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Hook point types
Hook points are also described relative to their offset from the hook shank. A kerbed hook point is offset to the left, a straight point has no offset and a reversed point is offset to the right.
Hook points are commonly referred to by these names: needle point, rolled-in, hollow, spear, beak, mini-barb, semi-dropped and knife edge. Some other hook point names are used for branding by manufacturers.
Eyes
The eye of the hook is the widened ring/loop at its proximal end, with a hole where the fishing line (typically the leader line) is passed through (threaded) for fastening via knot-tying. Hook eye design is usually optimized for either strength, weight and/or presentation. Typical eye types include:
Ringed eye or ball eye — a circular loop often with a closely opposed gap between the loop end and the loop base;
Brazed eye — like a ringed eye, but the loop end is welded shut fully without any gap;
Tapered eye — like a ringed eye, but with a pointed loop end;
Looped eye — the loop end is elongated with the extended portion laid parallel to the hook shank;
Needle eye — the eye hole is ellipsical, or just a narrow slit.
Most hook eyes are directly knotted to the fishing line and are responsible for relaying the pulling force from the line onto the hook body, but sometimes the line is passed cleanly through the eye and tied directly onto the shank instead of onto the eye loop — this is known as a snell knot or "snelling", and the eye does not take part in transferring any force, merely serving to restrict line wobbling and knot sliding. In fishing lures, it is also not uncommon to see the hook being linked to the lure via a split ring through the eye, which allows the hook more range of motion.
Hook eyes can also be categorized into three types according to the angulation of the loop plane against the shank, where hooks with bent/"turned" eyes being more optimized for snelling:
Straight — the eye is in-line with the shank;
Up-turned — the eye is angled away from the hook point;
Down-turned — the eye is angled towards the hook point. | Fish hook | Wikipedia | 473 | 989954 | https://en.wikipedia.org/wiki/Fish%20hook | Technology | Hunting and fishing | null |
Some hooks, such as the traditional Japanese Tenkara hooks, lack any opening for the line to be threaded, and are thus eyeless. Eyeless hooks instead have a widened "spade end" to help snelling the line onto the shank without slipping.
Size
There are no internationally recognized standards for hooks and thus size is somewhat inconsistent between manufacturers. However, within a manufacturer's range of hooks, hook sizes are consistent.
Hook sizes generally are referred to by a numbering system that places the size 1 hook in the middle of the size range. Smaller hooks are referenced by larger whole numbers (e.g. 1, 2, 3...). Larger hooks are referenced for size increases by increasing whole numbers followed by a "/" and a "0" (i.e. sizes over zero), for example, 1/0 (read as "one nought"), 2/0, 3/0.... The numbers represent relative sizes, normally associated with the gap (the distance from the point tip to the shank). The smallest size available is 32 and largest is 20/0. | Fish hook | Wikipedia | 230 | 989954 | https://en.wikipedia.org/wiki/Fish%20hook | Technology | Hunting and fishing | null |
Mental health encompasses emotional, psychological, and social well-being, influencing cognition, perception, and behavior. According to the World Health Organization (WHO), it is a "state of well-being in which the individual realizes his or her abilities, can cope with the normal stresses of life, can work productively and fruitfully, and can contribute to his or her community". It likewise determines how an individual handles stress, interpersonal relationships, and decision-making. Mental health includes subjective well-being, perceived self-efficacy, autonomy, competence, intergenerational dependence, and self-actualization of one's intellectual and emotional potential, among others.
From the perspectives of positive psychology or holism, mental health may include an individual's ability to enjoy life and to create a balance between life activities and efforts to achieve psychological resilience. Cultural differences, personal philosophy, subjective assessments, and competing professional theories all affect how one defines "mental health". Some early signs related to mental health difficulties are sleep irritation, lack of energy, lack of appetite, thinking of harming oneself or others, self-isolating (though introversion and isolation aren't necessarily unhealthy), and frequently zoning out.
Mental disorders
Mental health, as defined by the Public Health Agency of Canada, is an individual's capacity to feel, think, and act in ways to achieve a better quality of life while respecting personal, social, and cultural boundaries. Impairment of any of these are risk factor for mental disorders, or mental illnesses, which are a component of mental health. In 2019, about 970 million people worldwide suffered from a mental disorder, with anxiety and depression being the most common. The number of people suffering from mental disorders has risen significantly throughout the years. Mental disorders are defined as health conditions that affect and alter cognitive functioning, emotional responses, and behavior associated with distress and/or impaired functioning. The ICD-11 is the global standard used to diagnose, treat, research, and report various mental disorders. In the United States, the DSM-5 is used as the classification system of mental disorders.
Mental health is associated with a number of lifestyle factors such as diet, exercise, stress, drug abuse, social connections and interactions. Psychiatrists, psychologists, licensed professional clinical counselors, social workers, nurse practitioners, and family physicians can help manage mental illness with treatments such as therapy, counseling, and medication.
History
Early history | Mental health | Wikipedia | 498 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
In the mid-19th century, William Sweetser was the first to coin the term mental hygiene, which can be seen as the precursor to contemporary approaches to work on promoting positive mental health. Isaac Ray, the fourth president of the American Psychiatric Association and one of its founders, further defined mental hygiene as "the art of preserving the mind against all incidents and influences calculated to deteriorate its qualities, impair its energies, or derange its movements".
In American history, mentally ill patients were thought to be religiously punished. This response persisted through the 1700s, along with the inhumane confinement and stigmatization of such individuals. Dorothea Dix (1802–1887) was an important figure in the development of the "mental hygiene" movement. Dix was a school teacher who endeavored to help people with mental disorders and to expose the sub-standard conditions into which they were put. This became known as the "mental hygiene movement". Before this movement, it was not uncommon that people affected by mental illness would be considerably neglected, often left alone in deplorable conditions without sufficient clothing. From 1840 to 1880, she won the support of the federal government to set up over 30 state psychiatric hospitals; however, they were understaffed, under-resourced, and were accused of violating human rights.
Emil Kraepelin in 1896 developed the taxonomy of mental disorders which has dominated the field for nearly 80 years. Later, the proposed disease model of abnormality was subjected to analysis and considered normality to be relative to the physical, geographical and cultural aspects of the defining group.
At the beginning of the 20th century, Clifford Beers founded "Mental Health America – National Committee for Mental Hygiene", after publication of his accounts as a patient in several lunatic asylums, A Mind That Found Itself, in 1908 and opened the first outpatient mental health clinic in the United States.
The mental hygiene movement, similar to the social hygiene movement, had at times been associated with advocating eugenics and sterilization of those considered too mentally deficient to be assisted into productive work and contented family life. In the post-WWII years, references to mental hygiene were gradually replaced by the term 'mental health' due to its positive aspect that evolves from the treatment of illness to preventive and promotive areas of healthcare. | Mental health | Wikipedia | 470 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Deinstitutionalization and transinstitutionalization
When US government-run hospitals were accused of violating human rights, advocates pushed for deinstitutionalization: the replacement of federal mental hospitals for community mental health services. The closure of state-provisioned psychiatric hospitals was enforced by the Community Mental Health Centers Act in 1963 that laid out terms in which only patients who posed an imminent danger to others or themselves could be admitted into state facilities. This was seen as an improvement from previous conditions. However, there remains a debate on the conditions of these community resources.
It has been proven that this transition was beneficial for many patients: there was an increase in overall satisfaction, a better quality of life, and more friendships between patients all at an affordable cost. This proved to be true only in the circumstance that treatment facilities had enough funding for staff and equipment as well as proper management. However, this idea is a polarizing issue. Critics of deinstitutionalization argue that poor living conditions prevailed, patients were lonely, and they did not acquire proper medical care in these treatment homes. Additionally, patients that were moved from state psychiatric care to nursing and residential homes had deficits in crucial aspects of their treatment. Some cases result in the shift of care from health workers to patients' families, where they do not have the proper funding or medical expertise to give proper care. On the other hand, patients that are treated in community mental health centers lack sufficient cancer testing, vaccinations, or otherwise regular medical check-ups. | Mental health | Wikipedia | 311 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Other critics of state deinstitutionalization argue that this was simply a transition to "transinstitutionalization", or the idea that prisons and state-provisioned hospitals are interdependent. In other words, patients become inmates. This draws on the Penrose Hypothesis of 1939, which theorized that there was an inverse relationship between prisons' population size and the number of psychiatric hospital beds. This means that populations that require psychiatric mental care will transition between institutions, which in this case, includes state psychiatric hospitals and criminal justice systems. Thus, a decrease in available psychiatric hospital beds occurred at the same time as an increase in inmates. Although some are skeptical that this is due to other external factors, others will reason this conclusion to a lack of empathy for the mentally ill. There is no argument for the social stigmatization of those with mental illnesses, they have been widely marginalized and discriminated against in society. In this source, researchers analyze how most compensation prisoners (detainees who are unable or unwilling to pay a fine for petty crimes) are unemployed, homeless, and with an extraordinarily high degree of mental illnesses and substance use disorders. Compensation prisoners then lose prospective job opportunities, face social marginalization, and lack access to resocialization programs, which ultimately facilitate reoffending. The research sheds light on how the mentally ill—and in this case, the poor—are further punished for certain circumstances that are beyond their control, and that this is a vicious cycle that repeats itself. Thus, prisons embody another state-provisioned mental hospital.
Families of patients, advocates, and mental health professionals still call for increase in more well-structured community facilities and treatment programs with a higher quality of long-term inpatient resources and care. With this more structured environment, the United States will continue with more access to mental health care and an increase in the overall treatment of the mentally ill.
However, there is still a lack of studies for mental health conditions (MHCs) to raise awareness, knowledge development, and attitudes toward seeking medical treatment for MHCs in Bangladesh. People in rural areas often seek treatment from the traditional healers and MHCs are sometimes considered a spiritual matter.
Epidemiology | Mental health | Wikipedia | 455 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Mental illnesses are more common than cancer, diabetes, or heart disease. As of 2021, over 22 percent of all Americans over the age of 18 meet the criteria for having a mental illness. Evidence suggests that 970 million people worldwide have a mental disorder. Major depression ranks third among the top 10 leading causes of disease worldwide. By 2030, it is predicted to become the leading cause of disease worldwide. Over 700 thousand people commit suicide every year and around 14 million attempt it. A World Health Organization (WHO) report estimates the global cost of mental illness at nearly $2.5 trillion (two-thirds in indirect costs) in 2010, with a projected increase to over $6 trillion by 2030.
Evidence from the WHO suggests that nearly half of the world's population is affected by mental illness with an impact on their self-esteem, relationships and ability to function in everyday life. An individual's emotional health can impact their physical health. Poor mental health can lead to problems such as the inability to make adequate decisions and substance use disorders.
Good mental health can improve life quality whereas poor mental health can worsen it. According to Richards, Campania, & Muse-Burke, "There is growing evidence that is showing emotional abilities are associated with pro-social behaviors such as stress management and physical health." Their research also concluded that people who lack emotional expression are inclined to anti-social behaviors (e.g., substance use disorder and alcohol use disorder, physical fights, vandalism), which reflects one's mental health and suppressed emotions. Adults and children who face mental illness may experience social stigma, which can exacerbate the issues.
Global prevalence
Mental health can be seen as a continuum, where an individual's mental health may have many different possible values. Mental wellness is viewed as a positive attribute; this definition of mental health highlights emotional well-being, the capacity to live a full and creative life, and the flexibility to deal with life's inevitable challenges. Some discussions are formulated in terms of contentment or happiness. Many therapeutic systems and self-help books offer methods and philosophies espousing strategies and techniques vaunted as effective for further improving the mental wellness. Positive psychology is increasingly prominent in mental health.
A holistic model of mental health generally includes concepts based upon anthropological, educational, psychological, religious, and sociological perspectives. There are also models as theoretical perspectives from personality, social, clinical, health and developmental psychology. | Mental health | Wikipedia | 500 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
The tripartite model of mental well-being views mental well-being as encompassing three components of emotional well-being, social well-being, and psychological well-being. Emotional well-being is defined as having high levels of positive emotions, whereas social and psychological well-being are defined as the presence of psychological and social skills and abilities that contribute to optimal functioning in daily life. The model has received empirical support across cultures. The Mental Health Continuum-Short Form (MHC-SF) is the most widely used scale to measure the tripartite model of mental well-being.
Demographics
Children and young adults
As of 2019, about one in seven of the world's 10–19 year olds experienced a mental health disorder; about 165 million young people in total. A person's teenage years are a unique period where much crucial psychological development occurs, and is also a time of increased vulnerability to the development of adverse mental health conditions. More than half of mental health conditions start before a child reaches 20 years of age, with onset occurring in adolescence much more frequently than it does in early childhood or adulthood. Many such cases go undetected and untreated. | Mental health | Wikipedia | 236 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
In the United States alone, in 2021, at least roughly 17.5% of the population (ages 18 and older) were recorded as having a mental illness. The comparison between reports and statistics of mental health issues in newer generations (18–25 years old to 26–49 years old) and the older generation (50 years or older) signifies an increase in mental health issues as only 15% of the older generation reported a mental health issue whereas the newer generations reported 33.7% (18-25) and 28.1% (26-49). The role of caregivers for youth with mental health needs is valuable, and caregivers benefit most when they have sufficient psychoeducation and peer support. Depression is one of the leading causes of illness and disability among adolescents. Suicide is the fourth leading cause of death in 15-19-year-olds. Exposure to childhood trauma can cause mental health disorders and poor academic achievement. Ignoring mental health conditions in adolescents can impact adulthood. 50% of preschool children show a natural reduction in behavioral problems. The remaining experience long-term consequences. It impairs physical and mental health and limits opportunities to live fulfilling lives. A result of depression during adolescence and adulthood may be substance abuse. The average age of onset is between 11 and 14 years for depressive disorders. Only approximately 25% of children with behavioral problems refer to medical services. The majority of children go untreated.
Homeless population
Mental illness is thought to be highly prevalent among homeless populations, though access to proper diagnoses is limited. An article written by Lisa Goodman and her colleagues summarized Smith's research into PTSD in homeless single women and mothers in St. Louis, Missouri, which found that 53% of the respondents met diagnostic criteria, and which describes homelessness as a risk factor for mental illness. At least two commonly reported symptoms of psychological trauma, social disaffiliation and learned helplessness are highly prevalent among homeless individuals and families.
While mental illness is prevalent, people infrequently receive appropriate care. Case management linked to other services is an effective care approach for improving symptoms in people experiencing homelessness. Case management reduced admission to hospitals, and it reduced substance use by those with substance abuse problems more than typical care.
Immigrants and refugees
States that produce refugees are sites of social upheaval, civil war, even genocide. Most refugees experience trauma. It can be in the form of torture, sexual assault, family fragmentation, and death of loved ones. | Mental health | Wikipedia | 505 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Refugees and immigrants experience psychosocial stressors after resettlement. These include discrimination, lack of economic stability, and social isolation causing emotional distress. For example, Not far into the 1900s, campaigns targeting Japanese immigrants were being formed that inhibited their ability to participate in U.S life, painting them as a threat to the American working-class. They were subject to prejudice and slandered by American media as well as anti-Japanese legislation being implemented. For refugees family reunification can be one of the primary needs to improve quality of life. Post-migration trauma is a cause of depressive disorders and psychological distress for immigrants.
Cultural and religious considerations
Mental health is a socially constructed concept; different societies, groups, cultures (both ethnic and national/regional), institutions, and professions have very different ways of conceptualizing its nature and causes, determining what is mentally healthy, and deciding what interventions, if any, are appropriate. Thus, different professionals will have different cultural, class, political and religious backgrounds, which will impact the methodology applied during treatment. In the context of deaf mental health care, it is necessary for professionals to have cultural competency of deaf and hard of hearing people and to understand how to properly rely on trained, qualified, and certified interpreters when working with culturally Deaf clients.
Research has shown that there is stigma attached to mental illness. Due to such stigma, individuals may resist labeling and may be driven to respond to mental health diagnoses with denialism. Family caregivers of individuals with mental disorders may also suffer discrimination or face stigma.
Addressing and eliminating the social stigma and perceived stigma attached to mental illness has been recognized as crucial to education and awareness surrounding mental health issues. In the United Kingdom, the Royal College of Psychiatrists organized the campaign Changing Minds (1998–2003) to help reduce stigma, while in the United States, efforts by entities such as the Born This Way Foundation and The Manic Monologues specifically focus on removing the stigma surrounding mental illness. The National Alliance on Mental Illness (NAMI) is a U.S. institution founded in 1979 to represent and advocate for those struggling with mental health issues. NAMI helps to educate about mental illnesses and health issues, while also working to eliminate stigma attached to these disorders. | Mental health | Wikipedia | 455 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Many mental health professionals are beginning to, or already understand, the importance of competency in religious diversity and spirituality, or the lack thereof. They are also partaking in cultural training to better understand which interventions work best for these different groups of people. The American Psychological Association explicitly states that religion must be respected. Education in spiritual and religious matters is also required by the American Psychiatric Association, however, far less attention is paid to the damage that more rigid, fundamentalist faiths commonly practiced in the United States can cause. This theme has been widely politicized in 2018 such as with the creation of the Religious Liberty Task Force in July of that year. Also, many providers and practitioners in the United States are only beginning to realize that the institution of mental healthcare lacks knowledge and competence of many non-Western cultures, leaving providers in the United States ill-equipped to treat patients from different cultures.
Occupations
Occupational therapy
Occupational therapy practitioners aim to improve and enable a client or group's participation in meaningful, everyday occupations. In this sense, occupation is defined as any activity that "occupies one's time". Examples of those activities include daily tasks (dressing, bathing, eating, house chores, driving, etc.), sleep and rest, education, work, play, leisure (hobbies), and social interactions. The OT profession offers a vast range of services for all stages of life in a myriad of practice settings, though the foundations of OT come from mental health. Community support for mental health through expert-moderated support groups can aid those who want to recover from mental illness or otherwise improve their emotional well-being.
OT services focused on mental health can be provided to persons, groups, and populations across the lifespan and experiencing varying levels of mental health performance. For example, occupational therapy practitioners provide mental health services in school systems, military environments, hospitals, outpatient clinics, and inpatient mental health rehabilitation settings. Interventions or support can be provided directly through specific treatment interventions or indirectly by providing consultation to businesses, schools, or other larger groups to incorporate mental health strategies on a programmatic level. Even people who are mentally healthy can benefit from the health promotion and additional prevention strategies to reduce the impact of difficult situations.
