes of dementia depend on the age at which symptoms begin. In the elderly population (usually defined in this context as over 65 years of age), a large majority of dementia cases are caused by Alzheimer's disease, vascular dementia, or both. Dementia with Lewy bodies is another commonly exhibited form, which again may occur alongside either or both of the other causes. Hypothyroidism sometimes causes slowly progressive cognitive impairment as the main symptom, and this may be fully reversible with treatment. Dementia is much less common under 65 years of age. Alzheimer's disease is still the most frequent cause, but inherited forms of the disease account for a higher proportion of cases in this age group. Frontotemporal lobar degeneration and Huntington's disease account for most of the remaining cases. Vascular dementia also occurs, but this in turn may be due to underlying conditions. People who receive frequent head trauma, such as boxers or football players, are at risk of chronic traumatic encephalopathy. In young adults (up to 40 years of age) who were previously of normal intelligence, it is very rare to develop dementia without other features of neurological disease, or without features of disease elsewhere in the body. Most cases of progressive cognitive disturbance in this age group are caused by psychiatric illness, alcohol or other drugs, or metabolic disturbance. At all ages, a substantial proportion of patients who complain of memory difficulty or other cognitive symptoms have depression rather than a neurodegenerative disease. Vitamin deficiencies and chronic infections may also occur at any age; they usually cause other symptoms before dementia occurs, but occasionally mimic degenerative dementia. These include deficiencies of vitamin B12, folate or niacin, and infective causes including cryptococcal meningitis, HIV, Lyme disease, progressive multifocal leukoencephalopathy, subacute sclerosing panencephalitis, syphilis and Whipple's disease. Rapidly progressive dementia: Creutzfeldt-Jakob disease typically causes a dementia that worsens over weeks to months, being caused by prions. The common causes of slowly progressive dementia also sometimes present with rapid progression: Alzheimer's disease, dementia with Lewy bodies, frontotemporal lobar degeneration (including corticobasal degeneration and progressive supranuclear palsy). On the other hand, encephalopathy or delirium may develop relatively slowly and resemble dementia. Possible causes include brain infection (viral encephalitis, subacute sclerosing panencephalitis, Whipple's disease) or inflammation (limbic encephalitis, Hashimoto's encephalopathy, cerebral vasculitis); tumors such as lymphoma or glioma; drug toxicity (e.g. anticonvulsant drugs); metabolic causes such as liver failure or kidney failure; and chronic subdural hematoma. Return to Table of Contents 23 As a feature of other conditions: There are many other medical and neurological conditions in which dementia only occurs late in the illness. For example, a proportion of patients with Parkinson's disease develop dementia. When dementia occurs in Parkinson's disease, the underlying cause may be dementia with Lewy bodies or Alzheimer's disease, or both. Cognitive impairment also occurs in the Parkinson-plus syndromes of progressive supranuclear palsy and corticobasal degeneration. Chronic inflammatory conditions of the brain may affect cognition in the long term, including Behçet's disease, multiple sclerosis, sarcoidosis, Sjögren's syndrome and systemic lupus erythematosus. Dengue Fever: See Arbovirus Infection – Acute Additional Information Dengue fever is an infectious tropical disease caused by the dengue virus (Arbovirus). Dengue is transmitted by several species of mosquito within the genus Aedes and is the most common viral disease transmitted by arthropods. Symptoms include fever, headache, muscle and joint pains, and a characteristic skin rash that is similar to measles. In a small proportion of cases the disease develops into the life-threatening dengue haemorrhagic fever, resulting in bleeding, low levels of blood platelets and blood plasma leakage, or into dengue shock syndrome, where dangerously low blood pressure occurs. Most people with dengue recover without any ongoing problems. The mortality is 1–5% without treatment. Depression: See Mental Health Problems Dermatitis: Action Accept See Infection – Acute Steroid Therapy Acne Additional Information Skin donation is not appropriate for the infected areas of skin Diabetes Insipidus: Action Accept, If underlying cause does not exclude. See if relevant Neurosurgery Additional Information Diabetes insipidus is either a problem with the production of antidiuretic hormone (central diabetes insipidus) or kidney's response to antidiuretic hormone (nephrogenic diabetes insipidus). It is characterized by excessive thirst and excretion of large amounts of severely diluted urine, with reduction of fluid intake having no effect on the concentration of the urine. Return to Table of Contents 24 Diabetes Mellitus: Action Accept Additional Information Diabetes mellitus type 1: (also known as type 1 diabetes, or T1DM; formerly insulin dependent diabetes or juvenile diabetes) is a form of diabetes mellitus that results from autoimmune destruction of insulin-producing beta cells of the pancreas. The subsequent lack of insulin leads to increased blood and urine glucose. Diabetes mellitus type 2: (formerly noninsulin-dependent diabetes mellitus (NIDDM) or adult-onset diabetes) is a metabolic disorder that is characterised by high blood glucose in the context of insulin resistance and relative insulin deficiency. Type 2 diabetes makes up about 90% of cases of diabetes. Diabetic Retinopathy: Action Accept Additional Information: Diabetic retinopathy, is retinopathy (damage to the retina) caused by complications of diabetes, which can eventually lead to blindness. It affects up to 80% of all patients who have had diabetes for 10 years or more (although this could be reduced with more vigilance and monitoring). The retinopathy itself will not affect the cornea or suitability for transplant. There is a suggestion however that long-term diabetes may affect the cornea’s suitability for transplantation but generally this is not significant. Diphtheria: Action Accept, if the clinician caring for the potential donor thinks that therapy given for an infection has successfully cleared it. See Infection – Acute Additional Information Diphtheria is an upper respiratory tract illness caused by Corynebacterium diphtheriae, an anaerobic, Gram-positive bacterium. It is characterized by sore throat, low fever, and an adherent membrane on the tonsils, pharynx, and/or nasal cavity. It is considered a highly infectious disease spread by direct physical contact or breathing the aerosolized secretions of infected individuals. Diphtheria Immunisation: Action Accept See Immunisation – Non-live Return to Table of Contents 25 Diptheria Tetanus Immunisation: Action Accept See Immunisation – Non-live Diptheria Tetanus Pertussis Immunisation: Action Accept See Immunisation – Non-live Disease of Unknown Aetiology: Action Must not donate Additional Information When the cause of an illness is not clear, there is an unknown risk to any recipient Diverticulosis: Action Accept See if relevant Infection – General Down Syndrome: Action Do not donate Additional Information Eye disorders are common in people with DS. Almost half have strabismus. Refractive errors requiring glasses or contacts are also common. Cataracts (opacity of the lens), keratoconus (thin, cone-shaped corneas), and glaucoma (increased eye pressures) are common Drowning: Action Refer to the Medical Director Additional Information Need to determine a likelihood of damage to the corneas due to immersion in water. Type of water (fresh, stagnant, sea-water, chlorinated) and length of time immersed need to be considered Drug Abuse: See Addiction and Drug Abuse Drug Treatment: Action The taking of some drugs may make a donor ineligible. This could be due to the underlying disease or to the medication. However for eyes this is unlikely. Self-medication with some drugs e.g. vitamins, over the counter analgesics need not prevent a donation being accepted See if relevant Addiction and Drug Abuse Dry Eye Syndrome: See Keratoconjunctivitis Sicca DTP Immunisation (Triple Antigen): See Immunisation – Non-live Duodenal Ulcer: See Gastric Ulcer Dutasteride (Avodart): Action Accept for corneas only not for sclera Return to Table of Contents 26 Additional Information Dutasteride is used for the symptoms of benign prostatic hyperplasia (BPH); colloquially known as an "enlarged prostate". The FDA has added a warning to dutasteride about an increased risk of high-grade prostate cancer. Evidence has suggested it could mask the early detection of prostate cancer. Precaution for organ and tissue is warranted as residual amounts (especially in the blood) can cause birth defects. Ear Piercing: See Body Piercing Ebola Fever: Action Must not donate if less than 12 months following recovery a) Asymptomatic travellers or residents returning from an EVD affected area should be deferred from donation for six month after return, provided they have reported no EVD symptoms (eg. undiagnosed febrile illness) b) Anyone with a confirmed EVD exposure (eg. contact with an infected person), cannot donate within six months of exposure or if being monitored for exposure, within six months of the commencement of the monitoring period Additional Information Ebola virus disease (EVD) or Ebola haemorrhagic fever (EHF) is the human disease which may be caused by any of four of the five known ebola viruses. EVD is a viral haemorrhagic fever (VHF), and is clinically nearly indistinguishable from Marburg virus disease (MVD). Prognosis is generally poor (average case-fatality rate of all EVD outbreaks to date = 68%). If a patient survives, recovery may be prompt and complete, or protracted with sequelae, such as orchitis, arthralgia, myalgia, desquamation or alopecia. Ocular manifestations, such as photophobia, hyperlacrimation, iritis, iridocyclitis, choroiditis and blindness have also been described. Outbreaks of EVD have mainly been restricted to Africa. The virus often consumes the population. EVD is believed to occur after an ebolavirus is transmitted to a human case via contact with an infected animal host. Human-to-human transmission occurs via direct contact with blood or bodily fluids from an infected person (including embalming of a deceased victim) or by contact with contaminated medical equipment such as needles. There is a higher concentration of EVD in the blood, organs and tissues during the immediate death or recovery phase as the body excretes live and infective viruses. A person can remain infectious for up to 7 weeks. Bats are considered the most likely candidate as a natural reservoir of the disease. Eczema: See Dermatitis Ehlers-Danlos Syndrome: Action Must not donate Additional Information Ehlers–Danlos syndrome (EDS) is an inherited connective tissue disorder with different presentations. It is caused by a defect in the synthesis of collagen. Depending on the individual, the severity of the mutation can vary from mild to life-threatening. On concern with any form of donation is that the structural integrity of the tissue is likely to be compromised. Electrolysis: Action Accept Return to Table of Contents 27 Elliptocytosis: See Hereditary Elliptocytosis Emphysema: Action Accept Encephalitis: Action Accept If caused by bacterial infection and the corneas are to be stored by organ culture See Infection – General Endocarditis: Action a) Accept If infection resolved. b) Accept If the cause is bacterial and the corneas are to be stored by organ culture. See Infection – General Epilepsy: Action Must not donate if a) Recent onset and not fully investigated b) Secondary to degenerative neurological disease Accept all other cases Episcleritis: Action Must not donate if active See Inflammatory Eye Disease Additional Information Episcleritis is a benign, self-limiting inflammatory disease affecting the episclera (thin layer of tissue that lies between the conjunctiva and sclera). Symptoms of include mild eye pain, redness, and watery eyes. Discharge is absent and vision is unaffected. It does not cause the presence of cells or flare in the anterior chamber of the eye. Most of the time, episcleritis is idiopathic. Several diseases are associated with episcleritis, including systemic vasculitic diseases (polyarteritis nodosa, Wegener's granulomatosis), connective tissue diseases, rosacea, atopy, gout, and ulcerative colitis. Extended-Spectrum Beta-lactamase (ESBL): See Multi Resistant Organisms Return to Table of Contents 28 Eye Disease: Action Must not donate if: a) Active ocular inflammation or infection b) Congenital or acquired ocular disorders or previous ocular surgery that may preclude a successful transplant outcome c) History of malignant tumours of the anterior segment or retinoblastoma Accept: a) Laser refractive surgery to the cornea but not for penetrating or anterior keratoplasty b) If laser surgery to the retina or lens capsule c) If cataract surgery d) If glaucoma e) If refractive error f) If retinal disease See also Autoimmune Disorders Glaucoma Immunosuppression Infection – General Laser Treatment (Surgery) Malignancy Ocular Surgery Steroid Therapy Eye Drops: Action Determine what the drops are being used to treat and then see if there is any relevant entry See if relevant Autoimmune Disorders Glaucoma Infection – General Steroid Therapy Fever: Includes Febrile episodes Action If less than two weeks from an undiagnosed episode(s) – investigate further and refer to Medical Director See if relevant Infection – General Additional Information A fever can be caused by many different conditions ranging from benign to potentially serious Return to Table of Contents 29 Filariasis: Action Must not donate See Infection – Tropical Onchocerciasis River Blindness Filariasis is a parasitic disease (usually an infectious tropical disease) that is caused by thread-like nematodes belonging to the superfamily Filarioidea, also known as "filariae". These are transmitted from host to host by blood-feeding arthropods, mainly black flies and mosquitoes. The adult worms, which usually stay in one tissue, release early larval forms known as microfilariae into the host's bloodstream. These circulating microfilariae can be taken up with a blood meal by the arthropod vector; in the vector, they develop into infective larvae that can be transmitted to a new host. Finasteride (Proscar): Discretionary Accept for corneas only not for sclera Additional Information Finasteride (brand names Proscar and Propecia by Merck, among other generic names) is a synthetic drug for the treatment of benign prostatic hyperplasia (BPH) and male pattern baldness (MPB). The FDA has added a warning to Finasteride about an increased risk of high-grade prostate cancer. Evidence has suggested it could mask the early detection of prostate cancer. Precaution for organ and tissue is warranted as residual amounts (especially in the blood) can cause birth defects. Recommendation is not to accept if less than 4 weeks from completion of therapy. Fits: See Epilepsy Foreign Travel: See Travel Fungal Infection: See Infection – General G6PD Deficiency (Glucose-6-phosphate dehydrogenase deficiency): Action Accept Additional Information Glucose-6-phosphate dehydrogenase deficiency (G6PD) is an X-linked recessive hereditary disease characterized by abnormally low levels of glucose-6-phosphate dehydrogenase, a metabolic enzyme involved in the pentose phosphate pathway, especially important in red blood cell metabolism. G6PD deficiency is the most common human enzyme defect. Individuals with the disease may exhibit non-immune haemolytic anaemia in response to a number of causes, most commonly infection or exposure to certain medications or fava beans. Gall Bladder Disease: Includes Gall Stones Action Accept See if relevant Infection – General Malignancy Gastrointestinal Disease: Return to Table of Contents 30 Action Accept but see Crohn’s Disease and Ulcerative Colitis See if relevant Crohn’s Disease Malignancy Ulcerative Colitis Gastric Ulcer: Includes Duodenal Ulcer and Erosions Action Accept. If associated with malignancy see Malignancy. Genital Herpes Infection: See Herpes – Genital Genital Warts: Action Accept See if relevant Sexually Transmitted Diseases Additional Information Genital warts (or condylomata acuminata, venereal warts, anal warts and anogenital warts) are symptoms of a highly contagious sexually transmitted disease caused by some types of human papillomavirus (HPV). It is spread through direct skin-to-skin contact, usually during oral, genital, or anal sex with an infected partner. Warts are the most easily recognized symptom of genital HPV infection. Although some types of HPV are known to cause cervical cancer and anal cancers, these are not the same types of HPV that cause genital warts. German Measles: See Rubella Giardiasis: Action Accept Additional Information This is a local intestinal infection that does not affect donation. Gilbert Syndrome: Action Accept Additional Information Gilbert's syndrome, also called Gilbert–Meulengracht syndrome, is the most common hereditary cause of increased bilirubin. Mild jaundice may appear under conditions of exertion, stress, fasting, and infections, but the condition is otherwise usually asymptomatic and has no serious consequences. Glandular Fever: Action Accept See Infection – Acute Return to Table of Contents 31 Additional Information Infectious mononucleosis (IM; also known as mono, glandular fever, Pfeiffer's disease, Filatov's disease, and sometimes colloquially as the kissing disease from its oral transmission) is an infectious, widespread viral disease caused by the Epstein–Barr virus (EBV), one type of herpes virus, against which over 90% of adults are likely to have acquired immunity by the age of 40. It is most common among adolescents and young adults. Especially in adolescents and young adults, the disease is characterized by fever, sore throat and fatigue, along with several other possible signs and symptoms. It is generally a self-limiting disease, and little treatment is normally required. Glasses (Spectacles): Action Accept See Refractive error See if relevant Laser Therapy Refractive surgery Glaucoma: Includes Primary glaucoma and its variants Action Accept Additional TGA criteria Must not donate if: a) Received a transplant of sclera during glaucoma surgery within the past 6 months if outside of Australia (or 4 months if HCV NAT performed) b) Ever received a transplant of sclera during glaucoma surgery in the United Kingdom This is in accordance with TGA standards of deferment of donor who have received human tissue outside of Australia. See if relevant Tissue and Organ Recipients Additional Information Glaucoma is a term describing a group of ocular disorders with multi-factorial aetiology united by a clinically characteristic intraocular pressure-associated optic neuropathy. Glaucoma can be roughly divided into two main categories, "open-angle" and "closed-angle" (or "angle closure") glaucoma. The angle refers to the area between the iris and cornea, through which fluid must flow to escape via the trabecular meshwork. Closed-angle glaucoma can appear suddenly and is often painful; visual loss can progress quickly, but the discomfort often leads patients to seek medical attention before permanent damage occurs. Open-angle, chronic glaucoma tends to progress at a slower rate and patients may not notice they have lost vision until the disease has progressed significantly. Goitre: See Thyroid Disease Gonorrhoea: See Sexually Transmitted Diseases Return to Table of Contents 32 Additional Information Gonorrhea is a common human sexually transmitted infection caused by the bacterium Neisseria gonorrhoeae. The usual symptoms in men are burning with urination and penile discharge. Women, on the other hand, are asymptomatic half the time or have vaginal discharge and pelvic pain. In both men and women if gonorrhea is left untreated, it may spread locally causing epididymitis or pelvic inflammatory disease or throughout the body, affecting joints and heart valves. Gout: Action Accept Grand Mal: See Epilepsy Granuloma Inguinale: Action Must not donate Additional Information Granuloma inguinale is a bacterial disease caused by Klebsiella granulomatis characterized by ulcerative genital lesions. It is endemic in many less developed regions. The microorganism spreads from one host to another through contact with the open sores. Grave’s Disease: See Thyroid Disease Growth Hormone: Action Must not donate if: Has ever received human pituitary growth hormone Accept, if treated exclusively with recombinant-derived growth hormone. (in Australia and New Zealand this will be since 1987) See if relevant Prion Associated Diseases Additional Information Worldwide (and 5 cases within Australia) human pituitary-derived hormones have transmitted classical CJD. From 1963 to 1986 such hormones were administered in Australia and New Zealand until the practice was ceased in 1987. Since 1987 all growth and fertility hormones administered in Australia (but not necessarily overseas) have been exclusively recombinant-derived, and are acceptable for donation. Recipients of human pituitary growth hormone are considered to be a high-risk for CJD transmission. Guillain-Barre Syndrome: Action Must not donate if: a) Less than 24 months from resolution b) There has been any recurrence of symptoms c) The doctor who managed the donor cannot confirm a typical monophasic Guillian-Barre syndrome that recovered completely within 12 months. See if relevant If treated with immunoglobulin or plasma exchange: Transfusion Return to Table of Contents 33 Additional Information Guillain–Barré syndrome, sometimes Landry's paralysis or Guillain–Barré–Strohl syndrome, is an acute polyneuropathy, a disorder affecting the peripheral nervous system. Ascending paralysis, weakness beginning in the feet and hands and migrating towards the trunk, is the most typical symptom, and some subtypes cause change in sensation or pain, as well as dysfunction of the autonomic nervous system. It can cause life-threatening complications, in particular if the respiratory muscles are affected or if the autonomic nervous system is involved. The disease is usually triggered by an infection. Guillain–Barré syndrome is rare, at one to two cases per 100,000 people annually, but is the most common cause of acute non-trauma-related paralysis. Guillain–Barré syndrome is now considered to be due to an immune response to foreign antigens (such as infectious agents) that mis-targets host nerve tissues. The most described antecedent infection is the bacterium Campylobacter jejuni. In addition, cytomegalovirus has a known association with GBS. In many cases, identification of a specific cause is impossible. Some cases may be triggered by the influenza virus. Guillain–Barré, unlike disorders such as multiple sclerosis (MS) and Lou Gehrig's disease (ALS), is a peripheral nerve disorder and does not in general cause nerve damage to the brain or spinal cord. Haemotological Disease: Action Must not donate if: a) Malignant b) Clonal disorder such as primary polycythaemia (rubra vera), essential thrombocythaemia and monoclonal gammopathy of unknown significance (MGUS). Discretionary If polycythaemia is secondary to a non-malignant condition, accept. See if relevant Haemoglobin Disorders Immune Thrombocytopenia Pupura Polycythaemia Additional Information Clonal disorders result from the proliferation of a single cell. Because they have the potential to become malignant they are treated in the same way as malignancy. Haemochromatosis: Action Accept Additional Information Haemochromatosis (or hemochromatosis) type 1 (also HFE hereditary haemochromatosis or HFE-related hereditary haemochromatosis) is a hereditary disease characterized by excessive intestinal absorption of dietary iron resulting in a pathological increase in total body iron stores. Excess iron accumulates in tissues and organs disrupting their normal function. The late effects of iron accumulation can be wholly prevented by periodic phlebotomies (by venesection) comparable in volume to blood donations. Haemoglobin Disorders: Action Accept Return to Table of Contents 34 See if relevant Sickle-cell disease Thalassaemia Transfusion Haemolytic Anaemia: Action See: a) Is there an entry for the condition? b) If not refer to Medical Director See if relevant Autoimmune disorder G6PD Deficiency Haemoglobin Disorders Hereditary Elliptocytosis Hereditary Spherocytosis Pyruvate Kinase Deficiency Transfusion Haemophilus Influenzae Type B Immunisation: See Immunisation – Non-live Hand, Foot and Mouth Disease: Action Accept See Infection – Acute Additional Information Hand, foot and mouth disease (HFMD) is a human syndrome caused by intestinal viruses of the picornaviridae family. It usually affects infants and children, and is quite common. It is moderately contagious but self-limiting. There is no specific treatment and only the symptoms are treated. Hashimoto’s Disease: See Thyroid disease Hendra Virus (Henipavirus): Action Must not donate Return to Table of Contents 35 Additional Information Henipavirus is a genus of the family Paramyxoviridae, order Mononegavirales containing three established species: Hendra virus, Nipah virus and Cedar Virus. The henipaviruses are naturally harboured by Pteropid fruit bats (flying foxes) and some microbat species. Henipavirus is characterised by a large genome, a wide host range, and their recent emergence as zoonotic pathogens capable of causing illness and death in domestic animals and humans. Hendra virus: (originally Equine morbillivirus) was discovered in September 1994 when it caused the deaths of thirteen horses, and a trainer at a training complex in Hendra, a suburb of Brisbane in Queensland, Australia. As of September 2012, a total of thirty-nine outbreaks of Hendra virus have occurred in Australia, all involving infection of horses. As a result of these events, seventy-six horses have died or been euthanised, with a further four having died or been euthanised as a result of possible hendra infection. Four of these outbreaks have spread to humans as a result of direct contact with infected horses. There have been two human deaths. A horse trainer, Victory ('Vic') Rail, and a stable hand were involved in nursing the first outbreak in horses, and both fell ill with an influenza-like illness within one week of the first horse’s death. The stable hand recovered while Mr Rail died of respiratory and renal failure. The source of the virus was most likely frothy nasal discharge from the horse. A second outbreak occurred in August 1994 in Mackay resulting in the deaths of two horses and their owner who assisted in necropsies of the horses and within three weeks was admitted to hospital suffering from meningitis. He recovered, but 14 months later developed neurologic signs and died. Case fatality rate in humans is 60% and in horses 75%. There is no evidence of transmission to humans directly from bats, and, as such it appears that human infection only occurs via an intermediate host, a horse. Nipah virus: was identified in April 1999, when it caused an outbreak of neurological and respiratory disease on pig farms in peninsular Malaysia, resulting in 257 human cases, including 105 human deaths and the culling of one million pigs. Symptoms of infection from the Malaysian outbreak were primarily encephalitic in humans and respiratory in pigs. Later outbreaks have caused respiratory illness in humans, increasing the likelihood of human-to-human transmission and indicating the existence of more dangerous strains of the virus. Based on seroprevalence data and virus isolations, the primary reservoir for Nipah virus was identified as Pteropid fruit bats. Cedar Virus: (CedPV) was first identified in pteropid urine during work on Hendra virus undertaken in Queensland in 2009. Although the virus is reported to be very similar to both Hendra and Nipah, it does not cause illness in laboratory animals. Hepatitis: Action Note: Hepatitis has a number of causes including infection and hypersensitivity to drugs. Of concern is Viral Hepatitis. Accept If fully recovered from non-viral hepatitis Return to Table of Contents 36 See if relevant Hepatitis A Hepatitis B Hepatitis C Hepatitis E Hepatitis A - Infection: Action Must not donate if less than 6 months from recovery Additional Information Hepatitis A (formerly known as infectious hepatitis) is an acute infectious disease of the liver caused by the hepatitis A virus (HAV), an RNA virus, usually spread by the faecal-oral route; transmitted person-to-person by ingestion of contaminated food or water or through direct contact with an infectious person. HAV infection produces a self-limited disease that does not result in chronic infection or chronic liver disease. Hepatitis A – Sexual Partner of Confirmed Case: Action Must not donate if less than 6 months from recovery of partner Accept if donor is shown to be immune to Hepatitis A (immunised) Additional Information There is a risk of transmitting the disease through sexual activity. The 6-month exclusion allows any infection to run its course and for any risk of passing the illness on through donation to have passed. Hepatitis A – Person Sharing Home: Action Must not donate if less than 6 months from recovery of last affected person in the home. Accept if donor is shown to be immune to Hepatitis A (immunised) Hepatitis A – Immunisation: Action Accept See Immunisation – Non-live See if relevant Hepatitis B – Immunisation Additional Information There are two types of vaccines: one type contains inactivated Hepatitis A virus, the other contains a live but attenuated virus. Both types stimulate active immunity against a future infection – the type used in Australia and New Zealand is the inactivated type. May be combined with Hepatitis B immunisation. Hepatitis B - Infection: Action Must not donate if active infection Return to Table of Contents 37 Additional Information Hepatitis B is an infectious inflammatory illness of the liver caused by the hepatitis B virus (HBV). Originally known as "serum hepatitis", the disease has caused epidemics in parts of Asia and Africa, and it is endemic in China. About a third of the world population has been infected at one point in their lives. The virus is transmitted by exposure to infectious blood or body fluids such as semen and vaginal fluids, while viral DNA has been detected in the saliva, tears, and urine of chronic carriers. Perinatal infection is a major route of infection in endemic (mainly developing) countries. Other risk factors for developing HBV infection include working in a healthcare setting, transfusions, dialysis, acupuncture, tattooing, sharing razors or toothbrushes with an infected person, travel in countries where it is endemic, and residence in an institution. However, hepatitis B viruses cannot be spread by holding hands, sharing eating utensils or drinking glasses, kissing, hugging, coughing, sneezing, or breastfeeding. The hepatitis B virus is highly infectious (50 to 100 times more infectious than HIV) and there is evidence (2 reports) of transmission by corneal transplantation. Hepatitis B – History of Infection: Action If more than 12 months from recovery, obtain history and analyse blood samples Accept if: a) HBsAg negative, HBcAb negative or b) HBsAg negative, HBcAb positive, HBV-DNA negative and HBsAb documented at more than 100 mIU/ml at some time (i.e. past infection with immunity) HBV-DNA negative is required if HBcAb positive to exclude for Occult HBV. Hepatitis B – Current or Former Sexual Partners of infected Individuals: Action Obtain history (including time since last sexual contact, and the dates that HBV immunisation given Must not donate if less than 3 months from last sexual contact Accept if: a) More than 3 months since last sexual contact b) If less than 3 months since last sexual contact and the donor is shown to be naturally immune Additional TGA criteria The TGA mandates 12 month exclusion for increased risk sexual practice but does not define it Return to Table of Contents 38 Hepatitis B – Current or Former Sexual Partners of Person who has recovered from Hepatitis B: Action Obtain history (including time since last sexual contact, date that the partner was diagnosed with HBV and the dates that HBV immunisation of the donor was given Most not donate if less than 3 months from last sexual contact with the partner who has been diagnosed with HBV infection less than 12 months ago Accept if: a) More than 3 months since last sexual contact regardless of when partner was diagnosed with HBV b) If partner was diagnosed with HBV infection more than 12 months ago and has cleared the infection at the time of last sexual contact Hepatitis B – Person Sharing Home with a person with Hepatitis B infection: Action Obtain history to determine if they are still sharing a home, and if not, the time sharing ceased Must not donate if less than 3 months since sharing ceased Accept if: a) More than 3 months since sharing ceased b) If less than 3 months since sharing ceased, and the donor is shown to be naturally immune Hepatitis B – Immunisation: Action Accept See Immunisation – Non-live Additional Information Sensitive assays for HBsAg may be positive following recent immunisation. Full screening for Hepatitis may be required. Hepatitis C: Action Must not donate For individuals that have been successfully treated seek Medical Director opinion. Note that such individuals will be HCV NAT negative but may retain HCV antibody reactivity. Return to Table of Contents 39 Additional Information Hepatitis C is an infectious disease affecting primarily the liver, caused by the hepatitis C virus (HCV). The infection is often asymptomatic, but chronic infection can lead to scarring of the liver and ultimately to cirrhosis, which is generally apparent after many years. In some cases, those with cirrhosis will go on to develop liver failure or liver cancer (hepatoma). It has also been linked with malignant lymphomas and autoimmune disease. Individuals who are chronically infected are sometimes referred to as carriers. They often have no, or minimal, symptoms associated with their infection. Many cases are linked to previous drug use, and before the introduction of HCV screening of blood, to transfusion. It has the longest “window period” for testing of the mandatory tests (HIV, HBV, HCV). There is now sufficient evidence to establish that individuals who have a sustained virological response to treatment for Hepatitis C infection (defined as remaining Hepatitis C RNA negative six months after cessation of the treatment) are likely to have been cured and the chance of a relapse is less than 1%. It is not considered contagious and is usually only spread through a direct blood to blood route. There is no evidence HCV can be spread by corneal transplantation. Hepatitis C – Current and Former sexual partners of HCV positive individuals: Action Must not donate if less than 3 months from the last sexual contact Accept if the donor’s HCV positive partner has been successfully treated for Hepatitis C infection and has been free of therapy for 12 months. Additional Information There is sufficient evidence to establish that individuals who have a sustained virological response to treatment for Hepatitis C infection (defined as remaining hepatits C RNA negative six months after cessation of treatment) are likely to have been cured and the chance of relapse is less than 1%. The very low risk of relapse combined with the rarity of sexual transmission of HCV means the transmission from a successfully treated individual to a sexual partner is most unlikely. Hepatitis C – Person currently or formerly Sharing Home: Action Accept Additional Information Hepatitis C is neither contagious nor spread by faecal-oral route. It is usually only spread through a direct blood to blood route. For these reasons household contacts do not need to be deferred. Hepatitis E: Action Must not donate if less than 6 months from recovery Additional Information Hepatitis E is an infectious hepatitis that is usually spread through contaminated food or water. Infection may be associated with travel to countries with poor hygiene/sewage conditions but increasingly, cases of hepatitis E are being identified due to consumption of undercooked contaminated meat. Hepatitis E can affect non-human animals and has been found in pigs. There have been reports of transmission by transfusion and transplant. Infection in healthy individuals is often symptom free but in people with underlying problems in their immune systems it can be serious and occasionally fatal. Hereditary Elliptocytosis: Action Accept Additional Information Hereditary elliptocytosis, also known as ovalocytosis, is an inherited blood disorder in which an abnormally large number of the patient's erythrocytes are elliptical rather than the typical biconcave disc shape. Return to Table of Contents 40 Hereditary Spherocytosis: Action Accept Additional Information Hereditary spherocytosis (also known as Minkowski–Chauffard syndrome) is a genetic disorder characterised by the production of red blood cells that are sphere-shaped rather than bi-concave disk shaped, and therefore more prone to haemolysis. t is the most common disorder of the red cell membrane, affecting 1 in 2,000 people of Northern European ancestry. Herpes – Genital: Action Accept if lesions are healing and there is no history of other Sexually Transmitted Diseases. See if relevant Sexually Transmitted Diseases Additional Information Genital herpes is a common sexually transmissible infection (STI) caused by the herpes simplex virus. There are two forms of the herpes simplex virus – HSV1 and HSV2. HSV1 more commonly occurs around the mouth, but it can also occur on the genitals. HSV2 occurs mainly on and around the genital area. It is estimated that about one in eight people have the virus that causes genital herpes and about 80 per cent of those infected may be unaware they have this infection. The herpes virus is spread by skin-to-skin contact and can be transmitted during vaginal, oral or anal sex. The infection can occur anywhere on the genitals, in areas around the groin or pubic area, and in or around the anus. Cold sores on the mouth can cause genital infection during oral sex for those who do not already have the cold sore virus. Recurrences are usually less painful and shorter in duration than the first episode. Over time, episodes usually become less frequent and may eventually stop altogether. Infections caused by HSV1 are less likely to recur in the genital area than infections caused by HSV2. Herpes – Ocular: Includes Simplex and Zoster Infection Action Must not donate if active or past infection of eyes. Return to Table of Contents 41 Additional Information Simplex Caused by HSV-1, ocular herpes is a common, recurrent viral infection affecting the eyes. Ocular herpes of the eye can be transmitted through close contact with an infected person whose virus is active or through self-contact and contamination during an active herpes infection (such as a cold sore of the lip). Ranging from a simple infection to a condition that can possibly cause blindness, there are several forms of ocular herpes: 1. Herpes keratitis is the most common form. It generally only affects the epithelium and usually heals without scarring. 2. Stromal keratitis occurs when the infection goes deeper into the layers of the cornea. This can lead to scarring, loss of vision and, occasionally, blindness. Stromal keratitis is thought to be caused by a late immune response to the original infection. Although the condition is rare, the NEI reports that stromal keratitis is the leading cause of corneal scarring that subsequently causes blindness in the United States. 3. Iridocyclitis is a serious form where the iris and surrounding tissues inside the eye become inflamed, causing severe sensitivity to light, blurred vision, pain and red eyes. 4. Herpes retinitis is when infection occurs in the retina Zoster Herpes zoster ophthalmicus occurs when the varicella-zoster virus is reactivated in the ophthalmic division of the trigeminal nerve. It represents up to one fourth of all cases of herpes zoster. Most patients present with a periorbital vesicular rash distributed according to the affected dermatome. A minority of patients may also develop conjunctivitis, keratitis, uveitis, and ocular cranial-nerve palsies. Permanent sequelae of ophthalmic zoster infection may include chronic ocular inflammation, loss of vision, and debilitating pain. Herpes – Oral: Action Accept Oral herpes is commonly referred to as “cold sores” and “fever blisters” and is typically caused by herpes simplex virus type 1 (HSV-1; oral HSV-2 is rare and is more commonly associated with genital herpes. More than 50 percent of the adult population has oral herpes and more than 80% of the population are infected with HSV-1. Oral herpes is transmitted through direct contact between the contagious area and broken skin (a cut or break) and mucous membrane tissue (such as the mouth or genitals). Most people contract the virus when they are children by receiving a kiss from a friend or relative. While symptoms of oral herpes most commonly appear on or around the lips, oral herpes is not always limited to this area. For some, symptoms may appear between the upper lip, on or inside the nose, or on the chin or cheek. In these instances, herpes is referred to as oral-facial herpes. Herpes Simplex: See if relevant Herpes – Genital Herpes – Oral Herpes – Ocular Return to Table of Contents 42 Herpes Zoster: See if relevant Infection – Acute Infectious Disease – Contact with Herpes – Ocular HIV - Infection: Includes AIDS Action Must not donate HIV- Current or Former Sexual Partners of Confirmed Case: Action Must not donate if less than 3 months from last sexual contact Additional Information HIV infection can be spread through sexual activity, including oral and anal sex. Despite regular sexual contact transmission of infection may not happen. It may however not be transmitted for a long time into a relationship. This could be because the infection becomes more active in the infected partner, the uninfected partner acquires another infection or injury to a mucous membrane, or there is a change in the use of, or failure of, barrier contraceptives. Waiting 3 months from the last sexual contact will ensure that any infection is picked up by the NAT tests used in ANZ. Hodgkin Lymphoma (Disease): Action Must not donate See Lymphoma Additional Information Hodgkin's lymphoma is characterized by the orderly spread of disease from one lymph node group to another and by the development of systemic symptoms with advanced disease. When Hodgkins cells are examined microscopically, multinucleated Reed–Sternberg cells (RS cells) are the characteristic histopathologic finding. The cause is unknown but probably multifactorial. Risk factors include: male, family history, history of infection with Epstein-Barr virus, weakened immune system including AIDS, prolonged use of human growth hormone, exposure to exotoxins such as Agent Orange. Homosexual – Female: Action Bi-sexual and Homosexual Female Homosexual – Male: Action Do not donate if any oral or anal sex with another man in the past six months (3 months with HCV) Accept homosexuality in the absence of male-to-male sex Additional TGA criteria The TGA mandates 12 month exclusion for increased risk sexual practice but does not define it Return to Table of Contents 43 Additional Information Men who have had sex with other men have a higher chance of having an undiagnosed communicable disease infection. Therefore on an epidemiological population basis, male to male sex is considered high-risk behaviour. If one considers having a good knowledge of an individual’s behaviour, then male to male sex may not be considered high-risk behaviour (e.g. in the instance of a long-term monogamous relationship). The suggested six month exclusion period is consistent with that of other high-risk for infectious disease exclusions, based on “worse-case scenario” of hepatitis and HIV window periods. 3 months if NAT is performed Hormone Replacement Therapy: Action Must not donate if: a) A recipient of human gonadotrophin of pituitary origin b) A recipient of human pituitary growth hormone Accept if: a) Treated with gonadotrophins that were exclusively non-pituitary derived (post 1987) b) Treated with growth hormone that was exclusively recombinant (post 1987) See if relevant Prion Associated Diseases Thyroid Disease HTLV Infection: Action Must not donate Additional Information The Human T-lymphotropic virus Type I (HTLV-1) is a human RNA retrovirus that is known to cause adult T-cell leukemia and lymphoma, and a demyelinating disease called HTLV-I associated myelopathy/Tropical spastic paraparesis (HAM/TSP). HTLV-I and HTLV-II are sexually transmitted. The transmission of HTLV by lymphocyte cells is valid for blood donations, but uncertain for tissues and cells and the prevalence outside of Japanese populations is insignificant. Therefore HTLV-1 & 2 testing is most often only required for the donation of leucocyte rich tissues. HTLV- Former and Current Sexual Partners of Confirmed Case: Action Must not donate if less than 3 months from last sexual contact Additional Information There is no defined infectious window period for HTLV. The risk of missing recent infection with individual sample testing is low after 3 months HTLV- Persons currently or formerly sharing a home with a Confirmed Case: Action Accept Huntington Disease: Action Accept If the diagnosis can be confirmed Additional Information Huntington's disease (HD) is a neurodegenerative genetic disorder that affects muscle coordination and leads to cognitive decline and psychiatric problems. It typically becomes noticeable in mid-adult life. HD is the most common genetic cause of abnormal involuntary writhing movements called chorea, which is why the disease used to be called Huntington's chorea. Return to Table of Contents 44 Hyperthyroidism: See Thyroid Disease Hypothyroidism: See Thyroid Disease Immune Thrombocytopenia Pupura (Idiopathic Thrombocytopenia Pupura (ITP)): Action Must not donate if: a) Symptomatic b) Chronic c) Recovered but less the 5 years from recovery See if relevant If treated with immunoglobulin or plasma exchange: Transfusion If treated with immunosuppressive therapy: Immunosuppression Additional Information Idiopathic thrombocytopenic purpura (ITP), also known as primary immune thrombocytopenic purpura and autoimmune thrombocytopenic purpura, is defined as isolated low platelet count (thrombocytopenia) with normal bone marrow and the absence of other causes of thrombocytopenia. ITP is an autoimmune condition with antibodies detectable against several platelet surface antigens. The cause of ITP is unknown, but viral infections might make the immune system malfunction and start producing the ‘rogue’ antibodies. There are two broad categories of ITP: one that usually goes away by itself with time (usually in childhood), and the other that lingers beyond six months (more often in adults). ITP is more common among children than adults, most often occurring around two to four years of age, affecting boys and girls equally. Among adults, young women are more likely to develop ITP than any other group, for reasons unknown. ITP symptoms can include any of the following: Abnormally heavy periods in women Bleeding into the skin, often around the shins, causing a skin rash that looks like pinpoint red spots (petechial rash) Easy bruising Nosebleed or bleeding in the mouth Caution in acceptance is due to diagnosis being dependant on the exclusion of other causes of a low platelet count – including leukaemia, myelodysplastic syndromes and pre-neoplastic states i.e. there is a high potential for misdiagnosis. These recommendations are that of United Kingdom Tissue Donor Selection Guidelines. Immunisation – Live: Includes Live attenuated bacteria or viruses Action Must not donate if less than four weeks from vaccination (note for smallpox vaccination the exclusion period is 8 weeks) Return to Table of Contents 45 Additional TGA criteria The TGA mandates the exclusion times listed See BCG Smallpox Immunisation Additional Information Live immunisations use living viruses or living bacteria that will stimulate the immune system but do not normally cause a severe illness. They may however cause severe illness in people who are already unwell and have a weakened immune system. By four weeks, any infection caused by the immunisation should have been controlled and so should not be passed on through donated material. The following live vaccines are currently in use in Australia and New Zealand: Live attenuated parenteral vaccines Live attenuated oral vaccines Viral Bacterial Viral Bacterial Japanese encephalitis (Imojev) Measles-mumps rubella (MMR) Measles-mumps rubella- varicella (MMRV) Varicella Yellow fever Zoster BCG Oral rotavirus vaccine Oral typhoid vaccine Immunisation – Non-Live: Action Accept non-live immunisations. Additional Information “Non-live” immunisations do not use material that can cause infection. The means there is no risk to people receiving blood or tissues from a recently immunised donor. NOTE: Assays for HBsAg may be positive following recent immunisation. Full screening for Hepatitis B may be required. Return to Table of Contents 46 Immunosuppression: Action Accept only if NAT HIV, HBV and HCV testing is performed if on long-term corticosteroids equivalent to >30 mg/day Prednisolone (see Additional information) or lower doses in combination with immunosuppressant drugs such as Azathioprine or Mycophenolate. Accept if on the equivalent of >30 mg/day Prednisolone for not more than two weeks within the 6 months before death (For potential eye donors with a history of malignancy who are on chemotherapy, or autoimmune disease who are on immunosuppressive therapy this is a likely scenario) See if relevant Autoimmune Disorders Steroid Therapy Tissue and Organ Recipients Additional Information A dose greater than 30 mg/day prednisolone may be immunosuppressive and theoretically could have an impact on serological tests for antibodies to markers of transmissible disease. Assays which directly detect the virus are not affected adversely by immunosuppression and are appropriate in this situation. Other corticosteroids equivalent to 30 mg Prednisolone: Betamethasone (4.5 mg) Cortisone acetate (150 mg) Deflazacort (36 mg) Dexamethasone (4.5 mg) Hydrocortisone (120 mg) Methylprednisolone (24 mg) Triamcinolone (24 mg) Infection – Acute: Action See if there is a specific entry for the disease you are concerned about. Consider if the infection was systemic (eg. septicaemia, viraemia), or localised and unlikely to have affected the cornea or eyes. Accept if caused by bacterial infection and the corneas are to be stored by organ culture. Accept if the clinician caring for the potential donor thinks that therapy given for an infection has successfully cleared it. Accept if the infection was not considered systemic (e.g. not septicaemia, not viraemia), but localised and not likely to affect the eyes. NB. For some disease states the infective agent can persist in the body for quite some time after the signs and symptoms of infection have ceased or when the infection is considered to be resolved. Check deferral times for disease after the infection has resolved (e.g. Ebola is 12 months deferral after resolution of the infection) Return to Table of Contents 47 See if relevant Congo Fever Crimean Fever Dengue Fever Ebola Fever Herpes- Genital Herpes Oral Lassa Fever Marburg Fever MRSA Steroid Therapy West Nile Virus Additional Information Potential donors who have been cared for in an ICU may have a local chest infection as a result of ventilation – acceptable Donors who have bacterial pneumonia are acceptable as eye donors but would not be acceptable for other tissues Donors who have had a positive screening test for MRSA (carriers) are acceptable. Donors with active MRSA infection at the time of death should be referred to the Medical Director for opinion. Infection – Chronic: Action See if there is a specific entry for the disease you are concerned about. Consider if the infection was systemic, or localised and unlikely to have affected the cornea or eyes. Accept 1. Acne – Secondary infection of the lid margin (blepharitis) on its own should not preclude eye donation, but donations must not be taken if there is also ocular surface disease. 2. Chronic superficial fungal infections 3. Typhoid and Paratyphoid – if more than 7 days from completion of antibiotic course and last symptoms 4. If caused by bacterial infection and the corneas are to be stored by organ culture See if relevant Acne Steroid Therapy Additional Information Typhoid and paratyphoid are gastrointestinal infections which rarely have a chronic carrier state. It is usually caught while travelling overseas and passed by the faecal-oral route and is not transfusion transmitted. Infection – General: Action See if there is a specific entry for the disease See if relevant Decide if the infection is of short duration with no long lasting carrier state, e.g. flu: Infection – Acute Or if lasting a long time (more than a few weeks) or possibly with long lasting carriage of infecting organism e.g. malaria, typhoid. Infection - Chronic Return to Table of Contents 48 Infection – Tropical: Action Must not donate if Filariasis (River Blindness, Onchocerciasis) Leishmaniasis, accept for corneas only not for sclera See if relevant Congo Fever Crimean Fever Dengue Fever Ebola Fever Lassa Fever Leishmaniasis Marburg Fever Malaria Onchocerciasis / River Blindness South American Trypanosomiasis Yellow Fever Additional Information Leishmaniasis is a disease caused by protozoan parasites that belong to the genus Leishmania and is transmitted by the bite of certain species of sand fly. Most forms of the disease are transmissible only from non-human animals (zoonosis), but some can be spread between humans. Cutaneous leishmaniasis is the most common form, which causes a sore at the bite site, which heals in a few months to a year, leaving an unpleasant-looking scar. As corneas are avascular there is not considered to be a risk of transmitting protozoal infections of this type. Onchocerciasis (River Blindness) is second in the world only to trachoma as an infectious cause of blindness. It is a form of subcutaneous filariasis is caused by Loa loa (the eye worm). Infectious Diseases – Contact with: Action Is there a specific entry for the disease with which there has been contact? Must not donate if: Within the incubation period for the condition or, if this is not known, less than 4 weeks from last contact. See if relevant Hepatitis Meningitis Sexually Transmitted Disease Tuberculosis Return to Table of Contents 49 Inflammatory Bowel Disease: Includes Crohn’s Disease Ulcerative Colitis Irritable Bowel Syndrome Action Accept if no ocular involvement (see below) Additional Information The cause of these conditions is not fully understood and may include infection. Lesions caused by the disease can increase the risk of bacteria entering the blood stream. Therefore it may be relevant to only accept eyes if the corneas are to be preserved by organ culture. There is a chance that other organs and tissues may not be accepted for donation. Inflammatory Eye Disease: Active Must not donate if active See if relevant Autoimmune Disorders Influenza Immunisation: See Immunisation – Non-live Inherited Diseases: Action Is there a specific entry for the condition? If not: refer to Medical Director See Congenital Disorders Inoculation Injury: Includes Bite (Human) Needle-stick injury Action Must not donate less than six months from receiving injury or four months if NAT HCV is performed. Additional TGA criteria NB: The TGA criteria mandate that it does not matter whether or not the inoculation injury involved a person with a communicable disease. The six month exclusion period for these increased risks (mucosal splash with blood, needle stick injury, tattoo, body piercing) is mandated by the TGA. Additional Information The following are all considered to have similar risk of acquiring a blood borne transmissible infection: Human bites, mucosal splash with blood, needle stick injury, tattoo, body piercing. Irritable Bowel Syndrome: Action Accept Additional Information Irritable bowel syndrome (IBS, or spastic colon) is a symptom-based diagnosis characterized by chronic abdominal pain, discomfort, bloating, and alteration of bowel habits. As a functional gastrointestinal disorder, IBS has no known organic cause. Japanese Encephalitis: Action Accept for corneas (not sclera) after clinical opinion that the infection has resolved. See Arthropod Borne Encephalitis (Arbovirus infection) Return to Table of Contents 50 Additional Information Japanese encephalitis is a disease caused by the mosquito-borne Japanese encephalitis virus from the family Flaviviridae. Domestic pigs and wild birds (herons) are reservoirs of the virus; transmission to humans may cause severe symptoms although the disease is asymptomatic in the vast majority of infections. The disease is most prevalent in Southeast Asia and the Far East. Signs which develop during the acute encephalitic stage include neck rigidity, cachexia, hemiparesis, convulsions and a raised body temperature between 38–41 °C and eventually coma. Japanese Encephalitis Immunisation: See Immunisation – Live Jaundice: Action Accept: a) If the cause of the jaundice was non-viral (this includes but is not limited to gall stones and drug reactions), accept. b) If due to Gilbert’s Syndrome, accept Must not donate if: a) History of jaundice (and reason unknown) b) If the cause of the jaundice was viral see the specific entry for that condition See if relevant Gall Bladder Disease Gilbert’s Syndrome Hepatitis A Hepatitis B Hepatitis C Hepatitis E Additional Information Many things can cause jaundice. The concern is with infectious causes. Kala-Azar: See Leishmaniasis Keratoconjunctivitis Sicca: Action Do not donate in severe cases. Need to determine the severity of the condition to arrive at an estimation of how much the sicca may have affected the cornea. Accept in less severe cases, or those where the history is not clear. Accept for donation but with the knowledge that the corneas may have some pathology and therefore be less likely to be suitable for donation. These pathologies, if present, will be detected post-donation by the eye bank. Return to Table of Contents 51 Additional Information Keratoconjunctivitis sicca (KCS), also called keratitis sicca, xerophthalmia or dry eye syndrome is an eye disease caused by eye dryness, which, in turn, is caused by either decreased tear production or increased tear film evaporation. Typical symptoms of keratoconjunctivitis sicca are dryness, burning and a sandy-gritty eye irritation that gets worse as the day goes on. Usually both eyes are affected. Having dry eyes for a while can lead to tiny abrasions across the epithelium of the cornea. In advanced cases, the epithelium undergoes pathologic changes to squamous metaplasia and loss of goblet cells. Some severe cases result in thickening of the corneal surface, corneal erosion, punctate keratopathy, epithelial defects, corneal ulceration, corneal neovascularization, corneal scarring, corneal thinning, and even corneal perforation. Causes include idiopathic, congenital alacrima, xerophthalmia, lacrimal gland ablation, and sensory denervation. In rare cases, it may be a symptom of collagen vascular diseases, including rheumatoid arthritis, Wegener's granulomatosis, and systemic lupus erythematosus. Sjögren's syndrome and autoimmune diseases associated with Sjögren's syndrome are also conditions associated with aqueous tear deficiency. In addition, certain drugs such as isotretinoin, sedatives, diuretics, tricyclic antidepressants, antihypertensives, oral contraceptives, antihistamines, nasal decongestants, beta-blockers, phenothiazines, atropine, and pain relieving opiates such as morphine can cause or worsen this condition. Kidney Disease: Includes Acute Nephritis Chronic Nephritis Chronic Renal Failure Action Accept if not immunosuppressed If immunosuppressed see entry “Immunosuppression” See if relevant Immunosuppression Kidney Recipient: See Tissue and Organ Recipients Immunosuppression Klinefelter’s Syndrome: Action Accept See Congenital Disorders Additional Information Klinefelter syndrome or Klinefelter's syndrome, is the set of symptoms resulting from additional X genetic material in males. Also known as 47,XXY or XXY , is a genetic disorder in which there is at least one extra X chromosome to a standard human male karyotype, for a total of 47 chromosomes rather than the 46 found in genetically normal humans. This chromosome constitution (karyotype) exists in roughly between 1:500 to 1:1000 live male births. It is the most common sex chromosome aneuploidy in males and the second most common condition caused by the presence of extra chromosomes. Return to Table of Contents 52 Laser Treatment (Surgery): Action Accept if applied to other parts of the body (eg. for skin lesions, cosmetic purposes). Accept if laser on eyes was: a) for retinal surgery b) for cataract surgery or related Accept only for posterior keratoplasty purposes: c) if laser treatment was refractive surgery to the cornea (and now healed) If laser treatment was related to malignancy, see Malignancy See if relevant Refractive Surgery Additional Information Laser treatment (other than refractive laser surgery to the cornea) does not affect the cornea. This will include laser treatment used in diabetic retinopathy, macular degeneration and that used in cataract surgery. Laser treatment for refractive errors involves only the anterior segment of the cornea and will not affect the posterior segment. Corneas that have undergone refractive surgery should not be used for penetrating keratoplasty or anterior keratoplasty due to concerns about structural integrity of the cornea. However, these corneas may be suitable for posterior keratoplasty although there needs to be an awareness of the refractive procedure to avoid any concerns or complications during the donor cornea cutting process. Lassa Fever: Action Must not donate is less than 12 months following recovery. Additional Information Lassa fever or Lassa hemorrhagic fever (LHF) is an acute viral hemorrhagic fever caused by the Lassa virus and first described in 1969 in the town of Lassa, in Borno State, Nigeria. The Lassa Fever is a member of the arenaviridiae virus family and is similar to Ebola. The virus is probably transmitted by contact with the faeces or urine of animals accessing grain stores in residences (particularly rodents). Further human-to-human transmission is thought to be via infected faeces or urine. Given its high rate of incidence, lassa fever has become a major problem in the African region killing about 5,000 people per annum (15-20% fatal). Legionnaire’s Disease: See Infection – Acute Additional Information Legionnaires disease is a potentially fatal pneumonia caused most commonly by Legionella pneumophila, a gram negative, aerobic bacteria that is carried by amoeba that thrive in stagnant water. Leishmaniasis: Includes Kala-azar Action Accept for corneas not for sclera Return to Table of Contents 53 Additional Information Leishmaniasis is a disease caused by protozoan parasites that belong to the genus Leishmania and is transmitted by the bite of certain species of sand fly. Most forms of the disease are transmissible only from non-human animals (zoonosis), but some can be spread between humans. Cutaneous leishmaniasis is the most common form, which causes a sore at the bite site, which heals in a few months to a year, leaving an unpleasant-looking scar. As corneas are avascular there is not considered to be a risk of transmitting protozoal infections of this type. Leptospirosis: See Infection – Acute Additional Information Leptospirosis (also known as Weil's syndrome, canicola fever, canefield fever, nanukayami fever, 7-day fever, Rat Catcher's Yellows, Fort Bragg fever, black jaundice, and Pretibial fever, is caused by infection with bacteria of the genus Leptospira and affects humans as well as other animals. It is among the world's most common diseases transmitted to people from animals. The infection is commonly transmitted to humans by allowing water that has been contaminated by animal urine to come in contact with unhealed breaks in the skin, the eyes, or with the mucous membranes. The disease is not known to be spread from person to person. Symptoms of leptospirosis include high fever, severe headache, chills, muscle aches, and vomiting, and may include jaundice, red eyes, abdominal pain, diarrhoea, and rash. Initial presentation may resemble pneumonia. The symptoms in humans appear after a 4–14 day incubation period. More severe manifestations include meningitis, extreme fatigue, hearing loss, respiratory distress, azotaemia, and renal interstitial tubular necrosis, which results in renal failure and occasionally liver failure. Occupations at risk include veterinarians, slaughterhouse workers, farmers, sewer maintenance workers, waste disposal facility workers, land surveyors and people working on derelict buildings. Slaughterhouse workers may contract the disease through contact with infected blood or body fluids. Rowers, kayakers and canoeists are also sometimes known to contract the disease. Lesbian: Action Accept Return to Table of Contents 54 Leukaemia: Includes (but not limited to) Acute lymphoblastic leukaemia (ALL). Chronic lymphocytic leukaemia (CLL). Acute myeloid leukaemia (AML). Chronic myeloid leukaemia (CML). Hairy cell leukaemia (HCL). Adult T-cell leukemia is caused by human T-lymphotropic virus (HTLV). Action Must not donate See Malignancy Additional Information Leukaemia is a type of cancer of the blood or bone marrow characterised by an abnormal increase of immature white blood cells called "blasts". Leukaemia is a broad term covering a spectrum of diseases. The following list is not comprehensive: • Acute lymphoblastic leukaemia (ALL) is the most common type of leukaemia in young children. • Chronic lymphocytic leukaemia (CLL) most often affects adults over the age of 55. It sometimes occurs in younger adults, but it almost never affects children. • Acute myeloid leukaemia (AML) occurs more commonly in adults than in children. • Chronic myeloid leukaemia (CML) occurs mainly in adults. • Hairy cell leukaemia (HCL) is sometimes considered a subset of chronic lymphocytic leukaemia, but does not fit neatly into this pattern. About 80% of affected people are adult men. No cases in children have been reported. • T-cell prolymphocytic leukaemia (T-PLL) is a very rare and aggressive leukaemia. • Large granular lymphocytic leukaemia may involve either T-cells or NK cells, it is a rare and indolent (not aggressive) leukaemia. • Adult T-cell leukaemia is caused by human T-lymphotropic virus (HTLV), a virus similar to HIV. Human T cell lymphotropic virus types I and II (HTLV-I/II) are endemic in certain areas of the world. The eye/corneal donation contraindication appears to be for two reasons: a) Wide dissemination of the malignancy throughout the body, spread through the blood and lymphatic system b) Viruses have also been linked to some forms of leukaemia. Experiments on mice and other mammals have demonstrated the relevance of retroviruses in leukaemia, and human retroviruses have also been identified. The first human retrovirus identified was human T-lymphotropic virus, or HTLV-1, which is known to cause adult T-cell leukaemia. Virus aetiology relates to the virus damaging the cell or creating DNA mutations, either of which can trigger leukaemia by activating oncogenes or deactivating tumour suppressor genes, and thereby disrupting the regulation of cell death, differentiation or division. Listeriosis (Listeria infection): See Infection – Acute Return to Table of Contents 55 Additional Information Listeriosis is a bacterial infection most commonly caused by Listeria monocytogenes. It primarily causes infections of the central nervous system (meningitis, meningoencephalitis, brain abscess, cerebritis) and bacteraemia in those who are immunocompromised, pregnant women, and those at the extremes of age (newborns and the elderly). It can also result in gastroenteritis in healthy persons who have ingested a large inoculum of the organism. Listeria is ubiquitous in the environment and is primarily transmitted via the oral route after ingestion of contaminated food products, after which the organism penetrates the intestinal tract to cause systemic infections. Lyme Disease: See Borreliosis Lyssavirus: See Australian Bat Lyssavirus Lymphogranuloma Venereum: Action Must not donate See Chlamydia Additional Information Lymphogranuloma venereum (also known as Climatic bubo, Durand–Nicolas–Favre disease, Poradenitis inguinale, and Strumous bubo) is a sexually transmitted disease caused by Chlamydia trachomatis. Lymphoma: Includes Hodgkin’s Disease Action Must not donate Additional Information Lymphoma is a type of haematological neoplasm that occurs when B or T lymphocytes, divide faster than normal cells or live longer than they are supposed to. Lymphoma may develop in the lymph nodes, spleen, bone marrow, blood or other organs which eventually presents as a solid tumour. Lymphomas are closely related to lymphoid leukaemias. Macular Degeneration: See Age-related Macular Degeneration Malaria: Action Accept for corneas not for sclera Additional Information As corneas are avascular there is not considered to be a risk of transmitting protozoal infections. Cases of malaria transmission (by transfusion) have occurred many years after the donor was last at risk of becoming infected with malaria. This is mainly a problem in people who have had repeated episodes of infection with malaria, although this is uncommon. This means that is it safer to test for malaria antibodies rather than wait a specific length of time. If malaria is confirmed in the donor, (as distinct risk) caution should perhaps be exercised for cornea donation. Return to Table of Contents 56 Malignancy: Action Accept for all cornea and sclera – except the below exclusions Must not donate if: a) Haematological malignancy b) Malignant tumour of anterior segment c) Disseminated melanoma d) Retinoblastoma Any malignancy is an exclusion for kerato-limbal allograft use. See if relevant Basal Cell Carcinoma Immunosuppression Malignant Melanoma Additional Information Many malignancies spread by invading surrounding tissues – thus the exclusion for malignancies of the eye. Tissue from donors with any malignancy (disseminated or not) is not suitable for kerato-limbal procedures. This is based on the kerato-limbal region being considered vascularised. Transmission of malignant melanoma from a donor with disseminated melanoma has been reported following a kerato-limbal allograft procedure. Risk assessment following this transmission resulted in disseminated melanoma being an exclusion criteria for any form of transplantation. This does not exclude donors with diagnosed localised (non-disseminated) malignant melanomas that have been resected. Cornea and sclera are suitable for use from donors with non-melanoma malignancies as long as they are not local to the anterior segment. Viruses that can be spread by blood and tissues can also cause some malignancies – thus the exclusion of haematological malignancies. NB: Some (UK, EEBA) exclude the use of sclera when any malignancy is present Malignant (Disseminated) Melanoma: Action Must not donate. Ocular Tissue from donors with any history of disseminated melanoma may not be released for any surgical use. See Malignancy Additional Information Transmission of malignant melanoma from a donor with disseminated melanoma has been reported following a kerato-limbal allograft procedure. Risk assessment following this transmission resulted in disseminated melanoma being an exclusion criteria for any form of transplantation. This does not exclude donors with diagnosed localised (non-disseminated) malignant melanomas that have been resected. Cornea and sclera are suitable for use from donors with non-melanoma malignancies as long as they are not local to the anterior segment. Marburg Fever: Action See Viral Haemorrhagic Fever. See Ebola Infection - Tropical Infection - Acute Return to Table of Contents 57 Additional Information Marburg virus disease (MVD) is the name for the human disease caused by any of the two marburgviruses Marburg virus (MARV) and Ravn virus (RAVV). MVD is a viral haemorrhagic fever (VHF), and the clinical symptoms are indistinguishable from Ebola virus disease (EVD). The natural maintenance hosts of marburg viruses remain to be identified unequivocally. However, the isolation of both MARV and RAVV from bats and the association of several MVD outbreaks with bat-infested mines or caves strongly suggests that bats are involved in marburg virus transmission to humans. Avoidance of contact with bats and abstaining from visits to caves is highly recommended. Marfan’s Syndrome: Action Must not donate Additional Information Marfan's syndrome is a genetic disorder of the connective tissue. People with Marfan tend to be unusually tall, with long limbs and long, thin fingers. Marfan syndrome can also seriously affect the eyes and vision. Nearsightedness and astigmatism are common, but farsightedness can also result. Subluxation (dislocation) of the crystalline lens in one or both eyes (ectopia lentis) (in 80% of patients) also occurs. Sometimes eye problems appear only after the weakening of connective tissue has caused detachment of the retina. Early onset glaucoma can be another related problem. Measles: See Infection – Acute Infectious Diseases – Contact with Additional Information Measles is an infection of the respiratory system caused by a paramyxovirus of the genus Morbillivirus. It is spread through respiration (contact with fluids from an infected person's nose and mouth, either directly or through aerosol transmission), and is highly contagious—90% of people without immunity sharing living space with an infected person will catch it. Complications with measles are relatively common, ranging from mild and less serious complications such as diarrhea to more serious ones such as pneumonia, otitis media, acute encephalitis (and very rarely SSPE – subacute sclerosing panencephalitis), and corneal ulceration (leading to corneal scarring). Complications are usually more severe in adults who catch the virus. Measles Immunisation: Measles Mumps Rubella (MMR) Immunisation: Measles Rubella Immunisation: See Immunisation – Live Meniere’s Disease: Action Accept Additional Information Ménière's disease is a disorder of the inner ear that can affect hearing and balance to a varying degree. It is characterised by episodes of vertigo, low-pitched tinnitus, and hearing loss. Ménière's disease is idiopathic, but it is believed to be linked to endolymphatic hydrops, an excess of fluid in the inner ear. Meningitis: Action Accept: a) if caused by bacterial infection and the corneas are to be preserved by organ culture; b) if caused by non-infectious disorders. Return to Table of Contents 58 See Infection – Acute Discretionary Additional Information Meningitis is inflammation of the protective membranes covering the brain and spinal cord, known collectively as the meninges. The inflammation may be caused by infection with viruses, bacteria, or other microorganisms. Bacterial The types of bacteria that cause bacterial meningitis vary according to the infected individual's age group. Viral Viruses that cause meningitis include enteroviruses, herpes simplex virus type 2 (and less commonly type 1), varicella zoster virus, mumps virus, HIV, and LCMV. Fungal There are a number of risk factors for fungal meningitis, including the use of immunosuppressants (such as after organ transplantation), HIV/AIDS, and the loss of immunity associated with aging. Parasitic A parasitic cause is often assumed when there is a predominance of eosinophils in the CSF. The most common parasites implicated are Angiostrongylus cantonensis, Gnathostoma spinigerum, Schistosoma, as well as the conditions cysticercosis, toxocariasis, baylisascariasis, paragonimiasis, and a number of rarer infections and noninfective conditions. Non-infectious Meningitis may occur as the result of several non-infectious causes: spread of cancer to the meninges (malignant or neoplastic meningitis) and certain drugs (mainly non-steroidal anti-inflammatory drugs, antibiotics and intravenous immunoglobulins). It may also be caused by several inflammatory conditions, such as sarcoidosis (which is then called neurosarcoidosis), connective tissue disorders such as systemic lupus erythematosus, and certain forms of vasculitis. Meningococcal Meningitis Immunisation: See Immunisation Non-live Mental Health Problems: Action Accept If being treated or identified for depression, or bipolar disorder (manic-depression). If the donor has had a new mental health problem within the last 12 months, or their condition has rapidly deteriorated in the last 12 months, further investigation of possible underlying causes and differential diagnoses are appropriate. Return to Table of Contents 59 Additional Information Many people have mental health problems that can be controlled with regular medication. There is no reason why they cannot donate whether on medication or not provided a firm diagnosis has been made and their condition has not deteriorated in the last 12 months. It is important to exclude other central nervous system disease including prion disease and rabies, which could present as new or deteriorating mental health problems. Note for Schizophrenia. Schizophrenia is a mental disorder characterized by a breakdown of thought processes and by a deficit of typical emotional responses. Common symptoms are delusions and disorganized thinking including auditory hallucinations, paranoia, bizarre delusions, disorganized speech, and it is accompanied by significant social or occupational dysfunction. Diagnosis is based on observed behaviour and the patient's reported experiences. Further investigation as to the mental history of the potential donor and the exclusion of differential diagnoses of an insidious nature are warranted. Middle East Respiratory Syndrome coronavirus (MERS-CoV): Action Must not donate if: a) Less than 21 days from leaving Saudi Arabia, the United Arab Emirates (UAE), Qatar, Oman, Jordan, Kuwait, Lebanon or Yemen b) Less than 21 days from last contact with a person with MERS c) Less than 3 months since recovery form MERS or possible MERS Accept if more than 21 days has passed since return from a MERS endemic area, or from last contact with a person affected by MERS and the donor has remained well Additional Information Authorities currently recommend Infection control for managing suspected, probable and confirmed cases to be consistent with those recommended for SARS-CoV and pandemic influenza. In the absence of recognised donation advice (as of February 2015) to deferral periods above are the same as those for SARS. As of 10 November 2015, the World Health Organization (WHO) global case count for MERS was 1,618 laboratory-confirmed cases, including at least 579 deaths (case fatality rate 36%) since the first cases were reported in September 2012. All cases of MERS world-wide have had a history of residence in or travel to the Middle East (mainly Saudi Arabia, and United Arab Emirates (UAE), Qatar, Oman, Jordan, Kuwait, Lebanon and Yemen) or contact with travellers returning from these areas, or can be linked to an initial imported case. There have been no cases in Australia or NZ. Camels are suspected to be the primary source of infection for humans, but the exact routes of direct or indirect exposure are not fully understood, and further studies (particularly case control studies) are needed. There is no evidence of ongoing community transmission in any country and only occasional instances of household transmission. Transmission in health care settings has been a feature of the outbreak. Sporadic infections have typically presented with, or later developed severe acute lower respiratory disease and this has predominantly occurred in adult males with certain underlying medical conditions. Mild or asymptomatic secondary infections have occurred in people of all ages, and have most frequently been associated with healthcare settings. Return to Table of Contents 60 Motor Neuron(e) Disease: Includes Amytrophic lateral sclerosis / Lou Gehrig's disease Progressive muscular atrophy Primary lateral sclerosis Progressive bulbar palsy Action Must not donate Additional Information The motor neuron diseases (MND) are a group of neurological disorders that selectively affect motor neurons, the cells that control voluntary muscle activity including speaking, walking, swallowing, and general movement of the body. They are generally progressive in nature, and cause increasingly debilitating disability and, eventually, death. The cause of non-familial forms is unknown. MRSA (Methicillin Resistant Staphylococcus Aureus): See if relevant Infection – General Additional Information Staphylococcus aureus is a widely occurring skin commensal organism. The carrier status or exposure of the donor is not relevant to donation. (MRSA) is a bacterium responsible for several difficult-to-treat infections in humans. The evolution of antibiotic resistance does not cause the organism to be more intrinsically virulent than strains of Staphylococcus aureus that have no antibiotic resistance, but resistance does make MRSA infection more difficult to treat with standard types of antibiotics and thus more dangerous. Multiple Resistant Organisms (MRO), Multiple Drug Resistance (MDR): See if relevant Infection – General MRSA Extended Spectrum Beta Lactamase Additional Information Multiple resistant organisms (MRO) , Multiple drug resistance (MDR), multi-drug resistance or multiresistance is a condition enabling disease-causing microrganisms (bacteria, viruses, fungi or parasites) to resist distinct antimicrobials, first and foremost antibiotics, but also antifungal drugs, antiviral medications, antiparasitic drugs, chemicals of a wide variety of structure and function targeted at eradicating the organism. The most common types are (usually bacteria): Vancomycin-Resistant Enterococci (VRE) Methicillin-Resistant Staphylococcus aureus (MRSA) Extended-spectrum β-lactamase (ESBLs) producing Gram-negative bacteria Klebsiella pneumoniae carbapenemase (KPC) producing Gram-negatives MultiDrug-Resistant gram negative rods (MDR GNR)MDRGN bacteria such as Enterobacter species, E.coli,Klebsiella pneumoniae, Acinetobacter baumannii, Pseudomonas aeruginosa Generally antibiotic resistance does not cause the organism to be more intrinsically virulent, but resistance does make infection more difficult to treat with standard types of antibiotics and thus more dangerous. Return to Table of Contents 61 Multiple Sclerosis: Action Must not donate Additional Information Multiple sclerosis (MS), also known as disseminated sclerosis or encephalomyelitis disseminata, is an inflammatory disease in which the insulating covers of nerve cells in the brain and spinal cord are damaged. This damage disrupts the ability of parts of the nervous system to communicate, resulting in a wide range of signs and symptoms, including physical, mental, and sometimes psychiatric problems. While the cause is not clear, the underlying mechanism is thought to be either destruction by the immune system or failure of the myelin-producing cells. Proposed causes for this include genetics and environmental factors such as infections. Thus the contraindication to donation. Mumps: See Infection – Acute Infectious diseases – contact with Mumps Immunisation: See Immunisation-Live Murray Valley encephalitis virus: Action Must not donate if within 12 months of last infection or recurrence See Arthropod Borne Encephalitis (Arbovirus infection) Additional Information Murray Valley encephalitis virus (MVEV) is a zoonotic flavivirus endemic to northern Australia and Papua New Guinea. It is the causal agent of Murray Valley encephalitis (previously known as Australian encephalitis) and in humans can cause permanent neurological disease or death. MVEV is a mosquito-borne virus that is maintained in a bird-mosquito-bird cycle. Water birds from the Ciconiiformes order, including herons and cormorants, provide the natural reservoir for MVEV. Muscular Dystrophy: Action Accept If the eyes are unaffected Additional Information Muscular Dystrophy (MD) is a group of muscle diseases that weaken the musculoskeletal system and hamper locomotion. They include Duchenne, Becker, limb-girdle, congenital, facioscapulohumeral, myotonic, oculopharyngeal, distal, and Emery-Dreifuss muscular dystrophies. Most are caused by a mutation of a gene located on the X chromosome and therefore predominantly affect males. Different muscular dystrophies follow various inheritance patterns. Most types of MD are multi-system disorders with manifestations in body systems including the heart, gastrointestinal system, nervous system, endocrine glands, eyes and brain. Any concern regarding eyes relates to pathology to the eye (or cornea) itself rather than any transmissible disease process. For example, droopy eyelids, possible ptosis or poor eyelid movement could create corneal problems. Myasthenia Gravis: Action Accept if the eyes are unaffected Return to Table of Contents 62 Additional Information Myasthenia gravis is an autoimmune neuromuscular disease leading to fluctuating muscle weakness and fatigue. It is associated with various autoimmune diseases, including Thyroid diseases (including Hashimoto's thyroiditis and Graves' disease), Diabetes mellitus type 1,Rheumatoid arthritis, Lupus, and Demyelinating CNS disease. The hallmark of myasthenia gravis is fatigability. Muscles become progressively weaker during periods of activity and improve after periods of rest. Muscles that control eye and eyelid movement, facial expressions, chewing, talking, and swallowing are especially susceptible. Indeed, in most cases, the first noticeable symptom is weakness of the eye muscles with possible ptosis. Myelodysplastic Syndrome: Action Must not donate Additional Information The myelodysplastic syndromes (MDS, formerly known as preleukaemia) are a diverse collection of haematological medical conditions that involve dysplasia of the myeloid class of blood cells. There is some risk for developing acute myelogenous leukaemia that is notoriously resistant to treatment. There is currently no definitive cause. Myeloproliferative Disease (Syndrome): Includes Chronic myelogenous leukemia (CML) Essential thrombocythemia (ET) Polycythemia vera (PV) Primary myelofibrosis (PMF) Action Must not donate Additional Information The myeloproliferative diseases (MPDs) or myeloproliferative neoplasms (MPNs) are a group of diseases of the bone marrow in which excess cells are produced. They are related to, and may evolve into, myelodysplastic syndrome and acute myeloid leukaemia. Myocarditis: Action Accept: a) If infection resolved. b) If the cause is bacterial and the corneas are to be stored by organ culture. Additional Information Myocarditis or inflammatory cardiomyopathy is inflammation of myocardium. A large number of causes of myocarditis have been identified, but often a cause cannot be found. In developed countries, viruses are common culprits. Worldwide, however, the most common cause is Chagas' disease, endemic to Central and South America. Myxoedema: See Thyroid Disease Return to Table of Contents 63 Needle-Stick Injury: See Inoculation Injury Nephritis: See Kidney Disease Neurofibromatosis: Action If related to malignancy see Malignancy Additional Information Neurofibromatosis refers to a number of inherited conditions that are clinically and genetically distinct and carry a high risk of tumour formation, particularly in the brain. It is an autosomal dominant disorder. Neurosurgery: Action Accept if the reason for surgery itself is not an exclusion Must not donate If human derived Dura mater was used during surgery Nonsteroidal Anti-Inflammatory Drugs (NSAID): Action Assess the reason for treatment and see relevant entry. Ocular Surgery: Action Accept if the procedure is unlikely to prejudice quality and outcome of transplant See Eye Disease Laser Treatment Malignancy Onchocerciasis (River Blindness): Action Must not donate Additional Information Onchocerciasis (River Blindness) is second in the world only to trachoma as an infectious cause of blindness. It is a form of subcutaneous filariasis is caused by Loa loa (the eye worm). It is not the nematode, but its endosymbiotic bacteria, Wolbachia pipientis, that causes the severe inflammatory response that leaves many blind. The parasite is transmitted to humans through the bite of a black fly of the genus Simulium. The larval nematodes spread throughout the body. When the worms die, their Wolbachia symbionts are released, triggering a host immune system response that can cause severe itching, and can destroy optical tissue in the eye. Orf: See Contagious Pustular Dermatitis Organ Recipient: See Tissue and Organ Recipients Osteoarthritis: Action Accept Osteogenesis Imperfecta: Action Should not donate Return to Table of Contents 64 Additional Information Osteogenesis Imperfecta (OI), also known as brittle bone disease or Lobstein syndrome is a congenital bone disorder characterised by brittle bones that are prone to fracture. It is as a result of defective connective tissue due to defects in the genes relating to production of Collagen I or other connective tissue proteins. Pathology includes bones that fracture easily, loose joints, poor muscle tone and thin, discoloured sclera. Corneas may be compromised. Osteomalacia: Action Accept Additional Information Osteomalacia is the softening of the bones caused by defective bone mineralization secondary to inadequate amounts of available phosphorus and calcium, or because of overactive resorption of calcium from the bone as a result of hyperparathyroidism - Such a donor would not be acceptable for bone donation. Osteomyelitis: Action Accept Additional Information Osteomyelitis is infection and inflammation of the bone or bone marrow. Sometimes it is difficult to be certain that all infection has been eliminated, and for bone donation an exclusion period of 2 years after completing treatment and cure is often applied. For eye donation one needs to consider the potential for systemic infection effecting the eyes – and if concerned then the corneas should be stored by organ culture. Osteoporosis: Action Accept Additional Information Bone banks would not accept for structural bone but they may accept for non-weight bearing bone. Ovarian Cyst: Action Accept Paget’s Disease of Bone: Includes Osteitis Deformans Action Accept Paratyphoid: Action Accept: a) if corneas are to be stored by organ culture, or b) if more than seven days from completion of antibiotic course and last symptoms Additional Information Paratyphoid fevers are a group of enteric illnesses caused by serotypic strains of the Salmonella genus of bacteria, S. Paratyphi. It is transmitted by the faecal-oral route (usually contaminated water or food) and do not have a chronic carrier state The paratyphoid bears similarities with typhoid fever, and the two are referred to by the common name enteric fever. The course of paratyphoid is more benign. Parkinson’s Disease: Action Must not donate Return to Table of Contents 65 Additional Information Parkinson's disease (PD also known as idiopathic or primary parkinsonism, hypokinetic rigid syndrome/HRS, or paralysis agitans) is a degenerative disorder of the central nervous system. The motor symptoms of Parkinson's disease result from the death of dopamine-generating cells in the substantia nigra, a region of the midbrain; the cause of this cell death is unknown. Parkinson's disease in most people is idiopathic. However a small proportion of cases can be attributed to known genetic factors. Genetic and pathological studies have revealed that various dysfunctional cellular processes, inflammation, and stress can all contribute to cell damage. In addition, abnormal clumps called Lewy bodies, which contain the protein alpha-synuclein, are found in many brain cells of individuals with Parkinson's disease. A proportion of patients with Parkinson's disease develop dementia. When dementia occurs in Parkinson's disease, the underlying cause may be dementia with Lewy bodies or Alzheimer's disease, or both. Other environmental risk factors include pesticide exposure, head injuries, and living in the country or farming. Smoking appears to confer some protective effect. A theory of an infectious aetiology is also gaining some traction. Encephalitis lethargica a pandemic of which swept the world in the 1920s, caused Parkinson’s-like symptoms in many sufferers. There is strong evidence that it was a virus that ultimately caused neurological symptoms appearing many years after the initial infection. The theory remains controversial. Parkinson’s disease is a contraindication because of its unknown cause, association with Lewy bodies as in forms of Alzheimer’s disease and CJD, and its potential for co-existence with, but disguising, more insidious progressive neurological disorders. However there is no direct evidence that Parkinson’s disease can be transmitted from human-to-human. Peptic Ulcer: See Gastric Ulcer Pericarditis: Action Accept: a) if caused by bacterial infection and the corneas are to be preserved by organ culture, or b) if a non-infectious cause See also Infection-Acute Additional Information Pericarditis is an inflammation of the pericardium. Infective causes include viral, bacterial, or fungal infection. Recent data suggest that the common viral forms are cytomegalovirus, herpesvirus, and HIV. Pneumococcus or tuberculous pericarditis are the most common bacterial forms. Fungal pericarditis is usually due to histoplasmosis, or in immunocompromised hosts Aspergillus, Candida, and Coccidioides. The most common worldwide cause of pericarditis is infectious pericarditis with Tuberculosis. Non-infectious causes include: Idiopathic, immunologic conditions including systemic lupus erythematosus or rheumatic fever, myocardial infarction (Dressler's syndrome), trauma, uremic pericarditis, malignancy, side effect of some medications, e.g. isoniazid, cyclosporine, hydralazine, warfarin, heparin, tetracyclines. It can also follow after CABG surgery. Peritonitis: See Infection-General Return to Table of Contents 66 Additional Information Peritonitis is an inflammation of the peritoneum, the thin tissue that lines the inner wall of the abdomen and covers most of the abdominal organs. Peritonitis may be localised or generalised, and may result from infection or from a non-infectious process. As a complication sepsis may develop, so blood cultures are often performed to monitor for this complication. Perforation of part of the gastrointestinal tract is the most common cause of peritonitis. Disruption of the peritoneum may also cause infection simply by letting micro-organisms into the peritoneal cavity. Examples include trauma, surgical wound, and continuous ambulatory peritoneal dialysis. Leakage of sterile body fluids into the peritoneum, such as blood, gastric juice, bile, urine or pancreatic juice (pancreatitis) can cause peritonitis. However, while these body fluids are sterile at first, they can become infected once they leak out of their organ, leading to infectious peritonitis. Permanent Makeup: See Body Piercing Pituitary Extract – Human: Includes Adrenocorticotrophic hormone, Follicle Stimulating Hormone, Gonadotrophin, Growth Hormone, Luteinising Hormone, Thyroid Stimulating Hormone Action Must not donate if has ever received Human Pituitary Extract See if relevant Growth Hormone Prion Associated Diseases Additional Information Human Pituitary Extracts have been contaminated with abnormal prions and have caused the transmission of CJD. Between 1967 and 1985 cadaver derived pituitary hormones were officially supplied to 1,976 Australians as treatment for infertility and short stature. In addition, 187 people are known to have been treated with human pituitary derived hormones in Australia as part of special projects associated with metabolic growth problems, and between 1972 and 1978 on some IVF programs. A number of people also received treatment outside these programs, as early as 1963. A total of 2,163 people were treated with human pituitary derived hormones in Australia during this time. Their use in Australia ceased in May 1985 after overseas reports of deaths from CJD in people who received treatment with human growth hormone. In Australia the first case of CJD in a pituitary hormone recipient was reported in 1988, a second case in 1990, a third and fourth cases in 1993 and a fifth case in 1995. All five individuals died between 1988 and 1991. Three of the five cases had confirmed CJD. There have been no further cases reported to date. New Zealanders received hormone made in the United States from 1963 to 1985. Six people possibly contracted CJD out of 159 people treated. Donors that have been given only synthetic pituitary hormones or gonadotrophin made from urine may be accepted. Plague (Yersinia Pestis): Action Must not donate unless 3 months after recovery Must not donate unless more than 2 weeks after last contact with an infected individual Return to Table of Contents 67 Additional Information Plague is a deadly infectious disease that is caused by the enterobacteria Yersinia pestis. Primarily carried by rodents (most notably rats) and spread to humans via fleas. Until June 2007, plague was one of the three epidemic diseases specifically reportable to the World Health Organization (the other two being cholera and yellow fever). Depending on lung infection, or sanitary conditions, plague also can be spread in the air, by direct contact, or by contaminated undercooked food or materials. The symptoms of plague depend on the concentrated areas of infection in each person: such as bubonic plague in lymph nodes, septicaemic plague in blood vessels, pneumonic plague in lungs, and so on. It is treatable if detected early. Plasma Dilution: Action Must not donate: a) if there is significant blood loss together with intra-venous infusions that it is estimated have resulted in more than 50% plasma dilution AND, b) if pre-transfusion/infusion sample is not available for testing Use the FULL available algorithms. See FDA Appendices 1-4 for procedure: /guidances/tissue/ucm073964.htm#APPENDIX1 UK Appendix 4 for procedure: Please note that both FDA and UK standards define the criteria for plasma dilution to be – following significant blood loss. Plasma dilution is not possible in the absence of significant blood loss. Platelet Disorder: Action See: Is there an entry for the condition? Accept if not covered by a specific entry See if relevant Haematological Disease Immune Thrombocytopenia Thrombocytosis Pleurisy: See if relevant Infection – General Malignancy Pneumococcal Immunisation: Includes Prevenar 13 Pneumovax 23 See Immunisation Non-live Pneumonia: See Infection – Acute Return to Table of Contents 68 Poisoning: Action Must not donate if there is evidence that the individual (donor/or mother of cord blood donor) has ingested or been otherwise exposed to toxic substances that could be transmitted in donated material in dosages that could endanger the health of recipients. If the individual is being monitored following exposure and the levels of the agent in question are within safe limits, accept. See if Relevant Addiction and Drug Abuse Poliomyelitis: Action Must not donate unless longer than 2 weeks after recovery Must not donate unless more than 6 weeks after last contact with an infected individual Additional Information Poliomyelitis, often called polio or infantile paralysis, is an acute, viral, highly contagious infectious disease spread from person to person, primarily via the faecal-oral and oral-oral (oropharyngeal) routes. Although approximately 90% of polio infections cause no symptoms at all, affected individuals can exhibit a range of symptoms if the virus enters the blood stream. In about 1% of cases, the virus enters the central nervous system, preferentially infecting and destroying motor neurons, leading to muscle weakness and acute flaccid paralysis. Polio (Poliomyelitis) Immunisation: See For Oral immunisation – Immunisation - Live For injected immunisation – Immunisation - Non-live Polycythaemia: Action Must not donate if primary polycythaemia (rubra versa) Accept if secondary polycythaemia (but only secondary to a non-malignant condition) Return to Table of Contents 69 Additional Information Polycythaemia is a disease state in which the proportion of blood volume that is occupied by red blood cells increases. Primary Polycythaemia: Primary polycythemias are due to factors intrinsic to red cell precursors. Polycythemia vera (PCV), polycythemia rubra vera (PRV), or erythremia, occurs when excess red blood cells are produced as a result of an abnormality of the bone marrow. Often, excess white blood cells and platelets are also produced. Polycythemia vera is classified as a myeloproliferative disease. Secondary Polycythaemia: Secondary polycythemia is caused by either natural or artificial increases in the production of erythropoietin, hence an increased production of erythrocytes. Conditions which may result in a physiologically appropriate polycythemia include: • Altitude related • Hypoxic disease-associated - for example in cyanotic heart disease where blood oxygen levels are reduced significantly. • Iatrogenic - Secondary polycythemia can be induced directly by phlebotomy (blood letting) to withdraw some blood, concentrate the erythrocytes, and return them to the body. • Genetic - Heritable causes of secondary polycythemia also exist but are uncommon. Conditions where the secondary polycythemia is not as a result of physiologic adaptation and occurs irrespective of body needs include: • Neoplasms - Renal-cell carcinoma or liver tumors, von Hippel-Lindau disease, and endocrine abnormalities including pheochromocytoma and adrenal adenoma with Cushing's syndrome. • People whose testosterone levels are high because of the use of anabolic steroids, including athletes who abuse steroids. Prion Associated Diseases: Includes Sporadic, Familial and Variant Creutzfeldt-Jakob Disease (CJD), Gerstmann-Straussler-Scheinker Disease and Fatal Familial Insomnia Action Must not donate: 1. Diagnosed with any form of CJD, or other prion associated disorder 2. Identified at increased risk of developing a prion associated disorder Includes: a) Individuals at familial risk of prion-associated diseases (have had two or more blood relatives develop a prion-associated disease or had been informed they are at risk following genetic counselling) b) Recipients of dura mater grafts c) Recipients of human pituitary derived extracts Return to Table of Contents 70 Additional TGA criteria Must not donate: a) Individuals who have lived in or visited England, Scotland, Wales, Northern Ireland or the Isle of Man for a cumulative period of six months or more, between 1st January 1980 and 31st December 1996 inclusive b) individuals who have received a transfusion or injection of blood or blood components while in England, Scotland, Wales, Northern Ireland or the Isle of Man at any time since 1st January 1980 onwards This is the “UK exclusion rule” for risk of variant CJD See if relevant Pituitary Extract Tissue and Organ Recipients Transfusion Prisons & Psychiatric Institutions: Action Must not donate if donor has been held in prison or psychiatric institution during the previous 6 months (4 months HCV NAT). Accept if held in custody for a period of 72 hours or less. Similarly for short-term psychiatric episodes Additional TGA criteria The TGA mandates 12 months exclusion period Additional Information The purpose behind the deferral period is that incarceration increases the likelihood of “at risk behaviours” for acquiring infectious diseases that can be transmitted by blood, cells or tissues. Risk behaviour is unlikely to have occurred while held in police custody under police powers of arrest. Proctitis: Action Accept: a) if related to inflammatory bowel disease and there is no evidence of ocular involvement and the corneas are to be preserved by organ culture b) If due to other causes Must not donate if due to sexually transmitted disease. Additional Information Proctitis is an inflammation of the anus and the lining of the rectum, affecting only the last 6 inches of the rectum. Proctitis has many possible causes. It often occurs idiopathically. Other causes include 1. damage by irradiation (for example in radiation therapy for cervical cancer and prostate cancer) 2. as a sexually transmitted infection (usually through anal intercourse), as in lymphogranuloma venereum, herpes proctitis, gonorrhoea, syphilis and chlamydia. 3. Due to ulcerative colitis and Crohn’s disease (inflammatory bowel disease) Progressive Multifocal Leukoencephalaopathy: Action Must not donate Return to Table of Contents 71 Additional Information Progressive multifocal leukoencephalopathy (PML), also known as progressive multifocal leukoencephalitis, is a rare and usually fatal viral disease characterized by progressive damage (-pathy) or inflammation of the white matter (leuko-) of the brain (-encephalo-) at multiple locations (multifocal). It occurs almost exclusively in people with severe immune deficiency, such as transplant patients on immunosuppressive medications, those receiving certain kinds of drug therapy for multiple sclerosis and Hodgkin's Lymphoma, or those with AIDS. The cause of PML is a type of polyomavirus called the JC virus (JCV), after the initials of the patient from whose tissue the virus was first successfully cultured. Recent publications indicate 39% to 58% of the general population are seropositive for antibodies to JCV, indicating current or previous infection with virus. The reason for the exclusion is the differential diagnosis of a prion disease and also the populations in which it most frequently occurs. Psoriasis: Action Accept If no ocular surface disease See Autoimmune Disorders Additional Information Etretinate was approved by the FDA in 1986 to treat severe psoriasis. It was subsequently removed from the market in the late 1990s due to the high risk of birth defects. It has an extremely long half-life and tissue other than eyes must not be donated. Acitretin and Isotretinoin can cause birth defects but have a lower half-life than Etretinate. Tissues other than eyes should not be donated if less than 24 months from last dose of Acitretin and 4 weeks from last dose of Isotretinoin. Psoriasis is primarily a skin condition caused by an autoimmune process. About one in ten people with psoriasis may develop joint problems (psoriatic arthropathy). Sometimes the disease is treated with powerful drugs to suppress the underlying autoimmune response. This may alter the body’s defense mechanisms to infection. If concerned the corneas should be preserved by organ culture. Psychiatric Problems: See Mental Health Problems Prisons & Psychiatric Institutions Pyelonephritis: Action Accept: a) if the corneas are to be preserved by organ culture and there are concerns about developing systemic infection b) if no concerns about systemic infection See Infection – General Additional Information Pyelonephritis is an ascending urinary tract infection that has reached the pyelum or pelvis of the kidney. Severe cases of pyelonephritis can lead to pyonephrosis (pus accumulation around the kidney), urosepsis (a systemic inflammatory response of the body to infection), kidney failure and even death. Return to Table of Contents 72 Pyruvate Kinase Deficiency: Action Accept Additional Information Pyruvate kinase deficiency, also called erythrocyte pyruvate kinase deficiency, is an inherited metabolic disorder of the enzyme pyruvate kinase which affects the survival of red blood cells and causes them to deform into echinocytes on peripheral blood smears. It is the second most common cause of enzyme-deficient haemolytic anaemia, following G6PD deficiency. Most affected individuals do not require treatment. Q Fever: Action Must not donate Additional Information Q fever is a disease caused by infection with Coxiella burnetii, a bacterium that affects humans and other animals. This organism is uncommon, but may be found in cattle, sheep, goats and other domestic mammals, including cats and dogs. The infection results from inhalation of a spore-like small cell variant, and from contact with the milk, urine, faeces, vaginal mucus, or semen of infected animals. It was first described by Edward Holbrook Derrick in abattoir workers in Brisbane, Queensland, Australia. The "Q" stands for "query" and was applied at a time when the causative agent was unknown; it was chosen over suggestions of "abattoir fever" and "Queensland rickettsial fever", to avoid directing negative connotations at either the cattle industry or the state of Queensland. The pathogen of Q fever was discovered in 1937, when Frank Macfarlane Burnet and Mavis Freeman isolated the bacterium from one of Derrick’s patients. The pathogenic agent is to be found everywhere worldwide except New Zealand. The bacterium is extremely sustainable and virulent: a single organism is able to cause an infection. The United States investigated Q fever as a potential biological warfare agent in the 1950s, with eventual standardization as agent OU. "At risk" occupations include, but are not limited to: • veterinary personnel • stockyard workers • farmers • shearers • animal transporters • laboratory workers handling potentially infected veterinary samples or visiting abattoirs • people who cull and process kangaroos • hide (tannery) workers Treatment of the acute Q fever with antibiotics is very effective but the chronic form is more difficult to treat and can require up to four years of treatment. The chronic form can manifest itself as an endocarditis and is usually fatal if untreated. Rabies: Action Must not donate Immunisation Post-exposure: Must not donate until at least 12 months post exposure and fully cleared by treating physician. Accept immunisation pre-exposure See Immunisation - Non-live Return to Table of Contents 73 See if relevant Bite Additional Information Rabies is a viral disease that causes acute encephalitis in warm-blooded animals. The disease is zoonotic and almost invariably fatal if postexposure prophylaxis is not administered prior to the onset of severe symptoms. Rabies causes about 55,000 human deaths annually worldwide. 95% of human deaths due to rabies occur in Asia and Africa. Roughly 97% of human rabies cases result from dog bites. The British Isles, New Zealand and Australia are virtually the only areas worldwide where rabies is not found. It is the most common form of disease transmission by organ, tissue or eye donors. Early stage symptoms of malaise, headache and fever can be confused with the flu (and therefore often overlooked). The incubation period of the disease is usually a few months in humans, depending on the distance the virus must travel to reach the central nervous system - It is extremely important to consider foreign travel and any history of bites or scratches from animals so as to identify a risk of rabies as distinct from flu-like symptoms. Failure to do this in the past has been the key factor in the transmission of the disease from donor to recipient. Radiation Therapy: Action Accept if the eyes were not included in the field of irradiation If related to malignancy, see Malignancy See if relevant Basal Cell Carcinoma Malignancy Raynaud’s Syndrome: Action Consider if part of a multisystem disorder and See if there is an Entry See if relevant Systemic Lupus Erythematosus Additional Information Terminology is important: Raynaud's syndrome (secondary Raynaud's), is where Raynaud’s phenomenon (see below), is caused by a known instigating factor, most commonly connective tissue disorders such as systemic lupus erythematosus. Raynaud's phenomenon is excessively reduced blood flow in response to cold or emotional stress, causing discoloration of the fingers, toes, and occasionally other areas. Raynaud's phenomenon includes Raynaud's disease (also known as primary Raynaud's phenomenon) where the cause is unknown, and Raynaud's syndrome (secondary Raynaud's) discussed above. Refractive Error (Vision): Action Accept See if relevant Glasses Laser Treatment Refractive Surgery Return to Table of Contents 74 Additional Information A refractive error, or refraction error, is an error in the focusing of light by the eye and a frequent reason for reduced visual acuity. Types of refractive error: Myopia: When the optics are too powerful for the length of the eyeball one has myopia or nearsightedness. This can arise from a cornea with too much curvature (refractive myopia) or an eyeball that is too long (axial myopia). Myopia can easily be corrected with a concave lens which causes the divergence of light rays before they reach the retina. Hyperopia: When the optics are too weak for the length of the eyeball, one has hyperopia or farsightedness. This can arise from a cornea with not enough curvature (refractive hyperopia) or an eyeball that is too short (axial hyperopia). This can be corrected with convex lenses which cause light rays to converge prior to hitting the retina. Astigmatism: People with a simple astigmatic refractive error see contours of a particular orientation as blurred, but see contours with orientations at right angles as clear. When one has a cylindrical error, one has astigmatism. This is caused by a deviation in the shape of the cornea, a shape other than spherical. This defect can be corrected with refracting light more in one area of the eye than the other. Cylindrical lenses serve this purpose. Presbyopia: When the flexibility of the lens declines typically due to age. Individual would experience difficulty in reading etc. This causes the individual to need visual assistance such as bifocal lenses. Only Astigmatism involves the cornea but the degree of corneal shape deviation is usually not great enough to affect suitability for transplantation. Refractive Surgery: Action Must not donate if: a) Radial keratotomy (Russian operation) b) Intra-stromal rings (Intac surgery) Accept for only endothelial keratoplasty (posterior corneal transplantation) laser eye surgery for refractive error: a) Photorefractive keratectomy (PRK) b) Laser-Assisted Sub-Epithelial Keratectomy (or Laser Epithelial Keratomileusis) (LASEK or LASIK) Return to Table of Contents 75 Additional Information Radial keratotomy (RK) Is a refractive surgical procedure to correct myopia that was developed in 1974, by Svyatoslav Fyodorov, a Russian ophthalmologist. It has been largely supplanted by newer operations, such as photorefractive keratectomy and LASIK. In RK, incisions are made with a diamond knife. Incisions are made quite deep (up to 90% of corneal thickness). Thus these corneas are not even suitable for endothelial keratoplasty. Intrastromal corneal rings (or intracorneal rings) Are small crescent or semi-circular shaped ring segments between the layers of the corneal stroma to correct vision. Considering their placement in the cornea they make the cornea unsuitable for transplant purposes. Laser surgery Involves only the anterior segment of the cornea and will not affect the posterior segment. Corneas that have undergone refractive surgery should not be used for penetrating keratoplasty or anterior keratoplasty due to concerns about structural integrity of the cornea. However, these corneas should be suitable for posterior keratoplasty although there needs to be an awareness of the refractive procedure to avoid any concerns or complications during the donor cornea cutting process. Renal Disease: See Kidney Disease Respiratory Disease: Includes Asthma Chronic Obstructive Airways Disease Cystic Fibrosis See if relevant Infection – General Steroid Therapy Retinitis Pigmentosa: Action Accept Additional Information Retinitis pigmentosa (RP) is an inherited, degenerative eye disease that causes severe vision impairment and often blindness. RP is caused by abnormalities of the photoreceptors (rods and cones) or the retinal pigment epithelium (RPE) of the retina leading to progressive sight loss. It does not affect the cornea or sclera of the eye and thus donation is acceptable. Reyes Syndrome: Action Must not donate If more than 3 months after recovery accept Return to Table of Contents 76 Additional Information Reye's syndrome is a potentially fatal disease that has numerous detrimental effects to many organs, especially the brain and liver, as well as causing hypoglycaemia. The classic features are a rash, vomiting, and liver damage. The disease causes fatty liver with minimal inflammation and severe encephalopathy. The exact cause is unknown and, while it has been associated with aspirin consumption by children with viral illness, it also occurs in the absence of aspirin use. Rheumatic Fever: Action Accept Additional Information Rheumatic fever is an inflammatory disease that occurs following a Streptococcus pyogenes infection, such as streptococcal pharyngitis. Believed to be caused by antibody cross-reactivity that can involve the heart, joints, skin, and brain, the illness typically develops two to three weeks after a streptococcal infection. The illness is so named because of its similarity in presentation to rheumatism. Rheumatoid Arthritis: Action Accept if no ocular involvement Additional Information Rheumatoid arthritis (RA) is an autoimmune disease that results in a chronic, systemic inflammatory disorder that may affect many tissues and organs, but principally attacks flexible (synovial) joints. The eye can be affected in the form of episcleritis which when severe can very rarely progress to perforating scleromalacia. Rather more common is the indirect effect of keratoconjunctivitis sicca, which is a dryness of eyes and mouth caused by lymphocyte infiltration of lacrimal and salivary glands. When severe, dryness of the cornea can lead to keratitis or ulceration of the cornea. River Blindness: See Onchocerciasis Ross River Fever: See Infection – Acute Arthropod Borne Encephalitis (Arbovirus) Additional Information Ross River Fever is a mosquito-borne infectious disease caused by infection with the Ross River virus. The illness is typically characterised by an influenza-like illness and polyarthritis. The virus is endemic to Australia, Papua New Guinea, Fiji, Samoa, the Cook Islands, New Caledonia and several other islands in the South Pacific. The virus is not contagious and is spread only by mosquitoes. The main reservoir hosts are kangaroos and wallabies, although horses, possums and possibly birds and flying foxes play a role. Symptoms of the disease may vary widely in severity, but major indicators are arthralgia, arthritis, fever, and rash. The incubation period is 7–9 days. Rubella – Acute Infection: Action Must not donate until the infection is considered to have completely resolved See Infection – Acute Rubella - Congenital Return to Table of Contents 77 Additional Information Rubella, also known as German measles, is a disease caused by the rubella virus. The disease is often mild and attacks often pass unnoticed. The disease can last one to three days. Infection of the mother by Rubella virus during pregnancy can be serious; if the mother is infected within the first 20 weeks of pregnancy, the child may be born with congenital rubella syndrome (CRS), which entails a range of serious incurable illnesses. Spontaneous abortion occurs in up to 20% of cases. Acquired (i.e. not congenital) rubella is transmitted via airborne droplet emission from the upper respiratory tract of active cases. The virus may also be present in the urine, faeces and on the skin. There is no carrier state: the reservoir exists entirely in active human cases. The disease has an incubation period of 2 to 3 weeks. Rubella – Contact with: See Infection – Contact with Rubella – Congenital: Action Must not donate Additional Information Infection of the mother by Rubella virus during pregnancy can be serious; if the mother is infected within the first 20 weeks of pregnancy, the child may be born with congenital rubella syndrome (CRS), which entails a range of serious incurable illnesses. Spontaneous abortion occurs in up to 20% of cases. The classic triad for congenital rubella syndrome is: Sensorineural deafness (58% of patients) Eye abnormalities—especially retinopathy, cataract and microphthalmia (43% of patients) Congenital heart disease—especially patent ductus arteriosus (50% of patients) For organs and tissues other than eyes, and if there is no other contraindication, donation is acceptable. Rubella Immunisation: See Immunisation – Live Salpingitis: See if relevant Sexually transmitted diseases See Infection – General Additional Information Salpingitis is an infection and inflammation in the fallopian tubes. It is often used synonymously with pelvic inflammatory disease (PID), although PID lacks an accurate definition and can refer to several diseases of the female upper genital tract, such as endometritis, oophoritis, myometritis, parametritis and infection in the pelvic peritoneum Sanfilippo Syndrome: Action Do not donate See Congenital disorders Return to Table of Contents 78 Additional Information Sanfilippo syndrome, or Mucopolysaccharidosis III (MPS-III) is a rare autosomal recessive lysosomal storage disease. It is caused by a deficiency in one of the enzymes needed to break down the glycosaminoglycan heparan sulfate (which is found in the extra-cellular matrix and on cell surface glycoproteins). Sanfilippo syndrome does not demonstrate corneal clouding grossly, but may have slit lamp evidence of clouding (which would be difficult to discern in a donated cornea). Hunter syndrome (another mucopolysaccharidoses) is similar in this respect. Other mucopolysaccharidoses (Hurler, Scheiee, Morquio and Maroteaux-Lamy Syndromes) all demonstrate progressive corneal clouding. Sarcoidosis: Action Accept if no ocular involvement Additional Information Sarcoidosis, also called sarcoid, Besnier-Boeck disease or Besnier-Boeck-Schaumann disease, is a syndrome involving abnormal collections of chronic inflammatory cells (granulomas) that can form as nodules in multiple organs. Manifestations in the eye include uveitis, uveoparotitis, and retinal inflammation, which may result in loss of visual acuity or blindness. The combination of anterior uveitis, parotitis, VII cranial nerve paralysis and fever is called uveoparotid fever, and is associated with Heerfordt-Waldenstrom syndrome. Scleral nodule development associated with sarcoidosis has been observed. The exact cause of is not known. The current working hypothesis is, in genetically susceptible individuals, sarcoidosis is caused through alteration to the immune response after exposure to an environmental, occupational, or infectious agent. SARS (Severe Acute Respiratory Syndrome): Action Must not donate if: a) Less than 21 days from leaving as country to which the Department of Foreign Affairs and Trade or the Department of Health has advised deferring travel because there is, or is thought to be, ongoing transmission of SARS b) Less than 21 days from last contact with a person with SARS c) Less than 3 months since recovery form SARS or possible SARS Accept if more than 21 days has passed since return from a SARS endemic area, or from last contact with a person affected by SARS and the donor has remained well. Additional Information Severe acute respiratory syndrome (SARS) is a viral respiratory disease of zoonotic origin caused by the SARS coronavirus (SARS-CoV). Antibiotics are ineffective, as SARS is a viral disease. Treatment of SARS is largely supportive with antipyretics, supplemental oxygen and mechanical ventilation as needed. It is still regarded as a relatively rare disease. See: Schistosomiasis: See Infections - Acute Return to Table of Contents 79 Additional Information Schistosomiasis (also known as bilharzia, bilharziosis or snail fever) is a collective name of parasitic diseases caused by several species of trematodes belonging to the genus Schistosoma. Snails serve as the intermediary agent between mammalian hosts. Although it has a low mortality rate, schistosomiasis often is a chronic illness that can damage internal organs and, in children, impair growth and cognitive development. Schistosomiasis is readily treated using a single oral dose of the drug praziquantel annually. Scleritis: Action Must not donate See Inflammatory Eye Disease Additional Information Scleritis is a serious inflammatory disease of the sclera. The disease is often contracted through association with other diseases of the body, such as Wegener's granulomatosis or rheumatoid arthritis; it can also be attained through disorders of menstruation. For this reason, scleritis occurs frequently among young women. There are three types of scleritis: diffuse scleritis (the most common), nodular scleritis, and necrotizing scleritis (the most severe). Scleritis may be the first symptom of onset of connective tissue disease. In severe cases the cornea may be damaged. Secondary keratitis or uveitis may also occur. Sexually Transmitted Diseases - Infection: Action See: Is there a specific entry for the disease? Must not donate if less than 12 months from completing treatment See if relevant Chlamydia Genital Warts Gonorrhoea Herpes – Genital Syphilis Sexually Transmitted Disease – Sexual Partner: Action See if there is a specific entry for the disease with which there has been contact Must not donate if: a) Donor required treatment and is less than 12 months since completing that treatment b) Donor did not require treatment and it is less than 12 months from the last sexual contact with the infected partner. Accept if the donor did not require treatment and it is more than 12 months since the infected partner has completed treatment. Return to Table of Contents 80 See if relevant Chlamydia Genital Warts Gonorrhoea Herpes – Genital Syphilis Sex Worker: Action Must not donate if engaged in sex work within the past 6 months (3 months with NAT) Additional TGA criteria 12 month deferral is mandated by the TGA for a donor whose sexual practices put them at increased risk of acquiring infectious diseases that can be transmitted by blood, cells or tissues. See if relevant Addiction and Drug Abuse Homosexual and Bisexual Individuals Hepatitis of Viral Origin HIV HTLV Infection - General Additional Information In this context sex is defined as vaginal, oral or anal sex with or without a condom/protective. If received injectable drugs of addiction for sex, see ‘Addiction and Drug Abuse’ entry as a 12-month deferral may apply. Shingles: See if relevant Herpes – Ocular See Herpes Zoster Sicca: See Keratoconjunctivitis Sicca Sickle-Cell Disease: Action Accept Additional Information Sickle-cell disease (SCD), or sickle-cell anaemia (SCA) or drepanocytosis, is a hereditary blood disorder, characterized by red blood cells that assume an abnormal, rigid, sickle shape. Sideroblastic Anaemia: Action Must not donate See if relevant Myelodysplastic Syndrome Additional Information Sideroblastic anaemia or sideroachrestic anaemia is a disease in which the bone marrow produces ringed sideroblasts rather than healthy erythrocytes. It may be caused either by a genetic disorder or indirectly as part of myelodysplastic syndrome, which can evolve into haematological malignancies (especially acute myelogenous leukaemia). Skin Cancer: See Malignancy Return to Table of Contents 81 Skin Disease: Action Accept See if relevant Acne Dermatitis Infection – General Malignancy Psoriasis Sjögren's syndrome: Action Determine the extent of any sicca symptoms and see Keratoconjunctivitis sicca. See Autoimmune Disorders Keratoconjunctivitis sicca Additional Information Sjögren's syndrome is a systemic autoimmune disease in which immune cells attack and destroy the exocrine glands that produce tears and saliva. It is named after Swedish ophthalmologist Henrik Sjögren (1899–1986), who first described it. 90% of Sjogren’s patients are women. The hallmark symptom of Sjögren's syndrome is a generalized dryness, typically including xerostomia (dry mouth) and keratoconjunctivitis sicca (dry eyes), part of what are known as sicca symptoms. Sicca syndrome also incorporates vaginal dryness, chronic bronchitis and lacks signs of arthritis. Sjögren's syndrome may cause skin, nose, and vaginal dryness, and may affect other organs of the body, including the kidneys, blood vessels, lungs, liver, pancreas, peripheral nervous system (distal axonal sensorimotor neuropathy) and brain. Objective evidence of eye involvement relies on Schirmer's test and the Rose bengal score (or similar). Sjögren's can damage vital organs of the body with symptoms that may plateau or worsen, or go into remission as with other autoimmune diseases. Some people may experience only the mild symptoms of dry eyes and mouth, while others have symptoms of severe disease. Sleeping Sickness: See African Trypanosomiasis Smallpox Immunisation: Action Must not donate if: a) The inoculation site has not fully healed b) Any secondarily infected site has not healed c) Less than 8 weeks from inoculation or from the appearance of any secondarily infected site. Return to Table of Contents 82 Additional Information Small pox immunisation is with live virus. By eight weeks, the infection caused by the inoculation should have been controlled. If the wound has not healed it is possible that there still may be infection present. Close contacts of vaccines (household or direct bodily contact) may become secondarily infected from direct skin contact with an infected inoculation site or from virus on clothing, bedding, dressings etc. If infection occurs, a new skin rash, blister or sore appears at the site of contact, which could be anywhere on the body. The rash represents a secondary vaccination site and presents exactly the same potential risk to patients and staff as that of a person who has been intentionally immunised. South American Trypanosomiasis (and risk of): Action Accept for corneas not for sclera See Chagas’ Disease Spectacles: See Glasses Spherocytosis: See Hereditary Spherocytosis Squamous Cell Carcinoma: See Malignancy Steroid Therapy: Action See if there is an entry for the underlying condition. Accept if not on an immunosuppressive dose and the underlying condition being treated is acceptable. See Immunosuppression See if relevant Autoimmune Disorders Immunosuppression Skin Disease Tissue and Organ Recipients Additional Information Steroid therapy in high doses causes immunosuppression. At these doses there is a theoretical issue of whether immunosuppressive doses can mask the results of mandatory serological testing. If possible NAT testing in these instances may be warranted. Subacute sclerosing panencephalitis (SSPE): Action Must not donate. See Measles Return to Table of Contents 83 Additional Information Also known as Dawson Disease, Dawson encephalitis, and measles encephalitis is a rare and chronic form of progressive brain inflammation caused by a persistent infection with measles virus (which can be a result of a mutation of the virus itself). The condition primarily affects children and young adults. It has been estimated that about 1 in 10,000 people infected with measles will eventually develop SSPE. No cure for SSPE exists and the condition is often fatal. Characterized by a history of primary measles infection usually before the age of 2 years, followed by several asymptomatic years (6–15 on average), and then gradual, progressive psychoneurological deterioration, consisting of personality change, seizures, myoclonus, ataxia, photosensitivity, ocular abnormalities, spasticity, and coma. SSPE is a rare condition, although there is still relatively high incidence in Asia and the Middle East. However, the number of reported cases is declining since the introduction of the measles vaccine— eradication of the measles virus prevents the SSPE mutation and therefore the progression of the disease or even the initial infection itself. Syphilis: Action Must not donate Accept if fully treated in the past and tests exclude recent infection. Additional Information Syphilis is a sexually transmitted infection caused by the spirochete bacterium Treponema pallidum subspecies pallidum. The primary route of transmission is through sexual contact. It is transmissible by kissing near a lesion, as well as oral, vaginal, and anal sex. It is highly infectious with approximately 30 to 60% of those exposed to primary or secondary syphilis contracting the disease. The interpretation of syphilis testing is often difficult and may require an experienced microbiologist to help make a decision on safety. Blood tests are divided into nontreponemal and treponemal tests. Clinically, nontreponemal tests are used initially, and include venereal disease research laboratory (VDRL) and rapid plasma reagin (RPR) tests. However, false positives on the nontreponemal tests can frequently occur (especially with some viral infections such as varicella and measles, as well as with lymphoma, tuberculosis, malaria, endocarditis, connective tissue disease, and pregnancy). In these cases confirmation is required with a treponemal test, such as treponemal pallidum particle agglutination (TPHA) or fluorescent treponemal antibody absorption test (FTA-Abs). Treponemal antibody tests usually become positive two to five weeks after the initial infection. Syphilis – Sexual Contact: See Sexually Transmitted Diseases Systemic Lupus Erythematosus: Action Accept See Autoimmune Disorders Return to Table of Contents 84 Additional Information Systemic lupus erythematosus often abbreviated to SLE or lupus, is a systemic autoimmune disease (or autoimmune connective tissue disease) that can affect any part of the body. It may manifest as a systemic disease (SLE) which is the more common form, or in a purely cutaneous form also known as incomplete lupus erythematosus, more often referred to a just ”Lupus”. Sequelae arising from SLE are more serious and can be life-threatening. There is no one specific cause of SLE. There are, however, a number of environmental triggers and a number of genetic susceptibilities. Research has failed to find a connection between infectious agents (viruses and bacteria), with no pathogen that can be consistently linked to the disease. Whilst SLE has widespread systemic manifestations and pathologies it does it does not affect the cornea. Tattoo: See Body Piercing Temporal Arteritis: See Autoimmune Disorders Additional Information Temporal arteritis is inflammation and damage to blood vessels that supply the head area, particularly the large or medium arteries that branch from the neck and supply the temporal area. If the inflammation affects the arteries in your neck, upper body and arms, it is called giant cell arteritis. The cause is unknown, but is believed to be partly due to a faulty immune response. The disorder has been associated with severe infections and the use of high doses of antibiotics. Tetanus Immunisation: See Immunisation – Non-live Thalassaemia: Action Accept Additional Information Thalassaemia are forms of inherited autosomal recessive blood disorders that originated in the Mediterranean region. In thalassemia, the disease is caused by the weakening and destruction of red blood cells. People with thalassemia make less haemoglobin and fewer circulating red blood cells than normal, which results in mild or severe anaemia. Thalassemia major occurs when a child inherits two mutated genes, one from each parent. Children born with thalassemia major usually develop the symptoms of severe anaemia within the first year of life. Thrombocytosis: Action Accept if reactionary/secondary thrombocytosis Must not donate if essential / primary thrombocytosis (considered a myeloproliferative disorder) See also Haemotological Disease Polycythemia Return to Table of Contents 85 Additional Information Thrombocytosis (or thrombocythaemia) is the presence of high platelet counts in the blood, and can be either primary (also termed essential and caused by a myeloproliferative disease) or reactive (also termed secondary). Although often symptomless (particularly when it is a secondary reaction), it can predispose to thrombosis in some patients. Essential (primary) thrombocytosis is a form of myeloproliferative disorder such as chronic myelogenous leukemia, polycythemia vera, myelofibrosis. Reactive (secondary) thrombocytosis is usually due to an inflammation such as a reaction to surgery or some other inflammatory disorders including autoimmune disorders. Thyroid Disease: Action Accept Discretionary If related to malignancy see Malignancy Additional Information Hypothyroidism Is a state in which the thyroid gland does not produce enough of the thyroid hormones thyroxine (T4) and triiodothyronine (T3). Iodine deficiency is the most common cause of hypothyroidism worldwide. In iodine-replete areas of the world, hypothyroidism is most commonly caused by Hashimoto's thyroiditis (an autoimmune disorder). Some drugs such as lithium and amiodarone can also cause disfunction. Hyperthyroidism Often referred to as an overactive thyroid, is a condition in which the thyroid gland produces and secretes excessive amounts of the free thyroid hormones triiodothyronine (T3) and/or thyroxine (T4). The major causes in humans are Graves' disease (an autoimmune disease (usually, the most common aetiology) and Toxic thyroid adenoma (the most common aetiology in Switzerland, 53%, thought to be atypical due to a low level of dietary iodine in this country). Often iodine-131 radioisotope therapy is given orally (either by pill or liquid) on a one-time basis, to severely restrict, or altogether destroy the function of a hyperactive thyroid gland. Caution for donation if it is less than six months from treatment with radioactive iodine as it has a long half-life and is not readily cleared from the tissues. Tick-Borne Encephalitis: See Infection – Acute Additional Information Tick-borne encephalitis (TBE) is a viral infectious disease involving the central nervous system caused by tick-borne encephalitis virus, a member of the genus Flavivirus in the family Flaviviridae. It is transmitted by the bite of several species of infected ticks and most often manifests as meningitis, encephalitis, or meningoencephalitis. Long-lasting or permanent neuropsychiatric sequelae are observed in 10-20% of infected patients. Russia and Europe report about 5,000-7,000 human cases annually. Return to Table of Contents 86 Tissue and Organ Recipients: Action Accept recipient of solid human organs or tissue in Australia and New Zealand but note issues regarding immunosuppression See - Immunosuppression Must not donate if: a) ever a recipient of human dura mater b) ever a recipient of viable xenotransplant (cells or tissue) c) ever a recipient of human pituitary derived hormone Additional TGA criteria Must not donate if: a) a solid organ recipient within the last 12 months b) a solid organ recipient in England, Scotland, Wales, Northern Ireland or the Isle of Man See Neurosurgery Immunosuppression Pituitary Extract Transfusion Xenotransplantation Additional Information The TGA exclusion criteria for solid organ and tissue recipients relate to the TGA requirement - “A deceased donor who, within 12 months prior to asystole, has been a recipient of allogeneic organ(s), cells, or tissue that are not in accordance with the requirements of this Order (TGO 88)”. Tissues are acceptable if transplanted in Australia as they will either be implanted more than 12 months prior to donation or they will be in accordance with the TGO 88. The TGA exclusions for England, Scotland, Wales, Northern Ireland and Isle of Man are related to the risk of vCJD. There is evidence that human pituitary hormones and human dura mater have transmitted cCJD. Exclusions NB: Immunosuppression at the levels encountered with organ recipients is theoretically capable of suppressing antibody responses to the extent that serological antibody tests would be invalidated. In these cases, NAT testing may be indicated to overcome this restriction. Tissue Products / Tissue grafts (Animal derived): Action Accept if the product has been approved by the regulator for human use(TGA in Australia, MedSafe in NZ) Must not donate if the product was not regulator approved, or was a “viable” non-human animal cell or tissue Return to Table of Contents 87 Additional Information “Viable” in this sense means any non-human animal cell or tissue that is still considered in its natural living state and has not been treated in any way to inactivate its normal living functions. For example this would include the administration of animal blood or blood components and living organ, cell or tissue xenografts. The exclusionary purpose is to prevent the possible transmission of pathogenic zoonoses. Animal derived products or non-viable derived products are specifically not excluded. This includes animal derived insulin and, for example, non-viable animal heart valves. Other likely products (in Australia) are those used in the context of haemostatic control. These products (such as Duragen, FloSeal, and Spongostan) are commonly used in neurosurgical procedures but also in AAA surgery and dental applications. All are TGA-approved products and therefore have met the criteria for TSE in that they are derived from TSE-free countries. Duragen - dural regeneration matrix that is bovine derived (achilles tendon) FloSeal Hemostatic matrix - bovine derived matrix + human derived thrombin component Spongostan -Porcine derived gelatin product Toxoplasmosis: Action Accept Additional Information Toxoplasmosis is a parasitic disease caused by the protozoan Toxoplasma gondii. The parasite infects most warm-blooded animals, including humans, but the primary host is the cat family. In humans, contamination of hands with cat faecal material is a significant risk factor. It is a common parasitic infection with, for example, the Centers for Disease Control placing the seroprevalence in the US population at more than 10%. Usually it does not cause symptoms as the body’s immune system easily overcomes the parasite. During the first few weeks after exposure, the infection typically causes a mild, flu-like illness or no illness. However, those with weakened immune systems, such as those with AIDS and pregnant women, may become seriously ill. The parasite can cause encephalitis (inflammation of the brain) and neurologic diseases, and can affect the heart, liver, inner ears, and eyes (chorioretinitis). Recent research has also linked toxoplasmosis with attention deficit hyperactivity disorder, obsessive compulsive disorder, and schizophrenia. Numerous studies found a positive correlation between latent toxoplasmosis and suicidal behaviour in humans. Corneas are avascular and therefore there is considered to be no risk of transmitting a protozoal infection. Transfusion: Includes Treatment with Blood Components, Products and Derivatives Return to Table of Contents 88 Action Must not donate if a recipient of allogeneic blood, blood components or human derived clotting factors outside of Australia or New Zealand within the past six months (or 4 months if NAT HCV is performed). Additional TGA criteria The TGA considers any transfusion that is not in accordance with the requirements of Therapeutic Goods Order No. 88 - Standards for donor selection, testing and minimising infectious disease transmission via therapeutic goods that are human blood and blood components, human tissues and human cellular therapy products – must not donate if within a six or four month period. In practice all transfusions conducted within Australia will now be in accordance with TGO 85 and therefore acceptable. All other transfusions are only acceptable after a 6 month (4 month with HCV NAT) exclusionary period. There is additional exclusion of - must not donate if ever received treatment with blood, blood components, products and derivatives in England, Scotland, Wales, Northern Ireland or the Isle of Man. See if relevant Immunoglobulin Therapy Immunosuppression Malaria Prion Associated Diseases South American Trypanosomiasis Risk Plasma Dilution Additional Information For New Zealand, the blood supply (and blood products) of both New Zealand and Australia are considered “safe” and therefore acceptable. (the TGA however only considers Australian transfusion to be “safe”). Elsewhere the six or four month rule will apply. Treatment with blood, blood components, products and derivatives at in England, Scotland, Wales, Northern Ireland or the Isle of Man is an exclusion for Australia due to TGA mandatory assessment of vCJD risk. Note the rules for plasma dilution testing of donor samples. Transgender Individuals: Action Assessment of donor suitability should be according to the gender assigned See if relevant Bi-sexual and Homosexual individuals Additional Information A careful and sympathetic consideration of sexual risk factors needs to be undertaken. Men who have sex with other men have a higher chance of having an undiagnosed infection which could be passed to anyone receiving their blood, tissues or cells. Trauma: See Accident Return to Table of Contents 89 Travel: Action For those reporting any travel overseas in the past 12 months: 1. Consider if there is any relation between the signs and symptoms of any reported illness while travelling or upon return (even if resolved), and the risk profile of travel to particular geographical areas (the links below provide a good up to date resource) 2. Consider risk of exposure/contact with any significant current epidemiological situations (eg. Ebola, SARS, Dengue Fever). Accept if no relationship (as above) and the donor is not otherwise medically contraindicated. If a relationship to a particular disease is possible, cross-check the entry for that disease and act upon the advice. Additional TGA criteria The “UK exclusion rule for risk of vCJD applies”. Must not donate: a) Individuals who donors who have lived in or visited England, Scotland, Wales, Northern Ireland or the Isle of Man for a cumulative period of six months or more, between 1st January 1980 and 31st December 1996 inclusive. b) individuals who have received a transfusion or injection of blood or blood components while in England, Scotland, Wales, Northern Ireland or the Isle of Man at any time since 1st January 1980 onward. See if relevant: United Kingdom’s Geographical Disease Risk Index The Centers for Disease Control and Prevention (CDC) also have valuable current information on disease outbreaks by geography The Australian Travel Advisory Malaria South American Trypanosomiasis Infection – Tropical Prion Disease Trisomy 21 (Down’s Syndrome): Action Must not donate See Down’s Syndrome Trypanosoma Cruzi Infection: See Chagas’ Disease Tuberculosis: Action Must not donate if; a) Infected b) Less than 24 months from confirmation of cure c) Under follow up See if relevant BCG Return to Table of Contents 90 Additional Information Tuberculosis, MTB, or TB is a common, and in many cases lethal, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis. Tuberculosis typically attacks the lungs, but can also affect other parts of the body. It is spread through the air when people who have an active TB infection cough, sneeze, or otherwise transmit respiratory fluids through the air. Most infections are asymptomatic and latent, but about one in ten latent infections eventually progresses to active disease which, if left untreated, kills more than 50% of those so infected. One third of the world's population is thought to have been infected with M. tuberculosis. The most important risk factor globally is HIV; 13% of all TB cases are infected by the virus. Progression from TB infection to overt TB disease occurs when the bacilli overcome the immune system defenses and begin to multiply. In primary TB disease (some 1–5% of cases), this occurs soon after the initial infection. However, in the majority of cases, a latent infection occurs with no obvious symptoms. These dormant bacilli produce active tuberculosis in 5–10% of these latent cases, often many years after infection. Turner’s Syndrome: Advisory Do not donate Additional Information Turner syndrome or Ullrich–Turner syndrome (also known as "Gonadal dysgenesis" encompasses several conditions in human females, of which monosomy X (absence of an entire sex chromosome, the Barr body) is most common. Visual impairments sclera, cornea, glaucoma, etc. have been noted with this syndrome, although eye problems do not appear to be as common as in Down’s Syndrome. Typhoid: Action Must not donate Accept if past infection that has completely resolved Additional Information Typhoid is a common worldwide bacterial disease transmitted by the ingestion of food or water contaminated with the faeces of an infected person, which contain the bacterium Salmonella enterica enterica, serovar Typhi. It is primarily a gastrointestinal infection that has early manifestations of high fever and bradycardia, frequently progressing to delirium and diarrhoea. Late and often fatal manifestations include dehydration, Intestinal haemorrhage or perforation, encephalitis, metastatic abscesses, cholecystitis, endocarditis and osteitis. Typhoid Immunisation (Injected): See Immunisation Non-live Typhoid Immunisation (Oral): See Immunisation Live Ulcerative Colitis: Action Accept if eyes uninvolved See Inflammatory Bowel Disease Return to Table of Contents 91 Additional Information Ulcerative colitis is a form of colitis, a disease of the colon that includes characteristic ulcers or open sores. The main symptom of active disease is usually constant diarrhoea mixed with blood, of gradual onset. It has similarities to Crohn's disease. Ulcerative colitis is treated as an autoimmune disease, with anti-inflammatory drugs, immunosuppression, and biological therapy targeting specific components of the immune response. NB: Patients may present with ophthalmic comorbidities such as iritis, uveitis and episcleritis. There is a chance that other organs and tissues may not be accepted for donation. Vasculitis: Action Consider what the underlying cause may be a) Infectious b) Autoimmune Accept for corneas only not sclera if underlying cause is not contraindicated. See Autoimmune disorders Temporal arteritis Additional Information Vasculitis is a group of disorders that destroy blood vessels by inflammation. The underlying cause may be infectious. For example, the cause of syphilitic aortitis is infectious. However, the causes of many forms of vasculitis are poorly understood. There is usually an immune component, but the trigger is often not identified. Viral Haemorrhagic Fever: Action Affected individual must not donate if ever infected Contact or travel to endemic region must not donate if: a) Was present in an area during an active outbreak b) Under investigation for viral haemorrhagic fever c) Has been in contact with an individual who was present in an area during an active outbreak d) Was in contact with an individual infected with or was under investigation for viral haemorrhagic fever e) less than six months after return to ANZ from an endemic area when there was no active outbreak. See if relevant The Geographical Disease Risk Index for countries with a current endemic Viral Haemorrhagic Fever risk Additional Information These infections have very high death rates and there is evidence that the virus may persist for some time after recovery. The 2014-16 outbreak of Ebola in West Africa had increased understanding about the persistence of the virus in affected individuals and the number of asymptomatic individuals who may be able to transmit the virus to others. There is no routine screening test for EBOV currently available. There is an option to test donors serologically for the presence of anti-EBOV (antibodies) two months after the exposure event if a test becomes available. A reactive test would result in permanent deferral, a negative test would allow donation to proceed. Return to Table of Contents 92 Vitiligo: Action Accept See Autoimmune Disorders Vitiligo is a condition that causes depigmentation of parts of the skin. It occurs when melanocytes, the cells responsible for skin pigmentation, die or are unable to function. The cause of vitiligo is unknown, but research suggests that it may arise from autoimmune or genetic causes. Von Recklinghausen’s Disease: See Neurofibromatosis West Nile Virus: Definition West Nile Virus Outbreak Areas are shown and updated in the United Kingdom’s Geographical Disease Risk Index The Centers for Disease Control and Prevention (CDC) also have valuable current information on disease outbreaks by geography Action Accept 4 months after their return from an affected area. This may be reduced to 4 weeks if they have had no symptoms or evidence of infection. For donors under four weeks from return, accept if a validated NAT of WNV is negative. Must not donate if: a) It is less than 6 months from a donor’s return from a WNV outbreak area and the donor has been diagnosed with WNV b) It is less than 6 months from a donor’s return from a WNV outbreak area and the donor has either had a history of symptoms suggestive of WNV while there or within 28 days of their return c) In other cases it is less than 4 weeks from a donor’s return from a WNV outbreak area. Additional Information West Nile virus (WNV) is a mosquito-borne zoonotic arbovirus belonging to the genus Flavivirus in the family Flaviviridae (similar to Dengue Fever). Approximately 80% of West Nile virus infections in humans are subclinical, which cause no symptoms. In the cases where symptoms do occur—termed West Nile fever in cases without neurological disease—the time from infection to the appearance of symptoms (incubation period) is typically between 2 and 15 days. Symptoms may include fever, headaches, fatigue, muscle pain or aches, malaise, nausea, anorexia, vomiting, myalgias and rash. Less than 1% of the cases are severe and result in neurological disease when the central nervous system is affected. People of advanced age, the very young, or those with immunosuppression, either medically induced, such as those taking immunosupressive drugs, or due to a pre-existing medical condition such as HIV infection, are most susceptible to developing West Nile meningitis, West Nile meningoencephalitis and West Nile poliomyelitis. The problem can vary both in relation to geography and the time of the year so it is not possible to state areas from which donors need to be deferred. Whooping Cough: See Infection – Acute Infectious Diseases – Contact with Return to Table of Contents 93 Additional Information Pertussis — commonly called whooping cough is a highly contagious bacterial disease caused by Bordetella pertussis. Symptoms are initially mild, and then develop into severe coughing fits, which produce the namesake high-pitched "whoop" sound in infected babies and children when they inhale air after coughing. Common complications of the disease include pneumonia, encephalopathy, earache, or seizures. Most healthy older children and adults will have a full recovery from pertussis, however those with comorbid conditions can have a higher risk of morbidity and mortality. Infection in newborns is particularly severe. Wilson’s Disease: Action Accept Additional Information Wilson's disease or hepatolenticular degeneration is an autosomal recessive genetic disorder in which copper accumulates in tissues; this manifests as neurological or psychiatric symptoms and liver disease. It is treated with medication that reduces copper absorption or removes the excess copper from the body, but occasionally a liver transplant is required. Xenotransplantation: Action Must not donate if: The donor has been a recipient of viable non-human animal cells or tissue. See also Tissue Products / Tissue grafts (Animal derived) Additional Information Any procedure that involves the transplantation, implantation or infusion into a human recipient of either a) live cells, tissues, or organs from a non-human animal. b) Human body fluids, cells, tissues or organs that have had ex vivo contact with live, non-human animals cells, tissues and organs. Biological products, drugs, or medical devices sourced from non-living cells, tissues or organs from non-human animals (including but not limited to porcine heart valves and porcine insulin) are not considered xenotransplantation products. Advisory Sexual partners of xenotransplant recipients, current and former, should not donate. Particularly when the recipient is immunosuppressed there is a risk of passing on infections that would not normally affect humans. Yaws: Action Must not donate Additional Information Yaws (also frambesia tropica, thymosis, polypapilloma tropicum, pian or parangi, "Bouba", "Frambösie", and "Pian") is a tropical infection of the skin, bones and joints caused by the spirochete bacterium Treponema pallidum pertenue. The disease is transmitted by skin-to-skin contact with an infective lesion, with the bacterium entering through a pre-existing cut, bite or scratch. Within ninety days (but usually less than a month) of infection a painless but distinctive "mother yaw" appears, which is a painless nodule which enlarges and becomes warty in appearance. Sometimes nearby "daughter yaws" also appear simultaneously. This primary stage resolves completely within six months. The secondary stage occurs months to years later, and is characterised by widespread skin lesions of varying appearance which frequently ulcerate (and are then highly infectious), but heal after six months or more. About ten percent of people then go on to develop tertiary disease within five to ten years (during which further secondary lesions may come and go), characterised by widespread bone, joint and soft tissue destruction, which may include extensive destruction of the bone and cartilage of the nose. Return to Table of Contents 94 Yellow Fever: See Infection – Acute Additional Information Yellow fever, known historically as Yellow Jack, is an acute viral haemorrhagic disease. It was the first illness shown to be transmissible via filtered human serum and transmitted by mosquitoes, by Walter Reed around 1900. The yellow fever virus is transmitted by the bite of female mosquitoes (the yellow fever mosquito, Aedes aegypti, and other species) and is found in tropical and subtropical areas in South America and Africa, but not in Asia. The only known hosts of the virus are primates and several species of mosquito. Yellow fever presents in most cases in humans with fever, chills, anorexia, nausea, muscle pain (with prominent backache) and headache, which generally subsides after several days. In some patients, a toxic phase follows, in which liver damage with jaundice (inspiring the name of the disease) can occur and lead to death. Because of the increased bleeding tendency (bleeding diathesis), yellow fever belongs to the group of haemorrhagic fevers. Zieve's syndrome: Action Accept Additional Information Zieve's syndrome is an acute metabolic condition that can occur during withdrawal from prolonged alcohol abuse. It is defined by haemolytic anaemia, hyperlipoproteinaemia (excessive blood lipoprotein), jaundice, and abdominal pain. The underlying cause is liver delipidization. Zika Fever (Virus): Action Must not donate if donor is actively infected. Defer for six months after resolution of active infection See Arbovirus Infection – Acute Return to Table of Contents 95 Additional Information Zika virus (ZIKV) is a member of the Flaviviridae virus family and the flavivirus genus. In humans, it causes a disease known as "zika", "Zika disease" or “Zika fever”. It is related to dengue, yellow fever, West Nile and Japanese encephalitis, viruses that are also members of the virus family Flaviviridae. As of 2016, the illness cannot be prevented by medications or vaccines. It is transmitted by mosquitoes and has been isolated from a number of species in the genus Aedes. Importantly most of the morbidity caused by this virus is the spread from a pregnant woman to her foetus. This can result in microcephaly, severe brain malformations, and other birth defects (see below the connection in Brazil). Zika infections in adults may result rarely in Guillain–Barré syndrome. Zika can be transmitted from a man or a woman to their sexual partners. As of April 2016 sexual transmission of Zika has been documented in six countries – Argentina, Chile, France, Italy, New Zealand and the United States – during the 2015 outbreak. It appears that the Virus may persist for some time after the acute infection (detected in semen up to 10 weeks after acute infection) but ZIKV has not been identified in ocular tissue in primate animal models. The virus was first isolated in 1947 from a rhesus monkey in the Zika Forest of Uganda, Africa and was isolated for the first time from humans in 1968 in Nigeria. From 1951 through 1981, evidence of human infection was reported from other African countries such as Uganda, Tanzania, Egypt, Central African Republic, Sierra Leone and Gabon, as well as in parts of Asia including India, Malaysia, the Philippines, Thailand, Vietnam and Indonesia. Zika virus could be considered an emerging pathogen, as it spread outside Africa and Asia for the first time in 2007 (Micronesia). It had a large outbreak in Brazil in April 2015. In December 2015, the Pan American Health Organization (PAHO) / World Health Organization (WHO) noted that transmission of Zika virus infection had occurred within nine member states: Brazil, Chile (specifically Easter Island), Colombia, El Salvador, Guatemala, Mexico, Paraguay, Suriname, and Venezuela. PAHO/WHO made recommendations for surveillance, case management, and prevention. More recently locally acquired cases have been noted in Florida. The CDC provides up to date details of current spread . Thus far, it has been a relatively mild disease with limited scope, but its true potential as a virus and as an agent of disease is currently unknown. However, Brazilian authorities have confirmed the previously suspected connection between zika infection by pregnant women and newborn microcephaly, with a full 1,248 reported cases of microcephaly in Brazil during 2015, with 7 fatalities. Common symptoms of infection with the virus include mild headaches, maculopapular rash, fever, malaise, conjunctivitis, and arthralgia. Eye banks should be screening for symptomatic infection in their donors. Zollinger–Ellison syndrome: See Malignancy Additional Information Zollinger–Ellison Syndrome (ZES) is caused by a non–beta islet cell, gastrin-secreting tumour of the pancreas that stimulates the acid-secreting cells of the stomach to maximal activity, with consequent gastrointestinal mucosal ulceration. Return to Table of Contents 96 www.ebaanz.org |
11740 | https://www.sps.nhs.uk/articles/managing-peripheral-oedema-caused-by-calcium-channel-blockers/ | Published Time: 2022-09-26T12:00:00+01:00
Managing peripheral oedema caused by calcium channel blockers – SPS - Specialist Pharmacy Service – The first stop for professional medicines advice
SPS - Specialist Pharmacy Service =================================
The first stop for professional medicines advice
About
Log in
Register
NHS ===
Guidance
Guidance
Guidance index
PGDs
Administering
Cautions and contraindications
Dosing
Switching
Deprescribing
Interactions
Medication Safety
Safety in breastfeeding
Safety in pregnancy
Temperature management
Excipients
Swallowing difficulties
Aseptic services
Intrathecal administration
Manufacturing and preparation
Unlicensed medicines
Homecare
ATMPs
Clinical Trials
Medical gases
Record keeping
Search Guidance
Guidance by Care Setting
Adults
Care homes
Care of the elderly
Community Health Services
Dentistry
Emergency medicine and urgent care
Health and Justice
Neonates
Occupational Health
Paediatrics
Palliative and End of Life Care
Primary Care
Transfer of care
Trusts
Guidance by Specialty
Allergy and immunology
Anaesthesia and pain
Cancers
Cardiovascular system disorders
Diabetes
Endocrine system disorders
Eyes and vision
Gastrointestinal disorders
Haematological disorders
Infection and infectious diseases
Mental health and illness
Neurological disorders
Nutritional and metabolic disorders
Obstetrics and gynaecology
Public Health
Radiology
Renal and urologic disorders
Reproductive health
Respiratory disorders
Sexual health
Skin disorders
Smoking cessation
Vaccinating
Wounds and injuries
Events
Events
Events index
Next events
Past Events
Search Events
Events by Care Setting
Community Health Services
Health and Justice
Primary Care
Events by Guidance area
Adverse effects
Aseptic services
Medication Safety
Patient Group Directions
Polypharmacy
Safety in Pregnancy
Events by Specialty
Cardiovascular system disorders
Diabetes
Endocrine system disorders
Infection and infectious diseases
Mental health and illness
Neurological disorders
Planning
Planning
Planning index
Biosimilars
Annual medicines planning
Search Medicines planning
Planning by Care Setting
Primary Care
Transfer of care
Trusts
Planning by Specialty
Diabetes
Eyes and vision
Gastrointestinal disorders
Musculo-skeletal disorders
Neurological disorders
Skin disorders
Training
Training
Training index
All Training and Development
Breastfeeding
Cardiovascular disease
Complementary medicines
Kidney disorders
Liver disorders
Polypharmacy
Medication safety
Pregnancy
Search Training and Development
Publications
Publications
Publications index
Monthly digest
Bulletins and summaries
Research and audits
Search Publications
Publications by Care Setting
Emergency medicine and urgent care
Publications by Specialty
Infection and infectious diseases
Neurological disorders
Public Health
Radiology
Renal and urologic disorders
Reproductive health
Respiratory disorders
Sexual health
Skin disorders
Smoking cessation
Tools
Medicines Tools
Medicines Tools index
Medicines Supply
Monitoring
MCA Stability
Fridge stability tool
Aseptics Stability
KME
Search
Home
Managing peripheral oedema caused by calcium channel blockers
Published 26 September 2022
Topics: Adverse effects · Amlodipine · Cardiovascular system disorders · 6 moreDiltiazem · Felodipine · Lacidipine · Lercanidipine · Nifedipine · Verapamil · Less
All calcium channel blockers may cause peripheral oedema. There are different treatment options and the incidence can be affected by different factors.
Contents
About ankle oedema
Symptoms
Cause
Factors affecting incidence
Differences in chemical class
Dose
Duration of action
Effect on blood pressure
Treatment options
Non-Pharmacological methods
Dose adjustments
Switching to another CCB
Angiotensin Converting Enzyme Inhibitors (ACEi)
Angiotensin II Receptor Blockers (ARB)
Nitrates
Diuretics
Discontinuation of CCB
Bibliography
About ankle oedema
Peripheral oedema, including ankle oedema, is a recognised adverse effect of calcium channel blockers (CCBs). This may affect compliance but also limit their usefulness, particularly in an aging population who are more likely to have co-morbidities.
The risk of developing ankle oedema whilst on a CCB appears to be higher in women, older patients, those with heart failure, upright postures, and those in warm environments.
Ankle oedema can range from being mild and unnoticed to severely affecting quality of life.
Symptoms
Ankle oedema in patients who have been taking CCBs for a long time may be associated with a petechial rash and in some cases hyper pigmentation and discolouration. This is thought to be due to increases in capillary permeability, leading to leakage of erythrocytes into the surrounding fluid.
CCB-related oedema commonly worsens in the evening, and may resolve or improve following the patient lying down overnight.
The COHORT study of 828 elderly, hypertensive patients, reported that ankle oedema may have a delayed onset, with its incidence increasing gradually as treatment continues, meaning it is not likely to be a transient, self limiting effect.
Cause
The mechanisms by which CCBs give rise to ankle oedema are not currently understood.
Proposed mechanisms include:
an increase in capillary pressure, resulting in fluid loss from the capillaries and leakage into interstitial areas, or
by interference with local vascular control
blocking reflex increases in precapillary resistance which occur on standing, compounding oedema formation
Unlike peripheral oedema caused by fluid retention, CCB-induced oedema appears to be due to redistribution of fluid from capillaries to interstitial spaces.
It seems unaffected by administration of diuretics, supporting the idea that it may be due to fluid pooling rather than fluid retention.
Oedema occurs despite CCBs possessing inherent diuretic effects.
Factors affecting incidence
Differences in chemical class
CCBs are generally classified into DHP and non-dihydropyridines based on their chemical structure.
Whilst ankle oedema appears to be a class effect in all CCBs, there does appear to be differences in the incidence of ankle oedema between the different classes.
Dihydropyridine
Oedema appears to be more likely with the dihydropyridine agents.
The incidence of ankle oedema has been reported as ranging from 1 to 15% in patients treated with DHP agents. There are thoughts suggesting more lipophilic agents, such as lercanidipine and lacidipine, may be associated with lower incidence of ankle oedema.
Non-dihydropyridines
Verapamil
The rate of ankle oedema occurring with verapamil therapy is variable. Verapamil increases plasma volume whilst also reducing vasoconstriction in the lower extremities, similarly to amlodipine and nifedipine.
Diltiazem
Some post-marketing surveillance data has reported a reduced incidence of ankle oedema in patients treated with diltiazem compared to other CCB agents.
Dose
Ankle oedema is dose related and its incidence may exceed 80% in patients taking long-term high doses of DHP agents. However, this association may not occur in an exact dose-proportional relationship.
Duration of action
Whilst the longer-acting CCBs generally appear to have fewer adverse effects associated with them (such as flushing, headache, and palpitations), this is not thought to be the case when considering ankle oedema.
Effect on blood pressure
Differences in blood pressure lowering ability of different CCB agents do not seem to correlate with differences in ability to cause ankle oedema.
Treatment options
Treatment of ankle oedema will depend on the severity and other patient factors. Mild oedema that is not troublesome to the patient does not require specific intervention.
Whilst ankle oedema associated with CCBs is rarely clinically serious, it may significantly reduce adherence to these potentially useful agents.
Non-Pharmacological methods
Elevation of legs when in a prone position, or graduated compression stockings, may be an option in some patients with mild oedema. However, there is little evidence to suggest these methods may be effective in reducing oedema.
Dose adjustments
As ankle oedema is dose-related (although not necessarily in a dose-proportional manner), reducing the dosage of a CCB may lead to a reduction in severity of ankle oedema.
Switching to another CCB
Switching to another CCB class may reduce ankle oedema, although current evidence on the success rates of this strategy are conflicting.
Options include:
switching from a dihydropyridine (DHP) agent to a non-dihydropyridine agent if clinically suitable, such as verapamil, which may lead to resolution of the ankle oedema
switching to a third generation dihydropyridine, such a lercanidipine, which has a lower reported incidence of ankle oedema
Angiotensin Converting Enzyme Inhibitors (ACEi)
It has been demonstrated in several trials that adding an ACEi to a CCB reduces the incidence of ankle oedema.
Mechanism
The mechanism by which this occurs is not currently known. It may be due to the dilation of venous capacitance vessels, which may then lead to a reduction in capillary hypertension and therefore leakage of fluid into the surrounding tissues.
Evidence
It is currently unknown whether any ACEi is superior to any other in treating ankle oedema, but any ACEi initiated should be titrated according to blood pressure.
One randomised controlled trialfound a lower prevalence of oedema in the group randomised to amlodipine in combination with ramipril (7.6%) compared to the group taking amlodipine alone (18.7%, P=0.011).
A systematic review found that a combination of a calcium channel blocker along with an angiotensin system blocker combination led to 38% fewer incidences of peripheral oedema than with a calcium channel blocker alone. This review was limited to Medline and Cochrane databases only and did not differentiate between ACEi and angiotensin II receptor blockers (ARB).
Angiotensin II Receptor Blockers (ARB)
Mechanism
The mechanisms by which ARBs reduce incidence of CCB-induced ankle oedema remains unknown, but are likely to be similar to that involved when an ACEi is added to CCB therapy.
Evidence
One open-label, blinded end point study found a significant reduction in the incidence of markers of ankle oedema in the group treated with valsartan in addition to amlodipine versus amlodipine alone, along with significantly reduced blood pressure. This suggests that an ARB may be an effective alternative for treatment of oedema in patients unable to tolerate an ACEi.
A more recent study by the same authors compared valsartan and olmesartan effectiveness at reducing ankle oedema caused by amlodipine. Both ARBs reduced foot volume, but valsartan reduced oedema significantly more than olmesartan.
Nitrates
Nitrates, due to their venodilating action, may offer some useful effects in treating CCB-induced ankle oedema, but their use is limited by the practical considerations of having a stop-start regimen so tolerance does not develop.
Diuretics
Diuretics appear to have little effect on CCB-induced oedema, even where there is large natriuresis and a subsequent decrease in plasma volume.
This applies to both thiazide and loop diuretics, and is due to the fact that diuretics act by reducing water retention only. Diuretics do not affect vasodilatory-induced fluid pooling.
Discontinuation of CCB
If other treatment options fail, discontinuing the CCB and switching to an antihypertensive from another class of medicines may be required.
Bibliography
Show Hide
Messerli FH, and Grossman E. Pedal edema- not all dihydropyridine calcium antagonists are created equal. American Journal of Hypertension 2002; 15: 1019-1020
Fogari R, Zoppi A, Derosa G et al. Effect of valsartan addition to amlodipine on ankle oedema and subcutaneous tissue pressure in hypertensive patients. Journal of Human Hypertension 2007; 21: 220-224
Andresdottir MB, van Hamersvelt HW, ven Helden MJ et al. Ankle edema formation during treatment with the calcium channel blockers lacidipine and amlodipine: a single-centre study. Journal of cardiovascular pharmacology 2000: 35 S25-S30
Sica DA. Calcium channel blocker-related peripheral edema: Can it be resolved? The Journal of Clinical Hypertension 2003; 5(4): 291-297
Sirker A, Missouris CG and Macgregor GA. Dihydropyridine calcium channel blockers and peripheral side effects. Journal of Human Hypertension 2001; 15: 745-746
Epstein BJ and Roberts ME. Managing peripheral edema in patients with arterial hypertension. Americal Journal of Therapeutics 2009; 16(6): 543-553
Weir MR, Rosenberger C, and Fink JC. Pilot study to evaluate a water displacement technique to compare effects of diuretics and ACE inhibitors to alleviate lower extremity edema due to dihydropyridine calcium antagonists. American Journal of Hypertension 2001; 14(9): 963-968
Miranda R, Mion J, Rocha J et al. An 18-week, prospective, randomized, double-blind, multicentre study of amlodipine/ramipril combination versus amlodipine monotherapy in the treatment of hypertension: The assessment of combination therapy of amlodipine/ramipril (ATAR) study. Clinical Therapeutics. 2008; 30(9): 1618-1628.
Kakani H, Bangalore S, Romero J et al. Effect of Renin-Angiotensin System Blockade on Calcium Channel Blocker-Associated Peripheral Edema. The American Journal of Medicine 2011 Volume 124 , Issue 2 , 128 – 135
Fogari R, Malamani G, Corradi L et al. Effect of valsartan or olmesartan addition to amlodipine on ankle edema in hypertensive patients. Adv Ther 2010; 27(1)
Print this page
Related
Using calcium-channel blockers during breastfeeding
Nifedipine and verapamil are the calcium-channel blockers of choice during breastfeeding. Recommendations apply to full term and healthy infants only.
Verapamil · 2 June 2023
Identifying risk factors for developing a long QT interval
Some medicines (erythromycin, citalopram, ondansetron) can cause a long QT interval. Consider risk factors and follow MHRA recommendations for safe use.
Interactions · 29 March 2023
Example medicines to prescribe by brand name in primary care
Prescribe certain medicines by brand to ensure supply of the same product. Examples are grouped by therapeutic area. The list of products is not exhaustive.
Tramadol · 27 March 2022
Managing medicines for people with sulfonamide allergy
Points to consider when prescribing and supplying medicines to people with sulfonamide allergy.
Tamsulosin · 2 October 2024
Managing reactions to dental local anaesthetic injections
Reactions to dental local anaesthetics (LA) are usually psychogenic. We describe symptoms and management of allergy, adverse effects and psychogenic reactions.
Prilocaine + Felypressin · 29 July 2024
Reviewing medicines for people at risk of falls
This webinar provided information and guidance on the importance of medication reviews in people who have had a fall or are at risk of falling.
Past Events · 9 January 2024
Using antiplatelet medicines during breastfeeding
Antiplatelets can be used with caution in breastfeeding. Avoid exposure to aspirin if infant has a fever. Recommendations apply to full term, healthy infants.
Ticagrelor · 7 July 2023
Adverse drug reactions: resources to support answering questions
Suggested resources to help primary care healthcare professionals find information to answer questions about side-effects of medicines
Primary Care · 30 November 2022
Managing constipation in people taking clozapine
Constipation in people taking clozapine can be fatal. Guidance on preventing and managing constipation in these individuals is described.
Mental health and illness · 14 October 2022
Learn more about us or ask for help
About SPS
Learn about who we are and what we do, our purpose and values, and how to get the best from our advice and guidance.
Get in touch
Get in touch with the Medicines Information and Medicines Quality Assurance parts of our service if you have a question.
Medicines Tools
Medicines Supply Tool
Latest information on supply issues, actions to take, alternatives to use, and expected resolution dates. Content provided by DHSC and MVA team, NHS England.
Medicines supply
Medicines Monitoring
Summarised recommendations and guidance for medicines requiring monitoring to save you time when you're making monitoring decisions with patients
Monitoring
Medicines in Compliance Aids Stability Tool
Information on stability for tablets and capsules to help create MCAs for patients. Find entries and then build and print lists specific to your MCA.
Medicines in compliance aids
Refrigerated medicines stability tool
Advice on whether refrigerated medicines can or can’t be used after exposure to out-of-range temperatures. Find entries and build and print lists.
Fridge medicines
Stability Information for Aseptic Services
Information on drug stability to help users make an informed decision on the shelf life for products prepared within aseptic facilities.
Key Molecule Expenditure
View how your organisation is performing for a number of key molecules and compare your performance with others.
SPS – Specialist Pharmacy Service - The first stop for professional medicines advice ====================================================================================
Accessibility · Terms and Conditions · Copyright © 2025 SPS - Specialist Pharmacy Service. All Rights Reserved. NHS-SPS 1 |
11741 | https://mathbitsnotebook.com/Geometry/Probability/PBpermutations.html | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- |
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- | | | | | --- | | | Permutations MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts | | | | Info about Permutations | A permutation is an arrangement of objects in specific order. The order of the arrangement is important!! Order matters!! | | | | | | | --- --- --- | | Permutations: All possible ways of doing something! | Consider, four dogs arranged in a line as shown. How many different ways could the dogs be arranged in this side-by-side pattern? | | | | | --- --- | | 1,2,3,4 1,2,4,3 1,3,2,4 1,3,4,2 1,4,2,3 1,4,3,2 | 2,1,3,4 2,1,4,3 2,3,1,4 2,3,4,1 2,4,1,3 2,4,3,1 | 3,2,1,4 3,2,4,1 3,1,2,4 3,1,4,2 3,4,2,1 3,4,1,2 | 4,2,3,1 4,2,1,3 4,3,2,1 4,3,1,2 4,1,2,3 4,1,3,2 | | The number of different arrangements is 24 or 4! = 4 • 3 • 2 • 1. There are 24 different arrangements, or permutations, of the four dogs sitting side-by-side. | | | | | | | | | --- --- --- --- | | | | | The notation for a permutation: n P r n is the total number of objects r is the number of objects chosen (wanted) read: "n objects taken r at a time" | | | | | For the dogs, there are"4 dogs taken 4 at a time". 4 P4 = 4 • 3 • 2 • 1 = 24 arrangements | | | | | | --- | | | | --- | | | For working with permutations on your calculator, click here. | | | | | | Formulas for Permutations: | | | | | --- | Formula for a permutation: | OR | Formula for a permutation: Remember that 0! = 1. | | If n = r, as it did with the dogs, them nPr = n! for either formula. | | | Examples: | | | --- | | 1. Compute 6 P2 6• 5 = 30 (multiply two factors of the factorial starting with 6) OR use formula: | 2. How many ways can 5 objects be arranged that are chosen from a set of 7 different objects. 7 P5 = 7 • 6 • 5 • 4 • 3 = 2520 (seven objects taken five at a time) OR use formula: | | 3. Find the total number of possible 5-letter arrangements of the letters s, c, o, r, e if each letter is used only once in the arrangement. 5 P5 = 5 • 4 • 3 • 2 • 1 = 60 OR use formula: or simply 5! | 4. How many different 3-digit integers can be made from the digits 5, 6, 7, 8, 9 if a digit can appear only once in a number. 5P3 = 5 • 4 • 3 = 60 OR use formula: | | | | Dealing with "Special" Arrangements | Consider the following example where special conditions are placed on the arrangement. Task: Using the letters from the word "WIZARD", how many 6-letter arrangements, with no repetition, are possible, if: a) the first letter must be a vowel b) the first and last letters must be consonants Solution: a) Set up a visual of the six needed letters: ____ • ____ • ____ • ____ • ____ • ____ The first location must be a vowel (i, a), so there are only 2 ways to fill the first empty slot. _2_ • ____ • ____ • ____ • ____ • ____ There are now 5 letters left to fill the remaining slots. _2_ • _5__ • _4__ • _3__ • _2__ • _1__ = 240 b) Again, visualize the 6 needed locations: ____ • ____ • ____ • ____ • ____ • ____ The first and last letters are consonants (w, z, r, d) _4_ • ____ • ____ • ____ • ____ • _3__ The remaining locations are filled with the remaining 4 letters. _4_ • _4__ • _3__ • _2__ • _1__ • _3__ = 288 | | | Dealing with Repetition | When dealing with permutations with repetition, remember that order still matters. Task: How many 3 letter words can be made from the letters c, a, t, allowing for repetition of the letters? | | | | | | | | | | --- --- --- --- | Solution: Set up a visual of the three needed letters: ____ • ____ • ____ There are 3 letters that can be used to fill the first slot. But, now, because repetition is allowed, we also have 3 letters to fill the second and third slots. _3_ • _3__ • _3__ = 27 arrangements | | | | | | | | | --- --- --- | | | | | --- | 27 arrangements with repetition: | | | | ccc caa ctt cat cta cac ctc cca cct | aaa acc att act atc aca ata aac aat | ttt taa tcc tac tca tct tat ttc tta | | | Dealing with repetition can become confusing when you encounter repetition of indistinguishable objects. Indistinguishable objects are simply items (such as letters) that are repeated in the original set. For example, if the word MOM was used instead of CAT, in the example above, the two M's are indistinguishable from one another, since they repeat. Using MOM, some of the answers would have been duplicates of one another because of the repeating M. When looking for answers that are not duplicates (unique answers), the problem of any letters or objects repeating in the original set must be addressed. | | | The number of different permutations of n objects, where there are n1 indistinguishable objects of style 1, n2 indistinguishable objects of style 2, ..., and nk indistinguishable objects of style k, is ; | In general, repetitions are taken care of by dividing the permutation by the factorial of the number of objects that are identical. | | | | Examples: | | | --- | | 1. How many different 6-letter words can be formed from the word SCHOOL ? Divide by 2! because the letter O repeats twice. | 2. How many different 7-digit integers can be written using all of the following 7 digits: 3,3,4,4,4,5,8 ? The 3 repeats twice and the 4 repeats three times, so divide by 2! • 3! | | | | With Replacement; Without Replacement | We have seen how "with replacement" and "without replacement" affect probability. These same situations appear when working with permutations. Examples: | | | 1. Two cards are drawn from a standard deck of 52 cards, without replacement. What is the probability that both cards drawn are jacks? There are 4 jacks in a deck of cards. The first card will NOT be returned to the deck before picking the second card. | | 2. One card is drawn from a standard deck of 52 cards and then replaced into the deck. A second card is then drawn and replaced. What is the probability that a jack is drawn each time? On the first draw, the probability of getting one of the four jacks in the deck is 4 out of 52 cards. Because the jack is replaced into the deck, the probability of getting a jack on the second draw remains the same. Using the Counting Principle we have: | | | | NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". | Topical Outline | Geometry Outline | MathBitsNotebook.com | MathBits' Teacher Resources Terms of UseContact Person: Donna Roberts Copyright © 2012-2025 MathBitsNotebook.com. All Rights Reserved. | | |
11742 | https://math.stackexchange.com/questions/2307126/calculating-a-20-discount-by-multiplying-by-0-8333 | percentages - Calculating a 20% discount by multiplying by 0.8333? - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Calculating a 20% discount by multiplying by 0.8333?
Ask Question
Asked 8 years, 4 months ago
Modified6 years, 10 months ago
Viewed 19k times
This question shows research effort; it is useful and clear
5
Save this question.
Show activity on this post.
I've been looking at some discounted prices of goods.
They are listed with a 20%20% discount, so to work this out I did:
$25.45⋅0.8333=$21.21.$25.45⋅0.8333=$21.21.
But their total was 20.34 20.34, which I presume they got by doing 25.42⋅0.8 25.42⋅0.8.
To apply a 20%20% discount or to subtract 20%20%, which of the above is correct?
percentages
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Jun 2, 2017 at 19:04
Chappers
69.2k 11 11 gold badges 74 74 silver badges 153 153 bronze badges
asked Jun 2, 2017 at 15:55
samsam
221 1 1 gold badge 2 2 silver badges 5 5 bronze badges
5
2 25.42×(1−20/100)25.42×(1−20/100)Éric Guirbal –Éric Guirbal 2017-06-02 15:58:19 +00:00 Commented Jun 2, 2017 at 15:58
38 why multiply by 5/6 5/6?Dando18 –Dando18 2017-06-02 16:00:28 +00:00 Commented Jun 2, 2017 at 16:00
11 You tell us neither where the 25.45 25.45 comes from, nor the 0.8333 0.8333.Carsten S –Carsten S 2017-06-02 17:39:19 +00:00 Commented Jun 2, 2017 at 17:39
11 Care to check your figures. You quote 25.45 then 25.42. Can't both be accurate.Tim –Tim 2017-06-03 14:16:42 +00:00 Commented Jun 3, 2017 at 14:16
3 Umm, seriously? This is a school-level problem...nicael –nicael 2017-06-04 09:47:02 +00:00 Commented Jun 4, 2017 at 9:47
Add a comment|
6 Answers 6
Sorted by: Reset to default
This answer is useful
72
Save this answer.
Show activity on this post.
A 20%20% discount means that the price is 80%80% of what it was originally, so you multiply by 1−0.2=0.8 1−0.2=0.8.
0.8 3˙0.8 3˙ is 1/1.2 1/1.2, which is used to find out the original price when you have been given 120%120% of it (if there's 20%20% tax included in it, for example)
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 2, 2017 at 16:00
ChappersChappers
69.2k 11 11 gold badges 74 74 silver badges 153 153 bronze badges
1
4 So 1−r 1−r is not the same as 1 1+r 1 1+r. This is related to the fact that (1−r)(1+r)(1−r)(1+r) is not one exactly (physicists would say it is 1 1 only to the first order in r r). Another well-known way of thinking of it is if you first remove r 100 r 100 percent, then add r 100 r 100 percent of the resulting amount, then you're not back where you started.Jeppe Stig Nielsen –Jeppe Stig Nielsen 2017-06-04 07:17:42 +00:00 Commented Jun 4, 2017 at 7:17
Add a comment|
This answer is useful
17
Save this answer.
Show activity on this post.
Note that multiplying by 0.83 3¯0.83 3¯ is really dividing by 1.2 1.2. In order to know which one is right, you need to keep in mind which number represents 100%100%, or the origin. Calculating percentages from the origin is done by multiplying, while recovering the origin from some given percentage is done by dividing.
So finding 80%80% of a price is done by multiplying with 0.8 0.8. If you have the 80%80% price and want to find the original price, you divide by 0.8 0.8 (which becomes multiplying by 1.25 1.25).
So multiplying with 0.83 3¯0.83 3¯, when we're talking about 20%20% and not 16.6 6¯%16.6 6¯%, is, as mentioned above, really dividing by 1.2 1.2. That means that what we find is the original price when we're given the new price after a 20%20% price increase. That's not the same as finding 80%80%.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Jun 2, 2017 at 16:06
answered Jun 2, 2017 at 16:01
ArthurArthur
205k 14 14 gold badges 186 186 silver badges 332 332 bronze badges
Add a comment|
This answer is useful
15
Save this answer.
Show activity on this post.
An x%x% discount normally means you subtract x%x% of the original price.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 2, 2017 at 15:59
Robert IsraelRobert Israel
472k 28 28 gold badges 376 376 silver badges 714 714 bronze badges
6
28 @TheGreatDuck I disagree.Dan Henderson –Dan Henderson 2017-06-02 22:46:46 +00:00 Commented Jun 2, 2017 at 22:46
3 @TheGreatDuck What don't you understand about my answer?Robert Israel –Robert Israel 2017-06-03 02:13:43 +00:00 Commented Jun 3, 2017 at 2:13
4 @TheGreatDuck Not an ounce of multiplication is even mentioned here Well, "x% of the original price" counts as a multiplication in my books. You take the "original price", then multiply it by "x%" which is in this case "a 20% discount" according to the OP, meaning you multiply by 20%=0.2 20%=0.2. Once you subtract that from the original price x x, what's left to pay is x−0.2 x=0.8 x x−0.2 x=0.8 x. Not sure what your problem was with this answer.dxiv –dxiv 2017-06-03 06:20:34 +00:00 Commented Jun 3, 2017 at 6:20
2 @TheGreatDuck I would expect the OP to very easily connect the x% discount from this answer to a 20% discount from the posted question, as to reckon that x=20 x=20.dxiv –dxiv 2017-06-03 06:26:56 +00:00 Commented Jun 3, 2017 at 6:26
1 @TheGreatDuck Read the question again, what the OP wrote was to work this out I did ... x 0.8333. That is not equivalent to subtracting 20 20 or multiplying by 0.8 0.8, which all other answers pointed out, and so did this one. That is the point I was trying to drive, but don't know how to make any more clear, so I'll leave it at this.dxiv –dxiv 2017-06-03 06:41:55 +00:00 Commented Jun 3, 2017 at 6:41
|Show 1 more comment
This answer is useful
8
Save this answer.
Show activity on this post.
A 20%20%increase in price can be computed by dividing by 0.83333…0.83333… (with 3 3 repeating), but that does not mean a 20%20%decrease results from multiplying by that number. The reason is that in the latter problem we're dealing with 20%20%of a different quantity. A 20%20% decrease is computed by multiplying by 1−0.2=0.8.1−0.2=0.8.
For example, if you cut a $100$100 price by 50%50% and then increase it by 50%,50%, you don't get back $100,$100, but rather $75.$75.
(Multiplying by 1.2 1.2 is simpler than dividing by 0.83333…0.83333… and is more accurate unless you know how to take the infinite repetition of the 3 3 into account (and calculators that I've seen do not know how to do that).
Postscript:
80%of$25.45 80%of$25.45$25.45$25.45=$25.45×0.8=$20.36=$25.45×4 5=$5.09×4=$20.36=5 4×$20.36=5×$5.09=1.25×$20.36=125%of$20.36 80%of$25.45=$25.45×0.8=$20.36 80%of$25.45=$25.45×4 5=$5.09×4=$20.36$25.45=5 4×$20.36=5×$5.09$25.45=1.25×$20.36=125%of$20.36
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Jun 5, 2017 at 16:49
answered Jun 2, 2017 at 16:19
Michael HardyMichael Hardy
1
6
Whatcha mean infinite? You'd only need enough precision to figure out the partial penny (e.g., a tenth of a penny). Even the partial penny only affects rounding, if you do that. If you know the tenth-of-a-penny, there's no reason to figure out the hundredth or thousandth (or infinitth)TOOGAM –TOOGAM 2017-06-03 06:18:12 +00:00 Commented Jun 3, 2017 at 6:18
3 @TOOGAM Not necessarily, you might be buying a thousand units. Your suggestion could mean that you end up paying £833.00 for a thousand one pound items discounted by a sixth, where the correct amount would be £833.33. It's not a big difference, but it's a difference, and that kind of thing tends to make accountants unhappy.Mike Scott –Mike Scott 2017-06-03 12:14:11 +00:00 Commented Jun 3, 2017 at 12:14
1 This is a simple explanation, and makes good sense. +1 Lots of folk (in real life) seem to not have grasped your middle para.Tim –Tim 2017-06-03 14:15:21 +00:00 Commented Jun 3, 2017 at 14:15
@TOOGAM : Rounding before the last step is far more hazardous than you seem to realize.Michael Hardy –Michael Hardy 2017-06-04 23:38:38 +00:00 Commented Jun 4, 2017 at 23:38
@MikeScott and Michael Hardy: Obviously rounding too early can affect things. Why'd both of you think there's another step, such as multiplying the 25.45 b y a t h o u s a n d?B o t h o f y o u s e e m t o i n d i c a t e t h e r e i s a t l e a s t o n e m o r e s t e p,b u t t h e q u e s t i o n d i d n′t s a y a n y t h i n g a b o u t a t h o u s a n d u n i t s,n o r d i d t h e q u e s t i o n s p e c i f y t h a t t h e 20 25.45 b y a t h o u s a n d?B o t h o f y o u s e e m t o i n d i c a t e t h e r e i s a t l e a s t o n e m o r e s t e p,b u t t h e q u e s t i o n d i d n′t s a y a n y t h i n g a b o u t a t h o u s a n d u n i t s,n o r d i d t h e q u e s t i o n s p e c i f y t h a t t h e 20 25.45 "each unit". After discussing "the goods", it says "their total was 20.34", suggesting that we've already added (to handle quantities), and are looking at the more final totals.TOOGAM –TOOGAM 2017-06-05 02:07:47 +00:00 Commented Jun 5, 2017 at 2:07
|Show 1 more comment
This answer is useful
3
Save this answer.
Show activity on this post.
The trouble with percentages (and a major source of math anxiety when people are doing mental arithmetic) is that they are inherently ambiguous. The question most people are not trained to ask is percent of what. Most of the problems occur because when percentages are use the percent of what is not explicitly specified.
In this question a 20% discount probably means deduct 20% from the stated price. But, unless this is stated very explicitly there is still room for confusion. If it really is 20% off the stated price then the correct result is price0.8.
But it is a little ambiguous. Here is an example. In the UK we have a sales tax called VAT levied at 20%. But 20% of what? Legally it is 20% of the pre-tax price but shops have to quote the total price after tax has been applied (something the USA ought to mandate to avoid confusion for foreigners if not locals as well). So if a discount is described as "we pay your VAT" it sounds like a 20% discount to many but is actually a reduction of ~16.7% on the stated price (1/1.2 since the stated price is pre-tax price1.2).
So, if you want to avoid confusion with percentages, always ask percent of what?
And, if you are going to be a good scientist always specify explicitly what your percentages mean. If your drug "reduces deaths by 20%" be very explicit in saying from what base (and tell us what percentage of people die without the drug so we can judge the absolute risk: 10 deaths reduced to 8 is a significant result when only 100 people were in the trial but not much when there are 10,000 people in the trial).
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 4, 2017 at 10:34
matt_blackmatt_black
530 3 3 silver badges 8 8 bronze badges
Add a comment|
This answer is useful
2
Save this answer.
Show activity on this post.
You cannot multiply anything by 0.8333 to arrive at 80% of the original. All you end up with is 83.33% of the original. So to get a discount of 20%, you need to calculate 80% 0f that figure. Don't know where '.8333' came from! That only gives a discount of 16.7%. X times 0.8 does it right. To get that discounted figure back to its original, multiply by 1.25.
Please explain why you thought 0.8333 was a good idea, and also edit the question with one start point value - you wrote two different ones.
Basic answer, the 'less 20%' - aka times 0.8 can be the only way. Even if tax was applied to the item before and after discount, it would make no difference.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 4, 2017 at 12:37
TimTim
129 4 4 bronze badges
2
We have phantom downvoters on this site too! I need to know what the reason is, please!Tim –Tim 2017-06-04 16:44:31 +00:00 Commented Jun 4, 2017 at 16:44
1 I didn't downvote, though my guess would be that downvoters didn't feel that this answer added anything beyond what's already been covered in prior answers. Repetition makes for clutter and consumes readers' time for no return. You'd probably have gotten a lot of upvotes if this were the first answer.Nat –Nat 2017-06-04 21:53:42 +00:00 Commented Jun 4, 2017 at 21:53
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
percentages
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
1Dynamic weighed probability
2Optimal Way To Get Unique Results Given Two Options of Selecting
0Trouble working out course work module precentages.
1Adding a percentage to a number and then subtracting the same percentage to get the same number
7Calculating discount not working
1Can a value of a discount % be calculated by multiplying the item %, by the item value?
0Discount Problem
0Successive Percentage discount
Hot Network Questions
Spectral Leakage & Phase Discontinuites
alignment in a table with custom separator
Numbers Interpreted in Smallest Valid Base
Find non-trivial improvement after submitting
Do we need the author's permission for reference
Sign mismatch in overlap integral matrix elements of contracted GTFs between my code and Gaussian16 results
My dissertation is wrong, but I already defended. How to remedy?
Origin of Australian slang exclamation "struth" meaning greatly surprised
Is it ok to place components "inside" the PCB
Determine which are P-cores/E-cores (Intel CPU)
Implications of using a stream cipher as KDF
Bypassing C64's PETSCII to screen code mapping
Another way to draw RegionDifference of a cylinder and Cuboid
Are there any world leaders who are/were good at chess?
Identifying a movie where a man relives the same day
How do trees drop their leaves?
Interpret G-code
How to locate a leak in an irrigation system?
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
How to start explorer with C: drive selected and shown in folder list?
What is a "non-reversible filter"?
Does the Mishna or Gemara ever explicitly mention the second day of Shavuot?
Suspicious of theorem 36.2 in Munkres “Analysis on Manifolds”
Is existence always locational?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
11743 | https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/07%3A_Further_Applications_of_Trigonometry/7.04%3A_Vectors_in_Three_Dimensions | 7.4: Vectors in Three Dimensions - Mathematics LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
7: Further Applications of Trigonometry
MTH 165 College Algebra, MTH 175 Precalculus
{ "7.04e:_Vectors_in_3D" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "7.01:Non-right_Triangles-Law_of_Sines" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.02:_Non-right_Triangles-_Law_of_Cosines" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.03:_Vectors_in_2D" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.04:_Vectors_in_Three_Dimensions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.05:_The_Dot_Product" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.06:_The_Cross_Product" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "00:_Preliminary_Topics_for_College_Algebra" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Equations_and_Inequalities" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Functions_and_Their_Graphs" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Polynomial_and_Rational_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Exponential_and_Logarithmic_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Trigonometric_Functions_and_Graphs" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Analytic_Trigonometry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Further_Applications_of_Trigonometry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Mon, 12 Dec 2022 03:09:02 GMT
7.4: Vectors in Three Dimensions
34952
34952
Mary Cameron
{ }
Anonymous
Anonymous
2
false
false
[ "article:topic", "Vectors in R3", "sphere", "coordinate plane", "right-hand rule", "octants", "standard equation of a sphere", "three-dimensional rectangular coordinate system", "calcplot:yes", "license:ccbyncsa", "showtoc:yes", "source-math-2587", "source-math-7639", "source-math-7503", "source-math-9094", "source-math-2587" ]
[ "article:topic", "Vectors in R3", "sphere", "coordinate plane", "right-hand rule", "octants", "standard equation of a sphere", "three-dimensional rectangular coordinate system", "calcplot:yes", "license:ccbyncsa", "showtoc:yes", "source-math-2587", "source-math-7639", "source-math-7503", "source-math-9094", "source-math-2587" ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Expand/collapse global hierarchy
1. Home
2. Campus Bookshelves
3. Monroe Community College
4. MTH 165 College Algebra, MTH 175 Precalculus
5. 7: Further Applications of Trigonometry
6. 7.4: Vectors in Three Dimensions
Expand/collapse global location
7.4: Vectors in Three Dimensions
Last updated Dec 12, 2022
Save as PDF
7.3e: Exercises - Vectors in 2D
7.4e: Vectors in 3D
Page ID 34952
( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}})
( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{#1}}} )
( \newcommand{\id}{\mathrm{id}}) ( \newcommand{\Span}{\mathrm{span}})
( \newcommand{\kernel}{\mathrm{null}\,}) ( \newcommand{\range}{\mathrm{range}\,})
( \newcommand{\RealPart}{\mathrm{Re}}) ( \newcommand{\ImaginaryPart}{\mathrm{Im}})
( \newcommand{\Argument}{\mathrm{Arg}}) ( \newcommand{\norm}{\| #1 \|})
( \newcommand{\inner}{\langle #1, #2 \rangle})
( \newcommand{\Span}{\mathrm{span}})
( \newcommand{\id}{\mathrm{id}})
( \newcommand{\Span}{\mathrm{span}})
( \newcommand{\kernel}{\mathrm{null}\,})
( \newcommand{\range}{\mathrm{range}\,})
( \newcommand{\RealPart}{\mathrm{Re}})
( \newcommand{\ImaginaryPart}{\mathrm{Im}})
( \newcommand{\Argument}{\mathrm{Arg}})
( \newcommand{\norm}{\| #1 \|})
( \newcommand{\inner}{\langle #1, #2 \rangle})
( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\AA}{\unicode[.8,0]{x212B}})
( \newcommand{\vectorA}{\vec{#1}} % arrow)
( \newcommand{\vectorAt}{\vec{\text{#1}}} % arrow)
( \newcommand{\vectorB}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}})
( \newcommand{\vectorC}{\textbf{#1}})
( \newcommand{\vectorD}{\overrightarrow{#1}})
( \newcommand{\vectorDt}{\overrightarrow{\text{#1}}})
( \newcommand{\vectE}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} )
( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}})
( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{#1}}} )
(\newcommand{\avec}{\mathbf a}) (\newcommand{\bvec}{\mathbf b}) (\newcommand{\cvec}{\mathbf c}) (\newcommand{\dvec}{\mathbf d}) (\newcommand{\dtil}{\widetilde{\mathbf d}}) (\newcommand{\evec}{\mathbf e}) (\newcommand{\fvec}{\mathbf f}) (\newcommand{\nvec}{\mathbf n}) (\newcommand{\pvec}{\mathbf p}) (\newcommand{\qvec}{\mathbf q}) (\newcommand{\svec}{\mathbf s}) (\newcommand{\tvec}{\mathbf t}) (\newcommand{\uvec}{\mathbf u}) (\newcommand{\vvec}{\mathbf v}) (\newcommand{\wvec}{\mathbf w}) (\newcommand{\xvec}{\mathbf x}) (\newcommand{\yvec}{\mathbf y}) (\newcommand{\zvec}{\mathbf z}) (\newcommand{\rvec}{\mathbf r}) (\newcommand{\mvec}{\mathbf m}) (\newcommand{\zerovec}{\mathbf 0}) (\newcommand{\onevec}{\mathbf 1}) (\newcommand{\real}{\mathbb R}) (\newcommand{\twovec}{\left[\begin{array}{r}#1 \ #2 \end{array}\right]}) (\newcommand{\ctwovec}{\left[\begin{array}{c}#1 \ #2 \end{array}\right]}) (\newcommand{\threevec}{\left[\begin{array}{r}#1 \ #2 \ #3 \end{array}\right]}) (\newcommand{\cthreevec}{\left[\begin{array}{c}#1 \ #2 \ #3 \end{array}\right]}) (\newcommand{\fourvec}{\left[\begin{array}{r}#1 \ #2 \ #3 \ #4 \end{array}\right]}) (\newcommand{\cfourvec}{\left[\begin{array}{c}#1 \ #2 \ #3 \ #4 \end{array}\right]}) (\newcommand{\fivevec}{\left[\begin{array}{r}#1 \ #2 \ #3 \ #4 \ #5 \ \end{array}\right]}) (\newcommand{\cfivevec}{\left[\begin{array}{c}#1 \ #2 \ #3 \ #4 \ #5 \ \end{array}\right]}) (\newcommand{\mattwo}{\left[\begin{array}{rr}#1 \amp #2 \ #3 \amp #4 \ \end{array}\right]}) (\newcommand{\laspan}{\text{Span}{#1}}) (\newcommand{\bcal}{\cal B}) (\newcommand{\ccal}{\cal C}) (\newcommand{\scal}{\cal S}) (\newcommand{\wcal}{\cal W}) (\newcommand{\ecal}{\cal E}) (\newcommand{\coords}{\left{#1\right}_{#2}}) (\newcommand{\gray}{\color{gray}{#1}}) (\newcommand{\lgray}{\color{lightgray}{#1}}) (\newcommand{\rank}{\operatorname{rank}}) (\newcommand{\row}{\text{Row}}) (\newcommand{\col}{\text{Col}}) (\renewcommand{\row}{\text{Row}}) (\newcommand{\nul}{\text{Nul}}) (\newcommand{\var}{\text{Var}}) (\newcommand{\corr}{\text{corr}}) (\newcommand{\len}{\left|#1\right|}) (\newcommand{\bbar}{\overline{\bvec}}) (\newcommand{\bhat}{\widehat{\bvec}}) (\newcommand{\bperp}{\bvec^\perp}) (\newcommand{\xhat}{\widehat{\xvec}}) (\newcommand{\vhat}{\widehat{\vvec}}) (\newcommand{\uhat}{\widehat{\uvec}}) (\newcommand{\what}{\widehat{\wvec}}) (\newcommand{\Sighat}{\widehat{\Sigma}}) (\newcommand{\lt}{<}) (\newcommand{\gt}{>}) (\newcommand{\amp}{&}) (\definecolor{fillinmathshade}{gray}{0.9})
Table of contents
1. Three-Dimensional Coordinate Systems
2. Working with Vectors in (ℝ^3)
1. Key Concepts
Glossary
Contributors and Attributions
Learning Objectives
Locate points in space using coordinates.
Write the distance formula in three dimensions.
Perform vector operations in (\mathbb{R}^{3}).
Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.
Three-Dimensional Coordinate Systems
As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal (x)-axis and the vertical(y)-axis. We can add a third dimension, the (z)-axis, which is perpendicular to both the (x)-axis and the (y)-axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.
Definition: Three-dimensional Rectangular Coordinate System
The three-dimensional rectangular coordinate system consists of three perpendicular axes: the (x)-axis, the (y)-axis, and the (z)-axis. Because each axis is a number line representing all real numbers in (ℝ), the three-dimensional system is often denoted by (ℝ^3).
In Figure (\PageIndex{1a}), the positive(z)-axis is shown above the plane containing the (x)- and (y)-axes. The positive (x)-axis appears to the left and the positive (y)-axis is to the right. A natural question to ask is: How was this arrangement determined? The system displayed follows the right-hand rule. If we take our right hand and align the fingers with the positive (x)-axis, then curl the fingers so they point in the direction of the positive(y)-axis, our thumb points in the direction of the positive (z)-axis (Figure (\PageIndex{1b})). In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.
Figure (\PageIndex{1}): (a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the (z)-axis, that is perpendicular to both the (x)-axis and the (y)-axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.
In two dimensions, we describe a point in the plane with the coordinates ((x,y)). Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, (z), is appended to indicate alignment with the (z)-axis: ((x,y,z)). A point in space is identified by all three coordinates (Figure (\PageIndex{2})). To plot the point ((x,y,z)), go (x) units along the(x)-axis, then(y) units in the direction of the (y)-axis, then (z) units in the direction of the (z)-axis.
Figure (\PageIndex{2}): To plot the point ((x,y,z)) go (x) units along the (x)-axis, then (y) units in the direction of the (y)-axis, then (z) units in the direction of the (z)-axis.
Example (\PageIndex{1}): Locating Points in Space
Sketch the point ((1,−2,3)) in three-dimensional space.
Solution
To sketch a point, start by sketching three sides of a rectangular prism along the coordinate axes: one unit in the positive (x) direction, (2) units in the negative (y) direction, and (3) units in the positive (z)direction. Complete the prism to plot the point (Figure 3).
Figure (\PageIndex{3}): Sketching the point ((1,−2,3).)
Try It (\PageIndex{1})
Sketch the point ((−2,3,−1)) in three-dimensional space.
Hint
Start by sketching the coordinate axes. e.g., Figure (\PageIndex{3b}). Then sketch a rectangular prism to help find the point in space.Answer
Most work in three-dimensional space is a comfortable extension of the corresponding concepts in two dimensions. We begin by adapting the distance formula to three-dimensional space.
If two points lie in the same coordinate plane, then it is straightforward to calculate the distance between them. We that the distance (d) between two points ((x_1,y_1)) and ((x_2,y_2)) in the x(y)-coordinate plane is given by the formula
(d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}.)
The formula for the distance between two points in space is a natural extension of this formula.
Definition: The Distance between Two Points in Space
The distance (d) between points ((x_1,y_1,z_1)) and ((x_2,y_2,z_2)) is given by the formula
(d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2+(z_2−z_1)^2}. \label{distanceForm})
The proof of this theorem is left as an exercise. (Hint: First find the distance (d_1) between the points ((x_1,y_1,z_1)) and ((x_2,y_2,z_1)) as shown in Figure (\PageIndex{4}).
Figure (\PageIndex{4}): The distance between (P_1) and (P_2) is the length of the diagonal of the rectangular prism having (P_1) and (P_2) as opposite corners.
Example (\PageIndex{2}): Distance in Space
Find the distance between points (P_1=(3,−1,5)) and (P_2=(2,1,−1).)
Figure (\PageIndex{5}): Find the distance between the two points.
Solution
Substitute values directly into the distance formula:
[\begin{align} d(P_1,P_2) &=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2+(z_2−z_1)^2} \[4pt] &=\sqrt{(2−3)^2+(1−(−1))^2+(−1−5)^2} \[4pt] &=\sqrt{(-1)^2+2^2+(−6)^2} =\sqrt{41}. \end{align}]
Try It(\PageIndex{2})
Find the distance between points (P_1=(1,−5,4)) and (P_2=(4,−1,−1)).
Hint
(d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2+(z_2−z_1)^2})
Answer
(5\sqrt{2})
Working with Vectors in (ℝ^3)
Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow.
Three-dimensional vectors can also be represented in component form. The notation (\vecs{v}=⟨x,y,z⟩) is a natural extension of the two-dimensional case, representing a vector with the initial point at the origin, ((0,0,0)), and terminal point ((x,y,z)). The zero vector is (\vecs{0}=⟨0,0,0⟩). So, for example, the three dimensional vector (\vecs{v}=⟨2,4,1⟩) is represented by a directed line segment from point ((0,0,0)) to point ((2,4,1)) (Figure (\PageIndex{6})).
Figure (\PageIndex{6}): Vector (\vecs{v}=⟨2,4,1⟩) is represented by a directed line segment from point ((0,0,0)) to point ((2,4,1).)
Vector addition and scalar multiplication are defined analogously to the two-dimensional case. If (\vecs{v}=⟨x_1,y_1,z_1⟩) and (\vecs{w}=⟨x_2,y_2,z_2⟩) are vectors, and (k) is a scalar, then
(\vecs{v}+\vecs{w}=⟨x_1+x_2,y_1⟩+⟨y_2,z_1+z_2⟩)
and
(k\vecs{v}=⟨kx_1,ky_1,kz_1⟩.)
If (k=−1,) then (k\vecs{v}=(−1)\vecs{v}) is written as (−\vecs{v}), and vector subtraction is defined by (\vecs{v}−\vecs{w}=\vecs{v}+(−\vecs{w})=\vecs{v}+(−1)\vecs{w}).
The standard unit vectors extend easily into three dimensions as well, (\hat{\mathbf i}=⟨1,0,0⟩), (\hat{\mathbf j}=⟨0,1,0⟩), and (\hat{\mathbf k}=⟨0,0,1⟩), and we use them in the same way we used the standard unit vectors in two dimensions. Thus, we can represent a vector in (ℝ^3) in the following ways:
(\vecs{v}=⟨x,y,z⟩=x\hat{\mathbf i}+y\hat{\mathbf j}+z\hat{\mathbf k}).
Example (\PageIndex{3}): Vector Representations
Figure (\PageIndex{7}): The vector with initial point (P=(3,12,6)) and terminal point (Q=(−4,−3,2)).
Let (\vecd{PQ}) be the vector with initial point (P=(3,12,6)) and terminal point (Q=(−4,−3,2)) as shown in Figure (\PageIndex{7}). Express (\vecd{PQ}) in both component form and using standard unit vectors.
Solution
In component form,
[\begin{align} \vecd{PQ} =⟨x_2−x_1,y_2−y_1,z_2−z_1⟩ \[4pt] =⟨−4−3,−3−12,2−6⟩=⟨−7,−15,−4⟩. \end{align}]
In standard unit form,
[\vecd{PQ}=−7\hat{\mathbf i}−15\hat{\mathbf j}−4\hat{\mathbf k}. \nonumber]
Try It(\PageIndex{3})
Let (S=(3,8,2)) and (T=(2,−1,3)). Express (\vec{ST}) in component form and in standard unit form.
Hint
Write (\vecd{ST}) in component form first. (T) is the terminal point of (\vecd{ST}).
Answer
(\vecd{ST}=⟨−1,−9,1⟩=−\hat{\mathbf i}−9\hat{\mathbf j}+\hat{\mathbf k})
As described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space (Figure (\PageIndex{8})).
Figure (\PageIndex{8}): To add vectors in three dimensions, we follow the same procedures we learned for two dimensions.
We have already seen how some of the algebraic properties of vectors, such as vector addition and scalar multiplication, can be extended to three dimensions. Other properties can be extended in similar fashion. They are summarized here for our reference.
Properties of Vectors in Space
Let (\vecs{v}=⟨x_1,y_1,z_1⟩) and (\vecs{w}=⟨x_2,y_2,z_2⟩) be vectors, and let (k) be a scalar.
Scalar multiplication: (k\vecs{v}=⟨kx_1,ky_1,kz_1⟩)
Vector addition: (\vecs{v}+\vecs{w}=⟨x_1,y_1,z_1⟩+⟨x_2,y_2,z_2⟩=⟨x_1+x_2,y_1+y_2,z_1+z_2⟩)
Vector subtraction: (\vecs{v}−\vecs{w}=⟨x_1,y_1,z_1⟩−⟨x_2,y_2,z_2⟩=⟨x_1−x_2,y_1−y_2,z_1−z_2⟩)
Vector magnitude: (\|\vecs{v}\|=\sqrt{x_1^2+y_1^2+z_1^2})
Unit vector in the direction of( \vecs{v}: \dfrac{1}{\|\vecs{v}\|}\vecs{v}=\dfrac{1}{\|\vecs{v}\|}⟨x_1,y_1,z_1⟩=⟨\dfrac{x_1}{\|\vecs{v}\|},\dfrac{y_1}{\|\vecs{v}\|},\dfrac{z_1}{\|\vecs{v}\|}⟩, \quad \text{if} \, \vecs{v}≠\vecs{0} )
Example (\PageIndex{4}): Vector Operations in Three Dimensions
Let (\vecs{v}=⟨−2,9,5⟩) and (\vecs{w}=⟨1,−1,0⟩). Find the following vectors.
(3\vecs{v}−2\vecs{w})
(5\|\vecs{w}\|)
(\|5 \vecs{w}\|)
A unit vector in the direction of (\vecs{v})
Solution
a. First, use scalar multiplication of each vector, then combine:
(3\vecs{v}−2\vecs{w} =3⟨−2,9,5⟩−2⟨1,−1,0⟩ =⟨−6,27,15⟩+⟨−2,2,0⟩ =⟨−6−2,27+2,15+0⟩ =⟨−8,29,15⟩. )
b. Write the equation for the magnitude of the vector, then use scalar multiplication:
[5\|\vecs{w}\|=5\sqrt{1^2+(−1)^2+0^2}=5\sqrt{2}. \nonumber]
c. First, use scalar multiplication, then find the magnitude of the new vector. Notice the result is the same as in part b.:
[\|5 \vecs{w}\|=∥⟨5,−5,0⟩∥=\sqrt{5^2+(−5)^2+0^2}=\sqrt{50}=5\sqrt{2} \nonumber]
d. Recall that to find a unit vector in two dimensions, we divide a vector by its magnitude. The procedure is the same in three dimensions:
( \dfrac{\vecs{v}}{\|\vecs{v}\|} =\dfrac{⟨−2,9,5⟩}{\|\vecs{v}\|} =\dfrac{⟨−2,9,5⟩}{\sqrt{(−2)^2+9^2+5^2}}=\dfrac{⟨−2,9,5⟩}{\sqrt{110}} = \left \langle\dfrac{−2}{\sqrt{110}},\dfrac{9}{\sqrt{110}},\dfrac{5}{\sqrt{110}} \right \rangle = \left \langle− \dfrac{\sqrt{110}}{55},\dfrac{9\sqrt{110}}{110},\dfrac{\sqrt{110}}{22} \right \rangle . )
Try It(\PageIndex{4}):
Let (\vecs{v}=⟨−1,−1,1⟩) and (\vecs{w}=⟨2,0,1⟩). Find a unit vector in the direction of (5\vecs{v}+3\vecs{w}.)
Hint
Start by writing (5\vecs{v}+3\vecs{w}) in component form.
Answer
( \left \langle \dfrac{\sqrt{10}}{30},\; −\dfrac{\sqrt{10}}{6},\; \dfrac{4\sqrt{10}}{15} \right \rangle )
Key Concepts
The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples ((x,y,z)) are used to describe the location of a point in space.
The distance (d) between points ((x_1,y_1,z_1)) and ((x_2,y_2,z_2)) is given by the formula [d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2+(z_2−z_1)^2}.\nonumber]
In three dimensions, as in two, vectors are commonly expressed in component form, (v=⟨x,y,z⟩), or in terms of the standard unit vectors, (xi+yj+zk.)
Properties of vectors in space are a natural extension of the properties for vectors in a plane.
Glossary
Right-hand rulea common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the (z)-axis in such a way that the fingers curl from the positive (x)-axis to the positive (y)-axis, the thumb points in the direction of the positive (z)-axisthree-dimensional rectangular coordinate systema coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple ((x,y,z)) that plots its location relative to the defining axes
Contributors and Attributions
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at
7.4: Vectors in Three Dimensions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.
Back to top
7.3e: Exercises - Vectors in 2D
7.4e: Vectors in 3D
Was this article helpful?
Yes
No
Recommended articles
4.2: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
4.3: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
1.2: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
12.2: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
12.3: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
Article typeSection or PageLicenseCC BY-NC-SAShow Page TOCyes
Tags
calcplot:yes
coordinate plane
octants
right-hand rule
source-math-2587
source-math-7639
source-math-7503
source-math-9094
source-math-2587
sphere
standard equation of a sphere
three-dimensional rectangular coordinate system
Vectors in R3
© Copyright 2025 Mathematics LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status× |
11744 | https://www.medrxiv.org/content/10.1101/2025.07.03.25330795v1.full-text | Published Time: 2025-07-05
Platelet CKB expression is associated with elevated serum CK-BB and metastatic phenotype in lung cancer | medRxiv
Skip to main content
Home
Submit
FAQ
Blog
ALERTS / RSS
Resources
About
Search for this keyword
Advanced Search
Follow this preprint
Platelet CKB expression is associated with elevated serum CK-BB and metastatic phenotype in lung cancer
View ORCID ProfileLubei Rao, Chao Hu, Yan Yue, Yajun Luo, Bo Ye, Yue Liang, View ORCID ProfileGuanbin Zhang, Dongsheng Wang
doi:
[This article is a preprint and has not been peer-reviewed what does this mean?]. It reports new medical research that has yet to be evaluated and so should not be used to guide clinical practice.
Lubei Rao
1 Department of Clinical Laboratory, Sichuan Clinical Research Center for Cancer, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, University of Electronic Science and Technology of China, Chengdu, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
ORCID record for Lubei Rao
For correspondence: raolubei@126.com
Chao Hu
2 School of Basic Medical Sciences, Chengdu University of Traditional Chinese Medicine, Chengdu 611137, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
Yan Yue
1 Department of Clinical Laboratory, Sichuan Clinical Research Center for Cancer, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, University of Electronic Science and Technology of China, Chengdu, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
Yajun Luo
3 Department of Gastrointestinal Surgery, Sichuan Clinical Research Center for Cancer, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, University of Electronic Science and Technology of China, Chengdu, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
Bo Ye
1 Department of Clinical Laboratory, Sichuan Clinical Research Center for Cancer, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, University of Electronic Science and Technology of China, Chengdu, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
Yue Liang
4 Department of Information Center, Sichuan Clinical Research Center for Cancer, Sichuan Cancer Hospital & Instituter, Sichuan Cancer Center, University of Electronic Science and Technology of China, Chengdu, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
Guanbin Zhang
5 School of Intelligent Medicine, Chengdu University of Traditional Chinese Medicine, Chengdu 611137, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
ORCID record for Guanbin Zhang
Dongsheng Wang
1 Department of Clinical Laboratory, Sichuan Clinical Research Center for Cancer, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, University of Electronic Science and Technology of China, Chengdu, China
Find this author on Google Scholar
Find this author on PubMed
Search for this author on this site
Abstract
Full Text
Info/History
Metrics
Data/Code
Preview PDF
Abstract
The abnormal increase of serum creatine kinase-MB activity exceeding total creatine kinase activity is an uncommon and paradoxical laboratory finding often observed in patients with cancer. However, the underlying origin and clinical relevance of this phenomenon remain unclear. In this study, we analyzed a large cohort of cancer patients presenting with this biochemical abnormality. We first characterized the cancer types and stages most frequently associated with this pattern, finding enrichment in colorectal and lung cancers, particularly at advanced stages. Serum isoenzyme electrophoresis revealed that the abnormal elevation was not due to myocardial damage but instead driven by high levels of brain-type and mitochondrial creatine kinase isoforms. We introduced a new composite index integrating these non-cardiac isoenzymes, which was correlated with poor prognosis, especially in colorectal cancer. Notably, we identified an unexpected elevation of creatine kinase brain-type messenger RNA in platelets from lung cancer patients. This expression level was significantly higher than that found in serum and showed a strong positive correlation, suggesting that platelets may contribute to the circulating isoenzyme pool. This observation was supported by analysis of a public dataset of tumor-educated platelets, which confirmed higher creatine kinase expression in lung cancer patients. Additional single-cell sequencing data showed that distant metastatic lesions, but not primary tumors, exhibited elevated creatine kinase expression. Functional gene enrichment analysis revealed associated pathways involved in metabolism, oxidative stress, and intercellular signaling. These findings suggest that platelet-related creatine kinase may be linked to metastatic potential in lung cancer and represent a previously underrecognized component in cancer-associated biochemical alterations.
1. Introduction
Creatine kinase (CK) is a central regulator of cellular energy homeostasis (1). It catalyzes the reversible transfer of a phosphate group from adenosine triphosphate (ATP) to creatine, forming phosphocreatine and adenosine diphosphate (ADP) (2). Through this reaction, CK plays a vital role in tissues with high energy demands, such as skeletal muscle, heart, and brain. CK is encoded by multiple genes, forming tissue-specific isoenzyme subtypes. Among them, CKB encodes the creatine kinase brain-type isoenzyme (CK-BB); CKM encodes the creatine kinase muscle-type isoenzyme (CK-MM); and the hybrid creatine kinase-MB isoenzyme (CK-MB) is a heterodimer composed of CKB and CKM gene products. These isoforms are predominantly cytoplasmic (3, 4). In addition, mitochondrial creatine kinase (MtCK), encoded by the CKMT1 and CKMT2 genes, is predominantly located within mitochondria, and exists in the form of mitochondrial creatine kinase isoenzyme II (CKII, dimeric form) and mitochondrial creatine kinase isoenzyme I (CKI, monomeric form) (5, 6), playing a key role in ATP buffering in oxidative tissues.
In healthy individuals, the serum CK pool is predominantly composed of CK-MM, with negligible amounts of other isoenzymes. During myocardial injury, CK-MB is released into the circulation, and serum CK-MB activity measurement is widely used for the diagnosis of myocardial infarction (7–11). However, serum CK-MB activity detected by the immunoinhibition assay (12) may be influenced by the presence of non-CK-MB isoenzymes (13). In particular, elevated levels of CK-BB (14) and mitochondrial CK isoenzymes (CKII) (2, 15) in the serum may cause a falsely elevated CK-MB activity even in the absence of myocardial injury (14, 16).
Studies have shown that when tumor cells enter the blood circulation and face hypoxia, CKB can be released from the cell into the extracellular space and catalyze the production of extracellular creatine phosphate, increase ATP content, maintain the energy required for tumor cell survival, and thus promote metastasis (2, 13, 17–22). Elevated circulating CKB can also interfere with CK-MB and CK activity assays, potentially explaining paradoxical CK-MB > CK patterns observed in cancer patients (23). While sporadic case reports have documented such anomalies, the biological origin and clinical relevance remain poorly characterized (24–27). Our institution, the largest cancer-specialized hospital in Southwest China, frequently observes this abnormal pattern in cancer patients. However, it remains unclear whether this is primarily driven by elevated CKB or other isoforms (28, 29).
Currently, the regulatory mechanisms underlying hematogenous metastasis of tumor cells remain poorly understood. This process is highly inefficient—fewer than 0.1% of tumor cells that enter the bloodstream ultimately succeed in forming metastatic colonies, largely due to environmental shifts and stressors such as hypoxia (17, 30). As the bloodstream serves as the primary route for cancer cells to disseminate from the primary site to distant organs, understanding how tumor cells interact with cellular and acellular components in the circulation may offer novel strategies for identifying or intercepting metastasis (30). Recent studies underscore the importance of platelet–tumor cell interactions in this process (31–34). Circulating tumor cells (CTCs) can bind to and internalize platelet components, coopting the platelet transcriptome and proteome to gain metastatic advantage (35). Tumor educated platelets (TEPs)— platelets reprogrammed by the tumor microenvironment—may facilitate tumor metastasis by secreting or transferring bioactive molecules (36–39). As the second most abundant cell population in the bloodstream, it remains unclear whether platelets can promote tumor hematogenous metastasis through factors such as CKB.
Our previous preliminary research indicated that colorectal cancer patients who developed CK-MB > CK abnormalities after curative resection exhibited significantly poorer prognoses compared to contemporaneous patients without such anomalies, characterized by shorter overall survival and a higher risk of recurrence and metastasis (40). A deeper investigation into the factors contributing to the CK-MB > CK abnormality, and the role played by tumor educated platelets in this process, may offer new insights into the mechanisms of hematogenous tumor metastasis.
In this study, we conducted a comprehensive investigation integrating retrospective cohort analysis, serum CK isoenzyme electrophoresis, and transcriptomic profiling of paired serum and platelet samples. In addition, we leveraged public RNA sequencing datasets to validate the involvement of platelet-associated CKB in lung cancer and to explore its potential role in metastatic dissemination. Our goal was to elucidate the cellular source and clinical im-plications of CK-MB > CK abnormalities, with a particular focus on the interplay between CK isoenzyme dysregulation, platelet biology, and the metastatic phenotype in cancer.
2. Materials and methods
2.1. Patient enrollment and data collection
This study combined retrospective data analysis with prospective sample collection, both of which were approved by the Institutional Ethics Committee of Sichuan Cancer Hospital and exempted from informed consent (IRB No. SCCHEC-02-2022-017).
For the retrospective component, patients diagnosed with malignant tumors and hospitalized at our institution between January 1, 2019, and December 31, 2022, were reviewed. Inclusion criteria required that patients had undergone serum CK-MB and total CK activity testing and exhibited CK-MB > CK abnormalities. The following exclusion criteria were applied: (1) presence of severe skeletal muscle injury or major cardiovascular diseases (e.g., acute myocardial infarction, myocarditis); (2) serum samples showing hemolysis or severe lipemia; (3) age younger than 18 or older than 100 years; (4) incomplete clinical data. Clinical information including sex, age, diagnosis, TNM stage, laboratory indices (CK-MB and total CK activity, platelet-related test results), treatment modalities (chemotherapy, radiotherapy, immunotherapy, targeted therapy), metastasis status, and survival outcomes was extracted from the electronic medical record system. Overall survival (OS) was defined as the time from the date of diagnosis to the date of death or the last follow-up. Retrospective data were accessed for research purposes between January 1, 2023, and May 31, 2023. All data were anonymized prior to analysis, and no personally identifiable information was accessible to the researchers.
For the prospective component, fresh serum and platelet samples were collected from a subset of colorectal and lung cancer patients admitted between April 1, 2021, and May 31, 2022. Samples were residual clinical specimens obtained during routine diagnostic testing and were anonymized prior to research use.
This study was conducted in accordance with the Declaration of Helsinki and complied with institutional and international ethical guidelines for human research. All data were independently reviewed and verified by two researchers.
2.2. Serum sample collection and laboratory testing
Peripheral venous blood samples were collected using BD Vacutainer tubes with clot activator and gel separator (BD, USA). Samples were centrifuged at 4,000 rpm for 8 minutes, and the supernatant serum was carefully transferred into RNase-free Eppendorf tubes and immediately stored at −80°C. Hemolytic, icteric, or lipemic samples were excluded from the analysis.
Serum total CK and CK-MB activities were measured using the phosphocreatine substrate method and immunoinhibition assay, respectively, on a Mindray BS-2000M automated biochemical analyzer (Mindray, China). The reference range for CK-MB was ≤25 U/L. The reference range for total CK was 24–194 U/L in males and 24–170 U/L in females. An abnormal result was defined as a CK-MB/CK ratio greater than 1.
Quantitative analysis of CK isoenzyme subtypes in serum samples was performed using the semi-automated HYDRASYS electrophoresis system and the HYDRAGEL ISO-CK kit (Sebia, France). Electrophoresis was conducted on an agarose gel in an alkaline buffer (pH 8.4). Densitometric analysis was performed after electrophoresis to obtain the relative quantification of each CK isoenzyme fraction.
In this study, we defined a novel composite index termed CK-Sub, representing the sum of non-CK-MM and non-CK-MB isoenzymes detected by serum isoenzyme electrophoresis. Specifically, CK-Sub was calculated as the combined relative abundance of CK-BB, CKII, and CKI. This composite measure was designed to reflect the overall contribution of non-cardiac, non-skeletal muscle CK isoenzymes, and to better capture the atypical CK isoenzyme activation patterns observed in cancer patients with CK-MB > CK abnormalities (defined as CK-MB/CK ratio abnormalities).
2.3. Platelet testing, separation and purity verification
First, venous blood was collected using EDTA anticoagulation tubes (BD, USA), and platelet-related parameters, including platelet count (PLT), mean platelet volume (MPV), and platelet distribution width (PDW), were measured using BC-5390 and BC-6800 fully automatic hematology analyzers (Mindray, Shenzhen). The instrument was internally quality controlled daily and calibrated weekly, and all results met the acceptable standards.
Then, whole blood samples (1.5 mL), anticoagulated with EDTA, were transferred into 2 mL microcentrifuge tubes. To obtain platelet-rich plasma (PRP), samples were centrifuged at 120 × g for 20 minutes (Shuke Instruments, Sichuan, China), effectively separating platelets from nucleated cells. To reduce contamination from leukocytes and erythrocytes, only the upper 80% of the PRP layer was carefully aspirated for subsequent procedures. Platelets were then pelleted by a second centrifugation step at 360 × g for 20 minutes. All isolation steps were completed within 2 hours of venipuncture to prevent RNA degradation. Platelet purity was assessed using morphological examination and automated platelet counts, and samples were considered acceptable if leukocyte contamination was below 5 white blood cells per 1 million platelets.
2.4. Quantitative reverse transcription PCR
Primer sequences were designed and synthesized by Tsingke Biological Technology Co., Ltd. (Beijing, China). The relative gene expression was quantified using the ΔCT method, with GAPDH serving as the internal control. The primer sequences used in this study were as follows: GAPDH (forward: GGAGCGAGATCCCTCCAAAAT; reverse: GGCTGTTGTCATACTTCTCATGG), CKB (forward: GCTGCGACTTCAGAAGCGA; reverse: GGCATGAGGTCGTCGATGG), CKMT1 (forward: TTGATTGTGAACGGCGTCTG; reverse: ATGCTTGGTGTGGATGACAG), and CKMT2 (forward: AGGTGACACCCAACGGCTA; reverse: TGACGGGGTCAAAAAGGTCAG).
All RT-qPCR amplification experiments were performed on a LightCycler 480 system (Roche Diagnostics). The reactions utilized the TB Green Premix Ex Taq II kit (TaKaRa, Dalian, China) according to the manufacturer’s instructions. Each assay was carried out in triplicate for biological validation. Reactions were set up in a total volume of 20 μL, consisting of 10 μL of SYBR Green Master Mix, 0.8 μL of each forward and reverse primer, 2 μL of cDNA template, and 6.4 μL of RNase-/DNase-free water. The thermal cycling conditions were: initial denaturation at 95°C for 2 minutes and 30 seconds, followed by 40 cycles of denaturation at 95°C for 5 seconds and annealing/extension at 60°C for 30 seconds. Primer specificity was confirmed by observing a single peak in the melting curve analysis.
2.5. Data acquisition from public repositories
Transcriptomic datasets were obtained from the Gene Expression Omnibus (GEO) database ( The dataset GSE89843 contains RNA-sequencing profiles of tumor-educated platelets (TEPs) from 402 non-small cell lung cancer (NSCLC) patients and 377 control individuals. The dataset GSE123902 includes single-cell RNA-seq data from clinical specimens across different disease stages: non-tumor-involved lung tissue (n = 4), primary lung adenocarcinomas (n = 8), and distant metastatic lesions located in the brain (n = 3), bone (n = 1), and adrenal gland (n = 1). The GSE74639 dataset includes RNA-Seq data of primary lung tumor cells, circulating lung tumor cells, and peripheral blood leukocytes. Additionally, bulk RNA-seq data of primary lung tumors and corresponding clinical information were retrieved from The Cancer Genome Atlas (TCGA) via the Genomic Data Commons (GDC) portal ( to assess the clinical and prognostic relevance of CKB expression in tumor tissue.
2.6. Statistical analysis
All statistical analyses were performed using R software (version 4.3.1; R Foundation for Statistical Computing, Vienna, Austria. accessed on June 16, 2023). Continuous variables were presented as medians with interquartile ranges (IQRs), and categorical variables were summarized as counts and percentages. For group comparisons of continuous variables, the Wilcoxon rank-sum test was used due to non-normal data distributions.
Enrichment ratio was calculated as the proportion of each cancer type among CK-MB > CK abnormal cases divided by the proportion of that cancer type in the overall cancer patient cohort: Abnormality rate for each cancer type was calculated as: Multivariate linear regression analysis was conducted to identify independent clinical predictors of CK-MB/CK ratio, with CK-MB/CK ratio as the dependent variable, and age, sex, tumor stage, and treatment status as covariates. Treatment status was categorized as yes or no based on whether patients received any form of adjuvant therapy. Principal component analysis (PCA) was performed to explore the overall patterns of CK isoenzyme expression profiles, and PERMANOVA (permutational multivariate analysis of variance) was used to assess statistical differences between groups. Linear regression analysis was performed to assess the association between CK-Sub levels and CK-MB/CK ratio. Patients were stratified into high and low expression groups based on the optimal cutoff determined by the Youden index from receiver operating characteristic (ROC) curve analysis. Survival analyses were conducted using Kaplan-Meier curves, and differences between survival distributions were compared using the log-rank test.
To explore the potential involvement of platelets in the development of abnormal CK isoenzyme profiles, we evaluated the correlation between CK-Sub, CKBB, or CKII levels and platelet-related hematological indices, PLT, MPV, and PDW. Spearman correlation analysis was first performed to identify monotonic relationships between CK-Sub, CKBB, or CKII and each platelet parameter. Subsequently, multivariate linear regression models were constructed with CK-Sub, CKBB, or CKII as the dependent variable, adjusting for potential confounders such as tumor type, TNM stage, age, and sex. The variance inflation factor (VIF) was calculated to assess multicollinearity.
Gene expression levels were quantified as ΔCT values, and relative expression was calculated using the 2^-ΔCT method. For paired samples (i.e., matched serum and platelet samples from the same patient), the Wilcoxon signed-rank test was used to compare gene expression between compartments. If the data met normality and homogeneity assumptions, a paired t-test was applied instead. For unpaired comparisons (e.g., platelet or serum expression between colorectal and lung cancer patients), the Mann–Whitney U test or independent-samples t-test was employed depending on the distribution of the data.
Raw gene expression matrices were retrieved and processed using the GEOquery package (v2.66.0) and limma (v3.56.2). Normalized CKB mRNA expression levels were extracted and, where necessary, transformed into log₂(TPM + 1). Statistical comparisons between groups were conducted using the Wilcoxon rank-sum test or paired Wilcoxon signed-rank test, as appropriate.
TCGA-LUAD RNA sequencing data and corresponding clinical survival information were accessed via the UCSC Xena browser. Patients were stratified into high and low CKB expression groups based on the median CKB expression level. Kaplan-Meier survival curves were plotted, and log-rank tests were used to assess statistical significance. Hazard ratios (HRs) and 95% confidence intervals (CIs) were estimated using Cox proportional hazards regression analysis.
For differential gene expression (DEG) analysis of platelet RNA-seq data from lung cancer patients, the DESeq2 package (v1.36.0) was applied. Genes significantly correlated with platelet CKB expression (adjusted p < 0.05 and |log₂FC| > 1) were intersected with CKB-associated DEGs from tumor tissues. Functional enrichment analyses, including Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway analysis, were performed using the clusterProfiler package (v4.4.4). Data visualization, including expression distribution and group comparisons, was performed using ggplot2 (v3.4.2).
A two-tailed P-value < 0.05 was considered statistically significant for all analyses.
3. Results
At first, patients with CK-MB > CK anomalies were retrospectively identified from a large-scale cancer cohort, and cancer types with high prevalence or enrichment were characterized. Then, CK isoenzyme profiles in serum were analyzed using electrophoresis and qRT-PCR to identify cancer-specific patterns. Next, correlations between platelet parameters and CK isoenzymes were explored, and the expression of CKB in paired serum and platelet samples was quantitatively validated. Finally, public transcriptomic datasets were utilized to confirm CKB overexpression in lung cancer platelets, investigate its metastatic relevance, and elucidate potential functional pathways. The overall workflow and key findings were illustrated in Fig 1.
Download figure
Open in new tab
Fig 1.Study workflow and key findings.
Background: In cancer patients, the clinical association between serum CK-MB > CK abnormality and tumors is unclear. Methods: First, a large cancer cohort was screened to characterize the epidemiological distribution of CK-MB > CK abnormality. Then, isozyme analysis and qRT-PCR were used to identify cancer type-specific expression of CK isozymes in serum and platelets. Further, the expression of platelet CK was validated using a public TEP dataset (GSE89843), and single-cell RNA sequencing data (GSE123902) were analyzed to explore the role of CKB in lung cancer metastasis. Results: CKB was identified as a platelet-derived transcript enriched in lung cancer patients with elevated serum CK-BB levels. Platelet CKB expression was associated with distant metastasis and poor prognosis, suggesting its potential functional role in shaping the metastatic tumor microenvironment. Abbreviations: CK-MB, creatine kinase-MB isoenzyme; CK, creatine kinase; CRC, colorectal cancer; CK-BB, creatine kinase brain-type isoenzyme; CKII, mitochondrial creatine kinase isoenzyme II; PLT, platelet count; TEPs, tumor-enhanced platelets.
3.1 Patient characteristics and clinical correlates of CK-MB > CK abnormalities
A total of 1,651 cancer patients presenting with CK-MB > CK abnormalities were retrospectively enrolled between 2019 and 2022. Most patients were aged between 40 and 79 years (Fig 2A). Based on ICD codes (S1 Table), colorectal cancer (291 cases, 17.6%), lung cancer (265 cases, 16.1%), and female reproductive system tumors (214 cases, 13.0%) were the top three cancer types represented (Fig 2B). To correct for differences in overall hospitalization frequencies across cancer types, we normalized CK-MB > CK occurrence against the total number of patients per cancer type. This revealed that colorectal cancer exhibited the highest enrichment and abnormality rate, indicating a strong association with CK-MB > CK anomalies. In contrast, the high absolute case count in lung cancer likely reflected its high population prevalence rather than specific enrichment (Fig 2C).
Download figure
Open in new tab
Fig 2.Patient characteristics and clinical correlates of CK-MB > CK abnormalities.
(A) Age and sex distribution of patients with CK-MB > CK abnormality from 2019 to 2022. (B) Top 10 malignant tumor types among patients with CK-MB > CK abnormality, based on patient counts. (C) Adjusted distribution of tumor types after normalizing by total visits during the same period. Statistical methods: (D) Age and sex distribution of colorectal and lung cancer patients with CK-MB > CK abnormality. (E) Clinical characteristics (age group, sex, and tumor stage) of colorectal cancer patients. (F) Clinical characteristics (age group, sex, and tumor stage) of lung cancer patients. (G–I) CK-MB/CK ratio comparisons across tumor stages, adjuvant therapy status, and different treatment types in colorectal cancer patients. (J–L) CK-MB/CK ratio comparisons across tumor stages, adjuvant therapy status, and different treatment types in lung cancer patients. Enrichment ratio was calculated as the proportion of each cancer type among CK-MB > CK abnormal cases divided by the proportion of that cancer type in the overall cancer patient cohort. Abnormality rate was calculated as the proportion of CK-MB > CK abnormal cases within each cancer type divided by the total number of patients diagnosed with that cancer type. Statistical methods: Pairwise comparisons of CK-MB/CK ratio between groups were conducted using the Wilcoxon rank-sum test. Abbreviations: CK-MB, creatine kinase-MB isoenzyme; CK, creatine kinase; chemo, chemotherapy; radio, radiotherapy; targeted, targeted therapy; immuno, immunotherapy.
To further explore the clinical relevance of these abnormalities, we focused on colorectal and lung cancer patients. In colorectal cancer (n = 291), the majority of patients with CK-MB > CK anomalies were aged between 40–80 years (Fig 2D) and stage IV (89.4%) (Fig 2E). A significantly higher CK-MB/CK ratio was observed in stage IV compared to stage III patients (P = 0.023, Fig 2G), suggesting a potential link with tumor progression. However, no significant associations were found between CK-MB/CK ratio and treatment modality or adjuvant therapy status (Figs 2H–1I). Multivariate regression confirmed that age, sex, stage, and therapy were not independently associated with CK-MB/CK ratio levels (S2 Table).
Similarly, in the lung cancer cohort (n = 265), patients also tended to be older (Fig 2D), with a male predominance (74.7%) and high proportion of advanced-stage disease (88.6% stage IV, Fig 2F). No significant associations were observed between CK-MB/CK ratio and tumor stage, adjuvant therapy, or treatment modality (Figs 2J–1L), and multivariate analysis again showed no independent clinical predictors of CK-MB/CK ratio elevation (S2 Table), suggesting that additional factors may be involved in explaining the observed CK-MB > CK abnormalities.
3.2. Creatine kinase isoenzyme profile analysis in colorectal and lung cancer patients
Isoenzyme electrophoresis was performed prospectively on 92 serum samples (62 colorectal, 30 lung cancer patients) exhibiting CK-MB > CK abnormalities (Fig 3A). All patients were stage III/IV (Fig 3B), consistent with the retrospective cohort, further suggesting that CK-MB > CK abnormalities were closely associated with enhanced tumor aggressiveness. Electrophoretic analysis revealed consistent abnormal bands across samples, predominantly involving elevated CK-BB and CKII isoenzymes, with most patients displaying a bimodal peak pattern (Fig 3C). Further comparison between colorectal and lung cancer patients showed that CKII elevation predominated in colorectal cancer, whereas both CKII and CK-BB elevations were observed in lung cancer (Fig 3D).
Download figure
Open in new tab
Fig 3.Isoenzyme profile analysis in colorectal and lung cancer patients with CK-MB > CK abnormality.
(A) Age and sex distribution of colorectal and lung cancer patients included in the isoenzyme electrophoresis analysis. (B) Clinical characteristics of colorectal and lung cancer patients. (C) Stacked bar plot of CK isozyme composition for each patient. (D) Overall isoenzyme composition by cancer type. (E) Principal component analysis (PCA) showing separation of colorectal and lung cancer patients based on isoenzyme expression profiles. (F) PCA loading plot indicating the contribution of different CK isoenzymes (CK-BB, CKII, CK-MB, CK-MM, CKI) to the variance. (G-H) Comparison of CK-BB and CKII expression between lung cancer and colorectal cancer patients. Statistical methods: Pairwise comparisons were conducted using the Wilcoxon rank-sum test. Abbreviations: CK, creatine kinase; CK-MB, creatine kinase-MB isoenzyme; CK-BB, creatine kinase brain-type isoenzyme; CK-MM, creatine kinase muscle-type isoenzyme; CKII, mitochondrial creatine kinase isoenzyme II; CKI, mitochondrial creatine kinase isoenzyme I; PCA, principal component analysis.
PCA showed that colorectal cancer samples clustered along Dim1, associated with increased CKII expression, whereas lung cancer samples clustered along Dim2, associated with CK-BB elevation (PERMANOVA, P=0.001) (Figs 3E–2F). Comparative analysis confirmed that lung cancer patients had significantly higher CK-BB levels (P=1.3e-5) and lower CKII levels (P=0.031) compared to colorectal cancer patients (Figs 3G–2H), suggesting cancer type-specific CK isoenzyme heterogeneity.
3.3. Clinical relevance of CK isoenzyme alterations in colorectal and lung cancer
To investigate the clinical implications of serum CK isoenzyme abnormalities in cancer patients with CK-MB > CK patterns, we conducted a stratified analysis of colorectal cancer (n=62) and lung cancer (n=30) patients who underwent isoenzyme electrophoresis.
In colorectal cancer patients, linear regression analysis revealed a mild positive linear relationship between CK-Sub (CK-Sub =CK-BB + CKII + CKI) levels and CK-MB/CK ratio (R² = 0.098), although the explanatory power was limited, suggesting that variations in CK-Sub alone may be insufficient to effectively predict CK-MB/CK ratio abnormalities (Fig 4A). We found that stage IV patients exhibited significantly higher CKII (P = 0.023) and CK-Sub (P = 0.022) levels compared with stage III patients (Figs 4C–3D), indicating that specific isoenzyme upregulation may be linked to tumor progression. In contrast, no significant differences were found in CK-BB expression (P = 0.41) or CK-MB/CK ratio (P = 0.86) between stages (S1 Fig), suggesting that the CK-MB/CK ratio alone may not sensitively reflect tumor advancement. Kaplan-Meier survival analysis revealed that patients with high CKII expression had significantly shorter overall survival (Fig 4F). Similarly, elevated CK-Sub levels were associated with worse survival outcomes (HR, 0.17; 95%CI, 0.0421-0.688; P = 0.0049) (Fig 4G). Although the overall variation in CK-MB/CK ratio was limited, alterations in individual isoenzyme components, particularly CKII, or the total CK-Sub, may serve as potential biomarkers for tumor progression in advanced-stage colorectal cancer.
Download figure
Open in new tab
Fig 4.Serum CK isoenzyme alterations and clinical relevance in colorectal and lung cancer patients with CK-MB > CK abnormality.
(A) Linear regression analysis between CK-Sub levels and CK-MB/CK ratio in colorectal cancer patients. (B) Linear regression analysis between CK-Sub levels and CK-MB/CK ratio in lung cancer patients. (C-D) Comparison of CKII, and CK-Sub between stage III and stage IV colorectal cancer patients. (E) Comparison of CK-BB expression between stage III and stage IV lung cancer patients. (F–G) Kaplan-Meier survival analyses of overall survival according to CKII and CK-Sub levels in colorectal cancer patients. (H–I) Kaplan-Meier survival analyses of overall survival according to CK-BB and CK-Sub expression levels in lung cancer patients. Statistical methods: Linear regression analysis was used to assess the association between CK-MB/CK ratio and CK-Sub levels. R² value represents the proportion of variance in CK-Sub explained by CK-MB/CK ratio. Group comparisons were performed using the Wilcoxon rank-sum test. Overall survival was analyzed by Kaplan-Meier curves, and differences between groups were assessed with the log-rank test. Abbreviations: CK-MB, creatine kinase-MB isoenzyme; CK, creatine kinase; CKII, mitochondrial creatine kinase isoenzyme II; CK-BB, creatine kinase brain-type isoenzyme; CK-Sub, total CK isoenzyme activity (CK-BB + CKII + CKI); CRC, colorectal.
Unlike the mild positive linear trend observed in colorectal cancer patients, linear regression analysis in lung cancer patients revealed no significant linear relationship between CK-Sub levels and CK-MB/CK ratio (Fig 4B). In the comparison across tumor stages, CK-BB levels were higher in stage IV patients; however, the difference did not reach statistical significance (P = 0.44) (Fig 4E), possibly due to the limited sample size. Similarly, no significant differences were observed in CKII, CK-Sub levels, or CK-MB/CK ratio between different tumor stages (S1 Fig). Regarding survival outcomes, although CK-BB levels did not significantly predict overall survival (P=0.23) (Fig 4H), a general trend toward poorer survival in the high-expression group was noted. Notably, patients with high CK-Sub levels had significantly shorter overall survival (P = 0.012) (Fig 4I). These findings suggest that tumor progression in lung cancer may involve abnormal activation of multiple CK isoenzyme subtypes. Although individual isoenzyme markers may not independently predict prognosis due to sample size limitations, the overall increase in total CK isoenzyme activity (CK-Sub) appears to be closely associated with poorer clinical outcomes.
3.4 Association between platelet parameters and CK isoenzymes
To investigate whether platelets are involved in the aberrant expression patterns of CK isoenzymes observed in cancer patients, we first assessed the correlations between routine platelet-related hematological parameters (including PLT, MPV, and PDW) and serum CK isoenzyme indices (CK-Sub, CK-BB, and CKII) in patients with colorectal and lung cancers. After adjusting for potential confounding variables such as cancer type, tumor stage, age, and sex using multiple linear regression, we found a significant inverse association between PLT and serum CK-BB levels (β = –0.038, p = 0.034). Variance inflation factor (VIF) analysis confirmed the absence of multicollinearity among covariates (all VIFs < 2.5), supporting the robustness of the model (Table 1). These findings suggest that reduced PLT may be closely linked to elevated serum CK-BB levels. Notably, upon inclusion of an interaction term (PLT × Cancer) in the regression model, the interaction between PLT and lung cancer approached statistical significance (β = –0.055, p = 0.053), indicating a potentially stronger inverse association between PLT and CK-BB expression specifically in lung cancer patients.
View this table:
View inline
View popup
Download powerpoint
Table 1.Multiple Linear Regression Analysis with VIF of Serum CK-BB Expression and Platelet Parameters.
3.5 Quantitative RT-PCR detection of CK family genes in serum and platelets
Furthermore, we performed qRT-PCR analysis to quantify the mRNA expression levels of three creatine kinase genes—CKB, CKMT1, and CKMT2—in paired serum and platelet samples collected from cancer patients. The expression profiles in serum closely mirrored the protein isoenzyme patterns observed in previous electrophoretic analyses, exhibiting marked cancer-type specificity: mitochondrial isoenzyme CKMT1 was predominantly expressed in colorectal cancer, whereas CKB expression was markedly elevated in lung cancer (Figs 5A–4B). All three CK isoenzyme transcripts were detectable in the paired platelet samples, indicating active transcription of CK genes in platelets. Notably, platelet-derived CKB mRNA expression was significantly higher in lung cancer patients compared to those with colorectal cancer (p = 5.4 × 10⁻⁶; Fig 5C), suggesting potential cancer-type-specific upregulation in the lung cancer context. Paired sample analysis further revealed that, in the majority of cases, platelet CKB mRNA levels exceeded the corresponding serum expression levels (p = 0.03, Wilcoxon signed-rank test; Fig 5D). Additionally, correlation analysis demonstrated a significant positive association between serum and platelet CKB expression levels (R² = 0.33, p = 2.9 × 10⁻⁵; Fig 5E). Collectively, these findings suggest that platelets may serve as a critical source of serum CKB in cancer patients—particularly in lung cancer—potentially contributing to the observed CK-MB > CK isoenzyme elevation phenotype.
Download figure
Open in new tab
Fig 5.Differential expression and correlation of CK genes in serum and platelets.
(A) Expression of CKB, CKMT1, and CKMT2 mRNA in serum samples from lung and colorectal cancer patients. (B) Z-score heatmap displaying CK genes expression patterns in serum stratified by cancer type. (C) Comparison of platelet CK genes expression between colorectal and lung cancer patients. (D) Paired expression analysis of CK genes in platelet versus serum from the same patients. (E) Correlation analysis between serum and platelet CK genes expression. Statistical methods: Wilcoxon rank-sum test was used for comparisons between colorectal and lung cancer groups in panels. Paired Wilcoxon signed-rank test was used for within-patient comparisons between serum and platelet expression. Pearson correlation analysis was used to evaluate association between serum and platelet mRNA expression levels. Expression values were log₂-transformed prior to analysis. Abbreviations: CK, creatine kinase.
3.6 Potential role of platelet CKB in distant metastasis of lung cancer
To further validate whether CKB is upregulated in the platelets of lung cancer patients, we analyzed RNA sequencing data from the tumor-educated platelet (TEP) dataset GSE89843. The results showed that CKB mRNA expression was significantly elevated in the platelets of lung cancer patients compared to healthy individuals (P = 7.7 × 10⁻¹¹, Fig 6A). Using the single-cell RNA sequencing dataset GSE123902, we assessed the expression of CKB mRNA within the tumor microenvironment. Notably, tumor cells from distant metastatic lesions in stage IV lung cancer patients exhibited markedly higher CKB expression, whereas no elevation was observed in primary tumor lesions (Figs 6D–5G). This finding is consistent with TCGA data, which demonstrated that while CKB expression is generally low in primary lung tumor tissues, patients with relatively higher tumor CKB expression had significantly worse survival outcomes (Figs 6C, 5H). These results collectively suggest that elevated CKB expression may serve as a potential promoter of distant metastasis.
Download figure
Open in new tab
Fig 6.Platelet CKB expression and its potential association with lung cancer metastasis.
(A) CKB mRNA is significantly upregulated in platelets from lung cancer patients (GSE89843); (B) CKB mRNA expression is lower in circulating tumor cells (CTCs) and peripheral blood leukocytes (WBCs) compared to primary tumor cells (GSE74639); (C) CKB mRNA is expressed at low levels in primary lung tumor tissues (TCGA dataset); (D, F) Tumor cells from distant metastatic lesions in stage IV lung cancer patients exhibit high CKB mRNA expression (GSE123902); (E, G) Tumor cells from primary lesions in stage IV lung cancer exhibit low CKB mRNA expression (GSE123904); (H) Lung cancer patients with high tumor CKB expression show poorer overall survival (TCGA data); (I) Functional enrichment and interaction analysis of CKB-associated genes in platelets. Statistical methods: Expression comparisons were performed using Wilcoxon rank-sum tests; Survival analysis was conducted using Kaplan–Meier curves and log-rank tests; Differentially expressed genes were identified using |log₂FC| > 1 and FDR < 0.05 as thresholds; Functional enrichment was assessed using clusterProfiler with GO Biological Process and KEGG pathway databases. Abbreviations: CKB, brain-type creatine kinase; NSCLC, non-small cell lung cancer; LUSC, Lung Squamous Cell Carcinoma; LUAD, Lung Adenocarcinoma; TEP, tumor-educated platelet; CTC, circulating tumor cell; WBC, White Blood Cell; GO, Gene Ontology; TCGA, The Cancer Genome Atlas.
Download figure
Open in new tab
Fig 7.Proposed mechanism of platelet CKB in promoting hematogenous metastasis of lung cancer.
Upon vascular invasion, circulating tumor cells (CTCs) rapidly interact with platelets. In response to tumor-derived signals, platelets become tumor-educated platelets (TEPs), characterized by increased CKB mRNA expression. Through intercellular exchange, tumor cells may acquire of exogenous CKB, which facilitates the generation of phosphocreatine (PCr) from creatine (Cr), maintaining ATP supply under hypoxic or shear stress conditions. This metabolic support may enhance CTC survival in circulation and promote distant organ colonization. CKB expression is elevated in metastatic tumor lesions but not in primary tumors, suggesting a potential role in metastasis initiation. PETs: platelet-educated tumor cells.
To explore the origin of elevated circulating CKB, we used GSE74639 data to analyze CKB mRNA levels in circulating tumor cells (CTCs) and peripheral blood leukocytes. Both cell populations showed markedly lower CKB expression than primary tumors (Fig 6B). Taken together with the high expression of CKB mRNA observed in peripheral platelets from stage IV patients, we speculated that platelets may represent an external source driving CKB accumulation in metastatic lung cancer cells.
To further explore the potential mechanisms by which platelet-derived CKB contributes to lung cancer metastasis, we performed an integrative analysis of CKB-associated differentially expressed genes (DEGs) in lung tumor tissues and platelet RNA profiles. A total of 25 overlapping DEGs were identified. Gene Ontology (GO) and KEGG pathway enrichment analysis revealed that these genes were significantly enriched in pathways related to metabolic processes, oxidative stress responses, immune regulation, and intercellular signaling (Fig 6I). These findings provide preliminary mechanistic insights, suggesting that platelet-derived CKB may play a functional role in shaping the metastatic tumor microenvironment and promoting distant dissemination in lung cancer.
4. Discussion
In this study, we comprehensively analyzed the clinical, biochemical, and transcriptomic landscape of cancer patients exhibiting the paradoxical CK-MB > CK abnormality. By integrating large-scale retrospective data, serum isoenzyme profiling, qRT-PCR analysis of platelet and serum mRNA, and bioinformatics, we uncovered that elevated CKB mRNA levels in platelets correlated with metastatic features, suggesting a possible involvement of platelet-associated CKB in lung cancer progression.
Our results showed that colorectal and lung cancers as the most frequently associated malignancies with CK-MB > CK abnormalities, particularly at advanced stages. Contrary to classical myocardial injury interpretations, electrophoresis and mRNA profiling revealed that these anomalies were primarily driven by aberrant activation of non-cardiac CK isoenzymes - namely, brain-type CK (CK-BB) and mitochondrial CK isoenzymes (CKII and CKI). To better capture these atypical patterns, we developed a composite index termed CK-Sub, defined as the sum of CK-BB, CKII, and CKI activity. This index demonstrated prognostic relevance, particularly in colorectal cancer, where elevated CK-Sub levels were significantly associated with stage IV disease and shorter overall survival. Importantly, cancer-type– specific CK expression patterns emerged: CKII was the predominant isoenzyme in CRC, while CK-BB was selectively enriched in lung cancer patients(41). Previous studies have indicated that CK-BB promotes extracellular ATP production under hypoxic conditions, thereby supporting tumor cell survival, migration, and metastasis(17, 42, 43). Our findings are consistent with these biological roles, confirming the presence of high CK-BB levels in patients with advanced-stage lung cancer(44). Notably, our study observed the important contribution of CKII to CK-MB > CK discrepancies, especially in colorectal cancer(3, 45). This highlights the potential role of mitochondrial metabolic reprogramming in tumor progression(42, 46–54)
The extracellular presence of CK-BB and mitochondrial CK may not only serve as passive markers of tumor burden but may also actively contribute to establishing a metabolically supportive microenvironment that promotes tumor survival and metastasis(55, 56). Notably, although the CK-MB/CK ratio itself exhibited limited sensitivity in predicting tumor stage or prognosis, the underlying isoenzyme signatures - particularly the elevations in CKII - showed significant correlations with disease stage. By integrating signals from multiple non-traditional CK isoenzymes, CK-Sub demonstrated significant prognostic value in both colorectal and lung cancer patients, with elevated levels indicating worse survival outcomes. These findings suggest that analyzing CK isoenzyme patterns can provide deeper biological and clinical insights compared to traditional CK-MB/CK ratio analysis(13, 19, 40), and that the cumulative activity of CK isoenzymes beyond CK-MB and CK-MM may more effectively reflect tumor aggressiveness(52).
A key and novel insight from our study is the identification of platelets as a non-canonical source of circulating CKB. qRT-PCR analysis revealed that CKB mRNA expression was significantly higher in the platelets of lung cancer patients than in colorectal cancer patients. Paired sample analysis confirmed that platelet CKB levels exceeded those in serum, and a significant positive correlation was observed between the two, indicating that platelets may actively contribute to the pool of circulating CK-BB. This hypothesis was further substantiated by analysis of the GSE89843 TEPs RNA-seq dataset, which confirmed that CKB was significantly upregulated in platelets of lung cancer patients. Additional exploration using the GSE123902 single-cell RNA-seq dataset revealed that CKB expression was specifically enriched in tumor cells within distant metastatic lesions, but not in primary tumors. These observations, combined with TCGA survival analysis linking high tumor CKB expression to poor prognosis. Functional enrichment analysis revealed that 25 shared differentially expressed genes associated with TEPs and CKB tumor related genes in lung cancer were enriched in oxidative stress response, immune regulation, and metabolic reprogramming. All evidences suggest that CKB may serve as a functional facilitator of lung cancer metastasis.
Taken together, our findings highlight a previously underexplored role of platelets as potential metabolic enablers of tumor dissemination. The discovery of elevated CKB mRNA levels in TEPs, particularly in lung cancer patients, and their correlation with distant metastasis raises the possibility that platelets may help shape the metastatic phenotype through metabolic crosstalk. We propose a mechanistic model (Fig 6) in which platelet-associated CKB supports the survival and adaptability of CTCs during hematogenous transit, potentially by enhancing ATP production in energy-stressed environments. Although our study does not provide direct evidence for the transfer or translation of platelet CKB in vivo, the convergence of clinical, biochemical, and transcriptomic findings provides a compelling rationale for further investigation.
This study also proposed that platelets may represent an important source of abnormally elevated serum CKB in cancer patients, especially in lung cancer. This not only complements the mechanistic basis for the clinical findings (CK-MB > CK), but also offers a theoretical framework for future studies on how platelets contribute to the release, transport, and systemic dissemination of tumor-associated metabolic factors such as CKB. Future studies incorporating co-culture systems, metabolic tracing, and in vivo metastasis assays are warranted to elucidate the precise functional contributions of platelet CKB in lung cancer progression. This may provide new clues for blocking platelet-assisted hematogenous metastasis.
5. Conclusions
This study suggests that in lung cancer patients with serum CK-MB>CK abnormalities, the atypical CK pattern is driven by CK-BB specificity. Among them, platelet CKB is significantly elevated and closely related to distant metastasis and poor prognosis. Platelet CKB may be a potential participant in hematogenous dissemination of lung cancer, and its mechanism deserves further investigation.
Data Availability
All relevant data are within the manuscript and its Supporting Information files.
Supporting information
View this table:
View inline
View popup
Download powerpoint
S1 Table.Cancer Group Classification Based on ICD-10 Codes.
View this table:
View inline
View popup
Download powerpoint
S2 Table.Multivariate linear regression analysis of factors associated with CK-MB/CK ratio in colorectal and lung cancer patients.
Download figure
Open in new tab
S1 Fig.
Expression of serum CK isoenzymes in different disease stages. (A–B) Comparison of CK-BB, CK-MB/CK ratio between stage III and stage IV colorectal cancer patients. (C–E) Comparison of CKII, CK-Sub, and CK-MB/CK ratio between stage III and stage IV lung cancer patients. Statistical methods: Group comparisons were performed using the Wilcoxon rank-sum test. Abbreviations: CK-MB, creatine kinase-MB isoenzyme; CK, creatine kinase; CKII, mitochondrial creatine kinase isoenzyme II; CK-BB, creatine kinase brain-type isoenzyme; CK-Sub, total CK isoenzyme activity (CK-BB + CKII + CKI).
Download figure
Open in new tab
Acknowledgments
During the preparation of this work, the authors used ChatGPT (OpenAI, GPT-4 version) for the purpose of assisting in refining the English language expressions. All scientific content, analyses, interpretations, and conclusions were independently developed by the authors. After using this tool, the authors have carefully reviewed and edited all outputs and take full responsibility for the content and interpretation presented in this publication.
Footnotes
raolubei{at}126.com (L. R.); 2508385190{at}qq.com (Y. Y.); yebo521521{at}163.com (B. Y.); wangdongshengnc{at}163.com (D. W.); 2020zs399{at}stu.cdutcm.edu.cn (C. H.); luoyajun1989{at}126.com (Y. L.); liangyue{at}scszlyy.org.cn (Y. L.); zhangguanbin{at}cdutcm.edu.cn (G. Z.)
Abbreviations
ATP Adenosine Triphosphate CK Creatine Kinase CK-MB Creatine Kinase-MB Isoenzymes CK-BB Creatine Kinase Brain-Type Isoenzymes CK-MM Creatine Kinase Muscle-Type Isoenzymes MtCK Mitochondrial Creatine Kinase CKII Mitochondrial Creatine Kinase Isoenzyme II (dimer form)CKI Mitochondrial Creatine Kinase Isoenzyme I (monomer form)RC Colorectal cancer PCA Principal Component Analysis PERMANOVA Permutational Multivariate Analysis of Variance HR Hazard Ratio CI Confidence Interval OS Overall Survival CTC Circulating tumor cell TEP Tumor-educated platelet PET Platelet-educated tumor cell PLT Platelet count MPV Mean platelet volume PDW Platelet distribution width PRP Pplatelet-rich plasma NSCLC Non-small cell lung cancer ROC Receiver operating characteristic VIF Variance inflation factor DEG Differential gene expression WBC White blood cell
References
1.↵Walters J, Gailani G. Serum creatine kinase: requesting and interpreting results. Pract Neurol. 2025. Google Scholar
2.↵Soroida Y, Ohkawa R, Nakagawa H, Satoh Y, Yoshida H, Kinoshita H, et al.Increased activity of serum mitochondrial isoenzyme of creatine kinase in hepatocellular carcinoma patients predominantly with recurrence. J Hepatol. 2012;57(2):330–6. OpenUrlPubMedGoogle Scholar
3.↵Schlattner U, Kay L, Tokarska-Schlattner M. Mitochondrial Proteolipid Complexes of Creatine Kinase. Subcell Biochem. 2018;87:365–408. OpenUrlCrossRefPubMedGoogle Scholar
4.↵Kanemitsu F, Okigaki T. Characterization of human creatine kinase BB and MB isoforms by means of isoelectric focusing. Clin Chim Acta. 1994;231(1):1–9. OpenUrlPubMedGoogle Scholar
5.↵Zhang L, Han F, Liu X, Xie C, Tian K, Bi Q, et al.Macro creatine kinase in an asymptomatic patient: a case report. Clin Chem Lab Med. 2020;59(2):e61–e4. OpenUrlPubMedGoogle Scholar
6.↵Schlattner U, Tokarska-Schlattner M, Wallimann T. Mitochondrial creatine kinase in human health and disease. Biochim Biophys Acta. 2006;1762(2):164–80. OpenUrlCrossRefPubMedGoogle Scholar
7.↵Collinson P, Suvisaari J, Aakre KM, Baum H, Duff CJ, Gruson D, et al.How Well Do Laboratories Adhere to Recommended Guidelines for Cardiac Biomarkers Management in Europe? The CArdiac MARker Guideline Uptake in Europe (CAMARGUE) Study of the European Federation of Laboratory Medicine Task Group on Cardiac Markers. Clin Chem. 2021;67(8):1144–52. OpenUrlPubMedGoogle Scholar
8.Alvin MD, Jaffe AS, Ziegelstein RC, Trost JC. Eliminating Creatine Kinase-Myocardial Band Testing in Suspected Acute Coronary Syndrome: A Value-Based Quality Improvement. JAMA Intern Med. 2017;177(10):1508–12. OpenUrlPubMedGoogle Scholar
9.Grela-Wojewoda A, Püsküllüoğlu M, Sas-Korczyńska B, Zemełka T, Pacholczak-Madej R, Wysocki WM, et al.Biomarkers of Trastuzumab-Induced Cardiac Toxicity in HER2-Positive Breast Cancer Patient Population. Cancers (Basel). 2022;14(14). Google Scholar
10.Kanemitsu F, Okigaki T. Creatine kinase MB isoforms for early diagnosis and monitoring of acute myocardial infarction. Clin Chim Acta. 1992;206(3):191–9. OpenUrlPubMedGoogle Scholar
11.↵Panteghini M, Cuccia C, Malchiodi A, Calarco M, Pagnoni N. Isoforms of creatine kinase MM and MB in acute myocardial infarction: a clinical evaluation. Clin Chim Acta. 1986;155(1):1–9. OpenUrlPubMedGoogle Scholar
12.↵Neumeier D, Prellwitz W, Würzburg U, Brundobler M, Olbermann M, Just HJ, et al.Determination of creatine kinase isoenzyme MB activity in serum using immunological inhibition of creatine kinase M subunit activity. Activity kinetics and diagnostic significance in myocardial infarction. Clin Chim Acta. 1976;73(3):445–51. OpenUrlCrossRefPubMedWeb of ScienceGoogle Scholar
13.↵Kovar FM, Aldrian S, Endler G, Vécsei V, Hajdu S, Heinz T, et al.CK/CK-MB ratio as an indirect predictor for survival in polytraumatized patients. Wien Klin Wochenschr. 2012;124(7-8):245–50. OpenUrlPubMedGoogle Scholar
14.↵Ota T, Hasegawa Y, Murata E, Tanaka N, Fukuoka M. False-Positive Elevation of CK-MB Levels with Chest Pain in Lung Adenocarcinoma. Case Rep Oncol. 2020;13(1):100–4. OpenUrlPubMedGoogle Scholar
15.↵Eidizadeh A, von Ahsen N, Friedewald S, Binder L. Macro-CK type 2 in metastatic prostate cancer. Diagnosis (Berl). 2019;6(3):307–9. OpenUrlPubMedGoogle Scholar
16.↵Annesley TM, McKenna BJ. Ectopic creatine kinase MB production in metastatic cancer. Am J Clin Pathol. 1983;79(2):255–9. OpenUrlCrossRefPubMedGoogle Scholar
17.↵Loo JM, Scherl A, Nguyen A, Man FY, Weinberg E, Zeng Z, et al.Extracellular metabolic energetics can promote cancer progression. Cell. 2015;160(3):393–406. OpenUrlCrossRefPubMedGoogle Scholar
18.Demir M, Koepping L, Li Y, Fujimoto L, Bobkov A, Zhao J, et al.Structural basis for substrate binding, catalysis, and inhibition of cancer target mitochondrial creatine kinase by a covalent inhibitor. Structure. 2025;33(4):786–97.e3. OpenUrlGoogle Scholar
19.↵Chang CC, Liou CB, Su MJ, Lee YC, Liang CT, Ho JL, et al.Creatine Kinase (CK)-MB-to-Total-CK Ratio: a Laboratory Indicator for Primary Cancer Screening. Asian Pac J Cancer Prev. 2015;16(15):6599–603. OpenUrlPubMedGoogle Scholar
20.Uranbileg B, Enooku K, Soroida Y, Ohkawa R, Kudo Y, Nakagawa H, et al.High ubiquitous mitochondrial creatine kinase expression in hepatocellular carcinoma denotes a poor prognosis with highly malignant potential. Int J Cancer. 2014;134(9):2189–98. OpenUrlPubMedGoogle Scholar
21.Qian XL, Li YQ, Gu F, Liu FF, Li WD, Zhang XM, et al.Overexpression of ubiquitous mitochondrial creatine kinase (uMtCK) accelerates tumor growth by inhibiting apoptosis of breast cancer cells and is associated with a poor prognosis in breast cancer patients. Biochem Biophys Res Commun. 2012;427(1):60–6. OpenUrlCrossRefPubMedGoogle Scholar
22.↵Lin YC, Chen SC, Huang CM, Hu YF, Chen YY, Chang SL, et al.Clinical features and diagnosis of new malignancy in patients with acute pulmonary embolism and without a history of cancer. J Chin Med Assoc. 2020;83(3):245–50. OpenUrlPubMedGoogle Scholar
23.↵Stein W, Bohner J. Influence of autoantibodies to creatine kinase-BB on assays for MB isoenzyme. Clin Chem. 1985;31(7):1189–92. OpenUrlAbstract/FREE Full TextGoogle Scholar
24.↵Wong SS, Wu AH, Fritsche HA. Reassessment of creatine kinase BB as a marker for cancer of the prostate, breast, and lung. Clin Chem. 1987;33(6):809–11. OpenUrlAbstract/FREE Full TextGoogle Scholar
25.Huddleston HG, Wong KK, Welch WR, Berkowitz RS, Mok SC. Clinical applications of microarray technology: creatine kinase B is an up-regulated gene in epithelial ovarian cancer and shows promise as a serum marker. Gynecol Oncol. 2005;96(1):77–83. OpenUrlCrossRefPubMedGoogle Scholar
26.Wang HC, Lu JY, Ting YM. Elevated serum CK-MB and CK-BB isoenzymes in a patient with small cell lung cancer: a case report. Zhonghua Yi Xue Za Zhi (Taipei). 1995;55(3):270–3. OpenUrlPubMedGoogle Scholar
27.↵Li Y, Chen Y, Shao B, Liu J, Hu R, Zhao F, et al.Evaluation of creatine kinase (CK)-MB to total CK ratio as a diagnostic biomarker for primary tumors and metastasis screening. Pract Lab Med. 2023;37:e00336. OpenUrlGoogle Scholar
28.↵Wada Y, Tanigawa J, Ishizaka N. Macro-CK and CK-BB Contributing to Sham CK-MB Elevation. Intern Med. 2017;56(11):1449–50. OpenUrlPubMedGoogle Scholar
29.↵Lee KN, Csako G, Bernhardt P, Elin RJ. Relevance of macro creatine kinase type 1 and type 2 isoenzymes to laboratory and clinical data. Clin Chem. 1994;40(7 Pt 1):1278–83. OpenUrlAbstract/FREE Full TextGoogle Scholar
30.↵Labelle M, Hynes RO. The initial hours of metastasis: the importance of cooperative host-tumor cell interactions during hematogenous dissemination. Cancer Discov. 2012;2(12):1091–9. OpenUrlAbstract/FREE Full TextGoogle Scholar
31.↵Gil-Bernabé AM, Lucotti S, Muschel RJ. Coagulation and metastasis: what does the experimental literature tell us?Br J Haematol. 2013;162(4):433–41. OpenUrlCrossRefPubMedGoogle Scholar
32.Coupland LA, Chong BH, Parish CR. Platelets and P-selectin control tumor cell metastasis in an organ-specific manner and independently of NK cells. Cancer Res. 2012;72(18):4662–71. OpenUrlCrossRefPubMedGoogle Scholar
33.Schlesinger M. Role of platelets and platelet receptors in cancer metastasis. J Hematol Oncol. 2018;11(1):125. OpenUrlCrossRefPubMedGoogle Scholar
34.↵Erpenbeck L, Schön MP. Deadly allies: the fatal interplay between platelets and metastasizing cancer cells. Blood. 2010;115(17):3427–36. OpenUrlAbstract/FREE Full TextGoogle Scholar
35.↵Martins Castanheira N, Spanhofer AK, Wiener S, Bobe S, Schillers H. Uptake of platelets by cancer cells and recycling of the platelet protein CD42a. J Thromb Haemost. 2022;20(1):170–81. OpenUrlPubMedGoogle Scholar
36.↵Placke T, Örgel M, Schaller M, Jung G, Rammensee HG, Kopp HG, et al.Platelet-derived MHC class I confers a pseudonormal phenotype to cancer cells that subverts the antitumor reactivity of natural killer immune cells. Cancer Res. 2012;72(2):440–8. OpenUrlAbstract/FREE Full TextGoogle Scholar
37.Placke T, Salih HR, Kopp HG. GITR ligand provided by thrombopoietic cells inhibits NK cell antitumor activity. J Immunol. 2012;189(1):154–60. OpenUrlAbstract/FREE Full TextGoogle Scholar
38.Clar KL, Hinterleitner C, Schneider P, Salih HR, Maurer S. Inhibition of NK Reactivity Against Solid Tumors by Platelet-Derived RANKL. Cancers (Basel). 2019;11(3). Google Scholar
39.↵Stoiber D, Assinger A. Platelet-Leukocyte Interplay in Cancer Development and Progression. Cells. 2020;9(4). Google Scholar
40.↵Rao L, Xu P, Zhang G, Zu R, Luo Y, Zhang K, et al.Prognostic value of creatine kinase (CK)-MB to total-CK ratio in colorectal cancer patients after curative resection. BMC Cancer. 2024;24(1):543. OpenUrlPubMedGoogle Scholar
41.↵Usui A, Fujita K, Imaizumi M, Abe T, Inoue K, Matumoto S, et al.Determination of creatine kinase isozymes in sera and tissues of patients with various lung carcinomas. Clin Chim Acta. 1987;164(1):47–53. OpenUrlPubMedGoogle Scholar
42.↵Ben-Sahra I, Puissant A. HER2 Signaling Hijacks the Creatine Shuttle to Fuel Breast Cancer Cell Growth. Cell Metab. 2018;28(6):805–7. OpenUrlPubMedGoogle Scholar
43.↵Krutilina RI, Playa H, Brooks DL, Schwab LP, Parke DN, Oluwalana D, et al.HIF-Dependent CKB Expression Promotes Breast Cancer Metastasis, Whereas Cyclocreatine Therapy Impairs Cellular Invasion and Improves Chemotherapy Efficacy. Cancers (Basel). 2021;14(1). Google Scholar
44.↵Zarghami N, Giai M, Yu H, Roagna R, Ponzone R, Katsaros D, et al.Creatine kinase BB isoenzyme levels in tumour cytosols and survival of breast cancer patients. Br J Cancer. 1996;73(3):386–90. OpenUrlPubMedWeb of ScienceGoogle Scholar
45.↵Lee K, Csako G, Bernhardt P, Elin RJCc.Relevance of macro creatine kinase type 1 and type 2 isoenzymes to laboratory and clinical data. 1994;40:1278–83. OpenUrlGoogle Scholar
46.↵Bera S, Ray M. The transcriptional cascade associated with creatine kinase down-regulation and mitochondrial biogenesis in mice sarcoma. Cell Mol Biol Lett. 2009;14(3):481–96. OpenUrlPubMedGoogle Scholar
47.Zhang Y, Sun M, Chen Y, Li B. MiR-519b-3p Inhibits the Proliferation and Invasion in Colorectal Cancer via Modulating the uMtCK/Wnt Signaling Pathway. Frontiers in pharmacology. 2019;10:741. OpenUrlPubMedGoogle Scholar
48.Heinz JW, O’Donnell NJ, Lott JA. Apparent mitochondrial creatine kinase in the serum of a patient with metastatic cancer to the liver. Clin Chem. 1980;26(13):1908–11. OpenUrlAbstract/FREE Full TextGoogle Scholar
49.Wang Z, Wu H, Chang X, Song Y, Chen Y, Yan Z, et al.CKMT1 deficiency contributes to mitochondrial dysfunction and promotes intestinal epithelial cell apoptosis via reverse electron transfer-derived ROS in colitis. Cell Death Dis. 2025;16(1):177. OpenUrlPubMedGoogle Scholar
50.Kita M, Fujiwara-Tani R, Kishi S, Mori S, Ohmori H, Nakashima C, et al.Role of creatine shuttle in colorectal cancer cells. Oncotarget. 2023;14:485–501. OpenUrlPubMedGoogle Scholar
51.Zhang H, To KKW. Serum creatine kinase elevation following tyrosine kinase inhibitor treatment in cancer patients: Symptoms, mechanism, and clinical management. Clin Transl Sci. 2024;17(11):e70053. OpenUrlPubMedGoogle Scholar
52.↵Joseph J, Cardesa A, Carreras J. Creatine kinase activity and isoenzymes in lung, colon and liver carcinomas. Br J Cancer. 1997;76(5):600–5. OpenUrlCrossRefPubMedWeb of ScienceGoogle Scholar
53.Li M, Liu H, Li J, Guo S, Lv YJRr.Mitochondrial creatine kinase 1 in non-small cell lung cancer progression and hypoxia adaptation. 2021;22(1):190. OpenUrlGoogle Scholar
54.↵Soroida Y, Ohkawa R, Nakagawa H, Satoh Y, Yoshida H, Kinoshita H, et al.Increased activity of serum mitochondrial isoenzyme of creatine kinase in hepatocellular carcinoma patients predominantly with recurrence. 2012;57(2):330–6. OpenUrlGoogle Scholar
55.↵Cai S, Xia Q, Duan D, Fu J, Wu Z, Yang Z, et al.Creatine kinase mitochondrial 2 promotes the growth and progression of colorectal cancer via enhancing Warburg effect through lactate dehydrogenase B. PeerJ. 2024;12:e17672. OpenUrlPubMedGoogle Scholar
56.↵Amamoto R, Uchiumi T, Yagi M, Monji K, Song Y, Oda Y, et al.The Expression of Ubiquitous Mitochondrial Creatine Kinase Is Downregulated as Prostate Cancer Progression. 2016;7(1):50–9. OpenUrlGoogle Scholar
Back to top
PreviousNext
Posted July 05, 2025.
Download PDF
Print/Save Options
Download PDFFull Text & In-line FiguresXML
More Info
Author Declarations
Data/Code
Email
Share
Platelet CKB expression is associated with elevated serum CK-BB and metastatic phenotype in lung cancer
Lubei Rao, Chao Hu, Yan Yue, Yajun Luo, Bo Ye, Yue Liang, Guanbin Zhang, Dongsheng Wang
medRxiv 2025.07.03.25330795; doi:
[This article is a preprint and has not been peer-reviewed what does this mean?]. It reports new medical research that has yet to be evaluated and so should not be used to guide clinical practice.
Share This Article: Copy
Citation Tools
Get QR code
Subject Area
Oncology
Reviews and Context
0
Comment
0
TRIP Peer Reviews
0
Community Reviews
0
Automated Services
0
Blogs/Media
0
Author Videos
Subject Areas
All Articles
Addiction Medicine(467)
Allergy and Immunology(787)
Anesthesia(244)
Cardiovascular Medicine(3617)
Dentistry and Oral Medicine(385)
Dermatology(316)
Emergency Medicine(517)
Endocrinology (including Diabetes Mellitus and Metabolic Disease)(1271)
Epidemiology(13930)
Forensic Medicine(26)
Gastroenterology(950)
Genetic and Genomic Medicine(5596)
Geriatric Medicine(542)
Health Economics(843)
Health Informatics(3622)
Health Policy(1219)
Health Systems and Quality Improvement(1311)
Hematology(468)
HIV/AIDS(1091)
Infectious Diseases (except HIV/AIDS)(15053)
Intensive Care and Critical Care Medicine(962)
Medical Education(527)
Medical Ethics(137)
Nephrology(575)
Neurology(5401)
Nursing(283)
Nutrition(811)
Obstetrics and Gynecology(961)
Occupational and Environmental Health(860)
Oncology(2744)
Ophthalmology(797)
Orthopedics(297)
Otolaryngology(369)
Pain Medicine(349)
Palliative Medicine(107)
Pathology(575)
Pediatrics(1435)
Pharmacology and Therapeutics(588)
Primary Care Research(599)
Psychiatry and Clinical Psychology(4575)
Public and Global Health(8059)
Radiology and Imaging(1860)
Rehabilitation Medicine and Physical Therapy(1130)
Respiratory Medicine(1051)
Rheumatology(506)
Sexual and Reproductive Health(566)
Sports Medicine(453)
Surgery(593)
Toxicology(80)
Transplantation(246)
Urology(219)
Evaluation/discussion of this paper x
Comments 0 TRiP 0 Community 0 Automated 0 Blogs/Media 0 Video 0
Comments
medRxiv aims to provide a venue for anyone to comment on a medRxiv preprint. Comments are moderated for offensive or irrelevant content (this can take ~24 h). Please avoid duplicate submissions and read our Comment Policy before commenting. The content of a comment is not endorsed by medRxiv.
Share this comments tab (click to copy link)Copied!
TRiP
medRxiv partners with journals and review services to enable posting of peer reviews and editorial decisions related to preprints they are evaluating. Reviews are posted with the consent of the authors.
TRiP reviews are part of a peer review pilot project in which participating organizations post peer reviews of manuscripts they are evaluating
There is no TRiP material for this paper.
Community Reviews
medRxiv aims to inform readers about online discussion of this preprint occurring elsewhere. The content at the links below is not endorsed by either medRxiv or the preprint's authors.
Community reviews for this article:
There are no community reviews for this paper.
Automated Evaluations
Certain services provide automated analysis of preprints. Analyses invited by the authors are displayed at the top of this tab. Those done independently of authors are shown underneath . None of these analyses is endorsed by medRxiv.
Automated Evaluations:
There are no automated evaluations for this paper.
Blog/Media Links
medRxiv aims to inform readers about online discussion of this preprint occurring elsewhere. The content at the links below is not endorsed by either medRxiv or the preprint's authors.
Video
medRxiv partners with conferences and institutions to display recordings of talks and seminars related to preprints. These are posted with the consent of the authors.
Video:
There are no videos for this paper.
Powered by
Powered by
Follow this preprint
X
You can now receive automatic notifications when a preprint is revised, withdrawn, commented on, peer reviewed, or published in a journal. Select the events you would like to follow below and click "Submit". To see all of the preprints you are currently following, please go to the medRxiv Alerts Page.
Sign In to Follow this Preprint
Email
Email this Article close
Thank you for your interest in spreading the word about medRxiv.
NOTE: Your email address is requested solely to identify you as the sender of this article.
Your Email
Your Name
Send To
Enter multiple addresses on separate lines or separate them with commas.
You are going to email the following Platelet CKB expression is associated with elevated serum CK-BB and metastatic phenotype in lung cancer
Message Subject (Your Name) has forwarded a page to you from medRxiv
Message Body (Your Name) thought you would like to see this page from the medRxiv website.
Your Personal Message
CAPTCHA
This question is for testing whether or not you are a human visitor and to prevent automated spam submissions.
Citation Tools close
Platelet CKB expression is associated with elevated serum CK-BB and metastatic phenotype in lung cancer
Lubei Rao, Chao Hu, Yan Yue, Yajun Luo, Bo Ye, Yue Liang, Guanbin Zhang, Dongsheng Wang
medRxiv 2025.07.03.25330795; doi:
[This article is a preprint and has not been peer-reviewed what does this mean?]. It reports new medical research that has yet to be evaluated and so should not be used to guide clinical practice.
Citation Manager Formats
BibTeX
Bookends
EasyBib
EndNote (tagged)
EndNote 8 (xml)
Medlars
Mendeley
Papers
RefWorks Tagged
Ref Manager
RIS
Zotero
We use cookies on this site to enhance your user experience. By clicking any link on this page you are giving your consent for us to set cookies.
Continue Find out more |
11745 | https://www.arrl.org/files/file/Technology/tis/info/pdf/q1106037.pdf | From November 2006 QST © ARRL It sometimes seems that one of the most mysterious creatures in the world of Amateur Radio is standing wave ratio (SWR). I often hear on-air discussion of guys brag-ging about and comparing their SWR num-bers as if it were a contest. There seems to be a relentless drive to achieve the most coveted 1:1 SWR at any cost. But why? This article is written to help explain what SWR actually is, what makes it bad and when to worry about it. What is SWR?
SWR is sometimes called VSWR, for voltage standing wave ratio, by the technical folks. Okay, but what does it really mean? The best way to easily understand SWR is by example. In the typical ham station setup, a transmitter is connected to a feed line, which is then connected to the antenna. When you key the transmitter, it develops a radio fre-quency (RF) voltage on the transmission line input. The voltage travels down the feed line to the antenna at the other end and is called the forward wave. In some cases, part of the voltage is reflected at the antenna and propagates back down the line in the reverse direction toward the transmitter, much like a voice echoing off a distant cliff. SWR is a measure of what is happening to the forward and reverse voltage waveforms and how they compare in size. Let’s look at what happens when a trans-mitter is connected to 50 Ω coax and a 50 Ω antenna. For now, pretend that the coax cable doesn’t have any losses and the transmitter is producing a 1 W CW signal. If you were to look at the signal on the output of the transmitter with an oscilloscope, what you would see is a sine wave. The amplitude of the sine wave would be related to how much power the transmitter is producing. A larger amplitude waves means more power. This wave of energy travels down the transmission line and reaches the antenna. If the antenna impedance is 50 Ω, just like the cable, then all of the energy is transferred to the antenna system to be radiated. Anywhere on the transmission line you measured, the voltage waveform would measure exactly the same as the sine wave coming from the transmitter. This is called a matched condition and is what Understanding SWR by Example Take the mystery and mystique out of standing wave ratio.
Darrin Walraven, K5DVW Table 1 SWR vs Reflected Voltage or Power VSWR Voltage Power Reflected (%) Reflected (%) 1.0:1 0 0 1.1:1 5 0.2 1.2:1 9 0.8 1.3:1 13 1.7 1.4:1 17 2.8 1.5:1 20 4 1.6:1 23 5.3 1.7:1 26 6.7 1.8:1 29 8.2 1.9:1 31 9.6 2.0:1 33 11 2.5:1 43 18.4 3.0:1 50 25 4.0:1 56 36 5.0:1 67 44.4 10.0:1 82 67 happens with a 1:1 SWR.
For the case of resistive loads (see sidebar), the SWR can be easily calculated as equal to the (Load R)/Z0 or Z0/(Load R), whichever gives a result greater than or equal to 1.0.
The load or terminating resistance is the RF resistance of whatever is on the end of the transmission line. It could be an antenna, amplifier or dummy load. The line imped-ance is the characteristic impedance of the transmission line and is related to the physi-cal construction of the line. Conductor size, space between conductors, what plastic was used in the insulation — all affect line impedance. Generally, the cable manufac-turer will list the line impedance and there’s nothing you, as a user, can do to change it.
But, what if the antenna wasn’t 50 Ω? Suppose that the antenna is 100 Ω and the Figure 1 — A graph showing the additional loss in a transmission line due to SWR higher than 1:1.
From November 2006 QST © ARRL In the main article I used several examples of SWR based on a resistive load. A resistive load is the easiest to visualize, calculate and understand, but it’s not the most common type of load. In most cases, loads have some reactive impedance as well. That is, they contain a resistive part and an inductive or capacitive part in combination. For instance, your antenna might appear as a 50 Ω resistor with a 100 nH inductor in series, or perhaps some capacitance to ground. In this situation, the SWR is not 1:1 because of the reactance. Even antennas that show a perfect 1:1 SWR in mid-band will typically have some larger SWR at the band edges, often due to the reactance of the antenna changing with frequency. Fortunately, a given SWR behaves the same on a transmission line whether it’s reactive or resistive. If you have a handle on understanding the resistive case, the concept will get you pretty far.
To explore SWR further, it’s useful to look at the reactive load case, or what happens under the condition that loads are not simply resistive. Complex imaginary number math is the routine way to analyze the SWR of complex loads and can be done if you have access to a calculator or computer program that will handle it. Even so, the math gets tedious in a hurry. Fortunately, there’s a very easy way to analyze complex loads using graphical methods and it’s called the Smith Chart. See Figure A.
The concept behind the Smith Chart is simple. There is a resistive axis that is down the middle of the chart, left to right, and a reactive axis along the outer edge of the chart’s circumference. Inductive loads are plotted in the top half of the graph, and capacitive loads in the bottom. Any value of resistance and reactance in a series combination can be plotted on the chart. Then, with a ruler and compass the SWR can be determined. Advanced users of the chart can plot a load and use graphic techniques to design a matching structure or impedance transformer without rigorous math or com-puter. It’s a very powerful tool. Here’s a simple example showing how to determine SWR from a known load using the chart.
Suppose you have just measured a new antenna with an impedance bridge and you know that the input impedance is 35 Ω in series with 12 Ω reactive. The coax cable feeding it has a Z0 of 50 Ω. What is the SWR at the antenna end of the coax? This impedance can be written as the complex num-ber Z=35 +j12.5 Ω. The “j” is used to indicate the reac-tive part from the real part and they can’t simply add together. To use a “normalized” Smith Chart we divide the impedance by 50 Ω, to normalize the impedance. [Smith Charts are also available designed for 50 Ω with 50 at the center instead of the 1.0 that we show for the normalized chart. — Ed.] We now have Z=0.7 +j0.25. Smith Chart numbers are normalized, which means that they have been divided by the system impedance before being plotted. In most cases, the system impedance is the transmission line impedance and is represented on the chart by the dot in the center. Now plot Z on the chart. Along the horizontal line fi nd the 0.7 marker and move upward (inductors or positive reactances are upward, capacitors or negative reactances are down-Adding Reactance to the Picture Figure A — The normalized Smith Chart, ready to simplify transmission line analysis.
Figure B — An impedance of 35 +j12.5 Ω normalized to 0.7 +j0.25 Ω to use on the normalized chart.
From November 2006 QST © ARRL Figure D — A line drawn from the chart center through the impedance of Figure B to the edge showing the distance from the pure resistive points on the line.
ward from center) until you cross the 0.25 reactance line (see Figure B). Draw a point here that notes your imped-ance of 35 +j12.5 Ω. Next draw a line from the center dot of the chart to your Z point. Measure the distance of that line, then draw the same length line along the bot-tom SWR scale. From here you can read the SWR for this load. The SWR is 1.6:1. Another useful attribute of the Smith Chart is called a constant SWR circle. The SWR circle contains all the possible combinations of resistance and reactance that equal or are less than a given SWR. The circles are drawn with a compass by using the distance from the SWR linear graph at the bottom and drawing a circle with that radius from the center of the chart, or use the numbers along the horizontal axis to the right of the center point. For instance, where you see 1.6 on the horizontal line, a circle drawn with radius distance from the center to that point is SWR 1.6:1 as shown in Figure C. Now, any combination of impedance on the circle will be equal to SWR 1.6:1 and anything inside the circle will be less than 1.6:1. The example impedance that was plotted before should lie on the 1.6:1 circle.
Also useful is to know that rotating around the outer diameter of the Smith Chart also represents a half wavelength of distance in a transmission line. That is, as you move along the outer circle it’s the same as moving along a transmission line away from the load. One time around the circle and you’re electrically λ/2 away. This is also very powerful feature of the chart since it allows you to see how your load impedance changes along a transmission line again, without doing the math.
Here’s another example. If your coax cable has a 1.6:1 terminating SWR, as you move along the trans-mission line away from the load, you move along the constant SWR circle on the chart. Remember from the article text that a 1.6:1 SWR can be equal to either 80 Ω or 31 Ω resistive? (1.6 = 80/50 or 50/31) From the Smith Chart, where the SWR circle crosses the hori-zontal axis, the impedance is purely resistive! Where the circle crosses 1.6, it’s equal to 80 Ω and crossing at 0.62, it’s equal to 31 Ω. This is the basis of how imped-ance transformers work. Remember to multiply any numbers from the chart by your working impedance (50 Ω in this example) to get their actual value.
Where do these purely resistive points lie on the transmission line? From the chart, looking at the line extended from the center dot through our Z point, fi nd where it crosses the “wavelengths toward generator curve” on the outside of the chart as shown in Figure D. The line crosses at approximately 0.07 wavelengths on the chart, which will be the starting point. Note that the 80 Ω point (1.6) is at 0.25 λ and the 31 Ω point (0.62) is at 0.50 λ on the chart. Subtracting the starting point of 0.07, the chart is telling us that at 0.18 λ (0.25 – 0.07) from the load, the impedance is 80 Ω and at 0.43 λ away, it’s 31 Ω.
To fi nd the distance in a real piece of cable, multiply the chart wavelengths by the free space wavelength by the cable velocity factor. For example, if your frequency is 144 MHz then a full wavelength in air would be 300/144 = 2.08 meters. Multiplying by the velocity fac-tor of 80% gives 1.67 meters. Multiplying by the chart wavelengths and then at 30 cm, you’d fi nd 80 Ω and at 72 cm it’s 31 Ω.
Figure C — A circle of constant SWR drawn through the impedance of Figure B. The SWR is 1.6.
From November 2006 QST © ARRL Figure 2 — A graph showing the actual SWR at an antenna based on measured SWR at the transmitter end of a transmission line with loss.
cable is still 50 Ω. The SWR for this setup is calculated as 100/50, or 2:1. Now the energy wave hits the antenna and part of it is radi-ated by the antenna, but part of it is reflected back down the line toward the transmitter. That is, the antenna is not matched to the line, so there is a reflection. It turns out that for a 2:1 SWR, 33 percent of the voltage wave is reflected like an echo back down the line. Table 1 lists how much voltage and power is reflected for various SWR values.
In the case of a mismatched condition, something interesting happens along the transmission line. Before, with the matched antenna, the same voltage existed anywhere along the line. Now as you move along the distance of the line, the voltage will change. It now has peaks and valleys. The 33 percent reflection from the antenna alternately adds to and subtracts from the forward voltage wave. At some places on the cable the reflec-tied voltage adds to 133 percent, and others it subtracts to 66 percent of the matched transmitter output. The voltage ratio is 133/66 or 2.0. That voltage ratio defines the SWR. The fact that the voltage along the line changes with position and is different from what the transmitter would produce is called a standing wave. Standing waves are only present when the line is mismatched.
Does Higher SWR Lead to Lower Power Being Transmitted? Not always so dramatically. Believe it or not, 100 percent of the power is actually trans-mitted in both of the previous examples. In the first case, with a 50 Ω antenna, it’s easy to see how all the power is transferred to the antenna to be radiated since there are no reflections. In the second case, the 33 percent voltage reflection travels back down to the transmitter where it doesn’t stop but is re-reflected from the transmitter back toward the antenna along with the forward wave. The energy bounces back and forth inside the cable until it’s all radiated by the antenna for a lossless trans-mission line. An important point to realize is that with extremely low loss transmission line, no matter what the SWR, most of the power can get delivered to the antenna. A later example will show how this can happen.
Is High SWR Bad, or Not?
Now that you have a sense of what SWR is, a few examples can show why under some conditions, high SWR can lead to less power radiating and in other cases, it’s no big deal. The easiest way to see how SWR affects an antenna system is to use a set of charts. Figures 1 and 2 are taken from The ARRL Handbook in the chapter discussing transmission lines. There is much more theory in the Handbook than I’m presenting here, so if you want to be an expert on trans-mission lines, that’s one place to learn more.
In the previous examples, the transmis-sion line had no loss and all our power was being delivered to the antenna. That’s a nice way to visualize what is happening with the reflections, but it doesn’t match the real world because all transmission lines have some loss. Here’s a more practical, straightforward situation. This time we have a length of 50 Ω cable with a total loss of 3 dB (50 percent power), and a 50 Ω antenna. SWR is therefore 1:1. Transmitting 1 W would result in 0.5 W applied to the antenna. Since the SWR is 1:1 there is no mismatch loss to worry about. A very simple situation and no charts are needed. If life were only that simple!
Next t ry t he 1 00 Ω a ntenna w ith t he s ame coax. The SWR is then 2:1 at the antenna since 100/50 = 2.0. According to Figure 1, From November 2006 QST © ARRL expect a mismatch loss of 0.35 dB in addi-tion to the cable loss. In this case we lose a total of 3.35 dB of our signal and send the antenna 0.46 W. Not much difference from a perfect SWR of 1:1.
How about a SWR of 3:1 with the same cable? According to Figure 1 again, we would have an additional loss of 0.9 dB, which makes a total loss of 3.9 dB and 0.41 W to the antenna. This is still not a lot of additional loss even with an SWR of 3:1. Under most conditions a power reduction of 0.9 dB is not noticeable over the air. Even at a 3:1 SWR, the signal it not significantly reduced.
The SWR vs loss tables make it easy to figure out what your additional loss might be for any given antenna system, as long as you know the matched cable loss. Is that the Whole Story?
No, not exactly. There’s even more to explore in the world of SWR. One very strange situation occurs on a long and lossy transmission line, which causes your SWR to appear good at your transmitter even if it’s terrible at the antenna. It’s entirely possible to have a good measured SWR, huge losses in your feed line and no power getting out. Here’s an example.
You’ve just set up your new 2 meter antenna and are feeding it with 120 feet of RG-8X cable. The manufacturer’s data says you should expect 4.5 dB of cable loss for this length of cable at 2 meter frequencies. Not great, but you accept it. You measure the SWR at your transmitter and record a 2:1, which isn’t great but not too terrible either. But wait — how bad is it? Remember all those reflections bouncing back and forth in the cable? Earlier I told you that they eventu-ally radiate, but that was without cable loss. The story is different now — we have loss. Each one of the reflections is attenuated in the cable by 4.5 dB every time it goes from one end to the other or 9 dB round trip. The cable loss is attenuating the reflections and they die out in the cable instead of being radiated. So, how bad is it really? Take a look at Figure 2. With a SWR at the transmitter of 2:1, and a 4.5 dB cable loss, the chart shows a 20:1 SWR at the antenna! Wow! That’s much worse than the 2:1 measured at the transmit-ter. Looking back at Figure 1, that 20:1 SWR at your antenna is costing you another 7 dB of mismatch loss. In reality, the antenna sys-tem that you thought has only 4 dB of loss has 11 dB. Less than 1/10th of your transmit-ter power is being radiated! Good grief!
If this same cable had an open circuit instead of an antenna, and was several hun-dred feet long, the SWR meter at the transmit-ter would read 1:1. Why? Because cable loss tends to make a very long cable appear like a virtual resistance to the transmitter as the reflections die out in the cable. Remember, no reflections looks like a SWR of 1:1. The value of that virtual resistance in this case is 50 Ω which is the definition of characteristic impedance, or why some cables are called 50 Ω and some are 75 or 300 Ω. That num-ber is the impedance the RF would see if the cable were infinitely long. Or, it is also the resistance of the load needed in order to cause no reflected energy and match the cable.
The moral of this story is to measure the SWR at the antenna, especially if you have a long cable run. SWR measurements at the transmitter can be deceiving. The second moral is to know that when a cable manufacturer quotes loss numbers, they are based on an SWR of 1:1, or a perfect match. Anything less than a perfect match can cause additional losses.
Why Ladder Line Works for High SWR Open wire line, window line or lad-der line has been used since the early days of radio. There is a good reason, since the loss of this type of cable is quite low at HF frequencies — lower than all but the very best coax cable. For instance, 300 feet of 450 Ω ladder line has a loss of less than 0.5 dB at 30 MHz when matched. A good quality expensive coax might have 1 dB of loss in the same length, but most high end amateur coax cable will have more than 2 dB attenuation under those conditions. It is because of this low loss that air dielectric (or mostly air in the case of window line) line can be used effectively on antennas that have high SWR, if the matching is provided at the transmitter. The lower loss of this type of line allows most of the reflections to radiate instead of being lost within the cable.
One last example shows how this works. You have just installed a full wave HF dipole. To feed it, you use 300 feet of 450 Ω ladder line with a loss of 0.5 dB at 30 MHz. You’ve modeled your antenna for 10 meters and you just happen to know that the impedance is 4500 Ω. That corresponds to a SWR of 4500/450 or 10:1 on your ladder line. Pretty bad, right? Not so fast. Consulting Figure 1 and knowing your matched loss is 0.5 dB shows an additional loss of 0.9 dB at an SWR of 10:1. The total loss of this antenna system is 1.4 dB. Not bad. Toss in a balanced line tuner and you’re ready to go! Your smart aleck buddy decides to install the same antenna but he springs for the best and most expensive coax figuring his antenna is only 40 feet away from his radio and he doesn’t like the look of ladder line. He boasts that his coax loss is specified at 0.25 dB, which is half that of your ladder line. He figures he can also use a tuner to take care of the mismatch. You quietly smile at him because you know that the 4500 Ω of the antenna will present an SWR of 90:1 on his 50 Ω cable resulting in a mismatch loss of 12 dB beyond the 0.25 dB cable loss. Sure he can tune his SWR to 1:1 with the tuner at his radio, but guess who will be working the DX?
Conclusion I hope I have been able to show by example that SWR can be a serious issue, or something not so important. With the help of the graphs and a little information about your transmission line and antenna, it’s easy to determine how much of your signal is actually getting on the air or how much is being lost in the transmission line.
Darrin Walraven, K5DVW, has been licensed since 1986 at the age of 18. He enjoys DX, CW and SSB. He has an engineering degree from Texas A&M University and is employed as an RF design engineer. You may contact him at k5dvw@hotmail.com.
New Products UNIVERSAL DIGITAL DIAL FROM ELECTRONIC SPECIALTY PRODUCTS The Model DD-103 Universal Digital Dial adds a digital frequency display to vintage receivers. It counts only the VFO (local oscillator) signal, and VFO offset and band information for many receivers is stored in the display’s memory. A manual program mode allows programming up to 32 bands for any receiver that’s not included on the preprogrammed list. Each band can be individually calibrated to allow for crystals that have aged or any other mis-alignment or error, to a rated accuracy of 10 Hz. Calibration information is also stored in memory, so each band only needs to be calibrated one time. The DD-103 can also serve as bench frequency counter up to 40 MHz. Power is from the receiver, and power and signal cables are supplied. The complete instruction manual, a list of preprogrammed receivers and current pricing information is available online at www.electronicspecialtyproducts.com. |
11746 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3999711/ | Genotype–phenotype correlations in neurogenetics: Lesch-Nyhan disease as a model disorder - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Brain
. 2013 Aug 22;137(5):1282–1303. doi: 10.1093/brain/awt202
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Genotype–phenotype correlations in neurogenetics: Lesch-Nyhan disease as a model disorder
Rong Fu
Rong Fu
1 Departments of Neurology, Human Genetics and Paediatrics; Emory University, Atlanta GA, USA
Find articles by Rong Fu
1, Irene Ceballos-Picot
Irene Ceballos-Picot
2 Paris Descartes University, Sorbonne Paris Cite and Department of Metabolic Biochemistry, Necker-Enfants Malades Hospital, Paris, France
Find articles by Irene Ceballos-Picot
2, Rosa J Torres
Rosa J Torres
3 Division of Clinical Biochemistry and Division of Internal Medicine Hospital Universitario La Paz, IdiPaz, Madrid, Spain
Find articles by Rosa J Torres
3, Laura E Larovere
Laura E Larovere
4 Centro de Estudio de las Metabolopatias Congenitas; Universidad Nacional de Cordoba, Argentina
Find articles by Laura E Larovere
4, Yasukazu Yamada
Yasukazu Yamada
5 Department of Genetics, Institute for Developmental Research, Aichi Human Service Centre, Aichi, Japan
Find articles by Yasukazu Yamada
5, Khue V Nguyen
Khue V Nguyen
6 Department of Paediatrics, University of California School of Medicine, San Diego CA, USA
Find articles by Khue V Nguyen
6, Madhuri Hegde
Madhuri Hegde
1 Departments of Neurology, Human Genetics and Paediatrics; Emory University, Atlanta GA, USA
Find articles by Madhuri Hegde
1, Jasper E Visser
Jasper E Visser
7 Department of Neurology, Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen Medical Centre, Nijmegen, The Netherlands; and Department of Neurology, Amphia Hospital, Breda, The Netherlands
Find articles by Jasper E Visser
7, David J Schretlen
David J Schretlen
8 Department of Psychiatry and Behavioural Sciences and Department of Radiology, Johns Hopkins University School of Medicine, Baltimore MD, USA
Find articles by David J Schretlen
8, William L Nyhan
William L Nyhan
6 Department of Paediatrics, University of California School of Medicine, San Diego CA, USA
Find articles by William L Nyhan
6, Juan G Puig
Juan G Puig
3 Division of Clinical Biochemistry and Division of Internal Medicine Hospital Universitario La Paz, IdiPaz, Madrid, Spain
Find articles by Juan G Puig
3, Patrick J O’Neill
Patrick J O’Neill
9 Department of Paediatrics, University of Vermont, Burlington, VT, USA
Find articles by Patrick J O’Neill
9, H A Jinnah
H A Jinnah
1 Departments of Neurology, Human Genetics and Paediatrics; Emory University, Atlanta GA, USA
Find articles by H A Jinnah
1,✉, for the Lesch-Nyhan Disease International Study Group
Author information
Article notes
Copyright and License information
1 Departments of Neurology, Human Genetics and Paediatrics; Emory University, Atlanta GA, USA
2 Paris Descartes University, Sorbonne Paris Cite and Department of Metabolic Biochemistry, Necker-Enfants Malades Hospital, Paris, France
3 Division of Clinical Biochemistry and Division of Internal Medicine Hospital Universitario La Paz, IdiPaz, Madrid, Spain
4 Centro de Estudio de las Metabolopatias Congenitas; Universidad Nacional de Cordoba, Argentina
5 Department of Genetics, Institute for Developmental Research, Aichi Human Service Centre, Aichi, Japan
6 Department of Paediatrics, University of California School of Medicine, San Diego CA, USA
7 Department of Neurology, Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen Medical Centre, Nijmegen, The Netherlands; and Department of Neurology, Amphia Hospital, Breda, The Netherlands
8 Department of Psychiatry and Behavioural Sciences and Department of Radiology, Johns Hopkins University School of Medicine, Baltimore MD, USA
9 Department of Paediatrics, University of Vermont, Burlington, VT, USA
✉
Correspondence to: H. A. Jinnah, M.D., Ph.D., Professor, Departments of Neurology, Human Genetics and Paediatrics, Emory University, Suite 6300 Woodruff Memorial Building, 101 Woodruff Circle, Atlanta GA, 30322 E-mail: hjinnah@emory.edu
✉
Corresponding author.
Received 2013 Mar 15; Revised 2013 May 1; Accepted 2013 May 21; Issue date 2014 May.
© The Author (2013). Published by Oxford University Press on behalf of the Guarantors of Brain. All rights reserved. For Permissions, please email: journals.permissions@oup.com
PMC Copyright notice
PMCID: PMC3999711 PMID: 23975452
Genotype-phenotype correlations for most monogenic neurological disorders are incompletely understood. Fu et al. draw upon data on 615 HPRT mutations, including 130 new cases, and their relationship to Lesch-Nyhan disease severity. The effect of mutations on hypoxanthine-guanine phosphoribosyltransferase activity accounts for much but not all of the phenotypic variability.
Keywords: Lesch-Nyhan disease, genotype–phenotype correlations, neurogenetics
Abstract
Establishing meaningful relationships between genetic variations and clinical disease is a fundamental goal for all human genetic disorders. However, these genotype–phenotype correlations remain incompletely characterized and sometimes conflicting for many diseases. Lesch-Nyhan disease is an X-linked recessive disorder that is caused by a wide variety of mutations in the HPRT1 gene. The gene encodes hypoxanthine-guanine phosphoribosyl transferase, an enzyme involved in purine metabolism. The fine structure of enzyme has been established by crystallography studies, and its function can be measured with very precise biochemical assays. This rich knowledge of genetic alterations in the gene and their functional effect on its protein product provides a powerful model for exploring factors that influence genotype–phenotype correlations. The present study summarizes 615 known genetic mutations, their influence on the gene product, and their relationship to the clinical phenotype. In general, the results are compatible with the concept that the overall severity of the disease depends on how mutations ultimately influence enzyme activity. However, careful evaluation of exceptions to this concept point to several additional genetic and non-genetic factors that influence genotype–phenotype correlations. These factors are not unique to Lesch-Nyhan disease, and are relevant to most other genetic diseases. The disease therefore serves as a valuable model for understanding the challenges associated with establishing genotype–phenotype correlations for other disorders.
Introduction
Understanding how abnormalities in specific genes cause disease is a fundamental goal in human genetics. Identifying benign sequence variants and linking specific pathological mutations with the clinical manifestations of a disease can be helpful for understanding variations in the clinical spectrum of the disease, and for elucidating the pathogenesis of different clinical features of a complex syndrome. Lesch-Nyhan disease historically has served as a model for exploring genotype–phenotype correlations (Jinnah et al., 2000, 2004). Lesch-Nyhan disease was the first neurogenetic disorder for which the responsible gene was identified (Jolly et al., 1983). The gene is designated HPRT1, and it is among the most intensively studied genes in human genetics (Keebaugh et al., 2006). The gene has only one functional messenger RNA transcript that encodes an enzyme involved in purine metabolism known as hypoxanthine-guanine phosphoribosyltransferase (HGprt). The disorder is inherited in an X-linked recessive manner, which means that the influence of genetic alterations of a single allele can be evaluated independent from a second allele.
Many different disease-causing mutations have been reported for the HPRT1 gene (www.lesch-nyhan.org). Spread through nearly the whole gene are missense mutations, nonsense mutations, splicing mutations, small and large coding and non-coding deletions or insertions, partial duplications, non-coding regulatory mutations, and more complex changes. A comprehensive study of genotype–phenotype correlations published more than a decade ago concluded that the most relevant factor for provoking disease was the effect of the HPRT1 mutation on residual HGprt enzyme activity (Jinnah et al., 2000). However, some apparent exceptions to this rule have been reported. In recent years, many new disease-causing mutations have been described, and there have been considerable advances in understanding the consequences of these mutations on the structure and biochemical kinetics of the HGprt enzyme (Duan et al., 2004; Fu and Jinnah, 2012).
There also has been increased appreciation for the need to accurately describe and quantify different aspects of the clinical phenotype, and a growing understanding of how the phenotype evolves with age. The classical clinical phenotype of Lesch-Nyhan disease is defined by uric acid overproduction, motor dysfunction, intellectual disability, and behavioural problems including recurrent self-injury (Lesch and Nyhan, 1964; Nyhan et al., 1965; Jinnah et al., 2006). However, there also are attenuated clinical variants in which some of these clinical features are clinically insignificant or even absent (Emmerson and Thompson, 1973; Puig et al., 2001; Jinnah et al., 2010; Torres et al., 2011). Collectively, patients with attenuated phenotypes are designed Lesch-Nyhan variants. The mildest variant is described by isolated overproduction of uric acid. These patients do not have clinically overt neurological or behavioural abnormalities, and most often are described as having HGprt-related hyperuricaemia (HRH). In between the two extreme phenotypes of Lesch-Nyhan disease and HRH is a spectrum of phenotypes with varying degrees of neurological and behavioural abnormalities, designated HGprt-related neurological dysfunction (HND). The patients with HND suffer from overproduction of uric acid along with some neurological or behavioural difficulties, but they do not exhibit the self-injurious behaviours seen in classic Lesch-Nyhan disease.
The present review provides a summary of 615 cases with disease-causing mutations, including 485 previously reported cases and 130 new cases. It also summarizes our current knowledge of the influence of these mutations on HGprt structure and function, and the relevance of this information for understanding the molecular pathogenesis of the disease and diagnostic testing. The analysis provides some valuable lessons for understanding challenges associated with establishing meaningful genotype–phenotype correlations in other neurogenetic disorders.
The HPRT1 gene and HGprt enzyme
The human HPRT1 gene is located on the X chromosome in the region q26-27 (Patel et al., 1984; Stout and Caskey, 1985; Melton, 1987; Keebaugh et al., 2006). It contains eight introns and nine exons that encode a single mature messenger RNA of ∼1.6 kb. The coding region contains 654 nucleotides and is translated into HGprt, a protein with a molecular weight of 24.6 kD. Aside from cleavage of the initial methionine from the protein and acetylation of the newly exposed alanine (Wilson et al., 1983), there are no known post-translational modifications. However, there are two slowly progressive alterations including spontaneous oxidation of C23 and deamidation of N107 to D107 (Johnson et al., 1982; Wilson et al., 1982; Keough et al., 1991).
The crystal structure of HGprt has been determined in different forms, including its free form (Keough et al., 2005), liganded with its inosine monophosphate or guanosine monophosphate (GMP) products (Eads et al., 1994; Xu et al., 1997), or complexed with a transition state analogue and pyrophosphate (Shi et al., 1999). The enzyme exhibits a core structure that consists of five parallel beta-sheets flanked by four alpha helices. HGprt assembles spontaneously as either a dimer or tetramer of identical subunits depending on the ionic strength of the solvent (Holden and Kelley, 1978; Johnson et al., 1979). The subunits are not covalently bound, but instead are linked by a network of hydrogen bonds, salt bridges, and hydrophobic interactions. Figure 1 shows a schematic predicted from the crystal structures. The subunits are designated A to D, and the most extensive interactions occur at the A–B interface (amino acids 23–28, 70–101 and 198–204) and the A–C interface (amino acids 7–23 and 38–51). The A–D interface involving amino acids 84-90 is minor. Most of the amino acids at the interface between subunits are far from the catalytic site.
Figure 1.
Open in a new tab
Ribbon diagram for human HGprt. (A) A monomer with guanosine monophosphate in the open loop state (PDB: 1hmp). (B) A monomer with transition-state analogue and PPi in the closed loop state (PDB: 1bzy). (A and B) The PPi loop is shown in red, the PRPP loop is shown in blue, the C-terminal hood domain is shown in green, and the long flexible loop is shown in yellow. K166 is displayed in purple and the ligands are black. (C) The tetrameric structure of HGprt with each of the four subunits in a different colour (A in grey, B in blue, C in red and D in green).
HGprt is one of a family of phosphoribosyltransferase enzymes that uses the high-energy phosphate bonds of phosphoribosylpyrophosphate (PRPP) rather than ATP to catalyze its reaction (Fox and Kelley, 1971; Musick, 1981). HGprt drives two similar reactions in the recycling pathway for purines. One involves the pyrophosphorylation of hypoxanthine into inosine monophosphate, a precursor for both adenine nucleotides (ATP, ADP, AMP) and guanine nucleotides (GTP, GDP, GMP). The other involves the pyrophosphorylation of guanine directly into GMP (Fig. 2). The reaction catalyzed by HGprt has been reported to proceed via an ordered mechanism, in which PRPP binds first and induces a conformational change for subsequent purine binding, followed by the release of pyrophosphate and nucleotide (Salerno and Giacomello, 1979).
Figure 2.
Open in a new tab
Purine metabolism.
AS = adenylosuccinate; asp = aspartic acid; ATP = adenosine triphosphate; GDP = guanosine diphosphate; gln = glutamine; gly = glycine; GMP = guanosine monophosphate; Gprt = guanine phosphoribosyltransferase; GTP = guanosine triphosphate; Hprt = hypoxanthine phosphoribosyltransferase; IMP = inosine monophosphate; XMP = xanthine monophosphate. In this diagram, HGprt is shown separately at its two distinct functional sites (Hprt and Gprt).
The catalytic reaction involves four domains of the HGprt protein including three loops that form the catalytic cleft and a long flexible loop that covers the reaction (Fig. 1). The PRPP binding loop between amino acids 130–140 is highly conserved across the phosphoribosyltransferase family, and serves to orient the ribose-phosphate moiety of PRPP in the catalytic cleft. Four consecutive amino acids from positions 68–71 constitute the PPi loop and also exhibit high structural homology across the phosphoribosyltransferase family. The PPi loop binds to the pyrophosphate moiety of PRPP in the catalytic cleft. A magnesium ion coordinates binding of hydroxyl groups and the 1-α-phosphate group of PRPP with other amino acids in active site. The hood domain at the carboxy terminus orients the purine ligand to PRPP bound within the cleft. Amino acid K166 influences substrate specificity toward the exocyclic 6-oxo group of the purine ring, and alterations here influence activity towards hypoxanthine, guanine, or xanthine. A long flexible loop from amino acids 100–128 exhibits open or closed conformations, moving up to 25 Å to shield the catalytic cleft from solvent during catalysis.
Studies of the crystal structure of HGprt have revealed multiple conformational changes during the catalytic cycle (Eads et al., 1994; Balendiran et al., 1999; Shi et al., 1999; Wang et al., 2001; Keough et al., 2005). The PRPP binding loop is relatively dynamic and opens when a 5’-phosphate group enters. This triggers a trans–cis isomerization between peptide bonds of L68–K69 to adopt a structure for optimal orientation of the ribose ring and pyrophosphate group. The binding of PRPP induces a rearrangement of the K166 side chain interaction, which turns to position the purine base via the 6-oxo group. The diacidic peptide E134–D135 of the PRPP binding loop moves to coordinate with the ribose ring, and the hood domain containing F187 folds over the ligands via stacking interactions. The long flexible loop then closes over the bound substrates. After the reaction, the flexible loop opens to liberate pyrophosphate followed by the nucleotide (Duan et al., 2004).
The precise knowledge of the structure and function of HGprt provides a rich background for delineating how mutations affect enzyme function and cause clinical manifestations. The four functional domains of the protein and the known subunit interactions lead to questions regarding their relative roles in disease pathogenesis. In addition, the dual roles of HGprt in recycling hypoxanthine and guanine raise questions regarding how mutations may affect one or both functions, and which function may be relevant for which aspects of the clinical phenotype.
The spectrum of the clinical phenotype
The classical phenotype of Lesch-Nyhan disease
The classical clinical phenotype of Lesch-Nyhan disease is relatively stereotyped. This phenotype includes overproduction of uric acid and an unusual neurobehavioural syndrome (Lesch and Nyhan, 1964; Nyhan et al., 1965). The overproduction of uric acid results in elevated serum levels of uric acid and increased excretion of uric acid in the urine (Torres et al., 2011). Because uric acid is near its limit of solubility in the body, it tends to precipitate in vulnerable body regions. Precipitation in the urogenital system, where it is concentrated by excretion, results in nephrolithiasis and associated problems such as urinary obstruction and renal failure. Precipitation in the subcutaneous tissues due to temperature gradients leads to solid masses known as tophi. Precipitation in the joints of the hands and feet with subsequent inflammation elicited by phagocytosis of the crystals by polymorphonuclear cells leads to gouty arthritis.
The neurobehavioural phenotype includes severe and recurrent self-injurious behaviours such as self-biting, self-hitting, eye poking, and others (Nyhan, 1976; Anderson and Ernst, 1994; Robey et al., 2003; Schretlen et al., 2005). Self-injurious behaviour is universal in Lesch-Nyhan disease. It usually emerges before 4 years of age, but may be delayed until the second decade of life. The neurobehavioural phenotype also includes severe motor handicap that resembles dystonic cerebral palsy (Watts et al., 1982; Jinnah et al., 2006). Motor function is dominated by generalized dystonia, a movement disorder that often is confused with spasticity or chorea. The motor disorder is sufficiently severe that all require wheelchairs and substantial assistance with activities of daily living. Most patients also have mild or moderate intellectual disability with IQ scores in the 60–80 range, but severe mental retardation is uncommon (Anderson et al., 1992; Matthews et al., 1995; Solan et al., 1997; Schretlen et al., 2001).
The milder Lesch-Nyhan variants
In the milder variant phenotypes, some clinical features are absent or sufficiently mild that they may escape clinical detection (Emmerson and Thompson, 1973; Puig et al., 2001; Jinnah et al., 2010; Torres et al., 2011). The mildest clinical phenotype of HRH includes only overproduction of uric acid and its associated problems. These patients do not have clinically overt neurological or behavioural abnormalities, although many have minor motor clumsiness or mild cognitive impairments that can be detected with appropriate neurological or psychometric testing.
In between the severe Lesch-Nyhan disease phenotype and the mild HRH phenotype is a broad spectrum of phenotypes with varying degrees of neurological and behavioural abnormalities known as HND (Jinnah et al., 2010). Patients with HND also suffer from overproduction of uric acid. They do not exhibit the self-injurious behaviours seen in classic Lesch-Nyhan disease, although they may have impulsivity and other undesirable behaviours. The motor features of HND vary from mild and clinically insignificant clumsiness to severe handicap similar to classic Lesch-Nyhan disease. Most also have some cognitive impairment, most notably difficulties with attention and concentration. Across the spectrum of HGprt deficiency, there is a rough correlation between the severity of motor and cognitive dysfunction (Jinnah et al., 2010), although rare cases may combine severe motor disability with little or no cognitive impairment.
Since Lesch-Nyhan disease is inherited in an X-linked recessive manner, the vast majority of cases are males. However, several manifesting females have been reported (Hooft et al., 1968; Hara et al., 1982; Ogasawara et al., 1989; Van Bogaert et al., 1992; Yukawa et al., 1992; Yamada et al., 1994; Aral et al., 1996; De Gregorio et al., 2000; Inokuchi et al., 2004; De Gregorio et al., 2005; Rinat et al., 2005; Jinnah et al., 2006, 2010; Sebesta et al., 2008). Eight of the manifesting females had the typical Lesch-Nyhan disease phenotype, and two were milder variants. The clinical features of the disease do not appear to be significantly influenced by sex. The clinical phenotype of these females does not result from haploinsufficiency. For those in whom the molecular defect was determined, all had a mutation in one allele together with skewed inactivation of the other allele. For females with the classic phenotype, skewed inactivation is thought to be complete, with corresponding severe loss of HGprt enzyme activity.
Limitations of clinical ascertainment
Although patients are clustered into three distinct sub-groups (Lesch-Nyhan disease, HND and HRH), the clinical spectrum actually is more continuous (Jinnah et al., 2010; Torres et al., 2011). Each of the major clinical problems associated with Lesch-Nyhan disease exhibits continuous grades of severity (Fig. 3). The existence of varying grades of severity means that some patients may fall between groups, and assignment to one group is sometimes arbitrary. For example, there are many cases initially published as HRH because of the absence of any overt neurological deficits. However, in some of these cases, careful neurological re-evaluation revealed mild impairments in motor skills (Jinnah et al., 2010). The cumulative severity of dystonia affecting all body regions can be estimated by the Fahn-Marsden dystonia rating scale, which goes from 0 (no dystonia) to 140 (severest dystonia). Some authors define a score of <5 as normal and compatible with HRH, to account for some minor clumsiness that may occur in otherwise normal children. Others have suggested a higher cut-off score of <10, highlighting the arbitrary nature of sub-grouping cases based on a disorder with a continuous spectrum of severity.
Figure 3.
Open in a new tab
Clinical spectrum in Lesch-Nyhan disease and its variants. The four major aspects of the clinical phenotype are shown in grey boxes, with overall severity and frequency depicted by tapering triangles. LND = Lesch-Nyhan disease.
Cognitive disability also varies considerably. Some cases considered cognitively normal based on adequate school performance had more obvious deficits on formal neuropsychological testing (Schretlen et al., 2001; Jinnah et al., 2010). In other cases, specific cognitive impairments have been documented, despite global IQ scores suggestive of ‘normal’ intelligence. In some of these cases, the selective cognitive deficits have been sufficiently severe to impair normal social or occupational activities. The classification of these cases as HRH or HND depends on the rigor of neuropsychological testing, criteria used to define normal intelligence, and whether selective but severe deficits in particular cognitive domains are considered clinically relevant.
Even the occurrence of self-injury can lead to diagnostic uncertainty. In many classic Lesch-Nyhan disease cases, self-injury is severe and obvious with serious tissue damage, and misclassification is unlikely (Marmattom, 2005). However in some cases, self-injury is very mild. An example is a case with a single small abrasion on one thumb caused by a ‘habit’ of rapidly putting his hand to his mouth when excited or stressed (Jinnah et al., 2006). The abrasion resulted from repeated contact of the thumb with the teeth, but he never bit the thumb and there was no bleeding. The family did not view this behaviour as self-injurious, although the clinical team questioned it. Other examples are cases with isolated episodes of self-injury occurring once or twice in a lifetime, often triggered by some stressful event, such as a painful kidney stone. There also are several reports of Lesch-Nyhan disease variants without overt self-injury but multiple ‘accidental’ injuries from ‘bad wheelchair driving’ (Jinnah et al., 2010). Because wheelchair accidents leading to multiple injuries are not typical of other patients with similar motor disabilities, the accidental nature of the injuries might be questioned. Obviously, the classification of patients as Lesch-Nyhan disease or HND according to the presence or absence of self-injury depends on the criteria used to define ‘self injury’.
In addition to the problem of clinical ascertainment, it is important to acknowledge that HGprt deficiency causes a developmental disorder in which the clinical features unfold over time. Normal motor skills vary dramatically with age, with the emergence of dystonic movements and associated disability most often occurring gradually during the first few years of life. Some problems can be difficult to measure in infants. One example involves the c.500 G>T mutation (p.R167M), which was reported to be associated with markedly different clinical phenotypes in three members of one family (Sarafoglou et al., 2010). However, the motor and cognitive differences can be partly attributed to comparing toddlers with older adults (Fu and Jinnah, 2011). There are several additional cases in which motor skills were considered normal in the first month of life, but obvious motor delay became apparent within the first year (Kim et al., 1997; Mak et al., 2000; Zamora et al., 2007; Jurecka et al., 2008; Kim et al., 2009). Cognitive skills also change dramatically with age, and cognitive deficits are difficult to document for infants and very young children. The emergence of self-injurious behaviour also is age-dependent. The average age of onset is 3.1 years but the range varies from <1 year to >20 years (Jinnah et al., 2006). The varying age at onset means that distinguishing Lesch-Nyhan disease from variant cases is subject to error, especially for cases reported as children. Several cases initially classified as HND because of the absence of self-injurious behaviour (Fujimori et al., 1991), were reclassified as Lesch-Nyhan disease when self-injury emerged later (Fujimori et al., 1994).
Finally, the clinical phenotype can reflect the quality of care and education available. Patients with Lesch-Nyhan disease are difficult to work with because of their behavioural abnormalities, and they often present both physical and social challenges in school. As a result, educational opportunities and socialization sometimes are limited, and the result is low scores on standardized tests of cognitive function and sometimes inappropriate social behaviour. The management of physical disabilities also is challenging, with dystonic movement limiting mobility and behavioural problems that inhibit physiotherapy to prevent the consequences of disuse, such as development of contractures. Thus, the clinical phenotype of an adult with Lesch-Nyhan disease or its variants can reflect an interaction between the biology of the disease and psychosocial factors that may modify the disease over time.
Any attempt to correlate the severity of the clinical phenotype with mutations in HPRT1 or biochemical measurements of HGprt function must be done with careful attention to the rigor of neurological and psychological assessments, the age of the patient at evaluation, and the quality of care and education. Inadequate clinical assessments readily lead to misclassification of the phenotype, and assessments in very young children may reflect an incompletely evolved phenotype. Careful clinical assessments are essential starting points for understanding genotype–phenotype correlations.
The heterogeneity of clinically relevant mutations
Types of mutations
Tables 1–5 provide lists of mutations for 130 new cases from unrelated families referred to one of the centres of the Lesch-Nyhan Disease International Study Group for testing. Mutations were determined at each test site according to published methods (Jinnah et al., 2000; Torres et al., 2000; Ea et al., 2009; Yamada et al., 2011 a). Most mutations were determined by reverse-transcriptase with PCR and direct sequencing of the resultant complementary DNA, using the reference sequence NM_000194.2. When complementary DNA could not be obtained, or contained an apparent deletion, genomic DNA was evaluated. Precise breakpoints in genomic DNA were not always determined as part of clinical diagnostic testing. Although all are new cases, some were found to have the same mutations previously described for other unrelated cases. All cases were males, except for one. The list includes 39 missense mutations, 13 nonsense mutations, 26 splicing mutations, 36 deletions, nine duplications, five indels and one with normal coding region. The female had a missense mutation with skewed X-inactivation.
Table 1.
New cases with missense mutations
| Case | Phenotype | Mutation | Result | Other cases reported |
--- ---
| CF | LND | c.47G>T | p.G16V | Yes |
| NS | HRH | c.59A>T | p.D20V | Yes |
| AC | LND | c.65T>G | p.F22C | Yes |
| MD | HND | c.85G>C | p.A29P | Novel |
| AP | HND | c.115C>G | p.H39D | Yes |
| AC | LND | c.118G>A | p.G40R | Yes |
| OA | LND | c.125T>C | p.I42T | Yes |
| MG | LND | c.130G>C | p.D44H | Novel |
| LA | HND | c.143G>A | p.R48H | Yes |
| AD | HND | c.143G>A | p.R48H | Yes |
| LN159, DP | LND | c.145C>T | p.L49F | Yes |
| MB | LND | c.146T>G | p.L49R | Novel |
| CM; SM | LND | c.148G>A | p.A50T | Novel |
| ML; AL | LND | c.158T>G | p.V53G | Novel |
| NS | HRH | c.179A>G | p.H60R | Yes |
| YG | HRH | c.200T>C | p.V67A | Novel |
| RT | LND | c.209G>A | p.G70E | Yes |
| LN139F, O | LND | c.220T>G | p.F74V | Yes |
| LN139C, H4755, B | LND | c.293A>G | p.D98G | Yes |
| HPRT Rumania | HND | c.295T>G | p.F99V | Yes |
| MC | HRH | c.374T>C | p.L125S | Novel |
| TP | LND | c.379G>T | p.G127V | Novel |
| EG | LND | c.392T>G | p.L131W | Novel |
| PA | HRH | c.404A>G | p.D135G | Yes |
| NG | HND | c.475A>G | p.K159E | Yes |
| LN167, RT | LND | c.476A>C | p.K159T | Novel |
| OB | HRH | c.485G>C | p.S162T | Novel |
| TA | LND | c.530A>T | p.D177V | Yes |
| LN139A, G8081, JG | LND | c.539G>A | p.G180E | Yes |
| LN216, KD | LND | c.541T>C | p.F181L | Yes |
| India II | LND | c.550C>A | p.P184T | Novel |
| OA; MA | HND | c.554A>T | p.D185V | Novel |
| MB | HND | c.554A>T | p.D185V | Novel |
| LN203C, NP | LNV | c.584A>C | p.Y195S | Yes |
| PB; AB | LND | c.595T>G | p.F199V | Yes |
| DG; MG | HND | c.597C>G | p.F199L | Novel |
| LN155, KH | LND | c.611A>T | p.H204L | Novel |
| BW | LND | c.611A>G | p.H204R | Yes |
| MD | LND | c.635G>A | p.G212E | Yes |
Open in a new tab
This table includes a summary of previously unreported cases. In some cases, the molecular mutation is novel, while in others it is similar to a previously reported case. Previously reported cases are listed at www.lesch-nyhan.org. Abbreviations follow standard nomenclature for genetic diseases.
Table 2.
New cases with nonsense mutations
| Case | Phenotype | Mutation | Result | Other cases reported |
--- ---
| NL | LND | c.84T>A | p.Y28 | Novel |
| JA | LND | c.151C>T | p.R51 | Yes |
| LN131, AJ | LND | c.151C>T | p.R51 | Yes |
| LN208 | LND | c.151C>T | p.R51 | Yes |
| SB | LND | c.151C>T | p.R51 | Yes |
| OS; MS | LND | c.430C>T | p.Q144 | Yes |
| JF | LND | c.454C>T | p.Q152 | Novel |
| LN148, SC | LND | c.508C>T | p.R170 | Yes |
| LN150, DP | LND | c.508C>T | p.R170 | Yes |
| LN169, LSZ | LND | c.508C>T | p.R170 | Yes |
| LN171, KM | LND | c.508C>T | p.R170 | Yes |
| TF; LS | LND | c.508C>T | p.R170 | Yes |
| VV | LND | c.508C>T | p.R170 | Yes |
Open in a new tab
This table includes a summary of previously unreported cases. In some cases, the molecular mutation is novel, while in others it is similar to a previously reported case. Previously reported cases are listed at www.lesch-nyhan.org. Abbreviations follow standard nomenclature for genetic diseases.
Table 3.
New cases with splicing mutations
| Case | Phenotype | Mutation | Result | Other cases reported |
--- ---
| HH | LND | c.27+1G>A | Splice error | Yes |
| LN128, H6214, DC | LND | c.27+5G>A | Splice error | Yes |
| TH | LND | c.135-1G>A | Splice error | Novel |
| LN175 | LND | c.135-2A>G | Exon 3 excluded | Novel |
| LN132, JC | LND | c.318+1G>T | Exon 3 excluded | Yes |
| LN204, AJZ | LND | c.384+1G>T | Exon 4 excluded | Yes |
| LN122, R | LND | c.384+1G>T | Exon 4 excluded | Yes |
| BP | LND | c.385-1G>A | Exon 5 excluded | Yes |
| MP | LND | c.402+1G>A | Splice error | Yes |
| VP | LND | c.402+5G>A | Splice error | Novel |
| LN174 | LND | c.403-1G>C | Exon 5 excluded | Yes |
| LN203B, JS; DS | LNV | c.485+5G>A | Exon 6 excluded | Yes |
| AQ | LND | c.486-1G>A | Splice error | Novel |
| LN126, BM | LND | c.486-3C>G | Exon 7 excluded | Novel |
| LN158, JCG | LND | c.532+1G>A | Exon 7 excluded | Yes |
| LN217, TM | LND | c.532+1G>A | Exon 7 excluded | Yes |
| PMK; WBK | LND | c.532+2G>T | Exon 7 excluded | Novel |
| LN139B, H8475, H | LND | c.532+5G>A | Exon 7 excluded | Yes |
| NS | LND | c.532+5G>T | Exon 7 excluded | Novel |
| BV | LND | c.533-13T>G | Exon 8 excluded | Novel |
| KD | LND | c.533-1G>A | Splice error | Yes |
| LN172, LN203A, MB; LB | LND | c.609+6T>G | Exon 8 excluded | Yes |
| LN162, II | LND | c.610-1G>A | 17 bp of exon 9 excluded | Yes |
| FG | LND | c.610-1G>A | Splice error | Yes |
| GD | LND | c.610-1G>T | Splice error | Novel |
| LN136, KB | LND | c.610-2A>T | 610–626 bp of exon 9 excluded | Yes |
Open in a new tab
This table includes a summary of previously unreported cases. In some cases, the molecular mutation is novel, whereas in others it is similar to a previously reported case. Previously reported cases are listed at www.lesch-nyhan.org. Abbreviations follow standard nomenclature for genetic diseases.
Table 4.
New cases with deletion mutations
| Case | Phenotype | Mutation | Result | Other cases reported |
--- ---
| ML | LND | 5'UTR deleted | No protein | Novel |
| LN130, ML | LND | Exon 1 deleted | 1 exon deleted | Yes |
| LN139D, H2339, JM, JHG, LN160 | LND | Exon 1 deleted | 1 exon deleted | Yes |
| LN161, OLH | LND | Exon 1 deleted | Macro del | Yes |
| LN201, JL | LND | Exon 1 deleted | Macro del | Yes |
| LN215, WM | LND | Exon 1 deleted | Macro del | Yes |
| Leganes | LND | Exon 1 deleted | 1 exon deleted | Yes |
| Colombia | LND | c.53delA | Frame shift in exon 2 | Novel |
| FM | LND | c.82_84delTAT | p.Y28del | Yes |
| LN170, RS | LND | c.105_108delGTTT | Frame shift in exon 2 | Novel |
| LN211, CN | LND | c.124_127delATTA | Frame shift in exon 2 | Novel |
| JC, AC | LND | c.131delA | Frame shift in exon 2 | Novel |
| LN139E, H4672, H | LND | Exons 2–3 deleted | 2 exons deleted | Yes |
| LN137, RC | LND | Exons 2–4 deleted | 3 exons deleted | Novel |
| FM | LND | c.212delG | Frame shift in exon 3 | Novel |
| LN165, EV | LND | c.269_270delAT | Frame shift in exon 3 | Yes |
| Bilbao | LND | c.275_298del | 16 bp deleted in exon 3 | Novel |
| LN205, RC | LND | c.288_289delTG | Frame shift in exon 3 | Novel |
| LN209, JB | LND | c.289_290delGT | Frame shift in exon 3 | Yes |
| LN164 | LND | c.337delG | Frame shift in exon 4 | Yes |
| DA | LND | Exon 4 deleted | 1 exon deleted | Yes |
| LN213, DR | LND | Exons 4–6 deleted | 3 exons deleted | Yes |
| GB | LND | Exons 4–8 deleted | No protein | Yes |
| ML | LND | Exons 4–9 deleted | No protein | Yes |
| LN124, ES | LND | c.402+1_402+4delGTAA | Splice error, frame shift in exon 5 | Novel |
| LN218, DD | LND | c.402+5delG | Splice error, frame shift in exon 5 | Novel |
| CA; AA | LND | c.512delG | Frame shift in exon 7 | Novel |
| VW | LND | Exon 7 deleted | 1 exon deleted | Yes |
| LN144, CS | LND | Exons 7–9 deleted | 3 exons deleted | Yes |
| PL | LND | Exon 8 deleted | 1 exon deleted | Yes |
| LN145, MP | LND | c.577delC | Frame shift in exon 8 | Novel |
| LD | LND | c.585_586delTA | Frame shift in exon 8 | Novel |
| LN146, KB | LND | c.6095_6098delGATT | Splice error, frame shift in exon 8 | Novel |
| LN127, GQ | LND | c.643_664del | Exon 9 excluded | Novel |
| Las Palmas | LND | Exon 9 deleted | 1 exon deleted | Yes |
| LN152, RR | LND | Exons 1–9 deleted | 9 exons deleted | Yes |
Open in a new tab
This table includes a summary of previously unreported cases. In some cases, the molecular mutation is novel, while in others it is similar to a previously reported case. Previously reported cases are listed at www.lesch-nyhan.org. Abbreviations follow standard nomenclature for genetic diseases, except for large deletions where actual breakpoints were not defined.
Table 5.
Other new cases with mutations
| Case | Phenotype | Mutation | Result | Other cases reported |
--- ---
| Duplication | | | | |
| Valencia | LND | c.50_51dupAT | Frame shift in exon 2 | Novel |
| Brussels | LND | c.212dupG | Frame shift in exon 3 | Yes |
| LN133, AC | LND | c.212dupG | Frame shift in exon 3 | Yes |
| LN135 | LND | c.212dupG | Frame shift in exon 3 | Yes |
| LN149, NY | LND | c.212dupG | Frame shift in exon 3 | Yes |
| LN153, DC | LND | c.212dupG | Frame shift in exon 3 | Yes |
| LN173, CD | LND | c.212dupG | Frame shift in exon 3 | Yes |
| DD | LND | c.212dupG | Frame shift in exon 3 | Yes |
| BT | LND | c.297dupT | Frame shift in exon 3 | Yes |
| Others | | | | |
| LN210, G | LND | c.-4_4delinsGCC | Frame shift in exon 1 | Novel |
| DK | NA | c.52_54delinsTAA | p.D18 | Novel |
| GB | HND | c.238_239delinsTT | p.D80F | Yes |
| ID | LND | c.320_326delinsCTTTTTTAT | Frame shift in exon 4 | Novel |
| NS | LND | c.456delinsTT | Frame shift | Novel |
| NL | LND | normal | No enzyme activity | Novel |
| Female | | | | |
| LN143 | LND | c.[580G>T]; [XIC] | p.D194Y; non-random inactivation of other allele | Novel |
Open in a new tab
This table includes a list of previously unreported cases. In some cases, the molecular mutation is novel, whereas in others it is similar to a previously reported case. Previously reported cases are listed at www.lesch-nyhan.org. Abbreviations follow standard nomenclature for genetic diseases.
XIC = non-random X-chromosome inactivation.
Double asterisk denotes presumed non-coding mutation in regulatory regions of the gene.
Table 6 summarizes these 130 new cases combined with 485 cases reported through to the end of December 2012. Together, the mutations were quite heterogeneous. Single base substitutions were most common, with 381 cases accounting for 62.0% of the total. Of the substitutions, 242 (39.3%) were missense mutations producing a single amino acid substitution, 90 (14.6%) caused splicing errors, and 49 (8.0%) were nonsense codons leading to premature termination of protein translation. Deletions were the second most common type of mutation, with 158 cases representing 25.7% of the total. Approximately half were large intragenic deletions with loss of one or more exons, whereas the other half were smaller deletions of up to seven bases. Ten deletions affected splice sites predicted to cause splicing errors. There were also 40 partial gene duplications (6.5%) and 26 (4.2%) other mutations including 13 complex deletions, six non-coding sequence alterations, four insertions, two double mutations, and one translocation. Ten females were reported with a mutation in one allele and skewed inactivation of the other allele.
Table 6.
Summary of new and previously reported HPRT1 mutations
| Mutation | LND (n = 461) | LNV (n = 142) | NA (n = 12) | Total (n = 615) | % of total |
--- --- --- |
| Single base substitutions | 255 | 121 | 5 | 381 | 62.0 |
| Missense | 131 | 107 | 4 | 242 | |
| Nonsense | 47 | 1 | 1 | 49 | |
| Splice error | 77 | 13 | 0 | 90 | |
| Deletions | 146 | 6 | 6 | 158 | 25.7 |
| Coding sequences | 138 | 5 | 5 | 148 | |
| Splice error | 8 | 1 | 1 | 10 | |
| Duplications | 37 | 3 | 0 | 40 | 6.5 |
| Coding sequences | 35 | 3 | 0 | 38 | |
| Splice error | 2 | 0 | 0 | 2 | |
| Others | 15 | 10 | 1 | 26 | 4.2 |
| Translocations | 1 | 0 | 0 | 1 | |
| Regulatory changes | 2 | 4 | 0 | 6 | |
| Double mutations | 0 | 2 | 0 | 2 | |
| Insertions | 3 | 1 | 0 | 4 | |
| Complex deletions | 9 | 3 | 1 | 13 | |
| Females | 8 | 2 | 0 | 10 | 1.6 |
| % of total | 75.0 | 23.1 | 2.0 | 100 | |
Open in a new tab
This table includes a list of all cases with HPRT1 mutations including the new cases from Tables 1–5 combined with all previously reported cases. A complete list of mutations is available at www.lesch-nyhan.org.
NA = not available; LND = Lesch-Nyhan Disease; LNV = Lesch-Nyhan variant.
Locations of mutations
Although the mutations are widely distributed throughout the HPRT1 gene, there are some isolated hot spots, where the same mutation has arisen multiple times in unrelated patients. Two of the most common hot spots are c.151C>T (n = 20) and c.508 C>T (n = 19). Both of these mutations result in the conversion of a codon encoding arginine to a stop signal (p.R51 and p.R170), and both probably arise through a similar mechanism involving methylation of cytosine in CpG motifs, with subsequent deamination of 5-methylcytosine to produce thymine (O'Neill and Finette, 1998). Another hot spot mutation is insertion of an extra guanine nucleotide in the string of guanine nucleotides from positions 212 (c.212dupG), leading to a frame shift and premature stop (n = 15). Here the molecular mechanism probably involves insertion of an extra base due to strand slippage.
Although most mutations affect coding regions or splice sites, there were 12 mutations reported to affect non-coding regions among patients where a clinical phenotype consistent with Lesch-Nyhan disease or an attenuated variant was confirmed by demonstrating biochemical evidence for reduced HGprt activity. Some of these mutations may affect regulatory regions of the gene and reduce messenger RNA transcription or stability. In fact, three cases were reported to have promoter region deletions up to 33 Kb from the start site, resulting in no detectable messenger RNA (Taniguchi et al., 2011; Yamada et al., 2011 b). Another two cases had intronic mutations, in which a single base substitution (c.485+2775T>A; and c.402+1229A>G) formed a functional splice site creating a pseudo-exon (Sege-Peterson et al., 1992; Corrigan et al., 2011). A third case was reported to have a translocation, involving intron 3 (Mizunuma et al., 2004; Yamada et al., 2007). Six more cases had unknown, presumably non-coding mutations with reduced messenger RNA and enzyme activity (Dawson et al., 2005; Garcia et al., 2008; Nguyen et al., 2012). Thus some cases have mutations that are difficult to detect with methods that focus on exons and splice junctions.
Genotype–phenotype correlations
Correlations with types of genetic mutations
Overall, 461 cases (75%) were associated with the typical clinical phenotype of Lesch-Nyhan disease, whereas 142 cases (23%) were associated with the less severe variant phenotypes. Twelve cases (2%) could not be clinically categorized due to insufficient clinical information in the original reports. As previously noted, placing patients into these clinical subgroups based on published reports is imprecise. The literature varies considerably in how carefully the clinical phenotype was ascertained, and some cases were reported at a very young age before the full clinical phenotype had time to develop. Despite these caveats, the large number of reported cases is sufficient to permit recognition of some general genotype–phenotype trends.
A comparison of Lesch-Nyhan disease versus Lesch-Nyhan variant reveals that specific categories of mutations are not equally represented in both groups. Virtually all of the deletion mutations involving coding regions were associated with the more severe clinical phenotype of Lesch-Nyhan disease, with only six reported to have the Lesch-Nyhan variant phenotype. Duplications also were only rarely associated with milder phenotypes, occurring in only three cases. Similarly, nonsense mutations leading to premature termination of protein translation were associated with classic Lesch-Nyhan disease in all cases except one. Only one Lesch-Nyhan variant case had an insertion. The majority of Lesch-Nyhan variant cases (75.4%) had missense mutations leading to single amino acid substitutions. Mutations leading to splicing errors were the second most common type of mutation in Lesch-Nyhan variants, occurring in 13 cases (9.2%, Tables 6 and 7).
Table 7.
Discrepancies in genotype-phenotype correlations
| Genotype | Type | Phenotype | Reasons for discrepancy | References |
--- ---
| c.508C>T | Nonsense | HND | 10 yrs old | Burgemeister et al., 1994 |
| c.213delC | Deletion | HND | 1 yr old | Sharma et al., 2012 |
| c.-12_1del | Deletion | LNV | AUG start deleted, functional upstream GUG start | Gibbs and Caskey, 1987; Gibbs et al., 1989; Davidson et al., 1994 |
| Introns 1–2 deleted | Deletion | HND | <1 yr old | Tvrdik et al., 1998 |
| c.403_452del | Deletion | HRH | Splice junction, multiple mRNA, 9% activity in IF | Sege-Peterson et al., 1992 |
| Exon 7 deleted | Deletion | HND | 5 yrs old | Davidson et al., 1991 |
| c.648_698del | Deletion | HRH | C-terminal loss of 2 amino acids, 9% activity in LL | Igarishi et al., 1989 |
| Exons 2–3 duplicated | Duplication | HND | Gene reversion | Wilson et al., 1986; Monnat et al., 1992 |
| Exons 2–3 duplicated | Duplication | HND | Gene reversion | Yang et al., 1984, 1988; Sege-Peterson et al., 1992; Jinnah et al., 2010 |
| Exons 7–8 duplicated | Duplication | HND | Gene reversion | Marcus et al., 1993b |
| c.429_430insGCA | Insertion | HND | One amino acid added, 7.5% activity in IF | Sege-Peterson et al., 1992 |
| c.176_180delins ATAGATGTGAAA | Indels | HND | 3 yrs old | Serrano et al., 2008 |
| c.238_239delinsTT | Indels | HND | Single amino acid substitution, 3.4 yrs old | Jinnah et al., 2010 |
| c.238_239delinsTT | Indels | HND | Single amino acid substitution | New case |
| c.27+1G>A | Splice error | HND | 7 yrs old | Marcus et al., 1993 a |
| c.27+1G>T | Splice error | HND | 5% activity in EL, 1% activity in LF | Jinnah et al., 2000, 2010 |
| c.27+5G>A | Splice error | HND | Multiple mRNAs | Yamada et al., 2007 |
| c.27+47C>T | Splice error | HRH | Inclusion 15 amino acids without frame shift, 7% activity in LE | Gaigl et al., 2001 |
| c.134+1G>A | Splice error | HND | Unknown | Jinnah et al., 2000 |
| c.385-1G>A | Splice error | HND | 7 yrs old | Jinnah et al., 2010 |
| c.385-2A>G | Splice error | HND | Multiple mRNAs | Jinnah et al., 2010; Torres et al., 2010 |
| c.402+1G>A | Splice error | HND | Aberrant mRNA | Jinnah et al., 2000 |
| c.402+1229A>G | Splice error | HRH | Multiple mRNAs, 50% activity in IF | Sege-Peterson et al., 1992 |
| c.485+2T>C | Splice error | HND | Unknown | Hladnik et al., 2008; de Gemmis et al., 2010 |
| c.485+5G>A | Splice error | LNV | Unknown | New case |
| c.485+5G>A | Splice error | HND | Unknown | Corrigan et al., 2011 |
| c.532+2T>C | Splice error | HND | Multiple mRNAs | Yamada et al., 2004 |
Open in a new tab
This table includes a list of cases where the reported phenotype and genotype are not consistent with the concept that mutations influence disease severity by their influence on HGprt enzyme activity.
EL = erythrocyte lysates; IF = intact fibroblasts; LL = lymphocyte lysates; LND = Lesch-Nyhan disease; LNV = Lesch-Nyhan variant; NA = not available.
These differences between Lesch-Nyhan disease and its variants suggest that mutations resulting in null enzyme function are more likely to cause the more severe Lesch-Nyhan disease phenotype, while mutations that may permit some residual enzyme function more frequently underlie the milder Lesch-Nyhan variant phenotypes (Jinnah et al., 2000). Thus deletions, insertions and duplications are uncommon in Lesch-Nyhan variants because they most often result in a structurally abnormal protein with no functional activity. Nonsense mutations are similarly uncommon in Lesch-Nyhan variants because they result in premature termination of translation and absent enzyme activity. On the other hand, missense mutations may be over-represented in Lesch-Nyhan variants, because a single amino acid substitution is more likely to permit some residual enzyme function. Nevertheless, there are several exceptions where cases with the Lesch-Nyhan variant phenotype were associated with mutations predicted to cause complete loss of enzyme activity. In addition, the high frequency of splicing errors seems unexpected since they are predicted to lead to exon skipping and often frameshifts, resulting in a dysfunctional protein. These exceptional cases are addressed in more detail below in the section on genotype–phenotype discrepancies.
Correlations with locations of genetic mutations
Taken together, the mutations have a broad but unequal distribution throughout the HPRT1 gene (Fig. 4). Because the emphasis of this review is to understand how mutations affect protein structure and enzyme activity, the following analysis focuses on missense mutations that lead to a single amino acid change. Although early studies suggested that mutations might cluster in highly conserved regions of the gene associated with the active site of the enzyme, it is now clear that many disease-causing mutations occur far from this site, where they may instead interfere with subunit interactions, conformational changes required in the catalytic cycle, or structural stability (Duan et al., 2004; Fu and Jinnah, 2012). In fact, it is possible to recognize five ‘hot clusters’ of mutations defined as 10 adjacent amino acids with a high frequency of reported cases (Fig. 4). One of these hot clusters involves amino acids 42–51, coinciding with the A–C subunit interface. The second hot cluster involves amino acids 65–74, overlapping the PPi binding loop. A third hot cluster covers positions 131–140, spanning the PRPP binding loop. The fourth cluster from 158–167 encompasses K166, which plays a key role in base recognition. The last cluster has a relatively broad range from 188–204, which contributes to the hood domain for purine base binding. No substitution mutations have been reported for 126 of 218 amino acids.
Figure 4.
Open in a new tab
Distribution of cases with missense mutations in the HPRT1 gene. The top panel shows the numbers of cases with missense mutations reported according to codon number. Cases with Lesch-Nyhan disease (LND) are shown as black lines and Lesch-Nyhan variant (LNV) as dark grey lines. The height of each bar reflects the total numbers of independent cases reported for unrelated families. The bottom panel shows the same data as in the top panel, with a line showing a running average of 10 codons at each point. This analysis more readily permits the visualization of five hot clusters of mutations with peaks of at least 10 cases. These hot clusters show similar distributions for Lesch-Nyhan disease (solid line) and Lesch-Nyhan variant (dashed line). The total cases combined are shaded in light grey. The different symbols above the graph depict the subunit interfaces (circles for A–B; triangles for A–C; squares for A–D) and the ligand binding sites (diamonds). Cases contributing to this figure are listed individually at www.lesch-nyhan.org.
There are 11 highly conserved amino acids among HGprt protein sequences from different species that are thought to be important for the active site (Craig and Eakin, 2000). Seven of these amino acids (L68, G70, S104, D135, G190, D194 and R200) have been linked with 28 cases. Among these, 21 (75%) had Lesch-Nyhan disease and seven had attenuated phenotypes. In addition to these conserved amino acids, adjacent amino acids might be expected to disrupt important functional domains. The hot cluster of mutations from amino acids 65–74 flanks the four amino acid PPi loop. This cluster has a total of 41 cases including 31 Lesch-Nyhan disease and 10 Lesch-Nyhan variant cases. The second hot cluster occurs at amino acids 131–140, corresponds to the PRPP loop and was affected in 16 cases, including eight patients with Lesch-Nyhan disease, seven milder variants and one case for which clinical information was limited. The third hot cluster at amino acids 188–204 corresponds to the hood domain at the carboxy terminus of the HGprt and overlaps with the A–B subunit interface. This region was affected in 40 cases, with 22 Lesch-Nyhan disease cases and 18 Lesch-Nyhan variants. The only hot cluster distant from the active site of the enzyme is at the A–C interface region spanning amino acids 42-51. In this region there were 19 cases with Lesch-Nyhan disease, 18 Lesch-Nyhan variant cases and two cases with limited clinical data. Aside from the hot clusters noted above, substitution of the residues around K166 resulted in eight Lesch-Nyhan disease and 10 Lesch-Nyhan variant phenotypes, consistent with the critical role of this amino acid and its surrounding network in purine binding. There also are several cases scattered through the N-terminal region along the A–C subunit interface region from positions 7–29.
There were no obvious differences in the locations of the mutations for Lesch-Nyhan disease versus Lesch-Nyhan variants (Fig. 4). Instead, differences in phenotypic severity with mutations at the same amino acid may be explained by the fact that non-conservative mutations involving substitution of physicochemically very different amino acids cause severe phenotypes whereas more conservative mutations cause less severe phenotypes. For example, G71 is associated with Lesch-Nyhan disease with non-conservative mutations to arginine or aspartic acid, which involve replacing a small neutral amino acid with a larger charged one. In contrast, G71 is associated with milder Lesch-Nyhan variant phenotypes when more conservatively mutated to valine. Another example is D194, which is associated with the mild phenotype of HND with the conservative mutation to glutamic acid, which is physicochemically similar to aspartic acid. However, D194 is associated with the severe phenotype of Lesch-Nyhan disease when mutated to physicochemically very different amino acids histidine, tyrosine, or asparagine. Similarly, the conservative p.I136T mutation exhibited a mild clinical phenotype, whereas the less conservative p.I136K mutation showed a severe clinical phenotype.
Although many mutations may disrupt HGprt enzyme activity by falling in specific functional domains of the protein, some do not. Presumably, these mutations may affect conformational changes during the catalytic cycle from a distance, or interfere with protein folding or stability. The region between codons 80–120 has an unusually low frequency of mutations. This low frequency may reflect a less vulnerable area of the gene. Alternatively, it may reflect an area where mutations may have no significant functional consequences, and thus may never come to clinical attention. Evaluation of the HPRT1 gene in 1000 apparently normal individuals revealed only a single variant resulting in p.H60R (Fujimori et al., 1997). Whether this case should be considered a rare sequence variant or asymptomatic mutation carrier is not clear, because residual HGprt enzyme activity was 37–46% of normal and the same mutation was found for a case with the mild phenotype of HRH (Table 1).
Predicting pathogenicity of mutations
Several software algorithms have been designed to predict the pathogenicity of individual sequence variants. The most commonly employed include SIFT ( and PolyPhen-2 ( SIFT is based on sequence comparisons from homologous or orthologous proteins and PolyPhen-2 uses both structure and sequence information (Adzhubei et al., 2010; Sim et al., 2012). Although they were designed to predict the pathogenicity of novel sequence variants of unclear functional significance, the large database of mutations in the HPRT1 gene associated with different grades of clinical severity provides an opportunity to test their performance.
SIFT correctly classified 92.8% of pathogenic missense mutations in the HPRT1 gene, but incorrectly classified 7.2% of the mutations as benign. Among those predicted to be benign, 11.8% were associated with the Lesch-Nyhan disease phenotype, and 88.2% with the Lesch-Nyhan variant phenotype. PolyPhen-2 correctly classified 83.1% of known pathogenic mutations. Among those predicted to be benign, 25% were associated with the Lesch-Nyhan disease phenotype, and 75% with the Lesch-Nyhan variant phenotype. These results demonstrate these software algorithms to have some predictive value. However, there is a significant failure rate, suggesting caution in relying on their predictions for clinical prognosis of mutations in the HPRT1 gene. These failure rates are comparable to those reported for other genes (Hicks et al., 2011).
Enzyme-phenotype correlations
The majority of the known HPRT1 mutations are consistent with the concept that the most relevant factor for provoking disease is the effect of the mutation on residual HGprt enzyme activity. This concept is supported by several biochemical studies addressing the consequences of the mutations on enzyme function. The best results have been obtained when HGprt is studied under naturalistic conditions in living cells, rather than in artificial in vitro conditions. In one assay, skin biopsies are taken from patients to grow fibroblast cultures, which make physiological levels of PRPP (Page et al., 1981; Hersh et al., 1986; Page and Nyhan, 1989). A pulse of radiolabelled hypoxanthine or guanine is given, and the rate of incorporation into nucleotides is used as an estimate of actual activity. With this assay, patients with the most severe phenotype of Lesch-Nyhan disease typically have <1.5% of normal enzyme activity, whereas the mildest phenotype of HRH is generally associated with >10% of more activity. Patients with intermediate phenotypes of HND have activity that normally falls in between (Fig. 5).
Figure 5.
Open in a new tab
Correlations between residual enzyme activity and clinical severity in patient cells. Panel A is a box-whisker plot showing residual activities for Hprt as measured by the intact fibroblast assay for 56 patients previously reported (Itiaba et al., 1978; Bakay et al., 1979; Gottlieb et al., 1982; Warzok et al., 1982; Fattal et al., 1984; Lorentz et al., 1984; Mitchell and McInnes, 1984; Hersh et al., 1986; Tarle et al., 1991; Gilbert et al., 1992; Sege-Peterson et al., 1992; Jenkins et al., 1994; Erhard et al., 1997; Puig et al., 2001; Zoref-Shani et al., 2003, 2004; Cherian and Crompton, 2005; Cossu et al., 2006; Jinnah et al., 2010; Sarafoglou et al., 2010; Torres et al., 2010). Panel B is a box-whisker plot showing residual activities for Hprt as measured by the intact erythrocyte assay for 54 patients previously reported (Shnier et al., 1972; Ludman et al., 1992; Sege-Peterson et al., 1992; Davidson et al., 1994; Willers et al., 1999; Puig et al., 2001; Augoustides-Savvopoulou et al., 2002; Bertelli et al., 2004; Kassimatis et al., 2005; Cossu et al., 2006; Garcia et al., 2008; de Gemmis et al., 2010; Corrigan et al., 2011). Corresponding values for Gprt activity were not available. In these plots, the whiskers show the whole data range, while the boxes show the 25th and 75th percentiles of the data. The horizontal line depicts the median. These data must be interpreted with caution, because they were collected from published reports from different laboratories, each with different methods for the assay and different strategies for normalization of the results.
Another good assay is based on intact erythrocytes isolated from fresh blood samples from patients. These cells also have endogenous HGprt and PRPP, and pulsing with radiolabelled hypoxanthine or guanine can be used as an estimate of enzyme activity (Fairbanks et al., 1987). Studies using the live erythrocyte assay also provide good correlations between residual enzyme function and clinical severity (Fig. 5) (Fairbanks et al., 1987; Torres et al., 2000). However, two unusual features of erythrocytes cause artefacts in some cases. First, erythrocytes have a lifespan of ∼120 days. Second, they lack a nucleus and therefore also lack ongoing messenger RNA and protein synthesis. Thus blood samples reflect a mixture of old and new erythrocytes, so that residual HGprt activity is dependent upon the structural stability of the enzyme. The live erythrocyte assay is prone to giving artificially low activity for HGprt mutants that are structurally unstable.
Although the live cell assays may be the best choices for estimating residual HGprt enzyme activity in vivo, they are not suitable for understanding the kinetic mechanisms responsible for the loss of enzyme activity. A recent study employed an alternative approach for examining enzyme–phenotype correlations for 44 clinically relevant HGprt mutants associated with a wide spectrum of disease severity (Fu and Jinnah, 2012). The HPRT1 mutations were re-created by site-directed mutagenesis, the mutant proteins were expressed and purified in vitro, and their biochemical kinetics were studied with a sensitive spectrophotometric assay. The results confirmed a good correlation between clinical severity and residual HGprt enzyme activity (Fig. 6). There was a parallel loss of hypoxanthine and guanine recycling for most mutant enzymes, although a few mutants showed skewed loss of activity towards one or the other substrate. Clinical severity seemed to correlate best with loss of hypoxanthine recycling, although there were too few cases with skewed loss of activity to be sure. This study also pointed to the existence of several biochemical mechanisms responsible for reducing enzyme activity. These mechanisms included alterations in the affinity of the enzyme towards one or more of its substrates, altered catalytic efficiency towards one or more substrates, and reduced structural integrity of the enzyme.
Figure 6.
Open in a new tab
Correlations between residual enzyme activity and clinical severity using purified engineered mutant proteins in vitro. The two different functions of HGprt are shown separately as Hprt and Gprt. The panels show box-whisker plots for residual Hprt activity with a fixed high concentration of hypoxanthine (A) and a fixed high concentration of PRPP (C), residual Gprt activity with a fixed high concentration of guanine (B) and a fixed high concentration of PRPP (D) as previously described (Fu and Jinnah, 2012). In these plots, the whiskers show the whole data range, while the boxes show the 25th and 75th percentiles of the data. The horizontal line depicts the median.
The live cell assays and the recreation of mutant enzymes in vitro provide powerful tools for measuring residual enzyme activity and understanding mechanisms for reduced activity, but they are technically demanding and expensive. These qualities make them difficult to use for routine clinical diagnostic testing. As a result, many other studies have used simpler assays. Some of these studies have suggested the existence of rare cases that conflict with the concept that disease severity is related to residual enzyme function (Holland et al., 1976; Rijksen et al., 1981; Hersh et al., 1986; Cossu et al., 2002). These discrepancies have led some investigators to question the relationship between enzyme activity and clinical phenotype. Although it is important to acknowledge such exceptional cases, it also is important to avoid placing too much weight on unconfirmed and isolated reports, especially when they involve the use of biochemical assays that may provide inaccurate results. There are multiple reports showing that the same case may have significant differences in residual enzyme activity depending on the type of biochemical assay used (Fujimoto and Seegmiller, 1970; Holland et al., 1976; Page et al., 1984; Hersh et al., 1986; Fairbanks et al., 1987; Augoustides-Savvopoulou et al., 2002; Zoref-Shani et al., 2003; Cossu et al., 2006). These exceptional cases are addressed in detail below under the section on genotype–phenotype discrepancies.
Discrepancies in genotype–enzyme–phenotype correlations
Genotype–phenotype discrepancies
As noted above, differences in the types of mutations associated with severe and mild phenotypes suggest that those that result in null enzyme function are more likely to cause the severe Lesch-Nyhan disease phenotype, whereas mutations that may permit some residual enzyme function more frequently underlie the milder Lesch-Nyhan variant phenotypes. However, there are several exceptions that are not consistent with this rule (Table 7).
For example, there are six cases in which milder Lesch-Nyhan variant phenotypes were reported with deletion mutations affecting coding regions. In principle, deletion mutations should cause complete enzyme deficiency and the severe Lesch-Nyhan disease phenotype because they result in significant disruption of protein structure, often combined with frameshift. Additional mutations expected to cause substantial disruption to protein structure and function include three Lesch-Nyhan variant cases with partial gene duplications, three with indels, one with an insertion, and one with a nonsense mutation. There also are multiple reports describing Lesch-Nyhan variant in association with either deletions or point mutations affecting intron-exon splice junctions, which are expected to result in exon skipping often combined with frame shifts.
Detailed evaluation of each of these apparent exceptions reveals several explanations. First, several of these cases were described at a very early age (Table 7). It is conceivable that these cases might have been mislabelled as Lesch-Nyhan variants because the full phenotype had not yet developed. In fact, two of these cases (Serrano et al., 2008; Sharma et al., 2012) began to exhibit self-biting a few years after the original report (personal communication). Another explanation for the apparent exceptions involves unusual molecular mechanisms that permit residual enzyme activity. For example, one case had a 13-bp deletion, which eliminated the AUG start codon, but a small amount of residual protein with functional activity was transcribed from an upstream GUG start site (Davidson et al., 1994). Another case had a deletion at the extreme C-terminus, with loss of only two amino acids, presumably allowing residual enzyme function (Igarishi et al., 1989). Three more Lesch-Nyhan variant cases with large duplications were shown to exhibit spontaneous reversions in vitro resulting in small amounts of residual HGprt enzyme activity (Yang et al., 1988; Monnat et al., 1992; Marcus et al., 1993 b). Another case had insertion of three bases, resulting in the insertion of one additional amino acid without frameshift, again presumably allowing residual enzyme activity (Sege-Peterson et al., 1992). Finally, several cases with splice site mutations were shown to have residual enzyme function. The mechanism responsible for residual enzyme activity involves variations in the fidelity of the splicing mechanism, with generation of multiple messenger RNA transcripts (Sege-Peterson et al., 1992; Yamada et al., 1992; Hunter et al., 1996). In some cases, these mutations yield a small proportion of transcripts encoding the entire normal protein (Yamada et al., 2007; Torres et al., 2010). In one example, a splice site mutation resulted in the addition of 15 amino acids to the normal HGprt protein, with 7% residual enzyme activity (Gaigl et al., 2001). Thus splicing mutations can yield unpredictable results.
In summary, there are multiple reports of mild clinical phenotypes associated with mutations that predict elimination of HGprt enzyme function. At first glance, these cases seem to disprove the concept that clinical severity is determined by the amount of residual enzyme activity. However, most of these cases can be explained by an incompletely ascertained clinical phenotype or specific molecular mechanisms permitting some residual enzyme function. As a result, these apparent exceptions do not provide strong evidence against the concept that relates clinical severity to residual enzyme function.
Enzyme–phenotype discrepancies
As noted above, there are also a small number of reported cases in which the measured HGprt enzyme activity does not correlate with the clinical phenotype. Specifically, there are a few reported cases with a severe phenotype associated with relatively high residual enzyme activity (Holland et al., 1976; Rijksen et al., 1981), and a few reported cases with attenuated phenotypes associated with non-detectable enzyme activity (Hersh et al., 1986; Cossu et al., 2002). Although uncommon, these exceptions are important to address, because they seem to provide evidence against the concept that residual HGprt enzyme activity is the primary determinant of clinical severity. Careful examination of these exceptions reveals that all involved the use of biochemical assays that may provide misleading results under certain conditions.
For example, one mechanism responsible for reduced HGprt enzyme function involves altered affinities towards hypoxanthine, guanine or PRPP (McDonald and Kelley, 1971; Benke et al., 1973; Richardson et al., 1973; Rijksen et al., 1981; Wilson et al., 1983; Snyder et al., 1984; Zanic et al., 1985; Fujimori et al., 1988; Lightfoot et al., 1994; Hikita et al., 1998; Zoref-Shani et al., 2000; Fu and Jinnah, 2012). For such mutants, in vitro assays employing artificially high substrate concentrations may yield high apparent activities, while concentrations of substrates normally available in vivo may result in little or no actual activity. An example is p.D194N, which has >200-fold increase in the K m for PRPP in association with the Lesch-Nyhan disease phenotype. This mutant had almost no activity using a live lymphoblast cell assay (<0.7%), which relies on endogenous production of physiological amounts of PRPP (Wilson et al., 1986). However, the same mutant displayed normal residual activity when PRPP was saturating, because of high PRPP concentrations used to drive the reaction in vitro (Wilson and Kelley, 1984; Wilson et al., 1986). Thus the in vitro assay incorrectly predicted a mild phenotype whereas the more naturalistic assay correctly predicted the severe phenotype.
Another mechanism leading to reduced HGprt enzyme function involves reduced structural stability of the enzyme, with rapid loss of enzyme function over time or loss of enzyme function outside of the normal cellular milieu (Richardson et al., 1973; Snyder et al., 1984; Sampat et al., 2011; Fu and Jinnah, 2012). Such mutants are likely to yield little or no activity in lysate-based assays or cell-based assays where continual messenger RNA transcription does not occur (such as erythrocytes, which lack a nucleus and ongoing messenger RNA synthesis) yet significantly higher activity in cells where ongoing messenger RNA transcription and translation will generate a constant supply of newly formed enzyme (such as live fibroblasts, where synthesis of messenger RNA and new protein are ongoing). This mechanism explains several cases where no activity was found in erythrocytes, yet significant activity was found in live fibroblasts. For example, p.R48H repeatedly has been associated with relatively mild clinical phenotypes with 20–39% residual activity in live fibroblast assays (Jinnah et al., 2010; Sampat et al., 2011). However, same mutation was reported to cause non-detectable residual activity in other assays (Jinnah et al., 2010; Sampat et al., 2011). Similar mechanisms could account for the differences between in vitro assays using purified protein and assays that rely on existing HGprt levels in live cells or lysates. There are examples of several mutants (p.D44G, p.E47G, p.F74L, and p.H204D) with no activity with assays based on lysates from human cell materials, yet high activity when the mutant enzyme is expressed in vitro and tested immediately (Davidson et al., 1988; Tarle et al., 1991; Sculley et al., 1992; Jinnah et al., 2006; Fu and Jinnah, 2012).
These observations emphasize the importance of careful attention to the biochemical assay used for the occasional reports of cases in whom residual enzyme activity does not appear to correlate with the clinical phenotype. In vitro assays are prone to give artificially high residual activity when they are conducted with non-physiological substrate concentrations or artificially low residual activity if the enzyme is structurally unstable outside of the normal cellular milieu. The live cell assays may yield results that are closer to those occurring in vivo, but erythrocyte-based assays are prone to give artificially low residual activity for mutant enzymes that are structurally unstable because of the lack of ongoing synthesis of new enzyme. The live fibroblast assay may be least susceptible to inaccurate results, although the phenomenon of somatic mosaicism may yield variable results in rare instances, such as mutations capable of reversion.
Different clinical phenotypes associated with the same mutation
If residual enzyme function is the most important factor for determining the clinical phenotype, then identical HPRT1 mutations should have the same biochemical consequences for HGprt function and thereby cause the same clinical phenotype. A total of 85 mutations were reported with at least two cases from unrelated families, for a total of 272 cases with shared mutations (Table 8). The majority of recurrent mutations (93.7%) produced a similar phenotype even in unrelated patients. However, there were a few exceptions where the same mutation caused discordant phenotypes (Table 9). For all discordant phenotypes, only one case for each mutation was discordant. Discordant phenotypes were most often observed for missense mutations, splicing errors, or duplications. There also were several reports of different phenotypes in the same family presumably carrying the same mutation (Hladnik et al., 2008; Jinnah et al., 2010; Sarafoglou et al., 2010). These cases argue that factors other than HGprt activity may influence clinical outcomes.
Table 8.
Summary of HPRT1 mutations with discordant clinical phenotypes
| Type of Mutation | Concordant | Discordant | % Discordant |
--- --- |
| Missense | 116 | 8 | 6.9 |
| Splicing errors | 48 | 5 | 10.4 |
| Nonsense | 42 | 1 | 2.4 |
| Microdeletions | 26 | 0 | 0.0 |
| Microduplications | 19 | 0 | 0.0 |
| Large duplications | 3 | 2 | 66.7 |
| Indels | 2 | 0 | 0.0 |
| Total | 256 | 16 | 6.3 |
Open in a new tab
This table summarizes the numbers of cases with the same (concordant) or different (discordant) reported phenotype associated with an identical molecular mutation.
Table 9.
Specific HPRT1 mutations with discordant clinical phenotypes
| | LND | LNV | Total |
--- --- |
| Missense | | | |
| c.115C>G | 1 | 1 | 2 |
| c.125T>C | 1 | 1 | 2 |
| c.134G>A | 3 | 1 | 4 |
| c.203T>C | 2 | 1 | 3 |
| c.212G>T | 1 | 3 | 4 |
| c.293A>G | 1 | 1 | 2 |
| c.486C>G | 1 | 1 | 2 |
| Splicing errors | | | |
| c.27+1G>A | 1 | 1 | 2 |
| c.27+1G>T | 2 | 1 | 3 |
| c.27+5G>A | 1 | 1 | 2 |
| c.385-1G>A | 3 | 1 | 4 |
| c.402+1G>A | 1 | 1 | 2 |
| c.485+2T>C | 1 | 1 | 2 |
| Nonsense | | | |
| c.508C>T | 18 | 1 | 19 |
| Microdeletion | | | |
| c.213delC | 2 | 1 | 3 |
| Duplications | | | |
| Exons 2-3 | 1 | 2 | 3 |
| Exons 7-8 | 1 | 1 | 2 |
Open in a new tab
This table provides a list of the cases where different (discordant) phenotypes were reported to be associated with an identical molecular mutation.
LND = Lesch-Nyhan disease; LNV = Lesch-Nyhan variant.
Several potential mechanisms may account for these discordant cases. There are both genetic and non-genetic modifiers that can affect the amount of residual HGprt enzyme activity associated with some HPRT1 mutations. As noted above, variations in the fidelity of splicing mechanisms can result in multiple messenger RNAs including some normal messenger RNA transcripts and associated enzyme activity (O'Neill et al., 1998). Since the mechanisms responsible for splicing are inherited independently from HPRT1, the amount of normal messenger RNA encoding HGprt may vary among individuals carrying the same splicing mutation. This mechanism may explain why discordant phenotypes are commonly associated with splicing mutations (Table 8). One example involves the c.485+2T>C splicing mutation, which was reported to be associated with different phenotypes in five members of one family (Hladnik et al., 2008). Only one had self-injurious behaviour and therefore had the Lesch-Nyhan disease phenotype. Two others did not have overt self-injury, but they suffered repeated injuries to the chin from falls. This observation raises questions regarding whether these injures were entirely accidental since they were not random but limited to the chin. Two more family members were reported to have HRH because of normal intelligence, but both had abnormal gaits. The gait disorder suggests these cases may be better classified as HND. This family provides an example of phenotypic variation associated with a splicing mutation, as well as the challenges of classifying a continuously graded spectrum of clinical phenotypes into distinct categories.
A related mechanism where a single mutation may variably affect enzyme activity in different cases involves non-coding mutations. In this situation, regulatory mechanisms that control messenger RNA transcription from the mutant gene or protein translation from messenger RNA with aberrant regulatory signals may operate with varying fidelity among different cases with the same mutation. The outcome could be differences in residual HGprt enzyme activity and clinical phenotype. Here, quantitative studies of messenger RNA levels or enzyme function may be useful (Nguyen et al., 2012).
Another mechanism that may alter enzyme activity independently from the associated mutation involves structural instability of the HGprt protein (Richardson et al., 1973; Snyder et al., 1984; Sampat et al., 2011; Fu and Jinnah, 2012). For unstable proteins, enzyme function depends on the balance between elimination of damaged proteins and ongoing synthesis of new functional protein. Since mechanisms of degradation and synthesis are inherited independently from HPRT1, different individuals with the same mutation may have different steady-state levels of HGprt activity. In the case of structurally unstable enzymes, environmental influences may also play a role. For example, thermal effects associated with frequent or lengthy febrile episodes may result in an increased structural damage to HGprt proteins, especially if they occur during a critical period of early brain development. This mechanism predicts a worse clinical outcome for patients who have experienced many fevers due to infections during early childhood (Sampat et al., 2011).
A fourth mechanism could involve biochemical pathways that compensate for the loss of HGprt-mediated purine recycling. Although there are no known HGprt isoenzymes that can replace its function, the products of HGprt can be produced by an alternative pathway known as the de novo synthesis pathway. This pathway is accelerated in HGprt-deficient cells, suggesting metabolic regulatory changes in synthesis that may compensate for the lack of recycling (Nyhan et al., 1965; Rosenbloom et al., 1968; Sorensen, 1970). The ability of this alternative pathway to compensate for HGprt deficiency may vary from case to case. As the mechanisms responsible for the de novo pathway are inherited independently from HPRT1, it is feasible that the degree of compensation may vary among patients inheriting the same mutation. This mechanism for explaining discordant phenotypes with the same HPRT1 mutation is theoretically attractive, but currently there are no studies comparing the extent to which the de novo pathway is influenced by different HPRT1 mutations or across different patients. Therefore, this possibility remains speculative.
These observations suggest several molecular mechanisms by which the same HPRT1 mutation may result in different levels of residual HGprt enzyme activity and thereby discordant clinical outcomes. There also are important non-molecular explanations for discordant clinical phenotypes associated with the same HPRT1 mutation. These non-biochemical mechanisms relate to limitations in the characterization of the clinical phenotype and assignment of patients to specific disease subgroups.
Beyond biochemical genetics
Many years of research in genetics and biochemistry have led to a rich understanding of how the clinical phenotype relates to HPRT1 gene mutations and associated HGprt enzyme activity. A major unsolved question is how the loss of HGprt enzyme function affects the brain to cause the neurobehavioural syndrome in Lesch-Nyhan disease and its attenuated variants. Although HGprt is expressed throughout the brain (Jinnah et al., 1992 a), many of the clinical features point to prominent dysfunction of a specific region known as the basal ganglia (Visser et al., 2000). However, autopsy studies have not revealed any consistent anatomical defects anywhere in the Lesch-Nyhan disease brain, including the basal ganglia. Notably, there is no evidence for a degenerative process (Jinnah et al., 2006). There is no evidence for major developmental malformations, and the multifocal atrophy of cerebellar Purkinje neurons reported for some cases (Del Bigio and Halliday, 2007) has not been found in other cases (unpublished observations). Structurally, there are no serious defects.
However, neurochemical studies of Lesch-Nyhan disease brains collected at autopsy have revealed 60–80% loss of dopamine, a critically important neurotransmitter in the basal ganglia (Lloyd et al., 1981; Saito et al., 1999; Saito and Takashima, 2000). Profound dysfunction of dopamine neurons also has been documented in imaging studies of patients with Lesch-Nyhan disease, which show significant loss of fluoroDOPA uptake into the basal ganglia and major reductions in the binding of WIN 35,428 to dopamine transporters in the basal ganglia (Ernst et al., 1996; Wong et al., 1996). The neurochemical and imaging studies are corroborated by studies of CSF from patients with Lesch-Nyhan disease, which show reduced monoamine neurotransmitter metabolites (Silverstein et al., 1985; Edwards et al., 1986; Manzke et al., 1986; Jankovic et al., 1988). A selective disruption of basal ganglia dopamine systems also has been documented in genetically engineered mouse models that lack HGprt (Jinnah et al., 1992 b, 1994, 1999; Egami et al., 2007), and even in tissue culture models involving neuron-like cells that make dopamine as a neurotransmitter (Bitler and Howard, 1986; Yeh et al., 1998; Lewers et al., 2008). The reasons for relatively prominent disruption of dopamine neurons remain unknown, but may relate to the selective vulnerability of this neuronal population to the loss of HGprt (Egami et al., 2007; Lewers et al., 2008; Ceballos-Picot et al., 2009; Guibinga et al., 2011; Kang et al., 2011). Additional work to clarify how the loss of HGprt affects these and other neurons is still needed.
Conclusions
Lesch-Nyhan disease and its attenuated variants are associated with characteristic clinical features, including overproduction of uric acid, motor and cognitive disability, and behavioural abnormalities. Although patients are divided into three distinct subgroups based on overall severity, the reality is a more continually graded spectrum, which evolves during early development. There are many different clinically relevant mutations in the HPRT1 gene, and genotype–phenotype correlations suggest that the most relevant aspect of the mutation is its influence on residual function of HGprt enzyme activity. The mutations are distributed through all functional domains of the HGprt protein. The mutations may affect residual enzyme dysfunction by disrupting the active enzyme site, by interfering with conformational changes required during the catalytic cycle, by blocking essential subunit interactions, or by reducing protein stability. There are also examples where residual enzyme activity may be influenced by factors other than the mutation itself.
These observations have direct implications for genetic testing and counselling regarding prognosis. New cases with mutations predicted to cause null enzyme activity are likely to develop the most severe clinical phenotype of Lesch-Nyhan disease, while those with mutations that permit residual enzyme activity have a higher probability of developing a milder phenotype. New cases found to have mutations already described have a high probability of developing a phenotype similar to the previously reported case. Predicting the outcome of new cases with novel mutations is more difficult, because available predictive algorithms such as SIFT and PolyPhen-2 are not always accurate. Here, biochemical tests for residual HGprt enzyme function may be better for predicting prognoses, and it is important to recognize that the live erythrocyte or fibroblast assays are more accurate than lysate-based assays, which are prone to multiple artefacts.
The basic principles of genotype–phenotype relationships associated with mutations in the HPRT1 gene provide a valuable model for understanding genotype–phenotype relationships in many other genetic disorders. The complexities associated with clinical ascertainment of the phenotype are relevant to all genetic disorders, and the mechanisms by which different mutations can alter protein function are likely to occur with many other gene products . Although Lesch-Nyhan disease often is considered a relatively ‘simple’ disorder because it is monogenic and inherited in an X-linked recessive manner permitting evaluation of single allelic defects, the unexpected and sometimes unusual mechanisms that influence genotype–phenotype relationships provide useful principles for other more complex disorders.
Funding
This work was supported in part by the Lesch-Nyhan Syndrome Children’s Research Foundation, Lesch-Nyhan Action and Malaury Associations, the Harold A. and Madeline R. Jacobs Fund at the San Diego Foundation, the Fondo de Investigaciones Sanitarias (Health Research Fund, FIS) 11/00598 from Spain, a Gout Research Foundation Grant of Japan, and grants from the National Institutes of Health of the USA (HD053312 and DK082840).
Glossary
Abbreviations
HGprt
hypoxanthine-guanine phosphoribosyltransferase
HND
HGprt-related neurological dysfunction
HRH
HGprt-related hyperuricaemia
PRPP
phosphoribosylpyrophosphate
References
Adzhubei IA, Schmidt S, Peshkin L, Ramensky VE, Gerasimova A, Bork P, et al. A method and server for predicting damaging missense mutations. Nat Methods. 2010;7:248–9. doi: 10.1038/nmeth0410-248. [DOI] [PMC free article] [PubMed] [Google Scholar]
Anderson LT, Ernst M. Self-injury in Lesch-Nyhan disease. J Autism Dev Disord. 1994;24:67–81. doi: 10.1007/BF02172213. [DOI] [PubMed] [Google Scholar]
Anderson LT, Ernst M, Davis SV. Cognitive abilities of patients with Lesch-Nyhan disease. J Autism Dev Disord. 1992;22:189–203. doi: 10.1007/BF01058150. [DOI] [PubMed] [Google Scholar]
Aral B, de Saint B, Al-Garawi S, Kamoun P, Ceballos-Picot I. Novel nonsense mutation in the hypoxanthine guanine phosphoribosyltransferase gene and nonrandom X-inactivation causing Lesch-Nyhan syndrome in a female patient. Hum Mutat. 1996;7:52–8. doi: 10.1002/(SICI)1098-1004(1996)7:1<52::AID-HUMU7>3.0.CO;2-R. [DOI] [PubMed] [Google Scholar]
Augoustides-Savvopoulou P, Papachristou F, Fairbanks LD, Dimitrakopoulos K, Marinaki AM, Simmonds HA. Partial hypoxanthine-guanine phosphoribosyltransferase deficiency as the unsuspected cause of renal disease spanning three generations: a cautionary tale. Pediatr. 2002;109:E17. doi: 10.1542/peds.109.1.e17. [DOI] [PubMed] [Google Scholar]
Bakay B, Nissinen E, Sweetman L, Francke U, Nyhan WL. Utilization of purines by an HPRT variant in an intelligent, nonmutilative patient with features of the Lesch-Nyhan syndrome. Pediatr Res. 1979;13:1365–70. doi: 10.1203/00006450-197912000-00013. [DOI] [PubMed] [Google Scholar]
Balendiran GK, Molina JA, Xu Y, Torres-Martinez J, Stevens R, Focia PJ, et al. Ternary complex structure of human HGPRTase, PRPP, Mg2+, and the inhibitor HPP reveals the involvement of the flexible loop in substrate binding. Protien Sci. 1999;8:1023–31. doi: 10.1110/ps.8.5.1023. [DOI] [PMC free article] [PubMed] [Google Scholar]
Benke PJ, Herrick N, Hebert A. Hypoxanthine-guanine phosphoribosyltransferase variant associated with accelerated purine synthesis. J Clin Invest. 1973;52:2234–40. doi: 10.1172/JCI107409. [DOI] [PMC free article] [PubMed] [Google Scholar]
Bertelli M, Randi D, Micheli V, Gallo S, Andrighetto G, Parmigiani P, et al. Molecular basis of hypoxanthine-guanine phosphoribosyltransferase deficiency in Italian Lesch-Nyhan patients: identification of nine novel mutations. J Inherit Metab Dis. 2004;27:767–73. doi: 10.1023/B:BOLI.0000045799.78633.23. [DOI] [PubMed] [Google Scholar]
Bitler CM, Howard BD. Dopamine metabolism in hypoxanthine-guanine phosphoribosyltransferase-deficient variants of PC12 cells. J Neurochem. 1986;47:107–12. doi: 10.1111/j.1471-4159.1986.tb02837.x. [DOI] [PubMed] [Google Scholar]
Burgemeister R, Gutensohn W, Van den Berghe G, Jaeken J. Genetic and clinical heterogeneity in hypoxanthine phosphoribosyltransferase deficiencies. Adv Exp Med Biol. 1994;370:331–5. doi: 10.1007/978-1-4615-2584-4_71. [DOI] [PubMed] [Google Scholar]
Ceballos-Picot I, Mockel L, Potier MC, Dauphinot L, Shirley TL, Torero-Ibad R, et al. Hypoxanthine-guanine phosphoribosyl transferase regulates early developmental programming of dopamine neurons: implications for Lesch-Nyhan disease pathogenesis. Hum Mol Genet. 2009;18:2317–27. doi: 10.1093/hmg/ddp164. [DOI] [PMC free article] [PubMed] [Google Scholar]
Cherian S, Crompton CH. Partial hypoxanthine-guanine phosphoribosyltransferase deficiency presenting as acute renal failure. Pediatr Nephrol. 2005;20:1811–3. doi: 10.1007/s00467-005-2065-8. [DOI] [PubMed] [Google Scholar]
Corrigan A, Arenas M, Escuredo E, Fairbanks L, Marinaki A. HPRT deficiency: identification of twenty-four novel variants including an unusual deep intronic mutation. Nucleosides Nucleotides Nucleic Acids. 2011;30:1260–5. doi: 10.1080/15257770.2011.590172. [DOI] [PubMed] [Google Scholar]
Cossu A, Micheli V, Jacomelli G, Carcassi A. Kelley-Seegmiller syndrome in a patient with complete hypoxanthine-guanine phosphoribosyltransferase deficiency. Clin Exp Rhematol. 2002;19:851–3. [PubMed] [Google Scholar]
Cossu A, Orru S, Jacomelli G, Carcassi C, Contu L, Sestini S, et al. HPRT-Sardinia: a new point mutation causing HPRT deficiency without Lesch-Nyhan disease. Biochim Biophys Acta. 2006;1762:29–33. doi: 10.1016/j.bbadis.2005.03.012. [DOI] [PubMed] [Google Scholar]
Craig SP, III, Eakin AE. Purine phosphoribosyltransferases. J Biol Chem. 2000;275:20231–4. doi: 10.1074/jbc.R000002200. [DOI] [PubMed] [Google Scholar]
Davidson BL, Golovoy N, Roessler BJ. A 13 base pair deletion in exon 1 of HPRTIllinois forms a functional GUG initiation codon. Hum Genet. 1994;93:300–4. doi: 10.1007/BF00212027. [DOI] [PubMed] [Google Scholar]
Davidson BL, Pashmforoush M, Kelley WN, Palella TD. Genetic basis of hypoxanthine guanine phosphoribosyltransferase deficiency in a patient with the Lesch-Nyhan syndrome (HPRTFlint) Gene. 1988;63:331–6. doi: 10.1016/0378-1119(88)90536-7. [DOI] [PubMed] [Google Scholar]
Davidson BL, Tarle SA, Van Antwerp M, Gibbs DA, Watts RW, Kelley WN, et al. Identification of 17 independent mutations responsible for human hypoxanthine-guanine phsophoribosyltransferase (HPRT) deficiency. Am J Hum Genet. 1991;48:951–8. [PMC free article] [PubMed] [Google Scholar]
Dawson PA, Gordon RB, Keough DT, Emmerson BT. Normal HPRT coding region in a male with gout due to HPRT deficiency. Mol Genet Metab. 2005;85:78–80. doi: 10.1016/j.ymgme.2005.01.005. [DOI] [PubMed] [Google Scholar]
de Gemmis P, Anesi L, Lorenzetto E, Gioachini I, Fortunati E, Zandona G, et al. Analysis of the HPRT1 gene in 35 Italian Lesch-Nyhan families: 45 patients and 77 potential female carriers. Mutat Res. 2010;692:1–5. doi: 10.1016/j.mrfmmm.2010.07.003. [DOI] [PubMed] [Google Scholar]
De Gregorio L, Jinnah HA, Nyhan WL, Trombley L, O'Neill JP. Lesch-Nyhan disease in one member of a female monozygotic twin pair heterozygous for a mutation in HPRT. Mol Genet Metab. 2005;85:70–7. doi: 10.1016/j.ymgme.2004.11.009. [DOI] [PubMed] [Google Scholar]
De Gregorio L, Nyhan WL, Serafin E, Chanoles NA. An unexpected affected female patient in a classical Lesch-Nyhan family. Mol Genet Metab. 2000;69:263–8. doi: 10.1006/mgme.2000.2967. [DOI] [PubMed] [Google Scholar]
Del Bigio MR, Halliday WC. Multifocal atrophy of cerebellar internal granular neurons in Lesch-Nyhan disease: case reports and review. J Neuropathol Exp Neurol. 2007;66:346–53. doi: 10.1097/nen.0b013e3180515319. [DOI] [PubMed] [Google Scholar]
Duan J, Nilsson L, Lambert B. Structural and functional analysis of mutations at the human hypoxanthine phosporibosyl transferase (HPRT1) locus. Hum Mutat. 2004;23:599–611. doi: 10.1002/humu.20047. [DOI] [PubMed] [Google Scholar]
Ea HK, Bardin T, Jinnah HA, Aral B, Liote F, Ceballos-Picot I. Severe gouty arthritis and mild neurological symptoms due to a new variant of the hypoxanthine-guanine phosphoriboysltransferase (F199C) Arthritis Rheum. 2009;60:2201–4. doi: 10.1002/art.24617. [DOI] [PMC free article] [PubMed] [Google Scholar]
Eads JC, Scapin G, Xu Y, Grubmeyer C, Sacchettini JC. The crystal structure of human hypoxanthine-guanine phosphoribosyltransferase with bound GMP. Cell. 1994;78:325–34. doi: 10.1016/0092-8674(94)90301-8. [DOI] [PubMed] [Google Scholar]
Edwards NL, Silverstein FS, Johnston MV. Purine and monoamine metabolites in cerebrospinal fluid. Adv Exp Med Biol. 1986;195 (Pt B):53–6. doi: 10.1007/978-1-4684-1248-2_9. [DOI] [PubMed] [Google Scholar]
Egami K, Yitta S, Kasim S, Lewers JC, Roberts RC, Lehar M, et al. Basal ganglia dopamine loss due to defect in purine recycling. Neurobiol Dis. 2007;26:396–407. doi: 10.1016/j.nbd.2007.01.010. [DOI] [PMC free article] [PubMed] [Google Scholar]
Emmerson BT, Thompson L. The spectrum of hypoxanthine-guanine phosphoribosyltranferase deficiency. Quart J Med. 1973;166:423–40. [PubMed] [Google Scholar]
Erhard U, Herkenrath P, Benz-Bohm G, Querfeld U. Lesch-Nyhan syndrome: clinical diagnosis and confirmation by biochemical and genetic methods. Pediatr Nephrol. 1997;11:124–5. [PubMed] [Google Scholar]
Ernst M, Zametkin AJ, Matochik JA, Pascualvaca D, Jons PH, Hardy K, et al. Presynaptic dopaminergic deficits in Lesch-Nyhan disease. N Engl J Med. 1996;334:1568–72. doi: 10.1056/NEJM199606133342403. [DOI] [PubMed] [Google Scholar]
Fairbanks LD, Simmonds HA, Webster DR. Use of intact erythrocytes in the diagnosis of inherited purine and pyrimidine disorders. J Inherit Metab Dis. 1987;10:174–86. doi: 10.1007/BF01800045. [DOI] [PubMed] [Google Scholar]
Fattal A, Spirer Z, Zoref-Shani E, Sperling O. Lesch-Nyhan syndrome: biochemical characterization of a case with attenuated behavioral manifestation. Enzyme. 1984;31:55–60. [PubMed] [Google Scholar]
Fox IH, Kelley WN. Phosphoribosylpyrophosphate in man: biochemical and clinical significance. Ann Int Med. 1971;74:424–33. doi: 10.7326/0003-4819-74-3-424. [DOI] [PubMed] [Google Scholar]
Fu R, Jinnah HA. Different phenotypes among Lesch-Nyhan variants: clinical reality or limitation of ascertainment? Arch Neurol. 2011;68:270. doi: 10.1001/archneurol.2010.360. [DOI] [PMC free article] [PubMed] [Google Scholar]
Fu R, Jinnah HA. Genotype-phenotype correlations in Lesch-Nyhan disease: moving beyond the gene. J Biol Chem. 2012;287:2997–3008. doi: 10.1074/jbc.M111.317701. [DOI] [PMC free article] [PubMed] [Google Scholar]
Fujimori S, Hidaka Y, Davidson BL, Palella TD, Kelley WN. Identification of a single nucleotide change in a mutant gene for hypoxanthine-guanine phosphoribosyltransferase (HPRTAnn Arbor) Hum Genet. 1988;79:39–43. doi: 10.1007/BF00291707. [DOI] [PubMed] [Google Scholar]
Fujimori S, Sakuma R, Yamaoka N, Hakoda M, Yamanaka H, Kamatani N. An asymptomatic germline missense base substitution in the hypoxanthine phosphoribosyltransferase (HPRT) gene that reduces the amount of enzyme in humans. Hum Genet. 1997;99:8–10. doi: 10.1007/s004390050300. [DOI] [PubMed] [Google Scholar]
Fujimori S, Tagaya T, Yamaoka N, Kamatani N, Akaoka I. Molecular analysis of hypoxanthine-guanine phosphoribosyltransferase deficiency in Japanese patients. Adv Exp Med Biol. 1991;309B:101–4. doi: 10.1007/978-1-4615-7703-4_22. [DOI] [PubMed] [Google Scholar]
Fujimori S, Tagaya T, Yamaoka N, Saito H, Kamatani N, Akaoka I. Direct evidence for a hot spot of germline mutation at HPRT locus. Adv Exp Med Biol. 1994;370:679–82. doi: 10.1007/978-1-4615-2584-4_141. [DOI] [PubMed] [Google Scholar]
Fujimoto WY, Seegmiller JE. Hypoxanthine-guanine phosphoribosyltransferase deficiency: activity in normal, mutant, and heterozygote cultured human skin fibroblasts. Proc Natl Acad Sci USA. 1970;65:577–84. doi: 10.1073/pnas.65.3.577. [DOI] [PMC free article] [PubMed] [Google Scholar]
Gaigl Z, Shin YS, Gathof BS. Novel splice mutation in a patient with partial hypoxanthine-guanine phosphoribosyltransferase (HPRT) deficiency. J Inherit Metab Dis. 2001;24(Supp 1):142. [Google Scholar]
Garcia MG, Torres RJ, Prior C, Puig JG. Normal HPRT coding region in complete and partial HPRT deficiency. Mol Genet Metab. 2008;94:167–72. doi: 10.1016/j.ymgme.2008.01.006. [DOI] [PubMed] [Google Scholar]
Gibbs RA, Caskey CT. Identification and localization of mutations at the Lesch-Nyhan locus by ribonuclease A cleavage. Science. 1987;236:303–5. doi: 10.1126/science.3563511. [DOI] [PubMed] [Google Scholar]
Gibbs RA, Nguyen PN, McBride LJ, Koepf SM, Caskey CT. Identification of mutations leading to the Lesch-Nyhan syndrome by automated direct DNA sequencing of in vitro amplified cDNA. Proc Natl Acad Sci USA. 1989;86:1919–23. doi: 10.1073/pnas.86.6.1919. [DOI] [PMC free article] [PubMed] [Google Scholar]
Gilbert RD, Wiggelinkhuizen J, Harley EH, Marinaki A. The Lesch-Nyhan syndrome–an under-recognised condition in South Africa? A case report. S Afr Med J. 1992;81:375–7. [PubMed] [Google Scholar]
Gottlieb RP, Koppel MM, Nyhan WL, Bakay B, Nissinen E, Borden M, et al. Hyperuricaemia and choreoathetosis in a child without mental retardation or self-mutilation - a new HPRT variant. J Inherit Metab Dis. 1982;5:183–6. doi: 10.1007/BF02179135. [DOI] [PubMed] [Google Scholar]
Guibinga GH, Hrustanovic G, Bouic K, Jinnah HA, Friedmann T. MicroRNA-mediated dysregulation of neural developmental genes in HPRT deficiency: clues for Lesch-Nyhan disease? Hum Mol Genet. 2011;21:609–22. doi: 10.1093/hmg/ddr495. [DOI] [PMC free article] [PubMed] [Google Scholar]
Hara K, Kashiwamata S, Ogasawara N, Ohishi H, Natsume R, Yamanaka T, et al. A female case of the Lesch-Nyhan syndrome. Tohoku J Exp Med. 1982;137:275–82. doi: 10.1620/tjem.137.275. [DOI] [PubMed] [Google Scholar]
Hersh JH, Page T, Hand ME, Seegmiller JE, Nyhan WL, Weisskopf B. Clinical correlations in partial hypoxanthine guanine phosphoribosyltransferase deficiency. Pediatr Neurol. 1986;2:302–4. doi: 10.1016/0887-8994(86)90025-1. [DOI] [PubMed] [Google Scholar]
Hicks S, Wheeler DA, Plon SE, Kimmel M. Prediction of missense mutation functionality depends on both the algorithm and sequence alignment employed. Hum Mutat. 2011;32:661–8. doi: 10.1002/humu.21490. [DOI] [PMC free article] [PubMed] [Google Scholar]
Hikita M, Hosoya T, Ichida K, Okabe H, Saji M, Ohno I, et al. Partial deficiency of hypoxanthine-guanine-phosphoribosyltransferase manifesting as acute renal damage. Int Med. 1998;37:945–9. doi: 10.2169/internalmedicine.37.945. [DOI] [PubMed] [Google Scholar]
Hladnik U, Nyhan WL, Bertelli M. Variable expression of HPRT deficiency in 5 members of a family with the same mutation. Arch Neurol. 2008;65:1240–3. doi: 10.1001/archneur.65.9.1240. [DOI] [PubMed] [Google Scholar]
Holden JA, Kelley WN. Human HPRT: evidence for tetrameric structure. J Biol Chem. 1978;253:4459–64. [PubMed] [Google Scholar]
Holland MJ, DiLorenzo AM, Dancis J, Balis ME, Yu TF, Cox RP. Hypoxanthine phosphoribosyltransferase activity in intact fibroblasts from patients with X-linked hyperuricemia. J Clin Invest. 1976;57:1600–5. doi: 10.1172/JCI108430. [DOI] [PMC free article] [PubMed] [Google Scholar]
Hooft C, Van Nevel C, De Schaepdryver AF. Hyperuricosuric encephalopathy without hyperuricaemia. Arch Dis Child. 1968;43:734–7. doi: 10.1136/adc.43.232.734. [DOI] [PMC free article] [PubMed] [Google Scholar]
Hunter TC, Melancon SB, Dallaire L, Taft S, Skopek TR, Albertini RJ, et al. Germinal HPRT splice donor site mutation results in multiple RNA splicing products in T-lymphocyte cultures. Somat Cell Mol Genet. 1996;22:145–50. doi: 10.1007/BF02369904. [DOI] [PubMed] [Google Scholar]
Igarishi T, Minami M, Nishida Y. Molecular analysis of hypoxanthine-guanine phosphoribosyltransferase mutation in five unrelated Japanese patients. Acta Paediatr Jpn. 1989;31:303–13. doi: 10.1111/j.1442-200x.1989.tb01306.x. [DOI] [PubMed] [Google Scholar]
Inokuchi T, Moriwaki Y, Takahashi S, Tsutsumi Z, Ka T, Yamamoto A, et al. Identification of a new point mutation in hypoxanthine phosphoribosyl transferase responsible for hyperuricemia in a female patient. Metabolism. 2004;53:1500–2. doi: 10.1016/j.metabol.2004.04.016. [DOI] [PubMed] [Google Scholar]
Itiaba K, Melancon SB, Dallaire L, Crawhall JC. Adenine phosphoribosyl transferase deficiency in association with sub-normal hypoxanthine phophoribosyl transferase in families of Lesch-Nyhan patients. Biochem Med. 1978;19:252–9. doi: 10.1016/0006-2944(78)90027-3. [DOI] [PubMed] [Google Scholar]
Jankovic J, Caskey CT, Stout JT, Butler IJ. Lesch-Nyhan syndrome: a study of motor behavior and cerebrospinal fluid neurotransmitters. Ann Neurol. 1988;23:466–9. doi: 10.1002/ana.410230507. [DOI] [PubMed] [Google Scholar]
Jenkins EA, Hallett RJ, Hull RG. Lesch-Nyhan syndrome presenting with renal insufficiency in infancy and transient neonatal hypothyroidism. Br J Rheumatol. 1994;33:392–6. doi: 10.1093/rheumatology/33.4.392. [DOI] [PubMed] [Google Scholar]
Jinnah HA, Ceballos-Picot I, Torres RJ, Visser JE, Schretlen D, Verdu A, et al. Attenuated variants of Lesch-Nyhan disease. Brain. 2010;133:671–89. doi: 10.1093/brain/awq013. [DOI] [PMC free article] [PubMed] [Google Scholar]
Jinnah HA, DeGregorio L, Harris JC, Nyhan WL, O'Neill JP. The spectrum of inherited mutations causing HPRT deficiency: 75 new cases and a review of 196 previously reported cases. Mutat Res. 2000;463:309–26. doi: 10.1016/s1383-5742(00)00052-1. [DOI] [PubMed] [Google Scholar]
Jinnah HA, Harris JC, Nyhan WL, O'Neill JP. The spectrum of mutations causing HPRT deficiency: an update. Nucleosides Nucleotides Nucleic Acids. 2004;23:1153–60. doi: 10.1081/NCN-200027400. [DOI] [PubMed] [Google Scholar]
Jinnah HA, Hess EJ, Wilson MC, Gage FH, Friedmann T. Localization of hypoxanthine-guanine phosphoribosyltransferase mRNA in the mouse brain by in situ hybridization. Mol Cell Neurosci. 1992a;3:64–78. doi: 10.1016/1044-7431(92)90010-y. [DOI] [PubMed] [Google Scholar]
Jinnah HA, Jones MD, Wojcik BE, Rothstein JD, Hess EJ, Friedmann T, et al. Influence of age and strain on striatal dopamine loss in a genetic mouse model of Lesch-Nyhan disease. J Neurochem. 1999;72:225–9. doi: 10.1046/j.1471-4159.1999.0720225.x. [DOI] [PubMed] [Google Scholar]
Jinnah HA, Langlais PJ, Friedmann T. Functional analysis of brain dopamine systems in a genetic mouse model of Lesch-Nyhan syndrome. J Pharmacol Exp Ther. 1992b;263:596–607. [PubMed] [Google Scholar]
Jinnah HA, Visser JE, Harris JC, Verdu A, Larovere L, Ceballos-Picot I, et al. Delineation of the motor disorder of Lesch-Nyhan disease. Brain. 2006;129:1201–17. doi: 10.1093/brain/awl056. [DOI] [PMC free article] [PubMed] [Google Scholar]
Jinnah HA, Wojcik BE, Hunt MA, Narang N, Lee KY, Goldstein M, et al. Dopamine deficiency in a genetic mouse model of Lesch-Nyhan disease. J Neurosci. 1994;14:1164–75. doi: 10.1523/JNEUROSCI.14-03-01164.1994. [DOI] [PMC free article] [PubMed] [Google Scholar]
Johnson GG, Eisenberg LR, Migeon BR. Human and mouse hypoxanthine-guanine phosphoribosyltransferase: dimers and tetramers. Science. 1979;203:174–6. doi: 10.1126/science.569362. [DOI] [PubMed] [Google Scholar]
Johnson GG, Ramage AL, Littlefield JW, Kazazian HH., Jr Hypoxathine-guanine phsophoribosyltransferase in human erythroid cells: posttranslational modification. Biochemistry. 1982;21:960–6. doi: 10.1021/bi00534a022. [DOI] [PubMed] [Google Scholar]
Jolly DJ, Okayama H, Berg P, Esty AC, Filpula D, Bohlen P, et al. Isolation and characterization of a full-length expressible cDNA for human hypoxanthine phosphoribosyltransferase. Proc Natl Acad Sci USA. 1983;80:477–81. doi: 10.1073/pnas.80.2.477. [DOI] [PMC free article] [PubMed] [Google Scholar]
Jurecka A, Popowska E, Tylki-Szymanska A, Kubalska J, Ciara E, Krumina Z, et al. Hypoxanthine-guanine phosphoribosylotransferase deficiency-The spectrum of Polish mutations. J Inherit Metab Dis. 2008;31(Suppl 2):S447–51. doi: 10.1007/s10545-008-1013-8. [DOI] [PubMed] [Google Scholar]
Kang TH, Guibinga GH, Jinnah HA, Friedmann T. HPRT deficiency coordinately dysregulates canonical Wnt and presenilin-1 signaling: a neuro-developmental regulatory role for a housekeeping gene? PLoS One. 2011;6:e16572. doi: 10.1371/journal.pone.0016572. [DOI] [PMC free article] [PubMed] [Google Scholar]
Kassimatis TI, Simmonds HA, Goudas PC, AMarinaki AM, Fairbanks LD, Diamandopoulos AA. HPRT deficiency as the cause of ESRD in a 24-year-old patient: a very rare presentaion of the disorder. J Nephrol. 2005;18:447–51. [PubMed] [Google Scholar]
Keebaugh AC, Sullivan RT, Thomas JW. Gene duplication and inactivation in the HPRT gene family. Genomics. 2006;89:134–42. doi: 10.1016/j.ygeno.2006.07.003. [DOI] [PubMed] [Google Scholar]
Keough DT, Brereton IM, de Jersey J, Guddat LW. The crystal structure of free human hypoxanthine-guanine phosphoribosyltransferase reveals extensive conformational plasticity throughout the catalytic cycle. J Mol Biol. 2005;351:170–81. doi: 10.1016/j.jmb.2005.05.061. [DOI] [PubMed] [Google Scholar]
Keough DT, Emmerson BT, de Jersey J. Localization of the 5-phospho-a-D-ribosyl-1-pyrophosphate binding site of human hypoxanthine-guanine phosphoribosyltransferase. Biochim Biophys Acta. 1991;1096:95–100. doi: 10.1016/0925-4439(91)90045-b. [DOI] [PubMed] [Google Scholar]
Kim KJ, Yamada Y, Suzumori K, Choi Y, Yang SW, Cheong HI, et al. Molecular analysis of hypoxanthine guanine phosphoribosyltransferase (HPRT) gene in five Korean families with Lesch-Nyhan syndrome. J.Korean Med Sci. 1997;12:332–9. doi: 10.3346/jkms.1997.12.4.332. [DOI] [PMC free article] [PubMed] [Google Scholar]
Kim SJ, Sohn YB, Park SW, Jin DK, Paik KH. Chronic peritoneal dialysis in a 4-year-old boy with Lesch-Nyhan disease. Pediatr Nephrol. 2009;24:1089–90. doi: 10.1007/s00467-008-1081-x. [DOI] [PubMed] [Google Scholar]
Lesch M, Nyhan WL. A familial disorder of uric acid metabolism and central nervous system function. Am J Med. 1964;36:561–70. doi: 10.1016/0002-9343(64)90104-4. [DOI] [PubMed] [Google Scholar]
Lewers JC, Ceballos-Picot I, Shirley TL, Mockel L, Egami K, Jinnah HA. Consequences of impaired purine recycling in dopaminergic neurons. Neuroscience. 2008;152:761–72. doi: 10.1016/j.neuroscience.2007.10.065. [DOI] [PMC free article] [PubMed] [Google Scholar]
Lightfoot T, Lewkonia RM, Snyder FF. Sequence, expression, and characterization of HPRTMoose Jaw: a point mutation resulting in cooperativity and decreased substrate affinities. Hum Mol Genet. 1994;3:1377–81. doi: 10.1093/hmg/3.8.1377. [DOI] [PubMed] [Google Scholar]
Lloyd KG, Hornykiewicz O, Davidson L, Shannak K, Farley I, Goldstein M, et al. Biochemical evidence of dysfunction of brain neurotransmitters in the Lesch-Nyhan syndrome. N Engl J Med. 1981;305:1106–11. doi: 10.1056/NEJM198111053051902. [DOI] [PubMed] [Google Scholar]
Lorentz WB, Burton BK, Trillo A, Browning MC. Failure to thrive, hyperuricemia, and renal insufficiency in early infancy secondary to partial hypoxanthine-guanine phosphoribosyl transferase deficiency. J Pediatr. 1984;104:94–6. doi: 10.1016/s0022-3476(84)80600-9. [DOI] [PubMed] [Google Scholar]
Ludman CN, Dicks-Mireaux C, Saunders AJ. Renal ultrasonographic appearances at presentation in an infant with Lesch-Nyhan syndrome. Br J Radiol. 1992;65:724–5. doi: 10.1259/0007-1285-65-776-724. [DOI] [PubMed] [Google Scholar]
Mak BS, Shi CS, Tsai CR, Lee WJ, Lin HY. New mutations of the HPRT gene in Lesch-Nyhan syndrome. Pediatr Neurol. 2000;23:332–5. doi: 10.1016/s0887-8994(00)00199-5. [DOI] [PubMed] [Google Scholar]
Manzke H, Gustmann H, Koke HB, Nyhan WL. Hypoxanthine and tetrahydrobiopterin treatment of a patient with features of the Lesch-Nyhan syndrome. Adv Exp Med Biol. 1986;195(Pt A):197–204. doi: 10.1007/978-1-4684-5104-7_31. [DOI] [PubMed] [Google Scholar]
Marcus S, Christensen E, Malm G. Molecular analysis of the mutations in five unrelated patients with the Lesch Nyhan syndrome. Hum Mutat. 1993a;2:473–7. doi: 10.1002/humu.1380020608. [DOI] [PubMed] [Google Scholar]
Marcus S, Hellgren D, Lambert B, Fallstrom SP, Wahlstrom J. Duplication in the hypoxanthine phosphoribosyl-transferase gene caused by Alu-Alu recombination in a patient with Lesch-Nyhan syndrome. Hum Genet. 1993b;90:477–82. doi: 10.1007/BF00217444. [DOI] [PubMed] [Google Scholar]
Marmattom BV. Self-mutilation in the Lesch-Nyhan syndrome. Neurology. 2005;65:25. doi: 10.1212/01.wnl.0000184611.55051.f3. [DOI] [PubMed] [Google Scholar]
Matthews WS, Solan A, Barabas G. Cognitive functioning in Lesch-Nyhan syndrome. Dev Med Child Neurol. 1995;37:715–22. doi: 10.1111/j.1469-8749.1995.tb15017.x. [DOI] [PubMed] [Google Scholar]
McDonald JA, Kelley WN. Lesch-Nyhan syndrome: altered kinetic properties of mutant enzyme. Science. 1971;171:689–90. doi: 10.1126/science.171.3972.689. [DOI] [PubMed] [Google Scholar]
Melton DW. HPRT gene organization and expression. Oxf Surv Euk Genes. 1987;4:35–75. [PubMed] [Google Scholar]
Mitchell G, McInnes RR. Differential diagnosis of cerebral palsy: Lesch-Nyhan syndrome without self-mutilation. Can Med Assoc J. 1984;130:1323–4. [PMC free article] [PubMed] [Google Scholar]
Mizunuma M, Yamada Y, Yamada K, Sonta S, Wakamatsu N, Kaneko K, et al. Disruption of the hypoxanthine-guanine phosphoribosyl-transferase gene caused by a translocation in a patient with Lesch-Nyhan syndrome. Nucleosides Nucleotides Nucleic Acids. 2004;23:1173–6. doi: 10.1081/NCN-200027441. [DOI] [PubMed] [Google Scholar]
Monnat RJ, Jr, Chiaverotti TA, Hackmann AF, Maresh GA. Molecular structure and genetic stability of human hypoxanthine phosphoribosyltransferase (HPRT) gene duplications. Genomics. 1992;13:788–96. doi: 10.1016/0888-7543(92)90154-k. [DOI] [PubMed] [Google Scholar]
Musick WD. Structural features of the phosphoribosyltransferases and their relationship to the human deficiency disorders of purine and pyrimidine metabolism. CRC Crit Rev Biochem. 1981;11:1–34. doi: 10.3109/10409238109108698. [DOI] [PubMed] [Google Scholar]
Nguyen KV, Naviaux RK, Paik KK, Nyhan WL. Lesch-Nyhan syndrome: mRNA expression of HPRT in patients with enzyme proven deficiency of HPRT and normal HPRT coding region of DNA. Mol Genet Metab. 2012;106:498–501. doi: 10.1016/j.ymgme.2012.06.003. [DOI] [PubMed] [Google Scholar]
Nyhan WL. Behavior in the Lesch-Nyhan syndrome. J Autism Child Schizophren. 1976;6:235–52. doi: 10.1007/BF01543464. [DOI] [PubMed] [Google Scholar]
Nyhan WL, Oliver WJ, Lesch M. A familial disorder or uric acid metabolism and central nervous system function II. J Pediatr. 1965;67:257–63. doi: 10.1016/0002-9343(64)90104-4. [DOI] [PubMed] [Google Scholar]
O'Neill JP, Finette BA. Transition mutations at CpG dinucleotides are the most frequent in vivo spontaneous single-base substitution mutation in the human HPRT gene. Environ Molec Mutagen. 1998;32:188–91. doi: 10.1002/(sici)1098-2280(1998)32:2<188::aid-em16>3.0.co;2-y. [DOI] [PubMed] [Google Scholar]
O'Neill JP, Rogan PK, Cariello N, Nicklas JA. Mutations that alter RNA splicing of the human HPRT gene: a review of the spectrum. Mutat Res. 1998;411:179–214. doi: 10.1016/s1383-5742(98)00013-1. [DOI] [PubMed] [Google Scholar]
Ogasawara N, Stout JT, Goto H, Sonta SI, Matsumoto A, Caskey CT. Molecular anaylsis of a female Lesch-Nyhan patient. J Clin Invest. 1989;4:1024–7. doi: 10.1172/JCI114224. [DOI] [PMC free article] [PubMed] [Google Scholar]
Page T, Bakay B, Nissinen E, Nyhan WL. Hypoxanthine-guanine phosphoribosyltranferase variants: correlation of clinical phenotype with enzyme activity. J Inherit Metab Dis. 1981;4:203–6. doi: 10.1007/BF02263652. [DOI] [PubMed] [Google Scholar]
Page T, Bakay B, Nyhan WL. Kinetic studies of hypoxanthine-guanine phosphoribosyltransferase in intact cells. Adv Exp Med Biol. 1984;165(Pt B):27–31. doi: 10.1007/978-1-4757-0390-0_6. [DOI] [PubMed] [Google Scholar]
Page T, Nyhan WL. The spectrum of HPRT deficiency: an update. Adv Exp Med Biol. 1989;253A:129–133. doi: 10.1007/978-1-4684-5673-8_20. [DOI] [PubMed] [Google Scholar]
Patel PE, Nussbaum RL, Framson PE, Ledbetter DH, Caskey CT, Chinault AC. Organization of the HPRT gene and related sequences in the human genome. Somat Cell Mol Genet. 1984;10:483–93. doi: 10.1007/BF01534853. [DOI] [PubMed] [Google Scholar]
Puig JG, Torres RJ, Mateos FA, Ramos TH, Arcas JM, Buno AS, et al. The spectrum of hypoxanthine-guanine phosphoribosyltransferase deficiency: clinical experience based on 22 patients from 18 Spanish families. Medicine. 2001;80:102–12. doi: 10.1097/00005792-200103000-00003. [DOI] [PubMed] [Google Scholar]
Richardson BJ, Ryckman DL, Komarnicki LM, Hamerton JL. Heterogeneity in the biochemical characteristics of red blood cell hypoxanthine-guanine phosphoribosyl transferase from two unrelated patients with the Lesch-Nyhan syndrome. Biochem Genet. 1973;9:197–202. doi: 10.1007/BF00487450. [DOI] [PubMed] [Google Scholar]
Rijksen G, Staal GE, van der Vlist MJ, Beemer FA, Troost J, Gutensohn W, et al. Partial hypoxanthine-guanine phosphoribosyl transferase deficiency with full expression of the Lesch-Nyhan syndrome. Hum Genet. 1981;57:39–47. doi: 10.1007/BF00271165. [DOI] [PubMed] [Google Scholar]
Rinat C, Zoref-Shani E, Ben-Neriah Z, Bromberg Y, Becker-Cohen R, Feinstein S, et al. Molecular, biochemical, and genetic characterization of a female patient with Lesch-Nyhan disease. Mol Genet Metab. 2005;87:249–52. doi: 10.1016/j.ymgme.2005.09.025. [DOI] [PubMed] [Google Scholar]
Robey KL, Reck JF, Giacomini KD, Barabas G, Eddey GE. Modes and patterns of self-mutilation in persons with Lesch-Nyhan disease. Dev Med Child Neurol. 2003;45:167–71. doi: 10.1017/s001216220300032x. [DOI] [PubMed] [Google Scholar]
Rosenbloom FM, Henderson JF, Caldwell IC, Kelley WN, Seegmiller JE. Biochemical bases of accelerated purine biosynthesis de novo in human fibroblasts lacking hypoxanthine-guanine phosphoribosyltransferase. J Biol Chem. 1968;243:1166–73. [PubMed] [Google Scholar]
Saito Y, Ito M, Hanaoka S, Ohama E, Akaboshi S, Takashima S. Dopamine receptor upregulation in Lesch-Nyhan syndrome: a postmortem study. Neuropediatrics. 1999;30:66–71. doi: 10.1055/s-2007-973462. [DOI] [PubMed] [Google Scholar]
Saito Y, Takashima S. Neurotransmitter changes in the pathophysiology of Lesch-Nyhan syndrome. Brain Dev. 2000;22(Suppl 1):S122–31. doi: 10.1016/s0387-7604(00)00143-1. [DOI] [PubMed] [Google Scholar]
Salerno C, Giacomello A. Human hypoxanthine-guanine phosphoribosyltransferase: IMP-GMP exchange stoichiometry and steady state kinetics of the reaction. J Biol Chem. 1979;254:10232–6. [PubMed] [Google Scholar]
Sampat R, Fu R, Larovere LE, Torres RJ, Ceballos-Picot I, Fischbach M, et al. Mechanisms for phenotypic variation in Lesch-Nyhan disease and its variants. Hum Genet. 2011;129:71–8. doi: 10.1007/s00439-010-0901-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
Sarafoglou K, Grosse-Redlinger K, Boys CJ, Charnas L, Otten N, Broock R, et al. Lesch-Nyhan variant syndrome: variable presentation in 3 affected family members. Arch Neurol. 2010;67:761–4. doi: 10.1001/archneurol.2010.116. [DOI] [PubMed] [Google Scholar]
Schretlen DS, Harris JC, Park KS, Jinnah HA, Ojeda del Pozo N. Neurocognitive functioning in Lesch-Nyhan disease and partial hypoxanthine-guanine phosphoribosyltransferase deficiency. J Int Neuropsychol Soc. 2001;7:805–12. doi: 10.1017/s135561770177703x. [DOI] [PubMed] [Google Scholar]
Schretlen DS, Ward J, Meyer SM, Yun J, Puig JG, Nyhan WL, et al. Behavioral aspects of Lesch-Nyhan disease and it variants. Dev Med Child Neurol. 2005;47:673–7. doi: 10.1017/S0012162205001374. [DOI] [PubMed] [Google Scholar]
Sculley DG, Dawson PA, Emmerson BT, Gordon RB. A review of the molecular basis of hypoxanthine-guanine phosphoribosyltransferase (HPRT) deficiency. Hum Genet. 1992;90:195–207. doi: 10.1007/BF00220062. [DOI] [PubMed] [Google Scholar]
Sebesta I, Stibarkova B, Dvorakova L, Hrebicek M, Minks J, Stolnaja L, et al. Unusual presentation of Kelley-Seegmiller syndrome. Nucleosides Nucleotides Nucleic Acids. 2008;27:648–55. doi: 10.1080/15257770802143863. [DOI] [PubMed] [Google Scholar]
Sege-Peterson K, Chambers J, Page T, Jones OW, Nyhan WL. Characterization of mutations in phenotypic variants of hypoxanthine phosphoribosyltransferase deficiency. Hum Mol Genet. 1992;1:427–32. doi: 10.1093/hmg/1.6.427. [DOI] [PubMed] [Google Scholar]
Serrano M, Perez-Duenas B, Ormazabal A, Artuch R, Campistol J, Torres RJ, et al. Levodopa therapy in a Lesch-Nyhan disease patient: pathological, biochemical, neuroimaging, and therapeutic remarks. Mov Disord. 2008;23:1297–300. doi: 10.1002/mds.21786. [DOI] [PubMed] [Google Scholar]
Sharma S, Jimenez RT, Aneja S, Garcia MG, Sethi GR. Lesch-Nyhan Syndrome in an Indian Family with Novel Mutation in the HPRT1 Gene. Indian J Pediatr. 2012;79:1520–2. doi: 10.1007/s12098-011-0657-9. [DOI] [PubMed] [Google Scholar]
Shi W, Li CM, Tyler PC, Furneaux RH, Grubmeyer C, Schramm VL, et al. The 2.0 A structure of human hypoxanthine-guanine phosphoribosyltransferase in complex with a transition-state analog inhibitor. Nat Struct Biol. 1999;6:588–93. doi: 10.1038/9376. [DOI] [PubMed] [Google Scholar]
Shnier MH, Sims F, Zail S. The Lesch-Nyhan syndrome: first case description in a South African family. S Afr Med J. 1972;46:947–9. [PubMed] [Google Scholar]
Silverstein FS, Johnston MV, Hutchinson RJ, Edwards NL. Lesch-Nyhan syndrome: CSF neurotransmitter abnormalities. Neurology. 1985;35:907–11. doi: 10.1212/wnl.35.6.907. [DOI] [PubMed] [Google Scholar]
Sim NL, Kumar P, Hu J, Henikoff S, Schneider G, Ng PC. SIFT web server: predicting effects of amino acid substitutions on proteins. Nucleic Acids Res. 2012;40:W452–7. doi: 10.1093/nar/gks539. [DOI] [PMC free article] [PubMed] [Google Scholar]
Snyder FF, Chudley AE, MacLeod PM, Carter RJ, Fung E, Lowe JK. Partial deficiency of hypoxanthine-guanine phosphoribosyltransferase with reduced affinity for PP-ribose-P in four related males with gout. Hum Genet. 1984;67:18–22. doi: 10.1007/BF00270552. [DOI] [PubMed] [Google Scholar]
Solan A, Matthews W, Barabas G, Robey K. Cognition in LND: a two-year follow-up study. Dev Med Child Neurol. 1997;39:492–3. [PubMed] [Google Scholar]
Sorensen LB. Mechanism of excessive purine biosynthesis in hypoxanthine-guanine phosphoribosyltransferase deficiency. J Clin Invest. 1970;49:968–78. doi: 10.1172/JCI106316. [DOI] [PMC free article] [PubMed] [Google Scholar]
Stout JT, Caskey CT. HPRT: gene structure, expression, and mutation. Ann Rev Genet. 1985;19:127–48. doi: 10.1146/annurev.ge.19.120185.001015. [DOI] [PubMed] [Google Scholar]
Taniguchi A, Yamada Y, Hakoda M, Sekita C, Kawamoto M, Kaneko H, et al. Molecular characterization of a deletion in the HPRT1 gene in a patient with Lesch-Nyhan syndrome. Nucleosides Nucleotides Nucleic Acids. 2011;30:1266–71. doi: 10.1080/15257770.2011.608396. [DOI] [PubMed] [Google Scholar]
Tarle SA, Davidson BL, Wu VC, Zidar FJ, Seegmiller JE, Kelley WN, et al. Determination of the mutations responsible for the Lesch-Nyhan syndrome in 17 subjects. Genomics. 1991;10:499–501. doi: 10.1016/0888-7543(91)90341-b. [DOI] [PubMed] [Google Scholar]
Torres RJ, Garcia MG, Puig JG. Partial HPRT deficiency phenotype and incomplete splicing mutation. Nucleosides Nucleotides Nucleic Acids. 2010;29:295–300. doi: 10.1080/15257771003730250. [DOI] [PubMed] [Google Scholar]
Torres RJ, Mateos FA, Molano J, Gathoff BS, O'Neill JP, Gundel RM, et al. Molecular basis of hypoxanthine-guanine phosphoribosyltransferase deficiency in thirteen Spanish families. Hum Mutat. 2000;15:383. doi: 10.1002/(SICI)1098-1004(200004)15:4<383::AID-HUMU17>3.0.CO;2-2. [DOI] [PubMed] [Google Scholar]
Torres RJ, Puig JG, Jinnah HA. Update on the phenotypic spectrum of Lesch-Nyhan disease and its attenuated variants. Curr Rheumatol Rep. 2011;14:189–94. doi: 10.1007/s11926-011-0231-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
Tvrdik T, Marcus S, Hou SM, Falt S, Noori P, Podlutskaja N, et al. Molecular characterization of two deletion events involving Alu-sequences, one novel base substitution and two tentative hotspot mutations in the hypoxanthine-phosphoribosyltransferase (HPRT) gene in five patients with Lesch-Nyhan syndrome. Hum Genet. 1998;103:311–318. doi: 10.1007/s004390050822. [DOI] [PubMed] [Google Scholar]
Van Bogaert P, Ceballos I, Desguerre I, Telvi L, Kamoun P, Ponsot G. Lesch-Nyhan syndrome in a girl. J Inherit Metab Dis. 1992;15:790–1. doi: 10.1007/BF01800022. [DOI] [PubMed] [Google Scholar]
Visser JE, Baer PR, Jinnah HA. Lesch-Nyhan syndrome and the basal ganglia. Brain Res Rev. 2000;32:449–75. doi: 10.1016/s0165-0173(99)00094-6. [DOI] [PubMed] [Google Scholar]
Wang F, Shi W, Nieves E, Angeletti RH, Schramm VL, Grubmeyer C. A transition-state analogue reduces protein dynamics in hypoxanthine-guanine phosphoribosyltransferase. Biochemistry. 2001;40:8043–54. doi: 10.1021/bi010203f. [DOI] [PubMed] [Google Scholar]
Warzok R, Schwesinger G, Knapp A, Seidlitz F. Neuropathologische befunde beim Lesch-Nyhan Syndrom. Zbl Allg Pathol U Pathol Anat. 1982;126:95–104. [PubMed] [Google Scholar]
Watts RWE, Spellacy E, Gibbs DA, Allsop J, McKeran RO, Slavin GE. Clinical, post-mortem, biochemical and therapeutic observations on the Lesch-Nyhan syndrome with particular reference to the neurological manifestions. Q J Med. 1982;201:43–78. [PubMed] [Google Scholar]
Willers I, Bolz H, Wehnert M, Gal A. Eighteen novel mutation in patients with Lesch-Nyhan syndrome or partial hypoxanthine phosphoribosyltransferase deficiency. J Inherit Metab Dis. 1999;22:845–6. doi: 10.1023/a:1005522527689. [DOI] [PubMed] [Google Scholar]
Wilson JM, Kelley WN. Molecular genetics of the HPRT-deficiency syndromes. Hosp Pract. 1984;19:81–9, 93–7, 100. doi: 10.1080/21548331.1984.11702819. [DOI] [PubMed] [Google Scholar]
Wilson JM, Landa LE, Kobayashi R, Kelley WN. Human hypoxanthine-guanine phosphoribosyltransferase: tryptic peptides and post-translational modification of the erythrocyte enzyme. J Biol Chem. 1982;257:14830–4. [PubMed] [Google Scholar]
Wilson JM, Stout JT, Palella TD, Davidson BL, Kelley WN, Caskey CT. A molecular survey of hypoxanthine-guanine phosphoribosyltransferase deficiency in man. J Clin Invest. 1986;77:188–95. doi: 10.1172/JCI112275. [DOI] [PMC free article] [PubMed] [Google Scholar]
Wilson JM, Young AB, Kelley WN. Hypoxanthine-guanine phosphoribosyltransferase deficiency. The molecular basis of the clinical syndromes. N Engl J Med. 1983;309:900–10. doi: 10.1056/NEJM198310133091507. [DOI] [PubMed] [Google Scholar]
Wong DF, Harris JC, Naidu S, Yokoi F, Marenco S, Dannals RF, et al. Dopamine transporters are markedly reduced in Lesch-Nyhan disease in vivo. Proc Natl Acad Sci USA. 1996;93:5539–43. doi: 10.1073/pnas.93.11.5539. [DOI] [PMC free article] [PubMed] [Google Scholar]
Xu Y, Eads J, Sacchettini JC, Grubmeyer C. Kinetic mechanism of human hypoxanthine-guanine phosphoribosyltransferase: rapid phosphoribosyl transfer chemistry. Biochemistry. 1997;36:3700–12. doi: 10.1021/bi9616007. [DOI] [PubMed] [Google Scholar]
Yamada Y, Goto H, Suzumori K, Adachi R, Ogasawara N. Molecular analysis of five independent Japanese mutant genes responsible for hypoxanthine guanine phosphoribosyltransferase (HPRT) deficiency. Hum Genet. 1992;90:379–84. doi: 10.1007/BF00220463. [DOI] [PubMed] [Google Scholar]
Yamada Y, Goto H, Yukawa T, Akazawa H, Ogasawara N. Molecular mechanisms of the second female Lesch-Nyhan patient. Adv Exp Med Biol. 1994;370:337–40. doi: 10.1007/978-1-4615-2584-4_72. [DOI] [PubMed] [Google Scholar]
Yamada Y, Nomura N, Yamada K, Wakamatsu N. Molecular analysis of HPRT deficiencies: An update of the spectrum of Asian mutations with novel mutations. Mol Genet Metab. 2007;90:70–6. doi: 10.1016/j.ymgme.2006.08.013. [DOI] [PubMed] [Google Scholar]
Yamada Y, Wakamatsu N, Taniguchi A, Kaneko K, Fujimori S. Hypoxanthine guanine phosphoribosyltransferase (HPRT) mutations in the Asian population. Nucleosides Nucleotides Nucleic Acids. 2011a;30:1248–55. doi: 10.1080/15257770.2011.603714. [DOI] [PubMed] [Google Scholar]
Yamada Y, Yamada K, Nomura N, Yamano A, Kimura R, Naiki M, et al. Molecular analysis of X-linked inborn errors of purine metabolism: HPRT1 and PRPS1 mutations. Nucleosides Nucleotides Nucleic Acids. 2011b;30:1272–5. doi: 10.1080/15257770.2011.597369. [DOI] [PubMed] [Google Scholar]
Yamada Y, Yamada K, Sonta S, Wakamatsu N, Ogasawara N. Mutations in the hypoxanthine-guanine phosphoribosyltransferase gene (HPRT1) in Asian HPRT-deficient families. Nucleosides Nucleotides Nucleic Acids. 2004;23:1169–72. doi: 10.1081/NCN-200027439. [DOI] [PubMed] [Google Scholar]
Yang TP, Patel PI, Chinault AC, Stout JT, Jackson LG, Hildebrand BM, et al. Molecular evidence for new mutation at the hprt locus in Lesch-Nyhan patients. Nature. 1984;310:412–4. doi: 10.1038/310412a0. [DOI] [PubMed] [Google Scholar]
Yang TP, Stout JT, Konecki DS, Patel PI, Alford RL, Caskey CT. Spontaneous reversion of novel Lesch-Nyhan mutation by HPRT gene rearrangement. Somat Cell Mol Genet. 1988;14:293–303. doi: 10.1007/BF01534590. [DOI] [PubMed] [Google Scholar]
Yeh J, Zheng S, Howard BD. Impaired differentiation of HPRT-deficient dopaminergic neurons: a possible mechanism underlying neuronal dysfunction in Lesch-Nyhan syndrome. J Neurosci Res. 1998;53:78–85. doi: 10.1002/(SICI)1097-4547(19980701)53:1<78::AID-JNR8>3.0.CO;2-G. [DOI] [PubMed] [Google Scholar]
Yukawa T, Akazawa H, Miyake Y, Takahashi Y, Nagao H, Takeda E. A female patient with Lesch-Nyhan syndrome. Dev Med Child Neurol. 1992;34:543–6. doi: 10.1111/j.1469-8749.1992.tb11477.x. [DOI] [PubMed] [Google Scholar]
Zamora A, Escarcega RO, Vazquez R, O'Neill JP. HPRT deficiency in a two-month-old child presenting acute renal failure and gout with a new deletion of two bases in exon 3 of the HPRT gene. Arch Med Res. 2007;38:460–2. doi: 10.1016/j.arcmed.2006.10.015. [DOI] [PubMed] [Google Scholar]
Zanic T, Gamulin V, Lipovac K. A case of severe hypoxanthine-guanine phosphoribosyl transferase deficiency. J Inherit Metab Dis. 1985;8:79. doi: 10.1007/BF01801670. [DOI] [PubMed] [Google Scholar]
Zoref-Shani E, Bromberg Y, Hirsch J, Feinstein S, Frishberg Y, Sperling O. A novel point mutation (I137T) in the conserved 5-phophorylribose-1-pyrophosphate binding motif of hypoxanthine-guanine phosphoribosyltransferase (HPRTJerusalem) in a variant of Lesch-Nyhan syndrome. Mol Genet Metab. 2003;78:158–61. doi: 10.1016/s1096-7192(03)00002-7. [DOI] [PubMed] [Google Scholar]
Zoref-Shani E, Bromberg Y, Hirsh J, Feinstein S, Frishberg Y, Sperling O. Clinical and biochemical manifestations and molecular characterization of the mutation in HPRT Jerusalem. Nucleosides Nucleotides Nucleic Acids. 2004;23:1165–8. doi: 10.1081/NCN-200027436. [DOI] [PubMed] [Google Scholar]
Zoref-Shani E, Feinstein S, Frishberg Y, Sperling O. Kelley-Seegmiller syndrome due to a unique variant of hypoxanthine-guanine phosphoribosyltransferase: reduced affinity for 5-phosphoribosy1-pyrophosphate manifested only at low, physiological substrate concentrations. Biochim Biophys Acta. 2000;1500:197–203. doi: 10.1016/s0925-4439(99)00103-9. [DOI] [PubMed] [Google Scholar]
Articles from Brain are provided here courtesy of Oxford University Press
ACTIONS
View on publisher site
PDF (801.2 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
Introduction
The HPRT1 gene and HGprt enzyme
The spectrum of the clinical phenotype
The heterogeneity of clinically relevant mutations
Genotype–phenotype correlations
Discrepancies in genotype–enzyme–phenotype correlations
Beyond biochemical genetics
Conclusions
Funding
Glossary
References
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
11747 | https://penrose.uri.edu/Gerhard/PHY525N/wtex154.pdf | [tex154] Ultrarelativistic classical ideal gas in two dimensions Consider a classical ideal gas of N particles confined to a two-dimensional box of area V in thermal equilibrium at extremely high temperature T. Most particles are moving at speeds close to the speed of light c. We describe this system by a Hamiltonian of the form, H = N X l=1 q p2 x + p2 y c.