The interventions focus on positive functioning, sensory strategies, managing emotions, interpersonal relationships, sleep, community engagement, and other cognitive skills (i.e. visual-perceptual skills, attention, memory, arousal/energy management, etc.).
Mental health in social work | Mental health | Wikipedia | 503 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Social work in mental health, also called psychiatric social work, is a process where an individual in a setting is helped to attain freedom from overlapping internal and external problems (social and economic situations, family and other relationships, the physical and organizational environment, psychiatric symptoms, etc.). It aims for harmony, quality of life, self-actualization and personal adaptation across all systems. Psychiatric social workers are mental health professionals that can assist patients and their family members in coping with both mental health issues and various economic or social problems caused by mental illness or psychiatric dysfunctions and to attain improved mental health and well-being. They are vital members of the treatment teams in Departments of Psychiatry and Behavioral Sciences in hospitals. They are employed in both outpatient and inpatient settings of a hospital, nursing homes, state and local governments, substance use clinics, correctional facilities, health care services, private practice, etc.
In the United States, social workers provide most of the mental health services. According to government sources, 60 percent of mental health professionals are clinically trained social workers, 10 percent are psychiatrists, 23 percent are psychologists, and 5 percent are psychiatric nurses.
Mental health social workers in Japan have professional knowledge of health and welfare and skills essential for person's well-being. Their social work training enables them as a professional to carry out Consultation assistance for mental disabilities and their social reintegration; Consultation regarding the rehabilitation of the victims; Advice and guidance for post-discharge residence and re-employment after hospitalized care, for major life events in regular life, money and self-management and other relevant matters to equip them to adapt in daily life. Social workers provide individual home visits for mentally ill and do welfare services available, with specialized training a range of procedural services are coordinated for home, workplace and school. In an administrative relationship, Psychiatric social workers provides consultation, leadership, conflict management and work direction. Psychiatric social workers who provides assessment and psychosocial interventions function as a clinician, counselor and municipal staff of the health centers.
Risk factors and causes of mental health problems
There are many things that can contribute to mental health problems, including biological factors, genetic factors, life experiences (such as psychological trauma or abuse), and a family history of mental health problems. | Mental health | Wikipedia | 455 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Biological factors
According to the National Institute of Health Curriculum Supplement Series book, most scientists believe that changes in neurotransmitters can cause mental illnesses. In the section "The Biology of Mental Illnesses" the issue is explained in detail, "...there may be disruptions in the neurotransmitters dopamine, glutamate, and norepinephrine in individuals who have schizophrenia".
Demographic factors
Gender, age, ethnicity, life expectancy, longevity, population density, and community diversity are all demographic characteristics that can increase the risk and severity of mental disorders. Existing evidence demonstrates that the female gender is connected with an elevated risk of depression at different phases of life, commencing in adolescence in different contexts. Females, for example, have a higher risk of anxiety and eating disorders, whereas males have a higher chance of substance abuse and behavioral and developmental issues. This does not imply that women are less likely to suffer from developmental disorders such autism spectrum disorder, attention deficit hyperactivity disorder, Tourette syndrome, or early-onset schizophrenia. Ethnicity and ethnic heterogeneity have also been identified as risk factors for the prevalence of mental disorders, with minority groups being at a higher risk due to discrimination and exclusion. Approximately 8 in 10 people with autism suffer from a mental health problem in their life time, in comparison to 1 in 4 of the general population that suffers from a mental health problem in their lifetimes.
Unemployment has been shown to hurt an individual's emotional well-being, self-esteem, and more broadly their mental health. Increasing unemployment has been shown to have a significant impact on mental health, predominantly depressive disorders. This is an important consideration when reviewing the triggers for mental health disorders in any population survey. According to a 2009 meta-analysis by Paul and Moser, countries with high income inequality and poor unemployment protections experience worse mental health outcomes among the unemployed. | Mental health | Wikipedia | 386 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Emotional mental disorders are a leading cause of disabilities worldwide. Investigating the degree and severity of untreated emotional mental disorders throughout the world is a top priority of the World Mental Health (WMH) survey initiative, which was created in 1998 by the World Health Organization (WHO). "Neuropsychiatric disorders are the leading causes of disability worldwide, accounting for 37% of all healthy life years lost through disease. These disorders are most destructive to low and middle-income countries due to their inability to provide their citizens with proper aid. Despite modern treatment and rehabilitation for emotional mental health disorders, "even economically advantaged societies have competing priorities and budgetary constraints".
Unhappily married couples suffer 3–25 times the risk of developing clinical depression.
The World Mental Health survey initiative has suggested a plan for countries to redesign their mental health care systems to best allocate resources.
"A first step is documentation of services being used and the extent and nature of unmet treatment needs. A second step could be to do a cross-national comparison of service use and unmet needs in countries with different mental health care systems. Such comparisons can help to uncover optimum financing, national policies, and delivery systems for mental health care."
Knowledge of how to provide effective emotional mental health care has become imperative worldwide. Unfortunately, most countries have insufficient data to guide decisions, absent or competing visions for resources, and near-constant pressures to cut insurance and entitlements. WMH surveys were done in Africa (Nigeria, South Africa), the Americas (Colombia, Mexico, United States), Asia and the Pacific (Japan, New Zealand, Beijing and Shanghai in the People's Republic of China), Europe (Belgium, France, Germany, Italy, Netherlands, Spain, Ukraine), and the Middle East (Israel, Lebanon). Countries were classified with World Bank criteria as low-income (Nigeria), lower-middle-income (China, Colombia, South Africa, Ukraine), higher middle-income (Lebanon, Mexico), and high-income. | Mental health | Wikipedia | 419 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
The coordinated surveys on emotional mental health disorders, their severity, and treatments were implemented in the aforementioned countries. These surveys assessed the frequency, types, and adequacy of mental health service use in 17 countries in which WMH surveys are complete. The WMH also examined unmet needs for treatment in strata defined by the seriousness of mental disorders. Their research showed that "the number of respondents using any 12-month mental health service was generally lower in developing than in developed countries, and the proportion receiving services tended to correspond to countries' percentages of gross domestic product spent on health care".
"High levels of unmet need worldwide are not surprising, since WHO Project ATLAS' findings of much lower mental health expenditures than was suggested by the magnitude of burdens from mental illnesses. Generally, unmet needs in low-income and middle-income countries might be attributable to these nations spending reduced amounts (usually <1%) of already diminished health budgets on mental health care, and they rely heavily on out-of-pocket spending by citizens who are ill-equipped for it".
Stress
The Centre for Addiction and Mental Health discusses how a certain amount of stress is a normal part of daily life. Small doses of stress help people meet deadlines, be prepared for presentations, be productive and arrive on time for important events. However, long-term stress can become harmful. When stress becomes overwhelming and prolonged, the risks for mental health problems and medical problems increase." Also on that note, some studies have found language to deteriorate mental health and even harm humans.
The impact of a stressful environment has also been highlighted by different models. Mental health has often been understood from the lens of the vulnerability-stress model. In that context, stressful situations may contribute to a preexisting vulnerability to negative mental health outcomes being realized. On the other hand, the differential susceptibility hypothesis suggests that mental health outcomes are better explained by an increased sensitivity to the environment than by vulnerability. For example, it was found that children scoring higher on observer-rated environmental sensitivity often derive more harm from low-quality parenting, but also more benefits from high-quality parenting than those children scoring lower on that measure.
Poverty
Environmental factors
Prevention and promotion | Mental health | Wikipedia | 457 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
"The terms mental health promotion and prevention have often been confused. Promotion is defined as intervening to optimize positive mental health by addressing determinants of positive mental health (i.e. protective factors) before a specific mental health problem has been identified, with the ultimate goal of improving the positive mental health of the population. Mental health prevention is defined as intervening to minimize mental health problems (i.e. risk factors) by addressing determinants of mental health problems before a specific mental health problem has been identified in the individual, group, or population of focus with the ultimate goal of reducing the number of future mental health problems in the population."
In order to improve mental health, the root of the issue has to be resolved. "Prevention emphasizes the avoidance of risk factors; promotion aims to enhance an individual's ability to achieve a positive sense of self-esteem, mastery, well-being, and social inclusion." Mental health promotion attempts to increase protective factors and healthy behaviors that can help prevent the onset of a diagnosable mental disorder and reduce risk factors that can lead to the development of a mental disorder. Yoga is an example of an activity that calms one's entire body and nerves. According to a study on well-being by Richards, Campania, and Muse-Burke, "mindfulness is considered to be a purposeful state, it may be that those who practice it belief in its importance and value being mindful, so that valuing of self-care activities may influence the intentional component of mindfulness." Akin to surgery, sometimes the body must be further damaged, before it can properly heal
Mental health is conventionally defined as a hybrid of the absence of a mental disorder and the presence of well-being. Focus is increasing on preventing mental disorders.
Prevention is beginning to appear in mental health strategies, including the 2004 WHO report "Prevention of Mental Disorders", the 2008 EU "Pact for Mental Health" and the 2011 US National Prevention Strategy. Some commentators have argued that a pragmatic and practical approach to mental disorder prevention at work would be to treat it the same way as physical injury prevention.
Prevention of a disorder at a young age may significantly decrease the chances that a child will have a disorder later in life, and shall be the most efficient and effective measure from a public health perspective. Prevention may require the regular consultation of a physician for at least twice a year to detect any signs that reveal any mental health concerns. | Mental health | Wikipedia | 497 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
Additionally, social media is becoming a resource for prevention. In 2004, the Mental Health Services Act began to fund marketing initiatives to educate the public on mental health. This California-based project is working to combat the negative perception with mental health and reduce the stigma associated with it. While social media can benefit mental health, it can also lead to deterioration if not managed properly. Limiting social media intake is beneficial.
Studies report that patients in mental health care who can access and read their Electronic Health Records (EHR) or Open | Mental health | Wikipedia | 104 | 990505 | https://en.wikipedia.org/wiki/Mental%20health | Biology and health sciences | Health, fitness, and medicine | null |
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.
The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set.
Definition
Given a vector space over a subfield of the complex numbers a norm on is a real-valued function with the following properties, where denotes the usual absolute value of a scalar :
Subadditivity/Triangle inequality: for all
Absolute homogeneity: for all and all scalars
Positive definiteness/positiveness/: for all if then
Because property (2.) implies some authors replace property (3.) with the equivalent condition: for every if and only if
A seminorm on is a function that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if is a norm (or more generally, a seminorm) then and that also has the following property:
Non-negativity: for all
Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
Although this article defined "" to be a synonym of "positive definite", some authors instead define "" to be a synonym of "non-negative"; these definitions are not equivalent.
Equivalent norms | Norm (mathematics) | Wikipedia | 498 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
Suppose that and are two norms (or seminorms) on a vector space Then and are called equivalent, if there exist two positive real constants and such that for every vector
The relation " is equivalent to " is reflexive, symmetric ( implies ), and transitive and thus defines an equivalence relation on the set of all norms on
The norms and are equivalent if and only if they induce the same topology on Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.
Notation
If a norm is given on a vector space then the norm of a vector is usually denoted by enclosing it within double vertical lines: Such notation is also sometimes used if is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation with single vertical lines is also widespread.
Examples
Every (real or complex) vector space admits a norm: If is a Hamel basis for a vector space then the real-valued map that sends (where all but finitely many of the scalars are ) to is a norm on There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
Absolute-value norm
The absolute value
is a norm on the vector space formed by the real or complex numbers. The complex numbers form a one-dimensional vector space over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.
Any norm on a one-dimensional vector space is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces where is either or and norm-preserving means that
This isomorphism is given by sending to a vector of norm which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.
Euclidean norm
On the -dimensional Euclidean space the intuitive notion of length of the vector is captured by the formula
This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem.
This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares. | Norm (mathematics) | Wikipedia | 466 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
The Euclidean norm is by far the most commonly used norm on but there are other norms on this vector space as will be shown below.
However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces.
The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis.
Hence, the Euclidean norm can be written in a coordinate-free way as
The Euclidean norm is also called the quadratic norm, norm, norm, 2-norm, or square norm; see space.
It defines a distance function called the Euclidean length, distance, or distance.
The set of vectors in whose Euclidean norm is a given positive constant forms an -sphere.
Euclidean norm of complex numbers
The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane This identification of the complex number as a vector in the Euclidean plane, makes the quantity (as first suggested by Euler) the Euclidean norm associated with the complex number. For , the norm can also be written as where is the complex conjugate of
Quaternions and octonions
There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers the complex numbers the quaternions and lastly the octonions where the dimensions of these spaces over the real numbers are respectively.
The canonical norms on and are their absolute value functions, as discussed previously.
The canonical norm on of quaternions is defined by
for every quaternion in This is the same as the Euclidean norm on considered as the vector space Similarly, the canonical norm on the octonions is just the Euclidean norm on
Finite-dimensional complex normed spaces
On an -dimensional complex space the most common norm is
In this case, the norm can be expressed as the square root of the inner product of the vector and itself:
where is represented as a column vector and denotes its conjugate transpose.
This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation:
Taxicab norm or Manhattan norm
The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point | Norm (mathematics) | Wikipedia | 503 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1.
The Taxicab norm is also called the norm. The distance derived from this norm is called the Manhattan distance or distance.
The 1-norm is simply the sum of the absolute values of the columns.
In contrast,
is not a norm because it may yield negative results.
p-norm
Let be a real number.
The -norm (also called -norm) of vector is
For we get the taxicab norm, for we get the Euclidean norm, and as approaches the -norm approaches the infinity norm or maximum norm:
The -norm is related to the generalized mean or power mean.
For the -norm is even induced by a canonical inner product meaning that for all vectors This inner product can be expressed in terms of the norm by using the polarization identity.
On this inner product is the defined by
while for the space associated with a measure space which consists of all square-integrable functions, this inner product is
This definition is still of some interest for but the resulting function does not define a norm, because it violates the triangle inequality.
What is true for this case of even in the measurable analog, is that the corresponding class is a vector space, and it is also true that the function
(without th root) defines a distance that makes into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic analysis.
However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional.
The partial derivative of the -norm is given by
The derivative with respect to therefore, is
where denotes Hadamard product and is used for absolute value of each component of the vector.
For the special case of this becomes
or
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)
If is some vector such that then:
The set of vectors whose infinity norm is a given constant, forms the surface of a hypercube with edge length
Energy norm
The energy norm of a vector is defined in terms of a symmetric positive definite matrix as
It is clear that if is the identity matrix, this norm corresponds to the Euclidean norm. If is diagonal, this norm is also called a weighted norm. The energy norm is induced by the inner product given by for . | Norm (mathematics) | Wikipedia | 509 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
In general, the value of the norm is dependent on the spectrum of : For a vector with a Euclidean norm of one, the value of is bounded from below and above by the smallest and largest absolute eigenvalues of respectively, where the bounds are achieved if coincides with the corresponding (normalized) eigenvectors. Based on the symmetric matrix square root , the energy norm of a vector can be written in terms of the standard Euclidean norm as
Zero norm
In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm
Here we mean by F-norm some real-valued function on an F-space with distance such that The F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.
Hamming distance of a vector from zero
In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory.
In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero.
However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness.
When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous. | Norm (mathematics) | Wikipedia | 356 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
In signal processing and statistics, David Donoho referred to the zero "norm" with quotation marks.
Following Donoho's notation, the zero "norm" of is simply the number of non-zero coordinates of or the Hamming distance of the vector from zero.
When this "norm" is localized to a bounded set, it is the limit of -norms as approaches 0.
Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous.
Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument.
Abusing terminology, some engineers omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the norm, echoing the notation for the Lebesgue space of measurable functions.
Infinite dimensions
The generalization of the above norms to an infinite number of components leads to and spaces for with norms
for complex-valued sequences and functions on respectively, which can be further generalized (see Haar measure). These norms are also valid in the limit as , giving a supremum norm, and are called and
Any inner product induces in a natural way the norm
Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article.
Generally, these norms do not give the same topologies. For example, an infinite-dimensional space gives a strictly finer topology than an infinite-dimensional space when
Composite norms
Other norms on can be constructed by combining the above; for example
is a norm on
For any norm and any injective linear transformation we can define a new norm of equal to
In 2D, with a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram of a particular shape, size, and orientation.
In 3D, this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base).
There are examples of norms that are not defined by "entrywise" formulas. For instance, the Minkowski functional of a centrally-symmetric convex body in (centered at zero) defines a norm on (see below). | Norm (mathematics) | Wikipedia | 507 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
All the above formulas also yield norms on without modification.
There are also norms on spaces of matrices (with real or complex entries), the so-called matrix norms.
In abstract algebra
Let be a finite extension of a field of inseparable degree and let have algebraic closure If the distinct embeddings of are then the Galois-theoretic norm of an element is the value As that function is homogeneous of degree , the Galois-theoretic norm is not a norm in the sense of this article. However, the -th root of the norm (assuming that concept makes sense) is a norm.
Composition algebras
The concept of norm in composition algebras does share the usual properties of a norm since null vectors are allowed. A composition algebra consists of an algebra over a field an involution and a quadratic form called the "norm".
The characteristic feature of composition algebras is the homomorphism property of : for the product of two elements and of the composition algebra, its norm satisfies In the case of division algebras and the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In the split algebras the norm is an isotropic quadratic form.
Properties
For any norm on a vector space the reverse triangle inequality holds:
If is a continuous linear map between normed spaces, then the norm of and the norm of the transpose of are equal.
For the norms, we have Hölder's inequality
A special case of this is the Cauchy–Schwarz inequality:
Every norm is a seminorm and thus satisfies all properties of the latter. In turn, every seminorm is a sublinear function and thus satisfies all properties of the latter. In particular, every norm is a convex function.
Equivalence
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square. For any -norm, it is a superellipse with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and for a -norm). | Norm (mathematics) | Wikipedia | 512 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors is said to converge in norm to if as Equivalently, the topology consists of all sets that can be represented as a union of open balls. If is a normed space then
Two norms and on a vector space are called if they induce the same topology, which happens if and only if there exist positive real numbers and such that for all
For instance, if on then
In particular,
That is,
If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.
Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.
Classification of seminorms: absolutely convex absorbing sets
All seminorms on a vector space can be classified in terms of absolutely convex absorbing subsets of To each such subset corresponds a seminorm called the gauge of defined as
where is the infimum, with the property that
Conversely:
Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct such a basis is to use a family of seminorms that separates points: the collection of all finite intersections of sets turns the space into a locally convex topological vector space so that every p is continuous.
Such a method is used to design weak and weak* topologies.
norm case:
Suppose now that contains a single since is separating, is a norm, and is its open unit ball. Then is an absolutely convex bounded neighbourhood of 0, and is continuous.
The converse is due to Andrey Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely:
If is an absolutely convex bounded neighbourhood of 0, the gauge (so that is a norm. | Norm (mathematics) | Wikipedia | 460 | 990534 | https://en.wikipedia.org/wiki/Norm%20%28mathematics%29 | Mathematics | Linear algebra | null |
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.
This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems and bizarre systems.
This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.
Overview
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that do not change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.
Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. | Dynamical systems theory | Wikipedia | 496 | 990632 | https://en.wikipedia.org/wiki/Dynamical%20systems%20theory | Mathematics | Other | null |
Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.
History
The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.
Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems.
Some excellent presentations of mathematical dynamic system theory include , , , and .
Concepts
Dynamical systems
The dynamical system concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.
A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule may be deterministic (for a given time interval one future state can be precisely predicted given the current state) or stochastic (the evolution of the state can only be predicted with a certain probability).
Dynamicism
Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic cognition, is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues that differential equations are more suited to modelling cognition than more traditional computer models.
Nonlinear system
In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known. | Dynamical systems theory | Wikipedia | 511 | 990632 | https://en.wikipedia.org/wiki/Dynamical%20systems%20theory | Mathematics | Other | null |
Related fields
Arithmetic dynamics
Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/or algebraic points under repeated application of a polynomial or rational function.
Chaos theory
Chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.
Complex systems
Complex systems is a scientific field that studies the common properties of systems considered complex in nature, society, and science. It is also called complex systems theory, complexity science, study of complex systems and/or sciences of complexity. The key problems of such systems are difficulties with their formal modeling and simulation. From such perspective, in different research contexts complex systems are defined on the base of their different attributes.
The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neurosciences, social sciences, meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation, economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves.
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics, in part it deals with influencing the behavior of dynamical systems.
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics. | Dynamical systems theory | Wikipedia | 446 | 990632 | https://en.wikipedia.org/wiki/Dynamical%20systems%20theory | Mathematics | Other | null |
Functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.
Graph dynamical systems
The concept of graph dynamical systems (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
Projected dynamical systems
Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.
Symbolic dynamics
Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.
System dynamics
System dynamics is an approach to understanding the behaviour of systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system. What makes using system dynamics different from other approaches to studying systems is the language used to describe feedback loops with stocks and flows. These elements help describe how even seemingly simple systems display baffling nonlinearity.
Topological dynamics
Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
Applications | Dynamical systems theory | Wikipedia | 420 | 990632 | https://en.wikipedia.org/wiki/Dynamical%20systems%20theory | Mathematics | Other | null |
In biomechanics
In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance and efficiency. It comes as no surprise, since dynamical systems theory has its roots in Analytical mechanics. From psychophysiological perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems. There is no research validation of any of the claims associated to the conceptual application of this framework.
In cognitive science
Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the neo-Piagetian theories of cognitive development. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and AI. It also believed that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.
In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. Self-organization (the spontaneous creation of coherent forms) sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process. These links form the structure of a new state of order in the mind through a process called scalloping (the repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, idiosyncratic and unpredictable.
Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error. | Dynamical systems theory | Wikipedia | 461 | 990632 | https://en.wikipedia.org/wiki/Dynamical%20systems%20theory | Mathematics | Other | null |
Further, since the middle of the 1990s cognitive science, oriented towards a system theoretical connectionism, has increasingly adopted the methods from (nonlinear) “Dynamic Systems Theory (DST)“. A variety of neurosymbolic cognitive neuroarchitectures in modern connectionism, considering their mathematical structural core, can be categorized as (nonlinear) dynamical systems. These attempts in neurocognition to merge connectionist cognitive neuroarchitectures with DST come from not only neuroinformatics and connectionism, but also recently from developmental psychology (“Dynamic Field Theory (DFT)”) and from “evolutionary robotics” and “developmental robotics” in connection with the mathematical method of “evolutionary computation (EC)”. For an overview see Maurer.
In second language development
The application of Dynamic Systems Theory to study second language acquisition is attributed to Diane Larsen-Freeman who published an article in 1997 in which she claimed that second language acquisition should be viewed as a developmental process which includes language attrition as well as language acquisition. In her article she claimed that language should be viewed as a dynamic system which is dynamic, complex, nonlinear, chaotic, unpredictable, sensitive to initial conditions, open, self-organizing, feedback sensitive, and adaptive. | Dynamical systems theory | Wikipedia | 258 | 990632 | https://en.wikipedia.org/wiki/Dynamical%20systems%20theory | Mathematics | Other | null |
A sol is a colloidal suspension made out of tiny solid particles in a continuous liquid medium. Sols are stable, so that they do not settle down when left undisturbed, and exhibit the Tyndall effect, which is the scattering of light by the particles in the colloid. The size of the particles can vary from 1 nm - 100 nm. Examples include amongst others blood, pigmented ink, cell fluids, paint, antacids and mud.
Artificial sols can be prepared by two main methods: dispersion and condensation. In the dispersion method, solid particles are reduced to colloidal dimensions through techniques such as ball milling and Bredig's arc method. In the condensation method, small particles are formed from larger molecules through a chemical reaction.
The stability of sols can be maintained through the use of dispersing agents, which prevent the particles from clumping together or settling out of the suspension. Sols are often used in the sol-gel process, in which a sol is converted into a gel through the addition of a crosslinking agent.
In a sol, solid particles are dispersed in a liquid continuous phase, while in an emulsion, liquid droplets are dispersed in a liquid or semi-solid continuous phase. | Sol (colloid) | Wikipedia | 261 | 991054 | https://en.wikipedia.org/wiki/Sol%20%28colloid%29 | Physical sciences | Mixture | Chemistry |
Rodenticides are chemicals made and sold for the purpose of killing rodents. While commonly referred to as "rat poison", rodenticides are also used to kill mice, woodchucks, chipmunks, porcupines, nutria, beavers, and voles.
Some rodenticides are lethal after one exposure while others require more than one. Rodents are disinclined to gorge on an unknown food (perhaps reflecting an adaptation to their inability to vomit), preferring to sample, wait and observe whether it makes them or other rats sick. This phenomenon of poison shyness is the rationale for poisons that kill only after multiple doses.
Besides being directly toxic to the mammals that ingest them, including dogs, cats, and humans, many rodenticides present a secondary poisoning risk to animals that hunt or scavenge the dead corpses of rats.
Classes of rodenticides
Anticoagulants
Anticoagulants are defined as chronic (death occurs one to two weeks after ingestion of the lethal dose, rarely sooner), single-dose (second generation) or multiple-dose (first generation) rodenticides, acting by effective blocking of the vitamin-K cycle, resulting in inability to produce essential blood-clotting factors—mainly coagulation factors II (prothrombin) and VII (proconvertin).
In addition to this specific metabolic disruption, massive toxic doses of 4-hydroxycoumarin, 4-thiochromenone and 1,3-indandione anticoagulants cause damage to tiny blood vessels (capillaries), increasing their permeability, causing internal bleeding. These effects are gradual, developing over several days. In the final phase of the intoxication, the exhausted rodent collapses due to hemorrhagic shock or severe anemia and dies. The question of whether the use of these rodenticides can be considered humane has been raised.
The main benefit of anticoagulants over other poisons is that the time taken for the poison to induce death means that the rats do not associate the damage with their feeding habits. | Rodenticide | Wikipedia | 446 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
First-generation rodenticidal anticoagulants generally have shorter elimination half-lives, require higher concentrations (usually between 0.005% and 0.1%) and consecutive intake over days in order to accumulate the lethal dose, and are less toxic than second-generation agents.
Second-generation anticoagulant rodenticides (or SGARs) are far more toxic than those of the first generation. They are generally applied in lower concentrations in baits—usually on the order of 0.001% to 0.005%—are lethal after a single ingestion of bait and are also effective against strains of rodents that became resistant to first-generation anticoagulants; thus, the second-generation anticoagulants are sometimes referred to as "superwarfarins".
Phylloquinone has been suggested, and successfully used, as antidote for pets or humans accidentally or intentionally exposed to anticoagulant poisons. Some of these poisons act by inhibiting liver functions and in advanced stages of poisoning, several blood-clotting factors are absent, and the volume of circulating blood is diminished, so that a blood transfusion (optionally with the clotting factors present) can save a person who has been poisoned, an advantage over some older poisons. A unique enzyme produced by the liver enables the body to recycle vitamin K. To produce the blood clotting factors that prevent excessive bleeding, the body needs vitamin K. Anticoagulants hinder this enzyme's ability to function. Internal bleeding could start if the body's reserve of anticoagulant runs out from exposure to enough of it. Because they bind more closely to the enzyme that produces blood clotting agents, single-dose anticoagulants are more hazardous. They may also obstruct several stages of the recycling of vitamin K. Single-dose or second-generation anticoagulants can be stored in the liver because they are not quickly eliminated from the body.
Metal phosphides | Rodenticide | Wikipedia | 426 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
Metal phosphides have been used as a means of killing rodents and are considered single-dose fast acting rodenticides (death occurs commonly within 1–3 days after single bait ingestion). A bait consisting of food and a phosphide (usually zinc phosphide) is left where the rodents can eat it. The acid in the digestive system of the rodent reacts with the phosphide to generate toxic phosphine gas. This method of vermin control has possible use in places where rodents are resistant to some of the anticoagulants, particularly for control of house and field mice; zinc phosphide baits are also cheaper than most second-generation anticoagulants, so that sometimes, in the case of large infestation by rodents, their population is initially reduced by copious amounts of zinc phosphide bait applied, and the rest of population that survived the initial fast-acting poison is then eradicated by prolonged feeding on anticoagulant bait. Inversely, the individual rodents that survived anticoagulant bait poisoning (rest population) can be eradicated by pre-baiting them with nontoxic bait for a week or two (this is important to overcome bait shyness, and to get rodents used to feeding in specific areas by specific food, especially in eradicating rats) and subsequently applying poisoned bait of the same sort as used for pre-baiting until all consumption of the bait ceases (usually within 2–4 days). These methods of alternating rodenticides with different modes of action gives actual or almost 100% eradications of the rodent population in the area, if the acceptance/palatability of baits are good (i.e., rodents feed on it readily).
Zinc phosphide is typically added to rodent baits in a concentration of 0.75% to 2.0%. The baits have strong, pungent garlic-like odor due to the phosphine liberated by hydrolysis. The odor attracts (or, at least, does not repel) rodents, but has a repulsive effect on other mammals. Birds, notably wild turkeys, are not sensitive to the smell, and might feed on the bait, and thus fall victim to the poison. | Rodenticide | Wikipedia | 478 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
The tablets or pellets (usually aluminium, calcium or magnesium phosphide for fumigation/gassing) may also contain other chemicals which evolve ammonia, which helps reduce the potential for spontaneous combustion or explosion of the phosphine gas.
Metal phosphides do not accumulate in the tissues of poisoned animals, so the risk of secondary poisoning is low.
Before the advent of anticoagulants, phosphides were the favored kind of rat poison. During World War II, they came into use in United States because of shortage of strychnine due to the Japanese occupation of the territories where the strychnine tree is grown. Phosphides are rather fast-acting rat poisons, resulting in the rats dying usually in open areas, instead of in the affected buildings.
Phosphides used as rodenticides include:
aluminium phosphide (fumigant and bait)
calcium phosphide (fumigant only)
magnesium phosphide (fumigant only)
zinc phosphide (bait only) | Rodenticide | Wikipedia | 223 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
Cholecalciferol (vitamin D3) and ergocalciferol (vitamin D2) are used as rodenticides. They are toxic to rodents for the same reason they are important to humans: they affect calcium and phosphate homeostasis in the body. Vitamins D are essential in minute quantities (few IUs per kilogram body weight daily, only a fraction of a milligram), and like most fat soluble vitamins, they are toxic in larger doses, causing hypervitaminosis D. If the poisoning is severe enough (that is, if the dose of the toxin is high enough), it leads to death. In rodents that consume the rodenticidal bait, it causes hypercalcemia, raising the calcium level, mainly by increasing calcium absorption from food, mobilising bone-matrix-fixed calcium into ionised form (mainly monohydrogencarbonate calcium cation, partially bound to plasma proteins, [CaHCO3]+), which circulates dissolved in the blood plasma. After ingestion of a lethal dose, the free calcium levels are raised sufficiently that blood vessels, kidneys, the stomach wall and lungs are mineralised/calcificated (formation of calcificates, crystals of calcium salts/complexes in the tissues, damaging them), leading further to heart problems (myocardial tissue is sensitive to variations of free calcium levels, affecting both myocardial contractibility and action potential propagation between the atria and ventricles), bleeding (due to capillary damage) and possibly kidney failure. It is considered to be single-dose, cumulative (depending on concentration used; the common 0.075% bait concentration is lethal to most rodents after a single intake of larger portions of the bait) or sub-chronic (death occurring usually within days to one week after ingestion of the bait). Applied concentrations are 0.075% cholecalciferol (30,000 IU/g)<ref name=usda2006>{{cite conference |last1=Rizor |first1=Suzanne E. |last2=Arjo |first2=Wendy M. |last3=Bulkin |first3=Stephan |last4=Nolte |first4=Dale L | Rodenticide | Wikipedia | 479 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
|title=Efficacy of Cholecalciferol Baits for Pocket Gopher Control and Possible Effects on Non-Target Rodents in Pacific Northwest Forests |url=https://naldc.nal.usda.gov/download/39036/PDF |conference=Vertebrate Pest Conference (2006) |publisher=USDA |quote= 0.15% cholecalciferol bait appears to have application for pocket gopher control.' Cholecalciferol can be a single high-dose toxicant or a cumulative multiple low-dose toxicant.' |access-date=27 August 2019 |archive-date=14 September 2012 |archive-url=https://web.archive.org/web/20120914083512/http://naldc.nal.usda.gov/download/39036/PDF |url-status=dead }}</ref> and 0.1% ergocalciferol (40,000 IU/g) when used alone, which can kill a rodent or a rat | Rodenticide | Wikipedia | 234 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
There is an important feature of calciferols toxicology, that they are synergistic with anticoagulant toxicant. In other words, mixtures of anticoagulants and calciferols in same bait are more toxic than a sum of toxicities of the anticoagulant and the calciferol in the bait, so that a massive hypercalcemic effect can be achieved by a substantially lower calciferol content in the bait, and vice versa, a more pronounced anticoagulant/hemorrhagic effects are observed if the calciferol is present. This synergism is mostly used in calciferol low concentration baits, because effective concentrations of calciferols are more expensive than effective concentrations of most anticoagulants.
The first application of a calciferol in rodenticidal bait was in the Sorex product Sorexa D (with a different formula than today's Sorexa D), back in the early 1970s, which contained 0.025% warfarin and 0.1% ergocalciferol. Today, Sorexa CD contains a 0.0025% difenacoum and 0.075% cholecalciferol combination. Numerous other brand products containing either 0.075-0.1% calciferols (e.g. Quintox) alone or alongside an anticoagulant are marketed. | Rodenticide | Wikipedia | 306 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
The Merck Veterinary Manual states the following:
Although this rodenticide [cholecalciferol] was introduced with claims that it was less toxic to nontarget species than to rodents, clinical experience has shown that rodenticides containing cholecalciferol are a significant health threat to dogs and cats. Cholecalciferol produces hypercalcemia, which results in systemic calcification of soft tissue, leading to kidney failure, cardiac abnormalities, hypertension, CNS depression and GI upset. Signs generally develop within 18-36 hours of ingestion and can include depression, anorexia, polyuria and polydipsia. As serum calcium concentrations increase, clinical signs become more severe. ... GI smooth muscle excitability decreases and is manifest by anorexia, vomiting and constipation. ... Loss of renal concentrating ability is a direct result of hypercalcemia. As hypercalcemia persists, mineralization of the kidneys results in progressive renal insufficiency."
Additional anticoagulant renders the bait more toxic to pets as well as humans. Upon single ingestion, solely calciferol-based baits are considered generally safer to birds than second generation anticoagulants or acute toxicants. Treatment in pets is mostly supportive, with intravenous fluids and pamidronate disodium. The hormone calcitonin is no longer commonly used.
Other | Rodenticide | Wikipedia | 302 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
Other chemical poisons include:
ANTU (α-naphthylthiourea; specific against Brown rat, Rattus norvegicus'')
Arsenic trioxide
Barium carbonate (sometimes called Witherite)
Chloralose (a narcotic prodrug)
Crimidine (inhibits metabolism of vitamin B6)
1,3-Difluoro-2-propanol ("Gliftor")
Endrin (organochlorine insecticide, used in the past for extermination of voles in fields)
Fluoroacetamide ("1081")
Phosacetim (a delayed-action acetylcholinesterase inhibitor)
Phosphorus allotropes
Pyrinuron (a urea derivative)
Scilliroside and other cardiac glycosides like oleandrin or digoxin
Sodium fluoroacetate ("1080")
Strychnine (A naturally occurring convulsant and stimulant)
Tetramethylenedisulfotetramine ("tetramine") - Deadly toxic to humans so use should be avoided
Thallium sulfate
Mitochondrial toxins like bromethalin and 2,4-dinitrophenol (cause high fever and brain swelling)
Zyklon B/Uragan D2 (hydrogen cyanide gas absorbed in an inert carrier)
Combinations
In some countries, fixed three-component rodenticides, i.e., anticoagulant + antibiotic + vitamin D, are used. Associations of a second-generation anticoagulant with an antibiotic and/or vitamin D are considered to be effective even against most resistant strains of rodents, though some second generation anticoagulants (namely brodifacoum and difethialone), in bait concentrations of 0.0025% to 0.005% are so toxic that resistance is unknown, and even rodents resistant to other rodenticides are reliably exterminated by application of these most toxic anticoagulants.