(a) Show that the canonical partition function is ZN = 1 N!
" 2πV kBT hc 2#N .
(b) Find the Helmholtz free energy A(T, V, N), the entropy S(T, V, N), the pressure p(T, V, N), and the internal energy U(T, N).
(c) Find the adiabate (for constant N) and express it in the form pνV = const.
(d) Infer from the given canonical partition function ZN(T, V ) an explicit expression for the grand partition function Z(T, V, µ), where µ = kBT ln z is the chemical potential and z is the fugacity.
Use R ∞ 0 dxxne−ax = n!a−n−1, ln n! ≃n ln n −n, P∞ n=0 xn/n! = ex.
Solution: |
11748 | https://mathoverflow.net/questions/44266/rooks-on-a-lifeline | Skip to main content
Rooks on a lifeline
Ask Question
Asked
Modified
9 years, 10 months ago
Viewed
1k times
This question shows research effort; it is useful and clear
8
Save this question.
Show activity on this post.
The short version of this question is:
If G is a graph whose nodes are associated with squares of a chessboard, such that no two nodes in the same row or column of the board are adjacent, we want to associate rooks with the vertices of G, such that at most one rook appears in each row and column of the chessboard under the constraint that the vertices containing the rooks induce a connected subgraph of G (thus, the rooks are connected to each other with a lifeline, or lifegraph if you want to be specific).
A maximal configuration of rooks is such that no rooks can be added to the chessboard without violating the constraint that each column/row contain only one rook.
Question: A back-tracking depth first search will find all maximal configutions. Will a back-tracking breadth-first search do the same?
Let's consider an m×n chessboard that will be inhabited by rooks. As is usual with chess problems in graph theory, each square is represented by a vertex: let vs,t represent the square at row s and column t of the board. Now, let G be an arbitrary graph on the set of vertices V that comprise the squares of the chessboard.
We want to fill the chessboard with rooks (said another way: we want to associate rooks with vertices), such that:
Every row and every column contain at most one rook (more formally, if a rook is associated with a vertex vi,j, then no rook will be associated with vi,s for s∈{1,…,n}) or vt,j for t∈{1,…,m}),
The vertices with rooks must induce a connected subgraph of G (hence the reference to a "lifeline" that must connect all rooks).
An example of a chessboard with a single rook (indicated by the black vertex) is shown here
The gray squares show the area covered by the rook - no other rook can be placed on any of the gray squares. Note that I have neglected to color the squares themselves black and white as they should appear on a real chessboard. Edges are indicated by red lines.
Let's consider how we might extend the neighborhood of the black node - call it v. There are three vertices adjacent to v, but we can only place rooks on at most two of them at a time without violating the constraint that only a single rook cover a given row/column. In fact, there are two ways of placing the rooks, as shown here:
and here (broken link).
A maximal configuration is a valid placement of rooks (according to our constraints) that cannot be extended by the addition of another rook without violating the constraints.
We can enumerate all maximal configurations by performing a back-tracking depth-first search from each node (i.e. each square on the chessboard). With a depth-first search, we only add one rook to the chessboard at a time.
Suppose that we perform a back-tracking breadth-first search instead. At each step, we add as many rooks the board as possible. Of course there possibly are many different ways of adding as many rooks as possible at each step. This is exactly what is done in the two images above: the maximum number of rooks are added in both possible configurations.
Will this strategy also enumerate all maximal configurations?
co.combinatorics
graph-theory
chess
Share
CC BY-SA 3.0
Improve this question
Follow this question to receive notifications
edited Jun 15, 2020 at 7:27
CommunityBot
122 silver badges33 bronze badges
asked Oct 30, 2010 at 18:56
Wynand WinterbachWynand Winterbach
8366 bronze badges
3
3
I may be out of my depth (and breadth?) here, but my understanding is that if you have a (finite) tree then depth-first and breadth-first searches will both eventually visit every node in the tree, just not in the same order. So if depth-first eventually finds everything you're looking for, so will breadth-first, unless you're using those terms with non-standard meanings. Now I'll stop to catch my breadth.
Gerry Myerson
–
Gerry Myerson
10/30/2010 22:44:51
Commented
Oct 30, 2010 at 22:44
1
That was my first guess too. But the fact that we're adding as many neighbors as possible with each "breadth-first search" step, means that we might commit ourselves to rook placements that would make us miss maximal configurations. If, instead of only adding as many neighbors as possible with each BFS step, we considered all possibilities of adding {1,…,n} neighbors (under our constraints), where n is the maximum number of neighbors, then we would definitely find all maximal configurations. So, we're being "greedy" in a sense here. I am worried that this might bite us.
Wynand Winterbach
–
Wynand Winterbach
10/31/2010 17:37:49
Commented
Oct 31, 2010 at 17:37
2
@winterstream It's been noted in a recent edit that the imageshack link is broken. Could you fix it, please?
Todd Trimble
–
Todd Trimble
10/21/2015 14:12:21
Commented
Oct 21, 2015 at 14:12
Add a comment
|
1 Answer 1
Reset to default
This answer is useful
7
Save this answer.
Show activity on this post.
The answer is that you have gotten too greedy! I give my squares coordinates in R^2. Place vertices on the points (z,z) for integers z with absolute value less than 100 and also at (-2,2), (1,2) and (-1,-2). Place edges between (z,z) and (z+1,z+1) for all z and also place edges between ((-2,2) and (0,0)) , ((-1,-2) and (0,0)) , ((-1,-2) and (1,1)), ((2,1) and (0,0)), ((2,1) and (-1,-1)). So 3 non diagonal vertices and 5 non diagonal edges.
Obviously the best we can do is place rooks on the diagonal from (-100,-100) to (100,100).
Now if you start your search at a vertex on the diagonal at or left of (-1,-1) (resp. at or right of (1,1)) then you won't get to the other side of the diagonal since at (-1,-1) (resp (1,1)) you take the edge to (2,1) (resp (-1,-2)) which cuts you off from (2,2) (resp (-2,-2)). That also explains why you can't start at (2,1) or (-1,-2). If you start at (-2,2) you are cut off from both diagonals as you can't cross (-2,-2) and (2,2). But if you start at the origin, you take the edge to (2,2) which shows you can't start at the origin either.
Thus you got too greedy. Close though, and interesting.
Share
CC BY-SA 2.5
Improve this answer
Follow this answer to receive notifications
answered Nov 4, 2010 at 4:39
OrangeOrange
12499 bronze badges
2
I can use some reputation to post a bounty ... this is a very nice construction.
Orange
–
Orange
11/05/2010 08:01:18
Commented
Nov 5, 2010 at 8:01
Hope that you got the reputation boost with the acceptance of the answer! Thanks Orange.
Wynand Winterbach
–
Wynand Winterbach
01/19/2011 10:11:32
Commented
Jan 19, 2011 at 10:11
Add a comment
|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
co.combinatorics
graph-theory
chess
See similar questions with these tags.
Related
35
Is there a good argument for why you can't place 4 queens which cover a chessboard?
37
Rooks in three dimensions
23
Hexagonal rooks
7
Collisions between rooks taking random flights on an N by M chessboard
2
Algorithm to solve Sokoban-like game on graphs - move chips from one set of vertices to another
7
Four Dimensional Rook Domination
2
Limited rook moves
P-rook polynomial and its generalization
Question feed
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. |
11749 | https://ximera.osu.edu/math/calc1Book/calcBook/concavity/concavity | Warning
You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
Delete my work and update to the new version.
Mathematical Expression Editor
In this section we learn about the two types of curvature and determine the curvature of a function.
1 Concavity
In this section we will discuss the curvature of the graph of a given function. There are two types of curvature: concave up and concave down. The main tool for discussing curvature is the second derivative, .
Concavity Suppose is differentiable on an open interval, . If is increasing on , then is concave up on and if is decreasing on , then is concave down on .
The following theorem helps us to determine where a function is concave up and where it is concave down.
Concavity If on an interval , then is concave up on and if on an interval , then is concave down on .
example 1 Find intervals where the function is concave up or concave down.The second derivative is given by Since for all values of , is concave up on the interval . This is clear based on the graph of as well.
example 2 Find intervals where the function is concave up or concave down.The second derivative is given by Since for all values of , is concave up on the interval . This is clear based on the graph of as well.
example 3 Find intervals where the function is concave up or concave down.The second derivative is given by Since has the same sign as itself, is concave up on the interval and concave down on the interval . This is clear based on the graph of as well.
The last example brings up a new concept. The function changes concavity at . We call this an inflection point of the function.
If is continuous at and changes concavity at , then we say that has an inflection point at .
In the next two examples, we will discuss the curvature of the given functions and find their inflection points.
example 4 Determine where the cubic polynomial is concave up, concave down and find the inflection points.The second derivative of is . To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for .
We have so . This value breaks the real number line into two intervals, and . The second derivative maintains the same sign throughout each of these intervals. To determine whether it is positive or negative, we choose a test point in each interval. For the interval, , we choose . Plugging this into the second derivative, we get . Next, we choose the test point for the interval . Plugging this into the second derivative gives . We can use this information to create a sign chart for the second derivative, as shown below.
We can now use the concavity theorem to conclude that the original function, is concave down on the interval . and concave up on the interval . Finally, noting that the original polynomial is continuous everywhere and changes concavity at , we can say that has an inflection point at .
(problem 4a) Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
The function has an inflection point at
(problem 4b) Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
The function has an inflection point at
example 5 Determine where the cubic polynomial is concave up, concave down and find the inflection points.The second derivative of is To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for .
We have so . This value breaks the real number line into two intervals, and . The second derivative maintains the same sign throughout each of these intervals. To determine whether it is positive or negative, we choose a test point in each interval. For the interval, , we choose . Plugging this into the second derivative, we get .
Next, for the interval , we choose the test point . Plugging this into the second derivative gives .
We can now use the concavity theorem to conclude that the original function, is concave down on the interval , and concave up on the interval . Finally, noting that the original polynomial is continuous everywhere and changes concavity at , we can say that has an inflection point at .
(problem 5a) Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
List multiple answers in ascending order.The function has inflection points at and .
(problem 5b) Find the inflection point(s) of the function
Find the second derivative,
Where does change sign?
Is continuous there?
If there are no inflection points, type “none”.The function has inflection point(s) at
(problem 5c) Find the intervals of increase/decrease, local extremes, intervals of concavity and inflection points for the function
example 6 Determine where the function is concave up, concave down and find the inflection points.To find , we will need to use the product rule twice. First, and second Now that we have the second derivative, we set it equal to zero. Solve for . Since the exponential is never equal to zero, the only solutions come from setting the quadratic to zero: This quadratic does not factor, so we need to use the quadratic formula. The solutions are For simplicity, we will call these two roots and . So and
These two values break the real number line into three intervals: and with test points and , respectively. Plugging these into the second derivative gives and
We can now use the concavity theorem to conclude that the original function, is concave up on the intervals and , and it is concave down on the interval . Finally, we can infer from this that the continuous function has inflection points at
(problem 6a) Find the inflection point(s) of the function
Find the second derivative,
Use the product rule to compute the derivatives
and is similar
Where does change sign?
Is continuous there?
The function has an inflection point at
(problem 6b) Find the inflection point(s) of the function
Find the second derivative,
Use the product rule to compute the derivatives
and is similar
Where does change sign?
Is continuous there?
List multiple answers in ascending order.The function has inflection points at and
(problem 6c) Find the inflection point(s) of the function in the interval
Find the second derivative,
Where does change sign?
Is continuous there?
List multiple answers in ascending order.On the interval , the function has inflection points at and .
example 7 Determine where the function is concave up, concave down and find the inflection points.By the chain rule, its derivative is and by the product rule (and the chain rule again, as well), its second derivative is Setting the second derivative equal to zero and noting that the exponential is always positive, we get Solving for gives
These two values of break the real number line into three intervals: and . The sign of will remain the same on each of these intervals. To determine the sign on each interval, we use the test points and respectively. Plugging the test points into the second derivative gives and
We can now use the concavity theorem to conclude that the original function, is concave up on the intervals and and concave down on the interval . Finally, since the original function is continuous everywhere, we can say that are inflection points for .
The function is the basis for the important Normal Distribution in Statistics.
(problem 7) Find the inflection point(s) of the function
Find the second derivative,
Use the Quotient Rule to compute the derivatives
Where does ?
To solve the equation , let and solve for first
Is continuous there?
List multiple answers in ascending order.The function has inflection points at and .
example 8 Determine where the cubic polynomial is concave up, concave down and find the inflection points.From the power rule with , we have and its second derivative is (power rule, ) Now we make the observation that has the same sign as and so will have the same sign as in this example. Thus, on the interval and on the interval We conclude that is concave up on and concave down on the interval Since is not continuous at (it has an infinite discontinuity there), there is no inflection point there.
(problem 8) Find the inflection point(s) of the function on the interval .
Find the second derivative,
The derivative of is
Use the Chain Rule to find the derivative of
Where does ?
On the interval , the function has an inflection point at .
2 Using the graph of the derivative
In this section, we are given the graph of the derivative, , and we are asked to make conclusions about the original function, .
example 9 Use the graph of , shown below, to answer the following questions about the graph of . Where is increasing/decreasing, and where are its local extremes? Where is concave up/down, and where are its inflection points?
To answer the first question, recall that when , then is increasing, and when , then is decreasing. From the graph, we see that is positive on the intervals and . Hence, the graph of is increasing on these intervals. We can also see that is negative on the interval and therefore is decreasing on this interval. We can now use the first derivative test to determine the nature of the local extremes. Since and changes sign from positive to negative at , the function, , has a local maximum at . Similarly, Since and changes sign from negative to positive at , the function, , has a local minimum at .To determine the concavity of ,recall that is concave up when is increasing and is concave down when is decreasing. From the graph, we see that is increasing on the interval , and decreasing on the interval . Hence, the graph of is concave up on and concave down on . Finally, has an inflection point at due to the change in concavity there.
(problem 9) Use the graph of the derivative, , given below, to answer the following questions.
Where is increasing? Select all that apply.
Where is decreasing? Select all that apply.
Describe the local extremes of . Select all that apply.
local maximum at local maximum at local minimum at local minimum at local minimum at
Where is concave up? Select all that apply.
Where is concave down? Select all that apply.
Where are the inflection points of ? Select all that apply.
inflection point at inflection point at inflection point at no inflection points
example 10 Use the graph of , shown below, to answer the following questions about the graph of . Where is increasing/decreasing, and where are its local extremes? Where is concave up/down, and where are its inflection points?
To answer the first question, recall that when , then is increasing, and when , then is decreasing. From the graph, we see that is positive on the intervals and . Hence, the graph of is increasing on these intervals. We can also see that is negative on the interval and therefore is decreasing on this interval. We can now use the first derivative test to determine the nature of the local extremes. Since and changes sign from negative to positive at , the function, , has a local minimum at . The situation at is different. From the graph, we see that but does not change sign at - it is positive on either side. Thus, does not have a local extreme at the critical number, .To determine the concavity of ,recall that is concave up when is increasing and is concave down when is decreasing. From the graph, we see that is increasing on the intervals and and decreasing on the interval . Hence, the graph of is concave up on and and concave down on . Finally, has an inflection points at and due to the changes in concavity at these points.
(problem 10) Use the graph of the derivative, , given below, to answer the following questions.
Where is increasing? Select all that apply.
Where is decreasing? Select all that apply.
Describe the local extremes of . Select all that apply.
local maximum at local maximum at local minimum at local minimum at local minimum at
Where is concave up? Select all that apply.
Where is concave down? Select all that apply.
Where are the inflection points of ? Select all that apply.
inflection point at inflection point at inflection point at no inflection points
3 Video lessons
Here is a detailed, lecture style video on concavity:
We next explore a special relationship between concavity and local extremes.
Second Derivative Test Suppose . If , then has a local maximum at , and if , then has a local minimum at
The following figure should convince the reader of the validity of the Second Derivative Test.
example 11 Use the Second Derivative Test to find the local extremes of The critical numbers of are and (verify). Next, (verify). Plugging the critical numbers into the second derivative gives, By the Second Derivative Test, has a local maximum at and a local minimum at .
(problem 11) Use the Second derivative Test to find the local extremes of The critical numbers are (list in ascending order) and .has a local maximum at and a local minimum at
2025-03-27 21:49:36 |
11750 | https://www.chasingtheunexpected.com/chinese-words-directions/ | Chinese words and basic expressions to use when asking for directions
Skip to content
ContactPrivacyAngela corrias
Menu
Travel Destinations
Travel Guides
Travel Tips
Travel Resources
Travel Essentials
Blog
Essential Chinese words and basic expressions to use when asking for directions
April 9, 2021
By Angela Corrias
Even though in Chinese cities the names of the streets are written in both Mandarin language and English and Google map covers pretty much everywhere, when you travel to China you might need to ask for directions. If your interlocutor doesn’t speak English, you might find useful some of the Chinese words and basic Mandarin phrases I will give you in this article to get by anywhere in China.
One of the most important topics and one of the first to be studied in every language course is the one that enables new learners to ask for directions (and understand the answer).
At the beginning of my Mandarin course in Shanghai, I was very excited to learn how to ask for directions, as this is one of the best reasons to start talking to people. The only problem, I couldn’t manage to understand their answer. This is why here I will write both questions and possible answers with the most common words used in this case. Mandarin pronunciation can be very tricky, especially due to the many ethnic groups living in Shanghai, each of them bearer of a particular accent and way of speaking, but if you know roughly what word they might be saying, you have ore chance to get where you want!
Asking for directions in Mandarin Chinese
The first thing you can say using Chinese words when approaching someone and to capture their attention is “Qing wen!” (请问!), which corresponds to the English “Excuse me, can I ask you a question?”
There are many ways to ask for direction using Chinese words, whether you will need to know where the place you want to go to is, or how to get there.
As an example, you might want to go to Hengshan Road, very popular street downtown Shanghai with a wide selection of great foreign restaurants, and ask about the way: “Qing wen, qù Hengshan lu zenme zou?” (请问,去衡山路怎么走?).
Hengshan Lu, Shanghai
In China, Facebook and other websites are filtered. This is why it’s good to have a VPN service to be able to access all websites by connecting from foreign servers. I useExpressVPN because it’s the most reliable among the ones I have tried, paid and free. ExpressVPN gives you a fast connection, a big choice of servers and you can install it on more than one device. You can do a monthly or annual subscription, obviously the annual is cheaper: the last time I bought it was one month at the cost of $12.95, while the annual package costs $8.32 per month.
Click here for more information on different packages and latest prices.
Other places of interest in Shanghai where you might need to go (and related Chinese words and phrases to get there) are:
– People’s Square, or Ren Min Guang Chang, where you can visit Shanghai Museum: “Qing wen, qù Ren Min Guang Chang zenme zou?” (请问,去人民广场怎么走?)
– Nanjing Road East, or Nanjing Dong Lu, the Mecca of shopping malls: “Qing wen, qù Nanjing Dong Lu zenme zou?” (请问,去南京东路怎么走?)
– the French Concession, or Shanghai Fa Zujie, a former foreign concession in Shanghai from 1849 to 1946: “Qing wen, qù Shanghai Fa Zujie zenme zou?” (请问,去上海法租界怎么走?)
– Xintiandi, trendy area with two pedestrian blocks that host many foreign brands of restaurants and cafés, maybe a little awkward since it has very little to do with traditional Chinese style, but by all means very popular among foreigners: “Qing wen, qù Xintiandi zenme zou?” (请问,去新天地怎么走?)
– Yu Yuan garden, or Yu Yuan, in the Old City, reproducing the traditional Chinese gardening style, with abundance of stones and water: “Qing wen, qù Yu Yuan zenme zou?” (请问,去豫园怎么走?)
– “Where is the closest metro station?”: “Zui jin de ditie zhan zai nali?” (最近的地铁站在哪里?)
Yu Yuan garden in the Old City, Shanghai
On another note, if you are in a restaurant or café or museum and you need the toilet, you will ask for the “wèi sheng jian” (卫生间): Qing wen, wèi sheng jian zai nali?(请问,卫生间在哪里? or “zai nar” 在哪儿 if you are in Beijing or in the North in general).
The main answers locals can give you in Chinese language are:
– Go straight: “Yizhi zou” (一 直走)
– Turn left: “zuo guai” (左拐) or “da guai” (大拐) if you are in Shanghai. “Da” means “big”, and they use it because since they drive on the right side, the left turn is usually wider.
– Turn right: “you guai” (右拐) or “xiao guai” (小拐), small turn.
– Take a taxi: “Da che” (打车)
– In front: “Qian mian” (前面)
– Crossroad: “Lu kou” (路口)
An example using Chinese words and phrases: Go straight, in front of the crossroad turn left: “Yizhi zou, qianmian de lu kou zuo guai” (一 直走, 前面的路口左拐).
If you take a taxi, the driver will ask you where you want to go (Qu nali?去哪里?) – and your reply will start with: I want to go… “Wo yao qu…” (我要去…) plus any destination. Once at the place, either the driver will ask you where to stop: “Ting zai nali?” (停在哪里?) and you might answer “here” “Zheli” (这里), or you will tell the driver to stop in the first place: “Ting yi ting” (停一停). Then, you ask for the price “Duo shao qian?” (多少钱?), and if you want the receipt: “Gei wo fapiao” (给我发票).
Of course, there are countless situations where you need to ask for directions and use Chinese words. It’s impossible to list them all but I tried to reproduce the ones where I used the most basic Chinese phrases. If you feel there are other important expressions I have overlooked, please leave them in a comment and I will either reply through the comment form or update the post.
Useful tools to master your Chinese
These are some of the tools I used when I was living in China and studying Chinese in Shanghai.
Study Mandarin. Better arriving with a bit of grasp of Chinese to avoid awkward moments and also to make things much easier and locals more prone to help you. Mandarin Blueprint promises to teach you how to read and understand basic Mandarin in 2 months, how to read (graded) books and stories in under 6 months and to have complete conversations in under a year. Pretty ambitious but possible. They offer a 30-day trial so it’s worth trying. Check it out now!
Oxford Beginner’s Chinese Dictionary. Very complete and useful once you learn to read and to look up in the dictionary itself!
Easy Chinese Glossary. If you are traveling to China and need a bit of quick help to get by in Mandarin Chinese, you will find a small vocabulary very helpful and easy to carry around. This is a guide divided into tables that show Chinese characters, the pinyin (transliteration of Chinese characters into the Latin alphabet) and the English translation.
Mandarin Chinese Flash Cards. If you are on a business trip or are planning to stay in China a little longer, improve your Mandarin Chinese with these easy-to-use Flash Cards to learn the most frequently used Chinese characters quickly and easily.
Other situations you might need to speak Chinese:
Traveling around China
Renting a house
Dealing with banks and money
Going to the restaurant
Getting sick on the road
Share
Post
Pin
Copy
Share
Share
Mail
Share
About the author
I'm Angela Corrias, an Italian journalist, photographer, and travel writer located in the heart of Italy's capital. Welcome to my website, your comprehensive source for your travels and expert guidance for crafting your dream travel experience.
Essential Tips and Phrases in Mandarin to Use if You Get Sick in China!
Top Essential Mandarin Words and Phrases to Use When Traveling in China
20 thoughts on “Essential Chinese words and basic expressions to use when asking for directions”
Marie May 31, 2011 at 2:20 pm C’est vachement interessant! J’en tiendrai compte la prochaine fois en Cine!
Reply
Angela Corrias May 31, 2011 at 8:28 pm Oui, commence à étudier maintenant :P
Reply
Domenico May 31, 2011 at 2:21 pm Brava Angela, sto prendendo nota!!!
Reply
Angela Corrias May 31, 2011 at 8:28 pm Xiexie! :D
Reply
Sasha May 31, 2011 at 2:11 pm This is great!!! Far more helpful then all my Mandarin textbooks!!! Keep these posts coming!!! I’m gonna practice these this week. :)
Reply
Angela Corrias May 31, 2011 at 2:15 pm Thanks! I have other topics scheduled, but if you have any idea that you don’t see here, let me know and I’ll write a post on that :)
Reply
jill- Jack and Jill Travel The World May 31, 2011 at 8:15 pm very useful, indeed. Can’t wait to get to China but the language intimidates me to no end (and I”m chinese by ethnicity, you’d think I have the genes for it).
Reply
Angela Corrias May 31, 2011 at 8:28 pm Of course Chinese is intimidating, is one of the hardest languages EVER! Don’t be put off though, when you manage to have a 5-minute conversation with locals after one year of full time course, it’s very rewarding! :P
Reply
Cathy Sweeney June 1, 2011 at 12:44 am Another great lesson. So true about understanding the responses when you ask a question. That’s hardest for me. Thanks for the Mandarin lessons – hope I get to use it someday.
Reply
Angela Corrias June 1, 2011 at 2:40 pm Come to China Cathy, it’s very exciting :) And so you can tell me if my tips make sense :P
Reply
Don Faust June 2, 2011 at 3:53 am We’d love to go to China for a number of reasons, but also to sample all the food! I work with a number of Chinese immigrants to the US, and learning the various words or phrases for me is easier than seeing the English written from and trying to say it. If I can hear someone say it a few times for me, I can normally mimic it – remembering is a whole other issue.
Reply
Angela Corrias June 2, 2011 at 9:46 am It’s difficult to remember the words because they have a very particular pronunciation. And also because to a foreigner they sound all the same! There are four tones in Mandarin, if you get a tone wrong you might be saying a different word, and of course it’s very easy to forget which is the tone for the word you want to say…
Reply
Dina June 2, 2011 at 10:58 am I have to admit, I don’t even dare to pronounced those words now! I’m Chinese descent that can’t speak Chinese. I learned how to write simple stuff though. I have some Japanese background, it helps.
Reply
Angela Corrias June 2, 2011 at 12:50 pm Ah! If you are Chinese descent I believe for you it will be easier, especially if your parents speak Mandarin, take advantage and practice with them!
Reply
Erica June 4, 2011 at 11:08 pm Helpful expressions! As a vegan, I’d love to learn how to say “Do you serve food without meat, fish and diary?” or “vegan food”.
Any idea how to say this?
Reply
Angela June 5, 2011 at 7:40 am Hi Erica, have a look at the first part of this series, where I wrote about ordering food in restaurants. I also wrote tips for vegetarians, as in China is not very easy being one… Let me know if it’s not complete, here is the link:
Reply
Leave a Comment Cancel reply
Comment
Name Email Website
Δ
This site uses Akismet to reduce spam. Learn how your comment data is processed.
About Me
Hi, my name is Angela Corrias, a Journalist, photographer, and blogger, I have a soft spot for the lesser visited corners of the world. After almost ten years of expat and semi-nomadic life around Brazil, Europe, China, and Iran among others, I came back to my homeland, Italy, to settle in Rome.
Chasing The Unexpected
Discover the most up-to-date and trustworthy travel tips and information with us. At our Chasing The Unexpected, we offer a wide range of travel guides and tips. From Italy travel tips and Iran travel guides to Afghanistan travel advice, we aim to provide comprehensive resources for travelers.
Home
Disclosure
Work with us
Destinations
Travel Tips
About
Blog
Contact
Privacy Policy
Seen on
© 2025 Chasing The Unexpected.
Chasing The Unexpected participates in several affiliate programs. This means that for any purchase you make through one of my links, I receive a small percentage – which however does not affect the final price for the customer. As an Amazon Associate, I earn from qualifying purchases. As a Booking Associate, I earn when you book through certain links. Chasing The Unexpected is by no means liable for any purchase of tours, tickets, and any other online reservations that are made without having fully read the terms and conditions of any third-party provider. | Angela Corrias P. Iva 12343591009
Close
Travel Destinations
Travel Guides
Travel Tips
Travel Resources
Travel Essentials
Blog
Search for:
Post
Share
Pin
Copy
Share
Mail
Share
Pin |
11751 | https://med.libretexts.org/Bookshelves/Anatomy_and_Physiology/Anatomy_and_Physiology_(Boundless)/27%3A_Human_Development_and_Pregnancy/27.4%3A_Third_Week_of_Development/27.4C%3A_Somite_Development | Published Time: 2018-07-24T22:50:19Z
27.4C: Somite Development - Medicine LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
27.4: Third Week of Development
27: Human Development and Pregnancy
{ }
{ "27.4A:_Gastrulation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.4B:_Neurulation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.4C:_Somite_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.4D:_Development_of_the_Cardiovascular_System" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.4E:_Chorionic_Villi_and_Placental_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "27.10:Changes_to_the_Mother\'s_Body_During_Pregnancy" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.1:_Overview_of_Pregnancy_and_Human_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.2:_First_Week_of_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.3:_Second_Week_of_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.4:_Third_Week_of_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.5:_Fourth_Week_of_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.6:_Fifth_Through_Eighth_Weeks_of_Development" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.7:_Birth(Parturition)" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.8:_Lactation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "27.9:_The_Fetal_Period" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Sat, 05 Oct 2024 22:51:52 GMT
27.4C: Somite Development
8279
8279
Delmar Larsen
{ }
Anonymous
Anonymous User
2
false
false
[ "article:topic", "license:ccbysa", "showtoc:no" ]
[ "article:topic", "license:ccbysa", "showtoc:no" ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Contents
1. Home
2. Bookshelves
3. Anatomy and Physiology
4. Anatomy and Physiology (Boundless)
5. 27: Human Development and Pregnancy
6. 27.4: Third Week of Development
7. 27.4C: Somite Development
Expand/collapse global location
Anatomy and Physiology (Boundless)
Front Matter
1: Introduction to Anatomy and Physiology
2: The Chemical Building Blocks of Life
3: Organization at the Cellular Level
4: Organization at the Tissue Level
5: Integumentary System
6: Skeletal System
7: Skeletal System - Parts of the Skeleton
8: Joints
9: Muscular System
10: Overview of the Nervous System
11: Central Nervous System
12: Peripheral Nervous System
13: Special Senses
14: Autonomic Nervous System
15: Endocrine System
16: Cardiovascular System - Blood
17: Cardiovascular System: The Heart
18: Cardiovascular System: Blood Vessels
19: Lymphatic System
20: Immune System
21: Respiratory System
22: Digestive System
23: Nutrition and Metabolism
24: Urinary System
25: Body Fluids and Acid-Base Balance
26: The Reproductive System
27: Human Development and Pregnancy
28: Reproduction, Chromosomes, and Meiosis
30: APPENDIX B: Development and Aging of the Organ Systems
29: APPENDIX A: Diseases, Injuries, and Disorders of the Organ Systems
Back Matter
27.4C: Somite Development
Last updated Oct 5, 2024
Save as PDF
27.4B: Neurulation
27.4D: Development of the Cardiovascular System
Page ID 8279
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Learning Objectives
2. Key Points
3. Key Terms
4. Intraembryonic Coelom Development
5. Somite Development
6. Dermatome
7. Myotome
8. Sclerotome
9. Syndetome
Somites develop from the paraxial mesoderm and participate in the facilitation of multiple developmental processes.
Learning Objectives
Describe the functions of somites
Key Points
The paraxial mesoderm is distinct from the mesoderm found more internally in the embryo.
Alongside the neural tube, the mesoderm develops distinct paired structures called somites that develop into dermis, skeletal muscle, and vertebrae.
Each somite has four compartments: the sclerotome, myotome, dermatome, and the syndetome. Each becomes a specific tissue during development.
Key Terms
neural crest cells: A transient, multipotent, migratory cell population that gives rise to a diverse cell lineage including melanocytes, craniofacial cartilage, bone, smooth muscle, peripheral and enteric neurons, and glia.
conceptus: The fetus or embryo, including all the surrounding tissues protecting and nourishing it during pregnancy.
somite: One of the paired masses of mesoderm, distributed along the sides of the neural tube, that will eventually become dermis, skeletal muscle, or vertebrae.
Intraembryonic Coelom Development
In the development of the human embryo the intraembryonic coelom (or somatic coelom) is a portion of the conceptus that forms in the mesoderm. During the second week of development the lateral mesoderm splits into a dorsal somatic mesoderm (somatopleure) and a ventral splanchnic mesoderm (splanchnopleure).
By the third week of development, this process gives rise to a cavity between the somatopleure and splanchnopleure referred to as the intraembryonic celom. This space later gives rise to both the thoracic and abdominal cavities.
Somite Development
In the developing vertebrate embryo, somites are masses of mesoderm that can be found distributed along the two sides of the neural tube. They will eventually become dermis (dermatome), skeletal muscle (myotome), vertebrae (sclerotome), and tendons and cartilage (syndetome).
The mesoderm found lateral to the neural tube is called the paraxial mesoderm. It is separate from the chordamesoderm underneath the neural tube. The paraxial mesoderm is initially called the unsegmented
mesoderm in vertebrates, but is called the segmented mesoderm in chick embryos.
Somites: A dorsal view of a human embryo. The repetitive somites are marked with the older term primitive segments.
As the primitive streak regresses and the neural folds gather preceding the formation of the neural tube, the paraxial mesoderm divides into blocks called somites. Somites play a critical role in early development by participating in the specification of the migration paths of neural crest cells and spinal nerve axons.
Later in development, somites separate into four compartments:
Dermatome
The dermatome is the dorsal portion of the paraxial mesoderm somite. In the human embryo it arises in the third week of embryogenesis.
The dermatomes contribute to the skin, fat, and connective tissue of the neck and of the trunk, though most of the skin is derived from the lateral plate mesoderm.
Myotome
The myotome is that part of a somite that forms the muscles. Each myotome divides into an epaxial part (epimere), at the back, and a hypaxial part (hypomere) at the front.
The myoblasts from the hypaxial division form the muscles of the thoracic and anterior abdominal walls. The epaxial muscle mass loses its segmental character to form the extensor muscles of the neck and trunk of mammals.
Sclerotome
The sclerotome forms the vertebrae and the rib cartilage and part of the occipital bone. It forms the musculature of the back, the ribs, and the limbs.
Syndetome
The syndetome forms the tendons and some blood vessels.
27.4C: Somite Development is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts.
Back to top
27.4B: Neurulation
27.4D: Development of the Cardiovascular System
Was this article helpful?
Yes
No
Recommended articles
27.4A: Gastrulation
27.4B: Neurulation
27.4D: Development of the Cardiovascular System
27.4E: Chorionic Villi and Placental Development
Article typeSection or PageLicenseCC BY-SAShow TOCno
Tags
This page has no tags.
© Copyright 2025 Medicine LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status×
contents readability resources tools
☰
27.4B: Neurulation
27.4D: Development of the Cardiovascular System |
11752 | https://soundamerican.org/issues/thirtieth/nico-muhly | Nico Muhly | Sound American
Skip to main content
Sound American
Breadcrumb
The Thirtieth Issue
Nico Muhly
Interview
Nico Muhly
I saw Nico Muhly only once, in 2018. I was coming out of the subway near Lincoln Center, and as I reached the top step, I saw a tall flash of all black sweep toward my left. It was like being a witness to pure energy. At the time, the Metropolitan Opera was performing Muhly’s opera based on Alfred Hitchcock’s film, Marnie, so his presence made sense. But something wonderful happened in that moment. Catching a glimpse of the composer as he rushed down Broadway renewed a small part of my romance with New York City.
Growing up, I imagined the city as a place where you traveled amidst creativity made flesh and bone: John Zorn zipping through the Lower East Side; Phil Glass towering in mid-town; Cecil Taylor drinking at the 55 Bar; Ornette Coleman holding court on Prince Street. Since moving here in 2001, I have—in fits and starts—experienced that kind of exhilaration of proximity. But it has always been tinged with a sort of nostalgia—sighting as a re-creation of my teenage imagination.
That’s what made the glimpse of Muhly so special. He—as someone of my generation engaged in affecting a world larger than my insular musical circle—made the creative energy of New York contemporary. He was hurrying to continue a tradition and push it further, and he had no time to spare. It would be hyperbole to say that Manhattan seemed to stand still in that moment if I hadn’t felt it so vividly.
Muhly does not need me to make him known to the world; he is that rare composer of interesting music that you can bring up in small talk at a dinner party. But he still immediately agreed to talk to me and made the time to be fully present through the entire process of our discussion. I would email him questions and receive his ecstatic, rapid-fire, and hilarious answers via the memo recorder on his phone, the perfect discursive analog to that flash I experienced on 66th Street. I was most interested to find out about how he tapped into the energy necessary to negotiate his incredibly packed creative life.
NW It seems like you simultaneously work in three moments: dealing with the promotion of something just released, premiering a new piece, and preparing upcoming compositions. What are you finishing, and what’s next on the horizon?
NM I’ve just come out of a very elaborate few months of conducting and performing and travel. I had a couple of weeks in Japan and then a few things in the city. I was in Doha for a month at the same time as the World Cup doing a series of dance pieces in collaboration with Benjamin Millepied, the choreographer, who’s an old, old friend. December was relatively calm and just a lot of writing. After that I was in London for about a week just dealing with various things, and then I was in Freiburg, where an older opera of mine, Marnie, was being staged again.
Now I’m home for two weeks before I’m an artist-in-residence in Hobart, Tasmania as part of a festival called Mona Foma. And that’s all old pieces, and one little, tiny new thing for choir. All the while I am orchestrating Monteverdi’s L’Orfeo for the Santa Fe Opera, which is happening this summer. As we speak, I am in the middle of Act Four, so we’re about to get to Pluto’s big aria. I’ve just finished a set of three piano miniatures for Yuja Wang, and she will be performing those in San Francisco; I’ll go join her there in early March. This summer is a bunch of teaching: I’m composer-
in-residence at Sidney Sussex College, Cambridge in the UK, and I’m teaching at Tanglewood at the Boston University Tanglewood Institute and then two weeks in Aspen. In the middle of that is L’Orfeo, and somewhere in there I need to finish this children’s ballet for the National Ballet of Hong Kong and their artistic director Septime Webre.
So, it kind of goes on and on like this. And it’s a combination of performing and writing and this and that. There’s an awful lot going on. I travel with my desktop computer and an eighty-eight-key keyboard, and I try on the road to be as diligent as possible about writing at least every day. So, for example, I fly to Singapore next Thursday. Friday is a wash, obviously, because it’s an eighteen-hour flight and that day just vanishes (although one gets it back on the return trip). I get into Singapore early in the morning on Saturday, and the first thing I will do is set up my stuff and write for at least an hour before anything else just to get myself kind of ready to go.
NW From the outsider’s perspective, you arrived on the scene with your compositional aesthetic fully formed. What was your early musical life like?
NM I would argue that my place in the scene—and I use that term warily—happened after a lot of time in conservatory and as an undergraduate. I also had a bunch of different day jobs, but most importantly, working as a kind of assistant/editor/engraver for Philip Glass. Growing up, my parents had music in the home all the time. Eclectic tastes, too; they listened to everything from Joni Mitchell to [Henry] Purcell. Neither of them is a musician, but they both had enormous exposure to classical music, and particularly contemporary music, because my mother was a fellow at the American Academy in Rome and remained very involved in that institution. So, I grew up kind of knowing what composers were. Her older brother, in fact, was a composer. That having been said, my music sounds very different from that world, which, at least when I was a kid, was more often than not linked to academia—not necessarily in a bad way, but it was just a different form of music making.
The music that grabbed me early was that of Steve Reich and Philip Glass and John Adams. When I was about nine—after playing the piano for a year or so—I joined an Anglican boys choir in Providence, Rhode Island, and that was the major, kind of seismic, event for me, both musically and personally. [Thomas] Tallis and [William] Byrd and [Orlando] Gibbons. I mean, that music is unbelievably great, right?
In hindsight, what I was most drawn to was that music’s lack of romanticism. If you’re a pianist around that age—or a teenage pianist—you’re playing Chopin and aspiring to Liszt and all this music with a constant sort of ocean-flow of emotions, with climaxes and this performative kind of shape. With Renaissance choral music, it’s the opposite of that. It’s, of course, meant to be beautiful, but it’s never giving too much away. I’m specifically thinking of the Tudor tradition, where these moments of emotional tumescence happen rather than one vertical climactic moment. Or, when you do get one of those moments, the force of the drama is so much more overwhelming than in Bruckner or Wagner; thinking about the ending of Byrd’s Infelix Ego or those gnarly Purcell cadences which give way to a radiant major chord. So, I think a combination of listening to a bunch of that kind of early American minimalism or post-minimalism was combined with the ethos of choral music. No one claps. No one is checking the credits to see who’s singing. It’s all part of a larger ritual, the liturgy of the day, but also of the liturgical year. So, all those things, I think, inform my practice now.
NW You talk about those early influences with such distinction and love. It makes me think of how my own early listening is still present in the way I think about music. Do you have any music that you’ve only become aware of recently that is affecting you in a similar way?
NM I had a weird thing during COVID where I could only deal with things I already knew; I couldn’t read new books, even. I just completely checked out. And in terms of music, I’m actually trying to have a pretty closed system right now, because I’m writing so many different things. But I’ve been listening a lot to [Francois] Couperin and [Jean-Philippe] Rameau, specifically Alexandre Tharaud’s recordings of the same, because I’m going to write for him. I’ve been listening to basically new recordings of old things. I wish I could say that I’ve been doing a lot more adventurous listening, but it’s mainly things from friends. So I just listened to the Totally Enormous Extinct Dinosaurs record. There’s a new recording of Steve Reich’s Cello Counterpoint I’ve been getting into. I don’t know. I regret to inform you that I’m not very interesting in that way.
NW Listening to new music is one of those things that falls by the wayside when I get busy. Lots of work means time and energy has to be borrowed from somewhere else. With the demands of your schedule, do you ever feel like you’re missing something?
NM Yes, of course, but I think I’m kind of in denial about that. I mean, one of the things that I missed most during COVID were the serendipitous encounters that you have on the road. Like, I’m in Köln, but amazingly, my friend’s band is in town, and it’s great to meet up with them by accident. I can go to their sound check just to say hi, and they can come to my dress rehearsal. I hadn’t realized how meaningful those things were to me until they were gone during the early two years of COVID.
I miss what used to be, when I was traveling, the tacit assumption that you would just run into people at the bar or at dinner or at the opera or whatever. Of course, that still happens when I’m home, but there’s more of an effort, and it’s more concentrated into two weeks at a time. There are a lot of things I feel like I miss outside of the intensities of COVID restrictions, but, again, I wouldn’t trade my life for the world. I had a crazy moment just the other day when I was flying back from Germany and changing planes at Charles de Gaulle in Paris. I ran into my friend Victoria James, a brilliant sommelier from New York and sat with her and her colleagues at the bar of the Yo! Sushi in terminal 2. That’s the kind of chance encounter that became completely impossible during COVID, but which is the sort of thing that gives one a sense of interconnectivity and bliss.
NW Have you had any moments of doubt in your musical life?
NM The answer, of course, is yes. “Am I doing too much right now?” I worry about that every day. I haven’t had a day off in months—like a real day off.
NW What’s a real day off?
NM For me, it’s just being able to write without any other thing to do, and that’s so rare. I find myself almost petulantly jealous of my friends who manage, somehow, to take genuine time off, particularly my English friends who seem to be perpetually off somewhere sunny, most decidedly not working, posting pictures of their eerily prehensile toes against a backdrop of pristine Spanish beachfront, or, in the case of the gays, that of Mykonos. Oftentimes for me, a day off will be a couple of hours of writing followed by not having to write a recommendation or deal with some schedule nonsense or vaguely combative admin. Although, sometimes I actually really enjoy booking plane tickets. It’s fun because it’s like sports, particularly when you get really frothy with miles and points and vouchers.
I did a lot of high-profile work very early in my life, which felt fun. But it’s also a little crazy. I always tell students the only way to get better is by messing up, by having your piece sound great in your computer, and then hearing it with real instruments and an audience and realizing it doesn’t work at all. It also helps you grow, I think, to get a terrible review or harsh criticism. Because I sort of bloomed early, (again, none of this made me particularly happy because I was so worried about other things, to say the least) success itself became a source of anxiety. It’s as if I skipped a couple of steps in how you’re “meant to do it,” and as a result, it’s tricky to situate where I am now at forty-one years old. I’m not worried about how things are received by anyone other than my friends and my collaborators, but it isn’t not a worry, if that makes sense.
NW Is that worry based solely on physical and mental health, or is there a musical or career component that comes up?
NM I mean the worry about doing too much is just that; you lose focus and can’t do quality control. I’ve learned that doing one thing at a time can be, for me, kind of mentally oppressive, and that it’s better to have a couple of different things on at once. I can always turn something off and then turn it back on. For example, I’m writing the ballet right now, which I took a day off from yesterday. But I’ll take half the day today to work on it. But then this afternoon will be L’Orfeo. On this upcoming trip, I want to be mainly focused on L’Orfeo for a week, and then send it off for a rough zhoosh from my publisher, during which point I’ll turn back to early mornings on the ballet and then performing, teaching, and so on. It just keeps me focused to have things bouncing off each other.
I was diagnosed as being Bipolar II a bunch of years ago, and one of the things that was tricky about that is that I had already been so “successful” by many metrics—as viewed from the outside—but, in a lot of ways, I was so out of it. I knew something was really off when a friend of my parents came to an opera I had at the Met, and she said, “You must have felt ten feet tall when you came out to take a bow.” And I thought: “Wow. No. I feel about as accomplished as if I’d done all my errands for the day and tidied up the studio.” All that imagined joy, all the adrenaline, was completely absent. That isn’t to say that you should write to get applause, or to have really high-profile work, but realizing that I had so few genuine emotions and living in a sea of grey, or occasionally the sparkler-bright, but frighteningly dangerous, beauty of a hypomanic episode.
There were a lot of pieces that I wrote during that time that I don’t remember writing, because I was in either a state of mania or a state of worrying about all that, about my mental health. And interestingly—I don’t want to say upsettingly—that work is actually really good. A lot of the pieces of mine that are played come from that period. And it was a very fecund period, both on the piece-to-piece level but also in a more macro-level of finding a harmonic language. There’s that. But I refuse to glamorize mental illness; I refuse to do it, and I can’t stress this enough.
I think part of my discipline pre-existed before my current equilibrium, which is the result of a big medication reset in 2015. But before, even when things were crazy, I had a good routine. So, getting up early, taking my meds, going to therapy, exercising often, all those things come from an awareness that this kind of routine—which you can bring with you anywhere—is really important. I’d like to think that even when I was the most checked out, or dangerously keyed-up, I held onto that core set of rituals.
NW There are a lot of misunderstandings about neurodivergent ways of being, and often they’re seen as a path to creativity.
NM I’ve had strangers say, “I read your essay about [being diagnosed Bipolar II], but part of it must be a gift to be so (now we would say) neurodivergent.” I find that offensive. I would really love to not be afflicted this way. I always jokingly say that if that’s a gift, I wish I’d kept the receipt. There was a long period of being on the wrong medication and the usual things that accompany these forms of illnesses—I don’t know if we still call them illnesses—which are tricky to navigate, particularly as an artist. There was a version of medication which definitely dampened the mania but left very little spark. I say this cautiously, because, again, I have no interest in glamorizing this or making it seem like people should not take medication if they need it; there are enough out there for BPD [Bipolar Disorders] and many other things where you should be able to find one for you that strikes a balance between overall health and your ability to create.
My state of mind caused me to have a lot of sort of difficult interpersonal relationships, particularly as it related to people in my professional life who weren’t as intense as I was, who weren’t working nineteen hours a day, and who were able to find time to take vacation. I was, of course, foully jealous and furious, and that was not a great way to be. However, I’ve been on a really stable set of medications for a long time, and I’ve yet to experience the same sense of insecurity that I had about it maybe seven years ago.
NW So, a little thought experiment: It’s thirty years from now, and everything has worked out just as you desired. Where do you see yourself in this best-of-all-possible-worlds? How and what are you composing? How are you living? How would you hope people see you?
NM I hope I would be doing exactly the same thing, but with one more day off every sixteen days and a slightly bigger apartment. I don’t know; I’m pretty happy with how everything’s looking structurally. I think I’m not taking too much work that I don’t want to take because of money. I’m really proud that I can live in New York and have a perfect, fabulous studio which I share with dear friends, and I pay for it by writing and playing music. And whether or not that’s film music, or ballet or whatever, I think maintaining that sense of constant work is important to me. What am I composing? I don’t know. How would I hope people see me? I can’t control that!
It’s interesting: a lot of your questions are about perception. And that’s just something I had to give up when I was in my late twenties. I push back on my friends, for example, when they describe someone’s music as underrated or overrated. First of all “overrated” is how Trump thinks and talks. Also, thinking about how something is perceived gives that perception dominance over the work itself, which I refuse to do to my own work, and the work of others. It’s a kind of unseen perception; it’s a fun house mirror. Or I’d like to say human centipede, but that’s a little bit nastier.
I always recoil slightly when I see people thanking reviewers for a good review, because you think: “What do you say if it’s bad? Or mixed?” I hate the idea that how something is “received” or perceived should have any bearing on what the actual-thing-itself is. When I have friends who have a big piece or small piece or any kind of piece, instead of reading a review and then congratulating them on a good one or ignoring it if it’s bad, I would prefer to call them up and say, “I would love to see a score, and I would love to hear a recording.” Then you can think about it in a way that has nothing to do with how it was received, which I think is much more honest.
NW Where do you think the music world would be headed if we expand that thought experiment?
NM I think things have changed so much in the last five years that it’s unimaginable to think what will happen in thirty, and I can’t begin to foresee it. I think the trajectory that we’re on now is an interesting one. I mean, there’s only good that can come out of having very difficult conversations in classical music about who has access to the spaces; and who has access to the mechanism of commissioning; and what those spaces and mechanisms are on the most basic level. I think we’re having all of those conversations as a result of years of ignoring really crucial discussions about race, about money, and about music’s value in society. And I should say that, as someone who works equally in the States and in the UK, the recent arts funding cuts in the UK have really exposed how the sausage is made in a way that is making people pay attention.
Something interesting about COVID is that it made us all look at how we’re supporting ourselves and each other—both financially and emotionally—as a community, and what the connection is between the big organizations and individuals. I think it’s necessary to be kind of uncomfortable, to make one another think about difficult topics and to sit in that discomfort for as long as it takes. I hope that in thirty years, our world will feel at least a little bit more equitable, and that what work there is to do will feel more joyful.
NW Do you see any examples of an institution really getting it right?
NM I worry about the changes that are happening now in all sorts of different ways. Are they structural or are they cosmetic? Some organizations are involved in real positive upheaval: “Let’s remake this thing; let’s reconsider this thing.” And then there are others where it’s as if nothing happened. Honestly, with these questions about equity—both race and class and all their intersections—I am very glad that it is not my job to deal with this on an institutional level. Obviously, I can do it on an interpersonal level and on the level as a friend and a colleague and on the level as someone who teaches younger people, either formally or as an informal mentor. I’m sure that there are organizations that are killing it, but I can’t single any out. One thing that often goes unspoken in these conversations is the built-in differences between how far in advance large institutions need to plan versus a small ensemble, for instance, who can pivot a lot faster than the giant cruise ships of big opera houses and orchestras. I think large and small institutions can put good pressure on one another to make different kinds of changes.
I’m not sure it will ever be resolved. Look at the criticism of the mode of protests for equality. At first, it was like, “Why can’t you just be peaceful?” But when people take a knee, it’s like, “We didn’t mean like that.” The perception of it is always wrong. And I think institutions are going to have to realize that you’re never going to nail it, right? It’s never going to be like, “Okay, cool. Done. All fixed.” It’s a continual process that has to happen every day, every year, every season. It’s sort of an asymptotic thing that we need to think about all the time to keep undoing prejudices, and to find all the individual ways in which we can all do better.
Nico Muhly
Facebook
Twitter
Email
Where to find SA
Past Issues
Contact
Imprint
Subscribe to the Sound American Newsletter
Email Address
Follow Sound American
Instagram
Twitter
Bandcamp
℗ & © 2011–2022 Anthology of Recorded Music, Inc.
Site by Hemidemisemi
Print edition by Remake Design
Privacy |
11753 | https://www.pw.live/school-prep/exams/isotopes-isobars-and-isotones | Understanding Isotopes, Isobars, and Isotones
Isotopes, isobars, and isotones are three fundamental concepts that shed light on how atoms differ and how they influence the behaviour of matter in the world of science and chemistry.
Ranvijay Singh28 Oct, 2023
Isotopes, Isobars, and Isotones
are fundamental concepts in nuclear physics and chemistry. They help us understand the diversity of atomic structures and the various ways in which atoms can differ from one another. Let's delve into these terms to gain a better grasp of their definitions and examples.
Define Isotopes, Isobars, and Isotones
Isotopes -
Atoms of the same element with the same number of protons but different numbers of neutrons are called isotopes. Depending on the number of neutrons in the atom, atomic masses can differ. For example, carbon has two stable isotopes, carbon-12 and carbon-13, each with 6 and 7 neutrons.
Isobars -
Isobars are atoms of different elements with the same mass number, meaning they have the same number of protons and neutrons. They may have different atomic numbers, indicating they are distinct elements. Examples are calcium-40 (20 protons and 20 neutrons) and argon-40 (18 protons and 22 neutrons).
Isotones -
Isotones are atoms with the same number of neutrons but different atomic numbers and, consequently, different chemical elements. An example is carbon-14 and nitrogen-14, both having 7 neutrons but different numbers of protons, 6 for carbon and 7 for nitrogen.
What are Isotopes?
Isotopes are different versions of an element, distinguished by the number of neutrons they contain. For instance, consider Uranium isotopes 23592 U and 23992 U. While they have the same number of protons, their neutron counts differ, with 143 for the former and 147 for the latter. This additional neutron dramatically alters the properties of the atom. Isotopes can be classified as stable or radioactive. Stable isotopes remain intact without any spontaneous disintegration. At the same time, radioactive isotopes are too unstable to sustain themselves and eventually break down into lighter elements through the emission of particles like alpha, beta, and gamma rays.
What are Isobars?
A series of elements with 40 mass numbers serves as a good example of an isobar;
4016
S,
4017
Cl,
4018
Ar,
4019
K, and
4020
Ca.
are all elements with the same number of nucleons (protons and neutrons). There are the same number of particles in the nucleus of all of the above-mentioned elements, but their protons and neutrons vary in number.
Also Read –
Malic Acid Formula
What are Isotones?
It is an atom containing the same neutron number but a different proton number that is called an isotone. For example,
3616
S,
3717
Cl,
3818
Ar,
3919
K, and
4020
Ca
contain all 20 neutrons and are all isotones of 20. In order to tap the explosive power of the nucleus, nuclear energy and radioactivity utilize unstable isotopes of heavy elements.
Also Check –
Iron (III) Hydroxide Formula
Examples of Isotopes, Isobars, and Isotones
Isotopes Example
-: Let's consider hydrogen. It has three isotopes: protium (1H), deuterium (2H), and tritium (3H). All three have one proton, but they differ in the number of neutrons (0, 1, and 2, respectively).
Isobars Example
Take calcium-40 and potassium-40. Calcium-40 has 20 protons and 20 neutrons, while potassium-40 has 19 protons and 21 neutrons. Despite different atomic numbers, they have the same mass number, 40.
Isotones Example
- Boron-10 and carbon-14 can serve as examples. Boron-10 has 5 protons and 5 neutrons, while carbon-14 has 6 protons and 8 neutrons. These isotones share the same number of neutrons, which is 5.
Also Read:
Molar Volume Formula
Identifying Isotopes, Isobars, and Isotones
Identifying isotopes is often straightforward since they share the same atomic number (same element) but have different atomic masses due to varying neutron counts.
Identifying isobars can be done by comparing the mass number. If two elements have the same mass number but different atomic numbers, they are isobars.
Identifying isotones requires looking at the neutron count. If two elements share the same number of neutrons but have different atomic numbers, they are isotones.
Isotopes, Isobars, and Isotones FAQs
Q1. What are Isotope, Isotones?
Ans- Isotopes are atoms of the same element with the same number of protons but different numbers of neutrons, resulting in varying atomic masses. For example, carbon-12 and carbon-14 are isotopes of carbon.
Isotones are atoms of different elements with the same number of neutrons but different atomic numbers, meaning they have distinct chemical properties. An example is carbon-14 and nitrogen-14.
Q2. What are Isotopes and Isobars with an Example?
Ans. Isotopes are atoms of the same element with different atomic masses due to varying neutron counts. For example, carbon-12 and carbon-14 are isotopes of carbon, with 6 and 8 neutrons, respectively.
Isobars are atoms of different elements with the same total number of protons and neutrons. An example is calcium-40 with 20 protons and 20 neutrons and argon-40 with 18 protons and 22 neutrons.
Q3. What are 5 Examples of Isotones?
Ans. 1. Carbon-14 and nitrogen-14
Boron-10 and carbon-14
Nitrogen-15 and oxygen-15
Fluorine-19 and neon-19
Sodium-23 and magnesium-23.
Q4. What are Isotones?
Ans. Isotones are atoms of different elements that share the same number of neutrons but have distinct atomic numbers, leading to different chemical properties. For instance, carbon-14 and nitrogen-14 are isotones because they have 7 neutrons but differ in the number of protons, with carbon having 6 and nitrogen having 7.
🔥 Trending Blogs
PW Marathon Class 9 Half-Yearly Science Analysis: Endgame 2025
PW Marathon Class 9 Half-Yearly Science Analysis: Endgame 2025
CBSE Class 11 Biology Sample Paper, PDF Download
CBSE Class 11 Biology Sample Paper, PDF Download
CBSE Class 11 Maths Sample Paper, PDF Download
CBSE Class 11 Maths Sample Paper, PDF Download
CBSE Class 10 Science Half Yearly Important Questions 2025-26 PDF Download
CBSE Class 10 Science Half Yearly Important Questions 2025-26 PDF Download
Class 10 Half-Yearly Marathon Endgame 2025 Live Revision Classes
Class 10 Half-Yearly Marathon Endgame 2025 Live Revision Classes
Talk to a counsellorHave doubts? Our support team will be happy to assist you!
08448982616
008448982616
Check out these Related Articles
CLASS 10 EXAM
CBSE Subject Codes for Class 10 2026 Subject Wise Class 10 Code List
Read Full Story
11 Sept 2025, 02:16 pm
SCHOOL PREPARATION
Class 9 Half-Yearly Marathon: Endgame 2025, Live Revision Classes
Read Full Story
11 Sept 2025, 03:39 pm
CLASS 9 EXAM
CBSE Class 9 Maths Notes Chapter 10 Solutions PDF
Read Full Story
11 Sept 2025, 02:17 pm
SCHOOL PREPARATION
CBSE Board Exam 2026 Registration for Private Candidates Class 10, 12 Started
Read Full Story
9 Sept 2025, 06:37 pm
CLASS 11 EXAM
CBSE Sample Paper Class 11 Chemistry 2025-26 PDF Download Set Wise
Read Full Story
8 Sept 2025, 02:04 pm
CLASS 11 EXAM
CBSE 11 Class Physics Sample Paper 2025-26 PDF Download
Read Full Story
4 Sept 2025, 01:32 pm
CLASS 12 EXAM
CBSE Class 12 Maths Weightage 2026, Chapter Wise Marks Distribution
Read Full Story
3 Sept 2025, 05:48 pm
ICSETEN
ICSE Class 10 Date Sheet 2026 PDF Soon @cisce.org
Read Full Story
9 Sept 2025, 02:08 pm
ISC CLASS 12
ISC Class 12 Date Sheet 2026 Soon, Check Tentative Exam Dates
Read Full Story
9 Sept 2025, 02:19 pm
Join 15 Million students on the app today!
Live & recorded classes available at ease
Dashboard for progress tracking
Millions of practice questions at your fingertips
Free Learning Resources
PW Books
NEET Books
IIT JEE Books
NEET Books
PW Stationery
UPSC CSE Books
PW Merchandise
NDA/Defence Books
GATE Books
Device Courses
CUET Books
CA/Finance Books
UPPSC Books
RRB Books
BPSC Books
SSC Books
MPSC Books
MPPSC Books
Banking Books
Notes (Class 10-12)
Class 10 Math's Notes
Class 10 Chemistry Notes
Class 10 Physics Notes
Class 10 Biology Notes
Aptitude & Reasoning
Class 10 Geography
Physics Class 11 Notes
Class 11 Chemistry Notes
Maths Notes Class 11
Zoology Class 11
Class 11 Botany Notes
Physics Class 12 notes
Chemistry Class 12
Maths Notes Class 12
Zoology class 12
Class 12th Botany Notes
PW Study Materials
Arjuna JEE Study Material
JEE 7 PYQs
JEE Mind Maps
Yakeen Study Material
NCERT Punch
Arjuna Neet Study Material
NEET PYQs
CBSE Sample Paper
CBSE Question Bank
Udaan For Class 10th Study Material
UPSC Study Material
PW Stationery
PW Merchandise
PW Device Courses
Notes (Class 6-9)
Class-6 Theory & Notes
Math's Notes for class 7
Science Notes for class 7
Class 8 Math Notes
Class 8 Chemistry Notes
Class 8 Physics Notes
Class 8 Biology Notes
Class 8 SST Notes
Class 9 Math's Notes
Class 9 Physics Notes
Class 9 Chemistry Notes
Class 9 Biology Notes
Ncert Solutions
NCERT Solutions For Class 6
NCERT Solutions For class 7
NCERT Solutions for class 8
NCERT Solutions for class 9
NCERT Solutions for class 10
NCERT Solutions for class 11
NCERT Solutions for Class 12
Govt Exams
SSC Exam
Banking Exam
Teaching Exam
JAIIB CAIIB Exam
Railway Exam
Defence Exams
State PSC Exam
SSC Exam
Entrance Exams
Gate Exam
IIT JEE Exam
NEET Exam
CUET Exam
CSIR-NET Exam
MBA Exam
Nursing Exam
IIT JEE Exam
Our Other Websites
PW Store
PW Skills
PWOnlyIAS
MedEd
PW Gulf
PW IOI
CuriousJr
PW Skills Blogs
Class 6th to 12th Online Courses
Class 12 Online Course
Class 11 Online Course
Class 10 Online Course
Class 9 Online Course
Class 8 Online Course
Govt Job Exams Courses
SSC Online Coaching
Bank Exam Online Coaching
TET Online Coaching
JAIIB & CAIIB Online Coaching
Bihar Exam Online Coaching
UPSC Coaching
UPSC Online Coaching
UPSC Offline & Hybrid Coaching
State PSC Online Coaching
UPPSC Online Coaching
BPSC Online Coaching
Defence Exam Coaching
NDA Online Coaching
CDS Online Coaching
AFCAT Online Coaching
Agniveer Online Coaching
Gate Exam Coaching
Civil Engineering Courses
Mechanical Engineering Courses
CS & IT Courses
ECE Courses
EE Courses
Science formulas
Chemistry formulas
Physics formulas
Math's Formulas
Other Exams
CA Exam
Company Secretary Exam
Know about Physics Wallah
Physics Wallah is an Indian edtech platform that provides accessible & comprehensive learning experiences to students from Class 6th to postgraduate level. We also provide extensive NCERT solutions, sample paper, NEET, JEE Mains, BITSAT previous year papers & more such resources to students. Physics Wallah also caters to over 3.5 million registered students and over 78 lakh+ Youtube subscribers with 4.8 rating on its app.
We Stand Out because
We provide students with intensive courses with India’s qualified & experienced faculties & mentors. PW strives to make the learning experience comprehensive and accessible for students of all sections of society. We believe in empowering every single student who couldn't dream of a good career in engineering and medical field earlier.
Our Key Focus Areas
Physics Wallah's main focus is to make the learning experience as economical as possible for all students. With our affordable courses like Lakshya, Udaan and Arjuna and many others, we have been able to provide a platform for lakhs of aspirants. From providing Chemistry, Maths, Physics formula to giving e-books of eminent authors like RD Sharma, RS Aggarwal and Lakhmir Singh, PW focuses on every single student's need for preparation.
What Makes Us Different
Physics Wallah strives to develop a comprehensive pedagogical structure for students, where they get a state-of-the-art learning experience with study material and resources. Apart from catering students preparing for JEE Mains and NEET, PW also provides study material for each state board like Uttar Pradesh, Bihar, and others
Privacy Policy
Terms of use
Copyright © 2025 Physicswallah Limited All rights reserved. |
11754 | https://www.khanacademy.org/standards/CCSS.Math/HSG.C | Use of cookies
Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account.
For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session.
Functional Cookies
These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences.
Targeting Cookies
These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission.
We do not use cookies to serve third party ads on our Service.
Performance Cookies
These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates.
For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users.
We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails. |
11755 | https://arxiv.org/pdf/2312.12440 | %PDF-1.4 %Çì¢ %%Invocation: path/gs -P- -dSAFER -dCompatibilityLevel=1.4 -q -P- -dNOPAUSE -dBATCH -sDEVICE=pdfwrite -sstdout=? -sOutputFile=? -P- -dSAFER -dCompatibilityLevel=1.4 ? 5 0 obj <> stream xå=iÇuÊ×µAü9óM=¦Y÷AÙ#YIKÐ @ 1ÉɱKÒ÷ê|U]=3»\Ùf«§«êÕ«wµßmØÌ7 ÿþ}üüÍzóôì»3ÞoÒ?o>¾vçke7mî?9_6;$ÅF(¹¹ÿülzøê¿÷oî ÉÅÌ
Rì ßl<øúÙü¯³Ù¹ùñk/äöþÎ>½öLs-·áv\+\0ÈáÝÆ7{
«>îmwl6\Jå§Ëð3SÌO¯·;>+¯¬¾ÝòÙrnùôª¾|³Ujv©éáv'f¦3Óþî¥å~úÞz3 !è0�$ít¾ÝI¦gÄô¼dÒëé\üªíúê¹{3}0Kï4ù,ËY \_Ö9\_Ñ9873á¦à¾¹ÿï?Í(þгPû糯qÞ§OÅÃÆ5ü ³q�$sïÉã½K:é´@Õ¬.,ÌlþàF�j#ÈB¾¾Åß-þ¹ü~¹e³·+@'>ÁÁ9øRÎ0¹3tx]¬^§]¨
ÿfabR³-¸®¬vÃmÃq{íqß~ÖÛ8FõÒ CWѳsuË�Îiça[éi´ @É»fä¸c Jn§Ç ¯+Èæ! Àys=ÆØp d«éÑbýýXÊ$rpzæf³ÓH2LFH9¬Äá@ (cp:[X .�j Kdbµüð»"F°�®½ª,&NN{xÒÒãHI'0¾ñ³7DjG¦BÊýììþ¦û¡eL(7=C¢ÊwnÚ#;x"p¥÷á5@:û�6äÓ>@fæ &ÿ´aµéi·³¸yÍÚóǸH+¥}bk§9æQÏ· ,²àe2A£Ï£
ßwB¡XpcTÜ0§¦=ù Ãbè\Oð% ßåjeNÆé(fÞlAH-tÆÌtç°ö9ApYÄ= ¤C¤Q8gøè0=ЦwEvÕõY A¤&kf/ N\wqU?¨¿§'Øi0/éwñðpõðÙÅéÉÔ1êï·qî¢Æ [D!qE ¨ñSäMä6PsD©
e³6"SÄ3>Óá¸@ÁÌNFQÒêOë gפ_![ÉYK3ý)(gïÔ ç=Þ ÏÜà[øØ¡ #ÜF> ºÈCË° Yí²~K> ø²Ék\Ì0tä� ì,JÞ^Õy̸Mx¨%ü;Áb%³«!qd_ ¨{Ä@@(³S'rgÀç¯×(£q¦Q¥©ó»[�£ÕP1h
ºn¼YféT[Ýú¤"©Þ/rüKìbJ¤gð³Ê6äÈ.3YßJKYhÌÆË=4,Hf%,øCÀ=PJ0Qvp�[e·4iv\[ÝùN¬',6owÎ�çK5å\_a/ÊeÁô! ¬KT�£�[Ý ¥ÕA¨ÃÁD³ä õ?E9È©Z 'óóúörñÎ�Ey¤o¹ªx°h¨4¦UÚ¶Êì|¡Ó¾n&íP2AÆ×å÷e[¾åe¤ )ÿÒfèÄ¥MÊ G4ʺÑÉã('#%Ó½h8(à+Yñú{ ú´00q©dg=¦õ\±Ú&H¬A kZ1öÐE.²óôóQ3߬{A"£Iú rd ÞÑU.êy =läwÐ÷9$ 4 ̳ÊR£ãT"¤srDW'ç\x29ÎÉÓVhÏ)£T³, òÌ\x JPþg%[Np¬ízEU{ ~\×Dý¶Ò÷¥Ídyᮣ²¨¡;µ-¥cÝ£dé÷u
êöëË¡VKd¸Ï¢^TÅ¥ýö»ÏDÞß&ÃBRà ð³tÆ÷=Ñ[Ùøÿ+\xMÑÄ<TÊA>)0f£Ô é° Ä:S1+Ö#á½ûÕPì;;®º²ö-ø_ÒXKó¿8ðëæ£ìÌÇrOój£ÐáÐÊ]¡:ñÅwÙ¶'þ"Äý\yµaµWe ÁÒPµÊ-èÖ\fY(1Ùó=[é(·c\Èeÿf ÒºÑÃB!Åý³¯ªÉ·ÆÎ4íÙ[ì;Eh+ãrcèjh ²¡¨hG öÅÏæÔ#ðÖý%ö!a<'Ä"Ìuú g5ù3×ÔoÁfà5BÐ9yðRÍ,:R #=}WuU?iÁÊæVçh¥§HPAF5G¤Ç\ÅN©]þ¡³3¦õ ŧ=k¹doÓ¸f½¨ÔQðV¾:¨òåR'q%¨R±\×ø8D»µæ2àÒè |«væÕC" Ùía¯ßZ-ëQÁiýqgõZ´L®(CÄR)¦oAª1Ñج®485K u¾WOOÿ2·ºQ;À±Êøi¤ Sà.ýiEѤxÈc7i0S¼¿8ÐhaÐë±~#r-QÙ¢¬ü·»fç ,ã¢Ï('Vó5}FiöòcûºµåÀõBG�Ä$#¢\d¼êĤÝ.¼rË£Ð!c!h¿¢x4L
@Í ¼X\¬¬1¬©¶ æ]4�3At³'è¬RÆà2¬VlÔ1M££ÄZ¿ÉG õkEa°e¬:$F¶ª½Äjyÿ»$¥A-Ü&"§ªü?¸]´JX&ÿ ¬ç×(<Àh3b;]³ Á[f6a;¾{ô½M¿K[l5×^õR·
#õqPúË7ðÌo¿A#lù ¦O²¦¨ê¨ª'²Yîð(}¶àEóÚ]¦g#6÷(^}qÕMjü¬åzXQZ&½ ?-¢ôM²E²KÓJ¯DÅ#ZyÈÆ�î.²þ¼z/"·öCR®ÛOê2CTNªÛÀ4\ØÐ(F~q9VÐLÆ4q³ ÖVcçlT1tVãÛaûFÌÀéÙª¼À(6uqW¹.¥,5}{Ø"ñQJòÎ Éç^mÿ«rĹ9J¹OéÎÕVp«Õ\ã´Z¢þÏÌÚwbĵ\_4¢Jf[ݨEdp ò²?QCä]° I«¼Ldc!ú0Å\_ó±cÆZ5Z®p Éw>D4ð>oµuR®?>SÿéØç(<ó )hüý°[Cß]iÃb·½³CDî85ûa ÏIÕº$
#P>Y°Ü!ÚgX=±?8è©<ϪÍà#Iw\_ØHCg»9åxFÄDÊACøv%F¦XpJgÎ/ÈÉx©:s'];}uWZßÔLyàCè¼Äkn.Eºª#ùEw´iT±Ç¨¹'LÕ7+RAëD$]¼ñÖÆêw Éå´�¡�÷ T%JΣKÖÔC^áj døýjåíôyÜ)rt]®)îûPGÿ²uA¯Ç2þ9øTd( ÈåÄÖ�K@ @« <àödÄpãÕz0&ÝÚ�VH¼¡u kOuH[7;{!v-�ÚJçkIQn¡ÖÌ|㨽>´h;-;ßhB«^Hr]¢·Ü
ßÙhuÎR>ýcÄ�Aø@V¹já\ç$y¢~[Å5?ga± s:[ÎpóÛ¡ØÑÉúÒiò î)±åS¿m»µ|&ºäba?Yêr
ÉéwSýúÒ¶¶è¿°þÃj:[À©XRèS>Þ2
ÚBH@PÐ[,¨Ï¶èJZ)øè§d÷dgù÷ßmËË¡øú2&´³/¨Ú+Ôy:uuk±äHé8n8t÷ ëä,(ÊÓ"£e[ro <ïÊl@"Im@W;qÃê¦v8)-4y8/ìÝÊrõIøÜbÑ$_òª[ÙÓí|E¹àzÅAoÍ4Õß6 ±íÒXÔ(¦=®VÊîtè\«A¡g<|~zÇ&%44rÜ/ @dG%§ý:ð[î7¾ã£IØáá»Eí;a|,ª&m«I-0MTëpR2rsìVLräåø5¾ùeó_oÑ.z·ù±GSè 93)ÐGA2r/Wx½áá÷:ô´wl÷iÆÃð¡¶ç:o¡ÀYÏÌ)+dLÙòí¯R5¨ XÒhÇKTñn<-©2tm±\Ä㫬¶øIk- sßjÇVyg½EB¤ ÁÑ VãZ[àꦹh¡|ÛÍ¥Öj®Æª¨ ¤,àT5nt"�$%¨Q÷KÏGÚ¨«»6Z =µê¯ã^ú¼WeO³ ×ÖQÿýA©Ö$8È5û¡FXÒãÃÆJð«ø Ñn}ð-Ë"ײËþÚ"òÝÞÏ@æÈ2gèq×6Õ®D0VØfÞz³ØCL6¥B'1�ÒaÿÚ_ ߤ¬æ·,¢ýMÙÛäÎܱP«oZBN® ¹Îíû©oÍR;Q×ÅhYhð R ^F¤bý]ºãÈK(ýè¤ ®Dÿ±É²¤\¯ .:«u²iªSL}4?.Ü-2$¥ 3Vf=NÁé.ÙAð:ý 8¤Ø¸úû"Z§ÅþUõ�îèN±¶ØèreÑ;é¾gB$ñuµyÐxç[<²eIYábtù:ÈÁ]/Ðì,³·+¦ð üÌß»~«]pÉQí»¬zëÊ&vhö í&)ÚK@h±ÀúP{+%½ñ8Gxr>ÃO;¸Fvý]wp[Qæ8ÊJ¡TV©À&5ÊE9.k9ÓùÁ°pçμv¨½8ÊNÒ@ät¯ûmUqìñ&^~ü>ZªÌ -Ù<Ëü ì¾hpEy;oGIffe¸ o¢$±FÉQ7Ai¶±±«$5bÄÕ¤dHF¥~ð Cjép Ak&)$²¤ò»zýÐ á¾pÖçÀ©ÙL8¢(,S v]î8ãXqÊÄÐA¶vÓ(¦ºÆ8çÊ£6³¶~ÛQu¶ÌmÃÖÌ>æYt6CÖ»0xlÓ rrmÖ
SM}}í¸åÔBÝxEÖÝÍeRðP ô£$C¸VhÞ r½4±¼Í®¥#´ÇEEuaÄÈÙCÝîì\ {1%ûÍ Ó?Ц¹±êoª Ü<®kKRXuÉ¿~Ýqüær)¾@ð$F¯Rüaå¤ïiÞçÝZA0"TíQãçK´AH56³= ð<ë® ûàeBåûC;ºË®Ç±$½ø(®\ö^ õ\´ë¯x,Ýëî¤ìªR¨Êã¶2{Ö Ê| ¦ywì¹°mqÔó Á^+ª1Ã9�a! Ç89¼¦mX9BNöÕOÅHMï/oGë.¸ì×<Z¬1æn¸ ÝQfṦ^zÙOw SIàìÛ^¿Y³vïÐöáæ²�´nçGò ØI×Ï,̼H£¿ãTCº·T¾yaHlFü¨ô©\ÖÏT;XüJhÓ�´k¯nµAÚØÉáaØê?ýH\_xq6m ®Í/tÍ Å^®CY3éÙ&ÅgôÍPK[ý̶gÜC E¢Å»;>¡Ñt®f)·�m7WìäcÓª µuô¶µk´A®% æÒ0E¥µ]ïs¨ax|Åf÷ô´ý·$%f¿ÆØnÝ£× Ñûþq=± ¯ÚY ï²ÑØÇÚ! Lú«Í0º@£^X0ÚÝDCÆ70Hrã]/jP?Æq®£|4ÇZÎýe¯ÛY9Ä/WB´¥ÛD1V
6 2'Q¸gsQQ�0¦ÜHw¡È_!ÃùÓKýjzÙ×8êi yV-Ò|m¹ªMh\¥oE䢱[áYuÌ»n¨;ÉÛã:F¯êïü¶ïCвi0iejì,³7Qú:RÄé÷jõc¥¶²piGL8pµQ©òS ¬ ñv÷m¼¤J²}ü/¹N0êr2óHêùp
¦oÞE/0¸�Á$Ð) (g«ÓmÛµ£Å]=tÍÃ$ýQ ®>ËÍÄ ¬ë]fßèf
r¹6® !døeLDùª>Ep4CºÝBÇLFÊÏsãcÁCÊÑ;ÚóÅê[ÈÐ;¶ÑØí¥ÍÐ{i» MÎѰż+Ý^ÍÁ ]O¥Ì·øij|Å®s°l,P ûXûͱ»þáëI±gm8�¢wÜåÇ»xU6 ¦<8°h=\móÖ©ÃãÛ±JÔ?¶G 6J±x?<Ðd±ÌË;6xúuø+K^
ÍRbQ% <+ c{5}1Oã®}i®úkÄÀk¥>J¾ºÅ þ"²hyÌ£²ÌݤÔ]~íæÕhAJÞéÐÉ:tfA&ô·v¶8xeίÎþyendstream endobj 6 0 obj 6303 endobj 89 0 obj <> stream xå\KÅ6× í³Ç·£i×ûÁÁØ8Ø#ÁÀj!óïY®¬êê^í CÍtWWee~ùå£úûùá¿ô÷Ù3¶ûêìû3¾Ý¥?Ï^ì>My¶¶åAuÏ¿ù�ø¨<èKöû[²Re|×B¢)KÐBzò ÏAÀh×ûlìc± åCÀ&P>ÁÉr¯Ë]h¼ªA é%£àÇ+|L|>Ï Ù çãM6|É{u7|r§6ü ¬ÕhÒµ¾ÊFòè+¹Á/ÆtÉSFØ¥AnjøÇô0I¶îMÚggøä¡ÐÝ«{SÖïdÊnrH=!ºMx
züÜ__µ=oöBÒJM]í×Åÿ=ÙIÌZp?ಸ ÍWFQÑ:pÉ(¨è^à0>¾£ÈøÄÕ½¤ZÙh>þNÓ¥£Ì¬Lj|ýSDd·0,¬õÈ+ÑÊ8ص«E u\ ¸oð7§Á¿õ]Õ\Úµ3ü?Ø9Ð×áÝu1À_ϳ{v» «>vqNÖûÉfÏñ+ÖöÎÕÞoµô¯8ð/Ù¿uy¼!>íøcÆ}êÞW^ÆgjVÍÅÑÒ3?Ãñ (Ô½Ú! vÜ8ÓÓáEò¶eºOJ ñmZÈ̯æx;tWU|¸<¾�3u^ vpJáOèsP bÚeí,ݱߣÅ3ËDDß[À|#)Òפ.ÓÓÄW¤¿¼ðì T£6Ì}欷q¦üNJÿ캵OlHô÷fwPðÐä-dÔýe õ²�Ad¾«DÜ~0fë»hb²b÷DÃ]Pº«.Gc%30èTÉkß.AýEó] qT³¡M²E2eÁÞ´_ T¯VyÕº7®©q:Ét]w¢à1¯O8jtÇ DTx @a 'Pï>éÀcG¦ëG²þ¾ÇCãÝälÌW¤Q803iéZ/ȪÚüP¢N~H8´HèEî»8ËéXÏq�ð >Ä ã5®ý2OжñÙ ÚðüÖ2~}/(¾^[$1(¨ô<ÓjXþ�AÕ©8@i/Ìæ&f ;äwgÿuûà'ébSØk@) Óa Ü)¿þ] 0йÄdÙÌòéø´XæeAèC^,,5©O?f¨$oÝ´�úîhPàãK~þ²ÜV°áX!´2ÒÄJ öð:¾¬ç ÏÕ«&wÁªçwÁ¯µ4ÑoL¶³Õá¾"ñ§ñÞnÉÃFõ rK $7$3 Æû.É<ÈC" KRG6Yî H ¯0}o$L^ymFÜÖÁî¿cd=çIudÏu¤¡uQ&FöðÿÞluy{5lµë³«baauש¹Ë®®úV'ÿ¸V¼X%~¡·¯
ó-ñãõµæó9bÔª¬.'-ZÍ)9¡zÖõÞ|\!GÌí-åwθll7+|ô¸Ëw³§lÖ ;¨ÈÆ�åBü.¼{oÀþÒ°«ì/úx,àNë£G]åïût©ÜåÏq 1iJRÓ0$aL ùB¦ï?r
ÅÚ<,TwÁA?K ïù$)G+/ Ieaaài7EçõºX» Ørz7ò£i¬yà>ÍÃôh�M½æÜ¢( b�nÇK«[�[ÃôÍÍ?%gJé° 4ªö Ç-5Z± ¾
"¾q¥@À ̨sÓÄÝ¢ÄÝ13Bë0'ª·'Z¤E~Yo!'>·DrddÕ£>H%7@Ë õê6Ll+§ÿï:쯻 7q:ëf¨£IµþZdíV ¨Ze.ôv¸ß0ßàR°;ÃÖÎ}¼u²5º)Õmº_ì6׿+Àe3ôË]ÏöÀMÿkxªg¿YSÀÝF²32Ö÷Ðà#×ú=mxYaÊ<øÞ1 ïo«!ÎÃêNõ(êÏNw¦p®u}o~)²É÷Õ¨^#|¨sk� ¡Õ¶3 ~>Ê]XªéN}Ký¾¤öiZvÿ±³
p¿-æA:[þkÓ>^°¢¹ØíJ å-7f BÑ\7%÷LÞ¤+$ikh Ñü®ú%¾Ês\³Ø[u@©4hM²;ÑmºÚoàb+àÍíuH²ÀÊ!±n),¼òq±·g)m ?ã4Ù0çÐ0¡ê^4¸¿\w=oÞqk©çÄ>[0,õÚäbn%µ¶}BN ©6þ§@O}¶¤óñ·Ê\®´2°pßÐð¼iǯÊ×G2ɺ14´\_v¢!N;ÐHÏ+·Ðüíbý÷ .ÑæÙVNvVG\ <(Õ4ÙÐ\Í&{Êûqi æÄúÒötML Mèïä± oIùb¯zJÜû{¢,¶µç+b$VÛïOÑå-t5ÿ ?ÈÞôWºýÖÓÔE¥(¶ Qñ v¶ëjjQ¾¨±¦õúª ¢äx¹m.),ÌT¼fJ¬Wͺ]2¸D:M±2nɱ¥{pVb6 ¼Ñk/ª®´ÒúþöîB"-rÿÍ_&~OgþLÎMöÓ3ä8ÁEµÄ®H¦fõè-Ó¾StØ' 7 aì;Ö> ¯ú czÁki\®0êO�èòt¦¶µµú5$¹¡ÀÒÐðó\ÔwÔ/;ÉïA¦3Ùõ{NüJ<
vý^Z"iBÙ ¼nã@jVbGoù¶Äõo=¾}íeä³²½nõ~ÊǹþïO»\нkêà7ÅÉy Ú ÔX»aÄp>+Õ/§ ÐLïÀØV ôÒ«ðØ ¦N²Å Ú §D cGôÖõþÀÓqÃTï£ÓÞu3U¬ò!Þ<´ÙVÞaÐtE�_«ùÛ%¥¸Õ·[t¦~¢Sa´;°¨$¥_)öH~²1J/~÷2ÕZÚhů0w¥Ó>õÒ j+/¦Xy1Òû©¥~=@ý.'MÑç+~³3ò¯ÌiL}W0¯xbKEÍ]8>i Ê´7mÇ 0Ð6St¹i¯uhÕ=g Ô«7åòª
³05Õ(hÞù'ò® ²1¯[4Ý©²i4Bǵ}óÿ.4øEàÒ¹ß'.iÝyQjë ò1KkYà[o¥57ÒmKÿ ]Y:ÒbaÇoÊ©Ðó[ø&Ì\BëðkÐ:�´g.¢Ì°Å¼°.RDiE~0&hÊ1®
þ0�TG:´Of0:Ã�¹p C9§úõ\Yw�]7¡ª¡~w°£ÛÐn
É.¬}ÞÃ=º£ öxZh)ûPBQYb~I·ÎÏÛÁK§)Vª´·ÆL°Ì1ç}æÕbdÓ9Y¦R© Sü=Êné6CIeª×U¾BØÊu¥¢B]Ç]å,@uËoÛ6Þ6¢¡SbeX6ÖÌ°Ûqym Á\@»§÷<.Tm7f0éD]»+^:B¼#õ÷ MÎw¡AÆÔ´óñã5ÿÛ"¹ñ´íõÚ'ÄÆÈ®NèÛ=PqÍt_H)ObãÁô¢FÛÜû;Ô6^v³3ùJTniý bóú³kz3ð,¿�cbvü³í <ñ:W^pÄû%ýó� ±:¹ $ß´ËÈ. Óõ«3Ô4½L¹¬D DR:mÊ�%ÏOAùùGçg¿ÿT-êÄendstream endobj 90 0 obj 4520 endobj 105 0 obj <> stream xí]IÇu¶®:Éþ}S·)æ¾ÈÖÁ¤h ¥ADø�ñ�0ä Ãø÷þ^fVå˪¬^fh!N³;+×÷¾÷½%Ko6bAÿ¿O¯ÎÄæÅÙ3¾Ý?O¯6>:ûä¡8D§ÙÒôºg«¬ÚßûN¡§À{b;çй³vçz0J¨oÌÔÓWBKw´Ñ;7 7{~.½¶sÉiê趢tcõP?GõPS= Ô8õ¸Ö¦îÕ@/£ýIhàêaܯª6k=W´Ú@¶& 24Òö·»Gß¡;êë ú×gfi}ãCuU0fD0Ç?¸@o¥$Iá@Um@ï0ô eÄP"·~µTXr×�§Áç\º>GC4:: BöíÈ¥ö¾¶dÄ¡x4�oÉÿÛç>:#ÐáÆÊÙ.<|þèìË4ý¸yñãYZxܼ/üáaå½ñ^ÚÄú |Y|tq°fsyöUiþà üèÌ~j�Rsa (¶Cst qªôÈȨBþÆ¡|óòìy Îõýöú[¸2íç B/±íÊ1ÎVÁJ64Ó´7uh|õX\_ÐÝõÚY Ð^E}J ¿Çn<ð~:¯tRj|G3°©z
«Vqpi¸PÐúFåçðIßÈhi¾õÜY¯\_í÷e×Kà§ÃàMöÇB:áeT1°}¥6rûv§PܾÜ) ®ÀlÞÕ?¢¥Á^kÖõùýîKðÔò"÷¼ÓÛÔMÿS¯ßwL2bügõÛÒqðÔ>E)íözú2 yGkËKt
UKeÓ\wÐO@ÎiLñM»ÚkÀº}¿Å¦) 6çô¥$GË¥íÒÞä%:ÌØ+=X$ì52nÿ{g!É0¡|Ú4YéÐØðy¿¬s¹Þ5üÞÄ{@öðM�¢BçOÊÛq(ê-ëYíê7ÔÖJxþ2ÙîR=7ÚÌ?Q¿Hѵ¶;QÝ´K£ÜÊ)á¦iN6GC 0åqÇ6%Vëôah»qãä¸ßZÅ"R«ä[_;ѹ²õI¢ØÑPiõöj§FÉOÖQÞÍåvÐn//?$É8Éò¡#,&KwKvÒw0òÁ8 6©Q&Ò½·¤fð®Ce {ú¬VDÈÅw{ÑÓÈ0D
.Ò°?Ú$jÍL^ÏZO ø $$�ËË<@³A3àù«mB÷e²ý/Îýó ñw´G8p3FØ21èÏF©%iÓ´D%fÒ$z. 87yáL3ù]Àùb\Ñ£° ¥V1àÊ3õ|W¼¢ça[½xNßz¬ÏÜ^Т(4·þÌæTDgÌa\þ/ëjHß M×ç88GëBÃò)´Èv+k§áÞïp�Úëd× >+ÕÌ,á%ÐÝcél¡ã®§
£Úà¹ý}G¦§
2ùáXMühêGö+6Æ÷T [(&ÇgõóÆv \Ð[lðdý\eýjs²~T g¸Í·ç%ä85qðs²7NLÚdÆGa
²\_ÏE|Ý2}ä¢1®¥qo3J²5NÚL°ê#Oc£[¼I¾w° é|Χ¹ìÏ@G y"ÊQ/DZH³#$ò0|)KvÛûTIÎTØmð¿Vàsû/{5ài[±íÇî°ð"ÁÆGãÄ=JtùYê" )ü£ëTNÿäèÿ{÷0$QxÃ~8ÝЫ&k á¬<0·Ö ¾µF&§y<5TpÈѯ~Ø5áàãÈär©hèmöؤô< xE_FòX¿Ù_)½S}?®SK¬Aý"ÔÎÓÑ^ÔÓÄZ¡§Ùoÿ éêÿ)O'ïÞ¹J@RhAqX¾.¢?ö,:Ø¡$ZfnÃèh§ÐôóNOüí\ëÚÈßøûé [=5ýºÆë^ra²WB 2fÏq{VC/mÀ!kã{ba@¡íëÚªF<Ï·£ç1,ú1·ÌÚY'¨-T¦mqÕ)ÔB:Ëm 5$¼<"¬ë(¬Ix5ÎL,±Ù&Y÷ ÒÃáVÙ©,Åò'm) f×ò{ Çáwó ·Ru1P0íDr úw3{Øø± )æÏ" ¥°^x ØIf J l ÃÃz�3çýähÆ
áÙlƽ÷óô àZ̼¬úÊÖWÍU$ifY ²·bÞVoY¿e¾ÅE=NÀ¦Ð,¦Û ¥íþ Í «Ë±v1ͽ¥¹@BÜZüeE&´O áWÝ&È;sê¬WÚ3>Æ)½¨B κcUö®Î¼%{©>52VÃèÊn cè¼nMϨ´Ãêl¤äSÎë0±ëjv¡EÛarUÿBè(äJ½ÔhÄXHÔð´<;þR绳y«k }ÓË[2Å»¨æº·d£hr Jõiæò³sy3åC;)Óìte'Æ(å¡u&6%AÑ·Ç)3u÷&^IY«½å]K"ú]¬ 0¨fÙªLlÌTÔ\ÓH0V%4©×55±ÎOØù\ÑïàÿÖßO¡CÎÛCbnNiHjmÀ@&bü©W£õ3þµ§%qØ&^%A»çQ¼Tã=6¸lDú×¥æÌ Èjldþ;»ý°Ð¹íïw ûÎÆBèlÜ@o¢µ)¼ ]ÉZg¹ÏhÐtb×2Rm�[%¯/1 =äAE²æ¬?ö¿û�= R½4
ÌûF¿LÐ#ß[ÊRI!¥lÌç,ͽ#tLÉw«² ú>wÕo´I²)Óî¼&Oö|T~\|²æLÁ.K²CBÚlÇÔë38øØ\B}pöÉ?oÞþðîÛ³Oþk#Ï>ùýëÓ¿~?~¿ù³Ïl¨EÁû¡[ô =~ºë²þÔuÝ ªÒÐRõòUR|¡¶$¦ñJ`¢E¼|ZGf¢ò6ETm;(Dp¨ÚG·¬t»)IÞ^w5{å.Ìtõ5¤1sÖQÇÙ×ÜFóÆ(- çtSÅиT¾c~LäCÓ¶Å2Sé3%JÁ#s90ª<KÜõE"u7
¦Lò«ýÉß\gSc«äþÚ@e\zpÊíÿãQÐäÙANÁªgI@·Sá©ò<®Èà%¦ÄM£RçHÊ|Y ?äoÛ.U½ÉôàCEù®7ò¹ÑØúñ6×%Jµ¨'?¡O¸kÖÓËí¼f1EXß(Så'ozI»³VbØßC>È^°±Å8ld´úD´I:~:ÚfAîm¦¢Ü=h#¢Íûô' MhÑfgê%Z¥h@ÃÜÉ<\»÷¶ Tm
nîu¿µ«¥XåüÔþ½óÃCòTkÁ¢îÅ?:?îͦ4??Û3LBrçÉ\ Ï ÐÚK¿WxåR©0¼ï3 ¨ò®¶òåËñI¸1ï}ÕFcRÁê&~Ã�w¡¤\XÛ>Ýxº£w#f,\\_[Öz^¾#ðz8Î ï\0ÏmÛ±2ÅIåºJ¹nºñÁm=Ô¥AÉ T~J«ç-åp%<%o¦ùÎÀÚ© ®HKÑU7A×}IKK§0!Ð�[V.¶ô«)=Wc©æÈæõ¤møNàÙ<ñ¯~{è#k»xè¢8YÈSÒ}¡ÖBÔG¬±B¯nfQа{Ç$ÿåpð£p6¸"@êÖAã. h×ñT´Ò·Ö¥íjrè@mã|¯çiìy«¾9Zó¬úĽò¬=¥C7¸üdçwö'¹m¸9/%.v û%ZÅîtÊ©ðÒ×ÏSÎÐ-¯-j¤)Kv Þ¦¥, æ;ï%èÌó;T0o¸Ê}¼í]RÃùöîgö¶ÐéBBJ-w®æÌiwÈõNªYù=ÜÿhÍæÞp+vÝ̱¡\Ù£Y°"Usa,± A1P|±M=Ø>ØîB¥oÒA±tвÄF^ÚÕ^#Vмïºý¿ÜåscéjÆáû[iÒÉ÷·T#\_èÈtç0ú£Ökº¦Éx§ÄLêzìÜÈd¢)ì]«oÒçÞª#5)ݤU"Ò|§{rT®´¼99^-Nµ¥òïú¡EÍLkb¹£ c+p|¿Ïb
ý-×K(O7DMd¹eLÁo¾ÃeÉräv,!GhV�ú;e{pZ 5Ì/Lýð¿C9Ò¡/ºbË¿þ'Í\Ýjã4Û»Ô»ë ò²ëZV&[#3Þ=˸Êåñ�Ãn8rTWÜ8æc¤¤øå¡òÛª±ßûÕ³õ©g ½,qcÔÝ-"ÚcS�rà}ö= ÄÛØØa¹×:f%�ntµ%]7²85ìùXºGä7³àS\«j¢7~løéþ¡ �û½8t¯·Ü¢/CÑoÞSÕPÆaÒ1ÊFõ ;Ú÷
Vw%¸æ¾ßÞÐ4ã ÓË BSÀK¯0 Ñøp>®$j¦Î\ Hùé:ÃñLx8¬ÁÏ"ãÐYÈOÚ¨ §hÜ¥¢\¶_=ç&3?ÌÜTöfNuçþëúxGÌÿÆ{ýwõã6I.FO²ÿ($c%·^ÑOFz:2?¿'ï/Mn�yY¡4u©¥Jar_j ÐÇòÃ^HÔ]bÜ$ÐH 9Þ Wî7³«êùmͧ×-¬>1ïVRßÓ Õ¯
9tk¸VîiY
\½Ç2»®XP (ñ¡¦{Jö¬Û2´N¯ûå5½'@3³r°PtÓ| VveUS9ÁÞCw £zWMQý¡ìç³ÆR f± ÿº^ j6)ßÁ\¿;7®]E\Mgä´q+$¿È9önÿT|0Ò|¿¢%Õézie$õ ã¼øO9ÝGR¡¿'{0,yó¬B+«ÙY&ɯÇBCß÷ÒYÝé.½¦Xþ¨W8cCØ KëA/í8¹ äÈh¶ïåêæïÞËòèÓí´\Õ×D!./ØY ö>½²lƾgqÇæÕwçÚZYÙ-#£Ý¢³Ý¥º¼¸Ä]ÍMxý"õws¦ì´Ý×V$éè×½ºéÊ,i5µ(il©7Ïýk&½Zª ^Ð]n3ÖbµvkÉ×Ä®X\_°ñ6J3ÖNÙ[<çWr²úæ¬|{I¤6è\_Ùeª^ø+eûi20U^zAª\_CÖ µééèÃÞLsò¿Çði½r 4)ï cD4 Z}×ønÌÞ6IZ©s«£±³ÔqrÝÔ²\ü¯Ë/£ø5l¦\¯KÁ+o(wåmh ÇÙÇWµ«lõmL©£0úÑ á"¨Óߦp¤ñóèê5°ò>Øþ3 µ·ÌKEO7^ ¿]~w¶~>»¦Wï{Ïo&wæð;1ot5®¾qÇ>vǼêXÿPIX¾´ZÌE¹íxß³-f)~O»bzeÆ«D£@@gÛäÑ> stream xí]I¯Éq¶¯N}v«mv9÷ŬÁäØÀC>óFÍGΣñü"ýMEäRÙUÍ× 6»«rý"%\_ØÌ ÿ¤¿\_¾yƯ}ýoé¯o?»{ö/¿ÕþÀùìµ»WÏ+üà̬ÖëÙ«ÃÝgMxcFOóÍNZëÌä'Î
åtx¡³ÓG1«Åôû#á ¥¦·ñ}Îà¥o'1[¡'¼9ÊYCSöww¿a~ð³7¨0,7KíÝáîWÏîþé³é¿Ø÷ÎMßGÉ ãÓ»òí;èAÎ\ ëÙ9ïåôÇãIÎÊ) ÝLN 3=<
T~zÃÖÚÖÏÆhkð%î§?µ
zú¢4z>¦{:ëf!aÀìpw«÷£¥c§geõìì¹÷"?ûwØ6,¤2½9Õ¸akföV:϶&ùæ¢2¾aɹ÷ÆbfÆR½ñ¬&¬nú[XãfÃeóQpð,l#L÷/9®o3x®d¼Ûü#ì±fÖpÓ ìèzôÏa=>$qÛ³�M5ý$à ýcùø S D8>oóYh, àþå³ ¶ ,ÊËøQúå[Yàh¹¦¿X%È{8}Xî¦{2Þé¯áA¤JóKàÖLefãx·ýe M<1mø~ êôJ©#_ Pchè\f0OÉYZe ,8(\r#£ÑövÊÏLxDÁ{á¨z¸AXUÆeÜoT]ÎÉÃÄÖ«ÐùÞPToUCwL;f§nÕC%¿Ü:e<nQ]ìêX9[ïM~uÕO«h?iyU_{?yy[ë{¥X¯m=F. ò[ Ò%~ðigÇÁ.C.sáÏ×æþÄaÈhnfxç× ÙðO $ÙiPç(úÍ{KÀ+2rÝn:ÎFûøh ÔpzF] ÓY¹.<¨�-©O¯qoþ²z+íÞBj]av YÁÔÐúÃQÁîJ®],TΣFÔQÏÆAFu ÚðE$)Ö:ÇêÉåã}±;ÄÜãz¸KMÔûÝG/ëK'yz±ëÝè½Êä_\t¸ÓZ]ZФI~zY]ZEj9,übLW¸¸sÉCMHAÞ£;"óÙ.£¤Øæóû4#S¡ð3iWb|ãÐÀ\H'oñ[Ø}éÉúÌç È=tàdTE3ÁË£³CÞU tmM Ñ#p¯e'ʲpú�'ÓW¾ gáy7ÁÙb!i/,N´<Ðå&@pfi 7©ÄB?I óÌ8·åiûò@4>mm~
oHÅ�nïq$èðäLi&+·aQÜSlä¸ÃÓ ÆÁ´¢¤ó¸pîTôW%?½Ò1Wxl+¨\Õúyð¦ô@i§1ìfJIµ¶§ ,Òª7jÍÕEwÒ¢ãÿ.j%ÔÿýÕ\îAë ÜÛY¯°ÀÃ]ö6,ûË"yµ¿ì?ª¿¼ øe²càÉ^&Ï0ä�j¶#;(
m3¾|Õ¸³Ä!NOJpÐ+ý×®Î�f Ø-+í¢ß&Ï\ØC æ°µ5Ú[{õ¸æ ª¿SíRñöÖàþY¢%¦³³ låã%ý¦åmMCOuh~tôÈÈr7säT¢¯ãõz_Ð#7ÞsаÃå+3P8=4¬ÃMçMÿ1ÁúZáÝa<8¨>pÅok<®ÚÇví>Öí«ÚÄÿM,yM½¦ö/·,ìNGÀrô¶ß<Ë0#£å ]WCo¥E÷üR°Ó°]5 ÷.½Ý±µKIS¶ÆLpå5Þª¨.@XmDH }>IÍîÒh\Óh4pÍxî{d9\iÐÏÔ) λAø)ÄàîB~£HïGî@i)í¬Øésøc\ÒÑ_êã}z" ÜßǸ&ÀUxX_ÅmÁ߶gZdÚÎæA«é»#þleà\�#tøi\_a^Y ê)$RóAìAñU8![D÷\ °Lºkú \èd¸FýHpM-Saw;¬%ÄmR:¨ôV\G:Ü&À2H2Òûà>1£1Ûµ¹ #D°Íé"bÞ X¿F"G¦%íf+²bé,þþó ùäd]öÇWöf3 Á
¨Æ3�Ç,2j± »yUaÈÔÈVà$=H2w¼ðºÆátã ÿ'�~3²gÔì
À,îq JÁb>m^¡§d3êm è<µdÙÚPV÷$§¥.X!ånÆä)N¾4|ÿVØ%®ÕµoAHælæÆhÍ~ ÆhÅ:1Xƨ$Á>Vîÿ¦ªßM9sAÍ^5ÏÑ&¾z1zÅ<^²T0/~x¤Pnÿê½z;Ìymû ÁPuקðz^×ÃkµéÚMàD®[lä¤[²Ãúе92ÍrQgÝAYí\r½0\_AãÃÊëÁ°äæµL²Ñ¨ø¦¼H'#nÊÚ»%éÊh=µ=QrÉkÛbß_Ê·ñrQÔ%�$óxÀ$Ó£î¶uUÀW èeRõV)ÓEßU¢jJ$³à§ª»Êì1 ÝCëùï:î«ÃödÀÄå×ã)A겤'XT
Þﳤ\_FÎ!mcYᥤ±¾·x½¼]u8îC\µ$c3÷Ä%ÊøbÆĪT^å¨\E-Á¸U¶Äb¦P¿®,ÏUyÖy\eUF)= P¥V¬r¤²§9¸4Ð%0LO Ä~0Wi³JÎ{;óNýgcPÌõüzfáÿì XOÆÑ¿ú¨Ä' 9©ýá¦N¤éIg-þ?
ÜjS©ýjð õ1 ktí¯ÿ�ZÀZAhe NtÃ@õt±nù¯ÊáÇÖjËM¶BZ©Td�«>WÅ
1\o SÐÔ°ò¥Ôai¥è¿dKÿ µO{dgÑsNõ
NïDwf¯4\®?Ü, Û¨V!µsp°x¼°Ëé¨ ñ[)] ø~³hú9 ¢Fé? n·»9°&qL8Ũ.Hðf[NÃs¹è!yEÎÿñÈÃ)4Ý%VÖãr6/EgGDh<«ªôüü Kb¼Å®ñkt?ÁÃ£É ¸ð(GUE¢°®nñûÖ° æ]TGÈp\_/¶�ͺY{Ox¿Îåeq2Ëûuñ 0Ù;ÒØ>9×l©³?ò\L0¢t¼XøÃ[ßí8ãp¤µwåÙwäØyñ>ϧ»q
1< uõRùäó;x8 /¼#¯}]póm/ZrÆÐ
~ÿ6LcOZE!´6ãJ¨| pG%Ôòír¾ÎÕpV¢Å»Íá\«:ίÖò6¥j(IK\ :¢ÃÿPª²ÞMXpBQaêNUH@:Ó%Î#MFmVsßÓÀM«¢®ceöcNÉua{42u
\aÇ\_. P%å)¯ ®éiãt$'ø[µ Kû²Û
÷1xÌ+=õ¿]Ï>jËéO¾V<+¬=PZi¦§wç ©¹
è|jN¢v¡Í¨Óܬ%<ú ¦@}±B}x\/j2XR念°P T9O� P^³±<±( ÙB<}¿áQ0lÎö»">PóßGgaªMhxÊOºö¬BÆÞE²FOP':Ç-Ó}=©ûî[1pÉ|ó[Þتc:xí®s®2¶:K%d£VJy¦ÔV,fkTåb@W2F¢ °a�RáýÃá]u]çÀe Õù1õBþ«Üÿª)Àkñ¸ÃJ¶¯IÿÈÏ1Ç«Ü _( 6o¸Öãì&NS$| kجQ¤ÔÜ¥à;]D¢äÏèñ¬ô;Bý|PlÆôô«Ó4NC7<èª3¨ Øn"p|ºQ6Î !m^¾¡O>OSF&jÛ$ÄþJËô$Äl^ðuù½ïľ¸óÞø¾XEQ 9TuÝAcÅ-skéàsZ9ÛÀÖr¢\O?N%¿ìJTe"ÛÎ!>ynXxËBãQ4:&®üD:î©ÔOÄMEÍÓç!w§¨Eá)/gÃ|EÞ¨íþ§|2+À0+¡>'»<
ÑÉé¨Ô
æIHm±îobLkÝ}ý,Ó¦lªàk¤Uv.×NP¨ÈÔÇ-¬vëë""6 £SÚ¥¸j¨V
o"M1$ý;8Fþt4Ölusw!£Ú\Ëñ¢Ú
ïËVá÷k7~ 6¥UlqH¨Õ-ü n£¨íË´%£+ òïe3sÞN2¹Îr.áëYã\jb¬ºÃ ËxÌF¥ü- /:ä�EÍ l ¬\_/å÷í¾ ÞSÿvJè(ÊbpMº1ô¤ûÝ@éú,N bLÎK\smW>,®)ð±h(»ð3Þµ3D§+9»p33·áìØ¿h% w£,\PÕ)éo"GØ�ì&)Ò¼K êý¹=X CµÚoaµT"ÀZ¬³DýÀPì)'°¢sC°T¸Z¨¤Ï1Ö£Ëäó¨T%9¸«LÍÈJ0Ìæ[1¯ u;¢b´Û!{OÁ8kϹ3J
~Íu¸Ëu\_5×»Õ¡'<èeƦ$úJ@A{( 'J})çmÑÉ©U©<,¡¸1Éû}ÃR»Äkx¥ê+^ñøzÙl"3}}ž?ÇrÁ~D÷_²äa}]Ä÷¢Åo_[³ËÀ¿¿ÒÕ(h"¤(Çãur&ñü¹
å Bj_ºµò^Ì¥¢ö× Ø ©^]©¶:Úò¿ìBúÿ·E«Ò¹ÍÍyÆ»ÚFÝ î=4Lmì0´o[Nûxëfoýxë¿/gþ·Àé\пé3«~kÒ f1úlªÏRªZGWèús¨4D N¬WwØѨZÀó;5Ýaè5åD
¨FεÁþ«BôÐ>UÆqçòó»g¿?ª täendstream endobj 124 0 obj 5009 endobj 151 0 obj <> stream xí]YÇu~g =Éû¯YÔõzñêêæÕÛØ^à cFo-[´Öß^s.Ô"77x9»y¹±Zlo¯¥cTzóÛkµï-«|ÛJ¿gp§uÕ_m½Z,÷róùGü»Z8÷ÿÄá×m¾Ë}o7ÊûüN§ }æß·raBÑ¿ÞÁ%o¥÷oàTÞéú>xT,V(ÁëßüìáW~ñFöË-R{wuóÛ7w7Á2ÉÁ&à~xïnŸ&½8çáçwõoa\½ÁÑ®ùbæaúBûçð æà¼.ÿ]ówõUßÀJåM/uÖH|ÞàEX>NÀ|nó¶nÐô%¸çßã½ãovÏäi}Ó礧ù¤Ìði¼h¤Ûì×߯·B/ ÎÏJ/pyÞë¼1@=÷a .ãÆæÔfWox]ÿ1nQ°sõÖ7¸Ý@#>ÿ²nL:õ÷@@RÑñïãn8m6Jê÷E븡ñjÚPcèëÔ¿ß?ß" oáÏngÕ³-;+p¬¼]ßo_¸Ýïôæ}ÏÉÙÄy cyi·8áÊ+O» E91Eî;¨5§Ä}·zWbü8E§l¼7Rov_Ò,R6ôO~×'©ì,Í;
ðCSqÀi/+!uòp®ÏFÈ´l£¤ZRQ%ÒãYT}8Ìbó0Ù§ªà
Lãµ÷%ìt°%@Á8yÁ²ÝîÀê8¾Ä¡~^�ãæ�ê
YA«@Vfi¡Ð ÓáßmÊUQOõwÛG1DÖ:Í+E§cP3yïq¯¥æ®PÈ9o&c/.ñ¦tö.>0å©Lfö$ eÑppÚ@!¶,S1 DbLì
EªååMÖ-á9Yô.´\LÝ9×h:yiQGj: $Þ©£p$;ñ´¾7òØ}^LÙêµ ÞÁø\Ú£î«\?zNY7DÅÕ˦¬Å8Ô#Ðb¦¢4yýËúÎ<)ÕªÕ;ÈvUÔ©\
ªT%òqh Ï 4cáNê3,vðÈÄCC-; û G:CyÇ¢Ôó !Íæa績º.Ò f § 3Ù!XÕ7v<÷þ7[ Rt£uW =§#OÃØaËsGp³5Ȥ± AiAñÕÒT×Í]#Ô®ÆWý�e'-@>¢Þ»'佦';Öô¢a&|¿�{q2b ·Æ¡Ùp®ähó\Ú¼tocPuüx4n3Ú{Ç ãý.ÜNÖ6:3_\ÙE)&d|ä,ÝSú êö@°i]á -drDrÈ¢LÅÛÓ5-}öÓÇ)Jü¸Od°+9Ü¢÷ç °¦å¼Ê«=ÕvóñT·íæÿ:r 5Úý|^'ìþ¶å÷ÁÓöLjKDi[7z3fäæC¨ë4¦ÖLÏVJÏ0ÝíÑßeÍ:Çã5&ðp3Éßßਰiàr½×u:1õ[ÉúÚ275ô$êbp.'ªfÜ®zkÞ2aZ׺ëìs´D&¾ÑXI½b|2°ÏªõÑã6ÌèHýErðõÄÒ+'\ùa=D« fVxÿ»õ¬ÕÜ û¨E<±ÜKn õ¶ÞvÚøæ ÉöNlãüØ^¿>²ìEY'ÒbµvZ@^3¿h0ÇàrØÇ[½
0]Ñ(æAÚÍø õ¥ ´7XÝàÃNg¾ùqt¼ vr̦¸Lö©Ï#©@CQòI\&.d¦Ðê©e£5#ö¬F¢UÞý÷ﶨ@ÖÉýV3»\_¿&H?º&HK ©>¾ñ^1¡¤6ÊPç+Í\_{½·p¨]j ;vj¨½;áO¶³{E ^ïi²Ò@i5R<¶ª\_Ȩ³ºãõ'¸ÑHn¯¥2þëüwcqñ,- Lå'p¹1-ðÁñë ÞÖ÷YáMÅÙ ·w1DóRmÑ¿îeOÃG@µ\ ÐxË!;ZÍèðæFr¯Ëjàú¡dæ5}Æf ì1¥·JT¾Àô?Vc+ g)$C9Ï")úìXÚóT^IRp=´CíõPjÀáPÐ4͵ >P 0ËÙç¦TÐ&PtÙ?$®43Ìäì»E 3ª,¼6ço'0Ô¢Rze$lm4É>dÑF"39 ày&[5cNYPGØkÄMxóN!¦Û9VÆYQüãµQæB½½
×"ªlÎ?î} |±»>=T® )3&.1 TµôòEVc_HE÷¯ÒGÝcèÃrÈÃoÙ¿ß!jÖåuÐÕqÊ"-LC- }@Éäæ¶ÃGÊ íe# ]µ> cg¬¥=Ê<©åÁhB~/Úª1<Ëñ "ß[rr4êlH&kóN(´F+8E¶æÉKq»d!®òspÜ]ªG°ª@¨9átÆ 0;WɶQÃÄm"mƿбVÁ xIj°8Û´Å f#=%"±udþÍÈrãZ.hæò®ÿ{ñ'¬- yW@²Ú×j;òúM 4¹-\»\DBí ¾OÆ¥ê±úp/Ì�%sÛÔ" i¸îÔ§À°ZcÊûæK°¡:Oä$[»a�ÁV¾aP�6å'2Å=ædìnü´/Ó½mÙGBûÁN¥kü&úÍ¿ía'$S#)mE$ °/TêHðH+#ÆhÒ)Xþ»ØÄ/¸D#h¬8Þl<ò+ ø¢1~¯aȦ0S#a¶ %:ßáE°
WR/ÚãÉ êlì¯ÄquÍgÚéa8~á(©¶êÛoU3ÑÝÉÊ8uÆ(8°´Òé½¾ù'¬Ó{ÛétælDÓtþv?@)>:lòeÃᤶq 3ÀIÌ«Þ ")äÆMQtc{ô¥ãSx ó@À_[è+öx ÓØÌÌåóX ¾}/ôø)Å÷¯Ij¨Ñ´4KI±�:¶NÌãQZ]ãÁêRíÏ?ÊZ-Êðs©[iÝcÿèH?اea(ÐéÞuweáxà0_&Æ|øFÐì$ p<¦}H9±S(®ÁÚlÕ¦ü"©!aRn6L QF[ql ÒÚ8'{ 7sc6!]Õî#+ Ò}>7áÇæ$¹¼áY½Jw¸GÔÆË¥Þò»r5U¸Äj¸_¨$2PÚ$Yräù&¤ï]¥óµå'ãyÕÀϺÔ=YwA6aÐ> ½×céò?C�®ÏFòr1\Ìá�Ðþ M\ùÙg©Í[E ó Zµ$$¯³úÜáazóÐM@I9g]Ñqù³ùêü æ ¹Övu1Í<9Ká°qò1/lJP¸
=80Êßìµ÷- s}X·nÿÎóÇßÙ£RvªotBÅûe²c¨!x°IüÔzTíפ~ð@>¾ä·KEßÓYMÇ$oศNMñ¾@Õz IÈÅ´=Y=y§@ÈìÅ!B¢ÇhvõmÇ̬Ýõ e¬ÕQÔ§6TÕ?ξoͨ´áåú\Oа¼'jË íBÌ^àÎѬ¥^ÿ>D)Åý¡6oéÓÅQêq'&)wvNp<¢Ã´^Ò÷ÄG\_µuiÜØEÏx¶Y
|÷Cyæ{ß8 ÿbo¿¬F¾ê DÐõ<÷äqsÈkÒ¶å¥áQ5XXÙ°þØö%É Ö4j\ñǸTØ®CøëÔÑÍ~(_&ê~úˤÍPGÓ¸évN½ÑW²³ó¡M¾Í9;ªcxÍ vȸVZ ©¬ÊókýÙZÝJÆ°VpË߬{ÿo}È1Õ¨îr{nö5Ãæ{ZK
>Cý¼cbì¡rï2ÅF©!ùHTrZU¾®dìK¤ÉTöOö.EÓ5¥«YùIVPñ±^ñhßL$àT³ÌÎ\^y°gÓ²+ú ¯ä¢\ykMè´öYÐ C¤x
Þ®[úýôZÊíúå~ T0Ìù¬¼Ôïr xi2ØT5Y®h j;B=쥺Ïl"»º¸!]Iõ»6ë¢Sm®¼Í)Bø6 ®¼«rh!ïÁKÁ& Øæj/QkÎÄ�KûiO³Ô¦ÚNÄTûaZ¬=x%õäà/ÓCrG~MÐ 0iLzè ðJ>zÕUwÎ>Íè gº¯ð¢·w-£ÞBçM¢²4×wI)%ÊÊÅtTÑÞÒvÜa@êÛ~ϪÆV¾î²\_îaù y§¶ê?ìÔ) 8íéÛú¾a§3ìðùl×NpÑ5XRhf´ À-é¬ß°Ömbøá/çZÜö¾_Z' Åï±ÇÛ j« ó&põ͸۸4ëýTn:2iéy¼4IÄAPµk0ì}ßzt·O|Ñ3~9±0pÁ#Së4¼ë¹µûxx%¨åyóMܸç] ã+.;G3¤<0?u4¶¶IÈÕÈE0[H"Ùõ,.Ø óÄ\){Èv\_¦¤O²¼{p8¥dAca²w{óÉFÚ°FÑi|ÓS2åU¤OjwÆ<ª&K9¬!ÞmEëÐ=ò=ïÕY"á»R SäIÁ">mtäþ¶Ó CvrNm4gÕ£õAÕ3ÔbO;ob)ý)Ô©Ú¿ ·å5V¬?Nl¡Ò¦S°©ze³¨ÃnV+q7À?Iû9\_»À$¤Waõ¸p3ÀJÁ KBѤYj¢XMÜ7Á \ù®>OìEEª|¤9xÀÈ©Ä%ªg44Å
¢= õ¥Í{KOèdR%LÎw¸Ñ®TiR³£Çì1ª=£ 1DÆVMøÖ¤Mà\µ9�¦Ô1}0.h/SÙæd;ò÷ ½Ü\q§Úï£$îd{R,S¬væÁÖ)µ{¿Ç+ÏÚêû8®m .¥µýö]¼Q^D
å§>Å ¹TW«r?ÒËm%Ú´f²HûÏä{üD+Ū ¶Kî¤5m$Jõ0XW("¶vÞõñ§§÷ÚÎÞD«Ûð£�!, ¢fËHs7©c
1«àÅÔw\°H«ÍUUh\=ÝDy�Ø9Þãki;/öYºOÇùÌÚ,$ét¯Õ¥Ã°®hÕ×bGNÕ«$fDg\ \©ðG4ÁnµÎ7Lä\ln:Æbÿ4óÏÀzLIûÝÕËÅ uW[%M\X«¡ýñW¦ÆmÑwVµÎ- RYrcó¶s%içþëVaöj0ô½<Æ=?õ"÷m§«q7ß ÑqJq!$ïí¡ ¡©ÎñC À»Ní4ÆÇäÅTÓÐ9M+QïóÊM¯ølÍËZëßà}?ã7Ïï~Òûl×ÍÓ$ß}4 µ'MB&8úE< Ŧ¾Ê!?Öi'Ù?ù#æ¹>~g¥ BOáÞÅeÁaáªû³cKc.ÚʱNaß}²)t[ÈÈ0yïç@ ³Äy!P^ÉIKó»¸sM[ Uó¡aH¾]ø¨"G$_[Îá{¾ =I¦Êîjú\ÑÔg:®Ò4ôvóïͤïÐd°îT×v =åA}1Ô¸%9û%Ñé0±4¯r-]'0eÅÒþcÁ>L6g&QÏ¢óô+%güQ¡z®JT5ùTY¶_íWAõÅ:¯Ã¿!_}_}ÖÝ5ªÁµMqvÑ®ý|OûÒQüúóT$}Wùyô¹ ß/dlìûpp+ÌItöàÇ6øÁ¦> .¦| p7ëK¹nóAÀʨE#_ÀQ¯rè38MóïVgó fùès9''~¸55ùÈ\Ð ïc;q ¨vN_È öÒ9Ù·üÁ/T øóá¿®Òë#¡Moh>,é_ÓÚéÚ0ùaåiR<:�>_½Nëu÷_m^_¦îyLÓ%§�f|.ËRWÁ# û´ÖÓÃvãsàÀ´½÷ÁÐ>3#È eÚÀJ³¢t5T2 Njk½©ºÉtyKhLj3ÍVÞ«äQµm>}6bMÀgÖHÊ7MÛT½ü|ïm©Áï9zØù2äµÞ¦ÊÃÁÀz8%ÅÕ¶½§£ ¶ÇûËÅ^Ö+é¡ôÞocæ&]´^¬÷çJh©1± çîP |:K¾tÐ}BðßÍn³_]ãGÙa¡YË.VPÀ>¼çóÿ ÿý$Ñxendstream endobj 152 0 obj 7117 endobj 190 0 obj <> stream xÝ\[ÇuN^z äeÞ<hu¿ÈñC(+19¶D HäÊË]Iëåoæ;U]]§º«ç²\Æp ;ê©®ªsê\¾s©ùq#¹ôßø÷é« ±yqñã
LO7㧯6÷]ÜûZ³CtÊÍ£çù¹ n0íñøÕÅ7ÛÿÙíå µòñ£Ç{Nò×Ìt ¯>zÁj··CíÓç£Þü1øí§»½ò~ÚnØ!ªC(ßÓPY?>Ç\¿ý¶>½®¯=ÃéÜö¹sø ¥Ãy£3½ÃkBÞ¬´üÀ>ë#dÜÒnÝ4ôñw4+ÄíÍôôz$V íeûf·× ¼¥iµTRo/Ú©ÁÚ cá &xR¿¿úf×~ûÈtNc·Wu0÷ -,÷ênÓC;ÈùÐïê\iN3½¯ÃöòzÚCæ©y S
¥\_ïöØ¡Áu@A!Ëè?n§+ñ/+Éûø88ëd½Þí!×Á9é¸OM&S«é{ãÇÍj¾Yg e³÷ÓbÞèУ bète»ia·7JÖÛí£ÊL¦h«Jæ(eéa¦ÍG-F²ØÑAz=HçOÓñ:F![æ2Sëc"¯uÜì¥Êûe=2Õ27©ý :ï #-t² §±¢hýUþ={ëG:Êèi$þ\xñï~g jzzzSµõmÝùõNY»ô£4E°'u$éö Òªþ<«»yÖÕËËwuë#ÅÊ&a´7²i¡A©Õ˺í @JC/1~Ô侦°Ò«ÄÙûþÛ hsÓvQÍ©ÇÞÌÁrN5éÁÕMÏàZdÏ5³ÆÌt¶2\3DZöZ¦ØùLX'L&Ø4¥×ªÍrY4uøB3N±ï§}%]9°Ø>3¢/yÖJ®SWìGÝ 7sNßÓ¨¬q\7ê´þu¤|¨²ÅöϾÿ¤±ôeÑzãnÉÛÀÈ6Ù lÓ=ïiÕÝh¼jw×Ùÿ4ê x:¡V÷Ë^8_R ÇXæ}ÞJÛsKÓ¨'SÔ½þ¨¨cûÒ>ÓWtîp:®cY?'§§´iìº;ÎFéædïµ"ëP}Öa[|Y¬ ð7\ÅZMà}ÅF,ì Sé¼ê̵r¶ÍéOK» äjif1$}ý¡òz0ÅZ²Gôeif²X' k}±-^ô/;=()ÍÅ"¬hÞ3kT?ä 1èh9ø#]8³ê»h» }èVÞqi+"xii^w½Ð$'stÚô#óm#XؤÄ/Îtyelr -TµxÎ DË¥{z;ùíüôA¸1SaúÌ�pª$k O>b¯øk®x±Ù,ÛË=~Qä¥g^¤CµçØ\fõäº+@O &&a)¨ªrþt§TaytÉNäÝ(@Dcf5aè5(#�lª,Ïzå5:^'¤·ïçþeÜðYº\cK2YÛ
&Íô~Ì$ë~Ä°ÿÓ®§kpdqÝ-07ÆloÒJ ¢ d¤d¨² wÌw ¨û9Á£2#6$IùÐÌÈovë{<µÑ̼_Ue|Ùu¬ÌÈL ¡ÊARW£&¥=/8RA§ë.Ь5¸©ÔkTÑM@±eòlÞòE_? lè%Ê.: w²åQÚÊf{"¶í7qoH?.ë÷Ù9G±
9þÒÕU¦%¸ª«YúòÑ?¥ÍA !ü ÷8$h�¡ÀjV)£HtH/Q°$°jÞ©fñÉ-§J½Iy{ÃÞä$fßû]³.ȲÃPR&p4X´C®ÃÕè3ï]F=xTËPÛ~¿{ô= Ç!±oÍÞt¾µ ßòkVÙRH±Ò7¾eãÀØÎ?êÍöWãVõ.Á|ãúniÊ>®Î»kD.9-qÙlÜ@8Yõï á/§ø÷ï¦v·�ÙËçùág ¥bõÀ'Sz
T¦´$pq°$yw/wÓt õ¹¢9ÉƯGHWsÉRhjÏ«ø60%ÀnXVlMôí¾¢Ü8̪M,¤cOv;ºXäÆPEz@´h@ GÿñÛ17>ðÈVJ¡@6$ é¥?!tõÀ
Ç\_2IçcÉv8< k#að|OK<ßÄõäÞØà©Üp]'øeد7?XÏ?ØÁ8itpÐ N5]=²çð.&²7¦ÞįíVrWB>eÒb+dóSèPpð�Ì8²yãWTÕ<¸^ êýl=õ!ë}¶ò±ËFG¥\_X3OH6'ÈX®¥bføKd 5!P£¢1z,»Ì+Põ{']ÛðngáHãEjJi²q@JȤ¦Tm|OÐ�hÎyDvTfn([ë"½lÍ×?òÑ\_<ºøêÇË}U@Gac 5zJ¿êÁºÿàÞßm~zýö»{ÿ¹÷~KÿÜÿÃçøóà7¿¹øÁæ1É[c¢(ø<Ôóó ôT·§K"¹3éZi?kñkÓ}füØ}ö:ý¼,P3ZF3~õÆt8|ÐQÔi°³A§!-ßèÔ1I0É¡ZëVjOg�'JÿÞà¨Ý]3p©óÞ±ÚêuÙÎCªúÌz-I@e¥ÌúkaÏ
JÅúð69¶2Á³4¬Ø ì©ùt"Jô|~uTJµ| V®ØÝl
]Iív[õ(ñDʶÉVÐæEI¶0MÑåH\_ôR³/óÉãXj¶ß63ÏBçvZÈ·Î5¼á]É�%ñ©dL®Û/ÀWølÒÞ1Ç úÔºÓîOæê·(º´J§MHîwÖKªESb4º\æ3 tëuÒÑÓì0µor¨=ýÛ^l&ç#åP?ÔilÕ+÷\ªC%ezGMð´D±Twº®l\4¯,¸´h]ô®êz9à ð´gY) ò¡y/Ö÷JaÈ7ì½åü>ã¸L§J¢¢FTVà¯~§)¸¥ ¥/BX·ÝÒܪ¹dLÑ×uÇØ7B½Þ$sx̾§ó Yti7¤,¼PpÏóºpw9³È>dzU¬áÐ:r-ÛSà ´öÙ¼i/©¦TdÒNø4½cÞquMê¶WFI6àUN±FpF}
§dfMpþ»µ>b'&Ú.>k&K3¨ðhò\Ðå§,ú´!²Ùäïvpt[Éi3ÚÍrª^åFÁeS&\È8v%½ÓèfChb¡¶ùôDÖ»u¦mæÓÛLÛý0×Æô³ªó3xÙÁ[³qµ³¶<íVƧ"ÄGÉû÷\~Qólßïº Ûg-d¯æ ~¢¯Ù÷^K »ñZ\ÛM]Í£>øT¨IbnÎW�7E°=J@l¿É®æ[Ë9mÙ6uλÓݯø\ËLF©³"W}à±LÏö¿R&ÃÀÚyÉYÄÅ°Õ)ô.¸t¥Æ BÅëRÓ$dljúrêºûlR:yR6ivMrd°O;KkLÔí{îÇ9]ÛùðòÇ:¢Lí«n¥#{R)tê¼]ïyãnLx{iùº5zN1$)|ãeU7¹[®·]uoIvÆ&ÉÁlvÁZÖAÔ3N6ùïw¹ ×®ÙVSÃþÌÜÁ§ã¸Úé8>u¼éH ù"V"ÕU¤Ø©û³§mË}K+QXaKÌìJç»[T»Ùº]M§í˜Pðk{¢åcð\GII#¸íÝJ¨Ãµ•^ƒ‹ÈÀ£DŽÎ$˜PÇ%lËr¼f >Lg€‚Ñîoë][ó©ò±r“o®ÍfTr–@yQ(µwm|ÓÔ{oÎe2C u9ú±]{\)BÜ�,?[{ÃÄMÆdF¸9µêú§¬[\·yxDÖ²÷U½þ4v )7&lË3 T¦ÐpöH£?vóyj ïN;x[¸ÞQÿ 2jèñ¤»ªGºi(RÎäaù:ϧËÕI¦éÙ4Å1ýèæ>?J©v,¹qnLMßyxKÉ¿îξiºd#;äv@.±LøúWPû~SÖÎT3=mWÜN[ÄõTWt\$.XáÝ[Íêø¶Tqë+5Ùo\=pϱ\éIqîXJÏD¢|=§7k aÀ^õI\þ8iÓä®tûxùdÞV Á»w¡{;Ñ6îÒUð FP"{çѹT¾W²&óRv¼mCA\fÑß)¿rLbÒ©?óB8U38ÇØÝ;Ùýø |Bì³ø~Ö�BÆW¤Ò'T·ÿ¶©°pÑ�Ãíûg¥{1;«Ûä nsíF: -¨ÓHN''X¾:wóR´ë©oRf¿ã,쩵Ü/ssr¬Q_¾¬ÔÔ8 ×þ5U~rå'�Úwq¸\Qfsä¹j"¸ë\Ë´rYÀHKL7Æ.ßäKW u7&ý9kua?5ø[PβðNµ;<4¶t-%6㬽X(ÙLTàÖnÒ_é&bcc»ÁÎÍäüW NÆdDú©÷ mßÐÁíô«¬O;Gøµq¢ýñNÏI Kku÷º7Z=ëuÛf¸ç0dà?Ä6²ìX¦µ}Ö-$ìéGl ôE<£Mî8 ÌpÝn¾X#%
vÝàN)ÌS0áÔ ²§ª$Ð6y¿PºÉIªaN
õþý¿¥R¦A
¢_NQªä_è> h7l¡èÎQ-¨Zéï?Hö_#¥UtBsªæw:T¿>ªôQóØÜöÀj[ìØßÖëÍÇ®2Iú ª�F@9¥··W&õ× Lêÿ¥2©ÿceº ±£ÈSÊtSÌ\§>Õþ\_Ò\_æ ¢Ý0\òè:do!²î0)7V.>Jr¤Uç2^ôçrbÃ(dÈT!ÒYê¥�êluXçÌ(ø3E§ú9¦RG8s§\<çÆÛ¬3%azTR£ó!éÞÕ½¯¥o.[æýxèë£n%=îíVq)¾¬ ¶ḐÛ1ÔëÉDUHømÆ8)+Û8[¸ÿ:?«¶>üÆÀÜHÇÁ WøB÷ñáÆ7AhÖÄ$ çó2?úRøC«ÑÑF³BÚ@9o}{Ldì{áû ñ÷ Ù4¶´TÔó°Zn-eç
³ÃÿÌz5òµ°~Hg�~Ö²¨®|ç¼uAü7·HÌ"DÅôUõ]ZýÊ¥· t9ç'DÞ¦üênÊ
YßÆ9ªÔj³OÕ?9KÈ1ÊÚõÕ:gè¤NÊ ¾ ~�Juu ]¡.¡D~ÎÖ.~¿¡0±dÆbÇYBðùÛͣȱJ ©
BÿÆå.Z{Ýl[_Il(Y+>O0øUsqùÄ"£'O.ôÔ7ÕeÆíÒcÔ±z ®sÉãOömÑUÅøt1º±+Û5qãýb+Bc[è¡tÊL^&r2Äj2 Uô]Wù?2(²-nr\eGJõRîþ»Vo±°vúôò HªZþfÕ?±îçz+¥ÌÌyu²Ìòp¡|ùï¾öõfg\ÊK)XdEæÜßyÅ}\_\e£¾Wo|Q½UÜ7È'ürm¥"þæn¬KÁ ©²Ý(ÄϪ«.ÛõAùèóh4$¦èô3"^¬Æì
#µ±põþªÓ¶¡úrÈÖà\,·T S¬Ù(ZªxuŸ£2\rãUìJ¹±¢m\_ý@ÕH}ü3©gÜÖ&ÁøK²'îû±²\H©íÞõê NGDÚB\_íÑܹÓbu07;;¹îy4Ù %>ìu:èV5±&K#Z´Äx^V.¹ù²©×,FYy1«÷¯Wñ|yÄþ,Eùùàç³øÝV«Îxý¾PÏÞ=cë@s\_ù Ç]ýÒ�7ò NoÛöñWÔ¢; þ [¿Ëê+ZÇrîy4Ö¤s|pÛ|6a? AíÓ³®Éñ+e¹» 8ud¼Ly&ë ð%¬sÖ>ÃÅ2T@ªë&þÇiÍ©AîW×¢áëÀHη8~U\_¯)0¦rË=R
ìѽ¥ë|mHt�R¥ÄØ˼÷�J ½xq¨Ê=æèܱÈÏê\6>/ÎNssëk¾¢(tݳ(µÞwèÝLt&R]×ìåîý¾xµ«¶ñfb¹ë¹u3Ñß.ÍKvÅ÷#j@.äÑoÇàîÖoZöÖÔ),~ýÇ_%³4öêÊÓ¬Ôat9Ï h~ÿlÕ ØË^ʯÞÈWö(Óª0PvÎp.@BoÑÄC鮪½¼tk3±:Á[u l,9sJ\DäiáÕ+×}mïè£pÖI0Özo#ý×Tç´bá$Ü,FüTÕØ^±,WWNdÑii ¥ýÕYº@\r3U5;Ä¥î vÁçÄ/Ú6áÅØk¿0M©U 55P¼y09·T35 ÈêÔRV¤©}êÔ¬Vòòå©Qåµök0iÙÈij
Ì9Öñ.j#tJZW¼2²á÷O®)ÐÛG§ Kú"u·\¤i¹¥½x!Ãfl#jÉÚðCüßñKí¾�2]n=4aÞ2ooaVìæõ¡éqÙ¸?ËDÌí|i#´Xñ\mù¤§g!\#¸X:ðDµ«+ øRlLõ£wÝ8^-)}< ݬú,\ê ÆDd>7© ÚÄ¡\T²ÅîfÑÊî{ÄßDÛþ$oh±eTeo ÷b2þÕüq\_ôÎÜ~-õð 8ê ßàÕBÈÀ¹ÀM?PPjú ûàs6¹ý×ßq >¾ßMû\_30û½0Úíî$h¯Ñ'·»û;î°È%ò¼ÃÍþféwBCßXëå¿Zû7r åü¹øwßHOÀk0lz°NÆ7éo|#Ì¢´HóN=Ôëx¸{²Ë3Ík{ü¶^8_¼§Í]ozfW#2t¹ÎËÉ,0^ÞŦq¦ñõ¼8�YkAi¦;hàfIö/e©÷wBa)NÊÛùIªP\ý²6øD"kJø"Å¥Hv|#÷ÿÎs)OlSi5i²Ü4Ò"Í;õP¯×gJö/k½÷¼±Ñ"¡ È9m÷h-ÉIo\¿ívDùõÆzê YeUWã®ëLØ?½òÓ¾µ²6JÃ2PY^OqzEøïïÀ+µG>ï¿æV}þëÌM0A6ª $Ìb�º¸õüΤ´ÊOH«±sT¶¹LÖÞÊÿC~o00ñ6I+?'máÏìVøy)7à@öÝãkuDV;ð9jÞ³ µZIØIz÷È{¹MNýz}]Ê'ÌNXN Æ\ä<ðÉøSØÎco¥l0ÿKïhãTÜLíþ0Zendstream endobj 242 0 obj 6643 endobj 257 0 obj <> stream xí]KGr¶¯Ä û칩ÛRóýØõ¼öÐB°%ÜÌ%EiäáP2EõïÊȬ¬G÷t÷H´ [5õÈÌxfÄßÝ°ß0ü/ýûòÍvóÕïðpõ&ýóòÍÍï=ùǧßøÁaÔͳ×O#nº±^®¿yò|÷ùÛA -v/ö=8ç½ÜýÏþå÷»ï÷bÐÚ¾»%?ß{ï÷)Ô ÞýqïøÓ³?à<WJïÏ^Á?ÝëA:&Ò½ÖÝÁ[ét¸wI~sàá¹ß³A3k¸Iï¯æ¨Í½7Õ#ÜÃðàËVÚÝíý^èIEgõÿ..¹ø5|È ïÛ} ×ðÖí^ïåà¬;ÓôÝÝݽÝs7p§Íî/û,cWÇê%\î¾JÐv÷~Z®wïË .?É ðe>#a¿Û ÇC#7¯½Wtô|¯Ñ¹ûõþ q9Õ#×Èôúfo¸Ò®G/3Hø9^jVt-¿.»}¹bPÜZ¸ 7»AZµ{³çóJT3þ²¬Ã÷K³4 Öñ<Íi Îê Ã@Çò½
ú ÜÝý¸w|Ü«Ý'0Di¡y^i)|o ¼÷)¼ëÐ[ksYkÅO =]ï8¼Þ°´ãzÛ´ÞÆæE¶¼,}Yï¹z¯<×ßá(/z?(º�'+è{¦UbzÑ]t±¢EÿÄþèê/º x^7ÞX|¢nò\Á.1¸æ<¾W£.9Ð5ÖQÒj Ô[Î1}0¼hÁ¡®yµø5î+XAÕeÎç»î¢JsÉî÷Ì Èpùàc:Ý)«u�ùCK7 X-¤ïòGýÁÜuù(O¹]yà|9GÈDð÷ee/å u¡Å3ô�SNôfßÁ)\_YÝi°Ô=ܪd¢É ·£¢#º,ú?Ö°KpðvS[
d\S2¤Å8 ±øZ]µ¸Ô|£¶tàÝàú¦»bé?Å%L ÝX ¶÷e0¢o¦êÇÑÙÀO¤óÀû@jXtëíD ãëWY\«©$§ÊR¯ªïv ÒÁ;ôîWå]±7÷ ä ïË./¨FÕzWñ©½+rõË{ãô3¯}«±ÅYIØæ .è¸Zü2îÑòôrµ6=Ó_äêmy.ÓIë#=SdØ�kVÜ¿¶Z è
¬ÈA:뢬úªÈÑ+2j"^_¥é2µû{|ØHÝGL?¶(²eÝ(\¯ÇÇzr«kI\¶60RjA¾jßÌÞ&%ÀQ4·ÅÔÀïÔzÁuʦx¦ ÊÊZj
5Pù¨�Gå²pixÍñVR\a\_ÊíÑJéCÍæÑùCg)(ØlõzNÑy·k©.¹%zéeÏAKs¬Îp'N çaM´÷iM<§k$8K oé~¶ÝY6Ï>{òì¦kûºj»×;ç·¸ ¤±åF¾C% ~VüW ¥l@%)ã.ó P
Þ¼9X°[Æ QcºaÔeÿ4\¾@í9®&À\_ÌøV^{zdËX«AA»{®öè!ÉÏúîqy©¨lþ;ZT|¼õOxQ°Pà á\_q§sôS/0 "8j?ìApV¾ó×ûKÀo°@: ë³tµ®ÌAD¯ÿÄñ«Q©Ö\+¼)ávÁû,q&!5D!ÀQ@/, 1sÎ0Ò3ñÍì¼Áýú¸áõF+y¸»MvõÉíToTt£.àÚ\Ö³¬«ðøgg%@�Á0m bûQT s3ÃR]C¯IÅ©SÜuÅq)R7OÅè NIîÌ%) Ûk®"A³\ÊAÕ¸ T,ÎùZõjògr\?Ñ-{^J\8'nðî ¶è½§9ÝAôÝzxxX%7¨ØÏãçJêWeƽ¹×sRZ?xyæÅøȬãdó>4JnwFÊ>mò ä©6ôC£ävè2dþ·ËtÆèØEçì0]Î1$ÃÕª©=p¥Ë´¡ÆÄhVEõCÅ,þ 3qGØÄAÖ¼v÷cïgRºÁ²\ã,WIpÓ8§Ë oݯ&I|/æ'ȵ'ÖÝ·ñµ5ÉpnàR«vUJ!ÈhØë³×$y·ï'qÉ\?ìʬ<òf/ MpdÈ÷8\_3hAC¿oÛÄ8"d H2Ä»1s¸³yfèUÖºÙ¢C¬rdºNÄ&ò£r¶NuÇkAÛ0yy´ìgÎ!Ïs¬iwS ÇÔ?é±åðÖ�õ¼ïÍCÝ>Ïöü}W½çÜÒ«áM&ú&Ø;³Fµ¶Éxß.èå´B¯Øm%mÅ¥ï-à%Ð#)íãiH}kP¢÷³>½v
¼YO§ %vì}¦h!N~t ²ä åfrOÓÈýZjM¡PóȦí+¬t×±9Ãϼzáçt6×
×oA7 6ØáD@\æ 5óöáF¦�F xÑèä7Q%Mòêå;t5 @xà gf³"¸#ó\ØlöF0 vüs+!pÒ£,Ù)Ø2ÜÕJË̳ÿ6 UÔ;y_@¥z=;Kß@Xzï}¹·
]OzQ�¨o@ôö«¢¾&ó R¡uWW{W¼Å¢±¸ïRç¹!|_ÒE4àõy@¥Q R:Xáèghe
_YVJªZ¤¬ïö.KøÅÍÀSÕõ©ÈÓã4óàõp«xÀÝG×h}¶àÓSýµ(S)Ê1¾{9ÊÁsFæµ7Ü.úù5j¬0_6³Éº,k- ÔèÊ°Z�9´v Ø0T³È)Qóá.ò6 ¼Vî'´k2[,((mté3§z èv}ªXZ!v?!I+\_¿&ÊUþ ñÄð×´9ÖÍÞ¤ªU=à4GÐ9\¸z$f¯Cç5ÒùôôÀÜ; O½ Ö³äIÔlòriÀ\T={K²¨§¯([¼RÒ{¿-ï%?ï º¾"bYb@"Äo|ÉÙ1~@ auñÃ/XS:|ka²»o01Ñ|h36ºjõîý7Ëv°XIØÌÃrå1a#Ë·)c^°WÍÎ] =xéÎˣΦäÜuùÙÐóÔ¼ù¹Í棺e°Û¯lgx?c°K½³}¤'ÑíMjlÅkö²£"3§+²3sþ#á >ο¢ÃØæ BiÊó·kÕD¶Ò+8¤ä¶?3jU¬]¬åÇ~)dµfjÒ»Åúû+Û#ìðãçjhÜSç1>{=j·n% j¬XøÐDJ7)5 QUåúóL®T?ÀÝòÇuWå¸ïÏ27'¢vàAøuu6©Ùg$>þØö¤KAH QM -Ñe·Ü0¢¤cÚøs_Cp>l7£ÿUÛ"#¢v wæçøº¥ºÚ WÊ®½ 'é½ìª')õÔñÏ~ÛvJ&Mð %/ïà�OÄúPÀÅÿ@ä®Á΢h¥ÛÕ0ö?È®±h£>rCÿÁðU�åK ·ô«ëIO©!gXqîiX{\Äy\Í¥ÝÈï1¯a0»Ýc¥ö!Ü|9Æt
×ñ¹ÏÞ WnKÆdC@¡y¸ ßÃS@IlĦ4s>B¸äü«òSÍÒE!bnt",¢jÇaØv½Ò&¿Á~³2~»Ãà®ê¼ó.Bt#¸Òý¯¶H©I;Æ\¨ ÿ{[éxÑË.Ã1cUð]f=àª\wB¯ëL UhìleÌä^ƶís-¥»¦ÃÖK6iÄ£t0à7í¦ºk^TcZ/ñéUã4í£9½$ê]´m¢Iÿ¨HáªWjÁÖk¾áó5¾Î , =bÑhѱÍ;ÛnÁk1ËÈzý°0Éi¶¡r%ÝñÑÔ×ZDÛnxè»cT¢Yä1£á9Ü«Y ªÀy2&a)Ãnt°¡Fpì@á êç̤6E(ÁÉ ~µÇ øÀ]5Êø70^ew·á«NøÀRñýäÚ÷ådYÐ_RrØöUð6ÃJÅ«KD0åK'W0M!D}/p´2^ãhÊõ:0jPddz(ytFî¸ aßø¯G6®SVµ)ðÀ§È?Ú'î÷h¦"#øaÔd Dòè$p¼F§òn®ê´þð&o×ã"ÌèºÅIÿ4@hTxå+p |Û°ÛK«oÄà)¾ªc%íÀü¦x£Áûî¨!vÚ2TõÜVåd ´(6]©BÀþquK©ÒÆ»3XÊÑøß [{@>6ìÍÊÕ2TÝaÔqÎ15ëé¢\_Òö¥z70LãÀøÍ3?5ÐÕzï¢Ó VM¥ÉÚ꧸ÄR´ømaðozmi3\qoÏMÏW£!yÈd´xf\G
¦©¬°í]n²7-×Ó¬2ETe©N|ZúÉuUN]v&©Õ±ÑFçé´ÎwªôZoJ°CªÐ¦¼3&v]¸þ)m¸eô¶äFòóûÆ- 4vÄÊæ+Íw
{ªtôN^MÀàPøóYÒvù\_/Ò²üX¿:iýÎÀª»xóÈIÆÅbÜ°Éq9 19úVÀü¡2
a«a^¹U¯Æàúè¾ccìäÅü )\¬ÎlèßöËÞ²w£ýþ ¥ÁFU
åTQ£Ý »(\_F{.µ@ÏrÐÙÚWJý®da4«Ýj #«Ø Xÿµ³þßs¹êïæ{M)zuÇt!eÔl%^¯ìZ£ ÖýBê½Þ'ÇXì3ü :Åê:Ú> 6ÿi¼Uê)ß:uo0³³©Í588¡ÐYØüúEúò»ÿ\Íu©cómñ\>y¨#Ka3eõqJut1TUhWà§ÃcMäX§E])Ã@PÛîêÞlO[°!6Å:Í ¯FÒ'« !Úþþ¿ÏÔMhwï¤ûªr6·^Ååο[B ÒÛ"-ÆFeà²çÁk/زb|¸¥&æÔs;o Wò£Î¥!/±1¡¿øæQëºJP1ÚõµøÀÔÕ ü\ÿóVfó]\~/×5O"jÁOÜûUÁOd#àÜ¡øuäql¨j-×Dm3Öpp§ñnÜuõ z Ç aÅ|,³éÙ9±«I\ú:7B{¼È^½äJ+RÉ;¿³1d[G|-H¶5¤]Kåäh/bÀ ûÙ\7Ðm¼Ñm V
ÍUê> øh´¾oø½+,ÐÉSøu~ÍÏÐê@ÖÕ'i5ºaV¬¬ázʬîVÏÖEEðce¾ë¢ÒãZV²®tûIS¡ôÔOcìÔ~Ã\NñQR!î¹yÊAÃvZM5uÔ"Á9Ô¨A f59îÏØéÍT=mÝäNfºbrgõØEMîõmÇäÒä©Òb[GW){¯Á?±çMehsZÿ¿ ¼ëm´¦LÒSκ¨GSÕGÇtæí¬6clUnõ\
Gu»ô¹KJ°í|°¦»¿6Ö�=þ:-ªù³ÈlÁvOzrýzN© .¢/JîåÔnRªþ
ó.êÆ>.eòiPXØæà ¾8Jµ Ö)Àd!»'5ø%'˽gú0ËéÀæYÖÃÄïjgí]S¾û´¼áLi%[)ÁJÆØuX3%ý\XÜù6å'ëÞÁd@mqNó ÜàõLzÖëÉùÙµy]´(æy^mgÁÇ öMÍxûçDÞö£¶ÄNGÎaÀêæ Ò<(à½eÔ Áü ÖÕÚ~~Eg»aúôöl8<@6æÝmu§C(³àS͵¥¬O,# I/h]êï°ÃI[Ó´jວ]·±æ.=»öT8Çd8ó¹ë绿kŨmzÀ2¯çI'ÓaÔÑI¹äfH ù>Ö»¶¥%¯jCË Æö ·2Îg<ñ ;ò¹¥ÄdU¡SwuJl.
U_·¨£AeÞ'Ë:~ç¤DÆ+ÐcMs±öáN!y_¤2ò@»)ô 98ý¾Ýû7ÊX|;º¯iVtÝôä»S:?û£àÎY¼ Üø[Õ¯X[
x×|[-2³ò &½c[æ|3\_Ð ØÝ~8{Ïm¥LpRÏ:¬ðº±íÕXîr¸-bnc'sF¥f§2Ø¿ù¦@M¾ZÑ4¥æ:aÔíIjþU¨11wóm¦V3+
^¶-:îèA³eKÁÍozÌ&Uu Vw'uþÚ µ2ÀáÖAQÏèÀ%²2RO£qÿæ|Ñ YΫ\±r ¼l^êA1+ àO%w9nv¥]9^ mÜöã³$IE²&ÓúøXbÞUíõHĤ\_¯QÒtßäýoKÙá'1¸¼óPl%×xªíÑÕ X/iïä,æKÚ£N4xJ"|NÇçæëßÓÏÍõïéTÇÔ£hmk
§U×Gsvãæ »DðD ÉB®õꬦFCmʤ$q;5þ3Áîg9~.æMôC+Ha~ÏQÄÌÍA¥% söÔrÿþÙ/à¿ÿ&°b
endstream endobj 258 0 obj 6209 endobj 270 0 obj <> stream xí\K·N®üÉ)=q¦Í÷#q 1866@¬µÒjÑÚ²e)ú÷©"ÙdÍÙYÍÊò:hÐËæ£XõÕWUd?[±¯þKÿ?xzÆV×gÏÎxxºJÿ=xºúàüìÝÏ \_ùÑaÔêüÑY|
C#>ªõzôðüéÙÅðÁz#F£5w÷k6zá½sÃóµµö¿[o¤Ð#÷þýx:\§½¶7åñ5Ñ´¼¯-¶´/D¡&,½è@¹Òj .àAÁïÔ¦HýIÝAø» ye©ëÙ=G> ?ZÑXIÏÊ f6ìÝPðçê;Tݲ \_#[ÍA¼Ù Í&áì°Û}¹æð¶§/qW,,¬>Èr=ÁwmP~@K©N¡hº(¹IFµä¤0£3jØ@Z\&§¬ÿ\)]\_½½|ø oY#èp0?, þ,-ìÃjËñ§.êäË.r0|BÃh7.^õÊeÞ¨¯Î9\í¦ÐÝ#&'ªnlئH¢¢69ÅD1e¢ßf!De?÷»+Po·G4æ°höBIÞ°zã Æ~ø9,иÑÀ<ë4 Dm%hô¼ëk9Õ)÷µO[¹GVÓtþÐúvï)wÜCKP¤dÃdÔ·_lU
¡|\Wx»=ÉÀ)ÐÜ >Ð�ðE/¬!ð·%}õ i8(¢xà öÐoØhßì\£²=¸bºùíA${Ð}ëÚCüĹ».|£¯ ]wbn·ÞÍet}óÝ#°YËè&vÝAt;ÆkÝ
O·ÔRLÌhHó£äÓË ÞëÉ 6OMþb Î8ÌàU¤tøË9¡ üöU¨èZãß1·°ÉÅ[¦£ ?=ç:P®T�fýý0
þ£&}§VËEËìzj9ZïÍw§þ÷e×?ÅÉÇÛ5úéE»Vl"{HS°"&WT¯Ä-½?(´?r ÑÃöÅZÀ"ç!ùDÐÔIÝܱϧ !hèªØô¡öX71ĵHj÷ä"[ÊGR'¥!'Ñëîa»|¤z[í]°ýPJbÛ 6v&4µÊÙ~UÆá¨?¨(9,Ózøi¶³áCDu º5üy@µ×Qµ_Á; ó7Þ~¤e&×õ3Æ ¦! kÌ °¯è_Ná1Ì×XY�>¥pð}þÝ (¿Pè]¬£ Aä¼ò ¿3 êSaßNܬ¥1 2Êï\Ñ;ö-&¶ÛÅ)c]�Eì;üßáÎ)ÛZƤ¾-ïÎÉÓð ¨Ûãu°RÉÀf¾ÌË Ç¬Ý}�tVÓ¸©g¥êK¯pÃ"èÅ!P¬Î;¾d©ipäÅ(Ï&9$Õ7ðÔLs·ÐD¨®ëµ´G~þñÙùo/O¢Ò)§g"ªÅ)¥Ä}\ᨠ«MíÕorYÎCÅ\b W´�d^7V¹íÇz$ܽòVÌÐU×è·sûæ,¼ÇÄMñ¥E£Nx
È=%Ó4!fdá}\'»B¸QÃ&lãQ7fÌñÅZÛQ@ÐG¸#P #oG
Ap0ÚÍF
kdvÒA Ô\_´ QÐq ò \r0¬°hIÈx¢=#"zFã-jAÜ<¸Í]ƧÊÓ¼ß"íbð©õr4Ì'å ¯mËÀOBRÆXOßJ»ñxÚE)|LÓÁ6;²OH3i¼gZÕå%VyÒBdÝQG?<Ó4ÝV¦Þ1ùVq>1 t[I>+Åà
W¥¼s¾Ææ|£ëPÕÔ«þñN®Ã\Ílõc\#VB»|÷3Ï© û)z°2'¤ììÍi2,Ö+¢}pvðn7ÇúÉol6ÏrßßÃr\<²W¼³OGË\ �Ååò§Ø¦½b¡m]?÷«XI©L±öï¡EÜ´¡R&Lrm69Gæ¾vQØÅÓ[{ z)^&ÒS °4%äô¸Åà©d¦WZHÄ> 1c¼bV@"À¿tÈØà4&Ý!']¦T ðºS2 RëòÒôpÕxø¹%Ìà¯ç<< }s)©@oÊßeÝèãëgöñ¶&V$¡<$_ø<_U9)b¯JMòUù$_%× BÞ Õ¸ä¸ñ#%7C Þ¾4 èô£q rZÌ)ªF]·G8Å®4Évt ÐK#üêÖns?? ·dü¦rpwL.r {Ùa
xI9¥¢j¦Ïl%ëÒöÿ$'§,"¶¼:«h#²2[¿e\8=}OwPI³Pôá V¢÷õ§+Jª^?eÜoNàn'Ó.mÓÖúõܽ~}s_µú-<½ÑìJsîä L
giGK3ÀÖ!�´E´ëy´¹¿º¯Øy¯|;^Áwù!ñ Gé3¾ÛíD=ù dQäzOBÛ{ÎÌÕæ¸ÿT"¸Ðú84 %'ìcÖOÚ1d;ÍKr¬S9aÞ¬ì¶%Ø>O¼8®¢Ùe¼ø¾å&ºèZß ]3H]3Tܺ~¿dީе9 ¨|(WVÐ6UWrq¶Öb4n¾ç|{¿Î)ú(ó¡ÕÈÁ±ôñ [ú^6Æ)á)/ã%AAa§b¨s!S,1eéôs%my#Õ°ú1�°%Ö-ôÖ?&¼+±T°N
æ©nñzµj¡ÔªýIkÕÚõòªVý ¥V¾¨SÖyQW%tów ¬ ¡tKé:Ô^w¤Ø]Íe# z+¢± &ó¬¤óK¥Çª°¯¯1aa%[.7\_):2\°r¡¨3©2r[\FÊ|Ð3ðTvðw¬NY :.§egôûõnªô0ofµ·Þm1G"Ôæ$'Z>ÔÇ3Ék !¾öÕmünZBG myJMoWȤÝBù ðþ+¬æÆäÞÒóDvç/Ѷåi¬M2¯5Á[s©¸Ð tÁ^fÀˬPÞ["è^Sį§¸A °ëY ¢ism¸Ê¦B/¾ìÐÅ®jÃíõ6FGFǦIéM#³Q}x~ö)üû?äbÐendstream endobj 271 0 obj 4042 endobj 287 0 obj <> stream xí]YG×Ño̧U÷A lmDd%Ù|+üïù²®ªêÕ¬° ÷ÖùåQYÅùÀèéßÏï6lx¹ùrÃÃ×!ýëùÝðáÕæÑ'Ú^k1\½ØÄ.|pfTõzôj¸ºÛ<ÝÊÝ2ÆÞ;6:i3[^}f»=çBo¯è³9»}µ£±Zl?ÛñÚ¾½8Ã\_íöb´B ^5¸Ûí1þè¹Üþ}»Û«Ñ(ïÕö7adÒN#y¿É¼ÞI å1ç¿Jg;9z+½ß~ÛÝFú
Õr/¥´Û\_SîÛëÜ2þ\ooJjðêë·ø !Ú¶BÊx½ý.ÎdÞ¾Q-£J~»ÔÛ8v~û?Gmi´faj²"#j¸ûíÎà+S¶bI5ü×è#5Çèß
4÷îW>øÑaTvÞ WÜ\ýüéöÏ»=¨ÊÑFЦ{ï{Ô9¢5-#Ñò9¯Ë^,äÇϯiÊÖ8¦³FRwC¹Û¦Ñ´½¡¶FNØíó2×2XEÖkÚƹjox=í§¥å-í£°.·ÀÎ\jOÉDô&æ¤ßÞ|»D;¬L¢Ý[TË ê<¯À¶[¬ K{] lo"´ÇÌ4ÒÕÜøv§í(4$«êëH¿µa¨LkÜPÍê ÅÞÜ@í¯®¡èz{»rËHI±M Pmãõ�c^vTÔ8-"êëQ¥®{G~¼Ó£t\%r ¬ÓÔVR0ÉL¨0·]hYÃMj=B½.$çÒB4o¤¨01nÀß4HVËfµ\_yKÅÍÝu×åëóò5Ê´R$õX¹¦}¸Ðí¦Lü9Bh}Ýë¦PF O¼ØK¨àëÎ¥u¥.JiI-áF+èCO°4µ÷d|º+lT¸ZÒ^¬¨| "¤r»í/wINd;òANLð§jph5¿ÚÏ´FÈ~ÿ>ñ¼D?Ú(U JHÓpÐ î0Ù£ ÔC lg= ôy>Ê{ò>>L³'dè÷ÞVIbÄÌ\_ë!¶{9ìí)"\k À@SÖ©c¤î¨:¶lp£\&ÆM³\bZ2#L43fÝÄ 2XZÞrÍð´håNï©z¨ÙþD66óy¶©,vRµ«r¶«&,ªì\0äêóí´ûÅ\ã[B:8¼(xhÒºª{ ¤ DMRô·+-$Á¤ÇãçøLs¤û~ ÄfQ{Lý¤:h¬F¤ÆôwÀô»ÉFtM a&ZÖ3Ñ
õAôV@0J M²)ÜOA¹5bÑy~V~PµÖk oÔxw½£9MáݼÝaZ8ÁWËf¼²ÁÏD�ðìEï«2ÅÏÉc ü~rúDØ
�EcW[öªéI|3'!9ùïwD'£Ôtt; T¢¤=¤0xx §òå^ð¼§j®ÚsÌÆßr]µî©0²IüÌóëËDjü¬ædبüäa°",1˽¦O¯»QùÚÓ äå5Ú^"-»Ú�yYШty6¤HóuýbØU¼©¤KÜù)°>£32 "¢D öW»^ѧSiã¡$RD«�G²´=}£É=Ve±f,«úg;ÄtY L4"Ù5(ô²YÉ£?g3KÒ@ÙDÄ,XÛËÆòÏÖ½©�czüÅôà úY-ºî°ÐûÀ2ÆZwv�^lòUóp¦U×ga$d yí:- ËÝÈ¢ÆíÜ óZ;ÃñÉ0Ú¨°QJ(cý(ÊÁTÆÖUPÐدHýáo¿(¬/ãg´-pLêUÓ kbÌíÍÜW¨>s
W!Á#wµãõÇú|yz>i$ßS 7ª®Bxëzti
ð q¯We»^ÇU¬V±£ø§Vé-ÿ:AY$HºàM¡ñ5c Ïuck'bMí &Iû |RÈJ1Ô'ì#mxfûÛÒh2³CzluéC´ÇY çÿϯË×ÄSBÇ8øe zfý® òÓ¿t?ÑùÄô,ÃgröôѤ¹õi(£DÞ'|Î3wÌusæRTÒCÙ|øOÅ´áO³ëIMÊÎJf2ÕS ¬H3�ÑvU¼D<o(¹¸D&>Çl!Túd.W��ÃÕý3ñÆ7UnÄ"<z1&Söe Qáj¾vôä+9«.R@q¿H~?Nßlði¾xÑû¤û#«Æ/2A}¤5ÒWÎt(ý@Tÿ|rÙ\>G5Ö§¥åmú¡ÏýX´ï«l'Ém>¥Ùc]OOþþåÑçƸΣ¿&ì°9ëåÙà±2ë\_#tñN´]í¤õf<�¤«í?;¹µ£k«¼Á¯±3¡ºóò&ã6{;O¡H°¬òÆdðì; Ñ;Ï -Ú %ù]Í yÚsÎÚ<Ù$kÛ³6l¥ô<Ö\h?8ÒÒÚô´l0ÔÁö'ª9D%=%ü ²Î#¶$Àò¼ÉbC¨=}{Cí<µ3Fïð(4d¤G,9¦yKXJ¯&
h]M)BÈtW6|üã¥$\ÒûÊRÆïÌg|î1¶uOAÅ^WѦfiP3®MV÷éè¦ÊæÓoKNE¬Ú+/=p^ÅæSØî ;Ô5ļüfhAömAqHS»fbÍëù¾ÖìN}©^C9ßûîûÑÑ8J¥ÊÛu£\ÁG#ÛEH%_¥¼Øp+(ôÊØÓé¦b&³ÏÁ=)sÒ¨P£ ti¦D¿XźϽê×sç²YêÿyîÍx?(wè¬Øbpºr2ßsO¹OóêòS~®CyGÄQçÿçóÛдhLG ùxmîªIxÁ[ªÚi\UÎ/®ÿÒð¨3fJ\H}µ %.ýHww02Ì/gw$ÒTé[&s÷ÚCrÊ«õ6ó!"¦pIà´J9çé)dÕþ~õ=¤°ÃÁZetjÀW;Ü_)ÅY>1¸îôðXðòá3Çdñ%ñDFxï Ä ©ãÔDµð1JÌEJ¬ËRPÅè~~é¬ÌðÒ¶RKíÊUº«zjù¡yý´ª©ñ
QvtiAµ· f=oI5EØlLbG²Æ$Cø-÷C+ÊÑ íè<\ZÊ6 Ê6~ø¢Mòy¸úcì÷h%åã?¤PLUsùê¶Ò¦ÿoSU¡5óÛ Ûe1TÞÓZeÍøô]ð¶-Ý{m9ëxÎqïìgzºOvñ©jë JY|©ÎѪú·tCMò
×yÖ½¹ñ+ÛagÿHJÛ²=
õ u²=Ú<éjº«;Éo(MÈi^ÞÕ Géê=ÃP×úĺqå-IrQd8 .çR«®Ù([UOvëqýı7
N.À[#ËhkðINröªÿ)±D ªÿPA÷ÒuI>®9X°àSô üªêþ¨µ»gÙ£=çíêÜ té°ëÃðýÇ°Ê 'Üê'ºË¶MNü0êÆÒµù´ð®ØÝNëxaißÔéÒr'(¼;¥TîºUÑ@¾(.wÁlº£+ùò]wµz~ª ÃýÌö P,.òKHZkQ?ÁR QÑ<|SnÏÞÉ�üN¯aPUûÁ)çCfù] káôBJ,[÷þü»c[÷ÌTäGÀV²{.Á³¡ã!TûU¸]"]¦«n4±§×|T1હ+X~Fm ½ÝðNOR5k;¬§Ët=[ÿ~GÇÖ|ª �îÒ1^¸ "ïó²¥3ÍóZ·|oßLÒuñ %¿(»õEu½@Ç;³}ª¦Æä
Ë£ ·%Àñôê×
7»P·é¬¬kÖ Ê FÉï¢>Ýïß¿sèÿCþÊY¾î-±fhÑ:h«Ë?þÚMÚ÷¿ªý[yäbd3)Ýwñ²´¯LbWß(åñõ£0çMy¤«ÞmÄéçE˲ÐE/¬}©ÊHµ÷|ïo-ßóEßlNÙÈVÚf ©2Vɶ¬ cתè¾9¦ÚÛ³=ÑÒ:¼þ"[}qvÉE¦7¸I2½ÉëÕr@L¤X05ÍÄ:®ËÏXHmváΡ ï¦TF Ûà/=¹(<«_º\NxÔï;P8-ÒV§35mÛëÄåëy7Ç×íª[Àýhó»ÈMf]ór!¾µ¨ç:õ6ÙôµOiï×õO ÒB(? ±ï?ÖY°§"]/:ûå¡u¦«ëNutç~]fÅ»M0iI7áª5é¹PEÎø
¨õ9zCìä³kÕ�Q¡²+ò~µ$²ùÁÅÙp>ñåîʲ^Ò$7p\ÕR?¿Û|øxóèñ}õÍgGøæÑïéþå7ø×ãß?Ú|ôxøxùÿÚ¡5)ÑyF:ôRÃ"æOúƱ2¿Tª)æíræ§æ(¿bÇ´¹væK'~¼½¸)Ý I³Ø¾YGN}?ùà!Ò3tô}Æö%<Û3öÞÊJø°úú÷à¨,»Ã÷©
N!½ »°lãêæöÜS²/££6"CO9ïÀãÍ¿JoÏendstream endobj 288 0 obj 5095 endobj 305 0 obj <> stream xÍ]KÇq¾#xòÁ\_Ö'Ï:°íz?ìàAä\xå=8BX q�" !ýÿMgÖ£+³ºº§gw :x঺ºùøòˬÚï®Ä$¯þWþÿúí3qõæÙwÏdúöªüïõÛ«/nýËïl¼rÖª«Û?ËÈ«à&s壢¹º}ûìÅA95ß@‡BgÓµ˜‚ö¸$_«ë)T<Ü×QYüáëk59oÕá®ot6öðÅõ9¼'ÏþéúFM^Y¥¡à9c'xÀÉ}ø<0Iø¾ÄÝá é ?;¼JïÓrîÔ(y¸»ÖSô:Æ·¿I;y§è3iÒaÒ6«Û¯ÝþóÃØ61ÈpxCjRÊ>@¿ÖF'aF8¹CHsRn ÎÀ+nìB| &?üåZé ÏÈÉ9+P ?9/d{^]»Ij|ü%¾_OÒÃñ¾ æfGìU:üxß^ulïz_¡¡¯©AðN±O^Ú:lz_&kÊáRÃ6Áú(áq0Z9åòb8LYÔÿ¼úðm{æK;S¡ø¶¦å¼£ MÛ ¤[ÉIEÛßáGËñ± á8nó A JmdÀIHÜÜQÜ! |«'>m [!a )·÷ ïÚÊ;è ZïiÄHQÜ9!k¿¨LÖÄ6LQþõúöØDÓ&v²2FU;ü%LÈHfwøG4rNºÆ@(ϽiQJ;¶JËÿa&²ljgróÍøqÔÃËçEÀ~dQw°zuq?kÿnØ =露7Råä¤ëh¾M;¯å$Û¢¶/tÉÔ=s·ðSÛQÜ)ÍxîîÖûö6JÖÃÍ~528ÄÎ4Û
?«0¹xÿ(ÞÖº<,+ùIé& ¿C©ôBñBãÚ÷Nÿ$úÓtáOy¸dÛ´Ð 5½yÉ^ød¯´è\¨{? (ÕFbæ÷ík"Å0¹ÈdIpJ¿"øÚönyÛÇ
ùëÞðOø;)IMÏ»yCGÛ ½´zÔ.Üѵ'QK3
ù÷´DNòÞ9é\NÉ/È ú
:ù¹Á"%
Sª]Ôî÷Y"Ý+)ÀÃþþúÄ\þý:�ø>©Î³áQ6À¿H:È;hݶcóÖ# 05ÐÊÇk0Ó0n@©°ÛîÇä$Ý¿~ q¸ÅÔÎï�@áFhO-Ú{Y@
�¬ÕØ»ci=t@ZNU¦-¶ÆÞжÞ8î©Jñçk´/VY²%§¦J¸� A7|¦³JèÇÀfYn³ßl6Ê}³úÛRIê"ÈùØ }¤~ CªHlQµWå@¨c+Pf KÀ"eõʤÿ-ÒqV³¿²M]¿åkTÍèÐ"Yî¸ì¤;iW'b´¾@é¬è¬�ç¬å£Öl%Vù5]ºÚÕiqgËIRÐ<ùY;£n+vÆCÀpiD¬vûý³¡ôôÀMÅÄwNeÆm ø;-ÏT¿¼F_$ÔÐÃZ^PE)P5[�ãt ±±÷y Ëh5'ÍpàîA> ïîáK£÷ w¬WÐDè.$ ¤¿6®nÀÓxn¾v¸Åtá3YÈ~ºÝ "uÌ×íP´3PëFº»:a ÙðR!+ûØ3ÅOLú±K& ÿô#R%nÒQÑÄß7#r̽¬¾X&
Ô#b~²BÒà{(ËJMæa:ØdÍë
{åKÔùÏêòQ^9¤sÍBg�]N¦Ú"ÉAñúÀÜivh-ðN1ð66\f>ÇéY§8 ãN©¡iJ ÎÏýñ5Úä'ÞL¦Û«W]sÅ£íA¢ !|o{øbOÊ÷·È&UHËW·ÃÎþIU»¶¬¶MùIöªBP3#dÁÒPo Ôôߢ¦&õ
XÄt,öp¬×&oÃöÎ\ -37rmæzZJ0ÁQHÆcdíÁêyJÜP¬ðÇQ4Z ƸÚP¶Ñr°"_@Ñn#}4Î løí°ÐR·î�� &ìn° É°�öÆ=y§V±¤±Ðs«¿|ΣÀ\[z£¶ËýOæpÅÌ"R9Ý9ãÎçCî=À¬ òÿ(òÆýüt?¡¾m?ÚÙ
ôqL´~¾¼¼·î;ðÚ8c'²VÆ"²IB8^0~"¯áo ÷§ÁüP$Qº@vP)×ä<ôM¢ÁA´Cã<¥74.Ä(0g ,øG8Ïqr²°X}U¸s� 7 èÿT9XB
v¶éV3÷}´¤;ó¨w3?± ôÆÚ} ÓÞλ§é·»Q/ãS+¬ñ²ÐÄJ Ïs8úS&qÖ #$˶aµ¦¤WòûgCÄ
\¬=ÒåÁÀFûõ5\«Azý¡eH4 +·B7äaI¦-¸\_0PÔqs~´~I~×hSA· �jTÓ0êJ»¸#$>õé?d+ØR-YqwyYÁÑÄæóë,Òn² ü/®"[].Yä5a^èJ~»Ú§ÈC/ ]'#1vnò«dêzwü˲Öý~ìX7 #~Yað£Z@Ì]f3O¾$s¢W0 îÇÇHظ±SëÌñp½n^ðð)Ü1ÏÐná.Kyt¢U~ ½:ilIo"Ö<úïBýnXW0zìà|6�3é�-.{9ïgºª28Ôæc E.5ð:YìÓ§{Ñ0X¬ÓÜ ¨´DðóoËb(@#_I;öÌ Éxô(Ø Íº¬¥ êSB¸XûG^õ>÷¯à)h¬ý#O¡(IY:I=©ÍîV2Ò¶ø¥ëÝuí|ü&ïVÄÇld#p2ñѯdæ åöÈVÖ¤å|p;Ø°²YÈq|Q:¤ÖRËZ,Í=/æó$J2Î":YXN=E¾¨hÔ?\¤pµÕ 2ÊIK#Óù#±s¯>á®DÙÖ2#ÆI?ãêéµ ¿øðA�×ÁþWJL xñJç%ÅaÕÒÜ ÚIDl´ }¤vu¼·y%mÌäg÷#C5L\r8}lÆll!N'Dv$o¥²Ú6osOR¿e¤êúߪ.ðEe[ÈÒXMÚVKGIoÇu ÷LItaøª>E
Áz¡öTPÌ£^å%ñ÷æ=v/Ê@°»£ )Ä8ÕáwpÑ/))èÄY. å©DiW9ªTHxRÜO2Z P5²£³µ¶²Ù(.xýj6MTMrùÚQ Á¸V_ñÉk}r¸A|,qçËsýÎlx%æ\ÌgÝ¢¯¼ä¦ÆßîÈ[¦K3k¨18cÇ,W~5Ò[YÁ}+ø!üXú1C>" ÙïFÚwLéx#RÍ%{Cyld¥oêrÌî(c +ºÃ mîAeD_ Õ¢ K}رÆò¢÷ÑÐdõq; 8ìCø±jȵ�Ýá8+À°NsÑbvȽ Ý!åFÏüÀ½æá5AtÄ \Î!ßò©\\_Z®X¤ÓéBëc¦c 7ÄÒ3+¤9ÐæûÀ1¨³(3 l¥jG1z¦dÈüf0ñÈq~Ͳ#M#mV]Ħ°?&[q·\_ÅݳdÈJýÏeÿg<ìWhN<}«½¿[Á¤ñ&ñ$ÊòÓ[fLq vZÀ.0>P°YdÃóÈEu�s7/ecÚT"£LM\Vß 3èô9&V|xº³QvÈdÎb¥ìÅZòÓäÔÙvÈ¡üô3xòËäeÝc^&/8Ç3yªy¶{4¾½°ü=WsÙ,¼,8ÕÃA-Siìåcx<òÅ$ÿQë9¼ãP~o.Uî>_M«6WÎW;Ó¯¦TûÏu¾úð2¢ÊzÅ«ú))¦ ¬Ùi.:_j:Kë£""w¹ÊD®¥Vô·õ|v¹¤Q&÷¥ªí)I9Ò§fyOQ9×QÏKoTÕtµÍhÉvUå'~¦Ré Ã]çÓ®[VUËN
®j¨ÆlZoÄί1XÈU+QçpåôÎ^×=ïeK!<HHôê£e²µtG·Ü~HZ£Õ||ÓÄ1Õ(¯OêèÙ? ÒXîÛQ.)º- µp)ô0Ny°Ã>CÔ>Y5º]T0s=N1ÆtH丨Äì jÞáSf~GÜ»qOJ,6Û)ÿå ÎZ©s|H×NBle¬ÛÈý§4ÕB�÷ÇD ´<þDd¬7Vå©Ûç<õ¥äÚTk«
àbRãسëç�ê´Ù[ó¬]ôdÖú̬oð|ä¨æRC2¬²÷î¡çs/BB^ ëÍû/
\úWeûé1�1äÚIY!äúémÒì§ÖÏ Óüu_÷$@ÆT ©/!½Î§Böaå|ì~¡ÏCç¿ÕxßC¤%¦o_¶¶xmþÏV©µ¢PÌW"' J¹¯Ûe±Ä|ËXÇÏrö´3Tn9îux÷ÓÇ>a2M¦#BçHÙéeHJÑ·Þ:ð X¥~D¬Á¸úí8C ·t¸Ý <QwÔ;¾¥\¾wt}æu»,³&àën¾p¬;yÙ$I~q{I";vV3I²ýS)Ùj[5º{w1UB¼¨V®Q«ª4ªYçøÅ ¹äg%΢Ði3W=¯iC¾MK¦3dV£®J¼,iª¹ »üF)3&áÊaÎF°ó²»ìuoä<ÿ}sõôéjËK¼g9" ¦F»"èÆ^ÔøÀ Y¿n¸±ÓäpGûrÜU¢Ä° V²SmNã3éÏWz¸!Ýôn¶eÕlx«e" gÀÉs@ÝÖÕÀ?½
ç²(
ÅÛÝ1³£÷T¾Ào²õv~Zé·vФÁ×ãô¬Y,) pýæ¬òÞ°·32Ýá$t/ç ©6/S×ß÷éþÆj"nL²ßÕNɶ¹@î+G8¨Ë,S°QÍ¥¦¿sy´ <¼3ÃÞPJcí>¤ùZsDã~å÷@ÞAÙvÍÙ[ÏxZ)ÉmØϬÍz öɱ\ZÒÁ%ô©¨[>ÎT®ÉÉù¨ò9!9;2ÚubCpApí|uóÕðc£r/4)>Ö'zgYÕPeÀzÕ¢eæÀh Òÿ-¶/~¯§æã éO±8̤PwvÁÓËm{ðE®}¶$Õ3¸³äÚíF/ç¦{®0ÇÖfS~VµRØ3äãìW>CÇ ;¬ìP1Ùv·¥$qUãR
«eÑ2NØ8Rkì hz¢3lòÉH?iר\r°XôÞ±c;nØÖíë?B/º,6;6n+PTû6ß?ºÊ=ý®Qå\_°4¦Î¾^zõ#ò"ÿ�Ô¥9ót#¤¶|×| ¸,Éo} Ì÷îÊt?ãºË,Rî»ZÌýWÐÔh»rüG¶bF~
Í7OSj¹àÝ!òéz9ßéi!y|ÜÓ²ñ}DÖÙÙ=p³Þ¹F¤ÌTÌÒ#vzDððY¨þ6 Î]u§uÉ
wW<ÍÝZ¾/<Ä^býþôGþÃcà }Ì'( xÞ^¶b¯Þ®øÁÕ+b ny4ºÄ(ëA.ûÜ: üÕ®ÔîA%137tõÆ©É2^jIj+Åà ©Ó-Ò¦$9¤ÓG»lº¬æ<7ïI Í7ÿ$ZÄ|欽ïvå$Iã6(r-ÖW aXPü¶±Ú¶|Äîp·|lÖò¯Ø¤éÛIpV5lV+oz³ÆXnÆé¸C9,Q)fï}ûì?à¿ÿúPendstream endobj 306 0 obj 6048 endobj 331 0 obj <> stream xÍ]Y¯$Åæùùa¤y/?MÕ¨oû1#Æ,æJ~h÷Ã¥º ªóæoÎ9±Ì¬»4Yru3c9qïlQßïØÄw ÿþ}òí]|Á÷»ôÏ»¯.Þ{¨ý«I#vWß\ÄWøÎIí¬×W»«ö.ÕÄf~ÿúp)à¡G©'xï¿>ÉîÝþeýûÓúñ×y»ÿ¼dÜä¥.£jþøê¯_]|+¸éZ¹óÍR/aR%'ïäþ£tÜèý«´U\Áú$87~ÿóA¢6dõ?.ùÄ4·ûgåÙÞ¸Qû'ø ³Æþ¾5jr^î¯pGx«÷Ïë³u¸ í¨þõ¬§¬ÙÿokCíÿrpnbÜêömÜ$¼µÿr4Ï÷K69§¼û7ýLv¢e¦O_ïÿ' %£~ÿ ~ ÿeÅþ!(NGË Ã\_icñd¹
W¼Ý]}zqõïÒ6½v oH¦]|ÖMVÚý'8·IÎMýOðSηÎ÷l@×2ö©,ðGO n¤.w[éÐjR"²\ä3kUf3ï3i±ÆfÌé5>CÁiùL NðÌg>ÃgoÇgRI øLrVfVùL XÓÏÖ$&ÿýÇ8¥$FcBkd4\7ÏóYÝAÒ/±p)WXIðÛ³2°ø{ ßùÉÐWôîRH 㻫§À#. .Nìü¨á}k
·\ ú¥ç=hoRãQÊO\ªöïJÃ#ð-î@ko8ð¼ðÞ9deAþ7"1 hXpN u é@òÖÓf�>£þ èk|[¶9öò ôÄàÏd52¿/ai³NãYy'ýݳY%Ç3xRß"<ÀøÉ·ÿ ÕYý8¿£#ÔUºæÔH"í'Tuÿôm]S Dî÷a×\Ä}_¥Å Ìó!GA k D�Ï+æg@;UjßNNúðÖN{¾?cC¡ ]×·Ì ®±fXtä «LEU.!¬s]¾<¥^ Ãà²ÈPÇ!oEÙýWÔB¨ÓÉßO ßÝ$û_ÈÑ?ÀøÁ¨a^Ëu<,]¨©}ûÖpú3AÀ "Ï+#ã<¼ Fì¬ÊÜ)oøäuöt'þ5Hdz. ÙØSºq< μË'ð¼òfYT£\ÞÔC8§Ê\_ü»å
]P\éݸ§s&ºUOÞ1I[.RRÇLkf°üÓÏá%Çy>ȤÏÐÆÁºEsÆÖÁYlq1YöÏmAb�!Ñ8ý-®çÇO/iEõëW-NìZQ=\Ú:¾îüÉPË+É;Ö �®ãËô¤ãjÿQ(N¯.eÈL1·¶ÕÃÞ¢®QREdÀ¢Ð�03yfßÿÅ5à|#�c\fÊâL1ù# ${°7I÷øä Ĥ¸µq直8"T:�RMZ)\z]¿'aádQBPÇÀæòþKÐaRÙÍiá°Qj²®ÈçéB¸ðäx¬[0oðEÀx¶ö ¨ V�KØNZÙùpD^E Ìð2\=3Yóà ëªü6V«(S UÍÕr$q¦°Q3 å+MØ&Iºq"r²-ÌÍßNCóvø 8FËæ7!^F£á+Ú5¢K¾�[-(ìHTzU×$N}Y¹ W£½õHÉg Ìuý6èQnpæý±Ð#´b ÞT>ªk|ä(¢ùV,Aྫྷ¿Ñ_ñÊwþHáî[ l´ý²Vç=9uö1È¿e&Þ÷©ã©¢% ô 5°Èf2¸yqÛádäþ(zóqT3OínG«¨X!A3½àÛ§í1¹Èµ½úÆÄ7> ²}à:+nzK 3ì¼VÐUì<¥Kc¯á\ûG°�+³çe�ç aV°_c30àl+)½Ss+)ª N¼~å´FÖÚwò+Úòé1¾Ì¨àLu;²6:Ü<îýÛÙªÊgAð¼¥y¶©þcÙôpf LKÏ
ÚY5ãå=:Åîrna
MjºÉ¨ ½ÆDrDWA²¯ß>.>ø²]·.çODÖÅÀC,é.óèúqÍÊ·yý¶Qײ!Î7ì·Í# LOÇO¸o {¤öþ) HThÕ9#-hþá,Pçp¿X©K(Æ®(0¥,½_�ª¢ï+'££×"-U§FaÛ¸M^b ]AL�0¡k\HÄËßfðn]ÁÇ:XKÒ°hdTK%3Ϋ¡Å@}ÃÛErb�£±6²�Q
½q ?õØ3q$Ãnù¡M´Gq +à/\_Ú�S\_÷LÙÃTËÑ4&+ö HF3óI 3ﵤ@Yðú
Ç=ÙþÍ xÍP÷h\_ÈgTñµY0Õf2ê©òè·ÅS$GÜ¥¸»hþýÚE¤Iý£ 9Ë&ø>ÔÒãÀÎXvíK¬õÙ\ÁüÁÞ òdU dzZEfùw³oÐÚÀäUFgÁG2ÑIÉÃ/E[Ǭù!ßsrÊ8&Û$WFØneûø[Ï-B¯¸b@rdFJÆ]õ:»úÚ· ¶g»©Æóm,ºÉV'?0@ÙÄÉ-ÇÑQ¾.55JW÷T¿}ã6à ,èÒêxªKóKDVé á>ܾ�Þ& ¡ãlØä=ß~©~m$¢Ë9GeÜbêÔX x\MË8.å+Hú.2�9þ ¾Ì�HÝÌÅè¹ Lоó ¸úí9e p×7ñʼZÒ7ÁoÆÜ�R2øàç°lJcÐ4>hH¬+̬ ¾ÔuVÆ;îµbH0pjðv)mOv}!ߦ´!¦+0mé6$A"¥ç#ú,+JcJ >I®óL²Áïì®ÇãSt4½
#㣵tt,<^3¦¡Bɳ®ópL&ßñ
ðgfÄ2¥ÔÚÀ±íéÓ0ô:}ZV)WÞÇÓñ¨4GÄÿ "l¦ïêh9<¹DE)#ù2z2ÚÚ¡,avÏE¢¶
?ËXп,V rõÛ!ÔMòö°R=&½c¢#ê2ò;åadÀF3Äs(ÿ$>¬äí tcÊ|\_!m ¸µÃIBðá;¡pt}ù¨fËz]\9¡Åª·]1zÒEKîDB¡Ì²¿3ªoI w:¿å<º%µU7[ªõ r¸.Ëç6üá¶0tQ n WᶸlFcÖIXÖ¡÷dQ{À)Øháá\»i¥K©^¯¸jµøB{ã ÇK"nÁ|):Ü@b¬.'Bàôf¹Ø:çyê·Æá!ðÇèÚa4üòKÐ@5I¯vøæ:äøC©.í4TN\_ç²Ç\IFúð
W2i¾+óÈ ÓßÀY´¿r@Çúøl=|e=è¦ÎÕá1;5T%\]½�}=ç¬0 ·jä¾Ä !3x´ÞÃЮ&µQblO m®¾\_Çø÷ÇçLX¹4á®:6dnÎÍÞ°°áùuJH¹hSÿkäÓ£õÝÕ&%Vºï(D5ÀAÉ8ÁÄÊ×6vóÉç¡¡¶ W¤opÀö^Ï7I= -÷l}áXÅO«)A@¿Æ©0D]TÚûQÆ\_ S>8¿Æª¨{r÷%¸/Nó·îîsü ß2J5D¹%2�¸ ÔVl»©uh%ÄF[jcg3õò[s¨u¡Ü£åê2È'ÐÆæßÀ. ¶½ÌF ã�A³SƺYdG²§a8<½¼éVا)LÛôºSÁW¿ Ã>#s×ñËä©;Yh{ !ÿ kà<03éÑ:<ݪÿNUs¥ÇîCç¶øþ«ÚÞß²ðØJÁG¿\Þ&JPÊvR¼K꣦0S¯B+þA{«M1Zk5K ¡Èì:Ié¼á<(-×y802ü.ÍaèéþÌð³»Ã§ (·o+3o4
%§Ê°P{WOP/¶u
·Ðü4ùÀ-$yö(?K©¶ë+î' #Я�û�:p túç¡®ÀÁfbÜ4±á«ôòX¿ С$DÎö¨,d %åþ 2D÷¶H
Ê%¯×1>´z«Ð)$®uº¸\ËbVÇã jÿ9¼Ã¬V¼×¨ 9fÜ
åÎçTT(dÍA(Ì�ØW¥õj�ûغLÙXw¡<ÞPDË3D¼\_ ºÿ³µ+y-½ÇrªX)ÉõЩôÑæ:8S²¦ó?FêiíÙº©oÃÀ3t\sÍ4'¸Þqr,¡ék5SY6 Ktöx5Õhä=\ÍÚxRìê! Ç16Zx¥þU"BPsBX¡ w «)Sù4rzn ÓÁ¶´°i !½P»Gux×ÿ8HzZÎýî0£xlÒJȾ|Yûñx_zˤW· é®6¥4íÀÙjÙWCAN
Ø Õ£ÅÜlñôR/(RFÒç}_¾Achðôjdº~ÎÐÄ·ZÖ}.>DaVZú ®¹ç¡1yx¹ÊSN±ÖKy9/\-<3a& iE#%ä\_{õÝÅlðvªöÓÆhÊTg(lãr3$ÿ²hëÑ®ÓÒa¦¢A f)¬ÏaxãP%]ÞV«(¡[àѧ tꬵ\_N,ÌLëÔë�©ÿÞ1KåzÄoCÄV\L´]ÈÇ=¿ÖQ:7ü¡ï ßVÆ·Iv¶; x
î6)D[\R
h+¹ts(ªÃofÆ«e2È8êoàG#Oä~ꦵ{Ð~¬a�& s®Ã[§zOWtäêJ.S«¶?"EÚeý;ãXÖTãá¢áuê"nEknbqÓbYéDºò5ÿeóÎ|P\Àh0L±<ËÕ1ãÊ'KZs¦ÚÒ¯# ËeAT7 Ì,b׶YXlc§«K]yz+ÅÓ|:¦Õ{H'\Îx'Õ+í\|Ô]|YÀØs¾¬ìó~Y\GàÙúj'ÖXôB(Ñóýä¦ÅÔ"ÞD KI¦yd¹äÁ4>CúûaØ«©~¨RÝÖGaF±WÁ× ¯Ì5öÅä{Æ \_hÎé�C ¶µp¢mnmbé欦ö§ð »Ãè,D§yáøÜ5P r2f! _ðtV=°õ ô5µßYǼj+¹Úª¬\úÒVLñ\5¶÷1Ilg_ÕdÍJÕ×û(3ÚXe_÷÷%L?8 ;/¹±PtÄûXáÁ}R·«Ó¤ÕK«8Ýê¬Ú««#Ëó|Ý°ê±ö+o¹&9T²©\©IcnäKCR9ÞRç3¾?ç
åk.FBÿ~éÊmäèvÎîÉÑÝø¾ÔÈm-7Ë¢®ÿrÀ+XÔø¤ccÉFyA}ðóò¾Å¾Üx{/ ´]ùHD©»øR ærT¦æb[,Ë>?f7ÍÔÄRÿ{H^(iAXÔÈóîà$Ü2<1ÌUó#F7¬É 2×Õ
¤rä¶Wr>¿ÚFGæ~^c®³ »ìkµB_¨>êì¦ÜÃÆ ¹ªb¢©¹¡2ÐÞFcWQ3¦Z¢¹ ¨h/7hxXÕá$½·eEkMq4> E®i¬hUçXQi£f´/ïy'Õ PçQ\þÙv¨ÕÞ³/o¶û¥w5²¢ªY¹µl¶ël{¦(!Jc}ä9Q·moo@² Ä¥6Øÿ¸R±ÌþÛé °´ºþæÝp¥"µQ¯¬í\$Æı¡fÝ�ß®ÿ ;GÏ?o¢²iôÅX]²ÿÿuË|³V;&MJwÉóÌÌý³èaôh5Ò¥®fp]\ &\ÈÛÕLy\95l¥BlÔÒê··C)깦³W1äç³×ÏrÙ¼òmB$LEéðÃðí¸¬é}ØZà9¾Ázéq»¸5ããñí¡«THhräJg9+øûh9rÉ(Üÿä$4¯lFh:._¸óy [ØÀ19§2xKLw -³wËÙø®õ3
=lOKymCO=å�¿-ß½ç^ ]»Ð_DnBj&6>"¥´MßÏV[íÜÕ®¸sÑIÕðn@#!ì&)b z½tx ó+tý½´#¬ynÈUßEæômI^5ºCa5¹tÙÖ
·R8? _ö5Ú3̵ ߪß4oA' kñgA¶ècp~_ë|×Ë0Z ®årøÅfþtýüÞ¤s~dØe¢~Ú6{3+ìV-
ÿѩص^Å-ÒP_"ÿ¨Ë°ixXÃ1|fÜåfß]NîùàªËÁ¥ËÚoèçHJ0Ý¡ú ¥/j?hmçÍ :×æõin¬¡ÀÞiI´û̪Iñ6«&Åy´Óej~p@VÀKñªeråÐÒz©¨|¼æjqH¼WQÔjÉùvñY?>³ñ²RXIFpõY[¿}\K7·k,C°$OcÜ× gLÁS㨤Õzi&ºxÿǺÎ5i¼¿{ÙB×ÝHxºæê?ÚÞ¾°¹{;¡Iõzws wî]§_Á¹§;X6q¯È·1Y¡=kôþ°3eöëPÆørÒ×õ×îÞRvpÖÏ6PÅÒÔXµ§o}\_gËÜzÿéèÒüs=ò©¡óäñbÏ}ó&;þ}?ñXGµÚd mò4ö§+i!õ2¶é~FfMºÝîpÓKðÇ¿w�Y±1 ïäf^7"JÔ8Êk{ i}¬þ)'\ éñe>3 ¸§Éé%¸×oTóæf|Võj{bå·'æ ýÔÕ¬õùE,Ò×Pºù]E·!S)óNh2¾ßº³-#6!üiSÚªMüöGÃp
©¢ãFpì³,Gv.·a1CϺV·$$eÊôñÆÀ;Uj«RG-õ?Ë÷çñézQ91±:åðÇæ°8½!ÑggÃUÛþFV-b ³ù§ÍÚ®¾)6ßC\}Åñ·è,^¾õ\ÿinnjl|Þþ¦Sí¾¿hö¬XÜPoe.p ÷ $Î32s§kº º1Ïð±+u
?Ò÷¶+u?7¦r¬í}.)|~µT°ý(/P ~\©ì ¦j"Hà!îéØàõö)ØT9ìþ£Öèè\ Õi¯ü¬É7Àìà¯l|$ÏÎ×&ôl ÀÝ½í¾½Ée~]AŹì7 ¾ZÖqçÖàÇÏ(4Lk^¸Õ¹Õ¨öÞ¹ÊÁÚ°0Ág8¿,gV5Û6C)Ëëð$¹~±´¼h?!éeB ðÓäßïTìYd;Ta;%døµ#Ш°{òÃO.Þûä³ÝO?¼ùú½¿íøÅ{ÿ
ÿ÷áçÁ?üy÷ÎÅÇì¾XüónUñ¯Tó ÈÒoË u#é u¿¿Æ-ª¢ü%R[TeÑbd3uw¢-UÊ6Ùø9L�¾ MoëË-ÄùH°ka>¢EHþÔ{-±×¯Ú>^Ðÿ¿ÉñÁê{8^®¹Þ!aèµÓÑ^ kêo?ÉõV¹Gbº.ÊeO5Qìïx§vûzºðR æMlø»¡¡ ¿»�èÈnV3F6¦3:�Üç=ôw¶:¾;Îi?>¤ä ¾!ðÆ pÍ<z¬ÎȬ¬¼±=îÝ°HzVwr2;C_2äCC¿¾ôÃ:ù;¯ÖϳoDïGöqí|ñÞÆË-hx; · Õ[º[ÐÖ®¶ÿ|n¤·cx÷gs¿ïïÒ ®ßºÝvcvöfs÷2ÛsTÙ® qûz¶óoù_ô÷i¹
ÒÚ3ÐÞjW»¸_ÞÙÚÔdª¦ïY-.ú7Ódj^&r+Mvæ}|(×áJêwùWnÒÆ/¥ÉÁ?úÓ'4@Õÿùz3endstream endobj 332 0 obj 7775 endobj 358 0 obj <> stream xÕ=Ù®Çuz¾ pÞÇo3¦Uû[L[e8%"iR#]BJÎù7sN-]§ª«;R @w8Ó]]uö½¿Ý±ïþþ>}yÃv/n¾½ááÛ]úóôåîÁÃ?3|çoQ»Ïo-ðÜY¯\_¾¼y´ÿëþ ½áûÓé¯ÃQs=Hÿz~8J)iíþÕ ^xïÜþ5~ËÎôþàã ²û·áwgÜ?=1(7}9Þôøáp;nÇÈÁ &aKÁ6|0Ü;/¶ngo¥Óx±$÷Zîà%½Sñ¯j°iÖ¯ëás»ê¿
§ÍÍþï£àNé ÚýaN
Ó 7XæãD\_¦µ¤�¤´éøB§K%\"ÌlÍ1õà³zk\Jr58¸ëñ÷W¡&¬Ùÿï¸Õ 9c°ã3àèÅ %]Þf Gq@%=ÈûAÆÓö¿ìY ¶\òÍÈ^ oêI+êѽ-<©îCìar.BL¿?h ûtÍÜ.)¯
$¡dW& ¥ $°ü4¼qÀþ]÷µ¼;eÇö>åE¢VgY OV o»zréi _µ¬ÖÌpUÄ+x"_Iå#_Ú-vÎç�'È"ÈZHϺH38¦¤~Ñ®Ì ¦|JD.<ÞdKm,äuNÄíÊ7ñÆzN£$ôBÑ L®pbEIq ZÔvÚÝ¢sGe§Á9@aTpgr\_ hR|ÇtS×$Ô°sÿÿEõ0ûhY»ÛʶlÕz«¼cy·N+¯ A;xüÇ|í¯í¨CcbiGÙ}¨îø�I$ê¯PÈYÐmÏØDÎ.À,¢9Fm9F ØÆçnÜ 8ø-Ùp(6qAUÑF©õ±~Æ}~£3ô®S²Mñ3ù, RëÍR!îðë¬F¶è>ü;dô³ìm}½ÝÛz4åìÚN/nß×õæi�ßw:AgP:Ù³r´Ëÿ ÑÚCĹý°b+´c8 :£)§Àì0©cX(Òy¡½ºEÕÕkªä»i
UM) í/\_卑oê§Vshßd¥2}ÈÇo>½ùv§XpcÙ©oÆ¸× 9L=øäæÃOþk÷æ»·»ùðó¿ùð÷ø¿ú üùä·»÷n>þd÷élþ¨ÙVÊ¡qÇ\�/Ú[»Éßh\ÐÃ8'�dMT¡ÇÉÌA®û$I¢çqd½5¬ä¦Ïÿ¶Ã>Àæ\ οw [y\_)AAûÞ59t4pæåEO«WpÇ£ Yá¯tF®¼oÜÁsо?w¶çp'hqVüÉøÑôã°Êdof#7ÇC³.{§ Vo' ÷#)®.|úñ\MS$©:ØÐ ®Ô·%íMT÷©ñ³E× ÎA¤ÁQ
Ç:"jÿqÄ. ¿7í¹®Hz ^³K1_q¡_s¡#Ë®À°7úÚô+ÀçhÃ80JPÁgðo°Y¡ø:ÝüFÒþçp{ðù ¥rÿ�Þ³ãÝoãÊhFùü|&ëb¼ØÏ/þ\3-ÒZÚ
b NÄ@Ð{ËF_ÈïôñOáìy¢-¡Åí[Xîü aÕ9èA{¯eð¨¬ÔGöÖIÑÁbÔÓVᥠ7ÙÒ:.ÉõÜWÚsÛ¥ ]§?Dy({Ò^ ÊGܼ±Äb¯@¬æj൤áÛÇeakHwBF¡t(:ù(Üð_zÛyeJ¯dѲ]<#GÚèïøòè^if ïËÓÐLØ´9¬eM ( V$ܲøbhCôùl<©b7M,¸w
®Ð3áä¼Maä$�ûlGÑÛv þ в |VòûKøÖy¬}hÒNq¥Wév¬Z Ú ¢eQþ½ÍYŵ¥bx;Á§^÷öxÑbp3^0Dü:Y[[?oT»fM5�Í~V<Ä®êeXJ4ªhæDÈò6pn <G$|pð±²^¬Îð q·¿=EeD×µ7YÔm5ZÑ -ÁcRcmÀã¶lçu©ÎöEü()×Nx[¸0г.!©6ÕðHë©vÖ½tY$v~+ å¯BìÖ²æ Ä7Èݾá�¾(h_lÌ{pô i~µ¯@BÛÑÌuÜã½"Vap]ö¹AUñÞǨÒ�ì5~Î;#Öh:LèÕ zmB#§ÚGtåc3Ð$4·Jðt\±B$Î&&Ú¤fÞþºÏSDà¼ÕJÔùnMð£VBm63i¢X¤3@î¼"ó¦QÝÜTªJJê[Õó ÓDÏw¨¦T,VÁ�Qo²JõE=\·ÑuuÕçrqH£æÒs%R nTF Ö£Á}�B\&k=X³|4}0·U
ÛÏaà}ûðü)=D6k ú2 ÷þeä~í\_ �Àáö�ÔËc¦¼1Z£\_ Ö \g2 OQÍ{²9] U·´)B,¢ dø OP|úÙk[i90.R6¸Zç³î±%ä¤ ÜeA%ÏKÇT/x«ª>ú4àÍ% é<üãÍÃîeÐBUye¢Úô;õïbÝm½\_nK"]+Þ» «fDev,}l²á´ðô©Ä8úÜ ¨]3GîÔ¥r¢+fûPÑù(NCÕ®ÒóhÊTçA'4"Tz¨é ÄÝÐz;G_¤^úE¿p>ºhÉ×'F Z3WÍ·tLÉÍÛ´|)ÃÿѸݦÆoªäãAÐF?rÀàË!ClÚoÊ·ð窣lû4UEZåÔ²Ò(£\²)'¬,ïÇÈ"ÑE¤Uák4e£2©Ä[ #çËÁô¦í&I´uM ÕKPÔ HCð&®d¢i¯XX_ã{Rïë^¨b
ÊñsÅ>·U0u¬¡üü�¸²Ã"!|¨4g0z09Ñ·Xs)c®!ÁR.hÅõ=
)!ÚP¢ÅJÆéÑG§qì¡ µªåõ^/Û+¾érTm�Þ¯¶Z®öA<]×xïéÊå´¨àîu2ç9w4ì-)æ ;ÄxwØ f75µä§ð¤æ@¾<ê i[8ª\"×ÚIÊXãeEáʼcÑÈXûÉ©PY?-8%&�q辡u|®'R(xÄé%Ò¬×.á Ó3Ò¶§öH0ÑÛ¸²ÑMYãô¢}¸Ã>A¾¬¶óÐ)Ø#TxnßvM@¼;j+³)En´÷d£kÑÑL©5S¼ÆSTê{ZX¨JmnçÖ÷¦ ÆêݤøX4VúI@¹úEË×eÿÅ=làíÕ[S+S8ë KxCµÔ®
KEQVÖ)V¼7á"±óJ�JM8z] d¶´ÿ7ñB.5RdLD2bLºw)'蝹ê@ξAHëLªW6$G[D®y;i µ"ñÅýWvj&Å~Dîõ}gSÝ8¶ :g869¯ z/¶$(Y $äêR! X±'zðüÅÍÖmë]}ªÒZÌ¥Ý0¤"²4׺qã©»MOzÉ?9F"Ò\�zÜ ¼lh^5´\ÃÔíàJÇ¥hnb
¿>p>æfJÜÝNÁyzIÅ\ H-COöy¸Ðú¥u·(õ)cç¡ÝÜ\_ãÖ7MþMêc0WÙÚ¤þúìØT!ÆÎD½:kb%ã¿ÞÅV¸2¹5öE ýV$·RUû¾QI0^ûą̈]Úú3VðÄNÌÐïwów+ݲ\_¹ÞªÁÆ6¹+Û?@ÒVw¬ÄŬÓ.ÍD@gÑ\§ôîqsh®«Ørl"¹/¨K>!DÑfÉÅüúQz ôÕ]èiïò%H-UxÛµ±ÉØyªXì°hÙ¿Nýë"l¡èõ£ÉפÆJFw\wÛMrÆ3V~Æ4·å¡©ªÅ1Í\±¹¿ÁÕlw~AÛÓ}7jó]òßäùÎ Nh\tòù iþÊû¦0Bg0a¬Á¯ÇVì¡Ä«qÀzhéuêËø!#«óAiyp®0ßRݱåë6:!q)p)e¸@Bʪ½tªjXrÄïU¯yå%~é±n¶TÕaF§Ê7Cia)~I²0ÝÁ×·Uj¡éD_c'2ØÆi~æ©ÑB» £=åI¤ürli±"ER(4>]EÊkÇèx¬
]ÖG>ãþ;Ò¤Õ·bËE¾HÞg&Í|c1óî3í )Úûy´\ ߢ÷^?_RpfÛÑUô¢=Æx@ kzYen§"¤ñØj3@"µ¬öÕ¡ ?¡en®£¸NKWmÄ (¤Pì°D'ATå(<Ý"-W}åÞ5¬õ#±>)LàÁìöaeÙÇb[qeD¨Ì¿îÇDmjHùÕ®]ßU A{a® æcWìÏaÔPÍÔ¦#üaÈù/ÈÇs?túk.WWË>-i$×K2 \B³YSDÒ~·gÅ(dîõ6sMÁ£ÉÅàx#ÝVÊ\ÝT5
³&flg�ÈX^ü¬{Ak¼Í& ˹ĤA×±ÒïÛ8)o1\®PènYµvQ±±tZoÙr7MÆôKºW Ïñò$\CûE.Q(6Þc,èbOÄÕ¾§ó³Èú¬½JÀéÌãV5©'(jÊÚ§uõePWñ´êl zÉa-KWE þÅów4U²sYix¹HjgHÏá1¼¦åß'·ý)Y¥ åúØ.!³ò3¾ñ5wvná\ 1Õ+ñ©Ú(tàÔ¯áVRyI¾%îñ[¥Tj;Óļ[¦;ÎûÓ]ÜKH¨É\¿R·C#¤tàMÔ¶X´ý»±h{&çkSME[|ýÌ¢ZfZ8teȲŦh(K
¡Õ¼£Üüµ>Ä(Wô[ºÚEsõ(F2¼{]ÃÕRcÅ¿:kjR+è¬&,OFÄÞ}_±¶óô.ø¯ãLîÇTñþqïî/ôg¼KnõNÚãy ÀsDsegèÄÕNZ¨¡V[å :ËsÜ»pø´Y�p@å(pë±öû¢ºì-éSk<µKp[o1Ñë54ñsÎDêÙ>é4ÍO"©¹Cÿ ½é;ø ppXÞ-m ±l=á(Ï©ý~ÿ³pVcÒбH$eî³=ÒFcp©´³5Û:Lrè«Íô= ãÅ4ùpÃz¡+·} °¹í£¢®Êéª ®/+|\Òö! óã¨[ks ººeµõ&I»§¾ Ô$Ð¥+L±Þý!fÐ0Ó¯+¹Oxîýº!¿'¯V'!}
îà³³ên neòÃ~)G7ªÕÏ\¹Zú0KÞ÷sæöm m½ ð,Ïk¤ÆYÐ\_ ®=%÷\ÕÃ&£ÜZ+¡5nm$LM8ëα$jË 52ú$cWi$$®2kÀ²ìX5@bßz&ÜÙdFÜBÜ Ç |-¬®¹MÚ\É ¾¢¹Sð2zls#\Ïõ«ø¸l ãX;n u´åÑõß0ÈYÚKü±·detNÒ¬ç¢Îæ6 ÁI+y¬í\_L3äܦ7C lÝðøßp2ª×®\¶¢]'X¤¤ïÌ3i¢ÐY Ì/¼C_ãë9íPé·°ÔS÷:óhnSdØêMï:(/LUØ67CkW oJÝ «Á÷QÙ©s!¦(%ný±Í{èßÅ,§§Ð¡ñV39#n.²¦®w²jáË|ËÃáh1[¬6MZ^Xh+@]%¶þ$&¶ ïçìw¡õª
jÞ{{¸¼øBæM¬zuHîÌ1Ô¦P¹5CѾzE¦$¬$u@\_Cj¶f,h5{g¬êZÔG´ûZÓªgà'ew°Wd»l¢÷ÖvG\I@¡ö²]ciåu Öf}WeÐg$ß/i®·Ð fÕk3ºØgÁºjPb÷ÅÑýbJ}ObÚöå½øB6m·kUáO8Iàa WQmn¤©?n§Ôîû¹a×&ÈFs-n½ÞìîßÜëU¢xTrðg#H¸tÔJ\©¥m,NOþG|µÏLËeê7
A>VY ôüîZÒåOîÓå£"Í6}ujXï%BW,«Êj¬ºå\ùQ1\_[W
ë~{ESùwù OkÍõdRBÛn3©&i|¶'E¸Ü¶Aÿø]%¡Ã¸¯úv©^Jr6úV¤ëi
ÏYñ¬Mz®o7tWa9µu÷ô¦JÖôT=¶çÔòü6¾,íý0úÃD{ÖäÙýÕ·:iËÞoP¹�õ¼tÉÏ¿¡ûÖ_ovã¥pÿúï.ÙDVö¨c¡ºÑ7j,h̪¢U[R7 Þ¤BIë.ͤ¯ûD~D#ÃÌÙº¶ @YÝ%iïÓv0a ÓwÏi³3Vì]L=ß´>yJÀÙy¸ËÈ«)?¼JÕËÌÊd ¶ÍaÆ
æ0Fí]uµ Úôóx§kÒ/føÏ}ÍGs=ðE³¾¡P¯Î{ÃÏ»P~ÃÏ/=ê¾~úî^¿Ær0MòñÖ\_½{ä8mßhÅ¢?yËÅ¡À«CÏæ§7ÿE~Hòendstream endobj 359 0 obj 7193 endobj 371 0 obj <> stream xí]I%·q¯Ðž·nïÙóÊØRd֢âæ¤Ñf±È&{"=Å\_ä¿éL �$P¨íõëéYÁ;Þ P@"7d~õÍøÃÿÆÿ¿|ý]}þèG<üz5þïåë«\_¯ÔõzðêêÉëGO??ÔÀUáÈ'uæ 'Î
$?|zZÃ×ǼqÞŬõ¯oq.£Ï?3fô8XÃOÒ±AJqø%¼a^[O+Ž2¿%<ÓÎÁoßOð¯B ~øLXa~ÑKöd\y§=<Á¼ÐÌÙÃeMizºdòª2ýkÊÚÁs~øÓ!nÕ{uxé&¬>p2ýûqòìÉoálßôpQPp>O^Áü/R&ØúÍÀ¹÷"ä%ôïa²&Á=èjM0ÚëÃ\_1ëðoæB<ùgeî?#¬k5üÊÞÀ~p'aðæêÄÝ µwqɿóÒ;
gÃc4¬#¬ÒH[1hí 'õtR 'üà²&¶îÊ ÞJ§q°ãfà ¥ñ7°Í¬á¦·j R ÇMµ ½�¾Î¥>wxàg}øøkPNYº·o¼õξ;":k$ð&üêqÀËò+¾@:dXdM=8ç½Ä K) äKÉÿ|M^f\pyxUþ\_+=þ Ë£1ÏÇAÆ?\ ¿:¯Ã óIß{~¸ÉÓQßáVN×H:>ȣȤ/ʤ7¸kaðwتD!\î\½5ÿù²¼ ¼õúëÞZñ\´Üác© pDªßÎ7ÎpmÇÑö@2ÆMþSx üIüªÌáèAÓàÏpnË@¡ðóq6Çxý}}¸{ò·D7ò@T¤°qÐN:gá<Ì´õd£6ãîpsÜXI^×# Ê®F)´¢»©Èb(øxô ÞÕ᦬äk\ [$ þäptDä|WDHÏ«®x!K3H°VÑ5²ì£ãÞÐ7�O¶.sºòÓãå¡áþ@¾eÄät8×/Pa§áÁ8Ti_ov oÿéÃð§qáÁvpØbWP¹èùg½g'¿h6$R¢üz&K´,H¼Þ¢A¢À²èÅ^À°H5Ø( 6¥3¢[&¼@Ä<ðEÓßTAó£Ö;qxSW'àe ^D\EzÿÞFp¯N¦¿IϺPÿ䧰B)à/©4\·Æqtþ7Ò[>JL«o ϼ·ðò ´NèÀ?8/¨7²Ai ڱ˻ Ö[øßáÜ î·¶æð=Z.X{ÒµJá-ËTRA¸wS UK%n 2ÛY®eí\=X<55Dxî¿Ðãµ@iJPK¨´ÑÚñÊ
?ae0=È·WºKÏÄBO~÷èÉ¿>ÅûEÐBUJ}46ÚLLÑÿ=õI~|ô]R�ô ·¨Ê&G)³/ÇõE] ÆS ÎQª ïkò²8óðU#Ò°M.Åf¼öÛsRéö%!^ ¤Ê©×±{OwSýP{¸^Àýª¹Ê=ZÕEÁ¤Í\²æ}}hìs¦v$¸®¢\$GhÙ=ùF½Ì
ºûÒ¼ÓêîLÜìKÑÁ.; Ý+uÙË«ÈW¦x¯àÁ3])D÷n })ÔÙ hJQàáËã/§FpÛHÐãé²w~hÍÄ_J,Å7kÔM§çzÁ³¿ýÀ´
ß&ö®Z:ÇyªócòÞ8oÕÄÈÈfG»fy¾Ënªð
ý¬yóf©Q®3zx¯rº"îöWi%u©,oW8¾á¤zbû60ÝíöWOlæ¦ä S¢3,¼ãPPCãzØêô ¼i:©gø¶¹4çúù»;m?$¡ÃããñcÖ\È iÝǯr$l;oçRÉú"8ý->uk«#Ùó]²»z3LÍÑ7ÓÈ0pÉ)+iD¹Æ±dÀÍwåçR Ï¿¤À|@!ÐèËÓóÇF&ÉAôFù¢|ØÎNq ¸ÜÜ|ùJ©Õ0Þ³o GÆ1ëqð¿,ë6ÜejþN|ÿTÌ<)°ã2eÀJ<¨H1¾ã¯Iº&g{ÆHxtæüÚªª9ÿæ6FèؼûDÅËÎH{7¨y]ô®c.{u
%üýF¢vQ¥D£læíûÊÞår8Ê¡CÇÜdûQ!_Éf6
£rZ·~è«ìs3Õp ÉÚ9¯¾<òÕÈPëÕË©W¯T_µ>"%AZ �B¾¨Ð/Ñ¥¹Ç>H ¸!¹r»®« ºä´R|ÔìëMñFÉ3gô1)3´®ùüÎ6{æsÌsæ ñïói~KÐ@úïÁ¹Ñ~¼6Ö¨Ëô¨®NÝR)EÞ
ÎÊýRV%Lr}üCÀµ9á V#P"°& ·2Ä4bPRáà XÄ vÛø¡õx½%c bë5M!´²
ÛÞBÛgÿP´@WØ'©Ëøë Ô¢ºòÖwO0\ ÌW&tIµº%ª£fø¹à.ßÀv$DÑ?c>è ãOÃ^ü¡Fí¡.UnFúQKPPt}Ûi)cóÚBú|Å'LupàÈ9¥¸ã8~D\/!NîJRðyU7=.3{os(¡<àc[ s g½JGÀ籶ipWDì¾-ÜGeWDcìEäÃr^FÝ&X7oË
#ÃçÝ(ºm'ád%MhXItÈÝ 'Ѽ(ÒzÓ»'ùÒÔÈ�M]B¦X&÷®½¦HSPÍ0¾©ñ¯»¹/#·[󵿬³·"!+®"Ý?ÒV"ògT8/GªfÝ+ )ãj%bùï·çÁîù ¾¥jD"Äy%èv¢7VÄÓv«$^d÷C,'Ïó£xAkÑ7@ÌæS ÚY×i�n3E$pýòÙ7Ô¤ ¨iG D¼ÊPê.UèÍ@ýpíIp{ºûºõÛc(B�GlFX1¯GîÀÖ¦}RJ¿à,wè¾.P6;ð×@ê;@4A¬\Ä}~"¼>ãμpOtÆ.§éÕ_^E7Äï?ud´ FÛøt×ݹJôA){µH!Iljº¢èàDMW øÏå/³n^±:-9_»-&M7Þí®Ì{¡ø×vOTUð�b6ê4¤ÏE{!»cL¾´³¯ksÿÀás§æµ»¿å
£ù°£v¼ß $LóPJ°d±d¨~/Å09[S¿¡ÞFU>ß±V£"ÅüV¿ø¡ÝV¤è|ZµåCm4L´¼£"K3¢Q
êßƲR;Q=IZˤÀôæíRÜT!·2øº O¿ d×ÿ¾ì^»ÐxÝÄ0îÎ×xÜÑZ yÔ/ó~æÔ¢õÏÏ°|ÃmØ3Ü'FP3¦}ÔKz OfnµMP¾3<þpºÃ\añcmf|ä"5q:(nk=K¼üsUñce@ |êÆÊ[¦ï±5©6úq[ z|&Ã×U
uR?»WASÁ}f¯cWÃzTL 7ÜÐÁT÷Ô«ù«zã̧\Ú¤^E|ÌñÀi»JµùÞ¶ä4÷x¬UiRO4ÉØ ¸Üüå\ÛÙì% Gnt¹¤£'HTûçå¡Êèp5¨MÊÁ4gZ]àpî.OWñÎáÁY7% Êë@ñÒ{><ϳ2Ò®cHG°¨g.ºä1Ö dÛ Ól§9FTNEFh,D$Tøò?t®3BI»HÏ\y/ifh³9¾¿Ç%>(²IqËVØ$V¯úéÈ&Þö½ê%6ÑÆé.à9°Ù&úUWëì:Rhª3"Ú³¨ÔFãão 0nÛžF´ø6\f»vûö3§f^9/Lêÿ%Ð .dLI«ÑX& Øýoyª²ÌÙñËêj¶ðet¿à"+ \² õëÙq¥çÓ¸ûå+î·nÆ.LÃ#,ÁøZ�1ll²gÝ(�mÙòj{³¡ÈUÜøaÝÛh¼©Ç,!
û?Q0k»Æ ·¥/ËEÎCúÔN¾¸³²YmîÄ+Y [¥ûV°Þ4²F¢Pݳiv,nkèt7é± ]'DAÇA�C6pEp·)MËTHÓZò=@Gc cµñe£ ã¶H¡lË}ë
£ø.Zi²ªvk'á°ÓB÷]öº>ÒoÓùr\_ݦÏi6OÛh;¢êñ:XHð¸µÌÜåËäëéy´z¿³üæ5îue)79ºm!& îųÛîÆ.áö¦ÓÚ³zdÿ1&e.\_�S\_ì"Ô¸©X¾Tû¨´²°\_H¢àývÔw!)®BF¬ÞvW\|Ý,±þqÀÄÎ8ÛÇx/·ÃæI¶HfÔ8.¿9:$³§p¦ æÐÌD§ÈkÒMkòòÎyå-§Ìi>xÊ<·GüÙõAk4¥="©tgÚÏóÕ´¸¡Á$'6M75jÛ£î÷æ¼Ï×·=�èKÍÛuè :s�²)úå. ±¼{°ÇcQNªÎ±9ÞÛIaò\W ]÷´rHçÑ3jNrã"Ù63z3úRº¥/
¾7Ðá¾Ô8í²£Ïhrá´{5ì$(x Àé<ÁÚô±7U\_iÃN¡Û/©ñ¶bø UU=Ô äW^UfæìßQ¡)frÎä¡ìr POv7Å÷£yµÔÖ=í}ÖmÞMiðv1¼ë«/)\¶Äv ¬©«Êk«õÔÕ÷i=X¤Ç?Ey ²oÕBa¾5CîÓW¸pm ËWú\VHÏgñ
ª+úE Ë[xZuA»·\_×¢ÌZFÏé¿ÉîB½&°ö®>îÅUÕ.ù)¼¦§"@ê~~Ì_8»eÐ3×ÀGäÃR<º£a$ihQÌ=éa;Òz8Ø@j¹ãFìJînÄ®Êå¿HÑ)µÜ Û¹C¹»!aëwb("û¯¯Q橽8fÓ"ÀT½as ¨Î®½b÷jϲ9úÜÜÀ,ȯ´¥U¥±f4YݨRVà¾ê:@.z %P
\öðqùªÅJy>~2;xÕ1¡FT8÷ûèôyK+ĪXVÜ©7d/ 7u~Ñb2>YZ¤Þ¼gl%@ T Ç{æËn0.Aï«ÑHTôº»FØa�×bxÎ$(H£\ý8�LÔi&} úuòC:"{lÄYPô$>ÚYüº,j'£w}bw[æ°U?ZTI§MÕàYTÛ'¿,ÕºWàQQVºåÙ_&:W¢ÊMØ«bñîr»÷q»SFÞùY%ë´ù» É:]sÒL þmscÝÄ®"Î7fÎáÜöbñû:Ѷ|gÉ'ÜÁ[nFç¶-52=6éµÅ-Zdþ'nU :òé
Õ[&ØPoedäz+lÔídÚ$©óxMRhÔF/hõ¸©lWÍCÿ[±jÄx,²ê½»e\=k\q/B´ÅY'jºÑJqu«xC¨æ²×Åîs yKTf³æ-å_·R¾LíV¾¢»ïßÊ «nwÚ۰ż®ÅDu»îYr¯.(ó²«(CéN6a¾íðÞ;(ؽÄÞXöÌ]s˱lýJ/�:©}W@Ã/#cYdØ2´îX·"oêx0û³« r;õ+Vìô}PÛ¢ÇÍ ÞwZMzg¤ CiEÿðEþá¬"KM(÷Ò t·ÇJû¢° a¶nòcü²~y]2
íÍi£ÖNYz sé²Óøimgl7 ßí´ðÜØ,EãA0uÛf) È3ÝÒÖ¢¡ÁdrV4°'Ü;k½ålê·ü"Ê²é ¸jT$½iAU¥ùc¯Ö@1-ú5Ä)K ñÓ×ÆúÐ~Ã1X ;^ññöu¿Þöu½Øç΢øXbWù¥j¤à ÀÌÇíÜ'¥©ÃÊ¢8RòÉ6VeÍ~ÒNIcx[µ>²Qf-e ÚtdÒ循F-ç=±o°aìëî£ ¼.áùßÕÑ{tø+5׬»Eç28ò7ÑbùöÚ·køa%TØXzF d¾ÝXÛævu]a׬VÊ%i42¹Æmy¯÷êÒ'»(få¢é²ñô·v ªOwüPÍÍñ,@ÅMý·^v
îAwGÑ«æËT?«P¾DE+E'¼3£#~©ë,?ìQ«¿ý«½Ünª Çf×åEÑuiÅå¿~Iæ&]H¼+ü7®|sÍ ¢N)¢Q¢°8C ô£ïn»réíý/ª.Ȥry ~5é³dÎ3¼Zo5³,²e1LÜÁ9 UºÅç vùj®]«9DæÓCû¶¦ìÌUíÓ©aTBbÝ&úÌoæcнTïp«3wX°ÏPm~TÑyïÛxRïîq±$dÒÃWuµ³ÃLSàÈ3xPPdZÑÄ@CS¤hw'ö¬}fÉÄ󫵪MþE+ìÊPk3³}yýMvÄR |¹ýö³·ý4ÿ¿$E[E¬ ÷%£@ñv+·K¯aôÙåûôëö®,Þ¬+"ÚàæNà°«ìvAp=à \_¡{>?¤êh%ç@¾}Ý%¾îÁý{A¯údtXÑ[P§ÕdgktÁîP O5ÞIê¹í?zIîÙñ¹Tl!7òáéLTÊHû§ÙÈþ|íà$@ ÕC?oL-Oä¾Õ§wô¯^oNUcZsý9céÇ¢¶Zcÿñô -yÍ<þ»Æoè.(7Uã,Eë·?³c7˱,ØÆjQ&ýÍðÁë[¯×¢¹Áã®VEêüÊF]Såãݾ)¡©§©¦N^ÞqßAQÿ ÓÐ$´ endstream endobj 77 0 obj <> endobj 456 0 obj <>stream x{LSWÇïåÞ+«Ì×½M|E2ÅÆ©CpÅÅZj©Å¶Xu¥>£úS¨Ô §ªDM£TT,KK)¨<ê"=>02 ;à.É´8ÅÅAÅåÀAQri°7\r_'¼×y¹/ñ/Å,új±Q°?S°zöêöT³Ë !^гeйõë¥öóª.6¾N8í¹çîè¬\¸D ëþûl°TxHîãÉõ4{P:ª,ò¨v5&ö +ÇrÆn¾åÒÕs?xC2;{õÊ´Yp¿NðÁ:eÑ+és±ù²wÿVË®t-kXLad�éÿ2 v¹ÏÔíYÜèbçU¸EùvZïC~ÍK´ñ°�´MÖ' pëdó³îÊ&øNç8õÕ_Öqïµù'ÿÛ\TÊ"]GºÑÿ¨ëü?ÕuX^ÎI\_xG3r Þàò=©çOLõβ/.H=æb.ý äv>æí·våµBÜávmK§½á$\ «êæïJÞÑ0fæ'XÕ³s´ÀùFùrîuñ]RÅ]õðûm[ܵ}pz¶: ÅÉõV°« ?LÐéu[°_ù188 tÏ[l5ýÜð�½ìvÑW^bÑ3F.ÃxÞ ,Í)ü"¿°hs,äùÐ=ÝGmÍ7[¯ÿO8ìz "!¢Ç/9]²£±i·«®øèbáàO×ö4÷üô)ѳVëE²lÜ60ÕJ¿£CȨÙ-MôhüÁ}ø¿s)Y"!q+çk'VµjÄ沪ÃÐÀ2;2Æ +Æý9û¢ø´ûù¹wHCè9~ü\æ<êVæä©=ë¬w8¬ eÑ3׫wöJN´&/uM¸Jg´--5ÙlPbçlE¶y°ûmxrÉ)>ÝwyNæ]بÈ)øò r¿wy cXL²õÊóeíe.ß h··ùUf³S úØÅNGñ½d7üFÎ3Üt§×x+ñ7¾Ö{Í/¾Ì7ÙäZÑA²ùFhÜk\ñu¸2×&s#Ôø¿")kxäc¯1ÞM> endobj 457 0 obj <>stream xmT PT×~
ݧAªÆÿ-jÄø×\VÒà"¨1ÔÅ]EXYvùÿ°ïì²Ê?ø³»k¦jýÕÔ¦u(Vª£t$t"6Lj'ç-÷Éô.F´ÎûÞ¹÷ûûÝ{ÎaÀ�eYÕu¿Øºx_+Íd%>@ U�ÿÝç|ÐÓA¬àÀ#]¿©FÝbUõy9¹þsPP}ÅJÞ¤Ë1 éú,Yoõ5 £QNþÙK:1Ò³6Ýà×Í9ú]/�ÇÐuƬtm~®é{ÃM¯®¯
bf²QÑ'¿>#>aëæíÌdXÖÁ:ÙZÖÅîc÷³ul=ÛÀ6²Ml3Û¶2üoȼÇcv]ú4iÀ1E¢'Px$P úRMÈpðÈ4ï7+MÁ
ä·©á¡7ãZÜÒèpÀ!®]t
B%ÙC¸h² È\NQe;Ä#_GÎ0 KWþãÌÞ7Nýª¥°w òhÅÐ8£ë¢#Kx_%Ùaòr¼Ot$hÊDò!ß!6UtÚz/üäÔr÷XùÏ<õå}±#SÕGû!ÑQ]Þ¾æòêþn» ó¿ð1 óîM¸¨ü8äS6q/P¿êLÿäé¢? f¥Hù ü~e¹hêée pHðÝR>BgYiþyìú·M
{×æ¹ÀÉ+8$®kpÖk:¸DÇG ä?MÒ(Q XPulà:K±$èÙ-àô°|$?ñH×Ü,Î9£À9òu!X=^p¤ªà±æ@ñYh½zøqoSO]¿·ÿî¶\v§½v÷ÉØXdÈRlÁ{ ©½'p#ñò} øðìÙÚZpB-×XÅÅv(Èä³p9äHC¿5XG®¸-¿MùÄÐ\êVàZë 1ìÕ ÒÆ~
ô1OòÔ
.4¤¯ØÚõ¤(m¶C1p¥P^íªvÚú =±=Á¥¤©ÜY Äk C$\^A-Ójí5ö¨áÊ ¹Õ mm~ýGJ) ñU¼dúý?¼P çáNÜÝ¥ø߯=I_ÇÙpúTð9àL¬wQm¹ai¶0o{Ä + ©±AÄrr Ò!I,«_¿(ã ýùV±µ²g÷à7ä²Ê8ù+6sæãÄÞ2 ÇÞ¢£r_X¨oZq¶½à"¥ zÛø´{ÙkýÒÔ~vªKÈ2Cr<@FõÐQW};´rGs[òl¹%\J½ñøÀ^¡åÊ <[ÃèÖq]ç Ù½y¦êJ s&oÉa··å׶uo¯{cUPÝd½£;#Eáð^öZ¯t¿WáKS±«· ÒS&H|$Q$oDA~ªziRáJ GFP '$Zá:øá>ðÛ ,øoª«qöÑ$×Íúñ»DaÝMh)¥ýÒ×uõ4Ë÷M÷çUU£¼Û7itN3à«ûÕÊûÛ: ÿÜEóLÀ® ¤(- 55eRç³Ôirü´¼öø¼°FyõÏrñz¦&+I8Q\áÑþó·B1Òü¼H"RD
rÆD?ÿ1{t
Çxoçc^µïq'}¥'¾÷(òXªþó :¸ãæsQêÖKÚ?áüëXgþ&ÜÅÞxû\Þ9Ƹ£ ÕÍéç> endobj 458 0 obj <>stream xcdabddä vs0420�qH3ýaîþýÃãg k7s7ËÒ茶³~gäÿ\ÀÀÄÈ(§ë[PZZ¤àZ§à¢àZQ&ÉÀÀÀ¸! ¤ýàƲ |ÿ^3ü^õÝýǦïîYkný^rN\xÍwý"¢QÞÝÑÝ¿:>ÏÞ^sº¼;Ã?\À;'fêJ¹ef/ê^!³¼|ifNqi¦ÙóÔ \_øz<ëXü&¹½óêÞÝýÅèªÞêøYÞó»wsßsìôª]yså ÚªKº8×äÏ/\»jñµô7ºë
)ú¯ Ø"·¡@´mwüw¾nîäÒü¢|ÿ¿9't\_èÞ¼pùå| ÓLùß»|Ûïèìï¹örËq1O³çád��"endstream endobj 70 0 obj <> endobj 459 0 obj <>stream xXxTUÚa¸´ºÑ{(¢+ÕU¤i�R¥
$¤÷I&2½ÜïNo´É¤gB1ô(ݶ² ¢®åG=7Þì>ÿ$°ë®ë³»Oè¢ÈÈå3g~Ìç³ bÈ9ËÏ =ê!0R�#7>2þË÷�ª$cxCùüôÑ,]!ËJODLy2bæìÙÏ?1kÆÙ Sã²c¢Ó""£% q©ÑüGJÄúôÄ8,bÊÜ$cÎôéR©tZtjö´ô¬øùO>!M$D¬ËËÊX&XÑ¿Ãiý?¥§fäH²""Ócã²Ò²ãS££S2¢cãR$ÑqÙ)éi3y<ÞÓ Ó6¿¾hñ¬¥å+¢Wæí^»zOTüÚuë^INI7vüÜóÃ^3ÇÈ[ÃÍÄ=Æ{7·÷oï)Þïy/ñæmá-Må-æMã-áMç½ÌÉ[ÆÅ{·÷ÞJÞ³¼U¼HÞjÞ^1ßÄ{<ÏðFñFóͼ©eÞ^Ó 8ÁPÁ6ÁÁñC y[X?tÜP1hvrxöðGGü4òÚ¨°YaG§6æÌmÎðîC¹¿û£èAQîØÔ±uã6Û>$Üþx·¸A|æáÇY:¬G A´=È.ñóÑËh¨(fYlÑNZaRä:ÞIÖAÐR{¡
nc#'Ù¡£f´ÄíðÚ=Nß»èwáÈ\4çú¸G@E¤n¢¥E\\½&_ý×ðÏÑ5}Ï{ÁïQ< 4µºgÑÖ|A«S\^ïrÚn4q7±_+ìÔ�ýÜ]r^pó.ÔI@OÈ|êo¹Óï ³÷°½&ðy¸=ت¬eÎÆZìºõ%]{"·r¿+ØE©%&(m- ï´¡ÉN·hÐ
zRúrÒú@äkÝU5®jWªèêDàñö®E3·®äú7v·ÃvÈUÉ ¢À@¨Üv3LÀN]öiï'åÒJù&¬ç"{øA>+c§G¯ÒN£1ØË:nÆÍ@áU�Mî]t4dA¤%+NsCq·]ã¢Ðp®ç¤råÖÛ 6ÓbyML &a¯%m°n½©[Å)5bz(-)õxáÿå;NcU2&0AkPÔ&È'°õp W¶ÎöWÆá
¯ 4Ô2ÆhÉïýb!é~çÒhDS}Cu{u·¥qT ¸åd$Ñ)°,ñ}»Tk£&Êe !rë1a] }¾ ?¼/RÅ5s1ä!¸ B=B4 GÃÿreíé§<ÔÝ®Áĵåç¹$÷0¯ÿæ 0QåF ä0Ñ´(ãÌ¢7ãj1[ö7+£v>óìÓ\ÅçnÀ:FÅÐ 8Áf[ùEÔÖ ¿ªX¸kʬ§¸áT×O²×®¡æ«6Õî.yk&÷0þÈMòö/G$Lr~®D´kgðý£µ×t:öÞ©z N7¾º&V:1n°öÄí½#'Ø35È?~KÐÃô¬Y5m2)äùjßöë�Ä\6Ãer
Ül\l>ú黺¨/>\»£NoÔIudñÒµKׯÊßÄÌ 'ÐSè¹oÑ0êìå;Ç>û3ó9"ÆÌ T¸®Å.·a=ïôÓ°gקôÌßQWBI¡? Q,î´x&<sOQ!¬ûÿí/.7¼T©ÑNr�ëL¢~¹± ßåþø7JÛ3w7nð_ñÆPäbtÑÞf~ÏfT$2Û,0H¤Pu:#¹²g·�/0ùWxÕ#ür³´ ЦRF½Ð[u.§ÅVb%O!ù»Þ¦Å)h2wL2©n¤sµ0Ö(¬1mv6 èLZ,!zÖåaìfÒZêô¸Å½f¡tkmõ 3Q/£¸ðÞÚ~ìÄ÷±cK
UÀ0R"ݤ¬Dg?+Æ"õÕ¿Qfk3zÄZîÜ[~É^g-®?\p:j÷C-(HÕçdC&QàÉ«««¯jmInܼbçÖôfø0Æn°»ÁÙq¥Z<Ølt©T&}Ì'=Ä áyä¸\¸®J/Aå¸8ô¯#'ÿó
jkì&7è½!ÕpaÍqêÌÀP4c ä�^Îf¡i vk þûSDüüóçh7ûèÐïî³õ°M´ùÁN)Üùtº)AMÆrÏ ÚP ~ê¦p×fȱ¡=îõïO
ô\_µVêÉ]6f'«\Þ[mvXË°¬Ðïáìÿ\÷@T;±6F Ò!©ð%5´¼ÿÐu§£QMqã{[Õ{h>´DÝg¡C_Ånktcè!gHh5#ÓYJå VðÓì4;0çÞ@Øü ÿ$ÛÙDνf×I J=PáÒY4T¬¦³ æXBË©C¾áÚ\7k¨jÚACèZ£Ú»3<û©7nLØè.8a1áÎb\_mscS;\_¶Uñrd׶nß½ðM >xM¿v\°ÃØ0¬-nîÊaÜî nòÔËß÷³^î¦Ñ-IG ªªJÊZÏnî3T¡&óWÈ¢vA×úf§ç�åé4»]í("ÜQ?ÝÚW¨ìw¾W¤±0ç+» ÒØRtrß±&̹CTðÌÁ¶ ÞòÍÆ¡3é1Ã?("2Y !Ê}'ZÆß¹¸ÏÚhqüµ°Ò Yt6ÄCl¿Ci÷té1F²TNßû0×þ!ºæj5»¯¸kKcÖQé°N4XlI(ô¿ÑEîáÞæðW\ñú˨§\!ß0Vá(ÆUCÞt(wØÊw¢¸«øzáàÞôSwÔJå@vêNK^áu:Ú-´ÔÔnFîlÁlyG(þ Eä§á6zû¶�}NìµU·/ ª¡R®fAv¼Ê;NDù×&l(Ȭ Üi@LZDzBº´]fªùÇ 5Í@T´¨×¦ä/ÛÃEP Ñ;A3·¸Æ¨z¢.ÇY¼ýôÀ
ÖKv éÝÓÛ ÄÿB¢ÔFÖH%E¼ DlAS3Ng aì>¡=üÃÝHyE4= D¡la&@×Ëï};|{®2bísþF2à굫bòçsdj;í¤ Å²hk¿ÐPWKc~¨c³xÐ#lg8#üä¯<³Á{Àî±9Ø)=£ðËܤÞKÿlõð¬x¤ \_Fî75#fXÙ}
t¬x^
O«ù^:=ÈïÓ¤¥k8U¿Ìÿwë÷¾ÿFÊ 3½.rw?´6/SÚ\fafa"$ôëûÀ\à¤N<¹Ø!ic¡æ÷® ç2Ù M¹àÝ?,ªûÙj;"DÐþuJSÖføÕ¸ÕZkÖÀ?Pp¶&ܪ£M
̽;ò-F¶'°ô´TÜd3èñÏÊü®ÛèÛV-ù{OýQXÖJße»Wìª(+©+Î:²UrEEmC ýÐöªÝ«dSRI].Êæ$Þ#YuAøb/%{\Æ&pÈ+?dqµB Q%óæe¦åÄí8ýÚÉ3íådÓ¹\4ÁÞÙ3' öZtã:o "4\_&"ê.üÞòkÜ8{ï·üßK¹Ýlcìz=x/%¼Ñ±4Vúll<µeÇÍNì alÎûñ'vbõMfGü88ú Q¡7²¶Úüs1{lÊÙ¨+gö¸)Æ;K±qu ILº«Ñå©ÅÃd©²"#'Y³»MúzàÊûÝ%dÚ$ÒeJ²ròre)2,ó[ Oti@ÃëSuG÷wtqÁ7G¯Çl46¹__¯A£lC ÿØÍìØv
]¦AO ¥6èÙ Ç=É¥p+æ ÁÑc5ÍVÏ>ÒMN$ühÄ´ÒãÐZ £ûìÅk³p-¬S½Kpk¢yÁÄ÷ás¸X&¾xW¿¿sèß^Ç=üÄfNÎÍJé[TW_bãÖ-¤v%¨7¢¢dØßÑÞÑÙY½ºÐdR¨Leª6=4iAN�×½+NUoø°kçög½Ú®°¸&Lm2ÀÄ¥±±TfæÆäÁB'}@#.ÁwÄ'3ÏsãH®ý~ý¶æËú£ê¡ëá2]?íþPù4Bx«|}B¡&E'¡´éê= %8 Å·|çÃÎcRµZ#µpW¾$m_Ð~ÜÚD}x>xoY0ÃpÿmûÿÃîýÝÔÖ3Dä/dddfäùò«««É½#EÞ<'Ô9ý6Ö&·mÇ=®îÌ¥6²îNG#vyg$çjs J¹# v©2åIKlkN8r§ =îúû5lUÒêÍ×$v_ÕY,´=xQÏG/_B
=)Wo2)ÈW×eÞT³ ÓCÜHn(7jN×öã
ÔOémê
0[~~¦#çÈ¥/º¾©!o³ðüáóԩè-¯@ÚÅ0N²æ졳8»ûÊ ÑZiÈǸý ,×Æg°wEz«Ýkq÷Öïæb ÉTδÌÎ3Y^áÙMRX²÷0L&mÞnq¸É¨Ç"tYçp3¶R;ù:^[¦à/¬FG+LtÈx.QF8^¥ [m]½ê0ëMu!ã9ÅX;Dd dfÉÁI¥Y4(ÅÙÆÈBº?÷ö,Úÿöi¬µ¿ã°Ïr÷^5ôå±Ä¯w¸¼vÒæ:"í5ÍôG°´xÈTÆÐE]TGÌƺҶ�YsÀsòÆ+}I²kjÜÓZf²RÖ2ì/nxkUÃsySSUõ®Cûï×lòqÈÒM2\_¥Õ± èËØö5JöîHß®yaW;ì¸2Ð#êäs¦ç\_OG.9¨5¡=ØKÓJ£ Äë$Þ[ão-#ËÛ½§ÿ)&äEaAO
&_5c³UPa?ê²zFÿ¯wYÚtMÎÒ©:ôгA{Mô½Ñ&»öY»ÐHÄb¿'1~]4'n¤ÔÙѦ5e4øáì;X]ïÚÑ~à?[½-T=YJÜíÖ± (/rê3Mr·F±EM«Ä&ý«3æwÍÞûz£ÞQOõdèo§ùío¢úó¶&\n
æê\Q\û='àfs#9³äVBÓÊàHÊKçþ&ÚÓz¥ÑûQ˪ób·KôËS¶Ig'n¦b£6dÆü®<¶möóo°HÐó<«yíxúp¤ ½Ç9R¯ÑB §Þ£gñ áÅZ35Ëåñhm ;Õ@Ì°îß~~[Àf"¥Ívt!÷(0¦u/øþܪ³ïÅ÷¦¢+x²Ô¡³è´4¥)ÙÖx <ïWW{©;hx%"p]Í&§ÓKÖç ¶EË^=3¾wÃÔç.P\_}½¢o7o¬5cüåQÈcsÚÀ ^Mië=|árh jµRmÓù´à^ü]¤Õë@ÝÇÅ=Q@ÏÌ&~çßÖ}{QÊØå"îÔ§¸±À"~~Ñ4ìÈ[1Çñ£®Ã"à¯niôË\I¿//¢¦¤º4PwREh{NXýñ§oåøÅ!7-n7èúÜßz«Ãc1yÔF,ÍjEº'k·tÞlq¶[¼TXN]äñ£Ôö ¿2>øyÿÈa5ö#y¼ÿ#WÉ endstream endobj 363 0 obj <> endobj 460 0 obj <>stream xcdabddä vs0400�qH3ýaîþó#ôgk7s7ËÒï÷¾³~gäÿÀÀÄÈ(§ë[PZZ¤àZ§à¢àZQ&ÉÀÀÀ¸! ¤ýêÆz[¾ÿL ¿c6¤}÷ù±ê»OÚ¡Û¾³}gøÎvQ\xåw½B¢Ñe
Ý1Ýê7Ìfí©¼WÕÍáä6mqpêIsæ/ZY¶<#7¿"ÅìyÒÕ§o|ÚZµd\ã¬Úé 3w/Xz¬{W÷{»ûk8gE͵³[êÄ#'×nh(^"WÔRUÞ]À!¼wAþºËçmz ¿ÅÑP+H%qN¢L¹©53ë§ÕrlÎía^¹7î»TTwbIQiaÀæßBSºOvoY¸xÑr¾Æi?¦|ëY9íwôDö÷{¹å¸§Ùóp20��À2' endstream endobj 45 0 obj << /Title(\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s) /Dest(section.71) /Parent 10 0 R /Prev 44 0 R >> endobj 8 0 obj << /Limits [ (Corollary.3) (subsubsection.62) ] /Names [ (Corollary.3) 111 0 R (Doc-Start) 46 0 R (Item.13) 132 0 R (Item.14) 135 0 R (Item.15) 142 0 R (Item.26) 222 0 R (Item.28) 227 0 R (Item.31) 243 0 R (Item.32) 246 0 R (Item.45) 308 0 R (Item.46) 309 0 R (Item.47) 310 0 R (Item.48) 311 0 R (Item.49) 312 0 R (Item.50) 313 0 R (Item.51) 314 0 R (Item.54) 345 0 R (Item.55) 346 0 R (Item.56) 347 0 R (Item.60) 353 0 R (Item.61) 357 0 R (Item.68) 389 0 R (Item.69) 390 0 R (appendix.53) 330 0 R (appendix.67) 385 0 R (cite.axler:measure-theory) 407 0 R (cite.bartle:ERA) 431 0 R (cite.bartle:IRA) 434 0 R (cite.conrad:stirlings-formula) 413 0 R (cite.knopp:infinite-series) 417 0 R (cite.mo:int-problems-ans1) 422 0 R (cite.mo:int-problems-ans2) 425 0 R (cite.mo:int-problems-qu) 420 0 R (cite.mo:obvious-hard-to-prove-ans) 429 0 R (cite.mo:obvious-hard-to-prove-qu) 427 0 R (cite.mse:countable-additivity) 432 0 R (cite.mse:series-rearrangements) 418 0 R (cite.weaver:measure-theory) 424 0 R (equation.11) 127 0 R (equation.12) 128 0 R (equation.21) 205 0 R (equation.22) 206 0 R (equation.23) 214 0 R (equation.24) 219 0 R (equation.25) 220 0 R (equation.27) 224 0 R (equation.29) 228 0 R (equation.35) 260 0 R (equation.38) 276 0 R (equation.42) 293 0 R (equation.43) 295 0 R (equation.65) 378 0 R (equation.66) 379 0 R (equation.70) 393 0 R (page.1) 9 0 R (page.10) 269 0 R (page.11) 285 0 R (page.12) 303 0 R (page.13) 328 0 R (page.14) 356 0 R (page.15) 369 0 R (page.16) 384 0 R (page.17) 403 0 R (page.2) 87 0 R (page.3) 104 0 R (page.4) 121 0 R (page.5) 149 0 R (page.6) 189 0 R (page.7) 210 0 R (page.8) 239 0 R (page.9) 256 0 R (section.52) 329 0 R (section.71) 404 0 R (section.1) 58 0 R (section.17) 198 0 R (section.2) 71 0 R (subsection.10) 122 0 R (subsection.16) 150 0 R (subsection.18) 199 0 R (subsection.19) 203 0 R (subsection.20) 204 0 R (subsection.3) 72 0 R (subsection.30) 240 0 R (subsection.33) 247 0 R (subsection.34) 252 0 R (subsection.36) 263 0 R (subsection.37) 274 0 R (subsection.39) 280 0 R (subsection.4) 75 0 R (subsection.40) 286 0 R (subsection.41) 291 0 R (subsection.44) 304 0 R (subsection.5) 88 0 R (subsection.57) 348 0 R (subsection.6) 94 0 R (subsection.63) 370 0 R (subsection.64) 376 0 R (subsection.7) 95 0 R (subsection.8) 98 0 R (subsection.9) 115 0 R (subsubsection.58) 350 0 R (subsubsection.59) 351 0 R (subsubsection.62) 360 0 R] >> endobj 478 0 obj <>stream GPL Ghostscript 10.01.2 2023-12-20T20:17:57-05:00 2023-12-20T20:17:57-05:00 LaTeX with hyperref Ross Ure Anderson endstream endobj 2 0 obj <>endobj xref 0 479 0000000000 65535 f 0000102671 00000 n 0000224134 00000 n 0000102409 00000 n 0000098297 00000 n 0000000182 00000 n 0000006555 00000 n 0000102839 00000 n 0000220243 00000 n 0000103028 00000 n 0000102596 00000 n 0000103083 00000 n 0000105419 00000 n 0000103237 00000 n 0000103486 00000 n 0000103706 00000 n 0000103926 00000 n 0000104146 00000 n 0000104484 00000 n 0000104366 00000 n 0000104740 00000 n 0000104960 00000 n 0000105181 00000 n 0000109756 00000 n 0000105678 00000 n 0000105918 00000 n 0000106201 00000 n 0000106489 00000 n 0000106862 00000 n 0000107280 00000 n 0000107608 00000 n 0000107976 00000 n 0000108359 00000 n 0000108787 00000 n 0000109155 00000 n 0000109558 00000 n 0000109966 00000 n 0000111549 00000 n 0000110756 00000 n 0000110115 00000 n 0000110301 00000 n 0000110500 00000 n 0000111035 00000 n 0000111296 00000 n 0000112029 00000 n 0000220107 00000 n 0000112518 00000 n 0000156314 00000 n 0000151784 00000 n 0000194183 00000 n 0000151097 00000 n 0000190776 00000 n 0000149757 00000 n 0000187303 00000 n 0000149347 00000 n 0000185646 00000 n 0000148836 00000 n 0000182116 00000 n 0000112575 00000 n 0000148008 00000 n 0000173814 00000 n 0000147088 00000 n 0000164735 00000 n 0000145760 00000 n 0000156381 00000 n 0000112633 00000 n 0000112783 00000 n 0000112937 00000 n 0000113088 00000 n 0000155537 00000 n 0000213330 00000 n 0000113243 00000 n 0000113301 00000 n 0000154180 00000 n 0000210291 00000 n 0000113359 00000 n 0000153595 00000 n 0000208468 00000 n 0000152683 00000 n 0000200827 00000 n 0000152055 00000 n 0000196135 00000 n 0000113417 00000 n 0000151445 00000 n 0000191925 00000 n 0000113600 00000 n 0000098496 00000 n 0000113786 00000 n 0000113843 00000 n 0000006575 00000 n 0000011167 00000 n 0000150339 00000 n 0000189752 00000 n 0000113901 00000 n 0000114038 00000 n 0000114097 00000 n 0000114156 00000 n 0000114293 00000 n 0000114430 00000 n 0000114489 00000 n 0000149591 00000 n 0000186332 00000 n 0000114626 00000 n 0000098677 00000 n 0000114760 00000 n 0000011188 00000 n 0000016743 00000 n 0000149188 00000 n 0000184903 00000 n 0000114819 00000 n 0000114865 00000 n 0000115021 00000 n 0000148610 00000 n 0000179822 00000 n 0000115082 00000 n 0000115220 00000 n 0000147767 00000 n 0000173213 00000 n 0000115280 00000 n 0000115315 00000 n 0000098875 00000 n 0000115488 00000 n 0000115547 00000 n 0000016765 00000 n 0000021848 00000 n 0000115607 00000 n 0000115745 00000 n 0000115883 00000 n 0000115948 00000 n 0000116013 00000 n 0000116151 00000 n 0000116288 00000 n 0000116425 00000 n 0000116486 00000 n 0000116624 00000 n 0000116763 00000 n 0000116824 00000 n 0000116962 00000 n 0000117101 00000 n 0000117240 00000 n 0000117379 00000 n 0000117518 00000 n 0000117657 00000 n 0000117718 00000 n 0000146927 00000 n 0000164071 00000 n 0000117869 00000 n 0000118008 00000 n 0000099150 00000 n 0000118155 00000 n 0000118214 00000 n 0000021870 00000 n 0000029061 00000 n 0000146533 00000 n 0000159839 00000 n 0000118274 00000 n 0000118407 00000 n 0000118546 00000 n 0000118684 00000 n 0000118833 00000 n 0000118984 00000 n 0000119136 00000 n 0000119287 00000 n 0000119426 00000 n 0000119565 00000 n 0000119704 00000 n 0000119863 00000 n 0000120021 00000 n 0000120172 00000 n 0000120311 00000 n 0000120450 00000 n 0000120589 00000 n 0000120740 00000 n 0000120881 00000 n 0000121032 00000 n 0000121183 00000 n 0000121334 00000 n 0000121485 00000 n 0000121634 00000 n 0000121789 00000 n 0000121926 00000 n 0000122063 00000 n 0000122202 00000 n 0000122341 00000 n 0000122480 00000 n 0000122619 00000 n 0000122770 00000 n 0000122921 00000 n 0000099561 00000 n 0000123081 00000 n 0000029083 00000 n 0000034557 00000 n 0000123140 00000 n 0000123291 00000 n 0000123430 00000 n 0000123567 00000 n 0000123718 00000 n 0000123857 00000 n 0000124008 00000 n 0000124069 00000 n 0000124129 00000 n 0000124267 00000 n 0000124406 00000 n 0000124545 00000 n 0000124606 00000 n 0000124667 00000 n 0000124732 00000 n 0000124797 00000 n 0000124832 00000 n 0000099815 00000 n 0000124979 00000 n 0000034579 00000 n 0000039326 00000 n 0000125038 00000 n 0000125177 00000 n 0000125240 00000 n 0000125377 00000 n 0000125514 00000 n 0000125651 00000 n 0000125788 00000 n 0000125853 00000 n 0000125915 00000 n 0000126071 00000 n 0000126132 00000 n 0000126269 00000 n 0000126334 00000 n 0000126471 00000 n 0000126608 00000 n 0000126668 00000 n 0000126733 00000 n 0000126870 00000 n 0000127003 00000 n 0000127140 00000 n 0000127277 00000 n 0000127410 00000 n 0000127547 00000 n 0000127684 00000 n 0000127719 00000 n 0000100117 00000 n 0000127829 00000 n 0000127888 00000 n 0000039348 00000 n 0000046065 00000 n 0000127948 00000 n 0000128008 00000 n 0000128147 00000 n 0000128286 00000 n 0000128347 00000 n 0000128408 00000 n 0000128547 00000 n 0000128686 00000 n 0000128825 00000 n 0000128964 00000 n 0000129024 00000 n 0000129059 00000 n 0000100347 00000 n 0000129193 00000 n 0000046087 00000 n 0000052370 00000 n 0000129252 00000 n 0000129408 00000 n 0000129473 00000 n 0000129629 00000 n 0000129766 00000 n 0000129824 00000 n 0000129963 00000 n 0000130102 00000 n 0000130241 00000 n 0000100550 00000 n 0000130375 00000 n 0000052392 00000 n 0000056508 00000 n 0000130434 00000 n 0000130573 00000 n 0000130712 00000 n 0000130773 00000 n 0000130912 00000 n 0000130976 00000 n 0000131115 00000 n 0000131254 00000 n 0000131391 00000 n 0000131452 00000 n 0000131591 00000 n 0000131730 00000 n 0000100769 00000 n 0000131864 00000 n 0000131923 00000 n 0000056530 00000 n 0000061699 00000 n 0000131983 00000 n 0000132122 00000 n 0000132261 00000 n 0000132322 00000 n 0000132460 00000 n 0000132525 00000 n 0000132664 00000 n 0000132729 00000 n 0000132866 00000 n 0000133005 00000 n 0000133161 00000 n 0000133298 00000 n 0000133333 00000 n 0000101015 00000 n 0000133467 00000 n 0000133526 00000 n 0000061721 00000 n 0000067843 00000 n 0000133586 00000 n 0000133741 00000 n 0000133802 00000 n 0000133863 00000 n 0000133924 00000 n 0000133985 00000 n 0000134046 00000 n 0000134107 00000 n 0000134168 00000 n 0000134305 00000 n 0000134438 00000 n 0000134571 00000 n 0000134722 00000 n 0000134855 00000 n 0000134988 00000 n 0000135121 00000 n 0000135254 00000 n 0000135405 00000 n 0000135556 00000 n 0000135707 00000 n 0000101266 00000 n 0000135854 00000 n 0000135913 00000 n 0000135973 00000 n 0000067865 00000 n 0000075714 00000 n 0000136033 00000 n 0000136172 00000 n 0000136327 00000 n 0000136468 00000 n 0000136609 00000 n 0000136750 00000 n 0000136891 00000 n 0000137032 00000 n 0000137173 00000 n 0000137314 00000 n 0000137455 00000 n 0000137596 00000 n 0000137737 00000 n 0000137798 00000 n 0000137859 00000 n 0000137920 00000 n 0000137978 00000 n 0000138117 00000 n 0000138178 00000 n 0000138239 00000 n 0000138380 00000 n 0000138441 00000 n 0000101541 00000 n 0000138601 00000 n 0000138660 00000 n 0000075736 00000 n 0000083003 00000 n 0000138720 00000 n 0000138780 00000 n 0000156153 00000 n 0000219434 00000 n 0000138921 00000 n 0000139063 00000 n 0000139205 00000 n 0000139346 00000 n 0000101736 00000 n 0000139521 00000 n 0000139580 00000 n 0000083025 00000 n 0000089546 00000 n 0000139640 00000 n 0000154865 00000 n 0000212678 00000 n 0000139781 00000 n 0000139842 00000 n 0000139983 00000 n 0000140048 00000 n 0000140113 00000 n 0000140250 00000 n 0000140387 00000 n 0000101923 00000 n 0000140547 00000 n 0000140606 00000 n 0000089568 00000 n 0000094922 00000 n 0000140666 00000 n 0000140817 00000 n 0000140878 00000 n 0000140939 00000 n 0000141078 00000 n 0000141217 00000 n 0000141282 00000 n 0000141433 00000 n 0000141572 00000 n 0000141710 00000 n 0000141847 00000 n 0000141986 00000 n 0000142123 00000 n 0000142262 00000 n 0000102166 00000 n 0000142396 00000 n 0000142455 00000 n 0000094944 00000 n 0000098275 00000 n 0000142516 00000 n 0000153057 00000 n 0000203503 00000 n 0000142574 00000 n 0000152299 00000 n 0000196794 00000 n 0000142737 00000 n 0000142798 00000 n 0000142995 00000 n 0000143191 00000 n 0000143387 00000 n 0000143448 00000 n 0000143509 00000 n 0000143694 00000 n 0000143752 00000 n 0000143930 00000 n 0000143991 00000 n 0000144169 00000 n 0000144230 00000 n 0000144288 00000 n 0000144466 00000 n 0000144527 00000 n 0000144704 00000 n 0000144765 00000 n 0000144942 00000 n 0000145003 00000 n 0000145064 00000 n 0000145249 00000 n 0000145310 00000 n 0000156824 00000 n 0000160209 00000 n 0000164296 00000 n 0000165246 00000 n 0000173432 00000 n 0000174247 00000 n 0000180080 00000 n 0000182415 00000 n 0000185118 00000 n 0000185865 00000 n 0000186563 00000 n 0000187578 00000 n 0000190002 00000 n 0000191004 00000 n 0000192241 00000 n 0000194437 00000 n 0000196354 00000 n 0000197172 00000 n 0000201084 00000 n 0000203832 00000 n 0000208784 00000 n 0000210719 00000 n 0000212903 00000 n 0000213674 00000 n 0000219659 00000 n 0000145391 00000 n 0000146196 00000 n 0000147617 00000 n 0000147916 00000 n 0000148495 00000 n 0000149496 00000 n 0000150088 00000 n 0000150727 00000 n 0000150857 00000 n 0000151350 00000 n 0000152204 00000 n 0000152969 00000 n 0000153496 00000 n 0000153929 00000 n 0000154536 00000 n 0000155026 00000 n 0000156007 00000 n 0000222651 00000 n trailer << /Size 479 /Root 1 0 R /Info 2 0 R /ID [<0272AD32BC270532662D237D784EE205><0272AD32BC270532662D237D784EE205>] >> startxref 224425 %%EOF~~~~~~~~ |
11756 | https://zhuanlan.zhihu.com/p/396740625 | 高考数学提分必备，知识整理⑤——立体几何平行与垂直的必备方法 - 知乎
关注推荐热榜专栏圈子 New付费咨询知学堂
直答
切换模式
登录/注册
高考数学提分必备，知识整理⑤——立体几何平行与垂直的必备方法
首发于高中数学重难点解析
切换模式
高考数学提分必备，知识整理⑤——立体几何平行与垂直的必备方法
爱洁老师
北京科技大学 应用数学硕士
收录于 · 高中数学重难点解析
7 人赞同了该文章
高中数学老师刘爱洁：江湖人称爱姐，北京科技大学数学系研究生。授课过程饱含激情又带有欢乐，只有亲身体验过才能知道其中的酸甜苦辣，所带学生单科成绩可进步20-80分，提倡快乐学习，爱上数学，变身数学学霸~
立体几何，其实主要的思路就两个字：转化。线面转化为面面，线线转化为线面，线线转化为面面。铁三角转化关系，平行和垂直也是一样的，都是互相转化。而且平行和垂直之间可以转化。如果不知道这个转化思想，可以用空间向量解决问题，只要可以建系，就没有啥问题。
先来个图，咱们看一下，立体几何的证明题转化思路基本都在这里了。
线线平行思路总结
1.利用平行四边形 ;
2.利用三角形或梯形的中位线 ;
3.线面平行: 如果一条直线和一个平面平行，经过这条直线的平面和这个平面相交，那么这条直线就和交线平行 ;
4.面面平行：如果两个平行平面同时和第三个平面相交，那么它们的交线平行 ;
5.线面垂直: 如果两条直线垂直于同一个平面，那么这两条直线平行 ;
6.平行于同一条直线的两条直线平行。
解决有关线面平行、面面平行的基本问题要注意：
(1)判定定理与性质定理中易忽视的条件，如线面平行的判定定理中条件线在面外易忽视．
(2)结合题意构造或绘制图形，结合图形作出判断．
(3)举反例否定结论或用反证法推断命题是否正确．
线面平行的方法总结
1.定义法：直线与平面没有公共点；
2.线面平行：平面外一条直线和这个平面内的一条直线平行，那么这条直线和这个平面平行；
3.两个平面平行，其中一个平面内的任何一条直线必平行于另一个平面；
4.反证法。
面面平行的方法总结
1.定义法：两平面没有公共点；
2.面面平行：如果一个平面内有两条相交直线都平行于另一个平面，那么这两个平面平行；
3.平行于同一平面的两个平面平行；
4.经过平面外一点，有且只有一个平面和已知平面平行；
5.垂直于同一直线的两个平面平行 。
线线垂直的方法总结
1.勾股定理；
2.等腰三角形；
3.菱形对角线；
4.圆所对的圆周角是直角；
5.点在线上的射影；
6.如果一条直线和一个平面垂直，那么这条直线就和这个平面内任意的直线都垂直；
7.在平面内的一条直线，如果和这个平面一条斜线的射影垂直，那么它也和这条斜线垂直；
8.在平面内的一条直线，如果和这个平面一条斜线垂直，那么它也和这条斜线的射影垂直；
9.如果两条平行线中的一条垂直于一条直线，则另一条也垂直于这条直线。
线面垂直的证明方法
1.定义法：直线与平面内任意直线都垂直；
2.点在面内的射影；
3.线面垂直:如果一条直线和一个平面内的两条相交直线垂直，那么这条直线垂直于这个平面；
4.面面垂直:如果两个平面互相垂直，那么在一个平面内垂直于它们交线的直线垂直于另一个平面；
5.两条平行直线中的一条垂直于平面，则另一条也垂直于这个平面；
6.一条直线垂直于两平行平面中的一个平面，则必垂直于另一个平面；
7.两相交平面同时垂直于第三个平面，那么两平面交线垂直于第三个平面；
8.过一点，有且只有一条直线与已知平面垂直；
9.过一点，有且只有一个平面与已知直线垂直。
面面垂直的证明方法
1.定义法：两个平面的二面角是直二面角；
2.面面垂直：如果一个平面经过另一个平面的一条垂线，那么这两个平面互相垂直；
3.如果一个平面与另一个平面的垂线平行，那么这两个平面互相垂直；
4.如果一个平面与另一个平面的垂面平行，那么这两个平面互相垂直。
以上就是咱们对于平行垂直这一块内容的梳理，基本的重点思路要记住，这一块的内容不是很难，大家一定不要着急，大家一定要要多总结多积累，多研究，需要练习题目的同学可以私信或者评论留言哈。预祝大家数学成绩芝麻开花节节高。
编辑于 2021-10-13 04:32
高考数学
高中数学
高考
赞同 7添加评论
分享
喜欢收藏申请转载
写下你的评论...
还没有评论，发表第一个评论吧
关于作者
爱洁老师
北京科技大学数学系
北京科技大学 应用数学硕士
回答 47文章 43关注者 209
关注她发私信
推荐阅读
高考数学提分必备，知识整理⑤——立体几何线线角、线面角、面面角的求解 ================================== 高中数学老师刘爱洁：江湖人称爱姐，北京科技大学数学系研究生。授课过程饱含激情又带有欢乐，只有亲身体验过才能知道其中的酸甜苦辣，所带学生单科成绩可进步20-80分，提倡快乐学习，爱上… 爱洁老师高中数学高分养成记③：立体几何中平行与垂直的解题技 ========================= 由爱老师详细讲解的系列课程“高中数学学习养成记”会逐日分享给大家，从基础入手，把高分带走！今天带来的内容分享是立体几何。立体几何是高中必修2中出现的内容，不同年级的学生可以做相… 有话说频道高考数学（3）——解析几何 ============= 上周事太多，咕了咕了（doge。在外面上大学上着上着家被偷了，吉林和九台那边的疫情好严重啊，希望所有人都没事，疫情快点过去吧，妖魔鬼怪快离开！！ 闲言少叙，进入正题： 1 解析几何是什… SuperbRain 高考数学立体几何问题，解决的关键到底是什么？应用逆向思维 ============================ 老黄知识生...发表于高等数学
想来知乎工作？请发送邮件到 jobs@zhihu.com
打开知乎App
在「我的页」右上角打开扫一扫
其他扫码方式：微信
下载知乎App
无障碍模式
验证码登录
密码登录
开通机构号
获取短信验证码
获取语音验证码
登录/注册
其他方式登录
未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》
扫码下载知乎 App
关闭二维码
打开知乎App
在「我的页」右上角打开扫一扫
其他扫码方式：微信
下载知乎App
无障碍模式
验证码登录
密码登录
开通机构号
中国 +86
获取短信验证码
获取语音验证码
登录/注册
其他方式登录
未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》
扫码下载知乎 App
关闭二维码 |
11757 | https://math.libretexts.org/Courses/Coalinga_College/Math_for_Educators_(MATH_010A_and_010B_CID120)/05%3A_Problem_Solving/5.06%3A_Recognizing_Patterns | Skip to main content
5.6: Recognizing Patterns
Last updated
: Aug 20, 2024
Save as PDF
5.5: Use a Problem-Solving Strategy
5.7: Beware of Patterns!
Page ID
: 157442
( \newcommand{\kernel}{\mathrm{null}\,})
Mathematics is often seen as a subject rooted in strict rules and procedures. However, one of the most powerful tools in a mathematician's toolkit is the ability to recognize patterns. Identifying patterns can simplify complex problems, reveal underlying structures, and lead to elegant solutions. This section will explore how recognizing patterns can enhance problem-solving skills and provide strategies for identifying and utilizing patterns in various mathematical contexts.
The Importance of Patterns in Mathematics
Patterns are fundamental to mathematics. They appear in numbers, shapes, algebraic expressions, and even in the way we solve problems. Recognizing patterns helps in:
Predicting Future Events: Understanding a pattern allows us to predict what comes next. This is crucial in sequences and series, as well as in real-world applications like predicting trends.
Simplifying Problems: Patterns can reduce the complexity of a problem by revealing relationships that are not immediately obvious.
Enhancing Understanding: Seeing the connections between different problems and solutions can deepen our understanding of mathematical concepts.
Developing Strategies: Recognizing patterns can lead to the development of general strategies for solving classes of problems.
Types of Patterns
Numerical Patterns
Numerical patterns involve sequences of numbers that follow a specific rule. For example:
Arithmetic Sequences: Each term is obtained by adding a constant difference to the previous term.
Example: 2, 5, 8, 11, 14 (difference of 3)
Geometric Sequences: Each term is obtained by multiplying the previous term by a constant factor.
Example: 3, 6, 12, 24, 48 (factor of 2)
Fibonacci Sequence: Each term is the sum of the two preceding terms.
Example: 1, 1, 2, 3, 5, 8, 13
Visual Patterns
Visual patterns appear in shapes and arrangements. These are often used in geometry and can include:
Symmetry: Objects or shapes that are mirror images.
Example: A square has four lines of symmetry.
Tessellations: Patterns of shapes that fit perfectly together without gaps.
Example: Hexagons tessellate to form a honeycomb pattern.
Fractals: Complex patterns that are self-similar at different scales.
Example: The Sierpinski triangle, where each smaller triangle is a reduced-scale copy of the whole.
Algebraic Patterns
Algebraic patterns involve expressions and equations. Recognizing these can simplify solving equations and understanding functions.
Binomial Patterns: Expansion of powers of binomials follows the binomial theorem.
Example:
Factoring Patterns: Certain forms of expressions can be factored in standard ways.
Example:
Function Patterns: Functions can exhibit specific types of behavior.
Example: Linear functions have a constant rate of change.
Strategies for Recognizing Patterns
Observation and Practice
The more you work with mathematical problems, the more you will start to notice patterns. Practice is essential. Consistently engaging with a variety of mathematical problems helps train your mind to recognize patterns and relationships more readily. When you work on diverse problems, make it a habit to pay attention to recurring themes and commonalities. Over time, this practice will sharpen your observational skills, allowing you to detect patterns more quickly and with greater accuracy.
Additionally, documenting the patterns you observe can be incredibly beneficial. Keep a journal or a log of the different types of problems you solve and note any patterns or strategies that emerge. This process not only reinforces your learning but also provides a valuable reference for future problem-solving tasks. Engaging with peers in discussions about problem-solving approaches and patterns can also enhance your ability to see different perspectives and uncover patterns you might have missed on your own.
Look for Regularities
Looking for regularities is a crucial aspect of recognizing patterns in mathematical problem-solving. Regularities are consistent and predictable aspects that appear repeatedly within problems, which can help simplify complex issues and provide insight into their solutions. To develop this skill, it's important to actively search for these regularities as you work through various mathematical tasks.
One effective strategy is to start by examining simpler versions of the problem at hand. Often, complex problems can be broken down into more manageable parts, and these simpler components may reveal underlying regularities. For example, when studying sequences, identifying the first few terms and their relationships can help you discern a pattern that applies to the entire sequence.
Another approach is to use visualization techniques. Drawing diagrams, graphs, or tables can make regularities more apparent. Visual representations often highlight relationships and patterns that might be less obvious in a purely numerical or algebraic form. For instance, plotting data points on a graph can reveal trends and regularities that facilitate the formulation of a general rule or function.
It's also beneficial to compare and contrast similar problems. By analyzing how different problems are solved and identifying common strategies, you can uncover regularities that apply across multiple contexts. This comparative approach not only reinforces your understanding of specific patterns but also broadens your ability to apply these patterns to new and unfamiliar problems.
Furthermore, consider using mathematical notation and symbols consistently. Standardized notation can help you see regularities more clearly and communicate your findings effectively. When you notice that a particular notation or symbolic representation is recurring, it can signal an underlying regularity worth exploring further.
Engaging in collaborative problem-solving can enhance your ability to identify regularities. Discussing problems with peers and sharing different perspectives can lead to new insights and uncover regularities that you might not have noticed on your own. Group work encourages a deeper exploration of patterns and often leads to a more comprehensive understanding of the problem-solving process.
Finally, regular reflection on your problem-solving methods and the regularities you encounter is crucial. After solving a problem, take the time to review your approach and identify any patterns or regularities that emerged. Reflecting on these aspects can consolidate your learning and improve your ability to recognize and apply regularities in future problems.
Looking for regularities involves actively seeking out consistent and predictable aspects of problems, using simplification, visualization, comparison, notation, collaboration, and reflection. By honing this skill, you can enhance your mathematical problem-solving abilities and approach complex problems with greater confidence and insight.
Use Tables and Diagrams
Tables and diagrams are powerful tools in mathematics for visualizing and identifying regularities in problem-solving. They provide structured ways to organize information, making patterns more apparent and aiding in the discovery of underlying relationships.
Tables:
Tables are useful for organizing data systematically, especially when dealing with numerical sequences, functions, or categorical data. By tabulating relevant variables or values, you can observe how different elements interact over time or across different conditions. For example, when analyzing the growth of a population over several years, a table can show how the numbers change annually and highlight any consistent rate of increase or decrease.
To effectively use tables to identify regularities, consider the following strategies:
Pattern Recognition: Arrange data in a table format and look for recurring numbers or sequences. For instance, in a multiplication table, noticing how each row and column follows a predictable pattern (e.g., increasing by the same multiple) can reveal underlying multiplication rules.
Comparative Analysis: Create multiple tables to compare different scenarios or conditions. By placing side-by-side tables of experimental results, you can identify trends or patterns that occur consistently across different sets of data. This comparative approach helps in drawing conclusions and formulating general rules.
Function Representation: Use tables to represent functions by mapping input values to corresponding output values. By examining how changes in input affect output, you can derive equations or formulas that describe the relationship between variables. This method is particularly useful in calculus and algebra, where tables can illustrate the behavior of functions graphically.
Diagrams:
Diagrams provide visual representations that can reveal patterns and relationships in a problem-solving context. They are especially effective for illustrating geometric properties, structural relationships, and process flows. Different types of diagrams include:
Graphs and Charts: Graphical representations like line graphs, bar charts, and pie charts visually display data relationships. For example, plotting points on a graph can show trends over time or correlations between variables, making patterns in data more apparent.
Geometric Figures: Diagrams such as geometric shapes (e.g., triangles, circles) or spatial arrangements (e.g., Venn diagrams) help in understanding spatial relationships and logical connections. These diagrams often highlight symmetries, proportions, or overlapping areas that signify regularities in geometric properties.
Flowcharts: Flowcharts depict sequences of actions or decision-making processes. They illustrate step-by-step procedures and help identify recurring steps or decision points. Analyzing flowcharts can reveal algorithmic patterns or procedural regularities in problem-solving strategies.
Strategies for Effective Use:
Label and Title: Clearly label elements in tables and diagrams to enhance clarity and understanding.
Analyze Trends: Look for trends or changes in data patterns as you study tables or interpret diagrams. Identifying consistent trends can lead to the discovery of underlying regularities.
Iterative Refinement: Iterate on your tables and diagrams by refining them based on new observations or insights. Adjusting variables or adding annotations can help in uncovering deeper patterns.
Tables and diagrams are versatile tools that complement mathematical reasoning by providing visual insights into problem structures and relationships. By leveraging these tools effectively, you can enhance your ability to recognize and utilize regularities in mathematical problem-solving, ultimately improving your proficiency and confidence in tackling complex problems.
Generalize from Examples
Generalizing from examples is a fundamental skill in mathematics that empowers elementary students to uncover patterns and rules from specific instances, fostering a deeper understanding of mathematical concepts.
Steps to Guide Students in Generalizing from Examples:
Explore Specific Instances: Start with concrete examples that students can easily grasp. For example, when introducing addition, use simple problems like and . Encourage students to solve these problems using manipulatives or drawings.
Identify Patterns: Guide students to identify patterns or similarities among the examples. Ask questions like, "What do you notice about these addition problems?" Prompt them to recognize that in each case, they are combining two numbers to find a total.
Formulate Simple Rules: Help students formulate simple rules or strategies based on their observations. For addition problems, they might notice that they always combine two numbers to get a larger number.
Apply to New Examples: Provide additional examples or variations of the same concept. Ask students to apply the rule they formulated to solve new problems. For instance, after understanding and , they can predict that .
Encourage Reflective Thinking: Foster reflective thinking by asking students to explain their reasoning. Encourage them to articulate why their rule works and how they know it applies to different examples.
Express Generalization: Once students have tested their rule across multiple examples and variations, guide them to express their generalized understanding in simple terms. For example, they might say, "When you add two numbers together, the total is always bigger."
Benefits of Generalizing from Examples for Students:
Conceptual Understanding: By exploring patterns and formulating rules, students develop a deeper conceptual understanding of mathematical operations.
Problem-Solving Skills: Generalized rules serve as tools for problem-solving, enabling students to approach new problems with confidence.
Critical Thinking: The process encourages critical thinking as students analyze patterns, make connections, and apply their understanding to different situations.
Building Mathematical Confidence: Successfully generalizing from examples builds students' confidence in their mathematical abilities and promotes a positive attitude towards learning.
Strategies for Effective Teaching:
Use Concrete Examples: Start with manipulatives, drawings, or real-world examples to make abstract concepts more tangible.
Provide Guided Practice: Scaffold learning by guiding students through progressively more complex examples.
Encourage Discussion: Foster collaborative learning by encouraging students to discuss their observations and reasoning with peers.
Connect to Real Life: Relate generalized rules to everyday situations to show relevance and application.
By guiding elementary students through the process of generalizing from examples, teachers empower them to not only solve mathematical problems but also to develop a deeper appreciation and mastery of mathematical concepts from an early age.
Verify and Refine
Verifying and refining mathematical understanding is a crucial step in ensuring that elementary students develop solid foundations in math concepts. It involves helping students confirm the accuracy of their solutions, identify errors, and improve their problem-solving strategies.
Steps to Guide Elementary Students in Verifying and Refining Mathematical Understanding:
Check Solutions for Accuracy: Teach students to check their solutions by verifying if their answers make sense in the context of the problem. For example, if solving encourage them to estimate the answer or use a different method (like subtraction or grouping) to confirm their result.
Identify and Correct Errors: Guide students to identify common types of errors, such as calculation mistakes, misinterpretation of the problem, or overlooking important details. Show them strategies to backtrack and find where mistakes occurred.
Use Multiple Approaches: Encourage students to solve problems using different methods or strategies. This not only reinforces understanding but also allows them to cross-verify their answers. For instance, after adding numbers, they could use subtraction to check their work.
Refine Problem-Solving Strategies: Help students reflect on their problem-solving processes. Ask questions like, "Was there a better or faster way to solve this problem?" Guide them to consider efficiency and accuracy when refining their strategies.
Seek Alternative Solutions: Promote creativity by encouraging students to find alternative solutions to the same problem. This fosters flexibility in thinking and deepens understanding of mathematical concepts.
Provide Constructive Feedback: Offer specific feedback that highlights both correct approaches and areas for improvement. Encourage peer feedback and discussions to promote a supportive learning environment.
Benefits of Verifying and Refining Mathematical Understanding for Students:
Enhanced Accuracy: Students develop skills to ensure their solutions are correct and logical.
Improved Problem-Solving Skills: The process encourages students to think critically and strategically about different ways to approach problems.
Building Resilience: Learning to identify and correct mistakes fosters resilience and perseverance in mathematical tasks.
Deepened Conceptual Understanding: By refining strategies and exploring alternatives, students gain a deeper understanding of mathematical concepts and relationships.
Strategies for Effective Teaching:
Model the Process: Demonstrate how to verify solutions and refine strategies through examples and think-alouds.
Encourage Peer Collaboration: Promote peer discussions where students can explain their reasoning and critique each other's solutions.
Use Real-World Examples: Relate mathematical problems to real-life situations to make the verification process more meaningful and applicable.
Celebrate Mistakes: Emphasize that mistakes are opportunities for learning and growth, encouraging students to embrace challenges.
By guiding elementary students through the process of verifying and refining their mathematical understanding, teachers empower them to become confident, reflective problem solvers who approach mathematics with curiosity and resilience.
Applying Pattern Recognition in Problem Solving
Example 1: Arithmetic Sequence
Problem: Find the 10th term of the sequence: 4, 9, 14, 19, ...
Solution
Identify the Pattern: Notice the difference between terms is 5.
Write the Rule: The term of an arithmetic sequence is given by .
Apply the Rule: For the 10th term, .
Example 2: Investigating Patterns in Bivariate Data
Problem: ou want to understand if there is a connection between students' favorite seasons and their preferred sports activities.
Solution
Data Collection: Ask each student in your class to share their favorite season (e.g., winter, spring, summer, fall) and their favorite sport (e.g., soccer, basketball, swimming, biking).
Organize Data: Create a table on the board or a worksheet with two columns: one for favorite seasons and another for favorite sports.
| Favorite Season | Favorite Sport |
--- |
| Winter | Soccer |
| Spring | Basketball |
| Summer | Swimming |
| Fall | Biking |
| ... | ... |
3. Plotting Data: Use a simple scatter plot on a large poster or interactive whiteboard. Place each student's favorite season along the x-axis and their favorite sport along the y-axis.
4. Observing Patterns: Look at the plotted points together as a class. Discuss what patterns or trends students notice. For example, are there groups of students who prefer winter and skiing?
5. Identifying Association: Guide students to identify any associations between favorite seasons and sports. Are there more students who prefer summer and swimming?
6. Interpreting Results: Help students interpret their findings. For instance, if many students who like fall also enjoy biking, discuss the association between these variables.
7. Discussion: Lead a class discussion on why these associations might exist. For example, why might students who like winter prefer skiing or snowboarding?
Example 3: Factoring Algebraic Expressions
Problem: Factor the expression .
Solution
Identify the Pattern: Recognize the expression as a perfect square trinomial.
Apply the Pattern: .
Conclusion
Recognizing patterns is a crucial skill in mathematics that enhances problem-solving abilities. By identifying and utilizing patterns, you can simplify complex problems, make accurate predictions, and develop deeper mathematical insights. Practice these strategies regularly to become proficient in recognizing patterns and applying them to various mathematical contexts.
Citations and Attributions
"Sierpinski Triangle (from L-System, 4 iterations).png" by LingoLoco is licensed under CC BY-SA 4.0
"202208 Honeycomb structure.svg" by DataBase Center for Life Science (DBCLS) is licensed under CC BY-SA 4.0
5.5: Use a Problem-Solving Strategy
5.7: Beware of Patterns! |
11758 | https://www.teacherspayteachers.com/Product/TEKS-43D-Comparing-Fractions-Unlike-Denominators-Print-Digital-6941484 | TEKS 4.3D - Comparing Fractions Unlike Denominators [Print & Digital]
Description
This resource is designed to assess the following Texas Essential Knowledge and Skills (TEKS) standard:
4.3D – compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <
This resource contains 3 assessments, for a total of 15 items, including multiple choice and multiple response. You can choose how you want to use this resource! The same assessments/items come in several versions including:
Each assessment contains a variety of problem situations to help you, the teacher, determine a student's level of understanding for this particular skill. This resource is designed to help you answer the following questions:
⭐️ Are students successful comparing values equal to and/or greater than ½ ?
⭐️ Are students successful comparing values equal to and/or less than ½ ?
⭐️ Are students successful comparing values greater than 1?
⭐️ Do students recognize equivalent values?
Understanding which part or parts of this skill a student masters or struggles with will help you plan your next move to either extend a student's learning or intervene if necessary.
Using the Google Slides™ or Task Card version of this resource will allow you to intentionally choose which problems students receive. The assessment version (the printable worksheet or Google Forms™) have several potential uses including:
➡️ pre/post tests
➡️ homework
➡️ data checkpoints
➡️ progress monitoring for RTI
➡️ STAAR prep, or
➡️ quick assessment of current skill level
The assessments that contain QR codes are a great way for students to self-check.
This file is a license for one teacher and their students.
You may:
✅ Print and copy this resource for your personal use.
✅ Post this only for your students on a password protected class website or LMS.
⭐️⭐️⭐️⭐️⭐️ Follow me to be notified of future products! ⭐️⭐️⭐️⭐️⭐️
If you have any questions please email me directly at KYDMathematics@gmail.com or use TPT's Ask a Questions feature.
2021 © KYD Mathematics
All rights reserved by the author.
Permission to copy for single classroom use only.
Electronic distribution limited to single class only.
Not for public display.
TEKS 4.3D - Comparing Fractions Unlike Denominators [Print & Digital]
Save even more with bundles
Reviews
Questions & Answers
Standards |
11759 | https://lipidworld.biomedcentral.com/articles/10.1186/1476-511X-9-27 | Lipids in Health and Disease
Evaluation of different formulas for LDL-C calculation
Download PDF
Download ePub
Download PDF
Download ePub
Research
Open access
Published:
Evaluation of different formulas for LDL-C calculation
Ana Vujovic1,
Jelena Kotur-Stevuljevic1,
Slavica Spasic1,
Nada Bujisic2,
Jelena Martinovic3,
Milica Vujovic4,
Vesna Spasojevic-Kalimanovska1,
Aleksandra Zeljkovic1 &
…
Dragoljub Pajic4
Lipids in Health and Disease volume 9, Article number: 27 (2010) Cite this article
33k Accesses
97 Citations
Metrics details
Abstract
Background
Friedewald's formula for the estimation of LDL-C concentration is the most often used formula in clinical practice. A recent formula by Anandaraja and colleagues for LDL-C estimation still needs to be evaluated before it is extensively applied in diagnosis. In the present study we validated existing formulas and derived a more accurate formula to determine LDL-C in a Serbian population.
Methods
Our study included 2053 patients with TG ≤ 4.52 mmol/L. In an initial group of 1010 patients, Friedewald's and Anandaraja's formulas were compared to a direct homogenous method for LDL-C determination. The obtained results allowed us to modify Friedewald's formula and apply it in a second group of patients.
Results
The mean LDL-C concentrations were 3.9 ± 1.09 mmol/L, 3.63 ± 1.06 mmol/L and 3.72 ± 1.04 mmol/L measured by a direct homogenous assay (D-LDL-C), calculated by Friedewald's formula (F-LDL-C) and calculated by Anandaraja's formula (A-LDL-C), respectively in the 1010 patients. The Student's paired t-test showed that D-LDL-C values were significantly higher than F-LDL-C and A-LDL-C values (p < 0.001). The Passing-Bablok regression analysis indicated good correlation between calculated and measured LDL-Cs (r > 0.89). Using lipoprotein values from the initial group we modified Friedewald's formula by replacing the term 2.2 with 3. The new modified formula for LDL-C estimation (S-LDL-C) showed no statistically significant difference compared to D-LDL-C. The absolute bias between these two methods was -0.06 ± 0.37 mmol/L with a high correlation coefficient (r = 0.96).
Conclusions
Our modified formula for LDL-C estimation appears to be more accurate than both Friedewald's and Anandaraja's formulas when applied to a Serbian population.
Background
The concentration of low-density lipoprotein cholesterol (LDL-C) is one of the strongest markers of atherosclerosis and predictor for assessing coronary heart disease (CHD) risk. Strong positive association between increased LDL-C and CHD has been well documented [1–3]. The National Cholesterol Education Programme's (NCEP) Adult Treatment Panel III (ATP III) deemed that LDL-C concentration was the primary basis for treatment and appropriate patients' classification in risk categories demonstrating that both accuracy and precision of LDL-C analysis are critically important.
The reference method for LDL-C concentration measurement, which combines ultracentrifugation-polianion precipitation, it is not readily available and also impractical in the routine laboratory . A new generation of direct homogenous assays [6, 7] for LDL-C determination in serum has been developed with a satisfactory degree of accuracy but at the same time they are expensive.
Despite some limitations, Friedewald's formula is still the most commonly employed procedure in clinical laboratories for the estimation of LDL-C concentration and it has been considered acceptable for patients' classification . It is often used in developing countries, including Serbia, due to its simplicity, convenience and low cost. The increase in cardiovascular disease in Serbian adults together with the fact that risk factors need to be established at an early stage of disease underline the necessity to obtain the most precise and reliable formula for LDL-C calculation.
As already reported by other authors the homogeneous methods and Friedewald's formula are not capable of providing identical results [10, 11]. Attempts have been made to evaluate and refine Friedewald's original formula. Recently, a new formula for LDL-C estimation was proposed by Anandaraja and colleagues as a substitute for Friedewald's formula in the Indian population.
The present study was aimed to assess the validity of LDL-C values calculated by the Friedewald's formula (F-LDL-C) and those derived from Anandaraja and colleagues (A-LDL-C) and to compare them to values obtained by the direct method (D-LDL-C) in order to determine if a new formula could be applicable to a Serbian population. We also examined correlations and concentration differences obtained by the calculations and the direct method. The results of the present study were used to derive a new formula for calculation of LDL-C concentration (S-LDL-C) that appears to be more accurate than the latter two in a Serbian population. We also examined the classification concordance in relation to the NCEP ATP III LDL-C cut-off points between measured LDL-C and LDL-Cs derived by the three formulas to determine whether different calculation methods could affect patients' classification of heart disease risk.
Materials and methods
Participants
The first studied population (initial group) consisted of 1010 patients (51.9% male, mean age 53.7 ± 14.6 years) who underwent routine lipid status estimation as a part of a regular annual medical check-up. It was performed in the "Belladonna" Clinical Chemistry Laboratory during the whole of 2007. The second studied population (validation group) consisted of 1043 patients (42.9% male, mean age 54.2 ± 13 years) who were attending regular health check-ups in the "Sava Stanojevic" Health Centre between February and August 2008.
Blood samples were obtained in the morning after an overnight fast from all subjects and were analysed on the day of blood collection. Patients (45 in the initial group and 53 in the validation group) with triglyceride (TG) levels ≥4.52 mmol/L were excluded from further analysis. All the study participants were free of any confirmed renal, hepatic or cardiovascular disease and diabetes mellitus.
All participants gave informed consent prior to enrolment in our study that was planned according to the ethical guidelines laid down by the Declaration of Helsinki.
Lipid-lipoprotein analyses
Total cholesterol (TC) and TG levels were measured enzymatically by CHOD-PAP and GPO-PAP methods (Roche Diagnostics GmbH, Mannheim, Germany), respectively according to the manufacturer's specifications. High-density lipoprotein cholesterol (HDL-C) was measured using a homogeneous assay without precipitation (Roche Diagnostics GmbH, Mannheim, Germany) .
A homogenous enzymatic colorimetric assay offered by Kyowa Medex and distributed by Roche Diagnostics, was used to measure LDL directly. The principle of D-LDL-C determination is as follows: At pH 6.75 and in the presence of magnesium ions, sulphated α-cyclodextrin and dextran sulphate the enzymatic reaction for cholesterol in very low-density lipoprotein (VLDL) and chylomicrons is markedly reduced. Polyoxyethylenepolyoxypropylene block polyether (POE-POP) blocks cholesterol, especially in HDL enabling LDL-C measurement by a conventional enzymatic reaction with cholesterol oxidase, cholesterol esterase and peroxidase (Roche Diagnostics GmbH, Mannheim, Germany) [6, 13]. Homogeneous assay has been shown to meet current NCEP criteria for precision (CV < 4%), accuracy (bias < 4%) and for total analytical error (<12%) [4, 13]. The intra-assay CVs for direct LDL-C were 1.8% at 2.0 mmol/L and 1.5% at 4.95 mmol/L and the inter-assay CVs were 2.3% at 1.27 mmol/L and 2.1% at 2.78 mmol/L.
All analyses in both laboratories were preformed on Roche Hitachi 911 Chemistry Analysers (Roche Diagnostics GmbH, Mannheim, Germany).
LDL-C concentrations were also calculated by Friedewald's formula : F-LDL-C (mmol/L) = TC - HDL-C - TG/2.2 and by Anandaraja's formula A-LDL-C (mg/dL) = 0.9TC - 0.9TG/5 - 28. Values in mg/dL were calculated and then expressed in mmol/L.
The percentage difference (%ΔLDL) defined as calculated LDL-C minus D-LDL-C compared to the direct measurement was calculated using the following formula: %Δcalculated LDL-C = [(calculated LDL-C)-(D-LDL-C)]/D-LDL-C100. Our study evaluated the ability of the three formulas to correctly classify subjects into the risk categories given by NCEP ATP III using the D-LDL-C concentrations as the true values. These cut-off points were <2.59, 2.60 - 3.35, 3.36 - 4.12, 4.13-4.89 and > 4.90 mmol/L. Additionally, in order to improve comparisons between the methods the samples were stratified according to cut-off points recommended by the NCEP ATP III for TC levels (≤4.13, 4.14 - 5.16, 5.17 - 6.20, 6.21 - 7.24 and ≥7.25 mmol/L) and for TG levels (≤1.13, 1.14-1.69, 1.7 - 2.25, 2.26-2.82, and 2.83-4.52 mmol/L).
Statistical analysis
Distribution of TC, TG, HDL-C, D-LDL-C, F-LDL-C, A-LDL-C and S-LDL-C was normal according to Kolmogorov-Smirnov test. Differences between values calculated with different formulas and from direct method were examined by the Student's paired t test. The Passing-Bablok linear regression was used to evaluate the degree of association between LDL-C values from different formulas and from the direct method . Two-tailed P values less than 0.05 were considered statistically significant. Statistic analyses were conducted using Microsoft® Office Excel 2003.
Results
Lipoprotein concentrations and their distributions in the initial group are given in Table 1.
The Student's paired t-test showed that D-LDL-C values were significantly higher than F-LDL-C and A-LDL-C values (p < 0.001). Directly measured LDL-C concentrations exceeded F-LDL-C and A-LDL-C concentrations in 82% and 65% of samples, respectively. The mean absolute bias and the mean %ΔLDL between calculated LDL-Cs compared to the direct method were - 0.27 ± 0.31 mmol/L and -6.9 ± 8.8% for Friedewald's formula and -0.18 ± 0.51 mmol/L and -3.9 ± 14.8% for Anandaraja's formula.
Mean percentage differences between Friedewald's formula and direct LDL-C values (%ΔF-LDL-C) were negative in all quartiles, the lowest in the first, which suggested that the whole distribution was shifted to negative values (Table 1). Mean percentage differences between Anandaraja's formula and direct LDL-C values (%ΔA-LDL-C) were negative in the first quartile but positive in the third and the whole distribution was also shifted towards negative values (Table 1).
A comparison of D-LDL-C (x) versus F-LDL-C (y) and D-LDL-C (x) versus A-LDL-C(y) values resulted in the following regression equations: y = -0.17 + 0.980×, r = 0.96 and y = 0.129 + 0,971×, r = 0.89, respectively (data not shown). In the whole initial group Friedewald's formula correctly classified 65% of the subjects and Anandaraja's formula only 55%. The same percentage of subjects (31%) was underestimated by both formulas (data not shown).
Derivation of the modified formula
The obtained unsatisfactory results led us to re-examine Friedewald's formula for LDL-C estimation. Following the procedure which led to Friedewald's formula derivation we re-calculated factor for VLDL-C concentration estimation. We used TC, TG, LDL-C and HDL-C concentration measurements in the initial group to calculate the VLDL-C/TG ratio for a Serbian population. We first subtracted the sum of HDL-C and LDL-C from TC for each person. This was estimation of VLDL-C concentration for each person. Thereafter, we divided the particular TG concentration with the corresponding calculated VLDL-C to determine the mean of the ratio. The TG/VLDL mean ratio was 3 compared with 2.2 according to Friedewald . Therefore, the modified formula should be stated as follows: S-LDL-C (mmol/L) = TC - TG/3 - HDL-C. The percentage difference for our modified formula (%ΔS-LDL-C) was calculated in the same way as for Friedewald's and Anandaraja's formulas.
Our observation was validated in the population consisting of 1043 patients. Lipoprotein concentrations and their distributions for the validation group are given in Table 2.
A significant difference between S-LDL-C and D-LDL-C values was not found. The absolute bias between these two methods was -0.06 ± 0.37 mmol/L and the mean %ΔS-LDL-C was -0.9 ± 9.3%. A high correlation (r = 0.96) was observed between calculated and measured values.
F- LDL-C and A-LDL-C values exhibited similar characteristics in both study groups and were compared with S-LDL-C values by calculating the percentage difference (%ΔLDL). A comparison between the values estimated by the three formulas is shown in Figure 1. The whole distribution of %SΔLDL-C values was almost symmetrical around the zero point indicating a similar number of negative and positive biases, all of which were less than those obtained by Friedewald's and Anandaraja's formulas.
Subgroups defined by cut-off values (ranges) for TC, TG and D-LDL-C values provided by NCEP ATPIII guidelines were analysed separately (Table 3). F-LDL-C was significantly lower (p < 0.001) compared to D-LDL-C in all TC, TG and D-LDL-C ranges. A-LDL-C showed no significant difference compared to D-LDL-C except for when D-LDL-C levels were less than 3.35 mmol/L. The optimal and closest results to D-LDL-C values were obtained by our modified formula (S-LDL-C) (Table 3). No differences between values were found at both the lowest and the highest TC levels (at TG concentrations 2.26-2.82 mmol/L and at LDL-C levels <4.12 mmol/L). Linear regression analyses demonstrated a high correlation of estimated LDL-Cs with the measured D-LDL-C (r > 0.91) in all TC, TG and D-LDL-C ranges.
The ability of the formulas to correctly classify subjects at the clinical decision cut-off points in specific subgroups is shown in Table 3. The percentages of samples correctly classified in risk categories for all formulas decreased with increasing TC, TG and LDL-C concentrations except at the highest concentrations of TC and LDL-C.
Mean %ΔLDL-C values steadily increased with increasing TG concentrations but decreased with increasing TC and D-LDL-C concentrations. Mean %ΔF-LDL-C values were negative in all ranges of TC, TG and D-LDL-C. Mean %ΔA-LDL-C values were negative in all ranges of TC but less than the same values obtained by Friedewald's formula. When TG concentrations were less than 1.14 mmol/L and D-LDL-C concentrations were less than 2.59 mmol/L mean %ΔA-LDL-C values were positive (Figure 2).
Figure 2 shows that mean %ΔS-LDL-C values were not different than ± 3% in whole TC, TG and D-LDL-C ranges and they were much lower than mean %ΔLDL-C values obtained from the other two formulas. This suggests a better agreement with D-LDL-C than that obtained from the two other formulas.
When TG concentrations were ≤1.13 mmol/L and between 2.83-4.52 mmol/L our modified formula led to the highest percentages of individuals having LDL-C values falling within ± 10%ΔLDL-C (88% and 60%, respectively) compared to Friedewald's (87% and 37%, respectively) and Anandaraja's (60% and 32%, respectively) (data not shown). In the whole validation group %ΔLDL-C differed by more than ± 10% in 20%, 36% and 55% of the subjects when our modified, Friedewald's and Anandaraja's formulas were employed, respectively.
Discussion
Strategies for treatment of lipid abnormalities are primarily based on LDL-C concentration. Therefore, LDL-C must be accurately determined to establish a personal CHD risk profile in order to initiate dietary adjustments, drug therapy and to monitor their effects .
In the past few decades attempts have been made to derive more accurate formulas for LDL-C calculation than the widely used Friedewald's formula [15–20]. Although the newer formulas offered few advantages over the Friedewald's, they have performed only marginally better, possibly due to diversity in terms of study populations and/or pathologies [21–23]. Some of them included apolipoprotein concentrations, apoA-I and/or apoB [18–20]. Anandaraja and colleagues described a new formula for LDL-C calculation in an Indian population of 1000 patients by applying multiple linear regression analysis and validated its accuracy in 1008 patients. In their study the mean LDL-C concentrations measured by a precipitation method and by their formula were 3.04 ± 1.04 mmol/L and 2.96 ± 0.96 mmol/L, respectively. The mean absolute difference between both methods was 0.1 ± 0.24 mmol/L and good correlation was found (r = 0.97). In addition, they confirmed a reduction in the false overestimation of LDL-C compared with Friedewald's formula. Anandaraja and colleagues called for the reliability of their formula to be tested in other populations.
On the other hand, Friedewald's formula has been shown to be relatively reliable and recommended by the NCEP as a routine method for estimation of LDL-C despite it having several well-established constraints. It cannot be applied to samples containing TG levels > 4.52 mmol/L (400 mg/dL), to non-fasting samples and to samples of patients with dysbetalipoproteinemia (Fredrickson Type III) [8, 13]. Some authors have demonstrated that the formula should not be used in certain groups of patients with diabetes, liver and renal dysfunction even with TG levels < 4.52 mmol/L [17, 22, 23]. The formula relies on the accuracy of the TC, TG and HDL-C assays and also on an additional mathematical term that is used to estimate the VLDL-C concentration. It assumes a fixed relationship between TC, TG, and HDL-C in fasting serum providing the TG/cholesterol ratio in the VLDL is constant and the assumption that TG is only present as VLDL. As well, the homogenous Roche method we have used has some limitations although it has been reviewed by Nauck et al to be precise and acceptably accurate. It gives an improvement in the measurement of LDL-C in samples with high TG and may assist better in classification of patients at risk categories for cardiovascular diseases than Friedewald's equation.
Anandaraja's team did not propose any limitations to their formula. Comparing the mean value of the direct LDL-C obtained in the first 1000 patients and that in the validation group of 1008 patients it seemed they did not exclude samples with high TG levels . In a study of over 10000 Brazilian patients Gasko and colleagues supported Anandaraja's formula. The mean LDL-C level measured by a direct method and that estimated by the new formula were similar to the Indian population (2.99 ± 0.57 mmol/L and 2.97 ± 0.59 mmol/L, respectively). The correlation coefficient between both methods was r = 0.97. Anandaraja's formula was also checked in 230 Greek patients (118 had metabolic syndrome and 112 were healthy) by Gazi and Elisaf . Friedewald's and Anandaraja's formulas gave similar results in the examined Greek population. The latter was approved for use in their laboratories.
In our study we investigated if Anandaraja's formula could be applied in the Serbian population by comparing the value obtained with that of the homogenous direct method for LDL-C determination. This is the first study of its kind where the reliability and accuracy of Friedewald's formula were tested in the Serbian population. In our initial group LDL-C values from the direct measurement and from Anandaraja's formula were both higher than the values in Indian, Brazilian and Greek populations by almost 1 mmol/L [12, 24, 25]. The A-LDL-C concentration was significantly lower than the D-LDL-C concentration (Table 1). The correlation coefficient between methods was good (r = 0.89) but lower than previously published (r = 0.97) [12, 24].
To the best of our knowledge only Paz and colleagues have performed a detailed systematic analysis of the reliability of Anandaraja's formula. They tested the new formula in schizophrenic patients treated with antipsychotic drugs. Their results demonstrated that LDL-CAnandaraja concentrations were underestimated or overestimated compared to LDL-CElectrophoresis and depended on the HDL-C concentrations. They found a higher correlation and a lower estimation error between LDL-CElectrophoresis and LDL-CFriedewald than LDL-CElectrophoresis and LDL-CAnandaraja. For that reason improved accuracy of Anandaraja's formula over Friedewald's formula was not claimed. Data from our study are in agreement with Paz and colleagues. We employed two apparently healthy populations from two different Serbian cities in which all analyses were completed with the same type of reagents on the same class of autoanalyser. In both populations the results were very similar and did not support Anandaraja's formula (Table 1 and 2). The percentages of patients properly classified in NCEP's risk categories and the percentage of patients that fell in the ± 10%ΔLDL-C group were smaller when compared to Friedewald's formula in all ranges of TC, TG and LDL-C (Table 3).
Figure 1 shows that %ΔLDL-C distribution for Anandaraja's formula was overrached and dismounted compared to Friedewald's. Our findings concerning Friedewald's formula are consistent with other published studies. As reported earlier [10, 27, 28], we found that calculated LDL-C values derived from Friedewald's formula often underestimate directly measured LDL-C concentrations. Tighe and colleagues found good correlation between LDL-C calculated by Friedewald's formula and directly measured LDL-C (r = 0.90). Only 48.1% of samples in Tighe's study showed similar results compared to 63% in our validation group.
Jun and co-workers revealed that F-LDL-C differed significantly from D-LDL-C over the concentration ranges of both TC and TG. They found that the mean %ΔLDL was -9.1% and assumed that this difference was critical for the evaluation of patients with hyperlipidemia. Their study demonstrated that higher TG resulted in a greater %ΔF-LDL-C and increased TC was associated with decreased %ΔF-LDL-C, which was also confirmed in our current study.
It would appear that calculated LDL-Cs by both Friedewald's and Anandaraja's formulas give unsatisfactory results compared to the direct homogenous method. It seems that the only advantage of Anandaraja's formula is the requirement of the concentration of only 2 variables (TC and TG) that reduces the analytical error. Our study in a Serbian population revealed that the HDL-C concentration should not be omitted from the formula, in agreement with Paz and colleagues .
In the course of the present study to investigate the reliability of Friedewald's and Anandaraja's formulas we derived a new modified formula. It resembles Friedewald's and it is based on the original study . Simple division of plasma TG by 2.2 for mmol/L or 5 for mg/dL does not give a very accurate estimation of VLDL-C even in a healthy population. Some authors have proposed alternative calculations including TG/4, TG/4.5, TG/5, TG/5.5, TG/6, TG/7 and TG/8 (mg/dL) [15, 16]. Nakanishi and colleagues demonstrated that the TG/5 formula correlated well with measured LDL-C and had the smallest mean difference between estimated and measured LDL-C in middle-aged Japanese men. Gonzales-Estrada concluded that the DeLong (TG/2.7 mmol/L or TG/6.17 mg/dL) calculation was more convenient than the original Friedewald's formula for most cases, despite a high error. On the contrary, according to some authors DeLong's formula did not improve LDL-C determination [21, 23].
In our initial population we calculated that to determine cholesterol in VLDL, TG should be divided by 3 (mmol/L) or 6.85 (mg/dL). Our modified formula was tested and its accuracy was validated in our second population. S-LDL-C showed no statistically significant difference compared to D-LDL-C in the validation group, despite the fact that there was still an underestimation. No significant difference between these two mean values was found in situations of TC < 4.13 mmol/L and TC > 7.24 mmol/L, TG 2.26 - 2.82 mmol/L and LDL-C < 4.12 mmol/L. The mean %ΔS-LDL-C was -0.9%, the smallest among these three calculations (Table 2). Our modified formula exhibited the lowest mean %ΔLDL-C in all ranges of TG, TC and D-LDL-C compared to Friedewald's and Anandaraja's formulas (Figure 2). As for the accuracy of the calculation method, the proportion of samples falling within a fairly broad ± 10%ΔLDL-C range was the highest. Our results indicate that LDL-C concentrations derived from the modified formula provide more accurate results compared to those derived from Friedewald's and Anandaraja's formulas for Serbian population.
According NCEP-ATP III guidelines, LDL-C concentrations < 3.36 mmol/L are considered desirable while those > 4.14 mmol/L are considered high. Medication should be administered to subjects falling into the latter group . Our study demonstrated that 45%, 45% and 26% of samples from Friedewald's, Anandaraja's and our modified formula, respectively underestimated the diagnostic LDL-C level of 4.14 mmol/L and were classified one cut-off point below that indicated for therapy (data not shown).
In conclusion, regarding patients' convenience, financial reasons and precision and accuracy we propose that our new modified formula should be used instead of Friedewald's formula for the estimation of LDL-C concentration in the Serbian population. We appeal to all laboratories principally in Serbia and in neighbouring countries with similar living conditions and habits to test our modified formula before its eventual implementation.
Abbreviations
CHD:
: coronary heart disease
NCEP:
: National Cholesterol Education Program
ATP III:
: Adult Treatment Panel III
F-LDL-C:
: LDL-C value calculated by Friedewald's formula
A-LDL-C:
: LDL-C value calculated by Anandaraja's formula
S-LDL-C:
: LDL-C value calculated by our new modified formula
%ΔLDL:
: percentage difference.
References
Scandinavian Simvastatin Survival Study Group: Randomised trial of cholesterol lowering in 4444 patients with coronary heart disease: the Scandinavian Simvastatin Survival Study (4S). Lancet. 1994, 344: 1383-1389.
Google Scholar
2. The Long-Term Intervention with Pravastatin in Ischaemic Disease (LIPID) Study Group: Prevention of cardiovascular events and death with pravastatin in patients with coronary heart disease and a broad range of initial cholesterol levels. N Engl J Med. 1998, 339: 1349-1357. 10.1056/NEJM199811053391902
Article Google Scholar
3. Ridker PM, Stampfer MJ, Rifai N: Novel risk factors for systemic atherosclerosis. J Am Med Assoc. 2001, 285: 2481-2485. 10.1001/jama.285.19.2481.
Article CAS Google Scholar
4. Executive summary of the third report of the National Cholesterol Education Programme (NCEP) Expert panel on detection, evaluation, and treatment of high blood cholesterol in adults (Adult Treatment Panel III). J Am Med Assoc. 2001, 285: 2486-2497. 10.1001/jama.285.19.2486.
5. Bachorick PS, Ross JW: For the National Cholesterol Education Program Working Group on Lipoprotein Measurements. National Cholesterol Education Program recommendations for measurement of low-density lipoprotein cholesterol: executive summary. Clin Chem. 1995, 41: 1414-1420.
Google Scholar
6. Bairaktari ET, Seferiadis KI, Elisaf MS: Evaluation of Methods for the Measurementof Low-Density Lipoprotein Cholesterol. J Cardiovasc Pharmacol Therapeut. 2005, 10: 45-54. 10.1177/107424840501000106.
Article CAS Google Scholar
7. Nauck M, Warnick GR, Rifai N: Methods for Measurement of LDL-Cholesterol: A Critical Assessment of Direct Measurement by Homogeneous Assays versus Calculation. Clin Chem. 2002, 48: 236-254.
CAS PubMed Google Scholar
8. Friedewald WT, Levy RI, Fredrickson DS: Estimation of the concentration of low-density lipoprotein cholesterol in plasma, without use of the preparative ultracentrifuge. Clin Chem. 1972, 18: 499-502.
CAS PubMed Google Scholar
9. Vukomirovic D: Causes of deaths. Statistical Yearbook of Serbia 2005. 2005, 90-(In Serbian), Statistical Office of the Republic of Serbia, Belgrade, Serbia,
Google Scholar
10. Tighe DA, Ockene IS, Reed G, Nicolosi R: Calculated low density lipoprotein cholesterol levels frequently underestimate directly measured low density lipoprotein cholesterol determinations in patients with serum triglyceride levels or <=4.52 mmol/l: an analysis comparing the LipiDirect® magnetic LDL assay with the Friedewald calculation. Clin Chim Acta. 2006, 365: 236-242. 10.1016/j.cca.2005.08.026
Article CAS PubMed Google Scholar
Esteban-Salan M, Guimon-Bardesi A, de La Viuda-Unzueta JM, Azcarate-Ania MN, Pascual-Usandizaga P, Amoroto-Del-Rio E: Analytical and clinical evaluation of two homogeneous assays for LDL-cholesterol in hyperlipidemic patients. Clin Chem. 2000, 46: 1121-1131.
CAS PubMed Google Scholar
12. Anandaraja S, Narang R, Godeswar R, Laksmy R, Talwar KK: Low-density lipoprotein cholesterol estimation by a new formula in Indian population. Int J Cardiol. 2005, 102: 117-120. 10.1016/j.ijcard.2004.05.009
Article CAS PubMed Google Scholar
13. Nauck M, Graziani MS, Bruton D, Cobbaert C, Cole TG, Lefevre F, Riesen W, Bachorik PS, Rifai N: Analytical and Clinical Performance of a Detergent-based Homogeneous LDL-Cholesterol Assay: A Multicenter Evaluation. Clin Chem. 2000, 46: 506-514.
CAS PubMed Google Scholar
14. Passing H, Bablok W: A new biometrical procedure for testing the equality of measurements from two different analytical methods. Application of linear regression procedures for method comparison studies in Clinical Chemistry, Part I. J Clin Chem Clin Biochem. 1983, 21: 709-720.
CAS PubMed Google Scholar
15. DeLong DM, DeLong ER, Wood PD, Lippel K, Rifkind BM: A comparison of methods for the estimation of plasma low- and very low-density lipoprotein cholesterol: the Lipid Research Clinics Prevalence Study. J Am Med Assoc. 1986, 256: 2372-2377. 10.1001/jama.256.17.2372.
Article CAS Google Scholar
16. Nakanishi N, Matsuo Y, Yoneda H, Nakamura K, Suzuki K, Tatara K: Validity of the Conventional Indirect Methods Including Friedewald Method for Determining Serum Low-Density Lipoprotein Cholesterol Level: Comparison with the Direct Homogeneous Enzymatic Analysis. J Occup Health. 2002, 42: 130-137. 10.1539/joh.42.130.
Article Google Scholar
17. Bairaktari ET, Tzallas C, Kalientzidou M, Tselepis AD, Siamopoulos KC, Seferiadis KI, Elisaf M: Evaluation of alternative calculation methods for determining low-density lipoprotein cholesterol (LDL-C) in hemodialysis patients. Clin Biochem. 2004, 37: 937-940. 10.1016/j.clinbiochem.2003.10.018
Article CAS PubMed Google Scholar
18. Planella T, Cortes M, Martinez-Bru C, Gonzalez-Sastre F, Ordonez-Llanos J: Calculation of LDL-cholesterol by using apolipoprotein B for classification of nonchylomicronemic dyslipemia. Clin Chem. 1997, 43: 808-815.
CAS PubMed Google Scholar
19. Hattori Y, Suzuki M, Tsushima M, Yoshida M, Tokunaga Y, Wang Y, Zhao D, Takeuchi M, Hara Y, Ryomoto KI, Ikebuchi M, Kishioka H, Mannami T, Baba S, Harano Y: Development of approximate formula for LDL-chol, LDL-apo B and LDL-chol:LDL-apo B as indices of hyperapobetalipoproteinemia and small dense LDL. Atherosclerosis. 1998, 138: 289-299. 10.1016/S0021-9150(98)00034-3
Article CAS PubMed Google Scholar
20. Walldius G, Jungner I, Holme I, Aastveit AH, Kolar W, Steiner E: High apolipoprotein B, low apolipoprotein A-I, and improvement in the prediction of fatal myocardial infarction (AMORIS study): a prospective study. Lancet. 2001, 358: 2026-2033. 10.1016/S0140-6736(01)07098-2
Article CAS PubMed Google Scholar
21. Warnick GR, Knopp RH, Fitzpatrick V, Branson L: Estimating Low-Density Lipoprotein Cholesterol by the Friedewald Equation Is Adequate for Classifying Patients on the Basis of Nationally Recommended Cutpoints. Clin Chem. 1990, 36: 15-19.
CAS PubMed Google Scholar
22. Rubies-Prat J, Reverter JL, Senti M, Pedro-Botet I, Salinas I, Lucas A, Nogues X, Sanmartì A: Calculated low-density lipoprotein cholesterol should not be used form management of lipoprotein abnormalities in patients with diabetes mellitus. Diabetes Care. 1993, 16: 1081-1086. 10.2337/diacare.16.8.1081
Article CAS PubMed Google Scholar
23. Matas C, Cabre M, La Ville A, Prats E, Joven J, Turner PR, Masana L, Campus J: Limitations of the Friedewald Formula for Estimating Low-Density Lipoprotein Cholesterol in Alcoholics with Liver Disease. Clin Chem. 1994, 40: 404-406.
CAS PubMed Google Scholar
24. Gasko R: Low-density lipoprotein cholesterol estimation by the Anandaraja's formula - confirmation. Lipids in Health and Disease. 2006, 5: 18- 10.1186/1476-511X-5-18
Article PubMed Central PubMed Google Scholar
25. Gazi IF, Elisaf M: LDL-cholesterol calculation formulas in patients with or without the metabolic syndrome. Int J of Cardiol. 2007, 119: 414-415. 10.1016/j.ijcard.2006.07.139.
Article Google Scholar
26. Paz E, Hermida J, Bouzas L, Brenlla J, Tutor JC: LDL cholesterol estimation using the Anandaraja's and Friedewald's formulas in schizophrenic patients treated with antipsychotic drugs. Clin Biochem. 2008, 41: 1002-1007. 10.1016/j.clinbiochem.2008.04.017
Article CAS PubMed Google Scholar
27. Jun KR, Park H, Chun S, Park H, Min WK: Effects of total cholesterol and triglyceride on the percentage difference between the low-density lipoprotein cholesterol concentration measured directly and calculated using the Friedewald formula. Clin Chem Lab Med. 2008, 46 (Supp 3): 371-375. 10.1515/CCLM.2008.064
CAS PubMed Google Scholar
28. Marniemi J, Maki J, Maatela J, Jarvisalo J, Impivaara O: Poor applicability of the Friedewald formula in the assessment of serum LDL cholesterol for clinical purposes. Clin Biochem. 1995, 28: 285-289. 10.1016/0009-9120(94)00095-D
Article CAS PubMed Google Scholar
29. Gonzalez Estrada M, Rodríguez Ferrer CR, Astarloa IR, Lahera EM: Use of Serum CholesteroliTriglyceride Ratio to Discern for Which Individuals the Friedewald Formula Can Be Used Confidently. Clin Chem . 1990, 36: 1673-1675.
CAS PubMed Google Scholar
Download references
Acknowledgements
This work was financially supported by a grant from the Ministry of Science and Environmental Protection, Republic of Serbia (project number 145036B). The study was also supported by COST B35 Action. We want to thank our colleague Dr. David R. Jones for help in editing the manuscript.
Author information
Authors and Affiliations
Institute of Medical Biochemistry, Faculty of Pharmacy, Belgrade, Serbia
Ana Vujovic, Jelena Kotur-Stevuljevic, Slavica Spasic, Vesna Spasojevic-Kalimanovska & Aleksandra Zeljkovic
2. "Belladonna" Clinical Chemistry Laboratory, Zemun, Belgrade, Serbia
Nada Bujisic
3. Belgrade Clinical Centre, Laboratory Department, Belgrade, Serbia
Jelena Martinovic
4. "Sava Stanojevic" Health Centre, Trstenik, Serbia
Milica Vujovic & Dragoljub Pajic
Authors
Ana Vujovic
View author publications
Search author on:PubMed Google Scholar
2. Jelena Kotur-Stevuljevic
View author publications
Search author on:PubMed Google Scholar
3. Slavica Spasic
View author publications
Search author on:PubMed Google Scholar
4. Nada Bujisic
View author publications
Search author on:PubMed Google Scholar
5. Jelena Martinovic
View author publications
Search author on:PubMed Google Scholar
6. Milica Vujovic
View author publications
Search author on:PubMed Google Scholar
7. Vesna Spasojevic-Kalimanovska
View author publications
Search author on:PubMed Google Scholar
8. Aleksandra Zeljkovic
View author publications
Search author on:PubMed Google Scholar
9. Dragoljub Pajic
View author publications
Search author on:PubMed Google Scholar
Corresponding author
Correspondence to Ana Vujovic.
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
AV wrote the manuscript and statistically analyzed data. JKS participated in the study design, statistically analysed data and critically revised the manuscript, SS carried out all aspects of the study design, statistically analysed data and critically revised the manuscript, NB performed experimental work and collected samples, MV performed experimental work and collected samples, JM performed experimental work and collected samples, VSK critically revised the manuscript, AZ critically revised the manuscript, DP collected samples. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Authors’ original file for figure 1
Authors’ original file for figure 2
Rights and permissions
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reprints and permissions
About this article
Cite this article
Vujovic, A., Kotur-Stevuljevic, J., Spasic, S. et al. Evaluation of different formulas for LDL-C calculation. Lipids Health Dis 9, 27 (2010).
Download citation
Received:
Accepted:
Published:
DOI:
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative
Keywords
Total Cholesterol
National Cholesterol Education Programme
Validation Group
Roche Diagnostics GmbH
Modify Formula
Lipids in Health and Disease
ISSN: 1476-511X
Contact us
General enquiries: journalsubmissions@springernature.com |
11760 | http://www.phy.olemiss.edu/~cremaldi/PHYS417/PhotoElectricEffectLab.pdf | Photo Electric Effect in a Metal When light of various wavelengths hits the surface of a metal electrons can be ejected with kinetic energy KE = hf - φ , where φ is the ionization energy (energy necessary to overcome the Coulomb attaction of the metal.) The most energetic electrons will correspond to φmin = W, defined as the work function of the metal. (also the Fermi energy). Photoelectric Effect 1) Record the stopping voltage at three frequencies R G B by zeroing the picoAmmeter. . 2) Graph VS vs f and perform a linear fit. 3) Determine Planck's constant h and the work function W of the metal photocathode w +-errors. 4) Identify the metal. 5) Explain why the photocurrent is zero and independent of light intensity when hf < W. KE = hf - φ φ W Conduction band EFermi If one applies a stopping voltage to these electrons on can measure the work function and Plank’s constant by plotting VS vs f . eVS = KEMAX = hf-W f Vs -W/e Slope=h/e = h/e -W/e hf A V a e- - + |
11761 | https://www.freemathhelp.com/forum/threads/finding-the-normal-or-tangent-line-to-an-ellipse.119613/ | New posts
Search forums
Menu
Log in
Register
Install the app
Forums
Free Math Help
Calculus
Finding the normal or tangent line to an ellipse
Thread starter
a_guy_and_he_does_stuff
Start date
A
a_guy_and_he_does_stuff
New member
Joined
: Dec 13, 2019
Messages
: 6
#1
Hello all, first time posting here. I'm currently chipping away at a rather tricky problem about mirrors and reflections, and I need help finding something - the equation of the normal/tangent line to an ellipse.
I have an ellipse with equation [MATH]k = \sqrt{a^2 + b^2} + \sqrt{a^2 + (AB - b)^2}[/MATH], where [MATH]k[/MATH] is some real number and [MATH]AB[/MATH] is the length of the line which connects the two centers of the ellipse.
Now, what I want to do is have a generalized equation that gives the vector function of the normal or tangent line to the surface of any ellipse of that specified form (OR the direction of that given line). My first attempt consisted of making the ellipse into a function of two variables and taking its gradient hoping to find the normal line to the surface (fresh out of multi-variable, first thing I thought of), and, though it produced a very interesting pattern, did not do what I wanted it to. It produced the gradient in reference to k.
I'm unsure how to proceed. My only other thought is the implicit function theorem or the simple, calculus 1 method for differentiating implicit functions, but I'm also unsure how to apply those two to this case.
I'm very tired right now, so I'll end it here.
Dr.Peterson
Elite Member
Joined
: Nov 12, 2017
Messages
: 16,821
#2
I'm not sure I understand what you are doing.
First, you mention a surface; are you really talking about an ellipsoid (surface) rather than an ellipse (curve in a plane)?
Second, are your a and b coordinates (in place of x and y), or something else? Are your "two centers" the foci, or something else?
Finally, is there a reason you haven't reworked the equation into the standard form?
A
a_guy_and_he_does_stuff
New member
Joined
: Dec 13, 2019
Messages
: 6
#3
My terminology is unclear probably because I haven't actually learned about or worked with ellipses before.
a and b are coordinates in place of x and y. The two centers are the foci of the ellipse, and AB is, as follows, the distance between the foci. I'm referring to a family of ellipses described by the above function, where k and AB are semi-arbitrary values which, when both filled in, create an ellipse. When I referred to the 'surface' I mean to say the points which describe the ellipse, not an ellipsoid.
The reason that I haven't put the equation into standard form is because I don't know how to get to that point (while also keeping in mind that expressing it in the particular variables I am already using will make the final steps easier). I've built this equation from the ground up using Pythagoras - it's only by coincidence that this happens to form an ellipse.
What I meant to say, revised, is that I want to find the normal or tangent vector to the curve of form [MATH]k=\sqrt{x^2 + y^2} + \sqrt{x^2 + (AB - y)^2}[/MATH], where k is some value and AB is the distance between the foci. I want to express it in terms of those variables, but I have no idea how.
Dr.Peterson
Elite Member
Joined
: Nov 12, 2017
Messages
: 16,821
#4
This is why we teach vocabulary in math classes, so it's possible to talk about something like an ellipse and be understood.
I don't have time at the moment to help directly, but I'll just ask: Have you tried looking up information on ellipses, so you can see what the equation of the curve is in a simpler form? And how much do you know about calculus that you will be able to apply to that equation? Finally, you mentioned that this is part of a larger problem involving reflections. It is entirely possible that if you told us that problem, we could suggest a quicker way to get where you want to go.
I'll try to get back to you when I can.
A
a_guy_and_he_does_stuff
New member
Joined
: Dec 13, 2019
Messages
: 6
#5
Okay. First, I will explain the problem and my approach to solve it. In simple terms, this is a problem about discovering if there exists a mirror such that all light starting from some given point will reach some other given point at the same time by reflecting off of the mirror.
What I found was that the only possible shape for such a mirror is an ellipse; this process was used to discover it:
The distance that light travels in a given time is assumed the speed of light in this problem, so the problem is now about distance as opposed to time.
Light starting at a point A reflects off of a point C and reaches a point B. Stipulating that line AB is the vertical axis, and that the line perpendicular to AB through A is the horizontal axis, we can make the following construction:
(Variables [MATH]a, b, c, d, f, g,[/MATH] refer to the length of the lines they represent.)
On a given mirror, the length of [MATH]f+g[/MATH] is the distance light travels to reach the observer at point B. This distance must remain constant through all starting angles. As such, the equation [MATH]k=f+g[/MATH] is created to represent this relationship, where k is a constant stipulating the distance light travels from A to C to B.
Note that [MATH]a=c[/MATH], and [MATH]d=AB-b[/MATH].
Then the distance [MATH]f=\sqrt{a^2+b^2}[/MATH] and [MATH]g=\sqrt{c^2+d^2}=\sqrt{a^2+(AB-b)^2}[/MATH]By substitution, we find [MATH]k=\sqrt{a^2+b^2}+\sqrt{a^2+(AB-b)^2}[/MATH].
Graphically, [MATH]x=a[/MATH], [MATH]y=b[/MATH]. Substituting values for k and AB yields an ellipse.
However, the process which reaches this point does not guarantee that this surface is necessarily a mirror. There is no stipulation in the premise which suggests that the normal line for reflection is actually perpendicular to the tangent of the mirror. Essentially, this curve may or may not actually obey the law of reflection.
If the mirror obeys the law of reflection, the tangent vector to the curve must always be perpendicular to the vector which bisects angle ACB. Based on the construction, the direction of the bisecting vector can be determined using trigonometry (the method to do so of which I will go into upon request, as it is a bulky process).
If these vectors are orthogonal, then their dot product will be zero. However, I am not sure how to find the tangent vector to the curve. I could also find the normal vector to the curve and compare its direction to the vector which bisects ACB, which would be equally valid. However, I don't understand how to find that vector in terms of this implicit function.
In the meanwhile, I will try to put the function into standard form.
D
Deleted member 4993
Guest
#6
a_guy_and_he_does_stuff said:
Okay. First, I will explain the problem and my approach to solve it. In simple terms, this is a problem about discovering if there exists a mirror such that all light starting from some given point will reach some other given point at the same time by reflecting off of the mirror.
What I found was that the only possible shape for such a mirror is an ellipse; this process was used to discover it:
The distance that light travels in a given time is assumed the speed of light in this problem, so the problem is now about distance as opposed to time.
Light starting at a point A reflects off of a point C and reaches a point B. Stipulating that line AB is the vertical axis, and that the line perpendicular to AB through A is the horizontal axis, we can make the following construction:
View attachment 15490
(Variables [MATH]a, b, c, d, f, g,[/MATH] refer to the length of the lines they represent.)
On a given mirror, the length of [MATH]f+g[/MATH] is the distance light travels to reach the observer at point B. This distance must remain constant through all starting angles. As such, the equation [MATH]k=f+g[/MATH] is created to represent this relationship, where k is a constant stipulating the distance light travels from A to C to B.
Note that [MATH]a=c[/MATH], and [MATH]d=AB-b[/MATH].
Then the distance [MATH]f=\sqrt{a^2+b^2}[/MATH] and [MATH]g=\sqrt{c^2+d^2}=\sqrt{a^2+(AB-b)^2}[/MATH]By substitution, we find [MATH]k=\sqrt{a^2+b^2}+\sqrt{a^2+(AB-b)^2}[/MATH].
Graphically, [MATH]x=a[/MATH], [MATH]y=b[/MATH]. Substituting values for k and AB yields an ellipse.
However, the process which reaches this point does not guarantee that this surface is necessarily a mirror. There is no stipulation in the premise which suggests that the normal line for reflection is actually perpendicular to the tangent of the mirror. Essentially, this curve may or may not actually obey the law of reflection.
If the mirror obeys the law of reflection, the tangent vector to the curve must always be perpendicular to the vector which bisects angle ACB. Based on the construction, the direction of the bisecting vector can be determined using trigonometry (the method to do so of which I will go into upon request, as it is a bulky process).
If these vectors are orthogonal, then their dot product will be zero. However, I am not sure how to find the tangent vector to the curve. I could also find the normal vector to the curve and compare its direction to the vector which bisects ACB, which would be equally valid. However, I don't understand how to find that vector in terms of this implicit function.
In the meanwhile, I will try to put the function into standard form.
Click to expand...
By definition, sum of the distances of any point on the ellipse from its foci is constant. This property is used to draw an ellipse.
F1P1 + P1F2 = F1P2 + P2F2 = F3P3 + P3F2
But be aware that many mirrors are presently manufactured using this property.
Dr.Peterson
Elite Member
Joined
: Nov 12, 2017
Messages
: 16,821
#7
As you may recognize from what you have looked up about ellipses, the equation simplifies more nicely if you define your coordinate system with the origin at the midpoint of AB; and the constants k and AB turn out to be what are traditionally called 2a (twice the major semiaxis) and 2c (twice the focal distance) respectively.
If we rename things so that your a becomes x, b becomes y, k becomes 2a, AB becomes 2c, the equation (still with the origin where you have it) will become after simplification
[MATH]\frac{x^2}{a^2 - c^2} + \frac{(y-c)^2}{a^2} = 1[/MATH]
and we can define [MATH]b^2 = a^2 - c^2[/MATH]. This is easier to work with, and you can implicitly differentiate it and continue as you plan.
Or, you can look up "reflection property of ellipse" to find proofs ... this is nothing new!
Incidentally, this all works for an ellipsoid as well. (Look up "whispering gallery".) And the property can be proved without calculus; there is a long, detailed development of this here.
A
a_guy_and_he_does_stuff
New member
Joined
: Dec 13, 2019
Messages
: 6
#8
Okay, thank you! The source you provided has a good explanation of a far more clever way to prove this property. I won't finish with my differentiation, as the point of all this was to prove the property I was trying to investigate. Next time I run into a wall, perhaps I'll step back instead of pushing through it.
Dr.Peterson
Elite Member
Joined
: Nov 12, 2017
Messages
: 16,821
#9
I was pondering whether it should be surprising that the ellipse has both the reflection property (all reflections from one focus pass through the other) and the equal-total-distance property, and realized that they are closely related. A reflection in a mirror takes the shortest path from one point, to the mirror, to the other point (just as light travels in straight lines, which are the shortest path). If every ray from one point, to a curve, to the other point is a reflection, then they must all be shortest paths, and therefore equal. There may be a way to prove the equivalence of these two properties directly.
A
a_guy_and_he_does_stuff
New member
Joined
: Dec 13, 2019
Messages
: 6
#10
If you mean to present the idea of proving this equivalence (as opposed to presenting the equivalence itself), then I can certainly see it being done. In fact, I decided to try myself, and I've got something here.
Trying to prove this is a bit wonky, because the properties are so closely intertwined that they trip over one another in the premises. So I had to define the properties more rigorously.
If we say that a curve has the equal-total-distance property, then the path from a fixed point to a point on the curve to another fixed point has the same length regardless of the point chosen on that curve. So, differentiating the reflection property, if a curve has the reflection property, then the path from a fixed point to any point on the curve to another fixed point is always a reflection, and therefore said path is always the shortest possible.
By using geometry, and a bit of cleverness, it should very well be possible to prove that each of these properties imply the other.
If we start with a very bold assumption, then the first half of this task is actually quite trivial! By previous result, it has already been proven that an ellipse has both the equal-total-distance property and the reflection property, meaning that, if one property implies an ellipse, and an ellipse implies both properties... well, the argument here is relatively easy to derive.
The backward direction is more complicated. Unless I'm mistaken, you can prove it through contradiction by thinking about the result: if an arbitrary point on the curve creates the shortest path between it and the fixed points, then all the paths from all the points must be equal. If we assume that they are not all equal, then there must exist some point on the curve which produces a path which is not equal to the others. But, this point cannot exist, because it must produce the shortest path - but it cannot, because if it does, then the other points do not produce the shortest path. This creates a contradiction.
In the forward direction, assume that there exists a curve and two points, [MATH]A[/MATH] and [MATH]B[/MATH], such that the equal-distance-property applies. Let [MATH]C[/MATH] be a point on the curve. Then by the equal-distance-property, the length of paths [MATH]AC[/MATH] and [MATH]BC[/MATH] sum to a fixed constant. Then the described curve is an ellipse. By previous result, the ellipse obeys the reflection property.
In the backward direction, we proceed by contradiction. Assume there exists a curve and two points, [MATH]A[/MATH] and [MATH]B[/MATH], such that the reflection property applies, but the equal-total-distance property does not apply. Let [MATH]C[/MATH] be a point on the curve. Let [MATH]k[/MATH] be equal to the length of [MATH]AC[/MATH] plus the length of [MATH]BC[/MATH]. By the reflection property, the path [MATH]A[/MATH] to [MATH]C[/MATH] to [MATH]B[/MATH] is a reflection, and [MATH]C[/MATH] is positioned such that the path [MATH]A[/MATH] to [MATH]C[/MATH] to [MATH]B[/MATH] is the shortest possible path.
Since the equal-total-distance property does not apply, then there must exist another point, [MATH]D[/MATH], on the curve, where the path [MATH]A[/MATH] to [MATH]D[/MATH] to [MATH]B[/MATH] has length [MATH]l[/MATH] equal to the length of [MATH]AD[/MATH] plus the length of [MATH]BD[/MATH], where [MATH]l\not=k[/MATH]. By the reflection property, the path [MATH]A[/MATH] to [MATH]D[/MATH] to [MATH]B[/MATH] is a reflection, and [MATH]D[/MATH] is positioned such that the path [MATH]A[/MATH] to [MATH]D[/MATH] to [MATH]B[/MATH] is the shortest possible path. Either [MATH]l>k[/MATH] or [MATH]l<k[/MATH], so we use cases.
Case 1: [MATH]l>k[/MATH]. Then the path [MATH]A[/MATH] to [MATH]D[/MATH] to [MATH]B[/MATH] is not the shortest path, because the path from [MATH]A[/MATH] to [MATH]C[/MATH] to [MATH]B[/MATH] is shorter.
Case 2: [MATH]l<k[/MATH]. Then the path [MATH]A[/MATH] to [MATH]C[/MATH] to [MATH]B[/MATH] is not the shortest path, because the path from [MATH]A[/MATH] to [MATH]D[/MATH] to [MATH]B[/MATH] is shorter.
...both of which are contradictions, giving us a grander contradiction as a result. So the equal-total-distance property must hold for any curve for which the reflection property holds (and also proven vice-versa). As long as the proof holds water, then a curve obeys the reflection property if and only if it obeys the equal-total-distance property as well, and additionally, we can extrapolate that only a curve with the properties of an ellipse can satisfy these requirements.
Thoughts?
You must log in or register to reply here.
Share:
Facebook
Twitter
Reddit
Pinterest
Tumblr
WhatsApp
Email
Share
Link
Forums
Free Math Help
Calculus
This site uses cookies to help personalise content, tailor your experience and to keep you logged in if you register.
By continuing to use this site, you are consenting to our use of cookies.
Accept
Learn more… |
11762 | https://med.libretexts.org/Bookshelves/Anatomy_and_Physiology/Anatomy_and_Physiology_(Boundless)/5%3A_Integumentary_System/5.4%3A_Functions_of_the_Integumentary_System/5.4E%3A__Blood_Supply_to_the_Epidermis | 5.4E: Blood Supply to the Epidermis - Medicine LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
5.4: Functions of the Integumentary System
5: Integumentary System
{ }
{ "5.4A:_Protection" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.4B:_Thermoregulation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.4C:_Cutaneous_Sensation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.4D:_Metabolic_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.4E:__Blood_Supply_to_the_Epidermis" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.4F:_Excretion_and_Absorption" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "5.1:_The_Skin" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.2:_Cell_Junctions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.3:__Accessory_Structures_of_the_Skin" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.4:_Functions_of_the_Integumentary_System" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.5:_Wound_Healing" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.6:_Overview_of_Cancer" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Tue, 17 Jan 2023 18:57:24 GMT
5.4E: Blood Supply to the Epidermis
7445
7445
Joshua Halpern
{ }
Anonymous
Anonymous User
2
false
false
[ "article:topic", "license:ccbysa", "showtoc:no" ]
[ "article:topic", "license:ccbysa", "showtoc:no" ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Contents
1. Home
2. Bookshelves
3. Anatomy and Physiology
4. Anatomy and Physiology (Boundless)
5. 5: Integumentary System
6. 5.4: Functions of the Integumentary System
7. 5.4E: Blood Supply to the Epidermis
Expand/collapse global location
Anatomy and Physiology (Boundless)
Front Matter
1: Introduction to Anatomy and Physiology
2: The Chemical Building Blocks of Life
3: Organization at the Cellular Level
4: Organization at the Tissue Level
5: Integumentary System
6: Skeletal System
7: Skeletal System - Parts of the Skeleton
8: Joints
9: Muscular System
10: Overview of the Nervous System
11: Central Nervous System
12: Peripheral Nervous System
13: Special Senses
14: Autonomic Nervous System
15: Endocrine System
16: Cardiovascular System - Blood
17: Cardiovascular System: The Heart
18: Cardiovascular System: Blood Vessels
19: Lymphatic System
20: Immune System
21: Respiratory System
22: Digestive System
23: Nutrition and Metabolism
24: Urinary System
25: Body Fluids and Acid-Base Balance
26: The Reproductive System
27: Human Development and Pregnancy
28: Reproduction, Chromosomes, and Meiosis
30: APPENDIX B: Development and Aging of the Organ Systems
29: APPENDIX A: Diseases, Injuries, and Disorders of the Organ Systems
Back Matter
5.4E: Blood Supply to the Epidermis
Last updated Jan 17, 2023
Save as PDF
5.4D: Metabolic Functions
5.4F: Excretion and Absorption
Page ID 7445
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. LEARNING OBJECTIVE
2. Key Takeaways
3. Key Points
4. Key Terms
The blood vessels in the dermis provide nourishment and remove waste from its own cells and from the stratum basale of the epidermis.
LEARNING OBJECTIVE
Identify the source of the blood supply for the integumentary system
Key Takeaways
Key Points
The epidermis contains no blood vessels, and cells in the deepest layers are nourished by diffusion from blood capillaries present in the upper layers of the dermis.
The papillary region of the dermis is composed of loose areolar connective tissue. This is named for its fingerlike projections called papillae, that extend toward the epidermis and contain terminal networks of blood capillaries.
The control of blood vessels within the dermis forms a key part of the body’s thermoregulatory capacity.
Key Terms
papillary region: The uppermost region of the dermis that is adjacent to the epidermis.
reticular region: The lower region of the dermis.
The epidermis does not contain blood vessels; instead, cells in the deepest layers are nourished by diffusion from blood capillaries that are present in the upper layers of the dermis. Diffusion provides nourishment and waste removal from the cells of the dermis, as well as for the cells of the epidermis.
The dermis: The distribution of the blood vessels in the skin of the sole of the foot. Corium—labeled at upper right—is an alternate term for dermis. Blood vessels that supply the capillaries of the papillary region are seen running through the reticular layer.
The dermis is the layer of skin beneath the epidermis that consists of connective tissue and cushions the body from stress and strain. The dermis is tightly connected to the epidermis by a basement membrane.
The dermis is structurally divided into two areas: a superficial area adjacent to the epidermis, called the papillary region, and a deep, thicker area known as the reticular region.
The papillary region is composed of loose areolar connective tissue. This is named for its fingerlike projections called papillae, that extend toward the epidermis and contain terminal networks of blood capillaries.
The reticular region lies under the papillary region and is usually much thicker. It is composed of dense, irregular connective tissue. The reticular region receives its name from the dense concentration of collagenous, elastic, and reticular fibers that weave throughout it.
These protein fibers give the dermis its typical properties of strength, extensibility, and elasticity. Blood vessels that supply the capillaries of the papillary region run through the reticular region.
Control of the blood supply to the dermis forms part of the body’s thermoregulatory capacity. Increasing blood flow, which makes the skin appear redder, will increase the loss of radiant heat through the skin, whereas constricting blood flow, making the skin appear paler, reduces heat loss.
5.4E: Blood Supply to the Epidermis is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts.
Back to top
5.4D: Metabolic Functions
5.4F: Excretion and Absorption
Was this article helpful?
Yes
No
Recommended articles
5.4A: Protection
5.4B: Thermoregulation
5.4C: Cutaneous Sensation
5.4D: Metabolic Functions
Article typeSection or PageLicenseCC BY-SAShow TOCno
Tags
This page has no tags.
© Copyright 2025 Medicine LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status×
contents readability resources tools
☰
5.4D: Metabolic Functions
5.4F: Excretion and Absorption |
11763 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3383522/ | Treatment and management of pseudogout: insights for the clinician - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Ther Adv Musculoskelet Dis
. 2012 Apr;4(2):121–131. doi: 10.1177/1759720X11432559
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Treatment and management of pseudogout: insights for the clinician
Paul MacMullan
Paul MacMullan
Find articles by Paul MacMullan
, Geraldine McCarthy
Geraldine McCarthy
Find articles by Geraldine McCarthy
✉
Author information
Article notes
Copyright and License information
✉
Geraldine McCarthy, MD, FRCPI, Consultant Rheumatologist and Professor of Medicine, Division of Rheumatology, Mater Misericordiae University Hospital, Dublin 7, Ireland gmccarthy@rcsi.ie
✉
Paul MacMullan, MB, BCh, BAO, MRCPI, Department of Molecular and Cellular Therapeutics, RCSI, Dublin and Rheumatology Department, Mater Misericordiae University Hospital, Dublin, Ireland
✉
Corresponding author.
Issue date 2012 Apr.
© The Author(s), 2011
PMC Copyright notice
PMCID: PMC3383522 PMID: 22870500
Abstract
Pseudogout and the associated calcium pyrophosphate dihydrate (CPPD)- crystal-related arthropathies are common conditions that present particular management problems in clinical practice as they often affect older patients with multiple medical comorbidities. The epidemiology, metabolic and endocrine disease associations, and routine investigations used in the diagnostic workup are briefly reviewed. Current treatment approaches that are mainly directed at relieving the symptoms of joint inflammation are outlined. Unlike gout, there are no agents available that have been shown to decrease crystal load in CPPD-related joint disease. Recent novel insights into the pathogenesis of crystal-induced joint inflammation and subsequent joint degeneration are also discussed. The potential of colchicine as a prophylactic agent in managing recurrent attacks and the likely mechanisms of its effects on the NACHT, LRR and PYD domains-containing protein 3 (NALP-3) inflammasome of the innate immune system are highlighted. The use of agents that directly target the inflammasome, in particular drugs which inhibit the interleukin 1 pathway, in the treatment of severe, refractory pseudogout is also discussed. Finally, there is particular emphasis on the likely pathogenic role of CPPD crystal deposition in degenerative joint disease and the use of targeted anticrystal therapies as potential disease-modifying drugs.
Keywords: pseudogout, CPPD, colchicine, osteoarthritis, inflammasome, crystal, interleukin-1
Introduction
The ‘pseudogout syndrome’ was first described by Kohn and colleagues in 1962 [Kohn et al. 1962] and accurately depicts acute attacks of synovitis induced by calcium pyrophosphate dihydrate (CPPD) crystals, which clinically resemble gouty arthritis due to monosodium urate (MSU) crystal deposition. CPPD deposition may also cause symptoms similar to septic arthritis, polyarticular inflammatory arthritis [often misdiagnosed as rheumatoid arthritis (RA)] or degenerative osteoarthritis (OA) [McCarthy, 2008]. The clinical manifestations, epidemiology, associated metabolic conditions and novel insights into the pathogenesis of CPPD-related joint inflammation are briefly discussed below. Unlike gout, the treatment of CPPD-related arthropathies is often difficult, partly due to patient characteristics, diagnostic uncertainty and the lack of any effective agent to decrease crystal load [Announ and Guerne, 2008]. The purpose of this article is to review current approaches that are used in the treatment of acute pseudogout and the management of chronic CPPD-related arthropathies, with a particular emphasis on the use of available agents to target the ‘inflammasome’ [Martinon et al. 2006], which plays a crucial role in crystal-induced inflammation and the theoretical potential of specific anticrystal therapies.
Clinical manifestations
CPPD crystal deposition disease has the potential to mimic most forms of inflammatory arthritis and a clinical classification of these phenomena has emerged from the studies of McCarty and colleagues [McCarty, 1970, 1976; Rosenthal, 2001]. There are at least six distinct presentations of this disorder, and along with acute pseudogout include asymptomatic or lanthanic chondrocalcinosis, pseudo-OA (with or without acute attacks), pseudo-RA, pseudo-polymyalgia rheumatica (pseudo-PMR) and pseudo-neuropathic arthropathy.
Acute pseudogout
Approximately 25% of patients with CPPD crystal deposition exhibit the classical pseudogout pattern of disease. Compared with true gout, pseudogout attacks may take longer to reach peak intensity and may persist for up to 3 months despite therapy [Masuda and Ishikawa, 1988]. Large joints are more commonly affected and the attacks are characterized by the cardinal signs of inflammation which may involve one or several hot, tender, red and swollen joints [Dieppe et al. 1982]. The knee is the most commonly involved joint, followed by the wrist, ankle, elbow, toe, shoulder and hip. As with gout, these attacks may occur spontaneously or can be provoked by trauma, surgery or severe medical illness [McCarthy, 2008; Rosenthal et al. 2005]. Flares of pseudogout after parathyroidectomy are well described [Bilezikian et al. 1973; Rosenthal et al. 2005] and may reflect abrupt changes in serum calcium levels post surgery. Pamidronate therapy for hypercalcaemia has also been implicated as a trigger for acute attacks of pseudogout [Malnick et al. 1997]. Patients are usually asymptomatic between acute attacks and differentiation from true gout or even a septic joint requires arthrocentesis and synovial fluid (SF) analysis for Gram stain, culture and cytology. Pseudogout is confirmed by the demonstration of CPPD crystals in SF which manifest as rhomboid-shaped rod-like structures that exhibit weakly positive or no birefringence by compensated polarized light microscopy, in contrast to the negatively birefringent needle-shaped MSU crystals found in gout (Figure 1). Gout and pseudogout may also coexist in a single inflammatory effusion, as 20% of patients with CPPD will have hyperuricaemia and a quarter of these will develop gout at some stage [Rosenthal et al. 2005].
Figure 1.
Open in a new tab
Compensated polarised light microscopy demonstrating coexistent monosodium urate (MSU) and (calcium pyrophosphate dihydrate) CPPD crystals. The MSU crystals (#) are needle shaped and negative birefringent whereas the CPPD crystals (♮) are rhomboid and weakly positively birefringent.
Asymptomatic chondrocalcinosis
Chondrocalcinosis specifically describes radiographic calcification in hyaline and fibrocartilage (see Figure 2). It is commonly present in patients with CPPD deposition disease, but is neither sensitive nor specific for affected patients. While CPPD crystal deposition accounts for approximately 95% of cases of chondrocalcinosis [McCarty et al. 1966], one study reported that a radiographic diagnosis was only made in about 40% of patients who had pathologically proven CPPD crystal deposition from samples obtained at arthroscopy [Fisseler-Eckhoff and Muller, 1992]. Some individuals with radiographic or pathologic evidence of articular chondrocalcinosis have no clinically apparent arthritis and this finding has been referred to as lanthanic CPPD deposition and is of uncertain significance [McCarthy, 2008]. These patients are often of advanced age and have not been rigorously studied to see whether they develop clinically relevant arthritis. However, there are reports that patients with apparently asymptomatic disease, when subjected to closer scrutiny, have a higher frequency of wrist and knee complaints and associated degenerative changes compared with age-matched controls without radiographic chondrocalcinosis [Ellman and Levin, 1975].
Figure 2.
Open in a new tab
Plain radiograph of right knee demonstrating meniscal chondrocalcinosis due to calcium pyrophosphate dihydrate deposition.
Pseudo-osteoarthritis
A large proportion of patients with clinically apparent CPPD crystal deposition follow a progressive course of articular degeneration in multiple joints in an oddly distributed pattern typically involving the knees, wrists, metacarpophalangeal (MCP) joints, hips, shoulders, spine, elbows and ankles. About half of these patients will experience acute attacks of pseudogout superimposed on their chronic symptoms [McCarthy, 2008], while the remainder will often present with complaints more typical of classical OA. While clinical examination findings are similar to OA, certain features such as the atypical distribution (especially involving the wrists and MCP joints), the presence of contractures and valgus knee deformities should highlight the possibility of CPPD deposition disease. However, when the degenerative process occurs in joints more typical for classical OA, such as the knee and hip, it can be difficult to differentiate from OA and consequently may be significantly under recognized. This is highlighted by the high prevalence of these crystals in samples taken from OA knees at the time of arthroplasty, with one study reporting a third of these specimens displaying definite evidence of CPPD deposition [Derfus et al. 2002]. Furthermore, there is an emerging body of evidence that these and other calcium-containing crystals, such as basic calcium phosphate (BCP), play a pathogenic role in joint degeneration and that calcification of articular cartilage is a feature of OA [Molloy and McCarthy, 2006]. Moreover, there is strong in vitro evidence which demonstrates that such calcium deposition is inhibited by phosphocitrate [Sun et al. 2010], and which is supported by complementary work using an animal model of OA [Cheung et al. 2006].
Pseudo-rheumatoid arthritis
A small proportion (<5%) of patients with CPPD deposition experience a nonerosive, asynchronous, inflammatory arthritis which mimics RA [Rosenthal et al. 2005]. The associated morning stiffness, malaise, synovial thickening, symmetrical involvement, flexion contractures and elevated acute phase inflammatory markers often lead to a misdiagnosis of RA. In addition, 10% of these individuals, who are often of older age, will have low titres of rheumatoid factor thus causing further diagnostic confusion [McCarthy, 2008]. Therefore, the presence of CPPD crystals in synovial fluid and radiographic changes more in keeping with OA favour a diagnosis of pseudo-RA, while elevated titres of anticyclic citrullinated peptide (anti-CCP) antibodies and the presence of bony erosions either on ultrasound or plain film radiographic imaging indicate a diagnosis of classical RA. However, these two conditions may occasionally coexist and the required differences in management approaches can be challenging. A rare subtype of pseudo-RA may also present with prominent systemic features such as fever, leucocytosis, altered mental status and polyarthritis and may mimic sepsis, particularly in older people [Bong and Bennett, 1981].
Pseudo-polymyalgia rheumatica
PMR is classically a disease of older people, presenting with a range of symptoms including fatigue, early morning stiffness and subjective proximal muscle weakness. It is often a considerable challenge for the clinician to distinguish older patients with pure PMR from others presenting with polymyalgia symptoms resembling this condition. Recent work in a large number of patients indicates that CPPD arthropathy should also be included in the wide spectrum of diseases mimicking classical PMR [Pego-Reigosa et al. 2005]. Knee OA, tendinous calcifications and episodic ankle arthritis are proposed as the variables that predict those older patients with PMR features who really have an atypical pattern of CPPD arthropathy [Pego-Reigosa et al. 2005].
Pseudo-neuropathic arthropathy
Some patients with CPPD deposition have a severe destructive monoarthritis that clinically resembles a neuropathic Charcot’s joint [Richards and Hamilton, 1974]. In contrast to the classical description of Charcot’s joints associated with diabetes mellitus, tabes dorsalis, syringomyelia and alcoholic neuropathy, these patients have no underlying neurological disorder yet present with a painful, destructive monoarthritis. The natural history of this condition is not well understood, management options are limited and surgical intervention may often be required.
Epidemiology
CPPD crystal deposition is largely a disease of older people and several radiographic surveys have demonstrated an age-related increase in the prevalence of joint calcification with almost half of all octogenarians displaying some evidence of articular chondrocalcinosis [Ellman and Levin, 1975; Wilkins et al. 1983]. The prevalence of CPPD deposition among younger people is unknown. Although there is no major gender predominance, attacks of acute pseudogout appear more commonly in men, with women more frequently exhibiting the pseudo-OA pattern of the disease [O’Duffy, 1976; Rosenthal et al. 2005].
Disease associations
Despite the fact that most cases of CPPD crystal deposition are idiopathic, several metabolic and endocrine disorders, a previous history of joint trauma and the well described hereditary condition of familial chondrocalcinosis are all associated with precocious articular calcification in younger patients [Jones et al. 1992]. Much has been learned about the likely pathogenic processes involved in pathological calcification from studying distinct pedigrees of autosomal dominant familial CPPD deposition and their association with a mutant ANKH gene located on the short arm of chromosome 5 [Pendleton et al. 2002]. This gene encodes a transmembrane transport protein critically involved in generating extracellular inorganic phosphate (PPi) which then acts as a substrate for CPPD crystal formation in the pericellular cartilage matrix [Williams et al. 2002]. This is discussed in more detail below.
Of the other endocrine and metabolic conditions that have been linked with chondrocalcinosis, only hereditary haemochromatosis is associated with the full spectrum of CPPD-related arthropathy [Jones et al. 1992]. Iron accumulation in joint tissues appears to be a key factor, as evidenced by reports of CPPD crystal deposition in patients with secondary iron overload due to transfusion haemosiderosis and haemophilia. Chronic hypomagnesaemia due to Gitelman’s and Bartter’s syndromes has also been associated with both chondrocalcinosis and acute pseudogout, thus forming the rationale for magnesium supplementation to prevent recurrent attacks of pseudogout [Cobeta-Garcia et al. 1998; Punzi et al. 1998].
A clear association with previous joint trauma, particularly involving the knee, lends weight to the likely role of chondrocalcinosis in contributing to the process of joint degeneration [Doherty et al. 1982; Settas et al. 1982]. Hyperparathyroidism, hypophosphatasia, gout and hypocalciuric hypercalcaemia are all associated with acute attacks of pseudogout, while the relationships between acromegaly, Wilson’s disease, diabetes mellitus and CPPD deposition are less clear [Jones et al. 1992].
Pathogenesis
Recent insights into the processes involved in both CPPD deposition and in the mechanisms whereby these crystals can interact with the inflammasome to cause joint inflammation have greatly increased our understanding of the pathogenesis of pseudogout. Several lines of evidence support the hypothesis that elevated levels of PPi in the immediate extracellular environment of chondrocytes provide the substrate for the formation of CPPD crystals, which are then deposited in the cartilaginous matrix [Rachow and Ryan, 1985; Ryan et al. 1984]. The nucleoside triphosphatase pyrophosphohydrolase (NTPPPH) group of ectoenzymes are strongly associated with this process as increased activity of these enzymes catalyses the production of extracellular PPi at the external surface of the chondrocyte cell membrane [Ryan et al. 1984], while deficiency of one of this group [the NTPPPH plasma cell membrane glycoprotein 1 (PC-1) enzyme] leads to excessive mineralization with BCP rather than CPPD crystals in a mouse model of pathological ossification [Okawa et al. 1998]. This is supported by data involving another mouse disorder, murine progressive ankylosis, in which a homozygous nonsense mutation of the aforementioned ANK gene results in reduced extracelluler PPi levels and excessive peripheral and axial skeleton ankylosis with BCP crystal material [Ho et al. 2000]. Thus, one function of extracellular PPi appears to be the inhibition of the growth of BCP crystals while excessive PPi levels, arising from NTPPPH overactivity or gain of function mutations in the ANK transmembrane transporter protein, could provide the substrate for CPPD crystal formation. Matrix vesicles that play a role in normal bone development are also implicated and studies have shown that these vesicles can generate CPPD crystals in vitro [Derfus et al. 1992, 1996]. These matrix vesicles are selectively rich in NTPPPH activity and gene transfer experiments involving the NTPPPH PC-1 enzyme and cultured chondrocytes demonstrate increased chondocyte apoptosis and matrix calcification [Johnson et al. 2001]. A schematic diagram representing some of the aforementioned processes involved in the formation of calcium-containing crystals, mechanisms of inflammation and subsequent joint degeneration is illustrated in Figure 3.
Figure 3.
Open in a new tab
Schematic diagram of calcium crystal formation [calcium pyrophosphate dihydrate (CPPD) versus basic calcium phosphate (BCP)], with subsequent inflammasome activation via CPPD and synovial inflammation via BCP, and how both can contribute to the vicious cycle of joint degeneration in calcium crystal-related arthropathies. ePPi, extracellular inorganic phosphate; HA, hydroxyapatite; IL-1β, interleukin 1β; MMP, matrix metalloproteinase; MV, matrix vesicle; NALP 3, NACHT, LRR and PYD domains-containing protein 3; NTPPPH, nucleoside triphosphatase pyrophosphohydrolase; PPi, inorganic phosphate.
Acute attacks of pseudogout are believed to represent a dose-related auto-inflammatory response to CPPD crystals shed from cartilaginous tissues into the synovial cavity [McCarthy, 2008]. Recent data describing how CPPD crystals (and indeed MSU crystals) interact with the caspase-1-activating NACHT, LRR and PYD domains-containing protein 3 (NALP-3) inflammasome of the innate immune system provide novel insights into the mechanisms by which these crystals cause episodic bouts of joint inflammation [Martinon et al. 2006]. This results in the production of several proinflammatory cytokines, particularly interleukin (IL)-1β, by macrophages and monocytes, thus initiating an inflammatory cascade with the subsequent influx of neutrophils, fibroblast-like synoviocytes and recruitment of other inflammatory pathways via tumour necrosis factor α (TNFα) [Meng et al. 1997]. Colchicine is used in the treatment of several auto-inflammatory conditions, including gout and pseudogout, and has recently been shown to completely block crystal-induced maturation of IL-1β in vitro, indicating that the drug acts upstream of inflammasome activation [Martinon et al. 2006]. Combining this observation with the known mode of action of colchicine as an inhibitor of microtubule assembly [Goldfinger et al. 1965], it is therefore likely that the drug acts at the level of crystal endocytosis and/or presentation to the inflammasome [Martinon et al. 2006]. Taken together, these data support a pivotal role for the inflammasome in pseudogout and provide a series of novel molecular targets for the treatment and prophylaxis of acute episodes of pseudogout and the other CPPD-related arthropathies.
Investigations
The definitive diagnosis of pseudogout is most commonly and accurately made by identifying CPPD crystals, by compensated polarized light microscopy, in the SF of affected joints [McCarthy, 2008]. Furthermore, all patients presenting with suspected CPPD deposition can be screened for chondrocalcinosis using certain key radiographs: a nonweight-bearing anteroposterior (AP) view of both knees, an AP view of the pelvis for visualization of the symphysis pubis and hips, and a posteroanterior (PA) view of each hand to include the triangular ligament of the wrists. Changes in the MCP joints, such as squaring of the bone ends, presence of subchondral cysts and hook-like osteophytes, are characteristic features of the arthropathy of haemachromatosis, but are also found in patients with CPPD deposition alone [McCarthy, 2008]. Thus, a probable diagnosis of CPPD arthropathy can be made in patients who present with acute or chronic arthritis of knee, hip, wrist, shoulder or MCP joints in the setting of radiographic features, even without the direct demonstration of CPPD crystals by micropscopy [McCarty, 1994].
Given the aforementioned disease associations, most patients with CPPD deposition, and particularly those who are relatively young, should undergo screening for abnormal serum calcium, phosphate, magnesium, alkaline phosphatase and thyroid-stimulating hormone levels, and any evidence of iron overload by haemetinic analyses (ferritin, iron saturation and transferrin) [Rosenthal et al. 2005]. This is especially true with regard to haemachromatosis in light of the strong association between this condition and all forms of CPPD-related arthropathy, and the deleterious consequences of delayed diagnosis [Jones et al. 1992].
Treatment and management
Available and potential treatment options in the management of pseudogout are summarized in Table 1. Strategies that are currently employed in ameliorating CPPD-related joint disease are broadly limited to the following; those directed against correcting underlying metabolic abnormalities and treating associated conditions, general treatment with nonsteroidal anti-inflammatory drugs (NSAIDs) and/or corticosteroids (either by local intra-articular injection or systemic therapy), and finally, low-dose oral colchicine [Choy, 2005]. Potential future therapies include agents targeted against crystal formation (such as probenecid and phosphocitrate), more potent anti-inflammatory medications such as methotrexate and anticytokine drugs which target the IL-1 pathway [Announ and Guerne, 2008].
Table 1.
Management strategies in the treatment of pseudogout.
| Strategy | Treatment | Comments |
:---
| Treating associated metabolic conditions | Decreasing iron overload, treating hypothyroidism | No hard evidence that this improves CPPD-related arthropathy |
| | Magnesium supplementation | Only indicated in hypomagnesemic states. Otherwise evidence is largely anecdotal |
| Conventional anti-inflammatory medications | NSAIDs | Symptomatic relief but may be relatively contraindicated in the older population |
| | Corticosteroids | Excellent symptomatic relief with intra-articular injections. Some evidence for short courses of oral/intramuscular steroids for polyarticular flares |
| | Colchicine | Acts upstream of the inflammasome and has an emerging role as a prophylactic agent |
| Anticrystal therapy | Probenecid | Good theoretical rationale but evidence lacking |
| | Phosphocitrate | Strong evidence from animal studies but no safety or efficacy data in humans |
| Targeting the inflammasome | Methotrexate | Emerging role in the prevention of recurrent attacks |
| | Inhibition of IL-1 pathway (anakinra, canakinumab, IL-1 Trap) | May become key agents for refractory polyarticular disease |
| | Anti-TNFα drugs | Poor efficacy in auto-inflammatory conditions and therefore unlikely to be successful in pseudogout |
Open in a new tab
CPPD, calcium pyrophosphate dehydrate; IL-1, interleukin 1; NSAID, nonsteriodal anti-inflammatory drug; TNFα, tumour necrosis factor α.
Associated conditions
Despite the value of establishing the presence of an associated disorder there is no evidence that targeted treatment of these conditions, such as decreasing iron load with phlebotomy in haemochromatosis, has any effect on CPPD crystal deposition [Hamilton et al. 1981]. However, recent data suggest that reversal of iron overload may at least lead to symptomatic relief in the long term [Harty et al. 2011].
As magnesium acts as a cofactor in the regulation of certain phosphatases, and coupled with the known association between chronic hypomagnesaemic states and chondrocalcinosis [Punzi et al. 1998], magnesium supplementation has often been recommended as a safe prophylactic agent to decrease the frequency of acute attacks of pseudogout. A 10-year follow-up study demonstrated reduced meniscal calcification in patients with familial hypokalaemia/hypomagnesaemia and chondrocalcinosis who were given magnesium supplementation [Smilde et al. 1994]. However, only one small placebo-controlled study in CPPD-related arthropathy, not specifically associated with hypomagnesaemia, using magnesium carbonate supplements showed a tendency to improvement at 6 months, but has never been replicated [Announ and Guerne, 2008].
Anti-inflammatory medications
NSAIDs, corticosteroids and colchicine have all been used clinically for managing acute pseudogout and the episodic inflammatory arthritis that is regularly seen in pseudo-OA [Choy, 2005]. NSAIDs still represent the mainstay of treatment, but are often of insufficient efficacy and also relatively contraindicated in a predominately older population due to the increased risk of gastrointestinal haemorrhage and associated renal impairment [Announ and Guerne, 2008]. Alternatively, and particularly when only one joint such as the knee is involved, aspiration and intra-articular injection of corticosteroids can be extremely effective [Rosenthal et al. 2005]. Short courses of oral steroids (0.5–1 mg/kg) with a rapid taper or even intramuscular injections are also useful for recurrent flares of polyarticular CPPD-related arthritis [Roane et al. 1997].
There is some evidence that hydroxychloroquine inhibits matrix metalloproteinase activity in an animal model of CPPD arthritis, and while it does not have a role in the management of acute pseudogout, current European League Against Rheumatism guidelines recommend its use as an adjunctive agent in patients with chronic CPPD arthropathy [Zhang et al. 2011].
In addition to the aforementioned inhibitory effects of colchicine on crystal-induced IL-1β production by the inflammasome, its disruption of normal cytoskeletal function leads to the impairment of a variety of cellular functions, including the secretion of other chemokines and mediators, and reduced cell migration and division [Nuki, 2008]. These effects, when coupled with its plasma half-life of 20–30 h, large volume of distribution, partial renal excretion (20%), and preferential accumulation in red blood cells and neutrophils result in a relatively narrow therapeutic index and significant potential for toxicity, especially regarding bone marrow suppression [Nuki, 2008]. Therefore, colchicine is nowadays only used in low-dose preparations (0.5 mg twice daily) for both gout and pseudogout. While there have been no major trials of colchicine in the treatment of acute pseudogout, it is commonly used clinically for this purpose [Choy, 2005]. However, there is some evidence for its role in the prevention of both acute attacks of pseudogout and recurrent episodes of the other CPPD-related arthritides [Alvarellos and Spilberg, 1986], and the recent mechanistic insights into its effects on crystal uptake into cells and inhibition of their subsequent interaction with the inflammasome have generated renewed interest in this medication as a potential anchor drug in the prophylaxis of pseudogout.
Anticrystal agents
Unlike gout, there are no available agents which directly target crystal load in CPPD deposition disease. Probenecid is a transmembrane-transport inhibitor which reduces the production of extracellular PPi in vitro [Rosenthal and Ryan, 1994], but to date has never been demonstrated to decrease CPPD crystal deposition in humans. The clear contribution of CPPD crystals to the acceleration of joint degeneration in the various CPPD-related arthropathies and the possible pathological role of these crystals in OA [Macmullan and McCarthy, 2010] has resulted in significant interest in the potential of phosphocitrate to directly target crystal deposition in both of these diseases [Sun et al. 2010]. Calcification of articular cartilage is now well recognized as an indissociable feature of OA [Molloy and McCarthy, 2006]. Furthermore, there is strong in vitro evidence that such calcium deposition is inhibited by phosphocitrate [Sun et al. 2010]. Taken together with previous work demonstrating that CPPD crystals accelerate joint degeneration in a rabbit model of OA [Fam et al. 1995], this suggests that agents which directly target articular calcification may not only have the potential to treat CPPD-related arthropathy but also OA itself.
Targeting the inflammasome
Alongside the recent advances in the understanding of the mechanisms involved in crystal-induced joint inflammation, specific attention has been focused on the ability of methotrexate to decrease the frequency and intensity of recurrent attacks of pseudogout, particularly in relation to polyarticular disease [Chollet-Janin et al. 2007]. Despite the fact that this agent is relatively contraindicated in patients with haemochromatosis-associated arthropathy (because of concomitant liver disease and the potential for increased toxicity), due to its broad-spectrum ability to inhibit most cytokine pathways involved in joint inflammation, this generally well tolerated molecule represents an attractive alternative in the treatment and prophylaxis of the entire spectrum of ‘pseudogout syndrome’ [Announ and Guerne, 2008]. The anti-inflammatory effect of methotrexate is mediated chiefly by an increase in extracellular adenosine levels [Cronstein et al. 1993], which activates membrane receptors on macrophages and other inflammatory cells, thereby inhibiting the release of several cytokines, most notably IL-1β [Morabito et al. 1998]. Thus, by acting in this proximal manner, methotrexate has a potent indirect effect on the ability of the inflammasome to generate IL-1β. Multicentre clinical trials are currently underway to further assess the potential impact of this widely available and affordable medication in this area.
Specific insights into the role of IL-1β in a variety of auto-inflammatory diseases have resulted in a number of case reports documenting the efficacy of anakinra, an IL-1 receptor antagonist, in difficult-to-treat polyarticular pseudogout [McGonagle et al. 2008]. Furthermore, a newly developed monoclonal antibody to IL-1β, canakinumab, has proven to be very effective in treating refractory cases of gouty arthritis [So et al. 2010], and a novel soluble fusion protein, IL-1 Trap, has recently been licensed for the treatment of several auto-inflammatory conditions [Ratner, 2008]. Taken together with the recent data on the role of the inflammasome in generating IL-1β in response to crystal endocytosis, these drugs may find a new place as disease-modifying agents in the treatment of CPPD-crystal-related arthropathy. However, their immunosuppressive effects in a largely older patient population and relative high cost ensure that, at this point, their use should be reserved for particularly difficult cases.
Although several reports exist of gout being successfully treated with anti-TNFα drugs, published cases of pseudogout successfully treated with these agents are lacking [Announ and Guerne, 2008]. As with many auto-inflammatory conditions, it seems that CPPD-related arthritis responds poorly to this class of medication. To date, the experimental data indicating that TNFα, unlike in RA, appears to exert its influence downstream of IL-1β in the acute inflammatory cascade associated with pseudogout would appear to be borne out clinically.
Conclusion
Pseudogout and the associated CPPD-related arthropathies are likely to increase in prevalence due to the demographic drift towards an aging population. Thus, the management of these conditions will continue to present particular challenges in the future. The use of regular colchicine as a prophylactic agent against acute attacks of pseudogout will probably become more widespread. Similarly, drugs targeting the IL-1 pathway offer considerable potential in severe refractory disease. However, in contrast to MSU crystals in gout, there is no practical way to remove CPPD crystals from the joint. Therefore, current management strategies remain symptomatic but not disease modifying. Further research into ways of reducing the CPPD crystal burden is very necessary.
Footnotes
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
The authors declare no conflicts of interest in preparing this article.
References
Alvarellos A., Spilberg I. (1986) Colchicine prophylaxis in pseudogout. J Rheumatol 13: 804–805 [PubMed] [Google Scholar]
Announ N., Guerne P.A. (2008) Treating difficult crystal pyrophosphate dihydrate deposition disease. Curr Rheumatol Rep 10: 228–234 [DOI] [PubMed] [Google Scholar]
Bilezikian J.P., Connor T.B., Aptekar R., Freijanes J., Aurbach G.D., Pachas W.N., et al. (1973) Pseudogout after parathyroidectomy. Lancet 1: 445–446 [DOI] [PubMed] [Google Scholar]
Bong D., Bennett R. (1981) Pseudogout mimicking systemic disease. JAMA 246: 1438–1440 [PubMed] [Google Scholar]
Cheung H.S., Sallis J.D., Demadis K.D., Wierzbicki A. (2006) Phosphocitrate blocks calcification-induced articular joint degeneration in a guinea pig model. Arthritis Rheum 54: 2452–2461 [DOI] [PubMed] [Google Scholar]
Chollet-Janin A., Finckh A., Dudler J., Guerne P.A. (2007) Methotrexate as an alternative therapy for chronic calcium pyrophosphate deposition disease: an exploratory analysis. Arthritis Rheum 56: 688–692 [DOI] [PubMed] [Google Scholar]
Choy G. (2005) An update on the treatment options for gout and calcium pyrophosphate deposition. Expert Opin Pharmacother 6: 2443–2453 [DOI] [PubMed] [Google Scholar]
Cobeta-Garcia J.C., Gascon A., Iglesias E., Estopinan V. (1998) Chondrocalcinosis and Gitelman’s syndrome. A new association? Ann Rheum Dis 57: 748–749 [DOI] [PMC free article] [PubMed] [Google Scholar]
Cronstein B.N., Naime D., Ostad E. (1993) The antiinflammatory mechanism of methotrexate. Increased adenosine release at inflamed sites diminishes leukocyte accumulation in an in vivo model of inflammation. J Clin Invest 92: 2675–2682 [DOI] [PMC free article] [PubMed] [Google Scholar]
Derfus B.A., Kurian J.B., Butler J.J., Daft L.J., Carrera G.F., Ryan L.M., et al. (2002) The high prevalence of pathologic calcium crystals in pre-operative knees. J Rheumatol 29: 570–574 [PubMed] [Google Scholar]
Derfus B.A., Kurtin S.M., Camacho N.P., Kurup I., Ryan L.M. (1996) Comparison of matrix vesicles derived from normal and osteoarthritic human articular cartilage. Connect Tissue Res 35: 337–342 [DOI] [PubMed] [Google Scholar]
Derfus B.A., Rachow J.W., Mandel N.S., Boskey A.L., Buday M., Kushnaryov V.M., et al. (1992) Articular cartilage vesicles generate calcium pyrophosphate dihydrate-like crystals in vitro. Arthritis Rheum 35: 231–240 [DOI] [PubMed] [Google Scholar]
Dieppe P.A., Alexander G.J., Jones H.E., Doherty M., Scott D.G., Manhire A., et al. (1982) Pyrophosphate arthropathy: a clinical and radiological study of 105 cases. Ann Rheum Dis 41: 371–376 [DOI] [PMC free article] [PubMed] [Google Scholar]
Doherty M., Watt I., Dieppe P.A. (1982) Localised chondrocalcinosis in post-meniscectomy knees. Lancet 1: 1207–1210 [DOI] [PubMed] [Google Scholar]
Ellman M.H., Levin B. (1975) Chondrocalcinosis in elderly persons. Arthritis Rheum 18: 43–47 [DOI] [PubMed] [Google Scholar]
Fam A.G., Morava-Protzner I., Purcell C., Young B.D., Bunting P.S., Lewis A.J. (1995) Acceleration of experimental lapine osteoarthritis by calcium pyrophosphate microcrystalline synovitis. Arthritis Rheum 38: 201–210 [DOI] [PubMed] [Google Scholar]
Fisseler-Eckhoff A., Muller K.M. (1992) Arthroscopy and chondrocalcinosis. Arthroscopy 8: 98–104 [DOI] [PubMed] [Google Scholar]
Goldfinger S.E., Howell R.R., Seegmiller J.E. (1965) Suppression of metabolic accompaniments of phagocytosis by colchicine. Arthritis Rheum 8: 1112–1122 [DOI] [PubMed] [Google Scholar]
Hamilton E.B., Bomford A.B., Laws J.W., Williams R. (1981) The natural history of arthritis in idiopathic haemochromatosis: progression of the clinical and radiological features over ten years. Q J Med 50: 321–329 [PubMed] [Google Scholar]
Harty L., Lai D., Connor S., Dunne A., Ali M., Ryan J., et al. (2011) Prevalence and progress of joint symptoms in hereditary hemochromatosis and symptomatic response to venesection. Clin Rheumatol 17: 220–222 [DOI] [PubMed] [Google Scholar]
Ho A.M., Johnson M.D., Kingsley D.M. (2000) Role of the mouse ank gene in control of tissue calcification and arthritis. Science 289: 265–270 [DOI] [PubMed] [Google Scholar]
Johnson K., Pritzker K., Goding J., Terkeltaub R. (2001) The nucleoside triphosphate pyrophosphohydrolase isozyme PC-1 directly promotes cartilage calcification through chondrocyte apoptosis and increased calcium precipitation by mineralizing vesicles. J Rheumatol 28: 2681–2691 [PubMed] [Google Scholar]
Jones A.C., Chuck A.J., Arie E.A., Green D.J., Doherty M. (1992) Diseases associated with calcium pyrophosphate deposition disease. Semin Arthritis Rheum 22: 188–202 [DOI] [PubMed] [Google Scholar]
Kohn N.N., Hughes R.E., McCarty D.J., Jr, Faires J.S. (1962) The significance of calcium phosphate crystals in the synovial fluid of arthritic patients: the ‘pseudogout syndrome’. II. Identification of crystals. Ann Intern Med 56: 738–745 [DOI] [PubMed] [Google Scholar]
Macmullan P.A., McCarthy G.M. (2010) The meniscus, calcification and osteoarthritis: a pathologic team. Arthritis Res Ther 12: 116. [DOI] [PMC free article] [PubMed] [Google Scholar]
Malnick S.D., Ariel-Ronen S., Evron E., Sthoeger Z.M. (1997) Acute pseudogout as a complication of pamidronate. Ann Pharmacother 31: 499–500 [DOI] [PubMed] [Google Scholar]
Martinon F., Petrilli V., Mayor A., Tardivel A., Tschopp J. (2006) Gout-associated uric acid crystals activate the NALP3 inflammasome. Nature 440: 237–241 [DOI] [PubMed] [Google Scholar]
Masuda I., Ishikawa K. (1988) Clinical features of pseudogout attack. A survey of 50 cases. Clin Orthop Relat Res (229): 173–181 [PubMed] [Google Scholar]
McCarthy G. (2008) Calcium phosphate dihydrate, hydroxyapatite, and miscellaneous crystals. In: Klippel J.H., Sone J.H., Crofford L.J., White P.H. (eds), Primer on the Rheumatic Diseases. 13th ed. Berlin: Springer, pp; 263–270 [Google Scholar]
McCarty D.J. (1970) Crystal-induced inflammation of the joints. Annu Rev Med 21: 357–366 [DOI] [PubMed] [Google Scholar]
McCarty D.J. (1976) Calcium pyrophosphate dihydrate crystal deposition disease – 1975. Arthritis Rheum 19(Suppl. 3): 275–285 [DOI] [PubMed] [Google Scholar]
McCarty D.J. (1994) Crystals and arthritis. Dis Mon 40: 255–299 [DOI] [PubMed] [Google Scholar]
McCarty D.J., Jr, Hogan J.M., Gatter R.A., Grossman M. (1966) Studies on pathological calcifications in human cartilage. I. Prevalence and types of crystal deposits in the menisci of two hundred fifteen cadavera. J Bone Joint Surg Am 48: 309–325 [PubMed] [Google Scholar]
McGonagle D., Tan A.L., Madden J., Emery P., McDermott M.F. (2008) Successful treatment of resistant pseudogout with anakinra. Arthritis Rheum 58: 631–633 [DOI] [PubMed] [Google Scholar]
Meng Z.H., Hudson A.P., Schumacher H.R., Jr, Baker J.F., Baker D.G. (1997) Monosodium urate, hydroxyapatite, and calcium pyrophosphate crystals induce tumor necrosis factor-alpha expression in a mononuclear cell line. J Rheumatol 24: 2385–2388 [PubMed] [Google Scholar]
Molloy E.S., McCarthy G.M. (2006) Calcium crystal deposition diseases: update on pathogenesis and manifestations. Rheum Dis Clin North Am 32: 383–400, vii [DOI] [PubMed] [Google Scholar]
Morabito L., Montesinos M.C., Schreibman D.M., Balter L., Thompson L.F., Resta R., et al. (1998) Methotrexate and sulfasalazine promote adenosine release by a mechanism that requires ecto-5’-nucleotidase-mediated conversion of adenine nucleotides. J Clin Invest 101: 295–300 [DOI] [PMC free article] [PubMed] [Google Scholar]
Nuki G. (2008) Colchicine: its mechanism of action and efficacy in crystal-induced inflammation. Curr Rheumatol Rep 10: 218–227 [DOI] [PubMed] [Google Scholar]
O’Duffy J.D. (1976) Clinical studies of acute pseudogout attacks: comments on prevalence, predispositions, and treatment. Arthritis Rheum 19(Suppl. 3): 349–352 [DOI] [PubMed] [Google Scholar]
Okawa A., Nakamura I., Goto S., Moriya H., Nakamura Y., Ikegawa S. (1998) Mutation in Npps in a mouse model of ossification of the posterior longitudinal ligament of the spine. Nat Genet 19: 271–273 [DOI] [PubMed] [Google Scholar]
Pego-Reigosa J.M., Rodriguez-Rodriguez M., Hurtado-Hernandez Z., Gromaz-Martin J., Taboas-Rodriguez D., Millan-Cachinero C., et al. (2005) Calcium pyrophosphate deposition disease mimicking polymyalgia rheumatica: a prospective followup study of predictive factors for this condition in patients presenting with polymyalgia symptoms. Arthritis Rheum 53: 931–938 [DOI] [PubMed] [Google Scholar]
Pendleton A., Johnson M.D., Hughes A., Gurley K.A., Ho A.M., Doherty M., et al. (2002) Mutations in ANKH cause chondrocalcinosis. Am J Hum Genet 71: 933–940 [DOI] [PMC free article] [PubMed] [Google Scholar]
Punzi L., Calo L., Schiavon F., Pianon M., Rosada M., Todesco S. (1998) Chondrocalcinosis is a feature of Gitelman’s variant of Bartter’s syndrome. A new look at the hypomagnesemia associated with calcium pyrophosphate dihydrate crystal deposition disease. Rev Rhum Engl Ed 65: 571–574 [PubMed] [Google Scholar]
Rachow J.W., Ryan L.M. (1985) Adenosine triphosphate pyrophosphohydrolase and neutral inorganic pyrophosphatase in pathologic joint fluids. Elevated pyrophosphohydrolase in calcium pyrophosphate dihydrate crystal deposition disease. Arthritis Rheum 28: 1283–1288 [DOI] [PubMed] [Google Scholar]
Ratner M. (2008) IL-1 trap go-ahead. Nat Biotechnol 26: 485. [DOI] [PubMed] [Google Scholar]
Richards A.J., Hamilton E.B. (1974) Destructive arthropathy in chondrocalcinosis articularis. Ann Rheum Dis 33: 196–203 [DOI] [PMC free article] [PubMed] [Google Scholar]
Roane D.W., Harris M.D., Carpenter M.T., Finger D.R., Jarek M.J., Alloway J.A., et al. (1997) Prospective use of intramuscular triamcinolone acetonide in pseudogout. J Rheumatol 24: 1168–1170 [PubMed] [Google Scholar]
Rosenthal A.K. (2001) Pathogenesis of calcium pyrophosphate crystal deposition disease. Curr Rheumatol Rep 3: 17–23 [DOI] [PubMed] [Google Scholar]
Rosenthal A.K., Ryan L.M. (1994) Probenecid inhibits transforming growth factor-beta 1 induced pyrophosphate elaboration by chondrocytes. J Rheumatol 21: 896–900 [PubMed] [Google Scholar]
Rosenthal A., Ryan L.M., McCarty D.J. (2005) Calcium pyrophosphate deposition disease, pseudogout, and articular chondrocalcinosis. In: Koopman W.J., Moreland L.W. (eds), Arthritis and Allied Conditions. 15th ed. Philadelphia: Lippincott Wiliams & Wilkins, p. 273 [Google Scholar]
Ryan L.M., Wortmann R.L., Karas B., McCarty D.J., Jr (1984) Cartilage nucleoside triphosphate (NTP) pyrophosphohydrolase. I. Identification as an ecto-enzyme. Arthritis Rheum 27: 404–409 [DOI] [PubMed] [Google Scholar]
Settas L., Doherty M., Dieppe P. (1982) Localised chondrocalcinosis in unstable joints. Br Med J (Clin Res Ed) 285: 175–176 [DOI] [PMC free article] [PubMed] [Google Scholar]
Smilde T.J., Haverman J.F., Schipper P., Hermus A.R., van Liebergen F.J., Jansen J.L., et al. (1994) Familial hypokalemia/hypomagnesemia and chondrocalcinosis. J Rheumatol 21: 1515–1519 [PubMed] [Google Scholar]
So A., De Meulemeester M., Pikhlak A., Yucel A.E., Richard D., Murphy V., et al. (2010) Canakinumab for the treatment of acute flares in difficult-to-treat gouty arthritis: results of a multicenter, phase II, dose-ranging study. Arthritis Rheum 62: 3064–3076 [DOI] [PubMed] [Google Scholar]
Sun Y., Mauerhan D.R., Honeycutt P.R., Kneisl J.S., Norton H.J., Zinchenko N., et al. (2010) Calcium deposition in osteoarthritic meniscus and meniscal cell culture. Arthritis Res Ther 12: R56. [DOI] [PMC free article] [PubMed] [Google Scholar]
Wilkins E., Dieppe P., Maddison P., Evison G. (1983) Osteoarthritis and articular chondrocalcinosis in the elderly. Ann Rheum Dis 42: 280–284 [DOI] [PMC free article] [PubMed] [Google Scholar]
Williams C.J., Zhang Y., Timms A., Bonavita G., Caeiro F., Broxholme J., et al. (2002) Autosomal dominant familial calcium pyrophosphate dihydrate deposition disease is caused by mutation in the transmembrane protein ANKH. Am J Hum Genet 71: 985–991 [DOI] [PMC free article] [PubMed] [Google Scholar]
Zhang W., Doherty M., Pascual E., Barskova V., Guerne P.A., Jansen T.L., et al. (2011) EULAR recommendations for calcium pyrophosphate deposition. Part II: management. Ann Rheum Dis 70: 571–575 [DOI] [PubMed] [Google Scholar]
Articles from Therapeutic Advances in Musculoskeletal Disease are provided here courtesy of SAGE Publications
ACTIONS
View on publisher site
PDF (1.1 MB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
Introduction
Clinical manifestations
Epidemiology
Disease associations
Pathogenesis
Investigations
Treatment and management
Associated conditions
Anti-inflammatory medications
Anticrystal agents
Targeting the inflammasome
Conclusion
Footnotes
References
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
11764 | https://steemit.com/mathematics/@drifter1/mathematics-all-about-complex-numbers | Mathematics - All about Complex numbers — Steemit
×
Sign in
Sign up
Welcome
FAQ
Switch to Night Mode
Stolen Accounts Recovery
Change Account Password
Vote for Witnesses
Steem Proposals
Third-party exchanges:
Poloniex
Advertise
Jobs at Steemit
Developer Portal
Steem Bluepaper
SMT Whitepaper
Steem Whitepaper
Privacy Policy
Terms of Service
LoginSign up
Explore
All Posts
Communities
My Profile
My Wallet
Mathematics - All about Complex numbers
drifter1(67)in#mathematics • 8 years ago(edited)
Hello it's a me again drifter1! Today we continue with Mathematics to talk about everything that you have to know about Complex numbers. I will give you all the theory with examples of this topic of maths. This knowledge will be enough in anything that needs Complex numbers in general. I hope that this post will be useful!