Low-toxicity/Eco-friendly rodenticides
Powdered corn cob and corn meal gluten have been developed as rodenticides. They were approved in the EU and patented in the US in 2013. These preparations rely on dehydration and electrolyte imbalance to cause death. | Rodenticide | Wikipedia | 494 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
Inert gas killing of burrowing pest animals is another method with no impact on scavenging wildlife. One such method has been commercialized and sold under the brand name Rat Ice.
Non-target issues
Secondary poisoning and risks to wildlife
One of the potential problems when using rodenticides is that dead or weakened rodents may be eaten by other wildlife, either predators or scavengers. Members of the public deploying rodenticides may not be aware of this or may not follow the product's instructions closely enough. There is evidence of secondary poisoning being caused by exposure to prey.
The faster a rodenticide acts, the more critical this problem may be. For the fast-acting rodenticide bromethalin, for example, there is no diagnostic test or antidote.
This has led environmental researchers to conclude that low strength, long duration rodenticides (generally first generation anticoagulants) are the best balance between maximum effect and minimum risk.
Proposed US legislation change
In 2008, after assessing human health and ecological effects, as well as benefits, the US Environmental Protection Agency (EPA) announced measures to reduce risks associated with ten rodenticides. New restrictions by sale and distribution restrictions, minimum package size requirements, use site restriction, and tamper resistant products would have taken effect in 2011. The regulations were delayed pending a legal challenge by manufacturer Reckitt-Benkiser.
Notable rat eradications
The entire rat populations of several islands have been eradicated, most notably New Zealand's Campbell Island, Hawadax Island, Alaska (formerly known as Rat Island), Macquarie Island and Canna, Scotland (declared rat-free in 2008). According to the Friends of South Georgia Island, all of the rats have been eliminated from South Georgia.
Alberta, Canada, through a combination of climate and control, is also believed to be rat-free. | Rodenticide | Wikipedia | 384 | 991169 | https://en.wikipedia.org/wiki/Rodenticide | Technology | Pest and disease control | null |
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.
Divisibility rules for numbers 1−30
The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means.
For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits.
To test the divisibility of a number by a power of 2 or a power of 5 (2n or 5n, in which n is a positive integer), one only need to look at the last n digits of that number.
To test divisibility by any number expressed as the product of prime factors , we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 is equivalent to testing divisibility by 8 (23) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 to prove divisibility by 24.
Step-by-step examples
Divisibility by 2
First, take any number (for this example it will be 376) and note the last digit in the number, discarding the other digits. Then take that digit (6) while ignoring the rest of the number and determine if it is divisible by 2. If it is divisible by 2, then the original number is divisible by 2.
Example
376 (The original number)
37 6 (Take the last digit)
6 ÷ 2 = 3 (Check to see if the last digit is divisible by 2)
376 ÷ 2 = 188 (If the last digit is divisible by 2, then the whole number is divisible by 2) | Divisibility rule | Wikipedia | 505 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Divisibility by 3 or 9
First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9).
Adding the digits of a number up, and then repeating the process with the result until only one digit remains, will give the remainder of the original number if it were divided by nine (unless that single digit is nine itself, in which case the number is divisible by nine and the remainder is zero).
This can be generalized to any standard positional system, in which the divisor in question then becomes one less than the radix; thus, in base-twelve, the digits will add up to the remainder of the original number if divided by eleven, and numbers are divisible by eleven only if the digit sum is divisible by eleven.
Example.
492 (The original number)
4 + 9 + 2 = 15 (Add each individual digit together)
15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large:
1 + 5 = 6 (Add each individual digit together)
6 ÷ 3 = 2 (Check to see if the number received is divisible by 3)
492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)
Divisibility by 4
The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that you know is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits. | Divisibility rule | Wikipedia | 476 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Alternatively, one can just add half of the last digit to the penultimate digit (or the remaining number). If that number is an even natural number, the original number is divisible by 4
Also, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4.
Example.
General rule
2092 (The original number)
20 92 (Take the last two digits of the number, discarding any other digits)
92 ÷ 4 = 23 (Check to see if the number is divisible by 4)
2092 ÷ 4 = 523 (If the number that is obtained is divisible by 4, then the original number is divisible by 4)
Second method
6174 (the original number)
check that last digit is even, otherwise 6174 can't be divisible by 4.
61 7 4 (Separate the last 2 digits from the rest of the number)
4 ÷ 2 = 2 (last digit divided by 2)
7 + 2 = 9 (Add half of last digit to the penultimate digit)
Since 9 isn't even, 6174 is not divisible by 4
Third method
1720 (The original number)
1720 ÷ 2 = 860 (Divide the original number by 2)
860 ÷ 2 = 430 (Check to see if the result is divisible by 2)
1720 ÷ 4 = 430 (If the result is divisible by 2, then the original number is divisible by 4)
Divisibility by 5
Divisibility by 5 is easily determined by checking the last digit in the number (475), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5.
If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8). | Divisibility rule | Wikipedia | 472 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
If the last digit in the number is 5, then the result will be the remaining digits multiplied by two, plus one. For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25).
Example.
If the last digit is 0
110 (The original number)
11 0 (Take the last digit of the number, and check if it is 0 or 5)
11 0 (If it is 0, take the remaining digits, discarding the last)
11 × 2 = 22 (Multiply the result by 2)
110 ÷ 5 = 22 (The result is the same as the original number divided by 5)
If the last digit is 5
85 (The original number)
8 5 (Take the last digit of the number, and check if it is 0 or 5)
8 5 (If it is 5, take the remaining digits, discarding the last)
8 × 2 = 16 (Multiply the result by 2)
16 + 1 = 17 (Add 1 to the result)
85 ÷ 5 = 17 (The result is the same as the original number divided by 5)
Divisibility by 6
Divisibility by 6 is determined by checking the original number to see if it is both an even number (divisible by 2) and divisible by 3.
If the final digit is even the number is divisible by two, and thus may be divisible by 6. If it is divisible by 2 continue by adding the digits of the original number and checking if that sum is a multiple of 3. Any number which is both a multiple of 2 and of 3 is a multiple of 6.
Example.
324 (The original number)
Final digit 4 is even, so 324 is divisible by 2, and may be divisible by 6.
3 + 2 + 4 = 9 which is a multiple of 3. Therefore the original number is divisible by both 2 and 3 and is divisible by 6. | Divisibility rule | Wikipedia | 446 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Divisibility by 7
Divisibility by 7 can be tested by a recursive method. A number of the form 10x + y is divisible by 7 if and only if x − 2y is divisible by 7. In other words, subtract twice the last digit from the number formed by the remaining digits. Continue to do this until a number is obtained for which it is known whether it is divisible by 7. The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7. For example, the number 371: 37 − (2×1) = 37 − 2 = 35; 3 − (2 × 5) = 3 − 10 = −7; thus, since −7 is divisible by 7, 371 is divisible by 7.
Similarly a number of the form 10x + y is divisible by 7 if and only if x + 5y is divisible by 7. So add five times the last digit to the number formed by the remaining digits, and continue to do this until a number is obtained for which it is known whether it is divisible by 7.
Another method is multiplication by 3. A number of the form 10x + y has the same remainder when divided by 7 as 3x + y. One must multiply the leftmost digit of the original number by 3, add the next digit, take the remainder when divided by 7, and continue from the beginning: multiply by 3, add the next digit, etc. For example, the number 371: 3×3 + 7 = 16 remainder 2, and 2×3 + 1 = 7. This method can be used to find the remainder of division by 7. | Divisibility rule | Wikipedia | 365 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
A more complicated algorithm for testing divisibility by 7 uses the fact that 100 ≡ 1, 101 ≡ 3, 102 ≡ 2, 103 ≡ 6, 104 ≡ 4, 105 ≡ 5, 106 ≡ 1, ... (mod 7). Take each digit of the number (371) in reverse order (173), multiplying them successively by the digits 1, 3, 2, 6, 4, 5, repeating with this sequence of multipliers as long as necessary (1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, ...), and adding the products (1×1 + 7×3 + 3×2 = 1 + 21 + 6 = 28). The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7 (hence 371 is divisible by 7 since 28 is).
This method can be simplified by removing the need to multiply. All it would take with this simplification is to memorize the sequence above (132645...), and to add and subtract, but always working with one-digit numbers. | Divisibility rule | Wikipedia | 251 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
The simplification goes as follows:
Take for instance the number 371
Change all occurrences of 7, 8 or 9 into 0, 1 and 2, respectively. In this example, we get: 301. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on.
Now convert the first digit (3) into the following digit in the sequence 13264513... In our example, 3 becomes 2.
Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So 301 becomes 21.
Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit: 6 + 1 = 7.
If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should simply be dropped. This is because that 7 would have become 0, and numbers with at least two digits before the decimal dot do not begin with 0, which is useless. According to this, our 7 becomes 0.
If through this procedure you obtain a 0 or any recognizable multiple of 7, then the original number is a multiple of 7. If you obtain any number from 1 to 6, that will indicate how much you should subtract from the original number to get a multiple of 7. In other words, you will find the remainder of dividing the number by 7. For example, take the number 186:
First, change the 8 into a 1: 116.
Now, change 1 into the following digit in the sequence (3), add it to the second digit, and write the result instead of both: 3 + 1 = 4. So 116 becomes now 46.
Repeat the procedure, since the number is greater than 7. Now, 4 becomes 5, which must be added to 6. That is 11.
Repeat the procedure one more time: 1 becomes 3, which is added to the second digit (1): 3 + 1 = 4. | Divisibility rule | Wikipedia | 492 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Now we have a number smaller than 7, and this number (4) is the remainder of dividing 186/7. So 186 minus 4, which is 182, must be a multiple of 7.
Note: The reason why this works is that if we have: a+b=c and b is a multiple of any given number n, then a and c will necessarily produce the same remainder when divided by n. In other words, in 2 + 7 = 9, 7 is divisible by 7. So 2 and 9 must have the same remainder when divided by 7. The remainder is 2.
Therefore, if a number n is a multiple of 7 (i.e.: the remainder of n/7 is 0), then adding (or subtracting) multiples of 7 cannot change that property.
What this procedure does, as explained above for most divisibility rules, is simply subtract little by little multiples of 7 from the original number until reaching a number that is small enough for us to remember whether it is a multiple of 7. If 1 becomes a 3 in the following decimal position, that is just the same as converting 10×10n into a 3×10n. And that is actually the same as subtracting 7×10n (clearly a multiple of 7) from 10×10n.
Similarly, when you turn a 3 into a 2 in the following decimal position, you are turning 30×10n into 2×10n, which is the same as subtracting 30×10n−28×10n, and this is again subtracting a multiple of 7. The same reason applies for all the remaining conversions:
20×10n − 6×10n=14×10n
60×10n − 4×10n=56×10n
40×10n − 5×10n=35×10n
50×10n − 1×10n=49×10n
First method example
1050 → 105 − 0=105 → 10 − 10 = 0. ANSWER: 1050 is divisible by 7.
Second method example
1050 → 0501 (reverse) → 0×1 + 5×3 + 0×2 + 1×6 = 0 + 15 + 0 + 6 = 21 (multiply and add). ANSWER: 1050 is divisible by 7. | Divisibility rule | Wikipedia | 482 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Vedic method of divisibility by osculation
Divisibility by seven can be tested by multiplication by the Ekhādika. Convert the divisor seven to the nines family by multiplying by seven. 7×7=49. Add one, drop the units digit and, take the 5, the Ekhādika, as the multiplier. Start on the right. Multiply by 5, add the product to the next digit to the left. Set down that result on a line below that digit. Repeat that method of multiplying the units digit by five and adding that product to the number of tens. Add the result to the next digit to the left. Write down that result below the digit. Continue to the end. If the result is zero or a multiple of seven, then yes, the number is divisible by seven. Otherwise, it is not. This follows the Vedic ideal, one-line notation.
Vedic method example:
Is 438,722,025 divisible by seven? Multiplier = 5.
4 3 8 7 2 2 0 2 5
42 37 46 37 6 40 37 27
YES
Pohlman–Mass method of divisibility by 7
The Pohlman–Mass method provides a quick solution that can determine if most integers are divisible by seven in three steps or less. This method could be useful in a mathematics competition such as MATHCOUNTS, where time is a factor to determine the solution without a calculator in the Sprint Round.
Step A:
If the integer is 1000 or less, subtract twice the last digit from the number formed by the remaining digits. If the result is a multiple of seven, then so is the original number (and vice versa). For example:
112 -> 11 − (2×2) = 11 − 4 = 7 YES
98 -> 9 − (8×2) = 9 − 16 = −7 YES
634 -> 63 − (4×2) = 63 − 8 = 55 NO
Because 1001 is divisible by seven, an interesting pattern develops for repeating sets of 1, 2, or 3 digits that form 6-digit numbers (leading zeros are allowed) in that all such numbers are divisible by seven. For example: | Divisibility rule | Wikipedia | 468 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
001 001 = 1,001 / 7 = 143
010 010 = 10,010 / 7 = 1,430
011 011 = 11,011 / 7 = 1,573
100 100 = 100,100 / 7 = 14,300
101 101 = 101,101 / 7 = 14,443
110 110 = 110,110 / 7 = 15,730
01 01 01 = 10,101 / 7 = 1,443
10 10 10 = 101,010 / 7 = 14,430
111,111 / 7 = 15,873
222,222 / 7 = 31,746
999,999 / 7 = 142,857
576,576 / 7 = 82,368
For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern.
This phenomenon forms the basis for Steps B and C.
Step B:
If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1000, apply Step A. This can be done by subtracting the first three digits from the last three digits. For example:
341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES
67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES
The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and follow with Step A.
Step C:
If the integer is larger than one million, subtract the nearest multiple of 999,999 and then apply Step B. For even larger numbers, use larger sets such as 12-digits (999,999,999,999) and so on. Then, break the integer into a smaller number that can be solved using Step B. For example: | Divisibility rule | Wikipedia | 482 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
22,862,420 − (999,999 × 22) = 22,862,420 − 21,999,978 -> 862,420 + 22 = 862,442
862,442 -> 862 − 442 (Step B) = 420 -> 42 − (0×2) (Step A) = 42 YES
This allows adding and subtracting alternating sets of three digits to determine divisibility by seven. Understanding these patterns allows you to quickly calculate divisibility of seven as seen in the following examples:
Pohlman–Mass method of divisibility by 7, examples:
Is 98 divisible by seven?
98 -> 9 − (8×2) = 9 − 16 = −7 YES (Step A)
Is 634 divisible by seven?
634 -> 63 − (4×2) = 63 − 8 = 55 NO (Step A)
Is 355,341 divisible by seven?
355,341 − 341,341 = 14,000 (Step B) -> 014 − 000 (Step B) -> 14 = 1 − (4×2) (Step A) = 1 − 8 = −7 YES
Is 42,341,530 divisible by seven?
42,341,530 -> 341,530 + 42 = 341,572 (Step C)
341,572 − 341,341 = 231 (Step B)
231 -> 23 − (1×2) = 23 − 2 = 21 YES (Step A)
Using quick alternating additions and subtractions:
42,341,530 -> 530 − 341 + 42 = 189 + 42 = 231 -> 23 − (1×2) = 21 YES
Multiplication by 3 method of divisibility by 7, examples:
Is 98 divisible by seven?
98 -> 9 remainder 2 -> 2×3 + 8 = 14 YES
Is 634 divisible by seven?
634 -> 6×3 + 3 = 21 -> remainder 0 -> 0×3 + 4 = 4 NO
Is 355,341 divisible by seven?
3 × 3 + 5 = 14 -> remainder 0 -> 0×3 + 5 = 5 -> 5×3 + 3 = 18 -> remainder 4 -> 4×3 + 4 = 16 -> remainder 2 -> 2×3 + 1 = 7 YES | Divisibility rule | Wikipedia | 493 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Find remainder of 1036125837 divided by 7
1×3 + 0 = 3
3×3 + 3 = 12 remainder 5
5×3 + 6 = 21 remainder 0
0×3 + 1 = 1
1×3 + 2 = 5
5×3 + 5 = 20 remainder 6
6×3 + 8 = 26 remainder 5
5×3 + 3 = 18 remainder 4
4×3 + 7 = 19 remainder 5
Answer is 5
Finding remainder of a number when divided by 7
7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, −1, −3, −2
Minimum magnitude sequence
(1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, 6, 4, 5
Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7.
Example: What is the remainder when 1036125837 is divided by 7?
Multiplication of the rightmost digit = 1 × 7 = 7
Multiplication of the second rightmost digit = 3 × 3 = 9
Third rightmost digit = 8 × 2 = 16
Fourth rightmost digit = 5 × −1 = −5
Fifth rightmost digit = 2 × −3 = −6
Sixth rightmost digit = 1 × −2 = −2
Seventh rightmost digit = 6 × 1 = 6
Eighth rightmost digit = 3 × 3 = 9
Ninth rightmost digit = 0
Tenth rightmost digit = 1 × −1 = −1
Sum = 33
33 modulus 7 = 5
Remainder = 5
Digit pair method of divisibility by 7 | Divisibility rule | Wikipedia | 393 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
This method uses 1, −3, 2 pattern on the digit pairs. That is, the divisibility of any number by seven can be tested by first separating the number into digit pairs, and then applying the algorithm on three digit pairs (six digits). When the number is smaller than six digits, then fill zero's to the right side until there are six digits. When the number is larger than six digits, then repeat the cycle on the next six digit group and then add the results. Repeat the algorithm until the result is a small number. The original number is divisible by seven if and only if the number obtained using this algorithm is divisible by seven. This method is especially suitable for large numbers.
Example 1:
The number to be tested is 157514.
First we separate the number into three digit pairs: 15, 75 and 14.
Then we apply the algorithm: 1 × 15 − 3 × 75 + 2 × 14 = 182
Because the resulting 182 is less than six digits, we add zero's to the right side until it is six digits.
Then we apply our algorithm again: 1 × 18 − 3 × 20 + 2 × 0 = −42
The result −42 is divisible by seven, thus the original number 157514 is divisible by seven.
Example 2:
The number to be tested is 15751537186.