For printing equations I will use the tool quicklatexwhich was recommended by @drdawud.
So, without further do, let's get started!
Getting into the need of Imaginary numbers
When solving equations in Mathematics we sometimes get in trouble, cause no real number solution(s) can be found.
This means that we have:
no natural (0, 1, 2, ...)
no integer(..., -2, -1, 0, 1, 2, ...)
no rational (ratio's of integers)
no real (numbers with decimals / digits after the decimal point) AND
no irrational (reals that are not rational) solution
For example think about:
To find the solution of such a 2nd-order polynomial we use the formula:
The subroot is negative and so we say that the equation has no solution. But, we are wrong!
The right thing to say is : "There is no solution in the real number set".
We know that in general the solutions depend on the subroot:
b^2 - 4ac > 0 => 2 solutions
b^2 - 4ac = 0 => 1 double-solution
b^2 - 4ac < 0 => no real solutions
Let's stay on that last line: "there are no real solutions".
We know that a polynomial equation of 2nd-order has 2 solutions.
For positive we have 2 solutions, for zero we have 1 double-solution and so the same solution twice, but either way they are 2.
Does this mean that there are 2 "hidden" solutions in the last case? Yes it does!
The Complex number set
But, where are those solutions at? Well, they are in an extension of the real number set: the complex number set.
Because, the problem of finding the solutions is in the negative subroot we define a number i, the so called imaginary unit or indeterminatethat will help us solve it.
i is a imaginary number with the following property:
Using it in the root of our previous example now gives us:
So, the solutionsare:
There it is! There are really 2 solutions for such equations that you could not solve before!
These solutions are not real numbers, but the problem was not finding real solutions, but finding 2 solutions that we knew existed, because of the polynomial equation's order that is 2!
Thinking about this formula and seeing the form of the solutions we can clearly see how an complex number is build up!
A complex number is of the form:
where:
z is the complex number notation
a and b are real numbers
i is the imaginary unity
Complex number properties
Some more things aboutz = a + bi:
a is called the real part and is also noted as Re(z)
b is called theimaginary part and is also noted as Im(z)
if b = 0 then z is of course a real number
if a = 0 then z is a "purely" imaginary number
The set of"purely" imaginary numbersis noted as I.
Because when setting b = 0 we get real numberswe say that:
the complex number set C of course containsthe real number set R, with R being a subset of C
all the propertiesof addition and multiplication of R apply to C too
if z is a complex number then: z+0 = z and z1 = z
Each complex number z can be written as: z = a + bi, where a and b are unique
The thing that C doesn't inherit from R is the arrangement and order relations of numbers.
We can't say that:
cause then -1 > 0, which can't be true!
Of course 2 complex numbers z = a + bi and w = c + di are equalif:
a = b
c = d
Of course this means that z = 0only if a + bi = 0 => a = b = 0
For this last equation we of course have to note that a AND b are BOTH real numbers!
Cartesian form/notation of Complex numbers
Each complex number is defined by the pair(a, b) = (Re(z), Im(z)).
Using the 2-dimensional Cartesian coordination system each complex number can so be represented by a point M(z) = M(a, b) on top of that complex plane, that contains a real(Re) and imaginary(Im) axis.
Because of that representation we can now do calculationson complex numbers like they where vectors.
So, if z = (OA) and w = (OB)
where z, w are complex numbers and OA, OB their corresponding vectors
Then we can say that:
kOA + λOB is a vector OP that corresponds to a new complex number v = kz + λw
We can clearly see that the addition of vectors is the same as the addition of the complex numbers directly!
Operations between Complex numbers
if z1 = a + bi and z2 = c + di are two complex numbers then:
If z = a + bi:
Everytime i is in the denominator, like in the divisions of above, we multiply the numerator and denominator with the conjugate of the denominator (more about that later on). For z1/z2 with (c - di) and for 1/z with (a - bi).
This should be a rule for you, cause it is needed everytime we have i in the denominator!
The additionin "vector-form" looks like this:
Of course because a complex number can be represented by vectorswe have that:
the vector z1 + z2 is the addition of the vectors z1 and z2
the vector z1 - z2 is the addition of the vectors z1 and -z2 (or the substraction with z2)
etc.
Exponentials of Complex numbers
Suppose z = a + bi is a complex number.
The powers z^k (with k being natural) are:
For the last one k-times.
Of course:
if z is not zero!
For the powers ofinteger exponents the same properties and rules apply as in real numbers.
Some propertiesof rational exponents can be defined, but there are also some exceptions.
This means that we will just remember that we can have integer exponents and use them like real numbers!
What about the powers of i?
Supposingn = 4q + r we have thatr = {0, 1, 2, 3}.
And so:
Which means that we end up with the cases:
For example:
Conjugate of Complex numbers and Properties
The complex numberz (or z with line on top) that has the same real part (a) but the opposite in sign imaginary part (b) as z is called the conjugateof z.
For exampleconjugate of z = 2 - 3i is z = 2 + 3i.
Thegeometric represenation of the conjugate is:
Geometrically we can see that the conjugate of z (z) and z are symmetrical to the x'x axis.
Which means that ifz = a + bi, which is the point M(z) = (a, b) then:
z = a - bi , which is the pointM(z) = (a, -b)
The following are some of the most important propertiesof the conjugate:
zz = (a + bi)(a - bi) = a^2 + b^2
(z1 + z2) = z1 + z2, which can be generalized to n numbers, cause we apply the conjugate for each term/number individually
(z1z2) = z1z2, which can again be generalized for n numbers.
(z^n) = (z)^n, for every natural n>=0 [so the exponent can be put out of the conjugate]
(z1/z2) = z1/z2, so the conjugate can be applied to the numerator and denominator independently
z + z = 2a = 2Re(z) AND z - z = 2bi = 2Im(z)i
From the last two properties we have that:
z is a real number if z = z, cause z - z = 0 and so there is no imaginary part.
z is a imaginary number if z = -z, cause then z + z = 0 and so there is no real part.
We use these exact properties to check if a complex number is real or imaginary!
The equation az^2 + bz + c = 0, with a!=0
So, to sum up, let's get back to the equation we started with!
We have the equation:
where a, b, c are real numbers.
If x is in the real number set then it will also be a real number and so we have the results that we had before which are:
b^2 - 4ac > 0 => 2 solutions
b^2 - 4ac = 0 => 1 double-solution
b^2 - 4ac < 0 => no real solutions
What if x is not a real number, but a complex number z?
Well, then:
b^2 - 4ac > 0 => 2 real solutions
b^2 - 4ac = 0 => 1 real double-solution
b^2 - 4ac < 0 => 2 complex solutions which are supplementally (one is the conjugate of the other)
And we get our final solution from:
Which will give us:
for each of the 3 cases.
Absolute value of Complex numbers and properties
Using the Cartesian form/notation for a complex number we can start talking about the absolute value of complex numbers.
The absolute value is defined as the distance of the origin point O(0, 0) and the point M(a, b) which represents the complex number in the complex plane.This means that the absolute value is also equal to the length of the vector OM.
In the Cartesian space this looks like this:
So, the absolute value can be found using the Pythagorean Theorem on the triangle to find the diagonal.
That way the absolute value equals:
If |z| = 0 then of course z = 0.
The following propertiesapply to the absolute value |z| of a complex number z:
|z| = |-z| = |z|, which means that the absolute value of z is equal to the absolute value of it's opposite and it's conjugate
|z| = zz [we already saw something similar before and this will of course be useful in calculations]
|z1z2...zn| = |z1||z2|...|zn|, which means that the absolute value of an product of complex numbers can be written as the product of the absolute values of each complex number individually [simplest case is: |z1z2| = |z1||z2|]
|z^n| = |z|^n for all naturals n >= 0, [the power can be found after finding the absolute value]
|z1/z2| = |z1|/|z2|, the same way as in multiplication the division's absolute value equals the quotient of the absolute values of the numerator and denominator individually.
Worth noting is the relation:
Which istrue for every real z, but not true for every complex number z
For example:
but,
and so:
But this makes it usefulin another way, cause:
If z is real then |z|^2 = z^2
If z is imaginary then |z|^2 = -z^2
The previous statement and the last 2 properties of the conjugate (as I mentioned before) are both useful when checking if a complex number is real or imaginary! So, to check this we use either one of these, depending on the data we have given to us.
Complex number argument and polar form
Using the Cartesian from again we can define another thing called the argument or phase of z.
The argumentof z or arg(z) is the angle φ of (OM) where O(0, 0) is the origin point and M(a, b) is the point that represents the complex number z = a + bi.
This looks like this:
A complex number z !=0 can of course have multiple arguments.
For two random arguments φ1, φ2 we have that:
φ2 - φ1 = 2kπ, with k being an integer and φ1, φ2 in radians
Because there are more then one arguments we use the principal argument in the interval [0, 2π)that defines the angle of (OM) to the x'x axis. That way:
Arg(z) = φ, where 0 <= φ < 2π
This one is of course unique.
So, if z != 0 is a complex number z = Re(z) + iIm(z) then:
if Arg(z) = 0 <=> Im(z) = 0 and Re(z) > 0
if Arg(z) = φ, where 0 < φ < π/2 <=> tanφ = Im(z)/Re(z) with Im(z) > 0 or Re(z) > 0
if Arg(z) = π/2 <=> Re(z) = 0 and Im(w) > 0
If Arg(z) = φ, where π/2 < φ < π <=> tanφ = Im(z)/Re(z) with Im(z) > 0 or Re(z) < 0
If Arg(z) = π <=> Im(z) = 0 and Re(z) < 0
Using those cases and with a simple diagram we can solve any case!
This means that we end up with the following very useful cases for z = a + bi:
If z is real and a > 0 then Arg(z) = 0
If a = 0 and b > 0 then Arg(z) = π/2
If z is real and a < 0 then Arg(z) = π
If a = 0 and b < 0 then Arg(z) = 3π/2
The argument of the conjugate of z is:
Arg(z) = 2π - φ
where Arg(z) = φ and φ in the interval (0, 2π)
Trigonometric/Polar Form of complex numbers
Because of the argument and the Cartesian form we can also talk about the trigonometric or polar form of a complex number.
So, if z = a + bi is a complex number z != 0, M(a, b) is the point in the complex plane that represents it and arg(z) = φ is it's argument then:
a = |z|cosφ and b = |z|sinφ
Because of that:
z = |z|(cosφ + isinφ)
All this is true only if |z| >0, which means that z != 0.
As mentioned previously the argument is not unique.
Suppose two complex numbers z1, z2 in polar form .
If z1 = |z1|(cos(φ1) + isin(φ1))and z2 = |z2|(cos(φ2) + isin(φ2)
Then we can say that:
z1 = z2 <=> |z1| = |z2| AND φ1 - φ2 = 2kπ, where k is an integer
This can be useful when checking if two complex numbers are equal!
We can again define propertiesfor operations that I will leave out for simplicity.
The polar form is useful when solving equationsof complex numbers that are in the forms:
z^n = 1
z^n = a, where a is complex
Polynomials of n-order/n-degree
and more...
I will not get into how we solve them...
Euler's formula
One very important formula is Euler's formula.
For any real number x we define:
Exercises and more information
I also wanted to do some exercises, but stepped on the following links that contain many many exercises that also contain things that we don't covered!
math-exercises
analyze math
And you can of course read more about Complex number in Khan's Academy:
Image sources:
And this is actually it and I hope that you learned something!
I also plan to make a post about trigonometry later on!
Bye!
#steemiteducation#steemstem#complex#numbers
8 years ago in#mathematicsbydrifter1(67)
$29.18
Past Payouts $29.18
- Author $22.26
- Curators $6.92
251 votes
+ hendrikdegrote
+ steemstem
+ zahnspange
+ knozaki2015
+ abigail-dantes
+ liberosist
+ justtryme90
+ steemiteducation
+ curie
+ romedog
+ b0y2k
+ meerkat
+ anyx
+ anwenbaumeister
+ muphy
+ aizensou
+ richman
+ kerriknox
+ joseph
+ yoshiko
and 231 more
Reply14
Sort: Trending
Trending
Votes
Age
[-]
muphy(61)8 years ago(edited)
Hi @Drifter1,
Thank you for this great lesson about complex numbers. Very well structured. I do not have time right now to read it fully till the end but I will!
Why am I so interested in your article? Because of the enthusiasm I felt when reading the first lines. Hey, maybe a life long block in my scientific life could be lifted here!
It hasn't.
Let me explain: Since high school I resisted i^2 = -1 So I just took it as a tool, but more I advanced in my studies up to the postdoc (Physics), realizing how complex numbers work so well in many domains, more I got confused.
I tried to understand it geometrically, like for instance two perpendicular lines will have the product of their slopes = to -1. As a complex number can be seen as a vector when drawn in a reference of two perpendicular axis, I tried to correlate this with that. Resulted in just further confusion.
For example, I though using the imaginary number i was just a practical way to write down coordinates, separating one coordinate from the other. So why don't we have 3D complex number hanging around, of the form z = a +bi +cj ? with j having special properties too...
So my question to you: can you convince me of the reality of i^2=-1. If I got that, that would really help me in quantum mechanics. A topic I love, but for which I can't deep dive in sometimes when equations involving complex numbers appear. Complex numbers prevent me getting a picture of what is really going on when reading the equations... Now the problem has become psychological... lol!
Off topic: I loved what I read up to now, would you be interested in joining me in my effort to promote education for all by making a video for the @openschool ? Maybe about complex numbers ;-)
$0.11
Past Payouts $0.11
- Author $0.11
- Curators $0.00
2 votes
+ muphy
+ drifter1
Reply
[-]
drifter1(67)8 years ago
Complex numbers are indeed a very interesting topic of mathematics.
They are used in many different science branches and are very useful.
The imaginary unit "i" is meant to be unreal and can't be real cause i^2 = -1 <=> i = root(-1) which can't be defined in the real number space, because the subroot is negative...
I guess that "3d complex numbers" could be interesting, but I can't think of an application right now.
Complex numbers are like vectors. They have a real and a imaginary part. It's not so complicated if someone thinks about it that way.
Lastly, I'm busy with my university and I already thought of making some videos, but there is so much uploaded to YouTube already that it isn't necessary I think..
$0.16
Past Payouts $0.16
- Author $0.12
- Curators $0.04
1 vote
+ muphy
Reply
[-]
muphy(61)8 years ago
Thank you for your kind answer.
Complex numbers are like vectors. They have a real and a imaginary part. It's not so complicated if someone thinks about it that way.
Yes, that is also how I see them, but what confuses me is why they are limited to 2D... not seeing complex numbers generalized to more dimensions (3,4 ,5 etc...) implies to me that it is just a lucky convenience, not a valid representation...
And I still do not know why i^2 = -1... (of course, I understand it mathematically as well as how to use it, but not conceptually). It is at the heart of the whole complex number tool box, but it sounds so 'invented'... For me, this very thought contradicts why complex numbers are so effective in Physics. If they are so effective as tools to describe the physical world, there must be an element of physical or geometrical truth in i^2=-1. Any clue?
Lastly, I'm busy with my university and I already thought of making some videos, but there is so much uploaded to YouTube already that it isn't necessary I think..
You didn't understand my request. This would not be any video, like there are thousands (with many good ones) on Youtube. This would be to directly teach kids (via Skype or Dlive) by getting invested into the @OpenSchool initiative. But of course, your studies should take precedent.
Be well
$0.00
1 vote
+ drifter1
Reply
[-]
rscalabrini(61)8 years ago
And now...complex analysis :D
$0.00
1 vote
+ drifter1
Reply
[-]
anevolvedmonkey(53)8 years ago
What's the point in studying complex number? They aren't even real! :)
Anywhoo, loved reading this! btw picture credit would look great under the picture itself.
$0.00
1 vote
+ drifter1
Reply
[-]
drifter1(67)8 years ago
Thank you.
I'm trying out different styles to see what looks better. But yes, I think you're right and crediting should be under the picture itself..
$0.00
Reply
[-]
dangtmaximun(41)8 years ago
excellent teacher information
$0.00
1 vote
+ drifter1
Reply
[-]
engrkomoo(43)8 years ago(edited)
wonderful post @drifter1. can't wait to see the post on trigonometry
$0.00
1 vote
+ drifter1
Reply
[-]
drifter1(67)8 years ago
Haha will look forward to it :P
$0.00
Reply
[-]
drdawud(42)8 years ago
This is a great post. I was planning to post on this, but unfortunately you beat me to the punch. Nevertheless, this is very well written. If it's ok with you, I want to build upon this and talk about the complex roots of unity, and their relation to the unit circle.
$0.00
1 vote
+ drifter1
Reply
[-]
drifter1(67)8 years ago
Thank you!
I would be happy to see a post about the unity circle from you!
And don't worry, I don't think that I will talk about more in complex numbers in a post specific only for that topic.
But, I may talk about more when they are needed in something else (like signal analysis), if it wasn't already been explained in this post!
$0.00
Reply
[-]
mcfarhat(70)8 years ago
Nice write up and summary, reminded me of my school years.
Thanks!
$0.00
1 vote
+ drifter1
Reply
[-]
dailypick(49)8 years ago
This post has been upvoted and picked by Daily Picked #12! Thank you for the cool and quality content. Keep going!
Don’t forget I’m not a robot. I explore, read, upvote and share manually :)
$0.00
1 vote
+ drifter1
Reply
[-]
steemstem-bot(57)8 years ago
| | | Guidelines | Project Update | | |
--- --- --- |
Being A SteemStem Member
$0.00
1 vote
+ drifter1
Reply
New to Steemit?
Welcome Guide
Announcements
Steemit Challenge Season 27 WK1
Learn More
Useful Links:
Newcomers Guide
Latest Updates from @dip.team
Coin Marketplace
STEEM 0.12
TRX 0.34
JST 0.032
BTC 111776.30
ETH 4114.74
USDT 1.00
SBD 0.78
↑ |
11765 | https://physics.stackexchange.com/questions/168915/phase-velocity-greater-then-the-speed-of-light | special relativity - Phase velocity greater then the speed of light? - Physics Stack Exchange
Join Physics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Physics helpchat
Physics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Phase velocity greater then the speed of light?
Ask Question
Asked 10 years, 6 months ago
Modified8 years, 6 months ago
Viewed 14k times
This question shows research effort; it is useful and clear
5
Save this question.
Show activity on this post.
It is my understanding the the phase velocity of a wave can be greater then the speed of light. So imagine we had a wave packet consisting of a single sinusoidal wave;
y=sin(ω t−k x)y=sin(ω t−k x)
Then from the definition of phase velocity, this wave will (if I am not mistaken) travel with the phase velocity, which as stated above can be greater then c c. But in this case information is been passed greater then the speed of light, is it not? If we start the wave packet of at a point A A say and let it propagate to another point B B it will travel to B B faster than the speed of light and pass of the information of its frequency and wavelength. Why is this not breaking special relativity?
special-relativity
waves
faster-than-light
phase-velocity
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Improve this question
Follow
Follow this question to receive notifications
edited Nov 18, 2015 at 1:06
Qmechanic♦
222k 52 52 gold badges 636 636 silver badges 2.6k 2.6k bronze badges
asked Mar 7, 2015 at 8:16
Quantum spaghettificationQuantum spaghettification
7,923 14 14 gold badges 77 77 silver badges 180 180 bronze badges
2
As for matter wave, phase velocity has no physical significance, because the motion of the wave group, not the motion of the individual waves that make up the group, corr. to the motion of the body, & group velocity is less than c c as it should be. So , the matter wave doesn't violate special relativity.user36790 –user36790 2015-03-07 10:31:15 +00:00 Commented Mar 7, 2015 at 10:31
Same is in the case for your concerned wave. +1 to Rennie's answer.user36790 –user36790 2015-03-07 10:34:39 +00:00 Commented Mar 7, 2015 at 10:34
Add a comment|
2 Answers 2
Sorted by: Reset to default
This answer is useful
6
Save this answer.
Show activity on this post.
The simple answer is that the wave packet travels at the group velocity not the phase velocity, and the group velocity is always less than or equal to c c.
You might argue that you aren't using a wave packet. For example you might argue that you are just turning the light on and waiting for it to get to the point B B. However any modulation of the wave intensity, including turning it on and off, will propagate at the group velocity.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Mar 7, 2015 at 8:24
John RennieJohn Rennie
369k 134 134 gold badges 792 792 silver badges 1.1k 1.1k bronze badges
5
Hi, but this wave consists of many 'smaller waves' travelling with the phase velocity, does this not therefore mean that the front of the wave packet will reach B B at a time faster then the speed of light.Quantum spaghettification –Quantum spaghettification 2015-03-07 08:28:20 +00:00 Commented Mar 7, 2015 at 8:28
@Joseph: I was just extending my answer when you commented. There's an animation of a wave packet here (and Googling will find many more). The top picture is a phase velocity greater than the group velocity. Note that the carrier wave passes through the packet but doesn't move faster than it.John Rennie –John Rennie 2015-03-07 08:31:35 +00:00 Commented Mar 7, 2015 at 8:31
On the top digram in the animation you have shown the wave-packet spreads out, this is due to dispersion. So this means that some waves are travelling faster then the group velocity. My point is the wave-packet is made by the superposition of lots of sinusoidal waves, some of which move faster then then group velocity and others that move slower. The once that move faster will reach then end (due to dispersion) in a shorter time then the slower waves and the wave packet as a whole.Quantum spaghettification –Quantum spaghettification 2015-03-07 08:36:55 +00:00 Commented Mar 7, 2015 at 8:36
@Joseph: yes, the packet does spread and therefore the leading edge moves faster than the average packet velocity. However the leading edge does not move at the phase velocity, it moves at a velocity ≤c≤c.John Rennie –John Rennie 2015-03-07 08:57:42 +00:00 Commented Mar 7, 2015 at 8:57
1 I thought only group velocity could exceed c. After reading your reply I was confused so I asked a question on this. Any chance you could help me clarify this? physics.stackexchange.com/questions/541321/…Alex P. –Alex P. 2020-04-03 22:16:15 +00:00 Commented Apr 3, 2020 at 22:16
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
You cant not modulate information on a single frequency wave so there exists no packet here. Turning on and off will add other frequencies to the signal and the group of frequencies will travel with the group velocity which is less than the speed of light.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Mar 22, 2017 at 23:28
Ali PM VafaAli PM Vafa
11 1 1 bronze badge
Add a comment|
Your Answer
Thanks for contributing an answer to Physics Stack Exchange!
Please be sure to answer the question. Provide details and share your research!
But avoid …
Asking for help, clarification, or responding to other answers.
Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Email and Password
Submit
Post as a guest
Name
Email
Required, but never shown
Post Your Answer Discard
By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
special-relativity
waves
faster-than-light
phase-velocity
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Linked
5What is the connection between the refractive index of a material and its transparency?
4Does phase velocity or group velocity change when light enters a new material? Contradictory sources
2Confusion in phase and group velocity
Related
7Controllable faster-than-light phase velocity
36In superluminal phase velocities, what is it that is traveling faster than light?
7Information relationship to Special Relativity
4Why group speed can contain information but phase speed can't?
1Why is group velocity greater than phase velocity in glass?
Hot Network Questions
How do you emphasize the verb "to be" with do/does?
With with auto-generated local variables
Bypassing C64's PETSCII to screen code mapping
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
Is encrypting the login keyring necessary if you have full disk encryption?
alignment in a table with custom separator
How to start explorer with C: drive selected and shown in folder list?
Does "An Annotated Asimov Biography" exist?
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
Why include unadjusted estimates in a study when reporting adjusted estimates?
Transforming wavefunction from energy basis to annihilation operator basis for quantum harmonic oscillator
Proof of every Highly Abundant Number greater than 3 is Even
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
In the U.S., can patients receive treatment at a hospital without being logged?
Implications of using a stream cipher as KDF
Vampires defend Earth from Aliens
On being a Maître de conférence (France): Importance of Postdoc
Calculating the node voltage
Alternatives to Test-Driven Grading in an LLM world
An odd question
My dissertation is wrong, but I already defended. How to remedy?
Storing a session token in localstorage
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Physics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
11766 | https://www.physicsclassroom.com/calcpad/momentum/problems | Momentum and Collisions Problem Sets
My Account
×
Custom Search
Sort by:
Relevance
Relevance
Date
TPC and eLearning
What's NEW at TPC?
Task Tracker
Edit Profile Settings
Classes
Users Voice
Tasks and Classes
Webinars and Trainings
Subscriptions
Subscription
Subscription Locator
Products and Plans
Request a Demo
Manual Order Form
Subscription Selection
Ad Free Account
My Cart
Read Watch Interact
Practice Review Test
Teacher-Tools
TPC and eLearningWhat's NEW at TPC?Task Tracker
Edit Profile Settings
Classes
Users Voice
Tasks and Classes
Webinars and Trainings
Subscriptions
Subscription Locator
Products and Plans
Request a Demo
Manual Order Form
Subscription Selection
Ad Free Account
My Cart
Read Watch InteractChemistry Tutorial
Measurement and Calculations
Matter
Elements, Atoms, and Ions
Compounds,Names, and Formulae
The Modern Atomic Model
Chemical Bonding
The Mole and its Applications
Chemical Reactions
Stoichiometry
Gases and Gas Laws
Solids, Liquids, and Intermolecular Forces
Thermochemistry
Solutions
Kinetics and Equilibrium
Acids and Bases
Solution Equilibria
Reference
Physics Tutorial
1-D Kinematics
Newton's Laws
Vectors - Motion and Forces in Two Dimensions
Momentum and Its Conservation
Work and Energy
Circular Motion and Satellite Motion
Thermal Physics
Static Electricity
Electric Circuits
Magnetic Fields and Electromagnetism
Vibrations and Waves
Sound Waves and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
Video Tutorial
Kinematics
Newton's Laws
Vectors and Projectiles
Forces in Two Dimensions
Momentum and Collisions
Work, Energy, and Power
Circular Motion and Gravitation
Static Electricity
Electric Circuits
Vibrations and Waves
Sound Waves
Light and Color
Reflection and Mirrors
Refraction and Lenses
Multimedia Studios
1-Dimensional Kinematics
Newton's Laws
Vectors and Projectiles
Momentum and Collisions
Work and Energy
Circular, Satellite, and Rotational Motion
Einstein's Theory of Special Relativity
Static Electricity
Waves, Sound and Light
Ray Optics
QuickTime Movies
Interactives
About the Physics Interactives
Task Tracker
Usage Policy
Kinematics
Newtons Laws
Vectors and Projectiles
Forces in 2D
Momentum and Collisions
Work and Energy
Circular Motion and Gravitation
Balance and Rotation
Fluids
Static Electricity
Electric Circuits
Electromagnetism
Waves and Sound
Light and Color
Reflection and Mirrors
Refraction and Lenses
Atomic Physics
Chemistry
Practice Review TestConcept Builders
About the Concept Builders
Version 2
Pricing For Schools
Directions for Version 2
Measurement and Units
Relationships and Graphs
Kinematics
Newton's Laws
Vectors and Projectiles
Forces in 2-Dimensions
Momentum and Collisions
Work and Energy
Circular and Satellite Motion
Rotation and Balance
Fluids
Static Electricity
Electric Circuits
Vibrational Motion
Waves and Sound
Light and Color
Reflection and Refraction
Chemistry
Minds On
About MOPs
Features
Teacher Accounts
Pricing For Schools
Task Tracker Directions
Kinematic Concepts
Kinematic Graphing
Newton's Laws
Vectors and Projectiles
Forces in 2-Dimensions
Momentum and Collisions
Work and Energy
Circular and Satellite Motion
Static Electricity
Electric Circuits
Wave Motion
Sound and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
Calc Pad
About CalcPad
Teacher Accounts
Pricing For Schools
Directions
1D Kinematics
Newton's Laws
Vectors and Projectiles
Vectors and Forces in 2D
Momentum and Collisions
Work, Energy, and Power
Circular Motion and Gravitation
Simple Harmonic Motion
Rotational Kinematics
Rotation and Torque
Rotational Dynamics
Fluids
Static Electricity
Electric Fields, Potential, and Capacitance
Electric Fields, Potential, and Capacitance with Calculus
Electric Circuits
Transient RC Circuits
Electromagnetism
Vibrations and Waves
Sound Waves
Light Waves
Reflection and Mirrors
Refraction and Lenses
Units and Measurement
Mole Conversions
Stoichiometry
Molarity and Solutions
Gas Laws
Thermal Chemistry
Acids and Bases
Kinetics and Equilibrium
Solution Equilibria
Oxidation-Reduction
Nuclear Chemistry
Science Reasoning
Task Tracker
NGSS Alignments
1D-Kinematics
Newton's Laws
Projectiles
Momentum
Energy
Circular Motion
Static Electricity
Circuits
Magnetism and Electromagnetism
Thermal Physics
Waves
Sound Waves
Light Waves
Reflection and Mirrors
Refraction and Lenses
Chemistry
The Review Session
1-D Kinematics
Newton's Laws of Motion
Vectors and Projectiles
Forces in Two Dimensions
Momentum and Collisions
Work and Energy Packet
Circular Motion and Gravitation
Static Electricity Review
Electric Circuits
Waves
Sound and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
ACT Test Center
About the ACT
ACT Preparation
ACT Tips
For Teachers
Other Resources
Physics Help
Graphing Practice
Recognizing Forces
Vector Direction
Vector Addition
Teacher-ToolsCurriculum Corner
Solutions Guide
Solutions Guide Digital Download
Usage Policy
Solution Guide Corrections
Motion in One Dimension
Newton's Laws
Vectors and Projectiles
Forces in Two Dimensions
Momentum and Collisions
Work, Energy and Power
Circular Motion and Gravitation
Static Electricity
Electric Circuits
Wave Basics
Sound and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
Chemistry of Matter
Measurement and the Metric System
Early Atomic Models
Names and Formulas
The Modern Model of the Atom
Chemical Bonding
The Mole and Its Applications
Chemical Reactions
Stoichiometry
Gases and Gas Laws
Solids, Liquids, and Intermolecular Forces
Thermal Chemistry
Solutions
Kinetics and Equilibrium
Lesson Plans
About
Algebra Based On-Level Physics
Honors Physics
Conceptual Physics
Task Tracker
Other Tools
Teacher Pres'n Pack
Contents
Frequently Asked Questions
Purchasing the Download
The Laboratory
About
Teacher Guide
Using Lab Notebooks
Share
CP Course Pack
Contents
Our Flavor of Conceptual Physics
Frequently Asked Questions
Purchasing the Download
Question Bank
Contents
Purchasing the Digital Download
Teacher Toolkits
About the Toolkits
Position-Velocity-Acceleration
Position-Time Graphs
Velocity-Time Graphs
Free Fall
Newton's First Law
Newton's Second Law
Newton's Third Law
Terminal Velocity
Vectors
Projectile Motion
Forces in 2 Dimensions
Impulse and Momentum Change
Momentum Conservation
Work-Energy Fundamentals
Work-Energy Relationship
Circular Motion
Roller Coaster Physics
Universal Gravitation
Satellite Motion
Charge and Charging
Coulombs Law
Electric Fields
Circuit Concepts
Series Circuits
Parallel Circuits
Vibrational Motion
Describing-Waves
Wave Behavior Toolkit
Standing Wave Patterns
Sound Waves
Resonating Air Columns
Wave Model of Light
Color
Plane Mirrors
Curved Mirrors
Snells Law
Total Internal Reflection
Lenses
The Photo Gallery
1-D Kinematics
Newton's Laws
Vectors - Motion and Forces in Two Dimensions
Momentum and Its Conservation
Work, Energy, and Power
Circular Motion and Satellite Motion
Thermal Physics
Static Electricity
Current Electricity
Waves
Sound Waves and Music
Light Waves and Color
Reflection and Ray Model of Light
Refraction and Ray Model of Light
NGSS Corner
About the NGSS Corner
NGSS Search
Force and Motion DCIs - High School
Energy DCIs - High School
Wave Applications DCIs - High School
Force and Motion PEs - High School
Energy PEs - High School
Wave Applications PEs - High School
Crosscutting Concepts
The Practices
Physics Topics
NGSS Corner: Activity List
NGSS Corner: Infographics
TPC and eLearning
What's NEW at TPC?
Task Tracker
Edit Profile Settings
Classes
Users Voice
Tasks and Classes
Webinars and Trainings
Subscriptions
Subscription Locator
Products and Plans
Request a Demo
Manual Order Form
Subscription Selection
Ad Free Account
My Cart
Read Watch Interact
Chemistry Tutorial
Measurement and Calculations
Matter
Elements, Atoms, and Ions
Compounds,Names, and Formulae
The Modern Atomic Model
Chemical Bonding
The Mole and its Applications
Chemical Reactions
Stoichiometry
Gases and Gas Laws
Solids, Liquids, and Intermolecular Forces
Thermochemistry
Solutions
Kinetics and Equilibrium
Acids and Bases
Solution Equilibria
Reference
Physics Tutorial
1-D Kinematics
Newton's Laws
Vectors - Motion and Forces in Two Dimensions
Momentum and Its Conservation
Work and Energy
Circular Motion and Satellite Motion
Thermal Physics
Static Electricity
Electric Circuits
Magnetic Fields and Electromagnetism
Vibrations and Waves
Sound Waves and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
Video Tutorial
Kinematics
Newton's Laws
Vectors and Projectiles
Forces in Two Dimensions
Momentum and Collisions
Work, Energy, and Power
Circular Motion and Gravitation
Static Electricity
Electric Circuits
Vibrations and Waves
Sound Waves
Light and Color
Reflection and Mirrors
Refraction and Lenses
Multimedia Studios
1-Dimensional Kinematics
Newton's Laws
Vectors and Projectiles
Momentum and Collisions
Work and Energy
Circular, Satellite, and Rotational Motion
Einstein's Theory of Special Relativity
Static Electricity
Waves, Sound and Light
Ray Optics
QuickTime Movies
Interactives
About the Physics Interactives
Task Tracker
Usage Policy
Kinematics
Newtons Laws
Vectors and Projectiles
Forces in 2D
Momentum and Collisions
Work and Energy
Circular Motion and Gravitation
Balance and Rotation
Fluids
Static Electricity
Electric Circuits
Electromagnetism
Waves and Sound
Light and Color
Reflection and Mirrors
Refraction and Lenses
Atomic Physics
Chemistry
Practice Review Test
Concept Builders
About the Concept Builders
Version 2
Pricing For Schools
Directions for Version 2
Measurement and Units
Relationships and Graphs
Kinematics
Newton's Laws
Vectors and Projectiles
Forces in 2-Dimensions
Momentum and Collisions
Work and Energy
Circular and Satellite Motion
Rotation and Balance
Fluids
Static Electricity
Electric Circuits
Vibrational Motion
Waves and Sound
Light and Color
Reflection and Refraction
Chemistry
Minds On
About MOPs
Features
Teacher Accounts
Pricing For Schools
Task Tracker Directions
Kinematic Concepts
Kinematic Graphing
Newton's Laws
Vectors and Projectiles
Forces in 2-Dimensions
Momentum and Collisions
Work and Energy
Circular and Satellite Motion
Static Electricity
Electric Circuits
Wave Motion
Sound and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
Calc Pad
About CalcPad
Teacher Accounts
Pricing For Schools
Directions
1D Kinematics
Newton's Laws
Vectors and Projectiles
Vectors and Forces in 2D
Momentum and Collisions
Work, Energy, and Power
Circular Motion and Gravitation
Simple Harmonic Motion
Rotational Kinematics
Rotation and Torque
Rotational Dynamics
Fluids
Static Electricity
Electric Fields, Potential, and Capacitance
Electric Fields, Potential, and Capacitance with Calculus
Electric Circuits
Transient RC Circuits
Electromagnetism
Vibrations and Waves
Sound Waves
Light Waves
Reflection and Mirrors
Refraction and Lenses
Units and Measurement
Mole Conversions
Stoichiometry
Molarity and Solutions
Gas Laws
Thermal Chemistry
Acids and Bases
Kinetics and Equilibrium
Solution Equilibria
Oxidation-Reduction
Nuclear Chemistry
Science Reasoning
Task Tracker
NGSS Alignments
1D-Kinematics
Newton's Laws
Projectiles
Momentum
Energy
Circular Motion
Static Electricity
Circuits
Magnetism and Electromagnetism
Thermal Physics
Waves
Sound Waves
Light Waves
Reflection and Mirrors
Refraction and Lenses
Chemistry
The Review Session
1-D Kinematics
Newton's Laws of Motion
Vectors and Projectiles
Forces in Two Dimensions
Momentum and Collisions
Work and Energy Packet
Circular Motion and Gravitation
Static Electricity Review
Electric Circuits
Waves
Sound and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
ACT Test Center
About the ACT
ACT Preparation
ACT Tips
For Teachers
Other Resources
Physics Help
Graphing Practice
Recognizing Forces
Vector Direction
Vector Addition
Teacher-Tools
Curriculum Corner
Solutions Guide
Solutions Guide Digital Download
Usage Policy
Solution Guide Corrections
Motion in One Dimension
Newton's Laws
Vectors and Projectiles
Forces in Two Dimensions
Momentum and Collisions
Work, Energy and Power
Circular Motion and Gravitation
Static Electricity
Electric Circuits
Wave Basics
Sound and Music
Light and Color
Reflection and Mirrors
Refraction and Lenses
Chemistry of Matter
Measurement and the Metric System
Early Atomic Models
Names and Formulas
The Modern Model of the Atom
Chemical Bonding
The Mole and Its Applications
Chemical Reactions
Stoichiometry
Gases and Gas Laws
Solids, Liquids, and Intermolecular Forces
Thermal Chemistry
Solutions
Kinetics and Equilibrium
Lesson Plans
About
Algebra Based On-Level Physics
Honors Physics
Conceptual Physics
Task Tracker
Other Tools
Teacher Pres'n Pack
Contents
Frequently Asked Questions
Purchasing the Download
The Laboratory
About
Teacher Guide
Using Lab Notebooks
Share
CP Course Pack
Contents
Our Flavor of Conceptual Physics
Frequently Asked Questions
Purchasing the Download
Question Bank
Contents
Purchasing the Digital Download
Teacher Toolkits
About the Toolkits
Position-Velocity-Acceleration
Position-Time Graphs
Velocity-Time Graphs
Free Fall
Newton's First Law
Newton's Second Law
Newton's Third Law
Terminal Velocity
Vectors
Projectile Motion
Forces in 2 Dimensions
Impulse and Momentum Change
Momentum Conservation
Work-Energy Fundamentals
Work-Energy Relationship
Circular Motion
Roller Coaster Physics
Universal Gravitation
Satellite Motion
Charge and Charging
Coulombs Law
Electric Fields
Circuit Concepts
Series Circuits
Parallel Circuits
Vibrational Motion
Describing-Waves
Wave Behavior Toolkit
Standing Wave Patterns
Sound Waves
Resonating Air Columns
Wave Model of Light
Color
Plane Mirrors
Curved Mirrors
Snells Law
Total Internal Reflection
Lenses
The Photo Gallery
1-D Kinematics
Newton's Laws
Vectors - Motion and Forces in Two Dimensions
Momentum and Its Conservation
Work, Energy, and Power
Circular Motion and Satellite Motion
Thermal Physics
Static Electricity
Current Electricity
Waves
Sound Waves and Music
Light Waves and Color
Reflection and Ray Model of Light
Refraction and Ray Model of Light
NGSS Corner
About the NGSS Corner
NGSS Search
Force and Motion DCIs - High School
Energy DCIs - High School
Wave Applications DCIs - High School
Force and Motion PEs - High School
Energy PEs - High School
Wave Applications PEs - High School
Crosscutting Concepts
The Practices
Physics Topics
NGSS Corner: Activity List
NGSS Corner: Infographics
The Physics Classroom » Calc Pad » Momentum and Collisions
Mechanics: Momentum and Collisions
Calculator Pad, Version 2
You are viewing the Legacy Version of The Calculator Pad. We have recently revised and improved The Calculator Pad. Version 2 is now LIVE! We have more than tripled the number of problems, broken each unit into several smaller, single-topic problem sets, and utilized a random number generator to provide numerical information for each problem. Student answers are automatically evaluated and feedback is instant. And we've maintained the same commitment to providing help via links to existing resources. While the FREE version does all the above, teachers with a Task Tracker subscription can take things a step further. They can modify our pre-made problem sets, write their own problems with our easy-to-use Problem Builder, and use the Calculator Pad to design their own program that expresses their emphasis on the use of mathematics in Physics.
Return to the Main Page to link into Version 2. Learn more About Version 2. Or visit the Store to make a Task Tracker purchase.
Momentum and Collisions: Problem Set
Problem 1:
Determine the momentum of …
a. … an electron (m= 9.1 x10-31 kg) moving at 2.18 x 10 6 m/s (as if it were in a Bohr orbit in the H atom).
b. … a 0.45 Caliber bullet (m = 0.162 kg) leaving the muzzle of a gun at 860 m/s.
c. … a 110-kg professional fullback running across the line at 9.2 m/s.
d. … a 360,000-kg passenger plane taxiing down a runway at 1.5 m/s
Audio Guided Solution
Show Answera. 2.0 x 10-24 kg•m/s
b. 140 kg•m/s (rounded from 139 kg•m/s)
c. 1.0 x 10 3 kg•m/s
d. 5.4 x 10 5 kg•m/s
Problem 2:
A bicycle has a momentum of 24 kg•m/s. What momentum would the bicycle have if it had …
a. … twice the mass and was moving at the same speed?
b. … the same mass and was moving with twice the speed?
c. … one-half the mass and was moving with twice the speed?
d. … the same mass and was moving with one-half the speed?
e. … three times the mass and was moving with one-half the speed?
f. … three times the mass and was moving with twice the speed?
Audio Guided Solution
Show Answera. 48 kg•m/s
b. 48 kg•m/s
c. 24 kg•m/s
d. 12 kg•m/s
e. 36 kg•m/s
f. 144 kg•m/s
Problem 3:
According to the Guinness Book of World Records, the fastest recorded baseball pitch was delivered by Nolan Ryan in 1974. The pitch was clocked at 100.9 mi/hr (45.0 m/s). Determine the impulse required to give a 0.145-kg baseball such a momentum.
Audio Guided Solution
Show Answer6.52 N•s (or 6.52 kg•m/s)
Problem 4:
Jerome plays middle linebacker for South's varsity football team. In a game against cross-town rival North, he delivered a hit to North's 82-kg running back, changing his eastward velocity of 5.6 m/s into a westward velocity of 2.5 m/s.
a. Determine the initial momentum of the running back.
b. Determine the final momentum of the running back.
c. Determine the momentum change of the running back.
d. Determine the impulse delivered to the running back.
Audio Guided Solution
Show Answera. 460 kg•m/s, east (rounded from 459 kg•m/s)
b. 210 kg•m/s, west (rounded from 205 kg•m/s)
c. 660 kg•m/s, west (rounded from 664 kg•m/s)
d. 660 N•s, west
Problem 5:
Kara Less was applying her makeup when she drove into South's busy parking lot last Friday morning. Unaware that Lisa Ford was stopped in her lane 30 feet ahead, Kara rear-ended Lisa's rented Taurus. Kara's 1300-kg car was moving at 11 m/s and stopped in 0.14 seconds.
a. Determine the momentum change of Kara's car.
b. Determine the impulse experienced by Kara's car.
c. Determine the magnitude of the force experienced by Kara's car.
Audio Guided Solution
Show Answera. -1.4 x 10 4 kg•m/s
b. -1.4 x 10 4 N•s
c. 1.0 x 10 5 N
Problem 6:
An interesting story is often told of baseball star Johnny Bench when he was a rookie catcher in 1968. During a Spring Training game, he kept signaling to star pitcher Jim Maloney to throw a curve ball. Maloney continuously shook off Bench's signal, opting to throw fastballs instead. The rookie catcher walked to the mound and told the veteran Maloney that his fastball wasn't fast enough and that he should throw some curve balls. Bench again signaled for a curve. Maloney shook of the signal and threw a fastball. Before the ball reached the plate, Bench took off his glove; he then caught the pitch barehanded.
a. Determine the impulse required to stop a 0.145-kg baseball moving at 35.7 m/s (80.0 mi/hr).
b. If this impulse is delivered to the ball in 0.020 seconds, then what is magnitude of the force acting between the bare hand and the ball?
Audio Guided Solution
Show Answera. -5.18 N•s
b. 260 N (rounded from 259 N)
Problem 7:
While playing basketball in PE class, Logan lost his balance after making a lay-up and colliding with the padded wall behind the basket. His 74-kg body decelerated from 7.6 m/s to 0 m/s in 0.16 seconds.
a. Determine the force acting upon Logan's body.
b. If Logan had hit the concrete wall moving at the same speed, his momentum would have been reduced to zero in 0.0080 seconds. Determine what the force on his body would have been for such an abrupt collision.
Audio Guided Solution
Show Answera. 3.5 x 10 3 N
b. 7.0 x 10 4 N
Problem 8:
NASA's Langley Research Center has been experimenting with the use of air bags to soften the landings of crew exploration vehicles (CEV) on land. What stopping time will be required in order to safely stop a 7250 kg CEV moving at 7.65 m/s with an average force of 426000 N (an average force of 6 Gs)?
Audio Guided Solution
Show Answer0.130 seconds
Problem 9:
In a study conducted by a University of Illinois researcher, the football team at Unity High School in Tolono, IL was equipped for an entire season with helmets containing accelerometers. Information about every impact in practice and in games was sent to a computer present on the sidelines. The study found that the average force on a top of the head impact was 1770 N and endured for 7.78 milliseconds. Using a head mass of 5.20 kg and presuming the head to be a free body, determine the velocity change experienced in such an impact.
Audio Guided Solution
Show Answer2.65 m/s
Problem 10:
Cassie has just finished her session on the trampoline during PE. As she prepares to exit the trampoline, her vertical momentum is reduced by a series of three resistive impulses with the bounce mat. Just prior to this series of impulses, her 48.5-kg body is moving downward at 8.20 m/s. On the first impulse, Cassie experiences an average upward force of 230 N for 0.65 seconds. The second impulse of 112 N•s lasts for 0.41 seconds. The last impulse involves an average upward force of 116 N which cases a 84 kg•m/s momentum change. What vertical velocity does Cassie have after these three impulses?
Audio Guided Solution
Show Answer1.08 m/s
Problem 11:
Aaron Agin nodded off while driving home from play practice this past Sunday evening. His 1500-kg car hit a series of guardrails while moving at 19.8 m/s. The first guard rail delivered a resistive impulse of 5700 N•s. The second guard rail pushed against his car with a force of 79000 N for 0.12 seconds. The third guard rail collision lowered the car's velocity by 3.2 m/s. Determine the final velocity of the car.
Audio Guided Solution
Show Answer6.5 m/s
Discover more
Science
Physics-Themed Art Prints
Physics E-books
Science-Themed Gifts
Physics-Themed Games
Physics-Themed Apparel
Teacher Training Workshops
Teacher resources for physical science
Physics-Themed Home Decor
Affordable learning platform
Science Camps
Educational Videos
Physics-Themed Puzzles
STEM Toys
Science-Themed Movies
Science-Themed Music
Interactive Simulations
Science-Themed Books
scientific
Science Kits
Homework Helper App
Physics tutorials
Science Subscription Boxes
Interactive learning modules for students
Physical science eLearning courses
Virtual Labs
Online physics classroom
Interactive teaching tools
Online Courses
Science educational resources
Problem 12:
Mr. H ignites the enthusiasm of the class with a home-made cannon demonstration. The 1.27-kg cannon is loaded with a 54-gram tennis ball and placed on the floor. Mr. H adds the fuel, waits for its vapors to fill the reaction chamber and then brings a match nearby. The explosion stuns the crowd and propels the ball forward. A photogate measurement determines that the cannon recoiled backwards with a speed of 7.8 m/s. Determine the speed of the ball.
Audio Guided Solution
Show Answer180 m/s
Problem 13:
An 82-kg male and a 48-kg female pair figure skating team are gliding across the ice at 7.4 m/s, preparing for a throw jump maneuver. The male skater tosses the female skater forward with a speed of 8.6 m/s. Determine the speed of the male skater immediately after the throw.
Audio Guided Solution
Show Answer6.7 m/s
Problem 14:
A candy-filled piñata is hung from a tree for Matthew's birthday. During an unsuccessful attempt to break the 4.4-kg piñata, Hayden cracks it with a 0.54-kg stick moving at 4.8 m/s. The stick stops and the piñata undergoes a gentle swinging motion. Determine the swing speed of the piñata immediately after being cracked by the stick.
Audio Guided Solution
Show Answer0.59 m/s
Problem 15:
During an in-class demonstration of momentum change and impulse, Mr. H asks Jerome (102 kg) and Michael (98 kg) to sit on a large 14-kg skate cart. Mr. H asks Suzie (44 kg) to sit on a second 14-kg skate cart. The two carts are placed on low friction boards in the hallway. Jerome pushes off of Suzie's cart. Measurements are made to determine that Suzie's cart acquired a post-impulse speed of 9.6 m/s. Determine the expected recoil speed of Jerome and Michael's cart.
Audio Guided Solution
Show Answer2.6 m/s
Problem 16:
A 70.9-kg boy and a 43.2-kg girl, both wearing skates face each other at rest on a skating rink. The boy pushes the girl, sending her eastward with a speed of 4.64 m/s. Neglecting friction, determine the subsequent velocity of the boy.
Audio Guided Solution
Show Answer2.83 m/s, West
Problem 17:
To Mr. H's disgust, a 450-g black crow is raiding the recently-filled bird feeder. As Mr. H runs out the back door with his broom in an effort to scare the crow away, it pushes off the 670-gram feeder with a takeoff speed of 1.5 m/s. Determine the speed at which the feeder initially recoils backwards.
Audio Guided Solution
Show Answer1.0 m/s
Problem 18:
Jaclyn plays singles for South's varsity tennis team. During the match against North, Jaclyn won the sudden death tiebreaker point with a cross-court passing shot. The 57.5-gram ball hit her racket with a northward velocity of 26.7 m/s. Upon impact with her 331-gram racket, the ball rebounded in the exact opposite direction (and along the same general trajectory) with a speed of 29.5 m/s.
a. Determine the pre-collision momentum of the ball.
b. Determine the post-collision momentum of the ball.
c. Determine the momentum change of the ball.
d. Determine the velocity change of the racket.
Audio Guided Solution
Show Answera. 1.54 x 10 3 g•m/s, North (1.54 kg•m/s, North)
b. 1.70 x 10 3 g•m/s, South (1.70 kg•m/s, South)
c. 3.23 x 10 3 g•m/s, South (3.23 kg•m/s, South)
d. 9.76 m/s, North
Problem 19:
Anna Litical and Noah Formula are doing The Cart and the Brick Lab. They drop a brick on a 2.6 kg cart moving at 28.2 cm/s. After the collision, the dropped brick and cart are moving together with a velocity of 15.7 cm/s. Determine the mass of the dropped brick.
Audio Guided Solution
Show Answer2.1 kg
Problem 20:
A Roller Derby exhibition recently came to town. They packed the gym for two consecutive weekend nights at South's field house. On Saturday evening, the 68-kg Anna Mosity was moving at 17 m/s when she collided with 76-kg Sandra Day O'Klobber who was moving forward at 12 m/s and directly in Anna's path. Anna jumped onto Sandra's back and the two continued moving together at the same speed. Determine their speed immediately after the collision.
Audio Guided Solution
Show Answer14 m/s
Problem 21:
Ima Rilla Saari rushes to her car in order to hurry home and get dressed for work. Failing to realize the dangers of driving under slick and icy conditions, she collides her 940-kg Mazda Miata into the rear of a 2460-kg pick-up truck which was at rest at the light on Lake Avenue. Ima's pre-collision speed was 12.5 m/s. Determine the post-collision speed of the two entangled cars as they slide across the ice.
Audio Guided Solution
Show Answer3.5 m/s
Problem 22:
In a goal line stand against New Greer Academy, South's linebackers Jerome (m=102 kg) and Michael (m=98 kg) meet the 84-kg halfback moving southward through the air at 6.4 m/s. Upon contact, Jerome and Michael are both moving at 3.6 m/s in the exact opposite direction. Determine the post-collision speed and direction of the collection of three players. Assume they move together after the collision.
Audio Guided Solution
Show Answer0.64 m/s, North
Problem 23:
Hayden (m=24.3-kg) is gliding along the sidewalk on his skateboard at 8.6 ft/s. As he travels under a low-hanging branch of the tree where Matthew is sitting, he grabs the 4.5-kg bag of soccer balls from Matthew's hands. Determine the speed of Hayden immediately after grabbing the bag of soccer balls.
Audio Guided Solution
Show Answer7.3 ft/s
Problem 24:
Rex (m=86 kg) and Tex (92 kg) board the bumper cars at the local carnival. Rex is moving at a full speed of 2.05 m/s when he rear-ends Tex who is at rest in his path. Tex and his 125-kg car lunge forward at 1.40 m/s. Determine the post-collision speed of Rex and his 125-kg car.
Audio Guided Solution
Show Answer0.61 m/s
Problem 25:
Abbey and Mia are in the basement playing pool. On Abbey's recent shot, the cue ball was moving east at 82 cm/s when it struck the slower 5-ball moving in the same direction at 24 cm/s. The 5-ball immediately speeds up to 52 cm/s. Determine the post-collision speed of the cue ball.
Audio Guided Solution
Show Answer54 cm/s
Problem 26:
Polly Ester and Ray Ahn are doing the Elastic Collision lab on a low-friction track. Cart A has a mass of 1.00 kg and is moving rightward at 27.6 cm/s prior to the collision with Cart B. Cart B has a mass of 0.50 kg and is moving leftward with a speed of 42.9 cm/s. After the magnetic repulsion of the two carts, Cart A is moving leftward at 10.1 cm/s. Determine the post-collision speed and direction of cart B.
Audio Guided Solution
Show Answer32.5 cm/s, Right
Problem 27:
Bailey is on the tenth frame of her recent bowling competition and she needs to pick up the last pin for a spare and the first place trophy. She rolls the 7.05-kg ball down the lane and it hits the 1.52-kg pin head on. The ball was moving at 8.24 m/s before the collision. The pin went flying forward at 13.2 m/s. Determine the post-collision speed of the ball.
Audio Guided Solution
Show Answer5.39 m/s
Problem 28:
Jack D. Ripper flipped out after missing a Must-Do-It question for the third time on his Minds On Physics assignment. Outraged by the futility of his efforts, he flings a 4.0-gram pencil across the room. The pencil lodges into a 221.0-gram Sponge Bob doll which is at rest on a countertop. Once in motion, the pencil/doll combination slide a distance of 11.9 cm across the countertop before stopping. The coefficient of friction between the doll and the countertop is 0.325. Determine the speed at which the pencil is moving prior to striking Sponge Bob.
Audio Guided Solution
Show Answer49 m/s
Problem 29:
A physics student hurls a 315-gram ball directly into a 3.54-kg box which is at rest on a table top. The baseball strikes the box with a pre-impact speed of 54.1 m/s. The box is filled with towels to help absorb the blow and effectively catch the ball. The coefficient of friction between the box and the table is 0.714. Determine the distance which the ball and box slide across the table after the collision.
Audio Guided Solution
Show Answer1.40 m
Problem 30:
A 72-kg boy and a 48-kg girl, both wearing ice skates face each other at rest on a skating rink. The boy pushes the girl, sending her eastward with a speed of 6.8 m/s. When the impulse is completed, the boy and girl are a distance of 1.4 meters apart. Determine the distance of separation between the boy and the girl 5.0 seconds after the impulse is completed.
Audio Guided Solution
Show Answer58.1 m
Problem 31:
The city police are in pursuit of Robin Banks after his recent holdup at the savings and loan. The high speed police chase ends at an intersection as a 2080-kg Ford Explorer (driven by Robin) traveling north at 32.6 m/s collides with a 18400-kg garbage truck moving east at 12.4 m/s. The Explorer and the garbage truck entangle together in the middle of the intersection and move as a single object. Determine the post-collision speed and direction of the two entangled vehicles.
Audio Guided Solution
Show Answer11.6 m/s, 16.6° N of E
Problem 32:
A 92-kg fullback moving south with a speed of 5.8 m/s is tackled by a 110-kg lineman running west with a speed of 3.6 m/s. Assuming momentum conservation, determine the speed and direction of the two players immediately after the tackle.
Audio Guided Solution
Show Answer3.3 m/s, 53° S of W
Return to Overview
View Audio Guided Solution for Problem:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32
Privacy Manager
privacy
contact
home
about
terms
© 1996-2025 The Physics Classroom, All rights reserved.
By using this website, you agree to our use of cookies. We use cookies to provide you with a great experience and to help our website run effectively.
Freestar.comReport This Ad |
11767 | https://link.springer.com/content/pdf/10.1007/s40544-013-0002-6.pdf | Friction 1(1): 24–40 (2013) DOI 10.1007/s40544-013-0002-6 ISSN 2223-7690 REVIEW ARTICLE Energy dissipation in atomic-scale friction Yuan-zhong HU , Tian-bao MA, Hui WANG State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China Received: 09 November 2012 / Revised: 25 January 2013 / Accepted: 01 February 2013 © The author(s) 2013. This article is published with open access at Springerlink.com Abstract: The mechanisms of energy dissipation are discussed in this paper by reviewing the models and research in atomic-scale friction. The study is undertaken to answer a fundamental question in the study of friction: How is frictional work dissipated, particularly in cases where material damage and wear are not involved. The initiation of energy dissipation, the role of structural commensurability, and the estimation of the interfacial shear strength have been examined in detail by introducing the Tomlinson model, the Frenkel– Kontorova model, and the cobblestone model, respectively. The discussion is extended to energy dissipation progress described in terms of phononic and electronic damping. The contributions from other mechanisms of dissipation such as viscoelastic relaxation and material wear are also included. As an example, we analyzed a specific process of dissipation in multilayer graphene, on the basis of results of molecular dynamics (MD) simulations, which reveal a reversible part of energy that circulates between the system and the external driver. This leads us to emphasize that it is crucial in future studies to clearly define the coefficient of dissipation. Keywords: energy dissipation; atomic-scale friction; phononic damping; viscoelastic relaxation 1 Introduction Why does friction exist universally between surfaces in relative motion? How is it created in the first place? These are fundamental questions that Leonardo da Vinci considered more than 500 years ago, and the questions that scientists and engineers have asked over the centuries. Researchers such as G. Amontons, C. A. Coulomb, F. P. Bowden, I. V. Kragelskii, and D. Tabor have tried to provide answers to these questions, but satisfactory explanation has not been presented thus far. In early studies, friction was assumed to result from engaged surface asperities or from interacting surface molecules . As an explanation for the origin of friction, however, both models are unsatisfactory in the sense that they are unable to describe precisely how the engagement of asperities or interaction of molecules lead to energy dissipation in the process of friction. In the case of sliding, friction used to be regarded as the force acting in the opposite direction of motion, and a certain amount of work would need to be done by the force to sustain the sliding motion. Frictional work is eventually converted into heat, suggesting that friction is in fact a process of energy transformation. This makes energy dissipation a crucial scientific issue in understanding friction. Historically, energy dissipation was not considered a problem difficult to explain as long as the frictional work was assumed to be dissipated through plastic deformation and material damage . This is true for clean metal surfaces in contact and sliding. However, if the surfaces are contaminated or oxidized, or if surface films/lubricants are artificially applied as in boundary lubrication, very little plastic deformation occurs and the friction is then mainly attributed to interfacial effects. Similar effects also appear in the sliding of materials like polymer and ceramics. Friction without plastic deformation and wear raises many problems, still unsolved, concerning the nature of energy dissipation. Corresponding author: Yuan-zhong HU. E-mail: huyz@tsinghua.edu.cn Friction 1(1): 24–40 (2013) 25 The problem of explaining energy dissipation has generally been recognized by scientists through the end of the 20th century , but it was David Tabor who clearly identified the problem and brought it to the attention of the tribology community twenty years ago in a NATO-sponsored conference on the Fundamentals of Friction. In a lecture presented at the conference, he suggested a mechanism of energy dissipation for the friction in elastic or near elastic sliding, which can be summarized as follows. (i) Atoms at the interface are displaced from their equilibrium position. (ii) The displaced atoms reach an unstable configuration, and at this point they flick back to a new equilibrium position. (iii) The strain energy is lost in the form of atomic vibrations and these in turn are degraded into heat . As pointed out by Tabor himself, the basic ideas listed above can be found in a paper by Tomlinson , who recognized the role of mechanical instability in frictional energy dissipation, but for some reason did not emphasize the important role of atomic vibrations. There has been an increase in research on friction in the years following the NATO conference. This is accompanied by rapid development of molecular dynamics (MD) simulations and successful applications of new scientific instruments, such as atomic force microscope (AFM), surface force apparatus (SFA), and quartz crystal microbalance (QCM). The research has led to the emergence of new terms in the tribology community, e.g., interfacial friction to emphasize the friction that occurs only at the contact interface, wearless friction to distinguish the friction without wear from the wear-induced process, and atomic-scale friction to identify the dimensions of the system and contributions from atomic interactions. In this article, we review recent progress in studies of interfacial or atomic-scale friction, including models, experiments, and simulations, but the focus is on the theoretical developments in this field, in particular, on the discussions pertaining to energy dissipation. The discussions have been limited to kinetic friction when two solid surfaces slide past each other. The issue of frictional dissipation by purely viscous effect is not included in this paper, but Klein and coworkers [6, 7] have published remarkable results on this issue, which reveal the mechanism of lubrication provided by hydration layers and polymer brushes with very low friction. For more details, readers are referred to the review article of “Hydration lubrication” by Klein in the current issue of “Friction” . This report is arranged in the following manner. The next three sections are dedicated to descriptions of three models for atomic-scale friction, the Tomlinson model, the Frenkel–Kontorova (FK) model, and the cobblestone model. Section 5 examines the progress of dissipation in terms of phononic and electronic damping. Contributions from other dissipation mechanisms such as viscoelastic relaxation and material wear are discussed in Section 6. Section 7 describes a specific process of energy dissipation based on MD simulations for multilayer graphene, which reveals a reversible component of dissipation energy. Finally, concluding remarks are given in Section 8. 2 Initiation of energy dissipation—the Tomlinson model The idea that mechanical instability of an atomic system would result in energy dissipation was attributed to a paper by Tomlinson in 1929, but it was discovered recently that a similar idea was published independently by Ludwig Prandtl in German, one year before Tomlinson . It has been suggested that the model be renamed as the Prandtl–Tomlinson model. In this paper, however, we still use the traditional term Tomlinson model for brevity. In spite of the idea being proposed a long time ago, it was more than 50 years later when McClelland presented a clear description of the model and a comprehensive analysis of the dynamics of the system in which a row of independent oscillators slide over a periodic potential, as shown in Fig. 1. The oscillators, representing interfacial atoms, do not interact with each other but each of them is connected by a single spring to a rigid support B, and the periodic potential describes the interactions from the opposite solid A. Fig. 1 Schematic of the Tomlinson model. 26 Friction 1(1): 24–40 (2013) Dynamic analysis for the system results in a criterion of stability that if support B is slid infinitesimally slowly relative to solid A, the sliding will become unstable under the following condition, 2 2 V k x (1) where k is the spring stiffness, V is the potential function, and x denotes the oscillator position in the sliding direction. The essence of the Tomlinson model is to clarify that energy dissipation initiates from mechanical instability, i.e., the energy is dissipated by the instability induced stick–slip motion and consequent atomic vibrations. One should keep in mind that the Tomlinson stick–slip is different from the usual macroscopic stick–slip, and the mechanism of energy dissipation discussed in the Tomlinson model is applied only to atomic-scale friction. There are several versions of the Tomlinson model, presented in slightly different ways [10, 11]. These versions have been employed to explain the atomic stick–slip behavior between the tip and sample, and the velocity dependence of friction observed in AFM experiments [12, 13]. The Tomlinson model extended to a finite temperature shows that if there is an energy barrier E between two neighboring minima in potential energy, the tip may still be able to jump from one energy valley to the next due to thermal activation. The probability for the tip to jump (or not to jump) depends on the sliding velocity and the exponential of E/kBT, which leads to a logarithmic dependence of friction on velocity. This confirms Tabor’s proposal in his lecture at the NATO conference that the behavior of speed-dependent friction may be understood in terms of the stress-aided rate theories developed by Eyring. A similar logarithmic dependence between the force needed to break a molecular bond and the velocity to pull the bond is known to scientists studying adhesion. Suda revealed the connection between the two processes, where velocity dependence in friction can be derived independently from the rupture dynamics . This means that the Tomlinson model is essentially consistent with rupture dynamics when modeling atomic-scale friction. More importantly, Suda’s analysis offered an insight into the relation between friction and adhesion hysteresis, which will be further discussed in Section 4. Another extension to the Tomlinson model was proposed recently by Huang and coworkers . In this so-called composite oscillator model, there are two rows of oscillators connected with each other in both vertical and lateral directions, and the lower harmonic potential is replaced by oscillators arranged similarly to those in the top system. Researchers are still able to learn important lessons from the Tomlinson model today. For example, it is natural to infer from the model that the extent of energy dissipation and the magnitude of average friction force depend on two factors, the corrugation of interfacial potential and the stiffness of the surface bond. It also sheds light on the study of superlubricity where a frictionless state can be achieved because of lack of energy dissipation if the spring stiffness is strong enough in comparison to the interfacial interaction, as will be discussed further in later sections. In addition to the Tomlinson model, there are two independent developments that contribute greatly to the study of atomic-scale friction, as summarized in the following sections. 3 Role of commensurability—the Frenkel–Kontorova model 3.1 Frenkel–Kontorova model The FK model involves a system shown in Fig. 2 where a one-dimensional chain of atoms connected by springs of average length a interacts with a harmonic potential in period b . The model was employed at first to study dislocation in crystals, but it soon proved useful in studying the mechanism of friction such as the origin of static friction and effect of structural commensurability. Fig. 2 Schematic of the Frenkel–Kontorova model. Friction 1(1): 24–40 (2013) 27 The dimensionless equation of motion for the atomic chain can be written as π 1 1 2 sin2 1,2,..., j j j j j j x x x x x x F j N (2) where xj is the position of the jth atom, denotes the internal damping coefficient in the atomic chain, V/k represents a normalized potential strength equal to the magnitude of the sinusoidal potential divided by spring stiffness k, and F is an external driving force. The chain is at the ground state when all the atoms are resting in the energy valleys, i.e., their equilibrium positions, which deviate from their initial position by a small shift j, 0 j B j x x a j , where a0 is the initial length of the spring and xB is the mass center position of the chain. Aubry defined a hull function g(x) for describing the shift, 0 j B g x a j (3) If the normalized strength is small, the hull function is analytic where an infinitesimal force would be able to make the chain slide adiabatically. However, when the strength exceeds a critical value S c , g(x) is no longer analytic and becomes a discontinuous function. This is called the breaking of analytics or Aubry transition. The transition leads to a pinning state where the atomic chain is locked by the harmonic potential so that a finite force must be applied in order to initiate a slide of the chain. This is the first time researchers were able to predict static friction mathematically in terms of the breaking of analytics of a well-defined function. Great efforts have been devoted to explore the dynamic response of the system by solving the equation of motion at a finite sliding velocity. In the over-damping condition when the damping coefficient is large, Eq. (2) gives rise to a periodic solution , and the system exhibits rich dynamics as damping decreases. In addition to the periodic solutions, there are quasi-periodic or even chaotic solutions, depending on the parameters , , and the sliding velocity v [19, 20]. It is important to note from the solutions that the vibrations of the atomic chain and particularly resonance at certain velocities may introduce a possible mode of energy dissipation for sliding at a finite velocity. Compared to a real system, both the FK model and the Tomlinson model seem to miss something important. In the former, the atoms are not connected to a support representing the bulk of a sliding body whereas in the latter, the lateral interactions between the oscillators have been ignored. In consideration of the atomic interactions in a real interfacial system, it would make more sense to couple the two models together, leading to the Frenkel–Kontorova–Tomlinson (FKT) model proposed by Weiss and Elmer [21, 22]. Because the FK and FKT models involve two periodic structures interacting with each other, they represent an effective approach for analyzing the role of structural commensurability in energy dissipation. 3.2 Role of commensurability Commensurability describes a length-scale correlation between two interacting objects composed of periodic structures with periods a and b (see Fig. 2, for example). The structure is called commensurate when the period a is a simple rational fraction of b, or the ratio c = a/b is a rational number; otherwise, the structure is incommensurate. The effects of commensurability on static friction have been studied using the FK or FKT model [17, 21]. Numerical solutions of the static friction force FS under different values of a/b, for example, were obtained by Weiss , as plotted in Fig. 3 against the strength . It can be seen that in the incommensurate case where the ratio a/b equals the irrational number 2 / 1 5 ） － （ (the golden mean), FS remains zero until the strength exceeds a critical value, S c , as discussed in Section 3.1. The critical strength decreases, accompanied by a monotonous rise in friction as the ratio takes the values of 3/5, 2/3, 1/2 and so forth until 0 S c for a/b = 1 when the system becomes perfectly commensurate. Similar dependence on commensurability was observed for kinetic friction in the quasi-static limit of v→0. In the framework of the FK or FKT model, kinetic friction can be determined by solving the equation of motion. As long as is small, there is a unique solution for the equation. In this case, the 28 Friction 1(1): 24–40 (2013) Fig. 3 Static friction as a function of and for a sequence of rational values of c (thin lines), which converges to the golden mean (thick line) . friction is a conservative force fluctuating around zero. Its average is zero (i.e., FK = 0). There is a second critical value for the potential strength, K c , above which the solution is no longer uniquely defined and metastable states appear. As the system approaches a point of bifurcation, the atoms have to jump from one metastable state to a more stable position so that the system is rearranged into a new configuration, accompanied by energy dissipation. The average friction and the value of K c were numerically calculated by Weiss and Elmer , which revealed the dependence of kinetic friction on the commensurability. The diverse frictional behavior in a commensurate or incommensurate system results from the difference in energy dissipation, which may be understood in terms of the Tomlinson model discussed in Section 2. For two lattices in contact and in relative motion, each atom on one lattice surface is subjected to interaction from the opposite body, described by a harmonic potential, and the total potential energy applied to the entire surface is obtained by superposing the interaction on a single atom. In the incommensurate case, the superposed potentials cancel each other, leading to a total energy with little corrugation. It is the disappearance of energy corrugation that prevents energy dissipation and gives rise to very low friction. As an example, Fig. 4 gives the plot of shear stress vs. sliding distance obtained from MD simulations for two sets of self-assembled monolayers (SAMs) in relative sliding . Fig. 4(a) shows the results from a Fig. 4 Shear stress (dark line) and molecule tilt angle (gray line) from MD simulations for commensurate (a) and incommensurate SAMs (b) . commensurate system where the upper and lower monolayers are perfectly aligned, and Fig. 4(b) shows results from the case where one of the monolayers has been turned by 90°so that the system becomes incommensurate. The shear stress in the commensurate system presents a distinct stick–slip pattern, while in the incommensurate case, the stick–slip disappears with only thermal fluctuations left in the curves, and the average shear stress is five times smaller than that in the commensurate case. The first experimental evidence for the effect of commensurability was found in a study using a QCM in 1991 when Krim reported that solid monolayers of krypton sliding on gold exhibited five times less friction than liquid monolayers of krypton sliding on gold did . The observed behavior indicating solids are more slippery than liquids was difficult to explain at first. It was discovered a few years later that the higher friction of liquid layers can be attributed to the fact that the liquid layers are more Friction 1(1): 24–40 (2013) 29 flexible and therefore slightly more commensurate with the underlying surface than the solid layers are . More direct and convincing evidence came from an experiment measuring the friction between an AFM tip and a highly oriented pyrolytic graphite (HOPG) sample. The experiment revealed a dependence of friction on sliding direction, as shown in Fig. 5, where the measured friction is plotted as a function of rotation angle of the sample . During the sliding, a small piece of graphite was transferred to the tip so that friction is in fact occurring between the graphite sheet adhering to the tip and the HOPG substrate. A rotation of the sample would cause a change in commensurability between the sheet and substrate. As a result, the friction remains very low when they are incommensurate unless the rotation causes the substrate to be commensurate with the sheet, resulting in high friction. There is an unsolved mystery regarding the effect of commensurability. The predicted frictionless state in incommensurate systems does not exist in macroscopic reality although almost all surfaces in macroscopic contacts will always be incommensurate . Even two surfaces of the same crystal are incommensurate unless they are aligned perfectly. Several hypotheses have been proposed to explain the mystery. For example, the strong interfacial interaction may cause local transitions from incommensurate to commensurate state, which is energetically more favorable, and the presence of lubricant molecules confined between two surfaces may modulate the system into a commensurate condition. The role of lubricant molecules will be discussed in the next section. Fig. 5 Friction force between an AFM tip and a HOPG sample plotted as a function of rotation angle of the sample . 4 Evaluating shear strength at the interface—the cobblestone model A cobblestone model was proposed first by Tabor and developed further by others to explain the friction of two solid surfaces sliding past each other in the absence of wear. The model suggests that the process of friction is similar to pushing a cart over a road of cobblestones, as shown in Fig. 6, where the cartwheels represent the molecules of lubricant or of the upper solid body, and the cobblestones represent atomic roughness of the lower surface . A certain lateral force F is required to raise the cartwheel against the attractive surface interactions in order to initiate motion. Assume that the normal distance between the two surfaces increases by a small amount D, meanwhile the upper surface moves forward a distance d along the lateral direction. The work done by the lateral force F to initiate sliding has to be balanced with the work required to overcome the force of adhesion Fad, which can be estimated in terms of the surface energy change when the normal distance increases from D = D0 to D = (D0 + D). 2 0 ad 2 0 0 2 1 4 D D d F D F A A D D D (4) where denotes the surface energy and A is the area of contact. From the point of view of energy dissipation, however, it is inappropriate to balance the frictional work only with the energy required for surface separation because the process of sliding consists of a series of events of surface separation and approach. A Fig. 6 Schematics for the cobblestone model. 30 Friction 1(1): 24–40 (2013) part of the work expended on separating the surfaces will be regained during the process of approach. If the energy to separate surfaces is fully recovered when they approach, there will be no energy dissipated and the average friction force will be zero. With this consideration, the work done by the friction force should equal to the energy loss in a cycle of surface separation/approach, and Eq. (4) is be rewritten as 0 4 D d F A D (5) where = (s − r) is the difference in energies for surface separation and approach. As a result, the shear strength at the interface can be evaluated by the following equation. 0 0 4 4 c F D D S A D d D d (6) In the above equation, a coefficient of dissipation defined as =/ is introduced to specify that a fraction of the energy is lost every time the surfaces experience a cycle of separation/approach. Some important outcomes of the cobblestone model are discussed below. (1) The model gives an estimation of the shear strength for the monolayers at the interface or for thin liquid films confined between two surfaces. For a typical hydrocarbon surface, 3 2 25 10 J / m , and other typical values are set as D = 0.5 Å, D0 = 2 Å, d = 1 Å, and = 0.1. Eq. (6) predicts 7 2 2.5 10 N/ m c S . This compares very well with the typical experimental value of 2 × 107 N/m2 calculated for a hydrocarbon surface sliding in air or separated by a one-molecule layer of cyclohexane . (2) The model provides a theoretical frame for understanding the relation between friction and adhesion hysteresis. In contrast to common belief, the experiments in SFA conducted by Israelachivil and coworkers reveal that friction is correlated with the adhesion energy hysteresis rather than the adhesion energy itself. The cobblestone model explains that due to the presence of atomic-scale roughness, a sliding motion is always accompanied by surface separation and approach, and it is the energy difference between the two motions (or the energy hysteresis) that determines the magnitude of friction. Consequently, the shear strength predicted by Eq. (6) is in direct proportion to . As revealed by Suda , the relationship between friction and adhesion hysteresis can be derived from rupture dynamics. If friction is considered as the force needed to break interfacial bonds, it can be expressed as Ff = (1/d) (NkBT ln1 − NkBT ln), where 1 and are the bond lifetime at equilibrium and at the transition state, respectively. Assuming that A and R are equivalent to –(N/r2))kBln1 and –(N/r2))kBln, respectively, one finally obtains Ff = (r2/d) (R − A). (3) Introduction of the factor represents progress toward clarifying the amount and mode of frictional energy dissipation. It has been long recognized that friction is accompanied by accumulation and release of mechanical energy but there are few clues as to what percentage of the energy is converted into heat. The factor gives an estimation for the fraction of the energy to be lost irreversibly. Moreover, the cobblestone model seems to suggest that the energy fraction is lost through the impact between the cartwheel and cobblestone (see Fig. 6). Tabor himself mentioned once that he was not totally pleased with this model because it gave no indication of the mode of energy dissipation. In reviewing this model, however, we realized that the model has in fact related the energy dissipation to a specific mechanism—adhesion hysteresis, which has not been expressed explicitly in other models. The adhesion hysteresis as a mechanism of frictional dissipation may be useful in describing applications such as boundary lubrication and friction of viscoelastic materials. This explains why the cobblestone model is widely accepted in the community of polymer tribology. 5 Progress of energy dissipation—phononic and electronic damping With the revelation that energy dissipation initiates from mechanical instability, the next questions involve the dissipation progress after interfacial atoms start to vibrate. Is the energy dissipated permanently into heat or is it transformed into different forms of energy? These questions are discussed in the following sections. Friction 1(1): 24–40 (2013) 31 From the phononic model system shown in Fig. 7(a), a general description of the process of energy dissipation can be summarized as follows : A nanoparticle consisting of N atoms (representing, for example, an AFM tip or a lubricant molecule) is coupled to the surface of object 1. The nanoparticle and the object 1 surface slide together over the surface of a lower sample (object 2) under a constant loading force FN. In the initial stick stage, the nanoparticle is elastically deformed, namely, its atoms are displaced whereby the potential energy of the particle is increased by VT. Once the nanoparticle arrives at a critical point M, slip occurs when the nanoparticle jumps to an adjacent energy valley. In the course of slip, the displaced atoms of the nanoparticle are suddenly released and they start to vibrate around their equilibrium positions. 5.1 Generation of excess phonons In the framework of phonon theory, the release of accumulated energy VT at the time of slip means that the energy is expended irreversibly to excite phonons above the occupation number present at ambient temperature T0. The excess phonons will be damped or will decay into a new state of equilibrium at a higher temperature T. Denoting the occupation numbers at T and T0 as nq and o q n , respectively, the excess phonon distribution is expressed as, 0 , , o q q q n n T n T (7) Here, q denotes the mode of the phonon, q is the phonon frequency in mode q, and T>T0. The total number of excess phonons covering all possible modes can be written as qnq. Since the nanoparticle Fig. 7 System used for modeling phononic damping . contains only a small number of atoms, it has a discrete frequency distribution with relatively large spaces between the frequencies, Da() = q( − q); meanwhile, the frequency distributions (or the densities of state) of the objects, D1() and D2(), are quasi- continuous as shown in Fig. 7(b). The decay or damping of the excess phonons, qnq, may occur in various ways. For example, the phonons may be damped by (1) phonon–phonon interactions in the nanoparticle, (2) interactions between the nanoparticle and adjacent objects, or (3) phonon– electron interactions. In the following, we start with the first and second mechanisms, i.e., phononic damping, and electronic damping will be discussed in Section 5.4. 5.2 Internal damping If the nanoparticle is isolated from all adjacent objects, the phonons are damped solely by the phonon– phonon interaction occurring inside the particle, i.e., by the internal damping, in which the anharmonic coupling between phonons causes the vibration energy of a phonon to be translated to other phonon modes. If the nanoparticle is small enough or the anharmonic terms are weak, there is a possibility that the phonons will not be damped by internal anharmonic interactions so they remain in discrete modes with no energy dissipation. Sokoloff examined energy dissipation in a mesoscopic solid represented by a linear atomic chain with its atoms interacting through harmonic and anharmonic forces with one end of the chain forced to vibrate harmonically. This model represents the interactions when the solid is sliding over another substrate . The results show a possible frictionless sliding for the atomic chain in the sense that the internal energy of the system does not increase with time. The study also reveals a critical size of the system beyond which there is a transition from nondissipative to dissipative behavior. A similar size dependence of friction was reported by Kajita and coworkers in analyzing a two dimensional crystal where friction was found to increase with the number of atomic layers, which was explained in terms of the contributions from bulk atoms to internal damping activity. 32 Friction 1(1): 24–40 (2013) 5.3 Interfacial damping If the nanoparticle is coupled weakly with an adjacent object, the coupling will also cause phonon decay, known as interfacial damping. The number of phonons in mode q will vary with time, following the equation below. d d q q n t (8) q is the decay rate of phonons from mode q to the object, also known as the coupling constant. The deformation energy of the nanoparticle accumulated during stick, VT, is converted into excitation energy of the phonons during slip, so the energy can be written in phononic form as 1 2 T q q q q q V V n (9) where Vq denotes the phononic energy in mode q. Combining Eqs. (8) and (9) leads to an estimation that the energy VT decays in exponential manner with a dissipation rate depending on the factors related to the number of phonons, the strength of coupling, and the density of state of the coupled objects. For the system shown in Fig. 7 and a given damping coefficient, the energy VT almost vanishes within 100 ps. Phononic description theory has not advanced sufficiently to provide an accurate prediction of energy dissipation. However, an important lesson we have learned from the analysis is that dissipation is a process in which the state density of phonons transits from a few discrete modes into a widespread and continuous distribution. Both internal and interfacial damping contribute greatly to the energy dissipation in the sense that they help phonons to transit from one particular mode to other modes. The question as to whether phononic friction is dominated by internal or interfacial damping remains unclear, but is expected to depend on the specific frictional system. 5.4 Electronic damping Electronic damping may contribute significantly to frictional dissipation for sliding on conductive or metallic surfaces. The mechanism involves excitation of electron–hole pairs as a result of electron–phonon coupling, and progress has been made in evaluating the electronic damping by measuring the “slip time” or the increase in electrical resistivity. When a microscopic particle or a molecule slides on a substrate, the friction force acting on the particle can be characterized by a “slip time” , corresponding to the time elapsed before the speed of the particle falls to 1/e of its original value. Slip time is inversely proportional to the amount of frictional damping. If both phononic and electronic damping are considered, the slip time can be written in an additive form, including the separate contributions from phonon ph and electron el slip times. ph el 1 1 1 (10) Slip times for various monolayers and bilayers adsorbed on metal substrates have been measured using a QCM by Krim and coworkers . It is reported that the measured slip times of Xe on Ag are in perfect agreement with the results of MD simulations with no electronic damping . This seems to suggest that the phononic damping is dominant, but simulations conducted by other investigators have lead to a different conclusion . This has caused a debate regarding the importance of the electronic contribution to friction, as will be discussed later in this section. The effect of electronic damping has been observed in an experiment involving N2 adsorbed on lead , which reveals a temperature dependence of the slip times as shown in Fig. 8. As the temperature decreases below the superconducting transition temperature Tc, the shear stress drops rapidly and the slip time shows a sharp increase. The sharp decrease of friction was considered a proof of the fact that electronic friction exists in normal conditions but disappears at temperatures below Tc due to superconductivity. It has been shown by Persson that the damping of molecules moving on a metal substrate is equivalent to an increase in the electrical resistivity of the substrate that occurs when molecules are adsorbed on the surface. As a result, the electronic slip time of a sliding particle can be calculated from the increase in the resistivity caused by adsorption of the particle. This enables one to evaluate the coefficient of electronic Friction 1(1): 24–40 (2013) 33 Fig. 8 Temperature dependence of slip time and shear stress for N2 adsorbed on lead . damping el in terms of the changes to resistivity as a function of adsorption coverage. The values of el for several different molecules adsorbed on Ag range from 3 × 108 to 3 × 109 s–1, which are much smaller in comparison to the coefficient of phononic damping ph deduced from the He-atom scattering measurement for hydrocarbons adsorbed on Cu (100). However, the role of electronic friction remains an open question. Different investigators have carried out several studies but results have been inconsistent. As mentioned above, MD simulations without considering any electronic effect produce friction curves that agree well with those from the QCM experiments, suggesting that the contribution from electronic damping is insignificant . Experiments by Fois and coworkers for a system similar to that in Ref. found that the N2 adsorbate sticks to the Pb substrate at the superconductivity temperature and does not show any slip, which is not consistent with Krim’s observations. Besides, the abrupt transition in friction coefficient observed in Ref. is in conflict with the prediction of the Bardeen–Cooper–Schrieffer (BCS) theory commonly accepted to describe super- conductivity at low temperatures. This has caused a debate regarding the reproducibility of the results reported in Ref. . Recently, a measurement of friction between a pendulum-type AFM probe and Nb films was carried out . The results showed a smooth transition of friction across the critical temperature of superconductivity, which is consistent with the BCS theory. A close examination confirms that the dependence of friction on tip–film distance at the temperature above the critical point Tc is consistent with the prediction for a two-dimensional electron system while dependence below Tc follows the law of phononic friction. This clearly shows the existence of electronic friction but further studies are certainly needed to clarify the contribution of electronic damping to friction. Below we discuss two types of energy dissipation that appear to be different from those discussed above, but at a fundamental level they belong to the same category as phononic dissipation. (1) Friction-induced sound The energy of elastic deformation accumulated in the stick phase may be released by emitting elastic or acoustic waves directly into the air instead of being converted into heat. As an example, while playing a violin, the stick–slip friction between the bow and the string is transformed to sound without significant heat generation. In contrast, brake noise in automobiles is an extreme example where sound is accompanied by a large amount of heat generated at the contact surfaces. Friction-induced sound and noise has become a major research subject due to its importance in engineering applications . Generally, the energy release through sound wave emissions should be cataloged as a special form of phononic dissipation , but the acoustic radiation in dynamic friction can be analyzed more specifically by an approach similar to that in electromagnetism . Radiative damping will produce a force to resist motion where amplitude increases with frequency according to a cubic law, much faster than that of viscous damping, and the power spectrum of surface displacements will decay at a rate proportional to 6 as confirmed in a recent experiment, where is the frequency of dynamic friction . (2) Thermoelastic damping The vibrations of an elastic beam or a microscopic resonator, no matter how carefully designed, will decay to a complete stop as a result of energy dissipation. The mechanism of energy dissipation involved here is called thermoelastic damping, which couples the elastic deformation directly to a local temperature rise. The strength of the coupling can be described by 34 Friction 1(1): 24–40 (2013) a single microscopic parameter, namely, the material’s thermal expansion coefficient . In atomic-scale friction accompanied by stick–slip motion, there is considerable elastic strain accumulated in the stage of stick so that thermoelastic damping has to be a possible mechanism of energy dissipation. In the phononic description, thermoelastic damping is caused by interactions between the acoustic mode (vibrations) and a surrounding bath of thermal phonons, so it is a special form of the internal damping discussed in Section 5.2. 6 Contributions from other mechanisms of energy dissipation The subjects we have discussed so far concern a specific mechanism of energy dissipation that transforms mechanical energy into heat when two solid surfaces slide past each other. The mechanism involves the accumulation and release of deformation energy, atomic vibration at the interface, and decay of the vibration energy into heat. However, other mechanisms, such as the viscoelastic relaxation and material wear, may also contribute to the energy transformation in the course of sliding, as discussed in this section. 6.1 Viscoelastic relaxation For a solid obeying linear elasticity subjected to a constant elastic strain, the strain will be constant with time, but in the case of a liquid obeying linear viscosity, the strain will be released immediately due to its incapability to store energy. The situation becomes more complicated for a viscoelastic material in which the elastic strain will be gradually released within a relaxation time that depends on material properties. The reason for raising this question is a concern regarding the possibility that viscoelastic relaxation may serve as a mechanism of frictional dissipation, especially for viscoelastic materials such as rubbers, polymers, and organic lubricant films. Evidence for such a mechanism of dissipation was found in a fretting friction test involving polymers , where the origin of the dissipation was attributed to viscoelastic loss within the contacting asperities. This was based on observations that the ratio between the out-of-phase and in-phase displacements was of the same order as the loss angle tan for the bulk polymer under test. Lei and Leng also found dissipation associated with viscoelastic relaxation in MD simulations involving liquid argon confined between solid walls. They reported that about 60% of total frictional work was converted into heat at the time of slip and the remaining 40% was released more slowly into the solidified films through various channels, including relaxation. While frictional dissipation for incompliant materials has been described by the Tomlinson model in terms of excitation of atomic vibrations, the sliding of compliant and viscoelastic materials can be also viewed as an activated process, in which molecules experience shear-induced de-bonding and relaxation that result in energy dissipation. The frictional processes involving molecule activation and relaxation can be described by a simple model based on Eyring’s theory of reaction rates. By plotting the friction force measured between an AFM tip and a polymer substrate as a function of logarithmic sliding velocity, one obtains a bell-shaped curve as shown in Fig. 9 if the shear rate matches the material intrinsic relaxation time . Note that for curves measured at different tem- peratures, there are horizontal shifts aT from the curve obtained at the reference temperature as shown in Fig. 9. If the data are treated based on the theory of time–temperature equivalence, one will get a master curve independent of temperature. The velocity and temperature dependence of friction and associated energy dissipation are a central theme in the study of friction for viscoelastic materials. Fig. 9 Friction–velocity curves showing a bell-shaped dependence with horizontal and vertical shifts . Friction 1(1): 24–40 (2013) 35 Due to the time–temperature equivalence, the horizontal shifts provide the means to determine the apparent activation energy Ea, ln( ) (1/ ) T a P a E R T (11) The thermal shift and the shape of the friction– velocity curve provide an energetic signature for the dissipation mechanism. The bell-shaped dependence originates from the competition between two time scales, namely, (i) the extrinsic drive time, and (ii) the intrinsic material response time. The friction force increases or decreases with increasing sliding velocity depending on whether the extrinsic time leads or trails the material response time. Friction force FF(v) peaks at a critical velocity vp when the two processes occur on comparable time scales. The product between vp and the relaxation time leads to a “dissipation length” that provides an estimation for the size of the region in which energy dissipation occurs . 6.2 Contributions from wear The concept of friction without wear confuses engineers somewhat, wondering how it could happen in reality. On the contrary, the relation between friction and wear is easy to understand. There is a common belief in the tribology community that plastic deformation and wear consume energy so they are responsible for the generation of friction. This is true in some cases. The ploughing of a hard indenter in a soft material is a simple example that relates friction force directly to the plastic deformation of the ploughed material . In general cases of sliding, friction is accompanied by material damage so that in addition to the energy dissipation via atomic vibrations, a portion of frictional work is expended to cause plastic deformation, microcracks, and wear . The evidence for a linear relationship between measured wear volume and energy dissipation has been found in experiments , as illustrated in Fig. 10. The wear volume measured in a test for a TiN monolayer sliding against alumina balls is plotted as a function of dissipated energy. A number of other models have been proposed to relate wear rate with energy dissipation . Fig. 10 Linear relation between wear volume and dissipated energy from sliding wear tests of a TiN monolayer against alumina balls . However, the energy-based wear models seem to suggest that all or most of frictional work is dissipated in the form of wear. This implicit assumption may be acceptable for fretting wear, but in normal sliding it is estimated that only a small portion of frictional work (less than 10%) will be stored in material in the form of structural distortion, and then be converted into wear [50, 51]. The estimate is supported by the fact that the wear rate in normal sliding is about 10 times lower than that observed in fretting tests for the same material combinations . Nevertheless, the big question regarding the division between the two dissipation mechanisms, namely wearless heating and wear-related energy loss, remains unanswered. The above discussions also explain why we have to focus on wearless friction. There are two reasons: (i) Wearless friction is a major part of the energy dissipation in normal sliding, and (ii) wearless dissipation is more difficult to understand and resolving the mystery is a challenging mission for us in tribology to accomplish. While focusing on energy dissipation in wearless friction, we noted that research on atomistic wear has increased recently because of its connection to nanofabrication. It is reported that in processes where material is removed atom by atom, wear does not 36 Friction 1(1): 24–40 (2013) obey the Archard equation but can be described by a model based on the breaking of individual bonds . 7 Energy dissipation in multilayer graphene—An example As an example, we will examine a specific process of energy dissipation on the basis of the results of MD simulations performed recently by Xu for a system consisting of multilayer graphene, with the top graphene layer being forced to move at constant velocity while the bottom layer remains fixed. The energy components and their time evolutions are displayed in Fig. 11. De denotes the work done by the external lateral force that drives the system, Pae is the potential energy of the system, Kae defines the vibrational energy of all atoms, and Ke is the kinetic energy caused by the center-of-mass motion of graphene layers. The left and right parts of the figure correspond to the results from a three- and a five-layer system, respectively. The results shown in Fig. 11 allow us to analyze the details of energy dissipation occurring in the multilayer graphene system, as summarized below. (1) Energy input and accumulation In the stick stage, the top graphene layer is pulled by an external lateral force to move it forward, and the work done by the external force is input to the system and stored in the form of potential energy Pae. As a result, both De and Pae increase in a synchronous manner, as shown in Fig. 11. (2) Transformation of energy to atomic vibration and layer translation In the slip stage, the potential energy drops rapidly while Kae and Ke increase, indicating that a part of the stored potential energy is converted to atomic vibrations and the mass-center motion of the graphene layers. The increase in Kae and Ke occurs in very short time, a few picoseconds according to the curves in Fig. 11, and is accompanied by a sharp decrease of Pae in the same time. The mechanism for the transformation of energy from Pae to Kae and Ke can be well understood in terms of the Tomlinson model and phononic damping discussed in previous sections. (3) Further transformation of the translational energy Ke It can be seen from Fig. 11 that during the time interval from the instant of slip to the full development of the next stick, there is a visible decrease in the translational energy Ke, especially for the five-layer system, but no detectable increase in vibration energy Kae. This indicates that the kinetic energy for the mass-center motion of graphene layers has been further transformed into other forms of energy with at least two possibilities: either Ke is further converted to atomic vibrations and heat, or a part of it is returned Fig. 11 Energy components and time evolution in multilayer graphene, (a) three-layer system, (b) five-layer system . Friction 1(1): 24–40 (2013) 37 to potential energy. Since no increase in vibration energy is observed as Ke decreases, we expect that the latter would dominate the transformation. (4) Slow release of Pae and reversible energy A careful examination of Fig. 11 reveals an important feature of the energy dissipation. In the course of slip, the decrease in Pae consists of two stages, a fast drop in a few picoseconds followed by a slow relaxation that may last for 20–30 ps. After Pae reaches the minimum, it increases again, preparing for a new stick event. As a result, the potential energy exhibits a U-type variation in the time interval between the slip and the next stick, which can be seen more clearly in the 3-layer system. The slow decrease in stored energy was also reported by Lei and Leng in their MD simulations and was attributed to material relaxation. While it is true that material relaxations do release energy in such a manner, we found that a cyclic energy exchange between graphene and the external driver is responsible for the slow release and the U-type variation of Pae. More specifically, a part of the potential energy is fed back to the driver, leading to a decrease in external work De, as can be seen in Fig. 11, indicating that the external driver receives a part of the energy from the graphene system. The exchange constitutes a conservative part of energy that circulates reversibly in a frictional system. The larger this reversible energy, the less is the energy dissipation. There would be no dissipation at all if the energy is fully reversible. This is a frictionless state predicted by the Tomlinson model when stick–slip disappears. It is interesting to compare the modes of dissipation discussed in this section with those in stick–slip friction between surfaces separated by confined liquids, as examined by Klein . It was demonstrated in that the major part of the stored energy is dissipated as viscous heating of the confined film during slip while the rest is dissipated as mechanical oscillations due to momentum transfer to the external system. This is consistent with what we see in Fig. 11, where there is a sudden increase in atomic thermal energy (Kae) at the time of slip, accompanied by significant oscillations of the graphene layers (Ke). However, we note that the slip-induced fast dissipation is followed by a relatively slow release of potential energy prior to the onset of the next stick, which was not observed in . The discrepancy may result from the difference in system setup in Klein’s case, where the top surface is connected to a spring whose end is pulled at a constant velocity. In our simulation, the top graphene layer is directly pulled by the driver so the elastic energy is stored mostly in the form of atomic deformation within the graphene layer. As a result, a part of the internal energy will be released in a relatively slow process of relaxation. The most significant aspect of this comparison is that in addition to the momentum transfer reported in , we show that there is an exchange of energy between the internal and external systems and the amount of this reversible energy has a crucial impact on atomic- scale friction. Here, we have shown that the potential energy stored in the stick phase may not be completely dissipated, raising the question as to how to determine the coefficient of dissipation, i.e., the irreversibly dissipated energy divided by the total stored energy. In fact, the concept of the dissipation coefficient was introduced in the cobblestone model (see Eq. (6) in Section 4), but little work has been undertaken to clarify it. It is simply assumed in most studies that all the stored energy has been dissipated. This leads us to emphasize that it is crucial in future studies of friction to clearly define the coefficient of energy dissipation. 8 Concluding remarks There is a prevalent mystery in the fundamental study of friction between two solid surfaces sliding past each other. How is frictional work dissipated, especially when material damage and wear are not involved? To answer this question, the mechanisms of energy dissipation have been discussed in this paper by reviewing the models and research that apply to atomic-scale friction, leading to the following concluding remarks. (1) According to the Tomlinson model, energy dissipation in atomic-scale friction initiates from the mechanical instability that induces stick–slip motions and atomic vibrations. The occurrence of the stick–slip motion can be predicted by a criterion that involves 38 Friction 1(1): 24–40 (2013) two features of the system, namely, the corrugation of surface energy and the stiffness of surface bonds. (2) The models lead to a logical conclusion that the magnitude of friction and energy dissipation would be greatly affected by the structural commensurability between two surfaces in contact, and a frictionless state (superlubricity) can be achieved in a perfectly incommensurate system. The prediction has been confirmed by a few carefully designed experiments, such as those using QCM and AFM, but it does not match the behavior of macroscopic friction encountered in engineering. (3) The cobblestone model suggests a slightly different picture of energy dissipation: Sliding consists of a series of events of surface separation and approach, and it is the energy loss in the cycle of surface separation–approach that determines the magnitude of friction. In this way, the mechanism of energy dissipation has been related to adhesion hysteresis, which may be appropriate for boundary lubrication. The model provides an estimate for the interfacial shear strength on the basis of the energy loss in adhesion hysteresis. (4) In the phononic description, the sudden release of stored energy corresponds to the excitation of excess phonons that will decay consequently to a new state of equilibrium by internal and interfacial damping. As a consequence, it is theoretically possible to achieve a nearly frictionless state for an isolated system of sufficiently small size because phonons will remain at isolated discrete modes with no interaction, implying no energy dissipation occurs. (5) In friction systems involving metallic materials, electronic damping may play an important role. Several carefully designed experiments reveal that there is a considerable drop in frictional force when the temperature goes below the superconductivity transition temperature, illustrating a dominance of electronic friction, but other experiments do not confirm the observations. The contribution from electronic damping remains an open question. (6) In addition to atomic vibration, other mechanisms also contribute to energy dissipation. Molecule activation and relaxation, for example, are particularly important to the frictional dissipation of viscoelastic materials. In most engineering applications, friction is thought to be accompanied by wear. However, the study of wearless friction should be emphasized because the majority of frictional work is converted into heat instead of wear debris, and wearless dissipation is more difficult to understand and resolving the mystery is a challenging mission. (7) MD simulations for multilayer graphene show that there is a portion of potential energy circulating reversibly between the system and the external driver. In other words, the dissipated component takes only a portion of the total stored energy. This creates a possibility of achieving a nearly frictionless state by minimizing energy dissipation. Acknowledgement The authors would like to express their appreciations to the financial support from the National Natural Science Foundation of China under the grant Nos. of 51075526, 51005129, and 51021064. The authors would like to thank Enago (www.enago.cn) for the English language review. Open Access: This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distri- bution, and reproduction in any medium, provided the original author(s) and source are credited. References Dowson D. History of Tribology. London (UK): Professional Engineering Publishing Limited, 1998. Bowden F P, Tabor D. Friction and Lubrication of Solids. Oxford (UK): Oxford University Press, 1954. McClelland G M. Friction between weekly interacting smooth surfaces. In Adhesion and Friction. Kreuzev and Grinze, Ed. Berlin: Series in Surface Science, Springer Verlag, 1989: 1–16. Tabor D. Friction as a dissipative process. In Fundamentals of Friction: Macroscopic and Microscopic Processes. Singer I L and Pollock H M, Ed. Dordrecht: Kluwer, 1992: 3–20. Tomlinson G A. A molecular theory of friction. Phil Mag 7: 905–939 (1929) Klein J, Kumacheva E, Mahalu D, Perahla D, Fetters L J. Reduction of frictional forces between solid surfaces bearing polymer brushes. Nature 370: 634–636 (1994) Raviv U, Klein J. Fluidity of bound hydration layers. Science 297: 1540–1543 (2002) Friction 1(1): 24–40 (2013) 39 Klein J. Hydration lubrication. Friction 1: in press (2013) Prandtl L, Gedankenmodell E, zur kinetischen Theorie der festen Körper. ZAMM 8: 85–106 (1928) Tomanek D, Zhong W, Thomas H. Calculation of an atomically modulated friction force in atomic-force microscopy. Europhys Lett 15: 887–892 (1991) Gnecco E, Bennewitz R, Gyalog T, Meyer E. Friction experiments on the nanometer scale. Journal of Physics: Condensed Matter 13: R619–642 (2001) Gnecco E, Bennewitz R, Gyalog T, Loppacher, Ch, Bammerlin M, Mayer E, Guntherodt H J. Velocity dependence of atomic friction. Physical Review Letters 84(6): 1172–1175 (2000) Sang Y, Dube M, Grant M. Thermal effects on atomic friction. Physical Review Letters 87(17): 174301 (2001) Suda H. Origin of friction derived from rupture dynamics. Langmuir 17: 6045–6067 (2001) Xu Z M, Huang P. Composite oscillator model for the energy dissipation mechanism of friction. ACTA PHYSICA SINICA 55(5): 2427–2432 (2006) Frenkel J, Kontorova T. On the theory of plastic deformation and twinning. J Phys USSR 1: 137 (1939) Aubry S. Critical behavior at the transition by breaking of analyticity in the discrete Frenkel-Kontorova model. J Phys C 16: 1593–1608 (1983) Floria L M, Mazo J J. Dissipative dynamics of the Frenkel- Kontorova model. Adv Phys 45(6): 505–598 (1996) Braun O M, Dauxois T, Paliy M V, Peyrard M. Nonlinear mobility of the generalized Frenkel-Kontorova model. Phys Rev E 55(3): 3598–3612 (1997) Strunz T, Elmer F J. On the sliding dynamics of the Frenkel-Kontorova model. In Physics of Sliding Friction. B N J Persson B N J and Tosatti E, Ed. Dordrecht: Kluwer, 1996: 149–162. Weiss M and Elmer, F J. Dry friction in the Frenkel- Kontorova-Tomlinson model: Static properties. Phys Rev B 53(11): 7539–7549 (1996) Weiss M and Elmer, F J. Dry friction in the Frenkel- Kontorova-Tomlinson model: dynamical properties. Zeitschrift Fur Physik B 104: 55–69 1(997) Hu Y Z, Zhang T, Ma T B, Wang H. Molecular dynamics simulations on atomic friction between self-assembled monolayers: Commensurate and incommensurate sliding. Computational Material Science 38(1): 98–104 (2006) Krim J, Solina D H, Chiarello R. Nanotribology of a Kr monolayer: a quatz crystal microbalance study of atomic- scale friction. Physical Review Letters 66: 181–184 (1991) Cieplak M, Smith E D, Robbins M O. Molecular origins of friction. Science 265: 1209–1212 (1994) Dienwiebel M, Verhoeven G S, Pradeep N, Frenken J W, Heimberg J A, Zandbergen H W. Superlubricity of graphite. Physical Review Letters 92(12): 126101 (2004) Tabor D. The role of surface and intermolecular forces in thin film lubrication. In Microscopic Aspects of Adhesion and Lubrication. Georges J M, Ed. Amsterdam: Elsevier, 1980: 651–679. Homola A M, Israilachvili J N, Gee M L, McGuiggan P M. Measurements of and relation between the adhesion and friction of two surfaces separated by molecularly thin liquid- films. Journal of Tribology, Trans ASME 111(4): 675–682 (1989) Berman A D, Israelachivili J N. Microtribology and Microrheology of Molecularly Thin Liquid Films. In Modern Tribology Handbook. Bhushan B, Ed. Boca Raton: CRC Press, 2001: 568–610. Chen Y L, Helm C, Israelachivili J N. Molecular mechanisms associated with adhesion and contact angle hysteresis of monolayer surfaces. Journal of Physical Chemistry 95: 10736–10747 (1991) Buldum A, Leitner D M, Ciraci S. Model of phononic energy dissipation in friction. Physical Review B 59(24): 16042–16046 (1999) Sokoloff J B. Possible nearly frictionless sliding for mesoscopic solids. Physical Review Letters 71(21): 3450– 3453 (1993) Kajita S, Washizu H, Ohmori T. Deep bulk atoms in a solid cause friction. Epl-Europhys Lett 87(6): 66002 (2009) Persson B N J, Nitzan A. Linear sliding friction: On the origin of the microscopic friction for Xe on silver. Surface Science 367(3): 261–275 (1996) Dayo A, Alnasrallah W, Krim J. Superconductivity- dependent sliding friction, Physical Review Letters 80: 1690–1693 (1998) Persson B N J. Applications of surface resistivity to atomic scale friction, to the migration of hot adatoms, and to electrochemistry. Journal of Chemical Physics 98: 1659– 1672 (1993) Fois G, Bruschi L, d’Apolito L, Mistura G, Torre B, de Mongeot F B, Boragno C, Buzio R, Valbusa U. Low- temperature static friction of N2 monolayers on Pb(111). Journal of Physical: Condensed Matter 19: 305013 (2007) Kisiel M, Gnecco E, Gysin U, Marot L, Rast S, Meher E. Suppression of electronic friction on Nb films in the superconducting state. Nature Materials 10: 119–122 (2011). Acay A. Acoustics of friction. J Aoust Soc Am 111(4): 1525–1548 (2002) 40 Friction 1(1): 24–40 (2013) Bruesch P. Phonons: Theory and Experiments I, Lattice dynamics and models of interatomic forces. Berlin: pringer- Verlag, 1982. Johansen A, Sornette D. Acoustic radiation controls dynamic friction: evidence from a spring-block experiment. Physical Review Letters 82(25): 5152–5155 (1999) Lifshitz R, Roukes M L. Thermoelastic damping in micro- and nanomechanical systems. Physical Review B 61(8): 5600–5609 (2000) Bureau L, Caroli C, Baumberger T. Elasticity and onset of frictional dissipation at a non-sliding and multi-contact interface. Proc R Soc Lond A 459: 2787–2805 (2003) Lei Y J, Leng Y S. Stick-slip friction and energy dissipation in boundary lubrication. Physical Review Letters 107: 147801 (2011) Knorr D B, Widjaja P, Acton O, Overney R M. Molecular friction and mode coupling in organic monolayers and polymer films. Journal of Chemical Physics 134: 104502 (2011) Scott S, Gray T, Overnet R M. Molecule dissipation phenomena of nanoscopic friction in the heterogenerous relaxation regime of a glass former. Journal of Chemical Physics 123: 134902 (2005) Amiri M, Khonasri M M. On the thermodynamics of friction and wear – a review. Entropy 12: 1021–1049 (2010) Huq M Z, Celis, J P. Expressing wear rate in sliding contacts based on dissipated energy. Wear 252: 375–383 (2002) Uetz H, Fohl J. Wear as an energy transformation process. Wear 49: 253–264 (1978) Fleisher G. Energetische methode der bestimmung des verschleibes. Schmierungstechnik 4: 9–22 (1973) Wen S Z, Huang P. Principle of Tribology. Beijing: Tsinghua University Press, 2008. Gotsmann B, Lantz M A. Atomistic wear in single asperity sliding contact. Physical Review Letters 101: 125501 (2008) Xu L. Research on the friction and energy dissipation behavior of few-layer graphene. Ph.D. Thesis. Beijing (China): Tsinghua University, 2012. Klein J. Frictional dissipation in stick-slip sliding. Physical Review Letters 98: 056101 (2007) Yuan-zhong HU graduated in 1968 from Tsinghua University, and received his PhD in 1985. Since then he joined the SKLT, Tsinghua University, as an associate professor and later a pro- fessor. During 1989–1992, he worked in Norwegian Technical University, Norway, and in Northwestern University, US, as a postdoctoral fellow. He stayed in the US again during 1997.5–1998.12, as a visiting scholar. His research interests include EHL, Mixed lubrication, Wear dynamics, Nano-Tribology, Thin film rheology, Molecular dynamics simulations, etc., and in these areas he has more than 100 papers published on international journals. Prof. Hu received the Edmond E. Bisson Award from STLE in 2003 due to his contributions to the numerical analysis of mixed lubrication, and he is now an Associate Editor of the Proceedings of IMechE, Part J, Journal of Engineering Tribology. |
11768 | https://files.ele-math.com/articles/jmi-07-08.pdf | Journal of Mathematical Inequalities Volume 7, Number 1 (2013), 93–96 doi:10.7153/jmi-07-08 OPERATOR MONOTONE FUNCTIONS, A > B > 0 AND logA > logB TAKAYUKI FURUTA To the memory of my familiar persons passed away by tsunami disaster at Tohoku district on 2011.3.11 with deep sorrow (Communicated by Y. Seo) Abstract. Let f(t) be any non-constant operator monotone function on [0,∞) and also let A and B be strictly positive operators on a Hilbert space H .
(i) If A > B, then the following inequality holds: f(Aα) > f(Bα) for all α ∈(0,1].
(ii) If logA > logB, then there exists β ∈(0,1] and following inequality holds: f(Aα) > f(Bα) for all α ∈(0,β].
1. A > B > 0, logA ⩾logB and logA > logB A capital letter means a bounded linear operator on a complex Hilbert space H .
An operator T is said to be positive (denoted by T ⩾0) if (Tx,x) ⩾0 for all x ∈H and an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. Chaotic order is defined by logA ⩾logB for strictly positive operators A and B, and also strictly chaotic order is defined by logA > logB for strictly positive operators A and B. The well known celebrated L¨ owner-Heinz inequality asserts that if A ⩾B ⩾0, then Aα ⩾Bα for any α ∈[0,1]. This means that t →tα is operator monotone. Another well known example of operator monotone is t →logt on (0,∞), that is, logA ⩾logB is weaker than the usual order A ⩾B ⩾0.
We consider the following two operator monotone functions on (0,∞) in [8, p.
151] and [10, p. 131]: ϕ(t) = t −1 logt and ψ(t) = t log−t + 1 log2t .
(1.1) Mathematics subject classification (2010): 47A63.
Keywords and phrases: L¨ owner-Heinz inequality, chaotic order, strictly chaotic order, operator mono-tone function.
c ⃝ , Zagreb Paper JMI-07-08 93 94 T. FURUTA THEOREM A. Let A and B be strictly positive operators on a Hilbert space H .
If log > logB, then there exists β ∈(0,1] and the following inequalities hold: ϕ(Aα) > ϕ(Bα) for all α ∈(0,β) (1.2) and ψ(Aα) > ψ(Bα) for all α ∈(0,β) (1.3) where ϕ(t) and ψ(t) are defined in (1.1).
THEOREM B. There exist strictly positive operators A and B such that logA ⩾ logB: ϕ(Aα) ̸⩾ϕ(Bα) for any α > 0 (1.4) and ψ(Aα) ̸⩾ψ(Bα) for any α > 0 (1.5) where ϕ(t) and ψ(t) are defined in (1.1).
(1.2) and (1.4) are shown in , and (1.3) and also (1.5) are shown in .
In §2, we shall show Theorem 2.2 which is a further extension of Theorem A.
2. Relations among A > B, logA > logB for A,B > 0 and operator monotone functions We study relations among A > B, logA > logB for A,B > 0 and operator mono-tone functions for strictly positive operators A and B.
THEOREM 2.1. Let A and B be strictly positive operators on a Hilbert space H .
If A > B, then the following inequality holds: f(A) > f(B) (2.1) for any non-constant operator monotone function f on [0,∞).
Proof. Let A > B. Then there exists some ε > 0 such that A−B ⩾εI (2.2) so that A+ s ⩾B+ s+ ε > B+ s for s ⩾0, so that there exists some δ > 0 such that (B+ s)−1 −(A+ s)−1 ⩾δI.
(2.3) It is well known (for examples, , ) that a function f on [0,∞) is an operator monotone if and only if it has the representation f(t) = a + bt + ∞ 0 ts t + sdm(s) = a + bt + ∞ 0 s−s2 t + s dm(s) (2.4) OPERATOR MONOTONE FUNCTIONS, A > B > 0 AND logA > logB 95 with a ∈R and b ⩾0 and a positive measure m on [0,∞).
Then (2.2), (2.3) and (2.4) ensure the following (2.5) f(A)−f(B) = b(A−B)+ ∞ 0 {(B+ s)−1 −(A+ s)−1}s2dm(s) > 0 (2.5) so that we have (2.1) by (2.5).
□ THEOREM 2.2. Let f(t) be any non-constant operator monotone function on [0,∞) and also let A and B be strictly positive operators.
If logA > logB, then there exists β ∈(0,1] and the following inequality holds: f(Aα) > f(Bα) for all α ∈(0,β].
(2.6) Proof. Recall the following obvious relation (2.7): X > Y > 0 = ⇒Xγ > Y γ for any γ ∈(0,1].
(2.7) In fact X > Y > 0 ensures X ⩾Y +εI > Y > 0 for some ε > 0, then Xγ ⩾(Y +εI)γ > Y γ for any γ ∈(0,1] by L¨ owner-Heinz inequality and we have (2.7).
[3, Corollary 2] asserts that logA > logB ⇐ ⇒ there exists β ∈(0,1] such that Aβ > Bβ.
(2.8) Applying (2.7) for γ = α β ∈(0,1] to (2.8), then we have Aα > Bα for any α ∈(0,β], so that we have (2.6) by Theorem 2.1.
□ REMARK 2.1. We remark that Theorem 2.2 is a further extension of Theorem A since Theorem 2.2 can be available for any non-constant operator monotone functions on [0,∞).
REMARK 2.2. For a simple proof of (2.7), we have only to put f(t) = tγ for γ ∈(0,1] in Theorem 2.1 since f(t) is a typical well known operator monotone. In fact (2.7) is cited in [2, p. 477, Corollary 8.6.11].
3. Concluding remark and a conjecture It is interesting to point out that an interesting contrast between A > B and logA > logB for A,B > 0 as follows.
REMARK 3.1. Let f(t) be any non-constant operator monotone function on [0,∞) and also let A and B be strictly positive operators.
(i) If A > B, then the following inequality holds: f(Aα) > f(Bα) for all α ∈(0,1].
(3.1) (ii) If logA > logB, then there exists β ∈(0,1] and the following inequality holds: f(Aα) > f(Bα) for all α ∈(0,β].
(3.2) 96 T. FURUTA Proof. If A > B, then Aα > Bα for any α ∈(0,1] by (2.7) and we have (3.1) by Theorem 2.1 and (ii) is already shown in Theorem 2.2.
It is reasonable understanding that the condition logA > logB in (ii) is weaker then A > B > 0 in (i), the corresponding result (3.2) is weaker than (3.1).
□ Theorem 2.2, Theorem A and Theorem B suggest the following conjecture.
CONJECTURE. There exist strictly positive operators A and B such that logA ⩾ logB, but f(Aα) ̸⩾f(Bα) for any non-constant operator monotone function f(t) on [0,∞) and for any α > 0.
We remark that useful and interesting results associated with §2 are discussed in , and .
Acknowledgement. We would like to express our cordial thanks to the referee for useful advice to improve the first version.
R E F E R E N C E S R. BHATIA, Matrix Analysis, Springer Verlag, New-York, 1997.
D. BERNSTEIN, Matrix Mathematics, Second Edition, 2009, Princeton University Press.
M. FUJII, J. F. JIANG AND E. KAMEI, Characterization of chaotic order and its application to Furuta inequality, Proc. Amer. Math, Soc. 125 (1997), 3655–3658.
M. FUJII, J. F. JIANG, E. KAMEI AND K. TANAHASHI, A Characterization of chaotic order and a problem, J. Inequality Appl. 2 (1998), 149–156.
T. FURUTA, Logarithmic order and dual logarithmic order, Operator Theory: Advances and Applica-tions 127 (2001), 279–290.
T. FURUTA, An operator monotone function t logt −t +1 log2t and strictly chaotoc order, Linear Algebra Appl. 341 (2002), 101–109.
J. F. JIANG, H. ZUO AND M. WANG, Two sequences of operator monotone functions under strictly chaotic order, Math. Inequal. Appl. 15 (2012), 371–380.
F. HIAI AND K. YANAGI, Hilbert space and linear operators, Makinoshoten, 1995, (in Japanese).
S. IZUMINO AND N. NAKAMURA, Operator monotone functions induced from L¨ owner-Heinz inequal-ity and strictly chaotic order, Math. Inequal. Appl. 7 (2004), 103–112.
M. K. KWONG, Some results on matrix monotone functions, Linear Algebra Appl. 118 (1989), 129– 153.
X. ZHAN, Matrix Inequalities, Springer Verlag, Berlin, 2002.
(Received March 5, 2012) Takayuki Furuta, Emeritus Professor Graduate School of Science and Technology Hirosaki University 1 Bunkyo-cho, Hirosaki, Aomori-ken 036-8560 Japan and 1-4-19 Kitayamachou Fuchu city Tokyo 183-0041 Japan e-mail: furuta@rs.kagu.tus.ac.jp Journal of Mathematical Inequalities www.ele-math.com jmi@ele-math.com |
11769 | https://en.wikipedia.org/wiki/45-45-90%20triangles?redirect=no | 45-45-90 triangles - Wikipedia
Jump to content
[x] Main menu
Main menu
move to sidebar hide
Navigation
Main page
Contents
Current events
Random article
About Wikipedia
Contact us
Contribute
Help
Learn to edit
Community portal
Recent changes
Upload file
Special pages
Search
Search
[x] Appearance
Appearance
move to sidebar hide
Text
Small Standard Large
This page always uses small font size
Width
Standard Wide
The content is as wide as possible for your browser window.
Color (beta)
Automatic Light Dark
This page is always in light mode.
Donate
Create account
Log in
[x] Personal tools
Donate
Create account
Log in
Pages for logged out editors learn more
Contributions
Talk
45-45-90 triangles
[x] Add languages
Add links
Article
Talk
[x] English
Read
Edit
View history
[x] Tools
Tools
move to sidebar hide
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Add interlanguage links
Print/export
Download as PDF
Printable version
In other projects
From Wikipedia, the free encyclopedia
Redirect to:
Special right triangle
Retrieved from "
This page was last edited on 28 December 2015, at 19:18(UTC).
Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Code of Conduct
Developers
Statistics
Cookie statement
Mobile view
Edit preview settings
Search
Search
45-45-90 triangles
Add languagesAdd topic |
11770 | https://books.google.com/books/about/Wheeler_s_Dental_Anatomy_Physiology_and.html?id=zPlpAAAAMAAJ | Wheeler's Dental Anatomy, Physiology, and Occlusion - Major M. Ash - Google Books
Sign in
Hidden fields
Try the new Google Books
Books
Add to my library
Try the new Google Books
Check out the new look and enjoy easier access to your favorite features
Try it now
No thanks
Try the new Google Books
My library
Help
Advanced Book Search
Get print book
No eBook available
Amazon.com
Barnes&Noble.com
Books-A-Million
IndieBound
Find in a library
All sellers»
### Get Textbooks on Google Play Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. Go to Google Play Now »
My library
My History
Wheeler's Dental Anatomy, Physiology, and Occlusion
Major M. Ash
W.B. Saunders, 1993 - Medical - 478 pages
This 7th Edition provides comprehensive, detailed coverage of the morphology of teeth relevant to the practice of dentistry. Approaches to data concerning standards of teeth formation and dental age assessment are included, as well as helpful radiographs and illustrations of tooth development. This text is practical, complete, and a standard for board study and continuing practical reference.
More »
From inside the book
Contents
Introduction 1
Development and Eruption of the Teeth 24
The Primary Deciduous Teeth 46
Copyright
15 other sections not shown
Other editions - View all
Wheeler's Dental Anatomy, Physiology, and Occlusion
Major M. Ash
Snippet view - 1993
Common terms and phrases
alveolaralveoliapexapical foramenAspect Figsbonebuccal aspectbuccal cuspcementoenamel junctioncentric occlusioncervical linecervical thirdconvexcrest of curvaturecrown and rootcusp ridgedeciduousdevelopmental grooveDiameter of Crowndistal aspectdistal cuspdistal marginal ridgedistobuccal cuspdistolingual cuspembrasureeruptionfunctiongingivalincisal aspectincisal ridgelabiolingualline anglelingual aspectlingual cusplingual surfacelobemandiblemandibular caninemandibular central incisormandibular first molarmandibular lateral incisormandibular molarsmandibular second molarmaxillamaxillary caninemaxillary central incisormaxillary first molarmaxillary first premolarmaxillary molarsmaxillary second molarmesial and distalmesial aspectmesial marginal ridgemesial or distalmesial rootmesial surfacemesiobuccal cuspMesiodistalMesiodistal sectionmesiolingualmesiolingual cuspmuscleocclusal aspectocclusal surfaceoutlineportionposterior teethprimary teethpulp cavitypulp chamberpulp hornsroot canalroot Figthird molartooth
References to this book
Illustrated Dental Embryology, Histology, and Anatomy
Mary Bath-Balogh,Margaret J. Fehrenbach
Snippet view - 2006
Pattern Biology and the Complex Architectures of Life
Michael Jay Katz
Snippet view - 1988
All Book Search results »
Bibliographic information
Title Wheeler's Dental Anatomy, Physiology, and Occlusion
AuthorMajor M. Ash
Edition 7, illustrated
Publisher W.B. Saunders, 1993
Original from the University of Michigan
Digitized Jul 18, 2008
ISBN 0721643744, 9780721643748
Length 478 pages
SubjectsMedical
›
Dentistry
›
General
Medical / Anatomy
Medical / Dentistry / General
Medical / Veterinary Medicine / General
Export CitationBiBTeXEndNoteRefMan
About Google Books - Privacy Policy - Terms of Service - Information for Publishers - Report an issue - Help - Google Home |
11771 | https://gmatclub.com/forum/how-many-integers-between-100-and-299-inclusive-have-a-units-digit-401329.html | How many integers between 100 and 299 (inclusive) have a units digit : Problem Solving (PS)
gmatclub
FORUMS
GMAT
MBA
RESOURCES
DEALS
REVIEWS
CHAT
All Forums Index Start a New Discussion General GMAT Questions Quantitative PS Verbal CR RC Data Insights DS G&T MSR TPA Ask GMAT Experts Share GMAT Experience
All Business School Discussions BSchool Application Questions Admitted - Which School to Choose? Profile Evaluations by Experts Student Loans & Financing Master's Programs (MiM, MFin, MSc) PhD in Business
GRE Forums General GRE Discussions Tech Support Downloads Blog
Diagnostic Quiz Question Banks GMAT Focus Tests Forum Quiz Error Log Question of the Day Study Plan
Business School Hub Decision Tracker What are my Chances Tool Request a Profile Evaluation Interview Debriefs Sample Essays Ask Admissions Consultants
GMATScore Calculator Study Plan Math Book Flashcards Practice Tests Overview Timing Strategies
MBAMBA Rankings 2025 - 2026 Deadlines All Application Fee Waivers Interview Release Dates Average GMAT Focus Scores School Stats Waitlists Scholarships & Negotiation Best MBA for Consulting, Finance, Tech MBA ROI Calculator
MBA ProgramsTuck (Dartmouth) Yale SOM Kenan-Flagler (UNC) Kelley (IU) Harvard Kellogg (Northwestern) Wharton (Upenn) Booth (Chicago) Sloan (MIT) Stanford Fuqua (Duke) ISB INSEAD
Goizueta (Emory) UC Riverside NTU CBS (Columbia) Haas (Berkeley) Ross (Michigan) Anderson (UCLA) Darden (UVA) Stern (NYU) LBS Johnson (Cornell) See All
GMATMarketplace - Compare all GMAT Deals e-GMAT - Save up to $525 Magoosh - Save up to $215 Manhattan Prep - Save up to $225 Target Test Prep - Save up to $579 GMATWhiz - Save up to $725 Experts' Global - Save up to $418 GMAT Tutoring Deals Compare All $500 Prodigy Loan Cashback
ADMISSION CONSULTINGAccepted.com Admissionado ApplicantLab Gatehouse Admissions mbaMission Square One Prep Stratus Admissions Counseling The Red Pen Admit Expert Amerasia ARINGO MBA Admissions Consulting ARLee Consulting Avanti Prep Embark MBA Experts' Global Compare All
Fortuna Admissions Gurufi Ivy Groupe Leland MBA Admit MBA and Beyond MBA Prep School Menlo Coaching OnCourse Vantage Personal MBA Coach Prep MBA Sia Admissions Stacy Blackman Consulting UniAdvise Vantage Point MBA Vikram Shah Consulting Compare All
STUDENT LOANSProdigy Finance Juno Financing Earnest Financing
See All Verified & Authentic Reviews GMAT COURSE REVIEWSe-GMAT Reviews Magoosh Reviews Manhattan Prep Reviews Target Test Prep Reviews Experts' Global Reviews GMATWhiz Reviews GMAT Tutors' Reviews See All Student Loan Reviews
Admissions Consulting ReviewsAccepted.com Reviews Admissionado Reviews ApplicantLab Reviews Gatehouse Reviews mbaMission Reviews Square One Prep Reviews Stratus Admissions Reviews The Red Pen Reviews Admit Expert Reviews Amerasia Reviews ARINGO MBA Admissions Consulting Reviews ARLee Consulting Reviews
Avanti Prep Reviews Embark MBA Reviews Experts' Global Reviews Fortuna Admissions Reviews Gurufi Reviews Ivy Groupe Reviews Leland Reviews MBA Admit Reviews MBA and Beyond Reviews MBA Prep School Reviews Menlo Coaching Reviews OnCourse Vantage Reviews Personal MBA Coach Reviews
Prep MBA Reviews Sia Admissions Reviews Stacy Blackman Consulting Reviews UniAdvise Reviews Vantage Point MBA Reviews Vikram Shah Consulting Reviews See All
Business School ReviewsHBS Reviews Wharton Reviews Stanford GSB Reviews Kellogg Reviews Booth Reviews Sloan Reviews ISB Reviews INSEAD Reviews Columbia Reviews Yale Reviews LBS Reviews Fuqua Reviews Ross Reviews Anderson Reviews See All
MAIN CHATROOM
- General GMAT Chat
- GMAT Questions Chat
- Manage WhatsApp Connection
- MBA Applicant WhatsApp Group
- GMAT Prep WhatsApp Group
- More GMAT & MBA Chatrooms
Sign InJoin now
Forum HomeGMATQuantitative QuestionsProblem Solving (PS)
GMAT Problem Solving (PS) Questions
It is currently 28 Sep 2025, 17:25
Watch on YouTube!GRE Quiz (20 Questions) @9:30am ET / 7pm ISTGMAT Quiz (30 Questions) @10:30am ET
Toolkit
Forum QuizPractice TestsQuestions BankDecision TrackerError LogMy PostsHot TopicsChat
Forum Home
GMAT
Quantitative Questions
Problem Solving (PS)
HomeGMATMessagesTestsSchoolsEventsRewardsChatMy Profile
Ask QuestionGMAT ForumsMBA ForumsExecutive MBAMasters ProgramsGRESupportHome
Make PostGeneral QuestionsQuantitativeVerbalData InsightsAsk GMAT ExpertsGMAT Experience
Start TestGMAT Focus TestsFree TestForum Quiz 3.0EA TestsGRE TestsError Log
Update StatusDecision Tracker
AndersonBoothCarlsonColumbiaCox (SMU)DardenFosterFuquaGoizuetaHaasHarvardHEC ParisIESEIIM AhmedabadIIM BangaloreIIM CalcuttaINSEADISBJohnson (Cornell)Jones RiceJudge CambridgeKelleyKelloggKenan-FlaglerLBSMarshall (USC)McCombsMcDonoughMendozaMerageNUS SingaporeOlin St. LouisOwen VanderbiltRoss (Michigan)RotmanSaidSchellerSchulichSchulichSimonSloan MITStanford GSBSternTepperTerry GeorgiaTuckUCR BusinessW.P. CareyWhartonYale
JoinPrize Fund ($30,000)RulesMy pointsQuestions
MBA Spotlight Fair Nov 13-14, 2024Sign upBenefitsApp Fee waiversActivate Tests
Latest VideoLatest PostLatest PostLatest TweetLatest PostLatest PostLatest PostLatest Post
Special Offer: Get 25% Off Target Test Prep GMAT Plans
Tier StatusPointsRedeemActivityTop MembersHow to Use?
New ChatMain ChatroomGeneral GMAT ChatPractice Questions ChatStudy Buddy ChatWhatsApp for GMATWhatsApp for MBA Applications
EditMy BookmarksMy PostsFollow FeedMy GMAT ScoreMy SchoolsNotificationsSettingsLog-In
Send PMInbox:SentDrafts
Ask QuestionGMAT ForumsMBA ForumsExecutive MBAMasters ProgramsGRESupportHome
Make PostGeneral QuestionsQuantitativeVerbalData InsightsAsk GMAT ExpertsGMAT Experience
Start TestGMAT Focus TestsFree TestForum Quiz 3.0EA TestsGRE TestsError Log
Update StatusDecision Tracker
AndersonBoothCarlsonColumbiaCox (SMU)DardenFosterFuquaGoizuetaHaasHarvardHEC ParisIESEIIM AhmedabadIIM BangaloreIIM CalcuttaINSEADISBJohnson (Cornell)Jones RiceJudge CambridgeKelleyKelloggKenan-FlaglerLBSMarshall (USC)McCombsMcDonoughMendozaMerageNUS SingaporeOlin St. LouisOwen VanderbiltRoss (Michigan)RotmanSaidSchellerSchulichSchulichSimonSloan MITStanford GSBSternTepperTerry GeorgiaTuckUCR BusinessW.P. CareyWhartonYale
JoinPrize Fund ($30,000)RulesMy pointsQuestions
MBA Spotlight Fair Nov 13-14, 2024Sign upBenefitsApp Fee waiversActivate Tests
Latest VideoLatest PostLatest PostLatest TweetLatest PostLatest PostLatest PostLatest Post
Special Offer: Get 25% Off Target Test Prep GMAT Plans
Tier StatusPointsRedeemActivityTop MembersHow to Use?
New ChatMain ChatroomGeneral GMAT ChatPractice Questions ChatStudy Buddy ChatWhatsApp for GMATWhatsApp for MBA Applications
EditMy BookmarksMy PostsFollow FeedMy GMAT ScoreMy SchoolsNotificationsSettingsLog-In
Send PMInbox:SentDrafts
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
Your Progress
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error LogLearn more
JOIN GMAT CLUB
Become part of our 1M+ member community!
×
GMAT Registration Benefits:
Vast Question Bank:
Practice thousands of GMAT questions with top expert solutions.
Error Tracking:
Identify and improve upon mistakes efficiently using our Error Log.
Expert Support:
Get the latest tips and news from our top GMAT professionals.
Sing up now
it’s free and easy!
Join now
Sign In
?
GMAT Club Timer Informer
Hi GMATClubber!
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
We’ll give you
an estimate of
your score
learn more
We’ll provide
personalized question
recommendations
learn more
Your score will improve
and your results
will be more realistic
learn more
Is there something wrong with our timer?Let us know!
Thanks! I'll try it now
Add Mistakes & Note
[x] Calculation Error
[x] Careless Mistake
[x] Anxiety
[x] Conceptual Gap
[x] Time Mismanagement
[x] Vocabulary Void
Add Note
Cancel Add
Request Expert Reply
Please wait...
ConfirmCancel
Events & Promotions
Sep 22 Special Offer: Get 25% Off Target Test Prep GMAT Plans
Sep 28 GRE Practice Test #41 - Free GRE Practice Covering Quant, Text Completion, Sentence Equivalence
Sep 28 GMAT Quiz #44 - Live GMAT Practice Covering Quant, Verbal, RC, & Data Insights
Sep 30 5 Quant & DI GMAT Hacks That Can Save You 10+ Minutes and Add 50+ Points
Sep 30 $325 Off TTP OnDemand Masterclass
Sep 30 Free Master’s in Europe? Scholarships & Hidden Hacks Explained | Masters (Ep4)
Oct 01 MBA INTERVIEW PREP SERIES
Oct 04 Free Webinar – Achieve 90th%ile score on GMAT CR and TPA
Oct 05 Ace Arithmetic on the GMAT Focus Edition
September 2025
SunMonTueWedThuFriSat
31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Special Offer: Get 25% Off Target Test Prep GMAT Plans
23
24
25
26
27
28
GRE Practice Test #41 - Free GRE Practice Covering Quant, Text Completion, Sentence Equivalence GMAT Quiz #44 - Live GMAT Practice Covering Quant, Verbal, RC, & Data Insights
29
30
5 Quant & DI GMAT Hacks That Can Save You 10+ Minutes and Add 50+ Points$325 Off TTP OnDemand Masterclass Free Master’s in Europe? Scholarships & Hidden Hacks Explained | Masters (Ep4)
1
MBA INTERVIEW PREP SERIES
2
3
4
Free Webinar – Achieve 90th%ile score on GMAT CR and TPA
Register Oct 05 Ace Arithmetic on the GMAT Focus Edition 11:00 AM IST
01:00 PM IST Attend this session to evaluate your current skill level, learn process skills, and solve through tough Arithmetic questions.
Save Now! Sep 22 Special Offer: Get 25% Off Target Test Prep GMAT Plans 12:00 PM EDT
11:59 PM EDT The Target Test Prep GMAT Flash Sale is LIVE! Get 25% off our game-changing course and save up to $450 today! Use code FLASH25 at checkout. This special offer expires on September 30, so grab your discount now!
Set a Reminder Sep 28 GRE Practice Test #41 - Free GRE Practice Covering Quant, Text Completion, Sentence Equivalence 06:30 AM PDT
07:15 AM PDT Solve GRE practice problems covering Quantitative reasoning, Verbal Reasoning, Text Completion, Sentence Equivalence, and Reading Comprehension Problems. Take this GRE practice quiz live with peers, analyze your GRE study progress, and more.
Set a Reminder Sep 28 GMAT Quiz #44 - Live GMAT Practice Covering Quant, Verbal, RC, & Data Insights 07:30 AM PDT
07:15 AM PDT Join us in a live GMAT practice session and solve 30 challenging GMAT questions with other test takers in timed conditions, covering GMAT Quant, Data Sufficiency, Data Insights, Reading Comprehension, and Critical Reasoning questions.
Set a Reminder Sep 30 5 Quant & DI GMAT Hacks That Can Save You 10+ Minutes and Add 50+ Points 08:30 AM PDT
09:30 AM PDT Do you constantly run out of time in the GMAT Quant and Data Insights sections? Losing 10+ minutes can cost you your target GMAT score. In this session, we reveal 5 powerful GMAT hacks that top scorers use to boost efficiency, accuracy, and confidence.
Save Now! Sep 30 $325 Off TTP OnDemand Masterclass 10:00 AM EDT
11:59 PM EDT Get a massive $325 off the TTP OnDemand GMAT masterclass by using the coupon code FLASH25 at checkout. If you prefer learning through engaging video lessons, TTP OnDemand GMAT is exactly what you need.
Watch Now! Sep 30 Free Master’s in Europe? Scholarships & Hidden Hacks Explained | Masters (Ep4) 10:00 AM PDT
11:00 AM PDT What if studying in Europe didn’t mean spending €40,000–50,000? What if you could study at HEC Paris, Bocconi, or ESSEC—and pay close to zero?
Register Now! Oct 01 MBA INTERVIEW PREP SERIES 09:00 AM PDT
11:00 AM PDT If you’re aspiring to join one of the top 20 MBA programs, we’ve got an exciting opportunity just for you! Welcome to the GMAT Club's Free Interview Prep Event
Register Today! Oct 04 Free Webinar – Achieve 90th%ile score on GMAT CR and TPA 11:00 AM IST
01:00 PM IST Attend this session to learn how to Deconstruct arguments,Pre-Think Author’s Assumptions, and apply Negation Test.
Register Oct 05 Ace Arithmetic on the GMAT Focus Edition 11:00 AM IST
01:00 PM IST Attend this session to evaluate your current skill level, learn process skills, and solve through tough Arithmetic questions.
Save Now! Sep 22 Special Offer: Get 25% Off Target Test Prep GMAT Plans 12:00 PM EDT
11:59 PM EDT The Target Test Prep GMAT Flash Sale is LIVE! Get 25% off our game-changing course and save up to $450 today! Use code FLASH25 at checkout. This special offer expires on September 30, so grab your discount now!
Back to Forum
Create Topic
Reply
How many integers between 100 and 299 (inclusive) have a units digit
Bunuel
Bunuel
Math Expert
Joined: 02 Sep 2009
Last visit: 28 Sep 2025
Posts: 104,347
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 104,347
Post URL10 Nov 2022, 09:03
Show timer
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
a 4%
b 4%
c 6%
d 87%
e 0%
A
B
C
D
E
Hide
Show
History
My Mistake
Official Answer and Stats are available only to registered users.Register/Login.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
15% (low)
Question Stats:
87% (01:18) correct 13%(01:56) wrong based on 52 sessions
History
Date
Time
Result
Not Attempted Yet
How many integers between 100 and 299 (inclusive) have a units digit between 3 and 9 (inclusive)?
A. 70
B. 90
C. 120
D. 140
E. 160
Show Hide Answer
Official Answer
Official Answer and Stats are available only to registered users.Register/Login.
New to the GMAT Club?
• Questions: Question Bank | Bunuel's Signature Collection
• Tests: GMAT Club Tests | Forum Quiz
• Guides: Quantitative | Verbal | Ultimate Quantitative Collection | All You Need for Quant
• Rules: Quantitative | Verbal
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
gmatophobia
gmatophobia
Quant Chat Moderator
Joined: 22 Dec 2016
Last visit: 28 Sep 2025
Posts: 3,169
Location: India
Concentration: Strategy, Leadership
Posts: 3,169
Post URL10 Nov 2022, 13:38
There are two cases possible
Case 1
The three digit integer starts with 1
1 _ _
The units digit can take any digit between 3 and 9, both inclusive , so the unit digit can be filled in 7 ways.
The tens place can take any digit between 0 and 9, both inclusive and hence can be filled in 10 ways.
Total number of ways = 7 10 = 70 ways.
Case 2
The three digit integer starts with 2
2 _ _
The units digit can take any digit between 3 and 9, both inclusive , so the unit digit can be filled in 7 ways.
The tens place can take any digit between 0 and 9, both inclusive and hence can be filled in 10 ways.
Total number of ways = 7 10 = 70 ways.
Total = 140 ways
Option D
Want to discuss quant questions and strategies : Join the quant chat group today
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
SaquibHGMATWhiz
SaquibHGMATWhiz
GMATWhiz Representative
Joined: 23 May 2022
Last visit: 12 Jun 2024
Posts: 623
Location: India
GMAT 1:760 Q51 V40
Expert
Expert reply
GMAT 1:760 Q51 V40
Posts: 623
Post URL11 Nov 2022, 01:14
Bunuel
How many integers between 100 and 299 (inclusive) have a units digit between 3 and 9 (inclusive)?
A. 70
B. 90
C. 120
D. 140
E. 160
Show more
Solution:
Let us divide this set like this:
100-110: 7 such integers (103, 104, 105, 106, 107, 108 and 109)
111-120: 7 such integers (113, 114, 115, 116, 117, 118 and 119)
121-130: 7 such integers
..
..
281-290: 7 such integers
291-299: 7 such integers
20 such subsets of 7 integers each =7×20=140=7×20=140
Hence the right answer is Option D
Register for Free GMAT Live SessionsEvery week we conduct 3 FREE live sessions on various topics in Quant, Verbal, Strategy and Admission Consulting. You can register for the webinars - HERE
GMATWhiz FREE Trial: Prepare from an application driven course that serves real-time improvement modules along with a well-defined adaptive study plan. Start a free trial to experience it yourself and get access to 25 videos and 300 GMAT styled questions.
Score Improvement Strategy: How to score Q50+ on GMAT | 5 steps to Improve your Verbal score
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
NEW TOPIC
POST REPLY
Question banks
Downloads
My Bookmarks
Important topics
Reviews
Similar topics
Similar Topic
Author
Kudos
Replies
Last Post
How many integers between 100 and 299 (inclusive) have a units digit
Bunuel
by: Bunuel
28 Oct 2022, 05:03
1
1
How many integers between 100 and 400, inclusive, contain the digit 2?
Bunuel
by: Bunuel
08 Jun 2025, 05:31
3
18
How many integers between 100 and 999, inclusive, have the property th
Bunuel
by: Bunuel
17 Nov 2024, 01:10
5
36
Of the integers between 100 and 699, inclusive, how many do not have
kingbucky
by: kingbucky
21 Feb 2025, 10:46
2
4
How many integers between 1 and 200, inclusive, have a 2 in the units
Bunuel
by: Bunuel
13 Dec 2022, 09:38
2
7
Moderators:
Bunuel
Math Expert
104347 posts
Krunaal
Tuck School Moderator
795 posts
Prep Toolkit
Announcements
How to Analyze your GMAT Score Report
Monday, Sep 29, 2025
11:30am NY / 3:30pm London / 9pm Mumbai
CLOSE SAVE
SUNDAY Quizzes! GRE Quiz @9:30am ET & GMAT Quiz @10:30am ET
Sunday, Sep 28, 2025
9:30am NY / 1:30pm London / 7pm Mumbai
Sunday, Sep 28, 2025
10:30am NY / 2:30pm London / 8pm Mumbai
CLOSE SAVE
What Makes a Great MBA Interview? Find Out from MBA Consultants and Students
CLOSE SAVE
An Emory MBA Powers Growth
Emory powers growth—professionally and personally. Built on a foundation of interdisciplinary learning, leadership development, and career readiness, our top-20 MBA program delivers a high return on investment with top-5 career outcomes.
CLOSE SAVE
Tuck at Dartmouth
Step Inside the Tuck Experience: Discover Tuck Events This Fall Current Tuck MBA students and Discover Tuck co-chairs share advice for prospective students and reflect on why they chose Tuck for their MBA.
Read More
CLOSE SAVE
AGSM at UNIVERSITY OF CALIFORNIA RIVERSIDE
CLOSE SAVE
Latest Posts
More Latest Posts
A company makes 40% of its revenue from product A at a profit margin
by:ajain31
1 hr
If the average (arithmetic mean) of 4 numbers is 50, how man
by:bumpbot
1 hr
If x,y,and z are positive numbers,what is the value of the average (ar
by:bumpbot
1 hr
Calling all MIT Sloan MBA Applicants: 2026 Intake Class of 2028
by:AWasey
1 hr
The 35-millimeter (mm) format for movie production became a de facto
by:VerbalBot
1 hr
Copyright © 2025 GMAT Club
Terms & ConditionsForum Rules
Get our application on:
App StorePlay Store
Follow us on:
Contact us:
SupportAdvertise
GMAT® is a registered trademark of the Graduate Management Admission Council ™
GMAT Club's website has not been reviewed or endorsed by GMAC®
Terms & ConditionsForum Rules
The post is bookmarked successfully
view my bookmarksreturn to the post
[x] don't show this message again
by Problem Solving (PS)
JOIN NOW
Check Answer
×
You must be logged in to check and save answers
LoginRegister
Or continue with
GoogleApple |
11772 | https://www.sciencedirect.com/science/article/abs/pii/S0966842X04001180 | Skip to article
My account
Sign in
Access through your organization
Purchase PDF
Patient Access
Article preview
Abstract
Section snippets
References (48)
Cited by (710)
## Trends in Microbiology
Volume 12, Issue 7, July 2004, Pages 317-324
The distinct morphogenic states of Candida albicans
Author links open overlay panel, ,
rights and content
Abstract
The human fungal pathogen, Candida albicans can grow in at least three different morphologies: yeast, pseudohyphae and hyphae. Further morphological forms exist during colony switching, for example, opaque phase cells are oblong, rather than the oval shape of yeast cells. Pseudohyphae and hyphae are both elongated and sometimes there has been little attempt to distinguish between them, as both are filamentous forms of the fungus. We review here the differences between them that suggest that they are distinct morphological states. We argue that studies on filamentous forms should always include a formal analysis to determine whether the cells are hyphae or pseudohyphae and we suggest some simple experimental criteria that can be applied to achieve this.
Section snippets
Cell shape
The traditional criterion for distinguishing between different forms is cell shape . Hyphae that develop from an unbudded yeast cell (also termed a blastospore) have no constriction at the neck of the mother cell and have parallel sides along their entire length. The formation of unconstricted filaments in response to serum is the basis of the germ tube test, which is used in clinical diagnoses to distinguish C. albicans from other Candida species ; although Candida dubliniensis, the
Cell cycle
The most fundamental differences between hyphae and pseudohyphae relate to the organization of their cell cycles. In the following discussion, we shall consider an unbudded yeast cell that is induced to form hyphae or pseudohyphae by appropriate stimuli as discussed previously, or that is allowed to continue growing in the yeast form. In C. albicans the terms hypha and germ tube are often used as synonyms. The hyphal projection that forms in the first cell cycle, before septation, is called a
Polarized growth
In both hyphae and yeast, polarized growth is associated with polarization of the actin cytoskeleton (Figure 2a) 27, 38. As in S. cerevisiae, the actin cytoskeleton of C. albicans is composed of cortical actin patches and cables. The actin patches are thought to organize endocytosis, whereas actin cables direct the transport of secretory vesicles driven by the Myo2 motor protein. In the yeast form, buds first grow in a polarized fashion until they reach approximately two-thirds of their final
Genetic determination of hyphae and pseudohyphae formation
Interestingly, there have been several recent reports of mutations that result in a constitutive pseudohyphal phenotype. These include tup1Î/tup1Î , nrg1Î/nrg1Î , fkh2Î/fkh2Î , gin4Î/gin4Î and hsl1Î/hsl1Î . An example of one of these, hsl1Î/hsl1Î is shown in Figure 1h. Note how the regular pattern of branching conforms to the theoretical pattern for a mature pseudohypha (Figure 2b). Interestingly, except for nrg1Î/nrg1Î, none of these mutants was able to respond to serum by
Experimental guidelines for distinguishing hyphae and pseudohyphae
The literature is replete with descriptions of filamentous growth, which do not distinguish between hyphae or pseudohyphae. Without more rigor in the discrimination of hyphae and pseudohyphae, there is a danger of increasing confusion. This is particularly true with the use of microarray studies that are used to study gene expression during morphological development. In Table 2 several features are listed that distinguish hyphae and pseudohyphae. It would be unrealistic to address all of these
Concluding remarks
We have documented here fundamental differences in the organization of the first and subsequent cell cycles as pseudohyphae and hyphae develop from unbudded yeast cells. Despite these well-documented differences, analysis of morphology is hindered by several inter-related problems. The C. albicans morphology is enormously plastic and is very sensitive to different culture regimes, as has been carefully documented . A related problem is the tendency for the morphological composition of in
Acknowledgments
P.E. Sudbery and N.A.R. Gow thank the BBSRC and Wellcome Trust for financial support of work relating to this review.
References (48)
M.G Miller et al.
### White-opaque switching in Candida albicans is controlled by mating-type locus homeodomain proteins and allows efficient mating
### Cell
(2002)
N.A.R Gow
### Fungal morphogenesis and host invasion
### Curr. Opin. Microbiol.
(2002)
H.P Liu
### Transcriptional control of dimorphism in Candida albicans
### Curr. Opin. Microbiol.
(2001)
M Whiteway
### Transcriptional control of cell type and morphogenesis in Candida albicans
### Curr. Opin. Microbiol.
(2000)
C.J Gimeno
### Unipolar cell divisions in the yeast Saccharomyces cerevisiae lead to filamentous growth: Regulation by starvation and RAS
### Cell
(1992)
A.S Kao
### The epidemiology of candidemia in two United States cities: Results of a population-based active surveillance
### Clin. Infect. Dis.
(1999)
M.B Edmond
### Nosocomial bloodstream infections in United States hospitals: A three-year analysis
### Clin. Infect. Dis.
(1999)
C.M Beck-Sague et al.
### Secular trends in the epidemiology of nosocomial fungal infections in the United states 1980-1990
### J. Infect. Dis.
(1993)
L.G Miller
### Estimating the cost of nosocomial candidemia in the United States
### Clin. Infect. Dis.
(2001)
L.A Merson-Davies et al.
### A morphology index for cell shape in Candida albicans
### J. Gen. Microbiol.
(1989)
Odds, F.C. (1988) Candida and Candidosis. Balliere Tindall,...
F.C Odds
### Morphogenesis in Candida albicans
### Crit. Rev. Microbiol.
(1985)
- J Berman et al.
### Candida albicans: A molecular revolution built on lessons from budding yeast
### Nat. Rev. Genet.
(2002)
- Gow, N.A.R. (1994) The growing fungus. (Gadd, G.M., ed.), pp. 404431, Chapman and...
- Gow, N.A.R. (2002) Candida and Candiasis. (Calderone, R., ed.), pp. 145158, American Society for Microbiology,...
- N.A.R Gow
### Germ tube growth of Candida albicans
### Curr. Top. Med. Mycol.
(1997)
- B Slutsky
### High-frequency switching of colony morphology in Candida albicans
### Science
(1985)
- Soll, D.R. (2002) Candida and Candidiasis. (Calderone, R., ed.), pp. 123142, ASM Press,...
- B Slutsky
### White-opaque transition: a second high frequency transition in Candida albicans
### J. Bacteriol.
(1987)
- J.M Hornby
### High phosphate (up to 600mM) induces pseudohyphal development in five wild type Candida albicans
### Journal of Microbiological Methods
(2003)
- Brown, A.J.P. (2002) Candida and Candidiasis. (Calderone, R., ed.), pp. 95106, American Society for Microbiology,...
- D.M Heddon et al.
### A re-emphasis germ tubes diagnostic for Candida albicans have no constrictions
### Mycopathologia
(1980)
- G.D Gilfillan
### Candida dubliniensis: phylogeny and putative virulence factors
### Microbiolology
(1998)
- M.J Sevilla et al.
### Development of Candida albicans hyphae in different growth media variations in growth-rates, cell dimensions and timing of morphogenetic events
### J. Gen. Microbiol.
(1986)
Cited by (710)
Neutrophil extracellular traps contain calprotectin, a cytosolic protein complex involved in host defense against Candida albicans
2009, Plos Pathogens
Neutrophils are the first line of defense at the site of an infection. They encounter and kill microbes intracellularly upon phagocytosis or extracellularly by degranulation of antimicrobial proteins and the release of Neutrophil Extracellular Traps (NETs). NETs were shown to ensnare and kill microbes. However, their complete protein composition and the antimicrobial mechanism are not well understood. Using a proteomic approach, we identified 24 NET-associated proteins. Quantitative analysis of these proteins and high resolution electron microscopy showed that NETs consist of modified nucleosomes and a stringent selection of other proteins. In contrast to previous results, we found several NET proteins that are cytoplasmic in unstimulated neutrophils. We demonstrated that of those proteins, the antimicrobial heterodimer calprotectin is released in NETs as the major antifungal component. Absence of calprotectin in NETs resulted in complete loss of antifungal activity in vitro. Analysis of three different Candida albicans in vivo infection models indicated that NET formation is a hitherto unrecognized route of calprotectin release. By comparing wild-type and calprotectin-deficient animals we found that calprotectin is crucial for the clearance of infection. Taken together, the present investigations confirmed the antifungal activity of calprotectin in vitro and, moreover, demonstrated that it contributes to effective host defense against C. albicans in vivo. We showed for the first time that a proportion of calprotectin is bound to NETs in vitro and in vivo.
### Candida albicans pathogenicity mechanisms
2013, Virulence
### Growth of Candida albicans hyphae
2011, Nature Reviews Microbiology
### Staphylococcus aureus biofilms: Properties, regulation and roles in human disease
2011, Virulence
### Regulatory circuitry governing fungal development, drug resistance, and disease
2011, Microbiology and Molecular Biology Reviews
### Systematic screens of a Candida albicans homozygous deletion library decouple morphogenetic switching and pathogenicity
2010, Nature Genetics
View all citing articles on Scopus
View full text
Copyright © 2004 Elsevier Ltd. All rights reserved. |
11773 | http://labman.phys.utk.edu/phys222core/modules/m5/faraday.html | Faraday's law
Faraday's law is one of Maxwell's equations. Faraday's law states that the absolute value or magnitude of the circulation of the electric field E around a closed loop is equal to the rate of change of the magnetic flux through the area enclosed by the loop. The equation below expresses Faraday's law in mathematical form.
ΔΦB/∆t (through a fixed area) = -Σaround loop E∙∆r (at a fixed time)
The minus sign in this equation tells us about the direction of the circulation. (See below.)
When the magnetic flux through the area enclosed by the loop changes, Σaround loop E∙∆r is not zero, the electric field E circulates.E∙∆r is the work done per unit charge by the electric field in moving the charge a distance ∆r.If the loop is an actual wire loop, then there is actual work done by the induced field on free charges. Σaround loop E∙∆r is the work per unit charge by the field in moving the charge once around the loop.This is an induced emf, and it is measured in Volts.The induced emf causes a current to flow without a potential difference due to separated charges.
ΔΦB/∆t (through a fixed area) = induced emf
The induced electric field is NOT a conservative field. When you move a charge against the induced field once around the loop, you have to do work. But your work is NOT stored as potential energy. You cannot let the electric field do work to recover the energy you expended in moving the charge. The induced electric field disappears as soon as the magnetic flux is no longer changing. The work you do on a charge against the induced field is not locally stored. The energy may be transported away in the form of an electromagnetic wave. Electromagnetic waves carry energy through free space.
What is the direction of the dynamic (induced) field?
The minus sign in the equation expressing Faraday's law tells us about the direction of the induced field. There is an easy way to remember this direction. The circulation of the induced field is equal to an emf. Any current flowing as the result of that emf produces a magnetic field that opposes the flux changes that produce it. This is called Lenz's law.
An induced emf acts as to oppose the CHANGE in flux that produce it.
Example:
A magnet is moved quickly towards a wire loop as shown. The flux through the wire loop is increasing in the downward direction. A current starts flowing in the loop in the direction indicated by the arrow. The magnetic field produced by this current points upward, it opposes the flux changes that produce it.The magnetic force due to the loop on the magnet acts to slow down the approaching magnet.
The beauty of Lenz's law is that you do not have to look at the details. If the magnetic flux through a conductor changes, currents will flow to oppose whatever caused the change. If some relative motion causes a flux change, the current will try to stop that relative motion. If a change in a current in a circuit is responsible for the flux change, then the induced emf will try to prevent the change in the current of that circuit.
Please watch: Electromagnetic Induction and Faraday's Law (Youtube)
Problem:
Consider a flat square coil with N = 5 loops. The coil is 20 cm on each side, and has a magnetic field of 0.3 T passing through it. The plane of the coil is perpendicular to the magnetic field: the field points out of the page. (a) If nothing is changed, what is the induced emf? (b) The magnetic field increases uniformly from 0.3 T to 0.8 T in 1 s. While the change is taking place, what is the induced emf in the coil? (c) While the magnetic field is changing, the emf induced in the coil causes a current to flow. Does the current flow clockwise or counter-clockwise around the coil?
Solution:
Reasoning:If the magnitude of the magnetic field B changes, then the flux Φ = BA changes, and an emf is induced.
Details of the calculation:(a) An emf is induced by a changing magnetic flux. If nothing changes, the induced emf is zero. (b) The coil has 5 turns. Each turn has area A = (0.2 m)2. The initial magnetic flux through each turn of the the coil is Φ0 = B0A = 0.3(0.2)2 Tm2 = 0.012 Tm2. The final magnetic flux through each turn of the coil is Φf = BfA = 0.8(0.2)2 Tm2 = 0.032 Tm2. The total change of the flux through the coil N(Φf - Φ0), with N = 5. The induced emf is emf = -N∆Φ/∆t = -N(Φf - Φ0)/∆t = [-5(0.032 -0.012)/1.0] V = -0.1 V. (c) While the magnetic field is changing, the magnetic flux increased out of the page. According to Lenz's law, the emf induced in the loop by this changing flux produces a current that sets up a field opposing the change. The field set up by the current in the coil, points into the page, opposite to the direction of the increase in flux. To produce a field into the page, the current must flow clockwise around the loop according to the right-hand rule.
Embedded Question 1
A bar magnet is positioned in front of a horizontal loop of wire with its north pole pointing toward the loop. Then the magnet is pulled away from the loop. Does the induced current in the loop flow clockwise or counterclockwise?
Discuss this with your fellow students in the discussion forum! Visualize the magnetic field of a bar magnet. How does the flux of this field through the wire loop change?
Self induction
If a long coil of wire of cross sectional area A and length ℓ with N turns is connected or disconnected from a battery, the changing magnetic flux through the coil produces an induced emf. The induced current produces a magnetic field, which opposes the change in the magnetic flux. The magnitude of the induced emf can be calculated using Faraday's law.
The magnetic field inside the long coil is B = μ0(N/ℓ)I.
The flux through the coil is NBA = μ0(N2/ℓ)IA.
The change in flux per unit time is μ0(N2/ℓ)A ∆I/∆t = L∆I/∆t, since I is the only quantity changing with time. L = μ0(N2/ℓ)A is called the self inductance of the coil. The units of inductance are Henry (H). 1 H = 1 Vs/A.
The induced emf is emf = -L∆I/∆t, where the minus sign is a consequence of Lenz's law.
The induced emf is proportional to the rate of change of the current in the coil. It can be several times the power supply voltage. When a switch in a circuit carrying a large current is opened, reducing the current to zero in a very short time interval, this can result in a spark. All circuits have self inductance, and we always have emf = -L∆I/∆t. The self inductance L depends only on the geometry of the circuit.
Problem:
A coil has an self inductance of 3 mH, and a current through it changes from 0.2 A to 1.5 A in a time of 0.2 s. Find the magnitude of the average induced emf in the coil during this time.
Solution:
Reasoning:The emf due to self induction is emf = -L∆I/∆t.
Details of the calculation:L = 3 mH, ∆I/∆t = (1.5 A - 0.2 A)/0.2 s = 6.5 A/s. emf = -L∆I/∆t = -(0.003 Vs/A)(6.5 A/s) = -0.0195 V.The minus sign indicates that the induced emf opposes the flux changes that produced it.
Problem:
A 25 turn circular coil of wire has a diameter of 1 m. It is placed with its axis along the direction of the Earth's magnetic field (magnitude 50 microT), and then, in 0.2 s, it is flipped 180o. What is the average emf generated
Solution:
Reasoning:ΦB = B∙A is the flux of B through the area A. Initially B and A are aligned, finally they are anti-aligned. The dot product changes sign.
Details of the calculation:emf = -∆ΦB/∆t. ΦB(initial) = NAB = 25π(0.5 m)25010-6 T = 9.8210-4Tm2.ΦB(final) = -ΦB(initial) since the coil is flipped.|∆ΦB| = 2ΦB(initial).|∆ΦB/∆t| = 2(9.8210-4 Tm2)/(0.2 s) = 9.8210-3 V.
Problem:
A 0.5 m radius, 500-turn coil is rotated one-fourth of a revolution in 4.17 ms, originally having its plane perpendicular to a uniform magnetic field. Find the magnetic field strength needed to induce an average emf of 10,000 V.
Solution:
Reasoning:emf = -∆ΦB/∆t. ΦB = NABcosθ changes from NAB to 0 in 4.17 ms, since θ changes from 0 to 90o in 4.17 ms.
Details of the calculation:|∆ΦB| = NAB = 500π(0.5 m)2B = (393 m2) B.We want |emf| = |∆ΦB/∆t| = (393 m2)/(4.1710-3 s) B = (94174 m2/s)B = 10000V.B = 0.1 Vs/m2 = 0.1 T.
If you miss having regular lectures, consider this video lecture.
Lecture 16: Electromagnetic Induction |
11774 | https://www.educationworld.com/teachers/step-step-success-breaking-down-process-solving-multi-step-word-problems | Published Time: Sun, 24 Aug 2025 05:52:43 GMT
Step-by-Step Success: Breaking Down the Process of Solving Multi-step Word Problems | Education World
Jump to Navigation
Enter your email for FREE Newsletter!
By signing up, you agree to our privacy policy.
Search
Search
☰
Teacher Essentials »
Featured Articles
Best Practices
Report Card Comments
Needs Improvement Comments
Teacher's Lounge
New Teachers
Our Bloggers
Lesson Plans »
Featured Lessons
Every-Day Edits
Lesson Library
Emergency Sub Plans
Character Education
Lesson of the Day
5-Minute Lessons
Learning Games
Lesson Planning
Standards
Subjects Center
Teaching Grammar
Admin »
Featured Articles
Hot Topics
Leadership Resources
Parent Newsletter Resources
Advice from School Leaders
Programs, Strategies and Events
Principal Toolbox
Administrator's Desk
Interview Questions
Professional Learning Communities
Teachers Observing Teachers
Tech »
Featured Articles
Article Library
Tech Lesson Plans
Science, Math & Reading Games
WebQuests
Tech in the Classroom
Tech Tools
Web Site Reviews
Creating a WebQuest
Digital Citizenship
PD Tips »
PD Tips & Ideas
Backpack-tivities
Behavior Management Tips of the Week
Bulletin Board Ideas
Book Report Makeover
Classroom Management
Responsive Classroom
Dr. Ken Shore: Classroom Problem Solver
Professional Development Resources
Dr. Fred Jones: Tools for Teaching
Worksheets »
Worksheet Library
Highlights for Children
Venn Diagram Templates
Reading Games
Word Search Puzzles
Math Crossword Puzzles
Friday Fun
Geography A to Z
Holidays & Special Days
Internet Scavenger Hunts
Student Certificates
Templates
Newsletter Sign Up
WHAT'S NEW
Making the Most of Preservice- Five Actions for New Te...
Probability Playground: Games of Chance!
The 5-Minute Planning Protocol: Streamline Your Week
Home>>Teachers>>Step-by-Step Success: Breaking Down the Process of Solving Multi-step Word Problems
Search form
Search
Step-by-Step Success: Breaking Down the Process of Solving Multi-step Word Problems
Solving multi-step word problems can be a scary task for your students. However, you can transform this challenge into a manageable and enjoyable process with the right strategies and tools.
Process to Solve Multi-step Word Problems
Here's how to break down multi-step word problems to help your students navigate them confidently and skillfully.
Step #1 Understand the Problem
To solve any word problem, your student needs to understand what is being asked. Encourage your students to:
Read the problem thoroughly:Your students should read it at least twice to ensure they understand all the details.
Highlight key information:This includes numbers, units of measurement, and keywords that indicate mathematical operations (e.g., "total," "difference," "product.")
Identify the question: What is the problem asking for? Understanding the final goal is crucial for planning the steps to reach it.
Example:
"Sarah has 5 apples. She buys 7 more apples and then gives 3 to her friend. How many apples does she have in total?"
Step #2 Plan the Solution
Once your students understand the word problem, they will plan their approach. The plan will involve identifying the operations needed and the order in which they should be performed.
Break the problem into smaller steps: Each step should involve a single or small set of operations.
Write down intermediate goals:This helps keep track of progress and makes the process less overwhelming.
Example Continued:
Determine the total number of apples Sarah has after buying more.
Subtract the number of apples she gives to a friend.
Step #3 Execute the Plan
Now, your students are ready to perform the calculations step by step. This stage requires careful calculation and attention to detail.
Perform each operation sequentially:Stick to the plan made in the previous step.
Keep work organized:Writing each step helps prevent mistakes and makes it easier to review the work.
Example Continued:
5 apples + 7 apples = 12 apples
12 apples - 3 apples = 9 apples
Sarah has 9 apples in total.
Step #4 Check the Work
After solving the problem, teach your students to review their work to ensure accuracy.
Re-read the problem:Make sure the solution addresses the question asked.
Recalculate: Double-check the calculations to confirm the results.
Estimate:Use estimation to check if the answer is reasonable. This is a good way to catch errors.
Additional Tools for Teaching Multi-Step Word Problems
To engage your students and promote understanding, try using these tools:
Visual Aids
Diagrams and charts:Drawing a diagram or a chart can help students visualize the word problem. For example, a bar graph can represent quantities and their relationship to one another.
Graphic organizers:These help break down the problem into manageable parts, such as a flow chart or a step-by-step organizer.
Interactive Activities
Math games:Incorporate games that require solving word problems in multiple steps. Games make learning fun and reinforce concepts through practice.
Group work:Have students work in pairs or small groups to solve word problems. Groups promote discussion and collaborative learning in your classroom.
Real-life Scenarios
Relatable examples:Use and create problems based on relevant real-life situations. Refer to pop culture topics, like Taylor Swift. (Or Paw Patrol for younger students.) Using examples your students are familiar with allows them to see the practical application of their math skills.
Project-based learning:Create projects where students solve multi-step problems related to a larger task, such as planning a school event or creating a budget for a classroom activity.
Teaching Example: Planning a Class Party
Let's use the example of planning a class party to show how to break down a multi-step word problem:
Problem:
Your class must buy decorations, snacks, and drinks for an end-of-year party. Decorations cost $25, snacks cost $30, and drinks cost $15. If the class has $80, how much money will be left after buying everything?
Step-by-Step Solution:
Identify the costs:
Decorations: $25
Snacks: $30
Drinks: $15
Total cost:
$25 + $30 + $15 = $70
Calculate the remaining money:
$80 - $70 = $10
Final answer: The class will have $10 left after buying everything.
Encourage a Growth Mindset
Fostering a growth mindset in students emphasizes that making mistakes is a part of learning. As your students develop their skills, they will learn that persistence is key.As their teacher, celebrate their progress, no matter how small. And in no time, they will be able to solve multi-step word problems.
Written by Brooke Lektorich
Education World Contributor
Copyright© 2024 Education World
EW Lesson Plans
More than 1,000 FREE Lessons Reading, Writing, Math, Science, History & Every-Day Edits!
EW Professional Development
Online PD Courses with dozens of topics to choose from for your CEU credits and building new skills!
EW Worksheets
1000's of Free Ready-to-Print Student Worksheets, Templates, Certificates & Every-Day Edits!
Chatter
Trending
125 Report Card Comments It's report card time and you face the prospect of writing constructive, insightful, and original comments on a couple dozen report cards or more. Here are 125 positive report card comments for you to use and adapt! Struggling Students? Check out our Needs Improvement Report Card Comments for even more comments! You've reached the end of another grading period, and what could be more daunting than the task of composing insightful, original, and unique comments about every child in your class? The following positive statements will help you tailor your comments to specific children and highlight their strengths. You can also use our statements to indicate a need for improvement. Turn the words around a bit, and you will transform each into a goal for a child to work toward. Sam cooperates consistently with others becomes Sam needs to cooperate more consistently with others, and Sally uses vivid language in writing may instead read With practice, Sally will learn to use vivid language in her writing. Make Jan seeks new challenges into a request for parental support by changing it to read Please encourage Jan to seek new challenges. Whether you are tweaking statements from this page or creating original ones, check out our Report Card Thesaurus [see bottom of the page] that contains a list of appropriate adjectives and adverbs. There you will find the right words to keep your comments fresh and accurate. We have organized our 125 report card comments by category. Read the entire list or click one of the category links below to jump to that list. AttitudeBehaviorCharacterCommunication SkillsGroup WorkInterests and TalentsParticipationSocial SkillsTime ManagementWork Habits Attitude The student: is an enthusiastic learner who seems to enjoy school. exhibits a positive outlook and attitude in the classroom. appears well rested and ready for each day's activities. shows enthusiasm for classroom activities. shows initiative and looks for new ways to get involved. uses instincts to deal with matters independently and in a positive way. strives to reach their full potential. is committed to doing their best. seeks new challenges. takes responsibility for their learning. Behavior The student: cooperates consistently with the teacher and other students. transitions easily between classroom activities without distraction. is courteous and shows good manners in the classroom. follows classroom rules. conducts themselves with maturity. responds appropriately when corrected. remains focused on the activity at hand. resists the urge to be distracted by other students. is kind and helpful to everyone in the classroom. sets an example of excellence in behavior and cooperation. Character The student: shows respect for teachers and peers. treats school property and the belongings of others with care and respect. is honest and trustworthy in dealings with others. displays good citizenship by assisting other students. joins in school community projects. is concerned about the feelings of peers. faithfully performs classroom tasks. can be depended on to do what they are asked to do. seeks responsibilities and follows through. is thoughtful in interactions with others. is kind, respectful and helpful when interacting with his/her peers is respectful of other students in our classroom and the school community demonstrates responsibility daily by caring for the materials in our classroom carefully and thoughtfully takes his/her classroom jobs seriously and demonstrates responsibility when completing them is always honest and can be counted on to recount information when asked is considerate when interacting with his/her teachers demonstrates his/her manners on a daily basis and is always respectful has incredible self-discipline and always gets his/her work done in a timely manner can be counted on to be one of the first students to begin working on the task that is given perseveres when faced with difficulty by asking questions and trying his/her best does not give up when facing a task that is difficult and always does his/her best is such a caring boy/girl and demonstrates concern for his/her peers demonstrates his/her caring nature when helping his/her peers when they need the assistance is a model citizen in our classroom is demonstrates his/her citizenship in our classroom by helping to keep it clean and taking care of the materials in it can always be counted on to cooperate with his/her peers is able to cooperate and work well with any of the other students in the class is exceptionally organized and takes care of his/her things is always enthusiastic when completing his/her work is agreeable and polite when working with others is thoughtful and kind in his/her interactions with others is creative when problem solving is very hardworking and always completes all of his/her work is patient and kind when working with his/her peers who need extra assistance trustworthy and can always be counted on to step in and help where needed Communication Skills The student: has a well-developed vocabulary. chooses words with care. expresses ideas clearly, both verbally and through writing. has a vibrant imagination and excels in creative writing. has found their voice through poetry writing. uses vivid language in writing. writes clearly and with purpose. writes with depth and insight. can make a logical and persuasive argument. listens to the comments and ideas of others without interrupting. Group Work The student: offers constructive suggestions to peers to enhance their work. accepts the recommendations of peers and acts on them when appropriate. is sensitive to the thoughts and opinions of others in the group. takes on various roles in the work group as needed or assigned. welcomes leadership roles in groups. shows fairness in distributing group tasks. plans and carries out group activities carefully. works democratically with peers. encourages other members of the group. helps to keep the work group focused and on task. Interests and Talents The student: has a well-developed sense of humor. holds many varied interests. has a keen interest that has been shared with the class. displays and talks about personal items from home when they relate to topics of study. provides background knowledge about topics of particular interest to them. has an impressive understanding and depth of knowledge about their interests. seeks additional information independently about classroom topics that pique interest. reads extensively for enjoyment. frequently discusses concepts about which they have read. is a gifted performer. is a talented artist. has a flair for dramatic reading and acting. enjoys sharing their musical talent with the class. Participation The student: listens attentively to the responses of others. follows directions. takes an active role in discussions. enhances group discussion through insightful comments. shares personal experiences and opinions with peers. responds to what has been read or discussed in class and as homework. asks for clarification when needed. regularly volunteers to assist in classroom activities. remains an active learner throughout the school day. Social Skills The student: makes friends quickly in the classroom. is well-liked by classmates. handles disagreements with peers appropriately. treats other students with fairness and understanding. is a valued member of the class. has compassion for peers and others. seems comfortable in new situations. enjoys conversation with friends during free periods. chooses to spend free time with friends. Time Management The student: tackles classroom assignments, tasks, and group work in an organized manner. uses class time wisely. arrives on time for school (and/or class) every day. is well-prepared for class each day. works at an appropriate pace, neither too quickly or slowly. completes assignments in the time allotted. paces work on long-term assignments. sets achievable goals with respect to time. completes make-up work in a timely fashion. Work Habits The student: is a conscientious, hard-working student. works independently. is a self-motivated student. consistently completes homework assignments. puts forth their best effort into homework assignments. exceeds expectations with the quality of their work. readily grasps new concepts and ideas. generates neat and careful work. checks work thoroughly before submitting it. stays on task with little supervision. displays self-discipline. avoids careless errors through attention to detail. uses free minutes of class time constructively. creates impressive home projects. Related: Needs Improvement Report Card Comments for even more comments! Student Certificates! Recognize positive attitudes and achievements with personalized student award certificates! Report Card Thesaurus Looking for some great adverbs and adjectives to bring to life the comments that you put on report cards? Go beyond the stale and repetitive With this list, your notes will always be creative and unique. Adjectives attentive, capable, careful, cheerful, confident, cooperative, courteous, creative, dynamic, eager, energetic, generous, hard-working, helpful, honest, imaginative, independent, industrious, motivated, organized, outgoing, pleasant, polite, resourceful, sincere, unique Adverbs always, commonly, consistently, daily, frequently, monthly, never, occasionally, often, rarely, regularly, typically, usually, weekly Copyright© 2022 Education World
125 Report Card Comments
Back to Geography Lesson Plan Where Did Foods Originate? (Foods of the New World and Old World) Subjects Arts & Humanities --Language Arts Educational Technology Science --Agriculture Social Studies --Economics --Geography --History ----U.S. History ----World History --Regions/Cultures Grade K-2 3-5 6-8 9-12 Advanced Brief Description Students explore how New World explorers helped change the Old World's diet (and vice versa). Objectives Students will learn about changes that occurred in the New World and Old World as a result of early exploration. use library and Internet sources to research food origins. (Older students only.) create a bulletin-board map illustrating the many foods that were shared as a result of exploration. Keywords Columbus, explorers, origin, food, timeline, plants, map, New World, Old World, colonies, colonial, crops, media literacy, products, consumer Materials Needed: library and/or Internet access (older students only) outline map of the world (You might print the map on a transparency; then use an overhead projector to project and trace a large outline map of the world onto white paper on a bulletin board.) magazines (optional) Lesson Plan The early explorers to the Americas were exposed to many things they had never seen before. Besides strange people and animals, they were exposed to many foods that were unknown in the Old World. In this lesson, you might post an outline map of the continents on a bulletin board. Have students use library and/or Internet resources (provided below) to research some of the edible items the first explorers saw for the first time in the New World. On the bulletin board, draw an arrow from the New World (the Americas) to the Old World (Europe, Asia, Africa) and post around it drawings or images (from magazines or clip art) of products discovered in the New World and taken back to the Old World. Soon, the explorers would introduce plants/foods from the Old World to the Americas. You might draw a second arrow on the board -- from the Old World to the New World -- and post appropriate drawings or images around it. Adapt the Lesson for Younger Students Younger students will not have the ability to research foods that originated in the New and Old World. You might adapt the lesson by sharing some of the food items in the Food Lists section below. Have students collect or draw pictures of those items for the bulletin board display. Resources In addition to library resources, students might use the following Internet sites as they research the geographic origins of some foods: Curry, Spice, and All Things Nice: Food Origins The Food Timeline Native Foods of the Americas A Harvest Gathered: Food in the New World We Are What We Eat Timeline (Note: This resource is an archived resource; the original page is no longer live and updated.) Food Lists Our research uncovered the Old and New World foods below. Students might find many of those and add them to the bulletin board display. Notice that some items appear on both lists -- beans, for example. There are many varieties of beans, some with New World origins and others with their origins in the Old World. In our research, we found sources that indicate onions originated in the New and sources that indicate onions originated in the Old World. Students might create a special question mark symbol to post next to any item for which contradictory sources can be found Note: The Food Timeline is a resource that documents many Old World products. This resource sets up a number of contradictions. For example: Many sources note that tomatoes originated in the New World; The Food Timeline indicates that tomatoes were introduced to the New World in 1781. The Food Timeline indicates that strawberries and raspberries were available in the 1st century in Europe; other sources identify them as New World commodities. Foods That Originated in the New World: artichokes, avocados, beans (kidney and lima), black walnuts, blueberries, cacao (cocoa/chocolate), cashews, cassava, chestnuts, corn (maize), crab apples, cranberries, gourds, hickory nuts, onions, papayas, peanuts, pecans, peppers (bell peppers, chili peppers), pineapples, plums, potatoes, pumpkins, raspberries, squash, strawberries, sunflowers, sweet potatoes, tobacco, tomatoes, turkey, vanilla, wild cherries, wild rice. Foods That Originated in the Old World: apples, bananas, beans (some varieties), beets, broccoli, carrots, cattle (beef), cauliflower, celery, cheese, cherries, chickens, chickpeas, cinnamon, coffee, cows, cucumbers, eggplant, garlic, ginger, grapes, honey (honey bees), lemons, lettuce, limes, mangos, oats, okra, olives, onions, oranges, pasta, peaches, pears, peas, pigs, radishes, rice, sheep, spinach, tea, watermelon, wheat, yams. Extension Activities Home-school connection. Have students and their parents search their food cupboards at home; ask each student to bring in two food items whose origin can be traced to a specific place (foreign if possible, domestic if not). Labels from those products will be sufficient, especially if the products are in breakable containers. Place those labels/items around a world map; use yarn to connect each label to the location of its origin on the map. Media literacy. Because students will research many sources, have them list the sources for the information they find about each food item. Have them place an asterisk or checkmark next to the food item each time they find that item in a different source. If students find a food in multiple sources, they might consider it "verified"; those foods they find in only one source might require additional research to verify. Assessment Invite students to agree or disagree with the following statement:The early explorers were surprised by many of the foods they saw in the New World. Have students write a paragraph in support of their opinion. Lesson Plan Source Education World Submitted By Gary Hopkins National Standards LANGUAGE ARTS: EnglishGRADES K - 12NL-ENG.K-12.2 Reading for UnderstandingNL-ENG.K-12.8 Developing Research SkillsNL-ENG.K-12.9 Multicultural UnderstandingNL-ENG.K-12.12 Applying Language Skills SOCIAL SCIENCES: EconomicsGRADES K - 4NSS-EC.K-4.1 Productive ResourcesNSS-EC.K-4.6 Gain from TradeGRADES 5 - 8NSS-EC.5-8.1 Productive ResourcesNSS-EC.5-8.6 Gain from TradeGRADES 9 - 12NSS-EC.9-12.1 Productive ResourcesNSS-EC.9-12.6 Gain from Trade SOCIAL SCIENCES: GeographyGRADES K - 12NSS-G.K-12.1 The World in Spatial TermsNSS-G.K-12.2 Places and Regions SOCIAL SCIENCES: U.S. HistoryGRADES K - 4NSS-USH.K-4.1 Living and Working together in Families and Communities, Now and Long AgoNSS-USH.K-4.3 The History of the United States: Democratic Principles and Values and the People from Many Cultures Who Contributed to Its Cultural, Economic, and Political HeritageNSS-USH.K-4.4 The History of Peoples of Many Cultures Around the WorldGRADES 5 - 12NSS-USH.5-12.1 Era 1: Three Worlds Meet (Beginnings to 1620)NSS-USH.5-12.2 Era 2: Colonization and Settlement (1585-1763)NSS-WH.5-12.6 Global Expansion and Encounter, 1450-1770 TECHNOLOGYGRADES K - 12NT.K-12.1 Basic Operations and ConceptsNT.K-12.5 Technology Research Tools Find many more great geography lesson ideas and resources in Education World's Geography Center. Click here to return to this week's World of Learning lesson plan page. Updated 10/11/12
Where Did Foods Originate? (Foods of the New World and Old World)
50 "Needs Improvement" Report Card Comments Having a tough time finding the right words to come up with "areas for improvement" comments on your students' report cards? Check out our helpful suggestions to find just the right one! The following statements will help you tailor your comments to specific children and highlight their areas for improvement. Be sure to check out our 125 Report Card Comments for positive comments! Needs Improvement- all topics is a hard worker, but has difficulty staying on task. has a difficult time staying on task and completing his/her work. needs to be more respectful and courteous to his/her classmates. needs to listen to directions fully so that he/she can learn to work more independently. is not demonstrating responsibility and needs to be consistently reminded of how to perform daily classroom tasks. works well alone, but needs to learn how to work better cooperatively with peers. does not have a positive attitude about school and the work that needs to be completed. struggles with completing his/her work in a timely manner. gives up easily when something is difficult and needs extensive encouragement to attempt the task. gets along with his/her classmates well, but is very disruptive during full group instruction. has a difficult time using the materials in the classroom in a respectful and appropriate manner. has a difficult time concentrating and gets distracted easily. is having a difficult time with math. Going over _ at home would help considerably. is having a very difficult time understanding math concepts for his/her grade level. He/she would benefit from extra assistance. could benefit from spending time reading with an adult every day. is enthusiastic, but is not understanding . Additional work on these topics would be incredibly helpful. is having difficulty concentrating during math lessons and is not learning the material that is being taught because of that.understands math concepts when using manipulatives, but is having a difficult time learning to _ without them. is a very enthusiastic reader. He/she needs to continue to work on to make him/her a better reader. needs to practice reading at home every day to help make him/her a stronger reader. needs to practice his/her sight words so that he/she knows them on sight and can spell them. needs to work on his/her spelling. Practicing at home would be very beneficial. can read words fluently, but has a difficult time with comprehension. Reading with __ every day would be helpful. could benefit from working on his/her handwriting. Slowing down and taking more time would help with this. is having difficulty writing stories. Encouraging him/her to tell stories at home would help with this. has a difficult time knowing when it is appropriate to share his/her thoughts. We are working on learning when it is a good time to share and when it is a good time to listen. needs to work on his/her time management skills. _is able to complete his/her work, but spends too much time on other tasks and rarely completes his/her work. needs reminders about the daily classroom routine. Talking through the classroom routine at home would be helpful. is having a difficult time remembering the difference between short and long vowel sounds. Practicing these at home would be very helpful. is struggling with reading. He/she does not seem to enjoy it and does not want to do it. Choosing books that he/she like and reading them with him/her at home will help build a love of reading. frequently turns in incomplete homework or does not hand in any homework. Encouraging _____to complete his/her homework would be very helpful. does not take pride in his/her work. We are working to help him/her feel good about what he/she accomplishes. does not actively participate in small group activities. Active participation would be beneficial. has a difficult time remembering to go back and check his/her work. Because of this, there are often spelling and grammar mistakes in his/her work. does not much effort into his/her writing. As a result, his/her work is often messy and incomplete. is struggling to understand new concepts in science. Paying closer attention to the class discussions and the readings that we are doing would be beneficial. is reading significantly below grade level. Intervention is required. does not write a clear beginning, middle and end when writing a story. We are working to identify the parts of the stories that he/she is writing. is struggling to use new reading strategies to help him/her read higher level books. is wonderful at writing creative stories, but needs to work on writing nonfiction and using facts. has a difficult time understanding how to solve word problems. needs to slow down and go back and check his/her work to make sure that all answers are correct. is not completing math work that is on grade level. Intervention is required. is struggling to understand place value. is very enthusiastic about math, but struggles to understand basic concepts. has a difficult time remembering the value of different coins and how to count them. Practicing this at home would be helpful. would benefit from practicing math facts at home. is very engaged during whole group math instruction, but struggles to work independently. is able to correctly answer word problems, but is unable to explain how he/she got the answer. is having a difficult time comparing numbers. Related: 125 Report Card Comments for positive comments! Student Award Certificates! Recognize positive attitudes and achievements with personalized student award certificates! Copyright© 2020 Education World
50 "Needs Improvement" Report Card Comments
Back to Top
75 Mill St. Colchester, CT 06415
Receive timely lesson ideas and PD tips
Enter your email for FREE Newsletter!
Sitemap
Close Sitemap
About Us
Contact Us
Help Center
Advertise
Reprint Rights
Privacy PolicyLesson Plans ------------ LessonArchive Newsletter Articles Box Cars Math Games Every-Day Edits Five-Minute Fillers Holiday Lessons Learning Games Lesson of the Day News for Kids Show-Biz Science Work Sheet LibraryProfessional Development ------------------------ Clip Art Gallery Math Corner New Teacher Advisor Reader's Theater Reading Coach Responsive Classroom Strategies That Work Teacher Feature Voice of Experience Web Site ReviewsTechnology ---------- The Math Machine The Reading Machine The Science Machine Online Safety Sites to See WebQuest Technology Experts Tech Team Using Technology Keyboarding Techtorials Wire Side Chats On the InternetAdministrators -------------- How I Handled… Wire Side Chats Partners for Student Success Principal Profiles Principal Toolbox Parent Involvement Programs Goals Best Books for School Leaders Exceptional Events Great Meetings Instant Meetings Morning Math School Resources ---------------- Career Center Classroom Management Early Childhood Channel Ed World's Columnist Team Site Reviews for Educators State/Nat'l Education Standards Subjects & Specialties Substitute Teacher Resources Teacher-Submitted Lesson PlansSpecialties ----------- Counseling Early Childhood Preservice Ed Special Ed Sub Station Vocational EdSubject Resources ----------------- The Arts Foreign Languages History Language, Literature Math PE and Health Science Social Sciences TechnologyFun Activities -------------- Animals A-Z Edits Box Cars Math Games Every-Day Edits Fact Monster Hunts Five-Minute Fillers Friday Fun Ideas Education World @ Home ---------------------- Ed World@Home for Parents Ed World@Home for StudentsLifestyle --------- LifeStyle CenterEarly Childhood --------------- Castaldo's Corner Activities Early Childhood Activity Bank
Sign up for our free weekly newsletter and receive
top education news, lesson ideas, teaching tips and more!
No thanks, I don't need to stay current on what works in education!
COPYRIGHT 1996-2016 BY EDUCATION WORLD, INC. ALL RIGHTS RESERVED.
Also of Interest:
Random Calling Strategies That Work Anywhere
6 Steps To Make The Most Of Virtual Learning In K-12
4 Summer Travel Opportunities For Teachers
Go to Full Site
COPYRIGHT 1996 - 2025 BY EDUCATION WORLD, INC. ALL RIGHTS RESERVED.
SchoolNotes.com
The Educator's Network
75 Mill Street, Colchester, CT 06415 |
11775 | https://link.springer.com/article/10.1078/1431-7613-00050 | Skip to main content
Ernst Mayr through time on the biological species concept — a conceptual analysis
Published:
Volume 121, pages 81–98, (2002)
Cite this article
Theory in Biosciences
Aims and scope
Submit manuscript
Peter J. Beurton1
756 Accesses
9 Citations
3
Altmetric
Explore all metrics
Summary
An analysis is made of Ernst Mayr’s treatment of the biological species definition over time. To avoid misunderstandings, it is important to distinguish between the varying conceptual assumptions which Mayr associates with this definition and the unchanged wording of the definition.
A polytypic species, that is a geographical aggregate of allopatric populations, does not fit the biological species definition very well. There is an undeniable tension between these two aspects [the polytypic and nondimensional] of the word species and from 1942 until the present time, I have never ceased to struggle with this problem (Mayr 1992b, p. 9).
... indeed, I pride myself on having changed my mind on frequent occasions (Mayr 1982a, p. 9).
This is a preview of subscription content, log in via an institution
to check access.
Access this article
Log in via an institution
Subscribe and save
Springer+
from $39.99 /Month
Starting from 10 chapters or articles per month
Access and download chapters and articles from more than 300k books and 2,500 journals
Cancel anytime
View plans
Buy Now
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Instant access to the full article PDF.
Institutional subscriptions
Similar content being viewed by others
On the Impossibility and Dispensability of Defining ‘‘Biodiversity’’
Chapter
© 2019
A Phylogenetic Approach to Conservation: Biodiversity and Ecosystem Functioning for a Changing Globe
Chapter
© 2022
Taxonomic Concepts and Taxonomic Evolution in European Characeae
Chapter
© 2024
Explore related subjects
Discover the latest articles, books and news in related subjects, suggested using machine learning.
Biological metamorphosis
Biological Taxonomy
Conceptual Analysis
Macroecology
Philosophy of Biology
Theoretical and Statistical Ecology
References
Dobzhansky, T. (1937) Genetics and the Origin of Species. Columbia University Press. New York.
Google Scholar
Dobzhansky, T. (1950) ‘Mendelian populations and their evolution’. The American Naturalist 84: 401–418.
Article
Google Scholar
Mayr, E. (1940) ‘Speciation Phenomena in Birds’. The American Naturalist 74: 249–278.
Article
Google Scholar
Mayr, E. (1942) Systematics and the Origin of Species. Columbia University Press. New York.
Google Scholar
Mayr, E. (1946) ‘The Naturalist in Leidy’s Time and Today’. Proceedings of the National Academy of Science Philadelphia 98: 271–276.
Google Scholar
Mayr, E. (1948) ‘The Bearing of the New Systematics on Genetical Problems. The Nature of Species’, Advances in Genetics 2: 205–237.
Article
Google Scholar
Mayr, E. (1949) ‘The Species Concept: Semantics Versus Semantics’. Evolution 3: 371–372.
Article
Google Scholar
Mayr, E. (1955) ‘The Species as a Systematic and as a Biological Problem’. Annual Biology Colloquium 16: 3–12.
Google Scholar
Mayr, E. (1957) ‘Species Concepts and Definitions’. In Mayr, E. (ed) The Species Problem. American Association for the Advancement of Science, Publication No. 50: 1–22.
Mayr, E. (1963) Animal Species and Evolution. Belknap Press. Cambridge, Mass.
Google Scholar
Mayr, E. (1969) Principles of Systematic Zoology. McGraw Hill. New York.
Google Scholar
Mayr, E. (1970) Populations, Species, and Evolution. Cambridge, Harvard University Press.
Google Scholar
Mayr, E. (1982a) The Growth of Biological Thought. Belknap Press. Cambridge, Mass.
Google Scholar
Mayr, E. (1982b) ‘What Use Are Subspecies?’ Auk 99: 593–595.
Google Scholar
Mayr, E. (1986) ‘The Species as Category, Taxon, and Population’. In Roger, J., Fischer, J. L. (eds) Histoire du concept d’espèce dans les sciences de la vie. Fondation Singer-Polignac. Paris. 294–311 (quoted from E. Mayr, 1988b, 315–334).
Google Scholar
Mayr, E. (1987a) ‘The Ontological Status of Species: Scientific Progress and Philosophical Terminology’. Biology and Philosophy 2: 145–166 (slightly, but in the present context importantly, revised version in E. Mayr 1988b, 335–358).
Article
Google Scholar
Mayr, E. (1987b) ‘Answers to These Comments’. Biology and Philosophy 2: 212–220.
Article
Google Scholar
Mayr, E. (1988a) ‘The Why and How of Species’. Biology and Philosophy 3: 431–441.
Article
Google Scholar
Mayr, E. (1988b) Toward A New Philosophy of Biology. Belknap Press. Cambridge, Mass.
Google Scholar
Mayr, E. (1988c) ‘Recent Historiographical Developments’. In Hawksworth, D. L. (ed). Prospects in Systematics. Clarendon Press. Oxford. 31–43.
Google Scholar
Mayr, E. (1992a) ‘A Local Flora and the Biological Species Concept’. American Journal of Botany 79: 222–238.
Article
Google Scholar
Mayr, E. (1992b) ‘Controversies in Retrospect’. Oxford Surveys of Evolutionary Biology 8: 1–34.
Google Scholar
Mayr, E.; Linsley, E. G.; Usinger, R. L. (1953) Methods and Principles of Systematic Zoology. McGraw-Hill Book Company. New York.
Google Scholar
Download references
Author information
Authors and Affiliations
Max Planck Institute for the History of Science, Wilhelmstraße 44, 10117, Berlin, Germany
Peter J. Beurton
Authors
Peter J. Beurton
View author publications
Search author on:PubMed Google Scholar
Rights and permissions
Reprints and permissions
About this article
Cite this article
Beurton, P.J. Ernst Mayr through time on the biological species concept — a conceptual analysis.
Theory Biosci. 121, 81–98 (2002).
Download citation
Received:
Accepted:
Issue Date:
DOI:
Share this article
Anyone you share the following link with will be able to read this content:
Provided by the Springer Nature SharedIt content-sharing initiative
Key words
Biological species definition
local and polytypic species
nondimensional and multidimensional species concept
species taxon
species category
Ernst Mayr |
11776 | https://web.pdx.edu/~caughman/Combinatorics%20Ed%20Textbook%20Research.pdf | Allenby & Slomson (2011) The Principle of Multiplication of Choices. If there are r successive choices to be made, and for 1 ≤i ≤r, the ith choice can be made in ni ways, then the total number of ways of making these choices is n1 × n2 × ... × nr.
Anderson, I. (2001) The Multiplication Principle. Suppose that an activity consists of k stages, and that the ith stage can be carried out in αi different ways, irrespective of how the other stages are carried out.
Then the whole activity can be carried out in α1α2...αk ways.
Anderson, J. (2001) Multiplication Counting Principle. Given a sequence of events E1, E2, E3, ..., Em such that E1 occurs in n1 ways and if E1, E2, E3, ...Ek−1 have occurred, then Ek can occur in nk ways. There are n1 × n2 × · · · × nm ways in which the entire sequence of events can occur.
Balakrishnan (1995) Sequential or Product Rule. If an event can occur in m ways and a second event can occur in n ways, and if the number of ways the second event occurs does not depend upon how the first event occurs, then the two events can occur simultaneously in mn ways.
More generally, if Ei (i = 1, 2, ..., k) are k events and if E1 can occur in n1 ways, E2 can occur in n2 ways (no matter how E1 occurs), E3 can occur in n3 ways (no matter how E1 and E2 occur),...,Ek can occur in nk ways (no matter how the previous k −1 events occur), then the k events can occur simultaneously in n1n2n3 · · · nk ways.
Bender & Williamson (2006) Rule of Product. Suppose structures are to be constructed by making a sequence of k choices such that, (i) the ith choice can be made in ci ways, a number independent of what choices were made previously, and (ii) each structure arises in exactly one way in this process. Then, the number of structures is c1 × ... × ck.
Cartesian Product. If C1, ..., Ck are sets, the Cartesian product of the sets is written C1 ×···×Ck and consists of all k-lists (x1, ..., xk) with xi ∈Ci, for 1 ≤i ≤k.
Berge (1971) The Cardinality of the Cartesian Product A × X.
Consider the set A of m elements a1, a2, . . . , am and the set X of n elements 1, 2, . . . , n, then A × X has exactly m × n elements.
Symbolically: |A × X| = |A| × |X|.
Berman & Fryer (1972) The Multiplication Principle. If a set of objects can be separated or partitioned into k nonempty disjoint subsets, and if each of these subsets can be separated into m nonempty disjoint subsets, then the original set can be separated into km nonempty disjoint subsets.
1 Biggs (1989) The Multiplication Principle. The cardinality of X × Y is given by |X × Y | = |X| × |Y | .
Exercise 4. Use the principle of induction to prove that |X1 × X2 × ... × Xn| = |X1| × |X2| × ... × |Xn| .
Bogart (1990) Product Principle. The size of a union of m disjoint sets, each of size n, is mn.
Bogart, Stein, & Drysdale (2006) The Product Principle. A union of m disjoint sets, each of size n, has m · n elements.
Another more concrete statement of the product principle suggested by our example says that if the lists in a set of two-element lists have m possible first elements, and if for each first element, n different second elements are paired with it in lists in the set, then the set has m · n lists.
General form of the product principle. If the lists in a set S of lists of length k have the properties that 1. there are m1 different first elements of lists in S; 2. for each way of specifying the first i −1 entries of a list, there are mi ways to specify the ith entry in the list; then S contains m1 · m2 · · · mk lists.
Bona (2007) Product Principle. Let X and Y be two finite sets. Then the number of pairs (x, y) satisfying x ∈X and y ∈Y is |X| × |Y |.
Generalized Product Principle. Let X1, X2, ..., Xk be finite sets. Then, the number of k-tuples (x1, x1, ...xk) satisfying xi ∈Xi is |X1| × |X2| × ... × |Xk|.
Bose & Manvel (1984) The multiplication principle says that if one task can be performed in m ways and then another task can be performed in n ways, the pair of tasks, first one and then the other, can be performed in m · n ways.
The generalized multiplication principle says that if one task can be done in m1 ways, a second in m2 ways, and so on until the kth task can be done in mk ways, then the succession of tasks, one after the other, can be done in exactly m1 · m2 · m3 · · · mk ways.
Britton & Bello (1984) Sequential Counting Principle. If one thing can occur in m ways and a second thing can then occur in n ways, the the sequence of two things can occur in m × n ways.
Sequential Counting Principle. If one thing can occur in m ways and a second thing can then occur in n ways, and a third thing can then occur in r ways, and so on, then the sequence of things can occur in m × n × r × · · · ways.
2 Brualdi (2004) Multiplication Principle. Let S be a set of ordered pairs (a, b) of objects, where the first object a comes from a set of size p, and for each choice of object a there are q choices for object b. Then the size of S is p × q: |S| = p × q.
Useful Formulation. If a first task has p outcomes and, no matter what the outocme of the first task, a second task has q outcomes, then the two tasks performed consecutively have p×q outcomes.
Chen & Koh (1992) (MP). Assume that an event E can be decomposed into r ordered events E1, E2, . . . , Er, and that there are: · n1 ways for the event E1 to occur, · n2 ways for the event E2 to occur, · .
.
.
· nr ways for the event Er to occur.
Then the total number of ways for the event E to occur is given by: n1 × n2 × · · · × nr = r Y i=1 ni.
Multiplication Principle. Let Πr i=1Ai = A1 × A2 × · · · × Ar = {(a1, a2, ..., ar) : ai ∈Ai, i = 1, 2, ..., r} denote the cartesian product of the finite sets A1, A2, ..., Ar. Then |Πr i=1Ai| = |A1| × |A2| × · · · × |Ar| = Πr i=1 |Ai| Cohen (1978) Rule of product. If X can happen in x ways and a distinct event Y in y ways then (X and Y ) can happen in (xy) ways.
The rule of product. If X1 can happen in x1 ways, X2 in x2 ways, X3 in x3 ways... then (X1 and X2 and X3 and ...) can happen at once in (x1x2x3...) ways.
DeTemple & Webb (2014) Multiplication Principle for a Cartesian Product. For any two sets A and B, the number of ordered pairs in the Cartesian product S = A × B is given by |S| = |A × B| = |A| |B| .
Multiplication Principle for a Multistage Process. Suppose that a multistage experiment consists of a sequence of n steps, for which there are: · p1 ways to perform step 1 · p2 ways to perform step 2 (independently of how step 1 was performed) · p3 ways to perform step 3 (independently of how steps 1 and 2 were performed) · .
.
.
· pn ways to perform step n (independently of how the preceding steps were performed) Then there are p1p2 · · · pn ways to carry out all n steps of the multistage experiment.
3 Dossey, Otto, Spence, & VandenEynden (2006) Multiplication Principle.
Consider a procedure that is composed of a sequence of k steps.
Suppose that the first step can be performed in n1 ways, and for each of these the second step can be performed in n2 ways, and in general, no matter how the preceding steps are performed, the ith step can be performed in ni ways (i = 2, 3, ..., k). Then the number of different ways in which the entire procedure can be performed is n1 · n2 · · · nk.
Eisen (1969) Theorem. If an event A can occur in t different ways, and for each of these ways a subsequent event B can occur in s different ways, then the event A and B can occur in ts different ways.
Corollary.
If the first event A1 can occur in r1 different ways, and for each of these ways a subsequent event A2 can occur in r2 different ways, and for each of these ways a following event A3 can occur in r3 different ways, . . ., and for each of these ways the last event An can occur in rn different ways, then the event “A1 and A2 and . . . and An” can occur in r1r2 · · · rn ways.
Epp (2010) The Multiplication Rule. If an operation consists of k steps and · the first step can be performed in n1 ways, · the second step can be performed in n2 ways [regardless of how the first step was performed], · the kth step can be performed in nk ways [regardless of how the preceding steps were performed], then the entire operation can be performed in n1n2...nk ways.
Ferland (2009) Multiplication Principle. Let A be a set of outcomes we wish to count. If there is a set of outcomes A1, and, for each outcome in A1, there is a set of outcomes A2 such that 1. each outcome from A can be uniquely characterized by a pair of outcomes, the first from A1 and the second from its corresponding set A2, and 2. for each outcome from A1, the number |A2| is the same, then |A| = |A1| · |A2| .
In this general case, the requirement is that, for each 2 ≤k ≤n, the number |Ak| must not depend on any of the sets Ai for 1 ≤i ≤k −1. Under those conditions, |A| is given by the product |A| = |A1| · |A2| · · · |An| .
Gerstein (1987) Product Rule. Suppose that n consecutive choices are to be made. Suppose further that there are m1 possibilities for the first choice and, for each k satisfying 2 ≤k ≤n, that there are mk possibilities for the kth choice, no matter how the first k −1 choices have been made. Then there are exactly m1m2 · · · mn possibilities for the sequence of n choices.
4 Gersting (1999) Fundamental Counting Principle. If there are n1 possible outcomes for a first event and then n2 possible outcomes for a second event, there are n1 · n2 possible outcomes for the sequence of the two events.
Goldstein & Schneider (1984) Multiplication Principle. Suppose that a task is composed of two consecutive operations. If operation 1 can be performed in m ways and, for each of these, operation 2 can be performed in n ways, then the complete task can be performed in m · n ways.
Generalized Multiplication Principle. Suppose that a task consists of t operations performed consecutively. Suppose that operation 1 can be performed in m1 ways; for each of these, operation 2 in m2 ways; for each of these, operation 3 in m3 ways; and so forth. Then the task can be performed in m1 · m2 · m3 · · · mt ways.
Goodaire & Parmenter (1998) Observation. If A and B are finite sets, then the Cartesian product A × B contains |A| × |B| elements. More generally, if A1, A2, ..., An are finite sets, |A1 × A2 × ... × An| = n Y i=1 |Ai| .
Multiplication Rule. The number of ways in which a sequence of events can occur is the product of the number of ways in which each individual event can occur.
Gosset (2002) General Counting Principle 1. If a project can be decomposed into two independent tasks with n1 ways to accomplish the first task and n2 ways to accomplish the second task, then the project can be completed in n1 · n2 ways.
Grimaldi (1994) The Rule of Product. If a procedure can be broken down into first and second stages, and if there are m possible outcomes for the first stage and if, for each of these outcomes, there are n possible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in mn ways.
Harris, Hirst, & Mossinghoff(2008) Product Rule. Suppose S1, S2, ..., Sm are finite sets, and |Si| = ni for 1 ≤i ≤m. Then the number of ways to select one element from S1, followed by one element from S2, and so on, ending with one element from Sm, is the product n1n2 ···nm, provided that the selections are independent, that is, the elements chosen from S1, ..., Si−1 have no bearing on the selection from Si, for each i.
Hein (2003) Product Rule.
|A × B| = |A| |B| .
5 Hillman, Alexanderson, & Grassl (1987) Product Rule. If an event A can occur in m ways and each possibility for A allows exactly n ways for an event B to occur, then the event (A and B) can occur in mn ways.
General Product Rule. Let A1, A2, ..., Ak be events. Let there be n1 ways for A1 to occur. For each way that A1 can occur let there be n2 ways for A2 to occur. For each way that A1 and A2 can occur, let there be n3 ways for A3 to occur, and so on. Then (A1 ∩A2 ∩... ∩Ak) can occur in n1n2 · · · nk ways.
Jackson & Thoro (1990) The Multiplication Principle. For any two sets A and B, |A × B| = |A| |B|.
The Multiplication Principle for Counting Procedures. If a procedure can be described as a sequence of m independent steps, with i1 possible outcomes for the first step, i2 possible outcomes for the second step, ..., and im possible outcomes for the mth step, then the total number of composite outcomes of the procedure is given by the product i1i2 · · · im.
Johnson & Mowry (1999) The Fundamental Principle of Counting. The total number of possible outcomes of a series of decisions (making selections from various categories) is found by multiplying the number of choices for each decision (or category) as follows: 1. Draw a box for each decision.
2. Enter the number of choices for each decision in the appropriate box and multiply.
Johnsonbough (2001) Multiplication Principle. If an activity can be constructed in t successive steps and step 1 can be done in n1 ways, step 2 can be done in n2 ways, ..., and step t can be done in nt ways, then the number of different possible activities is n1 · n1 · · · nt Koh & Tay (2013) The Multiplication Principle. Suppose that an event E can be split into two events E1 and E2 in ordered stages. If there are n1 ways for E1 to occur and n2 ways for E2 to occur, then the number of ways for the event E to occur is n1n2.
MP. Suppose that an event E can be split into k events E1, E2, . . . , Ek in ordered stages. If there are n1 ways for E1 to occur, n2 ways for E2 to occur,. . . , and nk ways for Ek to occur, then the number of ways for the event E to occur is n1n2 . . . nk.
MP. |A1 × A2 × · · · × Ak| = |A1| |A2| · · · |Ak|.
Kolman, Busby, & Ross (2008) The Multiplication Principle. Suppose that two tasks T1 and T2 are to be performed in sequence.
If T1 can be performed in n1 ways, and for each of these ways T2 can be performed in n2 ways, then the sequence T1T2 can be performed in n1n2 ways.
6 The Multiplication Principle. Suppose that tasks T1, T2, . . . , Tk are to be performed in sequence.
If T1 can be performed in n1 ways, and for each of these ways T2 can be performed in n2 ways, and for each of these n1n2 ways of performing T1T2 in sequence, T3 can be performed in n3 ways, and so on, then the sequence T1T2 · · · Tk can be performed in n1n2 · · · nk ways.
Krussel (2011) Multiplication Principle. Suppose there is a task, T, to be completed, but now the task can be completed in stages. That is, T can be broken down into subtasks, t1, t2, ..., tm so that T is accomplished only after all of the subtasks have been completed. If t1 can be done in b1 ways, t2 in b2 ways, ..., and tm in bm ways, then T can be completed in b1b2 · · · bm ways.
Lial, Greenwell, & Ritchey (2012) Multiplication Principle. Suppose n choices must be made, with m1 ways to make choice 1 m2 ways to make choice 2 and so on, with mn ways to make choice n.
Then there are m1 · m2 · · · mn different ways to make the entire sequence of choices.
Liu (1968) Rule of product.
If one experiment has m possible outcomes and another experiment has n possible outcomes, then there are m × n possible outcomes when both of these experiments take place.
Lovasz, Pelikan, & Vesztergombi (2003) Theorem. Suppose that we want to form strings of length n by using any of a given set of k1 symbols as the first element of the string, any of a given set of k2 symbols as the second element of the string, etc., any of a given set of kn symbols as the last element of the string. Then the total number of strings we can form is k1 · k2 · · · kn.
Maki & Thompson (2004) Multiplication Principle. Consider a multistage experiment consisting of k stages. Suppose that the first stage has n1 possible results, the second stage has n2 possible results regardless of the result of stage 1, the third stage has n3 results regardless of the results of the first two stages, ..., and the kth stage has nk results regardless of the results of the earlier stages. Then there are n1 × n2 × n3 × · · · × nk elements in the sample space of the experiment.
Marcus (1998) Product Rule. In a string of length k, if the ith term can be filled in a particular number of ways (ni), which does not depend on how any of the previous terms have been filled, then the total number of possible strings is the product n1n2n3 · · · nk.
7 Martin (2001) The Multiplication Principle. If one thing can be done in m ways and a second thing can be done in n ways independent of how the first thing is done, then the 2 things can be done in mn ways.
Mazur (2009) The Product Principle. In counting k-lists of the form (l1, l2, ..., lk), if 1. there are c1 ways to specify element l1 of the list, and each such specification ultimately leads to a different k-list; and 2. for every other list element li, there are ci ways to specify that element no matter the specification of the previous elements l1, ..., li−1, and that each such specification of li ultimately leads to a different k-list, then there are c1c2 · · · ck such lists.
Mott, Kandel, & Baker (1986) Product Rule: the principle of sequential counting. If S1, ..., Sn are nonempty sets, then the number of elements in the Cartesian product S1 × S2 × · · · × Sn is the product Πn i=1 |Si|. That is, |S1 × S2 × · · · × Sn| = Πn i=1 |Si| .
We can reformulate the product rule in terms of events. If events E1, E2, ..., En can happen e1, e2, ..., and en ways, respectively, then the sequence of events E1 first, followed by E2, ..., followed by En can happen e1 · e2 · · · en ways.
Polya, Tarjan, & Woods (1983) Suppose the elements of A are a1, a2, ..., am and the elements of B are b1, b2, ..., bn. We wish to select two elements, one from each set, in a specific order (say, first one from A and then one from B).
This operation is known as the Cartesian product of the two sets, due to its relationship with the rectangular (Cartesian) coordinate system. (...) In general, there are m · n possibilities.
Finally, take a more general case of the Cartesian product. Suppose that, having chosen a1, we then have a choice among a set of elements b11, b12, ..., b1n. If we start by choosing a2, we then have a choice from a different set: b21, b22, ..., b2n, and so on. In general, the possibilities for b differ depending upon our choice for a, but there are always n of them. As long as the number of possibilities for b is constant, the total number of pairs (ai, bj) is still m · n.
Richmond & Richmond (2009) The Fundamental Principle of Counting. If one task can be completed in m ways and another task can be completed in n ways, then the sequence of the two tasks can be completed in m·n ways.
m # ways to complete first task × n # ways to complete second task = m · n # ways to complete the sequence of tasks 8 The Fundamental Theorem of Counting or the Multiplication Principle. If A is the set of outcomes of the first task and B is the set of outcomes of the second task, then the set of outcomes of the sequence of tasks would be {(a, b) : a ∈A, b ∈B} = A × B the Cartesian product of A and B. Observe that the Fundamental Principle of Counting is simply a convenient interpretation of the set-theoretic fact of Theorem 1.2.6 that |A × B| = |A| · |B|.
The number of outcomes for any sequences of tasks is the product of the numbers of outcomes for the individual tasks.
If we have n tasks and the set of outcomes for the kth task is Sk, then the set of outcomes for the sequence of tasks is {(s1, s2, ..., sn) : si ∈Si} = S1 × S2 × ... × Sn. Our extended version of the Fundamental Principle of Counting simply says |S1 × S2 × ... × Sn| = |S1| · |S2| · · · |Sn|.
Riordan (2002) Rule of Product. If object A may be chosen in m ways, and thereafter B in n ways, both “A and B” may be chosen in this order in mn ways.
Roberts & Tesman (2005) Product Rule. If something can happen in n1 ways, and no matter how the first thing happens, a second thing can happen in n2 ways, then the two things together can happen in n1 × n2 ways.
Product Rule. If something can happen in n1 ways, and no matter how the first thing happens, a second thing can happen in n2 ways, and no matter how the first two things happen, a third thing can happen in n3 ways, and..., then all the things together can happen in n1 × n2 × n3 × · · · ways.
A more precise version. Suppose that A is a set of a objects and B is a set of b objects. Then the number of ways to pick one object from A and then one object from B is a × b.
Rolf & Williams (1988) Multiplication Rule. Two activities A1 and A2 can be performed in n1 and n2 different ways, respectively. The total number of ways in which A1 followed by A2 can be performed is n1 × n2.
Corollary. Activities A1, A2, ..., Ak can be performed in n1, n2, ..., nk different ways, respectively.
The number of ways in which one can perform A1 followed by A2 ... followed by Ak is n1 × n2 × ... × nk Rosen (2007) The Product Rule. Suppose that a procedure can be broken down into tasks. If there are n1 ways to do the first task and n2 ways to do the second task after the first task has been done, then there are n1n2 ways to do the procedure.
An extended version of the product rule. Suppose that a procedure is carried out by performing the tasks T1, T2, ..., Tm. If task Ti can be done in ni ways after tasks T1, T2, ..., and Ti−1 have been done, then there are n1 · n2 · · · nm ways to carry out the procedure.
The product rule is often phrased in terms of sets in the following way: If A1, A2, ..., Am are finite sets, then the number of elements in the Cartesian product of these sets is the product of the number of elements in each set. To relate this to the product rule, note that the task of choosing an element in the Cartesian product A1 × A2 × · · · × Am is done by choosing an element in A1, an element in A2, ..., and an element in Am. From the product rule it follows that |A1 × A2 × · · · × Am| = |A1| · |A2| · · · |Am|.
9 Ross & Wright (1992) Product Rules.
1. For finite sets S1, S2, ..., Sk we have |S1 × S2 × · · · × Sk| = Πk j=1 |Sj| .
2. More generally, suppose that a set of ordered k-tuples (s1, s2, ..., sk) has the following structure.
There are n1 possible choices of s1. Given s1 there are n2 possible choices of s2, given s1 and s2 there are n3 possible choices of s3, and in general, given s1, s2, ..., sj−1 there are nj choices of sj. Then the set has n1n2 · · · nk elements.
Ryser (1968) Initial Terminology. Let S and T denote two sets and let (s, t) be an ordered pair with s ∈S and t ∈T. Two pairs (s, t) and (s∗, t∗) are equal if s = s∗and t = t∗. The set of all of these ordered pairs is called the product set of S and T and is denoted by S × T.
Rule of Product.
Let M(S, T, n) denote a set of ordered pairs of the form (s, t), where s is arbitrary in S but each s ∈S is paired with exactly n elements t ∈T. Distinct elements of S need not be paired with elements of the same n-subset of T. The notation implies T contains at least n elements. Moreover, M(S, T, n) = S × T if and only if T is an n−set. Now let S be an m-set. Then M(S, T, n) is an (mn)-set.
Generalized Rule of Product. If T1 is an n1-set and if M2 = M(T1, T2, n2), M3 = M(M2, T3, n3), and finally if Mr = M(Mr−1, Tr, nr), then Mr is an (n1n2...nr)-set.
Scheinerman (2000) Multiplication Principle. Consider two-element lists for which there are n choices for the first element, and for each choice of the first element there are m choices for the second element. Then the number of such lists is nm.
Slomson (1991) The Principle of Multiplication of Choice. If there are k successive choices to be made, and, for 1 ≤i ≤k, the ith choice can be made in ni ways, then the total number of ways of making these choices is n1 × n2 × ... × nk, that is, Πk i=1nk.
Tan (2008) The Multiplication Principle. Suppose there are m ways of performing a task T1 and n ways of performing a task T2. Then there are mn ways of performing the task T1 followed by the task T2.
Tucker (2002) 10 The Multiplication Principle. Suppose a procedure can be broken into m successive (ordered) stages, with r1 different outcomes in the first stage, r2 different outcomes in the second stage,..., and rm different outcomes in the mth stage. If the number of outcomes at each stage is independent of the choices in the previous stages and if the composite outcomes are all distinct, then the total procedure has r1 × r2 × ... × rm different composite outcomes.
Vilenkin (1971) The Rule of Product. If an object A can be selected in m ways and if, following the selection of A, an object B can be selected in n ways, then the pair (A, B), A first, B second, can be selected in mn ways.
Wallis & George (2011) The Multiplication Principle states that if we are building an arrangement of objects in stages, and the number of choices at each stage does not depend on the choice made at any earlier stage, then the number of arrangements is the product of the number of choices at each stage.
11 |
11777 | http://math.uchicago.edu/~may/REU2023/REUPapers/Tan,Baichen.pdf | INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS BAICHEN TAN Abstract. This paper discusses classical techniques in dimensionality reduction methods. We study three important linear high dimensional reduction methods: principal component analysis, classical multidimen-sional scaling, and random projection. Detailed explanations will be given to illustrate the three methods’ intuition and theoretical foundations.
Contents 1.
Introduction 1 2.
Principal Component Analysis (PCA) 1 2.1.
Review of Linear Algebra 2 2.2.
The PCA Algorithm 3 2.3.
Statistical interpretation of PCA 8 3.
Classical Multidimensional Scaling (CMDS) 8 3.1.
Distance and Gram Matrices 8 3.2.
The CMDS algorithm 10 3.3.
Comparison between CMDS and PCA 14 4.
Random Projection 14 4.1.
Random Matrices 15 4.2.
Dimensionality Reduction with Random Projection 16 4.3.
Random Projection Algorithm 18 Acknowledgments 18 References 18 1. Introduction In recent years, dimensionality reduction has been gaining more and more attention in the realm of statistics. Dimensionality reduction is a technique used to reduce the number dimensions of a dataset while retaining essential information, such as pairwise distances or other geometric features. High-dimensional datasets can lead to challenges in visualization, computational complexity, and overfitting. For example, in genetics, we often have thousands of gene snippets to test disease effects, while we often only have access to a limited number of patients as observations.
In such a high-dimensional setting, traditional statistical tools, such as linear regression, would fail since there are many more variables than equations (observations). Moreover, many basic statistics such as correlation matrices scale quadratically (or worse) in size with dimensionality, making their computation intractable for the dimensionality of many real-world datasets. Therefore, dimensionality reduction is a very important technique in statistical analysis. This paper examines three linear methods for dimensionality reduction: principal component analysis (PCA), classical multidimensional scaling (CMDS), and random projections. Readers of this paper should be familiar with linear algebra. We will give detailed proofs of the important theorems and lemmas in this paper; proofs of some lemmas and propositions are relegated to or to .
2. Principal Component Analysis (PCA) Suppose we have a high dimensional dataset with a very large dimension p, but our data actually approx-imately lies in a hyperplane of dimension k ≪p. The goal of PCA is to project the original p dimensional data onto the k dimensional hyperplane to reduce the dimensionality of our original dataset. In Figure 1, we 1 2 BAICHEN TAN Figure 1. Intuition of PCA project a two dimensional dataset to an one dimensional line. As we can see, after the projection, the overall geometry of our new dataset is roughly preserved. The idea of PCA is to find the “correct” hyperplane on which to project the data as to minimize any distortion to the relative geometry.
2.1. Review of Linear Algebra. We first review some prerequisite linear algebra.
Theorem 2.1 (Singular Value Decomposition). Let A be an m × n matrix with rank r. Then there always exists a singular value decomposition A = UΣV T where U, Σ, V T are given by: U = u1, u2, . . . , ur ∈Rm×r, Σ = σ1 0 . . .
0 0 σ2 . . .
0 .
.
.
.
.
.
...
.
.
.
0 0 . . .
σr ∈Rr×r V T = vT 1 vT 2 .
.
.
vT r ∈Rr×n where ui and vi are called the left and right singular vectors, respectively. The values σi are called the singular values with the ordering σ1 ≥σ2 ≥... ≥σr. Importantly, each of the ui’s are orthogonal to each other and each of the vi’s are orthogonal to each other.
We now introduce the Frobenius norms on matrices.
Definition 2.2 (Frobenius norm). The Frobenius norm of a matrix A ∈Rm×n is defined as ∥A∥F = m X i=1 n X j=1 |aij|2 1 2 = tr(AT A) 1 2 INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 3 We introduce two important properties of the Frobenius norm. The proof of the following two propositions can be found in .
Proposition 2.3. The Frobenius norm is invariant under rotation by an orthogonal matrix U. That is, if U is orthogonal, then ∥AU∥F = ∥UA∥F = ∥A∥F for any matrix A.
Proposition 2.4. Let A ∈Rm×n be a matrix with rank r. Then the Frobenius norm of A is ∥A∥F = r X i=1 σ2 i ! 1 2 where σi are singular values of A 2.2. The PCA Algorithm. Now we will explain the PCA method. Let X = [x1, ..., xn]T ∈Rn×p be our dataset with n observations. The a-shift of a dataset X is denoted as Xa = X −a where a is a row vector in Rp. In practice, we often shift data by the sample mean, given by ¯ X = 1 n n X i=1 xi.
As can see in Figure 2, shifting a dataset doesn’t change the relative geometry.
(a) Dataset before shift by sample mean (b) Dataset after shift by sample mean Figure 2. Shifting a dataset For simplicity, we shall assume in later part of this text that our dataset is already centered, that is, X = b X = X −¯ X.
Definition 2.5. We define the energy of a dataset E(X) = n X i=1 ∥xi∥2 2 = ∥X∥2 F .
Loosely speaking, PCA is a technique that identifies the subspace in which a dataset clusters. We first consider the maximization problem of finding a one dimensional subspace S1 that maximizes the energy of X after we project it onto S1. We use a unit vector v1 to represent the direction of S1. We denote the projection transformation Rp →S1 by Tv1, and the set of all directions in Rp by the unit sphere Sp−1. Thus, our new coordinate v1 is the solution of the following maximization problem: (2.6) v1 = arg max a∈Sp−1 E (Ta(X)) 4 BAICHEN TAN (a) projection with low energy (b) projection with high energy Figure 3. Projection onto the first coordinate Figure 4. Illustration of PCA on single data points Figure 3 provides an intuitive visualization of why we want to perform this maximization. Because the energy is the sum of the norms of the projected vectors, the projection in plot B preserves more energy than that in plot A, and the projected data points in plot B represents the original data clustering better than plot A does.
We call v1 ∈Sp−1 the first principal direction. Now we take a closer look at what happens to individual data points in PCA. Figure 4 provides a visual explanation. Suppose we have two data points x1, x2 in the original dataset X, and we want to project the them onto the line where the direction vector v1 lies. Then the projected points y1, y2 are called the first principal components of x1, x2 respectively. They represent the locations of the projected points in the original coordinate system. To generalize, for each xi ∈X, its first principal component is denoted as (2.7) y1,i = Tv1 (xi) ∈Rp We can also write equation (2.7) in the form of matrix. We denote Y1 = [y1,1, · · · , y1,n]T = [Tv1(x1), · · · , Tv1(xn)]T = Tv1(X) ∈Rn×p Now suppose we want to project our dataset to a second basis vector.
This means finding a second principal direction v2 onto which to project the data that maintains as much as possible the information that is not in the span by the first coordinate v1. In other words, for each data point xi, we want to project INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 5 xi −v1y1,i onto a second subspace that maximizes the energy. Because xi −v1y1,i is the part of xi that is not in the span of the first principal direction, it is linearly independent from v1y1,i, and therefore v2 and v1 are also independent and thus they are orthogonal to each other. We denote the span(v1) as S1 and the subspace of Sp−1 that is orthogonal to S1 as S⊥ 1 ∩Sp−1. Therefore, the second principal component v2 is the solution of the following problem: (2.8) X1 = [x1 −v1y1,1, · · · , xn −v1y1,n]T v2 = arg max a∈S⊥ 1 ∩Sp−1 E (Ta(X1)) and for each xi −v1y1,i ∈X1, we call (2.9) y2,i = Tv2 (xi −v1y1,i) the second principal component of xi −v1y1,i. Similarly, we can write equation (2.9) in matrix form Y2 = [y2,1, · · · , y2,n]T = [Tv2(x1 −v1y1,1), · · · , Tv2(xn −v1y1,n)]T = Tv2(X1) ∈Rn×p Then, using mathematical induction, we can calculate the successive principal directions as follows. As-sume that the first s −1 principal directions Vs−1 = {v1, · · · , vs−1} ⊂Sp−1 and the corresponding s −1 principal components {Y1, · · · , Ys−1} ⊂Rp of X are well defined.
We denote the subspace spanned by v1, · · · , vs−1 as Ss−1 and the subspace of Sp−1 that is orthogonal to Ss−1 as S⊥ s−1 ∩Sp−1. Now, we want to project onto the sth coordinate the information in X that is not spanned by the previous s −1 principal components, which we denote as (2.10) Xs−1 = xi − s−1 X j=1 vjyj,i, 1 ≤i ≤n Similarly, vs is orthogonal to v1 · · · vs−1. Then the sth principal direction is defined as vs = arg max a∈S⊥ s−1∩Sp−1E (Ta (Xs−1)) and the sth principal component of X is Ys = Tvs (Xs−1).
While it is hard to solve the maximization problem described above directly, we can convert it into the following minimization problem which is easy to solve: Theorem 2.11. The projection Ta that maximizes the energy of a dataset X is also the projection that minimizes the energy of the orthogonal vector (I −Ta)(X). Namely, (2.12) v1 = arg max a∈Sp−1 E (Ta(X)) = arg min a∈Sp−1 E ((I −Ta) (X)) , and (2.13) vs = arg max a∈S⊥ s−1∩Sp−1E (Ta (Xs−1)) = arg min a∈S⊥ s−1∩Sp−1E ((I −Ta) (Xs−1)) .
Proof. This is the direct result of the Pythagorean theorem.
Recall that the energy is the sum of the distances from each projected data point to the origin. Consider for now a single point. We want to find the direction on the unit sphere that maximizes the distance between the projected data point and the origin. Notice in Figure 5 that the distance from the original data point to the origin (i.e. the norm of the vector c) is unchanged as the coordinate vector a is rotated around the origin. By the Pythagorean theorem, ∥c∥2 = ∥a∥2 + ∥b∥2. Thus, for fixed ∥c∥, maximizing ∥a∥is equivalent to minimizing ∥b∥. The norm of b can be represented as ∥xi −Ta(xi)∥where Ta is our desired transformation that maximizes the energy.
6 BAICHEN TAN Figure 5. Illustration of the Pythagorean theorem □ The solutions of principal directions can be obtained with the aid of the best k−rank approximation for matrices.
Theorem 2.14 (Mirsky (1960), Eckart and Young (1936)). Let X ∈Rn×p be our dataset, where each column is a data point and each row is an observation. Let r = rank(X) and the SVD decomposition of X be: X = UΣV T = r X i=1 σiuivT i where U = [u1, u2, ..., ur] ∈Rn×r, Σ = diag(σ1, ..., σr) ∈Rr×r with σ1 ≥σ2 ≥... ≥σr, and V = [v1, ..., vr] ∈ Rp×r Denote e X = Pk i=1 σiuivT i , k ≤r, then we have the following statement: e X is the rank k matrix that is closest in Frobenius norm to X, i.e.
∥X −e X∥F = min rank(A)=k∥X −A∥F We call e X the best k-rank approximation of X.
A proof can be found in .
INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 7 Remark 2.15. We can also write the best k-rank approximation in the form of matrix multiplication. Let the SVD of X be given by X = UΣV T and rank(X) = r. We denote Uk = [u1, · · · , uk] , Σk = diag (σ1, · · · , σk) , Vk = [v1, · · · , vk] , for some k ≤r Then the best k-rank approximation of X is e X = UkΣkV T k By the best k-rank approximation theorem, now we can then construct kth principal directions and components of a high dimensional dataset X ∈Rn×p using the following theorem.
Theorem 2.16. Let X ∈Rn×p be our centered dataset, and let Uk, Σk, Vk be defined as remark 2.15. Then columns v1, v2, · · · , vk of Vk are the k principal directions of X. Namely, they are the directions that minimize the energy of (I −Ta) (X) , (I −Ta) (X1) , ..., (I −Ta) (Xk−1) respectively, where X1 and Xk−1 are defined as (2.8) and (2.10). The k principal components of X are the column vectors of the matrix Y = [y1, · · · , yn] = UkΣk, Proof. Let r = rank(X) and assume k ≤r. We prove the theorem by induction. For 1 ≤s ≤k, denote Us = u1 · · · us , Σs = diag (σ1, · · · , σs) , Vs = v1 · · · vs and Ys = UsΣs.
Note Ys consists of the first s rows of Y . Define the matrix Bs = YsV T s ∈Rn×p. Suppose s = 1, then by theorem 2.14, B1 = Y1V T 1 is the best 1 rank approximation of X. Namely, (2.17) ∥X −B1∥F = min rank(B)=1 ∥X −B∥F = ∥X −U1Σ1V T 1 ∥F = ∥X −XV T 1 ∥F Recall that in theorem 2.11, we want to minimize E ((I −Ta) (X)) , a ∈Sp−1 to find the first principal direction v1. Rewriting the energy sign in the form of Frobenius norm, we have min E ((I −Ta) (X)) = min ∥(I −Ta) (X) ∥F = min ∥X −Ta(X)∥F Notice that Ta(X) has rank 1, and therefore the minimization problem of finding the first principal direction is equivalent to the best 1-rank approximation in equation (2.17). Namely, v1 is the first principal direction and U1Σ1 is the corresponding first principal component of X.
Now suppose v1, · · · , vs−1 ∈Vs−1 are the first s −1 principal directions of X and Us−1Σs−1 are the corresponding principal components, we want to prove that vs ∈Vs is the sth principal direction of X. By theorem 2.11, we want to minimize E ((I −Ta) (Xs−1)) , a ∈S⊥ s−1 ∩Sp−1 to find the sth principal direction of X. Rewriting the energy in the form of the Frobenius norm, we have min E ((I −Ta) (Xs−1)) = min ∥(I −Ta) (Xs−1) ∥F = min ∥Xs−1 −Ta(Xs−1)∥F = min ∥X −Us−1Σs−1V T s−1 −Ta(Xs−1)∥F = min ∥X − Us−1Σs−1V T s−1 + Ta(Xs−1) ∥F where the third equation holds because Xs−1 = X −Us−1Σs−1V T s−1 by the way Xs−1 is defined in (2.10).
We denote As = Us−1Σs−1V T s−1 + Ta(Xs−1). Notice that As is a rank s matrix because Us−1Σs−1V T s−1 has rank s −1 and Ta(Xs−1) has rank 1 and Ta(Xs−1) is not in the span of v1, . . . , vs−1. Then, the energy minimization problem in finding the sth principal direction becomes min ∥X −As∥F , rank(As) = s which is equivalent to the best s rank approximation problem defined in 2.14.
Therefore, vs is the sth principal direction of X.
To sum up, {v1, . . . , vk} ⊂Rk are successive principal directions of X and the columns of Ys = UsΣs are the corresponding d principal components of X.
□ 8 BAICHEN TAN 2.3. Statistical interpretation of PCA. Now we are going to discuss a statistical interpretation of PCA.
Let X = [x1, · · · , xn]T ∈Rn×p be our dataset. We now consider each column vector xi ∈X as an observation of the dataset, and each row the sample space of a variable.
For simplicity, we assume that our dataset X is already shifted by its sample mean so that all of its components have zero mean, i.e. X = b X. Recall that the variance of a random variable Z with n samples {zi}n i=1 is computed by Var(Z) = E (Z −¯ Z)2 = 1 n n X i=1 zi −¯ Z 2 , where ¯ Z is the mean of Z. Particularly, when ¯ Z = 0, Var(Z) = 1 n n X i=1 (zi)2 = 1 n∥z∥2 2.
We can think of each data point xi as a realization of some random variable following a certain distribution such as Gaussian or Bernoulli depending on the data assumption. Remember that when trying to find the first principal direction, we try to find the projection of X onto a lower dimension that maximizes the energy arg max E (Ta(X)). Because energy of a dataset is defined as the sum of the square of the norm of each data point, we can rewrite the formula in 2.6 as arg max a∈Sp−1 E (Ta(X)) = arg max a∈Sp−1 n X i=1 ∥(Ta(xi)) ∥2 2 We can see that maximizing Pn i=1 ∥(Ta(xi)) ∥2 is equivalent of maximizing the variance of Ta(X).Therefore, the first principal direction of the dataset X is defined as the direction of maximum variance.
3. Classical Multidimensional Scaling (CMDS) Classical multidimensional scaling (CMDS) is a dimensionality reduction method that preserves local features between the data points. Specifically, given a set of high dimensional data points X, we wish to project them onto some lower dimensional subspace in a way that preserves the pairwise distances between them as much as possible.
3.1. Distance and Gram Matrices. In this section we will introduce the distance matrix and gram matrix.
Definition 3.1 (Euclidean distance and square-distance matrix). Let X = [x1, x2, · · · , xn]T ∈Rn×p be our dataset. The Euclidean distance matrix is an n × n matrix D = [Dij] whose each entry Dij is the Euclidean distance between the data point xi and xj: (3.2) [Dij] = [∥xi −xj∥2]n i,j The Euclidean square-distance matrix S is defined as (3.3) [Sij] = [∥xi −xj∥2 2]n i,j Notice that both D and S are symmetric and invariant under shifts and rotations, since these transfor-mations do not affect pairwise distances.
From the definition 3.1 above, we can see that any matrix that can be written in the form (3.2) can be considered as a Euclidean distance matrix. The goal of CMDS is to recover the points xi from knowledge of D alone.
Definition 3.4. Let X = [x1, x2, ..., xn]T ∈Rn×p be a dataset. We define a Gram matrix G on dataset X as [Gij] = [⟨xi, xj⟩]n i,j=1 = XXT where ⟨·, ·⟩is the standard Euclidean inner product. If the Gram matrix is defined on a centered dataset b X, then it is called the centering matrix, which we often denote as Gc.
INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 9 Because the entries of a Gram matrix G are vector inner products, Gram matrix is not shift invariant.
We also notice that G is a positive semi-definite (p.s.d) matrix because for any nonzero real column vector z, (3.5) zT Gz = zT XXT z = zT (XT )T XT z = ∥XT z∥2 2 ≥0 For any dataset, we can easily determine its Gram matrix. Inversely, we can also think of any p.s.d matrix as a Gram matrix of some dataset. This is because if an n × n p.s.d matrix G has rank m, then it has a Cholesky decomposition (3.6) G = XXT where X = [x1, · · · , xn]T is an n×m upper triangular matrix, which we can view as the matrix representation of some sort of dataset {x1, x2, ..., xn} ⊂Rm. Given a p.s.d. matrix, we now want to investigate if it is a centering Gram matrix for some set of points in Rm.
Definition 3.7. Write 1 = [1, 1, · · · , 1]T ∈Rn, E = 11T , and let I denote the n × n identity matrix. Then the n × n matrix H = I −1 nE is called the n-centralizing matrix.
The following lemma helps us to check whether a dataset is centered and whether a positive semi-definite matrix can be viewed as a centering Gram matrix.
Lemma 3.8. The n-centralizing matrix H have the following properties.
(1) H2 = H (2) 1T H = H1 = 0, (3) X is a centered data set, if and only if HT X = X, (4) A positive semi-definite matrix C is a centering Gram matrix, if and only if HCH = C.
The proof of this lemma can be found as Lemma 1 in Chapter 6 of Remark 3.9. Let X ∈Rn×p be a data matrix and G be its Gram matrix. Then the centered data set of X is HT X, and the centering Gram matrix of X is Gc = HGH.
Just like PCA, we always shift our dataset by the sample mean in CMDS because the centering Gram matrix has the following nice property.
Lemma 3.10. Let Gc be a centering Gram matrix. Then Pn i=1 Gc ij = 0 for all j. That is, the sum of each row is equal to zero.
Proof. We can rewrite the summation in Lemma 3.10 in the form of inner products, n X i=1 Gc ij = n X i=1 ⟨xi, xj⟩= n X i=1 xT i , xj + Notice that because xi’s are centered, their summation Pn i=1 xT i adds up to 0.
□ Now we want to explore the relationship between the Gram matrix and the distance matrix.
Lemma 3.11. Let X = [x1, x2, ..., xn]T ∈Rn×p be our dataset. The Gram matrix G and the Euclidean distance matrix D has the following relation: Dij = ∥xi −xj∥2 = p Gii + Gjj −2Gij.
Proof. We compute Dij = q ∥xi −xj∥2 2 = q ⟨xi, xi⟩+ ⟨xj, xj⟩−2 ⟨xi, xj⟩ = p Gii + Gjj −2Gij □ Theorem 3.12. For a given centered dataset X, its Euclidean square-distance matrix S and Gram centering matrix Gc have the following relation.
Gc = −1 2S 10 BAICHEN TAN Proof. By lemma 3.10, Gc has the property Pn i=1 Gc ij = 0. Hence, the relation in lemma 3.11 immediately gives us n X i=1 D2 ij = nGc jj + n X i=1 Gc ii and n X j=1 D2 ij = nGc ii + n X j=1 Gc jj.
Therefore, the (i, j)-entry of S is given by Sij = D2 ij −1 n n X i=1 D2 ij + n X j=1 D2 ij −1 n n X i,j=1 D2 ij = D2 ij −Gc ii −Gc jj = −2Gc ij, completing the proof of the theorem.
□ Theorem 3.12 shows the importance of centering our dataset in CMDS. Given a centered dataset, we can always find its corresponding Euclidean square-distance matrix by its centering Gram matrix through the equation Gc = −1 2S.
By Theorem 3.12, we have the following corollary.
Corollary 3.13. Let A be a symmetric matrix. Then (1) A is a Gram matrix of a data set if and only if it is a p.s.d matrix.
(2) A is a centering Gram matrix if and only if it is a centering p.s.d matrix. A centering p.s.d matrix is a p.s.d matrix subtracted by its sample mean.
(3) A is a Euclidean square-distance matrix if and only if −1 2A is a centering p.s.d matrix.
Proof. (1) is already proved in equation (3.5) and (3.6). (2) can be easily proved by lemma 3.8 (4). We now want to prove (3). For the backwards direction, write Gc = −1 2A. If Gc is a centering p.s.d matrix, by equation (3.6), Gc has a Cholesky decomposition G = XXT where A can be seen as the Euclidean square-distance matrix of the dataset X. For the forwards direction, if A is a Euclidean square-distance matrix, then by theorem 3.12, Gc = −1 2A is the centering Gram matrix of X so that it is a centering p.s.d matrix.
□ The relationship between the centering Gram matrix and the (square) distance matrix is important in CMDS. We can observe that Corollary 3.13 helps us determine whether a given matrix can be viewed as a (centering) Gram matrix or distance matrix. Recall that the intuition of CMDS is that given a distance matrix, we want to construct a configuration of points that reflects this distance matrix. The Gram matrix can then help us find the formula for finding the low dimensional configuration set such that the distance matrix of this low dimensional configuration set is not distorted much from the original distance matrix.
3.2. The CMDS algorithm. Let X = [x1, x2, ..., xn]T ∈Rn×p be a centered dataset, and let D = [Dij] be the distance matrix of our dataset. CMDS tries to find a configuration Y = [y1, · · · , yn]T on a lower k dimensional subspace such that the distance matrix of Y is as close as possible to the matrix D. Namely, ∥yi −yj∥2 ≈Dij, ∀i, j.
Lemma 3.14. Assume that an n × n matrix D is a Euclidean distance matrix and S is the corresponding square-distance matrix. Let Gc = −1 2S. If the rank of Gc is r. Then there is an r-dimensional centered vector set X = [x1, · · · , xn]T ∈Rn×r such that ∥xi −xj∥2 = Dij, 1 ≤i, j ≤n.
INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 11 Proof. By theorem 3.12, Gc is a centering Gram matrix. Since the rank of Gc is r, there exists an n × r centered data matrix X such that Gc = XXT by Cholesky decomposition. Then the centered data set X satisfies the above relation.
□ We call r in Lemma 3.14 the intrinsic configuration dimension of D and X the exact configuration of D.
In section 3.1, we have shown how to determine if a p.s.d matrix is a distance matrix. Now for any distance matrix D, we can find the corresponding Gram matrix Gc by theorem 3.12 and the decomposition Gc = XXT gives you the exact dataset X for the original distance matrix. If we wish to reduce the dimensionality of the data, then we can try to find a lower rank Y = [y1, · · · , yn]T such that the Gram matrix of Y is close to Gc. We use the following loss function to judge whether a certain configuration Y ’s distance matrix is close to the original dataset’s distance matrix.
η(Y ) = n X i,j=1 ∥xi −xj∥2 2 −∥yi −yj∥2 2 , s.t. yi = Pxi, where P is an orthogonal projection from Rp to a k-dimensional subspace Sk ⊂Rr and X is an exact configuration of D. Our goal is to find the Y that minimizes the loss function η, namely, (3.15) Y = arg min Y ∈Rn×k η(Y ), s.t. Y = PX We need to show that each term of the loss function is non-negative so that they won’t cancel each other out. To prove this, we need the following theorem.
Theorem 3.16. Let x be a vector and Px be an orthogonal projection. Then the norm of x is always greater than or equal to Px, namely, ∥Px∥2 ≤∥x∥2 Before the proof of this theorem, we first take an intuitive look at why this is true. As we can see in Figure 6, when we project two vectors to a lower dimension, their distance always shrinks. The length of the projected vector is always shorter than that of the original vector when we project the data point to a lower dimension.
Figure 6. Illustration of Orthogonal Projection Proof of Theorem 3.16. We compute ∥x∥2 2 = ∥Px + x −Px∥2 2 = ∥Px + (I −P)x∥2 2 = ∥Px∥2 2 + ∥(I −P)x∥2 2 + 2(Px)T (I −P)x = ∥Px∥2 2 + ∥(I −P)x∥2 2 12 BAICHEN TAN where the last equation holds because 2(Px)T (I −P)x) = 2xT P T (I −P)x = 2xT (P T −P T P)x = 0 because the orthogonal projection matrix P is symmetric and P 2 = P Then, since ∥x∥2 2 = ∥Px∥2 2 + ∥(I −P)x∥2 2 and ∥(I −P)x∥2 2 ≥0, we have ∥Px∥2 2 ≤∥x∥2 2 and thus ∥Px∥2 ≤∥x∥2, as desired.
□ We can then denote x = xi −xj for xi, xj ∈X and Px = yi −yj. By theorem 3.16, each term of ∥xi −xj∥2 2 −∥yi −yj∥2 2 is always larger or equal to 0.
In order to find a solution for the above problem (3.15), we introduce the following 3 lemmas.
Lemma 3.17. If X is a centered dataset, and A is an orthogonal projection transformation to a lower dimension, then the transformed dataset XA is still centered.
Proof. Because X is centered, by Lemma 3.8 (3), X = HT X and so XA = HT XA where H is the n-centralizing matrix. Then by lemma 3.8 (3) again, XA is a centered dataset.
□ Lemma 3.18. Let X be a centered dataset with corresponding Euclidean square-distance matrix S = [Sij], and let Gc be its centering Gram matrix. Then tr (Gc) = 1 2n n X i=1 n X j=1 Sij The proof of this lemma is expanding the entrywise summation and can be found in Chapter 6 Lemma 6.4 in , and it immediately yields lemma 3.19.
Lemma 3.19. Let X = [x1, · · · , xn]T be the centered dataset and D to be its distance matrix. Then ∥X∥F = 1 √ 2n ∥D∥F .
The proof of this lemma can be found in Chapter 6 Lemma 6.5 in .
We now explain the method for finding the solution of the minimization problem in the equation (3.15).
Theorem 3.20. Let X ∈Rn×r be the configuration of D = [Dij]. Let rank(X) = r and the SVD of X be given by X = UΣV T For a given k ≤r, let Vk = [v1, · · · , vk] and Y = XVk. Then Y is a solution of the minimization problem in (3.15) with η(Y ) = r X i=k+1 σ2 i Proof. Let Sk be a k-dimensional subspace of Rr and B ∈Rr×k be a matrix whose columns form an orthonormal basis on Sk. Then BBT represents the corresponding orthogonal projection from Rr to Sk.
Now observe I −BBT (xi −xj) 2 2 = I −BBT (xi −xj) T I −BBT (xi −xj) = (xT i −xT j )(I −BBT )(I −BBT )(xi −xj) = (xT i −xT j )(I −2BBT + BBT BBT ) (xi −xj) (BT B = I) = (xT i −xT j )(I −BBT )(xi −xj) = xT i xj −xT i BBT xi −xT j xi + xT j BBT xi −xT i xj −xT i xj −xT i xj + xT i BBT xj + xT j xj −xT j BBT xj where the last term is exactly the expansion of the term ∥xi −xj∥2 2 −∥BBT xi −BBT xj∥2 2 INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 13 and so we have (3.21) I −BBT (xi −xj) 2 2 = ∥xi −xj∥2 2 −∥BBT xi −BBT xj∥2 2 Then the loss function η(Y ) becomes (3.22) η(Y ) = n X i,j=1 ∥xi −xj∥2 2 −∥BBT xi, BBT xj∥2 2 = n X i,j=1 I −BBT (xi −xj) 2 2 Therefore, the minimization problem is transformed to the problem of finding a B∗∈Rr×k such that minimizes η(Y ): (3.23) B∗= arg min B∈Rr×k n X i,j=1 I −BBT (xi −xj) 2 Notice that X I −BBT ∈Rn×r. We write ˆ Z = [z1, · · · , zn] = (X I −BBT )T such that each z1, · · · , zn ∈ Rr. Notice that ˆ Z is still centered by Lemma 3.17. We use D ˆ Z to denote the distance matrix of ˆ Z. By Lemma 3.19, we have (3.24) D ˆ Z 2 F = 2n∥ˆ Z∥2 F By equation (3.22), we have (3.25) η(Y ) = n X i,j=1 I −BBT (xi −xj) 2 2 = D ˆ Z 2 F and (3.26) ∥ˆ Z∥F = ∥ˆ ZT ∥F = X −XBBT F Therefore, minimizing η(Y ) is equivalent of minimizing D ˆ Z 2 F by equation (3.25), and is then equivalent to minimizing ∥ˆ Z∥F by equation (3.24). Namely, min η(Y ) = min D ˆ Z 2 F = min ∥ˆ Z∥F .
Because ∥ˆ Z∥F = ∥ˆ ZT ∥F = X −XBBT F by equation (3.26), we have min η(Y ) = min ∥ˆ Z∥F = min X −XBBT F Then our goal of minimizing η(Y ) is equivalent to minimizing X −XBBT F . Notice that XBBT has rank k because we are projecting X to a k-dimensional subspace by BBT . Therefore, minimizing X −XBBT F becomes the best k-rank approximation problem. Letting A = XBBT , we have min X −XBBT F = min rank(A)=k ∥X −A∥F By Theorem 2.14, the solution of the above minimization problem is A = UkΣkV T k = XV T k where Uk, Σk, V T k are defined as remark 2.15. Then we have that B∗BT ∗= Vk is the solution of our minimization problem in equation (3.23), so our lower dimensional configuration is Y = XVk, and the error term follows η (XVk) = ∥X −XVk∥2 F = r X i=k+1 σ2 i by Proposition 2.4.
□ 14 BAICHEN TAN 3.3. Comparison between CMDS and PCA. The starting point between PCA and CMDS are different: in PCA, we are given the dataset and we want to project onto a lower dimensional subspace: to achieve this goal we identify the most important components in a given dataset X that retain the largest variance. In CMDS, we are given the distance matrix of a dataset, and we focus on finding a set of points in a lower-dimensional Euclidean space that maintain the greatest similarities, using an Euclidean distance matrix for n data points. Figure 7 provides an intuitive understanding of the difference between PCA and CMDS.
e (a) Illustration of CMDS (b) Illustration of PCA Figure 7. Comparison between CMDS and PCA As we can see in Figure 7, in subplot 7a, CMDS tries to minimize the difference between the square of the length of the green line segment and that of the blue line segments, i.e. ∥xi −xj∥2 2 −∥yi −yj∥2 2. In subplot 7b, PCA tries to minimize the sum of the length of the green line segments. Therefore, the starting point of PCA and CMDS are different.
However, PCA and CMDS yield the same results. As we can see in Theorem 2.16, the way we find the k principal directions and components is to find the k-rank approximation Pk i=1 σiuivT i = UkΣkV T k . Similarly, in CMDS, the configuration is Y = XVk = UΣV T Vk which, if we write in the form of summation of row and column vectors, gives us the same result as finding the k-rank approximation in the PCA method. We can see that while PCA gives us the principal components Vk which form a basis that spans the subspace in which the transformed data lies, CMDS gives us the coordinates XVk of the coordinates of the transformed data.
4. Random Projection CMDS is a dimensionality reduction method without the direct access to the original dataset. In this section, we are going to introduce another method called random projection which, similar to PCA, operates on the dataset itself. While PCA is good at preserving global properties of a dataset, it may not maintain local relationships. For example, notice that in Figure 8, while the distance between the two yellow points is well preserved after PCA, the distance between the two green points is poorly preserved. This is because PCA projects the data points to a lower dimensional hyperplane in the direction of maximal variance. This means that when we project the data points onto the lower dimensional subspace, the subspace will tend to align in a way that fits points far away from the center and so the greatest pairwise distance will be preserved well while the smaller ones are less so. Thus, for points that are are close to the center but are relatively far away from each other (like the yellow points in the figure), their distances may shrink to be arbitrarily small after PCA. To preserve local data distances in dimensionality reduction, we can use random projection. We will first introduce Lipschitz embeddings, which are important in the application of random projection.
Definition 4.1. A mapping f : V ⊂Rp →Rk is called a Lipschitz mapping on V if there exist two positive constants A and B such that for each pair of vectors u, v ∈V , A∥u −v∥2 2 ≤∥f(u) −f(v)∥2 2 ≤B∥u −v∥2 2 INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 15 Figure 8. Illustration that PCA doesn’t preserve local distances From this, we can see that Lipschitz embeddings have a nice property of giving a lower and upper bound to the distance between any two vectors in the range of f. Random projections seek to employ this property of Lipschitz embeddings to preserve pairwise distances of the data points. We use random matrices to represent these random projections.
4.1. Random Matrices. A random matrix is a matrix such that each entry is an i.i.d random variable following a certain distribution. Here we introduce the most common type of random matrix.
Definition 4.2. A matrix R is Gaussian random matrix if each of its entry is a random variable following normal Gaussian distribution N(0, 1). Namely, R = [rij], f(rij) = 1 √ 2π exp −r2 ij 2 !
where f is the probability density function of normal distribution.
We shall note that while we introduce Gaussian random matrices, there exist other types of random matrices. For example, we can have a Bernoulli random matrix where each entry follows a Bernoulli random variable. For this paper, we shall assume in later part of this text that all random matrices are Gaussian random matrices.
Next we will introduce some interesting properties about random matrices. First, we shall notice that since each entry of a random matrix R ∈Rm×n is an i.i.d Gaussian random variable, if a ∈Rn is a vector, then Ra ∈Rm is a vector of linear combinations of Gaussians. Therefore, each entry of Ra is also a Gaussian random variable. We now explain the relationship of the mean and variances between the random matrix R and that of the random vector Ra.
Lemma 4.3. Let R = [rij] be a k × p matrix, in which all entries are i.i.d. N(0, 1). Let the mapping f : Rp →Rk be defined by (4.4) f(a) = 1 √ k Ra, a ∈Rp.
Then we have E ∥f(a)∥2 2 = ∥a∥2 2.
Proof. We shall first notice that f(a) is a random vector in Rk, of which all components are i.i.d. random variables with 0 mean and variance 1 k∥a∥2 2. We denote the ith row vector of R by ri and denote the inner 16 BAICHEN TAN product ri, a by ci. Then f(a) = c = [c1, · · · , ck]T . We have E [ci] = 1 √ k p X j=1 ajE [rij] = 0, E [cicj] = 1 k p X m=1 p X l=1 alamE [rilrjm] = 0, i ̸= j, and E c2 i = E p X j=1 aj (rij) 2 = 1 k p X j=1 a2 i E r2 ij + 2 X l̸=m alamE [ril] E [rim] = 1 k p X j=1 a2 i , which yields that all entries of f(a) are i.i.d. random variables with 0 mean and variance 1 k∥a∥2 2. Finally, we have E ∥f(a)∥2 2 = k X i=1 E c2 i = k ∥a∥2 2 k = ∥a∥2 2.
□ This lemma shows that for any random matrix R = [ri,j] with E[rij] = 0 and Var(ri,j) = 1, we have E[∥Ra∥2 2] = ∥a∥2 2 for any vector a. In particular, let R = [rij] be a k × D random matrix. If a is a unit vector, then √ kRa is a k-dimensional random vector whose entries are i.i.d random variables with 0 mean and variance 1.
4.2. Dimensionality Reduction with Random Projection. Let X = [x1, x2, · · · , xn]T ∈Rn×p be our high dimensional dataset.
Random projection tries to project our high dimensional dataset to a lower dimension by a linear mapping f defined in (4.4).
We should first notice that for any random projection f : Rp →Rk where k < p, the projection f cannot be a Lipschitz embedding. This is because by the rank-nullity theorem, f has nontrivial null space. Namely, there exists nonzero x1 ̸= x2 such that f(x1) = f(x2) = 0. Therefore we cannot find positive constants A and B such that A∥x1 −x2∥2 2 ≤∥f(x1) −f(x2)∥2 2 ≤B∥x1 −x2∥2 2 because A∥x1 −x2∥2 2 is larger than 0 while ∥f(x1) −f(x2)∥2 2 is equal to 0.
Therefore, random projection method does not aim to find a Lipschitz embedding f : Rp →Rk. Rather, it seeks to find an embedding that is Lipschitz on a subset X ⊂Rp with a high probability such that this embedding, after projecting our original dataset to lower dimension k with some restrictions, preserves the pairwise distances between data points by restricting the pairwise distances between a lower and an upper bound.
Lemma 4.5 (Johnson and Lindenstrauss Lemma (JL Lemma)). For any 0 < ε < 1 and any integer n > 0, let k be a positive integer such that (4.6) k ≥4 ε2/2 −ε3/3 −1 ln n Then for any dataset X ⊂Rp of n points, let f : Rp →Rk be defined as (4.4). Then for any two xi, xj ∈X, (4.7) (1 −ε)∥xi −xj∥2 2 ≤∥f(xi) −f(xj)∥2 2 ≤(1 + ε)∥xi −xj∥2 2 holds with probability 1 −1 n.
INTRODUCTION TO LINEAR DIMENSIONALITY REDUCTION METHODS 17 We shall notice the intuition behind the lower bound on k. In the inequality (4.6), ε is the error tolerance, which measures how tight the bound is on ∥f(xi) −f(xj)∥2, as stated in the inequality (4.7). As ε gets smaller, we allow for less error and 4 ε2/2 −ε3/3 −1 ln n becomes bigger, and thus k has a higher lower bound. Similarly, as n gets larger, 4 ε2/2 −ε3/3 −1 ln n also becomes bigger, and thus k has a higher lower bound. Intuitively, with less error or with more data points, our dimensionality reduction problem becomes harder, so we cannot project k to a very low dimension while still preserving pairwise distances well.
To prove the JL lemma, we need the following lemma first.
Lemma 4.8. Let R be a k × p (k ≤p) Gaussian random matrix and a ∈Rp be a unit vector. Let y = Ra and 1 < β. Then (4.9) Pr ∥y∥2 2 ≤k β < exp k 2 1 −1 β −ln β (4.10) Pr ∥y∥2 2 ≥kβ < exp k 2(1 −β + ln β) The proof of this lemma can be found in Chapter 7 Lemma 7.3 of the book .
Proof of JL Lemma. Let R be a k×p random matrix whose entries are i.i.d. random variables ∼N(0, 1). We want to show that the linear map R satisfies the inequality (4.7). For each pair of xi and xj in X with xi ̸= xj, we set a = xi −xj ∥xi −xj∥2 , z = R(a) y = √ kz Then Pr ∥f(xi) −f(xj)∥2 2 ∥xi −xj∥2 2 ≤(1 −ε) = Pr ∥z∥2 2 ≤(1 −ε) = Pr ∥y∥2 2 ≤(1 −ε)k .
Similarly, Pr ∥f(xi) −f(xj)∥2 2 ∥xi −xj∥2 2 ≥(1 + ε) = Pr ∥y∥2 2 ≥(1 + ε)k .
Applying formula (4.9) in Lemma 4.5 to y with 1 β = 1 −ε, we have Pr ∥y∥2 2 ≤(1 −ε)k < exp k 2(1 −(1 −ε) + ln(1 −ε)) ≤exp k 2 1 −(1 −ε) − ε −ε2 2 = exp −kε2 4 ≤exp(−2 ln n) = 1 n2 , where the inequality ln(1 −ε) ≤−ε −ε2/2 is used in second line, and the condition k ≥4 ε2/2 −ε3/3 −1 ln n is applied to deriving the inequality in the last line. Similarly, we have Pr ∥y∥2 2 ≥(1 + ε)k < exp k 2(1 −(1 + ε) + ln(1 + ε)) ≤exp k 2 −ε + ε −ε2 2 + ε3 3 = exp −k ε2/2 −ε3/3 2 !
≤exp(−2 ln n) = 1 n2 .
Therefore, for each pair xi and xj in X, Pr ∥f(xi) −f(xj)∥2 2 ∥xi −xj∥2 2 / ∈[1 −ε, 1 + ε] < 2 n2 18 BAICHEN TAN Since the cardinality of X is n, there are a total of n(n −1)/2 pairs. The probability of the event that at least one pair does not satisfy Lemma 4.6 is less than 1 −n(n−1) 2 2 n2 = 1/n > 0.
□ From the proof above, we can see that a random matrix R, while not a Lipschitz embedding on the whole vector space Rp, can help preserve local properties of a dataset X ⊂Rp with high probability. Specifically, if we have a dataset X with a very large number of observations n, then the probability that we can bound the distances of all pairs of f(xi), f(xj), xi, xj ∈X is 1 −1 n.
With the information above, now we can construct the algorithm for random projections.
4.3. Random Projection Algorithm. The random projection algorithm consists of two steps: first, we create a random matrix R; and second, we multiply the created random matrix with our high-dimensional dataset X and output the projected dataset RX.
Let xi, xj be two data points from X. Recall the formula (4.7) in JL lemma, (1 −ε)∥xi −xi∥2 2 ≤∥f(xi) −f(xj)∥2 2 ≤(1 + ε)∥xi −xj∥2 2 We can rewrite the formula as (1 −ε)∥xi −xj∥2 2 ≤ 1 √ k Rxi −1 √ k Rxj 2 2 ≤(1 + ε)∥xi −xj∥2 2 We can take the constant 1 √ k out of the norm operator, and move 1 k the the other sides of the inequations, (1 −ε)∥xi −xj∥2 2 ≤1 k ∥Rxi −Rxj∥2 2 ≤(1 + ε)∥xi −xj∥2 2 k(1 −ε)∥xi −xj∥2 2 ≤∥Rxi −Rxj∥2 2 ≤k(1 + ε)∥xi −xj∥2 2 Therefore, the JL lemma guarantees that the random projection algorithm can bound the pairwise distances of our projected dataset RX with an upper bound k(1+ε) and lower bound k(1−ε) with a high probability of 1 −1 n.
Acknowledgments I give sincere thanks to my mentor Phillip Lo, who gives great guidance in helping me understand the materials and write the paper.
References Severn, Katie. Multivariate Statistics Wang, Jianzhong. Geometric Structure of High-Dimensional Data and Dimensionality Reduction. Higher Education Press, 2012. |
11778 | https://www.mathway.com/popular-problems/Algebra/745006 | Algebra Examples
f(x)=-(x-1)(x+2)(x+1)2f(x)=−(x−1)(x+2)(x+1)2
Step 1
Identify the degree of the function.
Tap for more steps...
Step 1.1
Simplify and reorder the polynomial.
Tap for more steps...
Step 1.1.1
Simplify by multiplying through.
Tap for more steps...
Step 1.1.1.1
Apply the distributive property.
(-x--1)(x+2)(x+1)2(−x−−1)(x+2)(x+1)2
Step 1.1.1.2
Multiply -1−1 by -1−1.
(-x+1)(x+2)(x+1)2(−x+1)(x+2)(x+1)2
(-x+1)(x+2)(x+1)2(−x+1)(x+2)(x+1)2
Step 1.1.2
Expand (-x+1)(x+2)(−x+1)(x+2) using the FOIL Method.
Tap for more steps...
Step 1.1.2.1
Apply the distributive property.
(-x(x+2)+1(x+2))(x+1)2(−x(x+2)+1(x+2))(x+1)2
Step 1.1.2.2
Apply the distributive property.
(-x⋅x-x⋅2+1(x+2))(x+1)2(−x⋅x−x⋅2+1(x+2))(x+1)2
Step 1.1.2.3
Apply the distributive property.
(-x⋅x-x⋅2+1x+1⋅2)(x+1)2(−x⋅x−x⋅2+1x+1⋅2)(x+1)2
(-x⋅x-x⋅2+1x+1⋅2)(x+1)2(−x⋅x−x⋅2+1x+1⋅2)(x+1)2
Step 1.1.3
Simplify and combine like terms.
Tap for more steps...
Step 1.1.3.1
Simplify each term.
Tap for more steps...
Step 1.1.3.1.1
Multiply xx by xx by adding the exponents.
Tap for more steps...
Step 1.1.3.1.1.1
Move xx.
(-(x⋅x)-x⋅2+1x+1⋅2)(x+1)2(−(x⋅x)−x⋅2+1x+1⋅2)(x+1)2
Step 1.1.3.1.1.2
Multiply xx by xx.
(-x2-x⋅2+1x+1⋅2)(x+1)2(−x2−x⋅2+1x+1⋅2)(x+1)2
(-x2-x⋅2+1x+1⋅2)(x+1)2(−x2−x⋅2+1x+1⋅2)(x+1)2
Step 1.1.3.1.2
Multiply 22 by -1−1.
(-x2-2x+1x+1⋅2)(x+1)2(−x2−2x+1x+1⋅2)(x+1)2
Step 1.1.3.1.3
Multiply xx by 11.
(-x2-2x+x+1⋅2)(x+1)2(−x2−2x+x+1⋅2)(x+1)2
Step 1.1.3.1.4
Multiply 22 by 11.
(-x2-2x+x+2)(x+1)2(−x2−2x+x+2)(x+1)2
(-x2-2x+x+2)(x+1)2(−x2−2x+x+2)(x+1)2
Step 1.1.3.2
Add -2x−2x and xx.
(-x2-x+2)(x+1)2(−x2−x+2)(x+1)2
(-x2-x+2)(x+1)2(−x2−x+2)(x+1)2
Step 1.1.4
Rewrite (x+1)2(x+1)2 as (x+1)(x+1)(x+1)(x+1).
(-x2-x+2)((x+1)(x+1))(−x2−x+2)((x+1)(x+1))
Step 1.1.5
Expand (x+1)(x+1)(x+1)(x+1) using the FOIL Method.
Tap for more steps...
Step 1.1.5.1
Apply the distributive property.
(-x2-x+2)(x(x+1)+1(x+1))(−x2−x+2)(x(x+1)+1(x+1))
Step 1.1.5.2
Apply the distributive property.
(-x2-x+2)(x⋅x+x⋅1+1(x+1))(−x2−x+2)(x⋅x+x⋅1+1(x+1))
Step 1.1.5.3
Apply the distributive property.
(-x2-x+2)(x⋅x+x⋅1+1x+1⋅1)(−x2−x+2)(x⋅x+x⋅1+1x+1⋅1)
(-x2-x+2)(x⋅x+x⋅1+1x+1⋅1)(−x2−x+2)(x⋅x+x⋅1+1x+1⋅1)
Step 1.1.6
Simplify and combine like terms.
Tap for more steps...
Step 1.1.6.1
Simplify each term.
Tap for more steps...
Step 1.1.6.1.1
Multiply xx by xx.
(-x2-x+2)(x2+x⋅1+1x+1⋅1)(−x2−x+2)(x2+x⋅1+1x+1⋅1)
Step 1.1.6.1.2
Multiply xx by 11.
(-x2-x+2)(x2+x+1x+1⋅1)(−x2−x+2)(x2+x+1x+1⋅1)
Step 1.1.6.1.3
Multiply xx by 11.
(-x2-x+2)(x2+x+x+1⋅1)(−x2−x+2)(x2+x+x+1⋅1)
Step 1.1.6.1.4
Multiply 11 by 11.
(-x2-x+2)(x2+x+x+1)(−x2−x+2)(x2+x+x+1)
(-x2-x+2)(x2+x+x+1)(−x2−x+2)(x2+x+x+1)
Step 1.1.6.2
Add xx and xx.
(-x2-x+2)(x2+2x+1)(−x2−x+2)(x2+2x+1)
(-x2-x+2)(x2+2x+1)(−x2−x+2)(x2+2x+1)
Step 1.1.7
Expand (-x2-x+2)(x2+2x+1)(−x2−x+2)(x2+2x+1) by multiplying each term in the first expression by each term in the second expression.
-x2x2-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x2x2−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8
Simplify terms.
Tap for more steps...
Step 1.1.8.1
Simplify each term.
Tap for more steps...
Step 1.1.8.1.1
Multiply x2x2 by x2x2 by adding the exponents.
Tap for more steps...
Step 1.1.8.1.1.1
Move x2x2.
-(x2x2)-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−(x2x2)−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.1.2
Use the power rule aman=am+naman=am+n to combine exponents.
-x2+2-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x2+2−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.1.3
Add 22 and 22.
-x4-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.2
Rewrite using the commutative property of multiplication.
-x4-1⋅2x2x-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x2x−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.3
Multiply x2x2 by xx by adding the exponents.
Tap for more steps...
Step 1.1.8.1.3.1
Move xx.
-x4-1⋅2(x⋅x2)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2(x⋅x2)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.3.2
Multiply xx by x2x2.
Tap for more steps...
Step 1.1.8.1.3.2.1
Raise xx to the power of 11.
-x4-1⋅2(x1x2)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2(x1x2)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.3.2.2
Use the power rule aman=am+naman=am+n to combine exponents.
-x4-1⋅2x1+2-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x1+2−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-1⋅2x1+2-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x1+2−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.3.3
Add 11 and 22.
-x4-1⋅2x3-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x3−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-1⋅2x3-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x3−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.4
Multiply -1−1 by 22.
-x4-2x3-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.5
Multiply -1−1 by 11.
-x4-2x3-x2-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.6
Multiply xx by x2x2 by adding the exponents.
Tap for more steps...
Step 1.1.8.1.6.1
Move x2x2.
-x4-2x3-x2-(x2x)-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−(x2x)−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.6.2
Multiply x2x2 by xx.
Tap for more steps...
Step 1.1.8.1.6.2.1
Raise xx to the power of 11.
-x4-2x3-x2-(x2x1)-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−(x2x1)−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.6.2.2
Use the power rule aman=am+naman=am+n to combine exponents.
-x4-2x3-x2-x2+1-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x2+1−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-2x3-x2-x2+1-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x2+1−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.6.3
Add 22 and 11.
-x4-2x3-x2-x3-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-2x3-x2-x3-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.7
Rewrite using the commutative property of multiplication.
-x4-2x3-x2-x3-1⋅2x⋅x-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−1⋅2x⋅x−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.8
Multiply xx by xx by adding the exponents.
Tap for more steps...
Step 1.1.8.1.8.1
Move xx.
-x4-2x3-x2-x3-1⋅2(x⋅x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−1⋅2(x⋅x)−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.8.2
Multiply xx by xx.
-x4-2x3-x2-x3-1⋅2x2-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−1⋅2x2−x⋅1+2x2+2(2x)+2⋅1
-x4-2x3-x2-x3-1⋅2x2-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−1⋅2x2−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.9
Multiply -1−1 by 22.
-x4-2x3-x2-x3-2x2-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−2x2−x⋅1+2x2+2(2x)+2⋅1
Step 1.1.8.1.10
Multiply -1−1 by 11.
-x4-2x3-x2-x3-2x2-x+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−2x2−x+2x2+2(2x)+2⋅1
Step 1.1.8.1.11
Multiply 22 by 22.
-x4-2x3-x2-x3-2x2-x+2x2+4x+2⋅1−x4−2x3−x2−x3−2x2−x+2x2+4x+2⋅1
Step 1.1.8.1.12
Multiply 22 by 11.
-x4-2x3-x2-x3-2x2-x+2x2+4x+2−x4−2x3−x2−x3−2x2−x+2x2+4x+2
-x4-2x3-x2-x3-2x2-x+2x2+4x+2−x4−2x3−x2−x3−2x2−x+2x2+4x+2
Step 1.1.8.2
Simplify by adding terms.
Tap for more steps...
Step 1.1.8.2.1
Combine the opposite terms in -x4-2x3-x2-x3-2x2-x+2x2+4x+2−x4−2x3−x2−x3−2x2−x+2x2+4x+2.
Tap for more steps...
Step 1.1.8.2.1.1
Add -2x2−2x2 and 2x22x2.
-x4-2x3-x2-x3-x+0+4x+2−x4−2x3−x2−x3−x+0+4x+2
Step 1.1.8.2.1.2
Add -x4-2x3-x2-x3-x−x4−2x3−x2−x3−x and 00.
-x4-2x3-x2-x3-x+4x+2−x4−2x3−x2−x3−x+4x+2
-x4-2x3-x2-x3-x+4x+2−x4−2x3−x2−x3−x+4x+2
Step 1.1.8.2.2
Subtract x3x3 from -2x3−2x3.
-x4-3x3-x2-x+4x+2−x4−3x3−x2−x+4x+2
Step 1.1.8.2.3
Add -x−x and 4x4x.
-x4-3x3-x2+3x+2−x4−3x3−x2+3x+2
-x4-3x3-x2+3x+2−x4−3x3−x2+3x+2
-x4-3x3-x2+3x+2−x4−3x3−x2+3x+2
-x4-3x3-x2+3x+2−x4−3x3−x2+3x+2
Step 1.2
The largest exponent is the degree of the polynomial.
44
44
Step 2
Since the degree is even, the ends of the function will point in the same direction.
Even
Step 3
Identify the leading coefficient.
Tap for more steps...
Step 3.1
Simplify the polynomial, then reorder it left to right starting with the highest degree term.
Tap for more steps...
Step 3.1.1
Simplify by multiplying through.
Tap for more steps...
Step 3.1.1.1
Apply the distributive property.
(-x--1)(x+2)(x+1)2(−x−−1)(x+2)(x+1)2
Step 3.1.1.2
Multiply -1−1 by -1−1.
(-x+1)(x+2)(x+1)2(−x+1)(x+2)(x+1)2
(-x+1)(x+2)(x+1)2(−x+1)(x+2)(x+1)2
Step 3.1.2
Expand (-x+1)(x+2)(−x+1)(x+2) using the FOIL Method.
Tap for more steps...
Step 3.1.2.1
Apply the distributive property.
(-x(x+2)+1(x+2))(x+1)2(−x(x+2)+1(x+2))(x+1)2
Step 3.1.2.2
Apply the distributive property.
(-x⋅x-x⋅2+1(x+2))(x+1)2(−x⋅x−x⋅2+1(x+2))(x+1)2
Step 3.1.2.3
Apply the distributive property.
(-x⋅x-x⋅2+1x+1⋅2)(x+1)2(−x⋅x−x⋅2+1x+1⋅2)(x+1)2
(-x⋅x-x⋅2+1x+1⋅2)(x+1)2(−x⋅x−x⋅2+1x+1⋅2)(x+1)2
Step 3.1.3
Simplify and combine like terms.
Tap for more steps...
Step 3.1.3.1
Simplify each term.
Tap for more steps...
Step 3.1.3.1.1
Multiply xx by xx by adding the exponents.
Tap for more steps...
Step 3.1.3.1.1.1
Move xx.
(-(x⋅x)-x⋅2+1x+1⋅2)(x+1)2(−(x⋅x)−x⋅2+1x+1⋅2)(x+1)2
Step 3.1.3.1.1.2
Multiply xx by xx.
(-x2-x⋅2+1x+1⋅2)(x+1)2(−x2−x⋅2+1x+1⋅2)(x+1)2
(-x2-x⋅2+1x+1⋅2)(x+1)2(−x2−x⋅2+1x+1⋅2)(x+1)2
Step 3.1.3.1.2
Multiply 22 by -1−1.
(-x2-2x+1x+1⋅2)(x+1)2(−x2−2x+1x+1⋅2)(x+1)2
Step 3.1.3.1.3
Multiply xx by 11.
(-x2-2x+x+1⋅2)(x+1)2(−x2−2x+x+1⋅2)(x+1)2
Step 3.1.3.1.4
Multiply 22 by 11.
(-x2-2x+x+2)(x+1)2(−x2−2x+x+2)(x+1)2
(-x2-2x+x+2)(x+1)2(−x2−2x+x+2)(x+1)2
Step 3.1.3.2
Add -2x−2x and xx.
(-x2-x+2)(x+1)2(−x2−x+2)(x+1)2
(-x2-x+2)(x+1)2(−x2−x+2)(x+1)2
Step 3.1.4
Rewrite (x+1)2(x+1)2 as (x+1)(x+1)(x+1)(x+1).
(-x2-x+2)((x+1)(x+1))(−x2−x+2)((x+1)(x+1))
Step 3.1.5
Expand (x+1)(x+1)(x+1)(x+1) using the FOIL Method.
Tap for more steps...
Step 3.1.5.1
Apply the distributive property.
(-x2-x+2)(x(x+1)+1(x+1))(−x2−x+2)(x(x+1)+1(x+1))
Step 3.1.5.2
Apply the distributive property.
(-x2-x+2)(x⋅x+x⋅1+1(x+1))(−x2−x+2)(x⋅x+x⋅1+1(x+1))
Step 3.1.5.3
Apply the distributive property.
(-x2-x+2)(x⋅x+x⋅1+1x+1⋅1)(−x2−x+2)(x⋅x+x⋅1+1x+1⋅1)
(-x2-x+2)(x⋅x+x⋅1+1x+1⋅1)(−x2−x+2)(x⋅x+x⋅1+1x+1⋅1)
Step 3.1.6
Simplify and combine like terms.
Tap for more steps...
Step 3.1.6.1
Simplify each term.
Tap for more steps...
Step 3.1.6.1.1
Multiply xx by xx.
(-x2-x+2)(x2+x⋅1+1x+1⋅1)(−x2−x+2)(x2+x⋅1+1x+1⋅1)
Step 3.1.6.1.2
Multiply xx by 11.
(-x2-x+2)(x2+x+1x+1⋅1)(−x2−x+2)(x2+x+1x+1⋅1)
Step 3.1.6.1.3
Multiply xx by 11.
(-x2-x+2)(x2+x+x+1⋅1)(−x2−x+2)(x2+x+x+1⋅1)
Step 3.1.6.1.4
Multiply 11 by 11.
(-x2-x+2)(x2+x+x+1)(−x2−x+2)(x2+x+x+1)
(-x2-x+2)(x2+x+x+1)(−x2−x+2)(x2+x+x+1)
Step 3.1.6.2
Add xx and xx.
(-x2-x+2)(x2+2x+1)(−x2−x+2)(x2+2x+1)
(-x2-x+2)(x2+2x+1)(−x2−x+2)(x2+2x+1)
Step 3.1.7
Expand (-x2-x+2)(x2+2x+1)(−x2−x+2)(x2+2x+1) by multiplying each term in the first expression by each term in the second expression.
-x2x2-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x2x2−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8
Simplify terms.
Tap for more steps...
Step 3.1.8.1
Simplify each term.
Tap for more steps...
Step 3.1.8.1.1
Multiply x2x2 by x2x2 by adding the exponents.
Tap for more steps...
Step 3.1.8.1.1.1
Move x2x2.
-(x2x2)-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−(x2x2)−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.1.2
Use the power rule aman=am+naman=am+n to combine exponents.
-x2+2-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x2+2−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.1.3
Add 22 and 22.
-x4-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-x2(2x)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−x2(2x)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.2
Rewrite using the commutative property of multiplication.
-x4-1⋅2x2x-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x2x−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.3
Multiply x2x2 by xx by adding the exponents.
Tap for more steps...
Step 3.1.8.1.3.1
Move xx.
-x4-1⋅2(x⋅x2)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2(x⋅x2)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.3.2
Multiply xx by x2x2.
Tap for more steps...
Step 3.1.8.1.3.2.1
Raise xx to the power of 11.
-x4-1⋅2(x1x2)-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2(x1x2)−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.3.2.2
Use the power rule aman=am+naman=am+n to combine exponents.
-x4-1⋅2x1+2-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x1+2−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-1⋅2x1+2-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x1+2−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.3.3
Add 11 and 22.
-x4-1⋅2x3-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x3−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-1⋅2x3-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−1⋅2x3−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.4
Multiply -1−1 by 22.
-x4-2x3-x2⋅1-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2⋅1−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.5
Multiply -1−1 by 11.
-x4-2x3-x2-x⋅x2-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x⋅x2−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.6
Multiply xx by x2x2 by adding the exponents.
Tap for more steps...
Step 3.1.8.1.6.1
Move x2x2.
-x4-2x3-x2-(x2x)-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−(x2x)−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.6.2
Multiply x2x2 by xx.
Tap for more steps...
Step 3.1.8.1.6.2.1
Raise xx to the power of 11.
-x4-2x3-x2-(x2x1)-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−(x2x1)−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.6.2.2
Use the power rule aman=am+naman=am+n to combine exponents.
-x4-2x3-x2-x2+1-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x2+1−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-2x3-x2-x2+1-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x2+1−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.6.3
Add 22 and 11.
-x4-2x3-x2-x3-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−x(2x)−x⋅1+2x2+2(2x)+2⋅1
-x4-2x3-x2-x3-x(2x)-x⋅1+2x2+2(2x)+2⋅1−x4−2x3−x2−x3−x(2x)−x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.7
Rewrite using the commutative property of multiplication.
-x4-2x3-x2-x3-1⋅2x⋅x-x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.8
Multiply x by x by adding the exponents.
Tap for more steps...
Step 3.1.8.1.8.1
Move x.
-x4-2x3-x2-x3-1⋅2(x⋅x)-x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.8.2
Multiply x by x.
-x4-2x3-x2-x3-1⋅2x2-x⋅1+2x2+2(2x)+2⋅1
-x4-2x3-x2-x3-1⋅2x2-x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.9
Multiply -1 by 2.
-x4-2x3-x2-x3-2x2-x⋅1+2x2+2(2x)+2⋅1
Step 3.1.8.1.10
Multiply -1 by 1.
-x4-2x3-x2-x3-2x2-x+2x2+2(2x)+2⋅1
Step 3.1.8.1.11
Multiply 2 by 2.
-x4-2x3-x2-x3-2x2-x+2x2+4x+2⋅1
Step 3.1.8.1.12
Multiply 2 by 1.
-x4-2x3-x2-x3-2x2-x+2x2+4x+2
-x4-2x3-x2-x3-2x2-x+2x2+4x+2
Step 3.1.8.2
Simplify by adding terms.
Tap for more steps...
Step 3.1.8.2.1
Combine the opposite terms in -x4-2x3-x2-x3-2x2-x+2x2+4x+2.
Tap for more steps...
Step 3.1.8.2.1.1
Add -2x2 and 2x2.
-x4-2x3-x2-x3-x+0+4x+2
Step 3.1.8.2.1.2
Add -x4-2x3-x2-x3-x and 0.
-x4-2x3-x2-x3-x+4x+2
-x4-2x3-x2-x3-x+4x+2
Step 3.1.8.2.2
Subtract x3 from -2x3.
-x4-3x3-x2-x+4x+2
Step 3.1.8.2.3
Add -x and 4x.
-x4-3x3-x2+3x+2
-x4-3x3-x2+3x+2
-x4-3x3-x2+3x+2
-x4-3x3-x2+3x+2
Step 3.2
The leading term in a polynomial is the term with the highest degree.
-x4
Step 3.3
The leading coefficient in a polynomial is the coefficient of the leading term.
-1
-1
Step 4
Since the leading coefficient is negative, the graph falls to the right.
Negative
Step 5
Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.
Even and Positive: Rises to the left and rises to the right.
Even and Negative: Falls to the left and falls to the right.
Odd and Positive: Falls to the left and rises to the right.
Odd and Negative: Rises to the left and falls to the right
Step 6
Determine the behavior.
Falls to the left and falls to the right
Step 7
| | | |
---
| | | |
(
(
)
)
|
|
[
[
]
]
√
√
≥
≥
7
7
8
8
9
9
≤
4
5
6
/
^
×
∩
∪
1
2
3
-
+
÷
<
π
∞
,
0
.
%
=
[x2 12 √π ∫xdx ]
Please ensure that your password is at least 8 characters and contains each of the following:
a number
a letter
a special character: @$#!%?& |
11779 | https://www.ebay.com/itm/277113228039 | An Introduction to Human Evolutionary Anatomy by Aiello & Dean Paperback - GOOD 9780120455911| eBay
Skip to main content
Hi! Sign in or registerDaily DealsBrand OutletGift CardsHelp & Contact
Sell
WatchlistExpand Watch List
My eBayExpand My eBay
Summary
Recently Viewed
Bids/Offers
Watchlist
Purchase History
Buy Again
Selling
Saved Feed
Saved Searches
Saved Sellers
My Garage
Sizes
My Collection
Messages
PSA Vault
Expand Notifications
Please sign-in to view notifications.
Expand Cart
Loading...
Shop by category
### Motors
Parts & accessoriesCars & trucksMotorcyclesOther vehicles
### Clothing & Accessories
WomenMenHandbagsCollectible Sneakers
### Sporting goods
Hunting EquipmentGolf EquipmentOutdoor sportsCycling Equipment
### Electronics
Computers, Tablets & Network HardwareCell Phones, Smart Watches & AccessoriesVideo Games & ConsolesCameras & Photo
### Business & Industrial
Modular & Pre-Fabricated BuildingsTest, Measurement & Inspection EquipmentHeavy Equipment, Parts & AttachmentsRestaurant & Food Service
### Jewelry & Watches
Luxury WatchesWristwatchesFashion JewelryFine Jewelry
### Collectibles & Art
Trading CardsCollectiblesCoins & Paper MoneySports Memorabilia
### Home & garden
Yard, Garden & Outdoor Living ItemsTools & Workshop EquipmentHome ImprovementKitchen, Dining & Bar Supplies
### Other categories
Books, Movies & MusicToys & HobbiesHealth & BeautyBaby Essentials
All BrandsAll CategoriesSeasonal Sales & Events
SearchAdvanced
Find similar items from
MTW Youth
1.7M items sold
Shop store on eBay
People who viewed this item also viewed
### An Introduction to Human Evolutionary - Paperback, by Aiello Leslie; Dean - Good $13.30 Free shipping Sponsored
### An Introduction to Human Evolutionary Anatomy Leslie, Dean, Chris $12.74 Free shipping
### An Introduction to Human Evolutionary Anatomy Leslie, Dean, Chris $12.74 Free shipping
### An Introduction to Human Evolutionary Anatomy by Leslie Aiello: Used $20.52 Free shipping
### An Introduction to Human Evolutionary Anatomy by Aiello $19.99 + $6.99 shipping
(0) Detail(s) of 5 selected Apply
Picture 16 of 16
Gallery
Picture 16 of 16
Have one to sell?
Sell now
Share
Similar Items
4 S S p o n 6 7 s o r H 0 M e d Q 4 D A
See all
### Classic Human Anatomy: The Artist's Guide to Form, Function, and Movement, Winsl Pre-owned $14.99 Free shipping Seller with 99.6% positive feedback
### Human Anatomy & Physiology (9th - Hardcover, by Marieb Elaine N.; - Acceptable Pre-owned $17.57 Free shipping Top Rated Plus Seller with 99.4% positive feedback
### Sobotta Hammersen Histology Atlas Cytology Microscopic Anatomy 1980 Hardcover Pre-owned $27.99 previous price$39.99 30% off Free shipping Top Rated Plus Seller with 99.8% positive feedback
### Introduction to Animal and Veterinary Anatomy and Physiology - 9781800624047 New $43.57 previous price$50.26 13% off + $25.50 shipping Seller with 99.6% positive feedback
(0) Detail(s) of 5 selected Apply
An Introduction to Human Evolutionary Anatomy by Aiello & Dean Paperback - GOOD
Premium Picks
(3812)
99.8% positive
Seller's other items Seller's other items
Contact seller
US $14.99
or Best Offer
Condition:
Good Good
“Has a sturdy binding with some shelf wear. Cover has creases or "veins" from creases. Has original ”... Read more about condition
Good
A book that has been read but is in good condition. Very minimal damage to the cover including scuff marks, but no holes or tears. The dust jacket for hard covers may not be included. Binding has minimal wear. The majority of pages are undamaged with minimal creasing or tearing, minimal pencil underlining of text, no highlighting of text, no writing in margins. No missing pages. See the seller’s listing for full details and description of any imperfections.
Buy It Now An Introduction to Human Evolutionary Anatomy by Aiello & Dean Paperback - GOOD Sign in to check out Check out as guest
Add to cart
Make offer
Add to Watchlist
Oops! Looks like we're having trouble connecting to our server.
Refresh your browser window to try again.
Refresh Browser
Breathe easy. Returns accepted.
Shipping:
Free 2-4 day delivery
Get it between Wed, Oct 1 and Fri, Oct 3 to 94043. See details for shipping
Located in: San Diego, California, United States
Returns:
30 days returns. Buyer pays for return shipping. If you use an eBay shipping label, it will be deducted from your refund amount. See details- for more information about returns
Payments:
Special financing available. See terms and apply now- for PayPal Credit, opens in a new window or tab
Earn up to 5x points when you use your eBay Mastercard®.Learn more about earning points with eBay Mastercard
Shop with confidence
eBay Money Back Guarantee
Get the item you ordered or your money back.Learn more eBay Money Back Guarantee - opens new window or tab
About this item
Seller assumes all responsibility for this listing.
eBay item number:277113228039
Item specifics
Condition
Good
A book that has been read but is in good condition. Very minimal damage to the cover including scuff marks, but no holes or tears. The dust jacket for hard covers may not be included. Binding has minimal wear. The majority of pages are undamaged with minimal creasing or tearing, minimal pencil underlining of text, no highlighting of text, no writing in margins. No missing pages. See the seller’s listing for full details and description of any imperfections. See all condition definitions opens in a new window or tab
Seller Notes
“Has a sturdy binding with some shelf wear. Cover has creases or "veins" from creases. Has ... Read more about the seller notes“Has a sturdy binding with some shelf wear. Cover has creases or "veins" from creases. Has original owner's name written in book.Please see pics for condition.”Read Less about the seller notes
ISBN
9780120455911
Subject Area
Social Science, Science
Publication Name
Introduction to Human Evolutionary Anatomy
Publisher
Elsevier Science & Technology
Item Length
10 in
Subject
Life Sciences / Anatomy & Physiology (See Also Life Sciences / Human Anatomy & Physiology), Anthropology / Physical, Anthropology / General
Publication Year
1990
Type
Textbook
Format
Trade Paperback
Language
English
Author
Leslie Aiello, Christopher Dean
Item Weight
42.2 Oz
Item Width
6.9 in
Number of Pages
608 Pages
Category
breadcrumb
Books, Movies & Music
Books & Magazines
Books & Magazines
Textbooks, Education & Reference
Textbooks
See more Introduction to Human Evolutionary Anatomy by ...
About this product
Product Identifiers
Publisher
Elsevier Science & Technology
ISBN-10
0120455919
ISBN-13
9780120455911
eBay Product ID (ePID)
50134
Product Key Features
Number of Pages
608 Pages
Publication Name
Introduction to Human Evolutionary Anatomy
Language
English
Publication Year
1990
Subject
Life Sciences / Anatomy & Physiology (See Also Life Sciences / Human Anatomy & Physiology), Anthropology / Physical, Anthropology / General
Type
Textbook
Author
Leslie Aiello, Christopher Dean
Subject Area
Social Science, Science
Format
Trade Paperback
Dimensions
Item Weight
42.2 Oz
Item Length
10 in
Item Width
6.9 in
Additional Product Features
Intended Audience
College Audience
LCCN
95-185095
TitleLeading
An
Reviews
A highly innovative hybrid between textbook and extended review, that will be an essential resource for advanced undergraduates and researchers in any related field...Aiello and Dean have performed a major feat in bringing together in the compass of one volume such a mass of information. A couple of hours with Aiello and Dean will save the researcher days and weeks in the library. I believe it will come to be regarded as one of the most important resources for anyone framing a new research project in early hominid studies. --THE TIMES HIGHER EDUCATION SUPPLEMENT Its unique blend of anatomy and paleontology will be extremely useful for a wide range of students and researchers in anthropology, anatomy, and paleontology....After reading through the vast array of topics covered by this book, one can't help but feel that the authors and the illustrator have done us a great service. --Mark F. Teaford, The Johns Hopkins University School of Medicine, in AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY, "This outstanding contribution is anything but just another anatomy book. Aiello and Dean have filled a major gap in the literature of human evolution with this textbook of functional morphology for fossil hominids and modern great apes. Its uniqueness lies in a complete emphasis on subjects of concern in paleoanthropology, the use of an explicitly comparative approach, and a detailed discussion of the fossil record... Aiello and Dean have done a superb job. For human evolutionary anatomy, the state of the art has been measurably enhanced." --AMERICAN ANTHROPOLOGIST "This extraordinary book combines the two related studies of human anatomy and human paleontology... [Aiello and Dean] weave the two subjects together and produce a text that not only is filled with information but is easily usable... It is especially useful that the authors have included an introductory section on taxonomy and classification... It is a book that I believe everyone interested in human evolution should own. My final comment is that if I had to make a selection of limited items to take with me to the field where I would be studying or searching for hominid fossils, this book would be on my list." --BECKY A. SIGMON, University of Toronto "A highly innovative hybrid between textbook and extended review, that will be an essential resource for advanced undergraduates and researchers in any related field...Aiello and Dean have performed a major feat in bringing together in the compass of one volume such a mass of information. A couple of hours with Aiello and Dean will save the researcher days and weeks in the library. I believe it will come to be regarded as one of the most important resources for anyone framing a new research project in early hominid studies." --THE TIMES HIGHER EDUCATION SUPPLEMENT "Its unique blend of anatomy and paleontology will be extremely useful for a wide range of students and researchers in anthropology, anatomy, and paleontology....After reading through the vast array of topics covered by this book, one can't help but feel that the authors and the illustrator have done us a great service." --Mark F. Teaford, The Johns Hopkins University School of Medicine, in AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY
Illustrated
Yes
Table Of Content
An Introduction to the Fossil Record. Classification and Phylogenetic Reconstruction. Anatomical Nomenclature. The Microanatomy of Muscle and Bone. The Bones of the Skull. The Comparative Anatomy of the Hominoid Cranium. The Masticatory System of the Hominoids. The Microanatomy of Teeth. Hominoid Tooth Morphology. The Intracranial Region. The Anatomy of the Brain and Hominoid Endocasts. The Facial Skeleton of Hominoids. The Cervical Spine and Support of the Head. The Anatomy of the Vocal Tract. Bipedal Locomotion and the Postcranial Skeleton. The Comparative Anatomy of the Hominoid Thorax and the Vertebral Column. Bones. Muscles. Movements of the Upper Limb. The Hominoid Upper Limb (Excluding the Hand). The Hominoid Hand. Bones, Muscles and Movements of the Lower Limb. The Hominoid Pelvis. The Hominoid Femur. The Hominoid Knee Joint and Lower Leg. The Lower Leg. The Hominoid Foot.
Synopsis
An anthropologist and an anatomist have combined their skills in this book to provide students and research workers with the essentials of anatomy and the means to apply these to investigations into hominid form and function. Using basic principles and relevant bones, conclusions can be reached regarding the probable musculature, stance, brain size, age, weight, and sex of a particular fossil specimen. The sort of deductions which are possible are illustrated by reference back to contemporary apes and humans, and a coherent picture of the history of hominid evolution appears. Written in a clear and concise style and beautifully illustrated, An Introduction to Human Evolutionary Anatomy is a basic reference for all concerned with human evolution as well as a valuable companion to both laboratory practical sessions and new research using fossil skeletons.
LC Classification Number
GN282.A37 1990
Illustrated by
Cameron, Joanna
Item description from the seller
About this seller
Premium Picks
99.8% positive feedback•9.4K items sold
Joined Oct 2000
Usually responds within 24 hours
We’re a family-owned business specializing in books, from vintage gems to modern favorites, plus a mix of unique finds. Known for fast shipping and friendly service. Follow us for fresh titles and ...
See more
We’re a family-owned business specializing in books, from vintage gems to modern favorites, plus a mix of unique finds. Known for fast shipping and friendly service. Follow us for fresh titles and treasures added often — thanks for visiting!
See less
Visit storeContact
Save seller
Detailed seller ratings
Average for the last 12 months
Accurate description
5.0
Reasonable shipping cost
5.0
Shipping speed
5.0
Communication
5.0
Popular categories from this store
See all categories
CollectiblesCraftsEntertainment MemorabiliaEverything ElseMusical Instruments & GearCameras & PhotoClothing, Shoes & AccessoriesComputers/Tablets & NetworkingToys & Hobbies
Seller feedback (3,161)
Filter:All ratings
All ratings
Positive
Neutral
Negative
1f (59)- Feedback left by buyer. Past 6 months Verified purchase The experience was excellent. The Bible was exactly as listed, shipped quickly, and arrived in perfect condition. The seller was professional, and communication was clear throughout the process. I’d definitely recommend this seller based on this experience. Great value and no issues at all. Bible ESV English Standard Version Travel Size Antique Cross Crossway New Sealed (#277246098870)
tn (50)- Feedback left by buyer. Past 6 months Verified purchase This seller right here is superb. Communication was great, item shipped out lightning fast and received today. I’ve purchased items from other sellers on the same day as this seller and I’m still waiting for the shipping label to be printed, I wish I could buy everything from this seller. Item was packed phenomenal. Thank you!🤘👊 Metallica - Black Album CD - Elektra - Brand New & Sealed (#276973203037)
ao (900)- Feedback left by buyer. Past month Verified purchase Excellent seller!! Condition of book at least as good as described, even slightly better than expected, well packed (much appreciated!) and sent fast from USA to UK at very reasonable cost compared to many other international sellers. Highly recommended. Thank you!! Life Ahead by J. Krishnamurti (1963) On Education & Meaningful Living (#277299296921)
fa (24)- Feedback left by buyer. Past 6 months Verified purchase This seller is genuinely reliable. The photographs left little room for speculation, and my Bloomsbury edition of Harry Potter and The Half Blood Prince arrived as described and securely packaged. Harry Potter And The Half-Blood Prince J K Rowling 1st UK Edition pg 99 11 Owls (#276776901320)
55 (928)- Feedback left by buyer. Past 6 months Verified purchase How often does a Seller ensure a book will arrive undamaged by having it enclosed a "double wrapped" shipping package? This was, and both book and study guide were in perfect condition! Shipping was quick and everything as stated. Excellent Seller !!! Captured by Grace by Jeremiah, David Book And Study Guide - SIGNED - NEW (#277000311379)
id (170)- Feedback left by buyer. Past 6 months Verified purchase This is the second Oxford Companion that I purchased from this seller. As with the first, the seller carefully wrapped the book in bubble wrap to protect it during shipping. Another great experience with the seller. I would highly recommend to others. The Oxford Encyclopedia of World History- Easton Press- Leather Bound (#277149036203)
00 (8)- Feedback left by buyer. Past 6 months Verified purchase The purchased book is in perfect, sealed condition. It is totally as described looks like it literally just came from the publisher. The shipping packaging was quite exceptional, too! Better than Amazon! Great value for the purchase price. Moby Dick or The Whale By Herman Melville Easton Press - SEALED (#277240389634)
5y (30)- Feedback left by buyer. Past 6 months Verified purchase Arrived in a couple of days and perfectly packed. The books I received were exactly as described and at a great price. Thank you!! Hana Kimi For You In Full Blossom manga Lot Of 5 English Shojo Hisaya Nakajo (#276915287217)
See all feedback
Back to home page|See More Details about"Introduction to Human Evolutionary Anatomy by Leslie A..."Return to top
More to explore :
Anatomy Paperback Textbooks,
Nonfiction Anatomy Paperbacks Books,
Dean Koontz Paperbacks Books,
Dean Koontz Paperbacks Books,
Dean Koontz Paperbacks Books in English,
Anatomy Textbooks,
Paperbacks,
Paperbacks Books,
Nonfiction Anatomy Fiction & Nonfiction Books,
Dean Koontz Fiction & Books
Shop Top Sellers and Highly Rated Products in Textbooks
Best Sellers
Diagnostic and Statistical Manual of Mental Disorders DSM-5-TR by American Psychiatric Association (2022, Trade Paperback)
NFPA 70, National Electrical Code Handbook : 2023 Edition by National Fire Protection Association (NFPA) (2022, Hardcover)
Diagnostic and Statistical Manual of Mental Disorders : DSM-5-TR by American Psychiatric Association (2022, Hardcover)
National Electrical Code by National Fire Protection Association (2019, Trade Paperback)
Self Heal by Design : The Role of Micro-Organisms for Health by Barbara O'Neill (2018, Trade Paperback)
Top Rated
Constitution of the United States200 : With Index, and the Declaration of Independence by Madison (2005, ph)
Self Heal by Design : The Role of Micro-Organisms for Health by Barbara O'Neill (2018, Trade Paperback)
Diabetes Sin Problemas -Ver. Abrev. USA by Xiomara Acobes (2015, Trade Paperback)
Warning : Testimonies and Prophecies of the Illumination of Conscience by Christine Watkins (2019, Trade Paperback)
PPI FE Mechanical Practice Problems - Comprehensive Practice for the FE Mechanical Exam by Michael R. Lindeburg (2014, Trade Paperback)
Related Searches
Apologia Exploring Creation with Human Anatomy
Apologia Anatomy
Human Evolution Book
Essentials of Human Anatomy and Physiology
Essential Anatomy
Anatomy Animal
Fundamentals of Anatomy
Understanding Anatomy and Physiology
The Encyclopedia of Animals
Atlas of Human Anatomy
Animal Anatomy and Physiology
Anatomia Humana
Anatomy and Physiology Human
Anatomy of Domestic Animals
Human Anatomy 9th Edition
About this seller
Premium Picks
99.8% positive feedback•9.4K items sold
Joined Oct 2000
Usually responds within 24 hours
We’re a family-owned business specializing in books, from vintage gems to modern favorites, plus a mix of unique finds. Known for fast shipping and friendly service. Follow us for fresh titles and ...
See more
We’re a family-owned business specializing in books, from vintage gems to modern favorites, plus a mix of unique finds. Known for fast shipping and friendly service. Follow us for fresh titles and treasures added often — thanks for visiting!
See less
Visit storeContact
Save seller
Detailed seller ratings
Average for the last 12 months
Accurate description
5.0
Reasonable shipping cost
5.0
Shipping speed
5.0
Communication
5.0
Popular categories from this store
See all categories
CollectiblesCraftsEntertainment MemorabiliaEverything ElseMusical Instruments & GearCameras & PhotoClothing, Shoes & AccessoriesComputers/Tablets & NetworkingToys & Hobbies
Seller feedback (3,161)
Filter:All ratings
All ratings
Positive
Neutral
Negative
1f (59)- Feedback left by buyer. Past 6 months Verified purchase The experience was excellent. The Bible was exactly as listed, shipped quickly, and arrived in perfect condition. The seller was professional, and communication was clear throughout the process. I’d definitely recommend this seller based on this experience. Great value and no issues at all. Bible ESV English Standard Version Travel Size Antique Cross Crossway New Sealed (#277246098870)
tn (50)- Feedback left by buyer. Past 6 months Verified purchase This seller right here is superb. Communication was great, item shipped out lightning fast and received today. I’ve purchased items from other sellers on the same day as this seller and I’m still waiting for the shipping label to be printed, I wish I could buy everything from this seller. Item was packed phenomenal. Thank you!🤘👊 Metallica - Black Album CD - Elektra - Brand New & Sealed (#276973203037)
ao (900)- Feedback left by buyer. Past month Verified purchase Excellent seller!! Condition of book at least as good as described, even slightly better than expected, well packed (much appreciated!) and sent fast from USA to UK at very reasonable cost compared to many other international sellers. Highly recommended. Thank you!! Life Ahead by J. Krishnamurti (1963) On Education & Meaningful Living (#277299296921)
fa (24)- Feedback left by buyer. Past 6 months Verified purchase This seller is genuinely reliable. The photographs left little room for speculation, and my Bloomsbury edition of Harry Potter and The Half Blood Prince arrived as described and securely packaged. Harry Potter And The Half-Blood Prince J K Rowling 1st UK Edition pg 99 11 Owls (#276776901320)
55 (928)- Feedback left by buyer. Past 6 months Verified purchase How often does a Seller ensure a book will arrive undamaged by having it enclosed a "double wrapped" shipping package? This was, and both book and study guide were in perfect condition! Shipping was quick and everything as stated. Excellent Seller !!! Captured by Grace by Jeremiah, David Book And Study Guide - SIGNED - NEW (#277000311379)
id (170)- Feedback left by buyer. Past 6 months Verified purchase This is the second Oxford Companion that I purchased from this seller. As with the first, the seller carefully wrapped the book in bubble wrap to protect it during shipping. Another great experience with the seller. I would highly recommend to others. The Oxford Encyclopedia of World History- Easton Press- Leather Bound (#277149036203)
00 (8)- Feedback left by buyer. Past 6 months Verified purchase The purchased book is in perfect, sealed condition. It is totally as described looks like it literally just came from the publisher. The shipping packaging was quite exceptional, too! Better than Amazon! Great value for the purchase price. Moby Dick or The Whale By Herman Melville Easton Press - SEALED (#277240389634)
5y (30)- Feedback left by buyer. Past 6 months Verified purchase Arrived in a couple of days and perfectly packed. The books I received were exactly as described and at a great price. Thank you!! Hana Kimi For You In Full Blossom manga Lot Of 5 English Shojo Hisaya Nakajo (#276915287217)
See all feedback
About eBay
Announcements
Community
Security Center
Seller Center
Policies
Affiliates
Help & Contact
Site Map
Copyright © 1995-2025 eBay Inc. All Rights Reserved. Accessibility, User Agreement, Privacy, Consumer Health Data, Payments Terms of Use, Cookies, CA Privacy Notice, Your Privacy Choices and AdChoice
Copyright © 1995-2025 eBay Inc. All Rights Reserved. Accessibility, User Agreement, Privacy, Consumer Health Data, Payments Terms of Use, Cookies, CA Privacy Notice, Your Privacy Choices and AdChoice |
11780 | https://www.reddit.com/r/Mcat/comments/123w4cc/atomic_number_atomic_weight_mass_number_and/ | Atomic number, atomic weight, mass number and atomic mass difference? : r/Mcat
Skip to main contentAtomic number, atomic weight, mass number and atomic mass difference? : r/Mcat
Open menu Open navigationGo to Reddit Home
r/Mcat A chip A close button
Log InLog in to Reddit
Expand user menu Open settings menu
Go to Mcat
r/Mcat
r/Mcat
The #1 social media platform for MCAT advice. The MCAT (Medical College Admission Test) is offered by the AAMC and is a required exam for admission to medical schools in the USA and Canada. /r/MCAT is a place for MCAT practice, questions, discussion, advice, social networking, news, study tips and more. Check out the sidebar for useful resources & intro guides. Post questions, jokes, memes, and discussions.
311K Members Online
•3 yr. ago
SeaworthinessFit9214
Atomic number, atomic weight, mass number and atomic mass difference?
Question 🤔🤔
There are many terms that mean the same thing but I’m confused. So atomic number is the number of protons. Atomic weight is the weighted averages of the isotopes. Mass number and atomic mass is the same and that’s just the protons and neutrons added is that right
Read more
Share
Related Answers Section
Related Answers
Compare atomic mass and mass number for MCAT
How to get atomic weight for MCAT
Did Mendeleev believe in atoms MCAT context
Top MCAT study schedules for busy students
Effective MCAT CARS practice techniques
New to Reddit?
Create your account and connect with a world of communities.
Continue with Email
Continue With Phone Number
By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy.
Public
Anyone can view, post, and comment to this community
0 0
Top Posts
Reddit reReddit: Top posts of March 27, 2023
Reddit reReddit: Top posts of March 2023
Reddit reReddit: Top posts of 2023
Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved.
Expand Navigation Collapse Navigation |
11781 | https://www.bbc.co.uk/bitesize/guides/zscq6yc/revision/2 | Bitesize
GCSE
Edexcel
Approximation - EdexcelRounding to decimal places
Approximation includes estimation, rounding to powers of 10, decimal places and significant figures.
Part of MathsNumber
Rounding to decimal places
When rounding using decimal placesThe number of digits required after the decimal point when rounding a decimal answer. (dp), the degree of accuracy that is required is usually given. However, there are certain calculations where the degree of accuracy may be more obvious. For example, calculations involving money should be given to two decimal places to represent the pence.
To round to a decimal place:
look at the first digit after the decimal point if rounding to one decimal place or the second digit for two decimal places
draw a vertical line to the right of the place value digit that is required
look at the next digit
if it's 5 or more, increase the previous digit by one
if it's 4 or less, keep the previous digit the same
remove any numbers to the right of the line
Examples
Round 248.561 to 1 decimal place, then 2 decimal places.
248.5|61 to 1 decimal place is 248.6
248.56|1 to 2 decimal places is 248.56
Notice that your answer should have the same number of decimal places as the approximation asked for.
Round 0.08513 to 1 and 2 decimal places.
0.0|8513 to 1 decimal place is 0.1
0.08|513 to 2 decimal places is 0.09
Question
Round 249.5046 to 1, then 2 decimal places.
249.5|046 to 1 decimal place is 249.5
249.50|46 to 2 decimal places is 249.50
Notice that 249.50 = 249.5. However, as the question asked for 2 decimal places both numbers are left.
Question
Round 0.9583 to 1 decimal place, then 2 decimal places.
0.9|583 to 1 decimal place is 1.0
0.95|83 to 2 decimal places is 0.96
Question
Round 598.053 to 1 decimal place, then 2 decimal places.
598.0|53 to 1 decimal place is 598.1
598.05|3 to 2 decimal places is 598.05
Next page
Rounding to significant figures
Previous page
What is approximation?
More guides on this topic
NEW: Whole numbers
NEW: Order of operations and negative numbers
NEW: Decimals
NEW: Converting between fractions, decimals and percentages
NEW: Higher – How to convert recurring decimals
NEW: How to round numbers
NEW: What is accuracy in maths?
NEW: What are fractions?
NEW: How to add, subtract, multiply and divide fractions
NEW: Multiples and factors
NEW: Highest Common Factor and Lowest Common Multiple
NEW: Laws of indices
NEW: Negative and fractional indices
NEW: Standard form
NEW: Calculations using standard form
Whole numbers - Edexcel
Decimals - Edexcel
Multiples and factors - Edexcel
Standard form - Edexcel
Laws of indices - Edexcel
Fractions - Edexcel
Converting between fractions, decimals and percentages - Edexcel
Surds - Higher - Edexcel
Financial mathematics - Edexcel
Related links
Maths: Exam-style questions
Maths revision resources
Personalise your Bitesize!
Jobs that use Maths
Radio 4: Maths collection
Save My Exams Subscription
Quizlet
Pearson Education
Just Maths |
11782 | https://chemistry-europe.onlinelibrary.wiley.com/doi/10.1002/cctc.202401969 | Hydrogen Radical Detection in Catalytic Hydrogenation Reactions: Background C‐H Activation in Solvents - Rogers - 2025 - ChemCatChem - Wiley Online Library
Opens in a new window Opens an external website Opens an external website in a new window
This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy
Skip to Article Content
Skip to Article Information
Search within
Search term Advanced SearchCitation Search
Search term Advanced SearchCitation Search
Search term Advanced SearchCitation Search
Login / Register
Individual login
Institutional login
REGISTER
Journals
ChemistryEurope Open access
Chemistry—A European Journal
European Journal of Inorganic Chemistry
European Journal of Organic Chemistry
ChemistryOpen Open access
ChemistrySelect
Chemistry—Methods Open access
Analysis & Sensing
Analytical Science Advances Open access
Electrochemical Science Advances Open access
Batteries & Supercaps
ChemBioChem
ChemCatChem
ChemElectroChem Open access
ChemFoodChem Open access
ChemMedChem
ChemPhotoChem
ChemPhysChem
ChemPlusChem
ChemSusChem
ChemSystemsChem
About / Get Published
Diversity, Equity & Inclusion (DE&I)
Free Guide to Writing
Open Access Advantage
Contacts
Events
Virtual Events
EurJIC–Woehler Young Investigator Prize
Collections
Society Volumes
Reviews
Editors' Choice: Spotlights
Ulf Diederichsen Tribute
Open Science
Noncovalent Interactions (ICNI-III)
Perovskite Materials and Devices
Beyond Lithium-Ion Batteries
XXII International Symposium on Homogeneous Catalysis
Quantum Bioinorganic Chemistry (QBIC)
Celebrating 140 Years of the Swedish Chemical Society
Celebrating 120 Years of the Royal Netherlands Chemical Society
Celebrating 100 Years of the Association of Greek Chemists
Spotlighting Early-Career Chemistry Researchers in the Netherlands
All Special Collections
GET IN TOUCH WITH US
WeChat
ChemCatChem
Volume 17, Issue 9 e202401969
Research Article
Open Access
Hydrogen Radical Detection in Catalytic Hydrogenation Reactions: Background C-H Activation in Solvents
Hannah Rogers,
Hannah Rogers
Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 2AY UK
Search for more papers by this author
Timothy J. Woodman,
Timothy J. Woodman
Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 2AY UK
Search for more papers by this author
David J. Willock,
David J. Willock
Cardiff Centre on the Fundamentals of Heterogeneous Catalysis FUNCAT, Cardiff Catalysis Institute, School of Chemistry, Cardiff University, Main Building, Park Place, Cardiff, CF10 3AT UK
Search for more papers by this author
Andrea Folli,
Andrea Folli
Cardiff Centre on the Fundamentals of Heterogeneous Catalysis FUNCAT, Cardiff Catalysis Institute, School of Chemistry, Cardiff University, Main Building, Park Place, Cardiff, CF10 3AT UK
Search for more papers by this author
Simon J. Freakley,
Corresponding Author
Simon J. Freakley
s.freakley@bath.ac.uk
Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 2AY UK
E-mail: s.freakley@bath.ac.uk
Search for more papers by this author
Hannah Rogers,
Hannah Rogers
Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 2AY UK
Search for more papers by this author
Timothy J. Woodman,
Timothy J. Woodman
Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 2AY UK
Search for more papers by this author
David J. Willock,
David J. Willock
Cardiff Centre on the Fundamentals of Heterogeneous Catalysis FUNCAT, Cardiff Catalysis Institute, School of Chemistry, Cardiff University, Main Building, Park Place, Cardiff, CF10 3AT UK
Search for more papers by this author
Andrea Folli,
Andrea Folli
Cardiff Centre on the Fundamentals of Heterogeneous Catalysis FUNCAT, Cardiff Catalysis Institute, School of Chemistry, Cardiff University, Main Building, Park Place, Cardiff, CF10 3AT UK
Search for more papers by this author
Simon J. Freakley,
Corresponding Author
Simon J. Freakley
s.freakley@bath.ac.uk
Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 2AY UK
E-mail: s.freakley@bath.ac.uk
Search for more papers by this author
First published: 14 February 2025
About
Figures -------
References ----------
Related -------
Information -----------
PDF
Sections
Abstract
Graphical Abstract
1 Introduction
2 Results and Discussion
3 Conclusions
4 Experimental Methods
Acknowledgements
Conflict of Interests
Open Research
Supporting Information
References
PDF
Tools
Request permission
Export citation
Add to favorites
Track citation
ShareShare
Give access
Share full text access
Close modal
Share full-text access
Please review our Terms and Conditions of Use and check box below to share full-text version of article.
[x] I have read and accept the Wiley Online Library Terms and Conditions of Use
Shareable Link
Use the link below to share a full-text version of this article with your friends and colleagues. Learn more.
Copy URL
Share a link
Share on
Email
Facebook
x
LinkedIn
Reddit
Wechat
Bluesky
Abstract
Platinum group metal (PGM) nanoparticle catalysts are commonly used in a wide variety of liquid-phase hydrogenation reactions. Carbon-supported palladium nanoparticles are used extensively in many industrially applied hydrogenation reactions across the pharmaceutical industry and increasingly in the hydrogenolysis of bio-derived molecules. These reactions are often performed in solvents such as toluene or xylene which are considered inert towards mild hydrogenation conditions. Through a series of catalytic studies analysed using 1 H/2 H Nuclear Magnetic Resonance (NMR) and Electron Paramagnetic Resonance (EPR) spectroscopies, and Density Functional Theory (DFT) calculations, we present evidence that hydrogen is being constantly exchanged into the methyl groups of the solvent possibly via a radical mechanism at these mild conditions. These effects should be considered in the explanation of any model of catalytic hydrogenation reaction at metal surfaces.
Graphical Abstract
The study demonstrates that under mild hydrogenation conditions, C-H/D exchange occurs in common solvents toluene and xylene, especially at benzylic positions. Studies indicate that carbon-centered radicals bind strongly to Pd surfaces, suggesting their possible role in catalytic reactions. Spin-trapping EPR confirms H-radical adducts are present under these conditions, which may influence various catalytic processes.
1 Introduction
Hydrogenation reactions using heterogeneous catalysts are widely studied in industry and academia with applications in the production of pharmaceuticals, agrochemicals, fine chemicals, and hydrotreatments of petrochemical and bio-derived feedstocks. As such, hydrogenation reactions account for 10%–20% of all industrial chemical steps and will remain an essential catalytic technology to add value to renewable resources. Catalysts containing metallic palladium (Pd) nanoparticles (typically 1- 50 nm) and bimetallic particles containing Pd have been developed for a wide range of hydrogenative transformations including complete or selective hydrogenations of alkenes, alkynes, carbonyls, and nitro compounds in addition to hydrogenolysis of C─O/C─C bonds in biomass or polymer upgrading. The prevalence of Pd as a hydrogenation catalyst is clear in both the academic and patent literature due to H 2 dissociation on Pd proceeding with a low energy barrier.
Our understanding of hydrogenation on Pd surfaces developed from the mechanism proposed for alkene hydrogenation in 1934. The so-called Horiuti–Polanyi mechanism is described as a series of reaction steps leading to saturation of the alkene double bond: i) H 2 dissociation at the catalyst surface to form atomic surface hydrogen, H, ii) formation of a π-complex between the alkene and Pd surface, iii) sequential addition of H to the substrate, iv) saturated product desorption – this mechanism is almost universally recognized as the basis for many reactions on metal surfaces with low barriers to H 2 dissociation in both the gas and liquid phase. More recently evidence of an additional redox mechanism has been reported for the hydrogenation of O 2 to form H 2 O 2 and the hydrogenation of furanic compounds in protic media which draws an analogy to enzymatic hydrogenases or electrochemical reactions.[6-8] These proposed mechanisms concern proton-coupled-electron-transfer (PCET) of hydrogen oxidation (H 2 ⇌ 2H+ + 2e−) and C=C/C=O reduction via water-mediated proton transfer participating directly in the kinetically relevant reaction step.
An overlooked consideration in the description of surface hydrogenation processes is the presence and reactivity of weakly bound atomic H ad-atoms (H(w)) with free radical-like character in contrast to the strongly bound surface species. The study of free radicals in hydrogenation chemistry was initiated by Howe and co-workers in 1980 using spin trap experiments to demonstrate the radical character of certain adsorption modes of H 2 on ZnO catalysts used for gas phase alkene hydrogenation. Through isotopic exchange experiments, Howe and co-workers suggested an initial reversible weak adsorption of H 2 resulting in H atoms interacting with Zn sites. These species could react with spin traps and be detected by EPR suggesting radical-like character. The signatures of these radical species were absent in the presence of alkene substrates suggesting either blocking of the sites by alkenes or their involvement in the hydrogenation process rather than or in addition to strongly bound surface (H(s)) species.
These spin-trapping studies were extended by Carley et al. in 1994 to alkene hydrogenation using Pd/Al 2 O 3 catalysts. On exposure of the Pd catalyst to H 2 and either ethene or benzene – alkyl and aromatic free radical intermediates could be detected by spin trapping experiments using n-tert-butyl-phenylnitrone (PBN) as a spin trap. In the absence of hydrocarbons, the signature of H-PBN● free radicals was observed on the Pd catalysts. Supporting desorption and isotopic experiments suggested the H atoms reacting with the spin traps are not strongly chemisorbed hydrogen (H(s)) but those weakly adsorbed H-adatoms (H(w)) in equilibrium with H 2(g). The observation of ethyl radicals and the absence of H● radical signals in the presence of gas phase substrate support a proposed mechanism involving a gas-like overlayer composed of weakly adsorbed ethene reacting and consuming weakly adsorbed H adatoms (H(w)). This also supported other proposals that strongly adsorbed ethene has low activity in alkene hydrogenation reactions.Whilst these studies probed gas phase hydrogenation reactions, in all cases the spin trap molecule is introduced in either toluene or dichloromethane solution after reactant exposure. This suggests that H● radicals can be trapped from the catalyst surface in the liquid phase and could also be produced in liquid-phase hydrogenation reactions. Maher and co-workers performed analogous alkene hydrogenation studies to Carley using a liquid phase system by bubbling H 2-N 2 mixtures and showed the formation of trapped H● species by reaction of molecular H 2 with the Pd surface in the liquid phase when a flow of H 2 is introduced. The authors concluded that these surface hydrogen ad-atoms must have considerable radical character otherwise spin trapping would not be possible and they are unlikely to be interstitial or covalently bound hydrogen species. The signal of the hydrogen spin-adduct was either considerably diminished or lost completely in the presence of alkenes suggesting the removal of the hydrogen ad-atoms by reaction with the alkene in the liquid phase.
High-loaded Pd/C is frequently used to perform liquid-phase hydrogenations in organic fine synthesis in solvents such as p-xylene, THF, and toluene which are thought to be inert. The possible presence of H-ad atoms with significant radical character is often not mechanistically considered and the literature surrounding spin trapping of intermediates in hydrogenation chemistry is scarce since the publications of Carley and Maher, but their presence could have significant implications in H/D scrambling reactions and activation of functional groups. In the studies reported to date, the possible reaction of H(w) with radical character and the solvent molecules is not considered or conditions more relevant to liquid phase hydrogenations such as elevated temperature and pressure.
Our interest in the mechanisms of (de)hydrogenation reactions on Pd nanoparticles originated from the study of the acceptorless dehydrogenation of 1-phenylethanol to produce molecular H 2. This reaction was studied using commercial Pd/C catalysts by Nicolau et al. who reported an activation energy of 98 kJ mol−1 suggesting that C─H cleavage is involved in the rate determining step. Additional studies on benzyl alcohols with deuterated benzylic C─H bonds showed a kinetic isotope effect ∼1.6; however, the magnitude is lower than typically observed for C─H bond cleavage. This could indicate second-order effects where the deuterated bond is not directly involved in the rate-determining step, however, the reactions were conducted under a static inert atmosphere meaning ineffective removal of H 2 could also affect the measured rates due to re-hydrogenation of the formed carbonyl. Through a series of catalytic studies using isotopically labelled substrates analysed using 1 H/2 H Nuclear Magnetic Resonance (NMR) and Electron Paramagnetic Resonance (EPR) spectroscopies with supporting Density Functional Theory (DFT) calculations we probe the presence of these weakly bound H(w) species under commonly used mild hydrogenation conditions and study the stability of a range of C-H bonds under these conditions.
2 Results and Discussion
To conduct our initial studies, we prepared 1-phenylethanol deuterated at the α-carbon by reducing acetophenone with NaBD 4 followed by acidic workup, mono deuteration at the α-carbon was confirmed by solvent supressed 1 H in CDCl 3 and 2 H NMR in C 6 H 6 (Figure S1). Whilst the signals in 2 H NMR are broader than in 1 H NMR, coupling patterns can still be resolved and used to identify the presence and position of isotopic substitution which makes the technique a valuable tool in mechanistic studies. Using a 1% Pd/TiO 2 catalyst which we have previously reported for this reaction, we initially studied acceptorless dehydrogenation of the labelled 1-phenylethanol in p-xylene at 120°C under flowing Ar relative to the unlabelled substrate. Figure1 reveals a large kinetic isotope effect ∼7.5, (k H=0.0170 min−1 and k D=0.0023 min−1) suggesting that the breaking of the α-carbon-hydrogen bond is likely the rate-determining step to produce H 2 in this system. Generally, values between 2 and 7 (at 298 K) indicate the breaking of a C─H bond and is strongly effected by bond strength, steric effects, solvent effects, and variations in the reaction mechanism depending on the catalyst system. The values are in line with studies of acetone production from acceptorless dehydrogenation of IPA when the α-C-H is deuterated.
Figure 1
Open in figure viewerPowerPoint
Kinetic isotope effect of deuteration of the α-carbon in 1-phenylethanol acceptorless dehydrogenation. Conditions: 120°C, 1000 rpm, 0.355 M 1-phenylethanol, 20 mg catalyst (1%/Pd/TiO 2), 10 ml p-xylene, flowing Ar.
Figure2 shows analysis of the reaction mixture after 120 mins by 2 H NMR. Unexpectedly, in all reactions, a new 2 H resonance appeared at 2.1–2.2 ppm. The low chemical shift suggests this does not arise from hydrogenation or exchange at the aromatic C─H positions (6.8 ppm) of the substrate when compared to C 6 D 6 used as an internal reference (7.2 ppm), rather this suggests aliphatic C─D bonds which cannot result from the re-hydrogenation of the formed ketone. A 2 H NMR of p-xylene-d 10 shows a resonance in the same position as the unknown peak consistent with the methyl C─D environment of deuterated p-xylene-d.10 Contaminant C 6 D 6 resonances (7.2 ppm) was also present in all p-xylene-d 10 samples tested. Whilst the p-xylene used as a solvent in the KIE experiments was fully proton form, the position and splitting of the new resonance in the final reaction solutions suggest p-xylene with a single H/D substitution on the methyl group. Throughout our experiments, we did not observe H/D exchange by 2 H-NMR on aromatic or methyl groups of 1-phenylethanol.
Figure 2
Open in figure viewerPowerPoint
(a) 2 H NMR of mono deuterated 1-phenylethanol starting material, (b) post acceptorless dehydrogenation reaction solution, and (c) p-xylene d 10 reference. Top – 2 H NMR (77 MHz, no solvent) δ 4.61 (s, 1D, h). Middle – 2 H NMR (77 MHz, C 6 D 6), δ 4.61 (s), δ 2.19–1.98 (m). Bottom – 2 D NMR (77 MHz, C 6 H 6), δ 7.12 (C 6 D 6 standard) (s), δ6.89 (s, 4D, a) δ2.12 (s, 6D, b). Conditions for reaction in (b): 120°C, 1000 rpm, 0.355 M 1-phenylethanol-d20 mg catalyst (1%/Pd/TiO 2), 10 ml p-xylene, flowing Ar.
Various methods have been reported to prepare deuterated alkyl aromatics, typically including chlorinated starting materials, activation of C─H bonds by BF 3/D 2 O or homogeneous catalytic systems such as Ir pincer complexes by C─H insertion.[17-19] Approaches using hydrogenation catalysts with high pressures of D 2 and elevated temperatures often encounter the challenge of avoiding hydrogenation of the aromatic ring or C─C hydrogenolysis. This observation implies that under relatively mild conditions (120°C, flowing Ar 1 atm) with only the low concentration of HD/D 2 produced from acceptorless dehydrogenation, the catalyst can activate the methyl C─H bonds of p-xylene under reducing conditions and install a D atom – an observation that will be missed in all liquid phase hydrogenations that use proteo solvents and 1 H NMR as the H/C-H exchange is not observable.
A wide variety of metal surfaces have been shown to catalyse H/D exchange in C─H bonds since the early studies of Taylor on nickel-kieselguhr catalysts for methane deuteration.[20, 21] Typically, these reactions are carried in the gas phase at a temperature between 150 and 300°C, with collected experimental results showing that the reactivity of various alkanes can be understood based on C─H bond dissociation energy (BDE). Studies by Kembel on Pd surfaces report initial C─H activation being the rate-determining step with activation energies for CH 4 mono-deuteration between 92 and 140 kJ mol−1 and for propane 99 kJ mol−1 which is similar in magnitude to the measured activation energy of 1-phenylethanol acceptorless dehydrogenation, supporting C─H activation as the RDS.
We conducted a further set of experiments to identify if the observed C─H(D) activation of the p-xylene solvent during our catalytic reaction is related to 1-phenylethanol acceptorless dehydrogenation. In this set of experiments, we stirred p-xylene-d 10 under 3 bar H 2 at 120°C with no catalyst, bare TiO 2 and 1%Pd/TiO 2 (Scheme1) (Figure S2). This revelated that the H/D exchange was only observed when the Pd nanoparticles were present, suggesting that the C─D/H exchange with the gas phase was only possible with concurrent H 2 activation over the metal surface.
Scheme 1
Open in figure viewerPowerPoint
Exchange of methyl deuterium on xylene solvent molecules.
Analysis by 1 H-NMR spectroscopy of the reaction containing p-xylene-d 10 and 1% Pd/TiO 2 under 3 bar H 2 pressure at 120°C initially shows only small residual aromatic and methyl C─H from incomplete deuteration of the solvent as purchased as seen in Figure S3 and Pd free control reactions (Figure S2). Over the course of 3 h the residual aromatic C─H integral (6.95 ppm) remains constant whereas the methyl C─H integral (2.2 ppm) increases linearly over time (Figure3) indicating increasing H/D exchange of the methyl groups of the solvent over time. This result suggests that in the background of many liquid phase hydrogenation reactions solvent/gas phase exchange at methyl C─H bonds is occurring at the catalyst surface irrespective of substrate conversion.
Figure 3
Open in figure viewerPowerPoint
Ratio of aromatic to methyl signals as a function of reaction time determined by 1 H NMR of the reaction mixture in C 6 D 6. Aromatic C-H signal at 6.89 ppm, methyl C-H signal at 2.1 ppm. Conditions: 120°C, 1000 rpm, 3 bar H 2, 20 mg catalyst (1%/Pd/TiO 2), 10 ml p-xylene-d 10
After showing Pd/TiO 2 can facilitate this reaction, other nanoparticle catalysts commonly used in hydrogenation (Commercial 5% Pd/C, Synthesised – 5% Pt/C, 2.5% Au/C, 1% AuPd/TiO 2, and 1% Au/TiO 2) were investigated for H/D exchange of p-xylene-d 10 at 120°C under 3 bar H 2. The relative integral of the aromatic and methyl peaks was used as an indication of H/D exchange, which are summarised in Table1.
Table 1. Ratio of integrated methyl and aromatic peaks of 1 H NMR for p-xylene-d 10 following reaction at 120°C under 3 bar H 2 for 2 h in the presence of various catalysts.
| Catalyst | methyl / aromatic |
--- |
| 5% Pd/C | 112 |
| 5% Pt/C | 15 |
| 2.5% Au/C | 2 |
| 1% AuPd/TiO 2 | 35 |
| 1% Au/TiO 2 | 2 |
Both Au/C and Au/TiO 2 showed little to no H/D exchange over the course of the 3 h reaction, and only small amounts of 1,4-dimethylcyclohexane are detected (Figure S4) consistent with Au having a high barrier to H 2 activation. The Pt/C catalyst yielded mainly hydrogenation products, with only a small increase in the integral of the p-xylene methyl group as compared to the aromatic signal. The best catalysts for H/D exchange were 5%Pd/C and 1%AuPd/TiO 2; whilst both spectra show a high methyl/aromatic integral ratio compared to other catalysts suggesting Pd, the most used hydrogenation catalyst, is particularly effective for this H/D exchange of the solvent.
While H/D exchange occurs at the methyl C─H groups preferentially to the aromatic C─H in p-xylene, likely due to the differences in BDEs between the aromatic and benzylic C─H, other solvents were tested to determine if this observation is general across “inert” solvents commonly used in hydrogenation reactions. Initially, toluene-d 8 was tested and showed significant H/D exchange at the benzylic carbon in the presence of the Pd catalyst and 2 bar H 2 at 100°C (Figure4). Analysis of the 1 H NMR spectra of the residual protons in commercial toluene-d 8 before and after reaction shows a significant increase of C─H at the benzylic position relative to the aromatic C─H signal by ∼45x, while the ratio of the integrals between the three aromatic environments remained constant before and after reaction suggesting no H/D exchange on the aromatic ring.
Figure 4-
Open in figure viewerPowerPoint
1 H NMR spectra of tolune-d 8 before and after being stirred with a 1% Pd/ TiO 2 catalyst, 3 h, 2 bar H 2, 100°C.
Further experiments using D 2 and proteo solvents allowed us to probe a variety of solvents, not just those deuterated solvents made widely available for NMR such as p-xylene-d.10 Ethylbenzene was tested to determine selectivity towards benzylic (CH 2) or aliphatic (CH 3) position. The 2 H NMR (Figure S5a) shows deuterium substitution in only one resonance (3.44 ppm). The splitting pattern observed (doublet of quartets) suggests that the deuterium atom is situated on the benzylic carbon. The 3 methyl protons split the deuterium signal into a quartet with a coupling constant of 1.2 Hz, and the proton on the benzylic carbon splits the signal into a doublet with a larger coupling constant of 2.7 Hz. Further evidence of substitution on the benzylic carbon is shown in the 13 C NMR (Figure S5b). The singlet at 28.94 ppm indicates the benzylic carbon with no deuterium substitution. The adjacent 1:1:1 triplet at 28.59 ppm is representative of a single deuterium substitution, whilst the low-intensity quintet at 28.24 ppm is indicative of both benzylic protons being substituted for deuterium. Benzene was tested and as expected no exchange was observed before and after reaction (Figure S6a). Cyclohexane, which has a higher BDE (400 kJ mol−1) compared to either toluene or xylene methyl groups (313 and 323 kJ mol−1 respectively) and without benzylic carbons also showed no H/D exchange (Figure S6b).
As our results suggest the benzylic position of solvent molecules is particularly suspectable to this background H/D exchange in hydrogenation reactions, radical reaction pathways could be considered as a means of C─H activation via H-abstraction. The production of benzyl and xylyl radicals by photolysis or H-abstraction has been well studied in atmospheric chemistry through reaction with OH or Cl radicals. This photochemical gas phase C─H oxidation pathway has shown that H abstraction from benzylic carbon by OH has a low barrier.[24-26] It has been shown that weakly bound hydrogen species with a radical-like character can be formed on metal surfaces and are able to react with organic molecules adsorbed to the catalyst surface which could include H-abstraction of activated benzylic positions on solvent molecules. Further reactions were carried out in the presence of n-tert-butyl-phenylnitrone (PBN) a known spin trap for H-centred radicals. As any hydrogen radicals in our system would be short-lived, a spin trap (PBN) was used to create a persistent PBN-H aminoxyl radical adduct which could be subsequently detected by electron paramagnetic resonance. EPR experiments were carried out with the catalyst, toluene and spin trap (PBN) under 2 bar of H 2. As PBN is unstable at elevated temperatures, the reaction time was shortened so any radicals could be trapped and observed before the spin trap thermally decomposed.
Simulations of the expected EPR spectra for the PBN-H adduct show nine lines should be observed as the nitrogen splits the resonance into a 1:1:1 triplet, in which each peak is further split into a 1:2:1 triplet by adjacent hydrogen. The hyperfine coupling constant of the nitrogen is 1.499 mT, approximately double that of the hydrogen hyperfine coupling constant (0.749 mT). This means some lines overlap to give the seven-line spectra shown (Figure5a). These values are consistent with literature values for a PBN-H aminoxyl radical adduct in toluene. For each catalyst tested in our system an identical seven-line isotropic spectra of varying intensity was observed indicating the presence of H radicals in the system (Figure5b). The spectra obtained support data collected using NMR. Pt/C, AuPd/TiO 2, Pd/C, and Pd/TiO 2 were all active catalysts for hydrogen/deuterium exchange, evidenced by an increase in the ratio of methyl: aromatic peak integrals. While experimental evidence has been reported to support the presence of weakly absorbed H species on nanoparticle Au surfaces resulting from alcohol dehydrogenation under a H 2 atmosphere no such species were trapped suggesting that the radicals originate from H 2 cleavage consistent with the lack of H/D exchange. In our studies no PBN adducts of toluene species were detected, which could be due to multiple factors, such as the flux of hydrogen radicals could be significantly higher than the flux of carbon-centred radicals meaning that PBN adducts are statistically unlikely to form and are not observed. Another reason, which is also suggested by Lutterloh et al., is that when the toluene radical intermediate is formed it is adsorbed strongly onto the catalyst surface so cannot react with PBN in the solution. A hypothesis supported by the theoretical models of Scaiano who proposed a significant binding energy (−116 kJ mol−1) of toluene-CH 2● to Pd surfaces.
Figure 5
Open in figure viewerPowerPoint
(a) Simulation of the PBN-H aminoxyl radical adduct in the fast motion regime (red) compared to experimentally observed signal. (b) X band continuous wave (CW) RT EPR spectra recorded at 100 kHz field modulation frequency, 0.1 mT field modulation amplitude, and 3.17 mW microwave power with 5 spectra averaged per reported spectra.
Our own DFT calculations also considered the adsorption of toluene and the radical C 6 H 5 CH 2● to Pd surfaces. Scaiano used the Perdew-Burke-Ernzerhof(PBE) functional without the dispersion corrections included. In our calculations, the adsorption energies of toluene and the toluene radical with dispersion included (PBE-D3) are even more favourable with E ads(g) = −208 kJ mol−1 on the Pd(111) surface with respect to gas phase toluene and with reference to the liquid phase, we obtain E ads(l) = −130 kJ mol−1, which is more relevant to the experiments presented here.
In the adsorbed state the substituents on the toluene aromatic ring move out of plane (Figure6a). To quantify this effect, we calculate the mean plane of the aromatic carbon atoms and then measure the angle, Θ bp, between the bonds to ring substituents and the plane. For toluene on the Pd(111) surface Θ bp for the methyl group, Θ bp(C-CH 3) = 32°, is notably larger than for the hydrogen atoms on the ring which averages to Θ bp(C-H) = 18° (Table2). This is likely due to steric interactions between the methyl group and the surface. Figure6b shows the optimised structure for the Pd(111) adsorbed radical species, C 6 H 5 CH 2●. In this case, the carbon atom of the methylene radical forms a bond to a surface Pd atom and moves back into the plane of the aromatic ring (Θ bp(C-CH 2) = 3°) while the movement of the C-H bonds out of the plane of the aromatic ring is similar to that seen in toluene (av.Θ bp(C-H) = 15°, Table2). We also note that the C-C bond between the methylene group and the aromatic ring is significantly shorter than seen for the methyl group of toluene, suggesting a degree of conjugation between the CH 2 substituent and the ring in the radical.
Figure 6
Open in figure viewerPowerPoint
Calculated structures at the PBE-D3 level for (a) toluene and (b) a C 6 H 5 CH 2● radical, adsorbed to the Pd(111) surface. C-C bond distances are given in Å, angles between bonds and mean aromatic plane, Θ bp, are written at the end of bonds. Atom colours: Pd: grey, C: Brown, H: white.
Table 2. Calculated energies and geometric factors for toluene and toluene-CH 2● adsorption to Pd(111) and Pd(211).
| Surface | Site | Adsorbate | E ads(g) / kJ mol−1 | E ads(l) / kJ mol−1 | E act / kJ mol−1 | Θ bp(C-CH 3) / ° | Θ bp(C-H) / ° |
--- --- --- --- |
| Pd (111) | | C 6 H 5 CH 3 | −208 | −130 | 81 | 32 | 18 |
| | | C 6 H 5 CH 2● | −235 | −157 | − | 3 | 15 |
| | | | | | | | |
| Pd (211) | step | C 6 H 5 CH 3 | −214 | −137 | 87 | 25 | 18 |
| | terrace | C 6 H 5 CH 3 | −204 | −127 | 79 | 6 | 17 |
| | step | C 6 H 5 CH 2● | −198 | −121 | − | 22 | 19 |
| | terrace | C 6 H 5 CH 2● | −227 | −150 | − | | |
For the stepped Pd(211) surface, we also consider adsorption of toluene with the methyl group orientated towards the step edge (step, Figure S8 a) and with the aromatic ring to methyl group bond parallel to the step edge (terrace, Figure S8 c). The adsorption energy for toluene in the step site shows a small enhancement compared to the planar Pd(111) surface by 7 kJ mol−1 while adsorption on the terrace site has a calculated energy of some 3 kJ mol−1 less stable. These small energy differences suggest that toluene adsorption through the aromatic ring is largely site independent and we would expect toluene to adsorb to most of the available surface sites on a typical Pd nanoparticle. At the step site, when a C 6 H 5 CH 2● radical is formed the methylene carbon is co-ordinated to a Pd atom at the top edge of the step (Figure S8 b) and so the bond angle for the methylene group with respect to the aromatic ring, remains similar to that for the toluene molecule in the same adsorption site (Θ bp(C-CH 2) = 22° cf Θ bp(C-CH 3) = 25°). On the terrace site (Figure S8 d) the geometry of the radical adsorbate is quite similar to that seen for Pd(111) with the methylene carbon atom closer to the plane of the aromatic ring.
Figure7 shows the reaction barriers calculated for each adsorption site. The step edge is found to have the highest barrier to C 6 H 5 CH 2-H cleavage of any of the systems studied at 87 kJ mol−1 compared to around 80 kJ mol−1 for the Pd(211) terrace and Pd(111) surfaces. These barriers are notably lower than discussed earlier for the dehydrogenation of 1-phenyl ethanol suggesting that if hydrogen transfer involving the aromatic alcohol is taking place, then the exchange of H atoms with toluene solvent molecules should also be expected.
Figure 7
Open in figure viewerPowerPoint
Calculated barriers for C 6 H 5 CH 2-H bond cleavage over Pd(111) (circles), Pd(211) step (triangles) and Pd(211) terrace (squares). Energy is taken relative to the adsorbed toluene state in each case. Inset graphic shows the calculated transition state for Pd(211) terrace. Atom colours: Pd: grey, with terrace atoms blue, C: Brown, H: white.
3 Conclusions
Using isotopic labelling and 2 H NMR we show that under commonly used hydrogenation conditions (120°C, 3 bar H 2) in the presence of Pd/C catalysts C-H/D exchange occurs in solvent molecules (toluene, xylene), particularly at the benzylic positions. This observation is only possible when using combinations of H 2/D 2 with proteo/deuterated solvents and both 1 H and 2 H NMR. This process is likely occurring undetected in all liquid-phase hydrogen processes that utilize common proteo solvents which employ only 1 H-NMR for analysis. In addition, our DFT studies suggest that carbon-centered radicals derived from solvent activation can bind strongly to Pd surfaces and have reasonable activation barriers to C─H activation compared to many target substrates. This suggests that the presence of strongly bound solvent molecules with radical character should be considered in models of catalytic reactions at surfaces. We show that under these conditions in the presence of catalysts that are known for facile activation of H 2 we can trap H-radical adducts using spin-trapping EPR and these species could play mechanistic roles in numerous catalytic processes.
4 Experimental Methods
Chemicals were supplied by Sigma Aldrich and used without further purification unless otherwise stated.
4.1 Catalyst Synthesis – 1% Pd/TiO 2
Using a 500 mL round-bottomed flask Ar was bubbled through 400 mL of de-ionised water whilst vigorously stirring for 30 minutes. An acidic (0.5 M HCl) aqueous solution of PdCl 2(aq) (1.7 mL, 6 mg mL−1 Pd) was added using a syringe followed by an aqueous solution of polyvinyl alcohol (Mw 9000 – 10000, 1.2 mL, 1%wt.) so that the metal:PVA mass ratio was 1:1.2. After 2–3 minutes a freshly prepared aqueous solution of NaBH 4 (4.7 mL, 0.1 M) was added such that the metal:NaBH 4 molar ratio was 1:5. After stirring for a further 30 minutes TiO 2 (990 mg, Degussa p25) was added, followed by concentrated H 2 SO 4 (6 drops). The suspension was stirred for 1 h before being filtered and washed with water. The catalyst was then dried in air at 110°C for 16 h. 1% Au / TiO 2 and 1% AuPd/TiO 2 were prepared by a similar method using HAuCl 4 as the gold precursor. 5% Pt/C was prepared by a similar method using H 2 PtCl 6.2H 2 O and Norit GSX as the carbon support. 2.5% Au/C was prepared by a similar method using HAuCl 4 as the gold precursor and Norit ROX – as the carbon support. Commercial 5% Pd/C was obtained from Sigma Aldrich.
4.2 Catalyst Testing – Acceptorless Dehydrogenation in an Open System
A solution of 1-phenylethanol in p-xylene (10 mL, 0.4 M) was added to a 2-neck round bottomed ask equipped with a reflux condenser. The system was purged with Ar for 15 minutes, before being heated to 120°C and purged for a further 15 minutes. Catalyst (20 mg unless otherwise stated) was added to the flask to give a nominal Pd mol% of 0.05 based on a 1% theoretical metal loading. The reaction was stirred at 1000 rpm under flowing Ar for the duration of the reaction (typically 2 h). Aliquots removed at 30-minute intervals were centrifuged in 2.5 mL sample tubes at 10,000 rpm for analysis using gas chromatography.
4.3 Solvent H/D Exchange
Catalyst (0.05 mg active metal) and solvent (5 mL) were added to a 50 ml glass reactor (Radleys) and connected to a gas supply manifold. The flask was charged with H 2 (2 bar) and stirred at 120°C for 3 hours. Samples were filtered using a PTFE syringe filter for analysis by NMR.
4.4 1-Phenylethanol-d 1 Synthesis
Acetophenone (2 mL) was added dropwise to a solution of NaBD 4 (0.29 g) in EtOH (6 mL) at 30°C. This mixture was stirred at 30°C for 15 minutes before being removed from the heat and quenched with HCl (1 M, 6 mL). The mixture was then heated to 75°C to allow the EtOH to evaporate and the product to oil out. After cooling, the product was extracted with Et 2 O and washed with DI water, and the organic layer was then dried over MgSO 4 and concentrated in vacuo to give 1-phenylethanol-d 1 (1.7 g, 85% yield). Proton and deuterium NMR confirmed the identity of the product.
4.5 Product Analysis
An Agilent 6890 N GC fitted with a flame ionisation detector was used to analyse reaction mixture samples from the dehydrogenation of 1-phenylethanol and the hydrogenation of acetophenone. An HP-5 (5% phenyl methyl siloxane, capillary (30.0 m × 320 µm × 0.25 µm)) column was used with N 2 as a carrier gas. Samples were prepared for analysis by adding 0.5 mL of external standard (biphenyl, 0.2 M) to a 0.5 mL aliquot of reaction mixture. All NMR experiments were performed on a Bruker Avance III spectrometer operating at 500 MHz for 1 H. Samples of reaction mixture were prepared by filtering through a PTFE syringe filter. 2 H NMR samples were analysed without further dilution. To a 400 µl sample, 20 µl of a reference compound (CDCl 3, C 6 D 6, or MeOD was added). 1 H NMR and 13 C NMR samples were diluted prior to analysis. To 100 µl of reaction mixture, 400 µl of deuterated solvent (CDCl 3, C 6 D 6, or MeOD) was added. NMR spectra were processed using MNova software.
4.6 EPR
Samples were prepared by stirring catalyst (0.05 mg active metal), PBN spin trap (5 mg), and toluene-d 8 (10 mL) under an atmosphere of H 2 (2 bar) for 10 minutes at 120°C. Samples were filtered using a PTFE syringe filter before being loaded into a capillary tube for analysis. EPR experiments were carried out using a Bruker EMX (X-band) spectrometer. Xband continuous wave spectra were recorded at room temperature, with 100 kHz field modulation frequency, 0.1 mT field modulation amplitude, and 3.17 mW microwave power. For each sample the result was taken as the average of 5 spectra. EPR simulations were carried out in the fast motion regime by computing the Breit-Rabi formalism, i.e., the analytical diagonalization of the spin Hamiltonian for a S=1/2 electron spin system. The following parameters were used: g iso=2.0056, a iso(14 N)=1.44 mT, a iso(1 H β)=0.72 mT (x2).
4.7 DFT
Calculations were carried out using the VASP periodic DFT code with the PBE functional and Grimme D3 dispersion corrections (PBE-D3).[31-33] The core states of all atoms were represented using the projector augmented-wave (PAW) approach and a 500 eV plane wave cut-off was used throughout.[34, 35]The python based Atomic Simulation Environment (ASE) was used for management of calculation geometries.[36, 37]The fcc bulk unit cell of Pd was first optimised using 7×7×7 k-point sampling to give a cubic lattice parameter, a=3.906 Å, in excellent agreement with the experimentally determined value (a=3.890 Å). The optimised fcc cell was converted to a rhombohedral primitive cell and a p 4×4 Pd(111) slab model was created with surface cell vectors u=v=11.191 Å and a simulation cell vector perpendicular to the surface of 29.137 Å (Figure S7 a). The Pd(111) slab model contained 80 Pd atoms, i.e., 5 layers in the direction perpendicular to the surface so the chosen cell vector corresponds to a vacuum gap of 20.1 Å. To represent a stepped surface, a slab model for Pd(211) was created in the same way with surface cell vectors u=11.037 Å and v=13.517 Å and a simulation cell vector perpendicular to the surface of 47.790 Å in the direction perpendicular to the surface. This Pd(211) slab model contains 128 atoms, with at least 5 atom thickness at all points on the surface and a vacuum gap of 35.8 Å (FigureS7 b). Figure S6 c gives a top view of the Pd(211) surface showing that this Miller index choice gives a stepped surface with Pd(111) terrace and Pd(100) step features.
The adsorption energy relative to gas phase toluene, E ads(g), for a structure is calculated using:
(1)
where E struct is the calculated energy for the optimised adsorbate structure, E slab is the calculated energy of the slab model of the clean surface obtained using the same program settings. E tol(g) is the reference energy for gas phase toluene obtained from the optimisation of a single toluene molecule in the unit cell used for the Pd(111) surface slab model. This is the usual approach used in the literature and so E ads(g) is used for comparison with earlier work. Since the experimental work presented here is carried out in the liquid phase we have also attempted to estimate the adsorption energy relative to liquid phase toluene, E ads(l), In this case, the reference energy is taken from a calculation of toluene in the condensed phase starting from the P2 1/c crystal structure reported by Anderson et al. This structure was obtained at 165 K with a quoted density of 1.054 g cm−3. To obtain an approximation to the liquid phase we expanded the cell to give a density of 0.87 g cm−3 and then optimised atomic coordinates. This gives an estimate for the interaction energy per molecule in the liquid phase as
E tol (l) = -156 kJ mol−1 and so E ads(l) can be obtained using:
(2)
This accounts for the additional energy required for the molecule to move from the liquid state to the surface losing approximately half of the interactions with other molecules in the liquid phase. Note that both definitions imply that negative E ads values correspond to favourable adsorption energies.
Transition state searches were carried out using the ASE constrained bond facility. Starting from the optimised structure for toluene on each surface the methyl H atom closest to the surface was selected and the C─H bond extended by 0.1 Å. The structure was then re-optimized with the selected C─H bond constrained to maintain the same length. From the end point of this re-optimisation the same bond was again extended by 0.1 Å before a further optimisation. This process of bond elongation and optimisation was continued until the C.H distance was around 2 Å and a barrier could be seen in the energy profile. The point nearest the top of the barrier was then selected and a frequency calculation carried out to confirm a single negative mode at the transition state. The mode was also animated to confirm that it corresponded to C─H bond cleavage. In frequency calculations the Pd atoms were not included in the numerical evaluation of the Hessian based on the significant mass difference between Pd and C/H. This also reduced the computer time required for frequency calculations.
Acknowledgements
The authors have nothing to report.
Conflict of Interests
The authors declare no conflict of interest.
Open Research
Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Supporting Information
| Filename | Description |
--- |
| cctc202401969-sup-0001-SuppMat.docx4.5 MB | Supporting Information |
Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.
References
1 L. Zhang, M. Zhou, A. Wang, T. Zhang, Chem. Rev. 2020, 120, 683–733. 10.1021/acs.chemrev.9b00230 CASPubMedWeb of Science®Google Scholar
2 M. A. Stoffels, F. J. R. Klauck, T. Hamadi, F. Glorius, J. Leker, Adv. Synth. Catal. 2020, 362, 1258–1274. 10.1002/adsc.201901292 CASPubMedWeb of Science®Google Scholar
3 X. Zhao, Y. Chang, W. J. Chen, Q. Wu, X. Pan, K. Chen, B. Weng, ACS Omega 2022, 7, 17–31. 10.1021/acsomega.1c06244 CASPubMedWeb of Science®Google Scholar
4 W. Dong, G. Kresse, J. Hafner, J. Mol. Catal. A: Chem. 1997, 119, 69–76. 10.1016/S1381-1169(96)00470-0 CASWeb of Science®Google Scholar
5 B. Mattson, W. Foster, J. Greimann, T. Hoette, N. Le, A. Mirich, S. Wankum, A. Cabri, C. Reichenbacher, E. Schwanke, J. Chem. Educ. 2013, 90, 613–619. 10.1021/ed300437k CASWeb of Science®Google Scholar
6 J. S. Adams, M. L. Kromer, J. Rodríguez-López, D. W. Flaherty, J. Am. Chem. Soc. 2021, 143, 7940–7957. 10.1021/jacs.0c13399 CASPubMedWeb of Science®Google Scholar
7 N. M. Wilson, P. Priyadarshini, S. Kunz, D. W. Flaherty, J. Catal. 2018, 357, 163–175. 10.1016/j.jcat.2017.10.028 Web of Science®Google Scholar
8 Z. Zhao, R. Bababrik, W. Xue, Y. Li, N. M. Briggs, D.-T. Nguyen, U. Nguyen, S. P. Crossley, S. Wang, B. Wang, D. E. Resasco, Nat. Catal. 2019, 2, 431–436. 10.1038/s41929-019-0257-z CASWeb of Science®Google Scholar
9 T. Matsuzaki, T. Uda, A. Kazusaka, G. W. Keulks, R. F. Howe, J. Am. Chem. Soc. 1980, 102, 7511–7513. 10.1021/ja00545a019 CASWeb of Science®Google Scholar
10 A. F. Carley, H. A. Edwards, B. Mile, M. W. Roberts, C. C. Rowlands, F. E. Hancock, S. D. Jackson, J. Chem. Soc., Faraday Trans. 1994, 90, 3341–3346. 10.1039/ft9949003341 CASWeb of Science®Google Scholar
11 S. J. Thomson, G. Webb, J. Chem. Soc., Chem. Commun. 1976, 526–527. 10.1039/c39760000526 CASWeb of Science®Google Scholar
12 A. Burt, M. Emery, J. Maher, B. Mile, Magn. Reson. Chem. 2001, 39, 85–88. 10.1002/1097-458X(200102)39:2<85::AID-MRC805>3.0.CO;2-A CASGoogle Scholar
13 G. Nicolau, G. Tarantino, C. Hammond, ChemSusChem 2019, 12, 4953–4961. 10.1002/cssc.201901313 CASPubMedWeb of Science®Google Scholar
14 K. Kon, W. Onodera, T. Toyao, K.-i. Shimizu, Catal. Sci. Technol. 2016, 6, 5864–5870. 10.1039/C6CY00252H CASWeb of Science®Google Scholar
15 H. Rogers, I. T. Daniel, S. J. Freakley, Catal. Commun. 2022, 162, 106377. 10.1016/j.catcom.2021.106377 CASGoogle Scholar
16 M. Gómez-Gallego, M. A. Sierra, Chem. Rev. 2011, 111, 4857–4963. 10.1021/cr100436k CASPubMedWeb of Science®Google Scholar
17 J. W. Larsen, L. W. Chang, J. Org. Chem. 1978, 43, 3602-3602. 10.1021/jo00412a040 CASWeb of Science®Google Scholar
18 J. L. Rhinehart, K. A. Manbeck, S. K. Buzak, G. M. Lippa, W. W. Brennessel, K. I. Goldberg, W. D. Jones, Organometallics 2012, 31, 1943–1952. 10.1021/om2012419 CASWeb of Science®Google Scholar
19 J. D. Smith, G. Durrant, D. H. Ess, B. S. Gelfand, W. E. Piers, Chem. Sci. 2020, 11, 10705–10717. 10.1039/D0SC02694H CASPubMedWeb of Science®Google Scholar
20 A. Sattler, ACS Catal. 2018, 8, 2296–2312. 10.1021/acscatal.7b04201 CASWeb of Science®Google Scholar
21 K. Morikawa, W. S. Benedict, H. S. Taylor, J. Am. Chem. Soc. 1936, 58, 1445–1449. 10.1021/ja01299a040 CASWeb of Science®Google Scholar
22 C. Kemball, R. G. W. Norrish, Proc. R. Soc. London, Ser. A 1951, 207, 539–554. 10.1098/rspa.1951.0139 CASGoogle Scholar
23 B. Hammer, J. K. Norskov, Nature 1995, 376, 238–240. 10.1038/376238a0 CASWeb of Science®Google Scholar
24 V. H. Uc, J. R. Alvarez-Idaboy, A. Galano, I. García-Cruz, A. Vivier-Bunge, J. Phys. Chem. A 2006, 110, 10155–10162. 10.1021/jp062775l CASPubMedWeb of Science®Google Scholar
25 S.-X. Hu, J.-G. Yu, S.-M. Li, E. Y. Zeng, Computational and Theoretical Chemistry 2011, 977, 13–21. 10.1016/j.comptc.2011.08.032 CASWeb of Science®Google Scholar
26 S. Pan, L. Wang, J. Phys. Chem. A 2014, 118, 10778–10787. 10.1021/jp506815v CASPubMedGoogle Scholar
27 G. R. Buettner, Free Radic Biol Med 1987, 3, 259–303. 10.1016/S0891-5849(87)80033-3 CASPubMedWeb of Science®Google Scholar
28 M. Conte, H. Miyamura, S. Kobayashi, V. Chechik, J. Am. Chem. Soc. 2009, 131, 7189–7196. 10.1021/ja809883c CASPubMedWeb of Science®Google Scholar
29 C. Lutterloh, J. Biener, A. Schenk, J. Küppers, Surf. Sci. 1995, 331–333, 261–266. 10.1016/0039-6028(95)00225-1 CASGoogle Scholar
30 T. A. Gawargy, P. Costa, A. E. Lanterna, J. C. Scaiano, Org. Biomol. Chem. 2020, 18, 6047–6052. 10.1039/D0OB01227K PubMedWeb of Science®Google Scholar
31 G. Kresse, J. Hafner, Phys. Rev. B 1993, 47, 558–561. 10.1103/PhysRevB.47.558 CASPubMedWeb of Science®Google Scholar
32 G. Kresse, J. Furthmüller, Phys. Rev. B 1996, 54, 11169–11186. 10.1103/PhysRevB.54.11169 CASPubMedWeb of Science®Google Scholar
33 J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 1996, 77, 3865–3868. 10.1103/PhysRevLett.77.3865 CASPubMedWeb of Science®Google Scholar
34 S. Grimme, J. Antony, S. Ehrlich, H. Krieg, J. Chem. Phys. 2010, 132, 154104. 10.1063/1.3382344 CASPubMedWeb of Science®Google Scholar
35 G. Kresse, D. Joubert, Phys. Rev. B 1999, 59, 1758–1775. 10.1103/PhysRevB.59.1758 CASWeb of Science®Google Scholar
36 V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, M. Scheffler, Comput. Phys. Commun. 2009, 180, 2175–2196. 10.1016/j.cpc.2009.06.022 CASWeb of Science®Google Scholar
37 A. Hjorth Larsen, J. Jørgen Mortensen, J. Blomqvist, I. E. Castelli, R. Christensen, M. Dułak, J. Friis, M. N. Groves, B. Hammer, C. Hargus, E. D. Hermes, P. C. Jennings, P. Bjerre Jensen, J. Kermode, J. R. Kitchin, E. Leonhard Kolsbjerg, J. Kubal, K. Kaasbjerg, S. Lysgaard, J. Bergmann Maronsson, T. Maxson, T. Olsen, L. Pastewka, A. Peterson, C. Rostgaard, J. Schiøtz, O. Schütt, M. Strange, K. S. Thygesen, T. Vegge, L. Vilhelmsen, M. Walter, Z. Zeng, K. W. Jacobsen, J. Phys.: Condens. Matter 2017, 29, 273002. 10.1088/1361-648X/aa680e PubMedWeb of Science®Google Scholar
38 J. W. Arblaster, Platinum Met. Rev. 2012, 56, 181–189. 10.1595/147106712X646113 CASGoogle Scholar
39 M. Anderson, L. Bosio, J. Bruneaux-Poulle, R. Fourme, J. Chim. Phys. 1977, 74, 68–73. 10.1051/jcp/1977740068 CASWeb of Science®Google Scholar
Volume 17, Issue 9
May 8, 2025
e202401969
Figures -------
References ----------
Related -------
Information -----------
Recommended
DNA‐Mediated Palladium Nanoparticles as an Efficient Catalyst for Hydrogenation‐ and Suzuki–Miyaura Coupling Reactions
Hisanori Itoh,Hironori Maeda,Shinya Yamada,Yoji Hori, ChemCatChem
(Z)-Selective Partial Hydrogenation of Internal Alkynes by Using Palladium Nanoparticles Supported on Nitrogen-Doped Carbon Nanofiber
Youngjin Lee,Yukihiro Motoyama,Keita Tsuji,Seong-Ho Yoon,Isao Mochida,Hideo Nagashima, ChemCatChem
Catalytic Hydrogenation of Biomass-Derived Furoic Acid to Tetrahydrofuroic Acid Derivatives over Pd/CoO x Catalyst in Water
Bhanu Priya,Ankit Kumar,SK Nazmul Hasan Md Dostagir,Abhijit Shrotri,Sanjay Kumar Singh, ChemCatChem
The Simultaneous Total Oxidation of Toluene, Propene, and CO Environmental Pollutants Using Bimetallic Au-Pt/ZrO 2/UVM-7 Catalysts
Kieran J. Aggett,Tomás García,David J. Morgan,Mario Peláez-Fernández,Álvaro Mayoral,Raul Arenal,Benjamín E. Solsona,Pedro Amorós,Stuart H. Taylor, ChemCatChem
A Bifunctional Organic Photocatalyst for Efficient Single‐Electron and Energy Transfer Activation
Sumitava Mallik,Hailong Wang,Nunzio Matera,Bi-Xiao Li,Stefano Stagni,Paolo Melchiorre, Angewandte Chemie International Edition
Metrics
Full text views: 747
Full text views and downloads on Wiley Online Library.
More metric information
Details
© 2025 The Author(s). ChemCatChem published by Wiley-VCH GmbH
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Check for updates
Keywords
C-H Activation
Hydrogenation
Palladium
Radical
Publication History
Issue Online: 08 May 2025
Version of Record online: 28 April 2025
Accepted manuscript online: 14 February 2025
Manuscript accepted: 14 February 2025
Manuscript revised: 10 February 2025
Manuscript received: 28 November 2024
Close Figure Viewer
Previous FigureNext Figure
Caption
Download PDF
back
© 2025 Wiley-VCH GmbH, Weinheim
Additional links
About Wiley Online Library
Privacy Policy
Terms of Use
About Cookies
Manage Cookies
Accessibility
Wiley Research DE&I Statement and Publishing Policies
Help & Support
Contact Us
Training and Support
DMCA & Reporting Piracy
Sitemap
Opportunities
Subscription Agents
Advertisers & Corporate Partners
Connect with Wiley
The Wiley Network
Wiley Press Room
Copyright © 1999-2025 John Wiley & Sons, Inc or related companies. All rights reserved, including rights for text and data mining and training of artificial intelligence technologies or similar technologies.
Log in to Wiley Online Library
Email or Customer ID
Password
Forgot password?
NEW USER >INSTITUTIONAL LOGIN >
Change Password
Old Password
New Password
Too Short Weak Medium Strong Very Strong Too Long
Your password must have 10 characters or more:
a lower case character,
an upper case character,
a special character
or a digit
Too Short
Password Changed Successfully
Your password has been changed
Create a new account
Email
Returning user
Forgot your password?
Enter your email address below.
Email
Please check your email for instructions on resetting your password. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.
Request Username
Can't sign in? Forgot your username?
Enter your email address below and we will send you your username
Email
Close
If the address matches an existing account you will receive an email with instructions to retrieve your username
✓
Thanks for sharing!
AddToAny
More…
Close crossmark popup |
11783 | https://tisp.indigits.com/basic_real_analysis/real_line | 2.1. Real Line — Topics in Signal Processing
- [x]
Toggle navigation sidebar
[x]
Toggle in-page Table of Contents
Topics in Signal Processing ===========================
Foundation
1. Algebra- [x]
1.2. Sets
1.3. Relations
1.4. Functions
1.5. Integers
1.6. Cardinality
1.7. Sequences
1.8. General Cartesian Product
1.9. Matrices
1.10. Groups
1.11. Enumerative Combinatorics
2. Elementary Real Analysis- [x]
2.1. Real Line
2.2. Topology of Real Line
2.3. Sequences and Series
2.4. The Extended Real Line
2.5. Real Valued Functions
2.6. Real Functions
2.7. Differentiable Functions
2.8. Some Important Inequalities
3. Metric Spaces- [x]
3.1. Introduction
3.2. Metric Topology
3.3. Boundedness
3.4. Sequences
3.5. Subspace Topology
3.6. Functions and Continuity
3.7. Completeness
3.8. Compactness
3.9. Real Valued Functions
3.10. Discrete Metric Space
3.11. Special Topics
4. Linear Algebra- [x]
4.1. Vector Spaces
4.2. Matrices II
4.3. Linear Transformations
4.4. Normed Linear Spaces
4.5. Inner Product Spaces
4.6. Dual Spaces
4.7. The Euclidean Space
4.8. Matrices III
4.9. Eigen Values
4.10. Singular Values
4.11. Important Vector Spaces
4.12. Matrix Norms
4.13. Sequence Spaces
4.14. Affine Sets and Transformations
5. Multivariate Calculus- [x]
5.1. Differentiation
5.2. Differentiation in Banach Spaces
6. Geometry- [x]
6.1. Algebraic Geometry
7. Probability- [x]
7.1. Probability Spaces
7.2. Random Variables
7.3. Univariate Distributions
7.4. Basic Inequalities
7.5. Two Variables
7.6. Expectation
7.7. Random Vectors
7.8. Multivariate Gaussian Distribution
7.9. Subgaussian Distributions
8. Numerical Optimization- [x]
8.1. Mathematical Optimization
Convexity
9. Convex Sets and Functions- [x]
9.1. Real Vector Spaces
9.2. Convex Sets
9.3. Convex Subsets of R n
9.4. Cones
9.5. Cones II
9.6. Cones III
9.7. Generalized Inequalities
9.8. Convex Functions
9.9. Differentiability and Convex Functions
9.10. Function Operations
9.11. Topology of Convex Sets
9.12. Separation Theorems
9.13. Continuity
9.14. Recession Cones
9.15. Directional Derivatives
9.16. Subgradients
9.17. Conjugate Functions
9.18. Smoothness
9.19. Infimal Convolution
10. Convex Optimization- [x]
10.1. Convex Optimization
10.2. Projection on Convex Sets
10.3. Directions of Recession
10.4. Basic Duality
10.5. Constrained Optimization I
10.6. Linear Constraints
10.7. Constrained Optimization II
10.8. Lagrange Multipliers
10.9. Lagrangian Duality
10.10. Conjugate Duality
10.11. Linear Programming
10.12. Quadratic Programming
11. Subgradient Methods- [x]
11.1. Basic Subgradient Method
12. Proximal Algorithms- [x]
12.3. Proximal Mappings and Operators
Signal Processing
13. Signal Theory
14. Detection and Estimation Theory
15. Coding Theory
16. Wavelets
Machine Learning
17. Data Clustering- [x]
17.1. Introduction
17.2. Similarity Measures
17.3. Hierarchical Algorithms
17.4. K-Means Clustering
17.5. Graph Algorithms
17.6. Spectral Clustering
17.7. Expectation Maximization
17.8. Evaluation
Sparsity
18. Sparse Signal Models- [x]
18.3. Underdetermined Linear Systems
18.4. Sparsity in Orthonormal Bases
18.5. Sparse and Redundant Representations
18.6. Dictionaries
18.7. Compressive Sensing
18.8. Restricted Isometry Property
18.9. Dictionaries II
19. Compressive Sensing- [x]
19.1. Sensing Matrices
19.2. Quantization
20. Sparse Approximation with Dictionaries- [x]
20.1. Stability of the Sparsest Solution
20.2. Basis Pursuit
20.3. Orthogonal Matching Pursuit
21. Sparse Recovery from Compressive Measurements- [x]
21.1. Stability of the Sparsest Solution
21.2. Basis Pursuit
21.3. Orthogonal Matching Pursuit
21.4. Compressive Sampling Matching Pursuit
22. Dictionary Learning- [x]
22.1. Introduction
Applications
23. Subspace Clustering- [x]
23.5. Motion Segmentation
Epilogue
Notation
Bibliographic Notes
Index
Powered by Jupyter Book
repository
open issue
suggest edit
.md
.pdf
Contents
2.1.1. Prelude
2.1.2. Real numbers
2.1.3. Intervals
2.1.4. Completeness Axiom
2.1.5. Archimedian Property
2.1.6. Rational Numbers
2.1.7. Irrational Numbers
2.1.8. Uncountability
2.1.9. Algebraic Numbers
2.1.10. Transcendental Numbers
2.1.11. Intervals are Uncountable
Real Line
Contents
2.1.1. Prelude
2.1.2. Real numbers
2.1.3. Intervals
2.1.4. Completeness Axiom
2.1.5. Archimedian Property
2.1.6. Rational Numbers
2.1.7. Irrational Numbers
2.1.8. Uncountability
2.1.9. Algebraic Numbers
2.1.10. Transcendental Numbers
2.1.11. Intervals are Uncountable
2.1. Real Line#
The set of real numbers is also known as real line.
2.1.1. Prelude#
We look at some justification for why real numbers are needed.
Proposition 2.1
There is no rational number whose square root is 2.
Proof. Let us assume that there indeed is a rational number whose square root is 2:
(p q)2=2.
We assume that p and q have no common factor. Then we have: p 2=2 q 2. This means that p 2 is even. Thus, p is even. Let p=2 r. Then 4 r 2=2 q 2⟹q 2=2 r 2. This means that q 2 is even. Thus, q is even. But this contradicts our assumption that p and q have no common factor.
This argument shows that the set of rational numbers is not complete in the sense that we can posit the existence of numbers which are not rational.
We extend the set further to a larger number system known as real numbers. The completeness axiom plays a major role in the definition of real numbers.
2.1.2. Real numbers#
We present the axiomatic definition of real numbers.
Definition 2.1 (Real Numbers)
The real numbers are the members of a nonempty set R equipped with two operations + and ⋅ from R×R into R called addition and multiplication, that satisfy following axioms. We will denote arbitrary members of R as x,y and z. We will also abbreviate multiplication x⋅y as x y.
x+y=y+x and x y=y x; commutative laws.
x+(y+z)=(x+y)+z and x(y z)=(x y)z; associative laws.
x(y+z)=x y+x z; distributive law.
There exists an element 0∈R such that x+0=x∀x∈R; additive identity.
For each x∈R there exists an element in R denoted by −x such that x+(−x)=0; additive inverse.
There exists an element 1∈R with 1≠0 satisfying 1⋅x=x∀x∈R; multiplicative identity.
For each x≠0 there exists an element in R denoted by x−1 satisfying x x−1=1; multiplicative inverse.
For any x,y∈R either x≥y or y≥x holds; total order.
If x≥y, then x+z≥y+z holds for each z∈R.
If x≥y and z≥0 then x z≥y z.
Every nonempty set of real numbers that is bounded from above has a least upper bound; completeness axiom.
Axioms (1-7) are field axioms. Axioms (8-10) establish that R is an ordered field. Axiom (11) is discussed further in Axiom 2.1.
It can be shown that if R and S are two different sets satisfying these axioms then they are isomorphic and for all practical purposes identical.
Several remarks immediately follow from the definition.
The zero element (additive identity) is unique.
The one element (multiplicative identity) is unique.
The additive inverse −x is unique.
−x=(−1)x holds.
The multiplicative inverse x−1 for x≠0 is unique.
0⋅x=0.
−(−x)=x.
(−x)(−y)=x y.
x−y=x+(−y)=−(y−x).
(x−1)−1=x∀x≠0.
Remarks above are derived from field properties of real line.
Further remarks on order property
An alternative notation for x≥y is y≤x.
x>y means that x≥y and x≠y.
x<y means that x≤y and x≠y.
xx⟺y−x>x−x=0.
Definition 2.2 (Positive and negative numbers)
Any number x∈R satisfying x>0 is called a positive number.
Any number x∈R satisfying x<0 is called a negative number.
Any number x∈R satisfying x≥0 is called a non-negative number.
Any number x∈R satisfying x≤0 is called a non-positive number.
Proposition 2.2
If x+ϵ≥y for each ϵ>0, then x≥y holds.
Proof. If the result doesn’t hold, then x0. Let ϵ=1 2(y−x). Then x+ϵ=1 2(x+y)≥y. This implies 1 2(y−x)≥0⟺y−x<0, which is a contradiction.
Definition 2.3 (Absolute value)
The absolute value|a| of a real number a∈R is defined as follows. if a≥0, then |a|=a and if a<0, then |a|=−a. a∨b denotes the larger of the two numbers. a∧b denotes the smaller of the two numbers.
Remark 2.1
Thus |a|=a∨(−a)∀a∈R. Further |a|=|−a|.
Proposition 2.3
The absolute value satisfies following properties:
|a|≥0 for each a∈R, and |a|=0⟺a=0.
|a b|=|a|⋅|b|∀a,b∈R.
|a+b|≤|a|+|b|∀a,b∈R; the triangle inequality.
Proof. If a=0 then by definition |a|=0. Again from definition, for any non-zero value, |a| is not zero. Hence |a|=0⟺a=0.
Let a and b two non-negative numbers. Then |a|=a and |b|=b. Thus, |a||b|=a b=|a b|. Further |a|+|b|=a+b=|a+b|.
Now, let a and b two non-positive numbers. Then |a|=−a and |b|=−b. a b is non-negative, hence |a b|=a b. a+b is non-positive, hence |a+b|=−(a+b). Thus, |a||b|=(−a)(−b)=a b=|a b|. Further, |a|+|b|=(−a)+(−b)=−(a+b)=|a+b|.
Now lets consider when a and b are of opposite signs. WLOG (without loss of generality), let us assume that a is positive and b is negative. Further let us assume that |a|≥|b|. Clearly, then |a+b|<|a|+|b|. Also |a b|=−a b=|a||b|.
Corollary 2.1
For a,b∈R the following hold:
|a−b|≤|a|+|b|.
||a|−|b||≤|a−b|.
Proof. We know that
|a+b|≤|a|+|b|.
Replacing b with −b we get:
|a−b|≤|a|+|b|.
Again, replacing b with b−a, we get:
|a+b−a|≤|a|+|b−a|⟹|b|−|a|≤|a−b|.
Further, replacing a with a−b in |a+b|≤|a|+|b|, we get:
|a−b+b|≤|a−b|+|b|⟹|a|−|b|≤|a−b|.
Combining the two, we get our result.
Proposition 2.4 (Distance triangle inequality)
Let a,b,c∈R. Then:
|a−b|≤|a−c|+|c−b|.
This is straight-forward application of Proposition 2.3.
|a−c|+|c−b|≥|(a−c)+(c−b)|=|a−b|.
Proposition 2.5
Two real numbers a and b are equal if and only if for every real number ϵ>0, |a−b|<ϵ holds.
Proof. If a=b, then a−b=0⟹|a−b|=0<ϵ for every ϵ>0. Now for the converse, we are given that for every ϵ>0, |a−b|<ϵ. We have to show that a=b. Let as assume that a≠b. Choose ϵ 0=|a−b|. Clearly, for ϵ 0>0, |a−b|<ϵ 0 doesn’t hold. This contradicts our assumption that for every ϵ>0|a−b|<ϵ. Hence a=b.
2.1.3. Intervals#
Definition 2.4 (Interval)
A subset S of R is called an interval if for every a,b∈S such that a<b, S contains all the real numbers between a and b. In other words, if a<x<b, then x∈S.
An open interval does not include its endpoints and is denoted as (a,b)≜{x∈R|a<x<b}.
A closed interval does not include its endpoints and is denoted as [a,b]≜{x∈R|a≤x≤b}.
A half-open interval includes one of its its endpoints : [a,b)≜{x∈R|a≤x<b}; (a,b]≜{x∈R|a<x≤b}.
A degenerate interval is an interval of the form [a,a] which is a singleton containing a∈R.
2.1.4. Completeness Axiom#
Definition 2.5 (Upper and lower bounds)
Let A be a nonempty subset of R.
An upper bound of A is any u∈R such that x≤u∀x∈A.
A lower bound of A is any l∈R such that x≥l∀x∈A.
If A has an upper bound it is said to be bounded from above.
If A has a lower bound it is said to be bounded from below.
If A is both bounded from above and below, then A is said to be bounded.
A real number is called a least upper bound or supremum of A if it is an upper bound of A, and it is less than or equal to every other upper bound of A. The least upper bound is denoted by sup(A).
A real number is called a greatest lower bound or infimum of A if it is a lower bound of A, and it is greater than or equal to every other lower bound of A. The greatest lower bound is denoted by inf(A).
Remark 2.2
A set A can have at most one least upper bound and at most one greatest lower bound.
Remark 2.3
The infimum or supremum of a set A need not belong to the set itself.
The set S={x|0<x<1} has an infimum 0 and a supremum 1. Neither belong to the set.
The completeness axiom asserts that every nonempty set bounded from above has a least upper bound. We restate the axiom independently for emphasis.
Axiom 2.1 (Completeness axiom)
Every nonempty set of real numbers that is bounded from above has a least upper bound.
Example 2.1 (Least upper and greatest lower bounds)
Let
A={1 n:n∈N}.
The set is bounded from above with sup(A)=1. The set is bounded from below with inf(A)=0.
The open interval (0,2)={x∈R:0<x<2} has lower and upper bounds as 0 and 2 respectively. So does the closed interval [0,2]={x∈R:0≤x≤2}. But (0,2) has no maximum or minimum element. [0,2] has both. Further, (0,2] has maximum element but not minimum, and [0,2) has minimum element but not maximum.
Example 2.2 (Distinction with rational numbers)
The set S={x∈Q:x 2≤2} doesn’t have a least upper bound in the set of rational numbers. For every x∈S, it is possible to find a y∈Q such that y>x and y 2<2.
The axiomatic definition of real numbers claims that since the set S={x∈R:x 2≤2} is bounded from above, hence there exists a real number which is the least upper bound of this set. We simply denote this number as:
2≜sup{x∈R:x 2≤2}.
Corollary 2.2
Every nonempty set of real numbers that is bounded from below has a greatest lower bound.
Proof. Let A⊂R be a nonempty set bounded from below. Consider the set of lower bounds for A defined as
B={b∈R:b≤x∀x∈A}.
Since B is bounded from above by numbers in A, hence by completeness axiom, B has a least upper bound denoted by sup(B). This indeed is the greatest lower bound of A as sup(B)=inf(A).
This result demonstrates that we only need an axiom for the least upper bound. The idea of existence of greatest lower bound immediately follows from it. We could have easily started with the existence of greatest lower bound as an axiom and derived the existence of least upper bound from it.
Remark 2.4
If a set A has a maximum (resp. minimum) element, then max(A)=sup(A) (resp. min(A)=inf(A) ).
On the other hand, if the supremum of a set A exists and sup(A)∈A, then sup(A) is the maximum element of A. Similarly if the infimum of a set A exists and inf(A)∈A, then inf(A) is the minimum element of A.
Proposition 2.6
Assume that the supremum of a subset A of R exists. Then for every ϵ>0, there exists some x∈A such that
sup(A)−ϵ<x≤sup(A).
Conversely, let a be an upper bound of A⊆R such that for every ϵ>0, there exists an x∈A such that a−ϵ<x, then a is the least upper bound of A.
Proof. Clearly, if x∈A then x≤sup(A). Now, if for every x∈A we have x≤sup(A)−ϵ, then sup(A)−ϵ is an upper bound of A, which is less than the least upper bound. This is a contradiction. Thus, for every ϵ>0, there exists an x∈A such that sup(A)−ϵ<x≤sup(A).
Conversely, no matter how small ϵ>0 is chosen, a−ϵ cannot be an upper bound of A since there exists some x∈A such that x>a−ϵ. Hence a is indeed the least upper bound of A.
Proposition 2.7
Let A and B be two subsets of R bounded above with B⊆A. Then
sup(B)≤sup(A).
Proof. Let a be an upper bound of A. Then a≥x∀x∈A. Since B⊆A, hence a≥y∀y∈B too. Thus, every upper bound of A is also an upper bound of B. Thus, sup(A) is also an upper bound of B. Since sup(B) is least upper bound of B, it is smaller than or equal to any other upper bound of B. Thus sup(B)≤sup(A).
Proposition 2.8
The set of natural numbers N is unbounded.
Proof. Assume by contradiction that N is bounded. Since N⊂R, hence by completeness axiom, there exists a least upper bound of N. Let s=sup(N). Then n≤s∀n∈N. Then by Proposition 2.6, for ϵ=1 there exists some k∈N such that s−1<k. This implies s<k+1. Further since s is the supremum, hence k+1≤s. Thus, s<k+1≤s, which is impossible.
Proposition 2.9 (Nested interval property)
For each n∈N, assume that we are given a closed interval I n=[a n,b n]={x∈R:a n≤x≤b n}. Assume also that each I n contains I n+1. Then, the resulting nested sequence of closed intervals
I 1⊇I 2⊇⋯⊇I n⊇I n+1⊇…
has a nonempty intersection; i.e.
⋂n=1∞I n≠∅.
Proof. Consider the set of lower bounds
A={a n:n∈N}.
Due to completeness axiom, the set has a least upper bound. Let x=sup(A). Further, due to nested structure, each b n is also an upper bound of A. Since x is the least upper bound, hence x≤b n∀n∈N. Thus, we have a n≤x≤b n∀n∈N. Hence, x∈I n∀n⟹x∈⋂n=1∞I n .
2.1.5. Archimedian Property#
Proposition 2.10 (Archimedian property)
If x and y are two positive real numbers, then there exists some natural number n such that n x>y.
Proof. For contradiction assume that n x≤y holds for every n∈N. Since, x,y>0, n≤y x−1∀n∈N. This contradicts Proposition 2.8.
This also tells us that for any real number x>0, there exists an n∈N satisfying 1 n1⟹x>1 n.
Example 2.3
Let A={1 n:n∈N}. We show that inf(A)=0. Clearly 0 is a lower bound. Now if x>0 then we can find n 0∈N such that 1/n 00 cannot be a lower bound of A. Hence 0 is the greatest lower bound of A.
2.1.6. Rational Numbers#
Proposition 2.11 (Density of Q in R)
Between any two distinct real numbers there exists a rational number.
Proof. Let a and b be the two distinct real numbers. If the two numbers are of opposite sign, then 0 is an obvious rational number between them. If a and b are both non-positive, then the rational number can be chosen to be the negative of the rational number between −a and −b. We now consider the case where both a and b are non-negative. Without loss of generality, we assume that 0≤a<b. Consider the set
A={n∈N:n>max{1 b−a,1 b}}.
Since N is not bounded from above, A is nonempty. Fix an element q∈A. Clearly, 0<1 q<b−a and 1<b q. Now, let
B={n∈N:n<b q}.
Since 1∈B, B is nonempty. Also, B is a finite set. Let p=max(B). Clearly, p∈B and p+1∉B.
Finally, we will show that a<p q<b. Since p∈B, p<b q⟹p q<b. Further,
p+1≥b q⟹b≤p q+1 q<p q+b−a⟹a<p q.
Proposition 2.12
There exists a real number α∈R satisfying α 2=2.
Proof. Consider the set
A={x∈R:x 2<2}.
Clearly, A is bounded above. Now, choose α=sup(A). We show that α 2=2 by ruling out α 2<2 and α 2>2. Assume that α 2<2. Consider for some n∈N:
(α+1 n)2=α 2+2 α n+1 n 2<α 2+2 α n+1 n=α 2+2 α+1 n.
Now choose n 0∈N large enough so that
1 n 0<2−α 2 2 α+1.
This implies 2 α+1 n 0<2−α 2. Thus,
(α+1 n 0)2<α 2+(2−α 2)=2.
Thus α+1 n 0∈A contradicting the assumption that α 2 is an upper bound of A. Hence α 2<2 is not possible.
Now assuming α 2>2, we show that it cannot be the least upper bound of A. We start with
(α−1 n)2=α 2−2 α n+1 n 2>α 2−2 α n.
Choosing n 0>2 α/(α 2−2), we get 2 α n 0<α 2−2. Thus
(α−1 n 0)2>α 2−(α 2−2)=2.
Thus (α−1 n 0)2 is also an upper bound of A. But
(α−1 n 0)2<α 2.
Hence, α 2 cannot be the least upper bound of A. Thus, α 2>2 is not possible. Since for every real number α 2, exactly one of the possibilities α 2<2, α 2>2 or α 2=2 has to be true, hence α 2=2.
2.1.7. Irrational Numbers#
Definition 2.6 (Irrational number)
A real number which is not rational is known as irrational number. The set of irrational numbers is denoted by I.
Clearly, I=R∖Q. Also, R=Q∪I. Although Q forms a field, but I does not. For example 2×2=2 is rational. Thus I is not closed under multiplication. Further, 0 and 1 (the additive and multiplicative identities) don’t belong to I.
Proposition 2.13
The sum of a rational and an irrational number is irrational. The product of a non-zero rational number and an irrational number is irrational.
Proof. Let q∈Q and i∈I. Let r=q+i. Then i=r−q. If r were rational, then i would have to be rational (since Q is closed under addition). But i is irrational, hence r is irrational.
Now let q≠0. Let r=q i. Then i=r/q. If r were rational, then i would have to be rational since Q is closed under multiplication.
Proposition 2.14 (Density of I in R)
Between any two distinct real numbers there exists an irrational number.
Proof. Let a,b∈R. Without loss of generality, assume a<b. Consider the numbers: a−2,b−2. Since Q is dense in R (Proposition 2.11), there exists a rational number q such that
a−2<q<b−2.
Clearly a<q+2<b. But by Proposition 2.13, q+2 is irrational. This completes the proof.
Proposition 2.15 (Existence of roots)
For a real number a and any natural number n≥2, we have the following:
If a≥0 and n is even, there exists a unique b≥0 such that b n=a.
If a∈R and n is odd, then there exists a unique b∈R such that b n=a.
2.1.8. Uncountability#
Proposition 2.16
The set R is uncountable.
Proof. Assume that R is countable. Then it is possible to enumerate R as R={x 1,x 2,…}. Consider a closed interval I 1 which doesn’t contain x 1. Further, consider a second closed interval I 2⊆I 1 which doesn’t contain x 2. If I 1=[a,b], then by choosing c 1=(a+b)/3, and c 2=2(a+b)/3, we get disjoint closed sub-intervals [a,c 1] and [c 2,b]. x 2 can lie in at most one of them, hence it is always possible to choose I 2⊆I 1 as desired. Now given I n has been chosen, choose I n+1 such that:
I n+1⊆I n and
x n+1∉I n+1.
Now, consider the countable intersection of nested closed sets
I=⋂n=1∞I n.
If x i∈R (as an element in the enumeration of R), then x i∉I i. Thus x i∉I. Thus, if R is countable, then no real number lies in I. But by nested interval property, I is not empty. This contradicts our assumption that R is countable.
Corollary 2.3
The set of irrational numbers is uncountable.
Proof. If I were countable, then R=Q∪I would also be countable. But this is not true. Hence I is uncountable.
2.1.9. Algebraic Numbers#
Definition 2.7 (Algebraic real number)
A real number is called algebraic if it is the root of a polynomial with integer coefficients. i.e. x∈R is algebraic if there exist integers a 0,a 1,…,a n∈Z, not all zero, such that
a n x n+a n−1 x n−1+⋯+a 1 x+a 0=0.
Example 2.4 (Algebraic real numbers)
2 is algebraic since it is the root of the equation x 2−2=0.
2 3 is algebraic since it is the root of the equation x 3−2=0.
2+3 is algebraic since it is the root of the equation x 4−10 x 2+1=0.
Proposition 2.17
The set of algebraic numbers is countable.
Proof. Let n∈N. Let A n be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n given by
a n x n+a n−1 x n−1+⋯+a 1 x+a 0=0.
Each polynomial can be represented by an n+1-tuple given by
(a n,a n−1,…,a 1,a 0).
Thus, it has a one-one correspondence with the Cartesian product Z n+1. Since finite Cartesian product of countable sets is countable, hence Z n+1 is countable. This means that the number of polynomials of order n is countable. Since each polynomial of order n can have at most n roots, hence A n is countable.
Now consider the union
A=⋃n=1∞A n
as the set of algebraic numbers. Since union of a countable family of countable sets is countable, hence A is countable.
2.1.10. Transcendental Numbers#
Definition 2.8 (Transcendental numbers)
A real number is called transcendental if it is not algebraic.
Proposition 2.18 (Existence of transcendental numbers)
Transcendental numbers exist and they are uncountable.
Proof. Let A denote the set of algebraic numbers and B denote the set of transcendental numbers. Then R=A∪B. Since A is countable and R is uncountable, hence B is non-empty and is uncountable.
2.1.11. Intervals are Uncountable#
Proposition 2.19
The open interval (0,1)={x∈R:0<x<1} is uncountable.
Proof. Assume that (0,1) is countable. Then, the numbers x∈(0,1) can be enumerated. Let f:N→(0,1) be such an enumeration. Let f(m) be given by (in the decimal representation)
f(m)=.a m 1 a m 2….
Define a real number x∈(0,1) with the decimal expansion
x=.b 1 b 2…
using the rule:
b n={2 if a n n≠2;3 if a n n=2.
The number x cannot be f(1) since b 1≠a 11. The number x cannot be f(n)∀n∈N since b n≠a n n. Thus x doesn’t belong to the suggested enumeration of (0,1). Thus, (0,1) is uncountable.
Proposition 2.20
(0,1) is uncountable if and only if R is uncountable.
Proof. If R were countable, then (0,1) would be countable since (0,1)⊂R. Thus, (0,1) can be uncountable only if R is uncountable.
Now if (0,1) were countable then R would be countable since R can be written as
R=⋃n=−∞∞(n,n+1)∪Z.
Every open interval (n,n+1) has the same cardinality as (0,1) and Z is countable. Thus, R is uncountable only if (0,1) is uncountable. In other words, (0,1) is uncountable if R is uncountable.
previous 2. Elementary Real Analysisnext 2.2. Topology of Real Line
By Shailesh Kumar
© Copyright 2021-2022. |
11784 | https://www.theenglishisland.com/lessons/negative-questions/ | Skip to content
Need to translate this page?
Apply Now
PH 678-324-8760 WA 404-973-8811
Negative Questions
Negative Questions
January 5, 2016
A negative question is one that is worded in such a way as to require a “no” response for an affirmative answer and a “yes” response for a negative answer. In other words, negative questions switch the “yes/no” response order of regular, or positive, questions to a less intuitive “no/yes” order. The following is an example of a negative question:
Would you mind driving me to my class at the English Island on Tuesday?
For this negative question, a “no” answer would indicate that the person being asked the question would be able and/or willing to drive the person asking the question. A “yes” answer would mean that the askee is unable/unwilling to drive the asker.
Native English speakers frequently answer negative questions with positive answers. Likewise, many are used to receiving positive responses to negative questions. Because of this, you should always follow your “no/yes” answer with a clarification or explanation of your answer:
No, I don’t mind driving you. Yes, I do mind. I have an appointment at that same time.
In most cases, you can omit the “no/yes” entirely and simply answer with your clarification/explanation. This is generally a good idea when your answer would be “yes,” as some native speakers consider the type of response outlined above to be rude:
I don’t mind driving you. Sorry, I have an appointment at that same time.
You can also answer the question as though it were a positive question, just as many native speakers do. If you choose to answer in this manner, a clarification is absolutely essential to avoid a potential misunderstanding:
Sure! I’d be glad to drive you. Sorry, I can’t drive you. I have a doctor’s appointment.
Because negative questions can create the potential for misunderstandings, it is usually better to avoid using them. Here are some common negative questions and positive versions of those same questions:
Negative: Aren’t you going to English class tomorrow? Positive: Are you going to English class tomorrow?
Negative: Doesn’t she know that the report is due today? Positive: Does she know that the report is due today?
Negative: Do you mind if I drink the last can of Coke? Positive: May I drink the last can of Coke?
Negative: Didn’t you wash the dishes yesterday? Positive: Did you wash the dishes yesterday?
[service-plug]
Posted in Blog |
11785 | https://brainly.com/question/46119775 | [FREE] Let z be a complex number such that |z| = 1 . Find the largest possible value of |z^2 z - 1| . - brainly.com
Search
Learning Mode
Cancel
Log in / Join for free
Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions
Log in
Join for free
Tutoring Session
+88,3k
Smart guidance, rooted in what you’re studying
Get Guidance
Test Prep
+41,2k
Ace exams faster, with practice that adapts to you
Practice
Worksheets
+5,7k
Guided help for every grade, topic or textbook
Complete
See more
/
Mathematics
Textbook & Expert-Verified
Textbook & Expert-Verified
Let z be a complex number such that ∣z∣=1. Find the largest possible value of ∣z 2 z−1∣.
1
See answer Explain with Learning Companion
NEW
Asked by BIKRAMlawati3589 • 01/16/2024
0:04
/
0:15
Read More
Community
by Students
Brainly
by Experts
ChatGPT
by OpenAI
Gemini
Google AI
Community Answer
This answer helped 7173975 people
7M
0.0
0
Upload your school material for a more relevant answer
To maximize |z²z - 1| when |z| = 1, we note that the modulus of z²z is 1. The largest value occurs when z²z and -1 are opposites on the unit circle, yielding a maximum value of 2.
Explanation
To find the largest possible value of |z²z - 1| for a complex number z such that |z| = 1, we can use the following concepts from complex numbers and their properties.
First, since |z| = 1, z can be represented as eiϸ where ϸ is the argument of z. When we square z, we get e2iϸ which effectively doubles the argument. The modulus, however, remains 1 since multiplying a complex number by itself does not change its magnitude if the magnitude is 1. Therefore, |z²| is also 1.
Now, we examine the expression |z²z - 1|. Given that |z²| = 1 and |z| = 1, the modulus of z²z will also be 1 (since the modulus of a product is the product of the moduli). Hence, we are looking for the largest possible value of the modulus of a complex number of modulus 1 minus 1, which can be visualized in the complex plane as the furthest distance from the point -1. This happens when z²z is the point on the unit circle directly opposite to -1; since -1 has an argument of π, z²z should have an argument of 0 or 2π to maximize the distance. The largest possible value is then equal to the distance between the points -1 and 1 on the real line, which is 2.
Answered by sabareesanse47 •28.1K answers•7.2M people helped
Thanks 0
0.0
(0 votes)
Textbook &Expert-Verified⬈(opens in a new tab)
This answer helped 7173975 people
7M
0.0
0
Simple nature - Benjamin Crowell
Introductory Quantum Mechanics - Richard Fitzpatrick
Conceptual Physics - Benjamin Crowell
Upload your school material for a more relevant answer
The largest possible value of ∣z 2 z−1∣ for a complex number z such that ∣z∣=1 is 2. This occurs when the angle associated with the complex number leads the term z 2 z to be positioned directly opposite to −1 on the unit circle. Thus, maximizing the distance from −1 gives us this value.
Explanation
To find the largest possible value of ∣z 2 z−1∣ where ∣z∣=1, we start by clarifying what this condition implies. A complex number z on the unit circle can be expressed in the form z=e i θ, for some angle θ. Thus, we have z 2=e 2 i θ and z 2 z=e 3 i θ.
Next, we need to calculate ∣z 2 z−1∣=∣e 3 i θ−1∣. The magnitude of a complex number a+bi is given by a 2+b 2. Here, we can rewrite e 3 i θ as cos(3 θ)+i sin(3 θ), so:
∣e 3 i θ−1∣=∣cos(3 θ)+i sin(3 θ)−1∣=∣(cos(3 θ)−1)+i sin(3 θ)∣.
Thus, the magnitude becomes:
∣e 3 i θ−1∣=(cos(3 θ)−1)2+sin 2(3 θ).
Using the Pythagorean identity, we can simplify this further. Notice that:
sin 2(3 θ)=1−cos 2(3 θ),
and attentively substituting gives:
∣e 3 i θ−1∣=(cos(3 θ)−1)2+(1−cos 2(3 θ))=2−2 cos(3 θ).
By applying the identity for cosine, we get:
∣e 3 i θ−1∣=2(1−cos(3 θ))=2 sin(2 3 θ).
To maximize this expression, we realize that 2 sin(2 3 θ) achieves its maximum value of 2 when sin(2 3 θ)=1, which occurs when 2 3 θ=2 π+2 kπ for any integer k. This results in:
θ=3 π+3 4 kπ.
Consequently, the largest possible value of ∣z 2 z−1∣ is 2.
Examples & Evidence
For example, letting z=e i 3 π, we find ∣(e iπ−1)∣=∣(−1−1)∣=2, which demonstrates the maximum value attained under our conditions.
This conclusion is supported by the geometric interpretation of complex numbers on the unit circle and the properties of their magnitudes.
Thanks 0
0.0
(0 votes)
Advertisement
BIKRAMlawati3589 has a question! Can you help?
Add your answer See Expert-Verified Answer
### Free Mathematics solutions and answers
Community Answer Let z be a complex number such that |z| = 2, then maximum possible value of IZ+Z/2I is A)1 B)2 C)3 D)3
Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer
Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)?
Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD
Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four?
Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish?
Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit?
Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25
Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points.
New questions in Mathematics
The data shows poll ratings of a politician over time. | month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | :---: :---: :---: :---: :---: :---: | rating | 12 | 18 | 25 | 32 | 41 | 37 | 30 | 27 | 35 | 42 | 49 | 56 | a. Is the data Linear, Quadratic, or Cubic? b. Write the equation of the regression curve. c. Use your equation to find the average rate of change from May to October.
Write the fraction 40 4 in simplest form.
How many roots, real or complex, does the polynomial 7+5 x 4−3 x 2 have in all?
Subtract. $\begin{array}{r} -21-43=\square \ 1-(-4)=\square \end{array}$
The table represents a function. | x | f(x) | :--- | | -6 | 8 | | 7 | 3 | | 4 | -5 | | 3 | -2 | | -5 | 12 | Which value is an output of the function? A. 7 B. -6 C. -2 D. 4
Previous questionNext question
Learn
Practice
Test
Open in Learning Companion
Company
Copyright Policy
Privacy Policy
Cookie Preferences
Insights: The Brainly Blog
Advertise with us
Careers
Homework Questions & Answers
Help
Terms of Use
Help Center
Safety Center
Responsible Disclosure Agreement
Connect with us
(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)
Brainly.com
Dismiss
Materials from your teacher, like lecture notes or study guides,
help Brainly adjust this answer to fit your needs.
Dismiss |
11786 | https://www.cityofwarren.org/wp-content/uploads/2025/07/DDA-April-2-2025-Signed-Packet-1.pdf | he pv 4 Ee -YVARREN DDA OFFICERS Mayor Lori M. Stone Richard Fox, City Controller, Treasurer DDA MEMBERS Joseph Vicari, Vice Chair A REGULAR IN PERSON MEETING Gregory Jackson Hank Riberas OF THE Michael Wiegand DOWNTOWN DEVELOPMENT AUTHORITY Oscar Zamora Wednesday, April 2, 2025, 3:00 p.m. Tom Petzold Gina Cavaliere AGENDA 1. CALL TO ORDER 2, ROLL CALL >» Motion to excuse absent members 3: ADOPTION OF AGENDA >» Motion to adopt agenda 4, AUDIENCE PARTICIPATION 5. APPROVAL OF MINUTES (pg. 1) » Motion to approve the March 5, 2025 DDA meeting minutes 6. OLD BUSINESS (pg. 9) A. Request for Funding: Annual Membership Dues — Macomb County Chamber of Commerce (Tom Bommarito/Tiffany Nawrocki [pg. 9]) > Motion to approve funding for annual membership dues to Macomb county Chamber of Commerce in the amount of $1,285.00 7. NEW BUSINESS (pg. 11) A. Request for Funding: Furnish and Install Plants and Flowers — Jos. Kutchey & Sons, LLC (Anthony Casasanta [pg. 11]) >» Motion to approve funding to furnish and install plants and flowers by Jos. Kutchey & Sons, LLC with the amount not to exceed $18,237.00 B. Request for Funding: Portable Sound System — Pro Audio & Lighting (Anthony Casasanta [pg. 28]) » Motion to approve funding for portable sound system by Pro Audio & Lighting in the amount of $4,999.00.