(1 × 15 − 3 × 75 + 2 × 15) + (1 × 37 − 3 × 18 + 2 × 60) = −180 + 103 = −77
The result −77 is divisible by seven, thus the original number 15751537186 is divisible by seven.
Another digit pair method of divisibility by 7
Method
This is a non-recursive method to find the remainder left by a number on dividing by 7: | Divisibility rule | Wikipedia | 382 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Separate the number into digit pairs starting from the ones place. Prepend the number with 0 to complete the final pair if required.
Calculate the remainders left by each digit pair on dividing by 7.
Multiply the remainders with the appropriate multiplier from the sequence 1, 2, 4, 1, 2, 4, ... : the remainder from the digit pair consisting of ones place and tens place should be multiplied by 1, hundreds and thousands by 2, ten thousands and hundred thousands by 4, million and ten million again by 1 and so on.
Calculate the remainders left by each product on dividing by 7.
Add these remainders.
The remainder of the sum when divided by 7 is the remainder of the given number when divided by 7.
For example:
The number 194,536 leaves a remainder of 6 on dividing by 7.
The number 510,517,813 leaves a remainder of 1 on dividing by 7.
Proof of correctness of the method
The method is based on the observation that 100 leaves a remainder of 2 when divided by 7. And since we are breaking the number into digit pairs we essentially have powers of 100.
1 mod 7 = 1
100 mod 7 = 2
10,000 mod 7 = 2^2 = 4
1,000,000 mod 7 = 2^3 = 8; 8 mod 7 = 1
100,000,000 mod 7 = 2^4 = 16; 16 mod 7 = 2
10,000,000,000 mod 7 = 2^5 = 32; 32 mod 7 = 4
And so on.
The correctness of the method is then established by the following chain of equalities:
Let N be the given number .
Divisibility by 11
Method
In order to check divisibility by 11, consider the alternating sum of the digits. For example with 907,071:
so 907,071 is divisible by 11.
We can either start with or since multiplying the whole by does not change anything.
Proof of correctness of the method
Considering that , we can write for any integer:
Divisibility by 13
Remainder Test
13 (1, −3, −4, −1, 3, 4, cycle goes on.)
If you are not comfortable with negative numbers, then use this sequence. (1, 10, 9, 12, 3, 4) | Divisibility rule | Wikipedia | 479 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Multiply the right most digit of the number with the left most number in the sequence shown above and the second right most digit to the second left most digit of the number in the sequence. The cycle goes on.
Example: What is the remainder when 321 is divided by 13?
Using the first sequence,
Ans: 1 × 1 + 2 × −3 + 3 × −4 = −17
Remainder = −17 mod 13 = 9
Example: What is the remainder when 1234567 is divided by 13?
Using the second sequence,
Answer: 7 × 1 + 6 × 10 + 5 × 9 + 4 × 12 + 3 × 3 + 2 × 4 + 1 × 1 = 178 mod 13 = 9
Remainder = 9
A recursive method can be derived using the fact that and that . This implies that a number is divisible by 13 iff removing the first digit and subtracting 3 times that digit from the new first digit yields a number divisible by 13. We also have the rule that 10 x + y is divisible iff x + 4 y is divisible by 13. For example, to test the divisibility of 1761 by 13 we can reduce this to the divisibility of 461 by the first rule. Using the second rule, this reduces to the divisibility of 50, and doing that again yields 5. So, 1761 is not divisible by 13.
Testing 871 this way reduces it to the divisibility of 91 using the second rule, and then 13 using that rule again, so we see that 871 is divisible by 13.
Beyond 30
Divisibility properties of numbers can be determined in two ways, depending on the type of the divisor.
Composite divisors
A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. Note that checking 3 and 12, or 2 and 18, would not be sufficient. A table of prime factors may be useful. | Divisibility rule | Wikipedia | 436 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
A composite divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because they have no smaller factors.
Prime divisors
The goal is to find an inverse to 10 modulo the prime under consideration (does not work for 2 or 5) and use that as a multiplier to make the divisibility of the original number by that prime depend on the divisibility of the new (usually smaller) number by the same prime.
Using 31 as an example, since 10 × (−3) = −30 = 1 mod 31, we get the rule for using y − 3x in the table below. Likewise, since 10 × (28) = 280 = 1 mod 31 also, we obtain a complementary rule y + 28x of the same kind - our choice of addition or subtraction being dictated by arithmetic convenience of the smaller value. In fact, this rule for prime divisors besides 2 and 5 is really a rule for divisibility by any integer relatively prime to 10 (including 33 and 39; see the table below). This is why the last divisibility condition in the tables above and below for any number relatively prime to 10 has the same kind of form (add or subtract some multiple of the last digit from the rest of the number).
Generalized divisibility rule
To test for divisibility by D, where D ends in 1, 3, 7, or 9, the following method can be used. Find any multiple of D ending in 9. (If D ends respectively in 1, 3, 7, or 9, then multiply by 9, 3, 7, or 1.) Then add 1 and divide by 10, denoting the result as m. Then a number N = 10t + q is divisible by D if and only if mq + t is divisible by D. If the number is too large, you can also break it down into several strings with e digits each, satisfying either 10e = 1 or 10e = −1 (mod D). The sum (or alternating sum) of the numbers have the same divisibility as the original one. | Divisibility rule | Wikipedia | 503 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
For example, to determine whether 913 = 10 × 91 + 3 is divisible by 11, find that m = (11 × 9 + 1) ÷ 10 = 10. Then mq + t = 10 × 3 + 91 = 121; this is divisible by 11 (with quotient 11), so 913 is also divisible by 11. As another example, to determine whether 689 = 10 × 68 + 9 is divisible by 53, find that m = (53 × 3 + 1) ÷ 10 = 16. Then mq + t = 16 × 9 + 68 = 212, which is divisible by 53 (with quotient 4); so 689 is also divisible by 53.
Alternatively, any number Q = 10c + d is divisible by n = 10a + b, such that gcd(n, 2, 5) = 1, if c + D(n)d = An for some integer A, where
The first few terms of the sequence, generated by D(n), are 1, 1, 5, 1, 10, 4, 12, 2, ... .
The piece wise form of D(n) and the sequence generated by it were first published by Bulgarian mathematician Ivan Stoykov in March 2020.
Proofs
Proof using basic algebra
Many of the simpler rules can be produced using only algebraic manipulation, creating binomials and rearranging them. By writing a number as the sum of each digit times a power of 10 each digit's power can be manipulated individually.
Case where all digits are summed
This method works for divisors that are factors of 10 − 1 = 9.
Using 3 as an example, 3 divides 9 = 10 − 1. That means (see modular arithmetic). The same for all the higher powers of 10: They are all congruent to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1:
which is exactly the sum of the digits.
Case where the alternating sum of digits is used
This method works for divisors that are factors of 10 + 1 = 11. | Divisibility rule | Wikipedia | 486 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
Using 11 as an example, 11 divides 11 = 10 + 1. That means . For the higher powers of 10, they are congruent to 1 for even powers and congruent to −1 for odd powers:
Like the previous case, we can substitute powers of 10 with congruent values:
which is also the difference between the sum of digits at odd positions and the sum of digits at even positions.
Case where only the last digit(s) matter
This applies to divisors that are a factor of a power of 10. This is because sufficiently high powers of the base are multiples of the divisor, and can be eliminated.
For example, in base 10, the factors of 101 include 2, 5, and 10. Therefore, divisibility by 2, 5, and 10 only depend on whether the last 1 digit is divisible by those divisors. The factors of 102 include 4 and 25, and divisibility by those only depend on the last 2 digits.
Case where only the last digit(s) are removed
Most numbers do not divide 9 or 10 evenly, but do divide a higher power of 10n or 10n − 1. In this case the number is still written in powers of 10, but not fully expanded.
For example, 7 does not divide 9 or 10, but does divide 98, which is close to 100. Thus, proceed from
where in this case a is any integer, and b can range from 0 to 99. Next,
and again expanding
and after eliminating the known multiple of 7, the result is
which is the rule "double the number formed by all but the last two digits, then add the last two digits".
Case where the last digit(s) is multiplied by a factor
The representation of the number may also be multiplied by any number relatively prime to the divisor without changing its divisibility. After observing that 7 divides 21, we can perform the following:
after multiplying by 2, this becomes
and then
Eliminating the 21 gives
and multiplying by −1 gives
Either of the last two rules may be used, depending on which is easier to perform. They correspond to the rule "subtract twice the last digit from the rest".
Proof using modular arithmetic | Divisibility rule | Wikipedia | 460 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
This section will illustrate the basic method; all the rules can be derived following the same procedure. The following requires a basic grounding in modular arithmetic; for divisibility other than by 2's and 5's the proofs rest on the basic fact that 10 mod m is invertible if 10 and m are relatively prime.
For 2n or 5n
Only the last n digits need to be checked.
Representing x as
and the divisibility of x is the same as that of z.
For 7
Since 10 × 5 ≡ 10 × (−2) ≡ 1 (mod 7), we can do the following:
Representing x as
so x is divisible by 7 if and only if y − 2z is divisible by 7. | Divisibility rule | Wikipedia | 154 | 991210 | https://en.wikipedia.org/wiki/Divisibility%20rule | Mathematics | Basics | null |
An acidity function is a measure of the acidity of a medium or solvent system, usually expressed in terms of its ability to donate protons to (or accept protons from) a solute (Brønsted acidity). The pH scale is by far the most commonly used acidity function, and is ideal for dilute aqueous solutions. Other acidity functions have been proposed for different environments, most notably the Hammett acidity function, H0, for superacid media and its modified version H− for superbasic media. The term acidity function is also used for measurements made on basic systems, and the term basicity function is uncommon.
Hammett-type acidity functions are defined in terms of a buffered medium containing a weak base B and its conjugate acid BH+:
where pKa is the dissociation constant of BH+. They were originally measured by using nitroanilines as weak bases or acid-base indicators and by measuring the concentrations of the protonated and unprotonated forms with UV-visible spectroscopy. Other spectroscopic methods, such as NMR, may also be used. The function H− is defined similarly for strong bases:
Here BH is a weak acid used as an acid-base indicator, and B− is its conjugate base. | Acidity function | Wikipedia | 276 | 10851309 | https://en.wikipedia.org/wiki/Acidity%20function | Physical sciences | Concepts | Chemistry |
Comparison of acidity functions with aqueous acidity
In dilute aqueous solution, the predominant acid species is the hydrated hydrogen ion H3O+ (or more accurately [H(OH2)n]+). In this case H0 and H− are equivalent to pH values determined by the buffer equation or Henderson-Hasselbalch equation.
However, an H0 value of −21 (a 25% solution of SbF5 in HSO3F) does not imply a hydrogen ion concentration of 1021 mol/dm3: such a "solution" would have a density more than a hundred times greater than a neutron star. Rather, H0 = −21 implies that the reactivity (protonating power) of the solvated hydrogen ions is 1021 times greater than the reactivity of the hydrated hydrogen ions in an aqueous solution of pH 0. The actual reactive species are different in the two cases, but both can be considered to be sources of H+, i.e. Brønsted acids. The hydrogen ion H+ never exists on its own in a condensed phase, as it is always solvated to a certain extent. The high negative value of H0 in SbF5/HSO3F mixtures indicates that the solvation of the hydrogen ion is much weaker in this solvent system than in water. Other way of expressing the same phenomenon is to say that SbF5·FSO3H is a much stronger proton donor than H3O+. | Acidity function | Wikipedia | 314 | 10851309 | https://en.wikipedia.org/wiki/Acidity%20function | Physical sciences | Concepts | Chemistry |
A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which itself was a rediscovery of Allan Marquand's 1881 logical diagram (aka. Marquand diagram). It is also useful for understanding logic circuits. Karnaugh maps are also known as Marquand–Veitch diagrams, Svoboda charts -(albeit only rarely)- and Karnaugh–Veitch maps (KV maps).
Definition
A Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. It also permits the rapid identification and elimination of potential race conditions.
The required Boolean results are transferred from a truth table onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code, and each cell position represents one combination of input conditions. Cells are also known as minterms, while each cell value represents the corresponding output value of the Boolean function. Optimal groups of 1s or 0s are identified, which represent the terms of a canonical form of the logic in the original truth table. These terms can be used to write a minimal Boolean expression representing the required logic.
Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using the minimal number of logic gates. A sum-of-products expression (SOP) can always be implemented using AND gates feeding into an OR gate, and a product-of-sums expression (POS) leads to OR gates feeding an AND gate. The POS expression gives a complement of the function (if F is the function so its complement will be F'). Karnaugh maps can also be used to simplify logic expressions in software design. Boolean conditions, as used for example in conditional statements, can get very complicated, which makes the code difficult to read and to maintain. Once minimised, canonical sum-of-products and product-of-sums expressions can be implemented directly using AND and OR logic operators.
Example
Karnaugh maps are used to facilitate the simplification of Boolean algebra functions. For example, consider the Boolean function described by the following truth table. | Karnaugh map | Wikipedia | 485 | 10854684 | https://en.wikipedia.org/wiki/Karnaugh%20map | Mathematics | Mathematical logic | null |
Following are two different notations describing the same function in unsimplified Boolean algebra, using the Boolean variables , , , and their inverses.
where are the minterms to map (i.e., rows that have output 1 in the truth table).
where are the maxterms to map (i.e., rows that have output 0 in the truth table).
Construction
In the example above, the four input variables can be combined in 16 different ways, so the truth table has 16 rows, and the Karnaugh map has 16 positions. The Karnaugh map is therefore arranged in a 4 × 4 grid.
The row and column indices (shown across the top and down the left side of the Karnaugh map) are ordered in Gray code rather than binary numerical order. Gray code ensures that only one variable changes between each pair of adjacent cells. Each cell of the completed Karnaugh map contains a binary digit representing the function's output for that combination of inputs.
Grouping
After the Karnaugh map has been constructed, it is used to find one of the simplest possible forms — a canonical form — for the information in the truth table. Adjacent 1s in the Karnaugh map represent opportunities to simplify the expression. The minterms ('minimal terms') for the final expression are found by encircling groups of 1s in the map. Minterm groups must be rectangular and must have an area that is a power of two (i.e., 1, 2, 4, 8...). Minterm rectangles should be as large as possible without containing any 0s. Groups may overlap in order to make each one larger. The optimal groupings in the example below are marked by the green, red and blue lines, and the red and green groups overlap. The red group is a 2 × 2 square, the green group is a 4 × 1 rectangle, and the overlap area is indicated in brown.
The cells are often denoted by a shorthand which describes the logical value of the inputs that the cell covers. For example, would mean a cell which covers the 2x2 area where and are true, i.e. the cells numbered 13, 9, 15, 11 in the diagram above. On the other hand, would mean the cells where is true and is false (that is, is true). | Karnaugh map | Wikipedia | 493 | 10854684 | https://en.wikipedia.org/wiki/Karnaugh%20map | Mathematics | Mathematical logic | null |
The grid is toroidally connected, which means that rectangular groups can wrap across the edges (see picture). Cells on the extreme right are actually 'adjacent' to those on the far left, in the sense that the corresponding input values only differ by one bit; similarly, so are those at the very top and those at the bottom. Therefore, can be a valid term—it includes cells 12 and 8 at the top, and wraps to the bottom to include cells 10 and 14—as is , which includes the four corners.
Solution
Once the Karnaugh map has been constructed and the adjacent 1s linked by rectangular and square boxes, the algebraic minterms can be found by examining which variables stay the same within each box.
For the red grouping:
A is the same and is equal to 1 throughout the box, therefore it should be included in the algebraic representation of the red minterm.
B does not maintain the same state (it shifts from 1 to 0), and should therefore be excluded.
C does not change. It is always 0, so its complement, NOT-C, should be included. Thus, should be included.
D changes, so it is excluded.
Thus the first minterm in the Boolean sum-of-products expression is .
For the green grouping, A and B maintain the same state, while C and D change. B is 0 and has to be negated before it can be included. The second term is therefore . Note that it is acceptable that the green grouping overlaps with the red one.
In the same way, the blue grouping gives the term .
The solutions of each grouping are combined: the normal form of the circuit is .
Thus the Karnaugh map has guided a simplification of
It would also have been possible to derive this simplification by carefully applying the axioms of Boolean algebra, but the time it takes to do that grows exponentially with the number of terms.
Inverse
The inverse of a function is solved in the same way by grouping the 0s instead.
The three terms to cover the inverse are all shown with grey boxes with different colored borders:
:
:
:
This yields the inverse:
Through the use of De Morgan's laws, the product of sums can be determined:
Don't cares | Karnaugh map | Wikipedia | 463 | 10854684 | https://en.wikipedia.org/wiki/Karnaugh%20map | Mathematics | Mathematical logic | null |
Karnaugh maps also allow easier minimizations of functions whose truth tables include "don't care" conditions. A "don't care" condition is a combination of inputs for which the designer doesn't care what the output is. Therefore, "don't care" conditions can either be included in or excluded from any rectangular group, whichever makes it larger. They are usually indicated on the map with a dash or X.
The example on the right is the same as the example above but with the value of f(1,1,1,1) replaced by a "don't care". This allows the red term to expand all the way down and, thus, removes the green term completely.
This yields the new minimum equation:
Note that the first term is just , not . In this case, the don't care has dropped a term (the green rectangle); simplified another (the red one); and removed the race hazard (removing the yellow term as shown in the following section on race hazards).
The inverse case is simplified as follows:
Through the use of De Morgan's laws, the product of sums can be determined:
Race hazards
Elimination
Karnaugh maps are useful for detecting and eliminating race conditions. Race hazards are very easy to spot using a Karnaugh map, because a race condition may exist when moving between any pair of adjacent, but disjoint, regions circumscribed on the map. However, because of the nature of Gray coding, adjacent has a special definition explained above – we're in fact moving on a torus, rather than a rectangle, wrapping around the top, bottom, and the sides.
In the example above, a potential race condition exists when C is 1 and D is 0, A is 1, and B changes from 1 to 0 (moving from the blue state to the green state). For this case, the output is defined to remain unchanged at 1, but because this transition is not covered by a specific term in the equation, a potential for a glitch (a momentary transition of the output to 0) exists.