11. C. Request for Funding: HVAC Maintenance Agreement for BRAC Building (8777 Common Rd) — Johnson Controls (Steve Campbell [pg. 30]) >» Motion to approve funding for HVAC Maintenance Agreement for BRAC Building (8777 Common Rd) by Johnson Controls in the amount of $8,938.00. D. Request for Funding: Fire Station #1 Utility Pole & Guy Wire Relocation — DTE (Commissioner McAdams [pg. 60]) >» Motion to approve funding for Fire Station #1 utility pole & guy wire relocation by DTE Energy in the amount of $10,574.13. E. Request for Funding: Fire Station 5 Fiber Optic Installation — Comcast Cable (Commissioner McAdams [pg. 69]) » Motion to approve funding for Fire Station 5 fiber optic installation by Comcast Cable in the amount of $15,699.03. F. Request for Funding: Emergency Vehicles - EVP (Commissioner McAdams/Mark Knapp [pg. 71]) » Motion to approve funding for (7) 2025 Road Rescue Ambulances with the amount not to exceed $2,034,642.20 » Motion to approve issuance of payment or payments for the ambulances as received up to the amount authorized of $2,034,642.20 G. Request for Funding: Striping of the Warren Parking Garage — PK Contracting (Tina Gapshes [pg. 74]) >» Motion to approve funding for striping of the City Hall parking garage by PK Contracting with the amount not to exceed $45,750.00 APPROVAL OF THE LIST OF BILLS (pg. 76) » Motion to Approve the April 2, 2025 List of Bills GOOD OF THE ORDER Next DDA Regular Meeting is scheduled for Wednesday, May 7, 2025, at 3:00 PM. ADJOURNMENT > Motion to Adjourn DDA, TIFA, CED, CDBG and Brownfield CITY OF WARREN DOWNTOWN DEVELOPMENT AUTHORITY MINUTES OF THE BOARD Meeting held on March 5, 2025 A regular meeting of the City of Warren Downtown Development Authority was called for 3:00 pm on Wednesday, March 5, 2025. Present: Mayor Lori M. Stone Oscar Zamora Hank Riberas Michael Wiegand Tom Petzold Gina Cavaliere (arrived at 3:05 PM) Absent: Joseph Vicari Gregory Jackson 1. Call to Order Mayor Lori Stone called the meeting to order at 3:02 pm. 2. Roll Call MOTION: A motion was made by Mr. Petzold, supported by Mr. Wiegand, to excuse the absent members. No opposition, the motion passed 3. Adoption of Agenda MOTION: A motion was made by Mr. Petzold, supported by Mr. Riberas, to adopt the March 5, 2025 agenda. DDA Meeting Minutes — March 5, 2025 No opposition, the motion passed 4. Audience Participation None 5. Approval of Minutes (February 5, 2025) (pg. 1) MOTION: A motion was made by Mr. Riberas, supported by Mr. Zamora, to approve the minutes for the February 5, 2025 DDA meeting with the addition of Gina Cavaliere to the members present section. No opposition, the motion passed 6. Old Business (pg. 15) A. Request for Additional Funding: Parking Garage and Clock Tower Joint Sealing — RAM Construction Services (Tina Gapshes [pg. 15]) Tina Gapshes presented a request for additional funding due to unforeseen conditions in the parking garage and clock tower sealant replacement project. Zach Ellis, with RAM Construction Services, explained challenges with precast joint sealing, requiring more labor-intensive methods due to widened joints and existing conditions, He informed the board how the sealant process involves using hand tools instead of typical machinery due to joint conditions and addressed weather conditions affecting the timeline; estimated completion in about a month and half. MOTION: A motion was made by Mr. Zamora, supported by Mr. Riberas, to approve = of additional funding for parking garage and clock tower joint sealing by . RAM Construction Services in the amount of $101,135.00 ROLL CALL: The motion carried unanimously as follows: Mr. Riberas Yes Mr. Zamora Yes : Mr. Wiegand Yes Mr. Petzold Yes Ms. Cavaliere Yes Mayor Stone Yes : DDA Meeting Minutes — March 5, 2025 No opposition, the motion passed B. Request for Funding: Warren Community Center Pedestrian Bridge/Park TAP Grant Proposal — NFE (Tina Gapshes [pg. 20]) Tina Gapshes discussed the proposal to apply for a TAP grant for a pedestrian bridge. Two cost estimates were discussed: $1.5 million for a 10-foot bridge with city funds, $1.3 million for a 14-foot bridge with TAP grant funds. It’s noted that the proposal includes a pre-application phase ($4,500) and an application phase ($8,100). If pre-application fails, the process stops at $4,500. NFE has extensive experience with TAP grant applications, making them highly qualified for this project. The grant application will be submitted to MDOT and SEMCOG, meaning it is state-funded. TAP grants are accepted three times per year, with upcoming deadlines in June and October, and the estimated project timeline is approximately 18 months from approval. If the grant is denied, reapplication may be possible depending on the reason for rejection, with engineer Carol Thurber available to provide guidance. The project is not directly tied to the Iron Belle Trail but could be incorporated if necessary. Regardless of funding, a 14-foot-wide bridge is the preferred option, as it allows for safer two-way bicycle traffic and overall improved usability, making it a more effective long-term investment. MOTION: A motion was made by Mr. Wiegand, supported by Ms. Cavaliere, to hire NFE to create TAP applications with the amount not to exceed $12,600.00 ROLL CALL: The motion carried unanimously as follows: Mr. Wiegand Yes Ms. Cavaliere Yes Mr. Riberas Yes Mr. Zamora Yes Mr. Petzold Yes Mayor Stone Yes No opposition, the motion passed C. Request for Additional Funding/Change Order to PO 2528916 — J Eppink Partners Inc. (Tom Bommarito/Tiffany Nawrocki [pg. 23]) Tiffany Nawrocki presented a request to the board for additional funding to cover a change order reimbursing planners for supplies used at the Town Center Community Planning Sessions. The supplies, including easels, stickers, and posters, were essential in engaging the public and encouraging participation in discussions about the Town Center project. DDA Meeting Minutes — March 5, 2025 MOTION: A motion was made by Ms. Cavaliere, supported by Mr. Riberas, to approve additional funding for PO 2528916 in the amount of $136.74. ROLL CALL: . The motion carried unanimously as follows: Ms. Cavaliere Yes Mr. Riberas Yes Mr. Wiegand Yes Mr. Zamora Yes Mr. Petzold Yes Mayor Stone Yes No opposition, the motion passed 7. New Business (pg. 26) A. Request for Funding: Community Center Wi-Fi Project — Presidio Networked Solutions (Anthony Casasanta [pg. 26]) Anthony Casasanta presented a request for funding to support the Community Center Wi-Fi Project. The project aims to address the facility’s ongoing connectivity issues, which have posed challenges for both staff and visitors. The Parks and Recreation Department, City Council, and patrons will all benefit from improved Wi-Fi access, enhancing communication and usability within the building. The current infrastructure, with cinder block walls, has made it difficult to maintain a reliable signal. Community members have expressed the need for better connectivity, particularly for those attending City Council meetings, working out, or watching their children’s swim lessons. The proposed system includes strategically placed antennas throughout the building, ensuring Wi-Fi coverage in every room and location. Additionally, the upgraded network is expected to extend outside the facility, though the exact range is yet to be determined. There is hope that the signal will reach the two outdoor pavilions, further expanding public access. With no further questions or discussion, the request was put forward for consideration. MOTION: DDA Meeting Minutes — March 5, 2025 A motion was made by Mr. Zamora, supported by Mr. Petzold, to approve funding for Warren Community Center Wi-Fi Project by Presidio Networked Solutions with the amount not to exceed $164,738.11 ROLL CALL: The motion carried unanimously as follows: Mr. Zamora Yes Mr. Petzold Yes Mr, Riberas Yes Mr. Wiegand Yes Ms. Cavaliere Yes Mayor Stone Yes No opposition, the motion passed B. Procedure for Approval of Payment in Advance of List-of-Bills (Mark Knapp [pg. 42]) The procedure for payment on advance list bills was discussed, with a focus on formalizing a consistent process involving all relevant departments. Mark Knapp acknowledged past practices but emphasized the need for a clear procedure moving forward. Efforts have been made to refine the language to ensure alignment with board approval and bylaw incorporation. Additionally, the discussion included the establishment of a bylaw review committee. This initiative comes after the City Council rescinded bylaw revisions at a previous meeting(s) due to noncompliance with current regulations. The last comprehensive review of the bylaws took place 24 years ago, highlighting the need for an update. Mark Knapp presented a proposal for the updated procedure, which he believes aligns with best practices. Furthermore, the board was asked to appoint a member to participate in the review process, and Gina Cavaliere volunteered, given her prior involvement in drafting the original bylaws. With no further questions or discussion, the board considered the adoption of anew payment approval procedure for implementation. MOTION: A motion was made by Mr. Riberas, supported by Mr. Petzold, to adopt proposed procedure for approval of payment in advance of the List of Bills. DDA Meeting Minutes - March 5, 2025 ROLL CALL: The motion carried unanimously as follows: Mr. Riberas Yes Mr. Petzold Yes Mr. Wiegand Yes Mr. Zamora Yes Ms. Cavaliere Yes Mayor Stone Yes No opposition, the motion passed C. Access to DDA Brokerage Account with Fifth Third Bank (Mark Knapp [pg. 52]) Mark provided an update on managing the authority’s brokerage account, noting that with previous account managers now retired, he has been handling these tasks. To improve efficiency, he requested online access for reconciliation purposes, primarily to retrieve statements and monitor transactions. While this is not a checking account, the ability to manage transactions and provider payments online would streamline operations. It was requested that the DDA approve granting Rick Fox, City Controller, and Mark Knapp, Assistant City Controller, access to account information for the authority's brokerage account. MOTION: A motion was made by Mr. Riberas, supported by Mr. Petzold, to approve granting Richard Fox, City Controller, and Mark Knapp, Assistant City Controller, access to account information in the Authority’s Brokerage Account with Fifth Third Securities (F3C-701818). ROLL CALL: The motion carried unanimously as follows: Mr. Riberas Yes Mr. Petzold Yes Mr. Wiegand Yes Mr. Zamora Yes Ms. Cavaliere Yes Mayor Stone Yes No opposition, the motion passed DDA Meeting Minutes ~ March 5, 2025 8. Approval of the List of Bills (pg. 54) MOTION: A motion was made by Ms. Petzold, supported by Mr. Riberas, to approve the March 5, 2025 list of bills. ROLL CALL: The motion carried unanimously as follows: Mr. Petzold Yes Mr. Riberas Yes Mr. Wiegand Yes Mr. Zamora Yes Ms. Cavaliere Yes Mayor Stone Yes No opposition, the motion passed Good of the Order Mr. Bommarito stated Public outreach meetings for the Towne Center project have been ongoing, and over 400 survey responses have been received so far. Attendees are encouraged to take the survey, and share it with others to maximize participation. Several upcoming community events and updates were announced: 1, Empowering Civic Engagement — Water Department Focus (March 12"): The next session will focus on the Water Department, offering insights into drinking water quality, ongoing initiatives, and efforts to remediate copper and lead piping. This session aims to educate and engage residents in understanding local water systems. Registration is still open, and community members are encouraged to participate. 2. Mayor’s Community Conversation (March 28th): The next Mayor’s Community Conversation will take place on Thursday, March 28, 2025, at Civic Center South. This event provides residents with an opportunity to engage with the Mayor, discuss community concerns, and stay informed on city updates. DDA Meeting Minutes — March 5, 2025 3. Pitch Perfect (March 13th): In partnership with Velocity, the Pitch Perfect event will offer valuable insights into pitch deck development, stage presence, storytelling, and audience engagement. This session is designed to help individuals refine their presentation skills and deliver compelling pitches. The event is scheduled for Thursday, March 13, 2025, from 11:00 AM to 3:00 PM at the Burnette Branch Library. 10. Next DDA Regular Meeting is scheduled for Wednesday, April 2, 2025, at 3:00 PM, in the City Hall 1st Floor Conference Room. 11. Adjournment MOTION: A motion was made by Mr. Riberas, supported by Mr. Zamora to adjourn. The March 5, 2025 DDA meeting adjourned at 3:27 pm. z 4 Thomas Bommarito Lori M. Stone DDA Director Mayor DDA Meeting Minutes — March 5, 2025 Docusign Envelope ID: 5C6C1F2F-7D54-464C-BDD9-08CEE44E2148 3/21/2025 DDA City of Warren One City Square Warren, MI. 48093 Re: Funding Request for Macomb County Chamber Membership Dues Dear DDA Board, The DDA/TIFA Department is requesting the Downtown Development Authority to authorize funding is requesting funding for the annual membership dues to the Macomb County Chamber of Commerce. The Macomb County Chamber is the leading voice of business in Macomb County and provides a valuable network for economic growth, collaboration, and visibility for local municipalities and businesses. Membership benefits include access to networking groups, government and community forums, business marketing opportunities, and increased credibility and exposure within the community. These services align with the goals of the DDA to foster community and business development, support economic vitality, and enhance public-private partnerships. Project Details: Recommended Company: Macomb County Chamber of Commerce, 28 First Street, Suite B, Mount Clemens, MI 48043 Bid Number: N/A Requested Amount: $1,285.00 GL/Account Number: 494-9494-95800 The DDA/TIFA Department is asking for the Downtown Development Authority to provide funding for our annual membership dues. Please find all supporting documents attached to this cover letter. If you have any questions or require any additional information, please contact me. Thank you for your consideration, Sincerely, Read and Concur: Read and Concur: Signed by: DocuSigned by: Signed by: Tom Bommarito Mark beware (rain Trepp. 6F52A3F825A947D... 874BO9CBFD6E4E8... C9046D671F6C4BB... Tom Bommarito Mark Knapp Craig Treppa DDA Director Assistant Controller Purchasing Agent Docusign Envelope ID: 5C6C1F2F-7D54-464C-BDD9-08CEE44E2148 a Invoice a C O mm Invoice Date: 2/24/2025 2: > Invoice Number: 22258 SS ay am ¥ ECOUNTY a z CHAMBER 28 First Street, Suite B Mount Clemens, MI 48043 (586) 493-7600 City of Warren One City Square Ste. 215 Warren, MI 48093 Terms Due Date Net 30 3/26/2025 Description Quantity Rate Amount Membership Dues 1 $1,285.00 $1,285.00 Subtotal: $1,285.00 Tax: $0.00 Total: $1,285.00 Payment/Credit Applied: $0.00 Balance: $1,285.00 Thank you for your support of the Macomb County Chamber of Commerce Please return this portion with your payment, Member Name: City of Warren Payment Amount: $ Payment Method: L]check # CIcreait Card Make all checks payable to Macomb County Chamber of Commerce or enter credit card information below. Enter Credit Card Billing Address (inc. zip code) Address City/State/Zip Credit Card #: Exp. Date: CVV Code (3 digits on back of card) Name on Card: Signature: Invoice #: 22258 10 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 on PARKS AND RECREATION 5460 Arden Warren, MI 48092 (586) 268-8400 March 14, 2025 www.cityofwarren.or Tom Bommarito City of Warren One City Square Suite 215 Warren, MI. 48093 Re: ITB-W-1466 - Furnish and Install Plants and Flowers. Dear Mr. Bommarito, The Parks & Recreation Department is requesting the Downtown Development Authority to authorize funding to Furnish and Install Plants and Flowers. Jos. Kutchey & Sons, LLC. is the sole bidder of ITB- W-1466. A total cost of the project is $18,237.00. Please find the all supporting documents and backup information related to the project attached to this cover letter. Project Details: Recommended Company: Jos. Kutchey & Sons, LLC. 17110 26 Mile Rd, Macomb, MI 48042 Bid/Contract # ITB-W-1466 Requested Amount: $18,237.00 GL/Account Number: 494-9494-96123 If you have any questions or require any additional information, please contact me. Sincerely, Read and Concur: Signed by: DocuSigned by: Signed by: [ Autny Casasaunta. Mark bwage Crain Treppa DE63B7B76774B1... 874B09CBFDBEAES... ~C9048D871F6CABB... Anthony Casasanta, Mark k app Craig Teppa Director Parks & Recreation Assistant Controller Purchasing Agent Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 CORRECTED BID SUMMARY City of Warren BID: ITB-W-1466 One City Square Bid Opening Date: 3/12/2025 Warren MI 48093 Department: PARKS & REC Product of Service: FURNISH & INSTALL PLANTS & FLOWERS BIDDER GRAND TOTAL JOS. KUTCHEY & SONS, LLC. $ 18,237.00 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 JOS. KUTCHEY & SONS, LLC. CORRECTED BID DETAIL FURNISH AND INSTALL PLANTS AND FLOWERS ITEM DESCRIPTION U/M QTY UNIT PRICE EXTENDED PRICE ] Plants & Flowers for Stationary Containers Container 16 $ 362.00 | $ 5,430.00 2 JIFLOR Lamppost Containers Container 23 $ 110.00 | $ 2,530.00 3 Plants & Flowers for Hanging Plant Containers Container 100 $ 70.00 | $ 7,000.00 4 Plant & Flowers - Wrap Around Pot Each 55 $ 55.00} $ 3,025.00 5 Fertilizer 19-2-19 (20 pounds) Each 2 $ 51.00 | $ 102.00 6 Delivery Fee $ 150.00 GRANDTOTAL: |$ 18,237.00 JOS, Kutchey & Sons, LLC. bid an incorrect extended total for Item 4 in the amount of $6,050.00. The corrected extended total for Item 4, in the amount of $3,025.00 is shown above. Page 2 of 2 13 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 Lori M. Stone, Mayor /ARREN Purchasing Division Phone (586) 574-4639 Office of the Controller FAX (586) 574-4614 One City Square, 4" Floor, Suite 425 Warren, MI 48093-5289 ITB-W-1466 INVITATION TO BID FEBRUARY 24, 2025 ELECTRONIC BIDS FOR FURNISHING PLANTS AND FLOWERS (INCLUDING PLANTING FEE) FOR THE CITY OF WARREN ARE BEING ACCEPTED. ELECTRONIC BIDS MUST BE ENTERED INTO THE BIDNET (MITN) PROCUREMENT SYSTEM ON, OR PLEASE SEE SPECIAL INSTRUCTIONS ON PAGE TWO (2) OF THE BID FORM A PUBLIC BID OPENING WILL TAKE PLACE AT 1:00 PM EST ON MARCH 12, 2025 VIA THE “ZOOM APP” WHICH WILL BE HOSTED BY THE CITY COUNCIL OFFICE, PLEASE JOIN THE ZOOM MEETING BETWEEN 12:50 AND 1:00 PM EST, ON THE DAY OF THE BID OPENING IN ORDER TO VIEW OR LISTEN TO THE BID OPENING VIA THE ZOOM APP. LINK TO ZOOM MEETING: owd=hE7kaabXXHEgfsbxdcS5GFGHbO2URaR. | ZOOM CALL-IN #: 1 312 626 6799 ZOOM MEETING ID: 884 6977 9894 ZOOM MEETING PASSCODE: 209929 Addenda, clarifications and changes to the bid documents must be obtained on line by registering (free registration available) for the MITN system as_ follows: 1) go to www.BidNetDirect.com/MITN, 2) Click on “Register Now", 3) Activate your account & select your registration option. Call 800-835-4603 and press option two (2) to speak live with customer support. Please use the following NIGP commodity codes: 59540 - Nursery, Greenhouse and Floral Supplies Additional information regarding this bid or any questions can be answered by contacting the Assistant Buyer, Shanah Turner of the City of Warren Purchasing Division preferably by e-mail, sturner@cityofwarren.org., Subject: ITB-W-1466. Sincerely, SEL? Shanah Turner Assistant Buyer ITB-W-1466.docx Last printed 2/24/2025 1:46:00 PM 14 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 2 OF 14 SPECIAL INSTRUCTIONS FOR ELECTRONIC BID SUBMISSION: 4, 9. The bid opening shall be made available to the public via a "Zoom Meeting". Interested parties The City will require the vendor to perform the following, via the BidNet (MITN) system, within the bid solicitation, in order to have the bid considered for award: 1. The bidder shall complete the bid form in its entirety, sign Page 3 and all other appropriate areas (clauses, etc.), scan fhe document, and upload the completed document (pages 1-14) with your electronic bid submission. . If the Vendor has any questions regarding the steps needed to complete the electronic bid submission, they shall contact the BidNet (MITN) help desk at 1-800-835-4603. Select Option 2 when prompted. . Electronic bids must be entered no later than the bid due date and time. The BidNet (MITN) system will prohibit vendors from entering bid information after the scheduled due date/time. The City shall not have access to bid results until after the scheduled due date/time. will need to access the Zoom app and enter the Meeting ID # and the Password in order to obtain access to the public bid opening. Interested members of the public may view or listen to the results at that time. . The Zoom Meeting Call-in #, ID# and password for this bid opening can be found on the first page of this bid document. . The link to view the bid opening via “Zoom Meeting” can be found on the first page of this bid document. . The public shall have the ability to join the Zoom Meeting any time after 12:50 pm EST on the bid due date. ?. The public bid opening via "Zoom" shall occur at 1:00 pm EST on the bid due date. COMPANY NAME: 05. Kutehey + Sos LLC 15 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 3 OF 14 BID PROPOSAL FORM: The undersigned proposes to FURNISH PLANTS AND FLOWERS, INCLUDING PLANTING FEE, to the City of Warren, in accordance with the atfached specification requirements to be considered an integral part of this proposal at the following prices (SEE PAGE FIVE FOR SPECIFICATION): - GRAND TOTAL (From Page 4 of the bid form):§ 21,262. IT IS MANDATORY OF ALL BIDDERS TO RETURN THIS SHEET FULLY COMPLETED AND SIGNED BY AN AUTHORIZED REPRESENTATIVE WITH THEIR ELECTRONIC BID. FAILURE TO PROPERLY SIGN IN THE AREA PROVIDED BELOW MAY RESULT IN YOUR BID NOT BEING ACCEPTED. The undersigned has carefully checked the bid figures and understands that he shall be responsible for any error of omission in this bid offer and js in receipt of all addenda as issued. Signature Chupke _ DATE. 2-28-2025 Authorized Comp&ny Representative Joseph Kuctchey COMPANY NAMES. Kutch ey + Gons LLC (Print name of Signafure} IMC 26 Mile Rd, Maconb MI ASDY 2 ADDRESS CITY STATE ZIP CODE 5Ste-855-#8T)O _ Skutthe (@adomcast, net TELEPHONE NUMBER FAX NUMBER E-MAIL ADDRESS WARRANTY (If Applicable) ACKNOWLEDGMENT OF ADDENDA: In the event that any addenda are posted on the MITN system, acknowledge receipt of the addenda by indicating the addenda number below and sign in the space provided. A copy of each addenda posted should be included in your proposal: Addenda #1: Signed: Addenda #2: Signed: Addenda #3: Signed: EXCEPTIONS: Where an exception to any specification is taken, such exception shall be clearly indicated below and along with the specifications attached hereto, shall be considered an integral part of this bid proposal. oT COMPANY NAME: jos Kwichecr Sons LLC 16 Docusign Envelope ID: B1CD9903-F324-4030-B1 3D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 4 OF 14 PRICING DETAIL: ITEM | DESCRIPTION U/M QTY UNIT COST TOTAL COST pranis & Flowers for Stationary Container} 15 | $ Ail? oo |¢ 5HBO,2° 2 | JIFLOR Lamppost Containers Container | 23 | $|{O,C° $72.530.°° 3 pran's & Flowers for Hanging Plant Container | 100 | Foe? $ 10 Oo. oO 4 | Plant & Flowers — Wrap Around Pot Each 55 | $ 65° ° $ (50 SO oS 5 | Fertilizer 19-2-19 (20 pounds) Each 2 |$ 5],e° $ 1o2z.cC>° 6 | Delivery Fee $ 150°° GRAND TOTAL (Enter on page 3): | $ Z| 262 O° ESTIMATED QUANTITIES: Quaniities indicated are estimates for bid award purposes and are estimated on an annual basis. The City has provided the best estimate of quantities and actual usage may increase or decrease. Vendor shall supply items as per bid and hold the City harmless for increase or decreases in quantities. Authorized Signature forty \r g LM, 4 Estimated Quantities Clause 4 PRICE CLAUSE: The City of Warren requests that prices be held firm for 60 days or bid award, whichever comes first except for the successful ak whose prices shall be held firm for the entire contract period. Authorized signature toro Ke ad, ri At \ Price Clause ry WORK LOCATION: Work shall be performed around the Warren City Hall Building located at: Warren City Hall One City Square Warren, MI 48093 COMPLETION DATE: Work shall be completed no later than May 20, 2025. aa COMPANY NAME; JO%, Kuctney ¢ Gens LLCO 17 Docusign Envelope ID: B1CD9803-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 5 OF 14 DESCRIPTION/SPECIFICATION: ITEM: Include planting fee for all containers. Vendor will provide additional soil as needed, will provide growth established plants & flowers, pre-grown in larger containers according to designed placement to provide for a more productive display when transplanted on site in all above stationary containers. All plants & flowers should include: 7.5” pol size Purple Baron Millet, Pennisetum Fountain Grass, Cordyline (two-year growthsize), Baby Tut, Dahlia Mystic, Salvia Blue Emotion, along with various bright coleus, Acapulco Cascade ivy Geraniums, Ibomea Sweet Potato Vine (Goldfinger, Marguerite, Blacky), Lantanas, Verbena Lanai, Scaevola {Suntastic Yellow, Brilliant and Diamond), Petunia Ray Rim Purple, Proven Winner Supertunia Picasso, Raspberry Blast, Vista Bubblegum, Fuchsia, Silverberry, Fame varieties, Ball Floral Phantom, Pinstripe, Dragonwing Begonia & SunSpun, 2 - 25 pound bags of Jacks Petunia special. Brand Preference: All Brands are acceptable 23/800MM JIFLOR LAMPPOST CONTAINERS Plants &flowers, including soil for 26/800mm Lamppost Containers. Vendor will plant in Jiflor culture tub liners in a greenhouse environment where they will remain until delivered to final display location. Two-year growth size Jiflor Pots- More heat and drought tolerant choices Sure Fire waxed leaf begonias, Supertunia Mini petunia-multi colors Trailing Angelonia, Trailing Mezzo. Soil requirement is Canadian Peat with pine bark including perlite. Soil should include extra polymer with an additional wetting agent along with Jack's Classic Cote slow release fertilizer. Vendor will provide onsite training session for water crew & 2 -25 pound bags of Jacks Petunia special. Brand Preference: All Brands are acceptable 100/23” DIAMETER H2O HANGING PLANT CONTAINER LINER Plants & flowers including soil for 100/23" diameter H2O hanging plant container liner inserts. Vendor will plant liners in a greenhouse environment where they will remain until delivered to final display location. Two-yeer growth size Cordyline, Acapulco Cascade lvy Geraniums, Thunbergia (Lemon & Sunny Susy), lpomea Sweet Potato Vine (Goldfinger, Marguerite, Blacky), Lobularia, Vinca Vine, Verbena Lanai, Lantana, Bidens, Scaevola (Suntastic Yellow, Brilliant & Diamond), Petunia Glow series, Proven Winner Supertunias, Whispers, Sangunas, Surfinias, Rays & Famous varieties, Soil requirement is Canadian Peat with pine bark including perlite. Soil should include extra polymer with an additional wetting agent along with Jack's Classic Cote slow release fertilizer. Vendor will provide onsite training session for water crew. 2 - 25 pound bags of Jacks Petunia special. Brand Preference: All Brands are acceptable 8 - 28" WITH PLANTS AND FLOWERS Plants &flowers, including soil for 3/28" H2) Wrap Around Pot. Vendor will plant in H2O tub liners in a greenhouse environment where they will remain until delivered to final display location. Two- year growth size Cordyline, Acapulco Cascade Ivy Geraniums, Thunbergia (Lemon & Sunny Susy), lpomead Sweet Potato Vine (Goldfinger, Marguerite, Blacky}, Lobularia, Vinca Vine, Verbena Lanai, Lantana, Bidens, Scaevola (Suntastic Yellow, Brilliant & Diamond), Petunia Glow series, Proven Winner Supertunias, Whispers, Sangunas, Surfinias, Rays and Famous varieties. Soil requirement is Canadian Peat with pine bark including perlite. Soil should include extra polymer with an additional wetting agent along with Jack's Classic Cote slow release fertilizer. Vendor will provide onsite training session for water crew & 2 - 25 pound bags of Jacks Petunia special. Brand Preference: All Brands are acceptable COMPANY NAME: JOS. Kuchey Song LLC 18 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 REFERENCES: 1. Agency: City of Livonia ITB-W-1466 CITY OF WARREN PAGE 6 OF 14 Please list the municipalities/companies for which your company has provided similar services, Year: ZO6 -focurrey Address: Contact Name: Pot 2 Dowed 2. Agency: Cita of Centerline Phone: 2ID- NS- 13g Year: 2016 te euvrent Address: Contact Name: N n clk Phone: Be - 15'1- 6300 Year: 2Ol - corrent 3. agency: City of Fe rndete Address: Contact Name: Derk. Kael Phone: Qu8-SHL -2525 X26 4. Agency: Ge ot Se rosse p> nk LXeats Year: 202\ +0 cocre nt Address: Contact Name: di Mm Kowolski Phone: 2!B- 263-1257 COMPANY NAME: \J 06. Kudthey + Sons LLC 19 Docusign Envelope ID: B1CD9903-F324-4030-B 13D-7C6473F4E917 ITB-W-1464 CITY OF WARREN PAGE 7 OF 14 INSURANCE REQUIREMENTS: INSURANCE CERTIFICATE MUST BE SUBMITTED BY THE INSURANCE AGENCY, NOT THE PROPOSER. The awarded vendor shall be required to provide the City with certificates of insurance naming the City of Warren, City of Warren Municipal Building Authority, Downtown Development Authority, Tax Increment Finance Authority, the 37th District Court, all elected appointed officials, employees and volunteers as individuals acting within the scope of their authority, AS AN ADDITIONAL INSURED. It is Understood and agreed by naming the City of Warren as additional insured, coverage afforded is considered to be primary, and any other insurance the City of Warren may have in effect shall be considered secondary and/or excess. Additionally, the awarded vendor shall provide language, within the "Description of Operations” section, that “AGREES TO WAIVE THEIR INSURER'S RIGHT OF SUBROGATION UNDER ITS POLICIES”. The awarded vendor shall provide the following coverage and limits (Sub-contractors Utilized by the awarded bidder shall be subject to these same conditions). COMMERCIAL GENERAL LIABILITY: The following coverage is part of the General Liability policy: Policy should be on an OCCURRENCE BASIS WITH COMBINED SINGLE LIMITS. General Aggregate $2,000,000 Products/Completed Operations Aggregate $1,000,000 Personal & Advertising Injury $1,000,000 Each Occurrence $1,000,000 Fire Damage-Any one fire $ 50,000 Medical Expense-Any one person $ 5,000 Per project aggregate limit, Independent contractor's coverage, Broad form property damage Blanket contractual liability coverage AUTOMOBILE LIABILITY: Automobile liability insurance coverage shall be $1,000,000 combined single limit for any auto and include hired autos and non-owned autos. WORKERS’ COMPENSATION INSURANCE: Workers’ compensation insurance shall be statutory under tne State of Michigan Workers’ Compensation Act. Coverage shall be provided by a carrier(s) rated A- or better by A.M. Bests. Insurance certificates shall contain a provision to the effect that the insurance company SHALL NOTIFY the City at least thirty (30) days prior te CANCELLATION OR MATERIAL CHANGE of the Ingufanes: (“} Can meet insurance as indicated. { } Cannot meet but offer the following: } i Authorized signature fof px.) [ of \ ‘A Insurance Clause f /) — COMPANY wanes |05. Kuctehey 4 Sore LLC 20 Docusign Envelope ID: B1CD9903-F324-4030-B 13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 8 OF 14 GENERAL CONDITIONS (Effective January 17, 2025) SIGNATURE Bids and all information requested of the vendor shall be entered in the appropriate space on the bid form and Signature Page. Failure to do so may disqualify your offer. An authorized officer or employee of the vendor shall sign all bids. ELECTRONIC BID SUBMISSION Electronic bids shall be submitted by the date specified and at or prior to the time specified to be considered. Late bids, e-mail, sealed, telegraphic, or telephone bids will NOT be accepted. The bidder is required to submit their bid electronically via the BidNet (MITN) system in order to be considered for award. Bids received after 12:30 pm of the date they are due will not be accepted. RELATIONSHIP DISCLOSURE It is required that any relationship (business or personal) to a City employee or official be disclosed. This includes employment or other professional engagements. ALTERATION OF BID DOCUMENTS Vendor changes or alterations to the bid documents, including the specification, may result in the bid being considered non-responsive and/or the Bidder being debarred. The only authorized vendor changes to the bid documents will be in the areas provided for the Bidder's response including the “Exceptions” section of the bid and on separate attached sheets submitted by the vendor. Vendor shall clearly identify product offered and deviations from the specification. If a change or alteration to the bid document is undetected, and the bid is awarded the contract, the original terms, conditions, and specification in the authorized version of the bid document will be applicable during the terms of the contract. Bidders are responsible for ensuring they have obtained all relevant documents including amendments, clarifications, changes, drawings, etc. as made available by the City. PRICES Prices quoted shall be for new products in current production unless otherwise specified. Where refurbished or discontinued items are offered they shall be clearly identified as such. Prices quoted shall be exclusive of any rebates due the City. Any rebates the City may be entitled to should be shown as a separate line item and include expiration date. Corrections and/or modifications received after the bid closing time specified will not be accepied, Unit prices prevail. All information shall be entered in ink or typewritten. Mistakes may be crossed out and corrections inserted before submission of your bid. The person signing the bid shall initial corrections in ink. All prices will be proposed F.O.B. DESTINATION, INCLUDE ALL DELIVERY AND ANY ADDITIONAL CHARGES and remain in effect as specified in the bid. COMPANY NAME: 04, Audchey + Sms LLC 21 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1446 CITY OF WARREN PAGE 9 OF 14 AWARD Unless otherwise stated in the bid documents, the City cannot guarantee exclusivity of the contract for the proposed products or services. Award of the bids shall be based upon a combination of factors, including but not limited to, adherence to bid requirements, references and any other factors that may be in the City’s best interest, The City reserves the right to reject any and all bids, and to waive any defect or irregularity in bids. The City reserves the right to accept and separate items in the bid and to accept the bid that, in the opinion of the City, is to the best advantage and interest of the public we serve. The City also has the right to re-solicit bids if it is deemed to be in the best interest of the City. The City reserves the right to reject low bids which have major deviations from our specifications: to accept a higher bid which has only minor deviations. By signing the bid, Bidders agree to accept a split award unless the Bidder clearly indicates that it takes Exception. The bid will be awarded to that responsible, responsive firm whose bid, conforms to this solicitation and will be most advantageous to the City, with regard not only to price but also to availability of product, location and quality of product considered. The City reserves the right to award all line items, to make no award or to award on an individual line item basis, whichever is deemed to be in the best interest of the City. Time of delivery may be a consideration in the award. The City reserves the right to consider as unqualified to perform the contract any bidder who does not habitually perform with its own forces seventy-five (75%) of the work involved. TERMINATION 1. Failure to Perform. The City may terminate a bid award for the failure to perform a term of the bid specifications to the satisfaction of the City. The City shall provide ten (10) days advance written notice to the Awarded Vendor for the failure to perform services or for the violation of any other term of the bid specifications. Unless futile or the violation is recurring, the City shall provide notice and the opportunity to cure the violation prior to termination. Such notice to cure shall be given in writing by first-class mail. In the event of a dispute, or in order to avoid interruption of service, the City may engage another to perform the work and the Awarded Vendor shall be responsible for any costs the City incurs as a result of the Awarded Vendor's violation. The City may withhold payment to offset any damages the City incurs as a result of the Awarded Vendor's violation, 2. At Will. A bid award may be terminated at will by the City upon a minimum of thirty (30) days prior written notice to the Awarded Vendor. In the event of termination as provided in this subsection, the Awarded Vendor will be compensated for all services performed and approvable reimbursable expenses from the inception date to the termination date provided the services performed and the expenses were provided in accordance with the bid specifications. Payment shall be made upon the Awarded Vendor delivering to the City all information and materials retained by the Awarded Vendor, affiliates, or subcontractors in performing the services described in the bid specifications, whether completed or in progress. 3. MISREPRESENTATION. In addition, the City may reject this Bid, or cancel a contract with an Awarded Vendor, if there is evidence of any misleading or intentionally fraudulent information or documents provided in connection with this Bid. COMPANY NAME: \DS, Kudehex, aSons LLC 22 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 10 OF 14 SPECIFICATION Brand names and numbers, when used, are for reference to indicate the character or quality desired. Alternate items of equal quality will be considered, provided your offer clearly describes the product being offered. Offers for alternate items shall state the brand and number, or level or quality. When the bidder does not state brand, or level of quality, it is understood the offer is exactly as specified. The City shall have determine if alternate products are “Equal” to the product being specified. All products and services shall be in accordance with all applicable federal, state and local statutes, rules, ordinances, etc. All personnel shall have the appropriate licenses with endorsements for the work performed. In addition, any personnel driving a vehicle on City property shall have the appropriate valid driver's license and have or exceed minimum statutory insurance requirements. E-VERIFY Any bidder, attesting to his bid by signature, is affirming that the Bidder has regisiered with, participates in and utilizes the E-Verify Program (or any successor program implemented by federal Department of Homeland Security and Social Security Administration) to verify the work status of all newly hired employees employed by the Bidder. NON-IRAN LINKED BUSINESSES By signing below, Bidder certifies and agrees on behalf of Bidder and the company submitting this bid the following: (1) that the Bidder is duly authorized to legally bind the company submitting this bid; (2) that the company submitting this bid is not an “Iran linked business,” as defined in Section 2(e) of the iran Economic Sanctions Act, being Michigan Public Act No. 517 of 2012; and (3) that Bidder and the company submitting this bid will immediately comply with any further certifications or information submissions requested by the City in this regard. ASSIGNMENT OF AGREEMENT ~ OTHER CONTRACTORS. The Awarded Vendor shall not assign the contract or any part thereof without the written consent of the City. PERIOD AGREEMENTS No Exclusive Contract/Additional Services. The Awarded Vendor agrees and understands that the contract shall not be construed as an exclusive agreement and that the City may, at any time, secure similar or identical services at its sole option. Any contract executed pursuant fo this Bid, which is for a specific term shall include for an extension of the contract term, at the option of the City, as follows: The City shall have the sole option to extend the contract herein for a period of two months by written notice to the Awarded Vendor exercising the option served at least ten days prior to the expiration date of the contract, in the event such option is exercised by City, all of the provisions of the contract shall remain in full force and effect other than the date of expiration of the contract, The quantities have been estimated for bid award purposes and may be estimated based on past usage. The quantities may increase or decrease and the City makes no representation as to guarantee of usage. The quantities are estimated on an annval basis. COMPANY NAME: JOS. Kurtehexy Sons Le 23 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 11 OF 14 PAYMENT TERMS The City’s normal payment terms are 45 days in connection with cash discounts specified with this bid. Time will be computed from the date of complete delivery of services, supplies, or equipment, as specified, or from the date correct invoices are received in the Office of the City Controller, if the latter is later than the date of delivery. Prices will be considered as net if no cash discount is shown. Progress payments will be made on the basis of hours of work completed during the course of the engagement in accordance with the firm's fee bid. Interim billings shall cover a period of not less than a calendar month. MICHIGAN FREEDOM OF INFORMATION ACT (FOIA) All costs incurred in the preparation and presentation of this bid, in any way whatsoever, shall be wholly absorbed by the Bidder. All supporting documentation shall become the property of the City unless requested otherwise at the time of submission. Michigan FOIA requires the disclosure, upon request, of all public records that are not exempt from disclosure under Section 13 of the Act, which are subject to disclosure under the Act. Therefore, confidentially of information submitted in response to this REP is not assured, EQUAL OPPORTUNITY CLAUSE This contract requires adherence to the equal opportunity clause, 41 CFR § 60-1.4, EXCEPTIONS TO THE BID SOLICITATION Each individual/group shall provide a list of Exceptions taken to this bid. Any Exceptions taken shall be identified and explained in writing. An Exception is defined as the individual/group's inability to meet a mandatory requirement or exceed a requirement in the manner specified in the bid solicitation. If the Bidder provides an alternative solution when taking an Exception toa requirement, the benefits of this alternative solution shall be explained. The City reserves the right to accept or reject any Exception whichever is deemed to be in the best interest of the City. WITHDRAWAL OF BID Bidders may withdraw their bids by submitting a written request over the signature of an authorized individual to the Purchasing Department any time prior to the submission deadline. Bidders may thereafter submit a new bid prior to the deadline. Modification or withdrawal of the bid in any manner, oral or written, will not be considered if submitted after the deadline. DEFAULT TO CITY it is Understood that any Bidder who is in default to the City at the time of opening its bid shall have its bid declared null and void. BIDDER DISCLOSURE The Bidder declares that it has not, nor willit, provide gifts, gift certificates, entertainment, favors, or other gratuities to a City official, employee, agent, or volunteer, or to their families. The Bidder acknowledges that if it violates this policy then the City may terminate the contract with the Bidder. COMPANY NAME: Os, Aucchey a Sons LLC 24 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 12 OF 14 INDEMNITY CLAUSE To the extent permitted by law, the Contractor shall indemnify the City, its officers, employees, agents, and boards and commissions from and against any claim of liability; penalties: damages; attorney fees; professional advisors’ fees: settlements; or other fees or expenses arising from or in connection with goods provided and services performed under this Invitation to Bid. The Contractor shall provide the Warren City Attorney's Office with all documents filed in any proceeding related to this Invitation to Bid in which any of the above-listed people or entities are named. APPENDIX A OF TITLE VI PLAN During the performance of this contract, the contractor, for itself, its assignees, and successors, in interest (hereinafter referred to as the “contractor”) agrees, as follows: 1. COMPLIANCE WITH REGULATIONS, The contractor shall comply with Regulations relative to nondiscrimination in Federally-assisted programs of the Department of Transportation, Title 49, Code of Federal Regulations, Part 21, as they may be amended from time to time (hereinafter referred to as the Regulations), which are herein incorporated by reference and made a part of this contract. 2. NONDISCRIMINATION. The contractor, with regard to the work performed by if during the contract, shall not discriminate on the grounds of race, color, sex, or national origin in the selection, retention, and treatment of subcontractors, including procurements of materials in the discrimination prohibited by Section 21.5 of the Regulation, including employment practices when the contractor covers a program set for in Appendix B of the Regulations. 3. SOLICITATION FOR SUBCONTRACTS, INCLUDING PROCUREMENTS OF MATERIALS AND EQUIPMENT. In alll solicitations either by competitive bidding or negotiation made by the contractor for work fo be performed under a subcontract, including procurements of materials or leases of equipment, each potential subcontractor or supplier shall be notified by the contractor of the contractor's obligations under the contract and the Regulations relative to nondiscrimination on the grounds of race, color, sex, or national origin. 4. INFORMATION AND REPORTS. The contractor shall provide all information and reports required by the Regulations, or directives issues pursuant thereto, and shall permit access to its books, records, accounts, other sources of information, and its facilities as may be determined by the State Highway Department or the Federal Highway Administration to be pertinent to ascertain compliance with such Regulations or directives, Where any information required of a coniractor is in the exclusive possession of another who fails or refuses to furnish this information, the contractor shall so certify to the State Highway Department or the Federal Highway t Administration, as appropriate, and shall set forth what efforts it has made to obtain the information. 5. SANCTIONS FOR NONCOMPLIANCE. In the event the contractor's noncompliance with the nondiscrimination provisions of this contract, the State Highway Department shall impose such contract sanctions as it or the Federal Highway Administration may determine to be appropriate, including, but not limited to: a. Withholding payments to the contractor under the contract until the contractor complies : and/or b. Cancellation, termination or suspension of the contract, in whole or in part. COMPANY name: JOS, Kuchey +Sons LLC 25 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 13 OF 14 INCORPORATION OF PROVISIONS. The contractor shall include provisions of paragraphs (1) through (6) in every subcontract, including procurement of material and leases of equipment, unless exempt by the Regulations, or directives issued pursuant thereto. The contractor shall take such action with respect to any subcontract or procurement as the State Highway Department or the Federal Highway Administration may direct as a means of enforcing such provisions including sanctions for noncompliance: provided, however, that, in the event a contractor becomes involved in, or is threatened with, litigation with a subcontractor or supplier as a result of such direction, the contractor may request the State Highway Department to enter into such litigation to protect the interests of the State, and, in addition, the contractor may request the United States to enter into such litigation to protect the interests of the United States. EXTENSION OF AWARD TO THE MITN PURCHASING COOPERATIVE By signing below, all vendors awarded contracts from this Bid may, upon mutual agreement, extend pricing, terms, and conditions of this bid to local entities, which are part of the Michigan Intergovernmental Trade Network (MITN) Purchasing Cooperative. Each entity is responsible for its own payments and is fo be considered individually for billing and collection purposes. Each entity will provide their own purchase order and delivery location(s), and must be invoiced separately to the address indicated on their purchase order. If an award is made to your company, it is agreed that the pricing, terms, and conditions of this bid maybe extended to any local entity in the Michigan Intergovernmental Trade Network (MITN) Purchasing Cooperative. Ws. Kucehey + Sons LLC Joseph Kutthey [ (Company Name} (Authorized Representative) SAFETY DATA SHEETS IMPORTANT: All City purchases require SAFETY DATA SHEETS where applicable, in compliance with the Occupational Safely and Health Administration (OSHA) Hazard Communication Standard. THE ABOVE GENERAL CONDITIONS ARE CONSIDERED IN FORCE UNLESS SPECIFICALLY ADDRESSED IN ANOTHER SECTION OF THE BID DOCUMENT We have read and acknowledge the above GENERAL CONDITIONS Josh hak i ie a wanes / Gighature COMPANY NAME: Jos, Kuehne eSeoms LLC 26 Docusign Envelope ID: B1CD9903-F324-4030-B13D-7C6473F4E917 ITB-W-1466 CITY OF WARREN PAGE 14 OF 14 CHECK LIST FOR BIDDERS MISTAKES OR OMISSIONS CAN RESULT IN THE REJECTION OF YOUR BID. v Is your electronic bid PROPERLY SIGNED on the AUTHORIZED REPRESENTATIVE portion of the BID FORM, AND ALL CLAUSES AS INCLUDED IN THE BID DOCUMENT? f If required, have you entered a unit price for each bid item? (Unit Price governs) =< Have you uploaded the complete document via the BidNet (MITN) system with your electronic submission? vA Late bids will NOT be considered, Bids must be received by the Purchasing Division before 12:30 P.M., Wednesday, on the date specified. a COMPANY NAME: JOS, Kuctchevy a Sas LLC 27 Docusign Envelope ID: BECOB4E8-999D-4770-ABF1-C7F03552CEB7 PARKS AND RECREATION 5460 Arden Warren, MI 48092 (586) 268-8400 oo Tom Bommarito City of Warren One City Square Suite 215 , Warren, MI. 48093 Re: Portable Sound System Dear Mr. Bommarito, The Parks & Recreation Department is requesting the Downtown Development Authority to authorize funding for a new portable sound system to be used for the special events at City Hall and City Square. This new portable sound system will be replacing an old out dated system. This needed item are budgeted for FY 2025 DDA Capital Outlays. Project Details: Recommended Company: Pro Audio & Lighting 2400 East 14 Mile Rd. Warren, MI 48092 Bid: No Bid Needed Requested Amount: $4,999.00 GL/Account Number: 494-9494-97400 Please find the attached quote. If you have any questions or require any additional information, please contact mé. Sincerely, Read and Concur: Signed by: [td Casasanta | Mart banag Crain Treppa QDE53B7875774B1... 7 4ROSCREDCEAES... 9046067 1F6C4BB.., Anthony Casasanta, ark Knapp Craig Teppa Director Parks & Recreation Assistant Controller Purchasing Agent 28 PROPOSAL Docusign Envelope ID: BECOB4E8-999D-4770-ABF1-C7F03552CEB7 Pro Audio and Lighting PRC 2400 East 14 Mile Road AUDIO & LIGHTING ‘ae an wane PROAAL.COM (886) 682- 03/19/2025 3:15p TXNEVILL 04/02/2025 Bill To: Attn: Attn. Purchasing Department City Of Warren One City Square, Ste. 425 Warren MI 48093 United States . H(810) 574-4639 2 ETXLSPCVR PADDED COVER FOR ETX-15P, EV LOGO 89,00 12.72 165.28 1 BAG90D Speake» Stand Tote for Two Regular , 49,99 3,57 7 46.42 ; Sized Speaker S$ 2 TS80B Original Speaker Stand - Black 99,99 14,30 185,68 2 ETX15PUS ETX-L5P 2-WAY POWERED LOUDSPEAKER, 1699.00 242.93 3155.07 BLK, US L M8DX 15A Standard Power Conditioner 219,95 15.72 204.23 . W/Lights gamp; Digital Volt Met : 1 BLX24RSMS58H9 Vocal System with Rack Mount 429.00 30.67 398.33 Wireless Receiver and Handheld Tr 1 GRCLOX6 10U Top 6U Side Console Audio " 449,00 32,10 . 416,90 : Rack. 1 VLZ1402RACKKIT VLZ1402 Rackmount Kit 59,99 4,29 —-§5,70 1 204076400 VL2Z4 Series 14-Ch Compact Mixer 399,99 28,60 371,39 1 SUBTOTAL 4999,00 TOTAL 4999.00 Pg 1 of i . Acct: 80431 Ref: 66306 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 3/25/2025 DDA City of Warren One City Square Warren, MI. 48093 Re: Funding Request for HVAC Equipment Maintenance Agreement — BRAC Building Dear DDA Board, The DDA/TIFA Department is requesting the Downtown Development Authority to authorize funding for the HVAC equipment maintenance agreement for 8777 Common Rd. This project involves securing a planned service agreement with Johnson Controls to ensure proper maintenance and functionality of the building’s HVAC systems. Regular preventative maintenance will improve energy efficiency, extend equipment life, and help maintain a safe and comfortable environment for staff and visitors. Project Details: Recommended Company: Johnson Controls, 2875 High Meadow Cir, Auburn Hills, MI 48326 Bid Number: RFP-W-9103 Requested Amount: $8,938.00 (for the first year) GL/Account Number: Contractual Services: 494-9494-80100 Attached is the quote from Johnson Controls outlining the terms of the planned service agreement. The agreement is for a term of one year, is renewable annually, and paid in advance upon receipt of invoice. The DDA/TIFA Department is requesting funding approval to move forward with this important maintenance contract. Please find the supporting documentation attached for your review. If you have any questions or require additional information, please contact me. Thank you for your consideration. Sincerely, Read and Concur: Read and Concur: Slgned by: Tom Pommarito DocuSigned by: Signed by: Tom ommarito crusosooroned Crain Trego. DDA/TIFA Director Mark Knapp Craig Treppa DDA/TIFA Department Assistant Controller Purchasing Agent Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FASA35F3 Johnson Controls planned service proposal Prepared for CITY OF WARREN-BRAC Customer CITY OF WARREN-BRAC Local Johnson Controls Office 2875 HIGH MEADOW CIR AUBURN HILLS, Ml 483262773 Agreement Start Date: 04/01/2025 Proposal Date 03/08/2025 Estimate No: 1-1Q6QWBGX Partnering with you to deliver value-driven solutions Every day, we transform the environments where people live, work, learn and play. From optimizing building performance to improving safety and enhancing comfort, we are here to power your mission. A Planned Service Agreement with Johnson Controls provides you with a customized service strategy designed around the needs of your facility. Our approach features a combination of scheduled, predictive and preventative maintenance services that focus on your goals. As your building technology services partner, Johnson Controls delivers an unmatched service experience delivered by factory-trained, highly skilled technicians who optimize operations of the buildings we work with, creating productive and safe environments for the people within. By integrating our service expertise with innovative processes and technologies, our value-driven planned service solutions deliver sustainable results, minimize equipment downtime and maximize occupant comfort. With more than a century of healthy buildings expertise, Jonnson Controls leverages technologies to successfully deliver smart solutions to facilities worldwide. oh Johnson Controls was recognized by Frost & FROST SULLIVA . : re : Sullivan as the 2020 North American Company of the Year for innovation in the Smart 2020 North American Company of the Year for connected Chillers market Award | Innovation in Smart Connected Chillers 31 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 recii owe sone Planned Service Proposal Toe) ye Sait CITY OF WARREN-BRAC Executive summary Planned service proposal for CITY OF WARREN-BRAC Dave, We value and appreciate your interest in Johnson Controls as a service provider for your building systems and are pleased to provide a value-driven maintenance solution for your facility. The enclosed proposal outlines the Planned Service Agreement we have developed on your facility. Details are included in the Planned Service Agreement summary (Schedule A), but highlights are as follows: 2 In this proposal we are offering a service agreement for 1 Year starting 04/01/2025 and ending 04/30/2026. o The agreement price for first year is $8,938.00; see Schedule A, Supplemental Price and Payment Terms, for pricing in subsequent years. 2 The equipment options and number of visits being provided for each piece of equipment are described in Schedule A, Equipment list. As a manufacturer of both mechanical and controls systems, Johnson Controls has the expertise and resources to provide proper maintenance and repair services for your facility, Again, thank you for your interest in Johnson Controls and we look forward to becoming your building technology services partner. Please contact me if you have any questions. Sincerely, Michael Konczak HVAC Account AE (810)300-4809 The power behind yuur mtission johnson ‘Y) © 2025 Johnson Controls, Inc. onnson f Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 2 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 ‘tt rwdiet Planned Service Proposal vepy =r con CITY OF WARREN-BRAC Benefits of planned service A Planned Service Agreement with Johnson Controls will allow you to optimize your building's facility performance, providing dependability, sustainability and energy efficiency. You'll get a value-driven solution that fits your specific goals, delivered with the attention of a local service company backed by the resources of a global organization. With this Planned Service Agreement, Johnson Controls can help you achieve the following five objectives: 1. Identify energy savings Opportunities Since HVAC equipment accounts for a major portion of a building's energy usage, keeping your system performing at optimum levels may lead to a significant reduction in energy costs, 2. Reduce future repair costs 38 benefits 28 of a PSA Ze Routine maintenance may maximize the life of your equipment and may reduce equipment breakdowns. 3. Extend asset life Through proactive, factory-recommended maintenance, the life of your HVAC assets may be extended, maximizing the return on your investment. 4. Ensure productive environments Whether creating a comfortable place where employees can be productive or controlling a space to meet specialized needs, maintenance can help you achieve an optimal environment for the work that is being accomplished 5. Promote environmental health and safety When proper indoor conditions and plant requirements are maintained, business outcomes may be improved by minimizing sick leave, reducing accidents, minimizing greenhouse gas emissions and managing refrigerant requirements. All of the services we perform on your equipment are aligned with "The 5 Values of Planned Maintenance" and our technicians understand how the work they perform can help you accomplish your business objectives. © 2025 Johnson Controls, Inc. Customer Confidential Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Johnson ya Controls Page 3 of 26 33 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FASA35F3 Vt se ot Planned Service Proposal ets a CITY OF WARREN-BRAC Our partnership Personalized account management A Planned Service Agreement also provides you with the support of an entire team that knows your site and can closely work with you on budget planning and asset management. Your local Johnson Controls account management team can help guide planned replacement, energy retrofits and other building improvement projects. You'll have peace of mind that an entire team of skilled professionals will be looking out for what is best for your facility and budget. A culture of safety Johnson Controls technicians take safety seriously and personally, and integrate it into everything they do. All of our technicians participate in regular and thorough safety training. Because of their personal commitment, we are a leader in the HVAC service industry for workplace safety performance. This means that you do not have to worry about us when we are on your site. Commitment to customer satisfaction Throughout the term of your Planned Service Agreement, we will periodically survey you and use your feedback to continue to make improvements to our service processes and products. Our goal is to deliver the most consistent and complete service experience possible, To meet this goal, we've developed and implemented standards and procedures to ensure you receive the ultimate service experience — every time. Energy & sustainability A more sustainable world one building at a time — Johnson Controls is a company that started more than 125 years ago with a product that reduced energy use in buildings. We've been saving energy for customers ever since. Today, Johnson Controls is a global leader in creating smart environments where people live, work and play, helping to create a more comfortable, safe and sustainable world. The value of integrity Johnson Controls has a long, proud history of integrity. We do what we say we will do and stand behind our commitments. Our good reputation builds trust and loyalty. In recognition for our commitment to ethics across our global operations, we are honored to be named one of the World's Most Ethical Companies by Ethisphere Institute, a leading think tank dedicated to business ethics and corporate social responsibility. In addition, Corporate Responsibility Magazine recognizes Johnson Controls as one of the top companies in its annual "100 Best Corporate Citizens” list. AM. py | © 2026 Johnson Controls, Inc. Johnson ‘Zi Customer Confidential Controls Do nol copy (physically, electronically, or in any other media) withoul the express writlen permission of Johnson Controls, Inc. Page 4 of 26 34 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FAS9A35F3 © 2025 Johnson Controls, Inc. leap cafe aeccoina ol Planned Service Proposal rt. CITY OF WARREN-BRAC Service delivery As part of the delivery of this Planned Service Agreement, Johnson Controls will dedicate a local customer service agent responsible for having a clear understanding of the agreement scope, and your facility procedures and protocols. A high-level overview around our service delivery process is outlined below including scheduling, emergency service, on-site paperwork, communication and performing repairs outside of the agreement scope. Scheduling Preventative maintenance service will be scheduled using our automated service management system. In advance of the scheduled service visit, our technician is sent a notice of service toa smartphone. Once the technician acknowledges the request, your customer service agent will call or e-mail your on-site contact to let you know the start date and type of service scheduled. The technician checks in, wears personal protective equipment, performs the task(s) as assigned, checks out with you and asks for a screen capture signature on the smartphone device. A work order is then e-mailed, faxed or printed for your records. Emergency services Emergency service can be provided 7 days a week, 24 hours a day, 365 days a year. During normal business hours, emergency service will be coordinated by the customer service agent. After hours, weekends and holidays, the emergency service number transfers to the Johnson Controls after-hours call center and on-call technicians are dispatched as needed. Johnson Controls is committed to dispatching a technician within hours of receiving your call through the service line. A work order is e-mailed, faxed or printed for your records. Depending on the terms of your agreement, you may incur charges for after hour services. Communication A detailed communication plan will be provided to you so you know how often we will provide information to you regarding your Planned Service Agreement. The communication plan will also provide you with your main contacts at Johnson Controls. Approval process for non-covered items Johnson Controls will adhere to your procurement process, No work will be performed outside of the agreement scope without prior approval, Johnson Controls will work with you closely to ensure your procurement process is followed before any non-covered item work is started. Johnson ») ( Customer Confidential Controls Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 5 of 26 20 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 hata oot tol Planned Service Proposal oer we tat CITY OF WARREN-BRAC Summary of services and options Comprehensive and operational inspections During comprehensive and operational inspections, Johnson Controls will perform routine checks of the equipment for common issues caused by normal wear and tear on the equipment. Additional tests can be run to confirm the equipment's performance. Routine maintenance, such as lubrication, cleaning and tightening connections, can be performed depending on the type of equipment being serviced. Routine maintenance is one of the keys to the five values of maintenance — it can help identify energy saving opportunities, reduce future repair costs, extend asset life, ensure productive environments, and promote health and safety. Filter Replacement Clean air filters help maintain proper airflow throughout your building. Decreased airflow can impair the performance of the cooling coil and may lead to occupant discomfort and inefficient operation of the HVAC system. Johnson Controls will replace the filters on a regular basis to maintain airflow and maximize air quality. Customer Portal / Service Information Acces: The Johnson Controls customer portal is the online gateway to easily access various elements of your service information. This real-time, self-service mechanism is just one more way for you to stay in touch with our service within your facilities. Using the internet, you can view service call history by location, monitor agreements, as well as view asset and invoice information. df J Di © 2025 Johnson Controls, Inc. Jensen, ae Customer Confidential Controls Do not copy (physically, electronically, or in any other media) wilhoul the express writlen permission of Johnson Controls, Inc. Page 6 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 etna ‘ios outa Planned Service Proposal sya! su CITY OF WARREN-BRAC Summary Thank you for considering Johnson Controls as your building technology services partner. The following agreement document includes all the details surrounding your Planned Service Agreement. With planned service from Johnson Controls, you'll get a value-driven solution that can help optimize your building controls and equipment performance, providing dependability, sustainability and energy efficiency. You'll get a solution that fits your specific goals, delivered with the attention of a local service company backed by the resources of a global organization. (tie per top bebe oe tad fou Johnson Hi © 2025 Johnson Controls, Inc. Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 7 of 26 af Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 Joh one och Planned Service Proposal fo ey retain CITY OF WARREN-BRAC Planned Service Agreement Customer Name : CITY OF WARREN-BRAC Address: 8777 COMMON RD WARREN, MI 48093 Proposal Date: 03/08/2025 Estimate #: 1-1Q6QWBGX Scope of Service Johnson Controls, Inc. (“JCI”) and the Customer (collectively the “Parties”) agree Preventative Maintenance Services, as defined in Schedule A (“Services”), will be provided by JCI at the Customer's facility. This Planned Service Agreement, the Equipment List, Supplemental Price and Payment Terms, Terms and Conditions, and Schedules attached hereto and incorporated by this reference as if set forth fully herein (collectively the “Agreement’), cover the rights and obligations of both the Customer and JCI. Extended Service Options for Premium Coverage lf Premium Coverage is selected, on-site repair services to the equipment will be provided as specified in this Agreement for the equipment listed in the attached Equipment List. Equipment List Only the equipment listed in the Equipment List will be covered as part of this Agreement. Any changes to the Equipment List must be agreed upon in writing by both Parties. Term / Automatic Renewal This Agreement takes effect on 04/01/2025 and ending 04/30/2026 (“Original Term’). The Agreement will automatically renew and extend for successive terms equal to the Original Term unless the Customer or JCI gives the other written notice it does not want to renew prior to the end of the then-current term (each a “Renewal Term’). The notice must be delivered at least (90) days prior to the end of the Original Term or of any Renewal Term. The Original Term and any Renewal Term may be referred to herein as the “Term”. Renewal price adjustments are discussed in the Terms and Conditions. Refrigerant Charges Refrigerant is not included under this Agreement and will be billed separately to the Customer by JCI, h Di 4 i © 2025 Johnson Controls, Inc, Johnson i. Cuslomer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express writlen permission of Johnson Controls, Inc. Page 8 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 ferdite Joys “roorted Planned Service Proposal - it CITY OF WARREN-BRAC Price and Payment Terms The total Contract Price for JCI's Services during the first year of the Original Term is $8,938.00. This amount will be paid to JCI in advance in Annual installments, Pricing for each subsequent year of a multiyear Original Term is set forth in the Supplemental Price and Payment Terms. Unless otherwise agreed to by the parties, All payments will be due upon receipt. Renewal price adjustments are set forth in the Terms and Conditions. Any additional taxes, duties, tariffs or similar items imposed prior to shipment will be charged. Invoices will be sent to the following location: © CITY OF WARREN ATTN PURCHASING STE 425 1 CITY SQUARE WARREN, MI 48093 To ensure that JCI is compliant with your company's billing requirements, please provide the following information: PO is required to facilitate billing: [] No: This signed contract satisfies requirement {] number : AR Invoices are accepted via e-mail: [] YES: E-mail address to be used : {] No: Please submit invoices via mail {] No: Please submit via : This proposal is valid for thirty days from the proposal date. JOHNSON CONTROLS Inc. JCI Manager: Customer Manager: JC! Manager Signature: Customer Manager Signature: Title: Date: Title: Date: JCI Branch: JOHNSON CONTROLS DETROIT MI CB - ON16 Address:2875 HIGH MEADOW CIR AUBURN HILLS, MI 483262773 Branch Phone:(866) 866-0888 Branch Email: y) ‘fi © 2025 Johnson Controls, Inc. Johnson ))i Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express writlen permission of Johnson Controls, Inc. Page 9 of 26 39 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 ve (toon oot Planned Service Proposal rei ety 1 CITY OF WARREN-BRAC Schedule A - Equipment List CITY OF WARREN-BRAC 8777 COMMON RD WARREN, MI 48093 Product: Terminal Product, Fan Coil Unit (FCU), Vertical Air Flow Quantity: 4 Services Provided Coverage Level: — Basic 1 Operational 1 Belt Change Customer Tag Manufacturer Model # Serial # Unit Ventilators 1-4 Modine Product; Roof Top Unit (RTU), Cooling/Gas Heating, with Economizer Quantity: 4 Services Provided Coverage Level: Basic 2 Return Air Filter Change 1 Condenser Coil Cleaning 4 Operational (Mid Season - Cooling/Heating with Economizer) 1 Belt Change Customer Tag Manufacturer Model # Serial # RTUs 1-4 RHEEM & Carrier Product; Fan, Exhaust Fan, Roof Mounted Quantity: 5 Services Provided Coverage Level: — Basic 2 Operational 1 Belt Change Customer Tag Manufacturer Model # Serial # Exhaust Fans 1-5 Greenheck Product: Split System, Cooling with Electric Heat — HealthQuest Area Quantity: 1 Services Provided Coverage Level: Basic 1 Condenser Coil Cleaning 3 Operational (Mid Season - Cooling/Heating) Customer Taq Manufacturer Model # Serial # Ductless mini split 1 & 2 Mitsubishi © 2025 Johnson Controls, Inc. Customer Confidential Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Johnson Di Controls Page 10 of 26 40 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 t 4 in at: Planned Service Proposal ~ 4 CITY OF WARREN-BRAC Product: Contractor - Equipment Rental Quantity: 1 Services Provided Coverage Level: Basic 1 Contractor - Equipment Rental Lift rental to reach Gym Units - 1x per year for inspection Customer Tag Manufacturer Model # Serial # Product: Boiler, Gas-Fired, Fire Tube Quantity: 1 Services Provided Coverage Level: Basic 3 Operational 1 Government/Local Jurisdiction Inspect (performed during tube brushing - gaskets not Included) Customer Taq Manufacturer Model # Serial # Domestic HWT Boiler #1 And Storage Tank Product: Boiler, Gas-Fired, Fire Tube, <50 HP Quantity: 1 Services Provided Coverage Level: Basic 3 Operational 1 Government/Local Jurisdiction Inspect (performed during tube brushing - gaskets not included) Customer Tag Manufacturer Model # Serial # Main Heating Boiler #1 © 2025 Johnson Controls, Inc. Customer Confidential Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Johnson Di Controls Page 11 of 26 41 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 ee 1 “i ini bl Planned Service Proposal ot CITY OF WARREN-BRAC Equipment tasking Government/Local Jurisdiction Inspect (performed during tube brushing - gaskets not included) Operational Belt Change Operational Boiler, Gas-Fired, Fire Tube All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Lock and tag out unit Isolate and drain boiler Open covers Conduct inspection Replace covers Fill system Prepare unit for operation All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Blow down boiler Check for proper operation of low and high gas pressure cut-out switches Check factory supplied gas piping and components for leakage Check burner for proper sequence of operation Check flame quality Visually inspect combustion chamber, draft diverter and flue for accumulation of soot Check boiler relief valves for leakage Verify proper operation of low water cut-out control Check combustion blower motor operation Check hot water/steam temperature and pressure Check proper operation of make-up water valv Check overall condition of unit Document tasks performed during visit and report any observations to appropriate customer representative Fan, Exhaust Fan, Roof Mounted All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Perform belt change procedures Remove and dispose any debris from any maintenance activity Document tasks performed during visit and report any observations to appropriate customer representative Check with appropriate customer representative for operational deficiencies Check rain guard Check and tighten electrical connections Clean area around equipment Inspect starter Lubricate as required Motor operating amps Check belt guard Check belt(s) (if applicable) Check drive condition Check electrical contacts Check fan blades Check for proper rotation © 2025 Johnson Controls, Inc. Customer Confidential Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Johnson Pi Controls Page 12 of 26 42 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 Belt Change Return Air Filter Change Condenser Coil Cleaning Operational (Mid Season - Cooling/Heating with Economizer) Condenser Coil Cleaning © 2025 Johnson Controls, Inc. Customer Confidential 4 uite al! Planned Service Proposal “tetra CITY OF WARREN-BRAC Complete any required maintenance checklists, report observations to appropriate customer representative Roof Top Unit (RTU), Cooling/Gas Heating, with Economizer All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Perform belt change procedures Remove and dispose any debris from any maintenance activity Document tasks performed during visit and report any observations to appropriate customer representative All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Turn equipment off Remove dirty filters Install new filters Turn equipment on Dispose of dirty filter appropriately Document tasks performed during visit and report any observations to appropriate customer representative All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Rinse coil(s) thoroughly with water Remove and dispose any debris from any maintenance activity Document tasks performed during visit and report any observations to appropriate customer representative All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Review control panel for proper operation and recorded fault histories (if applicable) Check condition of condenser coil Check condition of evaporator coil Check condenser fan motors and blades Check blower motor operation Check economizer operation Check heating operation (when applicable) Check condition of pulley and belts Check condition of filters Check condensate drain Check for visual signs of refrigerant/oil leak(s) Visually inspect electrical connections Check for unusual noise and vibration Check overall condition of unit Document tasks performed during visit and report any observations to appropriate customer representative Split System, Cooling with Electric Heat All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Rinse coil(s) thoroughly with water Johnson y Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 13 of 26 43 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 be) tava seh Planned Service Proposal »oaty oe tae CITY OF WARREN-BRAC Operational (Mid Season - Cooling/Heating) Belt Change Operational Remove and dispose any debris from any maintenance activity Document tasks performed during visit and report any observations to appropriate customer representative All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Check condition of condenser coil Check condenser fan motors and blades Check condensing unit electrical connections and contactor(s) Check blower motor operation Check heating operation (when applicable) Check condition of filters Check condition of pulley and belts (if applicable) Check condensate system Check for visual signs of refrigerant/oil leak(s) Check for unusual noise and vibration Check overall condition of unit Document tasks performed during visit and report any observations to appropriate customer representative Terminal Product, Fan Coil Unit (FCU), Vertical Air Flow All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Perform belt change procedures Remove and dispose any debris from any maintenance activity Document tasks performed during visit and report any observations to appropriate customer representative All work must be performed in accordance with Johnson Controls safety policies Check with appropriate customer representative for operational deficiencies Check condition of pulley and belts (if applicable) Check condition of coils Check condition of filters Check condensate drain Check for unusual noise and vibration Check unit for proper operation Check overall condition of unit Document tasks performed during visit and report any observations to appropriate customer representative © 2025 Johnson Controls, Inc. J epnepn Cuslomer Confidential Controls Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 14 of 26 44 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 jot ety ood Planned Service Proposal i CITY OF WARREN-BRAC ire qd =" Supplemental Price & Payment Terms (Applies to Multi-Year Contracts Only) Year a Total Annual Dollar Amount | | Payment Frequency Year( a $8,938.00 Annually © 2025 Johnson Controls, Inc. Customer Confidential Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Johnson Di Controls Page 15 of 26 45 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 lofi ot tony Planned Service Proposal bray er, CITY OF WARREN-BRAC Special Additions and Exceptions © 2025 Jolinson Controls, Inc. Customer Confidential Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Johnson YG Controls Page 16 of 26 46 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FASA35F3 © 2025 Johnson Controls, Inc. Customer Confidential “or Planned Service Proposal be erces' yo CITY OF WARREN-BRAC TERMS AND CONDITIONS DEFINITIONS (REV 1/25) DIGITAL ENABLED SERVICES mean services provided hereunder that employ JCI software and related equipment Installed at Purchaser facilitles and JCI cloud-hosted software offerings and tools to Improve, develop, and enable such services, Digital Enabled Service may Include, but are not limited to, (a) remote servicing and inspection, (b) advanced equipment fault detection and diagnostics, and (c) data dashboarding and health reporting. CONTRACT PRICE means the price that Customer shall pay to JCI for the Services, COVERED EQUIPMENT means the equipment for which Services are to be provided under this Agreement. Covered Equipment Is set forth in Schedule A - Equipment List, EQUIPMENT FAILURE means the fallure, under normal and expected working conditions, of moving parts or electric or electronic components of the Covered Equipment that are necessary for Its operation, PREMISES means those Customer premises where the Covered Equipment is located or Services performed pursuant to thls Agreement. REMOTE MONITORING SERVICES means remote monitoring of Covered Equipment and/or systems including building automation, HVAC equipment, and fire alarm, intrusion, and/or other life safety systems for alarm and event notifications using a UL Certified Central Station. REMOTE OPERATIONS CENTER (ROC) Is the department at JCI that remotely monitors alarm and Industrial (HVAC) process signals. REMOTE OPERATING SERVICES means remote Interrogation, modification and/or operation of building automation, HVAC equipment, and/or other Covered Equipment. REPAIR LABOR is the labor necessary to restore Covered Equipment to working condition following an Equipment Failure, but does not Include services relating to total equipment replacement due to obsolescence or unavallabllity of parts. REPAIR MATERIALS are the parts and materials necessary to restore Covered Equipment to working condition f lowing an Equipment Failure, but excludes total equipment replacement due to obsolescence or unavallabillty of parts, unless excluded from the Agreement. At JCI’s option, Repair Materials may be new, used, or reconditioned. SCHEDULED SERVICE MATERIALS are the materials required to perform Scheduled Service Visits on Covered Equipment, unless excluded from the Agreement. SCHEDULED SERVICE VISITS are the on-site labor visits required to perform JCI recommended inspections and preventive maintenance on Covered Equipment, SERVICES are the work, materials, labor, service visits, and repairs to be provided by JCI pursuant to this Agreement except that the Services do not Include the Connected Equipment Services or the provision of other software products or digital or cloud services, which are provided under separate terms and conditions referenced In Sectlon P. A, JCI’S SERVICES FOR COVERED EQUIPMENT 1, BASIC COVERAGE means Scheduled Service Visits, plus Scheduled Service Materials (unless excluded from this Agreement). No parts, equipment, Repair Labor or Repair Materials are provided for under BASIC COVERAGE. 2, PREMIUM COVERAGE means BASIC COVERAGE plus Repair Labor, plus Repair Materlals (unless excluded from the Agreement), If Customer has ordered PREMIUM COVERAGE, JCI will Inspect the Covered Equipment within forty-five (45) days of the date of this Agreement, or as seasonal or operational conditions permit. JCI will then advise Customer If JCI finds any Covered Equipment not In working order or in need of repair. With Customer's approval, JCI will perform the work necessary to put the Covered Equipment in proper working conditlon, subject to the terms of this Agreement. Customer will pay for such work at JCI’s standard rates for parts and labor In effect at the time that the work Is performed. If Customer does not want JCI to perform the work identified as necessary by JCI, any equipment thereby affected will be removed from the list of Covered Equipment, and the Contract Price will be adjusted accordingly. Should Customer not make JCI’s recommended repairs or proceed with the modifled PREMIUM COVERAGE, JCI reserves the tight to Invoice Customer for the cost of the Initial equipment Inspection. 3, EXTENDED SERVICE means Services performed outside JCI’s normal business hours and \s avallable only if Customer has PREMIUM COVERAGE, Extended Service Is avallable either 24/5 or 24/7, at Customer's election. The price for Extended Service, if chosen by Customer, is part of the total Contract Price. 4, REMOTE MONITORING SERVICES OR REMOTE OPERATING SERVICES. If Remote Monitoring Services or Remote Operating Services are provided, Customer agrees to furnish JCI with a list of the names, titles, addresses, email addresses, and phone numbers of all persons authorized to be contacted by, or be able to contact the ROC to perform specific agreed upon actions with the appropriate authority. If JCI’s Services Include “Remote Monitoring Services with Open and Close,” Customer also agrees to furnish JCI with Customer's dally and hollday opening and closing schedules. Customer agrees to malntain and update the call lists with accurate Information. Customer further agrees to notify JCI of such changes as soon as possible. JCI/ROC Is not responsible to fInd new contacts/numbers if the contacts on the call lists cannot Do not copy (physically, electronically, or in any other media) withoul the express wrillen permission of Johnson Controls, Inc. Johnson Controls Page 17 of 26 47 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FASA35F3 viio Planned Service Proposal poe, oo CITY OF WARREN-BRAC be reached. A maximum of three contacts are allowed for any time of the day. If none of those contacts can be reached, then neither JCI nor the ROC are responsible for damages. Customer Is responsible for any and all costs and expenses arising from Customer's fallure to provide timely updates for any of the contact Information submitted to the ROC. 5, CUSTOMER SERVICE INFORMATION PORTAL, Customer may be able to utilize JCI’s Customer Service Information Portal during the term of the Agreement, pursuant to the then applicable Terms of Use Agreement. B, OUT OF SCOPE SERVICES If, during any Service Visit, JCI detects a defect In any of Customer's equipment that Is not Covered Equipment under this Agreement (an “Out of Scope Defect”), JCI may (but shall have no obligation to) notify Customer of such Out of Scope Defect. If Customer elects for JCI to repalr such Out of Scope Defect, or if JCI otherwise performs any Services or provides any materials, parts, or equipment outside the scope of the Services (collectively, “Out of Scope Services”), Customer shall direct JCI to perform such Out of Scope Services In writing, and Customer shall pay for such Out of Scope Services at JCI’s standard fees or hourly rates. If, after recelving notice of an Out of Scope Defect, Customer elects not to engage JCI to repair such Out of Scope Defect, Customer shall defend and Indemnify JCI from and agalnst any and all losses, damages, clalms, costs and expenses arising directly or indlrectly out of such Out of Scope Defect. Any Out of Scope Services performed by JCI at the direction of Customer pursuant to this Section shall be subject to the Customer Terms In effect as of the Effective Date (the “Customer Terms”), which Customer Terms are Incorporated Into thls Agreement by this reference, A copy of the Customer Terms currently in effect is found at www. johnsoncontrols.com/customerterns, Cc. EXCLUSIONS JCI’s Services and warranty obligations expressly exclude: (a) the repair or replacement of ductwork, casings, cabinets, structural supports, tower fill/slats/basin, hydronic and pneumatic piping, and vessels, gaskets, and piping not normally replaced or maintained on a scheduled basis, and removal of oil from pneumatic piping; (b) disposal of hazardous wastes (except as otherwise expressly provided herein); (c) disinfecting of chiller condenser water systems and other components for blohazards, such as but not Ilmited to, Legionella unless explicitly set forth In the scope of services between the partles. Unless explicitly provide for within the scope of services, this Is Out of Scope Services and the Customer's exclusive responsibility to make arrangements for such services with a provider other than JCI. Mentions of chiller tube cleaning, condenser cleaning, cooling tower cleaning or boller tube cleaning in any scope of services, only involve work to remove normal buildup of debris and scale using tube brush cleaning, pressure washing or acid flushing. Reference to such cleaning does not include chemical cleaning, disinfection or chemical water treatment required to eliminate, control or disinfect against blohazards such as but not limited to Legionella; (d) refrigerant; supplies, accessories, or any Items normally consumed during the use of Covered Equipment, such as ribbons, bulbs and paper; (e) the furnishing of materials and supplies for painting or refinishing equipment; (f) the repair or replacement of wire in condult, buried cable/transmission lines, or the like, If not normally replaced or maintained on a scheduled basis; (g) replacement of obsolete parts; and (h) damages of any kind, Including but not limited to personal injury, death, property damage, and the costs of repairs or service resulting from: @ abuse, misuse, alterations, adjustments, attachments, combinatlons, modifications, or repairs to Covered Equipment not performed, provided, or approved In writing by JCI; © — equipment not covered by this Agreement or attachments made to Covered Equipment; @ acts or omissions of the Customer, Including but not IImited to the failure of the Customer to fulfill the Customer Obligations and Commitments to JCl as described In Sectlon F of this Agreement, operator error, Customer's fallure to conduct preventive maintenance, Issues resulting from Customer's previous denial of JCI access to the Covered Equipment, and Customer's failure to keep the site clean and free of dust, sand, or other particles or debris, unless such conditions are previously expressly acknowledged by JC} in writing; © — use of the Covered Equipment In a manner or environment, or far any purpose, for which it was not designed by the manufacturer; @ site-related and environmental conditions, Including but not limited to power failures and fluctuations In electrical current (or “power surges”) and blohazards such as but not limited to Leglonella assoclated with condenser water, cooling tower systems and subcomponent systems; ® the effects of erosion, corrosion, acid cleaning, or damage from unexpected or especially severe freezing weather; @ —|ssues or failures not specifically covered by this Agreement; or @ — occurrences beyond JCI's reasonable contro! and without JCI's fault or negligence. D. PAYMENT TERMS; PRICE ADJUSTMENTS Unless otherwise agreed by the parties in writing, fees for Services to be performed shall be paid annually in advance. Fees and other amounts due hereunder are due thirty (30) days from the date of the Invoice, which shall be pald by Customer via electronic delivery via EFT/ACH . Such payment is a condition precedent to JCI’s obligatlon to perform Services under the Agreement. Any involce disputes are waived unless Identifled In writing by Customer within 21 days of the date of invoice. Payments of any disputed amounts are due and payable upon resolution. Customer acknowledges and agrees that timely payments of the full amounts listed on Invoices Is an essential term of this Agreement and that fallure by Customer to make payment in full when due Is a material breach of thls Agreement. Work performed on a time and material basls shall be at the © 2025 Johnson Controls, Inc. Johnson Customer Confidential Controls Do not copy (physically, electronically, or in any other media) wilhoul the express writlen permission of Johnson Controls, Inc. Page 18 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 snevot Planned Service Proposal it CITY OF WARREN-BRAC then-prevalling JCI rate for material, labor, and related Items, In effect at the time supplied under this Agreement. Customer shall provide financial Information requested by JCI to verify Customer's ability to pay for goods or Services, If Customer falls to provide financlal Information or If JCI, in Its sole discretion determines that reasonable grounds exist to question Customer's ability or willingness to make payments when due (e.g., not making payments when due, late payments, or a reduction In Customer's credit score), JCI may defer shipments, change payment terms, require cash in advance and/or require other security, without llabillty and without walving any other remedies JCI may have against Customer, JCI shall provide Customer with advance written notice of changes to Payment terms, Customer further acknowledges that if there Is any amount outstanding on an Invoice, It Is materlal to company and will give JCI, without prejudice to any other right or remedy, the right to, without notice: (i) suspend, discontinue or terminate performing any services and/or withhold further dellverles of equipment and other materials, terminate or suspend any unpald software licenses, and/or suspend JCI's obligatlons under or terminate this Agreement; (il) charge Customer Interest on the amounts unpald at a rate equal to the lesser of one and one half (1.5) percent per month or the maximum rate permitted under applicable law, until payment is made in full; and (itl) pay all of JCI’s costs of collection, including (1) actual out of pocket expenses and charge Customer actual costs, plus a collection fee of twenty-five percent (25%) of the past due amount if collected through a collection agency or attorney and thirty-five percent (35%) if litigation Is commenced to collect such past due amount, JCI’s election to continue providing future services does not, In any way diminish JCI’s right to terminate or suspend services or exercise any or all rights or remedies under this Agreement. JCI shall not be llable for any damages, clalms, expenses, or liabilities arising from or relating to suspension of services for non-payment. In the event that there are exigent circumstances requiring services or the JC! otherwise performs services at the premises following suspension, those services shall be governed by the terms of this Agreement unless a separate contract Is executed. If Customer disputes any late payment notice or JCI’s efforts to collect payment. Customer shall immediately notify JCI In writing and explain the basis of the dispute. Customer will pay all of JC's reasonable collection costs (including legal fees and expenses). In the event of Customer’s default, the balance of any outstanding amounts will be immediately due and payable. This Agreement is entered into with the understanding that the Services to be provided by JCI are not subject to any local, state, or federal prevalling wage statute. If It is later determined that local, state, or federal prevalllng wage rates apply to the Services to be provided by JCI, JCI reserves the right to Issue a modification or change order to adjust the wage rates to the required prevall ng wage rate, Customer agrees to pay for the applicable prevailing wage rates. JCI may increase prices upon notice to the Customer to reflect increases in materlal and labor costs. Prices do not Include taxes, fees, duties, tariffs, false alarm assessments, permits and levies or other charges Imposed and/or enacted by a government, however designated or Imposed (collectively, “Taxes”), All Taxes are the responsib! lity of Customer, unless Customer presents an exemption certificate acceptable to JCI and the applicable taxing authorities, If JCI Is required to pay any such Taxes or other charges, Customer shall reimburse JCI on demand, If any such exemption certificate Is Invalid, then Customer will immediately pay JCI the amount of the Taxes, plus penalties and Interest. Prices may be adjusted by JCI prior to shipment to take Into account Increases In the cost of raw materlals, component parts, third party products or labor rates or taxes; Trade Restrictions {as defined below); government actions; or to cover any unforeseen or other extra cost elements. “Trade Restrictions” means any additional or new tariff/duty, quota, tarlff-rate quota, or cost assoclated with the withdrawal of tarlff/duty concessions pursuant to a trade agreement(s), If this Agreement Is renewed, JCI will provide Customer with notice of any adjustments In the Contract Price applicable to any Renewal Term. Unless Customer terminates this Agreement In writing at least sixty (60) days prior to the start of the Renewal Term, the adjusted Contract Price shall be the price for the Renewal Term. Prices for products covered by this Agreement may be adjusted by JCI, upon notice to Customer at any time prior to shipment and regardless of Customer's acceptance of the JCI’s proposal or quotation, to reflect any Increase in JCl’s cost of raw materials (e.g, steel, aluminum) tnabllity to secure Products, changes or increases in law, labor, taxes, dutles, tariffs or quotas, acts of government, any similar charges, or to cover any extra, unforeseen and unusual cost elements. If the actual number of devices installed or services to be performed Is greater than that set forth In this Agreement, the price will be increased accordingly. If this Agreement extends beyond one year, Johnson Controls may Increase prices upon notice to the Customer. E, WARRANTIES JCI warrants Its Services will be provided in a good and workmanlike manner for 90 days from the date of Services. If JCI receives written notice of a breach of this warranty prior to the end of this warranty period, JCI will re-perform any non-conforming Services at no additional charge within a commercially reasonable time of the notification. If JCI installs or furntshes a plece of equipment under this Agreement, JCI warrants that equipment labeled JCI shall be free from defects In material and workmanshlp arising from normal usage for a perlod of 90 days. No warranty Is provided for third-party products and equipment installed or furnished by JCI. Such products and equipment are provided with the third-party manufacturer's warranty to the extent available, and JCI will transfer the benefits, together with all imitations, of that manufacturer's warranty to Customer. All transportation charges Incurred In connection with the warranty for equipment and/or materials not covered under this Agreement shall be borne by Customer. Except as provided herein, if JCI receives written notice of a breach of this warranty prior to the end of this warranty period, JCI will repair or replace (at JCI’s option) the defective equipment. These warranties do not extend to any Services or equipment that have been misused, altered, or repalred by Customer or third parties without the supervision of and prior written approval of JCI, or If JC! sertal numbers or warranty decals have been removed or altered. All replaced parts or equipment shall become JCI’s Property, This warranty Is not assignable, Warranty service will be provided during normal business hours, excluding holidays. The remedies set forth herein shall be Customer's sole and exclusive remedy with regards to any warranty claim under this Agreement. Any lawsult based upon the warranty must be brought no later than one (1) year after the expiration of the applicable warranty perlod. This IImitation is In lieu of any other applicable statute of limitations, CUSTOMER FURTHER ACKNOWLEDGES AND AGREES THAT THESE WARRANTIES ARE JCI'S SOLE WARRANTIES AND TO THE MAXIMUM EXT! ENT PERMITTED UNDER APPLICABLE LAW ARE IN LIEU OF ALL OTHER WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THOSE OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, JCI makes no and specifically disclaims all representations or warrantles that the services, products, software or third party product or software will be secure from cyber threats, hacking or other similar mallclous activity, or will detect the presence of, or eliminate, prevent, treat, or mitigate the spread, transmission, or outbreak of any pathogen, disease, virus or other contagion, including but not limited to COVID 19, F. CUSTOMER OBLIGATIONS AND COMMITMENTS TO JCI 1, Customer warrants It has given JC] all information concerning the condition of the Covered Equipment, The Customer agrees and warrants (1) operate the Covered Equipment according to the manufacturer's and/or JC's recommendations; © 2025 Johnson Controls, Inc. Johnson )) Customer Confidential Do not copy (physically, electronically, or in any other media) without the express wrilten permission of Johnson Controls, Inc. Page 19 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FAQA35F3 ot tol Planned Service Proposal bos ety CITY OF WARREN-BRAC (2) keep accurate and current work logs and Information about the Covered Equipment as recommended by the manufacturer and/or JCI; (3) provide an adequate environment for Covered Equipment as recommended by the manufacturer and/or JCI, Including, but not Ilmited to adequate space, electrical power, water supply, alr conditloning, and humidity control; (4) notify JCI Immediately of any Covered Equipment malfunction, breakdown, or other condition affecting the operation of the Covered Equipment; (5) provide JCI with safe access to Its Premlses and Covered Equipment at all reasonable and necessary times for the performance of the Services; (6) allow JCI to start and stop, periodically turn off, or otherwise change or temporarlly suspend equipment operations so that JCI can perform the Services required under this Agreement; (7) as applicable, provide proper condenser, cooling tower and boiler water treatment for the proper functloning of Covered Equipment and protect agalnst any environmental Issues and instances of blohazards such as but not limited to Leglonella; (8) carefully and properly set and test the Intrusion alarm system each night or at such other time as Customer shall close the Premises; (9) obtain all necessary licenses and permits required for and pay all taxes associated with the Services; (10) notify JCI immediately of any claimed inadequacy in, or fallure of, the Covered Equipment or other condition affecting the operation of the Covered Equipment; (11) furnish any necessary 110 volt A/C power and electrical outlets at its expense; (12) properly maintain, repair, service, and assure the proper operation of any other property, system, equipment, or device of Customer or others to which the Covered Equipment may be attached or connected, In accordance with manufacturer recommendations, Insurance carrler requirements, or the requirements of any fire rating bureau, agency, or other authorities having Jurisdiction thereof; (13) not tamper with, alter, adjust, disturb, Injure, remove, or otherwise interfere with any Covered Equipment (including any related software) and not permit the same to be done; (14) refrain from causing false alarms, and reimburse JCI for any fine, penalty, or fee pald by or assessed against JCI by any governmental or municipal agency as a result thereof; (15) be solely responsible for the establishment, operation, maintenance, access, security and other aspects of Its computer network (“Network”) and shall supply JCI secure Network access for providing Its services. Products networked, connected to the Internet, or otherwise connected to computers or other devices must be appropriately protected by Customer and/or end user against unauthorized access; and (16) take appropriate measures, Including performing back-ups, to protect Information, including without limit data, software, or files (collectively “Data”) prior to receiving the service or products. 2. Customer acknowledges and understands that unless water treatment for blohazards (such as Legionella) Is explicitly Included in the services JCI |s providing, it is Customer's responsibility to provide such treatment. Customer also acknowledges that its failure to meet the above obligations will relleve JCI of any responsibility for any Covered Equipment breakdown, or any necessary repair or replacement of any Covered Equipment. If Customer breaches any of these obligations, JCI shall have the right, upon written notice to Customer, to suspend its Services until Customer cures such breach, In addition, Customer shall be responsible for paying or relmbursing JCI for any costs associated with corrective work required as a result of Customer's breach of these obligations, G. INSURANCE Customer Is responsible for obtaining all Insurance coverage that Customer believes is necessary to protect Customer, Customer's property, and persons In or on the Premises, Including coverage for personal injury and property damage. THE PAYMENTS CUSTOMER MAKES UNDER THIS AGREEMENT ARE NOT RELATED TO THE VALUE OF THE PREMISES, CUSTOMER'S PROPERTY OR POSSESSIONS, OR THE PERSONS OCCUPYING OR AT ANY TIME PRESENT IN OR ON THE PREMISES, BUT RATHER ARE BASED ON THE COST OF THE SYSTEM AND THE SERVICES, AND TAKE INTO CONSIDERATION THE PROTECTION AFFORDED TO JCI UNDER THIS AGREEMENT, Customer hereby releases JCI from any liability for any event or condition customarlly covered by commercial Ilablllty Insurance. Customer understands that neither the Services nor the Covered Equipment are designed to reduce, but not eliminate, certain risks, JCI does not guaranty that neither the Services nor Covered Equipment will prevent personal injury, unauthorized entrances or fire and smoke damage to the Premises. Customer further agrees that Customer has read and understands the terms and conditions of this Agreement. H. INDEMNITY JCI and Customer shall each indemnify the other party and its officers, agents, directors, and employees, from any and all damages, losses, costs and expenses (including reasonable attorneys’ fees) arising out of third party claims, demands, or sults for bod ly Injury (including death) or damage to tangible property to the extent arising out of the negligence or intentional misconduct of the Indemnifying party or its employees or agents. Customer expressly agrees that JCI shall be responsible for Injury, damage, or loss only to the extent caused directly by JCI’s negligence or Intentional misconduct. The obligations of JCI and Customer under this section are further subject to sections | and K below. 1, LIMITATION OF LIABILITY TO THE MAXIMUM EXTENT PERMITTED BY LAW, IN NO EVENT SHALL JCI AND ITS AFFILIATES AND THEIR RESPECTIVE PERSONNEL, SUPPLIERS AND VENDORS (“JCI PARTIES”) BE LIABLE TO YOU OR ‘ANY THIRD PARTY UNDER ANY CAUSE OF ACTION OR THEORY OF LIABILITY EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGES, FOR ANY: (1) SPECIAL, INCIDENTAL, CONSEQUENTIAL, PUNITIVE, OR INDIRECT DAMAGES; (2) LOST PROFITS, REVENUES, DATA, CUSTOMER OPPORTUNITIES, BUSINESS, ANTICIPATED SAVINGS, OR GOODWILL; (3) BUSINESS INTERRUPTION; OR (4) DATA LOSS OR OTHER LOSSES ARISING FROM VIRUSES, RANSOMWARE, CYBER ATTACKS OR FAILURES OR INTERRUPTIONS TO NETWORK SYSTEMS. IN ANY CASE, THE ENTIRE AGGREGATE LIABILITY OF THE JCI PARTIES UNDER THIS AGREEMENT FOR ALL DAMAGES, LOSSES, AND CAUSES OF ACTION, WHETHER IN CONTRACT, TORT (INCLUDING NEGLIGENCE), OR OTHERWISE SHALL BE LIMITED TO $250,000, CUSTOMER UNDERSTANDS THAT JCI IS NOT AN INSURER REGARDING THE WORK OR THE SERVICES. JCI SHALL NOT BE RESPONSIBLE FOR ANY DAMAGE OR LOSS THAT MAY RESULT FROM FIRE SAFETY OR SECURITY EQUIPMENT THAT FAILS TO PERFORM PROPERLY OR FAILS TO PREVENT A CASUALTY OR LOSS, Ny, © 2025 Johnson Controls, Inc. Johnson SS Customer Confidential Controls Do not copy (physically, electronically, or in any other media) withoul the express written permission of Johnson Controls, Inc, Page 20 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FASA35F3 ‘1 Planned Service Proposal meyeis ti 1 CITY OF WARREN-BRAC J. FORCE MAJEURE JCI shall not be lable, nor In breach or default of Its obligations under thls Agreement, for delays, Interruption, fallure to render services, or any other fallure by JCI to perform an obligation under this Agreement, where such delay, Interruption or fallure Is caused, In whole or in part, directly or indirectly, by a Force Majeure Event. A “Force Majeure Event” is a condition or event that Is beyond the reasonable control of JCI, whether foreseeable or unforeseeable, Including, without limitation, acts of God, severe weather (Including but not limited to hurricanes, tornados, severe snowstorms or severe rainstorms), wildfires, floods, earthquakes, selsmic disturbances, or other natural disasters, acts or omlsslons of any governmental authority (Including change of any applicable law or regulation), epidemics, pandemics, disease, viruses, quarantines, or other public health risks and/or responses thereto, condemnation, strikes, lock-outs, labor disputes, an Increase of 5% or more In tariffs or other exclse taxes for materlals to be used on the project, fires, explosions or other casualties, thefts, vandalism, clvil disturbances, Insurrection, mob violence, riots, war or other armed conflict (or the serlous threat of same), acts of terrorism, electrical power outages, Interruptions or degradations In telecommunications, computer, network, or electronic communications systems, data breach, cyber-attacks, ransomware, unavallability or shortage of parts, materials, supplies, or transportatlon, or any other cause or casualty beyond the reasonable control of JCI. If JCI’s performance of the work is delayed, Impacted, or prevented by a Force Majeure Event or Its continued effects, JCI shall be excused from performance under the Agreement. Without limiting the generality of the foregoing, If JCI is delayed in achieving one or more of the scheduled milestones set forth in the Agreement due to a Force Majeure Event, JCI will be entitled to extend the relevant completion date by the amount of time that JCI was delayed as a result of the Force Majeure Event, plus such additlonal time as may be reasonably necessary to overcome the effect of the delay. To the extent that the Force Majeure Event directly or Indirectly Increases JCI’s cost to perform the services, Customer |s obligated to relmburse JCI for such Increased costs, Including, without Imitation, costs Incurred by JCI for additional labor, Inventory storage, expedited shipping fees, traller and equipment rental fees, subcontractor fees, compllance with vaccination requirements or other costs and expenses incurred by JCI in connectlon with the Force Majeure Event, K, ONE-YEAR CLAIMS LIMITATION; FORUM; CHOICE OF LAW JCI shall have the sole and exclusive right to determine whether any dispute, controversy or claim arising out of or relating to the Agreement, or the breach thereof, shall be submitted to a court of law or arbitrated, The laws of Delaware shall govern the validity, enforceability, and Interpretation of this Agreement, without regard to conflicts of law principles thereof, and the exclusive venue for any such litigation or arbitration shall be In Milwaukee, Wisconsin, The partles waive any objection to the exclusive jurisdiction of the specifled forums, Including any objection based on forum non conveniens. In the event the matter is submitted to a court, JCI and Customer hereby agree to walve thelr right to trlal by jury.. In the event the matter is submitted to arbitration by JCI, the costs of arbitration shall be borne equally by the parties, and the arbitrator’s award may be confirmed and reduced to judgment In any court of competent Jurisdiction. If Company prevails in any collection action, Customer will pay all of JCI's reasonable collection costs (including legal fees and expenses). Except as provided below, no clalm or cause of action, whether known or unknown, shall be brought by either party against the other more than one year after the claim first arose. Claims not subject to the one-year limitation include claims for unpald: (1) contract amounts, (2) change order amounts (approved or requested) and (3) delays and/or work Inefficiencles, L. TERM AND TERMINATION 1. The Initial Term of this Agreement shall commence on the date of this Agreement and continue as set forth herein. The Initial Term of this Agreement Is three (3) years. After the Initlal Term, this Agreement shall automatically extend for successlve terms equal to the same length as the Initlal Term unless the Customer or JCI gives written notice to the other that It does not want to renew at least sixty (60) days prior to the end of the then-current term (each a “Renewal Term”), The Initial Term and any Renewal Term may be referred to herein as the “Term.” Customer agrees to Issue and send a Purchase Order to JCI at least thirty (30) days prior to expiration of the Original Term or any Renewal Term if necessary for payments to be processed, but fallure to do so Is not a pre-condition to Renewal Term payments being due to JCI. No Purchase Order Is required for any emergency work requested by Customer, Customer shall have no right to reject such invoices due to the lack of a Purchase Order. 2, Remote Monitoring Services and Remote Operating Services may be immediately canceled by either party If JCI’s Remote Operations Center, connecting wires, or monitoring systems are destroyed by fire or other catastrophe, or where the Premises are so substantially damaged that It Is Impractical to continue Services. 3. If either party fails to perform any of its material obligations under this Agreement, the other party shall provide written notice thereof to the party alleged to be In default. Should the party alleged to be in default fail to respond in writing or take action to cure the alleged default within ten (10) days of receiving such written notice, the notifying party may terminate this Agreement by providing written notice of such termination. 4, If JCI’s performance of Its obligations becomes impracticable due to obsolescence or unavailability of systems, equipment, or products (Including component parts and/or materials) or because the JCI or its suppller(s) has discontinued the manufacture or the sale of the equipment and/or products or Is no longer Is in the business of providing the Services, JCI may terminate this Agreement, or the affected portions, at its sole discretion upon notice to Customer, 5. JCI may terminate this Agreement and discontinue any Services If JCI is unable to obtain or continue to support technologies, or for convenlence upon forty-five (45) days written notice. JC! will not be liable for any damages or subject to any penalty as a result of any such termination. 6, JCI may terminate this Agreement, or the affected portions, at its sole discretion upon notice to the Customer If JCI’s performance of Its obligations are prohibited because of changes In applicable laws, regulations or codes, 7, In the event JCI Is unable, after reasonable commercial efforts, to acquire and provide steel products, or products made from plastics or y) Q) © 2025 Johnson Controls, Inc. Johnson I) Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 21 of 26 Docusign Envelope 1D: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 sk Ny GF Planned Service Proposal -s i. ') 4 CITY OF WARREN-BRAC other commodities, if required to perform work required by this Agreement, Customer hereby agrees that JCI may terminate the Agreement, or the relevant portion of the Agreement, at no additional cost and without penalty. 8. JCI may terminate this Agreement and discontinue any Service(s), if JCI's central monitoring center (“CMC”) or remote operations center or elther of their respective systems Is substantially damaged by fire or catastrophe or If JCI Is unable to obtain any connectlons or privileges required to transmit signals between the Customer's premises, JCI's CMC or the Municipal Fire or Police Department or other first responder. If Customer falls to maintain any required licenses or permits, JCI shall not be responsible for performing the services and may terminate the services without notice to Customer, JCI may terminate the services immediately upon notice to the Customer if JCI, In its sole discretion, determines that the Customer's premises are unsafe to be accessed by JCI's employees or subcontractors, JCI may terminate the services upon notice to the Customer, If Customer does not follow JCI's recommendations for updates and upgrades to the equipment and systems. 9. Upon termination of this Agreement for any reason, Customer shall pay to JCI all undisputed amounts owed through the date of termination within thirty (30) days of such termination. For termination prior to the end of the Term, Customer agrees to pay JCI, in addition to any outstanding fees and charges for service(s) rendered prior to termination, 50% of the charges for services remaining to be pald for the unexplred term of the Agreement as liquidated damages but not as a penalty, Customer shall provide JCI with reasonable access to the premises to remove any JCI property and to un-program any controls, intrusion, fire, or life safety system, as applicable., Customer shall be liable for all fees, costs, and expenses that JCI may Incur In connection with the enforcement of thls Agreement, Including without limitation, reasonable attorney fees, collectlon agency fees, and court costs. M. ASBESTOS, MOLD, BIOAHAZARDS, AND HAZARDOUS MATERIALS “Hazardous Materials” means any material or substance that, whether by Its nature or use, Is now or hereafter defined or regulated as a hazardous waste, hazardous substance, pollutant, or contaminant under any local, state, or federal law, regulation, or ordinance relating to or addressing public and employee health and safety and protection of the environment, or which is toxic, explosive, corrosive, flammable, radioactive, carcinogenic or otherwise hazardous or which Is or contalns petroleum, gasoline, diesel, fuel, another petroleum hydrocarbon product or polychlorinated biphenyls, “Hazardous Materials” specifically Includes mold, lead-based paints, blohazards such as but not limited to Legionella and asbestos-containing materlals (“ACM”), Nelther Customer nor JCI desires to or Is licensed to undertake direct obligations relating to the Identification, abatement, cleanup, control, removal or disposal of ACM. JCI will be responsible for removing or disposing of any Hazardous Materlals that It uses In providing the Services (“JCI Hazardous Materlals”) and for the remediation of any areas affected by the release of JC! Hazardous Materlals, For other Hazardous Materlals that may be present at Its facilities (“Non-JC! Hazardous Materials"), Customer shall supply JCI with any Information In its possession relating to the presence of Hazardous Materials If their presence may affect JCI’s performance of the Services, If elther Customer or JC] becomes aware of or suspects the presence of Non-JCl Hazardous Materials that may Interfere with JCI's Services, it shall Immediately stop the Services In the affected area and notify the other party. As between Customer and JCI, Customer shall be responsible at its sole expense for removing and disposing of Non-JC! Hazardous Materials from its facilities and for the remediation of any areas impacted by the release of the Non-JCI Hazardous Materials and must provide a certificate of abatement before JCI will be obligated to perform or continue Its Services, unless JCI had actual knowledge that Non-JC] Hazardous Materials were present and acted in disregard of that knowledge, In which case (|) JCI shall be responsible at its sole expense for the remediation of any areas Impacted by its release of such Hazardous Materlals, and (il) Customer shall remain responsible at Its sole expense for the removal of Hazardous Materials that have not been released and for releases not resulting from JCI’s performance of the Services. Customer shall defend and indemnify JCI against any losses, costs, damages, expenses, and clalms arising out of its failure to comply with this Section M. N, CUSTOMER DATA Customer data obtained from the Services Is owned by and shall belong to Customer. JCI will access and use Customer data to provide Services to Customer. Except as set forth herein, JCI will not disclose to any third party any Individual Customer data acquired through performance of the Services without Customer's consent, Customer agrees that JCI and its subsidiaries, affillates and approved third party contractors and developers may collect and use Customer data for any reason, as long as any external use of the data is on a de-Identifled basis that does not personally Identify Customer or any Individual. Customer hereby grants JCI a perpetual, worldwide, irrevocable, royalty free license to use, modify, manipulate, sublicense, and create derivative works from such data, JCI shall retain all rights to any intellectual property, data, materials and products created as a result of its performance of Services. O. JCI'S INTELLECTUAL PROPERTY JCI shall retain ail right, title and Interest in any (a) work provided to Customer, including without {Imitation, all software source and object code, documentation, technical information or data, specifications and designs and any changes, Improvements or modifications thereto (“Deliverables”), and (b) Know-How (defined below) employed by JCI in the creation of the Deliverables or performance of the Services, whether known to JCI prior to, or developed or discovered or acquired in connectlon with, the performance of its obligations under this agreement. Ownership of all Deliverables and Know-How shall vest solely In JCI and no Deliverables shall be deemed “works made for hire.” Without limiting the generality of the foregoing, ownership of all source files used In the course of performing the Services shall remain the exclusive Property of JCI. For purposes of this Agreement, “Know-How” means any know-how, processes, techniques, concepts, methodologies, tools, analytical approaches, database models and designs, discoveries, and Ideas furnished, produced by, developed, or used by JCI In the creation or provision of the Deliverables or in the performance of the Services, and any changes, improvements, or modifications thereto or derivatives thereof, P. DIGITAL ENABLED SERVICES Digital Enabled Services; Data. If JCI provides Digital Enabled Services under this Agreement, these Digital Enabled Services require the collection, transfer and Ingestion of building, equipment, system time serles, and other data to JCI’s cloud-hosted software applications, Customer consents to and grants JCI right to collect, ingest and use such data to enable JCI and its affillates and agents to provide, maintaln, protect, develop and Improve the Digital Enabled Services and JCI products and services. Customer acknowledges that, while Digital Enabled Services generally Improve equipment performance and services, Digital Enabled Services do not prevent all potential malfunction, Insure against a loss, or guarantee a certain level of performance, Customer shall be solely responsible for the establishment, operation, maintenance, access, security and other — © 2026 Johnson Controls, Inc. Johnson Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 22 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FAQA35F3 «" Planned Service Proposal foe sty at CITY OF WARREN-BRAC aspects of Its computer network (“Network”), shall appropriately protect hardware and products connected to the Network and will supply JCI secure Network access for providing Its Digital Enabled Services, |f Customer accesses and uses Software that Is used to provide the Digital Enabled Services, the Software Terms (defined below) will govern such access and use. Q, JCI DIGITAL SOLUTIONS JCI Digital Solutions. Use, implementation, and deployment of the software and hosted software products (“Software”) offered under these terms shall be subject to, and governed by, JCI's standard terms for such Software and Software related professional services In effect from time to time at www.johnsoncontrols.com/techterms (collectively, the “Software Terms”), Specifically, the JCI General EULA set forth at www.Johasoncantrols.com/bultdings/legal/digital/generateuta governs access to and use of software Installed on Customer's premises or systems and the JCI Terms of Service set forth at www.johnsoncontrols.com/buildings legal/digital/generaltos govern access to and use of hosted software products, The applicable Software Terms are Incorporated herein by this reference, Other than the right to use the Software as set forth In the Software Terms, JCI and Its Ilcensors reserve all right, title, and Interest (Including all Intellectual property rights) In and to the Software and Improvements to the Software, The Software that Is Ilcensed hereunder Is licensed subject to the Software Terms and not sold, If there Is a conflict between the other terms herein and the Software Terms, the Software Terms shall take precedence and govern with respect to rights and responsibllities relating to the Software, its implementation and deployment and any Improvements thereto Notwithstanding any other provisions of this Agreement, unless otherwise agreed, the following terms apply to Software that Is provided to Customer on a subscription basis (i.e., a time limited license or use right), (each a “Software Subscription”): Each Software Subscription provided hereunder will commence on the date the initial credentials for the Software are made avallable (the “Subscription Start Date”) and will continue In‘effect until the expiration of the subscription term noted In the applicable statement of work, order or other applicable ordering document . At the expiration of the Software Subscription, such Software Subscription will automatically renew for consecutive one (1) year terms (each a “Renewal Subscription Term”), unless elther party provides the other party with a notice of non-renewal at least ninety (90) days prior to the expiration of the then-current term. To the extent permitted by applicable law, Software Subscriptions purchases are non-cancelable, and the sums paid nonrefundable. Fees for Software Subscriptions shall be pald annually in advance, invoiced on the Subscription Start Date and each subsequent anniversary thereof, Customer shall pay all invoiced amounts within thirty calendar days after the date of Invoice. Payments not made within such time perlod shall be subject to late charges as set forth in the Software Terms. Unless otherwise agreed by the partles In writing, the subscription fee for each Renewal Subscription Term will be priced at JCI's then-applicable Ilst price for that Software offering. Any use of Software that exceeds the scope, metrics or volume set forth In thls Agreement and applicable SOW will be subject to additlonal fees based on the date such excess use began, R, PRIVACY 1. JCl as Processor: Where JC! factually acts as Processor of Personal Data on behalf of Customer (as such terms are defined {n the OPA) the terms at www. johnsoncontrols.com/dpa (“DPA”) shall apply, 2, JC{ as Controller: JCI will collect, process and transfer certain personal data of Customer and its personnel related to the business relationship between it and Customer (for example names, emall addresses, telephone numbers) as controller and in accordance with JCI’s Privacy Notice at Customer acknowledges JCI's Privacy Notice and strictly to the extent consent Is mandatorily required under applicable law, Customer consents to such collection, processing and transfer, To the extent consent to such collection, processing and transfer by JCI Is mandatorily required from Customer's personnel under applicable law, Customer warrants and represents that it has obtained such consent. 5. MISCELLANEOUS PROVISIONS 1. All notices required to be glven hereunder shall be in writing and shall be considered properly given if; (a) delivered In person, (b) sent via the United States Postal Service, postage prepald, registered or certifled with return receipt requested, (c) sent by overnight delivery service (e.g., FedEx, UPS), or (d) sent by facsimile, email or other electronic means and confirmed by facsimile, return email or telephone, 2. This Agreement Is not assignable by the Customer except upon written consent of JCI first belng obtained. JCI shall have the right to assign thls Agreement, in whole or In part, or to subcontract any of its obligations under this Agreement without notice to Customer. This Agreement Inures to the benefit of and Is applicable to any assignees or subcontractors of JCI, and is binding upon Customer with respect to sald assignees or subcontractors with the same force and effect as it binds Customer to JCI, 3. If any provision of this Agreement Is found to be Invalid, illegal or unenforceable in any respect, the validity, legality and enforceability of the remaining provisions contained herein shall not In any way be affected or impaired thereby. 4. This Agreement Is the entire contract between JCI and Customer and supersedes any prior oral understandings, written agreements, proposals, or other communications between the parties, 5. Customer acknowledges and agrees that any purchase order Issued by Customer in connection with this Agreement Is intended only to establish payment authority for Customer's Internal accounting purposes and shall not be considered to be a counteroffer, amendment, modification, or other revision to the terms of this Agreement. No term or condition included or referenced {n Customers purchase order will have any force or effect and these terms and conditlons shall control. Customers acceptance of any Services shall constitute an acceptance of these terms and conditions. Any proposal for additional or different terms, whether In Customer's purchase order or any other document, unless expressly accepted In writing by JCI, Is hereby objected to and rejected. 6.JCI supplies “commercial items” within the meaning of the Federal Acquisition Regulations (FAR), 48 CFR Parts 1-53. As to any customer order for a U.S, Government contract, JCI will comply only with those mandatory flow-downs for commercial item and commercial services subcontracts Jisted elther at FAR 52.244-6, or 52.212-5(e)(1), as applicable, hnson ‘Ahi © 2025 Johnson Controls, Inc. Jo hnson ay Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express writlen permission of Johnson Controls, Inc. Page 23 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 ‘i+ | Planned Service Proposal t~ 1-4 CITY OF WARREN-BRAC 7. \f there are any changes to Customer's facilities or operations, or to applicable regulations, laws, codes, taxes, or ut ity charges, that materially affect JCI’s performance of the Services or its pricing thereof, JCI shall have the right to an equitable and appropriate adjustment to the scope, pricing, and other affected terms of this Agreement, ADDENDUM TO PSA TERMS AND CONDITIONS FOR MONITORING OF INTRUSION, FIRE AND OTHER SAFETY SYSTEMS If Remote Monitoring Services explicitly Includes remote flre alarm Monitoring, security alarm monitoring or video monitoring In the scope of work or customer charges, the Agreement Is hereby modified and amended to include the terms and provisions of thls Addendum to the PSA for Monitoring of Intrusion, Fire and Safety Systems (the “Addendum’), Capitalized terms that are not defined hereln, shall have the meaning given to them In the Agreement. In the event of a conflict between the terms and conditions of this Addendum and those appearing In the Agreement, the terms and conditions of this Addendum shall prevall, 1, Remote Monitoring of Alarm Signals, If JCI recelves an emergency alarm signal at JCI's ROC, JCI shall endeavor to notify the appropriate police or fire department, or other emergency response agency having Jurisdiction and JC! shall endeavor to notify Customer or Its designated representative by email unless instructed to do otherwise by Customer in writing and/or based on standard operating procedures for the ROC, JCI, upon recelpt of a non-emergency signal from the Premises, shall endeavor to notify Customer's representative pursuant to Customer's written Instructions, defaulting to emall or text notification. Customer acknowledges that if the signals transmitted from the Premises will be monitored In a monitoring facility not operated by JCI, the personnel In such monitoring facllitles are not the agents of JCI, nor does JCI assume any responsibility for the manner In which such signals are monitored or the response to such signal, 2, Remote Monitoring Services Pricing. Remote Monitoring Services shall be Provided by JCI if the Agreement includes a charge for such Service, If such Service is purchased, JC! will monitor the number of alarms for the Premises and the Initial charge is based on the pricing agreed to by the partles, subject to the terms and conditions of this Addendum. If the number of alarms produced at the Premises goes beyond the contracted number of alarms in a month, Customer will be billed an overage fee. 3, Communications Media, Customer acknowledges that monitoring of Covered Equipment requires transmission of signals over standard telephone lines and/or the Internet and that these modes of transmission may be Interrupted, circumvented, or compromised, in which case no slgnal can be transmitted from the Premises to the monitoring facillty, Customer understands that to allow the monitoring facility to be aware of such a condition, additional or alternative protection can be Installed, such as IIne security devices, at Customer’s cost and expense and for transmission via telephone line only. Customer acknowledges it Is aware that line security devices are avallable and, unless expressly Identlfled In Schedule A - Equipment List, has declined to purchase such devices, Customer further acknowledges that such additional protection Is not available for Internet transmission under this Agreement, 4. False/Unnecessary Alarms; Service Calls, At JCI's option, an additional fee may be charged for any false alarm or unnecessary Service Visit caused or necessitated by Customer. In addition, Customer shall be fully responsible and liable for fines, penaltles, assessments, taxes, fees or charges Imposed by a governmental body, telephone, communication, or signal transmission company as the result of any false alarm and shall reimburse JCI for any costs incurred by JCI In connection therewith. Customer shall Operate the system carefully so as to avold causing false alarms. False alarms can be caused by severe weather or other forces beyond the control of JCI. If an undue number of false alarms are recelved by JCI, In addition to any other available remedies available to JCI, JCI May terminate this Agreement and discontinue any Service(s) and seek to recover damages. If an agent Is dispatched, by a governmental authority or otherwise, to respond to a false alarm, where the Customer, or any other party has intentionally, accidentally or negligently activated the alarm signal, Customer shall be responsible for and pay any and all fees and/or fines assessed with respect to the false alarms and pay to JCI the additional charges and costs Incurred by It from a false alarm. If the Customer's system has a local audible device, Customer authorizes JCI to enter the Premises to turn off the audible device if JCI Is requested or ordered to do so by governmental authorities, nelghbors or anyone else and Customer will pay JCI its standard service call charge for each such visit, Police agencies require repair of systems which cause false dispatches. Customer shall maintain the equipment necessary for JCI to supply the Services and Customer shall pay all costs for such maintenance. At least monthly, Customer will test the system's protective devices and send test signals to the ROC for all monitoring equipment in accordance with Instructions from JCI or the ROC. Customer agrees to test the monitoring systems, Including testing any ultrasonic, mlcrowave, infrared, capacitance or other electronic equipment prior to the end of each month and will immedlately report to JCI if the equipment fails to respond to the test. Customer shall make any necessary repalrs as soon after recelpt of notice as Is reasonably practical. Customer shall at all times be solely responsible for maintaining any sprinkler system in good working order and provide adequate heat to the Premises. 5. Remote Monitoring of Video Monitoring Services, During the Term, JCI's sole and only obligation arising from the inclusion of Video Monitoring Services in any Service offering shall be to monitor the digital signals actually received by JC| at its ROC from means of the Video System and upon receipt of a digital signal Indicating that an alarm condition exists, to endeavor, as permitted by law, to notify the police or other municipal authority deemed appropriate In JCI's absolute discretion and to such persons Customer has designated in writing to JCI to receive notification of such alarm condition as set forth herein. . No alarm Installation, repair, maintenance or guard responses will be provided under this Video Monitoring Services option. JCI may, without prlor notice to Customer, In response to applicable law or Insurance requirements, revise, replace, discontinue and/or rescind its response policies and procedures, a. Inception and conclusion of service, Video Monitoring shall be provided by JCI if this Agreement Includes a charge for Video Monitoring Services. If such Video Monitoring Service Is purchased, Video Monitoring Services will begin when the Video System Is installed and operational, and when the necessary communications connection Is completed. No obligation for the provision of this Video Monitoring Service will commence until these requirements are met. b, Customer Equipment. Customer shall obtain, at Its own cost and expense: (a) the equipment necessary to connect to JCI’s ROC; and (b) whatever permission, permits or licenses that may be necessary from all persons, governmental authoritles, utility, and any other related service providers In connectlon with the Services. The video system to be used by the Customer Is Intended to produce and transmit video Images (the “Video System Images”) of the Premises to the ROC (the “Video System”). JCl makes no promise, warranty or representation that the video system will operate as Intended. Customer further agrees that, notwithstanding any role or y) it © 2025 Johnson Controls, Inc. Johnson )) Customer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc, Page 24 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 ‘ts tnicl Planned Service Proposal oe et CITY OF WARREN-BRAC participation by JCI In Video System and Video System Images, JCI shall have no responsibility or obligation with regard to Customer, the Video System or any other Customer equipment. c. System Location. The Video System related cameras shall be located and positioned by Customer along with attendant burglary digital alarm slgnal(s). Customer shall ensure that the Video System related cameras will be positioned and located such that It will only produce or capture Video System Images of areas of the Premises. Customer will provide adequate Illumination under all operating conditions for the proper viewing of the cameras. Customer acknowledges and agrees that JCI has exerclsed no control over, or participated in locating or positioning the Video Systern related camera Including, but not limited to selecting what areas, locations, things or persons that the Video System Images may depict or capture, d. Images. Customer shall be solely responsible for the Video System Images produced or captured by the Video System and Customer shall defend, indemnify and hold harmless JCI and Its officers, agents, directors, and employees, from any and all damages, losses, costs and expenses (including reasonable attorneys’ fees) arising out of third party clalms, demands, or sults In connection with the use, operation, location and position of the Video System, and the Video System Images resulting there from, including, but not Ilmited to, any claims of any person depicted In a Video System Image, Including but not limited to, any clalm by such person that his or her privacy has been invaded or Intruded upon or his or her likeness has been misappropriated, Any duty to obtain the consent or permission of any person depicted In a Video System Image to have his or her likeness to be depicted, recelved, transmitted or otherwise used, and the duty to determine and comply with any and all applicable laws, regulations, standards and other obligations that govern the legal, proper and ethical use of video capturing devices, such as the Video System, Including, but not Ilmited to, notification that the Video System is In use at the Premlses, shall be the sole responsibility of the Customer. JCI agrees to make Video System Images available to Customer and upon thelr respective request. JCI makes no promise, warranty or representation as to the length of time that it retains Video Images, or the quality thereof. e, Video System Signals. When a signal from the Video System Is recelved, JCI reserves the right to verify all alarm signals before notifying emergency personnel, and may choose not to notify emergency personnel If It has reason to belleve, In its sole discretion, that an emergency condition does not exist, JCI will first attempt to verify the nature of the emergency by using visual verification and/or the two-way volce system (if applicable) of the Video System Included in Customer's system. If JCI determines that an emergency condition exists, JCI will endeavor to notify the proper police or emergency contact on a notification call list provided In writing by Customer to JCI, or Its designee, When a non-emergency signal Is received, JCI will attempt to contact the first available Customer representative on the notification call list but will not notify emergency authorities, thls notification will be in the form of email or text and follow ROC processes, lf the customer requires phone calls to the call list for any emergency or non-emergency situation, the customer will need to make this request In writing, Customer authorizes and directs JCI, as its agent, to use its full discretion In causing the arrest or detention of any person or persons on or around the premises who are not authorized by Customer, JCI WILL NOT ARREST OR DETAIN ANY PERSON. f. Recordings. Customer consents to the tape recording of all telephonic communications between the Premises and JCI. JCI will have no liability arising from recording (or failure to record) or publication of any two-way volce communications, other video recordings or thelr quality, JCI shall have no Ilability In connection with Video System or the Video System Images, Including, but not limited to, any failure, omission, negligence or other act by JCI, or any of its officers, employees, representatlves, agents, contractors, or any other third party in connection with the receipt (or fallure of receipt), transmission, reading, interpreting, or response to any Video Image. 6. Risk of Loss is Customer's, JCI does not represent or warrant that the Services will detect or prevent the events they are Intended to help detect or prevent Including any burglary, holdup, fire or otherwise ("Detection Events"), or that the Services will be uninterrupted or error-free. Customer assumes all risk of loss or damage to Customer's person or property, or the property of others, if such an event occurs. Neither JCI nor its suppliers shall have any liability for loss, damage or injury due directly or Indirectly to Detection Events, or the consequences therefrom, Further, expressly excluded from thls Agreement are the warrantles of merchantabllity or fltness or sultabllity for a particular purpose NOTWITHSTANDING THE FOREGOING, IF JOHNSON CONTROLS SHOULD BE FOUND LIABLE FOR LOSS, DAMAGE OR INJURY DUE TO AN ACT OR OMISSION BY JOHNSON CONTROLS OR A FAILURE OF THE EQUIPMENT OR SERVICE IN ANY RESPECT, ITS LIABILITY WILL BE LIMITED TO A SUM EQUAL TO 10% OF THE ANNUAL SERVICE CHARGE OR $1,000, WHICHEVER 1S GREATER, AS THE AGREED UPON DAMAGES AND NOT ASA PENALTY, AND AS THE EXCLUSIVE REMEDY. THIS PARAGRAPH SHALL APPLY IF LOSS, DAMAGE OR INJURY, IRRESPECTIVE OF CAUSE OR ORIGIN, AND REGARDLESS OF LEGAL THEORY, RESULTS DIRECTLY OR INDIRECTLY TO PERSON OR PROPERTY OR FROM PERFORMANCE OR NONPERFORIMANCE OF OBLIGATIONS IMPOSED BY THIS AGREEMENT. IF ANY THIRD PARTY, INCLUDING CUSTOMER'S SUBROGATING INSURER, MAKES A CLAIM OR FILES A LAWSUIT AGAINST JOHNSON CONTROLS IN ANY WAY RELATING TO THE EQUIPMENT OR SERVICES, CUSTOMER AGREES TO INDEMNIFY AND HOLD HARMLESS JOHNSON CONTROLS FROM ANY AND ALL SUCH CLAIMS AND LAWSUITS INCLUDING THE PAYIMENT OF ALL DAMAGES, EXPENSES, COSTS AND ATTORNEVYS' FEES. 7. JCI’S RECEIPT OF ALARM SIGNALS, ELECTRONIC DATA, VOICE DATA OR IMAGES (COLLECTIVELY, “ALARM SIGNALS”) FROM THE EQUIPMENT OR SYSTEM INSTALLED IN THE PREMISES IS DEPENDENT UPON PROPER TRANSMISSION OF SUCH ALARM SIGNALS. JCV’S ROC CANNOT RECEIVE ALARM SIGNALS WHEN THE CUSTOMER’S TELCO SERVICE OR OTHER TRANSMISSION MODE (S NOT OPERATING OR HAS BEEN CUT, INTERFERED WITH, OR IS OTHERWISE DAMAGED, OR IF THE ALARM SYSTEM IS UNABLE TO ACQUIRE, TRANSMIT OR MAINTAIN AN ALARM SIGNAL OVER CUSTOMER’S TELCO SERVICE OR TRANSMISSION MODE FOR ANY REASON INCLUDING BUT NOT LIMITED TO NETWORK OUTAGE OR OTHER NETWORK PROBLEMS SUCH AS CONGESTION OR DOWNTIME, ROUTING PROBLEMS, OR INSTABILITY OF SIGNAL QUALITY. CUSTOMER UNDERSTANDS THAT SIGNAL TRANSMISSION FAILURE MAY OCCUR OVER CERTAIN TYPES OF TELCO SERVICES SUCH AS SOME TYPES OF DSL, ADSL, VOIP, DIGITAL PHONE, INTERNET PROTOCOL BASED PHONE OR OTHER INTERNET INTERFACE-TYPE SERVICE OR RADIO SERVICE, INCLUDING CELLULAR, WIRELESS OR PRIVATE RADIO, OR CUSTOMER'S PROPRIETARY TELCOMMUNICATION NETWORK, INTRANET OR IP-PBX, OR OTHER THIRD-PARTY EQUIPMENT OR VOICE/DATA TRANSMISSION NETWORKS OR SYSTEMS OWNED, MAINTAINED OR SERVICED BY CUSTOMER OR THIRD PARTIES, IF: (1) THERE 1S A LOSS OF NORMAL ELECTRIC POWER TO THE MONITORED PREMISES OCCURS (THE BATTERY BACK-UP FOR JCI’S ALARM PANEL DOES NOT POWER CUSTOMER’S COMMUNICATION FACILITIES OR TELCO SERVICE); OR (2) ELECTRONIC COMPONENTS SUCH AS MODEMS MALFUNCTION OR FAIL. CUSTOMER UNDERSTANDS THAT JCI WILL ONLY REVIEW THE INITIAL COMPATIBILITY OF THE ALARM SYSTEM WITH CUSTOMER'S TELCO SERVICE AT THE TIME OF INITIAL INSTALLATION OF THE ALARM SYSTEM AND THAT CHANGES IN THE TELCO SERVICE’S DATA FORMAT AFTER JCI’S INITIAL REVIEW OF COMPATIBILITY COULD MAKE THE TELCO SERVICE © 2025 Johnson Controls, Inc. Johnson ‘ Cuslomer Confidential Controls Do not copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 25 of 26 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 deo sail Planned Service Proposal Pues tet vt CITY OF WARREN-BRAC UNABLE TO TRANSMIT ALARM SIGNALS TO JCI'S ROC, IF JCI DETERMINES IN ITS SOLE DISCRETION THAT CUSTOMER'S TELCO SERVICE IS COMPATIBLE, JCI WILL PERMIT CUSTOMER TO USE ITS TELCO SERVICE AS THE PRIMARY METHOD OF TRANSMITTING ALARM SIGNALS, ALTHOUGH CUSTOMER UNDERSTANDS THAT JC| RECOMMENDS THAT CUSTOMER ALSO USE AN ADDITIONAL BACK-UP METHOD OF COMMUNICATION TO CONNECT CUSTOMER'S ALARM SYSTEM TO JCI’S ROC REGARDLESS OF THE TYPE OF TELCO SERVICE USED, CUSTOMER ALSO UNDERSTANDS THAT IF JCI DETERMINES IN ITS SOLE DISCRETION THAT CUSTOMER'S TELCO SERVICE IS, OR LATER BECOMES, NON-COMPATIBLE, OR IF CUSTOMER CHANGES TO ANOTHER TELCO SERVICE THAT IS NOT COMPATIBLE, THEN JCI WILL REQUIRE THAT CUSTOMER USE AN ALTERNATE METHOD OF COMMUNICATION ACCEPTABLE TO JCI AS THE PRIMARY METHOD TO CONNECT CUSTOMER'S ALARM SYSTEM TO JCI’S ROC, JCI WILL NOT PROVIDE FIRE OR SMOKE ALARM MONITORING FOR CUSTOMER BY MEANS OTHER THAN AN APPROVED TELCO SERVICE AND CUSTOMER UNDERSTANDS THAT IT IS SOLELY RESPONSIBLE FOR ASSURING THAT IT USES APPROVED TELCO SERVICE FOR ANY SUCH MONITORING AND THAT IT COMPLIES WITH NATIONAL FIRE ALARM STANDARDS AND LOCAL FIRE CODES, CUSTOMER ALSO UNDERSTANDS THAT IF CUSTOMER'S ALARM SYSTEM HAS A LINE CUT FEATURE, IT MAY NOT BE ABLE TO DETECT ALARM SIGNALS IF THE TELCO SERVICE IS INTERRUPTED, AND THAT JCI MAY NOT BE ABLE TO DOWNLOAD SYSTEM CHANGES REMOTELY OR PROVIDE CERTAIN AUXILIARY MONITORING SERVICES THROUGH A NON-APPROVED TELCO SERVICE, CUSTOMER ACKNOWLEDGES THAT ANY DECISION TO USE A NON-APPROVED TELCO SERVICE AS THE METHOD FOR TRANSMITTING ALARM SIGNALS [Ss BASED ON CUSTOMER’S OWN INDEPENDENT BUSINESS JUDGMENT AND THAT ANY SUCH DECISION IS MADE WITHOUT ANY ASSISTANCE, INVOLVEMENT, INPUT, RECOMMENDATION, OR ENDORSEMENT ON THE PART OF JCI. CUSTOMER ASSUMES SOLE AND COMPLETE RESPONSIBILITY FOR ESTABLISHING AND MAINTAINING ACCESS TO AND USE OF THE NON-APPROVED TELCO SERVICE FOR CONNECTION TO THE ALARM MONITORING EQUIPMENT. CUSTOMER FURTHER UNDERSTANDS THAT THE ALARM SYSTEM MAY BE UNABLE TO SEIZE THE TELCO SERVICE TO TRANSMIT AN ALARM SIGNAL IF ANOTHER CONNECTION HAS DISABLED, IS INTERFERING WITH, OR BLOCKING THE CONNECTION, CUSTONER ACCEPTANCE i cret op eigAgreement, Qustomer 8p Se tothe tems and on “cers atcha) he COPAY AND OFA Lerma BAT COND TONS 1S uNdEreodd tha tne yas vero ane an iret Casorrer mayisout, Any chaqoes requested ty Cute: merges sta | oe aumetzed ‘1 Wr7tg. ATTENTION S DIRECTED TO THE LIMITATION OF LIABILITY, WARRANTY, No THIS AGREEMENT. ins Agere end my stannerts o MYVAtSi on aariie and ond tons cn Bry Tent sal ce pad tor oy the Qustoner and such ITY AND OTHER CONDITIONS CONTAINED IN Perng spaced ujen thet of Comics ACH EFT b agb ng ad peynrert tenn: Invces Wi be dé ered V3 ene |, payntant due dx] oh NET 3, ard tyozes 2¢ 10 be ced VAACH bee tender ans 2213'S vs be $9 coming vac convedud agreement, This offer shall be void if not accepted in writing within thirty (30) days fromthe date first set forth above. [END OF DOCUMENT] © 2025 Johnson Controls, Inc. Johnson Customer Confidential Controls Do nol copy (physically, electronically, or in any other media) without the express written permission of Johnson Controls, Inc. Page 26 of 26 56 Docusign Envelope ID: BD031A57-4616-4E63-AAFF-C2C6FA9A35F3 CITY OF WARREN Office of the Council Secretary INTER-OFFICE COMMUNICATION Item 4j DATE: September 13, 2022 TO: James R. Fouts, Mayor SUBJECT: Request of the Building Maintenance Division to apptove a price increase and extend award for premium and basic heating, ventilation, and air conditioning (HVAC) maintenance services TO Johnson Controls. Inc., for a four (4) year estimated total cost of . $1,318,332,.68, CONSIDERATION AND OF A RESOLUTION, At a regular meeting of the City Council held Tuesday, September 13, 2022, Council made the formal motion to approve the above listed item. Trusting this information to be of value. PMamaley, 7 pee. Mindy Moore Council Secretary cc; Attorney Clerk Building 57 Docusign Envelope ID: BD031A57-4618-4E63-AAFF-C2C6FASA35F3 RESOLUTION ; Document No: RFP-W-9103 Product or Service: HVAC Services — Price Increase for Year 7 of Contract and iG Extension for Years 8-10 ; Requesting Department: Various At a Regular Meeting of the City Council of the City of Warren, County of 3 Macomb, Michigan, held on _ September 13 , 2022 at 7 p.m. Local Time, in the Council Chamber at the Warren Community Center Auditorium, 5460 Arden, Warren, Michigan. PRESENT: Councilmembers; Green, Kabacinski, Lafferty, Moore, Papandrea, Rogensues, Watts ABSENT: Councilmembers: _ None The following preamble and resolution were offered by Councilmember Moore and supported by Councilmember. Lafferty . On May 24, 2016 City Council approved an award to Johnson Controls, Ine. (JCI 611 Sterling Drive North, Sterling Heights, Mi 48312 to furnish both Premium Coverage and Basic Coverage HVAC Maintenance Services for the City of Warren. The award was approved for an initial three-year term with options to extend for three additional two-year periods, followed by on final option to extend for one additional year. | On April 23, 2019 Gity Council approved an extension of RFP-W-9103 for the period May 1, 2019 through April 30, 2021. On March 23, 2021 City Council approved an extension of RFP-W-9103 for the period May 1, 2021 through April 80, 2023. : Resolution RFP-W-9103 Price Increase and Extension ~ Page 4 58 Docusign Envelope ID; BD031A57-4616-4E63-AAFF-C2C6FAQA35F3 THEREFORE, !T IS RESOLVED, that the award for Premium and Basic HVAG i Maintenance and Repair Services with Johnson Controls is hereby accepted by City Council for the final extension period, which Is a three-year period, commencing on May i 1, 2023 and running through April 30, 2026 in an estimated annual amount of $329,583.17, based on the approval of a 8.58% price increase becoming effective on May 1, 2022, Funds are available in the varlous departmental accounts. IT {S FURTHER RESOLVED, that the Purchasing Agent and/or Mayor and City Clerk are authorized to execute any such documents that are necessary for this approval consistent with the terms of the: Proposal document X Contract X Resolution and in such form that meets with the satisfaction of the City Attorney if review Is | requited.In accordance with the City Council duly adopted budget resolution of May 12, 2022. AYES: Councilmembers: Moore, Lafferty, Kabacinski Papandrea Rogensues, Watts, Green NAYS: Councilmembers: None i RESOLUTION DECLARED ADOPTED this_13th day of September , 2022. Mindy oore é Secretary of the Council : Resolution RFP-W-9103 Price Increase and Extension ~ Page 2 Docusign Envelope ID: 9DF67EF7-9B0D-472F-942E-11FCE1DAA9FB (ARREN March 25, 2025 Tom Bommarito WARREN TN eee : choenherr City of Warren Warren, MI 48089 One City Square (586) 756-2800 Suite 215 WWw.cityofwarren.org Warren, MI. 48093 Re: Fire Station #1 Utility Pole and Guy wire relocation Dear Mr. Bommarito, The Fire Department is requesting the Downtown Development Authority to authorize funding for the relocation of a utility pole and guy wire from the North side of Continental to the South side of Continental. The movement of the utility pole and guy wire will eliminate the guy wire from creating an egress obstruction in front of the rear man door at new fire station #1. Currently the guy wire lands in the middle of the patio and in front of the rear man door exit to the patio creating an egress and trip hazard for employees exiting fire the station using this door. The relocation eliminates the code violation and egress hazard that requires unobstructed egress from a building. Additionally by relocating the utility pole and guy wire allows the patio to function as designed by the architect creating a clear unobstructed patio space and outdoor cooking and sitting area during summer months. The total cost of this project equals $10,574.13, Project Details: Recommended Company: DTE Energy, 41112 Concept Dr., Plymouth MI 48170 Bid Number: Sole Source Requested Amount: $10,574.13 GL/Account Number: 493-9493-97450 Please find attached the invoice and contract from DTE Energy for the relocation of a utility pole and guy wire from the North side of Continental to the South side of Continental. Total project cost equals $10,574.13. Please direct questions to my attention at Ext. 3100. Professionally, Uldurt Nef Laws [ art Mapp Craig Trgpa —GBEGB7ESSAUDABD... S7ABOOCBFDSEAES... COp46D871F6C4BB... Wilburt McAdams Mark Knapp Craig Treppa Fire Commissioner Assistant Controller Purchasing Agent Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-11FCE1DAASFB 43230 Elizabeth Rd Clinton Township, MI 48036 o¥ DYE Energy: _ Tuesday, February 25, 2025 City Of Warren Wilburt McAdams 23211 Van Dyke Ave Warren, Mi 48089 Regarding: 23211 Van Dyke Ave, Warren Enclosed are two (2) copies of the Accounts Receivable Agreement for your signature, The payment for this work is $10,574.13 based on: City of Warren request for sidewalk guy to be removed. DTE to install new pole with slack-span in order to support sidewalk guy removal. Please return the signed agreement to me with a check made payable to OTE Energy. Keep the “Customer Copy" document for your records. To ensure proper credit, the Agreement number should be indicated on your remitted check. When we receive the signed agreement and your check, we will proceed to schedule the work, lf you have any questions regarding this job, please feel free to contact me at the phone number or e-mail address indicated below. Sincerely, Danielle Lynn Clark Planner danielle.clark@dteenergy.com enclosures: Two copies of the Accounts Receivable Agreement 61 Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-11FCE1DAASFB o¥ DTE Energy: Accounts Receivable Agreement No, 74701168 “DTE Energy" and "Customer" make this agreement for consideration of the promises in the Agreement. "DTE Energy” Is: "Customer" Is: The DTE Energy Company City Of Warren 43230 Elizabeth Ad Wilburt McAdams Clinton Township, Mi 48036 23211 Van Dyke Ave Warren, MI 48089 Background Statement: Customer requests DTE Energy to perform the work Indicated below in the vicinity of 23211 Van Dyke Ave, Warren. To do this, DTE Energy requires that payment be made in the amount indicated below. Under Michigan Public Service Commission rules, DTE Energy is permitted to require payment before performing this work, DTE Energy and Customer agree to the following terms: See details of this contract under the Terms and Conditions section Payment for the requested work Is $10,574.13. The type of work to be performed: ° City of Warren request for sidewalk guy to be removed. OTE to install new pole with slack-span in order to support sidewalk guy removal. In return for the above payment, The DTE Energy Company agrees to perform the requested work, providing all necessary permits and rights-of-way can be secured, This job will not be scheduled until OTE Energy receives payment for the above work. Notwithstanding anything herein to the contrary, the installation, ownership, and maintenance of electric services and the rates, fees, and charges to be made shall be subject to and In accordance with the orders and rules and regulations adopted and approved trom time to time by the Michigan Public Service Commission, DTE Energy:(sign) Kokeb Waglae @ Title: ___ Supervisor Date: 2-25-2025 Rokib M Mylex Customer:{sign) (print) Date: (sign) (print) Date: o62012 W.O. 74701168 Page lat 3 62 Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-11FCE1DAASFB Terms and Conditions Accounts Receivable Agreement 1. MPSC Rules - This Agreement is subject to the Michigan Public Services Commission (“MPSC) Rules, including but not limited to, Rule C6.1, “Extension of Service", Rule C6.2, “Overhead Extension Policy", Rule C6.3, "Underground Distribution Systems"; and Rule C6.5 "Miscellaneous Customer Requests", which are Incorporated herein by reference. 2. Description of Work - DTE Energy or one of its contractors shail install or ramove the underground or ovarhaad conductors and any associated overhead or underground equipment required for the request, (collectively, the “Wark"), DTE Energy will only instal! electric service and is not responsible for any other utility service Including, bul not limited io, cable television, or olhar communication services. Customer shail contact those companies responsible for the installation of services other than electrical service. 3. Customer Staking Requlraments &. Prior to commencement of the Work, Customer shail visually identity, by ailhar exposing or clearly slaking thraugh the use of flags or other appropriate Identilication device, all private underground property, including but not limited to: a. privata electrical lines I. burlal sites of pets b. sprinkler systems }. geothermal systams c. Invisible fences x. private walter mains and lines d. swimming pool hardware solar power equipment @, septic flanks and fields m, privately owned gas {. fiber optic tines n. propane and petroleum lines 9. security systems 0, any other underground equipment not praviously listed. h. heated sidewalk and driveway equipment b. If Gustomar refuses the route suggested by OTE Energy for the Work and requests an alternative toute, which Is mutually agreed to by Customer and OTE Energy, Customer shalt stake the alternative route as provided in paragraph 3(a) above. ¢, If Customer falls to clearly stake all private underground property, then Customer releases OTE Energy trom any and all Ilabliily for properly damage related to the Installation, oparation or maintenance of the Work, including, but not limited to, toss of trees, shrubs or othor landscape. 4. Total Payment - By executing this Agresment, Customer agrees to pay DTE Energy the "Total Payment" calculated on page 1 of this agreament. " Termination prior to Commencement of Work - if Customer fails to complete any obligations under this Agreement within six (6) months from the dale DTE Energy raceives tull payment or the Total payment, then, upon written notice, DTE Energy may cancel this Agreement and a relund may be Issued ta Customer, less all reasonable costs incurred by DTE Energy, 6. Fallura to Execute Agreement; Changes to Agreement: If the Customer falls to execute this Agregmant and pay the Total payment due to DTE Energy within six (6) months of Ihe date of this Agreement, than this Agreamen! shall become null and void, Further, Customer shall not Make any changes fo this Agreement, including but not timiled to handwrilten changes or striking any language. In the event Customer makes any changes to this Agreement without the specitic written consent of DTE Energy, then this Agreement shall become null and void, 7, Damages and Limitation on Liability - f Customer, its contractors, agents, and/or employees cause damage to the Work, then Customer shall reimburse DTE Energy for all costs related to that damage. DTE Energy reserves the right to retain portions of the Refundable Construction Advance to offset such damages. OTE Energy's sola liability 1 Customer, ts employees, agenis, subcontractors and to ail olher parsons arising oul of or related to the performance of the Work, whether in centract, under any claims warranty, In tort, or otherwise shall be Simited to either DTE Energy rapairing or replacing the Work at iis own expense or, at DTE Energy's option, refund the money paid for the Work. The foregoing shall be Customer's sole remedy. In no event will DTE Energy or its contractors be table under this Agreement or under any cause of action relating to the subject mater of this Agreement, whether based on contract, warranty, tort (including negligence), strict lability, indemnity or otherwise, for any Incidental or consequential damages including but no} limited to loss of use, interast charges, inablily to operate full capacily, lost profits or other simllar claims of Customer. 8, Set Off - DTE Energy shall be entitled at any time to set off any sums owing by Customer or any of Customer's afliliated companies with common ownership, lo DYE Energy or any of DTE Energy's affiliated companies, agalnst sums payable by DTE Energy. 9. Assignment and Nofices - Customer shall not assign this Agreement without DTE Energy's prior wrilten consent. All notices required by \his Agreement must be in writing and sent by U.S. mail or delivered in person to the addresses listed on page 1 of this Agreement. 10. Saving Clauge - Each term and condition of this Agreement is deemed to have an independent effect and tha invalidity of any partial or whole paragraph or section shall not invalidate Ihe remaining paragraphs or sections, The obligation to perform alf of the terms and conditions shall remain in affect regardless of the perlormance of any invalid tarm by the other party, 8/2012 W.O, 74701165 Pagezol 3 Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-1 1FCE1DAASFB 11, Governing Law and Jurisdictlon - This Agreement shail be construed In accordance with the law of the State of Michigan, without regard ta conflict of law principals. The partlas agree that any action with respect to this Agreement shail be brought in a court of competent Jurisdiction located in the State of Michigan and the parties hereby submit themselves to {he exclusive jurisdiction and venus of such court for the purpose of such action. 12, Entire Agreement - This Agreement logather with the DTE Energy Rate Book on file with the MPSC, the Elecirical Service Installation Gulda, which is avallable at: www.dieenergy.com/businessCustomers/nuildersContractors/electricService/standards.html and, if applicabie, the Certificate of Grade (referred to herein collectively, as the "Contract Documents") constitutes the entire Agreement beiween the parlles regarding this transaction. Any agreements, negotiations or understanding of the parties prior to or contemporanaous to the date of the Agreement, whather written or oral, are superseded hereby. in the evant of a conflict between the Contract Documents, then the Contract Document shall control In the order stalad above. Ge2012 O. TAZ011 68 Pogsdola Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-11FCE1DAAQFB é DTE Energy: Accounts Receivable Agreement No. 74701168 "DTE Energy" and "Customer" make this agreement for consideration of the promises in the Agreement. "DTE Energy” is: "Customer" Is: The OTE Energy Company City Of Warren 43230 Elizabeth Rd Wilburt McAdams Clinton Township, MI 48036 23211 Van Dyke Ave Warren, Ml 48089 Background Statement: Customer requests DTE Energy to perform the work indicated below in the vicinity of 23211 Van Dyke Ave, Warren. To do this, DTE Energy requites that payment be made in the amount indicated below, Under Michigan Public Service Commission rules, DTE Energy Is permitted to require payment before performing this work. DTE Energy and Customer agree to the following terms: See details of this contract under the Terms and Conditions section Payment for the requested work is $10,574.13. The type of work to be performed: City of Warren request for sidewalk guy to be removed. DTE to install new pole with slack-span in order to support sidewalk guy removal, In return for the above payment, The DTE Energy Company agrees to perform the requested work, providing all necessary permits and rights-of-way can be secured. This job will not be scheduled until DTE Energy receives payment for the above work. Notwithstanding anything herein to the contrary, the instaflation, ownership, and maintenance of electric services and the rates, fees, and charges to be made shall be subject to and in accordance with the orders and rules and regulations adopted and approved from tima to time by the Michigan Public Service Commission. DTE Energy:(sign) Kaki Wyfrs @ Title; ___ Supervisor Date: 2-25-2025 Rokib M Mylés Customer:{sign) (print) Date: (sign) (print) Date: on2012 W.O. 74704168 Pega tots 65 Docusign Envelope ID: 9DF67EF7-9BOD-472F-942E-11FCE1DAASFB Terms and Conditions Accounts Recelvabie Agreement 1, MPSC Rules - This Agreement is subject to the Michigan Public Services Commission ("MPSC") Rules, Including but not limited to, Rule C6.1, "Extension of Service", Rule C6.2, “Overhead Extension Policy", Rule C8.3, “Underground Distbution Systems"; and Rule C6.5 “Miscellaneous Customer Requests", which are incorporatad herein by reference. 2, Description of Work - DTE Energy or one of lis contractors shail install or remove the underground or overhead conductors and any associated overhead or underground equipment required for the request, (collectively, the "Work"), DTE Energy will only Install electric service and is not responsible for any other ulllily service including, but not limited to, cable television, or other communication services, Customer shall contact those companies responsible for the installation of services other ihan electrical service. 3. Customer Staking Requirements a. Prlor to commencement of the Work, Customer shall visually Idenlily, by either exposing or clearly staking through the use of flags or ather appropriate identification device, all private underground property, including but not limited to: a. private electrical tines |. burlal sites of pats b. sprinkler systems j. geothermal systems 0. invisible fences k. private water mains and lines d. swimming pool hardware 1, solar power equipment @. septic tanks and lisids m. privately owned gas {. fiber optic tines n. propane and petroleum lines g. securily systems 0. any other underground equipment not previously listed. : h, heated sidewalk and driveway equipment b. HM Customer refuses the routa suggested by DTE Energy for the Work and requasts an allernative route, which is mutually agread to by Customer and DTE Energy, Customer shail stake the alternative route as provided in paragraph 3(a) above. e. If Customer fails to clearly stake all private underground property, (hen Customer releases OTE Energy from any and all Ilability for properly damage related to the installation, operation or malntanance of ihe Work, inciuding, bul nat limitad to, toss of trees, shrubs or other Jandscape, 4. Total Payment - By executing this Agreement, Customer agreas to pay DTE Energy the "Tata! Payment" calculated on page 1 of this agreement. 1 prior to t of Work - if Customer falls to complete any obligations under this Agreamant within six (6) months trom the date DTE Energy recelves full payment or the Totat payment, then, upon written notice, OTE Energy may cancel this Agreement and a refund may be issued to Customer, less all reasonable costs incurred by OTE Energy. Failure to E te Agr t; Ch to Agreement: If ihe Customer fails to execula this Agraement and pay the Totat payment due to OTE Energy within six (6) months of tha date of this Agreement, then this Agreement shall become nul and void, Further, Customer shall not make any changes fo this Agreement, including but not limited to handwrilten changas or striking any language. In the evant Customer makes any changes to this Agreement without the spacifle written consent of DTE Enargy, then this Agreament shall become null and void. Damages and Limitation on Liability - {{ Customer, its contractors, agents, and/or employees cause damage to Ihe Work, then Customer shall reimburse DTE Energy for all costs related to that damage. OTE Energy reserves the right to retaln portions of {he Refundable Construction Advance to offsel such damages. DYE Energy's sole liability to Customer, its employees, agents, subcontractors and to all other persons arising out of or related to the performance of the Work, whether in contract, under any claims warranty, in tort, or otherwise shall be limited to either OTE Energy repairing or replacing the Work at its own expense or, at DTE Energy's option, refund the money pald for the Work, The foregoing shall be Customer's sole remedy. In no event will DTE Energy or ils contractors be fiable under this Agraement or under any cause ol actlon relating lo tha subject matter of this Agreement, whether based on contract, warranty, tort (including negligence), strict labilily, indemnity or otherwise, for any incidental or consequential damages including but not limited to loss of use, interest charges, Inability to operate full capacity, lost profits or other similar claims of Customer. » ? = Set Off - DTE Energy shall be entijled al any time to set off any sums owing by Customer or any ol Customer's affillated companias wilh common ownarship, to OTE Energy or any of DTE Energy's alliliated companies, against sums payable by DTE Energy. Assignment and Notices - Customer shall not assign this Agreement without DTE Energy's prior written consent, All notices required by this Agreement must be in writing and sent by U.S. mail or delivered in person to the addresses listed on page 1 of this Agreement, 10. Saving Clause -» Each term and condition of this Agraement is deemed ta have an independant effect and the invalidity of any partial or whole paragraph or section shall not invalidate the remaining paragraphs or sections. The obligation to perform all of the terms and conditions shall remain in effect regardless of the performance of any Invalid term by the other parly, bad oa012 W.O, 74701168 Pagezot 9 66 Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-11FCE1DAASFB 41. Governing Law and Jurisdiction « This Agreement shall be construed In accordance with the law of the State of Michigan, without regard to contilc! of law principals. The partles agree that any action wilh respect to this Agreernent shall be brought in a court of competent jurisdiction located in the State of Michigan and the parties hereby submil themselves to the exclusive jurisdiction and venue of auch court for the purpose of such action. 12. Entire Agreement - This Agreement logether with the OTE Energy Rate Book on file with the MPSC, the Etectrical Service Installation Guide, which js available ai: www.dlesnergy.com/businessCuslomers/bulldersContraclors/electricService/standards.him! and, if applicable, tha Certificate ol Grade (rafarrad {o herein collectively, as the “Contract Documents") constitutes the entire Agreement between the partias regarding {his transaction. Any agreaments, negotiations or understanding of the parties prior to or contemporaneous to the date of the Agreement, whelhar written or oral, are superseded hereby. in the event of a contilel betwaen the Contract Documents, then the Contract Document shall control in the order stated above. ONO? W.O. 74701168 Page dof 3 P 67 Docusign Envelope ID: SDF67EF7-9B0D-472F-942E-11FCE1DAA9FB DTE Sil! SCOPE OF WORK: CITY WANTS SIDEWALK GUY REMOV MAKE ROOM FOR PATIO. INSTALLING SLACK SPAN POLE IN ORDER SUPPORT SWG REMOVAL ED TO SAC NWA J [Know what's below, Call before you tig MISS DIG FROM CENTERLINE WL1: 34'S CONTINENTAL & 230' W M53 WL2; 32' N CONTINENTAL & 230'W M53 23211 VAN DYKE AVE WARREN CITY FIRE STATION NAME-IDR { CONSTRUCTION NOTES in| BUILT AS DESIGNED SEE NOTED CHANGES FROM DESIGN SKETCH O 4.8 KV AREA m Oo Oo WL. 2: 352281-354288 LW M 53 1PN CONTINENTAL RM: 10IN HELIX ANC RM: SWG 42FT 5/16IN 40FT RM SIDEWALK GUY GOING NORTH CONTINENTAL AVE WL 1: 352284-354213 LW M 53 1PS CONTINENTAL IN: 45 3 PINE IN: SSS-DET 282 rs IN: DET 25X35B S$ 96 7 IN: DET 612A @ E QUAD IN: 3- SLEEVE 4.8KV IN: 20' 1/0 ACSR-C AS NEEDED 6 - OPERATING: RE-SLACK PRI SLACK SPAN BOTH WAYS O PAIGE AVE QE Oo BAY SAC NWA CUSTOMER CONTACT: SKIP (FIRE CHEIF) 586-756-2800 DTE ote ctectric - Distribution Engineering and Planning {— LEGEND ~ © EXISTING DECO POLE KE AVE, WARREN - INSTALL SLACK SPAN x PROPOSED POLE Phone) ra704 178 GUL CUG. pric 1) FORRIGN POLE 734.280,0062 f4i0 166 Cireult 2 PH SCMAT So EKISTING ANCHOR Supaivagt sd _ 74701174 | 74704477 @-—PROPOSED ANCHOR Sonica Canter] Worksde Clip Workrie twp County WARREN WRN Maco; mee wwenjU NOT REQUIRED ” . cosemeeme ss 420/240 V LINE WU Company Contact pera Phare waa mrenaio wenn 4, IN “ 4.8 KV LINE | JU Company Contact Email Phone iw ow emevmmems 43,2 KV LINE Py, one Soh etter mewetmeteees 40 KV LINE CUE Numbor Ver |Plot Date Seale Town Range Section Qh 1578942 | 41 hi 2/20/2025 NONE JOIN WIZE ef 68 Docusign Envelope ID: B1B51B9D-50F5-4A7F-B3E8-41358B9107E1 March 25, 2025 Tom Bommarito City of Warren One City Square Suite 215 aren WARREN FIRE DEPARTMENT 23295 Schoenherr Warren, MI 48089 (586) 756-2800 WWW, cityofwarren.org Warren, MI. 48093 Re: Fiber Optic Cable Installation — Fire Station #5 Dear Mr. Bommarito, The Fire Department is requesting the Downtown Development Authority to authorize payment of the invoice from Comcast Cable to connect new fire station #5 to the city’s existing fiber optic network. To accomplish this task Comcast cable will need to string approximately 3,500 feet of fiber optic cable on existing DTE Energy utility poles from near the intersection of Common and Dover to new fire station #5 located at 30619 Schoenherr. Connecting new fire station #5 to the city’s existing fiber optic network will allow the transmission of computer aided dispatch (CAD) information to the fire station and connect the station to the fire administration building IT room where our servers operate all software programs used by the department. Project Details: Recommended Company: Comcast Cable, 41112 Concept Dr., Plymouth MI 48170 Bid Number: Sole Source Requested Amount: $15,699.03 GL/Account Number: 493-9493-97450 Please find attached to the cover letter and invoice in the amount of $15,699.03 which depicts the total cost of project including fiber, permits and connection fees to the city’s existing fiber network and the underground connection to new fire station #5. Please direct questions to my attention at Ext. 3100. Professionally, Signed by: DocuSlgned by: DocuSigned by: i . try ‘ Mal daw Mark. bu ag a Wid aD4b.... , wporcaroecd ° mow Wilburt McAdams Mark Knapp Craig Treppa Fire Commissioner Assistant Controller Purchasing Agent 69 Docusign Envelope ID: B1B51B9D-50F5-4A7F-B3E8-41358B9107E1 Ihe cal COMCAST Comcast Cable 5700 Enterprise CT Warren Mi 49082 Customer Name Warren Fire Station 5 Date 3/24/25 Address 30619 Schoenherr Rd Order No. City Warren State Mi ZIP Rep Tim Frederick Phone FOB MI Qty Description Unit Price TOTAL 1 Project Design $719.49 $719.49 1 Pemnitting $3,379.18 $3,379.18 1 Fiber Construction Labor $8,570.87 $8,570.87 1 Materials $2,759.03 $2,759.03 1 Survey $240.46 $240.46 ~~ Subtotal $15,669.03 - Payment Details Shipping & Handling © Cash Sales Tax @ = Check © Credit Card TOTAL $15,669.03 Name CC# Office Use Only S Expires Payment required before work can begin. 70 WARREN LORI STONE - MAYOR i) 4 Wi chy olwarren org March 31, 2025 City of Warren Downtown Development Authority Board RE: Purchase of Fire Department Emergency Vehicles Dear Board Members: On October 10, 2023, Warren City Council authorized the purchase of up to seven (7) Emergency Medical Service Vehicles from Emergency Vehicle Plus (EVP) for a unit price of $302,817. Included with the approval was the authorization to issue a deposit payment of $121,126.80. It was known then that there is historically a time lag of up to 2 years between order and delivery and supply price changes could be a factor. Six (6) of the seven (7) vehicles are scheduled for delivery in the next 30 days. Economic reasons outside of the City’s and vendor’s control has increased the unit price to $307,967 per vehicle. Originally the purchase was to be funded with proceeds from capital equipment bond issues supported by General Fund revenue. On March 25, 2025, Administration requested and received authorization from City Council to amend the City of Warren Downtown Development Bonds, Series 2023 issue to allow the purchase of these items with unallocated principal and interest earned thus freeing the funding from the other bond issues to address City needs elsewhere. A schedule of the balance available is attached. Most of the cost is covered by interest earnings and there is still a substantial residual uncommitted amount for any situations that may arise regarding finalizing construction of the fire stations. 71 City of Warren Downtown Development Authority Board RE: Purchase of Fire Department Emergency Vehicles Page 2 The Board actions requested are as follows: 1. Approve the use of the City of Warren Downtown Development Bonds, Series 2023 to acquire seven (7) 2025 Road Rescue Ambulances at $307,967 per unit. The net cost to the bond issue is $2,034,642.20 ($307,957 x 7 = $2,155,769.00 less the deposit of $121,126.80 previously paid from other City resources) 2. Approve the issuance of payment or payments for the ambulances as received up to the amount authorized of $2,034,642.20 Copies of the original purchase authorizations from 2023 and the amendment to the City of Warren Downtown Development Bonds, Series 2023 resolution are attached. Concurred: : Welt Wl Richard Fox Wliburt McAdams City Controller Fire Commissioner 12 CITY OF WARREN, MICHIGAN DOWNTOWN DEVELOPMENT AUTHORITY EMS RESCUE VEHICLES TO BE FUNDED FROM FIRE STATION BOND PROCEEDS Bond Par Value $ 20,620,000.00 Bond Premium 2,301,908.35 Underwriter Discount (106,943.36) Closing Expenses (115,462.00) Net Bond Proceeds 22,699,502.99 Bond Payments & Commitments Thru 03.21.2025 (21,133,846.62) Unallocated Balance 1,565,656.37 Investment Earnings Through 12.31.2024 1,328,810.39 Available Funds 2,894,466.76 Emergency Vehicles Plus Unit Cost Total 2025 Road Rescue Ambulance 6 307,967.00 $ 1,847,802.00 Less Deposit - Check #660121 (103,822.97) Net Invoice 623457,.59.61-64 1,743,979.03 Uninvoiced 1 307,967.00 307,967.00 Less Deposit - Check #660121 (17,303.83) Net to billed 290,663.17 Total to be funded from 2023 DDA Revenue Bond Proceeds $ 2,034,642.20 Summary 2025 Road Rescue Ambulance 7 307,967.00 2,155,769.00 Less Deposit - Check #660121 Totat (121,126.80) 2,034,642.20 APPENDIX B at Wess ‘, ee WARREN PUBLIC SERVICE DEPARTMENT ENGINEERING DIVISION ; One City Square, Suite 300 April 2, 2025 Warren, Michigan 48093-2390 P: (586) 759-9300 Tom Bommarito F: (686) 759-9318 Economic Development Director www.cityofwarren.org RE: Request of the Engineering Division to approve funding for City Hall Parking Garage Striping, in an amount not to exceed $45,750. The scope of the project is to restripe the City Hall parking garage during the upcoming two week shut down to complete the parking garage sealing project. As indicated in the attached cost estimate, PK Contracting provided a total estimate amount of $45,750. This lump sum price is based on the current City Contract PM-24-814 2024 Pavement Marking City- Wide. It is the Engineering Division’s recommendation that the Downtown Development Authority (DDA) approve funding for the parking garage striping in the total amount not to exceed $45,750. Please place this item on the next available DDA agenda for consideration. If you have any questions regarding this matter, I can be reached in my office at (586) 759-9306. Sincerely, Read and Concurred, Recommended to DDA, Ape (seqeline Tina G. Gapshes, P.E. City Engineer Attached: Cost Estimate cc: Mark Knapp, Assistant Controller 74 MAIN GFFICE 1965 Barratt Drive Troy, Mi 48094-6372 Puone 248-362-2190 Fax 248-362-4909 To: _ESTIMATING DEPARTMENT Contact: ESTIMATING DEPT. Address: - Phone: (248) 362-2130 Fax: (248) 362-4969 Project Name: WARREN PARKING GARAGE Bid Number: 25-1341 Project Location: WARREN Bid Date: 4/2/2025 |Line # Item # Item Description Estimated Quantity Unit Unit Price Total Price] 1 PAVEMENT MARKINGS 1.00 LS $45,750.000 $45,750.00 Total Bid Price: $45,750.00 Notes: e QUOTE IS BASED ON ONE MOVE IN FOR FINAL MARKINGS AT COMPLETION OF PROJECT. ¢ QUOTE IS BASED ON APPLYING ONE APPLICATION OF WATERBORNE PAINT OVER EXISTING MARKINGS. ACCEPTED: CONFIRMED: The above prices, specifications and conditions are satisfactory and PK CONTRACTING, INC. are hereby accepted. Buyer: Signature: Authorized Signature: Date of Acceptance: Estimator: Page 1 of 1 79 WARREN DDA DDA LIST OF BILLS 4/212025 Required Formal Approval of the Following: Vendor DATE OF PAYEE Number INVOICE GL Account # AMOUNT Invoice # PO# DETAILS Dn 211412025, 02/25/2025, 6013525001114, 6023425000768, 02/28/2025, 6013525001495, 6013525001560, 03/05/2026, 6013525001463, 6013525001636, 03/12/2025, 6013525001645, 03/18/2025, 6013525001851, 6013525001985, FURNISH AND INSTALL VOICE AND PRESIDIO NETWORKED 03/24/2025, & 494-0000-09493 & 494- 6023425001092, & 2529962 & DATASYSTEMS FOR FIRE STATIONS 1 SOLUTIONS 004951 03/25/2025 9494-97400 78,100.68 6023425001184 2530361 & 5 LCOMMUNITY CENTER WIFI JOHNSON CONTROLS INC 013314 3/12/2025 494-9494-97400 90,792.40 CB10107584/APP06 2529158 | WCC FLOOD REPAIRS 212812025 & VIDEO SURVEILLANCE/CONTROL D/A CENTRAL INC 014619 03/26/2025 494-0000-09493 144,285.26 28939, 29079, & 29420 2529963 __ SYSTEMS FOR FIRE STATIONS 1&5 HALLAHAN & ASSOCIATES PC 015071 3/3/2025 494-9494-80100 815.49 22398 2529667 PROPERTY TAX APPEAL SERVICES CITY SQUARE PAVEMENT PK CONTRACTING 015551 21/6/2024 494-9494-97400 83,137.00 PROJECT NO, P-22-772 NON-PO | ENHANCEMENT GIBBS PLANNING GROUP INC 017229 3/12/2025 494-9494-80100 42,500,00 25-110 2528156 — TOWN CENTER DEVELOPMENT PLANTERRA CORPORATION 017360 3/1/2025 494-9494-80100 360.00 23835 2528917 STANDARD HORTICULTURE SERVICES GEOTECHNICAL INVESTIGATION - FIRE STATION 5 TRAFFIC SIGNALS | G2 CONSULTING MATERIAL TESTING FOR FIRE GROUP LLC. 017963 2/28/2025 494-00000493 8,026.75 250202 & 250107 NON-PO._ STATION 7 AND 23211 VAN DYKE MICHIGAN DOWNTOWN ASSOCIATION 018548 3/25/2025 494-9494-95800 750,00 3702 2530403. MEMBERSHIP DUES ADA IMPROVEMENTS - LOCKS AT THE JDN HARDWARE SALES 018989 3/12/2025 494-9494-97400 49,941.54 40712730-00 25293541 P.D. BRIVAR CONSTRUCTION COMPANY 019289 2/28/2025 494-0000-09493 866,378.59 ITB-W-0912 #18 NON-PO___FIRE STATION 1 & 5 - Payment No. 18 J EPPINK PARTNERS TOWN CENTER DEVELOPMENT INC 019584 41512025 494-9494-80100 5,136.74 1662 2528916 __ OUTREACH/PROFESSIONAL SERVICES EXERCISE EQUIPMENT - FIRE ROGUE FITNESS 019713 3/20/2025 494-9494-97400 19,606.97 13217477 2530134 STATIONS 1&5 FITNESS EQUIPMENT - FIRE STATIONS TITAN DISTRIBUTORS 019724 3/6/2025 494-9494-97400 7,479.68 iINV4344800 2530141 185 COMCAST FIBER FOR NEW FIRE COMCAST. 3/24/2025 493-9493-97450 15,669.03 NIA NON-PO STATION 5 TOTAL: _1,362,980.10 16 INVOICE: (6013525001114 ) DATE: 2/14/2025 > PAGE: 1 of 2 Presidia Networked Solutions Group, LLC EIN: 76-0515249, DUNS: 15-405-0959 For questions on this invoice please call: Dugini Kranthi Kumar (p) +1.781.970,6499, (f) dkranthikumar@presidio.com DDA SHIP TO; BILL TO: I City of Warren City of Warren - Information Systems Michelle Patterson ©) Laura Wilson a Attn Payables + One City Square _ One City Square, Suite 425, Purchasing Suite 420 || Warren, MI 48093 Warren, MI 48093 Customer #: CITYWo05s Customer PO#: 2829962 > Account Manager: Cassie Damer Order #: 3001222509990 Payment Terms: Net 30 Quote #: 2003524 108149-04 Title; Warren-New Fire Dept No. 5 Comments: Please make sure these emails are emailed to Michelle Patterson Contract Vehicle: Open Markel mpallerson@cityofwarren.org and Laura Wilson hvilson@cityolwarren.org Description Unit Price C9300-48P-A $5,917.010 1.0 1.0 $0,00 $5,917.01 Catalyst 9300 48-port PoE+, Network Advantage Serial #: FJC28481PKX | CON-LINCD- $3,012,930 1.0 1.0 $0.00 $3,012.93 C93004PA CX LEVEL 1 8X7NCDCatalyst 9300 48port PoE Network Adva CON-L1ISWT-C93A48 $546.480 1.0 1.0 $0.00 $546.48 CX LEVEL 1 SW SUB C9300 DNA Advantage €9300-DNA-A-48-5Y $4,369,720 1.0 1.0 $0.00 $4,369.72 C9300 DNA Advantage, 48-Por, 5 Year Term License PWR-C1-715WAC- $869,760 1.0 0 §0.00 $869.76 P/2 715W AC 86+ platinum Config 1 SecondaryPower Supply Sorial#: DCC2834CP5M MI Cooperan re Contract ® 421000000133 3 77 INVOICE: 6013525001114 DATE: 211412025 | PAGE: 2 of 2 ' Presidio Networked Solutions Group, LLC EIN: 76-0515249, DUNS: 15-405-0959 For questions on this invoice please call. Dugini Kranthi Kumar (p) +1.781.970,6490, (A) dkranthikumar@presidio.com Description UnitPrice Tax Extended | STACK-T1-50CM $69.590 1.0 1.0 $0.00 $69.59 50CM Type 4 Stacking Cable Serial #: MOG2849A5F6 C9300-NM-8X $1,596,890 1.0 1.0 i Calalyst 9300 8 x 10GE Network | Module Serial Wi: FJZ2848WSHS No return merchandise accepted without prior Retum Authorization. Sub Tolal: All retums subject ta a 20% restocking fee. £ _ _ - i a If not billed on this invoice, all taxes are to be pald by the buyer. : \ Past due balances are subject to 1.5% per month finance charge. Miscellaneous: ; $0.00 GST/HST# 75468 2292 RTO001 rT _ ~ \ \ Delivery of software licenses and software maintenance ara agreed to be accepted in electronic farm, _ Shipping & Handling. $0.00 1 Tax: | $0.00 | | Grand Total: $16,382.30 Discrepancies must be reported within 5 days of receipt of shipment of shipment will be considered complete, 1 Gy - 6000-09443 78 00445 | PRESIDIO. eae Send as INVOICE: Presidio Networked Solutions Graup, LLC DATE: PO Box 677638 PAGE: Presidio Networked Solutions Group, LLC Pag TA 2egnernSe8 EIN: 76-0515249, DUNS: 15-405-0959 Wire or ACH Payments: For questions on this invoice please call: PNC Bank Dugini Kranthi Kumar Acct: 8616159745 / ABA 031000053 (p) +1.781,970.6490, (f) Remit: remittanceadvice@presidio.com dkranthikumar@presidio.com BILL TO: | City of Warren ~ WORK LOCATION: q City of watreri Michelle Patterson DDF \ =| Laura Wilson Attn Payables -| 37TH DISTRICT COURT 8300 Common : One City Square, Suite 425, Purchasing Warren, MI 48093 : { Warren, MI 48093 Customer #: CITYWo00S Customer PO#: (2529962 ) Account Manager: Cassie Damer Order # 3001222509991 Payment Terms: NET30 Quote #: 2003525109307-01 Title: CITYW005 Fire Station Paging (3001222509991) [TM] Comments: 3001222509991 Services OT Rate ‘OT Hours ‘Jeff P Harris Collab Eng - Sr. $245.00 3.25 $245,00 0.00 $796.25 Jeff P Harris Collab Eng - Sr. - $245.00 0.75 $245.00 0,00 7 $183.75 Jeff P Harris _ ~ Collab Eng- Sr. 7 a 7 ~ $245.00 2.25 $245.00 0,00 | 7 $551.25 Jeff P Harris 7 ~~ Collab Eng - Sr. 7 7 ~ $245.00 0.50 $245.00 —-0,00 : $122.50 Total: $1,653.75 \Subtotal: $1,653.75 | — = - MICHIGAN, STATE OF Taxes $0.00 iAmount Due: ($1,853.75 44d .0000.09493 79 Invoice Details F248 AEN ATER 02/252025 City of Warren Easgemont CITYWO05 Fire Station Paging (300122250999 1) [TM] Committed five Caieay! USD Task hows Amount Tae Torat ask Rate es ap OT PA yettenr ftp 02/07/2025 JeffP Hanis CITYWO05 Fire Station Collab Eng - Sr. Interface with SA 0.000 245.0000 0.00 0,000 245,00 0,00 122.5000 Paging (3001222509991) [TM] and generate questions lo customer 02/11/2025 JeffP Haris CITYWO005 Fire Station Collab Eng - Sr. discuss O75 0.000 245.0000 0,00 9.000 245,00 0.00 483,7500 Paging (300122256899 1} (TM} requirements for PA systems 02/19/2025 JeffP Haris CITYWO05 Fire Station Collab Eng - Sr. setup and program 3.25 0.000 245.0000 0,00 0.000 245.00 0,00 796.2500 Paging (3001222509991) [TM] Algo units 02/14/2025 JeffP Hawis CITYW005 Fire Station Collab Eng - Sr. setup access lo fire 2.25 0.000 245.0000 0.00 0.000 245.00 0.00 551.2500 Paging (300122250999 1) [TM) stations from admin Total: 6.75 0,000 0.00 0,000 0.00 = 1659.7500 Time Total. 6.75 0.000 9,00 0.000 6.00 = 1653.7500 Sn De |e cee | Paes ? ) es | ‘@ aes Presidio Networked Solutions Group, LLC EIN: 76-0515249, DUNS: 15-405-0959 For questions on this invoice please call: Dugini Kranthi Kumar (p) +1.784.970.6490, (f) dkranthikumar@presidio.com \ Ms OO44S! Please send payments made payable to: Presidio Networked Sofullons Group, LLC PO Box 677638 Dallas, TX 75267-7638 Wire or ACH Payments: PNC Bank Acct; 8616159745 / ABA 031000053 Remit: remitlanceadvice@presidle,com DATE: INVOICE: \6013525001495 C 2/28/2025 ) PAGE: 1 of 1 BILL TO: City of Warren SHIP TO: City of Warren - Information Systems Laura Wilson Laura Wilson One City Square, Suile 425 One City Square Purchasing Dept Suite 420 Warren, MI 48093 Warren, MI 48093 DDA Customer #: CITYWo00s Customer PO#: 2003525109908-0380 WS ARGU AL Account Manager: Cassle Damer Order #: 3002552500008 Payment Terms: Net 30 Quote #: 2003525109908-03 Tile: Warren-New Fire Dept No, 4 -Juniper Comments; Please make sure these emails are emailed to Michelle Patterson Contract Vehicle; ‘Open Market mpallerson@cllyofwarren.org and Laura Wilson lwilson@cltyofwarren.org tenet? DoE SOUNOIy UT co ma aly ‘eax (OTE Te a] Silfqyadl AP45-US Premium Performance MulliGlgabit WiFi 6E Access Polnt ( $615.870 12.0 12.0 $0.00 $7,390.44 4x4:4) wilh Adaplive Bluelooth Low Energy Arra Sarlal #: SA1626240200FA, SA162624020174, SA1626240201D8, SA162624020162, SA162624020247, SA162824021838, SA162824021950, $A1630240207C7, SA163024020808, SA163124020347, SA1634240217F2, SA163424021806 9PX2000RT-L Ealon 9PX 2000VA 1800W 120V Online Double-Conversion $2,971.220 4.000 1,000 $0.00 $2,971.22 UPS 5-20P 6x 5-20R 1 L5-20R Oullets Lithlum-lon Serlal i: PGO1R47EL7 9PX1500RT-L Eaton 9PX 1500VA 1350W 120V Online Double-Conversion $2,688,730 1.0 1.0 $0.00 $2,688,73 UPS 5-15P, 8x 5-15R Oullets, Lithlum-lon Batlery Serlal#: PGOOR41NF6 NETWORK-M3 Ealon Cybersecure Gigabit NETWORK-M3 Card for UPS and $373,560 2.0 2.0 $0.00 $747.12 PDU Serlal #: P337R42KLP, P337R45VWM No retum merchandise accepled without prior Return Authorization, Sub Total: $13,797.51 Allrelurns subject to a 20% restocking fee, If nol billed on this Invoice, all taxes are to be pald by the buyer. . Past due balances are subject to 1.5% per month finance charge. es $0.00 GST/HST# 75468 2292 RTO001 a Delivery of software licenses and software maintenance are agreed to be accepted In electronic form. Shipping & Handling: $0,00 Tax: $0.00 Grand Total: $13,797,5 Discrepancles must be reported wilhin 5 days of recelpt of shipment or shipment will be consldered complete. - 4944 -0000-094-93 81 iy i) j Cs Presidio Networked Solutlons Group, LLC EIN: 76-0515249, DUNS; 15-405-0959 For questions on this invoice please call: Dugini Kranthi Kumar (p) +1.781.970.6490, (f) dkranthikumar@presidio.com 004951 ’ Please send payments made payable to; Presidio Networked Solullons Group, LLC PO Box 677638 Dallas, TX 75267-7638 Wire or ACH Payments: PNC Bank Acct; 8616159745 / ABA 031000053 Remit: remiltanceadvice@presidio,com INVOICE: (6013525001560 DATE: PAGE: 2/28/2025 1ofi BILL TO: City of Warren SHIP TO: City of Warren - Information Systems Michelle Patterson Laura Wilson ‘ Altn Payables One City Square One Gity Square, Suite 425, Purchasing Suite 420 Warren, MI 48093 Warren, MI 48093 Customer #: cITYwoos Customer PO#: (2529962 ) Account Manager: Cassie Damer Order #: 3001222510050 Payment Terms: Net 30 Quote #: 2003525109846-02 Title; Clty of Warren - Warren Fire 5 - 9800 WMK Wall mount kits Comments: Please make sure these emalls are emailed ta Michelle Patterson Contract Vehicle: “Open Market mpallerson@cityofwarren.org and Laura Wilson {wilson@cllyofwarren.org } lyon Wie RY ay ‘Ws Fttancdted Ovilarall Siijnpadl fitie DP-9800-WMIC= Clsco Desk Phone 9800 Wall Mount Icit- SPARE $69,160 8.0 8,0 $0.00 $553,28 No retum merchandise accepted wilhout prior Return Authorization. Sub Total: $553,28 All retums subject to a 20% restocking fee. If nol billed on this Invoice, all taxes are to be pald by the buyer. ‘ Past due balances are subject to 1.5% per month finance charge, Miscallaneous; $0.00 GST/HST# 75468 2292 RTO001 _ Delivery of software licenses and software malnlenance are agreed to be accepted In electronic form. Shipping & Handling: $0.00 Tax: $0.00 Grand Total; (= $553,28 D Discrepancies must be reported wilhin § days of receipt of shipment or shipment will be considered complele, H94-0000-094 93 82 PRESIDIO seoneers tele INVOICE: (6013525001463 Presidio Netaones Solutions Group, LLC DATE: 2/28/2025) ox 677638 i PAGE: 1ofi Presidio Networked Solutions Group, LLC Galles cS EIN: 76-0515249, DUNS: 15-405-0959 Wire or ACH Payments: For questions on this invoice please call: PNC Bank Dugini Kranthi Kumar Acct: 8616159745 / ABA 031000053 (p) +1.781.970,6490, (f) Remit: remittanceadvice@presidio.com dkranthikumar@presidio.com City of Warren SHIP TO: BILL TO: |} City of Warren - Information Systems | Laura Wilson Laura Wilson ‘| One City Square, Suite 425 One City Square | Purchasing Dept Suite 420 Warren, MI 48093 Warren, MI 48093 1 FO 99299G2L Customer #: CITYWwo05 Customer PO#: 2003525 109908-03SQ Account Manager; Cassie Damer Order #: 3002552500006 Payment Terms: Net 30 Quote #: 2003525 109908-03 Title: Warren-New Fire Dept No, 1 -Juniper Comments: Please make sure these emails are emailed to Michelle Patterson Contract Vehicle; ‘Open Market mpatterson@cityofvarren.org and Laura Wilson Iwilson@cityofwarren.org Desenption Unit Price Gly Extendad Onorad Price SFP-10G-LR-S= 10GBASE-LR SFP Module, Enterprise-Class $809.940 2.0 2,0 $0.00 $1,619.88 Serial #: OPM26420MEA, OPM28420P7T SFP-H10GB-CUIM= 10GBASE-CU SFP+ Cable 1 Meter $75.910 2.0 2.0 $0.00 $151.82 Serlal#: JPC282400AA, JPC2824008C SFP-10G-SR-S= 10GBASE-SR SFP Module, Enterprise-Class $263.610 4.0 4.0 $0.00 $1,134.44 Serlal#: ACW284810NV, ACW28482815, ACW284B283N, ACW284S08RZ DP-984 1-K9= Cisco Desk Phone 9841, Carbon Black $141,140 9.0 9.0 $0.00 $1,270.26 Sorial #: SW2ZP28399M59, SWZP28399P2C, SWZP28399P2V, SWZP28399P30, SWZP28399P38, SWZP283S9P3G, SWZP28399PE9, SW2P28399PEE, SWZP28399PFH CP-8865-K9= Cisco IP Phone 8865 $459,790 1.0 1.0 $0.00 $459.79 Serlal # ~=FVH28471BXD No retum merchandise accepted without prior Return Authorization. _ Sub Total: | $4,636.19 All relurns subject to a 20% restocking fee, i —— If not billed on this Involce, all taxes are lo be paid by the buyer. , . | Past due balances are subject to 1.5% per month finance charge. Miscellaneous: | $0.00 GST/HST# 75468 2292 RTO001 ; a ae Delivery of software licenses and saftware maintenance ara agreed to be accepted in electronic form. Shipping & Handling: | $0.00 _ 1 Tax: $0.00 ‘ Grand Total: ‘ $4,636.19 Discrepancies must be reported within 5 days of receipt of shipment or shipment vall be considered complete. #94 -C 00069493 83 PRESIDIO Presidio Networked Solutions Group, LLC EIN: 76-0515249, DUNS: 15-405-0959 For questions on this invaice please ca Dugini Kranthi Kumar (p) +1.781.970.6490, (f) dkranthikumar@presidio.com City of Warren 00445] Please send payments made payable to: INVOICE: 6013525001636 31512025 >) 1 of 2 DATE: PAGE: Presidio Networked Solutions Group, LLC PO Box 677638 Dallas, TX 75267-7638 Wire or ACH Payments: PNC Bank Acct: 8616159745 / ABA 031000053 Remit: remittanceadvice@presidio.com £0 F252 BILL TO: © SHIP TO: 'i City of Warren - Information Systems ; “Laura Wilson } Laura Wilson One City Square, Suite 425 One City Square Purchasing Dept | Suite 420 Warren, MJ 48093 Warren, Mi 48093 i PPA ‘| Customer #: CITY W005 Customer PO#: 2003525 109908-03SQ Account Manages: Cassie Damer Order #: 3062552500006 Payment Terms: Net 30 Quote #: 2003525 109908-03 Title: Warren-New Fire Dept No, 1 -Juniper Comments: Please make sure these emails are emailed to Michelle Patterson Contract Vehicle: Open Market mpatterson@cityohvarren.org and Laura Wilson hulson@cityofwarren.org Part? Boseription €9200-48P-E Serlaf#: FYH290518HT CON-SNT-C92048PE SNTC-8X5XNBD Catalyst 9200 48-port PoE+, Network Esse C9200-DNA-E-48-3Y C9200 Cisco DNA Essentials, 48-port - 3 Year Term License Calalyst 9200 4 » 10G Network Module Serlal#: FVH29010EDF C9200-NM-4X PWR-C6-1KWAC/2 Serial 4: OCI2809NS5NJ C€9200-48P-E Serial #: FVH290518SC C9200-DNA-E-48-3Y C9200 Cisca ONA Essentlafs, 48-porl - 3 Year Term License C9200-NM-4X Catalyst 9200 4 x 10G Network Module Serlal#: = FVH29010F6T Calalyst 9200 48-port PoE+, Network Essentials IKW AC Config 6 Power Supply - Secondary Power Supply Catalyst 9200 48-part PoE+, Network Essentials UnitPrice Gly Oty Tax Ordered Shipped $3,069,490 1,0 1.0 $0,00 $549,710 1.0 1.0 $0.00 $748.000 10 1.0 $6.00 $948,840 1.0 1.0 $0.00 $1,494,060 10 1.0 $0,00 $3,069,490 1.0 1.0 $0,00 $748,000 1.0 1.0 $0.00 $948,840 1.0 1.0 $0.00 MI Cooperative # MAct 1000 0001333 T96L Extended Price $3,069.49 $549.74 $748.00 | $948.84 $1,494.06 $3,069.49 $748.00 $948.84 84 PRESIDIO Pia a INVOICE: 6013525001636 Presidio Networked Solutions Group, LLG DATE: 3/5/2025 PO Box 677638 Dallas, TX 75267-7638 PAGE; Pof2 Presidio Networked Solutions Group, LLC ed y EIN: 76-0515249, DUNS: 15-405-0959 Wire or ACH Payments: For questions on this invoice please call: PNC Bank Dugini Kranthi Kumar Acct; 8616159745 / ABA 031000053 (p) +1,781.970.6490, (f) Remit: remittanceadvice @presidio.com dkranthikumar@presidio.com Description Unit Price aty Extendad Shipped Pnce PWR-C6-1KWAC/2 1KWAC Config 6 Power Supply - Secondary Power Supply $1,494.060 1.0 1.0 $0.00 $1,494.06 Serial 4: DCI2B09N5RQ 1 C6-PR-NB-1-ENC CAT6 PURPLE 1FT NO BOOT PATCH CABLE $1.200 124.0 124.0 $0,00 $148.80 C6-YL-NB-1-ENC CAT6 YELLOW 1FT NO BOOT PATCH CABLE $1.200 22.0 22,0 $0,00 $26.40 SCLC-SM-2M-ENC — SC-LC 2M OS1 9/125 SINGLE-MODE DUPLEX $12.380 3.0 3.0 $0.00 $37.14 LC2-OM4-1M-ENC LC-LC 1M OM4 50/125 MULTIMODE DUPLEX $15,800 5.0 5.0 $0.00 $79,00 C6-PR-NB-5-ENG CAT6 PURPLE 5FT NO BOOT PATCH CABLE $2.560 10.0 10.0 $0.00 $25.60 C6-BL-NB-10-ENC = CAT6 BLUE 10FT NO BOOT PATCH CABLE $3,260 4.0 4.0 $0.00 $13.04 No retum merchandise accepted wilhout prior Return Authorization. Sub Total: $13,400.47 All retums subjact to a 20% restocking fee. _ _ _— _ _ if not billed on this invoice, all taxes are lo be paid by the buyer. 5 : Past due balances are subject to 1.5% per month finance charge. Miseelaneolis: | $0.00 GST/HST# 75468 2292 RTO001 . ; _ a Delivery of software licenses and software maintenance are agreed to be accepted in electronic form. Shipping & Handling: $0.00 Tax: i $0.00 —=, Grand Total: ( $13,400.47) a - _ _ _ aes = Discrepancies must be reported within 5 days of receipt of shipment or shipment will ba considered complete. 494-0000 -04+ 4A 85 004495! Please send payments made payable to; INVOICE: (6013525001645 Presidio Networked Solutions Group, LLC DATE: PO Box 677638 D ; 5267-7638 PAGE: 1of 1 Presidio Networked Solutions Group, LLC allas, TX 75267-7 EIN: 76-0515249, DUNS: 15-405-0959 Wrre or ACH Payments: For questions on this invoice please call: PNC Bank Dugini Kranthi Kumar Acct: 8616159745 i ABA ness (p) +1.781.970.6490, (f) Remit: remittanceadvice@presidio.com = dkranthikumar@presidio.com POF AD AINIGL BILL TO: City of Warren SHIP TO: EI City of Warren - Information Systems + Laura Wilson = Laura Wilson One City Square, Suite 425 a One City Square H Purchasing Dept Suite 420 ! Warren, Ml 48093 1 Warren, MI 48093 DOA Customer #: CITYWo0s Customer PO#: 2003525109908-03SQ Account Manager: Cassie Damer Order #; 3002552500006 Payment Terms: Net 30 Quote #; 2003525109908-03 Title: Warren-New Fire Dept No. 1 -Juniper Comments: Please make sure these emails are emailed to Michelle Patterson Contract Vehicle; Open Market mpalterson@cityofwarren.org and Laura Wilson hvilson@cityofwarren.org Description Unit Price aty Qty Extended: SUB-2S-5Y Subscriptions for 2 services (specify from SUB-MAN, SUB- $433,090 ENG, SUB-AST, SUB-VNA, SUB-PMA) for one acces Mi Cooperative i All retums subject to a 20% restocking fee. l If not billed on this invoice, all taxes are to be paid by the buyer. ' Past due balances are subject to 1.5% per month finance charge. | GST/HST# 75468 2292 RTO001 j Delivery of software licenses and software maintenance are agreed to be accepted in electronic form. td. Mt will be ed i Discrepancies must be reported within 5 days of receipt of shipment or shi Omered Shipped Pricey 12.0 12.0 $0.00 $5,197.08 | # MA21000000133 3 Sub Total: ft $5,197.08 Miscellaneous: | $0.00 "Shipping & Handling: $0.00 be ee | Tax: i $0.00 | 4 -| | Grand Total: $5,197.08 | ) 494-GaaaA-d09 43 86 004495! PRESIDIO. en INVOICE: (601 3525001851 ) Presidio Networked Solutions Group, LLC DATE: 3/12/2025) PO Box 677638 PAGE: loft Presidio Networked Solutions Group, LLC Eagas TN TSEBYtGa8 : EIN: 76-0515249, DUNS: 15-405-0959 Wire ar ACH Payments: For questions on this invoice please call: PNC Bank Dugini Kranthi Kumar Acct: 8616159745 / ABA 031000053 (p) +1.781.970,6490, (f) Remit: remittanceadvice@presidio.com dkranthikumar@presidio.com BILL TO: | City of Warren SHIP TO: | City of Warren - Information Systems | Laura Wilson : Laura Wilson ', One City Square, Suite 425 | One City Square .1 Purchasing Dept « Db pb | Suite 420 || Warren, MI 48093 D i| Warren, MI 48093 | i po & 2529792 Customer #: CITYWo0s Customer PO#: 2003525 109908-03SQ Account Manager; Cassie Damer Order #; 3002552500006 Payment Terms: Net 30 Quote #; 2003525 109908-03 Title: Warren-New Fire Dept No. 1 -Juniper Comments: Please make sure these emails are emailed to Michelle Patterson Contract Vehicle: | Open Market mpalterson@cityofwarren.org and Laura Wilson hwilson@cityofwarren.org UnttPrice Gty Extended Oniered Price C9300-48P-A Catalyst 9300 48-port PoE+, Network Advantage $5,917,010 1,0 1.0 $0.00 $5,917.01 Serial#: FJC285117BQ CON-L1NCD- CX LEVEL 1 8X7NCDCatalyst 9300 48port PoE Network Adva $3,012,930 1.0 1.0 $0.00 $3,012.93 C93004PA CON-L1SWT-C93A4B CX LEVEL 1 SW SUB C9300 DNA Advantage $546.480 1.0 1.0 $0.00 $546.48 C9300-DNA-A-48-5Y C9300 DNA Advantage, 48-Port, 5 Year Term License $4,369.720 1.0 1.0 $0.00 $4,369.72 PWR-C1-715WAC- —715W AC 80+ plalinum Config 1 SecondaryPower Supply $869,760 1.0 1.0 $0.00 $869.76 Pi2 Serial#: DCC2834CKQV \ STACK-T 1-50CM 5OCM Type 1 Stacking Cable $69.590 1.0 1.0 $0.00 $69.59 Serial #; = MOG2846A3X9 C9300-NIM-8X Catalyst 9300 8 x 10GE Network Module $1,596,890 1.0 1.0 $0,00 $1,596.89 Serial #: FJ2284902UH 7 Mi Cotpere Crue Conts eet: # 4421000000134 3 7 | No relurm merchandise accepted without prior Retum Authorization, | Sub Total: $16,382.38 All returns subject to a 20% restocking fee, _ - | If not billed on this invoice, all taxes are to be paid by the buyer. . . Past due balances are subject to 1.5% per month finance charge. Miscallanieaus: __ i $0.00 GST/HST# 75468 2292 RT0001 . : — ~] ~ _ Delivery of software licenses and software maintenance are agreed lo be accepted In electronic form. Shipping & Handling: : $0.00 Tax: j $0.00 \ - _ - a, Grand Total: C $16,382.38 ) = _ sass = ~ See ee Oiscrapancies must be reported within 5 days of receipt of shipment or shipment will be considered complete. 494 - 0000-09443 87 OO44S5! PRESIDIO Please send payment INVOICE: (6013525001985 Presidio Networked Solutions Group, LLC DATE: (3/18/2025 > PO Box 677638 PAGE Vert las, TX 75267-7638 : Presidio Networked Solutions Group, LLC Palas ss EIN: 76-0515249, DUNS: 15-405-0959 Wire or ACH Payments: For questions on this invaice please call: PNC Bank Dugini Kranthi Kumar Accl: 8616159745 / ABA 031000053 (p) +1.781.970.6490, (f) Remit: remittanceadvice@presidio.com dkranthikumar@presidio.com BILL TO: City of Warren SHIP TO: q City of Warren - Fire Administration | Michelle Patterson = Bill Alter ‘Attn Payables =} 23295 SCHOENHERR | One City Square, Suite 425, Purchasing “i Warren, MI 48089 Warren, Ml 48093 : ‘ 1 y LDA d Customer #: CcITYWoos Customer PO#: (2529962 ) Account Manager: Cassie Damer Order #: 3002552500014 Payment Terms: Net 30 Quote #: 2003025001 205-01 Title: City of Warren - Phone Mounts Fire Contract Vehicle: ‘Open Markel Description. Unit Price: aly Ly Extendod Oniered Shipped Price Pant? DP-9800-WMK= Cisco Desk Phone 9800 Wall Mount Kil - SPARE $69,160 4.0 4.0 $0,00 $276.64 No retum merchandise accepted vathout prior Retum Authorization, Sub Total: i $276.64 All retums subject to a 20% restocking fee. If not billed on this invoice, all taxes are to be pald by the buyer, | Past due balances are subject to 1.5% per month finance charge, Miscellaneaus! | $0.00 GST/HST# 75468 2292 RTOOOI | . . | Delivery of software licenses and software maintenance are agreed to be accepted in electronic form. ' Shipping & Handling: | $0.00 Tax: | $0.00 | Grand Total: C $27664 ) Discrepancies must be reported within 5 days of recelpt of shipment or shipment will be considered complete. ' 494. 0000 . (4493 dod gsl Please send payments made payable to; Presidio Networked Solutions Group, LLC PO Box 677638 Dallas, TX 75267-7638 PRESIDIO Presidio Networked Solutions Group, LLC EIN: 76-0515249, DUNS: 15-405-0959 For questions on this invoice please call: Dugini Kranthi Kumar (p) +1.781.970.6490, (f) dkranthikumar@presidio.com Wire or ACH Payments: PNC Bank Acct: 8616159745 / ABA 031000053 Remit: remittanceadvice@presidio.com BILL TO: # City of Warren ~ i ft Michelle Patterson DDA 4 \ Attn Payables 4 ‘| One City Square, Suite 425, Purchasing : | Warren, MI 48093 Customer #: cITYW005 Customer PO#: Account Manager, Cassie Damer Order #: Payment Terms: NET45 Quote #: Title: CITYW005 City of Warren - Community Center Juniper MIST Deployment ( 3002552500023) [FF] Comments: 3002552500023 Fixed Fee INVOICE: (6023425001 092) DATE: (312412025 PAGE: 1of1 City of Warren Laura Wilson 5460 Arden Ave Warren, MI 48092 pee ( 2530361 3002552500023 2003025001 249-01 Amount: Project Initiation ‘Subtotal: ue 'amount Due: MICHIGAN, STATE OF Taxes - $5,576.00 Total: $5,576,00 $5,576.00 $0.00 ( $5,576.00 ee Gey pep a7 XN, zd 4A 89 PRESIDIO. Presidio Networked Solutions Group, LLC EIN: 76-0515249, DUNS: 15-405-0959 For questions on this invoice please call: Dugini Kranthi Kumar (p) +1,781,970.6490, (f) dkranthikumar@presidio.com BILL TO: — City of Warren Ti | | Michelle Patterson IID A { | Attn Payables ‘ i One City Square, Suite 425, Purchasing 'y | Warren, MI 48093 1 | 4 ‘| Customer #: CITYW005 Customer PO#: Account Manager: Cassie Damer Order #: Payment Terms: NET30 Quote #: Title: CITYW005 Fire Station Paging (3001222509991) [TM] Comments: 3001222509991 Services Resoures Jeff P Harris Collab Eng - Sr. ooHrds | Please send payments made payable to: Presidio Networked Solutions Group, LLC PO Box 677638 Dallas, TX 75267-7638 Wire or ACH Payments: PNC Bank Acct: 8616159745 / ABA 031000053 Remit: remittanceadvice@presidio,com $245,00 ‘Subtotal. MICHIGAN, STATE OF Taxes ‘Amount Due: WORK LOCATION: INVOICE: (6023425001184 DATE: (312512025 ) PAGE: 1of1 City of Warren Laura Wilson 37TH DISTRICT COURT 8300 Common Warren, MI 48093 os ~——— “Egy (2529962 ) 2529962 3001222509991 2003525 109307-01 OT Hours 0,00 OT Rate $245.00 $245,00 Total: $245,00 $245.00 $0.00 Cetas00 90 Invoice Details Big M2GUu Tt 18S City of Warren Committed 03/28/2025 CITYW005 Fire Station Paging (3001222509991) {TM] usp Thee Gate Rag OTR 4 ina Tat 02/24/2025 JetfP Harris CHTYW005 Fire Station Collab Eng - Sr. Paging prep 0.000 245.0000 0.00 245.00 0.00 245.0000 Paging (3001222509991) (TM} Total: 1.0 0,000 0.00 0,000 0.00 245.0000 Tima Total: 1.00 0.000 0.00 0.000 0.00 245.0000 91 ce cae he EP Ae c6 ¥ prs ” APPLICATION AND CERTIFICATION FOR PAYMENT AIA DOCUMENT G702 PAGE ONE OF 2 PAGES TO OWNER: CITY OF WARREN PROJECT: Warren Community Center Boiler Replac: APPLICATION NO: 7 Distribution to: ATTN PURCHASING STE 425 PROJECT MGR: MEADOWS. JOHN W FEDERAL ID: 39-0380010 OWNER 1 CITY SQUARE ARCHITECT WARREN. MI 48093 PERIOD TO: 31-Mar-25 CONTRACTOR FROM CONTRACTOR: VIA ARCHITECT: INVOICE NO: (€B10107584/4res6] Johnson Controls Inc. REMIT TO: Johnson Controls INVOICE DATE: 22-Mar-25 > 31831 Sherman Ave PO Box 730068 PROJECT NOS: MC4N160170 Madison Heights. MJ 48071 Dallas. TX 75373 P.O.NO: 2529158 } CONTRACT FOR: control systems CONTRACT DATE 16-Sep-24 CONTRACTOR'S APPLICATION FOR PAYMENT The undersigned Contractor certifies that to the best of the Contractor's knowledge, Application is made for payment. as shown below, in connection with the Contract. information and belief the Work ccevered by this Application for Payment has been Continuation Sheet, AIA Document G703. is attached. completed in accordance with the Contract Documents, that all amounts have been paid by the Contractor for Work for which previous Certificates for Payment were issued and payments received from the Owner, and that current payment shown herein is now due. I. ORIGINAL CONTRACT SUM $ 880,000.00 “t 2, Net change by Change Orders $ 0.00 CONTRACTOR: Johnson Controls, Inc. ‘ 3. CONTRACT SUM TO DATE (Line 1 4 2) $ 880,000.00 7 . 4. TOTAL COMPLETED & STORED TO $ 879,714.80 iA sh fo . : DATE — (Column G on G703) By: 2% WA. RAID hh Date: 3/13/2025. 5. RETAINAGE: a a. 0 % of Completed Work $ 0.00 State of: Massachusetts County of: Worcester = ‘ (Column D + E on G703) Subscribed and swomto before mc this 13 day of March 2025 ES b. 0 %of Stored Material f 0.00 Notary Public: ay = Column F on G703) My Commission expires: : . @ 4 Total Retainage (Lincs 5a + Sb or / i‘ : & I 9O3/ « a: Total in Column | of G703) S 0.00 ARCHITECT'S CERTIFICATE FOR PAYMENT es 6. TOTAL EARNED LESS RETAINAGE $ $79.714.80 In accordance with the Contract Documents. based on on-site observations and the data (Line 4 Less Line 5 Total) comprising the application. the Architect certifies to the Owner that to the best of the 7, LESS PREVIOUS CERTIFICATES FOR Architect's knowledge. information and belicf the Work has progressed as indicated. PAYMENT (Line 6 from prior Certificate) $ 788.922.40 the quality of the Work is in accordance with the Contract Documents, and the Contractor 8. CURRENT PAYMENT DUE $ 90,792.40 is entitled to payment of the AMOUNT CERTIFIED. a. 0.00% Tax $ — 0.00 ee ee b. Total Current Payment with Tax § Q90,792.405 Y4u- s4qu et yee (oP p= Ge sue 9. BALANCE TO FINISH, INCLUDING RETAINAGE $ 285.20 ; ; we #2 (Line 3 less Line 6) AMOUNT CERTIFIED..................-.02-.2-000005- $ CHANGE ORDER SUMMARY ADDITIONS DEDUCTIONS (-litach explanation if amount certified differs from the amount applied. Initial all figures on ths Total changes approved [pplication and onthe Continuation Sheet that are changed to conform with the amount certified.) _in previous months by Owner $0.00 $0.00 ARCHITECT: Total approved this Month $0.00 $0.00 By: Date: TOTALS $0.00 $0.00 This Certificate is not negotiable. The AMOUNT CERTIFIED is payable only to the Contractor named herein. Issuance, payment and acceptance of payment are without NET CHANGES by Change Order $0.00 prejudice to any rights of the Owner or Contractor under this Contract. AIA DOCUMENT G702 - APPLICATION AND CERTIFICATION FOR PAYMENT - 1992 EDITION - AIAO - © 1992 THE AMERICAN INSTITUTE OF ARCHITECTS 1735 NEW YORK AVE.. N.W., WASHINGTON, DC 20006-5292 PIS ARTHAS SAS RISE REPHBE SREB OREO? AUP aasROHE SURG SAREE UMS EBA A A SinSt for resale.