There is a second potential glitch in the same example that is more difficult to spot: when D is 0 and A and B are both 1, with C changing from 1 to 0 (moving from the blue state to the red state). In this case the glitch wraps around from the top of the map to the bottom. | Karnaugh map | Wikipedia | 506 | 10854684 | https://en.wikipedia.org/wiki/Karnaugh%20map | Mathematics | Mathematical logic | null |
Whether glitches will actually occur depends on the physical nature of the implementation, and whether we need to worry about it depends on the application. In clocked logic, it is enough that the logic settles on the desired value in time to meet the timing deadline. In our example, we are not considering clocked logic.
In our case, an additional term of would eliminate the potential race hazard, bridging between the green and blue output states or blue and red output states: this is shown as the yellow region (which wraps around from the bottom to the top of the right half) in the adjacent diagram.
The term is redundant in terms of the static logic of the system, but such redundant, or consensus terms, are often needed to assure race-free dynamic performance.
Similarly, an additional term of must be added to the inverse to eliminate another potential race hazard. Applying De Morgan's laws creates another product of sums expression for f, but with a new factor of .
2-variable map examples
The following are all the possible 2-variable, 2 × 2 Karnaugh maps. Listed with each is the minterms as a function of and the race hazard free (see previous section) minimum equation. A minterm is defined as an expression that gives the most minimal form of expression of the mapped variables. All possible horizontal and vertical interconnected blocks can be formed. These blocks must be of the size of the powers of 2 (1, 2, 4, 8, 16, 32, ...). These expressions create a minimal logical mapping of the minimal logic variable expressions for the binary expressions to be mapped. Here are all the blocks with one field.
A block can be continued across the bottom, top, left, or right of the chart. That can even wrap beyond the edge of the chart for variable minimization. This is because each logic variable corresponds to each vertical column and horizontal row. A visualization of the k-map can be considered cylindrical. The fields at edges on the left and right are adjacent, and the top and bottom are adjacent. K-Maps for four variables must be depicted as a donut or torus shape. The four corners of the square drawn by the k-map are adjacent. Still more complex maps are needed for 5 variables and more.
Related graphical methods | Karnaugh map | Wikipedia | 471 | 10854684 | https://en.wikipedia.org/wiki/Karnaugh%20map | Mathematics | Mathematical logic | null |
Related graphical minimization methods include:
Marquand diagram (1881) by Allan Marquand (1853–1924)
Veitch chart (1952) by Edward W. Veitch (1924–2013)
Svoboda chart (1956) by Antonín Svoboda (1907–1980)
Mahoney map (M-map, designation numbers, 1963) by Matthew V. Mahoney (a reflection-symmetrical extension of Karnaugh maps for larger numbers of inputs)
Reduced Karnaugh map (RKM) techniques (from 1969) like infrequent variables, map-entered variables (MEV), variable-entered map (VEM) or variable-entered Karnaugh map (VEKM) by G. W. Schultz, Thomas E. Osborne, Christopher R. Clare, J. Robert Burgoon, Larry L. Dornhoff, William I. Fletcher, Ali M. Rushdi and others (several successive Karnaugh map extensions based on variable inputs for a larger numbers of inputs)
Minterm-ring map (MRM, 1990) by Thomas R. McCalla (a three-dimensional extension of Karnaugh maps for larger numbers of inputs) | Karnaugh map | Wikipedia | 245 | 10854684 | https://en.wikipedia.org/wiki/Karnaugh%20map | Mathematics | Mathematical logic | null |
Flory–Huggins solution theory is a lattice model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.
Theory
The thermodynamic equation for the Gibbs energy change accompanying mixing at constant temperature and (external) pressure is
A change, denoted by , is the value of a variable for a solution or mixture minus the values for the pure components considered separately. The objective is to find explicit formulas for and , the enthalpy and entropy increments associated with the mixing process.
The result obtained by Flory and Huggins is
The right-hand side is a function of the number of moles and volume fraction of solvent (component ), the number of moles and volume fraction of polymer (component ), with the introduction of a parameter to take account of the energy of interdispersing polymer and solvent molecules. is the gas constant and is the absolute temperature. The volume fraction is analogous to the mole fraction, but is weighted to take account of the relative sizes of the molecules. For a small solute, the mole fractions would appear instead, and this modification is the innovation due to Flory and Huggins. In the most general case the mixing parameter, , is a free energy parameter, thus including an entropic component.
Derivation
We first calculate the entropy of mixing, the increase in the uncertainty about the locations of the molecules when they are interspersed. In the pure condensed phases – solvent and polymer – everywhere we look we find a molecule. Of course, any notion of "finding" a molecule in a given location is a thought experiment since we can't actually examine spatial locations the size of molecules. The expression for the entropy of mixing of small molecules in terms of mole fractions is no longer reasonable when the solute is a macromolecular chain. We take account of this dissymmetry in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on a lattice. Each site is occupied by exactly one molecule of the solvent or by one monomer of the polymer chain, so the total number of sites is
where is the number of solvent molecules and is the number of polymer molecules, each of which has segments. | Flory–Huggins solution theory | Wikipedia | 497 | 3010589 | https://en.wikipedia.org/wiki/Flory%E2%80%93Huggins%20solution%20theory | Physical sciences | Thermodynamics | Chemistry |
For a random walk on a lattice we can calculate the entropy change (the increase in spatial uncertainty) as a result of mixing solute and solvent.
where is the Boltzmann constant. Define the lattice volume fractions and
These are also the probabilities that a given lattice site, chosen at random, is occupied by a solvent molecule or a polymer segment, respectively. Thus
For a small solute whose molecules occupy just one lattice site, equals one, the volume fractions reduce to molecular or mole fractions, and we recover the usual entropy of mixing.
In addition to the entropic effect, we can expect an enthalpy change. There are three molecular interactions to consider: solvent-solvent , monomer-monomer (not the covalent bonding, but between different chain sections), and monomer-solvent . Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is
The total number of such contacts is
where is the coordination number, the number of nearest neighbors for a lattice site, each one occupied either by one chain segment or a solvent molecule. That is, is the total number of polymer segments (monomers) in the solution, so is the number of nearest-neighbor sites to all the polymer segments. Multiplying by the probability that any such site is occupied by a solvent molecule, we obtain the total number of polymer-solvent molecular interactions. An approximation following mean field theory is made by following this procedure, thereby reducing the complex problem of many interactions to a simpler problem of one interaction.
The enthalpy change is equal to the energy change per polymer monomer-solvent interaction multiplied by the number of such interactions
The polymer-solvent interaction parameter chi is defined as
It depends on the nature of both the solvent and the solute, and is the only material-specific parameter in the model. The enthalpy change becomes
Assembling terms, the total free energy change is
where we have converted the expression from molecules and to moles and by transferring the Avogadro constant to the gas constant .
The value of the interaction parameter can be estimated from the Hildebrand solubility parameters and
where is the actual volume of a polymer segment. | Flory–Huggins solution theory | Wikipedia | 456 | 3010589 | https://en.wikipedia.org/wiki/Flory%E2%80%93Huggins%20solution%20theory | Physical sciences | Thermodynamics | Chemistry |
In the most general case the interaction and the ensuing mixing parameter, , is a free energy parameter, thus including an entropic component. This means that aside to the regular mixing entropy there is another entropic contribution from the interaction between solvent and monomer. This contribution is sometimes very important in order to make quantitative predictions of thermodynamic properties.
More advanced solution theories exist, such as the Flory–Krigbaum theory.
Liquid-liquid phase separation
Polymers can separate out from the solvent, and do so in a characteristic way. The Flory–Huggins free energy per unit volume, for a polymer with monomers, can be written in a simple dimensionless form
for the volume fraction of monomers, and . The osmotic pressure (in reduced units) is
.
The polymer solution is stable with respect to small fluctuations when the second derivative of this free energy is positive. This second derivative is
and the solution first becomes unstable when this and the third derivative
are both equal to zero. A little algebra then shows that the polymer solution first becomes unstable at a critical point at
This means that for all values of the monomer-solvent effective interaction is weakly repulsive, but this is too weak to cause liquid/liquid separation. However, when , there is separation into two coexisting phases, one richer in polymer but poorer in solvent, than the other.
The unusual feature of the liquid/liquid phase separation is that it is highly asymmetric: the volume fraction of monomers at the critical point is approximately , which is very small for large polymers. The amount of polymer in the solvent-rich/polymer-poor coexisting phase is extremely small for long polymers. The solvent-rich phase is close to pure solvent. This is peculiar to polymers, a mixture of small molecules can be approximated using the Flory–Huggins expression with , and then and both coexisting phases are far from pure.
Polymer blends
Synthetic polymers rarely consist of chains of uniform length in solvent. The Flory–Huggins free energy density can be generalized to an N-component mixture of polymers with lengths by
For a binary polymer blend, where one species consists of monomers and the other monomers this simplifies to | Flory–Huggins solution theory | Wikipedia | 458 | 3010589 | https://en.wikipedia.org/wiki/Flory%E2%80%93Huggins%20solution%20theory | Physical sciences | Thermodynamics | Chemistry |
As in the case for dilute polymer solutions, the first two terms on the right-hand side represent the entropy of mixing. For large polymers of and these terms are negligibly small. This implies that for a stable mixture to exist , so for polymers A and B to blend their segments must attract one another.
Limitations
Flory–Huggins theory tends to agree well with experiments in the semi-dilute concentration regime and can be used to fit data for even more complicated blends with higher concentrations. The theory qualitatively predicts phase separation, the tendency for high molecular weight species to be immiscible, the interaction-temperature dependence and other features commonly observed in polymer mixtures. However, unmodified Flory–Huggins theory fails to predict the lower critical solution temperature observed in some polymer blends and the lack of dependence of the critical temperature on chain length . Additionally, it can be shown that for a binary blend of polymer species with equal chain lengths the critical concentration should be ; however, polymers blends have been observed where this parameter is highly asymmetric. In certain blends, mixing entropy can dominate over monomer interaction. By adopting the mean-field approximation, parameter complex dependence on temperature, blend composition, and chain length was discarded. Specifically, interactions beyond the nearest neighbor may be highly relevant to the behavior of the blend and the distribution of polymer segments is not necessarily uniform, so certain lattice sites may experience interaction energies disparate from that approximated by the mean-field theory.
One well-studied effect on interaction energies neglected by unmodified Flory–Huggins theory is chain correlation. In dilute polymer mixtures, where chains are well separated, intramolecular forces between monomers of the polymer chain dominate and drive demixing leading to regions where polymer concentration is high. As the polymer concentration increases, chains tend to overlap and the effect becomes less important. In fact, the demarcation between dilute and semi-dilute solutions is commonly defined by the concentration where polymers begin to overlap which can be estimated as
Here, m is the mass of a single polymer chain, and is the chain's radius of gyration. | Flory–Huggins solution theory | Wikipedia | 446 | 3010589 | https://en.wikipedia.org/wiki/Flory%E2%80%93Huggins%20solution%20theory | Physical sciences | Thermodynamics | Chemistry |
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1770).
The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size.
The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method.
Geometrical description
Purpose and why it works
Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.
The idea is that while the curve is initially unknown, its starting point, which we denote by is known (see Figure 1). Then, from the differential equation, the slope to the curve at can be computed, and so, the tangent line.
Take a small step along that tangent line up to a point Along this small step, the slope does not change too much, so will be close to the curve. If we pretend that is still on the curve, the same reasoning as for the point above can be used. After several steps, a polygonal curve () is computed. In general, this curve does not diverge too far from the original unknown curve, and the error between the two curves can be made small if the step size is small enough and the interval of computation is finite.
First-order process
When given the values for and , and the derivative of is a given function of and denoted as . Begin the process by setting . Next, choose a value for the size of every step along t-axis, and set (or equivalently ). Now, the Euler method is used to find from and : | Euler method | Wikipedia | 482 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
The value of is an approximation of the solution at time , i.e., . The Euler method is explicit, i.e. the solution is an explicit function of for .
Higher-order process
While the Euler method integrates a first-order ODE, any ODE of order can be represented as a system of first-order ODEs. When given the ODE of order defined as
as well as , , and , we implement the following formula until we reach the approximation of the solution to the ODE at the desired time:
These first-order systems can be handled by Euler's method or, in fact, by any other scheme for first-order systems.
First-order example
Given the initial value problem
we would like to use the Euler method to approximate .
Using step size equal to 1 ()
The Euler method is
so first we must compute . In this simple differential equation, the function is defined by . We have
By doing the above step, we have found the slope of the line that is tangent to the solution curve at the point . Recall that the slope is defined as the change in divided by the change in , or .
The next step is to multiply the above value by the step size , which we take equal to one here:
Since the step size is the change in , when we multiply the step size and the slope of the tangent, we get a change in value. This value is then added to the initial value to obtain the next value to be used for computations.
The above steps should be repeated to find , and .
Due to the repetitive nature of this algorithm, it can be helpful to organize computations in a chart form, as seen below, to avoid making errors.
{| class="wikitable"
|-
! !! !! !! !! !! !!
|-
| 0 || 1 || 0 || 1 || 1 || 1 || 2
|-
| 1 || 2 || 1 || 2 || 1 || 2 || 4
|-
| 2 || 4 || 2 || 4 || 1 || 4 || 8
|-
| 3 || 8 || 3 || 8 || 1 || 8 || 16
|} | Euler method | Wikipedia | 464 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
The conclusion of this computation is that . The exact solution of the differential equation is , so . Although the approximation of the Euler method was not very precise in this specific case, particularly due to a large value step size , its behaviour is qualitatively correct as the figure shows.
Using other step sizes
As suggested in the introduction, the Euler method is more accurate if the step size is smaller. The table below shows the result with different step sizes. The top row corresponds to the example in the previous section, and the second row is illustrated in the figure.
{| class="wikitable"
|-
! step size !! result of Euler's method !! error
|-
| 1 || 16.00 || 38.60
|-
| 0.25 || 35.53 || 19.07
|-
| 0.1 || 45.26 || 9.34
|-
| 0.05 || 49.56 || 5.04
|-
| 0.025 || 51.98 || 2.62
|-
| 0.0125 || 53.26 || 1.34
|}
The error recorded in the last column of the table is the difference between the exact solution at and the Euler approximation. In the bottom of the table, the step size is half the step size in the previous row, and the error is also approximately half the error in the previous row. This suggests that the error is roughly proportional to the step size, at least for fairly small values of the step size. This is true in general, also for other equations; see the section Global truncation error for more details.
Other methods, such as the midpoint method also illustrated in the figures, behave more favourably: the global error of the midpoint method is roughly proportional to the square of the step size. For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order.
We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps. This large number of steps entails a high computational cost. For this reason, higher-order methods are employed such as Runge–Kutta methods or linear multistep methods, especially if a high accuracy is desired. | Euler method | Wikipedia | 496 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
Higher-order example
For this third-order example, assume that the following information is given:
From this we can isolate y''' to get the equation:
Using that we can get the solution for :And using the solution for , we can get the solution for :We can continue this process using the same formula as long as necessary to find whichever desired.
Derivation
The Euler method can be derived in a number of ways.
(1) Firstly, there is the geometrical description above.
(2) Another possibility is to consider the Taylor expansion of the function around :
The differential equation states that . If this is substituted in the Taylor expansion and the quadratic and higher-order terms are ignored, the Euler method arises.
The Taylor expansion is used below to analyze the error committed by the Euler method, and it can be extended to produce Runge–Kutta methods.
(3) A closely related derivation is to substitute the forward finite difference formula for the derivative,
in the differential equation . Again, this yields the Euler method.
A similar computation leads to the midpoint method and the backward Euler method.
(4) Finally, one can integrate the differential equation from to and apply the fundamental theorem of calculus to get:
Now approximate the integral by the left-hand rectangle method (with only one rectangle):
Combining both equations, one finds again the Euler method.
This line of thought can be continued to arrive at various linear multistep methods.
Local truncation error
The local truncation error of the Euler method is the error made in a single step. It is the difference between the numerical solution after one step, , and the exact solution at time . The numerical solution is given by
For the exact solution, we use the Taylor expansion mentioned in the section Derivation above:
The local truncation error (LTE) introduced by the Euler method is given by the difference between these equations:
This result is valid if has a bounded third derivative.
This shows that for small , the local truncation error is approximately proportional to . This makes the Euler method less accurate than higher-order techniques such as Runge-Kutta methods and linear multistep methods, for which the local truncation error is proportional to a higher power of the step size.
A slightly different formulation for the local truncation error can be obtained by using the Lagrange form for the remainder term in Taylor's theorem. If has a continuous second derivative, then there exists a such that | Euler method | Wikipedia | 512 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
In the above expressions for the error, the second derivative of the unknown exact solution can be replaced by an expression involving the right-hand side of the differential equation. Indeed, it follows from the equation that
Global truncation error
The global truncation error is the error at a fixed time , after however many steps the method needs to take to reach that time from the initial time. The global truncation error is the cumulative effect of the local truncation errors committed in each step. The number of steps is easily determined to be , which is proportional to , and the error committed in each step is proportional to (see the previous section). Thus, it is to be expected that the global truncation error will be proportional to .
This intuitive reasoning can be made precise. If the solution has a bounded second derivative and is Lipschitz continuous in its second argument, then the global truncation error (denoted as ) is bounded by
where is an upper bound on the second derivative of on the given interval and is the Lipschitz constant of . Or more simply, when , the value (such that is treated as a constant). In contrast, where function is the exact solution which only contains the variable.
The precise form of this bound is of little practical importance, as in most cases the bound vastly overestimates the actual error committed by the Euler method. What is important is that it shows that the global truncation error is (approximately) proportional to . For this reason, the Euler method is said to be first order.
Example
If we have the differential equation , and the exact solution , and we want to find and for when . Thus we can find the error bound at t=2.5 and h=0.5:
Notice that t0 is equal to 2 because it is the lower bound for t in .
Numerical stability
The Euler method can also be numerically unstable, especially for stiff equations, meaning that the numerical solution grows very large for equations where the exact solution does not. This can be illustrated using the linear equation
The exact solution is , which decays to zero as . However, if the Euler method is applied to this equation with step size , then the numerical solution is qualitatively wrong: It oscillates and grows (see the figure). This is what it means to be unstable. If a smaller step size is used, for instance , then the numerical solution does decay to zero. | Euler method | Wikipedia | 501 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
If the Euler method is applied to the linear equation , then the numerical solution is unstable if the product is outside the region
illustrated on the right. This region is called the (linear) stability region. In the example, , so if then which is outside the stability region, and thus the numerical solution is unstable.
This limitation — along with its slow convergence of error with — means that the Euler method is not often used, except as a simple example of numerical integration. Frequently models of physical systems contain terms representing fast-decaying elements (i.e. with large negative exponential arguments). Even when these are not of interest in the overall solution, the instability they can induce means that an exceptionally small timestep would be required if the Euler method is used.
Rounding errors
In step of the Euler method, the rounding error is roughly of the magnitude where is the machine epsilon. Assuming that the rounding errors are independent random variables, the expected total rounding error is proportional to . Thus, for extremely small values of the step size the truncation error will be small but the effect of rounding error may be big. Most of the effect of rounding error can be easily avoided if compensated summation is used in the formula for the Euler method.
Modifications and extensions
A simple modification of the Euler method which eliminates the stability problems noted above is the backward Euler method:
This differs from the (standard, or forward) Euler method in that the function is evaluated at the end point of the step, instead of the starting point. The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
Other modifications of the Euler method that help with stability yield the exponential Euler method or the semi-implicit Euler method.
More complicated methods can achieve a higher order (and more accuracy). One possibility is to use more function evaluations. This is illustrated by the midpoint method which is already mentioned in this article:
.
This leads to the family of Runge–Kutta methods.
The other possibility is to use more past values, as illustrated by the two-step Adams–Bashforth method:
This leads to the family of linear multistep methods. There are other modifications which uses techniques from compressive sensing to minimize memory usage | Euler method | Wikipedia | 490 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
In popular culture
In the film Hidden Figures, Katherine Goble resorts to the Euler method in calculating the re-entry of astronaut John Glenn from Earth orbit. | Euler method | Wikipedia | 32 | 3011098 | https://en.wikipedia.org/wiki/Euler%20method | Mathematics | Differential equations | null |
Florigens (or flowering hormone) are proteins capable of inducing flowering time in angiosperms. The prototypical florigen is encoded by the FT gene and its orthologs in Arabidopsis and other plants. Florigens are produced in the leaves, and act in the shoot apical meristem of buds and growing tips.
Mechanism
For a plant to begin flowering, it must undergo changes in its shoot apical meristem (SAM). However, there are multiple environmental factors affecting the plant even before it begins this process — in particular, light. It is through "the evolution of both internal and external control systems that enables plants to precisely regulate flowering so that it occurs at the optimal time for reproductive success." The way the plant determines this optimal time is through day-night periods through the use of photoperiodism. Although it was originally thought that the accumulation of photosynthetic products controlled the flowering of plants, two men by the names of Wightman Garner and Henry Allard proved it was not. They instead found that it was a matter of day length rather than the accumulation of the products within the plants that affected their flowering abilities.
Flowering plants fall into two main photoperiodic response categories:
"Short-day plants (SDPs) flower only in short days (qualitative SDPs), or their flowering is accelerated by short days (quantitative SDPs)"
"Long-day plants (LDPs) flower only in long days (qualitative LDPs), or their flowering is accelerated by long days (quantitative LDPs)"
These types of flowering plants are differentiated by the whether the day has exceeded some duration - usually calculated by 24-hour cycles - known as the critical day length. It is also important to note that there is no absolute value for the minimum day length as it varies greatly amid species. Until the correct amount of day length is reached, the plants ensure no flowering results. They do so through adaptations like preventing immature plants from responding to inadequate day lengths. Plants also have the ability to prevent the response of the photoperiodic stimulus until a certain temperature is reached. Species like winter wheat that rely on just that. The wheat require a cold period before being able to respond to the photoperiod. This is known as vernalization or overwintering. | Florigen | Wikipedia | 475 | 3011436 | https://en.wikipedia.org/wiki/Florigen | Biology and health sciences | Plant hormone | Biology |
This ebb-and-flow of flowering in plants is essentially controlled by an internal clock known as the endogenous oscillator. It is thought that these internal pacemakers "are regulated by the interaction of four sets of genes expressed in the dawn, morning, afternoon, and evening hours [and that] light may augment the amplitude of the oscillations by activating the morning and evening genes." The rhythms between these different genes are generated internally in the plants, starts with the leaves, but requires an environmental stimulus such as light. The light essentially stimulates the transmission of a floral stimulus (florigen) to the shoot apex when the correct amount of day-length is perceived. This process is known as photoperiodic induction and is a photoperiod-regulated process that is also dependent on the endogenous oscillator..
The current model suggests the involvement of multiple different factors. Research into florigen is predominately centred on the model organism and long day plant, Arabidopsis thaliana. Whilst much of the florigen pathways appear to be well conserved in other studied species, variations do exist. The mechanism may be broken down into three stages: photoperiod-regulated initiation, signal translocation via the phloem, and induction of flowering at the shoot apical meristem.
Initiation
In Arabidopsis thaliana, the signal is initiated by the production of messenger RNA (mRNA) coding a transcription factor called CONSTANS (CO). CO mRNA is produced approximately 12 hours after dawn, a cycle regulated by the plant's circadian rhythms, and is then translated into CO protein. However CO protein is stable only in light, so levels stay low throughout short days and are only able to peak at dusk during long days when there is still some light. CO protein promotes transcription of another gene called (FT). By this mechanism, CO protein may only reach levels capable of promoting FT transcription when exposed to long days. Hence, the transmission of florigen—and thus, the induction of flowering—relies on a comparison between the plant's perception of day/night and its own internal biological clock.
Translocation
The FT protein resulting from the short period of CO transcription factor activity is then transported via the phloem to the shoot apical meristem. | Florigen | Wikipedia | 480 | 3011436 | https://en.wikipedia.org/wiki/Florigen | Biology and health sciences | Plant hormone | Biology |
Flowering
Florigen is a systemically mobile signal that is synthesized in leaves and the transported via the phloem to the shoot apical meristem (SAM) where it initiates flowering. In Arabidopsis, the FLOWERING LOCUS T (FT) genes encode for the flowering hormone and in rice the hormone is encoded by Hd3a genes thereby making these genes orthologs. It was found though the use of transgenic plants that the Hd3a promoter in rice is located in the phloem of the leaf along with the Hd3a mRNA. However, the Hd3a protein is found in neither of these places but instead accumulates in the SAM which shows that Hd3a protein is first translated in leaves and then transported to the SAM via the phloem where floral transition is initiated; the same results occurred when looked at Arabidopsis. These results conclude that FT/Hd3a is the florigen signal that induces floral transition in plants.
Upon this conclusion, it became important to understand the process by which the FT protein causes floral transition once it reaches the SAM. The first clue came with looking at models from Arabidposis which suggested that a bZIP domain containing transcription factor, FD, is somehow interacting with FT to form a transcriptional complex that activates floral genes. Studies using rice found that there is an interaction between Hd3a and OsFD1, homologs of FT and FD respectively, that is mediated by the 14-3-3 protein GF14c. The 14-3-3 protein acts as intracellular florigen receptor that interacts directly with Hd3a and OsFD1 to form a tri-protein complex called the florigen activation complex (FAC) because it is essential for florigen function. The FAC works to activate genes needed to initiate flowering at the SAM; flowering genes in Arabidopsis include AP1, SOC1 and several SPL genes, which are targeted by a microRNA and in rice the flowering gene is OsMADS15 (a homolog of AP1).
Antiflorigen
Florigen is regulated by the action of an antiflorigen. Antiflorigens are hormones that are encoded by the same genes for florigen that work to counteract its function. The antiflorigen in Arabidopsis is TERMINAL FLOWER1 (TFL1) and in tomato it is SELF PRUNING (SP). | Florigen | Wikipedia | 505 | 3011436 | https://en.wikipedia.org/wiki/Florigen | Biology and health sciences | Plant hormone | Biology |
Research history
Florigen was first described by Soviet Armenian plant physiologist Mikhail Chailakhyan, who in 1937 demonstrated that floral induction can be transmitted through a graft from an induced plant to one that has not been induced to flower. Anton Lang showed that several long-day plants and biennials could be made to flower by treatment with gibberellin, even when grown under a non-flower-inducing (or non-inducing) photoperiod. This led to the suggestion that florigen may be made up of two classes of flowering hormones: Gibberellins and Anthesins. It was later postulated that during non-inducing photoperiods, long-day plants produce anthesin, but no gibberellin, while short-day plants produce gibberellin, but no anthesin. However, these findings did not account for the fact that short-day plants grown under non-inducing conditions (thus producing gibberellin) will not cause flowering of grafted long-day plants that are also under noninductive conditions (thus producing anthesin).
As a result of the problems with isolating florigen, and of the inconsistent results acquired, it has been suggested that florigen does not exist as an individual substance; rather, florigen's effect could be the result of a particular ratio of other hormones. However, more recent findings indicate that florigen does exist and is produced, or at least activated, in the leaves of the plant and that this signal is then transported via the phloem to the growing tip at the shoot apical meristem where the signal acts by inducing flowering. In Arabidopsis thaliana, some researchers have identified this signal as mRNA coded by the FLOWERING LOCUS T (FT) gene, others as the resulting FT protein. First report of FT mRNA being the signal transducer that moves from leaf to shoot apex came from the publication in Science Magazine. However, in 2007 other group of scientists made a breakthrough saying that it is not the mRNA, but the FT Protein that is transmitted from leaves to shoot possibly acting as "Florigen". The initial article that described FT mRNA as flowering stimuli was retracted by the authors themselves. | Florigen | Wikipedia | 460 | 3011436 | https://en.wikipedia.org/wiki/Florigen | Biology and health sciences | Plant hormone | Biology |
Triggers of gene transcription
There are three genes involved in clock-controlled flowering pathway, GIGANTEA (GI), CONSTANS (CO), and FLOWERING LOCUS T (FT). Constant overexpression of GI from the Cauliflower mosaic virus 35S promoter causes early flowering under short day so an increase in GI mRNA expression induces flowering. Also, GI increases the expression of FT and CO mRNA, and FT and CO mutants showed later flowering time than GI mutant. In other words, functional FT and CO genes are required for flowering under short day. In addition, these flowering genes accumulate during light phase and decline during dark phase, which are measured by green fluorescent protein. Thus, their expressions oscillate during the 24-hour light-dark-cycle. In conclusion, the accumulation of GI mRNA alone or GI, FT, and CO mRNA promote flowering in Arabidopsis thaliana and these genes expressed in the temporal sequence GI-CO-FT.
Action potential triggers calcium flux into neurons in animal or root apex cells in plants. The intracellular calcium signals are responsible for regulation of many biological functions in organisms. For instance, Ca2+ binding to calmodulin, a Ca2+-binding protein in animals and plants, controls gene transcriptions.
Flowering mechanism
A biological mechanism is proposed based on the information we have above. Light is the flowering signal of Arabidopsis thaliana. Light activates photo-receptors and triggers signal cascades in plant cells of apical or lateral meristems. Action potential is spread via the phloem to the root and more voltage-gated calcium channels are opened along the stem. This causes an influx of calcium ions in the plant. These ions bind to calmodulin and the Ca2+/CaM signaling system triggers the expression of GI mRNA or FT and CO mRNA. The accumulation of GI mRNA or GI-CO-FT mRNA during the day causing the plant to flower. | Florigen | Wikipedia | 394 | 3011436 | https://en.wikipedia.org/wiki/Florigen | Biology and health sciences | Plant hormone | Biology |
Many types of glucose tests exist and they can be used to estimate blood sugar levels at a given time or, over a longer period of time, to obtain average levels or to see how fast the body is able to normalize changed glucose levels. Eating food for example leads to elevated blood sugar levels. In healthy people, these levels quickly return to normal via increased cellular glucose uptake which is primarily mediated by increase in blood insulin levels.
Glucose tests can reveal temporary/long-term hyperglycemia or hypoglycemia. These conditions may not have obvious symptoms and can damage organs in the long-term. Abnormally high/low levels, slow return to normal levels from either of these conditions and/or inability to normalize blood sugar levels means that the person being tested probably has some kind of medical condition like type 2 diabetes which is caused by cellular insensitivity to insulin. Glucose tests are thus often used to diagnose such conditions.
Testing methods
Tests that can be performed at home are used in blood glucose monitoring for illnesses that have already been diagnosed medically so that these illnesses can be maintained via medication and meal timing. Some of the home testing methods include
fingerprick type of glucose meter - need to prick self finger 8-12 times a day.
continuous glucose monitor - the CGM monitors the glucose levels every 5 minutes approximately.
Laboratory tests are often used to diagnose illnesses and such methods include
fasting blood sugar (FBS), fasting plasma glucose (FPG): 10–16 hours after eating
glucose tolerance test: continuous testing
postprandial glucose test (PC): 2 hours after eating
random glucose test
Some laboratory tests don't measure glucose levels directly from body fluids or tissues but still indicate elevated blood sugar levels. Such tests measure the levels of glycated hemoglobin, other glycated proteins, 1,5-anhydroglucitol etc. from blood.
Use in medical diagnosis
Glucose testing can be used to diagnose or indicate certain medical conditions. | Glucose test | Wikipedia | 413 | 3012322 | https://en.wikipedia.org/wiki/Glucose%20test | Biology and health sciences | Diagnostics | Health |
High blood sugar may indicate
gestational diabetes. This temporary form of diabetes appears during pregnancy, and with glucose-controlling medication or insulin symptoms can be improved.
type 1 and type 2 diabetes or prediabetes. If diagnosed with diabetes, regular glucose tests can help manage or maintain conditions. Type 1, is commonly seen in children or teenagers whose bodies are not producing enough insulin. Type 2 diabetes, is typically seen in adults who are overweight. The insulin in their bodies are either not working normally, or there is not being enough produced.
Low blood sugar may indicate
insulin overuse
starvation
underactive thyroid
Addison's disease
insulinoma
kidney disease
Preparing for testing
Fasting prior to glucose testing may be required with some test types. Fasting blood sugar test, for example, requires 10–16 hour-long period of not eating before the test.
Blood sugar levels can be affected by some drugs and prior to some glucose tests these medications should be temporarily given up or their dosages should be decreased. Such drugs may include salicylates (Aspirin), birth control pills, corticosteroids, tricyclic antidepressants, lithium, diuretics and phenytoin.
Some foods contain caffeine (coffee, tea, colas, energy drinks etc.). Blood sugar levels of healthy people are generally not significantly changed by caffeine, but in diabetics caffeine intake may elevate these levels via its ability to stimulate the adrenergic nervous system.
Reference ranges
Fasting blood sugar
A level below 5.6 mmol/L (100 mg/dL) 10–16 hours without eating is normal. 5.6–6 mmol/L (100–109 mg/dL) may indicate prediabetes and oral glucose tolerance test (OGTT) should be offered to high-risk individuals (old people, those with high blood pressure etc.). 6.1–6.9 mmol/L (110–125 mg/dL) means OGTT should be offered even if other indicators of diabetes are not present. 7 mmol/L (126 mg/dL) and above indicates diabetes and the fasting test should be repeated.
Glucose tolerance test
Postprandial glucose test
Random glucose test | Glucose test | Wikipedia | 464 | 3012322 | https://en.wikipedia.org/wiki/Glucose%20test | Biology and health sciences | Diagnostics | Health |
The Rheic Ocean (; ) was an ocean which separated two major paleocontinents, Gondwana and Laurussia (Laurentia-Baltica-Avalonia). One of the principal oceans of the Paleozoic, its sutures today stretch from Mexico to Turkey and its closure resulted in the assembly of the supercontinent Pangaea and the formation of the Variscan–Alleghenian–Ouachita orogenies.
Etymology
The ocean located between Gondwana and Laurentia in the Early Cambrian was named for Iapetus, in Greek mythology the father of Atlas (from which source the Atlantic Ocean ultimately gets its name), just as the Iapetus Ocean was the predecessor of the Atlantic Ocean. The ocean between Gondwana and Laurussia (Laurentia–Baltica–Avalonia) that existed from the Early Ordovician to the Early Carboniferous was named the Rheic Ocean after Rhea, sister of Iapetus.
Geodynamic evolution
At the beginning of the Paleozoic Era, about 540 million years ago, most of the continental mass on Earth was clustered around the south pole as the paleocontinent Gondwana. The exception was formed by a number of smaller continents, such as Laurentia and Baltica. The Paleozoic ocean between Gondwana, Laurentia and Baltica is called the Iapetus Ocean. The northern edge of Gondwana had been dominated by the Cadomian orogeny during the Ediacaran period. This orogeny formed a cordillera-type volcanic arc where oceanic crust subducted below Gondwana. When a mid-oceanic ridge subducted at an oblique angle, extensional basins developed along the northern margin of Gondwana. During the late Cambrian to Early Ordovician these extensional basins had evolved a rift running along the northern edge of Gondwana. The rift in its turn evolved into a mid-oceanic ridge that separated small continental fragments such as Avalonia and Carolina from the main Gondwanan land mass, leading to the formation of the Rheic Ocean in the Early Ordovician.
As Avalonia-Carolina drifted north from Gondwana, the Rheic Ocean grew and reached its maximum width () in the Silurian. In this process, the Iapetus Ocean closed as Avalonia-Carolina collided with Laurentia and the Appalachian orogeny formed. | Rheic Ocean | Wikipedia | 495 | 3012682 | https://en.wikipedia.org/wiki/Rheic%20Ocean | Physical sciences | Paleogeography | Earth science |
The closure of the Rheic began in the Early Devonian and was completed in the Mississippian when Gondwana and Laurentia collided to form Pangaea. This closure resulted in the largest collisional orogen of the Palaeozoic: the Variscan and Alleghanian orogens between Gondwana's West African margin and southern Baltica and eastern Laurentia and the Ouachita orogeny between the Amazonian margin of Gondwana and southern Laurentia.
Effects on life
The Prague Basin, which was an archipelago of humid volcanic islands in the Rheic Ocean on the outer edges of what was then the Gondwanan shelf during the Silurian, was a major hotspot of plant biodiversity during the early stages of the Silurian-Devonian Terrestrial Revolution. The geologically rapid environmental changes associated with the formation and erosion of volcanic islands and high rates of endemism associated with island ecosystems likely played an important role in driving the rapid early diversification of vascular plants.