€6 CONTINUATION SHEET AIA DOCUMENT G703 PAGE 2 OF 2 PAGES AIA Document G703, APPLICATION AND CERTIFICATION FOR PAYMENT, containing JCL PROJECT NO: MC4N 160170 APPLICATION NO: 7 Contractor's signed certification is attached. JCI INVOICE NO: CB10107584 APPLICATION DATE: 12-Mar-25 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 31-Mar-25 Use Column I on Contracts where variable retainage for line items may apply. MEADOWS. JOHN W ARCHITECT'S PROJECT NO: A B c D | E F G H { WORK COMPLETED MATERIALS TOTAL ITEM SCHEDULED FROM PREVIOUS PRESENTLY COMPLETED % BALANCE RETAINAGE NO. DESCRIPTION OF WORK VALUE APPLICATION THIS PERIOD STORED AND STORED (G+C) TO FINISH (IF VARIABLE (D+ E) (NOT IN TO DATE (C-G) RATE) DOR E) (D+E+F) Base Contract I [Demo $56,000.00 $56,000.00 $0.00 $0.00 $56,000.00 100% $0.00 $0.00 2 [Mobilization $47,000.00 347,000.00 $0.00 $0.00 $47,000.00 100% $0.00 $0.00 3 jEngineering $31,999.00 $31,999.00 $0.00 $0,00 $31,999.00 100% $0.00 $0.00 4 |Material $572,324.00 $518,792.60 $53,531.40 $0.00 $572,324.00 100% $0.00 $0.00 5 | Installation $137,500.00 $125.250.00 $12,250.00 $0.00 $137,500.00 100% $0.00 $0.00 6 [Commissioning $27,863.00 $2,566.80 $25,011.00 $0.00 $27,577.80 59% $285.20 $0,00 7 ~~ [Project Initiation $7,314.00 $7,314.00 $0.00 $0.00 $7,314.00 100% $0.00 $0.00 } GRAND TOTALS: $880,000.00 $788,922.40 $90,792.40 $0.00 $879,714.80 100% $285.20 $0.00 This document was produced under AIA Order No. 3104245240 which expires on T0/e 5, and is not for resale. AIA DOCUMENT G703 - CONTINUATION SHEET FOR G702 - 1992 EDITION - AIA® - © 1982 6703-1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C. 20006-5232 Johnson Y))( Controls PARTIAL CONDITIONAL WAIVER We the undersigned, Johnson Controls, Inc., have been employed by: CITY OF WARREN to furnish the system controls for the property described as: Warren Community Center Boiler Replaceme Therefore, we the undersigned, upon receipt of valuable consideration in the amount of: $90,792.40 , AND any previously submitted yet unpaid invoices, hereby waive any right to pursue a lien against the above-described premises for labor or materials furnished onorbefore: 31-Mar-2025 . The right to assert llen rights for labor and materials furnished after said date on said property is expressly reserved. JCI Invoice: CB10107584 JC! Contract: MC4N160170 Johnson Controls, Inc. BY: Onnw Thasuarl Annu Thakur MBC - Contract Accounting Intellion park 4th Floor, Tower 1 Khewat No 221/274 — off Golf Course Extn Road.Sector 58, village Berhampur, Gurugram , Haryana 122004 Signed on: 13-Mar-2025 OY] q | INVOICE D/A CENTRAL 28939 INVOICE DATE PURCHASE ORDER # CUSTOMER # TERMS PROJECT # PROJECT TITLE SALES PERSON New Fire Station 1 Security and Data VANWELJ SOLD TO: City of Warren SHIP TO: City of Warren Fire Station 1-New Purchasing Department 23211 Van Dyke Ave One City Square-Sulte 425 Warren, Ml) 48093-5289 Warren, MI 48089 DESCRIPTION Labor $25,214.30 Equipment $27,365.54 COMMENTS THIS INVOICE: $52,579.84 30% Start Up / Mobilization Invoice SALES TAX (EXEMPT): $0.00 INVOICE TOTAL: ( $52,579.84 , ; ah pe ei Yeates. } t ro tfepap « FORE = OP one ¥ You can now pay online! www.dacentral.com/pay-invoice or by ACH Acct #: 4100689456 Routing:072413829 Swift: MEMIUS33 D/A Central Inc. Thank you for your business! AR@dacentral.com | 13155 Cloverdale 248-399-0600 | Oak Park, MI 48237 intelligent technology solutions INVOICE DATE PURCHASE ORDER # HVA INVOICE CUSTOMER # (2128/2025 2529963 >) 2738 NET 45 PROJECT # New Fire Station 5 Security and Data PROJECT TITLE SALES PERSON VANWELJ SOLD TO: City of Warren Purchasing Department One City Square-Suite 425 Warren, MI 48093-5289 DESCRIPTION SHIP TO: City of Warren Fire Station 5-New 30619 Schoenherr Rd Warren, MI 48088 TOTAL PRICE 1.00 24C-H5A-3MH - Progress Billing February ($1,272.48) 3.00 24C-H5A-3MH - Progress Billing February $3,817.44 1.00 6.0C-H6A-D1-IR - Progress Billing February $341.94 1.00 6.0C-H6A-DO1-IR - Progress Billing February ($772.49) 3.00 6.0C-H6A-DO1-IR - Progress Billing February $2,317.47 1.00 8.0C-H6A-BO1-IR - Progress Billing February ($937.16) 3.00 8.0C-H6A-BO1-IR - Progress Billing February $2,811.48 4.00 ACC7-ENT - Progress Billing February ($835,59) 12.00 ACC7-ENT - Progress Billing February $1,823.16 1.00 FPO150-B100C8PD8PE4M1 - Progress Billing February $245.06 1.00 FPO75-B100C4PD8PE4M - Progress Billing February $167.13 1.00 H4AMH-AD-IRIL1 - Progress Billing February ($186.91) 3.00 H4AMH-AD-IRIL1 - Progress Billing February $560.73 1.00 LNL-1320-S3 - Progress Billing February $197.70 1.00 LNL-1320-S3 - Progress Billing February $201.09 1.00 LNL-R11320-05TB - Progress Billing February ($435.23) 7.00 LNL-R11320-05TB - Progress Billing February $1,015.49 4.00 LNL-R11325-05TB - Progress Billing February $420.68 1.00 LNL-X2220 - Progress Billing February $546.89 1.00 | MONITOR-SS - Progress Billing February $794.40 1.00 NVR6-PRM-FORM-D-96TB-S22-NA - Progress Billing February $11,208.00 1.00 ONETIMEPURCHASE - Progress Billing February ($20.94) 3.00 ONETIMEPURCHASE - Progress Billing February $62.85 41.00 ONETIMEPURCHASE - Progress Billing February $80,00 1.00 ONETIMEPURCHASE - Progress Billing February $102.98 1.00 ONETIMEPURCHASE - Progress Billing February $418.91 4.00 ONETIMEPURCHASE - Progress Billing February $450.00 1.00 ONETIMEPURCHASE - Progress Billing February $1,494.00 1.00 RM7-WKS-4MN-NA - Progress Billing February $912.00 1.00 FREIGHT - Progress Billing February $989.17 1.00 LS2-ELEMENTS-2RS-EX - Progress Billing February $384.48 1.00 BISCUIT - CAT6 - 1 PORT - Progress Billing February $38.85 1.00 ELEMENTS READER FEE - 3 YEAR - Progress Billing February $1,935.36 D/A Central Inc. Thank you for your business! AR@dacentral.com 13155 Cloverdale Oak Park, Ml 48237 248-399-0600 96 D/A CENTRAL INVOICE intelligent technology solutions 29079 1.00 J-HOOK - 2 - Progress Billing February $213.75 1.00 MISC PARTS - Progress Billing February $653.49 1.00 MISC PARTS - Progress Billing February $44.31 1.00 MISC PARTS - Progress Billing February $212.25 7.00 MISC PARTS - Progress Billing February $36.91 1.00 MISC PARTS - Progress Billing February , $202.17 1.00 MISC PARTS - Progress Billing February $16.15 1.00 MISC PARTS - Progress Billing February $316.77 1.00 PRIME SUPPORT - Progress Billing February $9,695.59 1.00 18/2-PLNM-SHLD - Progress Billing February $26.65 1.00 18/2-PLNM-UNSHLD - Progress Billing February $15.27 1.00 18/2-PVC-UNSHLD-DB - Progress Billing February $48.26 1.00 180-12-G - Progress Billing February $13.92 2.00 180-12-G - Progress Billing February ‘ $6.96 1.00 22/6-PVC-SHLD-DB - Progress Billing February $80.71 1.00 ACC-ENT-SMART-3YR - Progress Billing February $262.43 1.00 APTMMB - Progress Billing February $22.43 1.00 BATTERY-12V-7A - Progress Billing February $35.45 1.00 CAT6-PLNM-UNSHLD - Progress Billing February $1,819.86 1.00 CAT6-PLNM-UNSHLD - Progress Billing February $233.98 1.00 CAT6-PVC-UNSHLD-DB - Progress Billing February $36.00 1.00 COMPOSITE-PLNM - Progress Billing February : $967.50 4.00 CRNMT-1001 - Progress Billing February $139.20 1.00 DS160 - Progress Billing February $328.32 1.00 ETP-SMH-1 - Progress Billing February $126.00 1.00 GL36WDXM-B-SH-00 - Progress Billing February $340.40 4.00 HSAMH-AD-PEND1 - Progress Billing February $191.34 1.00 H5AMH-DO-COVR? - Progress Billing February ($95.65) 1.00 H5AMH-DO-COVRi - Progress Billing February $286.96 1.00 HUB2SA - Progress Billing February $12.27 4.00 LE-D - Progress Billing February $24.00 1.00 LEF-5C - Progress Billing February $108.00 1.00 MC-CS-10-E - Progress Billing February $118.03 1.00 POEGOU-1BTE - Progress Billing February $169.92 1.00 PS-1208UL - Progress Billing February $42.45 1.00 RG6-PLNM-SHLD - Progress Billing February $653.22 1.00 RY-PA- Progress Billing February $12.48 1.00 UNP-6-DM-1U-24 - Progress Billing February $85.96 1.00 UNP-6-DM-2U-48 - Progress Billing February $425.45 1.00 WLMT-1001 - Progress Billing February $116.47 1.00 LIFT RENTAL - Progress Billing February $313.87 BUCKET TRUCK LABOR - Progress Billing February $720.00 CABLING - Progress Billing February $5,712.00 CAD - Progress Billing February $150.00 ENGINEERING - Progress Billing February $675.00 D/A Central Inc. Thank you for your business! AR@dacentral.com 13155 Cloverdale 248-399-0600 Oak Park, Ml 48237 97 INSTALLATION - Progress Billing February $4,845.00 INSTALLATION - Progress Billing February $70.80 PROJECT MANAGEMENT - Progress Billing February $1,680.00 PROJECT MANAGEMENT - Pragress Billing February $72.00 COMMENTS THIS INVOICE: $61,160.73 Progress Billing February SALES TAX (EXEMPT): $0.00 fae he INVOICE TOTAL: ($61 160.73 3 Y9y orcs - 0999 You can now pay online! www.dacentral.com/pay-invoice or by ACH Acct #: 4100689456 Routing:072413829 Swift:MEMIUS33 Re & ale 46 D/A Central Inc. Thank you for your business! AR@dacentral.com 13155 Cloverdale 248-399-0600 Oak Park, Ml 48237 f4el 7 INVOICE D/A CENTRAL NVOICE intelligent technology solutions acule-iillelie=ib INVOICE DATE PURCHASE ORDER # CUSTOMER # TERMS PROJECT # PROJECT TITLE SALES PERSON New Fire Station 5 Security and Data VANWELJ SOLD TO: City of Warren SHIP TO: City of Warren Fire Station 5-New Purchasing Department 30619 Schoenherr Rd One City Square-Suite 425 Warren, MI 48093-5289 Warren, MI 48088 DESCRIPTION Labor $12,899.11 Equipment $16,688.38 COMMENTS THIS INVOICE: $30,544.69 Pregrasw Elling Mare SALES TAX (EXEMPT): $0.00 INVOICE TOTAL: ¢ $30,544.69] , an ea ‘exe runs IPYS< 4 fof ph be ie FE You can now pay online! www.dacentral.com/pay-invoice or by ACH Acct #: 4100689456 Routing:072413829 Swift:MEMIUS33 D/A Central Inc. Thank you for your business! AR@dacentral.com 13155 Cloverdale 248-399-0600 Oak Park, MI 48237 9g OVSO: 7/ Hallahan & Associates, P.C. Attorneys at Law 1750 S, Telegraph Road, Suite 202 Bloomfield Hills, Michigan 48302-0179 (248) 731-3089 Email ™ (March 3, 2025 >) City of Warren DD. ————--- One City ee Suite 425 - Purchasing Warren, Mi 48093 Please include Invoice No. with your payment Invoice No{22398 | $4,510.62- 815.49 Professional services rendered through February 28, 2025 v a Y- gy sf ¢ ¥- So lon Purchase Order #2529667 ) (09/15/2024 - 09/14/2025) 02/03/25 02/04/25 02/05/25 02/25/25 02/26/25 02/03/25 02/05/25 02/07/25 02/20/25 02/25/25 Name Lowe's Home Centers, Inc. - 24-001266 SAO KMM LMH SAO KMM LMH Received and reviewed documents from client for discovery responses. In-firm discussion with S. O'Loughlin regarding Respondent's responses to Petitioner's discovery requests. Telephone conference with Mike Shapiro. Prepare and file discovery answers with Petitioner; update case access chart. Prepare Respondent's Prehearing Statement. Communicate with appraiser. Subtotal: Wal-Mart Real Estate Business Trust - 24-001158 SAO SAO SAO LMH KMM Received and reviewed documents from client for discovery responses. Prepare and file discovery answers with Petitioner; update case access chart. Received and reviewed Petitioner's discovery answers; set aside documents for review. Telephone conference with opposing representative. Prepare Respondent's Prehearing Statement. Subtotal: Subtotal of charges Administrative fee (2%) Professional services rendered Timekeeper Summary Laura M. Hallahan Kelsea M. Melcher Seth A. O'Loughlin Previous balance Hours — Amount 0.10 20.50 0.10 20.50 0.30 61.50 0.10 20.50 1.00 205.00 0.30 61.50 1.90 389.50 0.20 41,00 0.10 20.50 0.20 41.00 0.30 61.50 1.20 246.00 2.00 410.00 $799.50 $15.99 3.90 ( $815.49 | Hours Rate 0.90 205.00 2.30 205.00 0.70 205.00 $3,695.03 100 City of Warren DDA Page 2 Amount AMOUNT DUE $4,510.52 Current 30 Days 60 Days 90 Days 120 Days 2,760.72 1,750.40 0:60 0.00 0.00 101 PAYMENT REQUEST Date: To: February 6, 2024 Mark Knapp, Assistant Controller, Controller's Office Sara Karpuk, Accountant, Controller's Office Re: Payment No. Project No, Location Improvement: Original Contract Amount Change Order No. | Change Order No. 2 Land Final Payee : P-22-772 DDA - City Square City Square Pavement Enhancement Current Contract Amount Total Work performed to Date Less Retainage Net Amount Earned to Date 10/7/2024 0.00% Amount of Previous Payment Requests Amount Due This Estimate Retainage Previously Withheld Retainage Change this Pay Estimate Chargeable to : DDA Account Fund PreparedSy Chukwudi Nnaji Civil Engineer cc; Payee WARREN PUBLIC SERVICE DEPARTMENT ENGINEERING DIVISION PK Contracting, LLC. 1965 Barrett Drive Troy, MI 48084 $83, Ui a i 137,00 $0.00 $0.00 $83,137.00 $83,137.00 $0,00 $83,137.00 $0.00 $83,137.00 $0.00 $0.00 $ 83,137.00 Approved for Payment: Tina Gapshes, P.E. City Engineer PAYMENT REQUEST Date: November 25, 2024 To: Mark Knapp, Assistant Controller, Controller's Office Sara Karpuk, Accountant, Controller's Office Re; Payment No, Project No. Location Improvement: Original Contract Amount Change Order No, | Change Order No, 2 Current Contract Amount Total Work performed to Date Less Retainage Net Amount Eamed to Date Amount of Previous Payment Requests Amount Due This Estimate Retainage Previously Withheld Retainage Change this Pay Estimate Chargeable to: DDA Account Fund Prepared by: DDA - City Square Cliy Square Pavement Enhancement IARREN PUBLIC SERVICE DEPARTMENT ENGINEERING DIVISION PK Contracting, LLC, 1965 Barrett Drive Troy, MI 48084 $83,137.00 $0.00 $0.00 $83,137.00 $0.00 $83,137.00 $0.00 $0,00 $0.00 Approved for Payment: $83,137.00 $83,137.00 $83,137.00 Chukwudl Nnaji Civil Engineer ce! Payee Tina Gapshes, P.E, City Engineer P-22-772 WVARREN Contractor: PAK Contracting, LLC, City Square Pavement Enhancement Engineering Division Address; 1968 Bareett Drive Estimate No, band Final CONSTRUCTION ESTIMATE Cltyt Troy AST 49084 ORIGINAL | CONTRACT | ORIGINAL | QUANTITY | AMOUNT | QUANTITY | AMOUNT ITEM DESCRIPTION UNITS BID vart BID PAID PAID yes THIS NO, QUANTITY PRICE AMOUNT | TODATE | TODATE | PAYMENT | PAYMENT 1 INSTALL DECORATIVE CROSSWALK, BRICK.COLOR AND PATTERN SF 1.0 $47.250 $17,250.00 1660.00 1 $17,250.00 1.00.00] § 17,250.00 TO MATCH EXISTING PAVERS (ALL PREP WORK TO DE INCLUDED IN THIS PAY ITEM) I COLOR, 2 JINSTALL DECORATIVE CROSSWALK, ORICK.COLOR AND PATTERN SF 4.0 $37.236 $51,730.00 3,692.00 | $ 63,687.00 2692.00 1 $63,687.00 YO MATCH ENISTING PAYERS (ALL PREP WORK TO BE INCLUDED 120 THIS PAY ITEM 2 COLORS 3 1BOND, INSURANCE AND INITIAL SET UP EXPENSES LS l $2,102,00 $2,200.00 LO] $ 2,200.00 LOOPS 2,200.40 (NOT TO EXCEED 344 OF CONSTRUCTION COST) Total Work Performed as of 10/7/2024 $74,200.00 $ $3,137.00 $ 83.137,00 Less Retsingge 0.00% $ . $ : Net Amount Eamd $ $3,937.00 $3,137.60 Less Previous Paymeats $ ‘ $ “ Tots Balance Dug thls Estimate $ $3,137.00 S_ $3,137.0 Leeetify that { have cheshed thi peciodic catimate: that te Ure best of my knowledges and ballet itis true ond comesd stdtemsnt of work performed by dhe contractor, that all work Included ln dils Riciodio estimate bas brea int pasted by! ime omy duly suthorized representative or sssislants and it has bron portonnied In full agcordance with the requirements of the contrast, Tina Gapshes, PLE. City Engineer Accouding to the bust of my bowledge and bellef, F certify that sii feats and amounts thown on this periodic estimates are Correct; that all work Aas basa perfonned fn full accor dew the téquirements of the Contract, that the foregalog (5 0 true and comest s(stemeat of the Contract eccount up to and Including the fast day of the period covered ” a part of the “balance duc this estimate® bas been received, That paymeal of same dus hecewith, ts withoo! collusion aed fraud in anyt respec vA ¢stim: (of PR Contrasting, Ine, 7 denisste 104 NOAA FRRREENI Cily Square Pavement Enhancement Contractor: PAR. Contracting, LLC. City Project No, P-22-772 Englneesing Diylsion Address: 1965 Barrett Drive Pay Estimate No. 1 and Final CONSTRUCTION ESTIMATE City: Troy, ME 48084 ORIGINAL AMOUNT AMOUNT DESCRIPTION CONTRACT PAID THIS AMOUNT TO DATE PAYMENT $ $3,137.00 Total Work Performed as of: 1024 $ $3,137.00 | $ $3,137.00 Less Retalnage 0.00% $ ~ 7s . Net Amount Eamed $ $3,137.00 1 $ $3,137.00 Less Previous Payments $ ‘ Total Amouat Due this Estimate $ 83,137.00 1 $ $3,137.00 Total 2024 DDA Fund Work Perfonned as of: lord $ $3,137.00] $ $3,137.00 | $ $3,137,00 Less Retalnage 0.00% $ . $ . Net Amount Eamed $ $3,137,00 | 5 $3,137,00 Less Previous Payments $ - 1s . Total 2023 DDA Fund Estimate: $ $3,137.00 | $ 33,137.00 1 certify that E have checked this periodic estimate; that (o the best of my knowledge and belief it is true and correct stalement of work performed by the contractor, that alf work included in this periodic estimate has been inspected by me or my duly authorized representative or assistants and it has been performed in full oecordance with the requirements of the contract, for the City of Warren, Tina Gapshes, PLE. City Engineer According to the best of my knowledge and belief, I certify that all {tents and amounts shown on this periodic estinials are correct; that all work has been performed In full accordance with the requirements of the Contracl, that the foregoing [5 a trve and correct slatement of the contract account Up fo and Including the last day of the period covered by thy periodie celmates that no part of the “balance due this estimate” hag been received, That payntens Of game dus herewith, if Without collusion and fraud In any respect ly ( a be hee vslor PK Contracting, LLC, wet” Contractor Oln22y G bbs en Invoice ———— March 12, 2025 > Date: Invoice: Project: Warren City Hall Plaza PO # 528156 ) Tom Bommarito, Director Community, Economic & Downtown Development City of Warren One City Square, Suite 215 Warren, Michigan 48093-6726 Services: Urban Planning and Engagement Services per July 1, 2024 Agreement 2528156 Professional Fees: 90% complete Town Ctr. Master Plan Design $25,000............22,500 Deduct previous payments:.... sstssessnssasseeseaeese($.10,000) Total Amount Due: semtitinnnseihe$42,500 194 VAY: BO (00 Sincerely, GIBBS PLANNING GROUP, INC. ¥ x x x, ; | New Mailing Address: - 3 at4—. Gibbs Planning Group 4611-F Sangamore Road PO Box 341 Bethesda, MD 20816 Robert J. Gibbs, FASLA, AICP President bob@gibbsplan.com Gibbs Planning Group, Inc. A Michigan Corporation Hl Ue aeey Fed Tax ID: 38 28-38903 bed) OVS New Mailing Address 4611-F Sangamore Road Box 341 Bethesda, MD 20816 248 642-4800 106 «3 \ “| Wo) @ 7315 Drake Road Date West Bloomfield, MI 48322 Terms PO # ( 2528917 ) Sales Rep Zach Pliska ba Amount Due $360.00 ra AW Bill To Property Address City of Warren City of Warren One City Square One City Square 4th Floor Warren, MI 48093-5289 Warren, MI 48093-5289 Description Amount #5542 - Standard Horticulture Service March 2025 $360.00 Subtotal $360.00 Sales Tax $0,00 a Total ( $360.00 ) Credits/Payments $0.00 Balance Due $360.00 LE Cuy - CALA Ce] a 6 sheeilo éurrseat 1-30 Days 31-60 Days 61-90 Days 90+ Days urren Past Due Past Due Past Due Past Due $360.00 $360.00 $360.00 $0.00 $0.00 p 2486611515 | f www.planterra.com 107 Date: To: From: Re: Payee: February 28, 2025 Mark Knapp, Assistant Controller Engineering Division Engineering Division Payment Request Environmental, Geotechnical, Testing Engineering Services RFP-W-0025 Payment No. 180 Contract: Invoice # 250202 Improvement: Fire Station 5 Traffic Signals Activity: Geotechnical Investigation (workscope approved 5/1/24) G2 Consulting Group, LLC 1866 Woodslee Street Troy, MI 48083 Original Contract Amount (approved 9/16/19) Contract Renewal - No Current Cap Current Amended Contract Amount Amount this Billing Previously Approved Work to Date Total Work Performed as of 01/31/25 Total Amount Due this Payment This Project Total Contract S 9,500.00 S 2,000.00 S 2,000.00 S 7,000.00 S 588,068.37 $ 9,000.00 S$ 590,068.37 $ 2,000.00 Chargeable to: 0.00% $ = Chargeable to: 494-00000493 100.00% $ 2,000.00 Prepared By: Auicleell— Tina G. Gapshes, PE, City Engineer cc: Payee Approved for Payment: Tom Bammarito Economic Development Director 108 G2 Consulting Group, LLC INVOICE 1866 Woodslee Street Invoice Number: 250202 Troy, MI 48083 Invoice Date: January 31,2025 Voice: 248.680.0400 Page Number: j Fax; 248.680.9745 Bill To: City of Warren One City Square Suite 300 Warren, Mi 48093-2390 eS eve =o Ship Date = 1 shi 1,00! Foundation Design Calculations, Lump Sum “2,000.00 2,000.00 Warren Fire Station No. 5 Traffic Signals, Schoenherr Road, Warren, Michigan Client Contact - Olivia Girimonte Total Invoice Amount $ 2,000.00 If you have any questions concerning this invoice, call Mark W. Smolinski, (248) 680-0400, Client agrees to pay a charge of 1.5 percent per month on accounts past due 30 days from invoice date, Make all checks payable to: G2 Consulting Group, LLC, 109 Engineering Division Payment Request Date: February 28, 2025 To: Mark Knapp, Assistant City Controller From: Engineering Division Re: Payment No. 181 Contract: : Environmental, Geotechnical, Testing Engineering Services RFP-W-0025 Invoice # 250107 Improvement: Fire Station 1, 23211 Van Dyke Activity: Material Testing Payee: G2 Consulting Group, LLC 1866 Woodslee Street Troy, Ml 48083 This Project Tota! Contract Original Contract Amount (approved 9/16/19) As Needed Contract Renewal - No Current Cap Current Amended Contract Amount Amount this Billing S 6,026.75 $ 6,026.75 Previously Approved Work to Date S 23,760.18 $ 590,068.37 Total Work Performed as of 01/27/25 S 29,786.93 S 596,095.12 Total Amount Due this Payment Ss 6,026.75 Chargeable to: 0.00% $ = Chargeable to: 494-00000493 100.00% $ 6,026.75 Prepared By: Aww Jevfo ~ Tina G. Gapshes, P.E. City Engineer Approved for Payment: Tom Bommarito Economic Development Director cc: Payee 110 G2 Consulting Group, LLC INVOICE 1866 Woodslee Street Invoice Number: 250107 Troy, MI 48083 Invoice Date: January 30, 2025 Voice: 248.680.0400 Page Number: 1 Fax: 248.680.9745 Bill To: City of Warren One City Square Suite 300 Watren, Mi 48093-2390 ciptio é = 45.00 Engineering Te hnician, Regular Hours, each — — 93.96 2,00| Engineering Technician, Overtime Hours, each 106.41 212.82 30.00 | Compressive Strength Test Cylinders, each 18.14 544,20 2,00 | Modified Proctor Test, each 241.85 483.70 7.00 | Troxler Nuclear Moisture/Density Gauge, each 60,46 423,22 4,00] Cylinder Pick-Up, per trip 120.93 483.72 3,00] Sample Pick-Up, each 102.79 308.37 1.00 | Project Manager, per hour 175.34 175.34 1.00} Administrative Assistant, per hour 76,18 76.18 Warren Fire Station 1, Warren, Michigan - Quality Control Observation and Testing Services on 1/3/25 through 1/27/25 Client Contact: Tina Gapshes Total Invoice Amount $ 6,026,75 If you have any questions concerning this invoice, call Mark W. Smolinski, (248) 680-0400, Client agrees to pay a charge of 1,5 percent per month on accounts past due 30 days from invoice date, Make all checks payable to; G2 Consulting Group, LLC. i 111 omade Michigan Downtown Association P.O. Box 3591 North Branch, MI 48461 248-838-9711 director@michigandowntowns.com MICHIGAN ; DOWNTOWN Membership Invoice ASSOCIATION Invoice € 3702 ) Thomas Bommarito City of Warren One City Square Suite 215 Warren, MI 48093 PNA Membership dues are based upon an organization's total budget for DOWNTOWN. Please refer to the tiers below and if necessary, adjust your annual membership rate accordingly. $249,000 & Under: $250 $250,000-$499,999: $400 ay Ue epi 4 $500,000+: $650 The $100 contribution to our Advocacy Fund has already been included in your membership dues renewal invoice and is 100% OPT-IN. MEMBER ITEM AMOUNT City of Warren Municipal or Public Agency 650.00 3/1/2025 - 3/1/2026 Membership Dues Contribution 400.00 Please remit payment to the address listed. Please do NOT Fax payments. TOTAL:( 750.00 ° 4g = Gy 4 } py ROO Payment Stub Member: Bommarito, Thomas City of Warren Invoice #: 3702 Please tear off this stub and include with your payment. Description: Membership Dues Send payment to: Date Due: 3/1/2025 Michigan Downtown Association Amt. Due: — 750.00 P.O. Box 3591 North Branch, MI 48461 Enclosed: h7S0.00 or login and pay online at www.michigandowntowns.com 112 MICHIGAN DOWNTOWN ASSOCIATION Michigan Downtown Association P.O. Box 3594 North Branch, Ml 48461 March 21, 2025 Dear Tom, Thank you for your continued support of the Michigan Downtown Association (MDA). On behalf of the MDA, the Board of Directors, and our 230 members, | want to thank you and the Warren community for your commitment to strong Michigan Downtowns. lam writing to inform you of a change to cur membership dues. After careful consideration and a thorough review of our financial requirements and the services the MDA provides, the Board of Directors has decided to increase the annual membership dues and reduce the number of membership tiers to three (previously five). This adjustment, the first in over eight years, is essential to ensure that we continue to deliver the high-quality programs, advocacy, and resources that our members expect and deserve. A downtown management organization’s annual dues are based upon the downtown’s annual budget. As your downtown's annual budget is over $500,000 effective October 1, 2024: New Dues Annual Amount Tier 3: $650 This increase will enable us to enhance our current offerings and introduce new initiatives aimed at providing even greater value to our members. We are committed to supporting you, your downtown, and organization and we are excited to share some of the specific benefits your membership will continue to provide: e« Access to member-only resources like reporting requirement templates, promotional materials, and shareable info graphics. « Discounted rates for our annual conference, workshops, and webinars. e Participation in podcasts and the Professional Development Certificate Program, Michigan’s only educational program dedicated to downtown management. « Opportunities to network with downtown stakeholders and peers from across the State. « Advocacy efforts at the local, state and national levels. These benefits, along with many others, are designed to provide substantial value and support to our members. We are committed to continually improving and expanding our offerings to meet your needs, elevate downtawn directors, and strengthen Michigan downtowns. 113 The MDA remains diligent and proactive with regards to Lansing and downtown legislation. To this end, the MDA respectfully asks each member to consider a $100 contribution to the MDA Legislative and Advocacy Fund added to their membership renewal dues. The $100 contribution has already been included in your membership dues renewal invoice and is 100% OPT-IN. Again, the contribution is OPT-IN and not required, but greatly appreciated. We understand that any increase in dues may be a concern, and we want to assure you that this decision was not made lightly. Our goal is to provide exceptional value for your membership, and we believe that the continued expansion of the quality resources we offer wiil significantly benefit you, your organization, and all our members. If you have any questions or would like to discuss this change further, please do not hesitate to contact me at director@MichiganDowntowns.com or 248-838-9711. ] am here to address any concerns you may have and to ensure a smooth transition. Thank you foryour continued support and involvement in the Michigan Downtown Association. We look forward to serving you in the coming year and working together to advance and strengthen Michigan downtowns. Sincerely, Dana Walker Dana Walker Director Michigan Downtown Association 114 OISIIG f= @m DN Hardware Sales AESINU SATIN — 3 55K Industrial Road ILJi | | Livonia MI 48150 INVOICE INVOICE DATE | INVOICE NO. SS ae : 734.591, —% Beene et eer dase 0312/25") 40712730-00)" 4 , _ P.O. NO. PAGE# cust. 20770 _ an (2527988. P snip to. CITY OF WARREN = corresponoence to: IDN Hardware Sales a ; ONE CITY SQUARE P.O. Box 510624 257935 / SUITE 425 Livonia, MI 48151 WARREN, MI 48098 aitto: CITY OF WARREN ONE CITY SQUARE SUITE 425 WARREN, MI 48098 ; - | INSTRUCTIONS SHIP POINT SHIP VIA SHIPPED TERMS _IDN Livonia Ground 03/12/25 Net 45 We ano Seon ee 1 .L9080BD-06N-626 27 0 27 ea 689.25000 18609,75 SCHLAGE STOREROOM MORTISE LOCK W/ SFIC HSG 2 .80-036.EV 626 R125 17 0 17 ea 54,70000 929.90 SCHLAGE SFIC CORES R125 KEYWAY 3. .35-264-EV 468 R125 50 0 50 ea 7.80000 390.00 SCHLAGE KEY BLANK 3 Lines Total Qty Shipped Total 94 Total 19929.65 Freight Out 11.86 Invoice Total ( 19941.51>| Ye TYPY. 97 Yo Last Page 115 old ) 44 GLL APPLICATION AND CERTIFICATE FOR PAYMENT AIA DOCUMENT G702 -— yy PAGE ONE OF 14 PAGES PROJECT: Fire Station No. 1 Fire Station No. 5 TO: City of Warren DDA One City Square Warren, Michigan 48093 FROM: BRIVAR Construction Company 9325 Maltby Road Brighton, Ml 48116 CONTRACT FOR: General Contracting APPLICATION NO: 018REV Distribution to: 7 OWNER PERIOD TO: _ 2/28/2025 CONSTRUCTION PO NO: ITB-W-0912 MANAGER [_]ARCHITECT CONTRACT DATE: []|CONTRACTOR CONTRACTOR'S APPLICATION FOR PAYMENT Application is made for payment, as shown below, in connection with the Contract. Centinuation Sheet, ALA Document G703, is attached. . ORIGINAL CONTRACT SUM 16,446,478.55 . Net change by Change Orders $10,164.52 ; CONTRACT SUM TO DATE (Line 1 +2) 17,256,043.07 yh AAWYW . TOTAL COMPLETED & STORED TO DATE 14,016,181.26 — (Column G on G703) . RETAINAGE: a 10 %of Completed Work $ (Column D + E on G703) b. 10 %ofStored Material (Column F on G703) ‘Total Retainage (Lines 5a+ 5b or Total in Column [ of G703) $ wa $ 67,142.98 1,269,386.59 12,746,794.67 6. TOTAL EARNED LESS RETAINAGE $ (Line 4 less Line 5 | otal) 3 7. LESS PREVIOUS CERTIFICATES FOR PAYMENT (Line 6 from prior Certificate) g 11,880,416.08 8. CURRENT PAYMENT DUE g $60,378.99 9. BALANCE TO FINISH, INCLUDING RETAINAGE $ 4,509,848.40 (Line 3 less Line 6) CHANGE ORDLRK SUMMAKY ADDITIONS DEDUCTIONS | Total changes approved in previous months by Owner: CO 001-007 $796,398.22 Total approved this Month: CO 008, 009 $13,766.30 TOTALS $810,164.52 $0.00 NET CHANGES by Change Order $810,164.52 The undersigned Contractor certifies that to the best of the Contractor's knowledge, information and belief the Work covered by this Application for Payment has been completed in accordance with the Contract Documents, that all amounts have been paid by the Contractor for Work for which previous Certificates for Payment were issued and payments received from the Owner, and that current payment shown herein is now due. naa CONTRACTOR: Date: SOF ZS Tt TN State of: Michigan ‘County play japon Subscribed and.s ism is ali 7 day of si. 2025 ON Notary Public: My Commission Wer 11/20/2028 Notary Public, State of Michigan County of Livingston SSiON CXDINES Acting im the County of Dis In accordance with the Contract Documents, b’ on on-site observations and the data comprising this application, the Contractor and Architect certity to the Owner that to the best of their knowledge, information and belief the Work has progressed as indicated, the quality of the Work is in accordance with the Contract Documents, and the Contractor is entitled to payment of the AMOUNT CERTIFIED. 2S SS Se AMOUNT CERTIFIED $ 866,378.59 (Attach explanation if amount certified differs from the amount applied for. Initial all Jigures on this Application and on the Continuation Sheet that changed to conform to the amount certified Bb ARTNERS in Architecture, PLC By: dul) ONL seeron A. Malone Date: 3/7/2025 This Certificate is not negotiable. The AMOUNT CERTIFIED is payable only to the Contractor named herein. Issuance, payment and acceptance of payment are without prejudice to any rights of the Owner or Contractor under this Contract. ALA DOCUMENT G702- APPLICATION AND CERTIFICATION FOR PAYMENT - 1992 EDITION - AIA®- © 1892 THE eae INSTITUTE ee es5 Lae N. WW. WASHINGTON, Be 20006-5292 acompleted AIA Document D401 - Certification of Document's Authenticity G702 1992 G703-1992 = 7 ~— re — ae ere ~~ LLL Continuation Sheet - Fire. Station No. 1 PAGE 20F 14 PAGES AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: ; A B Cc (3) E F 7 G H I PTTEM | DESCRI [OF K | SCHEDULED WORK COMPLETED MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+5 (NOT IN TO DATE DORE) (D+E+F) 1A|DEMOLITION/REMOVALS LABOR $22,900.00 $22,900.00 $22,900.00 | 100% $0.00 $1,145.00 1BIDEMOLITION/REMOVALS MATERIAL $4,800.00 $4,800.00 $4,800.00 | 100% $0.00 $240.00 ‘2A/SITE EARTHWORK/UTILITIES LABOR $210,000.00 $165,700.00 $165,700.00 79% $44,300.00 $16,570.00 2B/SITE EARTHWORK/UTILITIES MATERIAL $400,000.00 $319,600.00 $319,600.00 80% $80,400.00 $31,960.00 3A; ASPHALT PAVING/BASE LABOR $13,500.00 $0.00 $0.00 0% $13,500.00 $0.00 3BIASPHALT PAVING/BASE MATERIAL $0.00 $0.00 |7BRBAH $0.00 $0.00 4A|FENCING LABOR $14,683.00 $0.00 $0.00 0% $14,683.00 $0.00 4BiFENCING MATERIAL $53,047.00 $53,047.00 $53,047.00 | 100% $0.00 $5,304.70 S5A/LANDSCAPE LABOR $24,092.50 $0.00 $0.00. 0%; $24,092.50 $0.00 5B/LANSCAPE MATERIAL $46,307.50 $0.00 $0.00 0% $46,307.50 $0.00 6AIFOUNDATIONS LABOR $35,757.00 $35,757.00 $35,757.00 | 100% $0.00 $3,575.70 6B FOUNDATIONS MATERIAL $85,000.00 $85,000.00 $85,000.00 | 100% $0.00 $8,500.00 7A} CONCRETE/FLATWORK LABOR $219,426.00 $53,313.00 $53,313.00 24% $166,113.00 $5,331.30 7BICONCRETE/FLATWORK MATERIAL $148,000.00 $63,167.00 $63,167.00 43% $84,833.00 $6,316.70 8A|PRECAST/TILT-UP LABOR $15,000.00 $15,000.00 $15,000.00 | 100% $0.00 $0.00 8B/PRECAST/TILT-UP MATERIAL $34,150.00 $34,150.00 $34,150.00 | 100% $0.00 $0.00 9A] MASONRY LABOR $664,800.00 $651,504.00 $13,296.00 $664,800.00 | 100% $0.00 $66,480.00 9BIMASONRY MATERIAL $959,585.00 $940,393.30 $19,191.70 $959,585.00 | 100% $0.00 $95,958.50 SUBTOTAL THIS SHEET $2,951 ,048.00 $2,444,331 30 $32,487.70 $0.00 | $2,476,819.00 84% $474,229.00 $241,381.90 .Users-may obtain validation of this document by requesting of the'license a completed AIA Document D401 ~ Certification of | Document's Authenticity AJA DOCUMENT G703 - CONTINUATION SHEET FOR G702~ 1992 EDITION~ AIA® - @ 1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, NuW,, WASHINGTON, D.C, 20006-5232 G703-1992 SLL Continuation Sheet - Fire Station No. 1 PAGE 3 OF 14 PAGES LALA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV ‘Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. . PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A B Cc D | E F G H t ITEM DESCRIPTION OF WORK SCHEDULED WORK COMPL D “MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+E) (NOT IN TO DATE DORE) (D+E+F) 10A|STRUCTURAL STEEL LABOR $110,500.00 $110,500.00 $110,500.00 |} 100% $0.00 $11,050.00 10B)/ STRUCTURAL STEEL MATERIAL $253,500.00 $253,500.00 $253,500.00 } 100% $0.00 $25,350.00 11A;CARPENTRY LABOR $138,300.00 $28,400.00 $28,400.00 21% $109,900.00 $2,840.00 11B] CARPENTRY MATERIAL $102,700.00 $38,575.00 $38,575.00 38% $64,125.00 $3,857.50 12A} MILLWORK LABOR $10,030.00 $3,052.00 $3,052.00 30% $6,978.00 $305.20 12B| MILLWORK MATERIAL $49,970.00 $0.00 $0.00 0% $49,970.00 $0.00 12CISTAINLESS STEEL MILLWORK LABOR $3,767.00 $500.00 $500.00 13% $3,267.00 $50.00 12D] STAINLESS STEEL MILLWORK MATERIAL $27,593.00 $0.00 $0.00 0% $27,593.00 $0.00 13} WATERPROOFING $0.00 $0.00 $0.00 HSH $0.00 $0.00 Inlcuded in Masonry $0.00 $0.00 RRA $0.00 $0.00 14A|ROOFING/METAL PANELS LABOR $108,200.00 |} $108,200.00 $108,200.00 } 100% $0.00 $10,820.00 14B] ROOFING/METAL PANELS MATERIAL $226,300.00 $206,300.00 $206,300.00; 91% $20,000.00 $20,630.00 15A DOORS, FRAMES & HARDWARE MATERIAL $129,270.00 $7,078.00 $12,229.00 $19,307.00 18% $109,963.00 $1,930.70 16A/OVERHEAD DOORS LABOR $65,200.00 $7,710.00 $5,700.00 $13,410.00 21% $51,780.00 $1,341.00 16B!|OVERHEAD DOORS MATERIAL $109,162.00 $49,797.00 $27,046.00 $76,843.00 70% $32,319.00 $7,684.30 17A|GLASS & GLAZING LABOR $20,497.00 $11,836.00 $11,836.00 58% $8,667.00 $1,183.60 17BiGLASS & GLAZING MATERIAL $127,503.00 $127,503.00 $127,503.00 | 100% $0.00 $12,750.30 SUBTOTAL THIS SHEET $1,482,492.00 $952,951.00 | $44,575.00 $0.00: ~ $997,926.00 67% $484,566.00 $99,792.60 Users may- obtain validation. of this.document by requesting oF the license a completed AIA Document D401 = Certification of Document’s Authenticity AIA DOCUMENT G703 - CONTINUATION SHEET FOR G702- 1992 EDITION~ AIA® - @ 1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W,, WASHINGTON, D.C. 20006-5232 G703-1992 6LL Continuation Sheet - Fire Station No. 1 PAGE 4 OF 14 PAGES 41a Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing ‘APPLICATION NO: O18REV (Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 in tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A B c oD aL. E F G H f TEM |- DESCRIPTION OF WORK SCHEDULED WORK COMPLETED MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROMPREVIOUS [ ‘THIS PERIOD PRESENTLY COMPLETED (G+C} TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (O+5) (NOT IN TO DATE DORE) (D+E+F) 18AIFLOOR COVERING LABOR $116,923.00 $0.00 $0.00 0% $116,923.00 $0.00 18B]FLOOR COVERING MATERIAL $80,000.00 $59,110.22 $59,110.22 74% $20,889.78 $5,911.02 19A] PAINTING/CAULKING LABOR $35,000.00 $0.00 $8,000.00 $8,000.00:} 23% $27,000.00 $800.00 19B| PAINTING/CAULKING MATERIAL $20,000.00 $5,750.00 $5,750.00 29% $14,250.00 $575.00 20A)FIRE EXTINGUISHERS LABOR $2,850.00 $0.00 $0.00 | 0% $2,850.00 $0.00 20B|FIRE EXTINGUISHERS MATERIAL $0.00 |3eRHEE $0.00 $0.00 21 AITOILET PARTITIONS & ACCESSORIES LABOR $35,340.00 $0.00 $0.00} 0% $35,340.00 $0.00 2:1 BI TOILET PARTITIONS & ACCESSORIES MATERIAL $0.00 |##HRHHH $0.00 $0.00 224) APPLIANCES MATERIAL $59,642.00 $5,607.50 $51,522.96 $57,130.46 96% $2,511.54 $5,713.05 22B\ APPLIANCES LABOR $5,000.00 $0.00 $0.00 0% $5,000.00 $0.00 23AlFIRE SUPPRESSION LABOR $50,000.00 $7,000.00 $37,475.00 $44,475.00 89% $5,525.00 $4,447.50 23B| FIRE SUPPRESSION MATERIAL $28,200.00 $0.00 $28,200.00 $28,200.00 | 100% $0.00 $2,820.00 24A]/ CONVEYANCE SYSTEMS LABOR $20,627.00 $0.00 $0.00 0% $20,627.00 $0.00 24BICONVEYANCE SYSTEMS MATERIAL $0.00 $0.00 [3484HHF $0.00 $0.00 25A} PLUMBING & COMPRESSED AIR LABOR $229,141.00: $78,270.80 $43,468.50 $121,739.30 53% $107,401.70 $12,173.93 25BiPLUMBING & COMPRESSED AIR MATERIAL $286,694.00 $146,651.40 $51,682.95 $198,334.35 69% $88,359.65 $19,833.44 26AIHVAC LABOR $304,495.00: $75,110.00 $36,250.00 $111,360.00 37% $193,135.00 $11,136.00 26B/ HVAC MATERIAL $484,470.00 $102,829.53 $188,640.47 $291 470.00 60% $193,000.00 $29,147.00 SUBTOTAL THIS SHEET $1, 758,382.00 $415,469.23 $210,826.45 $299,273.65 $925,569.33 53% $832,812.67 $92 556.94 Users may obtain validation of this document by requesting of the license a completed AIA Document D401 - Certification of Document's Authenticity AIA DOCUMENT G703 + CONTINUATION SHEET FOR G702 - 1992 EDITION - AIA®@- © 4892 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, NW, WASHINGTON, D.C, 20006-5232 G703-1992 OL Continuation Sheet - Fire Station No. 1 PAGE 5 OF 14 PAGES AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A {| B c D E F G H { TTEM ‘DESCRIPTION OF WORK . SCHEDULED WORK COMPLETED MATERIALS “TOTAL To BALANCE RETAINAGE NO. VALUE FROM PREVIOUS — THIS PERIOD PRESENTLY COMPLETED (G+) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (G-G) RATE) (O+E) (NOT IN TO DATE DORE) (D+E+F) 27A JELECTRICAL LABOR $171,627.00 $28,828.60 $17,091.14 $45,919.74 27% $125,707.26 $4,591.97 27B |ELECTRICAL MATERIAL $570,332.00 $160,803.52 $38,116.40 $159,811.03 $358,730.95 63% $211,601.05 | $35,873.10 27C |ELECTRICAL GENERATOR $163,322.00 $0.00 $163,322.00 $163,322.00 | 100% $0.00 $16,332.20 28A {LOW VOLTAGE LABOR $103,502.50 $50,539.00 $8 900.00 $59,439.00 57% $44,063.50 $5,943.90 28B |LOW VOLTAGE MATERIAL $98,497.00 $49,473.95 $49,023.05 $98,497.00 | 100% $0.00 $9,849.70 29A |FLAGPOLES LABOR $14,250.00 $0.00 $0.00 0% $14,250.00 $0.00 29B |FLAGPOLES MATERIAL $0.00 SHAAHEE $0.00 $0.00 30 PERMITS (ALLOWANCE) $5,000.00 $1,818.00 $1,818.00 36% $3,182.00 $181.80 31 STAKING $18,600.00 $15,480.00 $1,860.00 $17,340.00 93% $1,260.00 $1,734.00 32A |SIGNAGE LABOR $16,375.00 $0.00 $0.00 | ; 0% $16,375.00 $0.00 32B SIGNAGE MATERIAL $0.00 $0.00 HRHHF $0.00 $0.00 383A iIWINDOW TREATMENTS LABOR $4,369.00 $0.00 $0.00 0% $4,369.00 $0.00 34 OWNER ALLOWANCE - $51,916.00 $35,350.16 $0.00 $0.00 | 0% $35,350.16 $0.00 34A OCOR 008 - Gates $4,406.13 $400.00 $400.00 9% $4,006.13 $40.00 34B OCOR 016 - Pre-Cast Plates $605.00 $605.00 $605.00 | 100% $0.00 $60.50 34C OCOR 017 - Millwork Handie $1,259.28. $0.00 $0.00 0% $1,259.28 $0.00 34D | OCOR 019 - Added Millwork Depth $1,711.69 $0.00 $0.00 0% $1,711.69 $0.00 34E | OCOR 050 - Partially Funded $4,587.14 $767.85 $767.85 17% $3,819.29 $76.79 34F OCOR 051 - Sign Demo $3,500.00 $3,500.00 $3,500.00 | 100% $0.00 $175.00 34G } OCOR 052 - Partially Funded $496.60 $0.00 $0.00 0% $496.60 $0.00 SUBTOTAL THIS SHEET $1,217,790.50 $312,215.92 $65,967.54 $372,156.08 $750,339.54 62% $467,450.96 | $74,858.96 Users may obtain validation of this document by requesting of ihe license a completed AIA Document D401 - Certification of Documents Authenticity Als, DOCUMENT G703 - CONTINUATION SHEET FOR G702-~ 1992 EDITION - AIAD- @ 1892 6703-1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W,, WASHINGTON, GC, 20006-5232 LoL Continuation Sheet - Fire Station No. 1 PAGE 6 OF 14 PAGES THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C. 20005-5232 lA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A 8 c D E F G H I ITEM: DESCRIPTION OF WORK SCHEDULED T ~ WORK COMPLETED MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THI f PRESENTLY COMPLETED (G=C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+E) (NOT IN TO DATE D ORE) (D+E+F) 35 |ALLOWANCE #1 - $11,287.50 OCOR 033 - Partially Funded $11,287.50 $11,287.50 $11,287.50 400% $0.00 $1,128.75 36 |ALLOWANCE #2 - $8,850.00 1 OCOR 033 - Partially Funded $8,850.00 $8,850.00) $8,850.00 100% $0.00 $885.00 37 {GENERAL CONDITIONS $31,175.00 $19,950.00 $1,700.00 $21,650.00 69% $9,525.00 $2,165.00 38 [BRIVAR LABOR $148,266.50 $95,000.00 $8,000.00 $103,000.00 69% $45,266.50 $10,300.00 PROJECT MANAGEMENT, SUPERVISION, ENGINEER, LABOR, ETC 39 fINSURANCE $12,000.00 $12,000.00 $12,000,00 100% $0.00 $1,200.00 40 }PERFORMANCE BOND $60,307.50 $60,307.50 $60,307.50 100% $0.00 $6,030.75 41 {WEATHER CONDITIONS - $60,000.00 $42,928.38 $0.00 $0.00 0% $42,928.38 $0.00 (ALLOWANCE) OCOR 055 $4,400.00 $0.00 $4,400.00 $4,400.00 100% $0.00 $440.00 OCOR 056 $12,671.62 $12,671.62 $12,671.62 100% $0.00 $1,267.16 42 jiOVERHEAD $174,484.00 $124,000.00 $124,000.00 71% $50,484.00 $12,400.00 43 =|FEE $325,850.50 $209,050.00 $17,000.00 $226,050.00 69% $99,800.50 $22,605.00 44 |iChange Order 002 - $275,185.94 444 | OCOR 010R2 $59,246.89 $39,200.00 $39,200.00 66% $20,046.89 $3,920.00 44B OCOR 022 $126,262.29 $126,262.29 $126,262.29 100% $0.00 $12,626.23 44C | OCOR 023 $89,676.76 $0.00 $0.00 0% $89,676.76 $0.00 “|SUBTOTAL THIS SHEET $1,107,406.94 $718,578.91 $31,100.00" $0.00 $749,678.91 68% $357,728.03 $74,967.89 Users may obtsin vabdation of this. document by tequesting of the license a completed AIA Document D401 - Certification of Documents Authenticity AIA DOCUMENT G703 - CONTINUATION SHEET FOR G702- 1992 EDITION - AIA®- © 1992 G703-1992 CCl Continuation Sheet - Fire Station No. 1 PAGE TOF 14 PAGES AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column I on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A | 8 c D | E F G H I ITEM DESCRIPTION OF WORK SCHEDULED WORK COMPLETED MATERIALS TOTAL BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE} (D+) (NOT IN TO DATE . DORE) (D+E+F) 45 |Change Order 004 - OCOR026 $33,536.90 $2,523.16 $2,500.00 $5,023.16 18% $28,513.74 $502.32 46 |Change Order 006 ; 46A OCOR 033 $31,634.00 $31,634.00 $31,634.00 | 100% $0.00 $3,163.40 47 |Change Order 007 - ATA OCOR 041 $29,481.45 30.00 $4,000.00 $4,000.00 14% $25,481.45 $400.00 47B OCOR 054 $11,897.35 $1,539.81 $1,539.81 13% $10,357.54 $153.98 47C OCOR 057 $20,107.64 $20,107.64 $20,107.64 | 100% $0.00 $2,010.76 48 |Change Order 008 48A OCOR 058 $1,013.65 $0.00 0% $1,013.65 $0.00 48B OCOR 059 $1,529.00 $0.00 0% $1,529.00 $0.00 48C OCOR 060 $2,860.00 $0.00 0% $2,860.00 $0.00 49 |Change Order 009 49A OCOR 025 - Generator Storage $1,480.50 $1,480.50 $1,480.50 ; 100% $0.00 $148.05 $0.00 HHABAHT $0.00 $0.00 $0.00 [fS8HHHE $0.00 $0.00 $0.00 (aHHHE $0.00 $0.00 $0.00 |##BHRA $0.00 $0.00 $0.00 |#HRAHHE $0.00 $0.00 $0.00 |35HHHHF $0.00 $0.00 SUBTOTAL THIS SHEET $1 33,540.49 $55,804.61 $7,980.50 $0.00 $63,785.11] 48% $69,755.38 $6,378.51 SUBTOTAL FIRE STATION 1 $8,650,659.93 $4,899,350.97 $393,337.19 $671,429.73 | $5,964,117.89 69%] $2,686,542.04 | $589,936.80 A DOCUMENT eyed SERRA bias: Val dahon tthisdagurmaagss by requesting of the license a completed AIA Document D401 - Cerfification of Documents Authenticity HE AMERIC, > OF ARGHH SUIS, TFS NE LAV NAA, WASHINGTON, Bx, . PZ eros A? Continuation Sheet - Fire Station No. 5 PAGE 8 OF 14 PAGES AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column I on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A B i c D E F G H t PTE DESCRIPTION OF WORK SCHEDULED WORK COMPLETED MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED {G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+) (NOT IN TO DATE DORE) (D+E+F). 51A |DEMOLITION/REMOVALS LABOR $29,500.00 $29,500.00 $29,500.00 | 100% $0.00 $1,475.00 | 51B |DEMOLITION/REMOVALS MATERIAL $20,000.00 $20,000.00 $20,000.00 100% $0.00 $1,000.00 52A ISITE EARTHWORK/UTILITIES LABOR $220,632.00 $199,854.15 $199,854.15 91% $20,777.85 $19,985.42 52B |SITE EARTHWORK/UTILITIES MATERIAL $400,000.00. $356,636.41 $356,636.41 89% $43,383.59 $35,663.64 53A |ASPHALT PAVING/BASE LABOR $19,580.00 $0.00 $0.00 0% $19,580.00 $0.00 53B [ASPHALT PAVING/BASE MATERIAL $0.00: $0.00 | HB $0.00 $0.00 54A |FENCING LABOR $18,035.00 $18,035.00 $18,035.00 | 100% $0.00 $1,803.50 54B |FENCING MATERIAL $60,022.00 $60,022.00 $60,022.00 | 100% $0.00 $6,002.20 554 |LANDSCAPE LABOR $19,044.50 $19,044.50 $19,044.50 | 100% $0.00 $1,904.45 55B |LANSCAPE MATERIAL $45,325.50 $45,325.50 $45,325.50 | 100% $0.00 |: $4,532.55 56A |FOUNDATIONS LABOR $30,504.00 $30,504.00 $30,504.00 |} 100% $0.00 $3,050.40 56B |FOUNDATIONS MATERIAL $57,025.00 $57,025.00 $57,025.00 | 100% $0.00 $5,702.50 57A |CONCRETE/FLATWORK LABOR $148,114.00 $148,114.00 $148,114.00 | 100% $0.00 $7,405.70 57B iCONCRETE/FLATWORK MATERIAL $224,168.00 $224,168.00 $224,168.00 | 100% $0.00 $11,208.40 58A |PRECAST/TILT-UP LABOR $15,000.00 $15,000.00 $15,000.00 | 100% $0.00 $0.00 58B {PRECAST/TILT-UP MATERIAL $29,100.00 $28,100.00 $29,100.00 | 100% $0.00 $0.00 594 {MASONRY LABOR $681,000.00 $68 1,000.00 $681,000.00 | 100% $0.00 $34,050.00 59B [MASONRY MATERIAL $881,000.00 $881,000.00 $881,000.00 | 100% $0.00 $44,050.00 SUBTOTAL THIS SHEET $2,898,050.00 | $2,814,328.56 $0.00 $0.00 $2,814,328.56 97% $83,721.44 { $177,833.76 Users may obtain validation of this document by requesting. of the license-a completed AIA Document D401 - Certification of Documents Authenticity AJA DOCUMENT G703 - CONTINUATION SHEET FOR G702- 1992 EDITION - AIA@> © 1992 G703-1982 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C. 20006-5232 vel Continuation Sheet - Fire Station No. 5 PAGE 9 OF 14 PAGES. ‘AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: 018REV Contractor's signed certification Is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A B Cc D E F G H i TEM DESCRIPTION OF WORK SCHEDULED WORK COMPLETED “|. MATERIALS TOTAL “He BALANCE RETAINAGE NO. VALUE FROM PREVIOUS | THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+E) (NOT IN TO DATE DORE) (D+E+F) 60A/STRUCTURAL STEEL LABOR $110,500.00 $110,500.00 $110,500.00 | 100% $0.00 $5,525.00 60B| STRUCTURAL STEEL MATERIAL $253,500.00 $253,500.00 $253,500.00 | 100% $0.00 $12,675.00 61AiCARPENTRY LABOR $130,800.00 $112,310.00 $142,310.00 86% $18,490.00 $11,231.00 61BICARPENTRY MATERIAL $84,700.00 $84,700.00 $84,700.00 | 100% $0.00 $8,470.00 62A| MILLWORK LABOR $10,030.00 $3,052.00 $3,052.00 30% $6,978.00 $305.20 62B/ MILLWORK MATERIAL $49,970.00 $0.00 $0.00 0% $49,970.00 $0.00 62C|STAINLESS STEEL MILLWORK LABOR $3,767.00 $3,767.00 $3,767.00 | 100% $0.00 $376.70 62D|STAINLESS STEEL MILLWORK MATERIAL $27,593.00° $27,593.00 $27,593.00 | 100% $0.00 $2,759.30 63) WATERPROOFING $0.00 $0.00 $0.00 |2HRHHEE $0.00 $0.00 Inlcuded in Masonry $0.00 $0.00 |38HHF $0.00 $0.00 64A| ROOFING/METAL PANELS LABOR $109,800.00 - $109,800.00 $109,800.00 | 100% $0.00 $10,980.00 64B] ROOFING/METAL PANELS MATERIAL $198,500.00 $178,500.00 $178,500.00 90% $20,000.00 $17,850.00 65AIDOORS, FRAMES & HARDWARE MATERIAL $126,770.00 $62,406.00 $64,364.00 $126,770.00 | 100% $0.00 $12,677.00 66A| OVERHEAD DCORS LABOR $65,200.00 $59,671.50 $59,671.50 92% $5,528.50 $5,967.15 66B] OVERHEAD DOORS MATERIAL $109,162.00 $109,162.00 $109,162.00 | 100% $0.00 $10,916.20 67A{GLASS & GLAZING LABOR $17,322.00 $14,661.00 $14,661.00 85% $2,661.00 $1,466.10 67BIGLASS & GLAZING MATERIAL $130,678.00 $130,678.00 $130,678.00 | 100% $0.00 $13,067.80 SUBTOTAL THIS SHEET ~ $1,428,292.00 $1,260,300.50 $64,364.00 $0.00 | $1,324,664.50 93% $103,627.50 | $114,266.45 Users may obtain validation. of this document by requesting of the license a completed. AIA Document:D401.-Certification.of Document's Authenticity AIA DOCUMENT G703 - CONTINUATION SHEET FOR G702- 1992 EDITION ~ AIA@- © 1992 G703-1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C, 20005-5222 Continuation Sheet - Fire Station No. 5 PAGE 10 OF 14 PAGES GCL AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 in tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: LOA B c D J E F G H t TEM DESCRIPTION OF WORK “SCHEDULED WORK COMPLET MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+) (NOT IN TO DATE DORE) (D+E+F) 68A}FLOOR COVERING LABOR $116,923.00 $0.00 $65,210.92 $65,210.92 56% $51,712.08 $6,521.09 68BIFLOOR COVERING MATERIAL $80,000.00 $59,110.21 $20,889.79 $80,000.00 } 100% $0.00 $8,000.00 69AIPAINTING/CAULKING LABOR $35,000.00 $26,250.00 $3,500.00 $29,750.00 85% $5,250.00 $2,975.00 698) PAINTING/CAULKING MATERIAL $20,000.00 $15,000.00 $2,000.00 $17,000.00 85% $3,000.00 $1,700.00 7OAIFIRE EXTINGUISHERS LABOR $2,850.00 $0.00 $0.00 | 0% $2,850.00 $0.00 | 7OBIFIRE EXTINGUISHERS MATERIAL $0.00 $0.00 |2ARHHHE $0.00 $0.00 7 1AITOILET PARTITIONS & ACCESSORIES LABOR $35,340.00 $0.00 $0.00; 0% $35,340.00 $0.00 71 BIToILer PARTITIONS & ACCESSORIES MATERIAL $0.00 $0.00 HAAHHHE $0.00 $0,00 72A;APPLIANCES MATERIAL $59,642.00 $53,951.54 $53,951.54 90% $5,690.46 $5,395.15 72B|APPLIANCES LABOR $5,000.00 $0.00 $0.00 0% $5,000.00 $0.00 73A\FIRE SUPPRESSION LABOR $50,000.00 $46,275.00 $3,525.00 $49,800.00 | 100% $200.00 $4,980.00 73B\FIRE SUPPRESSION MATERIAL $28 200.00 $28,200.00 $28,200.00 | 100% $0.00 $2,820.00 74A| CONVEYANCE SYSTEMS LABOR $20,627.00 $0.00 _ $0.00 0% $20,627.00 $0.00 74B| CONVEYANCE SYSTEMS MATERIAL $0.00 $0.00 [aR $0.00 $0.00 75A|PLUMBING & COMPRESSED AIR LABOR $220,675.00 $186,599.26 $34,075.74 $220,675.00 | 100% $0.00 $22,067.50 75B| PLUMBING & COMPRESSED AIR MATERIAL $295,160.00. $271,329.40 $23,830.60 $295,160.00 | 100% $0.00 | $29,516.00 76AIHVAC LABOR $304,495.00 $292,320.00 $6,000.00 $298,320.00 98% $6,175.00 $29,832.00 76B)HVAC MATERIAL $484,470.00 $462,756.00 $11,000.00 $473,756.00 98% $10,714.00 $47,375.60 SUBTOTAL THIS SHEET $1,758,382.00 $1 441,797.41 $170,032.05 $0.00 $1,611,823.46 92% $146,558.54 | $161,182.34 Users. may obtain validation of this document by-requesting of the license a completed AIA Document D401 Certification of Document's Authenticity AIA, DOCUMENT G703 - CONTINUATION SHEET FOR G702 « 1992 EDITION - AIA®- © 1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C. 20006-5232 G703-1992 QL Continuation Sheet PAGE 11 OF 14 PAGES THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C. 20006-5232 AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV |Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations belaw, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A B c D E F G H ; 1 TrEM DESCRIPTION OF WORK SCHEDULED WORK COMPLETED “MATERIALS JOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+) (NOT IN TO DATE DORE) (DIE+F) 77A {ELECTRICAL LABOR $169,427.00 $134,580.00 $24,064.30 $158,644.30 94% $10,782.70 $15,864.43 77B |ELECTRICAL MATERIAL $513,702.00 $464,393.80 $19,543.45 $483,937.25 94% $29,764.75 | $48,393.73 77C \|ELECTRICAL GENERATOR $163,322.00 $163,322.00 $163,322.00 | 100% $0.00 $16,332.20 78A |LOW VOLTAGE LABOR $103,502.50 $79,998.56 $8,117.19 $88,115.75 85% $15,386.75 $8,811.58 78B |LOW VOLTAGE MATERIAL $98,497.00 $98,497.00 $98,497.00 |} 100% $0.00 $9,849.70 79A |FLAGPOLES LABOR $14,250.00 $0.00 $0.00 0% $14,250.00 $0.00 79B |FLAGPOLES MATERIAL $0.00 $0.00 |235RHHF $0.00 $0.00 80 {PERMITS (ALLOWANCE) $5,000.00 $3,184.80 $3,184.80 64% $1,815.20 $318.48 81 STAKING $18,600.00 $13,560.00 $13,560.00 73% $5,040.00 $1,356.00 82A {SIGNAGE LABOR $16,375.00 $0.00 $0.00 0% $16,375.00 $0.00 828 |SIGNAGE MATERIAL $0.00 $0.00 |$8HAHHE $0.00 $0.00 83 |WINDOW TREATMENTS $4,369.00 $0.00 $0.00 0% $4,369.00 $0.00 84 [OWNER ALLOWANCE - $51,916.00 $1,573.53 $0.00 $0.00 0% $1,573.53 $0.00 184A | OCOROS- Site Debris/Bridge Over Footing $4,033.92 $4,033.92 $4,033.92 | 100% $0.00 $403.39 |84B OCOROS - Gates $4,406.12 $1,955.06 $1,955.06 44% $2,451.06 $195.51 |84c OCOROS - Temp Road $417.68 $417.68 $417.68 | 100% $0.00 $41.77 84D OCOR'1E6 - PreCast Plates $605.00 $605.00 $605.00 | 100% $0.00 $60.50 84E OCOR17 - Millwork Handle $1,259.28 $0.00 $1,259.28 $1,259.28 | 100% $0.00 $125.93 84F OCOR19 - Added Millwork Depth $1,711.69 $0.00 $1,711.69 $1,711.69 | 100% $0.00 $171.17 84G | OCOR20 -U/G Comm. Conduit $8,024.64 $8,024.64 $8,024.64 | 100% $0.00 $802.46 SUBTOTAL THIS SHEET $1,129,076.36 $972,572.46 $54,695.91 $0.00} $1,027,268.37 91% $101,807.99 | $102,726.85 Users may obtain validation of this document by requesting. of the license a completed AIA Document 5401:- Certification oF Document's Authenticity AIA DOCUMENT G703- CONTINUATION SHEET FOR G702 - 1992 EDITION » ALA®- © 1892 G703-1982 Continuation Sheet PAGE 12 OF 14 PAGES: AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. : APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A. . 8 , _ . C. D we _E F 2 G H I ITEM DESCRIPTION OF WORK SCHEDULED WORK COMPLETED MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS | FHIS PERIOD PRESENTLY COMPLETED (G+ C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED (C-G) RATE) (D+E) (NOT IN TO DATE DORE) (D+E+F) 84H | OCORO7 - Masonry Winter Conditions $18,842.16 $18,842.16 $18,842.16 | 100% $0.00 | $942.11 84] OCOR24 - ATT Permit $500.00 $500.00 $500.00 | 100% $0.00 $50.00 : 845 OCOR43 - Temp Heating $5,458.24 $5,458.24 $5,458.24 | 100% $0.00 $545.82 | 84K | OCORSO - Partially Funded $4,587.14 $3,745.28 $3,745.28 82% $841.86 | $374.53 841. OCORS2 - Partially funded $496.60 $0.00 $0.00 0% $496.60 $0.00 85 |ALLOWANCE #1 - $11,287.50 $0.00 $0.00 $0.00 |HHRE $0.00 $0.00 85A |OCOROG - Foundation Undercuts $11,287.50 $11,287.50 $11,287.50 | 100% $0.00 $1,128.75 86 jALLOWANCE #2 - $8,850.00 $3,195.00 $0.00 $0.00 0% $3,195.00 $0.00 86A |OCOROE - Foundation Undercuts $5,655.00 $5,655.00 $5,655.00 ; 100% $0.00 $565.50 87A |CONGCRETE RETAINING WALL LABOR $22,163.55 $22,163.55 $22,163.55 100% $0.00 $2,216.36 87B {CONCRETE RETAINING WALL MATERIAL $29,061.70 $29,061.70 $29,061.70 | 100% $0.00 $2,906.17 | 88 |GENERAL CONDITIONS $38,175.00 $33,767.00 $2,000.00 $35,767.00 94% $2,408.00 $3,576.70 89 BRIVAR LABOR $181,266.50 $159,800.00 $11,000.00 $170,800.00 94% $10,466.50 $17,080.00: PROJECT MANAGEMENT, SUPERVISION, ENGINEER, LABOR, ETC 90 INSURANCE $12,000.00 $12,000.00 $12,000.00 | 100% $0.00 $1,200.00 91 |PERFORMANCE BOND $60,307.50 $60,307.50 $60,307.50, 100% $0.00 $3,015.38 92 WEATHER CONDITIONS - $60,000.00 $0.00 $0.00 $0.00 HHBHHT $0.00 $0.00 92A |OCOR02 - Foundation Depth $35,254.69 $0.00 $0.00 0% $35,254.69 $0.00 92B j|OCOR04 - Concrete Mix, Blankets $6,586.56 $6,586.56 $6,586.56 | 100% $0.00 $658.66 92C {OCORDS - Temp Road $18,158.75 $18,158.75 $18,158.75 | 100% $0.00 $1,815.88 93 OVERHEAD $214,484.00 $189,250.00 $13,000.00 $202,250.00 | 94% $12,234.00 $20,225.00 SUBTOTAL THIS SHEET $667,479.89 $576,583.24 $26,000.00 $0.00 $602,583.24 90% $64,896.65 | $56,300.36 Users may obtain validatian of this document by requesting of the license a completed AIA Document D401 - Cerffiication of Documents Authenticity AJA DOCUMENT G703 - CONTINUATION SHEET FOR G702 - 1992 EDITION - AIA@~ © 7892 G703-1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, NLW., WASHINGTON, D.C. 20006-5232 RA! Continuation Sheet PAGE 13 OF 14 PAGES AIA Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O1BREV ‘Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar. PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECTS PROJECT NO: A B c D | E F G H t ITEM DESCRIPTION OF WORK SCHEDULED : “WORK COMPLETED MATERIALS : FOTAL % BALANCE RETAINAGE NO. VALUE FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G+C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED {C-G) RATE) (D+&) (NOT IN TO DATE DORE} . (D+E+F) j94 {FEE $323,264.80 $284,900.00 $20,000.00 $304,900.00 94% $18,364.80 $30,490.00 95 Change Order 001 $67,962.24 $67,962.24 $67,962.24 | 100% $0.00 $6,796.22 | Subsoil conditions 96 |Change Order 003 - $133,614.08 964A | OCOR 010R2 $87,891.08 $66,597.91 $66,591.91 76% $21,299.17 $6,659.19 '96B | OCOR 014 $30,050.67 $30,050.67 $30,050.67 | 100% $0.00 $3,005.07 96C | OCORO1SR $15,672.33 $15,672.33 $15,672.33 | 100% $0.00 $1,567.23 97 |Change Order 004 - $55,969.90 . 987A | OCOR 026 $33,536.90 $8,541.18 $24,995.72 $33,536.90 | 100% $0.00 $3,353.69 97B }| OCOR 034 $22,433.00 $22,433.00 $22,433.00'| 100% $0.00 $2,243.30 98 iChange Order 005 - $23,603.20 S8A | OCOR 038R2 $9,550.24 $9,550.24 $9,550.24 | 100% $0.00 $955.03 98B OCOR 039R2 $3,644.55 $3,644.55) $3,644.55 | 100% $0.00 $364.46 98C | OCOR 040R2 $6,290.43 $6,290.43 $6,290.43 | 100% $0.00 $629.05 98D | OCOR 042R2 $4,117.98 $4,117.98 $4,117.98 | 100% $0.00 $411.80 99 {Change Order 006 - $23,603.20 994} OCOR035 $58,987.33 $0.00 $58,987.33 $58,987.33 | 100% $0.00 $5,898.74 99B | OCOR 045 $973.50 $0.00 $0.00 0% $973.50 $0.00 100 |Change Order 007 - 400A OCOR 041 $33,359.17 $6,117.33 $15,585.31 $21,702.64 65% $11,656.53 $2,170.26 SUBTOTAL THIS SHEET $697 ,734.22 $525,871.86 $11 9,568.36 $0.00 $645,440.22 93% $52,294.00 $64,544.04 Users may obtain validation of this document by Tequesting-of the license: acompleted AIA Document D407 - Certification of Documents Authemicity 8cl AIA DOCUMENT G703 - CONTINUATION SHEET FOR G702+ 1992 EDITION - AIA®- © 1992 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, D.C. 20006-5232 G703-1992 6CL Continuation Sheet - Fire Station No. 5 PAGE 14 OF 14 PAGES AIA. Document G702, APPLICATION AND CERTIFICATION FOR PAYMENT, containing APPLICATION NO: O18REV Contractor's signed certification is attached. APPLICATION DATE: 3/6/2025 In tabulations below, amounts are stated to the nearest dollar PERIOD TO: 2/28/2025 Use Column | on Contracts where variable retainage for line items may apply. ARCHITECT'S PROJECT NO: A B . Cc D E FE G H i TEM DESCRIPTION OF WORK SCHEDULED ; WORK COMPLETED MATERIALS TOTAL % BALANCE RETAINAGE NO. VALUE {FROM PREVIOUS THIS PERIOD PRESENTLY COMPLETED (G=C) TO FINISH (IF VARIABLE APPLICATION STORED AND STORED {C-G) RATE) (D+E) (NOT IN TO DATE DOR &) (D+E+F) 1008 | OCOR 046 $7,028.01 $1,916.73 $5,111.28 $7,028.01 100% $0.00 $702.80 qooc | OCOR 054 $13,057.51 $5,707.82 $7,349.69 $13,057.51 100% $0.00 $1,305.75 101 {Change Order 008 101A; OCOR 058 $1,013.65 $0.00 0% $1,013.65 $0.00 101B{ OCOR 059 $1,529.00 $1,529.00 $1,529.00 100% $0.00 $152.90 101C; OCOR 060 $2,860;00 $2,860.00 $2,860.00 100% $0.00 $286.00 102 |Change Order 009 402A} OCOR 025 - Generator Storage $1,480.50 $1,480.50 $1,480.50 100% $0.00 $148.05 $0.00 | #DIV/0! $0.00 $0.00 $0.00 | #DIV/0! $0.00 $0.00 $0.00 | #DIV/0! $0.00 $0.00 $0.00 | #DIV/0! $0.00 $0.00 $0.00 | #DIV/0! $0.00 $0.00 $0.00 | #DIV/0! $0.00 $0.00 $0.00 | #DIV/0! $0.00 $0.00 SUBTOTAL THIS SHEET $26,968.67 $7,624.55 $18,330.47 $0.00 $25,955.02 96% $1,013.65 $2,595.50 SUBTOTAL FIRE STATION 5 $8,605,983.14| $7,599,072.58 $452,990.79 $0.00 $8,052,063.37| ~ 94% $553,919.77 $679,449.80 GRAND TOTALS $17,256,643.07 | $12,498,423.55 $846,327.98 $671,429.73 | $14,016,181.26 81%! $3,240,461.81 | $1,269,386.60 _ Users may obtain validation of this-decument by requesting of-the license a comp! eed AIA Document D401-— Certification Gi Documents Authenticity AIA DOCUMENT G703 - CONTINUATION SHEET FOR G702- 1982 EDITION - AIA®- © 1982 THE AMERICAN INSTITUTE OF ARCHITECTS, 1735 NEW YORK AVENUE, N.W., WASHINGTON, 0.C, 20006-5232 G703-1992 O€lL SWORN STATEMENT. 3/7/2025 (deponent), being swom, states the State of Michigan) )ss County of Livingston 3 Date: Anthony Goins following: Brivar Construction Company is the (contractor) (subscentracter) for an improvement to the following described real property in | Macomb County, Michigan, described as follows: Warren Fire Station No. 1 - 23345 Van Dyke Avenue Warren, MI 48089 Warren Fire Station No. 5,.-.30619 Schoenherr Road Warren, Ml 48088 (insert legal description of property or name and address of Project) That the following is a statement of each subcontractor and supplier and laborer, for whom payment of wages or fringe benefits and withholdings is due but unpaid, with whom the (contractor) (subcstitraster) has (contracted) (SubeGninacted) for performance under the contract with the owner or lessee of the property and the amounts due to the persons as of the date of this statement are correctly and fully set forth opposite their names: Names of Subcontractor, | Subcontractor or Type of Total Amount Amount Balance supplier or supplier address | Improvement | Base Contract Change Contract already currently ta laborer and phone number furnished Amount Orders amount. paid owing. complete Biue Star, Inc. | 21950 Hoover, Demolition/ | $76,500.00 $3,500.00 $80,000.00 $76,615.00 $- $3,385.00 Warren, Ml 48089; | Removals 586-427-9933 _ . Site 30850 Stephenson | Sitework/ $1,235,782.00 | $484,669.49 $1,720,451.49 | $1,274,944.25 | $58,354.85 $387,152.39 Development | Hwy., Madison Utilities Inc Heights, Ml 48071, (248) 583-1200 x223 . Nagle Paving | 39525 W.13 Mile { Asphalt $33,080.00 -$11,450.00 $21,630.00 $- $- $21,630.00 Road, Suite 300 Paving Novi, Mf 48377; — (248). 553-0600 . ; Justice Fence | 1276 E. Columbia | Fencing $148,287.00 $4,706.90 $152,993.90 $118,141.20 $- $34,852.70 Ave. Baitle Creek, Ml 49014 Warren Fire Station No 1 & 5 Sworn Statement 1 2/28/2025 Leb Salisbury 6295 Joy Road, Landscaping | $127,770.00 $127,184.50 $254,954.50 | $88,345.80 $- | $166,608.70 Landscape Dexter, MI 48130; : Group 734-216-0950 Poured Brick | 8001 Park Place, Foundations | $208,285.00 $78,287.75 $287,572.75 $224,281.51 $- $63,291.24 Walls, Inc. Brighton, Mi 48116; (248) 437- 4494 . Charles Sinelli | 4100 Prairie Rose | Concrete $739,708.00 $9,700.00 $749,408.00 $441,408.60 $27,310.00 $280,689.40 and Sons Dr, Howell, MI Flatwork 48843; (248)684- 7014 a Hollowcore 60 Haltiner Ave., PreCast $99,400.00 $1,100.00 $100,500.00 $100,500.00 $- $- Inc. River Rouge, MI Concrete 48218 Leidal & Hart / 12100 Globe ' Masonry $3,186,385.00 | $19,727.16 $3,206,112.16 | $2,870,486.37 | $104,743.92 $230,881.87 Street, Livonia, Mi 48150: 734-522- . . 4.2400 B&A 1 50775 Richard W. | Structural $729,800.00 $729,800.00 $673,941.50 $- $55,858.50 Structural Ne Creston Steel Steel 948-2220 New Hudson | 37693 Schoolcraft | Carpentry - $446,500.00 $2,500.00 $449,000.00 $237,901.50 $- $211,098.50 Construction | Rd, Livonia, Ml Fire Station 48150; 734-743- No.1&5 :3186 MICA-TEC 21325 Hoover Rd., | Millwork $120,000.00 $120,000.00 $5,493.60 $-— $114,506.40 Warren, Mi 48089; . 586-758-4404 J JD Candler 6111 Sterling Drive | Roofing/ $623,516.00 $623,516.00 $532,827.00 $- $90,689.00 Commercial Nort Sten ng 40 Metal Paneis et Ss, . Roofing 734 765-01 00 . Stafford 1235 Waterville- Doors, $258,540.00 | $258,540.00 | $62,535.60 $71,570.70 $124,433.70 Building Monclova Rad, Frames, & Products Waterville, OH; Hardware (419)878-3070 . International | 8001 Ronda Drive, | Overhead $286,266.00 $2,274.00 $288,540.00 $169,186.95 $- $119,353.05 Doors, Inc. Canton, Ml 48187; | Doors 800-231-9521 .X%203 . Daniels Glass | 21250 W. Seven Glass & $295,000.00 $295,000.00 $255,310.20 $- $39,689.80 Mile Road, Detroit, | Glazing : MI 48219; 313- et 538-2746 Warren Fire Station No 1 & 5 Sworn Statement 2 2/28/2025 cel SC 30610 Ecorse Floor ‘$200,000.00 | $1,843.00 $201,843.00 | $106,398.39 | $29,120.33 $66,324.28 Floorcovering | Road, Romulus, Ml ; Covering 48174; 248-359- ‘ 3500 . Concrete 38257 Abruzzi Dr., | Epoxy $193,846.00 $193,846.00 $- $50,850.00 $142,996.00 Floor yee rot Flooring Treatment 4640 ‘Thompson ——-| PO Box 858 Painting $110,000.00 $5,200.00 $115,200.00 $37,125.00 $19,665.00 $58,410.00 Painting Pinckney, MI 48169; 734-260- 40993 ws _ “D10 USA |'7702 Cooley Lake | Fire $5,700.00 $5,700.00 $- $- $5,700.00 Rd, Waterford, MI | Extinguishers 48327; (810)523- 8721 Di0 USA 7702 Cooley Lake | Toilet Room | $70,680.00 $70,680.00 $- $- $70,680.00 Rd, Waterford, MI | Partitions & , fo (810)523- | Accessories GreatLakes {24101 W. Nine Appliances | $135,284.00 | $993.20 $136,277.20 | $99,973.80 $- $36,303.40 Hotel Supply Mile Road, ; Southfield, MI 48033; 313-962- 9176 X224 : — Elite Fire 46620 Ryan Court, | Fire $156,400.00 $156,400.00 | $68,152.50 $58,725.00 $29,522.50 Safety Novi, Ml 48377; Suppression 248-834-4467 Stewart Conveyance | $41,254.00 $41,254.00 | $- $- $41,254.00 Engineering & Systems ; Sales . . SE Kalchik 47510 Avante Dr., | Plumbing & | $936,116.00 $17,126.25 | $953,242.25 $569,947.54 $136,500.14 $246 794.57 | Mechanical Wixom, M] 48393; 1 Compressed | 248-449-5070 Air 7 Sherman 4500 Morrish HVAC $1,567,021.00 | $27,880.00 $1,594,901.00 | $1,008,013.50 | $56,950.79 $529,936.71 Heating & Road, Swariz Cooling Creek, MI 48473; : 810-630-6032 : EL Electric eee Gentle Road, Electrical $1,761,732.00 | $114,966.42 $1,876,698.42 | $1,167,438.94| $134,427.49 $574,831.99 uburn Mids, 48326; 248-475- 4662 . : . Johnson 31831 Sherman | Low Voltage | $403,999.00 $7,036.12 $41 71,035.12 $294,778.39 | $15,315.47 $100,941.26 Controls Inc. | Ave., Madison Heights, MI 48071 Warren Fire Station No 1 & 5 Sworn Statement 3 2/28/2025 Cel Flagpoles Etc. | 51056 Century Flag Poles $28,501.00 $28,501.00 5- $- $28,501.00 Court, Wixom, MI! 48393; 248-634- : .| 7183 BRIVAR 9325 Maltby Road, | Permits $10,000.00 $500.00 $10,500.00 $4,952.52 $- $5,547.48 | Construction mH Te, GA8)446 (Allowance) Company 3000 . MEGA 298 Veterans Staking $37,200.00 $37,200.00 $26,136.00 $1,674.00 $9,390.00 Drive, Fowlerville, MI 48836; 517- | 223-3512 = . DMP Sign 20732 Negaunee, | Signage $32,750.00 $32,750.00 | $- $- $32,750.00 Company Southfield, MI 1 48033; 248-996- 9284 The Sheer 7393 23 Mile Window $8,738.00 $8,738.00 $- $- | $8,738.00 Shop Road, Shelby Treatments Township, Mi 48316; 586-731- . . 4499 City of Warren Owner $103,832.00 -$66,908.31 $36,923.69 | $- $- $36,923.69 Allowance Allowance #1 | $22,575.00 -$22,575.00 $- $- $- $- Allowance #2 | $17,700.00 -$17,700.00 $- $- $- $- Brick Faced 40469 W. 11 Mile | Retaining $51,225.25 $51,225.25 $46,102.75 $- $5,122.50 Concrete ‘Rd, Novi, Mi Wall ‘Wails Inc. 48375 BRIVAR 9325 Maltby Road, | General $69,350.00 $16,410.18 $85,760.18 $65,160.18 $3,330.00 $17,270.00 Construction BONO (eae) 446- Conditions ! Company 8000 . BRIVAR 9325 Maltby Road, | BrivarLabor | $329,533.00 $329,533.00 $231,120.00 $17,100.00 $81,313.00 Construction et or (a8) 446. ‘Company 8000 BRIVAR 9325 Maltby Road, | Insurances $24,000.00 $24,000.00 $21,600.00 $- $2,400.00 Construction | Brighton, MI Company 48116; (248)446- . 80006 = BRIVAR 9325 Maltby Road, | Bonds $120,615.00 | $4,801.25 $125,416.25 $111,568.88 $- $13,847.37 Construction EBTI6, (48) 446- Company g000. Warren Fire Station No 1 & & Sworn Statement 4 2/28/2025 vel BRIVAR 9325 Maltby Road, | Weather $120,000.00 -$77,071.62 $42,928.38 $- $- $42,928.38 Construction | Brighton, MI Conditions Company 481 16; (248)446- 8000 .. BRIVAR 9325 Maltby Road, | Overhead $388,968.00 $388,968.00 $281,925.00 $11,700.00 "$95,343.00 Construction | Brighton, Mi Company 461 i 6; (248)446- BRIVAR 9325 Maltby Road, | Fee $655,812.30 $68,521.29 $724,333.59 $488,721.45 $33,892.29 ) $201,719.85 Construction Bate (248)446 Company | g000 Detroit 1091 Centre Rd, Stainless $61,000.00 $5,941.94 $66,941.94 $28,674.00 $2,673.87 $35,594.07 Technical ate a rasa Millwork i IS, : Company | 2ag-232-804 Overhead PO Box 176, i Overhead $72,274.00 $72,274.00 $34,225.19 $29,471.40 $8,577.41 Door of Bancroft, Ml Doors Whitmore 48414 Lake : National 47510 Avante Dr., | Compressed | $95,554.00 $95,554.00 “$56,482.97 $3,003.34 $36,067.69 Water Wixom, MI 48393; | Air Solutions 248-449-5070 : $16,446,478.55 -| $810,164.52 $17,256,643.07 $11,880,416.08 | $866,378.59 $4,509,848.40 (Some columns are not applicable to all persons listed) Warren Fire Station No 1 & 5 Sworn Statement 5 2/28/2025 The contractor has not procured material from, or subcontracted with, any person other than those set forth, and owes no money for the improvement other than the sums set forth. 1 make this statement as the contractor to represent to the owner or lessee of the property and his or her agents that the property is free from claims of construction liens, or the possibility of construction liens, except as specifically set forth in this statement for claims of construction liens by laborers that may be provided under section 109 of the construction lien act, 1980 P.A. 497, MCL 570.1109. WARNING TO OWNER: AN OWNER OR LESSEE OF THE PROPERTY MAY NOT RELY ON THIS SWORN STATEMENT TO AVOID THE CLAIM OF A SUBCONTRACTOR, SUPPLIER, OR LABORER WHO HAS PROVIDED A NOTICE OF FURNISHING OR A LABORER WHO MAY PROVIDE A NOTICE OF FURNISHING UNDER SECTION 109 OF THE CONSTRUCTION LIEN ACT 1980 P.A. 497, MCL 570.1109 TO THE DESIGNEE OR TO THE OWNER OR LESSEE IF THE DESIGNEE IS NOT NAMED OR HAS DIED. oN WARNING TO DEPONENT: A PERSON, WHO GIVES A FALSE SWORN STATEMENT WITH THE INTENT TO DEFRAUD IS SUBJECT TO CRIMINAL PENALTIES AS PROVIDED IN SECTION 110 OF THE CONSTRUCTION LIEN ACT, ACT 1980 PA 497, MCL 570.1110. Subscribed and swom to before me on 7. dayof_ March 2 20 25 Sf If MARY..BETH FUSON. ..-- Notary Public; Stateof Michigan Notary Public : County of Livingston Y My Commission Expires 1 1-20-2029", NCSU? Livingston _ County Michigan Acting inthe County of LL OS GEL My commission expires: 11/20/2028 Warren Fire Station No 1 & 5 Sworn Statement 6 2/28/2025 PARTNERS March 7, 2025 RE: Warren Fire Station #1 and #5 Payment Application No, 18 PARTNERS Project #21-146 Craig Treppa Dear Craig: Purchasing Agent City of Warren - Purchasing Attached please find a copy of certified Payment Application No. 18 from One City Square - Suite 425 Brivar Construction Company. Please issue payment in the amount of | Warren, Mi 48093-2396 $866,378.59 to Brivar Construction Company at your earliest convenience. Email: ctreppa@cityofwarren.org ; Please do not hesitate to contact me if you have any questions or concerms. Sincerely, PARTNERS in Architecture, PLC - Michael A. amp Principal File Ref: 21-146 Enclosure: Certified Payment Application No.18 Ce: Marybeth Fuson PARTNERS in Architecture, PLC 65 Market Street Mount Clemens, Michigan 48043 P, 586.469.3600 F. 586,469,3607 136 oF) Outlook Re: Brivar Construction Payment Application No 018 From Tom Bommarito tbommarito@cityofwarren.org Date Mon 3/10/2025 9:46 AM To = Shanah Turner sturner@cityofwarren.org approve:) From: Shanah Turner sturner@cityofwarren.org Sent: Monday, March 10, 2025 9:43 AM To; Wilburt McAdams wmcadams@warrenfiredept.org; Tom Bommarito tbommarito@cityofwarren.org; Tina Gapshes tgapshes@cityofwarren.org Ce: Craig Treppa ctreppa@cityofwarren.org Subject: Brivar Construction Payment Application No 018 Hello Please authorize payment for the attached Brivar Construction payment application no 018, in the amount of $846,378.59, Thanks, Shanah Turner, Assistant Buyer City of Warren, Purchasing Division One City Square, Ste 425, Warren, MI 48093 1p'1586-574-4638 f: 586-574-4614 sturner@clityofwarren.org www. cityofwarren.org ¢ ‘ ARREN From: Craig Treppa ctreppa@cityofwarren.org Sent: Monday, March 10, 2025 8:55 AM To: Shanah Turner sturner@cityofwarren.org Subject: Fw: Warren Fire Stations #1 & #5 - Payment Application Craig J. Treppa Purchasing Agent 137 oF Outlook Re: Brivar Construction Payment Application No 018 From Tina Gapshes tgapshes@cityofwarren.org Date Mon 3/10/2025 11:55 AM To — Shanah Turner sturner@cityofwarren.org; Wilburt McAdams wmcadams@warrenfiredept.org; Tom Bommarito tbommarito@cityofwarren.org Ce Craig Treppa ctreppa@cityofwarren.org No site work on this pay app so my approval is not needed. Thanks Tina G. Gapshes, PE City Engineer One City Square, Suite 300 Warren, MI 48093-2390 Phone: 586,759,9300 Fax: 586.759.9318 From: Shanah Turner sturner@cityofwarren.org Sent: Monday, March 10, 2025 9:43 AM To: Wilburt McAdams wmcadams@warrenfiredept.org; Tom Bommarito ; Tina Gapshes tgapshes@cityofwarren.org Cc: Cralg Treppa Subject: Brivar Construction Payment Application No 018 Hello Please authorize payment for the attached Brivar Construction payment application no 018, in the amount of $866,378.59, Thanks, Shanah Turner, Assistant Buyer City of Warren, Purchasing Division One City Square, Ste 425, Warren, MI 48093 101586-574-4638 f: 586-574-4614 sturner@cityofwarren.org www.cityofwarren.org 138 fo 8} Outlook Re: Brivar Construction Payment Application No 018 From Wilburt McAdams wmcadams@warrenfiredept.org Date Mon 3/10/2025 3:48 PM To — Shanah Turner sturner@cityofwarren.org; Tom Bommarito tbommarito@cityofwarren.org; Tina Gapshes Cc Craig Treppa ctreppa@cityofwarren.org Shanah Based upon the recommendation of P..A. and my cursory review | authorize payment number 18 in the amount of $866,378.59 to be paid to Brivar. Skip Fire Commissioner From: Shanah Turner sturner@cityofwarren.org Date: Monday, March 10, 2025 at 9:43 AM To: Wilburt McAdams wmcadams@warrenfiredept.org, Tom Bommarito tbommarito@cityofwarren.org, Tina Gapshes tgapshes@cityofwarren.org Ce: Craig Treppa ctreppa@cityofwarren.org Subject: Brivar Construction Payment Application No 018 Hello Please authorize payment for the attached Brivar Construction payment application no 018, in the amount of $866,378.59. Thanks, Shanah Turner, Assistant Buyer City of Warren, Purchasing Division One City Square, Ste 425, Warren, MI 48093 p:586-574-4638 ft 586-574-4614 sturner@cityofwarren.org www.cityofwarren.org From: Craig Treppa ctreppa@cityofwarren.org Sent: Monday, March 10, 2025 8:55 AM | | 139 O19 SEY J Eppink Partners, Inc Invoice 27 S. Squirrel - Ste 104 J EPPINK Auburn Hills, Ml 48326 US 248-922-0789 4 PARTNERS i URBAN DESIGN STUDIO » Www. jeppink.com = a ea Jepp D Ds a BILL TO - City of Warren Attn: Purchasing Division One City Square Suite 425 Warren, MI 48093 INVOICE # DATE TOTAL DUE DUE DATE TERMS ENCLOSED ( 1662 ) (01/05/2025 “> $5,136.74 01/20/2025 Net 15 PROJECT Warren DDA - Town Genter foot? 289/60 | ACTIVITY DESCRIPTION AMOUNT PO NUMBER: 2528916 Bill as per the delivery of Land Planning & Design Services Project: Warren Town Center Period: October - December 2024 Authorization: per Tom Bommarito Warren DDA / Agreement dated 8/1/24 Town Center Land Planning & Community Engagement - Lead Community Engagement & Planning of Town Center Area - Coordinate project objectives and timeline with DDA & Gibbs Planning - Coordinate workshop objectives, presentations, and sessions - Coordinate activities and outcomes with DDA & Gibbs Planning - Coordinate and participate in Steering Committee meeting 10/3/24 - Coordinate workshop and planning meetings 12/9/24, 12/11/24, 12/13/24 - Design, create, and print workshop activity boards and related documents Landscape 65% Completion of Agreement 3,250.00 ($5,000 x 65% = $3,250) 5 Reproduction Reproduction of display boards and workshop supplies 1,886.74 - 9 display board and workshop activity tools | BALANCE DUE ($5,1 36.74) YVY-F919-4. 80/00 | 140 Bay Sap aey oy chee fet \t OFFICIAL INVOICE Bill To: ier Ship To: Product Total $20,095.68) CITY OF WARREN DLP. ORRIN FERGUSON Discount Sala sume ws £0819 SCHOENHERR FD Shipping (Frlght- RL) __sis1.48 WARREN MI 48089 WARREN Mil 48088 Grand Tota! C $19,606.97 » Tax ID # Tax Id Needed Outstanding Balance $19,606.97 hE FAG Thao Invoice #: (13217477 PO #: ( 2530131. Invoice Date{ 3/20/2025 ) | Rem Desenption | sku] ory [pace {tot | Hosp NG Burner 2:0» }O0CB Set (Choose yausbreakdown): | __. IROSI41900 | es eee ea = i 1.2 Ohio Bar (Red Shait/Black Sleeve) | ___RA2B89-H1B05-IL_ | 2 | _ sx6.a8 $673.78 | ..._Fingua Ohlo Power Bar45%p (Red Shall / Black Seow) | _pAasnsnteosi. L a oul | segs eeeg eg NOMUSERhO Slam Balla 20s) i peveleco | - Lt Pe | SBS _ ee ee OLE BC rao | AN. OTS |... $ehess | cs ee UE ave BAOMA. | ee eee | 105.62 | ___ Regue V2 BumperStackr — (RAGS | ath $09.74 | $89.74, | Make checks payable to: Rogue Fitness All POs are processed with Net 30 terms starting the date the order ships. POs over $25,000 will require a 50 % deposit to Initiate the order. Tha remaining 50 % of the balance will have Net 30 terms ° " starting the data the order ships. "If any invoiced amount Is not received by the due date, then without limiting Rogue's rights or remedies, (a) out standing amounts will accrue late interest at the rale 3/20/2025 10:36:41 AM of 1.5% of the out standingbalance per month, or tha maximum permitted by law, whichever Is lower, and/ or (b) futura orders may be conditioned with a reduction or elimination of terms. Upon confirmation of this Order, and/or acceptance of the ordered product(s), you hereby agree to thesa credit terms. +R. iW AY S pil hing Minu tes Page 1 of 3 2-5-2505 Atie died ¥ 141 OFFICIAL INVOICE Rogue 9 Bas Holder 2.0 RA0524 1 $162.09 $162.00 Rogue V2 Bumper Starker with Wheels RFO488 2 $112.97 $225,594 Rogue Vertical Plats Tree 2.0 RFOG44 1 $207.81 $207.81 2 Tier Universal Storage System 2.0 - MA/KB RFO771-MB8-KB t $481.46 $481.36 3 Tier Universal Storage Systam 2.0 - DB/OR/DB RF0772-DB-DB-0B 2 $864.19 $1,768.26 RFOS59-2095C-MG-TXT- | Rogue Ac Bench 3.1 (MG Rogue Red « Txt) - FP Pad FP 2 $585.00 $1,190,00 Monster Landmine 2.0 (il Bleck} RAT671-IL. 1 $141.38 $141.38 Rogue Monster Velocidor - TXT Black Frame - Smooth Handles | RA2842-BLACK-TXT-SM 4 $283.20 $283.20 Rogue Y-2 Yoke 2.0 RFO632 1 $638.54 $838.54 __ RF 0648-F8S6-C2476-75- Rogue RM~4S08T Monster Rack 2.0 (Satin Black) 772-717-518 1 $1,629.25 $1,629.25 LT-1 50 Cal Monster Trolley & Lever Arm Kit 2.0 (Standard) - MG RFOS42-BLACK-MG- Black STANDARD 1 $1,154.28 $1,164.28 28LB Rogue Kattlaball IP0670-28 2 $52.25 $104.50 35LB Rogue Ketifebed IPO670-35 2 $61.75 $123.50 SLB Rogue Keitiebel IPOB7D-53 2 $81.60 $163.20 7OLB Roque Keltlebel 1P0670-70 2 $105.60 $215.20 Rogus 5 - 50ibs Dumbbell Sat iPOOSo 1 $785.00 $795.00 Rogue 55 - 100%s Dumbbea Set IPOO61 4 $2,680.00 $2,880.00 i . 2 RAAY’ Games Box - Box Joint . RA0368 ee ee A a Make checks payable to: Rogue Fitness All POs are processed with Net 30 terms starting the date the order ships. POs over $25,000 will require a 50 % deposit to Initiate the order. The remaining 50 % of the balance will hava Net 20 terms starting tha date the ordar ships. ‘If any invoiced amount Is not received by the due date, then without limiting Rogue's rights or remedies, (a) out standing amounts will accrue late interest at the rate 9/20/2025 10:36:41 AM of 1.56 of the out standingbalance per month, or the maximum permitted by law, whichever Is lower, and/ or (b) futura orders may be conditioned with a reduction or elimination of terms. Page 2 of 3 Upon confirmatian of this Order, and/or acceptance of the ordered product(s), you hereby agree to these credit terms. 142 | Assault Alr Bike - Black LoFooo2 1 $499,00 { Rogue Echo Rower v1.1 1P0917 1 $895.00 $395.00 Rogue Battle Rope KAPS! 1 $139,99 $139.99 TANK™ Md Push Sled TQ0003 i $1,599.00 $1,599.00 TANK™ ¥-Strap TQ0005 1 $49.00 $49.00 TANK™ Tow Rape TQ0006 1 _ $149.00 $118.00 TANK™ Strap/riarness Tow Kit for M4/X TQ0010 . 1 $109.00 $109.00 i RF Pullup Package HBOOOS 1 $67.45 967.45 RF Pullop Package 2 HBOOOS 1 $83.85 $63.65 Strongman Bag - 100 Lbs _ RA1160-100 1 $84.79 $84.79 Rogua Sandbag 2.0 - Large RA22104. 1 $213,765 $213.75 Rogue Sandbag 2.0 - Medium FIA2210-M 1 $186.25 $185.25 Rogue Sandbag 2.0-Smal RA2210-S 1 $158.75 S15675 Make checks payable to: Rogue Fitness Alt POs are processed with Net 30 terms starting the date the order ships. POs over $25,000 will require a 50 % deposit to initiate the order. The ramaining 50 % of the balance will have Net 30 terms starting the date the order ships. ‘If any Invoiced amount is not received by the due date, then without limiting Rogue's rights or remedias, (a) out standing amounts will accrue fate Interest at the rate of 1.5% of the out standingbalance per month, or the maximum permitted by law, whichever is lower, and/ or (b) future orders may be conditioned with a reduction or elimination of terms. Upon confirmation of this Order, and/or acceptance of the ordered product(s), you hereby agree to these credit terms. OFFICIAL INVOICE 3/20/2025 10:36:41 AM Page 3 of 3 143 ol9T2! TITAN any © & Invoice BRANDS , 2 Date: 6/2 Sales: J 44800 3839 Forest Hill-Irene Rd Ship Date: 3/6/2025 Sulte 100 Sales Channel; Call In Orders Memphls TN 38125 Shipping Method: Oversized SDA Peper Me ; a had: POU: $00000126' — PO No.(2530141_) Bill To Ship To FIRE CHIEF ORRIN FERGUSON Warren Flre Department Warren Fire Department FIRE CHIEF ORRIN FERGUSON One City Square 30619 SCHOENHERR RD Suite 425 WARREN Mi 48088 Warren MI 48093 586) 756-2800 ext, 3200 Ptonpy be & " on a ‘ oa ye nae “i haf ENTLTNR 400883 - 2 Titan Functional Trainer $2,244.97 $4,489.94 FNTLTNR LEGPRSICtT 401486 1 LEG PRESS HACK SQUAT MACHINE In CRATE $1,494.87 $1,494.87 - LEGPRSKIT LEGPRSKIT 401486 4 LEG PRESS HACK SQUAT MACHINE In CRATE $1,494.87 $1,494.87 + LEGPRSIOT Oversized 4 $0.00 $0.00 handling fee i Subtotal $7,479.68 Tar (96) $0.00 TL INV4344800 Total (47479 68 %y = F494 «T1400 TRie W-1389 494-9494 144 of?» COMCAST Comcast Cable 9700 Enterprise CT Warren Mi 49082 Customer Name Warren Fire Station 5 Date 3/24/2025 Address 30619 Schoenherr Rd Order No. City Warren State Mi ZIP Rep Tim Frederick Phone FOB MI Qty Description a Unit Price TOTAL 4 Project Design $719.49 $719.49 1 Permitting $3,379.18 $3,379.18 4 Fiber Construction Labor $8,570.87 $8,570.87 1 Materials $2,759.03 $2,759.03 1 Survey $240.46 $240.46 Subtotal $15,669.03 Payment Details Shipping & Handling O Cash Sales Tax © Check O Credit Card TOTAL $15,669.03 Name CC# Office Use Only Expires Payment required before work can begin. 145 |
11787 | https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_25?srsltid=AfmBOor9ZtL84H7WTcerU8b3rHNLSvj2EdvQNNa4_oSf1q24eHb_Nfln | Art of Problem Solving
2016 AMC 12B Problems/Problem 25 - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki 2016 AMC 12B Problems/Problem 25
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
2016 AMC 12B Problems/Problem 25
Contents
1 Problem
2 Solution 1
3 Solution 2
4 Solution 3
5 Solution 4 (Using a formula)
6 Video Solution by CanadaMath (Problem 21-25)
7 See Also
Problem
The sequence is defined recursively by , , and for . What is the smallest positive integer such that the product is an integer?
Solution 1
Let . Then and for all . The characteristic polynomial of this linear recurrence is , which has roots and .
Therefore, for constants to be determined . Using the fact that we can solve a pair of linear equations for :
.
Thus , , and .
Now, , so we are looking for the least value of so that
.
Note that we can multiply all by three for convenience, as the are always integers, and it does not affect divisibility by .
Now, for all even the sum (adjusted by a factor of three) is . The smallest for which this is a multiple of is by Fermat's Little Theorem, as it is seen with further testing that is a primitive root .
Now, assume is odd. Then the sum (again adjusted by a factor of three) is . The smallest for which this is a multiple of is , by the same reasons. Thus, the minimal value of is .
Solution 2
Since the product is an integer, it must be a power of , so the sum of the base- logarithms must be an integer. Multiply all of these logarithms by (to make them integers), so the sum must be a multiple of .
The logarithms are . Using the recursion (modulo to save calculation time), we get the sequence Listing the numbers out is expedited if you notice .
The cycle repeats every terms. Notice that since , the first terms sum up to a multiple of . Since , we only need at most the first terms to sum up to a multiple of , and this is the lowest answer choice.
Note 1: To rigorously prove this is the smallest value, you will have to keep a running sum of the terms and check that it is never a multiple of before the th term.
Note 2: In response to note 1, it can be proven that , where . Since is a multiple of , it suffices to find the minimal such that . In this case, happens to be minimal such , so the answer would be .
The relation can be proven by rearranging the relation to for all integers , then adding those equations together. The LHS telescopes into , and the RHS becomes . Therefore, if you don't find a cleaner solution involving the relation , you can always solve the problem just by considering the value of rather than keeping a running sum.
Solution 3
Like in Solution 2, calculate the first few terms of the sequence, but also keep a running sum of the logarithms (not modulo here): Notice that for odd and for even . Since is relatively prime to , we can ignore even and calculate odd using (modulo ): is first a multiple of at . ~emerald_block
Solution 4 (Using a formula)
Consider the product (will finish tommorow)
Video Solution by CanadaMath (Problem 21-25)
Fast Forward to 26:01 for problem 25
~THEMATHCANADIAN
See Also
2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 24Followed by
Last Problem
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25
All AMC 12 Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
Retrieved from "
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
11788 | https://stackoverflow.com/questions/6637589/using-exponentiation-0-5-less-efficient-than-math-sqrt | python - Using exponentiation 0.5 less efficient than math.sqrt? - Stack Overflow
Join Stack Overflow
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
Sign up with GitHub
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Overflow
1. About
2. Products
3. For Teams
Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers
Advertising Reach devs & technologists worldwide about your product, service or employer brand
Knowledge Solutions Data licensing offering for businesses to build and improve AI tools and models
Labs The future of collective knowledge sharing
About the companyVisit the blog
Loading…
current community
Stack Overflow helpchat
Meta Stack Overflow
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Let's set up your homepage Select a few topics you're interested in:
python javascript c#reactjs java android html flutter c++node.js typescript css r php angular next.js spring-boot machine-learning sql excel ios azure docker
Or search from our full list:
javascript
python
java
c#
php
android
html
jquery
c++
css
ios
sql
mysql
r
reactjs
node.js
arrays
c
asp.net
json
python-3.x
.net
ruby-on-rails
sql-server
swift
django
angular
objective-c
excel
pandas
angularjs
regex
typescript
ruby
linux
ajax
iphone
vba
xml
laravel
spring
asp.net-mvc
database
wordpress
string
flutter
postgresql
mongodb
wpf
windows
xcode
amazon-web-services
bash
git
oracle-database
spring-boot
dataframe
azure
firebase
list
multithreading
docker
vb.net
react-native
eclipse
algorithm
powershell
macos
visual-studio
numpy
image
forms
scala
function
vue.js
performance
twitter-bootstrap
selenium
winforms
kotlin
loops
express
dart
hibernate
sqlite
matlab
python-2.7
shell
rest
apache
entity-framework
android-studio
csv
maven
linq
qt
dictionary
unit-testing
asp.net-core
facebook
apache-spark
tensorflow
file
swing
class
unity-game-engine
sorting
date
authentication
go
symfony
t-sql
opencv
matplotlib
.htaccess
google-chrome
for-loop
datetime
codeigniter
perl
http
validation
sockets
google-maps
object
uitableview
xaml
oop
visual-studio-code
if-statement
cordova
ubuntu
web-services
email
android-layout
github
spring-mvc
elasticsearch
kubernetes
selenium-webdriver
ms-access
ggplot2
user-interface
parsing
pointers
c++11
google-sheets
security
machine-learning
google-apps-script
ruby-on-rails-3
templates
flask
nginx
variables
exception
sql-server-2008
gradle
debugging
tkinter
delphi
listview
jpa
asynchronous
web-scraping
haskell
pdf
jsp
ssl
amazon-s3
google-cloud-platform
jenkins
testing
xamarin
wcf
batch-file
generics
npm
ionic-framework
network-programming
unix
recursion
google-app-engine
mongoose
visual-studio-2010
.net-core
android-fragments
assembly
animation
math
svg
session
intellij-idea
hadoop
rust
next.js
curl
join
winapi
django-models
laravel-5
url
heroku
http-redirect
tomcat
google-cloud-firestore
inheritance
webpack
image-processing
gcc
keras
swiftui
asp.net-mvc-4
logging
dom
matrix
pyspark
actionscript-3
button
post
optimization
firebase-realtime-database
web
jquery-ui
cocoa
xpath
iis
d3.js
javafx
firefox
xslt
internet-explorer
caching
select
asp.net-mvc-3
opengl
events
asp.net-web-api
plot
dplyr
encryption
magento
stored-procedures
search
amazon-ec2
ruby-on-rails-4
memory
canvas
audio
multidimensional-array
random
jsf
vector
redux
cookies
input
facebook-graph-api
flash
indexing
xamarin.forms
arraylist
ipad
cocoa-touch
data-structures
video
azure-devops
model-view-controller
apache-kafka
serialization
jdbc
woocommerce
razor
routes
awk
servlets
mod-rewrite
excel-formula
beautifulsoup
filter
docker-compose
iframe
aws-lambda
design-patterns
text
visual-c++
django-rest-framework
cakephp
mobile
android-intent
struct
react-hooks
methods
groovy
mvvm
ssh
lambda
checkbox
time
ecmascript-6
grails
google-chrome-extension
installation
cmake
sharepoint
shiny
spring-security
jakarta-ee
plsql
android-recyclerview
core-data
types
sed
meteor
android-activity
activerecord
bootstrap-4
websocket
graph
replace
scikit-learn
group-by
vim
file-upload
junit
boost
memory-management
sass
import
async-await
deep-learning
error-handling
eloquent
dynamic
soap
dependency-injection
silverlight
layout
apache-spark-sql
charts
deployment
browser
gridview
svn
while-loop
google-bigquery
vuejs2
dll
highcharts
ffmpeg
view
foreach
makefile
plugins
redis
c#-4.0
reporting-services
jupyter-notebook
unicode
merge
reflection
https
server
google-maps-api-3
twitter
oauth-2.0
extjs
terminal
axios
pip
split
cmd
pytorch
encoding
django-views
collections
database-design
hash
netbeans
automation
data-binding
ember.js
build
tcp
pdo
sqlalchemy
apache-flex
mysqli
entity-framework-core
concurrency
command-line
spring-data-jpa
printing
react-redux
java-8
lua
html-table
ansible
jestjs
neo4j
service
parameters
enums
material-ui
flexbox
module
promise
visual-studio-2012
outlook
firebase-authentication
web-applications
webview
uwp
jquery-mobile
utf-8
datatable
python-requests
parallel-processing
colors
drop-down-menu
scipy
scroll
tfs
hive
count
syntax
ms-word
twitter-bootstrap-3
ssis
fonts
rxjs
constructor
google-analytics
file-io
three.js
paypal
powerbi
graphql
cassandra
discord
graphics
compiler-errors
gwt
socket.io
react-router
solr
backbone.js
memory-leaks
url-rewriting
datatables
nlp
oauth
terraform
datagridview
drupal
oracle11g
zend-framework
knockout.js
triggers
neural-network
interface
django-forms
angular-material
casting
jmeter
google-api
linked-list
path
timer
django-templates
arduino
proxy
orm
directory
windows-phone-7
parse-platform
visual-studio-2015
cron
conditional-statements
push-notification
functional-programming
primefaces
pagination
model
jar
xamarin.android
hyperlink
uiview
visual-studio-2013
vbscript
google-cloud-functions
gitlab
azure-active-directory
jwt
download
swift3
sql-server-2005
configuration
process
rspec
pygame
properties
combobox
callback
windows-phone-8
linux-kernel
safari
scrapy
permissions
emacs
scripting
raspberry-pi
clojure
x86
scope
io
expo
azure-functions
compilation
responsive-design
mongodb-query
nhibernate
angularjs-directive
request
bluetooth
reference
binding
dns
architecture
3d
playframework
pyqt
version-control
discord.js
doctrine-orm
package
f#
rubygems
get
sql-server-2012
autocomplete
tree
openssl
datepicker
kendo-ui
jackson
yii
controller
grep
nested
xamarin.ios
static
null
statistics
transactions
active-directory
datagrid
dockerfile
uiviewcontroller
webforms
discord.py
phpmyadmin
sas
computer-vision
notifications
duplicates
mocking
youtube
pycharm
nullpointerexception
yaml
menu
blazor
sum
plotly
bitmap
asp.net-mvc-5
visual-studio-2008
yii2
electron
floating-point
css-selectors
stl
jsf-2
android-listview
time-series
cryptography
ant
hashmap
character-encoding
stream
msbuild
asp.net-core-mvc
sdk
google-drive-api
jboss
selenium-chromedriver
joomla
devise
cors
navigation
anaconda
cuda
background
frontend
binary
multiprocessing
pyqt5
camera
iterator
linq-to-sql
mariadb
onclick
android-jetpack-compose
ios7
microsoft-graph-api
rabbitmq
android-asynctask
tabs
laravel-4
environment-variables
amazon-dynamodb
insert
uicollectionview
linker
xsd
coldfusion
console
continuous-integration
upload
textview
ftp
opengl-es
macros
operating-system
mockito
localization
formatting
xml-parsing
vuejs3
json.net
type-conversion
data.table
kivy
timestamp
integer
calendar
segmentation-fault
android-ndk
prolog
drag-and-drop
char
crash
jasmine
dependencies
automated-tests
geometry
azure-pipelines
android-gradle-plugin
itext
fortran
sprite-kit
header
mfc
firebase-cloud-messaging
attributes
nosql
format
nuxt.js
odoo
db2
jquery-plugins
event-handling
jenkins-pipeline
nestjs
leaflet
julia
annotations
flutter-layout
keyboard
postman
textbox
arm
visual-studio-2017
gulp
stripe-payments
libgdx
synchronization
timezone
uikit
azure-web-app-service
dom-events
xampp
wso2
crystal-reports
namespaces
swagger
android-emulator
aggregation-framework
uiscrollview
jvm
google-sheets-formula
sequelize.js
com
chart.js
snowflake-cloud-data-platform
subprocess
geolocation
webdriver
html5-canvas
centos
garbage-collection
dialog
sql-update
widget
numbers
concatenation
qml
tuples
set
java-stream
smtp
mapreduce
ionic2
windows-10
rotation
android-edittext
modal-dialog
spring-data
nuget
doctrine
radio-button
http-headers
grid
sonarqube
lucene
xmlhttprequest
listbox
switch-statement
initialization
internationalization
components
apache-camel
boolean
google-play
serial-port
gdb
ios5
ldap
youtube-api
return
eclipse-plugin
pivot
latex
frameworks
tags
containers
github-actions
c++17
subquery
dataset
asp-classic
foreign-keys
label
embedded
uinavigationcontroller
copy
delegates
struts2
google-cloud-storage
migration
protractor
base64
queue
find
uibutton
sql-server-2008-r2
arguments
composer-php
append
jaxb
zip
stack
tailwind-css
cucumber
autolayout
ide
entity-framework-6
iteration
popup
r-markdown
windows-7
airflow
vb6
g++
ssl-certificate
hover
clang
jqgrid
range
gmail
Next You’ll be prompted to create an account to view your personalized homepage.
Home
Questions
AI Assist Labs
Tags
Challenges
Chat
Articles
Users
Jobs
Companies
Collectives
Communities for your favorite technologies. Explore all Collectives
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Collectives™ on Stack Overflow
Find centralized, trusted content and collaborate around the technologies you use most.
Learn more about Collectives
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Using exponentiation 0.5 less efficient than math.sqrt?
Ask Question
Asked 14 years, 2 months ago
Modified13 years, 9 months ago
Viewed 3k times
This question shows research effort; it is useful and clear
11
Save this question.
Show activity on this post.
A quote from "Python Programming: An Introduction to Computer Science"
We could have taken the square root using exponentiation . Using math.sqrt is somewhat more efficient.
"Somewhat", but to what extent, and how?
python
exponentiation
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Improve this question
Follow
Follow this question to receive notifications
edited Jul 9, 2011 at 22:00
RobertRobert
asked Jul 9, 2011 at 21:30
RobertRobert
2,282 2 2 gold badges 22 22 silver badges 36 36 bronze badges
1
3 You can always measure it yourself with timeit. For the record, math.sqrt is only roughly 5% faster for me.user395760 –user395760 2011-07-09 21:36:42 +00:00 Commented Jul 9, 2011 at 21:36
Add a comment|
5 Answers 5
Sorted by: Reset to default
This answer is useful
16
Save this answer.
Show activity on this post.
Theoretically, hammar's answer and duffymo's answer are good guesses. But in practice, on my machine, it's not more efficient:
```python
import timeit
timeit.timeit(stmt='[n 0.5 for n in range(100)]', setup='import math', number=10000)
0.15518403053283691
timeit.timeit(stmt='[math.sqrt(n) for n in range(100)]', setup='import math', number=10000)
0.17707490921020508
```
Part of the problem is the . operation. If you import sqrt directly into the namespace, you get a slight improvement.
```python
timeit.timeit(stmt='[sqrt(n) for n in range(100)]', setup='from math import sqrt', number=10000)
0.15312695503234863
```
Key word there: slight.
Further testing indicates that as the number gets larger, the benefit you get from using sqrt increases. But still not by a lot!
```python
timeit.timeit(stmt='[n 0.5 for n in range(1000000)]', setup='import math', number=1)
0.18888211250305176
timeit.timeit(stmt='[math.sqrt(n) for n in range(1000000)]', setup='import math', number=1)
0.18425297737121582
timeit.timeit(stmt='[sqrt(n) for n in range(1000000)]', setup='from math import sqrt', number=1)
0.1571958065032959
```
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
edited May 23, 2017 at 11:46
CommunityBot
1 1 1 silver badge
answered Jul 9, 2011 at 21:35
senderlesenderle
152k 36 36 gold badges 217 217 silver badges 243 243 bronze badges
1 Comment
Add a comment
zneak
zneakOver a year ago
I came to the same conclusions.
2011-07-09T21:40:41.343Z+00:00
0
Reply
Copy link
This answer is useful
11
Save this answer.
Show activity on this post.
No need to guess the implementation, we can read the code!
math.sqrt is a thin wrapper about sqrt from the standard C library: see mathmodule.c, line 956
The operator has multiple implementations depending on the types of the arguments, but in the case of a floating-point exponent, it eventually dispatches to pow from the standard C library (see floatobject.c line 783).
Modern CPUs often have special square root instructions which general exponentiation routines don't use (compare and contrast the implementations of pow and sqrt in glibc for x86-64, for example). But once all the interpreter overhead is added (byte codes, type checking, method dispatch etc), the difference in raw speed doesn't matter all that much, and can be dominated by issues like whether you call sqrt directly or look it up via the math module (as shown by the timings in other answers).
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
edited Jul 9, 2011 at 22:46
answered Jul 9, 2011 at 22:28
Gareth ReesGareth Rees
66.1k 10 10 gold badges 139 139 silver badges 166 166 bronze badges
Comments
Add a comment
This answer is useful
1
Save this answer.
Show activity on this post.
has to support any exponent while math.sqrt knows it's always 0.5. math.sqrt can therefore use a more specialized (and therefore probably more efficient) algorithm.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
answered Jul 9, 2011 at 21:33
hammarhammar
140k 18 18 gold badges 310 310 silver badges 388 388 bronze badges
3 Comments
Add a comment
zneak
zneakOver a year ago
An optimal implementation of could simply branch to math.sqrt if the exponent is smaller than 1. That would probably have a barely measurable impact.
2011-07-09T21:36:20.233Z+00:00
0
Reply
Copy link
Dietrich Epp
Dietrich EppOver a year ago
@zneak: Most implementations do.
2011-07-09T21:38:07.01Z+00:00
0
Reply
Copy link
hammar
hammarOver a year ago
@zneak: Even so, it has to make that test, so it's always going to be (however slightly) slower.
2011-07-09T21:45:35.413Z+00:00
0
Reply
Copy link
Add a comment
This answer is useful
1
Save this answer.
Show activity on this post.
My guess is that math.sqrt uses Newton's method, which converges quadratically, and exponentiation uses something else that is slower.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
answered Jul 9, 2011 at 21:35
duffymoduffymo
309k 46 46 gold badges 375 375 silver badges 570 570 bronze badges
2 Comments
Add a comment
user395760
user395760Over a year ago
As also noted by zneak in a comment: There's no reason expotentiation shouldn't use the same algorithm, or simply reuse the existing implementation, for expotentiation by 0.5.
2011-07-09T21:41:22.917Z+00:00
0
Reply
Copy link
zneak
zneakOver a year ago
math.sqrt is probably an alias for the C math function sqrt, which is implemented using the best algorithm for your platform. If your CPU supports SSE instructions, you get an sqrt instruction family, of which all members are as fast as can be.
2011-07-09T21:43:53.547Z+00:00
1
Reply
Copy link
This answer is useful
1
Save this answer.
Show activity on this post.
Here's a slightly different approach. We want an int just bigger than the square root. Two ways (which disagree for square numbers but that's OK):
```python
timeit.timeit(stmt='[int(n0.5)+1 for n in range(1000000)]', setup='', number=1)
0.481772899628
timeit.timeit(stmt='[ceil(sqrt(n)) for n in range(1000000)]', setup='from math import sqrt, ceil', number=1)
0.293844938278
timeit.timeit(stmt='[int(ceil(sqrt(n))) for n in range(1000000)]', setup='from math import sqrt, ceil', number=1)
0.511347055435
```
So the math functions are faster...until you convert the float to int. (I need to do a lot of comparisons with the value, and while I haven't tested it, comparing integers should be cheaper than comparing floats.)
But hey, it's Python. You're on top of too many abstractions to try to optimize performance with this level of granularity.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
answered Dec 18, 2011 at 18:13
mgoldmgold
6,396 5 5 gold badges 42 42 silver badges 41 41 bronze badges
Comments
Add a comment
Your Answer
Thanks for contributing an answer to Stack Overflow!
Please be sure to answer the question. Provide details and share your research!
But avoid …
Asking for help, clarification, or responding to other answers.
Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Email and Password
Submit
Post as a guest
Name
Email
Required, but never shown
Post Your Answer Discard
By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
python
exponentiation
See similar questions with these tags.
The Overflow Blog
The history and future of software development (part 1)
Getting Backstage in front of a shifting dev experience
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
New and improved coding challenges
New comment UI experiment graduation
Policy: Generative AI (e.g., ChatGPT) is banned
Community activity
Last 1 hr
Users online activity 20838 users online
1 question
0 answers
103 comments
391 upvotes
Popular tags
recursionc#phppythonc++sql
Popular unanswered question
Can I store JSON object in Azure key vault? just like I would in DB, I'd want to store it using a POST route and also GET it
azureazure-keyvaultlicensingvaultlicense-key
Ishwar R
1
2 hours ago
Linked
3Generating large prime numbers in python
2Is there any advantage using math.sqrt(num) over num0.5 in Python
Related
256Which is faster in Python: x.5 or math.sqrt(x)?
7How is exponentiation by squaring faster?
2Is there any advantage using math.sqrt(num) over num0.5 in Python
3How to raise an integer to fractional power efficiently?
8Why is numpy.power slower for integer exponents?
7Why is 2100 so much faster than math.pow(2,100)?
1Python: looking for a faster, less accurate sqrt() function
52Why is math.sqrt massively slower than exponentiation?
0Raising 2 to large exponents
1Python - "Fast Exponention Algorithm" performance outperformed by "worse" algorithm
Hot Network Questions
Are there any legal strategies to modify the nature and rules of the UN Security Council in the face of one, or more SC vetos?
Alternatives to Test-Driven Grading in an LLM world
Weird utility function
Who is the target audience of Netanyahu's speech at the United Nations?
Making sense of perturbation theory in many-body physics
On being a Maître de conférence (France): Importance of Postdoc
I have a lot of PTO to take, which will make the deadline impossible
Repetition is the mother of learning
Checking model assumptions at cluster level vs global level?
Is there a feasible way to depict a gender transformation as subtle and painless?
Another way to draw RegionDifference of a cylinder and Cuboid
Proof of every Highly Abundant Number greater than 3 is Even
Why is a DC bias voltage (V_BB) needed in a BJT amplifier, and how does the coupling capacitor make this possible?
In Dwarf Fortress, why can't I farm any crops?
How do you create a no-attack area?
A time-travel short fiction where a graphologist falls in love with a girl for having read letters she has not yet written… to another man
ConTeXt: Unnecessary space in \setupheadertext
Interpret G-code
Riffle a list of binary functions into list of arguments to produce a result
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Matthew 24:5 Many will come in my name!
My dissertation is wrong, but I already defended. How to remedy?
Languages in the former Yugoslavia
Gluteus medius inactivity while riding
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
lang-py
Why are you flagging this comment?
Probable spam.
This comment promotes a product, service or website while failing to disclose the author's affiliation.
Unfriendly or contains harassment/bigotry/abuse.
This comment is unkind, insulting or attacks another person or group. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Stack Overflow
Questions
Help
Chat
Products
Teams
Advertising
Talent
Company
About
Press
Work Here
Legal
Privacy Policy
Terms of Service
Contact Us
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.29.34589 |
11789 | https://puzzling.stackexchange.com/questions/11814/snow-white-and-the-secret-message-secret-sharing-problem | mathematics - Snow White and the Secret Message (Secret Sharing Problem) - Puzzling Stack Exchange
Join Puzzling
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Puzzling helpchat
Puzzling Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Snow White and the Secret Message (Secret Sharing Problem)
Ask Question
Asked 10 years, 5 months ago
Modified10 years, 4 months ago
Viewed 856 times
This question shows research effort; it is useful and clear
7
Save this question.
Show activity on this post.
Snow White needs to send a message to Prince Charming. She considers telling this message to each of the Seven Dwarves, then sending them to deliver it to the Prince. However, the road to the castle is treacherous, and there is risk that agents of the evil Queen might intercept some of the dwarves. So, Snow White wants to say something to the dwarves so that...
As long as any 4 dwarves reach Prince Charming, they can tell him the message, but
If at most 3 are captured, the Queen cannot learn anything about the message, no matter how hard she interrogates the poor dwarves.
What should Snow White tell the dwarves?
Remarks: Think of the message as a number. Snow White has access to a fair coin. You don't need any fancy cryptography protocols to solve this: there are several elegant solutions using very basic math.
mathematics
cryptography
information-theory
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Improve this question
Follow
Follow this question to receive notifications
edited Jun 3, 2015 at 2:13
Mike EarnestMike Earnest
asked Apr 10, 2015 at 7:18
Mike EarnestMike Earnest
33.9k 8 8 gold badges 97 97 silver badges 250 250 bronze badges
3
1 Does 'not learning anything about the message' include meta-information such as the fact that it exists, its probable length, its sender and intended recipient, etc?A E –A E 2015-04-10 07:51:15 +00:00 Commented Apr 10, 2015 at 7:51
1 @AE Let's say that the queen already knows all this meta-information, so it doesn't matter whether she can learn this form the dwarves.Mike Earnest –Mike Earnest 2015-04-10 07:59:16 +00:00 Commented Apr 10, 2015 at 7:59
1 Wouldn't this be equivalent to a Hamming code of length 7?qzx –qzx 2015-04-10 08:19:04 +00:00 Commented Apr 10, 2015 at 8:19
Add a comment|
2 Answers 2
Sorted by: Reset to default
This answer is useful
7
Save this answer.
Show activity on this post.
This is a classic problem in cryptography, known as:
The secret sharing problem - how can we divide a secret among n people so that selected subsets, eg: any group of at least r people - can reconstruct the secret, but ZERO information is contained in other subsets.
A standard solution to this problem is:
Snow White calls the secret A and picks 3 random numbers B,C,D as well as 7 different random numbers X 1,X 2,…,X 7 X 1,X 2,…,X 7, one for each dwarf. To dwarf i i, she gives the value of the polynomial A+B X+C X 2+D X 3 A+B X+C X 2+D X 3, evaluated at X i X i. Any four dwarves can recover the coefficients A,B,C,D A,B,C,D knowing the values at four different points, but knowing the values at 3 different points reveals no information about any coefficient. I believe this solution was first given by Adi Shamir.
Instead of using polynomials, we could have also chosen 7 random and linearly independent vectors in R 4 R 4 and given their inner product with (A,B,C,D)(A,B,C,D) to the dwarves.
Another known solution to this is:
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
answered Apr 10, 2015 at 10:20
AravindAravind
1,738 1 1 gold badge 14 14 silver badges 17 17 bronze badges
3
Very elegant! Does the polynomial one also work over a finite field, instead of R R? Because that would make the "choose a random number part" much more friendly.Mike Earnest –Mike Earnest 2015-04-10 15:42:49 +00:00 Commented Apr 10, 2015 at 15:42
Yes, it also works over a finite field, which I guess should be chosen to be large.Aravind –Aravind 2015-04-10 19:54:31 +00:00 Commented Apr 10, 2015 at 19:54
IIRC, Shamir's method was slightly different. Call the secret A, choose a modulus M (M > A, M has no small factors), then choose three random numbers B, C, and D, and find the unique cubic polynomial P such that P(0) = A (mod M), P(1) = B (mod M), P(2) = C (mod M), and P(3) = D (mod M). Give the dwarf i i the numbers M and P(i)m o d M P(i)m o d M. This uses only integers and gives a person no information until they have 4 dwarfs.ralphmerridew –ralphmerridew 2020-12-05 01:46:54 +00:00 Commented Dec 5, 2020 at 1:46
Add a comment|
This answer is useful
3
Save this answer.
Show activity on this post.
Encode the message as a binary string. Decompose this string into 4 parts that can be xored bitwise together to get back the original, using the coin to generate randomness (three flips are needed for each bit). Like so:
```
original: 1000111011110011...
part 1: 0000111011010100...
part 2: 1010111001100000...
part 3: 1111100011001010...
part 4: 1101011010001101...
```
Note that each three parts are worthless without the fourth one. Also note that the probabilities of finding 0s and 1s in all parts are 50%, regardless of the value of the original bit.
Now take four dwarves and give each of them one part.
There are 35 possible combinations of 4 different dwarves out of 7, so repeat the process described above 35 times.
The final output is highly redundant, but no information can be deduced from any incomplete decomposition that could be used to infer the missing information in another incomplete decomposition.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
edited Apr 10, 2015 at 13:56
answered Apr 10, 2015 at 13:46
GOTO 0GOTO 0
13.6k 6 6 gold badges 49 49 silver badges 95 95 bronze badges
3
This was the solution I had in mind, as it bears similarity to lock and key problems.Mike Earnest –Mike Earnest 2015-04-10 15:32:59 +00:00 Commented Apr 10, 2015 at 15:32
So, I understood how only 4 dwarves can be sent and if 3 or less get caught then the msg cannot be decrypted. But, I didn't understand how we can send all the 7 dwarves. Can you pls explain?Hemant Agarwal –Hemant Agarwal 2024-05-13 23:59:37 +00:00 Commented May 13, 2024 at 23:59
For each way to choose 4 of 7 dwarfs, make one of those four-packs, and give one piece to each dwarf in that set.ralphmerridew –ralphmerridew 2024-11-19 12:22:01 +00:00 Commented Nov 19, 2024 at 12:22
Add a comment|
Your Answer
Thanks for contributing an answer to Puzzling Stack Exchange!
Please be sure to answer the question. Provide details and share your research!
But avoid …
Asking for help, clarification, or responding to other answers.
Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Email and Password
Submit
Post as a guest
Name
Email
Required, but never shown
Post Your Answer Discard
By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
mathematics
cryptography
information-theory
See similar questions with these tags.
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
The USAMTS attracts a lot of cheating attempts
Linked
14The Puzzle of the Locked Box
Related
16Secret within the Gossiping Partiers
18Black and White
2Mr. Hilbert and the Problem of the Rogue Resistor
0Pipe and socket problem
1The unknown wheat-and-chessboard problem
33100 prisoners and a secret number
5100 prisoners and a secret number (sequel)
Hot Network Questions
Numbers Interpreted in Smallest Valid Base
Origin of Australian slang exclamation "struth" meaning greatly surprised
Bypassing C64's PETSCII to screen code mapping
Lingering odor presumably from bad chicken
Gluteus medius inactivity while riding
Suspicious of theorem 36.2 in Munkres “Analysis on Manifolds”
Do we need the author's permission for reference
Can you formalize the definition of infinitely divisible in FOL?
What’s the usual way to apply for a Saudi business visa from the UAE?
Matthew 24:5 Many will come in my name!
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Is encrypting the login keyring necessary if you have full disk encryption?
What were "milk bars" in 1920s Japan?
Overfilled my oil
I have a lot of PTO to take, which will make the deadline impossible
Is existence always locational?
ConTeXt: Unnecessary space in \setupheadertext
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Should I let a player go because of their inability to handle setbacks?
Is it ok to place components "inside" the PCB
Change default Firefox open file directory
What meal can come next?
Interpret G-code
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Puzzling
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
11790 | https://static1.squarespace.com/static/558c85a6e4b01a87e89f6ed0/t/5de7a5532aee26493f1779f6/1575462228081/Bayes+Theorem.pdf | Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 1 Warm Up: Urn A contains 3 red and 2 green marbles and Urn B contains 1 red and 4 green marbles. The probability that Urn A is chosen is !
". One marble is chosen. (a) Draw a tree diagram. (b) Find the probability that a red marble is chosen. (c) Given that a red marble was chosen, find the probability that it came from Urn A. Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 2 Notes & Model Problems We have already learned the formula for the probability that event 𝐴 occurs given that 𝐵 has occurred. We can also write the probability that event B occurs given that A has occurred. ⟹𝑃(𝐵 | 𝐴) = 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐴) And we can rearrange the second equation to solve for P(𝐴 ∩ 𝐵). If we substitute this expression for 𝑃(𝐴 ∩ 𝐵) in equation (2) back into equation (1), we get This result is known as Bayes’ Theorem 𝑃(𝐴 | 𝐵) = 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) 𝑃(𝐵) ⟹𝑃(𝐴 | 𝐵) = 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐵) (1) ⟹𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) (2) Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 3 Partitions of the Sample Space: For any event 𝐴 and its complement 𝐴. • 𝐴 and 𝐴. are mutually exclusive, so 𝐴∩𝐴. = {} • 𝐴∪𝐴. = 𝑈, the sample space. We say that 𝐴 and 𝐴. partition the sample space, and we can draw the following diagram. For any other event 𝐵 in the sample space 𝑈, ⟹𝑃(𝐵) = 𝑃(𝐵∩𝐴) + 𝑃(𝐵∩𝐴.) = 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) + 𝑃(𝐵 | 𝐴.) ∙𝑃(𝐴.) So, Bayes theorem can alternatively be re-written as 𝑃(𝐴 | 𝐵) = 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) 𝑃(𝐵) = 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) + 𝑃(𝐵 | 𝐴.) ∙𝑃(𝐴.) 𝑃(𝐴 | 𝐵) = 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) 𝑃(𝐵 | 𝐴) ∙𝑃(𝐴) + 𝑃(𝐵 | 𝐴.) ∙𝑃(𝐴.) Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 4 Model Problem #1: A can contains 4 blue and 2 green marbles. One marble is randomly drawn from the can without replacement, and its color is noted. A second marble is then drawn. (a) Draw a tree diagram. (b) Find the probability that the second marble is blue. (c) Find the probability that the first marble was green, given that the second marble is blue. Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 5 Model Problem #2: 54% of the students at a university are female. 8% of the male students are color blind, and 2% of the female students are color blind. (a) Draw a tree diagram (b) A randomly chosen student is color blind. Find the probability that the student is male. (c) A randomly chosen student is not color blind. Find the probability that the student is female. Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 6 HOMEWORK: (1) Coffee machines A and B produce coffee in identically shaped plastic cups. Machine A produces 65% of the coffee sold each day, ad machine B produces the remainder. Machine A underfills a cup 4% of time, while machine B underfills a cup 5% of time. (a) A cup of coffee is chosen at random. Find the probability that it is underfilled. (b) A cup of coffee is randomly chosen and is found to be underfilled. Find the probability that it came from machine A. (2) A marble is randomly chosen from a can containing 3 red and 5 blue marbles. It is replaced by two marbles of the other color. Another marble is then randomly chosen from the can. Given that the marbles chosen have the same color, what is the probability that they are both blue. Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 7 (3) A newspaper printer has three presses A, B, and C which print 30%, 40%, and 30% of daily production respectively. Due to the age of the machines, the presses will produce streaks on their output 3%, 5%, and 7% of the time, respectively. (a) Find the probability that a randomly chosen newspaper does not have streaks. (b) If a randomly chosen newspaper does not have streaks, find the probability that it was printed by press A (c) If a randomly chosen newspaper does not have streaks, find the probability that it was printed by either A or C. Bayes’ Theorem Assigned 12/4/2019 Due 12/4/2019 8 (4) A sample space 𝑈 is partitioned into three by the mutually exclusive events 𝐴!, 𝐴5, and 𝐴6. The sample space also contains another event B Write an expression for 𝑃(𝐵). |
11791 | https://www.cnn.com/2014/06/09/opinion/galdorisi-leave-no-man-behind | Opinion: Why we leave no man behind | CNN
CNN values your feedback
How relevant is this ad to you?
Did you encounter any technical issues?
[x] No
Video player was slow to load content - [x] Video content never loaded - [x] Ad froze or did not finish loading - [x] Video content did not start after ad - [x] Audio on ad was too loud - [x] Other issues - [x]
Ad never loaded - [x] Ad prevented/slowed the page from loading - [x] Content moved around while ad loaded - [x] Ad was repetitive to ads I've seen previously - [x] Other issues - [x]
Cancel Submit
Thank You!
Your effort and contribution in providing this feedback is much appreciated.
Close
Ad Feedback
US
World
Politics
Business
Health
Entertainment
Underscored
Style
Travel
Sports
Science
Climate
Weather
Ukraine-Russia War
Israel-Hamas War
Games
More
USWorldPoliticsBusinessHealthEntertainmentUnderscoredStyleTravelSportsScienceClimateWeatherUkraine-Russia WarIsrael-Hamas WarGames
WatchListenLive TVSubscribe
Sign in
My Account
Settings
Newsletters
Topics you follow
Sign out
Your CNN account Sign in to your CNN account
Sign in
My Account
Settings
Newsletters
Topics you follow
Sign out
Your CNN account Sign in to your CNN account
Live TVListenWatch
Edition
US
International
Arabic
Español
Edition
US
International
Arabic
Español
US
Crime + Justice
World
Africa
Americas
Asia
Australia
China
Europe
India
Middle East
United Kingdom
Politics
Trump
Facts First
CNN Polls
2025 Elections
Business
Tech
Media
Calculators
Videos
Markets
Pre-markets
After-Hours
Fear & Greed
Investing
Markets Now
Nightcap
Health
Life, But Better
Fitness
Food
Sleep
Mindfulness
Relationships
CNN Underscored
Electronics
Fashion
Beauty
Health & Fitness
Home
Reviews
Deals
Gifts
Travel
Outdoors
Pets
Entertainment
Movies
Television
Celebrity
Tech
Innovate
Foreseeable Future
Mission: Ahead
Work Transformed
Innovative Cities
Style
Arts
Design
Fashion
Architecture
Luxury
Beauty
Video
Travel
Destinations
Food & Drink
Stay
Videos
Sports
Pro Football
College Football
Basketball
Baseball
Soccer
Olympics
Hockey
Science
Space
Life
Unearthed
Climate
Solutions
Weather
Weather
Video
Climate
Ukraine-Russia War
Israel-Hamas War
Watch
Featured
Shows & Films
Network TV
Clips
CNN Headlines
CNN Shorts
TV Shows A-Z
CNN 10
CNN Max
TV Schedule
FlashDocs
Listen
CNN 5 Things
Chasing Life with Dr. Sanjay Gupta
The Assignment with Audie Cornish
One Thing
Tug of War
CNN Political Briefing
The Axe Files
All There Is with Anderson Cooper
All CNN Audio podcasts
Games
Daily Crossword
Jumble Crossword
Photo Shuffle
Sudoblock
Sudoku
5 Things Quiz
About CNN
Subscribe
Photos
Investigations
CNN Profiles
CNN Leadership
CNN Newsletters
Work for CNN
Follow CNN
Why we leave no man behind
George Galdorisi
5 min read
Updated 11:28 AM EDT, Mon June 9, 2014
Link Copied!
Subtitles
Off
Caption Styling Settings
Video Ad Feedback
History of 'No man left behind' motto
02:15 • Source: CNN
History of 'No man left behind' motto
02:15
Editor’s Note:Capt. George Galdorisi is a career naval aviator and co-author of “Leave No Man Behind: The Saga of Combat Search and Rescue.” He has also written eight other books, including “Tom Clancy Presents: Act of Valor” and “The Kissing Sailor,” which proved the identity of the two principals in Alfred Eisenstaedt’s famous photograph. His latest book is a reboot of the best-selling Tom Clancy’s Op-Center series – Book One, “Out of the Ashes.” The opinions expressed in this commentary are solely those of the author.
Story highlights
Capt. George Galdorisi co-wrote the book on why the military leaves no man behind
He says the concept goes back centuries and has modern roots in our armed services
Combat search and rescue is a special practice that saves downed or missing forces
The practice, he says, has been inconsistently applied throughout modern warfare
CNN—
“The United States has always had a pretty sacred rule, and that is: we don’t leave our men or women in uniform behind,” proclaimed a defiant President Barack Obamadays after he authorized the swap that brought Sgt. Bowe Bergdahl home after five years in the hands of the Taliban.
But has that always been the practice of the U.S. Armed Forces? And why is it worth risking the lives of more soldiers just to bring one lone man or woman in uniform home? You’d be surprised to know why it hasn’t always been that way.
The concept of the Latin phrase nemo resideo, or “leave no one behind,” is almost as old as warfare itself. Greek mythology portrayed heroes who rescued those captured by enemies.
Ad Feedback
Two millennia later, in early colonial America, this mythology morphed into legends such as that of James Fenimore Cooper’s Natty Bumppo and the Deerslayer in “The Last of the Mohicans,” who famously rescued the helpless from the savage’s camp. About a century later, images of the U.S. Cavalry began to appear on the silver screen, riding to the rescue of captured women and children.
Capt. George Galdorisi
U.S. Navy
But with the advent of mechanized warfare and early aviation, the military gained a practical ability to go around or over the enemy to bring back stranded comrades or liberate prisoners of war.
Today, the phrase is often attributed to the elite U.S. Army Rangers, who have the language as part of their creed: “I will never leave a fallen comrade to fall into the hands of the enemy.”
Similar language can be found in the Soldier’s Creed – “I will never leave a fallen comrade” – as well as the Airman’s Creed – “I will never leave an Airman behind.” The Latin version, nemo resideo, is even the motto of the Marine Corps’ Personnel Retrieval and Processing Company, the team charged with retrieving the remains of fallen service members from the battlefield.
And as conflicting media reports trickle in about numerous search and rescue missions trying to find Bergdahl in Afghanistan, one thing is clear: a dedicated, organized and trained combat search and rescue force is critical to finding and saving our lost soldiers, airmen, sailors and Marines.
Those brave soldiers who recovered Bergdahl from his Taliban captors last week and spirited him off were reportedly members of U.S. Special Forces. In fact, when the disoriented Bergdahl first boarded the Black Hawk helicopter, he was said to have scrawled on a paper plate “SF?” “Yes!” shouted at least one man over the roar of the rotors. “We’ve been looking for you for a long time!”
But that team was different from a dedicated combat search and rescue force – special troops, helicopters and other assets held at the ready – that has only recently become a permanent fixture in the U.S. military.
Opinion: An ugly decision, but the right call
On March 23, 2003, Pvt. Jessica Lynch was captured by Iraqi forces in Nasiriyah and rescued a week later by a dedicated CSAR team. Two years later in Afghanistan, nearly two dozen CSAR Special Forces operators were lost while failing to rescue SEAL Lt. Michael Murphy’s recon team. The story of that tragedy was told in the recent film “Lone Survivor.”
This undated image provided by the U.S. Army shows Sgt. Bowe Bergdahl, who has been held by insurgents in Pakistan since 2009. Extremely sensitive discussions are under way with intermediaries overseas to see if there is any ability to gain his release, a U.S. official told CNN on February 19, 2014. U.S. Army video Related video The Bowe Bergdahl story
Taliban Hostage: US Soldier Bowe Bergdahl From IntelCenter/Taliban video video Related video What happened the night he disappeared
Caption:HAILEY, ID - JUNE 01: A sign announcing the release of Sgt. Bowe Bergdahl sits in the window of the Hailey Paint and Supply store on Main Street June 1, 2014 in Hailey, Idaho. Sgt. Bergdahl was captured in Afghanistan in 2009 while serving with U.S. Armys 501st Parachute Infantry Regiment in Paktika Province. Yesterday he was released after a swap for 5 prisoners being held at Guantanamo Bay was arranged. Bergdahl was considered the only U.S. prisoner of war held in Afghanistan. (Photo by Scott Olson/Getty Images) Scott Olson/Getty Images video Related video Bergdahl's life before the Army
nr.brooke.bergdahl.release.video_00014218.jpg video Related video New video shows moment Bergdahl was freed
Even though the results were different, both actions define the risks and rewards of trying to rescue our warriors held in captivity or trapped behind enemy lines, with roots running through our national ethos for over a century.
What is CSAR and how is it different from other rescues?
It is a dedicated force fighting for a downed airman or missing soldier, exerting local dominance long enough to suppress hostile opposition and snatch the threatened person from the jaws of a sophisticated enemy. And hurry up doing it.
If that is not the situation, then those who argue against dedicated combat rescue organizations are right; send whoever is handy and available. It is not a combat rescue. Any competent helicopter crew should be able to do it.
The men and women who make up the CSAR cadre of the U.S. military willingly lay down their lives to rescue their comrades.
Opinion: Will Democrats pay a price for Bergdahl deal?
But in an era of declining military budgets, with two wars winding down, and with the American public weary of war, it is unclear whether we as a nation are willing to invest in the equipment and training needed to keep our CSAR forces primed and ready. Sadly, history suggests we may not be.
The history of rescuing downed airmen by other airmen is as old as the history of air combat. In the World War I, we had no combat search and rescue, although that war saw the birth of the concept of combat rescue. In World War II, we began without combat search and rescue, had unprepared forces attempting rescues on the fly – often with catastrophic results – then ended the war with a robust CSAR effort, only to dismantle it in the euphoria of demobilization.
In the Korean War, we had a strong CSAR but allowed it to atrophy again. In Vietnam, we finally established one of the strongest CSAR forces to date. For the first time, combat rescue as a mission survived.
Between the end of Vietnam in 1973 and Operation Desert Storm in 1991, the need for CSAR was demonstrated often, but the military’s CSAR forces subsisted on meager funding.
When Desert Storm broke out, the nation’s combat rescue capability had degraded to the point our aviators going into combat did not have the backup they sorely needed.
Of 87 airmen downed in that war, 39 survived, but only eight were rescued. Of the 31 others, 10 were deemed to have been rescueable but were not. Largely because of the short war and low losses compared with previous wars, however, few noticed this poor showing.
We argue sooner or later, somebody has to go down there and get the downed airman or missing soldier. And hurry up doing it.
It should be trained, dedicated and fully funded combat search and rescue assets who get there first.
Our men and women fighting for America deserve nothing less. We will never leave them behind.
Follow us on Twitter @CNNOpinion.
Join us on Facebook.com/CNNOpinion.
Up next
Marine veteran in custody after 3 killed, at least 8 injured in shooting at a waterfront bar in North Carolina, officials say Sep 28, 2025 4 minute read min read
Here’s how to get a refund from the historic Amazon Prime settlement Sep 26, 2025 3 minute read min read
Who is Ian Roberts, the Iowa school superintendent accused by ICE of being in the US illegally? Sep 28, 2025 8 minute read min read
Linda McMahon is dismantling the Education Department — but first she’ll help make school ‘patriotic’ again Sep 27, 2025 8 minute read min read
Bethpage Black comes close to boiling over as frustrated American fans crank up the heckling of Team Europe at Ryder Cup Sep 28, 2025 7 minute read min read
Most read
Marine veteran in custody after 3 killed, at least 8 injured in shooting at a waterfront bar in North Carolina, officials say
Here’s how to get a refund from the historic Amazon Prime settlement
Who is Ian Roberts, the Iowa school superintendent accused by ICE of being in the US illegally?
Linda McMahon is dismantling the Education Department — but first she’ll help make school ‘patriotic’ again
Bethpage Black comes close to boiling over as frustrated American fans crank up the heckling of Team Europe at Ryder Cup
Hurricane Humberto strengthened into a rare Cat 5, but there’s another storm the US needs to watch
Russia attacks Ukraine with one of the largest aerial assaults of the war, killing four and injuring dozens
BMW tells nearly 200,000 vehicle owners to park outside because their cars could short circuit and catch fire
Kilmar Abrego Garcia transferred to Pennsylvania detention facility
Scientists analyzed years of interviews with Taylor Swift to track how dialects evolve
Up next
Marine veteran in custody after 3 killed, at least 8 injured in shooting at a waterfront bar in North Carolina, officials say Sep 28, 2025 4 minute read min read
Here’s how to get a refund from the historic Amazon Prime settlement Sep 26, 2025 3 minute read min read
Who is Ian Roberts, the Iowa school superintendent accused by ICE of being in the US illegally? Sep 28, 2025 8 minute read min read
Linda McMahon is dismantling the Education Department — but first she’ll help make school ‘patriotic’ again Sep 27, 2025 8 minute read min read
Most read
Marine veteran in custody after 3 killed, at least 8 injured in shooting at a waterfront bar in North Carolina, officials say
Here’s how to get a refund from the historic Amazon Prime settlement
Who is Ian Roberts, the Iowa school superintendent accused by ICE of being in the US illegally?
More from CNN
Marine veteran in custody after 3 killed, at least 8 injured in ... Sep 28, 2025
Here’s how to get a refund from the historic Amazon Prime settlement Sep 26, 2025
Who is Ian Roberts, the Iowa school superintendent accused by ICE ... Sep 28, 2025
Ad Feedback
News & buzz
Marine veteran in custody after 3 killed, at least 8 injured in ... Sep 28, 2025
Here’s how to get a refund from the historic Amazon Prime settlement Sep 26, 2025
Who is Ian Roberts, the Iowa school superintendent accused by ICE ... Sep 28, 2025
Ad Feedback
Ad Feedback
Subscribe
Sign in
My Account
Settings
Newsletters
Topics you follow
Sign out
Your CNN account Sign in to your CNN account
Live TVListenWatch
US
Crime + Justice
World
Africa
Americas
Asia
Australia
China
Europe
India
Middle East
United Kingdom
Politics
Trump
Facts First
CNN Polls
2025 Elections
Business
Tech
Media
Calculators
Videos
Markets
Pre-markets
After-Hours
Fear & Greed
Investing
Markets Now
Nightcap
Health
Life, But Better
Fitness
Food
Sleep
Mindfulness
Relationships
CNN Underscored
Electronics
Fashion
Beauty
Health & Fitness
Home
Reviews
Deals
Gifts
Travel
Outdoors
Pets
Entertainment
Movies
Television
Celebrity
Tech
Innovate
Foreseeable Future
Mission: Ahead
Work Transformed
Innovative Cities
Style
Arts
Design
Fashion
Architecture
Luxury
Beauty
Video
Travel
Destinations
Food & Drink
Stay
News
Videos
Sports
Pro Football
College Football
Basketball
Baseball
Soccer
Olympics
Hockey
Science
Space
Life
Unearthed
Climate
Solutions
Weather
Weather
Video
Climate
Ukraine-Russia War
Israel-Hamas War
Watch
Featured
Shows & Films
Network TV
Clips
CNN Headlines
CNN Shorts
TV Shows A-Z
CNN 10
CNN Max
TV Schedule
FlashDocs
Listen
CNN 5 Things
Chasing Life with Dr. Sanjay Gupta
The Assignment with Audie Cornish
One Thing
Tug of War
CNN Political Briefing
The Axe Files
All There Is with Anderson Cooper
All CNN Audio podcasts
Games
Daily Crossword
Jumble Crossword
Photo Shuffle
Sudoblock
Sudoku
5 Things Quiz
About CNN
Subscribe
Photos
Investigations
CNN Profiles
CNN Leadership
CNN Newsletters
Work for CNN
WatchListenLive TV
Follow CNN
Subscribe
Sign in
My Account
Settings
Newsletters
Topics you follow
Sign out
Your CNN account Sign in to your CNN account
Terms of UsePrivacy PolicyManage CookiesAd ChoicesAccessibility & CCAboutSubscribeNewslettersTranscriptsHelp Center
© 2025 Cable News Network. A Warner Bros. Discovery Company. All Rights Reserved.
CNN Sans ™ & © 2016 Cable News Network.
Legal Terms and Privacy
By clicking "Agree", you have read and agree to the Terms of Use and agree to the collection and use of your information by cookies and similar technologies, as set forth in our Privacy Policy.
Agree
Privacy Policy
For privacy options, please see our privacy policy:
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Close |
11792 | https://www.youtube.com/watch?v=-xonjgXPerk | Distance on the Coordinate Plane Video
Simplify the Middle
11400 subscribers
242 likes
Description
50706 views
Posted: 29 Jun 2023
For a copy of the notes, vocabulary, and interactive activities, visit me at .
This video explains the difference between horizontal and vertical given 2 points, how to find distance of two points both graphically and by ordered pairs, how to find cardinal directions from a point.
Transcript:
[Music] hi and welcome to today's lesson distance on the coordinate plane here we have a couple of points today we will learn how to find the distance between points like these but first we need to identify if a line is horizontal or vertical this line is an example of a horizontal line let's look at why if we were to stack our two points on top of each other and we look at the x coordinates and the y-coordinates we notice something our y-coordinates are both four so whenever we have the same y coordinate we know we're talking about a horizontal line let's look at a couple of other points here we have negative 3 3 and negative 3 negative 4. this is an example of a vertical line because it is straight up and down if we take a closer look at our ordered pairs we notice something about our coordinates again in this one we can see our x coordinates are both negative three so anytime that two ordered pairs have the same x coordinate they make a vertical line now that we understand the difference between them let's practice trying to find the distance between them so here I have two points one four and four four I automatically know that they are horizontal because I see them graphed but what about if I don't see them graphed so because these ones are graphed I do notice that it is going from left to right and I also can see from my ordered pairs if I didn't have a picture of it I can see from my ordered pairs that the Y values the Y coordinates are the same this means horizontal now to find the distance in a picture I can just count the distance between them so this is three so let's try another way to find the distance when I don't have the picture so if I were to just have the ordered pairs to find distance we're going to look at the coordinates that are not shared in this case the x coordinates we will take the larger minus the smaller value so here I have 4 is bigger than one so four minus 1 is equal to 3. and I know that that answer is correct because I do have a graph picture and I counted but also I know this is correct because it is a positive number and my distance must always be positive so if you get to the end of your question and you see that you got a negative number for an answer then you know that you must have done something wrong because distance must always be positive let's take a look at those points from earlier negative 3 3 and negative 3 negative 4. so here again we can see that they have the same x coordinate and when you have the same x coordinate that is a vertical line and I have my non-shared values as my 3 and my negative 4. because this is graphed I'm going to go ahead and count the distance between these so as I count I notice that I get 7. and then I'm just going to go ahead and write out this problem the same way that I did the other one keep in mind that the higher up on the graph for a vertical line the bigger the number also this is a positive and this is a negative so that's how I can tell 3 is bigger than 4 as well so I'm going to say 3 minus negative 4. and I just put parentheses around it to keep it the problem a little bit more organized and then remember when you're subtracting integers you're actually adding the opposite so we don't subtract integers we just change it to add the opposite so I'm going to keep the 3 The Way It Is change this to addition so it's 3 plus the opposite of negative 4 is positive 4 so it's actually 3 plus 4. So my answer is 7. and I could confirm that from the picture of it being drawn in me counting those the distance of those units as well so let's take a look at one last type of problem that you might see and this is when you have cardinal directions so let's say here we are at our 0.21 and it tells us it wants us to move three miles and we're assuming that each unit here is a mile it wants us to move three miles to the east west north and south so I'm gonna have four different ordered pairs that represents each of those movements um the first thing that I'm gonna do I went ahead and put together a little table that makes this really easy for me to know what I'm doing for each of the coordinates and the second thing that I did is I just went ahead and wrote out my original point two comma one um my original ordered pair and I'm going to do my math down here so if I go to the east east is this direction as I'm going this Direction with my point I notice that my numbers for my x value my x coordinate is getting bigger but nothing is happening to the one so to go East I'm going to add that 3 and I'm going to keep my y-coordinate the same so my new point if I was moving from 2 1 to the east 3 miles would be five comma one to move to the west west goes this direction so I can see I'm moving um over here and I'm moving three so I'm actually going to be subtracting but I'm only moving on my x coordinate so I'm going to subtract 3 from 2 and I'll keep my y coordinate the same because it didn't go up or down it just went to the west and 2 minus 3 is going to give me negative 1 1. next to go north I'm going up so I'm going to go up three so I'm adding making that two bigger I'm sorry I'm making the one bigger in this case I'm not going to do anything to the two so notice how it doesn't have a color and the reason why is because um my X is going to stay the same I'm going to stay on this line I'm just going to go up on that line I'm going to go up three so it's my y then I'm going to add to so I end up with two comma 4 and then finally if I want to go south I'm going to be going down but I'm still staying on the same x coordinate so that stays the same and I will be subtracting the three so I end up with 2 comma negative 2. let's recap points that share the same y coordinate are horizontal and points that share the same x coordinate are vertical finally we know to find the distance of points that are vertical or horizontal we will do that by subtracting the larger value by the smaller value thanks so much for tuning in feel free to click to subscribe for this and other videos until next time |
11793 | https://www.nagwa.com/en/explainers/279181578290/ | Lesson Explainer: Indefinite Integrals: The Power Rule | Nagwa
Lesson Explainer: Indefinite Integrals: The Power Rule | Nagwa
Sign Up
Sign In
English
English
العربية
English
English
العربية
My Wallet
Sign Up
Sign In
My Classes
My Messages
My Reports
My Wallet
My Classes
My Messages
My Reports
Lesson Explainer: Indefinite Integrals: The Power Rule Mathematics • Second Year of Secondary School
In this explainer, we will learn how to find the indefinite integrals of polynomials and general power functions using the power rule for integration.
Recall that an antiderivative, also known as an inverse derivative or primitive, of a function is another function whose derivative is equal to the original function .
Definition: An Antiderivative of a Function
For any function defined on a subset and a differentiable function , if we have then we say that is an antiderivative of .
The antiderivative of a function is equivalent to the indefinite integral, which we define as follows.
Definition: The Indefinite Integral
The indefinite integral of with respect to can be written in terms of an antiderivative as where is also called the constant of integration.
Antiderivatives of always exist when is continuous, and there are infinitely many antiderivatives for , obtained by adding the arbitrary constant to . This constant, also known as the constant of integration, is very important, as it produces a family of antiderivatives paramaterized by . In other words, is the most general function that has a derivative , for all . For example, the derivative of is given by
Thus, we can say that is an antiderivative of 1, but is the most general antiderivative of 1, which means that , , , , and so on are all also antiderivatives of 1. This is what we call the indefinite integral and is expressed as
Similarly, the derivative of is which implies that the indefinite integral of is
Recall that the derivative satisfies the property
This means we can always take out a constant multiple outside the derivative. Thus, is an antiderivative of , or is the general antiderivative of , for all , which is the same as the indefinite integral. In other words, if the derivative gets multiplied by a constant, the antiderivative also gets multiplied by the same constant and vice versa. This implies the following property of indefinite integrals:
Using the indefinite integral of as written above, we can take out the factor of 2 using this property and divide both sides of the expression by 2 to obtain where we note that we have kept the constant the same since this is arbitrary and is just another constant. In this explainer, we will be particularly interested in determining indefinite integrals of the form using the power rule of integration. We can determine this rule directly from the power rule of differentiation. Suppose , for . The derivative of this function can be found by the power rule for differentiation as follows:
It will be useful to rewrite this as where we have divided by the constant , as a constant multiple of a function does not affect the derivative or antiderivative. But what if we want to work in reverse? (i.e., given , we want to determine the antiderivative). This means we want to find the most general function that differentiates to give .
We have already shown that the derivative of is , for . If we let , then we have
Thus, is an antiderivative of , provided . We can express this in terms of an indefinite integral in the following definition.
Rule: The Power Rule for Integration
The power rule for integration allows us to determine the indefinite integral of , provided , as follows:
For example, using this power rule, we can determine the indefinite integral of as follows: which can be verified directly by differentiating the right-hand side to obtain the integrand .
In the first example, we will determine the indefinite integral of a function involving a positive integer power of using the power rule along with the property that allows us to take a constant multiple outside the integral.
Example 1: The Power Rule of Integration
Determine .
Answer
In this example, we will determine the indefinite integral of a positive integer power of , in particular the function .
In order to determine the integral, we will make use of the following property of indefinite integrals:
We will also make use of the power rule:
We can use the property to take out the factor of outside the integral and determine the indefinite integral of using the power rule:
Now, let’s consider an example where we will determine the indefinite integral of a function involving a negative integer power of using the power rule along with the property that allows us to take a constant multiple outside the integral.
Example 2: Finding the Integration of a Function Using the Power Rule for Integration with a Negative Exponent
Determine .
Answer
In this example, we will determine the indefinite integral of a negative integer power of , in particular the function .
In order to determine the integral, we will make use of the following property of indefinite integrals:
We will also make use of the power rule:
We can use the property to take the factor of outside the integral and determine the indefinite integral of using the power rule:
This result is valid for all , since we require the integrand and integral to be continuous and well defined.
We can use the power rule to determine the indefinite integral of any power of and not just integers, provided that power is not equal to . In the next example, we will determine the indefinite integral of a function involving a positive fractional power of by rewriting the radical in terms of a power of and using the power rule along with the property that allows us to take a constant multiple outside the integral.
Example 3: Finding the General Antiderivative of a Function Using the Power Rule of Integration with Fractional Exponents
Determine .
Answer
In this example, we will determine the indefinite integral of a positive fractional power of , in particular the function .
Let’s first rewrite the integrand, by noting that , as
In order to determine the integral, we will make use of the following property of indefinite integrals:
We will also make use of the power rule:
We can use the property to take the factor of 7 outside the integral and determine the indefinite integral of using the power rule. Following this, we can rewrite the final answer back in terms of the square root:
This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the square root is only defined for nonnegative numbers.
Now, let’s consider an example where we will determine the indefinite integral of a function involving a negative fractional power of by rewriting the radical in terms of a power of and by using the power rule along with the property that allows us to take a constant multiple outside the integral.
Example 4: Finding the Integration of a Function Using the Power Rule with Roots
Determine .
Answer
In this example, we will determine the indefinite integral of a negative fractional power of , in particular the function .
First, let’s rewrite the integrand, by noting that , as
In order to determine the integral, we will make use of the following property of indefinite integrals:
We will also make use of the power rule:
We can use the property to take out the factor of 6 outside the integral and determine the indefinite integral of using the power rule:
This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the 8th root is only defined for nonnegative numbers.
Recall that the derivative is a linear operation, as it satisfies
This also implies a similar rule for indefinite integrals:
Therefore, in order to determine the indefinite integral of a sum of functions, we just find an indefinite integral of each part separately and add the results together, not forgetting the at the end. Usually, we would obtain multiple constants for each part from the process of integration, but we can combine these into one constant. We can also combine this property with the one that allows us to take constants outside the integral.
Property: Linearity Property of Integration
For any continuous functions and defined on a subset , we have the linearity property for .
The power rule for integration along with this linearity property allow us to determine the indefinite integral involving sums of different powers of including polynomial, reciprocal, and radical functions. For example, we can determine the indefinite integral of the linear function as follows:
In the next example, we will determine the indefinite integral of a polynomial function using the properties of linearity and the power rule for integrals.
Example 5: Finding the Integration of a Polynomial Function Using the Power Rule
Determine .
Answer
In this example, we will determine the indefinite integral of the polynomial function .
In order to determine the integral, we will make use of the following properties of indefinite integrals:
We will also make use of the power rule:
Using the first property, we can split the given integral into three parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:
Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .
Now, let’s consider an example where we will determine the indefinite integral of a polynomial function by distributing two brackets using the properties of linearity and the power rule for integrals.
Example 6: Finding the Integration of a Polynomial Involving Multiplying Two Brackets and Applying the Power Rule
Determine .
Answer
In this example, we will determine the indefinite integral of the polynomial function .
Let’s first simplify the integrand by distributing the brackets:
In order to determine the integral, we will make use of the following properties of indefinite integrals:
We will also make use of the power rule:
Using the first property, we can split the given integral into two parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:
Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .
In the next example, we will determine the indefinite integral of a rational function using factorization, the properties of linearity, and the power rule for integrals.
Example 7: Finding the Integration of a Rational Function Using the Factorization of the Difference of Two Squares
Determine .
Answer
In this example, we will determine the indefinite integral of the rational function .
Let’s first simplify the integrand by noting that the numerator is the difference of two squares and can be written as ; thus, the integrand becomes for . In order to determine the integral, we will make use of the following properties of indefinite integrals:
We will also make use of the power rule:
Using the first property we can split the given integral into two parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:
Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .
This result is valid for all , since we require the integrand and integral to be continuous and well defined.
Now, let’s consider an example where we determine the indefinite integral of a rational function with negative powers of using the properties of linearity and the power rule for integrals.
Example 8: Finding the Integration of a Function Using the Power Rule for Integration with a Negative Exponent
Determine .
Answer
In this example, we will determine the indefinite integral of the rational function .
Let’s first rewrite the integrand as powers of :
In order to determine the integral, we will make use of the following properties of indefinite integrals:
We will also make use of the power rule:
Using the first property, we can split the given integral into three parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:
Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .
This result is valid for all , since we require the integrand and integral to be continuous and well defined.
In the next example, we will determine the indefinite integral of a function with roots and negative exponents using the properties of linearity and the power rule for integrals.
Example 9: Finding the Integration of a Function Using the Power Rule for Integration with Roots and Negative Exponents
Determine .
Answer
In this example, we will determine the indefinite integral of the function .
Let’s first rewrite the integrand as powers of using : for . In order to determine the integral, we will make use of the following properties of indefinite integrals:
We will also make use of the power rule:
Using the first property, we can split the given integral into three parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:
Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .
This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the square root is only defined for nonnegative numbers.
In the last example, we will determine the indefinite integral of a function involving fractional exponents by using factorization, the properties of linearity, and the power rule for integrals.
Example 10: Finding the Integration of a Function Using Factorization
Determine .
Answer
In this example, we will determine the indefinite integral of the function .
We note that the numerator in the integrand can be written as the difference of two squares, as . Using this, we can simplify the integrand and rewrite the remaining terms as powers of using : for . In order to determine the integral, we will make use of the following properties of indefinite integrals:
We will also make use of the power rule:
Using the first property, we can split the given integral into two parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:
Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .
This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the 8th root is only defined for nonnegative numbers.
Let’s finish by considering the key points that we covered in this explainer.
Key Points
In order to determine the indefinite integrals of functions involving different powers of including polynomial, reciprocal, and radical functions, we make use of the following:
The linearity property of integrals: for .
The power rule for integration:
Lesson Menu
Lesson
Lesson Plan
Lesson Presentation
Lesson Video
Lesson Explainer
Lesson Playlist
Join Nagwa Classes
Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!
Interactive Sessions
Chat & Messaging
Realistic Exam Questions
Nagwa is an educational technology startup aiming to help teachers teach and students learn.
Company
About Us
Contact Us
Privacy Policy
Terms and Conditions
Careers
Tutors
Content
Lessons
Lesson Plans
Presentations
Videos
Explainers
Playlists
Copyright © 2025 Nagwa
All Rights Reserved
Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy
Accept |
11794 | https://instavm.org/wp-content/uploads/2021/05/H8.pdf | THE ORIGIN OF DIVIDING A CIRCLE INTO 360 PARTS P.Devaraj Abstract The circle is a fundamental geometrical shape and commonly seen in Nature. Naturally, the angular division of a circle into 360 degrees is also important and forms the basis of angular measurement. The history of the 360 degree division is so ancient that it is difficult to discover the real origin. This paper seeks to uncover some of the ancient references. It is possible that 360 relates to the Sun’s circuit, when observed from Earth. In this study we explore various measurements and systems used by ancient civilizations to calculate the time taken by sun to complete its circuit – the year. In the Rig veda - the oldest Vedic text, there are clear references (1.164.48) to a chakra or wheel of 360 spokes placed in the sky. “Twelve are its fellies. The wheel is one. It has three naves. Who has understood it?” It is these hymns, of the great Rishi Dirghatamas, that clearly show the ancient Indians dividing the year into twelve rasis (months), consisting of 30 days each, forming the 360 days - the number of spokes in this chakra (Wheel). Dirghatamas also prepared a table for dividing these 360 parts into smaller parts known as talpara, vikala (second), kala (minute), bhaga (degree) and more. Key words: Circle, Degrees, Rig Veda, Dirghatamas 1 Introduction The Circle is a fundamental geometrical shape which is commonly seen in nature. The angular division of circle into 360 parts, degrees forms the basis of measuring circular objects or angular measurement. The history of the 360 degree division is so ancient that it is difficult to pinpoint the real origin. This paper tries to highlight the ancient references about dividing a circle into 360 parts. 1.1 The need Ancient peoples tracked the movement of planets, sun, moon, and so on, to evaluate time, climatatic and religious festivals. This was a socio economic requirement for those people. Based on the position of these celestial bodies people assessed the time elapsed, since sunrise, moonrise, etc., and they charted the recurrence of events such as eclipses, full moon, and such like. With these recurring events people constructed calendars. People watched the Sun, Moon, and the five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn), primarily for omens. They tried to understand the motions physically. They noticed the circular track of the Sun's annual path across the sky and knew that it took about 360 days to complete one year's circuit. Consequently, they divided the circular path into 360 parts to track each day's passage of the Sun's annual journey. 1.2 Sumerians and Babylonians Babylonia was an ancient post-Sumer civilization that originated in the central-southern region of Mesopotamia. Normally it was stated that the Sumerians had a calendar, dating from 2400 BC that divided the year into 12 months of 30 days each, that is, 360 days. It was during the reign of Nebuchadnezzar (605-562 BC) in the Chaldean dynasty in Babylon that the circle was divided into 360 degrees. This was because the Chaldeans had calculated by observation and inference that a complete year consists 360 days. One more reason for the basis of angular measure for the Babylonians was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when performing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees, because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. 1.3 Egyptian The Egyptian calendar, established several thousand years before the common era, was the first calendar known to use a year of 365 days, approximately equal to the solar year. In addition to this civil calendar, the ancient Egyptians simultaneously maintained a second calendar based upon the phases of the moon. 1.4 Roman The Romans borrowed parts of their earliest known calendar from the Greeks. The calendar consisted of 10 months in a year of 304 days. The Romans seem to have ignored the remaining 61 days, which fell in the middle of winter. The 10 months were named Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November, and December. The last six names were taken from the words for five, six, seven, eight, nine, and ten. Romulus, the legendary first ruler of Rome, is supposed to have introduced this calendar in the 700s B.C.E. According to tradition, the Roman ruler Numa Pompilius added January and February to the calendar. This made the Roman year 355 days long. To make the calendar correspond approximately to the solar year, Numa also ordered the addition every other year of a month called Mercedinus. Mercedinus was inserted after February 23 or 24, and the last days of February were moved to the end of Mercedinus. In years when it was inserted, Mercedinus added 22 or 23 days to the year. Julius Caesar added a month of 31 days during 44BCE , July ,named after him and later, in 8BCE, Augustus Caesar added one more month, August, to make it a 12 month Calendar. 2 Indian system Twelve are the fellies, and the wheel is single; three are the naves. What man hath understood it? Therein are set together spokes three hundred and sixty, which in no-wise can be loosened. Dirghatamas - Rigveda 1.164.48; Samvalsara Kala Chakram The twelve fellies represent the twelve months in a year, and 360 spokes represent the days. Three naves represent three regions, heaven, mid-air and terrestrial. They also represent three periods of a day for Somayajna in which soma juice is offered three times during the period of one day and one night. With this it clearly explains a day period of one sunrise to next sunrise. The Indian classification is based on the apparent movement of celestial bodies as viewed from Earth. Indians used observational astronomy, as observed from Earth, to track celestial bodies, including the Sun moon, stars, etc. The Sun is called “Divakara” meaning, ‘maker of day’. It also causes changes of climate. As viewed from the Earth the Sun orbits around the earth in circle through 12 celestial segments called Rasis, or months, and so completes one year. Indians assigned different names to the Sun whilst in different Rasis, commencing from “Medam” or Aries. Brihat Samhita explains these twelve names as Dhata, Mitra, Aryama, Rudra, Varuna, Surya, Bhaga, Vivaswan, Pusa, Savita, Twasta and Visnu. In the Bhuta Sankhya system also the Sun means “twelve”. In each Rasi the Sun spends 30 days so that the full circle is completed in 360 days. Indians called it Savana Varsa, solistic motion of Sun. The Savana year is used particularly for sacrificial purpose, mainly for Soma Yajna.The period between two consecutive sunrises is known as a Savana day. During the Soma Yajna soma juice is offered three times during the period of one day and night. The Soma sacrifice which is completed in 24 hours is known as ‘Aha’. In the Vedas, according to the Bharatheya Jyotisha Sastra, a group of six such Aha is known as Shadaha. Five such Shadaha make one Masa or month. We can relate this division of the Sun's orbit to 12 rasis to 360 degrees. Our earlier calculation of a year was 360 days but this was later corrected based on further observations. Indian astronomers correlated measurement of time and space by the same designations which were later accepted in other world cultures. One year is 12 months, having 30 days in a month each. Nadi is one day and 60 Vinadikas is 1 Nadi. Aryabhatiyam: 3.1,2 2.1 Similarities of division of time and space 1 Year = 12 Months 1 Rotation = 12 Rasis 1 Month = 30 Days 1 Rasi = 30 Bhagas/Degrees 1 Day = 60 Nadikas 1 Degree=60 Kala/ Minutes 1 Nadika = 620 Vinadikas 1 Minute = 60 Vikala/ Seconds (1 Vikala = 60 Tatpara) 1 day was divided into 24 Horas. The Hora is derived from Ahorathram. 2.2 The concept of 7 days in a week and the logic of its naming Ancient astronomers considered the seven planets as masters of Hora. Saturn, Jupiter, Mars, Sun, Venus, Mercury, and the Moon, which are in order of distance from the Earth, are assigned lordship of one hora each in a block of Seven horas. The cycle is repeated at regular intervals of 7 horas each. In a day of 24 horas, the above cycle is repeated three times and the next day commences with fourth Horadhipa in the above order. This is repeated indefinitely. Aryabhateeyam 3.16 explains this. The days are named after the first Horadhipa after sunrise (6 to 7am) and will be the Dinadhipas (master of that day). On Saturday the first hora after sunrise will be Saturn, so it is known as Sanivara/Saturday. The fourth in the above order is Sun which will be the master of the first hora after sunrise the next day. Hence the day after Saturday is Ravi Vara, Sunday, as it is the Dinadhipa - master of the day. 2.3 An Interesting correlation Pranenithy kala bhu Aryabhatiyam 1.6 With the time of one respiration, the earth moves through 1 minute (angular) Or the earth rotates through 1 Kala/minute (angle) in 4 seconds. Time for 1 degree rotation of earth = 4 x 60 sec = 4 min Time for 360 rotation = 360 x 4 min = 24 hrs/1 day 2.4 Indian Calendar Based on the astronomical statements contained in the Vedic texts, the scholar Dr Jacobi states, “The age of Rig veda as estimated to be BCE 4500”. Vedanga Jyotisha is the oldest astronomical text in Sanskrit and reckoned to date from 1400 BCE. It forms the basis of calendars used by Indians. From the Vedic period itself Indians were using a 12 month calendar. They specifically divided the year into 12 months of 30 days and 24 hours in each day. Indian Lunar months are named after the fixed Nakshatras at or near which the moon is full during that month. For example, the month is named Chaitra or Karthika according as the full moon in the month is found to be near the star Chitra or Karthika respectively. The interval between two full moons is 29 days,12 hrs, 44 min, 2,87 sec (synodical lunar month). The other system was the Siderial lunar month, based on the time taken for the moon to arrive at a particular star again, which is 27 days, 7 hrs 43 min and 11.5 sec. There is a difference of 2 days 5hrs 0min and 51.37 sec. 2.5 Inspiration of Indian systems in Modern Gregorian Calendar When Pope Gregory XIII decided to revise Julian calendar he appointed a committee. The German Jesuit astronomer and mathematician Christopher Clavius (1538- 1555) was there in the committee. Clavius is the first person in Europe who introduced angular measurement tables in decimal notation. Clavius got his inspiration from works of the Portuguese mathematician Pedro Nunes (1502- 1578). He found mathematical solutions for ship navigation, etc. We should note that during these periods, according to historical evidence, the presence of Portuguese sailors, as well as Jesuit priests who were mathematicians, in Kerala, South West India, were well established. So the Kerala system was possibly a leading light in modern Calendar. 3 Conclusion Indians were using a systematic Kala Chakra, division of time in a samvalsara, to measure time. They were used as reference to measure duration, short, like the time for a breath, to long, such as a yuga or maha yuga. We can clearly find out the systematic division of time only in Indian Texts like Vedas and Astronomical texts like Vedanga Jyotisha. References Rig veda samhitha,Bharatheeya Vidya bhavan Aryabhateeyam Dr VB Panicker,Swadeshi Science Movement Astronomy in India Astronomy before telescope, C Walker, 1996,p122-123 Vedic Chronology and Vedanga Jyotisha, Tilak,Bal Gangadhar Lokamanya, Tilak Bros Bharatheeya Jyotisha Sastra –History of Indian Astronomy, Dikshit, Sankar Balakrishna,GOI Press. A History of Astronomy, A Pannekoek, Dover Publications Indian Astronomy Concepts and Procedures, Dr.S Balachandra Rao, MP Birla Inst.Bengaluru |
11795 | https://math-gpt.org/problems/question/a-sinusoidal-function-of-the-form-y-a-b-x-c-d-has-a-maximum-value-at | A sinusoidal function of the form y=a (b(x-c))+d has a max | MathGPT
A sinusoidal function of the form y=a (b(x-c))+d has a maximum value at (3 18, 3) and a subsequent minimum value at ...
Question:
A sinusoidal function of the form y=a (b(x-c))+d has a maximum value at (3 18, 3) and a subsequent minimum value at (9 18,-1) . The values of a, b, c , and d , respectively, are1. The amplitude a is the distance from the midline to the maximum or minimum value. Given the maximum value is 3 and the minimum value is -1, the midline d can be calculated as follows: d = 3 + (-1)2 = 1. Therefore, the amplitude a is: a = 3 - d = 3 - 1 = 2. 2. The period T of the sine function can be determined from the distance between a maximum and the next minimum. Here, the distance between 318 and 918 is: 918 - 318 = 618 = 3. Since this distance represents half the period, the full period T is: T = 2 3 = 23. 3. The value of b is related to the period by the formula: T = 2b b = 2T = 223 = 3. 4. The phase shift c is determined using the x-coordinate of the maximum value. The maximum occurs when the sine function equals 1, thus: b(x - c) = 2 3(318 - c) = 2. Solving for c : 318 - c = 6 c = 318 - 6 = 318 - 318 = 0. Thus, the values are:a = 2, b = 3, c = 0, d = 1. The final answer is (2, 3, 0, 1) .
Answer:
Given that the maximum value is at and the subsequent minimum value is at , we can deduce several things about the sinusoidal function .
Amplitude (): The maximum and minimum values of a sinusoidal function are determined by its amplitude and ve...
Unlock with MathGPT
rtical shift. The distance between the maximum and minimum values is . Since the amplitude is half this distance, .
Vertical Shift (): The vertical shift is the average of the maximum and minimum values, so .
Period: The distance (in ) between a maximum and the next minimum is half the period of the sinusoidal function. Since this distance is , the full period is . The period of a sinusoid is given by , so . Solving for gives .
Phase Shift (): The maximum of occurs at for the first time. Since the first maximum occurs at , .
Therefore, the values of and are , respectively.
MathGPTNext Problem
; |
11796 | https://www.youtube.com/watch?v=eb3PwvsSjho | Graph the Fraction 3/4 on a Number Line
MagnetsAndMotors (Dr. B's Other Channel)
84500 subscribers
86 likes
Description
33722 views
Posted: 1 Apr 2023
In order to represent 3/4 (three-fourths) on a number line we first need to draw our numberline.
Since 3/4 is a positive number (it doesn’t have a negative sign in front of it) it will be to the right of the “0” on our number line. 3/4 is greater than 0 but less than 1. So we’ll place it between 0 and 1 on the number line.
A number line is a visual representation of numbers in order from smallest to largest, usually arranged horizontally. It is a way of representing numbers in a linear format to help understand their relationship to each other.
On a number line, whole numbers are evenly spaced out and are represented by tick marks or points. The numbers to the right of zero are positive, and the numbers to the left of zero are negative. Zero is the midpoint of the number line, and it is neither positive nor negative.
Using a number line, we can perform basic math operations such as addition, subtraction, multiplication, and division. We can also compare and order numbers, find the distance between numbers, and identify patterns in numbers.
17 comments
Transcript:
let's find 3/4s on a number line so here's how we do it the red these numbers are negative on this side here positive over here with zero in the center we want to find 3/4s so three is smaller than four this number is going to be less than one that means it's going to be right here on our number line so let's zoom in on that so we're talking about 3/4s we know it's less than one here so let's divide this area right here into four equal pieces so we had 3/4s 1/4 24/4s 34s so right here this point right here that is 3/4s on our number line and 3 / 4 that's 0.75 so you can call that 0.75 as a decimal it's the same thing so that's it that's how you find 3/4s on a number line this is Dr be thanks for watching |
11797 | https://www.mathsisfun.com/definitions/radical.html | Radical Definition (Illustrated Mathematics Dictionary)
Show Ads
Hide Ads | About Ads
We may use Cookies
OK
Home
Algebra
Data
Geometry
Physics
Dictionary
Games
Puzzles
[x]
Algebra
Calculus
Data
Geometry
Money
Numbers
Physics
Activities
Dictionary
Games
Puzzles
Worksheets
Hide Ads
Show Ads
About Ads
Donate
Login
Close
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Definition of
Radical
A square root, cube root, etc.
The symbol is √
See: Square Root, Cube Root, Exponent
Donate ○ Search ○ Index ○ About ○ Contact ○ Cite This Page ○ Privacy
Copyright © 2025 Rod Pierce |
11798 | https://math.stackexchange.com/questions/3738109/sum-and-product-of-roots-of-a-polynomial | Skip to main content
Sum and product of roots of a polynomial
Ask Question
Asked
Modified
5 years, 2 months ago
Viewed
10k times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
Let's say we had a nth degree polynomial equation anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0=0, with a being real coefficient. What would the sum and product of its roots be(in terms of a)? I think I got the product one but not the sum.
For the product:
Let's say the polynomial's roots are r1,r2,r3,…,rn.
Then polynomial can be factored as:
an(x−r1an)(x−r2)(x−r3)…(x−rn)
We can set this equal to original polynomial:
an(x−r1an)(x−r2)(x−r3)…(x−rn)=anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0=0
Compare constant terms:
anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0 constant term=a0.
an(x−r1an)(x−r2)(x−r3)…(x−rn) constant term=(−1)n∗(r1an)∗r2∗r3∗⋯∗rn
a0=(−1)n∗(r1an)∗r2∗r3∗⋯∗rn
Multiply (−1)nan both sides:
r1∗r2∗r3∗⋯rn=(−1)na0an
Is this correct? Also, what can I do for the sum of the roots(I think we use the coefficients of xn−1)?
EDIT: J.W. Tanner has noted in his comment that this is Vieta's Formulas which is exactly what I was looking for but couldn't find.
algebra-precalculus
polynomials
roots
Share
CC BY-SA 4.0
Follow this question to receive notifications
edited Jun 28, 2020 at 20:19
Aiden ChowAiden Chow
asked Jun 28, 2020 at 20:03
Aiden ChowAiden Chow
2,8581212 silver badges3434 bronze badges
3
2
an(x−r1an) should be an(x−r1)
Alexey Burdin
–
Alexey Burdin
2020-06-28 20:06:05 +00:00
Commented
Jun 28, 2020 at 20:06
2
Cf. Vieta's formulas
J. W. Tanner
–
J. W. Tanner
2020-06-28 20:13:13 +00:00
Commented
Jun 28, 2020 at 20:13
@J.W.Tanner Oh thanks a lot! This is exactly what I am looking for.
Aiden Chow
–
Aiden Chow
2020-06-28 20:15:13 +00:00
Commented
Jun 28, 2020 at 20:15
Add a comment
|
1 Answer 1
Reset to default
This answer is useful
1
Save this answer.
Show activity on this post.
an is not a root, it is the leading coefficient.
Imagine the polynomial (x−2)(x−3)=x2−5x+6.
The leading coefficient is 1 but the roots are 2 and 3.
This part of your argument is incorrect.
Then polynomial can be factored as:
an(x−r1an)(x−r2)(x−r3)…(x−rn)
It should read this.
Then polynomial can be factored as:
an(x−r1)(x−r2)(x−r3)…(x−rn)
From this representation it gets obvious that the free term is
an(−1)nr1r2...rn
But on the other hand we know it is a0.
So the product of the roots must be:
a0(−1)nan=(−1)n⋅a0an
For calculating the sum of the roots just compare the coefficient before xn−1.
You will get that the sum of the roots equals
−an−1an
See also: Vieta's formulas
Share
CC BY-SA 4.0
Follow this answer to receive notifications
edited Jun 28, 2020 at 20:19
answered Jun 28, 2020 at 20:11
peter.petrovpeter.petrov
13k22 gold badges2323 silver badges4040 bronze badges
0
Add a comment
|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
algebra-precalculus
polynomials
roots
See similar questions with these tags.
Featured on Meta
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Related
10
Vieta's theorem
1
I have a question about Viete's formulas
5
On the sum of the reciprocals of the roots of a Polynomial
5
For a polynomial with integer coefficients, is it true that if constant term is prime then it cannot be the root of the polynomial.
8
Finding the sum of squares of the real roots
1
Powers of roots in terms of polynomial coefficients
4
Weird Problem on Polynomial Roots
Hot Network Questions
Regression model with a band of zeros
problems with siunitx and percent unit column
Conjunctive adverbs vs. transitional devices
Could there be a set of vectors that is linearly independent in the usual sense, but becomes linearly dependent if we allow infinite sums?
The new boss is insulting my appearance
How to auto adjust chart height to ensure minipages are the same height?
Reducing an integral to a Gamma function type
Is it OK to pray at breakneck speed?
A million and one constants
Implications of OP_RETURN changes in upcoming Bitcoin Core version 30.0
In the novel Dune, why were atomics banned but shield + lasgun explosions weren't?
Is it normal for postdoc to write grant proposals for PI, but then is denied the opportunity to work on the projects later on?
RSA variant with cyclotomic polynomial
How to Calculate Habitable Area of an Infinite Plane
Cryptic Family Reunion: It's Easy to Overlook Us
Do you need to do correction on multi-factor ANOVAs?
Help with piano fingering from B4 to F4. Left hand
How to cut a tetrahedron in half using only a pencil and a knife?
Examples of mathematically interesting numbers that can be expressed as an integer tetration (besides Graham’s number)
Regarding the moral impulses of average people
mate-panel not visible after ugprade, but process is started
What historical data prove the reliability of the Tripitaka?
Thread @ Join unexpected behavior
Ubuntu 25.04 sound not working
Question feed
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. |
11799 | https://physics.stackexchange.com/questions/786369/photocurrent-in-photoelectric-effect-and-its-misconceptions | electric current - Photocurrent in Photoelectric effect and its misconceptions - Physics Stack Exchange
Join Physics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Physics helpchat
Physics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Photocurrent in Photoelectric effect and its misconceptions
Ask Question
Asked 1 year, 11 months ago
Modified1 year, 11 months ago
Viewed 249 times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
This question has been asked many times on this site before with different doubts in each question but I am conflicted with the vast amount of sources stating contradictory things. There are many questions on this site that state "saturation current does not increase with frequency but if kinetic energy increases then velocity of electrons increases and so current should increase". All answers to these questions explain why current cannot increase with speed of electron but do not correct the statement "saturation current does not increase with frequency". Does that mean it is true?
In a lecture I saw on the MIT open courseware site, the professor said current is directly proportional to intensity, but did not mention anything about frequency. In other papers/lectures of different universities, I saw only the current dependency with intensity and not frequency anywhere (an MIT "photoelectric effect notes" pdf said the number of photoelectrons ejected is strictly dependent only on intensity).
It does seem somewhat contradictory when you consider that photoelectrons ejected per second are proportional to photons hitting the surface per second and intensity of light = number of photons in light per second energy of light (proportional to frequency). So if the frequency increases and intensity stays the same, photons in light should decrease and thus photoelectrons ejected should decrease and so saturation photocurrent should decrease? But then some sources state that increasing frequency increases probability to eject electrons and so both the factors (decrease in number of ejected electrons due to decrease in number of photons and increase in number of ejected electrons due to increase in frequency) cancel out and saturation current stays the same?
Let us consider we somehow got rid of all practical difficulties and consider an ideal situation (like energy loss and such). Then does changing frequency (but keeping intensity constant) change the photo/saturation current?
electric-current
wave-particle-duality
photoelectric-effect
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Improve this question
Follow
Follow this question to receive notifications
edited Oct 30, 2023 at 11:01
Ham LemonHam Lemon
asked Oct 30, 2023 at 5:11
Ham LemonHam Lemon
201 2 2 silver badges 11 11 bronze badges
2
Does this previous Q&A ( physics.stackexchange.com/questions/401526/… ) answer your question?GiorgioP-DoomsdayClockIsAt-89 –GiorgioP-DoomsdayClockIsAt-89 2023-10-30 05:28:12 +00:00 Commented Oct 30, 2023 at 5:28
I don't think I am able to draw a conclusion for the question "is photocurrent dependent on frequency" from the answer (which does not clearly state a conclusion and rather just explains the process of photoemission by using some terms of quantum mechanics(which I am not well versed in))Ham Lemon –Ham Lemon 2023-10-30 10:57:47 +00:00 Commented Oct 30, 2023 at 10:57
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
1
Save this answer.
Show activity on this post.
Your conclusion is correct. In an ideal situation, if we increase the frequency at constant intensity past the threshold where all photons extract electrons, the number of emitted electrons and, consequently, the current may decrease. The reason is that the intensity is proportional to the number of photons times their frequency. By increasing the frequency while holding the intensity constant, the number of photons must decrease.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Nov 1, 2023 at 23:33
GiorgioP-DoomsdayClockIsAt-89GiorgioP-DoomsdayClockIsAt-89
40.2k 9 9 gold badges 54 54 silver badges 120 120 bronze badges
5
If we now include the "probability of ejecting electrons increases due to increase in frequency", then can these both effects (increase in photoelectrons due to frequency increase and decreases in photoelectrons due to decrease in photons) cancel each other out? I am asking this because this also felt like a convincing explanation given hereHam Lemon –Ham Lemon 2023-11-02 06:01:44 +00:00 Commented Nov 2, 2023 at 6:01
@HamLemon There may be compensation only below the threshold where all photons extract electrons, Not above that threshold. At that point, there is a one-to-one correspondence between the number of photons and the number of extracted electrons.GiorgioP-DoomsdayClockIsAt-89 –GiorgioP-DoomsdayClockIsAt-89 2023-11-02 06:49:23 +00:00 Commented Nov 2, 2023 at 6:49
Oh ok, I get it now, so photocurrent may be compensated and may stay constant for varying frequency and constant intensity but saturation current will vary. Is my conclusion correct?Ham Lemon –Ham Lemon 2023-11-02 06:51:30 +00:00 Commented Nov 2, 2023 at 6:51
@HamLemon Your conclusion is correct.GiorgioP-DoomsdayClockIsAt-89 –GiorgioP-DoomsdayClockIsAt-89 2023-11-02 06:54:05 +00:00 Commented Nov 2, 2023 at 6:54
Ok, now I clearly understand the concept. Thank you for bearing with me and patiently explaining things to me🙏🏼Ham Lemon –Ham Lemon 2023-11-02 07:00:58 +00:00 Commented Nov 2, 2023 at 7:00
Add a comment|
Your Answer
Thanks for contributing an answer to Physics Stack Exchange!
Please be sure to answer the question. Provide details and share your research!
But avoid …
Asking for help, clarification, or responding to other answers.
Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Email and Password
Submit
Post as a guest
Name
Email
Required, but never shown
Post Your Answer Discard
By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
electric-current
wave-particle-duality
photoelectric-effect
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Linked
2Effect of collector plate potential on photoelectric current with varying frequencies?
1Effect Of Frequency on Photocurrent
Related
6Photoelectric effect: Why is the saturation current the same, as you vary the frequency of incident light but keep its intensity constant?
3In photoelectric effect experiment, why will the number of photoelectrons not change on increasing frequency?
1Effect Of Frequency on Photocurrent
2Relationship Between Saturation Current and Light Intensity According to Wave Theory of Light
2Effect of collector plate potential on photoelectric current with varying frequencies?
0Why photocurrent doesnt increase after saturation limit in Photoelectric effect on increasing Voltage
2Does photocurrent depend on intensity or on number of photons?
Hot Network Questions
My dissertation is wrong, but I already defended. How to remedy?
How to home-make rubber feet stoppers for table legs?
What can be said?
How many stars is possible to obtain in your savefile?
Overfilled my oil
Is there a specific term to describe someone who is religious but does not necessarily believe everything that their religion teaches, and uses logic?
What "real mistakes" exist in the Messier catalog?
Another way to draw RegionDifference of a cylinder and Cuboid
Do we declare the codomain of a function from the beginning, or do we determine it after defining the domain and operations?
I'm having a hard time intuiting throttle position to engine rpm consistency between gears -- why do cars behave in this observed way?
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Analog story - nuclear bombs used to neutralize global warming
Can a cleric gain the intended benefit from the Extra Spell feat?
Cannot build the font table of Miama via nfssfont.tex
What meal can come next?
The rule of necessitation seems utterly unreasonable
Why do universities push for high impact journal publications?
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
What were "milk bars" in 1920s Japan?
Does a Linux console change color when it crashes?
Exchange a file in a zip file quickly
Gluteus medius inactivity while riding
Is it ok to place components "inside" the PCB
What is the meaning of 率 in this report?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Physics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.