It is believed that the closure of the Rheic, alongside the simultaneous onset of the Late Palaeozoic Ice Age, may have sparked the Carboniferous-Earliest Permian Biodiversification Event, an evolutionary radiation of marine life dominated by increase in species richness of fusulinids and brachiopods. | Rheic Ocean | Wikipedia | 271 | 3012682 | https://en.wikipedia.org/wiki/Rheic%20Ocean | Physical sciences | Paleogeography | Earth science |
Teleosauridae is a family of extinct typically marine crocodylomorphs similar to the modern gharial that lived during the Jurassic period. Teleosaurids were thalattosuchians closely related to the fully aquatic metriorhynchoids, but were less adapted to an open-ocean, pelagic lifestyle. The family was originally coined to include all the semi-aquatic (i.e. non-metriorhynchoid) thalattosuchians and was equivalent to the modern superfamily Teleosauroidea. However, as teleosauroid relationships and diversity was better studied in the 21st century, the division of teleosauroids into two distinct evolutionary lineages led to the establishment of Teleosauridae as a more restrictive family within the group, together with its sister family Machimosauridae.
Amongst teleosauroids, teleosaurids were generally smaller and less common than machimosaurids, suggesting the two families occupied different niches, similar to modern species of crocodilians. However, teleosaurids were more diverse than machimosaurids, with generalist coastal predators (Mystriosaurus), long-snouted marine piscivores (Bathysuchus), and potentially even long-snouted, semi-terrestrial predators (Teleosaurus). Additionally, teleosaurids occupied a wider range of habitats than machimosaurids, from semi-marine coasts and estuaries, the open-ocean, freshwater, and potentially even semi-terrestrial environments.
Classification
Teleosauridae is phylogenetically defined in the PhyloCode by Mark T. Young and colleagues as "the largest clade within Teleosauroidea containing Teleosaurus cadomensis but not Machimosaurus hugii. Teleosauridae is split into two subfamilies, the Teleosaurinae and the Aeolodontinae, the former defined in the PhyloCode as "the largest clade within Teleosauroidea containing Teleosaurus cadomensis, but not Aeolodon priscus and the latter defined in the PhyloCode as "the largest clade within Teleosauroidea containing Aeolodon pricus, but not Teleosaurus cadomensis. | Teleosauridae | Wikipedia | 477 | 5508267 | https://en.wikipedia.org/wiki/Teleosauridae | Biology and health sciences | Other prehistoric archosaurs | Animals |
Palaeobiology
Teleosaurids were originally regarded as marine analogues to modern gharials, as they both typically share long, tubular snouts and narrow teeth. However, differences in the jaws, teeth, and skeleton of different teleosaurids suggest that they were more ecologically diverse than this. Earlier teleosaurids were coastal semi-aquatic generalists, while the two subfamilies were more specialised. Teleosaurines appear to have been semi-terrestrial, as they were more heavily armoured and had forward-facing nostrils. In contrast, aeolodontines have been found in deep marine waters and had reduced armour, implying that they were open water predators similar to metriorhynchoids (although the oldest aeolodontine, Mycterosuchus, appears to have been semi-terrestrial, similar to teleosaurines).
Palaeoecology
Distribution
Definitive fossils of teleosaurids are restricted to Laurasia, with material found in Europe(England, France, Germany, Italy, Portugal, Russia and Switzerland) and Asia (China and Thailand, and possibly India). | Teleosauridae | Wikipedia | 234 | 5508267 | https://en.wikipedia.org/wiki/Teleosauridae | Biology and health sciences | Other prehistoric archosaurs | Animals |
The Lygaeidae are a family in the Hemiptera (true bugs), with more than 110 genera in four subfamilies. The family is commonly referred to as seed bugs, and less commonly, milkweed bugs, or ground bugs. Many species feed on seeds, some on sap or seed pods, others are omnivores and a few, such as the wekiu bug, are insectivores. Insects in this family are distributed across the world.
The family was vastly larger, but numerous former subfamilies have been removed and given independent family status, including the Artheneidae, Blissidae, Cryptorhamphidae, Cymidae, Geocoridae, Heterogastridae, Ninidae, Oxycarenidae and Rhyparochromidae, which together constituted well over half of the former family.
The bizarre and mysterious beetle-like Psamminae were formerly often placed in the Piesmatidae, but this is almost certainly incorrect. Their true affiliations are not entirely resolved.
Distinguishing characteristics
Lygaeidae are oval or elongate in body shape and have four-segmented antennae. Lygaeidae can be distinguished from Miridae (plant bugs) by the presence of ocelli, or simple eyes. They are distinguished from Coreidae (squash bugs) by the number of veins in the membrane of the front wings, as Lygaeidae have only four or five veins.
Subfamilies and selected genera
An incomplete list of Lygaeidae genera is subdivided as:
subfamily Ischnorhynchinae Stål, 1872
Crompus Stål, 1874
Kleidocerys Stephens, 1829
subfamily Lygaeinae Schilling, 1829
Lygaeus Fabricius, 1794
Oncopeltus Stål, 1868
Melanocoryphus Stål, 1872
Spilostethus Stål, 1868
Tropidothorax Bergroth, 1894
subfamily Orsillinae Stål, 1872
Nysius Dallas, 1852
Orsillus Dallas, 1852
subfamily † Lygaenocorinae
Unplaced genera
Lygaeites Heer, 1853
The Pachygronthinae Stål, 1865 (type genus Pachygrontha Germar, 1840) may be placed here or elevated to the family Pachygronthidae.
Gallery | Lygaeidae | Wikipedia | 484 | 5512111 | https://en.wikipedia.org/wiki/Lygaeidae | Biology and health sciences | Hemiptera (true bugs) | Animals |
The Egyptian mongoose (Herpestes ichneumon), also known as ichneumon (), is a mongoose species native to the tropical and subtropical grasslands, savannas, and shrublands of Africa and around the Mediterranean Basin in North Africa, the Middle East and the Iberian Peninsula. Whether it is introduced or native to the Iberian Peninsula is in some doubt. Because of its widespread occurrence, it is listed as Least Concern on the IUCN Red List.
Characteristics
The Egyptian mongoose's long, coarse fur is grey to reddish brown and ticked with brown and yellow flecks. Its snout is pointed, its ears are small. Its slender body is long with a long black tipped tail. Its hind feet and a small area around the eyes are furless. It has 35–40 teeth, with highly developed carnassials, used for shearing meat. It weighs .
Sexually dimorphic Egyptian mongooses were observed in Portugal, where some females are smaller than males.
Female Egyptian mongooses have 44 chromosomes, and males 43, as one Y chromosome is translocated to an autosome.
Distribution and habitat
The Egyptian mongoose lives in swampy and marshy habitats near streams, rivers, lakes and in coastal areas. Where it inhabits maquis shrubland in the Iberian Peninsula, it prefers areas close to rivers with dense vegetation. It does not occur in deserts.
It has been recorded in Portugal from north of the Douro River to the south, and in Spain from the central plateau, Andalucía to the Strait of Gibraltar.
In North Africa, it occurs along the coast of the Mediterranean Sea and the Atlas Mountains from Western Sahara, Morocco, Algeria and Tunisia into Libya, and from northern Egypt across the Sinai Peninsula.
In Egypt, one individual was observed in Faiyum Oasis in 1993. In the same year, its tracks were recorded in sand dunes close to the coast near Sidi Barrani.
An individual was observed on an island in Lake Burullus in the Nile Delta during an ecological survey in the late 1990s.
In the Palestinian territories, it was recorded in the Gaza Strip and Jericho Governorate in the West Bank during surveys carried out between 2012 and 2016.
In western Syria, it was observed in the Latakia Governorate between 1989 and 1995; taxidermied specimens were offered in local shops.
In southern Turkey, it was recorded in the Hatay and Adana Provinces. | Egyptian mongoose | Wikipedia | 497 | 5512223 | https://en.wikipedia.org/wiki/Egyptian%20mongoose | Biology and health sciences | Other carnivora | Animals |
In Sudan, it is present in the vicinity of human settlements along the Rahad River and in Dinder National Park. It was also recorded in the Dinder–Alatash protected area complex during surveys between 2015 and 2018. In Ethiopia, the Egyptian mongoose was recorded at elevations of in the Ethiopian Highlands.
In Senegal, it was observed in 2000 in Niokolo-Koba National Park, which mainly encompasses open habitat dominated by grasses.
In Guinea's National Park of Upper Niger, the occurrence of the Egyptian mongoose was first documented during surveys in spring 1997. Surveyors found dead individuals on bushmeat markets in villages located in the vicinity of the park.
In Gabon's Moukalaba-Doudou National Park, it was recorded only in savanna habitats.
In the Republic of Congo, it was repeatedly observed in the Western Congolian forest–savanna mosaic of Odzala-Kokoua National Park during surveys in 2007.
In the 1990s, it was considered a common species in Tanzania's Mkomazi National Park.
Occurrence in Iberian Peninsula
Several hypotheses were proposed to explain the occurrence of the Egyptian mongoose in the Iberian Peninsula:
TraditionalIy, it was thought to have been introduced following the Muslim invasion in the 8th century.
Bones of Egyptian mongoose excavated in Spain and Portugal were radiocarbon dated to the first century. The scientists therefore suggested an introduction during the Roman Hispania era and use for eliminating rats and mice in domestic areas.
Other authors proposed a natural colonisation of the Iberian Peninsula during the Pleistocene across a land bridge when sea levels were low between glacial and interglacial periods. This population would have remained isolated from populations in Africa after the Last Glacial Period.
Behaviour and ecology
The Egyptian mongoose is diurnal.
In Doñana National Park, single Egyptian mongooses, pairs and groups of up to five individuals were observed. Adult males showed territorial behaviour, and shared their home ranges with one or several females. The home ranges of adult females overlapped to some degree, except in core areas where they raised their offspring. | Egyptian mongoose | Wikipedia | 428 | 5512223 | https://en.wikipedia.org/wiki/Egyptian%20mongoose | Biology and health sciences | Other carnivora | Animals |
It preys on rodents, fish, birds, reptiles, amphibians, and insects. It also feeds on fruit and eggs. To crack eggs open, it throws them between its legs against a rock or wall.
In Doñana National Park, 30 Egyptian mongooses were radio-tracked in 1985 and their faeces collected. These samples contained remains of European rabbit (Oryctolagus cuniculus), sand lizards (Psammodromus), Iberian spadefoot toad (Pelobates cultripes), greater white-toothed shrew (Crocidura russula), three-toed skink (Chalcides chalcides), dabbling ducks (Anas), western cattle egret (Bubulcus ibis), wild boar (Sus scrofa) meat, Algerian mouse (Mus spretus) and rat species (Rattus).
Research in southeastern Nigeria revealed that it also feeds on giant pouched rats (Cricetomys), Temminck's mouse (Mus musculoides), Tullberg's soft-furred mouse (Praomys tulbergi), Nigerian shrew (Crocidura nigeriae), Hallowell's toad (Amietophrynus maculatus), African brown water snake (Afronatrix anoscopus), and Mabuya skinks.
It attacks and feeds on venomous snakes, and is resistant to the venom of Palestine viper (Daboia palaestinae), black desert cobra (Walterinnesia aegyptia) and black-necked spitting cobra (Naja nigricollis).
In Spain, it has been recorded less frequently in areas where the Iberian lynx was reintroduced. | Egyptian mongoose | Wikipedia | 372 | 5512223 | https://en.wikipedia.org/wiki/Egyptian%20mongoose | Biology and health sciences | Other carnivora | Animals |
Reproduction
Captive males and females reach sexual maturity at the age of two years. In Doñana National Park, courtship and mating happens in spring between February and June. Two to three pups are born between mid April and mid August after a gestation of 11 weeks. They are hairless at first, and open their eyes after about a week. Females take care of them for up to one year, occasionally also longer. They start foraging on their own at the age of four months, but compete for food brought back to them after that age. In the wild, Egyptian mongooses probably reach 12 years of age. A captive Egyptian mongoose was over 20 years old.
Its generation length is 7.5 years. | Egyptian mongoose | Wikipedia | 145 | 5512223 | https://en.wikipedia.org/wiki/Egyptian%20mongoose | Biology and health sciences | Other carnivora | Animals |
Taxonomy
In 1758, Carl Linnaeus described an Egyptian mongoose from the area of the Nile River in Egypt in his work Systema Naturae and gave it the scientific name Viverra ichneumon.
H. i. ichneumon (Linnaeus, 1758) is the nominate subspecies. The following zoological specimen were described between the late 18th century and the early 1930s as subspecies:
Viverra cafra (Gmelin, 1788) − based on a description of a specimen from the Cape of Good Hope.
Herpestes ichneumon numidicus F. G. Cuvier, 1834 − two individuals from Algiers in Algeria kept in the menagerie of the Muséum d'Histoire Naturelle, France
Herpestes ichneumon widdringtonii Gray, 1842 − a specimen from Sierra Morena in Spain
Herpestes angolensis (Bocage, 1890) − a male specimen from Quissange in Angola
Mungos ichneumon parvidens (Lönnberg, 1908) − three specimens collected near the lower Congo River in Congo Free State
Mungos ichneumon funestus (Osgood, 1910) − a specimen from Naivasha in British East Africa
Mungos ichneumon centralis (Lönnberg, 1917) − two specimens from Beni, Democratic Republic of the Congo
Herpestes ichneumon sangronizi Cabrera, 1924 − a specimen from Mogador in Morocco
Herpestes caffer sabiensis (Roberts, 1926) − a specimen from Sabi Sand Game Reserve in Southern Africa
Herpestes cafer mababiensis (Roberts, 1932) − a specimen from Mababe in northern Bechuanaland
In 1811, Johann Karl Wilhelm Illiger subsumed the ichneumon to the genus Herpestes.
Threats
A survey of poaching methods in Israel carried out in autumn 2000 revealed that the Egyptian mongoose is affected by snaring in agricultural areas. Most of the traps found were set up by Thai guest workers.
Numerous dried heads of Egyptian mongooses were found in 2007 at the Dantokpa Market in southern Benin, suggesting that it is used as fetish in animal rituals.
Conservation
The Egyptian mongoose is listed on Appendix III of the Berne Convention, and Annex V of the European Union Habitats and Species Directive.
In Israel, wildlife is protected by law, and hunting allowed only with a permit.
In culture | Egyptian mongoose | Wikipedia | 501 | 5512223 | https://en.wikipedia.org/wiki/Egyptian%20mongoose | Biology and health sciences | Other carnivora | Animals |
Mummified remains of four Egyptian mongooses were excavated in the catacombs of Anubis at Saqqara during works started in 2009.
At the cemetery of Beni Hasan, an Egyptian mongoose on a leash is depicted in the tomb of Baqet I dating to the Eleventh Dynasty of Egypt.
The American poet John Greenleaf Whittier wrote a poem as an elegy for an ichneumon, which had been brought to Haverhill Academy in Haverhill, Massachusetts, in 1830. The long lost poem was published in the November 1902 issue of "The Independent" magazine.
The Sherlock Holmes canon also features an ichneumon the short story The Adventure of the Crooked Man, though due to Watson's description of its appearance and its owner's history in India it is likely to actually be an Indian grey mongoose. | Egyptian mongoose | Wikipedia | 179 | 5512223 | https://en.wikipedia.org/wiki/Egyptian%20mongoose | Biology and health sciences | Other carnivora | Animals |
Toxicodendron vernicifluum (formerly Rhus verniciflua), also known by the common name Chinese lacquer tree, is an Asian tree species of genus Toxicodendron native to China and the Indian subcontinent, and cultivated in regions of China, Japan and Korea. Other common names include Japanese lacquer tree, Japanese sumac, and varnish tree. The trees are cultivated and tapped for their toxic sap, which is used as a highly durable lacquer to make Chinese, Japanese, and Korean lacquerware.
The trees grow up to 20 metres tall with large leaves, each containing from 7 to 19 leaflets (most often 11–13). The sap contains the allergenic compound urushiol, which gets its name from this species' Japanese name urushi (); "urushi" is also used in English as a collective term for all kinds of Asian lacquerware made from the sap of this and related Asian tree species, as opposed to European "lacquer" or Japanning made from other materials. Urushiol is also the oil found in poison ivy and poison oak that causes a rash.
Uses
Lacquer
Sap, containing urushiol (an allergenic irritant), is tapped from the trunk of the Chinese lacquer tree to produce lacquer. This is done by cutting 5 to 10 horizontal lines on the trunk of a 10-year-old tree, and then collecting the greyish yellow sap that exudes. The sap is then filtered, heat-treated, or coloured before applying onto a base material that is to be lacquered. Curing the applied sap requires "drying" it in a warm, humid chamber or closet for 12 to 24 hours where the urushiol polymerizes to form a clear, hard, and waterproof surface. In its liquid state, urushiol can cause extreme rashes, even from vapours. Once hardened, reactions are possible but less common. | Toxicodendron vernicifluum | Wikipedia | 411 | 5512913 | https://en.wikipedia.org/wiki/Toxicodendron%20vernicifluum | Biology and health sciences | Sapindales | Plants |
Products coated with lacquer are recognizable by an extremely durable and glossy finish. Lacquer has many uses; some common applications include tableware, musical instruments, fountain pens, jewelry, and bows for archery. There are various types of lacquerware. The cinnabar-red is highly regarded. Unpigmented lacquer is dark brown but the most common colors of urushiol finishes are black and red, from powdered iron oxide pigments of ferrous-ferric oxide (magnetite) and ferric oxide (rust), respectively. Lacquer is painted on with a brush and is cured in a warm and humid environment.
The leaves, seeds, and the resin of the Chinese lacquer tree are sometimes used in Chinese medicine for the treatment of internal parasites and for stopping bleeding. Compounds butein and sulfuretin are antioxidants, and have inhibitory effects on aldose reductase and advanced glycation processes.
Buddhist monks who practiced the art of Sokushinbutsu would use the tree's sap in their ceremony.
Wax
The fruits of T. vernicifluum can also be processed to produce a waxy substance known as Japan wax used for numerous purposes including varnishing furniture and producing candles. The fruits of the trees are harvested, dried, steamed, and pressed to extract the wax, which hardens when cooled. | Toxicodendron vernicifluum | Wikipedia | 288 | 5512913 | https://en.wikipedia.org/wiki/Toxicodendron%20vernicifluum | Biology and health sciences | Sapindales | Plants |
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